- Email: [email protected]
Robust stabilization of discrete time-delay Markovian jump systems
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
14th World Congress of IFAC
Copyright «;) 1999 IFAC l4th Triennial World Congress~ Bcijing~ P.R. China
J-3d-Ol-3
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOVIAN JUMP SYSTEMS * Peng Shi * Rarnesh K. Agarwal ** and El Kebir Boukas ***
* To whom correspondence should be addressed: Centre for Industrial and Applicable
Mathematics, The University of South Australia;, The Levels Campus, Mawson SA 5095, A'ustralia. Fax.~ (+61-8) 89025785. Email: [email protected] ** National Institute for Aviation Research, Department of Aerospace Engine.ering, Vllichita State University; Wichita, KS 67260-0044; USA *** Departement de Genie Mecanique, Ecole Polytechnique de Montreal p~ O. Box 6D79, Station "Centre- Ville", Montreal, Quebec, Canada H3C 3A 7 Lakes~
1
Abstract. In this paper, v..~e investigate the stochastic stabilization problem for a class of linear discrete tilue-dclay systems \vith Markovian jump parameters. The jump parameters considered here is modelled by a discrete-tilne Markov chain. Our attention is focused on the design of linear state feedback memoryless controller such that stochastic stability of the resulting closed-loop system is guaranteed when the system under consideration is either with or without uncertainties. Sufficient conditions are proposed to solve the above problems, which are in terlns of a set of solutions of coupled matrix inequalities. (~opyright ch; 1999IJrAC Keywords .. Discrete-time systems,. !\-1arkovian jump parameters, l\1atrix inequalities, Parameter uncertainties, Time-delay.
1. INT'RODUCTION The dynamic. behaviour of IIlany industrial processes contains inherent time delays. Time delays may results from the distributed nature of the system, material transport, or from the time required to nleasure some of the variables. It has been known processes with time delays are inherently difficult to control, in the sense that it is difficult to achieve satisfactory performance. Control of time-delay systems has been a subject of great practical importance which has attracted a great deal of interest. for several decades; see, e.g., (~1alek-Zavarei
*
This \~rork was supported in part by the Australian Research Council under Grant A19532206 and in part by the Natural Science and Engin~ering Research Council of Canada under Grant OGP0036414.
and Jamshidi, 1987). In the past few years considerable attention has been given to both the problems of robust stabilization and robust control for linear systems with certain types of time-delays. rVlemoryless stabilization and H co control of uncertain continuous-time delay systems have been extensively investigated. For some representative prior v.. .ork on this general topic, v.re refer the reader to (Shen et al., 1991; Lee et al., 1994; ~1ahmoud and Al-]Vluthairi, 1994; Kirn et al., 1996; Li and de Souza, 1997). On the other hand, stochastic linear uncertain systems have been studied extensively, in particular, the linear uncertain systems "vith ~larko vian jumping parameters, to name a fev,t., (S'\vorder and Rogcrs, 1983; WilIsky, 1976; Chizeck et al., 1986; Ji and Chizeck, 1990; Boukas and Haurie, 1990; Ji et al., 1991; Boukas et al., 1995; Costa and Fragoso, 1995; Pan and
4935
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
Bar-Shalom, 1996a.; Pan and Bar-Shalom, 1996b; Shi and Boukas, 1997; de Souza and Fragoso~ }993). In particular, the problems of H oo control for linear systems ","ith both Markovian jump perturbations and parametric uncertainties have been tackled in continuous-time case (Shi and Boukas, 1997), and discrete..time case (Shi et at, 1998). The problem of optimal inventory-production control is studied in which the products depend on a stochastic deIIland rate governed by a ~larkov process. Also, a robust quadratic guaranteed cost control is designed in (Boukas and Sbi, 1998) such that a given cost function is bounded regardless of all admissible uncertainties. However, to the best of the authors' knowledge, the issue of stabilization of discrete time-delay I\farkovian jUlllP systeuls \vith or V\<~ithout parameter uncertainties has not been fully investigated and remains to be important and challenging. In this paper, \Ve first study the problenl of stochastic stabilization for a class of linear discrete time-delay systems possessing rvIarkovian jump parameters. The jump parauleters considered here is 11lodelled by a discretetillle l\tlarkov chain with constant probability transition matrix. A sufficient condition is proposed to stabilize the time-delay system via linear matrix inequality approach. Next, Vle deal with the design of robust linear state feedback mernoryless controller such that stochastic stability of the resulting closed-loop system is required to be achieved ,,~hen the system under consideration contains paranleter uncertainties. It is shown that the above problem can be solved if a set of coupled matrix inequalities have positive definite solutions.
Notation. The notations in this paper are quite standard. Z, ~n and ~nx m. denote, respectively, the set of integer numbers~ the n dimensional EucIidean space and the set of all n xm real matrices. The superscript HT" denotes the t.ranspose and the notation X "2:: Y(respectively, -,y > Y) where X and Y are symmetric matrices, means that X ~ Y is positive semi-definite(r€spectively~positive definite). I is the identity matrix with compatible dimension~ E{·} denotes the expectation operator wit.h respective to some probability measure p~ f 2 [Or ooJ is the space of square summable vector sequence over [O~ (Xl]. 11 . 11 "rill refer to the Euclidean vector IlOTln whereas 11 • II(o~ooJ denotes the £2 [0, oc·]-norm over [0, (X)] defined as Ilftl[o,co] = L:~ Itfk IP~·
14th World Congress of IFAC
Xk+l
4(T/k)Xk
:= ..
+ . 4 d (17k )Xk-d + B(1]k)Uk,
x(O) =xo;
1}o
== i; k E Z
(1)
vlhere Xk E Rn is the system state, Uk E R m is the control input, d is an unknoVv"n positive integer time delay of the system. {1]k, k E Z} is a tilne homogeneous 1\1"arkov chain taking values in a finite set S =:: {I, 2, .... s}, \vith transition probability from mode i at time k to mode j at time k + 1, k E Z 1
Pij
with
Pij
(2)
= Pr{71k+l :::: jl1Jk = i)
2: 0 for i, j
E S, and
2:;=1 Pij
= 1.
