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Stabilization of quasi-one-sided Lipschitz nonlinear systems

Journal: IMA Journal of Mathematical Control and Information

Manuscript ID: IMAMCI-2011-115.R1

Manuscript Type: Original Article

Date Submitted by theAuthor:

18-Mar-2012

Complete List of Authors: Fu, Fengyu; Harbin Institute of Technology , Center for ControlTheory and Guidance TechnologyHou, Mingzhe; Harbin Institute of Technology, Center for ControlTheory and Guidance TechnologyDuan, Guangren; Harbin Institute of Technology, Center for ControlTheory and Guidance Technology

Keywords:feedback control, quasi-one-sided Lipschitz condition, nonlinearsystems, linear matrix inequality

http://mc.manuscriptcentral.com/imamci

Manuscripts submitted to IMA Journal of Mathematical Control and Information

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Responses to the comments of the referees for the paper:


IMA Journal of Mathematical Control and Information IMAMCI-2011-115

Fu Fengyu, Hou Mingzhe and Duan Guangren

We thank both referees for their valuable comments. We have revised our paper to addresstheir concerns. The following is a detailed response to their specific comments. The page,equation and reference numbers refer to those in the revised manuscript, unless otherwiseindicated.

Responses to the comments of Referee #1

1. The first section does not meet my expectation regarding a proper introduction since itleaves (at least) three important question unanswered:

(a) Why do other contributions focus on the observer design and leave the designquestion unanswered? Do they really, i.e. for what purpose have the observers bedesigned / how are the reconstructed states used for control for the other systemclasses?

(b) What is the motivation for the quasi-one-sided Lipschitz condition? Are therecommon classes of systems / control problems when one encounters nonlinearitiesof this class?

(c) What gain do the authors see in their approach?

Further, I would have liked to see an interpretation of the quasi-one-sided condition,i.e. the matrix P, by means of geometry / in comparison with the one-sided condition(or at least a reference).

In total, the introduction lacks some thoroughness in setting the context and to providethe motivation of this contribution

Answer:

(a) Observer design is of paramount importance for nonlinear systems when it isimpossible to measure all the states. As far as we know, the quasi-one-sidedLipschitz condition was introduced only a few years ago. It is not difficult tofind, along the line of Hu [12],[16],[14], that the one-sided Lipschitz condition andthe quasi-one-sided Lipschitz condition are tractable for observer design, henceconsiderable attention has been paid to it. However, for the control design problemIn fact, both the observer design and the controller design have been investigated

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in the available literature for Lipschitz nonlinear systems. However, for the quasi-one-sided Lipschitz nonlinear systems considered in the study, the nonlinearitysatisfies a more relaxed condition than the Lipschitz condition, it is difficult toget elegant results for controller synthesis of this kind of nonlinear systems. Hence,there are few results for the stabilization problem of the quasi-one-sided Lipschitz

nonlinear systems appear in the literatures.

(b) The motivation of studying the quasi-one-sided Lipschitz condition is to formu-late a more relaxed condition and exploit more information of the nonlinearityin order to obtain a simple and practical control synthesis scheme. There area common class of systems/control problems. Indeed, there do exist nonlinearcontrol systems with the nonlinearity which does not satisfy Lipschitz conditionbut can be verified to satisfy quasi-one-sided Lipschitz condition, see Example 4.1in Section 4.

(c) The main contribution of this study is that the separation principle is provedto hold for the output feedback control design problem under a framework of amore relaxed Lipschitz condition, and the corresponding control design schemeis obtained with a relatively easy calculation. This can be seen in a series ofnumerical examples in Section 4.

On the one hand, compared to the one-sided Lipschitz condition introduced in the ref-erence [15], the introduction of the matrix P in the quasi-one-sided Lipschitz conditionis useful for the construction of the Lyapunov function to perform stability analysis,which has been pointed out in every proof of the results derived in reference [16]; Onthe other hand, if we choose the matrix P = I

nand the so-called one-sided Lipschitz

constant matrix M = vIn, where v is the one-sided Lipschitz constant, then the quasi-

one-sided Lipschitz condition becomes the one-sided Lipschitz condition in reference[15]. Hence, replacing the identity matrix by a general positive-definite matrix P inthis condition implies a more relaxed restriction for the nonlinear function. Unfortu-nately, there is no relevant reference introducing the geometric meaning of the matrixP. We hope, however, that the remarks 2.1-2.3 given in Section 2 would help to makethe definition of the quasi-one-sided Lipschitz condition (2.4) more clearer.

In total, we have rewritten the Introduction part as suggested to clearly provide themotivation of this contribution in the revised manuscript.

2. It seems to me, that proof of the results could- and should- be condensed in the presen-tation.

Answer: We have taken this advice and have condensed the proofs, on page 4-6.

3. Finally, an algorithm is given summing up the steps carried out in the previous discus-sion. At this point, the paper suffers from the fact, that the authors do not interprettheir result in any way, besides the numeric example in the following section.

