IMA Journal of Mathematical Control and Information Advance Access published August 11, 2012
IMA Journal of Mathematical Control and Information 2012 Page 1 of 16 doi:10.1093imamcidns023
Stabilization of quasionesided Lipschitz nonlinear systems
Fengyu Fu, Mingzhe Hou and Guangren Duan
Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China
Corresponding author: fufengyu2002163.com
Received on 25 October 2011; revised on 21 May 2012; accepted on 11 June 2012
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 onesided Lipschitz condition. Sufficient conditions for the existence of state feedback controller 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; quasionesided Lipschitz condition; nonlinear systems; linear matrix inequality.
1. Introduction
Stabilization is one of the most important issues currently under consideration by researchers in the nonlinear 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 very different approaches, due to its importance in many practical applications where measurement of all the state variables is not possible see AndrieuPraly, 2009 and references therein. Output feedback control design, utilizing estimated state and output as feedback, usually involves the first two problems. Generally, for nonlinear systems, stabilization by state feedback plus observability do not imply stabilization by output feedback. And it is well known that the observer design for nonlinear systems by itself is quite challenging. Therefore, the output feedback control problem for nonlinear systems is much more challenging than stabilization using fullstate feedback. Basically, one has to consider some special classes of nonlinear systems due to their practical significance, for example, the Lipschitzian nonlinear systems, 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 consider able attention in the past few decades. In fact, a major class of nonlinear systems do satisfy the Lipschitz condition either globally or locally. Moreover, incorporation of the Lipschitz condition into a linear matrix inequality offers a tractable formulation for an efficient solution of the observer design. Thus, many strategies concerning observer design have been developed for such systems see Aboky et al., 2002; KreisselmeierEngel, 2003; LuHo, 2006; RaghavanHedrick, 1994; Rajamani, 1998; RajamaniChao, 1998; Thau, 1998; Zak, 1990; ZhuHan, 2002. Most of the observer design tech niques proposed in those papers are based on quadratic Lyapunov functions and thus depend heavily on the existence of a positive definite solution to an algebraic Ricatti equation. In RajamaniChao 1998 and ZhuHan 2002, existence of a stable observer for Lipschitz nonlinear systems was addressed and a sufficient condition was given on the Lipschitz constant, respectively.
c The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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In practice, the nonlinear part may contribute to the stability of nonlinear systems ZhuHu, 2009. Thus, in order to fully make use of the useful information of the nonlinear part, the socalled onesided Lipschitz condition was recently introduced in Hu 2006, Zhao et al. 2010, and Xu et al. 2009 for the full order and the reduced order state observer design of nonlinear ordinary differential systems. In fact, the onesided Lipschitz condition, which plays an important role in the numerical stability analysis of nonlinear ordinary differential equations DekkerVerwer, 1984 has been widely used. Furthermore, a more relaxed condition, namely the quasionesided Lipschitz condition, was further proposed, soon afterwards Hu, 2006, to replace the onesided Lipschitz condition for observer design by Hu 2008. Due to the fact that it involves much more useful information of the nonlinear part, the quasionesided Lipschitz condition is shown to be an extension of onesided Lipschitz condition and the Lipschitz con dition Hu, 2008 and is less conservative than those two kinds of conditions. Hence, the control design schemes formulated in the available literatures see ChoiLim, 2005; PagillaZhu, 2004; Yu et al., 2000 references therein are not always applicable to the quasionesided Lipschitz nonlinear systems, especially for systems which are not Lipschitz but can be checked to satisfy the quasionesided Lips chitz condition for their nonlinear parts. In fact, it is an interesting research subject to study such physi cal systems that obey the quasionesided Lipschitz condition, for example, a simple inverted pendulum model, in which the nonlinear dynamics of the angular velocity actually satisfies the quasionesided Lipschitz condition locally; we refer the readers to Example 4.3 in Section 4 for a detailed simulation. In Fu 2010, the feedback control problem was considered for a class of nonlinear systems under a more relaxed condition by comparison to the quasionesided Lipschitz condition, which provides a sufficient condition for the existence of state feedback controller by using linear matrix inequalities, under which the state feedback controller and dynamic output feedback controller are gained. All of the above reasons motivate the authors to exploit a quasionesided Lipschitz condition for nonlinear systems, with the goal of obtaining less conservative and simple design schemes of stabilization.
