robust tracking control for a class of mimo nonlinear systems...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control (in press)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1204

Robust tracking control for a class of MIMO nonlinearsystems with measurable output feedback

Ya-Jun Pan1,*,y, Horacio J. Marquez2 and Tongwen Chen2

1Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada B3J 2X42Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4

SUMMARY

This paper proposes a robust output feedback controller for a class of nonlinear systems to track a desiredtrajectory. Our main goal is to ensure the global input-to-state stability (ISS) property of the tracking errornonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances.Our approach consists of constructing a nonlinear observer to reconstruct the unavailable states, and thendesigning a discontinuous controller using a back-stepping like design procedure to ensure the ISSproperty. The observer design is realized through state transformation and there is only one parameter tobe determined. Through solving a Hamilton–Jacoby inequality, the nonlinear control law for the firstsubsystem specifies a nonlinear switching surface. By virtue of nonlinear control for the first subsystem, theresulting sliding manifold in the sliding phase possesses the desired ISS property and to certain extent theoptimality. Associated with the new switching surface, the sliding mode control is applied to the secondsubsystem to accomplish the tracking task. As a result, the tracking error is bounded and the ISS propertyof the whole system can be ensured while the internal stability is also achieved. Finally, an example ispresented to show the effectiveness of the proposed scheme. Copyright # 2007 John Wiley & Sons, Ltd.

Received 8 September 2004; Revised 22 February 2007; Accepted 13 March 2007

KEY WORDS: tracking control; input-to-state stability; output feedback; nonlinear systems; uncertainties;nonlinear observers

1. INTRODUCTION

Output feedback tracking control has been the subject of constant research over the past severaldecades. Despite these efforts, robust tracking of general nonlinear systems remains an open

*Correspondence to: Ya-Jun Pan, Department of Mechanical Engineering, Dalhousie University, Halifax, NS, CanadaB3J 2X4.

yE-mail: [email protected]

Contract/grant sponsor: NSERC and Syncrude Canada

Copyright # 2007 John Wiley & Sons, Ltd.

problem. In the literature, several approaches have been proposed to deal with the outputfeedback control in the presence of structured or unstructured uncertainties: adaptive controlapproach [1, 2], variable structure control approach [3], and output dynamics controller withalmost disturbance decoupling [4], etc. However, the adaptive control approach can only dealwith systems with constant parametric uncertainties. Oh and Khalil considered a nonlinearsingle-input single-output (SISO) system that can be represented by an input–output model andapplied a tracking error estimator [3]. In [4], Marino and Tomei considered SISO systems withnonlinearities that depend on outputs only.

Assuming that it is not possible to have a sensor for each state variable, if the controller is inthe state feedback form, then it is necessary to design a state observer to estimate the internalstates. Several observers have been proposed for linear and nonlinear systems. In the presence ofdisturbances and model uncertainty, high-gain observer has the advantage that they can acquirethe state information while neglecting the influence of these effects. In [3, 5, 6], different types ofobservers based on high-gain estimation error feedback were proposed for SISO systems andmultiple-input multiple-output (MIMO) systems with both linear or highly nonlinear terms.Due to their high gains in the feedback form, these observers are effective in ensuring theconvergence of estimation errors so that asymptotic states can be obtained for the controllerdesign.

In sliding mode control (SMC), switching surface design and discontinuous reaching controllaw are two of the control issues. A common practice in SMC is to design a switching surfaceaccording to the null space dynamics, which must ensure a stable sliding manifold when thesystem is in the sliding mode [7]. For known linear time-invariant (LTI) systems, such design canbe done by pole placement [8] or LQR based approach [9]. For known nonlinear systems,nonlinear optimal design can be applied [10]. However, if there exist uncertainties in the nullspace nonlinear dynamics, switching surface design becomes extremely difficult. Traditionally,the reaching control law is to force the system to reach and stay on the switching surface.Nevertheless, this feature alone is no longer sufficient in the presence of unmatcheduncertainties. Due to the effect of the unmatched uncertainties, the nonlinear dynamics maybecome divergent in a period shorter than the reaching time, if the input-to-state stability (ISS)property does not hold during the reaching phase. Hence, ISS property should be guaranteedeither in the sliding phase or in the reaching phase.

In this paper, a class of nonlinear systems with null space dynamics and range space dynamicsare considered for the tracking control task. Assuming that the full state is not available formeasurement, the main objective of the paper is to ensure global ISS of the tracking errornonlinear dynamics while achieving a small tracking error bound. The features of our approachare the following: (i) a nonlinear observer is designed in which only one parameter needs to bedetermined; (ii) the resulting sliding manifold in the sliding phase possesses the desired ISSproperty and to certain extent the optimality through solving a Hamilton–Jacoby inequality;(iii) associated with the new switching surface, the SMC applied to the second subsystemachieves the desired tracking. As a result, the tracking error is bounded and the ISS property ofthe whole system can be ensured while the internal stability of the system states is also ensured.Usually, a discontinuous term is used to handle the matched L1½0;1Þ type system disturbancewhere the upper-bound knowledge is available [11, 12]. However, in this paper, a discontinuousterm is used to ensure convergence of the error dynamics resulting from the observer and thedesired trajectory in which there are no uncertain terms in the error dynamics. In the example, itis shown that the tracking error is small and bounded under the proposed controller.