In the
above, for 7Jk = i, i E S) A(1Jk) ~ Ai, Ad(1]k) g A di B(1]k) g Ri arc known rcal-"valued constant matrices of appropriate dilnensions that describe the nominal system. In system (1)-(2), "re define x(k) == 0 if k < 0, k E z~
Remark 2 .. 1 Note that system (1)-(2) is called a discretetime Markovian jump linear system. This kind of system can be used to represent many important physical systems subject to randorrt failures and structure changes, 8uch as electric power systern8 (Will.sky, 1976), control systems of a solar thermal central receiver (Sworder and Rogers, 1983), communications systems (Athans, 1987), aircraft jiight control (Moe1'der et al.~ 1989), control of nu.clear power plants (Petkovski, 1987) and manufacturing systems (Boukas and Haurie, 1990; Boukas et al~, 1995). It should be mentioned that one of the motivations to conduct this resear'ch i8 that most of the real dynamic physical processes contains inherent time delays and uncertainties, and can be modelled by an uncertain system with state delay ((Malek-Zavarei and Jamshidi, 1987)) which is frequently a cause for performance degradation and instability~ Thus, the system (1)-(2) encompasses many state space models of delay systems and have been widely used in systerns like cold rolling mills, wind tunnel and water resources, see, for example, (Malek Zavarei and Jamshidi, 1981) and the references therein.
o Let Xk (xo~ 7}o) denote the trajectory of the state Xk from the initial state (xo, 1]0). \Ve introduce the follo"\ving stability and stabilizability concepts for discrete-time jump linear systerIls.
Definition 2.1 (Ji et al., 1991) For system (1)-(2) with == 0, the equilihriurn point 0 is stochastically stable, if for every initial state (xo, TJo)
Uk
2~
PR,OBLEJ\1
FORNI1JLATIO~ AND
PRELI~:1INARIES
Consider t.he following class of dynarnical system in a fixed complete probability space (0, f, P):
hold8.
4936
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
Jefinition 2.2 (Ji et al., 1991) We say the system (1)2) is stochastically stabilizable,. if for every initial state xo, 'TJo), there exists a linear feedback control law Uk =::
14th World Congress of IFAC
}J"(i) == [Q(i) - Q(ri)B(i)(I + B T (i)Q(i)B) -2 B T (i)Q(i)] s
Q(i) == Po(i)
+ 1,
Po(i) ~ LPijP(j)
~(i)Xk ~ L(TJk == i) ~ L(i) (constant for each value of i) l1ch that the closed-loop system
j=l
then system (1)- (2) can be stochastically stabilized by a controller 'Uk = L(TJk)Xk
is stochastically stable. In this paper, \ve assume that all jump states 1Jk, k E Z and system stateR Xk, k E Z are accessible, that is, they arc perfect for feedback. The objective of this paper is to design a delay-independent memory-less state feedback controller of the form
(3)
based on the state Xk a.nd the mode 'T}k, such that the resulting closed-loop system of (1)-(2) with 9(T)k) is stochas-
L(1]k) ~ -2 [I
+ B T (11k)Q(Tlk)B(1]k)]-1
B T CTJk)Q(l1k)A(1]k).
Rem.ark 2.2 Theoreln £.1 provides a sufficient condition for the stochastic stability of the time-delay syste7n (1)-(2). Note that the control gain matrix L(nk} for the ,1jtabilizing controller does not depend upon the time delay d (It is not known in general). AL~o, the conditions of (4)-(4) are linear matrix inequalities (LMIs) type which can be resolved via the technique proposed in (Boyd et al., 1994). 0
tically stable.
Let us recall the follo'\ving lemmas which Vv'"ill be used in the proofs of our main results in this paper. Lemma 2.1 (Boyd et al., 1994) Given constant rnatrices M, L, Q of appropriate dimensions where l~f and Q are symmetric and Q > 0) then }.If + LTQL < 0 if and only if
Ai LT ] [ L _Q-l
3. ROBLTST STABILIZATION RESlTLTS In this section, on the basis of the result in previous section we will consider the robust stabilization on a class of uncertain systems in the folloVw·ing form
<0
(4) " . here .L\A(k, 17k), 6Ad(k~ 1}k) and 6.B(k, 7)k) are unknown matrices which represent time-varying parametric uncertainties and assumed to belong to certain bounded compact sets, and matrices A(llk), ..4d(1}k) , B(1]k) and all other variables are as in (1)-(2).
or equivalently
L] < o.
Q-1 [ LT 1\11
The admissible parameter uncertainties are assumed to be modelled as
Before ending this section, ",~e establish the following result of stochastic stabilization for system (1)-(2).
Theorem. 2.1 Consider system (1)-(2). If there exists a set of matrices {P(i) > 0, i =: 1,2, ... 1 s} J such that the following inequalities hold for i := 1, 2, ... , s
_Q-l (i) [ BT(i) _
B{i)
]
[I + BT('i)Q(i)B(i)]z <
where for 0
E d{1]k)
~
=
'T}k
:::=
i, i
E 5, .
E 1 (1}k) ~ E1(i) ~
E Rjxn, .
Ed(i) E RJxn, E 2 (1Jk) = E 2(i) E RJxn and
H(1Jk) ~ H(i) E R nxi a.re known real constant mat rices which characterize ho~~ the uncertain paranleter in F(k, T}k) £ F i (k) enters the nominal matrix A(17k), B(1Jk); and F(k, 'Y}k), fJk := i E S is a unknoVll"n time-varying ma-
T
-P(i) V2A (i)] [ V2A(i) _y-l(i) < 0
1
trix function satisfying
IIF(k,7Jk)11 :5 1i Vk E Z; r/k
where
==
i E S.
(6)
4937
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAYMARKOV...
14th World Congress ofIFAC
-P+Ql AT +ETFTDTJ [ A+DFE _p-l+Q2 <0
~ernark
3.1 In practical: it is almost always impossito get an exact mathematical model of a dyna'mical Istern (for example, manufacturing systems (Boukas nd Haurie, 1990j Boukas et al., 1995; Boukas et al. 1 998)) due to the complexity of the system, the difficulty f measuring 1Jarious parameters, environmental noises, ncertain and/or time-varying parameters, etc. Indeed, the model of the system to be controlled a17nost ai-ways contains some type of uncertainty. Thus} robustness requir'ement (robust .stability of the uncertain closed-loop system an,d the robust performance) is essential and important in control system design, which leads our .,r;tudy to the uncertain syste.m of (4). Note that, similarly to the deterministic system, the parameter '{J,ncerlainty St1~UC ture as in (5)-(6), when 1Jk =::: i, i E S, has been l1Jidely used in the problems of robust control and robust filtering of uncertain systems (see, e~g. (Boukas and Shi, 1998; Petersen; 1987; Shi et aI., 1998; Shi: 1998; Xie et at, 1993) and the refer"ences therein) and many pract2'cal systems possess paro:rneter uncertainties which can be either exactly modelled) or overbounded by (6). Observe that the unknou1n matrix F(k 1)k) in (5) can even be allowed to be state-dependent, i.e~ F(k, 'Tlk) == F(k Xk, 'lJk}, as long as (6) is satisfied. 0 ~e
if and only if there exists a constant e > 0 such that
NO\~l
'vc arc in the position to establish our main result
in this paper.
