Robust control of a class of uncertain nonlinear systems

Systems & Control Letters 19 (1992) 139-149 North-Holland

139

R o b u s t control of a class of uncertain nonlinear systems * Youyi Wang The School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263

Lihua Xie and Carlos E. de Souza Department of Electrical and Computer Engineering, The University of Newcastle, NSW, Australia

Received 23 November 1991 Revised 7 March 1992 Abstract: This paper considers the robust control of a class of nonlinear systems with real time-varying parameter uncertainty.

Interest is focused on the design of linear dynamic output feedback control and two problems are addressed. The first one is the robust stabilization and the other is the problem of robust performance in an H® sense. A technique is proposed for designing stabilizing controllers for both problems by converting them into 'scaled' H= control problems which do not involve parameter uncertainty. Keywords: Robust stabilization; nonlinear systems; robust performance; uncertain systems; output feedback.

I. Introduction In the past few years a great deal of interest has been devoted to the design of robust control of fixed structure for dynamic systems with parameter uncertainty. Considerable attention has been given to both the problems of robust stabilization and robust performance of uncertain linear systems and a number of significant results covering these issues have been reported in the literature; see [1,12-14,16-20] and the references therein. On the other hand, despite the significant advances in the theory of control of nonlinear systems (e.g. [11,21] and the references therein), to date the issue of designing robust output feedback control for nonlinear systems with parameter uncertainty has not been fully investigated. Robust stabilization of a class of uncertain nonlinear systems via memoryless state feedback controllers of fixed structure has been analysed in a number of papers (e.g. [2,4-6] and the references therein), whilst the problems of robust tracking for uncertain nonlinear systems has been studied by several researchers [7,9,10]. However, the issue of robust stabilization and robust performance of nonlinear systems with parameter uncertainty remains to be important and challenging. In this paper we consider the problem of robust output feedback control of a class of uncertain nonlinear systems with time-varying parameter uncertainty in both the state and output equations. The class of uncertain nonlinear systems considered here is described by an uncertain linear state space model with the addition of state dependent nonlinearities. These nonlinearities enter both the state and output equations via unknown nonlinear functions. Here, attention is focused on the design of linear dynamic output feedback controllers of fixed structure and two problems are addressed. The first one is the robust stabilization and the other is the problem of robust performance in an H~ sense. The main Correspondence to: Dr. C.E. de Souza, Dept. of Electrical & Computer Engineering, University of Newcastle, Rankin Drive, Newcastle, NSW 2308, Australia. * This work is supported by the Australian Research Council.

0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

Y. Wang et aZ / Robust control of uncertain nonlinear systems

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results of this paper establish that both the above problems can be converted into 'scaled' H= control problems which do not involve parameter uncertainty and unknown nonlinear functions.

2. P r o b l e m s t a t e m e n t and p r e l i m i n a r i e s

Consider the class of uncertain nonlinear systems described by a state space model of the form (~1):

~(t)=[A+AA(t)]x(t)+[G+AG(t)]g[x(t)]

+ [B+aB(t)]u(t),

y ( t ) = [C + a C ( t ) ] x ( t ) + [H + a H ( t ) ] h [ x ( t ) ] + [D + A D ( t ) ] u ( t ) ,

(2.1a) (2.1b)

where x(t) ~ E" is the state, u(t) ~ ~m is the control input, y(t) ~ Er is the measured output, A, B, C, D, G and H are real constant matrices of appropriate dimensions, AA(.), ,~IB(.), AC(.), AD(.), AG(.) and ~IH(-) are real-valued matrix functions which represent time-varying parameter uncertainties and the mappings g(-) : E" ---, E'~ and h(.) : E" ~ R "h are unknown nonlinear functions. The parameter uncertainties considered here are norm-bounded and of the following forms:

aA(.)

aB(.)

] [L,]