Y.-J. PAN, H. J. MARQUEZ AND T. CHEN

Copyright # 2007 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control (in press)DOI: 10.1002/rnc

The paper is organized as follows. In Section 2, the problem formulation is presented whilesome notation and preliminaries are listed. In Section 3, a nonlinear observer is designed indetail and the property of the observer is also analysed. In Section 4, a discontinuous controlleris proposed and even more the stability analysis is given. Section 5 shows an illustrativeexample. Section 6 draws the conclusions.

2. PROBLEM FORMULATION

2.1. Notation and preliminaries

In this section, we introduce our notation and collect some preliminary results that will beneeded later.

Rn denotes the n-dimension real vector space; jj # jj is the Euclidean norm or induced matrixnorm; lmaxðAÞ and lminðAÞ denote the maximum and minimum eigenvalue of the matrix A;respectively; fAg%n%%n represents the first n rows and n columns in A; and fAg

%m%

%n represents the last

m rows and n columns in A: Aþ denotes the left inverse of the matrix A 2 Rn%m such thatAþA ¼ I :

Positive definiteness ðP > 0Þ: Let P 2 Rn%n be a symmetric matrix. P is said to be positivesemidefinite if xTPx50;8x 2 Rn: If in addition xTPx=0;8x=0; then P is said to be positivedefinite. The matrix P is positive definite if and only if all its eigenvalues are positive.

Class K; class K1; and class KL functions [13]: A continuous function g: ½0; aÞ ! ½0;1Þ issaid to belong to class K if it is strictly increasing and gð0Þ ¼ 0: It is said to belong to class K1if a ¼ 1 and gðrÞ ! 1 as r ! 1: It is said to belong to class KL if, for each fixed s; themapping gðr; sÞ belongs to class K with respect to r and, for each fixed r; the mapping gðr; sÞ isdecreasing with respect to s and gðr; sÞ ! 0 as s ! 1:

ISS [14, 15]: Consider a nonlinear dynamical system of the form

’x ¼ fðx; uÞ ð1Þ

The system in (1) is said to be locally input-to-state stable if there exist a class KL function b; aclass K function g; and constants k1; k2 2 Rþ such that

jjxðtÞjj4bðjjx0jj; tÞ þ gðjjuT ð#ÞjjL1Þ 8t50; 04T4t ð2Þ

for all x0 2 D and u 2 Du satisfying: jjx0jj5k1 and supt>0jjuT ðtÞjj ¼ jjuT jjL15k2; 04T4t: It issaid to be input-to-state stable or globally ISS if D ¼ Rn; Du ¼ Rm; and (2) is satisfied for anyinitial state and any bounded input u:

ISS Lyapunov functions [14, 15]: A continuous function V : D ! R is an ISS Lyapunovfunction on D for (1) if and only if there exist class K functions c1; c2; c3; and c4 such that thefollowing two conditions are satisfied:

c1ðjjxjjÞ4VðxÞ4c2ðjjxjjÞ 8x 2 D; t > 0

DxVðxÞfðx; uÞ4( c3ðjjxjjÞ þ c4ðjjujjÞ 8x 2 D; u 2 Du

V is an ISS Lyapunov function if D ¼ Rn; Du ¼ Rm; and c1; c2; c3; and c4 2 K1:

ROBUST TRACKING CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS


2.2. Problem formulation

The following MIMO nonlinear cascade system with uncertainties is considered:

’x1 ¼ f1ðt;x1Þ þ B1ðtÞx2 þ G1ðt;x1Þd1ðtÞ

’x2 ¼ f2ðt;xÞ þ B2ðtÞ½uþ gðt;xÞ) þ G2ðt; xÞd2ðtÞ

y ¼ x1

ð3Þ

where x1 2 Rn is the null space dynamics and x2 2 Rm is the range space dynamics; u 2 Rm

denotes the control input; d1 2 Rp; and d2 2 Rq are the external disturbances. The mappingsf1ðt; x1Þ 2 Rn; f2ðt; xÞ 2 Rm; B1ðtÞ 2 Rn%m; B2ðtÞ 2 Rm%m; G1ðt;x1Þ 2 Rn%p; and G2ðt; xÞ 2 Rm%q

are known. gðx; tÞ 2 Rm is the matched uncertainty. We assume that m4n: System (3) isassumed to satisfy the following assumptions.

Assumption 1There exist two positive constants a1 and a2 such that 8x1 2 Rn; 8t > 0;

05a21Im4BT1 ðtÞB1ðtÞ4a

22Im ð4Þ

where Im is the identity matrix. The time derivative of B1ðtÞ is uniformly bounded. B2ðtÞ isassumed to be invertible and uniformly bounded. The functions G1ðt;x1Þ and G2ðt; xÞ areuniformly bounded.

Assumption 2The uncertainties d1ðtÞ; d2ðtÞ; and gðx; tÞ are bounded as

jd1ðtÞj4b1; jd2ðtÞj4b2; jgðx; tÞj4bZ ð5Þ

where b1; b2; and bZ are known positive constants.