Theorem 3al Consider system (4) and (2) (setting AB(1]k) == If there exist a set of matrices {P(i) > 0, i ~ 1,2, ... , s} and a set of scalars {Ci > 0, i .:;;;; 1, 2, H., 8} such that the following inequalities hold for i = 1,2, ... , S
O)~
-P(i) [
j
1
+ cl E[(i) El (i) V2A T (i)] < 0 J2~4(i) - ..¥ (i)
where
In this section, our aim is to design a controller 'ilk = L(7]k)Xk for the uncertain syRtem (4) with (2) such that the resulting closed-loop systerIl is stochastically stable for all admissible uncertainties satisfying (5)-(6). First, \ve shall, however, confine our study in the case when uncertainties only appear in the state matrices A( 7Jk ) and Ad (1]k) , i.e., 6.B(k, TJk) ~ o. To this end, we need some intermediate results which are introduced in the sequel as lemmas.
s
Q(i)
= Po(i) + I,
Po(i)
=
LPijP(j) j=1
then system (1)-(2) can be stQchastically stabilized by a controller
LernIIla 3 . 1 (Petersen and HaUot, 1986) Given 1natrices D, F, E of appropriate dimensions and \Ix E Rn and Vy E Rn, then
'Uk = L(1Jk)Xk
L(ruc)
==
-2
(1 + B T C71k)QCTJk )B(l1k)] -1 BT(rtk)Q('Tlk)A(
for all admissible parameter uncertainty ~A4{1]k) and D.A d(1Jk) satisfying (5)- (6). LelllDla 3.2 (Petersen, 1987) Let .oI¥' Y and Z be given k x k symmetric matT·ices such that X 2 0, Y < 0 and
Z
~
o.
Remark 3 ..2 In The.orem 3.1 it is shown that the robust stochastic stabilization of system (4) with (2) is guaranteed if the inequalities {7}- (7) are hold. Note that when time-delay term in (4) disappears and parameter uncertainty 6.A(?7k) equals to zero, Theorem 3.1 will reduce to the results in (Chizeck et al., 1986; Ji et aI., 1991). Furthermore, if the modes 'r}k are all equal to 1, Theorem 3.1 will recover those, for example: (F'lAruta and Phooy'aruenchanachai; 1990; de Souza et al., 1993; Garcia et al., 1994; Yuan et aI.) 1996)~ 0 j
Furthermore, assume that
for all non-zero € E Rk. Then, there exists a constant > 0 such that the matrix £2 X + eY + Z is negativedefinite.
€
LeIllD1a 3.3 Given matrice.c; ll, D, F, E, Q1 and Q2 of
Frorll Theorem 3.1, when parameter uncertainties appear in both state and control input matrices, v.re have the following result-
0 and Q2 ~ 0 and pT F ~ I, then there exists a matrix P > 0 such that appropriate dimensions w2'th Q1
~
4938
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
:'heorelll 3 . 2 Consider system (.4) and (2). If there ex~t a set of matrices {P(i) > 0, i == 1,2, ... , s} and a set f scala,"'s {c i > 0, i =: 1, 2 'I ~ •• ~ .~} l~'U.ch that the following ~equalities hold for i = 1,2, ... ) S
-P(i) [
+ ~ [E[{i) El (i) + Ef(i)E2(i)] v'2AT (i)] < 0 v'2..4 (i) -X(i)
14th World Congress ofIFAC
formance has been made for both H 2 (with known statistics noise) and H oo (with energy bounded noise only) cases, see, (Costa et al.:, 1997; Shi et al., 1998)7 which is beyond the intere..~t8 of this paper on stabilization issue. Howet.'er, the study on Markovian jump systems with timc- delay and input noise should be quite interesting. 0
4.
CONCLUSIO~
In this paper, we studied the problems of stochastic stabilization of discrete-time system containing 11arkovian jurnp paraIIleters and tinle-delay in state. Both the cases of being \vith and Vw'"1.thou t parameter uncertainties of the underlying system are considered. It is shown that the above problems can be solved if certain coupled matrix inequalities have positive definite solutions.
uJhere
X(i)
=:
[Q(i) - Q{i)B(i)(I
+
B T (i)Q(i)B(i))-2 BT(i)Q(i)]-l
+ v'2 H(i)HT(i) Ei B
Q(i)
=:
Po (i)
+ I,
Po(i)
==
LPijP(j) j=l
then system (1)-(2) can be controller
.~tochastically
stabilized by a
L('TJk)Xk L(TJk) =:= -2 [1 + B T (7Ik)Q(rlk)BC17k)]-1 B T (1]k)Q(r}k)A('1Jk) llk :=:
for all admissible parameter ?j,ncertainty ~A(l1k)) L\.. 4 d (r}k) and ~B(1}k) satisfying (5)-(6). Proof. It can be derived via the same technique as that used in Theorem 3.1. ~'Vv
Renlark 3.3 It should be noted that due to the existence of unknown time-delay and parameter uncertainties? Theorems 3.1 and 3.2 provide only sufficient C01l,ditions to guarantee the robust stabil1:zation of system (4). It may be conservative in the sense of the controller being over-designed. However) these conditions, i.e., the solvability of (7)- (7) and (1)-(7) can be easily verified by the techniques pr·oposed in (Abo'u-Kandil et al., 1995), that is) given a set of scalars {ci > 0, i == 1, 2) ... s} J to compute the solution {P(i) > 0, i = 1 7 2: ... ,s} of (7)(7) or (7)-(7) by employing the algorithm~ from (AbouKandil et al.] 1995)~ If it fail.r;, then decrease {ci > o, i == 1, 2, ... s} and try to so Ive the corresponding inequalities again. Finally, it is ·worthwh·ile to point o'ut that if the unknown time-delay teT1n Ad(1}k)Xk-d is t,'eated as a input noise} the analysis on robust stability and per-
References
Abou-Kandil, H., FreiIing, G., and Jank, G. 1995. On the solution of discrete-time Markovian jump linear quadratic control problems, Auto1natica 32(5),765768. Athans, l\1~ 1987. Command and control (C2) theory: A challenge to control science, IEEE Trans. Automat. Control 32(2), 286-293. Boukas, E. K. and Haurie, A. 1990. l\tlanufacturing flow control and preventive maintenance: a stochastic control approach~ IEEE Trans. Automat. Control 35(7),1024-1031.