In this paper, we provide a design method of stabilization, including both the state feedback con trol and the output feedback control problem, for quasionesided Lipschitz nonlinear systems under some sufficient conditions based on the onesided Lipschitz constant matrix. First, we derive a suffi cient condition for the state feedback control, under which the asymptotical stabilization is guaranteed and a design scheme of linear fullstate feedback control law is presented. Secondly, we propose a Luenbergerlike observer, which is shown to be an exponentially stable observer under a sufficient con dition based on a linear matrix inequality. Once the sufficient conditions of the controller and observer design problems are satisfied, the proposed controller with estimated state feedback from the proposed observer will achieve exponential stabilization for the overall system. That is, the state feedback con troller and observer proposed in this study can be performed separately, which is the main feature in dealing with linear systems. This implies that the output feedback control problem for this special class of nonlinear systems can be easily adopted in practice.
The organization of this paper is as follows. In Section 2, notation and some preliminaries are given. In Section 3, we state, prove and discuss our main results, for both the fullstate feedback control prob lem 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 concluding remarks.
2. Preliminaries
In this section, we begin by introducing some notation that will be used later in the following sections. Next, for clarity, a brief description of three kinds of Lipschitz nonlinear systems is consecutively listed,
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STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS 3 of 16
then the main objective of this paper, namely the quasionesided Lipschitz nonlinear systems, is fully described.
Throughout the current paper, In stands for the ndimensional unit matrix, and for a square matrix F, F0 F0 means that the matrix is symmetric positivedefinite negativedefinite. ,represents the Euclidean inner product on Rn andis the corresponding Euclidean norm of a vector or the spec tral norm of a matrix. We use minF and maxF to denote the minimal and the maximal eigenvalues of a symmetric matrix F, respectively.
We consider the following class of nonlinear systems formulated by
x A xB ux , u, yCx,
2.1
where xRn , uRr , yRm represent the state, input and output of the system, respectively. Here and below, ARnn, BRnr, CRmn and x, u is a nonlinearity with respect to x.
A nonlinear function x, u is called a Lipschitz function with a Lipschitz constant 0, if
x,ux,uxx foranyx,xRn,uRr. 2.2
The inequality 2.2 is called the Lipschtiz condition, i.e. the classical Lipschitz condition. Similarly, the socalled onesided Lipschitz condition introduced by Hu 2006 in the study of observer design is defined as follows
fx,ufx,u,xxvpxx foranyx,xRn,uRr, 2.3
where f x, uPx, u and P is some symmetric positivedefinite matrix to be determined later, the constant vp, which may be negative, is called a onesided Lipschitz constant for f x, u with respect to x. Note that vp depends on the particular choice of the symmetric positivedefinite matrix P. For more details of the condition 2.3, we refer the readers to DekkerVerwer 1984. The system 2.1 with the nonlinearity x, u satisfying the Lipschitz condition 2.2 and the onesided Lipschitz condition 2.3 are referred to as the Lipschitz nonlinear system and the onesided Lipschitz nonlinear system, respectively. For some related discussions on how to check a nonlinear function satisfying the condition 2.3, we refer the readers to Zhao et al. 2010 and Xu et al. 2009.
In this paper, we focus on the study of the quasionesided Lipschitz system, formulated by system 2.1 with the nonlinear function x, u satisfying the following condition:
Px,uPx,u,xxxxTMxx foranyx,xRn,uRr, 2.4
where P is some symmetric positivedefinite matrix to be determined later. Here, a real symmetric matrix M on the righthand side of 2.4, parallels the onesided Lipschitz constant vp in 2.3 that typically depends on the choice of P, is called a onesided Lipschitz constant matrix for f x, u with respect to x. Accordingly, the inequality 2.4 is called a quasionesided Lipschitz condition, which was first defined in Hu 2006 in the framework of observer design. We assume that0, u0, for any uRr , and x0 is the equilibrium point of the system 2.1.