Assumption 3The mapping functions f1ðt; x1Þ and f2ðt;xÞ are globally Lipschitz with respect to x1 and x;respectively.

2.3. Control objective

The system is required to track the known reference model: y ) yd ¼ x1d ; i.e. the x1 subsystemis required to track the desired reference model

’x1d ¼ fdðx1d ; rðtÞ; tÞ ð6Þ

where rðtÞ is a smooth reference input. Define the tracking error as z1 ¼ x1 ( x1d : The controlobjective is to obtain ISS stability with respect to the disturbances and attenuate the disturbanceinfluence d ¼ ½d1; d2;g)T on the tracking error z1: Furthermore, we have the following assumption.

Assumption 4There exists a function g1ð#Þ such that the system dynamics ’n ¼ g1ðn; tÞ is asymptotically stable.Then we can get the following equation:

f1ðt;x1Þ ( fdðt; x1d ; rðtÞÞ ¼ g1ðt; z1Þ þ B1ðtÞfðt;x1;x1d ; rðtÞÞ ð7Þ

where fð#Þ is a smooth function with respect to its arguments.



From (3) and (6), the system with the error dynamics of z1 can be expressed as

’z1 ¼ g1ðt; z1Þ þ B1ðtÞ½x2 þ fðx1;x1d ; rðtÞ; tÞ) þ G1ðt;x1Þd1ðtÞ

’x2 ¼ f2ðt;xÞ þ B2ðtÞ½uþ gðx; tÞ) þ G2ðt;xÞd2ðtÞ

y ¼ x1

ð8Þ

Note that z1 ¼ y( x1d is also available.

3. NONLINEAR OBSERVER DESIGN

For system (3), we can first construct a nonlinear observer according to the work in [6, 16]. Theidea is to construct an observer through a state transformation to convert the nonlinear systeminto a new form such that the observer gain can be designed in a straightforward manner. Theobserver design in this paper is an extension of the work in [6]. The extension of the observerconstruction is applied to the system with a more general representation of the disturbance termwhile B1ðtÞ is not a state-dependent function.

The system in (3) can be rewritten as follows:

’x ¼ fðt;xÞ þ hðt; uÞ þ BðtÞxþ Gðt; xÞdðx; tÞ

y ¼Cx :¼ x1ð9Þ

where

fðt;xÞ ¼f1ðt;x1Þ

f2ðt;xÞ

" #

; hðt; uÞ ¼0

B2ðtÞu

" #

; BðtÞ ¼0 B1ðtÞ

0 0

" #

Gðt; xÞ ¼G1ðt; x1Þ 0 0

0 G2ðt;xÞ B2ðtÞ

" #

; dðx; tÞ ¼

d1ðtÞ

d2ðtÞ

gðt;xÞ

2

664

3

775; CT ¼

In

0m

" #

Define the transformation matrix TðtÞ; the matrices Dy; A; and %C as

TðtÞj2n%ðnþmÞ ¼In 0

0 B1ðtÞ

" #

; Dy ¼In 0

0Iny

2

64

3

75; A ¼0 In

0n 0

" #

; %CT ¼In

0n

" #

Hence,

wðtÞ ¼ TðtÞx ¼x1

B1ðtÞx2

" #

; TðtÞBðtÞ ¼ ATðtÞ; DyAD(1y ¼ yA; %CT %CDy ¼ %CT %C



Denote TþðtÞ as the left inverse of the matrix TðtÞ: The w system can be written as

’wðtÞ ¼TðtÞ’xþ ’TðtÞx

¼TðtÞ½fðt;xÞ þ hðt; uÞ þ BðtÞxþ Gðt;xÞdðx; tÞ) þ ’TðtÞx

¼Awþ TðtÞ½fðt;xÞ þ hðt; uÞ þ Gðt;xÞdðx; tÞ) þ ’TðtÞx

¼Awþ T ½fðt;TþwÞ þ hðt; uÞ þ Gðt;TþwÞdðTþw; tÞ) þ ’TðtÞTþw

y ¼ %Cw

ð10Þ

Then the observer for the transformed w system in (10) can be constructed as

’#wðtÞ ¼A #wþ T ½fðt;Tþ #wÞ þ hðt; uÞ) þ ’TðtÞTþ #w

þ yD(1y P(1 %CTðy( %C #wÞ ð11Þ

where P is the symmetric positive-definite solution of the following algebraic Lyapunovequation:

Pþ ATPþ PA( %CT %C ¼ 0 ð12Þ

Theorem 1Assume that the system in (9) satisfies Assumptions 1 and 2. Then the estimation error of thestates has the following property:

jjexðtÞjj ¼ jjxðtÞ ( #xðtÞjj4kyjjexð0Þjjþ bdd ð13Þ

where

ky ¼ mþt y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilmaxðPÞlminðPÞ

s

e(ððy(c1Þ=2Þtmt; bd ¼ mþt

c2yðy( c1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilminðPÞ

p ; d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 þ b

22 þ b

2Z

q

and c15y; c2; mþt ; mt are positive constants.

