Abstract— This article proposes a new approach to stability analysis of nonlinear systems via numerical computation for the Lyapunov’s methods. The presentation includes the review of the Lyapunov’s methods, the quadratic Lyapunov function, the threshold accepting algorithms and illustrative examples, respectively. The proposed approach yields a larger stability region for a polynomial system than an existing method does.

Keywords—Stability, Lyapunov’s methods, Threshold accepting algorithms, Numerical computing.

I. INTRODUCTION

YAPUNOV’S stability methods have been successfully applied for long years by engineers and scientists [1],[2].

The direct method of Lyapunov’s is regarded as clean and concise. Nonetheless, for some systems, finding Lyapunov function is not straightforward. Doing this manually in some cases is very time consuming, perhaps not possible. Once the Lyapunov function is obtained for the system of interest, the next practical issue becomes the region of attraction. In order to find this, some computational approaches, e.g. geometrical, numerical methods etc., have to be applied. For instance, various previous works have been proposed for the construction of Lyapunov functions based on conventional method [3],[4], numerical methods [5],[6] and artificial intelligent methods [7],[8].

It is of interest to find a general tool to assist the construction of the Lyapunov function and the stability region for a nonlinear system. Threshold accepting algorithms are appropriate candidates because of its flexibility and easy to use [9],[10]. The results are compared with the previous work [11] considering the same systems as appeared in [12],[13].

II. LYAPUNOV’S METHODS

A nonlinear system can be represented by

),( txfx =ɺ (1)

for a non-autonomous one, and

Received April 2, 2010. This work has been supported by Suranaree

University of Technology and Rajamangkala University of Technology Isarn, Thailand.

The authors are with the School of Electrical Engineering, Suranaree University of technology, Thailand. (*correspondence : sarawut@sut.ac.th)

)(xfx =ɺ (2)

for an autonomous system. At the equilibrium 0=ex , the

following conditions hold : 0)( =exf and 0=exɺ .

The Lyapunov’s indirect method : consider an autonomous system, the Jacobian at the equilibrium point can be defined as

0

)(

=∂

∂=

exx

f xA (3).

- If all eigenvalues of A are strictly in the left-half

complex plane, then the asymptotic stability at the equilibrium point of the linearized system can be concluded.

- If at least one eigenvalue of A is strictly in the right-half complex plane, then the instability of the linearized system is concluded.

- If all eigenvalues of A are in the left-hand complex plane but at least one of them is on theωj axis, then the linearized system is said to be marginally stable but one cannot conclude anything from the linear approximation.

The Lyapunov’s direct method: By using the Lyapunov’s direct method, the Lyapunov

function )(xV must be found, and the stability can be

concluded without knowing the solutions of the equations governing the systems. )(xV must be scalar, positive definite

and differentiable. A nonlinear system can be said to have a globally

asymptotically stable equilibrium, if these exist

- 0)( >xV

- 0)( <xVɺ

- ∞→)(xV as ∞→x.

Consider a nonlinear mass-spring-damper system described by

03

10 =+++ xkxkxxbxm ɺɺɺɺ (4).

Numerical Approach to Lyapunov’s Stability Analysis of Nonlinear Systems

Using Threshold Accepting Algorithms

Suphaphorn Panikhom, Sarawut Sujitjorn*

L

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ISSN: 1792-4324 58 ISBN: 978-960-474-208-0

The energy function is used as the Lyapunov function candidate expressed by

dxxkxkxmV

x

)(2

1)(

010

2 3∫ ++= ɺx

4

1

2

0

2

4

1

2

1

2

1xkxkxm ++= ɺ (5).

It is clearly seen that )(xV is scalar, differentiable, positive

definite and unbounded. The function )(xV is, hence, the Lyapunov function. Next,

3)()( xbxxbxV ɺɺɺɺɺ −=−=x (6)

is negative definite. Therefore, the global asymptotic stability of the system can be concluded.

According to the indirect method, the quadratic Lyapunov function can be generally applied. It can be expressed by

0)( >= Pxxx TV (7)

,whereas x is the state vector and P is a symmetrically scalar matrix. The following equations must be satisfied

PxxPxxx TT +=)(Vɺ (8)

0)( <−= Qxxx TVɺ (9)

PAPAQ T+= (10)

,and

TQQ = (11).