AG(.) = L c F c ( . ) E c ,

AH(.) =L,FH(.)EI~ ,

(2.2b)

where L a ~ ~ x i , L2 ~ Erxi, E1 ~ E j×,, E2 c Ejxm, L c ~ E~xic, E c ~ Ei~×~, LH ~ Er×iH and E n R ~'×"~ are known real constant matrices, and F ( . ) ~ Ei×j, Fc(. ) ~ Ei~,×j~, and F n ( . ) ~ R iH×m are unknown matrix functions satisfying

r T ( t ) F ( t ) <~I, F c ( t ) F ~ ( t ) <~I, Fz4(t)Ft~(t ) <~I,

Vt,

(2.3)

with their elements being Lebesgue measurable. In the above, the superscript 'T' denotes the transpose and the notation X >1 Y (respectively, X > Y), where X and Y are symmetric matrices, means that X - Y is positive semi-definite (respectively, positive definite). Moreover, I1" II will denote the Euclidean vector norm. Note that F(.), F c ( . ) and FH(-) can even be allowed to be state-dependent, i.e. F(t)=F[t, x(t)], Fc(t) = Fc[t, x(t)] and FH(t)= Fl4[t, x(t)], as long as (2.3) is satisfied. The unknown vector functions g(-) and h(.) are assumed to satisfy the following boundedness conditions: A s s u m p t i o n 1. There exist known positive scalars k 1 and k 2 such that for all x ~ ~",

II g ( x ) I I ~
(2.4a)

II h(x)II < k2 II x II.

(2.4b)

R e m a r k 2.1. It should be noted that nonlinear models of the form (2.1) can be used to represent many

important physical systems. A typical example is a power system modelled in the form of single machine - infinite bus [3]. The parameter uncertainty structure in (2.2) has been widely used in the problem of

robust stabilization of uncertain linear systems [12] and can represent parameter uncertainty in many physical cases. This paper is concerned with the problem of designing a linear dynamic output feedback controller for the system (2.1) such that the closed-loop system is globally uniformly asymptotically stable about the origin for all admissible uncertainties. In this case we say that the closed-loop system is robustly stable. To motivate the technique used in this paper for solving the robust stabilization problem, in the sequel we recall the notion of H~ disturbance attenuation and a related result.

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Consider the system A(t) = A x ( t ) + B w ( t ) ,

(2.5a)

z( t) = Cx( t ) ,

(2.5b)

where x ( t ) ~ ~n is the state, w(t) ~ ~m is the disturbance input and z ( t ) ~ ~t, is the controlled output. Definition 2.1 [15]. Given a scalar 3' > 0, the system (2.5) is said to have disturbance attenuation 3' if the following conditions are satisfied: (i) A is stable; (ii) II C(sI - A ) - l B I[o~< 3'. [] In the above, [[ G(s)[[ = denotes the infinity norm of a stable transfer matrix function G(s), defined by [[ G(s) I[o~= sup,o ~ Rtrmax[G(jto)], where O'max(") stands for the maximum singular value of a matrix. Lemma 2.1 [8,15]. Given a scalar y > O, the following statements are equivalent: (i) the system (2.5) has disturbance attenuation 3'; (ii) there exists a symmetric matrix P > 0 such that

ATp + PA + T-ZPBBTp + c T C < 0; (iii) there exists a solution Q

=

ATQ + QA + T-2QBBTQ +

QT ~ 0 cTc

=

to

the algebraic Riccati equation

0

such that the matrix A + y-2BBTQ is stable.

[]

To conclude this section, we introduce two matrix inequalities which will be used in the proof of our main results. Lemma 2.2. Let G, L, E and F(t) be real matrices of appropriate dimensions with F(t) being a matrix function. Then, we have (a) for any e > 0 and F T ( t ) F ( t ) <~I,

(2.6)

L F ( t ) E + E T F T ( t ) L T <.~~ L L T "JrE2ETE, (b) for any e > 0 such that e2ETE < I and F ( t ) F T ( t ) <<.I,

1 [G + L F ( t ) E ] [G + L F ( t ) E ] T <~G( I - e Z E T E ) - tGT + - ~ L L T.

(2.7)

Proof. Part (a) is a well known fact. To prove part (b), introduce the matrix

W(t)= - ~ - E E T

EG T -

--~

Note that e2EE T < I since e2ETE < I. Then, the expansion of w T ( t ) W ( t ) >~0 leads to

GETFT(t)L T + LF(t)EG T + LF(t)EETFT(t)L T

(I

<.~GET - ~ - E E

T

)-' EGT + - ~1L F ( t ) F T ( t ) L

T.

By considering that F(t)FT(t)<<. I and using the matrix inversion lemma, the desired result follows immediately. []

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3. Robust stabilization results

This section will display our main findings on robust stabilization of the system (2.1). We will establish an interconnection between the problem of robust stabilization and a scaled H~ control problem. This will allow us to solve the robust stabilization problem via existing techniques of Ha control such as the interpolation theory approach or the Riccati equation approach. We first consider the problem of stability analysis for the system (2.1). For the sake of generality, in this analysis condition (2.4a) is replaced by the following: Assumption 2. There exists a known positive scalar k~ and a bounded matrix Wg such that

IIg(x) ll
Vx~l~ n.