Answer: We have removed the algorithm but added some discussions in remarks3.3-3.7 to interpret our results. We also provide several more numerical examples inSection 4 to further verify the main results.

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4. The third section presents a numerical example in a very brief fashion. However, inmy opinion, the paper does not benefit from it:1) The example is purely academic and the nonlinearity seems kind of trivial. Thevalidity of the example would dramatically increase, if a somehow motivated examplecould be given ( some simple model of a process / plant). This could also strengthen

the significance of the presented results.2) The results are not put into context. The performance is not compared to any otherapproach.Generally, the third section is completely unmotivated. (Besides to show that the proofswere correct and the numerics work.)Answer: We have now added a motivated numerical example concerning a physicalsystem plant in Example 4.3, which is devoted to illustrate that the method is indeedeasy to be applied to the classical Lipschitz nonlinear systems in practice. Another twoacademic examples are presented, in Example 4.1 and Example 4.2, to demonstratethat the proposed method is applicable to a larger class of systems compared with an

existing method in reference [17].

5. Typos: - After equation (1) the dimension of matrix B is incorrect.Answer: This has been corrected as suggested.

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Responses to the comments of Referee #2

1. The last reference in your paper may be corrected.

Answer: We have kept that same reference in [20] but with a different description, on

page 2.

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Fu Fengyu , Hou Mingzhe, Duan Guangren(Center for Control Theory and Guidance Technology,Harbin Institute of Technology, Harbin 150001, China)

March 18, 2012

Abstract

This paper is concerned with the problems of state feedback and output feedback control for a

class of nonlinear systems. The nonlinearity of this class of nonlinear systems is assumed to satisfy

a global quasi-one-sided Lipschitz condition. Sufficient conditions for the existence of state feedbackcontroller and output feedback controller are presented. Methods of calculating the controller gain

matrices are derived in terms of linear matrix inequalities. The effectiveness of our results is tested in

a series of numerical experiments.

Keywords: feedback control; quasi-one-sided Lipschitz condition; nonlinear systems; linear matrixinequality

1 Introduction

Stabilization is one of the most important issues currently under consideration by researchers in thenonlinear control field. It involves three related problems, that is, full state feedback controller design,observer design, and output feedback controller design. Among them, considerable attention has been

paid to the study of the output feedback control problem for nonlinear systems in the literature with verydifferent approaches, due to its importance in many practical applications where measurement of all thestate variables is not possible (see [1] and references therein). Output feedback control design, utilizingestimated state and output as feedback, usually involves the first two problems. Unlike linear systems,separation principle does not generally hold for nonlinear systems, and it is well known that the observerdesign for nonlinear systems by itself is quite challenging. Therefore, the output feedback control problemfor nonlinear systems is much more challenging than stabilization using full-state feedback. Basically, onehas to consider a special class of nonlinear systems, for example, the Lipschitzian nonlinear systems, inorder to solve the observer design problem as well as the output feedback control problem.

As an important class of nonlinear systems, the Lipschitzian nonlinear system has drawn considerableattention in the past few decades. In fact, a major class of nonlinear systems do satisfy the Lipschitzcondition either globally or locally. Moreover, incorporation of the Lipschitz condition into a linear

matrix inequality offers a tractable formulation for an efficient solution. Thus, many strategies concerningobserver design have been developed for such systems (see [2]-[9]). Most of the observer design techniquesproposed in those papers are based on quadratic Lyapunov functions and thus depend heavily on theexistence of a positive definite solution to an algebraic Ricatti equation. In [4] and [9], the existence of astable observer for Lipschitz nonlinear systems was addressed and a sufficient condition was given on theLipschitz constant. In practice, the nonlinear part may contribute to the stability of nonlinear systems([10]). Thus, in order to make fully use of the useful information of the nonlinear part, the so-called one-sided Lipschitz condition was introduced recently in [12, 13, 14] instead of the classical Lipschitz conditionfor the full order and the reduced order state observer design of nonlinear ordinary differential systems. Infact, the one-sided Lipschitz condition, which plays an important role in the numerical stability analysis ofnonlinear ordinary differential equations ([15]), has been widely used. It is shown, for many problems, that

Corresponding Author: E-mail Address: [email protected]. This work is supported by the National Natural Science

Foundation of China (No. 61021002 and No. 61074111).

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the one-sided Lipschitz constants are significantly smaller than the classical Lipschitz constants, whichmakes the one-sided Lipschitz constants much more appropriate for estimating the influence of nonlinearterms ([15]). Furthermore, a more relaxed condition, the quasi-one-sided Lipschitz condition, was furtherproposed, soon afterwards [12], to replace the one-sided Lipschitz condition for observer design by Hu in[16]. Due to the fact that it involves much more useful information of the nonlinear part, the quasi-one-

sided Lipschitz condition is shown to be an extension of one-sided Lipschitz condition and the Lipschitzcondition ([16]) and is less conservative than those two kinds of conditions. Hence, the control designschemes formulated in the available literature (see [17]-[19] references therein) are not always applicableto the quasi-one-sided Lipschitz nonlinear systems, especially for systems which are not Lipschitz but canbe verified to satisfy the quasi-one-sided Lipschitz condition for their nonlinear parts. In [20], the feedbackcontrol problem was considered for a class of nonlinear systems under a more relaxed condition comparedto the quasi-one-sided Lipschitz condition, which provides a sufficient condition for the existence of statefeedback controller by using linear matrix inequalities, under which the state feedback controllers anddynamic output feedback controllers are gained. All the above reasons motivate the authors to exploit aquasi-one-sided Lipschitz condition for nonlinear systems, which can lead to less conservative and simpledesign schemes of stabilization.

