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Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations
Automatica 36 (2000) 1033}1040
Brief Paper
Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations夽 A. El Bouhtouri , D. Hinrichsen, A. J. Pritchard * Universite& Chouaib Doukkali, Faculte& des Sciences, BP 20 El Jadida, Morocco Institut fu( r Dynamische Systeme, Universita( t Bremen, D-28334 Bremen, Federal Republic of Germany Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Received 10 February 1998; revised 11 March 1999; received in "nal form 10 November 1999
Abstract We consider stochastic discrete-time systems with multiplicative noise which are controlled by dynamic output feedback and subjected to blockdiagonal stochastic parameter perturbations. Stability radii for these systems are characterized via scaling techniques and it is shown that for real data, the real and the complex stability radii coincide. In a second part of the paper we investigate the problem of maximizing the stability radii by dynamic output feedback. Necessary and su$cient conditions are derived for the existence of a stabilizing compensator which ensures that the stability radius is above a prespeci"ed level. These conditions consist of parametrized matrix inequalities and a coupling condition. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Stability radius; Discrete-time stochastic systems; State-dependent noise; k-analysis; Scaling; Linear matrix inequalities
1. Introduction Consider a perturbed system of the form: , & : x(t#1)"Ax(t)# D * (E x(t))b (t), t3-, (1) H H H H H where (b (t)) - , j3N"+1,2, N, are N independent real H RZ random processes. M We view & as a stochastic perturbation of the nom inal discrete-time linear system & : x(t#1)"Ax(t). The family (D , E ) of matrix pairs describes the structure H H HZ,M of these perturbations whilst * , j3N are unknown "nite H M operators * are gain nonlinearities. If the disturbance H linear, then & describes a linear system with noisy parameters. Models of this type play an important role in applications where dynamical systems operate in a
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor F. Jabbari under the direction of Editor R. Tempo. The "rst author was supported by a grant from the German Academic Exchange Service DAAD. The hospitality of the University of Bremen is gratefully acknowledged. * Corresponding author. Tel.: #44-1203-523-128; fax: 44-1203-524182. E-mail address: [email protected] (A. J. Pritchard).
random environment. A survey of theoretical and experimental results on systems with parametric random vibrations is given in Ibrahim (1985). This book contains many references and discusses a number of mechanical examples and real-life control problems. Since the late 1960s there has been a good deal of research on the stability of systems with multiplicative noise, see Kushner (1967), Curtain (1972), Kubrusly (1986) and Kozin (1969) for an early survey. Most of these papers consider the continuous-time case. Frequency domain stability criteria for both continuous- and discrete-time parameter excited systems are derived in Willems and Blankenship (1971). Discrete-time systems with stochastic parameters also play a role in the convergence analysis of iterative stochastic algorithms, see Polyak (1977). In this paper we will study problems of robust stability and robust stabilization for discrete-time systems with blockdiagonal random parameter perturbations (N51 arbitrary) which, in the linear case, take the form A A# , D * E b (t) (see (1)). Thus the perturbaH H H H H tions considered in this paper are a (nonlinear) stochastic version of the blockdiagonal parameter perturbations which are studied in deterministic k-analysis. The largest o50 which guarantees mean square stability of all the perturbed systems & , ""* ""(o, is called the stability H radius of & .
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 1 3 - 3
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For deterministic systems with deterministic parameter perturbations the concept of stability radius was introduced in 1986; for a survey see Hinrichsen and Pritchard (1990), and for further development of the discrete-time case, see Hinrichsen and Son (1991). In the special case where all the E are equal the stability radius H of a continuous-time system with respect to stochastic perturbations was characterized in El Bouhtouri and Pritchard (1992). A discrete-time version of this result can be found in Morozan (1997). In the present paper we overcome the restrictive assumption that all the E are H equal, which essentially reduces the mathematical analysis to the single perturbation case (N"1). Thus, in a stochastic framework, this paper makes the step from `unstructureda to `structureda perturbations (k-analysis) and from H-theory to k-synthesis. The study of genuine multiperturbations requires the use of scaling techniques (i.e. multiplying E , by a positive H real a and D by a\, see Doyle (1982), Hinrichsen and H H H Pritchard (1990)). In deterministic k-analysis, scaling techniques have been used to obtain bounds for the k-function which are computationally feasible. But if there are more than three blocks * these bounds will H not, in general, be tight (Doyle, 1982). In the deterministic case the problem of determining the stability radius for arbitrary block diagonal perturbation structures is still far from being solved. This is in contrast with the stochastic case where we will show that a complete characterization of the stability radius can be obtained via scaling techniques. This is the main contribution of the paper. It solves the stability radius problem of stochastic k-analysis for discrete-time systems. For the continuous time case, see Hinrichsen and Pritchard (1996). In the second part of the paper we examine the problem of optimizing the stability radius by dynamic output feedback, which, in our framework, is a problem of ksynthesis.