(3.1)

In the sequel we develop an interconnection between the robust stability of the unforced system of (2.1) (setting u(t)-= 0) and the disturbance attenuation property of the following scaled linear timeinvariant systems: (X2):

~(t)=A~(t)+/~(vl,

v2)~(t),

(3.2a)

i ( t ) = C(v 1, v2)~(t),

(3.2b)

where so(t) ~ ~n is the state, ~(t) ~ ~ is the disturbance input which belongs t o L 2 [ 0 , oo), Z(t) ~ ~/3 is the controlled output, A is as in (2.1) and /~(ul, v 2) and C(ul, v 2) are such that 1

B(]"I, v2)J~T(pl, /~'2) ~ F~LILT-I-G(1-v2EgEG) -IGT-I- 1"~22 LGLG'I T

(3.3a)

c T ( p l , p2)C(/'Jl, /"2) /.~ v2E~E1 + kZWTWg

(3.3b)

where E 1, E a, G, L l and L c are the same as in (X1), k I and Wg are as in (3.1) and v I and v 2 are 2 T positive scaling parameters such that I - v2Ec, E c > O. Theorem 3.1. Consider the unforced system of (2.1) (setting u(t) - O) satisfying Assumption 2. Then, this system is robustly stable if there exist positive scaling parameters v 1 and v 2 such that v2E~Ec < I and the system (3.2) has unitary disturbance attenuation. Proof. Note that by Lemma 2.1 it follows that there exists a matrix P = pT :¢ 0 such that

ATe +PA + P/~(Vl, I/2)/~T(/~I, p2)P-I-cT(I.'I,

I.'2)C(/~l, /)2) < 0.

By considering (3.3), it results that A T p + P A + P [ '- ~ 1 L , LTI + G ( I - v ~ E g E c ) - ' G T

+ ~ 1L G L a f 1P + v ~ E ( E , + k ~ W : W ~ < O .

(3.4)

Hence, using Lemma 2.2, (3.4) implies

[ A + LIF( t ) E, I Tp + P[ A + L1F( t ) E, ]

+ P[G + LcFG(t)Ea] [G + LcFc(t)Ec]TP + k2wTwe < 0. Now, let the following Lyapunov function candidate for the system (2.1): 2 t V(x, t ) = x T ( t ) P x ( t ) + k , f °

[W,x(s)lV[%x(s)l d s -

fotgTfx(s)]gtx(s)] ds.

(3.5)

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Note that in view of (3.1), V(x, t) > 0 whenever x(t) ~ O. Furthermore, along any state trajectory of the unforced system of (-~1) we have that ~)

M(t)

P[G+LcF~(t)Ec]][

[G+LcFc(t)E~]wP

-I

[ x(t)]T[

- ~ V ( x , t) = [g[x(t)]

x(t)

J[g[x(t)]

] (3.6)

where

M( t) = [ A + LIF( t)E1]T P + P[ A + LIF( t)E1] + kEWgTWg.

(3.7)

Finally, considering (3.5)-(3.7) it follows immediately that OV(x, t)/Ot < 0 whenever x(t) ~ O. Therefore, V(x, t) is a Lyapunov function of the unforced system of (2.1) and thus the equilibrium state x = 0 of this system is globally uniformly asymptotically stable for all admissible uncertainties. [] Remark 3.1. The result of Theorem 3.1 shows that the problem of stability analysis of a nonlinear system of the form ('~1) with u(t) - 0 can be converted into a 'scaled' H= analysis problem for a system without parameter uncertainty. Therefore, in view of Lemma 2.1, the robust stability of (-~1) with u(t) =- 0 can be assessed in terms of an algebraic Riccati equation. In connection with the robust stabilization problem for the system (2.1), we introduce a scaled linear time-invariant system as below: (,~3):

-k-Bl(gl,

eE)ff(t ) + B u ( t ) ,

(3.8a)

y ( t ) = Cx(t) + D l ( e 1, e3)ff(t ) + D u ( t ) ,

(3.8c)

~(t) =Ax(t)

5(t)=

~11+k2 I

where x ( t ) ~ ~n is the state, u ( t ) ~ R" is the control input, i f ( t ) ~ R i+".+"h+iG+i" is the disturbance input, y(t) ~ R r is the measured output, 5(t) ~ R n+j is the controlled output, A, B, C, D, El, g2, k 1 and k 2 are the same as in (2.1)-(2.4), Bl(el, e 2) and Dx(el, e 3) are defined as follows:

-~2

a G)

--L a

O,

(3.9a)

e 2

with L1, L2, G, H, L a, L H, E a and E H as given in (2.1) and (2.2), e 1, e 2 and e 3 being positive scaling 2 T parameters such that e~EgE a < I and e3EHEH < I. The interconnection between the robust stabilization of (2.1) and H~ control of (3.8) is provided by the next theorem. Theorem 3.2. Consider the system (2.1) satisfying Assumption 1. Then, this system is robustly stabilizable via a linear strictly proper output feedback controller Gc(s) if there exist positive scaling parameters el, e 2 and e 3 such that (i) e2E~Ec < I and e. a2 E HT E H < I; and (ii) the closed-loop system of (3.8) with the controller Go(s) has unitary disturbance attenuation.

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Proof. Let the controller Gc(s) be of the following state space realization:

2c(t)=Acxc(t)+Bcy(t),

(~c):

xc(0)=0,

(3.10a)

u(t) = Kcxc(t),

(3.10b)

where the dimension of the controller, n~, and the matrices A c, B c and K c are to be chosen. Now, the closed-loop system of (2.1) and (3.10) is given by ( ( t ) = [_d + L F ( / ) f f ~ ] s c ( t ) + [G + L c F c ( t ) f f ~ a ] f [ ~ ( t ) ]

(3.11)

where T T Xc] ,

sc = [ x v

hV[x(t)]] T,

f[{:(t)] =[gT[x(t)]

BK c

A

(3.12a)

--L 1 F~1

~xT=

,

BcC

Z =

Ac + BcDK c

1 ' --BcL 2

(3.12b)

E1

ff~ = [elE 1 elE2Kc],

Bc H ,

1 I I: I [ 1

--L~ T'°=

G=

0

e2

0

1

E2

,

a

0

Ec=

--BcL H

Fa(t )

,

e3Etl

0

fro(t) =

.

0

(3.12d)

Fl4(t )

E3

Note that Pc(t)F~(t)~
II f(~)II ~ k~-~2+ k~ II W{: II, where W = [ I , × , 0]. On the other hand, the closed-loop system of (3.8) and (3.10) is of the form ( ( t ) =A-~(/) + B f f ( t ) ,

(3.13a)

zT(t) = CsC(t),

(3.13b)

where

and the other matrices are as in (3.12). Now, using (3.12) it is straightforward to verify that ~T

= ~ - ' T -I'- G ( I

- E T E c ) -1~-T + T'a L~c,

~ T ~ = ff.Tff. + (k 2 + k ~ ) w T w .

(3.14a) (3.14b)

Hence, by applying Theorem 3.1 to (3.11) and (3.13) and considering (3.14), the desired result follows immediately. []

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The result of Theorem 3.2 can be easily specialized to the case of state feedback as follows: Corollary 3.1. Consider the system (2.1) satisfying Assumption 1 and suppose that the state variables are available for feedback. Then, this system is robustly stabilizable via a control law u ( t ) = Kx(t), where K ~ ~m×n is a constant matrix, if there exist positive scaling parameters e 1 and e 2 such that (i) e 22ETrGLG< I; and (ii) The closed-loop system of (3.8a) and (3.8b) with u(t) = Kx(t) and k 2 = 0 has unitary disturbance attenuation. Remark 3.2. In view of Theorem 3.2 or Corollary 3.1, the robust stabilization problem of (2.1) can be converted into a scaled H~o control problem for which neither parameter uncertainties nor unknown nonlinear functions appear. The corresponding scaled H= control problem can then be solved via existing techniques such as those in [8]. More specifically, the scaled H= control problem can be solved in terms of algebraic Riccati equations (AREs). In particular, when the state of (2.1) is available for feedback, only one A R E needs to be solved. In the output feedback situation, two AREs are involved, one for constructing the state feedback law and another for the estimation of the state feedback control law. It should also be noted that in view of the well known characterization of all strictly proper stabilizing controllers for output feedback Hoo control [8], Theorem 3.2 allows us to characterize a class of robust strictly proper stabilizing controllers for (2.1). Finally, we point out that when there is no nonlinear term in (2.1), i.e. g[x(t)] - 0 and h[x(t)] = O, the result of Theorem 3.2 will recover that of [20].