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Remark 2.1 Note that when MvpIn with vp being a constant, the condition 2.4 becomes the one sided Lipschitz condition 2.3. Hence, it is an extension of the onesided Lipschitz condition. In addi tion, it is not difficult to check that a Lipschitz function with the Lipschitz constantis also a quasi onesided Lipschitz function for any nn positivedefinite matrix P with a onesided constant matrix M max PIn . We refer the readers to Fu 2010 for more details.
3. Main results
In this section, the stabilization of system 2.1 with the quasionesided Lipschitz condition 2.4 by using the methodology of linear fullstate feedback is first considered. Then, in combination with the obtained result, the output feedback control problem based on the observer design is further investigated.
3.1 State feedback control design
Stabilization of system 2.1 with the quasionesided Lipschitz condition 2.4 via linear fullstate feed
back means to design a state feedback control law
uKx, 3.1
such that the closedloop system
xAB Kxx , K x 3 . 2is asymptotically stable. We have the following result.
Theorem 3.1 Consider the nonlinear system 2.1 with the quasionesided Lipschitz condition 2.4. If a gain matrix K can be chosen such that the following inequality:
ABKTPPABK2maxMIn0 3.3
has a symmetric positivedefinite solution P satisfying the quasionesided Lipschitz condition 2.4, then the zero solution of the closedloop system 3.2 is asymptotically stable.
Furthermore,letQP1 andKWP.IfmaxM0,inequality3.3isequivalenttothefollow ing condition: there exist a symmetric positivedefinite solution QRnn and a real matrix WRrn satisfying the linear matrix inequality
QAT AQWTBT BW Q
1 0. 3.4 2maxMIn
Q
Else if maxM0, the following condition is sufficient to inequality 3.3: there exist a symmetric positivedefinite solution QRnn and a real matrix WRrn satisfying the linear matrix inequality
QAT AQWTBT BW 0. 3.5
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Proof. Consider the Lyapunov function candidate
VxTPx,
whose derivative along the solution of the closedloop system 3.2 is
V ABKxx, KxTPxxTPABKxx, Kx
xTABKTPPABKx2xTPx, Kx. By taking into account conditions 2.4 and 3.3, we obtain
VxTABKTPPABK2Mx
xTABKTPPABK2maxMInx 0, foranyx0,
which implies that the closedloop system 3.2 is asymptotically stable. We now let QP1 and K WQ1 to obtain
3.6
QABKTPPABK2maxMInQQATAQWTBTBW2maxMQQ. It follows from the positivedefiniteness of the matrix Q that
ABKTPPABK2maxMIn 0QAT AQWTBT BW 2maxMQQ0. 3.7
On the one hand, if max M 0, by applying the Schur complement lemma in Boyd et al. 1994, one has
QAT AQWTBT BW 2maxMQQ0 QAT AQWTBT BW Q
Q
1 0. 2maxMIn
On the other hand, if max M 0, it is easy to see that
QAT AQWTBT BW 0QAT AQWTBT BW 2maxMQQ0.
From 3.7, we have,
QAT AQWTBT BW 0ABKTPPABK2maxMIn 0.
Thus, the proof is completed.
Note that the proof of Theorem 3.1 is valid regardless of whether the matrix pair A, B of the system 2.1 is controllable or not. This fact will further be illustrated by simulative examples in Section 4.
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6of16 F.FUET AL. 3.2 Output feedback control design
In this section, by combining the linear fullstate feedback control design methodology in Section 3.1 and the following observer design, we provide a solution to the output feedback control problem for system 2.1 with the quasionesided Lipschitz condition 2.4.
First, we construct the observer as
xAxBux,uLyCx, x0 x0, 3.8
where x represents the estimate state and L is a gain matrix of the observer. Next, by introducing the estimation error as
xx with the initial condition 00x0x0, we obtain
AL Cx , u x , u.3 . 9The following result shows that the estimation error converges to zero exponentially.
Theorem 3.2 Assume that there exist a symmetric positivedefinite matrix SRnn and a real matrix RRnm such that the linear matrix inequality
ATSSARCCTRT2M0
holds, where S and M satisfy the following quasionesided Lipschitz condition
Sx,uSx,u,xxxxTMxx foranyx,xRn,uRr. Let LS1R, then the estimation errorconverges to zero exponentially.