In this paper, we provide a design method of stabilization, including both the state feedback control

and the output feedback control problem, for quasi-one-sided Lipschitz nonlinear systems under somesufficient conditions on one-sided Lipschitz constant matrix. First, we derive a sufficient condition forthe state feedback control under which the asymptotical stabilization is guaranteed and a design schemeof linear full-state feedback control law is presented. Second, we propose a Luenberger-like observer,which is shown to be an exponentially stable observer under a sufficient condition based on a linearmatrix inequality. Once the sufficient conditions of the controller and observer problem are satisfied,it is shown that the proposed controller with estimated state feedback from the proposed observer willachieve exponential stabilization, that is, the controller and observer designs proposed in this study satisfythe separation principle. This implies that the output feedback control problem for this special class ofnonlinear systems can be easily adopted in practice.

The organization of this paper is as follows. In section 2, notation, background and some preliminariesare given. In section 3, we state, prove and discuss our main results, for both the full-state feedbackcontrol problem and the output feedback control problem. Several numerical experiments are reported in

section 4 to verify the effectiveness of our proposed method. We end in section 5 with some concludingremarks.

2 Preliminaries

In this section, firstly, we give some notation that will be used later in the following sections. Then, forclarity, an introduction of three kinds of Lipschitz nonlinear systems is listed, then the main body of ourpaper, namely the quasi-one-sided Lipschitz nonlinear systems, are fully investigated.

Throughout the current paper, In stands for the n-dimensional unit matrix, and for a square matrix F,F > 0 (F < 0) means that the matrix is symmetric positive-definite (negative-definite). , representsthe Euclidean inner product on Rn and is the corresponding Euclidean norm of a vector or the spectralnorm of a matrix. We use min (F) and max (F) to denote the minimal and the maximal eigenvalue of asymmetric matrix F, respectively.

For the following class of nonlinear systems{x = Ax + Bu + (x, u) ,y = Cx,

(2.1)

where x Rn, u Rr, y Rm are the system state, input and output, respectively. A Rnn, B Rnr, C Rmn and (x, u) is a nonlinearity with respect to x.The nonlinear function (x, u) is called a Lipschitz nonlinear function with respect to the variable x

with a Lipschitz constant > 0, if there holds

(x, u) (x, u) x x, for any x, x Rn, u Rr. (2.2)The inequality (2.2) is called the Lipschtiz condition, i.e., the classical Lipschitz condition. Similarly, the

so called one-sided Lipschitz condition introduced in [12] by Hu in the study of observer design is defined

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as followsf(x, u) f(x, u) , x x vpx x, for any x, x Rn, u Rr, (2.3)

where f(x, u) = P (x, u) and P is some symmetric positive-definite matrix to be determined later, theconstant vp, which may be negative, is called a one-sided Lipschitz constant for f(x, u) with respect to

x. Note that vp depends on the choice of the particular symmetric positive definite matrix P. For moredetails of the condition (2.3), we refer the readers to [15]. The system (2.1) with the nonlinearity (x, u)satisfying the Lipschitz condition and the one-sided Lipschitz condition are regarded as the Lipschitznonlinear system and the one-sided Lipschitz nonlinear system, respectively.

In this paper, we are concentrate on the study of the quasi-one-side Lipschitz system, which can beformulated by system (2.1) with the nonlinear function (x, u) satisfying the following condition

P (x, u) P (x, u) , x x (x x)T M(x x) , for any x, x Rn, u Rr, (2.4)where P is some symmetric positive-definite matrix to be determined later. Here, a real symmetric matrixM, just as the one-sided Lipschitz constant vp in 2.3, typically depending on the choice of P is called asa one-sided Lipschitz constant matrix for f(x, u) with respect to x. Accordingly, the inequality (2.4) iscalled a quasi-one-sided Lipschitz condition. We assume that (0, u) = 0, for any u Rr, and x = 0 is theequilibrium point of the system. Let us stress several important points concerning this the quasi-one-sidedLipschitz condition.

Remark 2.1. The quasi-one-sided Lipschitz condition is first introduced in [12] in the framework ofobserver design. When M = vpIn, where vp is a constant, it becomes the one-sided Lipschitz condition(2.3). Hence, it is an extension of the one-sided Lipschitz condition. In addition, it is not difficult tocheck that a Lipschitz function with Lipschitz constant is also a quasi-one-sided Lipschitz function forany n n positive definite matrix P with a one-sided constant matrix M = maxIn. The readers arereferred to [20] for more details.