ProofSee Appendix. &

From (10) and ’#w ¼ TðtÞ’#xþ ’TðtÞ #x; the observer to the original coordinate is’#xðtÞ ¼Tþ½’#w( ’TðtÞ #x)

¼TþfA #wþ T ½fðt;Tþ #wÞ þ hðt; uÞ) þ ’TðtÞTþ #wþ yD(1y P(1 %CTðy( %C #wÞ ( ’TðtÞ #xg

¼TþAT #xþ fðt; #xÞ þ hðt; uÞ þ yD(1y P(1 %CTðy( C #xÞ

¼BðtÞ #xþ fðt; #xÞ þ hðt; uÞ þ yTþD(1y P(1 %CTðy( C #xÞ

y ¼ x1

ð14Þ



Hence, the estimation error dynamics in the x-coordinate with exðtÞ ¼ xðtÞ ( #xðtÞ becomes

’ex ¼ fðt;xÞ ( fðt; #xÞ þ BðtÞx( BðtÞ #xþ Gðt;xÞdðx; tÞ ( yTþðtÞD(1y P(1 %CTðy( C #xÞ

¼BðtÞex þ fðt;xÞ ( fðt; #xÞ þ Gðt; xÞdðx; tÞ ( yTþðtÞD(1y P(1 %CTCex

Remark 1From (13) in Theorem 1, it is obvious that the larger the value of the observer gain y; the smallerthe value ky: Hence, the influence of the initial error condition jjexð0Þjj on the state estimationerror bound jjexðtÞjj is smaller.

Based on the nonlinear observer designed in this section, the corresponding controller is thenproposed in the next section to ensure the tracking performance of the nonlinear system withmeasurable output feedback.

4. CONTROLLER DESIGN AND STABILITY ANALYSIS

Before the controller design, we would like to rewrite the observer dynamics in (14) as

’#x1 ¼ f1ðt; #x1Þ þ B1ðtÞ #x2 þ wn’#x2 ¼ f2ðt; #xÞ þ B2ðtÞuþ wm ð15Þ

where

wn ¼ fyTþðtÞD(1y P

(1 %CTg%n%%nðy( C #xÞ ¼4Enðy( %C #wÞ ¼ En %Ce ¼ En %CD(1y %e

denotes the n-vector with the first n elements in the vector yTþðtÞD(1y P(1 %CTðy( C #xÞ and

wm ¼ fyTþðtÞD(1y P

(1 %CTg%m%

%nðy( C #xÞ ¼

4Emðy( %C #wÞ ¼ Em %Ce ¼ Em %CD(1y %e

represents the m-vector with the last m elements in the vector yTþðtÞD(1y P(1 %CTðy( C #xÞ:

According to the structure in (8), and comparing the desired target in (6) and the observerdynamics in (15), we have the following error dynamics:

’#z1 ¼ g1ðt; #z1Þ þ B1ðtÞ½ #x2 þ fð #x1; x1d ; rðtÞ; tÞ) þ wn’#x2 ¼ f2ðt; #xÞ þ B2ðtÞuþ wm

ð16Þ

where #z1 ¼4#x1 ( x1d : For brevity, wn ¼

4H%e with H ¼ En %CD(1y :

The controller design is separated into two steps: (1) design a desired #xn2 for the errordynamics of the null space dynamics #z1 to facilitate the switching surface design; and (2) design acontroller u for the whole system in (16) with ISS stability.

Step 1: The task in this step is to find the desired #xn2 :

Theorem 2The tracking error norm, jjz1ðtÞjj; tends, in finite time, to a ball Br defined as

Br ¼ fz1ðtÞ : jjz1ðtÞjj4r23d2 ¼ rg



where r3 is a positive constant and d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 þ b

22 þ b

2Z

q; if the following sliding mode

holds:

rð #x; x1d ; tÞ ¼ #x2 ( #xn2 ¼ #x2 þBT1 ðtÞ

r1ðt; #x1;x1dÞðD#z1VÞ

T þ fð #x1; x1d ; rðtÞ; tÞ ð17Þ

where Vð#z1; tÞ; 8#z1 2 Rn; and t50 is a positive-definite smooth solution of the followingHamilton–Jacoby inequality:

DtV þ ðD#z1VÞg1 ( ðD#z1VÞB1B

T1

r1ðD#z1VÞ

T þ1

4r21ðD#z1VÞHH

TðD#z1VÞT þ #zT1 #z140 ð18Þ

with r1ð#z1;x1d ; tÞ > 0:

ProofConstruct a second Lyapunov function as V1 ¼ V0 þ V ; where Vð#z1; tÞ; 8#z1 2 Rn; t50 is apositive-definite smooth Lyapunov function to be determined. Design

#xn2 ¼ (BT1 ðtÞ

r1ðt; #x1;x1d ÞðD#z1VÞ

T ( fð#Þ

The derivative of Vð#Þ is’V ¼DtV þ ðD#z1VÞ½g1 þ B1ð #x2 þ fÞ þ wn)