Boukas) E. K. and Shi, P. 1998. Stochastic stability and guaranteed cost control of discrete--time uncertain systems with I\1arkovian jumping parameters, Int. J~ Robust fj Nonlinear ControL to appear. Boukas, E. K., Shi 1 P., and Andijani, .oLl\4 1998. Optimal inventory-production control problem with stochastic demand~ Optimal Control Applications €1 Methods. to appear. Haukas, E. K., Zhang, Q., and Y-in, G. 1995. Robust production and maintenance planning in stochastic manufacturing systems~ IEEE Trans~ A utornat. Control 40(6), 1098-1102. Boyd, S., Ghaoui, L. E., Feron, E., and Balakrishnan V. 1994. "Linear matrix inequalities in system and control theory," SIAlVl PhiladelphiaChizeck, H. J., Winsky~ A. S., and Castanon, D. 1986. Discrete-time markovian-jump linear quadratic opt.imal control, Int. J. Control 43(1), 213- 231. Costa, O. L. \;. and Fragoso, M. D. 1995. Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems, IEEE Trans. Automat. Control 40(12), 2076-2088. 1
j
4939
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
~osta! O~
L. \T., \Tal, J. B. Do, and Gcromel, J. C. 1997. convex programming approach to H z control of discrete-time markovian jump linear systems~ Int. J. Control 66(4), 557-579. e Souza, C. E., Fu, M., and Xie, L. 1993. H co analysis and synthesis of discrete-timesystems with timevarying uncertainty~ IEEE Trans.Automat. Control 38(3), 459-462. ~t\
de Souza, E. C. and Fragoso, ~1. D. 1993~ H oo control for linea.r systems with :rvlarkovian jumping parameters, Control-Theory and Advanced Technology 9(2),457466. FUruta, K. and Phoojaruenchanachai, S. 1990. An algebraic approach to discrete-time H(XJ control problems. In Proc. 1990 American Control Conf pages 3067-3072 San Diego, CA. Garcia~ G., Bernussou, J., and ArzeIier, D. 1994. Robust stabilization of discrete- time linear systems with norm- bounded time-varying uncertainty, Systems 8 Control Letters 22(5), 327-339. Ji, Y. and Chizeck, H. J. 1990. Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Tmns. Automat. Control 35(7),777- 788. Ji, Y., Chizeck, H. J., Feng, X., and Loparo, K. -L~. 1991. Stability and control of discrete-time jump linear systenlS, Control Theo1Y and Advanced Technology 7(2), 247-270. Kim~
J. H., Jeung, E. A., and Park, H. B. 1996. Ro-
Int. J. Control 64(4), 631- 661. Pan, G. and B ar-ShaJ om, Y. 1996b. A stabilizing controller for jump linear gallssian systems "vith noisy state observations, European J. Control 2(2), 227~ 238. Petersen, I. R. 1987. A stabilization algorithln for a class of uncertain linear systems, System f1 Control Letters 8(4), 351-357. Petersen 1 I. R. and Hollot, C. V. 1986. A Riccati equation approach to the stabilization of uncertainlinear systems, Automatica 22(4),397-411. Petkovski, D. B. 1987. iv1ultivariable control system design: a case study of robust control of nuclear power plants, Fault Detection and Reliability 9, 239- 246. Shen, .1., Chen, B.~ and Kung, F'. 1991. ~lemoryless stabilization of uncertain dynamic delay systems:Riccati equation approach, IEEE Trans.Automat. Control 36~ 638---640. Shi, P. 1998. Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control43(7), 1022-1027. Shi~ P. and Boukas, E. K. 1997. HOC) control for NIarkovian jumping linear systems with parametric uncertainty, J. Optimization Theory and Applications
95(1), 75--99~ P., Boukas, E. K., and -,-~garwal, R. K. 1999. Robust control for Nlarkovian jumping discret.e-time systcms~ Int. J. Systems Science. to appear. S",rorder, D. D. and R,ogers, R. O. 1983. An LQ-solution Shi~
to a control problem associated with a solar thermal central receiver, IEEE Trans. Automat. Control 28(8)t 971-978. \\l'illsky, A. S. 1976. ,A. survey of design methods for failure detection in dynamic systems, Automatica 12(5), 601-611. Xie, L. t Shi, P., and de Souza, C. E. 1993. On designing controllers for a class of uncertain sampled-data nonlinear systems, lEE Proc.-Control Theory Appl. 140(2), 119-126.
bust control for parameter uncertain delay systems in state and control input, A 1Ltornatica 32(9), 13371339. Lee, J. H., Kim, S. W., and Kwon, W. H. 1994. Memoryless Hoo controllers for state delayed systems, IEEE Trans. Automat. Control 39(1),159-162. Li, X. and de Souza, C. E. 1997. Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear rnatrix inequality approach, IEEE Trans.Automat. Control 42(8), 1144-1148. Mahmoud, M. S. and ..J \l-Muthairi, N. F. 1994. Quadratic stabilization of continuous time systems with state-delay and norm-bounded tilne-varying uncertainties~
14th World Congress ofIFAC
YUaD, L., Achenie, L., and Jiang,
'~lr
1996. Robust Hoc
control for linear discrete-time systems witl} Ilormbounded time-varying uncertaintY1 Systems & Control Letters 27(2), 199~208.