4. Robust Ha performance results In this section the results on robust stabilization will be extended for solving the problem of robust performance in an H~o sense. Consider the following class of uncertain systems: (~4)"

.~(t) = [A + A A ( t ) ] x ( t )

+ [G+AG(t)]g[x(t)]

z(t) = ClX(t) + O,2u(t),

+BlW(t ) + [B+AB(t)]u(t),

(4.1a) (4.1b)

y ( t ) = [C + A C ( t ) ] x ( t ) + [H + A H ( t ) ] h [ x ( t ) ] + D21w(t ) + [D + A D ( t ) ] u ( t ) , (4.1c) where z ( t ) ~ EP is the controlled output, w(t) ~ Eq is the disturbance input which belongs to Lz[O, oo) and the other variables are as in (2.1)-(2.4). Here, we address the following robust performance problem: Given a scalar 3' > O, design a linear time-invariant feedback control law u = Gc(s)y such that the closed-loop system is robustly stable and guarantees that under zero initial conditions, II z II 2 < 3" LIw II z for all nonzero w ~ L2[0, oo) and for all admissible uncertainties given in (2.2)-(2.4), where [1" II 2 stands for the usual L2[O, oo) norm. In this case, we say that the system (4.1) is robustly stabilizable with disturbance attenuation 3" and the closed-loop system of (4.1) with Gc(s) is robustly stable with disturbance attenuation 3". Initially, we will analyse the problem of robust stability with disturbance attenuation y of the unforced system of (4.1) (setting u(t) = 0). Similar to the case of robust stability of (2.1), we first replace condition (2.4a) by (3.1) and define the following auxiliary system: ('~5):

~(t) =A~Z(t) + [B 1 3 " n ( v l , v2)]~(t),

(4.2a)

2(t)

(4.2b)

=

~(Vl

'

/"2) ~:(t),

Y. Wanget al. / Robust control of uncertain nonlinear systems

146

where so(t) ~ ~n is the state, ~(t) E ~ + q is the disturbance input which belongs to L2[O, oo), £(t) ~ [~+P is the controlled output, /~(v 1, v 2) and C(v 1, v z) are as in (3.3) and A, B 1 and C, are the same as in (4.1). Theorem 4.1. Consider the unforced system of (4.1) (setting u(t) =- O) satisfying Assumption 2. Given a scalar y > O, this system is robustly stable with disturbance attenuation y if there exist positive scaling parameters v, and v 2 such that v 2 E ~ E c < I and the system (4.2) has disturbance attenuation ~. Proof. By Lemma 2.1, it follows that there exists a matrix P = pT > 0 such that

A T p + P A + Y - 2 P B I B T p + P I B ( U l , P2)BT(vI, v 2 ) P w C T C I + cT(b'l, v2)C(Vl, v2) < 0 .

(4.3)

Using the same technique as in the proof of Theorem 3.1, (4.3) implies that

[ A + L I F ( t )E1] T P + p[ A + L1F( t )E1] + y 2pBzBlrp

Tp + c T c I

+P[G+LGFG(t)EG][G+LGFG(t)EG]

+

wfwg< 0

(4.4)

and that the system (4.1) is robustly stable. To establish that PIz II 2 < y II w [I 2 whenever II w II 2 4: 0, let us introduce

J = foCe(zTz -- y2wTw) dt.

(4.5)

Note that the boundedness of J is guaranteed by the asymptotic stability of (4.1) and the fact that w ~ L2[0, oo). With zero initial condition and considering (4.1) it can be easily shown from (4.5) that

J=

xTcTc1 x +

=

fo~{XTQ(t)x-

xTpx)

--

y2wTw

dt

[g(x)--Gg(t)Px]T[g(x)-G g ( t ) P x ]

--~2[W--~-2BTpx]T[w--'~-2BTpx]

+ g T ( x ) g ( x ) - k l ( W g2 x

) T (WgX)} dt

(4.6)

where Q(t) denotes the left hand side of (4.4) and Ga(t) = G + AG(t). Finally, considering (3.1) and (4.6) we conclude that J < 0, i.e. ]1z ]12 ( Y [[ W I] 2 for all non-zero w(t) ~ L2[0, oo) and for all admissible uncertainties. [] We now show that the problem of robust Hoo control of system (4.1) can be converted into a scaled Ho~ control problem. To this end, associated with system (4.1), we introduce the following auxiliary system: (26):

~(t)=Ax(t)+[B

2(t) =

e,E, ~1+k22I

y(t)=Cx(t)+ where y(t) ~ bance Bl(el,

l

[D2,

yB,(el, eZ)]~(t ) + B u ( t ) ,

x ( t ) + elE z u ( t ) ,

yD,(e,,e3)]ff(t )+Du(t),

(4.7a)

(4.7b)

(4.7c)

x(t) ~ ~ is the state, if(t) ~ ~q+~+~e+-h+~o+~n is the disturbance input which belongs to L2[0 , oo), ~ is the measured output, 2 ( t ) ~ ~P+"+J is the controlled output, y > 0 is the desired disturattenuation level for (4.1), A, B, C, D, C l, D12, D21, E 1 and E 2 are the same as in (4.1), e 2) and Dl(el, e 3) are as in (3.9).