¼DtV þ ðD#z1VÞg1 ( ðD#z1VÞB1BT1r1

ðD#z1VÞT þ ðD#z1VÞH%e

¼DtV þ ðD#z1VÞg1 ( ðD#z1VÞB1B

T1

r1ðD#z1VÞ

T þ r21%eT%e

þ1

4r21ðD#z1VÞHH

TðD#z1VÞT (

1

2r1HT ðD#z1VÞ

T ( r1%e""""

""""

""""

""""2

4DtV þ ðD#z1VÞg1 ( ðD#z1VÞB1BT1r1

ðD#z1VÞT þ r21%e

T%e

þ1

4r21ðD#z1VÞHH

TðD#z1VÞT ð19Þ

If there is a solution of V such that the inequality in (18) is satisfied, then (19) becomes

’V4( #zT1 #z1 þ r21%e

T%e ð20Þ

Using (20) and (A1), the derivative of V1 becomes

’V1 ¼ ’V0 þ ’V

4 ( y%eTP%eþ %eTPDyT ½fðt;TþwÞ ( fðt;Tþ #wÞ) þ %eTPDyTGdþ %eTPDy ’TTþD(1y %e

( #zT1 #z1 þ r21%e

T%e

4 ( ylminðPÞjj%ejj2 þ lmaxðPÞðlf þ ltÞjj%ejj2 þ %eTPDyTGd( #zT1 #z1 þ r21%e

T%e



4 ( ½ylminðPÞ ( r21 ( lmaxðPÞðlf þ ltÞ)jj%ejj2 ( #zT1 #z1

þ1

4r22%eTPDyTGðPDyTGÞT%eþ r22jjdjj

2

4 ( ylminðPÞ ( r21 ( lmaxðPÞðlf þ ltÞ (1

4r22lmaxðPÞ2lg

# $jj%ejj2 ( #zT1 #z1 þ r

22jjdjj

2

4 ( #zT1 #z1 þ r22jjdjj

2 ð21Þ

where y is selected such that

ylminðPÞ ( r21 ( lmaxðPÞðlf þ ltÞ (1

4r22lmaxðPÞ2l2g50 ð22Þ

Using ex;1 ¼ x1 ( #x1 ¼ z1 ( #z1; jj#z1jj25jjz1jj2 ( jjex;1jj2; and ex;1 ¼ %e1 according to ex ¼ TþD(1y %e;(21) becomes

’V14( zT1 z1 þ jjex;1jj2 þ r22jjdjj

2 ¼ (zT1 z1 þ jj%e1jj2 þ r22jjdjj

24( zT1 z1 þ jj%ejj2 þ r22jjdjj

2 ð23Þ

From (A2), jj%ejj is bounded as

jj%ejj4lmaxðPÞlg

½lminðPÞy( lmaxðPÞðlf þ ltÞ)jjdjj ð24Þ

where lminðPÞy( lmaxðPÞðlf þ ltÞ > 0 according to (22). Hence, (23) becomes’V14( zT1 z1 þ r

23jjdjj

2 ð25Þ

where

r3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir22 þ kd

q

and

kd ¼l2g

lminðPÞlmaxðPÞ

y( ðlf þ ltÞ# $2

Note that limy!1 kd ¼ 0: Equation (25) shows that the tracking error norm in the z1 subsystem,jjz1ðtÞjj; tends, in finite time, to a ball Br defined by

Br ¼ fz1ðtÞ : jjz1ðtÞjj4r23d2 ¼ rg

where d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 þ b

22 þ b

2Z

q: &

Step 2: The task in this step is to design the controller to ensure the ISS stability.

Theorem 3With the switching surface in (17) and the following SMC law,

u ¼ uc þ us ð26Þ

uc ¼ (B(12 ½Dtrþ ðDx1drÞ ’x1d þ Sðf1 þ B1 #x2 þ wnÞ þ f2 þ wm) ð27Þ



us ¼ (kdBT2 r

jjBT2 rjjð28Þ

where Sð #x1; x1d ; tÞ ¼ D #x1r 2 Rm%n; and kd > 0 is a positive constant, the system is globally ISS

stable with respect to the external disturbance inputs and the tracking error norm, jjz1jj; isbounded in Br as defined in Theorem 2.

ProofConstruct a Lyapunov function V2 ¼ 12 r

Tr: Define Sð #x1;x1d ; tÞ ¼ D #x1r 2 Rm%n and D #x2r ¼ Im

holds. Then

’V2 ¼ rT ’r ¼ rT½Dtrþ ðDx1drÞ ’x1d þ ðD #x1rÞ’#x1 þ ðD #x2rÞ’#x2)

¼ rT½Dtrþ ðDx1drÞ’x1d þ ðD #x1rÞðf1 þ B1 #x2 þ wnÞ þ Imðf2 þ B2uþ wmÞ)

¼ rT½Dtrþ ðDx1drÞ’x1d þ Sðf1 þ B1 #x2 þ wnÞ þ f2 þ B2uþ wm)

4 ( kdjjBT2 rjj ð29Þ

Define a new Lyapunov function V3ðx; #x; x1d ; tÞ ¼ V1ðx; #x;x1d ; tÞ þ V2ð #x;x1d ; tÞ: From (23) and(29), we have