Proof. Define
V TS,
then V is positivedefinite and its derivative along system 3.9 satisfies
V TALCTSSALC2TSx, ux, uTALCTSSALC2MTATSSARCCTRT 2M,
where LS1R. If we now denote
TATSSARCCTRT 2M,
then we have T0 according to 3.10, and
VT Tm a xTV .
minS
3.10
Let cmaxTminS, obviously, c0. This implies that t converges to zero exponentially. The proof is thus completed.
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STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS 7 of 16 Note that the proof of Theorem 3.2 is valid regardless of whether the matrix pair C, A of the system
2.1 is observable or not.
Remark 3.1 We would like to remark that although the conclusion obtained in Theorem 3.2 is similar to that in Theorem 3.2 in Hu 2008, a sharp result is, however, given here, which shows that the error dynamics converges to zero exponentially not just asymptotically.
Next, for the problem of output feedback control, our objective here is to design a suitable control law uKx such 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 quasionesided Lipschitz condition 2.4, if condi tions 3.3 and 3.10 hold, then the zero solution of the system 2.1 is exponentially stable under the control law
uKx, 3.11 where x is the estimate state of x generated by 3.8, K is the gain matrix given by KWQ with W and
Q obtained from 3.4 or 3.5.
Proof. By the definition the estimation error xx, we obtain the closedloop system of system
2.1 under the control law 3.11 as follows
xAB KxB K x , Kx .3 . 1 2
The time derivative of the Lyapunov function candidate given by 3.6 along the trajectories of system 3.12 satisfies
V xTABKTPPABK2maxMInx2xTPBK. Since P satisfies the inequality 3.3, then we obtain
and
NABKTPPABK2maxMIn 0,
V maxNx22PBKx
aVb V,
where amaxNminP, b2PBKminP. As a consequence,
dVaVb. dt 2 2
3.13
It has been shown in Theorem 3.2 that the estimation error xx converges to zero exponentially, and notice that a0. It follows from 3.13 that the state x converges to zero exponentially. This com pletes the proof.
Remark 3.2 The above results given in Theorems 3.13.3 will be applicable locally or globally depending on whether x, u locally or globally satisfies the quasionesided Lipschitz condition 2.4; see Example 4.3.
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Remark 3.3 It is worthy to note that, in our scheme of the output feedback control design, the design process of the controller and the observer can be done separately similar to the one in the linear system case, which makes the design simple and tractable .
Remark 3.4 Since the Lipschitz condition and the onesided Lipschitz condition are two special cases of the quasionesided Lipschitz condition, the state feedback and output feedback control approaches proposed in this paper are also applicable to the Lipschitz and the onesided Lipschitz nonlinear systems after a minor modification.
4. Numerical examples
In this section, we begin by showing two examples to demonstrate that the conditions given in Theorems 3.1 and 3.2 make the applicable class larger, we then continue to present a practical numerical example verifying the effectiveness of our proposed method.
Example 4.1 Let us consider the system 2.1, where
0.710.4 010T A1.3 0 1 , B 0 0 1 ,
0.5 2.1 3
C0 1 1, x,u0 0 x13T.
By a similar argument as that in Example 4.2 in Zhao et al. 2010, it is easy to check that x, u is not a Lipschitz function, and hence the control design method proposed for Lipschitz nonlinear system such as PagillaZhu, 2004 cannot be directly used. However, it does obey the quasionesided Lipschitz condition 2.4 by taking M0, for every P in the form
where 0, and
p11 p12 0
Pp21 p22 0 , 4.1
00
p11 p12 0. p21 p22
In fact, for any xx1 x2 x3T and xx1 x2 x3TR3 with x3x3, the meanvalue theorem yields a nonzero minx3, x3, maxx3, x3, such that
Px, uPx, u, xxx13x13xx3333
23x 3x 320 . 3
Using Theorem 3.1, we obtain, by solving a linear matrix inequality 3.5, the following solutions
3
0.3462 Q0.5193
0
0.5193 1.0732 0
00.2597 1.2117 1.2132
0, W0.8343 0.2270 3.7563 1.0732
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and
1.5 1 0.5
state x 1
1
0 1 2
0
2
0 5 10
STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS
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0 3 012345012345
Time s
25
Time s
the control effort u1 the control effort u2
state x 3
15
Time s Time s
Fig. 1. The simulation results via state feedback control for Example 4.1.