III. THRESHOLD ACCEPTING ALGORITHMS FOR NONLINEAR

STABILITY ANALYSIS

A. The indirect method

As reviewed earlier, the stability of any linearized systems can be concluded via the quadratic Lyapunov function. Regarding this, the matrices P and Q must exist and satisfy the Lyapunov equation. Generic threshold accepting algorithms [9],[10] have been modified to search for these matrices. Considering the second-order systems, the procedural list below accommodates the task.

Example 1 Consider the system [2]

2111 2 xxx −−=xɺ

1212 xxx −=xɺ (12)

having

−

−=

∂

∂=

= 10

02

0e

xx

fA .

The search spent 4 iterations to obtain

=

5.07892.0793-

-2.07935.9006P and

=

5.9992-1.2575

1.2575-4.1585Q .

Therefore, )(xV and )(xVɺ are of the forms

2122

22

1 12.0793-2.0793-5.0789-5.9006)( xxxxxxV =x (13)

2111.8012-14.3162-14.3162)( 212

2 xxxxxV +=xɺ (14),

whose surface plots are illustrated in Figs. 2(a) and (b), respectively, and the asymptotic stability at the origin can be concluded.

-20-10

010

20

-20-10

010

200

1000

2000

3000

4000

5000

X1

Lyapunov funct ion

X2

-20

0

20

-20-1001020-14000

-12000

-10000

-8000

-6000

-4000

-2000

0

X1

dv

X2

(a) (b)

Fig.2 (a) The Lyapunov function )(xV , (b) the )(xVɺ of example 1.

STEP 0: Initialize solutions P and Q by zeroing all elements.

STEP 1: Generate randomly within the ranges [-10,+10] the elements of P . Determine

the positive definiteness of P . If P is not positive definite, go to STEP 1.

STEP 2: Compute Q based on P . Determine the negative definiteness of Q . If Q is not negative definite, go to STEP 1.

STEP 3: Accept solutions P and Q . Exit.

Fig.1 Threshold accepting algorithms to search for P and Q .

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Example 2 Consider the Van der pol system

21 x−=xɺ

2

2

112 )1( xxx −+=xɺ (15)

having

−

−=

∂

∂=

= 11

10

0e

xx

fA .

By 7 iterations, the search returned

=

8.7261.001-

-1.0013.149P and

=

17.453-3.004

3.004-12.597Q . Hence,

212

22

1 .00131-8.7265-1479.2)( xxxxV =x (16)

212

212 11.8012-14.3162-14.3162)( xxxxxV =xɺ . (17).

Fig. 3 depicts the surface plots of )(xV and )(xVɺ to confirm

the asymptotic stability at the origin of the system.

-20-10

010

20

-20

-10

0

10

200

1000

2000

3000

4000

5000

6000

X1

Lyapunov function

X2

-20-10

010

20

-20

-10

0

10

20

-6000

-4000

-2000

0

2000

X1

dv

X2

(a) (b)

Fig. 3 (a) The Lyapunov function )(xV , (b) the )(xVɺ of example 2.

B. The direct method

This section has an interest in the method proposed by Xiao-Lin and You-Lin [11]. The method considers the polynomial system of the form

=

=

),(

,

yxfdt

dy

ydt

dx

(18),

where

2)()()(),( yxRyxQxPyxf ++= (19)

∑∑∑===

===N

ii

N

ii

N

ii

iii xrxRxqxQxpxP001

)(,)(,)( (20).

This section demonstrates the usefulness of the threshold accepting algorithms to seek for the suitable coefficients of the

Lyapunov function candidate. Moreover, the search approach provides a larger stable region than that provided by the previous method [11]. Example 3 Consider the system

−+−=

=

yxxdt

dy

ydt

dx

3

, (21)

and the Lyapunov function candidate

2

210 )()()(),( yxQyxQxQyxV ++= (22).

Since

32

221

23

103

1

)()](2)([

)]()(2)()([])[(

yxQyxQxQ

yxQxxxQxQxxxQdt

dV

ɺɺ

ɺ

+−+

−−−+−−= (23),

let 0)(2 =xQɺ and ε−=− )(2)( 21 xQxQɺ , one can obtain

−=

=

xcxQ

cxQ

)2()(

)(

1

2

ε (24).

