Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 146137 10 pageshttpdxdoiorg1011552013146137

Research ArticleExact Asymptotic Stability Analysis and Region-of-AttractionEstimation for Nonlinear Systems

Min Wu1 Zhengfeng Yang1 and Wang Lin2

1 Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 200062 China2 College of Mathematics and Information Science Wenzhou University Wenzhou Zhejiang 325035 China

Correspondence should be addressed to Wang Lin linwangwzueducn

Received 14 November 2012 Revised 11 February 2013 Accepted 20 February 2013

Academic Editor Fabio M Camilli

Copyright copy 2013 Min Wu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems A hybridsymbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction ofnonlinear systems efficiently A numerical Lyapunov function and an estimate of region of attraction can be obtained by solvingan (bilinear) SOS programming via BMI solver then the modified Newton refinement and rational vector recovery techniques areapplied to obtain exact Lyapunov functions and verified estimates of regions of attraction with rational coefficients Experimentson some benchmarks are given to illustrate the efficiency of our algorithm

1 Introduction

Lyapunovrsquos stability theory which concern is with the behav-ior of trajectories of differential systems plays an impor-tant role in analysis and synthesis of nonlinear continuoussystems Lyapunov functions can be used to verify theasymptotic stability including locally asymptotic stability andglobally asymptotic stability In the literature there has beenlots of work on constructing Lyapunov functions [1ndash12]For example [4 5] used the linear matrix inequality (LMI)method to compute quadratic Lyapunov functions In [8 13]based on sum of squares (SOS) decomposition a methodis proposed to construct high-degree numerical polynomialLyapunov functions Reference [9] proposed a new methodfor computing polynomials Lyapunov functions by solvingsemialgebraic constraints via the tool DISCOVERER [14]Reference [15] constructed Lyapunov functions beyond poly-nomials by using radial basis functions In [11] the Grobnerbased method is used to choose the parameters in Lyapunovfunctions in an optimal way

Since the analysis of asymptotic stability is not sufficientfor safety critical systems the analysis of the region ofattraction (ROA) of an asymptotically stable equilibrium

point is a topic of significant importance The ROA is theset of initial states from which the system converges tothe equilibrium point Computing exact regions of attrac-tion (ROAs) for nonlinear dynamical systems is very hardif not impossible therefore researchers have focused onfinding estimates of the actual ROAs There are many well-established techniques for computation of ROAs [5 16ndash22]Among all methods those based on Lyapunov functionsare dominant in the literature These methods computeboth a Lyapunov function as a stability certificate and thesublevel sets of this Lyapunov function which give riseto estimates of the ROA In [17] the authors computedquadratic Lyapunov functions to optimize the volume of anellipsoid contained inROAby using nonlinear programmingReference [5] employed LMI based method to computethe optimal quadratic Lyapunov function for maximizingthe volume of an ellipsoid contained in the ROA of oddpolynomial systems In [22] based on SOS decompositiona method is presented to search for polynomial Lyapunovfunctions that enlarge a provable ROA of nonlinear polyno-mial system Reference [18] used discretization (in time) toflow invariant sets backwards along the flow of the vectorfield obtaining larger and larger estimates for the ROA

2 Abstract and Applied Analysis

Reference [21] employed quantifier elimination (QE)methodvia QEPCAD to find Lyapunov functions for estimating theROA

Taking advantage of the efficiency of numerical compu-tation and the error-free property of symbolic computationin this paper we present a hybrid symbolic-numeric algo-rithm to compute exact Lyapunov functions for asymptoticstability and verified estimates of the ROAs of continuoussystems The algorithm is based on SOS relaxation [23] ofa parametric polynomial optimization problem with LMIor bilinear matrix inequality (BMI) constraints which canbe solved directly using LMI or BMI solver in MAT-LAB such as SOSTOOLS [24] YALMIP [25] SeDuMi[26] PENBMI [27] and exact SOS representation recoverytechniques presented in [28 29] Unlike the numericalapproaches our method can yield exact Lyapunov functionsand verified estimates of ROAs which can overcome theunsoundness in analysis of nonlinear systems caused bynumerical errors [30] In comparison with some symbolicapproaches based on qualifier elimination technique ourapproach is more efficient and practical because parametricpolynomial optimization problem based on SOS relaxationwith fixed size can be solved in polynomial time theoreti-cally

The rest of the paper is organized as follows In Section 2we introduce some definitions and notions about Lyapunovstability and ROA Section 3 is devoted to transform theproblem of computing Lyapunov functions and estimatesof ROAs into a parametric program with LMI or BMIconstraints In Section 4 a symbolic-numeric approach viaSOS relaxation and exact rational vector recovery is proposedto compute Lyapunov functions and estimates of ROAsand an algorithm is described In Section 5 experiments onsome benchmarks are shown to illustrate our algorithm onasymptotic stability and ROA analysis At last we concludeour results in Section 6

2 Lyapunov Stability and Region of Attraction

In this section we will present the notion of Lyapunov stabil-ity and regions of attraction (ROAs) of dynamical systems

Consider the autonomous system

x = f (x) (1)

where f R119899 rarr R119899 is continuous and f satisfies theLipschitz condition Denote by 120601(119905 x

0) the solution of (1)

corresponding to the initial state x0= x(0) evaluated at time

119905 gt 0A vector x isin R119899 is an equilibrium point of the system (1)

if f(x) = 0 Since any equilibrium point can be shifted to theorigin 0 via a change of variables without loss of generalitywe may always assume that the equilibrium point of interestoccurs at the origin

Lyapunov theory is concerned with the behavior of thesolution 120601(119905 x

0) where the initial state x

0is not at the

equilibrium 0 but is ldquocloserdquo to it

Definition 1 (Lyapunov stability) The equilibrium point 0 of(1) is

(i) stable if for any 120598 gt 0 there exists 120575 = 120575(120598) such that1003817

1003817

1003817

1003817

x0

1003817

1003817

1003817

1003817

lt 120575 997904rArr

1003817

1003817

1003817

1003817

120601 (119905 x0)