Remark 2.2. (i) The requirement of the nonlinear function satisfying the quasi-one-sided Lipschitzcondition means that only for some positive definite matrix P determined later there exists a constantmatrix M such that (2.4) holds, but not necessarily for any positive definite matrix P. That is, satisfies

the quasi-one-sided Lipschitz condition (2.4) if and only if there exist a positive-definite matrixP

and aconstant matrix M such that (2.4) holds. Therefore, it is sufficient to find a pair of (P, M) such that satisfies (2.4). (ii) The constant matrix M is only required to be symmetric. It depends on the choice ofthe positive-definite matrix P and it can be positive-definite, negative-positive or even be indefinite.

Remark 2.3. Some technical discussions on how to verify a nonlinear function satisfying the condition(2.3) and to determine a one-sided Lipschitz constant was given in [13] and [14] for some special forms ofnonlinear function (x, u). For the quasi-one-sided Lipschitz condition, one can analyze it by a similarargument.

3 Main Results

In this section, the stabilization problem of system (2.1) with the quasi-one-sided Lipschitz condition (2.4)

by linear full-state feedback is first considered. Then, in combination with the obtained result, the outputfeedback control problem based on the observer design is further investigated.

3.1 State feedback control design

Stabilization of system (2.1) with the quasi-one-sided Lipschitz condition (2.4) via linear full-state feedbackmeans to design a state feedback control law

u = Kx, (3.5)

such that the closed-loop systemx = (A + BK) x + (x,Kx) (3.6)

is asymptotically stable. For such a problem, we have the following result.

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Theorem 3.1. Consider the nonlinear system (2.1) under the quasi-one-sided Lipschitz condition (2.4).If the gain matrix K can be chosen such that the following inequality

(A + BK)T P + P (A + BK) + 2max(M)In < 0 (3.7)

has a symmetric positive-definite solution P satisfying the quasi-one-sided Lipschitz condition (2.4), then

the zero solution of the closed-loop system (3.6) is asymptotically stable.Furthermore, let Q = P1 andK = W P, if max(M) > 0, inequality (3.7) is equivalent to the following

condition: there exist a symmetric positive-definite solution Q Rnn and a real matrix W Rrnsatisfying the linear matrix inequality

QAT + AQ + WTBT + BW QQ 1

2max(M)In

< 0. (3.8)

Else if max(M) 0, the following condition is sufficient to inequality (3.7): there exist a symmetricpositive-definite solution Q Rnn and a real matrix W Rrn satisfying the linear matrix inequality

QAT + AQ + WTBT + BW < 0. (3.9)

Proof. Consider the Lyapunov function candidate

V = xTP x, (3.10)

whose derivative along the solution of the system (3.6) is

V = xTP x + xTPx

= [(A + BK) x + (x,Kx)]T P x + xTP [(A + BK) x + (x,Kx)]

= xT

(A + BK)T

P + P (A + BK)

x + 2xTP (x,Kx) .

By taking into account conditions (2.4) and (3.7), we get

V xT

(A + BK)T P + P (A + BK) + 2M

x

xT (A + BK)

TP + P (A + BK) + 2max(M)Inx

< 0, for any x = 0,

which implies that the closed-loop system (3.6) is asymptotically stable.Let Q = P1, and K = W Q1, then

(A + BK)T

P + P (A + BK) + 2max(M)In

= ATQ1 + Q1WTBTQ1 + Q1A + Q1BW Q1 + 2max(M)In,

and hence,

Q

(A + BK)T

P + P (A + BK) + 2max(M)In

Q = QAT + AQ + WTBT + BW + 2max(M)QQ.

From the positive-definiteness of the matrix Q, we obtain

(A + BK)T P+P (A + BK)+2max(M)In < 0 QAT+AQ+WTBT+BW+2max(M)QQ < 0. (3.11)On the one hand, if max(M) > 0, by applying the Schur complement lemma [21], one has

QAT + AQ + WTBT + BW + 2max(M)QQ < 0

QAT + AQ + WTBT + BW QQ 12max(M)In

< 0.

On the other hand, if max(M) 0, it is easy to see thatQAT + AQ + WTBT + BW < 0 QAT + AQ + WTBT + BW + 2max(M)QQ < 0.

From (3.11), we have

QAT + AQ + WTBT + BW < 0 (A + BK)T P + P (A + BK) + 2max(M)In < 0.

Thus, the proof is completed.

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In what follows, we briefly discuss the results in Theorem 3.1.

Remark 3.1. Theorem 3.1 not only provides a sufficient condition for the existence of linear full-statefeedback controller of system (2.1) with the condition (2.4), but also presents a design scheme, by meansof linear matrix inequalities, for the gain matrix K of the control law (3.5).

Remark 3.2. The Theorem 3.1 derived here does not require that the matrix pair (A, B) is controllablecompared with the results in Theorem 1 in [17] where it does. That is, it is possible that Theorem 3.1 canstill be used as a guideline to design the state feedback controller for system (2.1) with the condition (2.4)even if the matrix pair (A, B) is uncontrollable. This fact will be illustrated by a simulative example insection 4.