’V3 ¼ ’V1 þ ’V24( zT1 z1 þ r23jjdjj

2 ( kdjjBT2 rjj4( zT1 z1 þ r

23jjdjj

2 ð30Þ

which implies that the system is globally ISS stable with respect to the external disturbance inputand the tracking error norm, jjz1jj; is bounded in Br in finite time. &

0 5 10 15 20 25

0

5

10

15

20

25

30

Time(sec)

Sta

tes

x11

x11d

x12

x12d

Figure 1. The evolution of the states x1ðtÞ and the desired trajectory x1d ðtÞ:



Generally, it may not be so straightforward to solve the HJI (18) for arbitrary nonlinearsystems [17]. As in [18], the nonlinear problem is simplified to a sequence of linear-quadratic andtime-varying approximating problems. Correspondingly, the method named ApproximatingSequence of Riccati Equations is applied to find the time-varying feedback controllers for theapproximated nonlinear systems. Hence, it is a sort of simplification for solving the nonlinearoptimal control problems. Specifically, for (18) in this paper, if we have further knowledge onthe g1 function as shown in Remark 2, then the inequality can be greatly simplified as well.

Remark 2In the nonlinear uncertain system (16), if g1ð#z1; tÞ can be expressed as F1ð#z1; tÞ#z1; where F1ð#z1; tÞ isa matrix-valued smooth function, then the HJ inequality in (18) can be simplified into thefollowing differential Riccati inequality:

1

2’Qþ

1

2ðQF1 þ FT1 QÞ (Q

B1BT1r1

(1

4r21HHT

# $Qþ In%n40 ð31Þ

0 5 10 15 20

0

0.5

1

1.5

2

(a) Time(sec)

x 11

and

desi

red

x 11d

x11

x11d

0 5 10 15 20

0

0.2

0.4

(b) Time(sec)

Trac

king

err

or z

11

0 5 10 15 20

0

1

2

3

4

(c) Time(sec)

x 12

and

desi

red

x 12d

x12

x12d

0 5 10 15 20

0

0.5

1

(d) Time(sec)

Trac

king

err

or z

12

Figure 2. (a) x11ðtÞ and desired x11d ðtÞ; (b) tracking error z11ðtÞ ¼ x11ðtÞ ( x11d ðtÞ; (c) x12ðtÞ anddesired #x12d ðtÞ; and (d) tracking error z12ðtÞ ¼ x12ðtÞ ( x12d ðtÞ:



where Qð#z1; tÞ 2 Rn%n is a symmetric positive-definite smooth matrix. The z1 subsystem warrantsthe tracking error norm, jjz1jj; to be bounded in Br (as defined in Theorem 2) by the nonlinearcontrol law

#xn2 ¼ (1

r1BT1Q#z1 ( f ð32Þ

which also specifies the switching surface as r ¼ #x2 ( #xn2 :

In Remark 2, the solution V is treated as a linear quadratic form 12 #zT1Q#z1: In (31), the matrixH is

dependent on B1ðtÞ matrix. Hence, for terms B1ðtÞTB1ðtÞ and HHT; the bound as denoted in (4) ofAssumption 1 can be applied to simplify the calculation of (31) for various specific examples.

0 1 2 3 4 5

0

0.5

1

1.5

2

(a) Time(sec)

x11 a

nd

estim

ate

d x

11

x11

estimated x11

0 1 2 3 4 5

0

5x 10

(b) Time(sec)

Estim

atio

n e

rro

r e

11

0 1 2 3 4 5

0

1

2

3

4

(c) Time(sec)

x1

2 a

nd

estim

ate

d x

12

x12

estimated x12

0 1 2 3 4 5

0

5x 10

(d) Time(sec)

Estim

atio

n e

rro

r e

12

Figure 3. (a) x11ðtÞ and estimated #x11ðtÞ; (b) estimation error e11ðtÞ ¼ x11ðtÞ ( #x11ðtÞ; (c) x12ðtÞ andestimated #x12ðtÞ; and (d) estimation error e12ðtÞ ¼ x12ðtÞ ( #x12ðtÞ:



Remark 3Note that the value of r3 determines the bound of the tracking error and it depends on r2 and y:According to the expression of r3 as shown in the proof of Theorem 2, for a fixed r2; a larger yvalue in the observer design results a smaller tracking error bound r3:

Remark 4In the observer design, the parameter y is the only key parameter to be determined. It should bedesigned to satisfy the two conditions in (22) and y ¼ maxf1; c1g simultaneously. Note that thematrix H in (18) is related to the parameter y which is a constant, as a result the solution V from(18) or (31) is dependent on the value y as well as other parameters r1 and r1: However, thedesign of y is independent on the inequalities (18) and (31). It is difficult to get the generalapproach in how to achieve the solutions from (18) and (31). Future investigation for thegeneral case to get the smooth solution V will be studied.