1003817

1003817

1003817

1003817

lt 120598 forall119905 gt 0 (2)here sdot denotes any norm defined on R119899

(ii) unstable if it is not stable(iii) asymptotically stable if it is stable and 120575 can be chosen

such that1003817

1003817

1003817

1003817

x0

1003817

1003817

1003817

1003817

lt 120575 997904rArr lim119905rarrinfin

120601 (119905 x0) = 0 (3)

(iv) globally asymptotically stable if it is stable and for allx0isin R119899

lim119905rarrinfin

120601 (119905 x0) = 0 (4)

Intuitively the equilibrium point 0 is stable if all solutionsstarting near the origin (meaning that the initial conditionsare in a neighborhood of the origin) remain near theorigin for all time The equilibrium point 0 is asymptoticallystable if all solutions starting at nearby points not only staynearby but also converge to the equilibrium point as timeapproaches infinity And the equilibrium point 0 is globallyasymptotically stable if it is asymptotically stable for all initialconditions x

0isin R119899 The stability is an important property

in practice because it means arbitrarily small perturbationsof the initial state about the equilibrium point 0 result inarbitrarily small perturbations in the corresponding solutiontrajectories of (1)

A sufficient condition for stability of the zero equilibriumis the existence of a Lyapunov function as shown in thefollowing theorem

Theorem 2 ([31 Theorem 41]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R such that

119881 (0) = 0 119881 (x) gt 0 in 119863 0 (5)

119881 (x) = 120597119881

120597xsdot f (x) le 0 in 119863

(6)

then the origin is stable Moreover if

119881 (x) lt 0 in 119863 0 (7)then the origin is asymptotically stable

A function 119881(x) satisfying the conditions (5) and (6) inTheorem 2 is commonly known as a Lyapunov function Andwe can verify globally asymptotic stability of system (1) byusing Lyapunov functions stated as follows

Theorem 3 ([31 Theorem 42]) Let the origin be an equilib-rium point for (1) If there exists a continuously differentiablefunction 119881 R119899 rarr R such that

119881 (0) = 0 119881 (x) gt 0 forallx = 0 (8)x 997888rarr infin 997904rArr 119881(x) 997888rarr infin (9)

119881 (x) lt 0 forallx = 0 (10)then the origin is globally asymptotically stable

Abstract and Applied Analysis 3

Remark that a function satisfying condition (9) is said tobe radially unbounded

It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition

Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin

R119899 | lim119905rarrinfin

120601(119905 x) = 0

The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo

Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x

0isin 119872 implies

120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a

positively invariant set119872 at some time instant then it belongsto119872 for all future time

In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem

Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω

119888= x isin R119899 | 119881(x) le 119888 with

119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained

in the ROA of the equilibrium 0

Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ

119888= x isin R119899 | 119881(x) le 119888 is

bounded and contained in119863 then every trajectory starting inΩ

119888remains inΩ

119888and approaches the origin as 119905 rarr infinThus

Ω

119888is an estimate of the ROA Remark that when the origin is

globally asymptotically stable the region of attraction is thewhole space R119899

3 Problem Reformulation

In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891

119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the

ring of real polynomials in the variables x

31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the

equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx isin 119863 0

119881 (x) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

(11)

In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum

119899

119894=1119909

2

119894minus 119903

2 for 1 le 119894 le 119899with a given constant 119903 isin R

+ for instance 119903 = 10

minus2Let us first predetermine a template of Lyapunov func-

tions with the given degree 119889 We assume that

119881 (x) = sum

120572

119888

120572x120572 (12)

where x120572 = 119909

1205721

1 119909

120572119899

119899 120572 = (120572

1 120572

119899) isin Z119899

ge0with

sum

119899

119894=1120572

119894le 119889 and 119888

120572isin R are parameters We can rewrite

119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909

1 119909

119899with total degree le 119889 and c isin R] with

] = (

119899+119889

119899) is the coefficient vector of 119881(x) In the sequel we

write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive

definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin

R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form

119897 (x) =119899

sum

119894=1

120598

119894119909

119896

119894 (13)

where 120598

119894isin R+and 119896 is assumed to be even Clearly the

condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897

1(x) 1198972(x) of the form of

(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication

119898

⋀

119894=1

(119901

119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)

holds where 119901

119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin

R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590

119894isin R[x] for 119894 = 0 119898 such that

119902 (x) = 120590

0(x) +

119898

sum

119894=1

120590

119894(x) 119901119894(x) (15)

4 Abstract and Applied Analysis

then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898

119894=1119901

119894(x) ge 0

Based on the above observation the problem (11) can befurther transformed into the following SOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) 119892 (x)

(16)

where 120590119895(x) and 120588

119895(x) are SOS inR[x] for 119895 = 1 2 Moreover

the degree bound of those unknown SOS polynomials 120590119895 120588119895

is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z

+ of the unknown

SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem

Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx = 0

x 997888rarr infin 997904rArr 119881(x) 997888rarr infin

119881 (x) lt 0 forallx = 0

(17)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form

(13) the condition that 119881(x c) minus 119897

1(x) is SOS guarantees

the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x)

(18)

where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The

decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897

119894(x) and 120588

119895(x)

32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899

Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set

x isin R119899

| 119881 (x) le 1 (19)

given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region

119875

120573= x isin R

119899

| 119901 (x) le 120573 (20)

where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875

120573sube 119863

Fix a template of 119881 of the form (12) The problem offinding an estimate 119875

120573of the ROA can be transformed into

the following polynomial optimization problem

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) gt 0 forallx isin 119863 0

119881 (x c) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

119901 (x) le 120573 ⊨ 119881 (x c) le 1

(21)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)

and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) (119881 (x c) minus 1)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) (119881 (x c) minus 1)

1 minus 119881 (x c) = 120588

3(x) minus 120590

3(x) (119901 (x) minus 120573)

(22)

where 120590

119894(x) and 120588

119894(x) are SOS in R[x] for 1 le 119894 le 3

In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590

119894(x)

and 120588

119894(x) Since 120573 and the coefficients of 119881(x c) and 120590

119894(x)

are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method

4 Exact Certificate of Sum ofSquares Decomposition

According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real

Abstract and Applied Analysis 5

function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573

41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method

Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that

120595 (x) = 119885(x)119879119876119885 (x) (23)

where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem

inf119882[119895]119876[119895]