2. Preliminaries Consider discrete-time systems of the form , & : x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D * (E x(t))b (t), t3-, H H H H H
probability space (), F, k) with Ew (t)"Eb (t)"0, E(w (s)w (t))"j d d , G H G H G GH QR E(b (s)b (t))"c d d , E(w (s)b (t))"0, G H H GH QR G H for all s, t3-, i, j3N. Here d , d are Kronecker symbols. The assumption GH QR that the w's and b's are mutually independent excludes the interesting case where w "b which would allow us H H to study the e!ect of disturbing the di!usion parameters A A#D * E . If Assumption 1 is not satis"ed it is H H H H H still possible to obtain a lower bound for the stability radius, but, as yet no complete characterization is available. However, our results can be applied to arbitrary deterministic discrete-time systems with random parameter disturbances as discussed in the introduction (A"0, i3N). We denote by F the sigma algebra G R F "p(+b (s), w (s); 04s4t, i, j3N,), R H G then b (t), w (t), i, j3N are independent of F . For H G norm is taken on *OH ,R\ each j3N the Euclidean *lH . We D (*) the vector space of all measurable denote by H "nite gain maps * : *OH P*lH with * (0)"0 and set H H D(*)" ,D (*). By de"nition each * 3D (*) H H H satis"es ""* (y)""*lH ""* "" " : sup H (R, j3N. (3) H ""y""*OH M W$ For any given family * 3D (*), j3N, let * denote the H H combined (blockdiagonal) perturbation , *" * : *OP*l, H where l"l #2#l , q"q #2#q , ,
and
""*"""max ""* "". (4) H HZ,M Without loss of generality we assume the initial state of (2), x(0)"x3*L is deterministic. For every *3D(*) there exists a unique solution of (2) with x(0)"x3*L, which we denote by x ( ) , x). Note that x (t, x) is F measurable for all t3- (we set F "+, ),). R\ \ De5nition 2. The zero equilibrium state of (2) is said to be l-stable if there exists a constant c'0 such that
(2)
where (A, A, D , E )3*L"L;*L"L;*L"lH ;*OH "L, G H H i, j3N, *"1 or " are given. We suppose Assumption 1. The w (t), b (t), t3-, i, j3N are sequences G H of real independent random variables on a complete
E(""x (t, x)"")4c""x"". R The nominal system is , x(t#1)"Ax(t)# Ax(t)w (t), t3-, G G G x(0)"x3*L.