IEEE Trans. Automat. Control 39(10),
2135-2139. :rvlalek-Zavarei~ ~f.
and Jaulshidi, M. 1987. Timedelay systems: analysis, optimization and applications. North-Holland Systems and Control Series. Moerder, D. D., Halyo, N., Broussard, .1. H.. , and Caglayan, A. K. 1989. Application of precomputed control la\vs in a reconfigurable aircraft flight control systern, J. of Guidance 7 Control and Dynamics 12(3), 325-333. Pan, G. and Bar-Shalom, Y. 1996a. Stabilization ofjump linear ga llssian systerns \vithou t mode observations,
4940
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
Copyright «;) 1999 IFAC l4th Triennial World Congress~ Bcijing~ P.R. China
J-3d-Ol-3
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOVIAN JUMP SYSTEMS * Peng Shi * Rarnesh K. Agarwal ** and El Kebir Boukas ***
* To whom correspondence should be addressed: Centre for Industrial and Applicable
Mathematics, The University of South Australia;, The Levels Campus, Mawson SA 5095, A'ustralia. Fax.~ (+61-8) 89025785. Email: [email protected] ** National Institute for Aviation Research, Department of Aerospace Engine.ering, Vllichita State University; Wichita, KS 67260-0044; USA *** Departement de Genie Mecanique, Ecole Polytechnique de Montreal p~ O. Box 6D79, Station "Centre- Ville", Montreal, Quebec, Canada H3C 3A 7 Lakes~
1
Abstract. In this paper, v..~e investigate the stochastic stabilization problem for a class of linear discrete tilue-dclay systems \vith Markovian jump parameters. The jump parameters considered here is modelled by a discrete-tilne Markov chain. Our attention is focused on the design of linear state feedback memoryless controller such that stochastic stability of the resulting closed-loop system is guaranteed when the system under consideration is either with or without uncertainties. Sufficient conditions are proposed to solve the above problems, which are in terlns of a set of solutions of coupled matrix inequalities. (~opyright ch; 1999IJrAC Keywords .. Discrete-time systems,. !\-1arkovian jump parameters, l\1atrix inequalities, Parameter uncertainties, Time-delay.
1. INT'RODUCTION The dynamic. behaviour of IIlany industrial processes contains inherent time delays. Time delays may results from the distributed nature of the system, material transport, or from the time required to nleasure some of the variables. It has been known processes with time delays are inherently difficult to control, in the sense that it is difficult to achieve satisfactory performance. Control of time-delay systems has been a subject of great practical importance which has attracted a great deal of interest. for several decades; see, e.g., (~1alek-Zavarei
*
This \~rork was supported in part by the Australian Research Council under Grant A19532206 and in part by the Natural Science and Engin~ering Research Council of Canada under Grant OGP0036414.
and Jamshidi, 1987). In the past few years considerable attention has been given to both the problems of robust stabilization and robust control for linear systems with certain types of time-delays. rVlemoryless stabilization and H co control of uncertain continuous-time delay systems have been extensively investigated. For some representative prior v.. .ork on this general topic, v.re refer the reader to (Shen et al., 1991; Lee et al., 1994; ~1ahmoud and Al-]Vluthairi, 1994; Kirn et al., 1996; Li and de Souza, 1997). On the other hand, stochastic linear uncertain systems have been studied extensively, in particular, the linear uncertain systems "vith ~larko vian jumping parameters, to name a fev,t., (S'\vorder and Rogcrs, 1983; WilIsky, 1976; Chizeck et al., 1986; Ji and Chizeck, 1990; Boukas and Haurie, 1990; Ji et al., 1991; Boukas et al., 1995; Costa and Fragoso, 1995; Pan and
4935
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
Bar-Shalom, 1996a.; Pan and Bar-Shalom, 1996b; Shi and Boukas, 1997; de Souza and Fragoso~ }993). In particular, the problems of H oo control for linear systems ","ith both Markovian jump perturbations and parametric uncertainties have been tackled in continuous-time case (Shi and Boukas, 1997), and discrete..time case (Shi et at, 1998). The problem of optimal inventory-production control is studied in which the products depend on a stochastic deIIland rate governed by a ~larkov process. Also, a robust quadratic guaranteed cost control is designed in (Boukas and Sbi, 1998) such that a given cost function is bounded regardless of all admissible uncertainties. However, to the best of the authors' knowledge, the issue of stabilization of discrete time-delay I\farkovian jUlllP systeuls \vith or V\<~ithout parameter uncertainties has not been fully investigated and remains to be important and challenging. In this paper, \Ve first study the problenl of stochastic stabilization for a class of linear discrete time-delay systems possessing rvIarkovian jump parameters. The jump parauleters considered here is 11lodelled by a discretetillle l\tlarkov chain with constant probability transition matrix. A sufficient condition is proposed to stabilize the time-delay system via linear matrix inequality approach. Next, Vle deal with the design of robust linear state feedback mernoryless controller such that stochastic stability of the resulting closed-loop system is required to be achieved ,,~hen the system under consideration contains paranleter uncertainties. It is shown that the above problem can be solved if a set of coupled matrix inequalities have positive definite solutions.
Notation. The notations in this paper are quite standard. Z, ~n and ~nx m. denote, respectively, the set of integer numbers~ the n dimensional EucIidean space and the set of all n xm real matrices. The superscript HT" denotes the t.ranspose and the notation X "2:: Y(respectively, -,y > Y) where X and Y are symmetric matrices, means that X ~ Y is positive semi-definite(r€spectively~positive definite). I is the identity matrix with compatible dimension~ E{·} denotes the expectation operator wit.h respective to some probability measure p~ f 2 [Or ooJ is the space of square summable vector sequence over [O~ (Xl]. 11 . 11 "rill refer to the Euclidean vector IlOTln whereas 11 • II(o~ooJ denotes the £2 [0, oc·]-norm over [0, (X)] defined as Ilftl[o,co] = L:~ Itfk IP~·
14th World Congress of IFAC
Xk+l
4(T/k)Xk
:= ..
+ . 4 d (17k )Xk-d + B(1]k)Uk,
x(O) =xo;
1}o
== i; k E Z
(1)
vlhere Xk E Rn is the system state, Uk E R m is the control input, d is an unknoVv"n positive integer time delay of the system. {1]k, k E Z} is a tilne homogeneous 1\1"arkov chain taking values in a finite set S =:: {I, 2, .... s}, \vith transition probability from mode i at time k to mode j at time k + 1, k E Z 1
Pij
with
Pij
(2)
= Pr{71k+l :::: jl1Jk = i)
2: 0 for i, j
E S, and
2:;=1 Pij
= 1.