Y. Wang et al. / Robust control of uncertain nonlinear systems

147

Theorem 4.2. Consider the system (4.1) satisfying Assumption 1 and let y > 0 be a prescribed level of disturbance attenuation. Then, the system (4.1) is robustly stabilizable with disturbance attenuation y via a linear strictly proper output feedback controller Gc(s) if there exist positive scaling parameters e l, e z and e 3 such that (i) eZzE~Ec < I and eZE~EI4 < I; and (ii) the closed-loop system of (4.7) with Go(s) has disturbance attenuation y.

Proof. Similarly to the proof of Theorem 3.2, the desired result can be established by applying Theorem 4.1 to the closed-loop system of (4.1) with the controller (3.10) and the closed-loop system of (4.7) with the same controller (3.10). [] Remark 4.1. Similar to Remark 3.1, Theorem 4.2 allows us to convert the problem of robust stabilization with H= performance into a scaled H= control problem for which neither parameter uncertainties nor unknown nonlinear functions appear. The corresponding scaled H= control problem can then be solved via existing techniques such as the interpolation theory approach and the Riccati equation approach. Moreover, a class of robust strictly proper stabilizing controllers for the problem of robust Hoo control can also be derived. Indeed, this class of controllers is made of all strictly proper stabilizing controllers for the scaled H= control problem. When there is no nonlinear term in (4.1), i.e. g [ x ( t ) ] - 0 and h[x(t)] - O, the result of Theorem 4.2 will recover that of [20].

5. An example

Consider the problem of robust stabilization of the uncertain nonlinear system i 1 =x2,

(5.1a)

-f2 = ( - 3 + a ) x 1 + x 2 +b{x~ +x~ sin(cxl) + (a + 2)u

(5.1b)

where a, b and c are unknown but bounded with a ~ [ - 1, 1] and b ~ [ - 1, 1] (see Figure 1), and the state is available for feedback. This system is of the form of (2.1) with A = [ 0-3

11]'

B=[~],

G = [ ~ l,

g ( x ) = x~1+x22 sin(cx,),

k1=1,

Note that the origin is not a stable equilibrium point of the unforced system of (5.1).

E

A

a

C

-1

B

Fig. 1. Unknown parameters a and b.

148

Y. Wang et al. / Robust control of uncertain nonlinear systems 3

,

i

i

,

i

,

2

....

1

0

\i ,

....

.:

-1

"'-.

i ""-

A

-2

B

-3

~:~:r D

-5

6

/

. "-~

-4

--,.5

;

•

i

-1

-0.5

0

i

i

i

0.5

1

1.5

"

'C

2.5

3.5

x(1) Fig. 2. The state-plane trajectories of the closed-loop system of (5.1) with the derived control law. By Corollary 3.1 the robust stabilization problem for system (5.1) is converted into a state feedback H~ control problem for the following system

~(t)=[i'

:tx(t)+[i']

(5.2b)

It has been found that suitable scaling parameters are e 1 = 1 and e 2 = 0.99. Hence, by solving the above Ha control problem with unitary disturbance attenuation, we obtain a feedback control law u ( t ) = [ - 1 . 1 9 6 2 - 2 . 8 6 4 1 I x ( t ) which is also a suitable stabilizing control law for (5.1). Figure 2 shows the state-plane trajectories of the closed-loop system associated with the parameters a and b at the vertices A,B, CandDand c=l.

6. Conclusion This p a p e r has developed a linear dynamic output feedback robust control technique for a class of uncertain nonlinear systems. We have shown that both the problems of robust stabilization and robust H~ p e r f o r m a n c e can be converted into standard H~ control problems which do not involve p a r a m e t e r uncertainty and unknown nonlinear functions. A numerical example is presented to demonstrate our results.

References

[1] B.R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, J. Optim. Theory Appl. 46 (1985) 399-408.

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