(5)
A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
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If '(t, s) is the fundamental matrix associated with (5),
3. The input}output operator
'(t, t)"I , '(t, s)"¸(t!1)¸(t!2)2¸(s), L s, t3-; t5s#1,
Let ¸(), *I) be the space of square integrable *I-valued functions on the probability space (), F,k) with norm "" ) "". For any positive ¹4R, we set - "+t3-; t4¹, (hence - "-) and de"ne the 2 space l (- ; *I) of k-dimensional nonanticipatory U @ 2 square integrable stochastic processes on - as the space 2 of all sequences y( ) )"(y(t)) -2 3l(- ; ¸(), *I)) such RZ 2 that y(t)3¸(), *I) is F -measurable for all t3- . So R\ 2 every y( ) )3l (- ; *I) is adapted to the shifted seU @ 2 quence of p-algebras (F ) -2 . The space l (- ; *I) U @ 2 R\ RZ is provided with the l-norm:
where , ¸(t)"A# Aw (t), G G G then '(t, s) is F -measurable for all s, t3-, t5s and R\ the solution of (5) is x(t, x)"'(t, 0)x. We have the following result; see Morozan (1983,1997). Proposition 3. The following are equivalent. (i) The nominal system (5) is l-stable. (ii) There exist constants M51, u'0, such that E ""'(t, s)""4Me\SR\Q, s, t3-, t5s. (iii) There exists P3H (*), PY0 such that L , AHPA!P# j AHPAO0. (6) G G G G Moreover if E3*O"L and (i) holds, then there exists a unique solution P)0 in H (*) of the discrete-time L Liapunov equation , AHPA!P# j AHPA#EHE"0, G G G G given by P" E['(t, 0)HEHE'(t, 0)]. R
(7)
(8)
De5nition 4. The stability radius of the system (5) with respect to the multiperturbation structure (D , E , b ( ) )) is de"ned to be H H H HZ,M r* "r@ U (A, (A) , (D , E ) ) * G GZ,M H H HZ,M , "inf ""*""; *" * 3D(*) and H G
(2) is not l-stable . Remark 5. (i) A stability radius with respect to linear perturbation can be de"ned by restricting the disturbance operators * : *OH P*lH to be linear. The characH terization of this radius is an open problem for stochastic systems. (ii) If the data A, A, D , E , i, j3N are real, two stabilG H H ity radii can be obtained depending on whether one chooses *"1 (real perturbations), or *"" (complex perturbations). In the deterministic framework, the real and the complex stability radii are in general distinct. We will show that the two stability radii are equal in our stochastic framework.
""y( ) )""lU @ -2 _
"E ""y(t)"" " E+""y(t)"",. (9) RZ-2 RZ-2 We associate with (2) the stochastic control system *I
, , x(t#1)"Ax(t)# Ax(t)w (t)# D v (t)b (t), G G H H H G H (10) x(0)"x3*L, z(t)"Ex(t), where E"[E?, E?,2, E? ]?3*O"L. Note that the per , turbed system (2) is obtained from (10) by the nonlinear output feedback v (t)"* (z (t))"* (E x(t)), j3N. For H H H H H M arbitrary (x, v)3*L;l (-; *l) there exists a unique U @ solution of (10) in l (- ; *L) for all ¹3-, which we U @ 2 denote by x(t, x, v). Lemma 6. Suppose P3H (*) satisxes the Liapunov L equation (7), then for (x, v)3*L;l (- ; *l) and U @ 2 x( ) )"x( ) , x, v), z( ) )"Ex( ) ), we have 2 E(""z(t)"")#E1x(¹#1), Px(¹#1)2 R 2 , "1x, Px2# c E1DHPD v (t), v (t)2. H H H H H R H
(11)
The lemma is proved by direct calculation. A detailed proof can be found in El Bouhtouri, Hinrichsen and Pritchard (1997). Applying Proposition 3 and Lemma 6 with E"I and ¹PR one obtains L Corollary 7. Suppose the nominal system (5) is l-stable. If (x, v)3*L;l (-; *l), then there exists c'0, such that U @ E(""x(t)"")4c[""x""#""v""lU @ - *l ]. _ R Lemma 6 and Corollary 7 lead to the following lemma. Lemma 8. Suppose the nominal system (5) is l-stable and P)0 in H (*) satisxes the Liapunov equation (7), then for L
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(x, v)3*L;l (-; *l), we have U @
and vice versa. The control system associated with (17) (compare (2) and (10)) is
E(""z(t)"")"1x, Px2 R , # c E1DHPD v (t), v (t)2. H H H H H R H
(12)
De5nition 9. The system (10) is said to be externally l-stable if z( ) )"Ex( ) , 0, v)3l (-; *O) for every U @ v3l (-; *l), and there exist c50 such that U @ ""z""lU @ - *O " E(""z(t)"")4c""v""lU @ - *l . _ _ R
(13)
, x?(t#1)"Ax?(t)# Ax?(t)w (t) G G G , # D?v?(t)b (t), z?(t)"E(a)x?(t), H H H H where
v? (t) E? x? (t) v?(t)" $ , E(a)" $ , x?(t)" $ . v? (t) ,
E?, ,
(18)
(19)
x? (t) ,
By Corollary 7 if the nominal system (5) is l-stable then (10) is externally l-stable.