2

sum

119895=1

Trace (119882[119895] + 119876

[119895]

)

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus 119901

1(x)119879 sdot 119876[1] sdot 119901

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

minus 119901

2(x)119879 sdot 119876[2] sdot 119901

2(x) 119892 (x)

(24)

where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2

119895=1Trace(119882[119895] + 119876

[119895]

) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function

Similarly the SOS programming (18) can be rewritten asthe following SDP problem

inf119882[1]119882[2]

Trace (119882[1]) + Trace (119882[2])

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

(25)

where matrices 119882

[1] 119882

[2] are symmetric and positivesemidefinite

Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently

Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem

Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and

minus119909

2

+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat

2119909

119889120593

119889119909

= 120601

0(119909) + 120601

1(119909) (1 minus 119909

2

) + 120601

2(119909) 120593 (119909)

(26)

where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1

deg(1206010) = 2 and deg(120601

1) = deg(120601

2) = 0 and that 120593(119909) = 119906

0+

119906

1119909 1206011= 119906

2and 1206012= V1 with 119906

0 119906

1 119906

2 V1isin R parameters

From (1) we have

120601

0(119909) = 119906

2119909

2

+ (2119906

1minus 119906

1V1) 119909 minus 119906

2minus 119906

0V1 (27)

whose square matrix representation (SMR) [33] is 1206010(119909) =

119885

119879

119876119885 where

119876 =

[

[

[

minus119906

2minus 119906

0V1

119906

1minus

1

2

119906

1V1

119906

1minus

1

2

119906

1V1

119906

2

]

]

]

119885 = [

1

119909

] (28)

Since all the 120601119894(119909) are SOSes we have 119906

2ge 0 V

1ge 0 and

119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint

B (119906

0 119906

1 119906

2 V1) = 119906

1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

]

]

]

]

+ 119906

2

[

[

[

[

1 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 1

]

]

]

]

+ V1

[

[

[

[

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

]

]

]

]

+ 119906

0V1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 0

]

]

]

]

+ 119906

1V1

[

[

[

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 minus

1

2

0 0 minus

1

2

0

]

]

]

]

]

]

]

⪰ 0

(29)

Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form

inf minus120573

st B (u k) = 119860

0+

119898

sum

119894=1

119906

119894119860

119894+

119896

sum

119895=1

V119895119860

119898+119895

+ sum

1le119894le119898

sum

1le119895le119896

119906

119894V119895119861

119894119895⪰ 0

(30)

where 119860

119894 119861

119894119895are constant symmetric matrices u =

(119906

1 119906

119898) k = (V

1 V

119896) are parameter coefficients of the

SOSpolynomials occurring in the original SOSprogramming(22)

Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Reference [21] employed quantifier elimination (QE)methodvia QEPCAD to find Lyapunov functions for estimating theROA

Taking advantage of the efficiency of numerical compu-tation and the error-free property of symbolic computationin this paper we present a hybrid symbolic-numeric algo-rithm to compute exact Lyapunov functions for asymptoticstability and verified estimates of the ROAs of continuoussystems The algorithm is based on SOS relaxation [23] ofa parametric polynomial optimization problem with LMIor bilinear matrix inequality (BMI) constraints which canbe solved directly using LMI or BMI solver in MAT-LAB such as SOSTOOLS [24] YALMIP [25] SeDuMi[26] PENBMI [27] and exact SOS representation recoverytechniques presented in [28 29] Unlike the numericalapproaches our method can yield exact Lyapunov functionsand verified estimates of ROAs which can overcome theunsoundness in analysis of nonlinear systems caused bynumerical errors [30] In comparison with some symbolicapproaches based on qualifier elimination technique ourapproach is more efficient and practical because parametricpolynomial optimization problem based on SOS relaxationwith fixed size can be solved in polynomial time theoreti-cally

The rest of the paper is organized as follows In Section 2we introduce some definitions and notions about Lyapunovstability and ROA Section 3 is devoted to transform theproblem of computing Lyapunov functions and estimatesof ROAs into a parametric program with LMI or BMIconstraints In Section 4 a symbolic-numeric approach viaSOS relaxation and exact rational vector recovery is proposedto compute Lyapunov functions and estimates of ROAsand an algorithm is described In Section 5 experiments onsome benchmarks are shown to illustrate our algorithm onasymptotic stability and ROA analysis At last we concludeour results in Section 6

2 Lyapunov Stability and Region of Attraction

In this section we will present the notion of Lyapunov stabil-ity and regions of attraction (ROAs) of dynamical systems

Consider the autonomous system

x = f (x) (1)

where f R119899 rarr R119899 is continuous and f satisfies theLipschitz condition Denote by 120601(119905 x

0) the solution of (1)

corresponding to the initial state x0= x(0) evaluated at time

119905 gt 0A vector x isin R119899 is an equilibrium point of the system (1)

if f(x) = 0 Since any equilibrium point can be shifted to theorigin 0 via a change of variables without loss of generalitywe may always assume that the equilibrium point of interestoccurs at the origin

Lyapunov theory is concerned with the behavior of thesolution 120601(119905 x

0) where the initial state x

0is not at the

equilibrium 0 but is ldquocloserdquo to it

Definition 1 (Lyapunov stability) The equilibrium point 0 of(1) is

(i) stable if for any 120598 gt 0 there exists 120575 = 120575(120598) such that1003817

1003817

1003817

1003817

x0

1003817

1003817

1003817

1003817

lt 120575 997904rArr

1003817

1003817

1003817

1003817

120601 (119905 x0)

1003817

1003817

1003817

1003817

lt 120598 forall119905 gt 0 (2)here sdot denotes any norm defined on R119899

(ii) unstable if it is not stable(iii) asymptotically stable if it is stable and 120575 can be chosen

such that1003817

1003817

1003817

1003817

x0

1003817

1003817

1003817

1003817

lt 120575 997904rArr lim119905rarrinfin

120601 (119905 x0) = 0 (3)

(iv) globally asymptotically stable if it is stable and for allx0isin R119899

lim119905rarrinfin

120601 (119905 x0) = 0 (4)