In the
above, for 7Jk = i, i E S) A(1Jk) ~ Ai, Ad(1]k) g A di B(1]k) g Ri arc known rcal-"valued constant matrices of appropriate dilnensions that describe the nominal system. In system (1)-(2), "re define x(k) == 0 if k < 0, k E z~
Remark 2 .. 1 Note that system (1)-(2) is called a discretetime Markovian jump linear system. This kind of system can be used to represent many important physical systems subject to randorrt failures and structure changes, 8uch as electric power systern8 (Will.sky, 1976), control systems of a solar thermal central receiver (Sworder and Rogers, 1983), communications systems (Athans, 1987), aircraft jiight control (Moe1'der et al.~ 1989), control of nu.clear power plants (Petkovski, 1987) and manufacturing systems (Boukas and Haurie, 1990; Boukas et al~, 1995). It should be mentioned that one of the motivations to conduct this resear'ch i8 that most of the real dynamic physical processes contains inherent time delays and uncertainties, and can be modelled by an uncertain system with state delay ((Malek-Zavarei and Jamshidi, 1987)) which is frequently a cause for performance degradation and instability~ Thus, the system (1)-(2) encompasses many state space models of delay systems and have been widely used in systerns like cold rolling mills, wind tunnel and water resources, see, for example, (Malek Zavarei and Jamshidi, 1981) and the references therein.
o Let Xk (xo~ 7}o) denote the trajectory of the state Xk from the initial state (xo, 1]0). \Ve introduce the follo"\ving stability and stabilizability concepts for discrete-time jump linear systerIls.
Definition 2.1 (Ji et al., 1991) For system (1)-(2) with == 0, the equilihriurn point 0 is stochastically stable, if for every initial state (xo, TJo)
Uk
2~
PR,OBLEJ\1
FORNI1JLATIO~ AND
PRELI~:1INARIES
Consider t.he following class of dynarnical system in a fixed complete probability space (0, f, P):
hold8.
4936
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
Jefinition 2.2 (Ji et al., 1991) We say the system (1)2) is stochastically stabilizable,. if for every initial state xo, 'TJo), there exists a linear feedback control law Uk =::
14th World Congress of IFAC
}J"(i) == [Q(i) - Q(ri)B(i)(I + B T (i)Q(i)B) -2 B T (i)Q(i)] s
Q(i) == Po(i)
+ 1,
Po(i) ~ LPijP(j)
~(i)Xk ~ L(TJk == i) ~ L(i) (constant for each value of i) l1ch that the closed-loop system
j=l
then system (1)- (2) can be stochastically stabilized by a controller 'Uk = L(TJk)Xk
is stochastically stable. In this paper, \ve assume that all jump states 1Jk, k E Z and system stateR Xk, k E Z are accessible, that is, they arc perfect for feedback. The objective of this paper is to design a delay-independent memory-less state feedback controller of the form
(3)
based on the state Xk a.nd the mode 'T}k, such that the resulting closed-loop system of (1)-(2) with 9(T)k) is stochas-
L(1]k) ~ -2 [I
+ B T (11k)Q(Tlk)B(1]k)]-1
B T CTJk)Q(l1k)A(1]k).
Rem.ark 2.2 Theoreln £.1 provides a sufficient condition for the stochastic stability of the time-delay syste7n (1)-(2). Note that the control gain matrix L(nk} for the ,1jtabilizing controller does not depend upon the time delay d (It is not known in general). AL~o, the conditions of (4)-(4) are linear matrix inequalities (LMIs) type which can be resolved via the technique proposed in (Boyd et al., 1994). 0
tically stable.
Let us recall the follo'\ving lemmas which Vv'"ill be used in the proofs of our main results in this paper. Lemma 2.1 (Boyd et al., 1994) Given constant rnatrices M, L, Q of appropriate dimensions where l~f and Q are symmetric and Q > 0) then }.If + LTQL < 0 if and only if
Ai LT ] [ L _Q-l
3. ROBLTST STABILIZATION RESlTLTS In this section, on the basis of the result in previous section we will consider the robust stabilization on a class of uncertain systems in the folloVw·ing form
<0
(4) " . here .L\A(k, 17k), 6Ad(k~ 1}k) and 6.B(k, 7)k) are unknown matrices which represent time-varying parametric uncertainties and assumed to belong to certain bounded compact sets, and matrices A(llk), ..4d(1}k) , B(1]k) and all other variables are as in (1)-(2).
or equivalently
L] < o.
Q-1 [ LT 1\11
The admissible parameter uncertainties are assumed to be modelled as
Before ending this section, ",~e establish the following result of stochastic stabilization for system (1)-(2).
Theorem. 2.1 Consider system (1)-(2). If there exists a set of matrices {P(i) > 0, i =: 1,2, ... 1 s} J such that the following inequalities hold for i := 1, 2, ... , s
_Q-l (i) [ BT(i) _
B{i)
]
[I + BT('i)Q(i)B(i)]z <
where for 0
E d{1]k)
~
=
'T}k
:::=
i, i
E 5, .
E 1 (1}k) ~ E1(i) ~
E Rjxn, .
Ed(i) E RJxn, E 2 (1Jk) = E 2(i) E RJxn and
H(1Jk) ~ H(i) E R nxi a.re known real constant mat rices which characterize ho~~ the uncertain paranleter in F(k, T}k) £ F i (k) enters the nominal matrix A(17k), B(1Jk); and F(k, 'Y}k), fJk := i E S is a unknoVll"n time-varying ma-
T
-P(i) V2A (i)] [ V2A(i) _y-l(i) < 0
1
trix function satisfying
IIF(k,7Jk)11 :5 1i Vk E Z; r/k
where
==
i E S.
(6)
4937
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAYMARKOV...
14th World Congress ofIFAC
-P+Ql AT +ETFTDTJ [ A+DFE _p-l+Q2 <0
~ernark
3.1 In practical: it is almost always impossito get an exact mathematical model of a dyna'mical Istern (for example, manufacturing systems (Boukas nd Haurie, 1990j Boukas et al., 1995; Boukas et al. 1 998)) due to the complexity of the system, the difficulty f measuring 1Jarious parameters, environmental noises, ncertain and/or time-varying parameters, etc. Indeed, the model of the system to be controlled a17nost ai-ways contains some type of uncertainty. Thus} robustness requir'ement (robust .stability of the uncertain closed-loop system an,d the robust performance) is essential and important in control system design, which leads our .,r;tudy to the uncertain syste.m of (4). Note that, similarly to the deterministic system, the parameter '{J,ncerlainty St1~UC ture as in (5)-(6), when 1Jk =::: i, i E S, has been l1Jidely used in the problems of robust control and robust filtering of uncertain systems (see, e~g. (Boukas and Shi, 1998; Petersen; 1987; Shi et aI., 1998; Shi: 1998; Xie et at, 1993) and the refer"ences therein) and many pract2'cal systems possess paro:rneter uncertainties which can be either exactly modelled) or overbounded by (6). Observe that the unknou1n matrix F(k 1)k) in (5) can even be allowed to be state-dependent, i.e~ F(k, 'Tlk) == F(k Xk, 'lJk}, as long as (6) is satisfied. 0 ~e
if and only if there exists a constant e > 0 such that
NO\~l
'vc arc in the position to establish our main result
in this paper.