3.2 Output Feedback Control Design

Output feedback for nonlinear systems is a problem of paramount importance, since in many applicationsit is not possible to measure all the states. Therefore, in this subsection, the output feedback controlproblem for system (2.1) based on observer is considered. Combining the linear full-state feedback controldesign of subsection 3.1 and the following observer design, we design an output feedback controller for

system (2.1) with the quasi-one-sided Lipschitz condition (2.4).First, we construct the observer as

x = Ax + Bu + (x, u) + L(y Cx), x0 = x(0), (3.12)where x represents the estimate state. Define the estimate error as

= x xwith the initial condition 0 = (0) = x(0) x(0), then one can obtain the error system as follows

= (A LC) + (x, u) (x, u). (3.13)The result that follows shows that the estimate error converges to zero exponentially.

Theorem 3.2. Assume that there exist a symmetric positive-definite matrix S Rnn

and a real matrixR Rnm such that the following linear matrix inequality

ATS + SA RC CTRT + 2M < 0 (3.14)holds, where S and M satisfy the following quasi-one sided Lipschitz condition

S (x, u) S (x, u) , x x (x x)T M(x x) , for any x, x Rn, u Rr.Then the estimate error converges to zero exponentially.

Proof. DefineV = TS,

then V is positive-definite and its derivative satisfies

V = T

(A LC)T S + S(A LC)

+ 2TS( (x, u) (x, u))

T

(A LC)T S + S(A LC) + 2M

= T[

ATS+ SA RC CTRT + 2M] ,where L = S1R. If we now denote

T = ATS+ SA RC CTRT + 2M,then we have T < 0 according to (3.14), and

V

TT

max(T)

min(S)

V.

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Let c = max(T)min(S)

, it is easy to show that c < 0 and V satisfies the differential inequality

V cV,V0 = V(0) = T0 S0.By the Comparison Lemma [22], we get

V V0ect, t 0.Consequently, the solution (t) of the error system (3.13) is well defined for all t 0 and it satisfies

(t) 1min(S)

V k1ek2t, t 0,

where k1 =

T0S0

min(S)> 0, k2 = 12c > 0. This implies that (t) converges to zero exponentially. This

completes the proof of Theorem 3.2.

Remark 3.3. We would like to remark that although the conclusion obtained in Theorem 3.2 is similarto that in Theorem 3.2 in [16], a sharp result is, however, given here, which shows that the error stateconverges to zero exponentially not just asymptotically. This implies that (3.12) is an exponentially stable

observer for system (2.1).

Remark 3.4. In [17], the matrix pair (C, A) is assumed to be observable, which guarantees that a gainmatrixL can be chosen such that matrix ALC is stable. Hence, when the matrix ALC is not stabilizedfor any gain matrix L, Theorem 2 in [17] can not be used to design the observer. But it is still possible toemploy Theorem 3.2 to design the observer (3.12). This shows that the condition (2.4) contributes morethan the Lipschitz condition to design an observer for system (2.1).

Next, for the problem of output feedback control, the objective is to give a suitable control law u = Kxsuch that the system (2.1) with the condition (2.4) is asymptotically stable. We have the following result.

Theorem 3.3. Consider the system (2.1) with the quasi-one-sided Lipschitz condition (2.4), if conditions(3.7) and (3.14) hold, then the zero solution of the system (2.1) is exponentially stable under the controllaw

u = Kx, (3.15)

where x is the estimate state of x generated by (3.12), K is the gain matrix given by K = W Q with Wand Q are obtained from (3.8) or (3.9).

Proof. Noting that the observation error = x x, we obtain the closed-loop system of system (2.1) underthe control law (3.15) as follows

x = (A + BK) x BK + (x, K(x )). (3.16)The time derivative of the Lyapunov function candidate given by (3.10) along the trajectories of (3.2)

satisfies

V

xT (A + BK)

TP + P (A + BK) + 2max (M) In x

2xTPBK.

Since P satisfies the inequality (3.7), then we get

N = (A + BK)T P + P (A + BK) + 2max(M)In < 0,

and

V max(N)x2 + 2P BKx aV + b

V,

where a = max(N)min(P)

, b = 2PBKmin(P)

. As a consequence,

d

V

dt a

2

V +

b

2

. (3.17)

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It has been shown in Theorem 3.2 that the observation error = x x converges to zero exponentially anda < 0, it follows from (3.17) and similar argument in the proof of Theorem 3.2 that the state x convergesto zero exponentially. The proof is thus completed.

Remark 3.5. The three results above given in Theorems 3.1-3.3 will be applicable locally or globally

depending on whether (x, u) is locally or globally satisfies the quasi-one-sided Lipschitz condition (2.4),see Example 4.3.

Remark 3.6. It is worthy to note that, in our scheme of the output feedback control design, the designprocess of controller and observer can be done separately similar to the one in the linear system case,which makes the design simple and tractable .