Intuitively the equilibrium point 0 is stable if all solutionsstarting near the origin (meaning that the initial conditionsare in a neighborhood of the origin) remain near theorigin for all time The equilibrium point 0 is asymptoticallystable if all solutions starting at nearby points not only staynearby but also converge to the equilibrium point as timeapproaches infinity And the equilibrium point 0 is globallyasymptotically stable if it is asymptotically stable for all initialconditions x

0isin R119899 The stability is an important property

in practice because it means arbitrarily small perturbationsof the initial state about the equilibrium point 0 result inarbitrarily small perturbations in the corresponding solutiontrajectories of (1)

A sufficient condition for stability of the zero equilibriumis the existence of a Lyapunov function as shown in thefollowing theorem

Theorem 2 ([31 Theorem 41]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R such that

119881 (0) = 0 119881 (x) gt 0 in 119863 0 (5)

119881 (x) = 120597119881

120597xsdot f (x) le 0 in 119863

(6)

then the origin is stable Moreover if

119881 (x) lt 0 in 119863 0 (7)then the origin is asymptotically stable

A function 119881(x) satisfying the conditions (5) and (6) inTheorem 2 is commonly known as a Lyapunov function Andwe can verify globally asymptotic stability of system (1) byusing Lyapunov functions stated as follows

Theorem 3 ([31 Theorem 42]) Let the origin be an equilib-rium point for (1) If there exists a continuously differentiablefunction 119881 R119899 rarr R such that

119881 (0) = 0 119881 (x) gt 0 forallx = 0 (8)x 997888rarr infin 997904rArr 119881(x) 997888rarr infin (9)

119881 (x) lt 0 forallx = 0 (10)then the origin is globally asymptotically stable

Abstract and Applied Analysis 3

Remark that a function satisfying condition (9) is said tobe radially unbounded

It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition

Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin

R119899 | lim119905rarrinfin

120601(119905 x) = 0

The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo

Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x

0isin 119872 implies

120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a

positively invariant set119872 at some time instant then it belongsto119872 for all future time

In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem

Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω

119888= x isin R119899 | 119881(x) le 119888 with

119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained

in the ROA of the equilibrium 0

Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ

119888= x isin R119899 | 119881(x) le 119888 is

bounded and contained in119863 then every trajectory starting inΩ

119888remains inΩ

119888and approaches the origin as 119905 rarr infinThus

Ω

119888is an estimate of the ROA Remark that when the origin is

globally asymptotically stable the region of attraction is thewhole space R119899

3 Problem Reformulation

In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891

119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the

ring of real polynomials in the variables x

31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the

equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx isin 119863 0

119881 (x) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

(11)

In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum

119899

119894=1119909

2

119894minus 119903

2 for 1 le 119894 le 119899with a given constant 119903 isin R

+ for instance 119903 = 10

minus2Let us first predetermine a template of Lyapunov func-

tions with the given degree 119889 We assume that

119881 (x) = sum

120572

119888

120572x120572 (12)

where x120572 = 119909

1205721

1 119909

120572119899

119899 120572 = (120572

1 120572

119899) isin Z119899

ge0with

sum

119899

119894=1120572

119894le 119889 and 119888

120572isin R are parameters We can rewrite

119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909

1 119909

119899with total degree le 119889 and c isin R] with

] = (

119899+119889

119899) is the coefficient vector of 119881(x) In the sequel we

write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive

definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin

R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form

119897 (x) =119899

sum

119894=1

120598

119894119909

119896

119894 (13)

where 120598

119894isin R+and 119896 is assumed to be even Clearly the

condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897

1(x) 1198972(x) of the form of

(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication

119898

⋀

119894=1

(119901

119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)

holds where 119901

119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin

R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590

119894isin R[x] for 119894 = 0 119898 such that

119902 (x) = 120590

0(x) +

119898

sum

119894=1

120590

119894(x) 119901119894(x) (15)

4 Abstract and Applied Analysis

then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898

119894=1119901

119894(x) ge 0

Based on the above observation the problem (11) can befurther transformed into the following SOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) 119892 (x)

(16)

where 120590119895(x) and 120588

119895(x) are SOS inR[x] for 119895 = 1 2 Moreover

the degree bound of those unknown SOS polynomials 120590119895 120588119895

is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z

+ of the unknown

SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem

Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx = 0

x 997888rarr infin 997904rArr 119881(x) 997888rarr infin

119881 (x) lt 0 forallx = 0

(17)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form

(13) the condition that 119881(x c) minus 119897

1(x) is SOS guarantees

the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x)

(18)

where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The

decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897

119894(x) and 120588

119895(x)

32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899

Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set

x isin R119899

| 119881 (x) le 1 (19)

given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region

119875

120573= x isin R

119899

| 119901 (x) le 120573 (20)

where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875

120573sube 119863

Fix a template of 119881 of the form (12) The problem offinding an estimate 119875

120573of the ROA can be transformed into

the following polynomial optimization problem

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) gt 0 forallx isin 119863 0

119881 (x c) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

119901 (x) le 120573 ⊨ 119881 (x c) le 1

(21)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)

and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) (119881 (x c) minus 1)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) (119881 (x c) minus 1)

1 minus 119881 (x c) = 120588

3(x) minus 120590

3(x) (119901 (x) minus 120573)

(22)

where 120590

119894(x) and 120588

119894(x) are SOS in R[x] for 1 le 119894 le 3

In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590

119894(x)

and 120588

119894(x) Since 120573 and the coefficients of 119881(x c) and 120590

119894(x)

are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method

4 Exact Certificate of Sum ofSquares Decomposition

According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real

Abstract and Applied Analysis 5

function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573

41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method

Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that

120595 (x) = 119885(x)119879119876119885 (x) (23)

where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem

inf119882[119895]119876[119895]

2

sum

119895=1

Trace (119882[119895] + 119876

[119895]

)

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus 119901

1(x)119879 sdot 119876[1] sdot 119901

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

minus 119901

2(x)119879 sdot 119876[2] sdot 119901

2(x) 119892 (x)

(24)

where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2

119895=1Trace(119882[119895] + 119876

[119895]

) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function

Similarly the SOS programming (18) can be rewritten asthe following SDP problem

inf119882[1]119882[2]