0 10 20 30

0

2

4

6

8

(a) Time(sec)

x 21

and

estim

ated

x21

x21

estimated x21

0 10 20 30

0

1

2

(b) Time(sec)

Est

imat

ion

erro

r e 2

1

0 10 20 30

0

5

10

15

20

25

(c) Time(sec)

x 22

and

estim

ated

x22

x22

estimated x22

0 10 20 30

0

1

2

3

(d) Time(sec)

Est

imat

ion

erro

r e 2

2

Figure 4. (a) x21ðtÞ and estimated #x21ðtÞ; (b) estimation error e21ðtÞ ¼ x21ðtÞ ( #x21ðtÞ; (c) x22ðtÞ andestimated #x22ðtÞ; and (d) estimation error e22ðtÞ ¼ x22ðtÞ ( #x22ðtÞ:



Remark 5In the controller design, the discontinuous unit vector control law us in (28) may causechattering when the system enters the sliding mode in a finite time. In order to eliminate thechattering phenomenon which is harmful for the implementation in hardware, us can bemodified as

us ¼ (kdGTr

jjGTrjjþ eð33Þ

where e is a positive constant.

5. NUMERICAL EXAMPLE

Consider a nonlinear uncertain cascaded system

’x1 ¼ f1ðt;x1Þ þ B1ðtÞx2 þ G1ðt;x1Þw1

’x2 ¼ f2ðt;xÞ þ B2ðtÞ½uþ gðt;xÞ) þ G2ðt; xÞw2ð34Þ

0 5 10 15 20 25 30

0

5

10

(a) Time(sec)

Con

trol

u1

0 5 10 15 20 25 30

0

50

(b) Time(sec)

Con

trol

u2

Figure 5. The evolution of the control profile uðtÞ: (a) u1ðtÞ and (b) u2ðtÞ:



where

f1ðt; x1Þ ¼ F1 # x1 ¼0 1

(2 (4

" #x11

x12

" #

; G1ðt; x1Þ ¼cosðx12Þ sinðx12Þ

sinðx11Þ cosðx11Þ

" #

G2ðt;xÞ ¼sinðx11Þ cosðx12Þ

cosðx21Þ sinðx22Þ

" #

; w1 ¼ ½sinð0:1tÞ;(sinð0:1tÞ)T

f2ðt;xÞ ¼ ½x11 sinðx21Þ;x12 sinðx22Þ)T; w2 ¼ ½( sinð0:1tÞ; sinð0:1tÞ)T

B1ðtÞ ¼ ð1þ 0:5 sinðtÞÞI2; B2ðtÞ ¼ ð1þ 0:5 cosðtÞÞI2

gðt;xÞ ¼ ½sinðx11Þ þ sinðx12Þ;( cosðx21Þ ( cosðx22Þ)T

The initial condition is as x1ð0Þ ¼ ½5; 5)T; x2ð0Þ ¼ ½3; 3)T:

0 2 4 6 8 10 12 14 16 18 20

0

50

100

150

(a) Time(sec)

u 1

0 2 4 6 8 10 12 14 16 18 20

0

50

100

150

(b) Time(sec)

u 2

Figure 6. The evolution of the control profile uðtÞ: (a) u1ðtÞ and (b) u2ðtÞ:



The observer is designed as in (14). The transformation matrix TðtÞ; the matrices Dy; A; and %Care as

TðtÞ ¼I2 0

0 B1ðtÞ

" #

; Dy ¼I2 0

0I2y

2

64

3

75; A ¼0 I2

02 0

" #

; %CT ¼I2

02

" #

According to (24), we have the symmetric positive-definite solution

P ¼

1 0 (1 0

0 1 0 (1

(1 0 2 0

0 (1 0 2

0

BBBBB@

1

CCCCCA

y ¼ 80 is selected according to Remark 4 with lf ¼ 4:5622; lt ¼ 1; lg ¼ 1:0028; and r2 ¼ 1:The target trajectory is x11d ¼ 0:2 sinðptÞ and x12d ¼ ’x11d ¼ 0:2p cosðptÞ: The error dynamics

of the x1 subsystem in ð16Þ can be expressed as

’z1 ¼ F1z1 þ B1ðtÞ½x2 þ fðtÞ) þ G1w1

where fðtÞ ¼ ½0;( ’x12d ( 2x11d ( 4x12d )T: The g1 function in Assumption 4 is then a linearfunction as g1ðz1; tÞ ¼ F1z1 where F1 is a stable matrix. In the #z1 subsystem, according to

0 0.5 1 1.5 2 2.5 3

0

1

(a) Time(sec)

σ 1

0 0.5 1 1.5 2 2.5 3

0

5

(b) Time(sec)

σ 2

Figure 7. The evolution of the switching surface rðtÞ: (a) s1ðtÞ and (b) s2ðtÞ:



Remark 2, we first choose Vð#z1; tÞ ¼ 12 #zT1Q#z1; where Q is determined by the differential Riccati

inequality (31). When ’Q ¼ 0; from the linear algebraic matrix inequality1

2ðQF1 þ FT1 QÞ (Q

B1BT1r1

(1

4r21HHT

# $Qþ I2%240 ð35Þ

and using the singular values of the matrices B1 andH ¼ fyTþðtÞD(1y P(1 %CTg%n%n %CD

(1y ; which are

0.5 and 160, respectively, we can get a symmetric positive-definite smooth matrix

Q ¼0:0106 0

0 0:0108

" #

r1 ¼ 0:9659; and r1 ¼ 0:001: Thus from (32), we have

#xn2ðx1;x1d ; tÞ ¼ (ð1=r1ÞBT1Q#z1 ( fðtÞ

The switching surface is r ¼ #x2 ( #xn2 : Then the controller is constructed according to (26) inTheorem 2.