Remark 3.7. Since the Lipschitz condition and the one-sided Lipschitz condition are two special cases ofthe quasi-one-sided Lipschitz condition, the state feedback and output feedback control approaches proposedin this paper are also applicable to the Lipschitz and the one-sided Lipschitz nonlinear systems after minormodification.

4 Numerical examples

In this section, we begin by showing an example to demonstrate that the conditions given in Theorem3.1 and Theorem 3.2 are applicable to a larger class of systems than the conditions given in [17], we thenmove on to present several more numerical examples verifying the effectiveness of our proposed method.

Example 4.1. Let us consider the system described by

{x = Ax + Bu + (x, u) ,y = Cx,

where x =[

x1 x2 x3]T R3 and

A =

0.7 1 0.41.3 0 1

0.5 2.1 3

, B =

0 1 00 0 1

T,

C =[

0 1 1]

, (x, u) =

0 0 x 133T

.

By a similar argument as that in Example 4.2 in [13], it is easy to check that (x, u) is not a Lipschitzfunction, and hence the control design method proposed for Lipschitz nonlinear system (such as [17]) cannot be used for it. However, it does obey the quasi-one-sided Lipschitz condition (2.4) by taking M = 0,for every P in the form

P =

p11 p12 0p21 p22 0

0 0

, (4.18)

where > 0, and p11 p12p21 p22

> 0.

In fact, for any x =[

x1 x2 x3]T

and x =[

x1 x2 x3]T R3 with x3 = x3, the mean-value

theorem provides a non-zero (min {x3, x3} , max {x3, x3}) , such that

P (x, u) P (x, u) , x x = x 133

x 133

(x3 x3)

= 3

2

3 |x3 x3|2 0

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0

5

10

time (sec)

0 1 2 3 4 5 6 7 8 9 1020

10

0

10

time (sec)

0 1 2 3 4 5 6 7 8 9 1020

10

0

10

time (sec)

State x1

State x2

State x3

Figure 4.1: The closed-loop system states via state feedback

0 10 20 30 40 502

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time (sec)0 10 20 30 40 50

15

10

5

0

5

time (sec)

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4

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time (sec)

State x1

Estimation of state x1

State x3

Estimation of state x3

State x2

Estimation of State x2

Norm of the estimation error

Figure 4.2: The closed-loop system states via output feedback

Using Theorem 3.1, we obtain, by solving a linear matrix inequality (3.9), the following solutions

Q =

0.3462 0.5193 00.5193 1.0732 0

0 0 1.0732

, W =

0.2597 1.2117 1.21320.8343 0.2270 3.7563

.

and

P = Q1 = 10.5345 5.0974 0

5.0974 3.3982 00 0 0.9318

.

Hence the gain matrix of the state feedback controller is given by

K = W Q1 =

8.9118 5.4412 1.13047.6321 3.4814 3.5000

.

The simulation result of the full-state feedback control with the initial condition x (0) =[

8 6 5 ]T isgiven in Figure 4.1, which shows that the zero solution of the closed-loop system is asymptotically stable.

Furthermore, using Theorem 3.2, we get the observer gain matrix as follows

P =

269.5272 159.8381 0159.8381 268.4175 0

0 0 3.8620

, R =

43.2147207.780859.5286

,

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0

2

time (sec)

0 1 2 3 4 5 60.5

0

0.5

1

time (sec)

0 1 2 3 4 5 64

2

0

2

time (sec)

state x1

state x2

state x3

Figure 4.3: The simulation result via state feedback

and

L = P1R =

0.46181.0491

15.4141

.

We plot, Figure 4.2, the simulation result of the output feedback control with the initial condition x (0) =[8 5 2 ]T and x (0) = [ 2 3 1 ]T . From Figure 4.2, we can see that all of the real states and the

estimate states converge to zero asymptotically, especially that the estimate error vanishes exponentially.In [17], Theorem 1 and Theorem 2 may not work when the parameter matrix pair ( A, B) and (C, A)

are uncontrollable and unobservable, respectively. However, our results in Theorem 3.1 and Theorem 3.2

are still effective. To better understand this point, let us consider the following example.Example 4.2. Consider the following system described by{

x = Ax + Bu + (x, u) ,y = Cx,

where x =[

x1 x2 x3]T R3 and

A =

1 1 10 2 1

0 0 3

,

B =[

0 0 1]T

,C =

[0 0 1

],

(x, u) =

0 0 x 133T

.

Obviously, the matrix pair (A, B) is uncontrollable and (C, A) is unobservable. According to Example4.1, we know that (x, u) is not a Lipschitz function. However, it does obey the quasi-one-sided Lipschitzcondition (2.4) for the matrix P of the form (4.18) by taking M = 0. Apparently, if the control u = 0,the system is unstable, so it needs to design the control law in order to stabilize the system. First, bysolving the linear matrix inequality (3.9) in Theorem 3.1, we arrive at

Q =

1.1357 0.1739 00.1739 0.4920 0

0 0 1.6610

, W = [ 1.6610 1.6610 5.8135 ] ,

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4

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estimation of state x1

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0

1

2

3

4

5

time (sec)



0 2 4 6 8 100

0.5

1

1.5

2

2.5

time (sec)

norm of the estimation error

Figure 4.4: The simulation result via output feedback

and

P = Q1 =

0.9309 0.3290 00.3290 2.1487 0

0 0 0.6020

.