Trace (119882[1]) + Trace (119882[2])

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

(25)

where matrices 119882

[1] 119882

[2] are symmetric and positivesemidefinite

Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently

Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem

Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and

minus119909

2

+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat

2119909

119889120593

119889119909

= 120601

0(119909) + 120601

1(119909) (1 minus 119909

2

) + 120601

2(119909) 120593 (119909)

(26)

where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1

deg(1206010) = 2 and deg(120601

1) = deg(120601

2) = 0 and that 120593(119909) = 119906

0+

119906

1119909 1206011= 119906

2and 1206012= V1 with 119906

0 119906

1 119906

2 V1isin R parameters

From (1) we have

120601

0(119909) = 119906

2119909

2

+ (2119906

1minus 119906

1V1) 119909 minus 119906

2minus 119906

0V1 (27)

whose square matrix representation (SMR) [33] is 1206010(119909) =

119885

119879

119876119885 where

119876 =

[

[

[

minus119906

2minus 119906

0V1

119906

1minus

1

2

119906

1V1

119906

1minus

1

2

119906

1V1

119906

2

]

]

]

119885 = [

1

119909

] (28)

Since all the 120601119894(119909) are SOSes we have 119906

2ge 0 V

1ge 0 and

119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint

B (119906

0 119906

1 119906

2 V1) = 119906

1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

]

]

]

]

+ 119906

2

[

[

[

[

1 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 1

]

]

]

]

+ V1

[

[

[

[

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

]

]

]

]

+ 119906

0V1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 0

]

]

]

]

+ 119906

1V1

[

[

[

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 minus

1

2

0 0 minus

1

2

0

]

]

]

]

]

]

]

⪰ 0

(29)

Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form

inf minus120573

st B (u k) = 119860

0+

119898

sum

119894=1

119906

119894119860

119894+

119896

sum

119895=1

V119895119860

119898+119895

+ sum

1le119894le119898

sum

1le119895le119896

119906

119894V119895119861

119894119895⪰ 0

(30)

where 119860

119894 119861

119894119895are constant symmetric matrices u =

(119906

1 119906

119898) k = (V

1 V

119896) are parameter coefficients of the

SOSpolynomials occurring in the original SOSprogramming(22)

Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Remark that a function satisfying condition (9) is said tobe radially unbounded

It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition

Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin

R119899 | lim119905rarrinfin

120601(119905 x) = 0

The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo

Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x

0isin 119872 implies

120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a

positively invariant set119872 at some time instant then it belongsto119872 for all future time

In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem

Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω

119888= x isin R119899 | 119881(x) le 119888 with

119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained

in the ROA of the equilibrium 0

Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ

119888= x isin R119899 | 119881(x) le 119888 is

bounded and contained in119863 then every trajectory starting inΩ

119888remains inΩ

119888and approaches the origin as 119905 rarr infinThus

Ω

119888is an estimate of the ROA Remark that when the origin is

globally asymptotically stable the region of attraction is thewhole space R119899

3 Problem Reformulation

In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891

119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the

ring of real polynomials in the variables x

31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the

equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx isin 119863 0

119881 (x) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

(11)

In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum

119899

119894=1119909

2

119894minus 119903

2 for 1 le 119894 le 119899with a given constant 119903 isin R

+ for instance 119903 = 10

minus2Let us first predetermine a template of Lyapunov func-

tions with the given degree 119889 We assume that

119881 (x) = sum

120572

119888

120572x120572 (12)

where x120572 = 119909

1205721

1 119909

120572119899

119899 120572 = (120572

1 120572

119899) isin Z119899

ge0with

sum

119899

119894=1120572

119894le 119889 and 119888

120572isin R are parameters We can rewrite

119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909

1 119909

119899with total degree le 119889 and c isin R] with

] = (

119899+119889

119899) is the coefficient vector of 119881(x) In the sequel we

write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive

definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin

R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form

119897 (x) =119899

sum

119894=1

120598

119894119909

119896

119894 (13)

where 120598

119894isin R+and 119896 is assumed to be even Clearly the

condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897

1(x) 1198972(x) of the form of

(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication

119898

⋀

119894=1

(119901

119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)

holds where 119901

119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin

R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590

119894isin R[x] for 119894 = 0 119898 such that

119902 (x) = 120590

0(x) +

119898

sum

119894=1

120590

119894(x) 119901119894(x) (15)

4 Abstract and Applied Analysis

then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898

119894=1119901

119894(x) ge 0

Based on the above observation the problem (11) can befurther transformed into the following SOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) 119892 (x)

(16)

where 120590119895(x) and 120588

119895(x) are SOS inR[x] for 119895 = 1 2 Moreover

the degree bound of those unknown SOS polynomials 120590119895 120588119895

is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z

+ of the unknown

SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem

Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx = 0

x 997888rarr infin 997904rArr 119881(x) 997888rarr infin

119881 (x) lt 0 forallx = 0

(17)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form

(13) the condition that 119881(x c) minus 119897

1(x) is SOS guarantees

the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x)

(18)

where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The

decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897

119894(x) and 120588

119895(x)

32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899

Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set

x isin R119899

| 119881 (x) le 1 (19)

given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region

119875

120573= x isin R

119899

| 119901 (x) le 120573 (20)

where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875

120573sube 119863

Fix a template of 119881 of the form (12) The problem offinding an estimate 119875

120573of the ROA can be transformed into

the following polynomial optimization problem

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) gt 0 forallx isin 119863 0

119881 (x c) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

119901 (x) le 120573 ⊨ 119881 (x c) le 1

(21)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)

and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) (119881 (x c) minus 1)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) (119881 (x c) minus 1)

1 minus 119881 (x c) = 120588

3(x) minus 120590

3(x) (119901 (x) minus 120573)

(22)

where 120590

119894(x) and 120588

119894(x) are SOS in R[x] for 1 le 119894 le 3

In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590

119894(x)

and 120588

119894(x) Since 120573 and the coefficients of 119881(x c) and 120590

119894(x)

are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method

4 Exact Certificate of Sum ofSquares Decomposition

According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real

Abstract and Applied Analysis 5

function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573

41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method

Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that

120595 (x) = 119885(x)119879119876119885 (x) (23)

where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem

inf119882[119895]119876[119895]

