J. Frank/in lnsr Vol
PII: S0016-0032(96)00132-9
Pergamon
-
3358, No 3, pp 517-124, 1998 IV’ 1997 The Frankhn lnshtute Published by Elwvier Scwnce Ltd Printed in Great Bntain 0016 0032,‘98 S19.00+(1.00
Robust H”O Controller De.signfor Linear Uncertain Systems with Delayed State and Control-T_ ~~JINGCHENGWANG,HONGYESU
and JIAN CHU
Institute of Industrial Process Control, Zhejiang People’s Republic qf China (Received
14 August
University,
1996; accepted in revised,form
Hangzhou,
18 November
310027,
1996)
: This paper mainly studies quadratic stabilization and robust H” controller design,Jcn linear uncertain dynamic systems with delayed state and control. The paper presents a static state feedback controller which stabilizes the plant and reduces the ejject qf the disturbance input on the controlled output to prescribed levels for all admissible uncertainties. The sufficient conditions ,Jin quadratic stabilization with a H” norm bound constraint are derived. The results can be regarded as an extension of existing results to robust H”’ controller design ,for linear uncertain dynamic, ‘“I 1997 The Franklin Institute. Published by Elsevier systems with delayed state and control. i( ABSTRACT
Science Ltd
I. Introduction There are two main branches in robust control theory: parameter robust control theory focusing on the known structure and bounds of uncertainties, and H” control theory focusing on the unknown structure of uncertainties but known H” norm bound constraint y from the disturbance input to the controlled output. In the last decade, significant results have been obtained in the parameter robust control theory [Edge Theorem etc, see e.g. (l)] and in the H” control theory [state-space two-Riccati approaches etc, see e.g. (2)]. This paper will consider H” norm bound constraint 1 (performance index in H” control theory) to deal with uncertainties whose structure is known (the controlled plant studied in parameter robust control theory). Over recent years, some results have been reported in this aspect. Zhou et al. (3) give a sub-optimal state feedback H” controller design method for time-invariant linear dynamic systems without uncertainties in system matrices, the H” controller can be easily obtained by solving an algebraic Riccati equation (ARE). Khargonekar et al. (4) propose a solution to a certain quadratic stabilization problem with uncertainties in the state and the control matrices, and gives out a relation between quadratic stabilization and N” control. Xie et al. (5) give necessary and sufficient conditions for quadratic stabilization with an H” norm bound on linear uncertain dynamic systems with time-varying normbounded uncertainties in both state and control matrices. Lee et al. (6) present a t Supported by the National Natural Science Foundation
517
of P.R. China
.I. Wuny et al.
518
memoryless H” controller for the state delayed system. Choi rt ~11. (7) present a memoryless H” controller design method for linear time-invariant systems without parameter uncertainties but with delays in the state and control. Choi et al. (8) consider the feedback stabilization of uncertain dynamic systems with time-varying delays in both states and controls. In this paper, we consider a robust H” controller design for a class of general linear systems with delayed state and control, uncertainties in all system matrices, but we know structures and bounds of uncertainties without the socalled matching condition. In the time domain, the idea of quadratic stabilization is used for a linear uncertain dynamic system. while in the frequency domain, the problem The memoryless linear is dealt with from the definition of H” norm bound constraint. time-invariant state feedback control law is obtained by solving an ARE or a Linear Matrix Inequality (LMI), which guarantees the quadratic stability of the closed-loop control system and reduces the effect of a disturbance input on the controlled output to a prescribed level ;‘.