Hence the gain matrix of the state feedback controller is given by

K = W Q1 =

[2.0926 4.1154 3.5000

].

The simulation result of the corresponding closed-loop system with the initial condition x (0) =[

1 0.8 2 ]Tis given in Figure 4.3, which shows that the zero solution of the closed-loop system is asymptotically stable.

On the other hand, solving the linear matrix inequality (3.14) results in

P =

0.9877 0.3218 00.3218 0.6400 0

0 0 1.6610

, R =

0.66590.3181

5.8135

,

and

L = P1R =

11

3.5

.

The simulation result of the output feedback control with the initial condition x (0) =[

3.5 0 1.5]T

and x (0) =[

3 2 0.5 ]T is plotted in Figure 4.4, from which we can see that all of the real states andthe estimate states converge to zero asymptotically. Meanwhile, the estimate error vanishes exponentiallyas proved in Theorem 3.2.

The last example is devoted to illustrate that our proposed method is effective for an important classof the Lipschitz nonlinear systems, which exist widespread in practice. This example deals with thetriangular Lipschitz nonlinear system studied in [19] and [23].

Example 4.3. Consider a simple inverted pendulum model which describes the mechanics of the limbmovement by the following state space representation:

= ,

= u + mglJ

sin ,

y = ,

. (4.19)

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0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

time (sec)

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

time (sec)

the angular position

the angular velocity

Figure 4.5: The inverted pendulum system with the control u = 0.

where is the angular position, is the angular velocity, J represents inertia of the pendulum, m is themass of the pendulum, g is acceleration of gravity, and l is the length from the pivot center to pendulumscenter of mass.

Taking the state x = [x1 x2]T

as x1 = , x2 = , the system (4.19) can be described by (2.1) with anappropriate parameter, where

A =

0 10 0

, B =

01

, C =

[1 0

], (x, u) =

[0 0.5sin x1

]T.

Here, the Lipschitz constant r is 0.5. For the inverted pendulum system model, we are interested in thecase that the state x1 is small enough to satisfies that x1 sin x1 > 0, i.e., 2 < x1 < 2 . Hence, forany positive-definite matrix P =

d

, with d, , are all positive numbers, and x = [x1 x2]

T,

x = [x1 x2]T, we have

P (x, u) P (x, u) , x x = r(sin x1 sin x1)(x1 x1) + r(sin x1 sin x1) (x2 x2)) r(x1 x1)2 + r

2

[(x1 x1)2 + (x2 x2)2

]= (x x)TM(x x),

where

M =r ( + 2 ) 0

0 r2

. (4.20)

Thus, (x, u) satisfies locally the quasi-one-sided Lipschitz condition (2.4). Using our approach, aftersolving the inequality (3.7), we obtain the following solution:

P =

3 0.1

0.1 0.6

, K =

[ 3 4 ] .Further, note that the matrix M is a linear representation of matrix P, solving the linear matrix inequality(3.14) results in:

P =

59.1492 27.2996

27.2996 45.4994

, R =

37.537059.1492

,

and

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0

0.2

0.4

0.6

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1

1.2

time (sec)

0 2 4 6 8 10

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0.5

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time (sec)


estimation of state x!



Figure 4.6: The simulation result via output feedback control

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2

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0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time (sec)

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Figure 4.7: The estimation error and the control effort.

L = P1R =

1.70742.3245

.

The simulation results for regulating the states of the inverted pendulum system (4.19) to zeros areshown in Figure 4.6 and Figure 4.7 with the initial values x(0) = [0.2 0.5]T and x(0) = [0.1 1.5]T.Figure 4.6 shows the pendulum angular position, angular velocity, and their estimates. Figure 4.7 showsthe estimate error of the states and the control effort. Comparing the Figure 4.6 with Figure 4.5, it is clearto see that, under the output feedback control, the two states of the pendulum converge to zero rapidly;whereas, without control, the states are divergence. Also, the convergence of the estimated states to theirtrue values and the corresponding control effort are observed in Figure 4.7.

5 Conclusions

In this paper, the full-state feedback control and the output feedback control problems of quasi-one-sidedLipschitz nonlinear systems, which is a general version of the classical Lipschitz nonlinear systems, are

investigated. We provide sufficient conditions for the existence of a linear full-state feedback controllerand a dynamical output feedback controller. The controller synthesis approaches based on a set of linearmatrix inequalities makes the design process simple and tractable. Generally, separation principle usuallydoes not hold for nonlinear systems. However, for the class of nonlinear systems considered in this paper,by using the proposed linear full-state feedback controller and the proposed nonlinear observer, we showthat the separation principle holds; that is, the same gain matrix obtained in the design of the linear full-state feedback controller can be used together with the estimated state that obtained from the proposedobserver. Besides, note that a Lipschitz nonlinear function with Lipschitz constant must be a quasi-one-sided Lipschitz nonlinear function with a corresponding constant matrix M = max(P)In for anyn n positive-definite matrix P and systems with Lipschitz nonlinearity are common in many practicalapplications, the nonlinear systems considered in this paper cover a fairly large number of systems inpractice. Furthermore, the design schemes presented in this work are still available even if the parametermatrix pairs (A, B) and (C, A) are uncontrollable or unobservable.