2

sum

119895=1

Trace (119882[119895] + 119876

[119895]

)

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus 119901

1(x)119879 sdot 119876[1] sdot 119901

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

minus 119901

2(x)119879 sdot 119876[2] sdot 119901

2(x) 119892 (x)

(24)

where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2

119895=1Trace(119882[119895] + 119876

[119895]

) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function

Similarly the SOS programming (18) can be rewritten asthe following SDP problem

inf119882[1]119882[2]

Trace (119882[1]) + Trace (119882[2])

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

(25)

where matrices 119882

[1] 119882

[2] are symmetric and positivesemidefinite

Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently

Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem

Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and

minus119909

2

+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat

2119909

119889120593

119889119909

= 120601

0(119909) + 120601

1(119909) (1 minus 119909

2

) + 120601

2(119909) 120593 (119909)

(26)

where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1

deg(1206010) = 2 and deg(120601

1) = deg(120601

2) = 0 and that 120593(119909) = 119906

0+

119906

1119909 1206011= 119906

2and 1206012= V1 with 119906

0 119906

1 119906

2 V1isin R parameters

From (1) we have

120601

0(119909) = 119906

2119909

2

+ (2119906

1minus 119906

1V1) 119909 minus 119906

2minus 119906

0V1 (27)

whose square matrix representation (SMR) [33] is 1206010(119909) =

119885

119879

119876119885 where

119876 =

[

[

[

minus119906

2minus 119906

0V1

119906

1minus

1

2

119906

1V1

119906

1minus

1

2

119906

1V1

119906

2

]

]

]

119885 = [

1

119909

] (28)

Since all the 120601119894(119909) are SOSes we have 119906

2ge 0 V

1ge 0 and

119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint

B (119906

0 119906

1 119906

2 V1) = 119906

1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

]

]

]

]

+ 119906

2

[

[

[

[

1 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 1

]

]

]

]

+ V1

[

[

[

[

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

]

]

]

]

+ 119906

0V1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 0

]

]

]

]

+ 119906

1V1

[

[

[

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 minus

1

2

0 0 minus

1

2

0

]

]

]

]

]

]

]

⪰ 0

(29)

Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form

inf minus120573

st B (u k) = 119860

0+

119898

sum

119894=1

119906

119894119860

119894+

119896

sum

119895=1

V119895119860

119898+119895

+ sum

1le119894le119898

sum

1le119895le119896

119906

119894V119895119861

119894119895⪰ 0

(30)

where 119860

119894 119861

119894119895are constant symmetric matrices u =

(119906

1 119906

119898) k = (V

1 V

119896) are parameter coefficients of the

SOSpolynomials occurring in the original SOSprogramming(22)

Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898

119894=1119901

119894(x) ge 0

Based on the above observation the problem (11) can befurther transformed into the following SOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) 119892 (x)

(16)

where 120590119895(x) and 120588

119895(x) are SOS inR[x] for 119895 = 1 2 Moreover

the degree bound of those unknown SOS polynomials 120590119895 120588119895

is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z

+ of the unknown

SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem

Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem

find 119881 (x) isin R [x]

st 119881 (0) = 0

119881 (x) gt 0 forallx = 0

x 997888rarr infin 997904rArr 119881(x) 997888rarr infin

119881 (x) lt 0 forallx = 0

(17)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form

(13) the condition that 119881(x c) minus 119897

1(x) is SOS guarantees

the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming

find c isin R]

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x)

(18)

where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The

decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897

119894(x) and 120588

119895(x)

32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899

Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set

x isin R119899

| 119881 (x) le 1 (19)

given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region

119875

120573= x isin R

119899

| 119901 (x) le 120573 (20)

where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875

120573sube 119863

Fix a template of 119881 of the form (12) The problem offinding an estimate 119875

120573of the ROA can be transformed into

the following polynomial optimization problem

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) gt 0 forallx isin 119863 0

119881 (x c) = 120597119881

120597xf (x) lt 0 forallx isin 119863 0

119901 (x) le 120573 ⊨ 119881 (x c) le 1

(21)

By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)

and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming

maxcisinR]

120573

st 119881 (0 c) = 0

119881 (x c) minus 119897

1(x) = 120588

1(x) minus 120590

1(x) (119881 (x c) minus 1)

minus

120597119881

120597xf (x) minus 119897

2(x) = 120588

2(x) minus 120590

2(x) (119881 (x c) minus 1)

1 minus 119881 (x c) = 120588

3(x) minus 120590

3(x) (119901 (x) minus 120573)

(22)

where 120590

119894(x) and 120588

119894(x) are SOS in R[x] for 1 le 119894 le 3

In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590

119894(x)

and 120588

119894(x) Since 120573 and the coefficients of 119881(x c) and 120590

119894(x)

are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method

4 Exact Certificate of Sum ofSquares Decomposition

According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real

Abstract and Applied Analysis 5

function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573

41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method

Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that

120595 (x) = 119885(x)119879119876119885 (x) (23)

where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem

inf119882[119895]119876[119895]

2

sum

119895=1

Trace (119882[119895] + 119876

[119895]

)

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus 119901

1(x)119879 sdot 119876[1] sdot 119901

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

minus 119901

2(x)119879 sdot 119876[2] sdot 119901

2(x) 119892 (x)

(24)

where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2

119895=1Trace(119882[119895] + 119876

[119895]

) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function

Similarly the SOS programming (18) can be rewritten asthe following SDP problem

inf119882[1]119882[2]

Trace (119882[1]) + Trace (119882[2])

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

(25)

where matrices 119882

[1] 119882

[2] are symmetric and positivesemidefinite

Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently

Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem

Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and

minus119909

2

+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat

2119909

119889120593

119889119909

= 120601

0(119909) + 120601

1(119909) (1 minus 119909

2

) + 120601

2(119909) 120593 (119909)