II. Problem Formulation We consider equation:
a linear uncertain i(t)
delay system described
by the following
differential
= A,~x(t)+A,x(t-~,)+B,,u(t)+B,Ll(t-~h*)+D~z.(t) = (A,,+AA,).u(t)+(A, +(B,,+AS,)u(r)+(B,
+AA,)x(t-11,) +AB,)I*(~--z~)+L)n~(t)
3 = .F.r(t) x(t) = 0,
tfz-d,O].
d= max(hl.h2~
(1)
where x(t) E R” is the state vector, u(t) E R”’is the control input vector, IV(~)E R” is the disturbance input vector which belongs to L,[O, ‘m), and z(t) E R” is the controlled output vector, &, A,, &, B,, AA,,, AA,, A& AB, are uncertain real-valued matrices with appropriate dimensions, AO, A,, B,,, B,. D and E are known constant real-valued matrices with appropriate dimensions, h, and h, are non negative constants denoting time delays in state and control, respectively. Suppose the static linear state feedback control law with constant gain matrix is given as follows:
K=
u(t) = KY(t)
(2)
-!&‘B;P E
(3)
where E is a positive constant to be chosen, RE R “rx’i7is a positive definite weighting matrix to be chosen and P E R” ’n IS a positive definite matrix to be determined later. Then the closed-loop transfer function H=),,(s) from LV(~)to z(t) is given by
Thus the robust
H’ controller
design can be formulated
as determining
the matrix
Robust H” Controller
519
Design
P such that the closed-loop control system is stable while guaranteeing the restriction //HJs) /I~ < y for a given positive constant y. III. Main Results Dejinition 1 (4): The system (1) (with u(t), w(t) = 0) is said to be quadratically stable if there exists a positive definite symmetric matrix P and a positive constant a such that for any admissible uncertainty the derivative of the Lyapunov function f
I xT(@KTKx(0) d0
x’(Qx(t3) de + 2
V(x(t), t) = x’(t)Px(t)+2
s f- h,
i
r-h?
with respect to time satisfies V d --cll/x112
(5)
for all pairs (x, t) E R” x R. The system of (1) and (2) (with w(t) = 0) is said to be quadratically stabilizable via linear state feedback if there exists a state feedback control u(t) = Kx(t) such that the closed-loop system is quadratically stable. Suppose the uncertain structures of the system (1) are given by, AA,
= H,F,,
AA,
AB, = GLF1,
= H2F2,
FiFT
i=
AB, = G2F4
(6)
1,2,3,4
(7)
then we have the following results, Theorem I: Suppose that the disturbance input is zero for all time and there exists a positive definite matrix P satisfying the following matrix inequality
+ PHI H;P+
PH2H;P+
PG,G;P+
PG2G;P+31
< 0
(8)
for a positive constant E and a positive definite matrix R. Then the closed-loop system of (1) and (2) is quadratically stabilizable. Proof: For simplicity, we denote xh, = x(t - h,) and xhZ= x(t - h2). Substituting the control input into system (1) with (2) and assuming zero disturbance input, the closedloop feedback system can be rewritten as, k(t)
=
li,X+A,Xh,
+B,KX+&KX,,
=
(A,+H,F,)x+(A,
+H~F~)x/,~+(B~+G~F~)Kx+(B~
+GzFdKxh?
The Lyapunov function for this system is chosen as follows, f v=
xTPx+2
I
x’(fI)x@) de + 2 s r-h,
XT(8)KTKX(B) de I r-h,
then its derivative can be obtained as follows, vi= 2xTP{(A,+H,F,)x+(A,+H,F,)
Xh,+(BO+GIF3)Kx+(BI
+
+G2FdKxh2)
2xTx - 2x;, x,,, + 2xTKT Kx - 2x;? KT Kx,, 2
520
J. Wang et al.
Consider (7) and the inequality 2x’_r d x’~x+~‘y, with appropriate dimensions, we can get.