4 012345012345
0
Hence, the gain matrix of the state feedback controller is
10.5345 PQ1 5.0974
5.0974 0
KWQ18.9118 5.4412 1.1304 . 7.6321 3.4814 3.5000
The simulation results of the fullstate feedback control with the initial condition x01 0.53T are given in Fig. 1, which show that the zero solution of the closedloop system is asymptotically stable. Furthermore, using Theorem 3.2, we obtain the observer gain matrix as follows:
and
0.4618
LP1R 1.0491.
15.4141
269.5272 P159.8381
0
159.8381 268.4175
0
43.2147
R207.7808
59.5286
0, 0 3.8620
3.3982 0
0 . 0.9318
state x 2
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F.FUET AL. 85
10 5 0 5
estimation of state x3
4
2
6 4 2 0
real value of state x1 estimation of state x1
0
5
10
2 15 0 10 20 30 40 50
Time s
15 6 real value of state x3
real value of state x2 estimation of state x2
0 10 20 30 40 50 Time s
norm of the estimation error
0 10 20 30 40 50 Time s
10 0 0 10 20 30 40 50
Time s
Fig. 2. The simulation results via output feedback control for Example 4.1.
We plot, in Fig. 2, the simulation results of the output feedback control with the initial condition x085 2T and x023 1T. From Fig. 2, we can see that all of the real states and the estimate states converge to zero asymptotically, moreover, the estimation errors vanish exponentially.
As mentioned before, our theoretical results do not depend on the controllability of the matrix pair A, B or the observability of the matrix pair C, A. To better understand this point, let us consider the following example:
Example 4.2 Consider the system 2.1, where
1 1 1 A 0 2 1 ,
003 B0 0 1T,
C0 0 1,
x, u0 0x13T.
Obviously, the matrix pair A, B is uncontrollable and C, A is unobservable. According to Exam ple 4.1, we know that x, u obeys the quasionesided Lipschitz condition 2.4, although it is not a Lipschitz function. Note that, when the control u0, the system is unstable, so it is necessary to design the control law to stabilize the system.
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STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS
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2
0
2 0123456
state x1
1
0.5
0
Time s
state x2
and
0.5 0123456
2
0
2
0.1739 0
0.4920 0
0, W1.66101.66105.8135 1.6610
0.9309 PQ1 0.3290
0.3290 2.1487 0
0
0 . 0.6020
3.5000.
0
Hence, the gain matrix of the state feedback controller is
KWQ12.09264.1154
Time s
state x3
4 0123456
Time s
Fig. 3. The simulation results for state x via state feedback for Example 4.2.
First, by solving the linear matrix inequality 3.5 in Theorem 3.1, we arrive at
1.1357 Q0.1739
0
The simulation results of the corresponding closedloop system with the initial condition x01 0.82T are given in Fig. 3, which show that the zero solution of the closedloop system is asymp totically stable.
On the other hand, solving the linear matrix inequality 3.10 results in
0.9877 P0.3218
0
0.3218 0.6400 0
0
0 , 1.6610
0.6659 R 0.3181
5.8135
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F.FUET AL.
42
real value of state x 1
estimation of state x 1
real value of state x 2
estimation of state x 2
3 2 1 0
1
0
1
1 2
0 2 4 6 8 10 0 2 4 6 8 10
Time s
5 2.5
Time s
real value of state x 3
estimation of
state x 3
norm of the
estimation error
4 3 2 1 0
2
1.5
1
0.5
and
Time s Time s Fig. 4. The simulation results via output feedback control for Example 4.2.
1 LP1R 1 .
3.5
1 0
0 2 4 6 8 10 0 2 4 6 8 10
The simulation results of the output feedback control with the initial condition x03.5 0 1.5T and x032 0.5T are plotted in Fig. 4, from which we can see that all of the real states and the estimate states converge to zero asymptotically. Meanwhile, the estimate error vanishes exponentially as proved in Theorem 3.2.
The last example is devoted to illustrate that our proposed method is effective for an important class of the Lipschitz nonlinear systems, which exist widespread in practice. This example deals with the triangular Lipschitz nonlinear system studied in ChoiLim 2005 and Zemouche et al. 2008.