Theorem 3al Consider system (4) and (2) (setting AB(1]k) == If there exist a set of matrices {P(i) > 0, i ~ 1,2, ... , s} and a set of scalars {Ci > 0, i .:;;;; 1, 2, H., 8} such that the following inequalities hold for i = 1,2, ... , S
O)~
-P(i) [
j
1
+ cl E[(i) El (i) V2A T (i)] < 0 J2~4(i) - ..¥ (i)
where
In this section, our aim is to design a controller 'ilk = L(7]k)Xk for the uncertain syRtem (4) with (2) such that the resulting closed-loop systerIl is stochastically stable for all admissible uncertainties satisfying (5)-(6). First, \ve shall, however, confine our study in the case when uncertainties only appear in the state matrices A( 7Jk ) and Ad (1]k) , i.e., 6.B(k, TJk) ~ o. To this end, we need some intermediate results which are introduced in the sequel as lemmas.
s
Q(i)
= Po(i) + I,
Po(i)
=
LPijP(j) j=1
then system (1)-(2) can be stQchastically stabilized by a controller
LernIIla 3 . 1 (Petersen and HaUot, 1986) Given 1natrices D, F, E of appropriate dimensions and \Ix E Rn and Vy E Rn, then
'Uk = L(1Jk)Xk
L(ruc)
==
-2
(1 + B T C71k)QCTJk )B(l1k)] -1 BT(rtk)Q('Tlk)A(
for all admissible parameter uncertainty ~A4{1]k) and D.A d(1Jk) satisfying (5)- (6). LelllDla 3.2 (Petersen, 1987) Let .oI¥' Y and Z be given k x k symmetric matT·ices such that X 2 0, Y < 0 and
Z
~
o.
Remark 3 ..2 In The.orem 3.1 it is shown that the robust stochastic stabilization of system (4) with (2) is guaranteed if the inequalities {7}- (7) are hold. Note that when time-delay term in (4) disappears and parameter uncertainty 6.A(?7k) equals to zero, Theorem 3.1 will reduce to the results in (Chizeck et al., 1986; Ji et aI., 1991). Furthermore, if the modes 'r}k are all equal to 1, Theorem 3.1 will recover those, for example: (F'lAruta and Phooy'aruenchanachai; 1990; de Souza et al., 1993; Garcia et al., 1994; Yuan et aI.) 1996)~ 0 j
Furthermore, assume that
for all non-zero € E Rk. Then, there exists a constant > 0 such that the matrix £2 X + eY + Z is negativedefinite.
€
LeIllD1a 3.3 Given matrice.c; ll, D, F, E, Q1 and Q2 of
Frorll Theorem 3.1, when parameter uncertainties appear in both state and control input matrices, v.re have the following result-
0 and Q2 ~ 0 and pT F ~ I, then there exists a matrix P > 0 such that appropriate dimensions w2'th Q1
~
4938
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
:'heorelll 3 . 2 Consider system (.4) and (2). If there ex~t a set of matrices {P(i) > 0, i == 1,2, ... , s} and a set f scala,"'s {c i > 0, i =: 1, 2 'I ~ •• ~ .~} l~'U.ch that the following ~equalities hold for i = 1,2, ... ) S
-P(i) [
+ ~ [E[{i) El (i) + Ef(i)E2(i)] v'2AT (i)] < 0 v'2..4 (i) -X(i)
14th World Congress ofIFAC
formance has been made for both H 2 (with known statistics noise) and H oo (with energy bounded noise only) cases, see, (Costa et al.:, 1997; Shi et al., 1998)7 which is beyond the intere..~t8 of this paper on stabilization issue. Howet.'er, the study on Markovian jump systems with timc- delay and input noise should be quite interesting. 0
4.
CONCLUSIO~
In this paper, we studied the problems of stochastic stabilization of discrete-time system containing 11arkovian jurnp paraIIleters and tinle-delay in state. Both the cases of being \vith and Vw'"1.thou t parameter uncertainties of the underlying system are considered. It is shown that the above problems can be solved if certain coupled matrix inequalities have positive definite solutions.
uJhere
X(i)
=:
[Q(i) - Q{i)B(i)(I
+
B T (i)Q(i)B(i))-2 BT(i)Q(i)]-l
+ v'2 H(i)HT(i) Ei B
Q(i)
=:
Po (i)
+ I,
Po(i)
==
LPijP(j) j=l
then system (1)-(2) can be controller
.~tochastically
stabilized by a
L('TJk)Xk L(TJk) =:= -2 [1 + B T (7Ik)Q(rlk)BC17k)]-1 B T (1]k)Q(r}k)A('1Jk) llk :=:
for all admissible parameter ?j,ncertainty ~A(l1k)) L\.. 4 d (r}k) and ~B(1}k) satisfying (5)-(6). Proof. It can be derived via the same technique as that used in Theorem 3.1. ~'Vv
Renlark 3.3 It should be noted that due to the existence of unknown time-delay and parameter uncertainties? Theorems 3.1 and 3.2 provide only sufficient C01l,ditions to guarantee the robust stabil1:zation of system (4). It may be conservative in the sense of the controller being over-designed. However) these conditions, i.e., the solvability of (7)- (7) and (1)-(7) can be easily verified by the techniques pr·oposed in (Abo'u-Kandil et al., 1995), that is) given a set of scalars {ci > 0, i == 1, 2) ... s} J to compute the solution {P(i) > 0, i = 1 7 2: ... ,s} of (7)(7) or (7)-(7) by employing the algorithm~ from (AbouKandil et al.] 1995)~ If it fail.r;, then decrease {ci > o, i == 1, 2, ... s} and try to so Ive the corresponding inequalities again. Finally, it is ·worthwh·ile to point o'ut that if the unknown time-delay teT1n Ad(1}k)Xk-d is t,'eated as a input noise} the analysis on robust stability and per-
References
Abou-Kandil, H., FreiIing, G., and Jank, G. 1995. On the solution of discrete-time Markovian jump linear quadratic control problems, Auto1natica 32(5),765768. Athans, l\1~ 1987. Command and control (C2) theory: A challenge to control science, IEEE Trans. Automat. Control 32(2), 286-293. Boukas, E. K. and Haurie, A. 1990. l\tlanufacturing flow control and preventive maintenance: a stochastic control approach~ IEEE Trans. Automat. Control 35(7),1024-1031.