Simulation results are shown as follows. In Figure 1, u ¼ 0 is first applied. It is shown that thetracking task cannot been realized without any controller and the system may be not stable. Hence,it is necessary to design an output feedback controller. In Figure 2, the tracking errors of the statesx11 and x12 are bounded with fast convergence. In Figures 3(a) and (b) and 4(a) and (b), theestimated states from the observer and the real states are compared. The estimation error convergesto a small bound as shown in Figures 3(c) and (d) and 4(c) and (d). The control profile is as shownin Figure 5. Compared with the control profile in Figure 6, we get a smoother control signal whichis easier to be implemented after introducing the smoothing strategy. Also the switching surfaceprofile is shown in Figure 7, in which we can see that it reaches zero in finite time.

6. CONCLUSIONS

We have considered the tracking control problem for a class of nonlinear systems with unknownsystem uncertainties and external disturbances, and proposed a robust output feedback controllaw based on a nonlinear observer that achieves input-to-state stability (ISS). The designprocedure is based on a back-stepping-like procedure. First a stable switching surface isallocated by solving a HJ inequality for the first subsystem. Then a discontinuous controller isconstructed to ensure the convergence of the Lyapunov function. As a result, the tracking erroris bounded.

APPENDIX

Proof of Theorem 1

Define eðtÞ ¼ wðtÞ ( #wðtÞ and consider a transformation on the error as %e ¼ Dye: According to(10) and (11), the estimation error dynamics %e becomes

’%e ¼ yðA( P(1 %CT %CÞ%eþ DyT ½fðt;TþwÞ ( fðt;Tþ #wÞ)

þ DyTGðt;TþwÞdðTþw; tÞ þ Dy ’TðtÞTþD(1y %e



Construct a Lyapunov candidate as V0 ¼ 12 %eTP%e; where P is the solution of (12). Then the

derivative ’V0 becomes

’V0 ¼ ( yV0 (y2%eT %CT %C%eþ %eTPDyT ½fðt;TþwÞ ( fðt;Tþ #wÞ)

þ %eTPDyTGðt;TþwÞdðTþw; tÞ þ %eTPDy ’TðtÞTþD(1y %e

4 ( yV0 þ %eTPDyT ½fðt;TþwÞ ( fðt;Tþ #wÞ)

þ %eTPDyTGðt;TþwÞdðTþw; tÞ þ %eTPDy ’TðtÞTþD(1y %e ðA1Þ

For any y51; we have jjDyT ½fðt;TþwÞ ( fðt;Tþ #wÞ)jj4lf jj%ejj; jjDy ’TTþD(1y jj4lt; jjDyTGjj4lg;and jjdðTþw; tÞjj4d; where lf ; lt; and lg do not depend on y:

Then (A1) can be

’V04( yV0 þ lmaxðPÞðlf þ ltÞjj%ejj2 þ lmaxðPÞlgjj%ejjjjdjj4( ðy( c1ÞV0 þ c2dffiffiffiffiffiffiV0

pðA2Þ

where

c1 ¼ 2lmaxðPÞlminðPÞ

ðlf þ ltÞ

and

c2 ¼ lmaxðPÞlg

ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilminðPÞ

p

If y > maxf1; c1g is selected, then (A2) becomes

dffiffiffiffiffiffiV0

p

dt4(

y( c12

% & ffiffiffiffiffiffiV0

pþ

c2d2

)ffiffiffiffiffiffiffiffiffiffiffiV0ðtÞ

p4e(ððy(c1Þ=2Þt

ffiffiffiffiffiffiffiffiffiffiffiV0ð0Þ

pþ

c2dy( c1

½1( e(ððy(c1Þ=2Þt)

) jj%eðtÞjj4


s

e(ððy(c1Þ=2Þtjj%eð0Þjjþc2d

ðy( c1ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilminðPÞ

p ðA3Þ

Using jj%eðtÞjj4jjeðtÞjj4yjj%eðtÞjj; (A3) becomes

jjeðtÞjj4y


s

e(ððy(c1Þ=2Þtjjeð0Þjjþc2dy


p 4k0yjjeð0Þjjþ b0dd

where

k0y ¼ y


s

e(ððy(c1Þ=2Þt

and

b0d ¼c2y


p



Hence, jjeðtÞjj ¼ jjwðtÞ ( #wðtÞjj4k0yjjeð0Þjjþ b0d : Furthermore, from Assumption 1, jjT

þjj4mþtand jjT jj4mt where mþt and mt are constants. According to wðtÞ ¼ TðtÞxðtÞ and #wðtÞ ¼ TðtÞ #xðtÞ;we have

jjexðtÞjj ¼ jjxðtÞ ( #xðtÞjj ¼ jjTþeðtÞjj4mþt k0ymtjjexð0Þjjþ m

þt b

0dd¼

4kyjjexð0Þjjþ bdd

where ky ¼ mþt k0ymt and bd ¼ m

þt b

0d :

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