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We would like to emphasize here that the conditions presented in our work for both the full-statefeedback and output feedback stabilization are all sufficient conditions, in view of this, there are somechallenging problems to be addressed in the future. For example, how to verify these two sufficientconditions is not a trivial problem. In addition, how to verify a nonlinear system, which is not Lipschitz,to be a quasi-one-sided Lipschitz nonlinear system is still an open problem, especially for physical systems

in practice, this also constitutes our future work.

References

[1] V. Andrieu, L. Praly. A unifying point of view on output feedback designs for global asymptoticstabilization. Automatica, 2009, 45: 1789-1798.

[2] Raghavan S., Hedrick J. K.. Observer design for a class of nonlinear systems. Int. J. Control. 1994,59: 515528.

[3] Rajamani R.. Observer for Lipschitz nonlinear systems. IEEE Trans. Autom. Control, 1998, 43:397401.

[4] Rajamani R., Chao Y. M.. Existence and design of observers for nonlinear systems: relation todistance to unobservability. Int. J. Control, 1998, 69: 717731.

[5] Thau F. E.. Observing the state of nonlinear dynamic systems. Int. J. Control. 1998, 17: 471480.

[6] C. Aboky G. Sallet and J.-C.Vivalda. Observers for Lipschitz non-linear systems. Int. J. Control,2002, 75(3): 204-212

[7] G. Lu, D. W. C. HO. Full-order and Reduced-order Observers for Lipschitz Descriptor Systems. IEEETrans. Circuits Syst., 2006, 2(53): 563-567.

[8] Gerhard Kreisselmeier and Robert Engel. Nonlinear Observers for Autonomous Lipschitz ContinuousSystems. IEEE Trans. Aotoma. Control, 2003, 48(3): 451-463.

[9] Zhu F., Han Z.. A note on observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control,2002, 47: 17511754.

[10] Qiao Zhu, Guangda Hu. Stability Analysis for Uncertain Nonlinear Time-daly Systems with Quasi-one-sided Lipschitz Condition. Acta Automatica Sinca, 2009, 35(7): 1006-1009.

[11] Zak S. H.. On the Stabilization and Observation of nonlinear dynamic systems. IEEE Trans. Autom.Control, 1990, 35: 604607.

[12] Guangda Hu. Observers for one-sided Lipschitz non-linear systems. IMA Journal of MathematicalControl and Information, 2006, 23: 395-401.

[13] Yanbin Zhao, Jian Tao, and Ning-Zhong Shi, A note on observer design for one-sided Lipschitznonlinear systems. Systems & Control Letters, 2010, 59: 66-71.

[14] Mingyue Xu, Guangda Hu and Yanbin Zhao.. Reduced-order observer design for one-sided Lipschitznon-linear systems. IMA Journal of Mathematical Control and Information, 2009, 26: 299-317.

[15] Dekker K, Verwer J. G.. Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations.Amsterdam: North-Holland, 1984.

[16] Guangda Hu. A note on observer for one-sided Lipschitz non-linear systems. IMA Journal of Mathe-matical Control and Information, 2008, 25: 297-303.

[17] Prabhakar R. Pagilla and Yongliang Zhu. Controller and Observer Design for Lipschitz NonlinearSystems. Proceeding of the 2004 American Control Conference Boston. Massachusetts June 30 -July2, 2004, 2379-2384.

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[18] Yu F., Su H. Y., and Chu J.. H Control for Lipschitz Nonlinear Systems via Output Feedback.Proceedings of the 3nd World Congress on Intelligent Control and Automaton, 2000, 3456-3460.

[19] Choi H. L., Lim J. T.. Output Feedback Stabilization for a Class of Lipschitz Nonlinear Systems[J].IEICE Trans. Fundamentals. 2005, 2: 602-605.

[20] Fu Qin. Feedback Control for Nonlinear Systems with Quasi-one-sided Lipschitz Condition. Proceed-ings of the 29th Chinese Control Conference July 29-31, 2010, Beijing, China, 309-311.

[21] S.P. Boyd, L.E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System andControl Theory, SIAM, Philadelphia, 1994.

[22] Hassian K. Khalil. Nonlinear systems. Prentice Hall Upper Saddle River. NJ07458. 2002. 84-85.

[23] Zemouche A., Boutayeb M. and Bara. G. I.. Observers for a class of Lipschitz system with extensionto H1 performance analysis. System and Control Letters, 2008, 57: 18-27.

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