(26)

where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1

deg(1206010) = 2 and deg(120601

1) = deg(120601

2) = 0 and that 120593(119909) = 119906

0+

119906

1119909 1206011= 119906

2and 1206012= V1 with 119906

0 119906

1 119906

2 V1isin R parameters

From (1) we have

120601

0(119909) = 119906

2119909

2

+ (2119906

1minus 119906

1V1) 119909 minus 119906

2minus 119906

0V1 (27)

whose square matrix representation (SMR) [33] is 1206010(119909) =

119885

119879

119876119885 where

119876 =

[

[

[

minus119906

2minus 119906

0V1

119906

1minus

1

2

119906

1V1

119906

1minus

1

2

119906

1V1

119906

2

]

]

]

119885 = [

1

119909

] (28)

Since all the 120601119894(119909) are SOSes we have 119906

2ge 0 V

1ge 0 and

119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint

B (119906

0 119906

1 119906

2 V1) = 119906

1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

]

]

]

]

+ 119906

2

[

[

[

[

1 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 1

]

]

]

]

+ V1

[

[

[

[

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

]

]

]

]

+ 119906

0V1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 0

]

]

]

]

+ 119906

1V1

[

[

[

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 minus

1

2

0 0 minus

1

2

0

]

]

]

]

]

]

]

⪰ 0

(29)

Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form

inf minus120573

st B (u k) = 119860

0+

119898

sum

119894=1

119906

119894119860

119894+

119896

sum

119895=1

V119895119860

119898+119895

+ sum

1le119894le119898

sum

1le119895le119896

119906

119894V119895119861

119894119895⪰ 0

(30)

where 119860

119894 119861

119894119895are constant symmetric matrices u =

(119906

1 119906

119898) k = (V

1 V

119896) are parameter coefficients of the

SOSpolynomials occurring in the original SOSprogramming(22)

Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573

41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method

Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that

120595 (x) = 119885(x)119879119876119885 (x) (23)

where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem

inf119882[119895]119876[119895]

2

sum

119895=1

Trace (119882[119895] + 119876

[119895]

)

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus 119901

1(x)119879 sdot 119876[1] sdot 119901

1(x) 119892 (x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

minus 119901

2(x)119879 sdot 119876[2] sdot 119901

2(x) 119892 (x)

(24)

where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2

119895=1Trace(119882[119895] + 119876

[119895]

) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function

Similarly the SOS programming (18) can be rewritten asthe following SDP problem

inf119882[1]119882[2]

Trace (119882[1]) + Trace (119882[2])

st 119881 (0) = 0

119881 (x c) minus 119897

1(x) = 119898

1(x)119879 sdot 119882[1] sdot 119898

1(x)

minus

120597119881

120597xf (x) minus 119897

2(x) = 119898

2(x)119879 sdot 119882[2] sdot 119898

2(x)

(25)

where matrices 119882

[1] 119882

[2] are symmetric and positivesemidefinite

Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently

Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem

Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and

minus119909

2

+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat

2119909

119889120593

119889119909

= 120601

0(119909) + 120601

1(119909) (1 minus 119909

2

) + 120601

2(119909) 120593 (119909)

(26)

where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1

deg(1206010) = 2 and deg(120601

1) = deg(120601

2) = 0 and that 120593(119909) = 119906

0+

119906

1119909 1206011= 119906

2and 1206012= V1 with 119906

0 119906

1 119906

2 V1isin R parameters

From (1) we have

120601

0(119909) = 119906

2119909

2

+ (2119906

1minus 119906

1V1) 119909 minus 119906

2minus 119906

0V1 (27)

whose square matrix representation (SMR) [33] is 1206010(119909) =

119885

119879

119876119885 where

119876 =

[

[

[

minus119906

2minus 119906

0V1

119906

1minus

1

2

119906

1V1

119906

1minus

1

2

119906

1V1

119906

2

]

]

]

119885 = [

1

119909

] (28)

Since all the 120601119894(119909) are SOSes we have 119906

2ge 0 V

1ge 0 and

119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint

B (119906

0 119906

1 119906

2 V1) = 119906

1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

]

]

]

]

+ 119906

2

[

[

[

[

1 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 1

]

]

]

]

+ V1

[

[

[

[

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

]

]

]

]

+ 119906

0V1

[

[

[

[

0 0 0 0

0 0 0 0

0 0 minus1 0

0 0 0 0

]

]

]

]

+ 119906

1V1

[

[

[

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 minus

1

2

0 0 minus

1

2

0

]

]

]

]

]

]

]

⪰ 0

(29)

Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form

inf minus120573

st B (u k) = 119860

0+

119898

sum

119894=1

119906

119894119860

119894+

119896

sum

119895=1

V119895119860

119898+119895

+ sum

1le119894le119898

sum

1le119895le119896

119906

119894V119895119861

119894119895⪰ 0

(30)

where 119860

119894 119861

119894119895are constant symmetric matrices u =

(119906

1 119906

119898) k = (V

1 V

119896) are parameter coefficients of the

SOSpolynomials occurring in the original SOSprogramming(22)

Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ

119901that penalizes the inequality constraints This

function satisfies a number of properties [35] such that forany 119901 gt 0 we have

B (u k) ⪰ 0 lArrrArr Φ

119901(B (u k)) ⪰ 0 (31)

This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem

inf minus120573

st Φ

119901(B (u k)) ⪰ 0

(32)

The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)

119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ

119901(B (u k))) (33)

where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]

Alternatively observing (30) B(u k) involves no cross-ing products like 119906

119894119906

119895and V

119894V119895 Taking this special form

into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice

42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints

Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem

find c isin R]

st 119881

1(x c) = m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) = m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

119882

[119894]

⪰ 0 119894 = 1 2 3

(34)

where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3

are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the

numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is

119881

1(x c) asymp m

1(x)119879 sdot 119882[1] sdotm

1(x)

119881

2(x c) asymp m

2(x)119879 sdot 119882[2] sdotm

2(x)

+ (m3(x)119879 sdot 119882[3] sdotm

3(x)) sdot 119881

3(x c)

(35)

as illustrated by the following example

Example 8 Consider the following nonlinear system

[

1

2

] =

[

[

[

[

[

[

[

[

[

minus05119909

1+ 119909

2+ 024999119909

2

1

+ 0083125119909

3

1+ 00205295119909

4

1

+ 00046875119909

5

1+ 00015191119909

6

1

minus119909

2+ 119909

1119909

2minus 049991119909

3

1+ 0040947119909

5

1

]