Setting *?" , *?H the scaled system (17) is obtained H H from (18) by the nonlinear feedback v?(t)"*?(E(a)x?(t)). Hence if x( ) )"x ( ) , x) is any solution of (2) or, equiva lently of (17), then
De5nition 10. Suppose (10) is externally l-stable. The operator +:l (-; *l)Pl (-; *O) de"ned by U @ U @
x (t, x)"x?(t, x , v?), where v?(t)"*?(E(a)x (t, x)).
R\ , (+v)(t)"Ex(t,0, v)"E '(t, s#1)D v (s)b (s), H H H Q H t3-, v3l (-;*l) U @
(14)
is called the input}output operator of the system (10). Its norm is de"ned as the minimal c50 such that (13) is satis"ed. Corollary 11. Suppose the nominal system (5) is l-stable and P)0 in H (*) satisxes the Liapunov equation (7), then L the input}output operator + has operator norm ""+"""max c ""DHPD "". H H H HZ,M
(15)
Theorem 12. Suppose the nominal system (5) is l-stable. If, for some p'0, there exist a3(0,#R), and P(a)3H>(*) satisfying L , , AHP(a)A!P(a)# j AHP(a)A# aEHE "0, G G G H H H G H (21)
Suppose that a3(0,R),, a"(a ,a ,2, a ), then the , nonlinear perturbed system (2) remains unchanged if we replace D , E , * with D?H , E?H , *?H , where H H H H H H *?H "a * (a\ ) ), H H H H
Although the solutions of (2) and (17) are the same and do not depend on a, the input}output operator of the scaled system, +? does change with a. We use this added freedom to characterize r* . First we obtain a lower bound.
p IlH !c DHP(a)D )0, j3N, H a H H M H
4. Characterization of r*
D?H "a\D , H H H
(20)
E?H "a E . (16) H H H
More precisely, every solution of the system (2) is also a solution of the scaled system , &? ? : x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D?H *?H (E?H x(t))b (t), H H H H H
t3-
(17)
(22)
then r* 5p. Proof. Suppose P(a) satis"es (21), (22) and let ""*""(p. We prove that (2) is l-stable. Let x(t)"x (t, x) be the solution of (2). By (20), x(t)"x (t, x) coincides with the state trajectory of the scaled system (18) generated by the input v?(t)"*?(E(a)x (t, x)). By Lemma 6 (with P re placed by P(a), E by E(a), etc.) for ¹'0, the corresponding output signal z?(t) of (18) satis"es 2 E(""z?(t)"")#E1x?(¹#1), P(a)x?(¹#1)2 R 2 , "1x, P(a)x2# c E1D?HP(a)D v?(t), v?(t)2. H H H H H R H
A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
Then P(a))3H>(*) is the unique solution of the L equation
But 2 2 ""v?(t)""4 E+""*?""""z?(t)"", R R 2 4""*?"" E""z?(t)"", R and since P(a))0, we have
, AHPA!P# j AHPA# aEHE "0. G G G I I I G IZ) Moreover
2 E(""z?(t)"")41x, P(a)x2 R c 2 , #max H ""DHP(a)D "" E""v?(t)"" H H a H H HZ,M R H 2 41x, P(a)x2#p\""*?"" E""z?(t)"". R Now ""*?H """""* "", so ""*?""(p and c " : p\""*?""(1. H H Thus 2 E""z?(t)""4(1!c)\1x, P(a)x2, ¹3-. R Therefore z?( ) )3l (-; *O) and there exists c '0, U @ such that ""z?""lU @ - *O 4c ""x"". Hence v?3l (-;*l) U @ and ""v?""lU @ - *l 4""*?"" ""z?""lU @ - *O 4pc ""x"". So by _ Corollary 7 there exists a c'0, such that E(""x (t, x )"")4c(""x""#""v?""lU @ - *l ) _ R 4c(1#pc )""x"". This shows that (2) is l-stable and concludes the proof. 䊐 It follows immediately from Theorem 12 that