where x and _r are any vectors
Ti~xT(PA,+A~P+PH,H~P+~)x+.~-‘(PA,A~P+PH,H~P)x+2s~,s,~, +2xTPBoKx+2xTPG,F,Kx+xT(PB, +2xTx-2x;,x,,,
B:P+
~~x~K~K,~-~x~,K~K~,,~
PG2G1P)x+2.urjlK’K.u,,,
PH, H; P
+PA,A:P+PH,H:P+2PBoK+PG,G:P+3K7K+PB,B~P+PG2G:P+3()s
Substituting
(3) into (9), we get
Thus if there exists a positive definite matrix P for a positive constant c and a positive definite matrix R, and the inequality (8) is satisfied, there must be some positive constant c such that li < - (.I/.‘cI/* < 0. From Definition I. the closed-loop system is quadratically stabilizable. H Theorem II: For a given positive constant 7, suppose matrix P satisfying the following matrix inequality PA,$A;PfPA,A:P--;PB,R-‘B;P+ (-:
that there exists a positive definite
ZPB,R~2B;P+PB,B;P cz
+PH,H~P+PH2HfP+PG,G~P+PG,G;P+31+~E’E+
i
;PDD-‘P i
< 0
(10)
for a positive constant e and a positive definite matrix R. Then the closed-loop system of (1) and (2) is quadratically stabilizable and satisfies the constraint I/H:,,(s) 11~< 7. Proqf The proof of this theorem suffices to prove the inequality Theorem I. Obviously, there exists a positive definite matrix Q such that: !ETE ‘/
= -&Q-
; PDDTP+ Y
r*(,jw)P+
-PA,-A~P-PA,A:P~~PB,R~~R;P-
I/H;,,.(s) 11 I, d y from
PT(,jo) -:-.PB~R~ ‘B;P-PB,B:‘P &?
-PH,HTP--PHzH:P-PG,G:P-PG2G:P-33-p(jw)P-PT(jw)
(11)
Robust H” Controller
T(j0.1) = jol-
l-B,Rp’B~Pe-J’“h~
le-lwhI + -B,R-‘ E BTP+
A, -A,
521
Design
E
=jd-(AO+HIF,)-(A,
+H2FZ)epfluhl
+ i(B,+G,F,)R-‘BtP+
f(B,
+GZF4)R-‘B;fPe-J”h2
Denote WT(jw)W, (jo) = (A:Pejuhl -Z)*(ATPeJUhI Wz(jo)
W,(jw)
= (FTH:Pe’““I
-I)
-Z)*(FrHTPel”‘hl
-I)
Wt(jw)W,(jo)
= (FTH:P-Z)*(FTHTP-I)
W,*(jo)W,(jw)
= (R-‘BTP/&+FTG:P)*(Rp’B;fP/~+FTG:P)
W:(jo)W,(jw)
= (Rp’B;fPI~+FTG:Peh”h~)*(R-‘B~PI~+FTGTPeJ”h2)
W$(jm)W,(jo)
= (R-‘B~PIE+B’~P~‘“~~)*(R-‘B~P/E+B:P~’”~~)
Then (11) becomes
iET_E=
Obviously,
-EQ-~PDD~P+T~(~oJ)P+PT(~~o-
T( jo)
is invertible,
(12)
thus we can denote U(jo)
Pre-multiplying
i Wf’(jo)W,(jo) i= I
(12) by yDTT-
= DTPTp’(jo)P
*( jo)
-yDTT-*(jo)
and post-multiplying
EQ+
i
it by T- ’(jw)D,
W?(jo)u/;(jw)
we get
Tp’(jo)D
r=l
??
Thus IIffzw(s)ll, Q Y.
IV. Example An example is given to illustrate the following matrices,
the preceding
results.
Consider
the system (1) with
522
J. Wuny et al.
A”=[;
_;j
A&=[;;
4=[0;,
Oylj
B”=[:]
A,,=[:]
B,=[“,‘l
;]
A,,;[:,
:,1
AB,=[;]
and
Id>14, Isl, Inl G 0.1. If AA, = H,F,
AA,
= r;
= H2F2 = L:,’ 0
ABO = G,F3 =
‘;jri”
(i,,]
“,‘Il’-‘:”
&j
s/o. 1
[ 0.1 ] AB, = G,F,
0
=
[ 0.1.
1
n/O. 1
where F,FT < I
i = 1,2,3,4.