]

]

]

]

]

]

]

]

(36)

We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2 When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 3112937368119909

2

1+ 3112937368119909

2

2

120573 = 032124

(37)

However119881(1199091 119909

2) and120573 cannot satisfy the conditions in (21)

exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore

Ω

119881= (119909

1 119909

2) isin R2

| 119881 (119909

1 119909

2) le 1 (38)

is not an estimate of the ROA of this system

In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows

We first convert 119882[3] to a nearby rational positive semi-definite matrix

119882

[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882

[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix

119882

[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882

[2]

simultaneously with respect to a given tolerance 120591

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and

119882

[2] that satisfy exactly

119881

1(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(39)

Since the equations in (39) are affine in entries of c and

119882

1

119882

2 one can define an affine linear hyperplane

L = c119882[1]119882[2] | 1198811(x c) minusm

1(x)119879 sdot 119882[1] sdotm

1(x) = 0

119881

2(x c) minusm

2(x)119879 sdot 119882[2] sdotm

2(x) minus 119881

3(x c)m

3(x)119879

sdot

119882

[3]

sdotm3(x) = 0

(40)

Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882

[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices

119882

[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le

119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]

43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)

Algorithm 9 Verified Parametric Optimization SolverInput

(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z

gt0 the bound of the common denominator

(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to

construct the SOS programming(iv) 120591 isin R

gt0 the given tolerance

Output

(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3

positive semidefinite

(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882

[119894]

⪸ 0 1 le 119894 le 3

(2) (Compute the verified solution c)

(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix

119882

[3] by non-negativetruncated PLDLTPT-decomposition

(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]

(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices

119882

[1]

119882

[2] can beobtained by orthogonal projection if 119882[1]119882

[2] are of full rank or by rational vectorrecovery method otherwise

(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and

119882

[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo

5 Experiments

Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems

Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system

x = f (x) =

[

[

[

[

[

[

[

[

1

2

3

4

5

6

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

minus119909

3

1+ 4119909

3

2minus 6119909

3119909

4

minus119909

1minus 119909

2+ 119909

3

5

119909

1119909

4minus 119909

3+ 119909

4119909

6

119909

1119909

3+ 119909

3119909

6minus 119909

3

4

minus2119909

3

2minus 119909

5+ 119909

6

minus3119909

3119909

4minus 119909

3

5minus 119909

6

]

]

]

]

]

]

]

]

]

]

]

]

]

(41)

Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909

2

1+119909

2

2+119909

2

3minus00001 When 119889 = 4 we obtain a feasible

solution of the associated SDP system Here we just list oneapproximate polynomial

119881 (x) = 081569119909

2

1+ 018066119909

2

2+ 16479119909

2

3+ 21429119909

2

4

+ 12996119909

2

6+ sdot sdot sdot + 067105119909

4

4+ 052855119909

4

5

(42)

Let the tolerance 120591 = 10

minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

the corresponding SOS polynomials Here we only list theLyapunov function

119881 (x) = 41

50

119909

2

1+

9

50

119909

2

2+

33

20

119909

2

3+

107

50

119909

2

4

+

69

100

119909

2

5+

13

10

119909

2

6+ sdot sdot sdot +

67

100

119909

4

4+

53

100

119909

4

5

(43)

Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic

stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain

119881 (x) = 078997119909

2

1+ 019119909

2

2+ 22721119909

2

3

+ 14213119909

2

6+ sdot sdot sdot + 14213119909

4

2+ 071066119909

4

5

(44)

By use of the rational SOS recovery technique described inSection 42 we then obtain

119881 (x) = 79

100

119909

2

1+

19

50

119909

1119909

2+

9

100

119909

1119909

5+

19

100

119909

2

2+

227

100

119909

2

3

+

199

100

119909

2

4+

9

100

119909

2119909

5+

71

100

119909

2

5

+

71

50

119909

2

6+

71

50

119909

4

2+

71

100

119909

4

5

(45)

which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied

Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10

minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds

Example 11 ([41 Example A]) Consider a nonlinear contin-uous system

x = f (x) = [

1

2

] = [

minus119909

2

119909

1+ (119909

2

1minus 1) 119909

2

] (46)

Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909

1 119909

2)

with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909

2

1+ 119909

2

2minus 00001 When 119889 = 2 we obtain

119881 (x) = 14957119909

2

1minus 07279119909

1119909

2+ 11418119909

2

2 (47)

Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)

Example 119899 Num deg(119881) Time(s)

1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830

Table 2 Algorithm performance on benchmarks (ROA)

Example119899 Num deg(119881)

120573 120573

Time(s)

1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915

Let the tolerance 120591 = 10

minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function

119881 (x) = minus

62

85

119909

1119909

2+

127

85

119909

2

1+

97

85

119909

2

2 (48)

Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified

estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909

1 119909

2) = 119909

2

1+ 119909

2

2

When 119889 = 2 we obtain

119881 (119909

1 119909

2) = 06174455119909

2

1minus 040292119909

1119909

2+ 043078119909

2

2

120573 = 13402225

(49)

Let the tolerance 120591 = 10

minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10

5By use of the rational SOS recovery technique described inSection 42 we obtain

119881 (119909

1 119909

2) =

6732

10903

119909

2

1minus

4393

10903

119909

1119909

2+

42271

10903

119909

2

2

120573 =

6701

5000

(50)

which exactly satisfy the conditions inTheorem 6Therefore

Ω

= (119909

1 119909

2) isin R2

|

119881 (119909

1 119909

2) le 1 (51)

is a certified estimate of the ROA of the given system

Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =

10

minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and

120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds

6 Conclusion

In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system

Acknowledgments

This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)

References

[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997

[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000

[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000

[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006

[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005

[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012

[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003

[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002

[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009

[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009

[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991

[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011

[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004

[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007

[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008

[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989

[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971

[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001

[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985

[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003

[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002

[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006

[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools

[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004

[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005

[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008

[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012

[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007

[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002

[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998

[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010

[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002

[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003

[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011

[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012

[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219

[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005

[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008

[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of