Hence, the Lyapunov candidate can be rewritten as

20 )2()(),( cyxycxQyxv +−+= ε (25)

, having

230

3 )2)4()(()()2( yycxxcxQxxxcdt

dVεεε −+−−+−−−= ɺ (26)

Let ∑ =

−=+−−N

i

ii xi

cxxcxQ1

21230 2

)2)4()(α

εɺ , then

∑ =

−+−−=N

i

ii xi

xc

xcxQ1

212420 22

)2

2()(αε , and

2

1

2123 )2

()()2( yyxi

xxxcdt

dV N

i

ii εα

ε −+−−−= ∑ =

− (27)

The function ),( yxV will be positive definite iff

30 αε ≤≤≤ c and 0,...,,, 12531 ≥−Nαααα . The threshold

accepting procedures for the direct method problem are declared in Fig. 4.

As a result, the dark solid line in Fig. 5 defines the stable region obtained from the search in which 47752.0=c ,

81243.03 =α , 71 10544.1 −×=α and 7101663.2 −×=ε . The dash-

dot line defines the stable region provided by the previous method.

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Fig.5 Stable regions of example 3.

I. CONCLUSION

The paper has demonstrated the usefulness of the search method, i.e. the threshold accepting algorithms, to stability analysis of nonlinear systems. The algorithms are simple, and provide a fast computation to reach for a meaningful solution (local solution). A larger region of attraction can be obtained by the proposed approach. Ongoing research is concentrated on developing more efficient algorithms for the purposes and applications to Lyapunov-based control design optimization.

REFERENCES [1] J.J.E. Slotine and W. Li. Applied Nonlinear Control System. Prentice

Hall Inc., Englewood Cliffs. 1991.

[2] H. K. Khalil. Nonlinear Systems. New York: MacMillan, 1992.

[3] G.H.Golub, S.Nash and C.Van Loan, “A Hessenberg-Schur method for

problem AX+XB=C,” IEEE Transactions on Automatic Control, Vol.

24, No.6, pp.909-913, 1979.

[4] X. A. Wang and W. Dayawansa. “On global Lyapunov functions of

nonlinear autonomous systems”, In Proceedings of The 38th Conference

on Decision & Control, pp.1635-1639, December 1999.

[5] T.Zhaolu and G.Chuanqing, “A numerical algorithm for Lyapunov

equations,” Applied Mathematics and Computation, 202, pp.44–53,

2008.

[6] D. C. Sorensen and Y. Zhou, “Direct method for matrix Sylvester and

Lyapunov equations,” Journal of Applied Mathematics, 2, pp.277–303,

2003.

[7] B. Grosman,, and D. R. Lewin, “Lyapunov-based stability analysis

automated by genetic programming,” Automatica, 45, pp.252-256,

December 2008.

[8] C. Banks, “Searching for Lyapunov functions using genetic

programming,” , http://www.aerojockey.com/files/lyapunovgp.pdf.

[9] G. Duech and T. Scheuer, “Threshold accepting: A general purpose

optimization algorithm appearing superior to simulated annealing.

Journal of Computational Physics, 90, pp.161-175, 1990.

[10] E-G. Talbi, Metaheuristics-From Design to Implementation, Wiley,

2009.

[11] L. Xiao-Lin and J. Yao-Lin, “Numerical algorithm for constructing

Lyapunov functions of polynomial differential systems,” J Appl Math

Comput , 29, pp.247–262, 2009.

[12] J.R. Hewit and C. Storey, “Numerical application of Szego’s method for

constructing Lyapunov functions”, IEEE. Transactions on Automatic

Control, 14, pp.106–108, February 1969.

[13] C.C. Hang and J.A. Chang, “An algorithm for constructing Lyapunov

functions based on the variable gradient method,” IEEE Transactions on

Automatic Control, 15, pp. 510–512, August 1970.

STEP 0: Initialize 0...,,, 12975 =−Nαααα .

STEP 1: Generate possible coefficients randomly:

]10[, 3 ∈αε and ]1010[, 81

−×∈αc . If

30 αε ≤≤≤ c is false, then go to STEP 1.

STEP 2: If Vɺ is not negative definite, then go to STEP 1.

STEP 3: Accept solutions 3 1, , ,cε α α . Exit.

Fig. 4 Threshold accepting algorithms for the direct method.

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x1

y 1

0.96 x12-0.036 x1

4+0.96 x1 y1+0.48 y12-1.0 = 0

COUNT = 2

c=0.47752

alpha3= 0.81243

alpha1= 1.5442e-007

e=2.1163e-007

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