If 1 R=2,
Q=
symmetric
1
y=l
i0 the positive-definite
0
solution
and
E= I,
1’
of the Riccati equation
PAo+A~P+PA,A~P-1PB,R~‘B;P+2PB,R~2B~P+PB,B’fP E t? +PH,HfP+PHzH;P+PG,G:P+PGzG:P+ is
1;ETE+ i
_;PDDTP+31+cQ /
= 0
Robust H” Controller 800 ,
;
1
600
1
z ‘$00
0
z
I
I
I
1
I
I
_____________i_____________i_____________i_____________:______
?
I
523
_____________~_____________~_____________~_____________~__________~i~
0
2
I
Design
____.
.
,.
_____________:______________~_____________;___-______~~~~~__-.____.
200
.d-
.. ..--__.--
t 0
___....-+--
.. .. .. ...
0
2
/1 4
:
-I
I
4
6
8
10
6
8
10
I
I
t(second)
0
2
4
t(second) FIG. 1
I
K m.l
a
2 ? e,
.o
I
I
.I__.____.____:____
0.5
.____.__..._.:_.____ _________ ??
------_
0.0
I
z I
-0.5 0
1
S 2
.______
A_----_____----$_--_-----_---_ I I
I
3
4
I
I
5
t(second)
I
1.0
I
L_.____.____._,_____________~____.____.___I.____._______L_.____.____. I ............................... . .
-0.5 0
I
I
I
I
1
2
3
4
t(second) FIG. 2
.
.
5
524
J. Wang et al. 2.0552 P=
- 0.0074
-0.0074 0.6694
and u = - 3.0643~~ - 1.6624x2 is a robust H” controller guaranteeing the H” performance IIHZW(s)I/u1d 1. Choose h, = 0.8 and h, = 1, the open-loop response of state variables is given in Fig. 1. It shows that the open loop time-delay system is unstable. By using robust H” controller, the closed-loop system is robustly stablizable and preserves good response shown in Fig. 2 (v = z! = s = n = 0.1, r = r = s = y1= 0. and r = 1’= s = n = -0.1 piecewise). V. Conclusion The H” control design method of Choi et al. (7) is extended to deal with linear delay systems including uncertainties in state and control without satisfying the so-called matching condition. A sufficient condition in terms of a ARE (or LMI) is obtained to ensure not only the quadratic stabilization, but also the H” norm bound constraint of the closed-loop system. References (1) A. C. Bartlett, C. V. Hollot and L. Huang, “Root locations for an entire polytope of polynomials: it suffices to check the edges”, Math. Contr. Signals Syst., Vol. 1, pp. 61-71, 1988. (2) J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis. “State-space solution to standard Hz and H” control problem”, IEEE Trans. Automat. Control.. Vol. 34(8), pp. 881-897, 1989. “An algebraic Riccati equation approach to H” opti(3) K. Zhou and P. P. Khargonekar, mization”, Systems & Control Lett, Vol. 11, pp. 85-91, 1988. (4) P. P. Khargonekar, I. R. Peterson and K. Zhou, “Robust stabilization of uncertain linear systems: quadratic stabilizability and H” control theory”, IEEE Trans. Automat. Control, Vol. 35(3), pp. 356-361, 1990. (5) L. Xie and C. E. Souza, “Robust H” control for linear systems with norm-bound timevarying uncertainty”, IEEE Trans. Automat. Control, Vol. 37(8), pp. 1188-l 191; 1992. (6) J. H. Lee, S. W. Kim and W. H. Kwon, “Memoryless H” controllers for state delayed systems”, IEEE Trans. Automat. Control, Vol. 39(l). pp. 1599162, 1994. (7) H. H. Choi and M. J. Chung, “Memoryless H” controller design for linear systems with delayed state and control”, Automatica, Vol. 31(6), pp. 917-919, 1995. (8) H. H. Choi and M. J. Chung, “Memoryless stabilization of uncertain dynamic systems with time-varying delayed states and controls”, Automatica, Vol. 31(9), pp. 1349-1351. 1995.