Example4.3 Considerasimpleinvertedpendulummodeldescribingthemechanicsofthelimbmove ment by the following state space representation:
,
u mgl sin, 4.2J
y,
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STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS 13 of 16
whereis the angular position,the angular velocity, J represents inertia of the pendulum, m the mass of the pendulum, g acceleration of gravity and l the length from the pivot center to pendulums center of mass.
Letting xx1 x2T T, we can rewrite the system 4.2 into the form of system 2.1 with
A 0 1 , B 0 , C1 0, x,u0 0.5sinx1T. 001
Here, the Lipschitz constant r is 0.5. For the inverted pendulum system model, we are interested in the case that the state x is small enough such that x sin x0, i.e. 2×2. Hence, for
1111
any positivedefinite matrix Pd, with d,andbeing positive constants, and xx1 x2T,
xx1 x2T, we have
Px, uPx, u, xxrsin x1sin x1x1x1rsin x1sin x1x2x2
where
r x 1x 12rx 1x 12 x 2x 222
xxT Mxx,
Mr 2 0. 4.3 0 r
2
Thus, x, u locally satisfies the quasionesided Lipschitz condition 2.4. Using our approach, after
solving the inequality 3.3, we obtain the following solution:
P 3 0.1 , K3 4. 0.1 0.6
Furthermore, note that the matrix M is a linear representation of matrix P, solving the linear matrix inequality 3.10 results in
and
P
59.1492 27.2996 , R27.2996 45.4994
LP1R1.7074 . 2.3245
37.5370 59.1492
The simulation results for regulating the states of the inverted pendulum system 4.2 to zeros are shown in Figs 5 and 6 with the initial values x00.20.5T and x00.11.5T, respectively. Specifically, Fig. 5 shows the dynamic transferred curves of the pendulum angular position, angular velocity, and their estimates, and Fig. 6 shows curves of the norm of estimation error and the control effort. It is clear to see that, under the output feedback control, two states of the pendulum converge to zero rapidly.
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11 0.5
real value of state x 1
estimation of state x 1
real value of state x 2
estimation of state x 2
0.5
0
0
0.5
1
0.5 1.5
0 2 4 6 8 10 0 2 4 6 8 10 Time s Time s
Fig. 5. The simulation results for state x via output feedback control for Example 4.3.
1.5 2
the control effort u
norm of the estimation error
1
0.5
0
2
4
0 6
0 2 4 6 8 10 0 2 4 6 8 10 Time s Time s
Fig. 6. The simulation results of estimation error and the control effort for Example 4.3.
5. Conclusions
In this paper, the fullstate feedback control and the output feedback control problems of quasione sided Lipschitz nonlinear systems are investigated. We provide sufficient conditions for the existence of a linear fullstate feedback controller and a dynamical output feedback controller. The controller syn thesis approaches are based on a set of linear matrix inequalities which makes the design process simple and tractable. We have proved that, for the class of nonlinear systems considered in this paper, the feed back control and observer design of output feedback can be processed separately by using the proposed approach and exponential stability follows for the overall system, that is, the same gain matrix obtained in the design of the linear fullstate feedback controller can be used together with the estimated states that obtained from the proposed observer. Furthermore, the controllability and observability of parame ter matrix pairs A, B and C, A are not the necessary conditions for the design schemes. Besides, note that systems with Lipschitz nonlinearity are common in many practical applications, thus the class of nonlinear systems considered in this study cover a fairly large number of systems in practice.
We would like to emphasize here that the conditions presented in our work for both the fullstate feedback and output feedback stabilization are all sufficient conditions, in view of this, there are some
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STABILIZATION OF QUASIONESIDED LIPSCHITZ NONLINEAR SYSTEMS 15 of 16
challenging problems to be addressed in the future. For example, how to verify these two sufficient con ditions is not a trivial problem. In addition, how to check a nonlinear system, which is not Lipschitz, to be a quasionesided Lipschitz nonlinear system is still an open problem, especially for pratical physical systems, this also constitutes our future work.
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that result in the improvements of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China No. 61021002 and No. 61074111.
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