Boukas) E. K. and Shi, P. 1998. Stochastic stability and guaranteed cost control of discrete--time uncertain systems with I\1arkovian jumping parameters, Int. J~ Robust fj Nonlinear ControL to appear. Boukas, E. K., Shi 1 P., and Andijani, .oLl\4 1998. Optimal inventory-production control problem with stochastic demand~ Optimal Control Applications €1 Methods. to appear. Haukas, E. K., Zhang, Q., and Y-in, G. 1995. Robust production and maintenance planning in stochastic manufacturing systems~ IEEE Trans~ A utornat. Control 40(6), 1098-1102. Boyd, S., Ghaoui, L. E., Feron, E., and Balakrishnan V. 1994. "Linear matrix inequalities in system and control theory," SIAlVl PhiladelphiaChizeck, H. J., Winsky~ A. S., and Castanon, D. 1986. Discrete-time markovian-jump linear quadratic opt.imal control, Int. J. Control 43(1), 213- 231. Costa, O. L. \;. and Fragoso, M. D. 1995. Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems, IEEE Trans. Automat. Control 40(12), 2076-2088. 1
j
4939
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST STABILIZATION OF DISCRETE TIME-DELAY MARKOV...
~osta! O~
L. \T., \Tal, J. B. Do, and Gcromel, J. C. 1997. convex programming approach to H z control of discrete-time markovian jump linear systems~ Int. J. Control 66(4), 557-579. e Souza, C. E., Fu, M., and Xie, L. 1993. H co analysis and synthesis of discrete-timesystems with timevarying uncertainty~ IEEE Trans.Automat. Control 38(3), 459-462. ~t\
de Souza, E. C. and Fragoso, ~1. D. 1993~ H oo control for linea.r systems with :rvlarkovian jumping parameters, Control-Theory and Advanced Technology 9(2),457466. FUruta, K. and Phoojaruenchanachai, S. 1990. An algebraic approach to discrete-time H(XJ control problems. In Proc. 1990 American Control Conf pages 3067-3072 San Diego, CA. Garcia~ G., Bernussou, J., and ArzeIier, D. 1994. Robust stabilization of discrete- time linear systems with norm- bounded time-varying uncertainty, Systems 8 Control Letters 22(5), 327-339. Ji, Y. and Chizeck, H. J. 1990. Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Tmns. Automat. Control 35(7),777- 788. Ji, Y., Chizeck, H. J., Feng, X., and Loparo, K. -L~. 1991. Stability and control of discrete-time jump linear systenlS, Control Theo1Y and Advanced Technology 7(2), 247-270. Kim~
J. H., Jeung, E. A., and Park, H. B. 1996. Ro-
Int. J. Control 64(4), 631- 661. Pan, G. and B ar-ShaJ om, Y. 1996b. A stabilizing controller for jump linear gallssian systems "vith noisy state observations, European J. Control 2(2), 227~ 238. Petersen, I. R. 1987. A stabilization algorithln for a class of uncertain linear systems, System f1 Control Letters 8(4), 351-357. Petersen 1 I. R. and Hollot, C. V. 1986. A Riccati equation approach to the stabilization of uncertainlinear systems, Automatica 22(4),397-411. Petkovski, D. B. 1987. iv1ultivariable control system design: a case study of robust control of nuclear power plants, Fault Detection and Reliability 9, 239- 246. Shen, .1., Chen, B.~ and Kung, F'. 1991. ~lemoryless stabilization of uncertain dynamic delay systems:Riccati equation approach, IEEE Trans.Automat. Control 36~ 638---640. Shi, P. 1998. Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control43(7), 1022-1027. Shi~ P. and Boukas, E. K. 1997. HOC) control for NIarkovian jumping linear systems with parametric uncertainty, J. Optimization Theory and Applications
95(1), 75--99~ P., Boukas, E. K., and -,-~garwal, R. K. 1999. Robust control for Nlarkovian jumping discret.e-time systcms~ Int. J. Systems Science. to appear. S",rorder, D. D. and R,ogers, R. O. 1983. An LQ-solution Shi~
to a control problem associated with a solar thermal central receiver, IEEE Trans. Automat. Control 28(8)t 971-978. \\l'illsky, A. S. 1976. ,A. survey of design methods for failure detection in dynamic systems, Automatica 12(5), 601-611. Xie, L. t Shi, P., and de Souza, C. E. 1993. On designing controllers for a class of uncertain sampled-data nonlinear systems, lEE Proc.-Control Theory Appl. 140(2), 119-126.
bust control for parameter uncertain delay systems in state and control input, A 1Ltornatica 32(9), 13371339. Lee, J. H., Kim, S. W., and Kwon, W. H. 1994. Memoryless Hoo controllers for state delayed systems, IEEE Trans. Automat. Control 39(1),159-162. Li, X. and de Souza, C. E. 1997. Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear rnatrix inequality approach, IEEE Trans.Automat. Control 42(8), 1144-1148. Mahmoud, M. S. and ..J \l-Muthairi, N. F. 1994. Quadratic stabilization of continuous time systems with state-delay and norm-bounded tilne-varying uncertainties~
14th World Congress ofIFAC
YUaD, L., Achenie, L., and Jiang,
'~lr
1996. Robust Hoc
control for linear discrete-time systems witl} Ilormbounded time-varying uncertaintY1 Systems & Control Letters 27(2), 199~208.
IEEE Trans. Automat. Control 39(10),
2135-2139. :rvlalek-Zavarei~ ~f.
and Jaulshidi, M. 1987. Timedelay systems: analysis, optimization and applications. North-Holland Systems and Control Series. Moerder, D. D., Halyo, N., Broussard, .1. H.. , and Caglayan, A. K. 1989. Application of precomputed control la\vs in a reconfigurable aircraft flight control systern, J. of Guidance 7 Control and Dynamics 12(3), 325-333. Pan, G. and Bar-Shalom, Y. 1996a. Stabilization ofjump linear ga llssian systerns \vithou t mode observations,
4940
Copyright 1999 IFAC
ISBN: 0 08 043248 4