\ c max H DHP(a)D , (23) r* 5 sup H H a H ?Z , HZ,M where P(a) is the unique solution of (21). By constructing a destabilizing perturbation whose norm is equal to the expression on the RHS of (23), we will show that equality holds. To do this we use a result on a minimax problem for quadratic forms which was proved in Hinrichsen and Pritchard (1996). Suppose the nominal system (5) is lstable. Let H "c E(DH'(t, 0)HEHE '(t, 0)D ), k, j3N. (24) IH H H I I H R For any KLN let (0,R)) denote the set of all families M reals and for any a)"(a ) 3(0,R)), (a ) of positive I IZ) I IZ) set P(a))" aE('(t, 0)HEHE '(t, 0)). I I I IZ) R
1037
(25)
c a H DHP(a))D " I H , j3N. H H IH a a H IZ) H Consider the minimax problem
(26)
(27)
, a G H Minimize max GH a HZ,M G H with respect to a3(0,R),. Let k( be the minimal value
, a I H . k( " inf max (28) IH a ?Z , HZ,M I H The next lemma follows directly from Theorem 2.4 in Hinrichsen and Pritchard (1996). Lemma 13. There exist a subset KLN and for every d'0, a vector a3(0,R),, such that
, a I H 4k( #d, j3N, IH a I H a I H "k( , j3K. IH a IZ) H
(29)
The following characterization of r* is the main result of the "rst part of this paper. Theorem 14. Suppose the nominal system (5) is l-stable, then (A, (A) , (D , E ) ) r@ U * G GZ,M H H HZ,M \ c " sup max H DHP(a)D , H H a H ?Z , HZ,M where P(a) is the unique solution of (21).
(30)
Proof. Let k( be de"ned by (28). If k( "0 then r* "R by (27) and Theorem 12, and so (30) is satis"ed. Now assume k( '0. Let p(a) be the largest p for which (21) and (22) have joint solution P(a)3H>(*). Then L sup p(a)"k( \. By Lemma 13 and (27), there ?Z >, exists KLN and a)3(0,R)) satisfying a c I k( " H " H DHP(a))D , j3K. (31) IH H a a H H IZ) H Let p( "k( \, then there exists v 3*lH such that ""v """1 H H and
a v, I H v H IH H a IZ) H c " v , H DHP(a))D v "k( , j3K. H a H H H H
(32)
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A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
De"ne * ( ) )3D (*) by H H * (z )"p( ""z ""v H H H H
if j3K, (33)
* (z )"0 if j3N!K, z 3*OH . H H H M Then ""* """p( , j3K and so ""*"""p( . We will show that H for this *, (2) cannot be stable. Assume the contrary then for arbitrary x3*L, the solution x ( ) , x) of (17) satis"es E(""x (t, x)"")(R. Now R *?H (z )"a * (a\z )"p( ""z ""v , z 3*lH , j3K. H H H H H H H H De"ne E(a)) analogously to E(a) in (19) and let z?)(t)"E(a))x (t, x). Then , x(t#1)"Ax(t)# Ax(t)w (t) G G G #p( D?H v ""z?H (t)""b (t). H H H H HZ) Applying Lemma 8 with P(a)) instead of P, E(a)) instead of E, D?H instead of D , j3K, D "0, j3N!K and H H H p( v ""z?H ( ) )"" instead of v ( ) ), we have by (32) M H H H E(""z?)(t)"")"1x, P(a))x2 R p( # c 1v , DHP(a))D v 2E(""z?H (t)"") H a H H H H H R HZ) H "1x, P(a))x2 # E(""z?)(t)""), x3*L. R So P(a))"0 which contradicts the fact that k( '0. Hence the system (2) is not stable for this *. Note that the (nonlinear) destabilizing disturbance constructed in the previous proof is real when the data A, (A) , (D , E ) are real. Thus G GZ,M H H HZ,M Corollary 15. If the data (A, A) , (D , E ) are real G GZ,M H H HZ, then the complex and the real stability radii of (2)M coincide, r" "r1 . In the next section we will use the following characterization of the stability radius in terms of strict linear matrix inequalities. It is proved from Theorem 14 by a perturbation argument, for details see El Bouhtouri et al. (1997). Proposition 16. The nominal system (5) is l-stable and r* 'p if and only if there exist a3(0,R), and
P3H>(*), PY0 such that L , , AHPA!P# j AHPA# aEHE O0, G G G H H H G H p IlH !c DHPD Y0, j3N. H a H H H
(34) (35)
5. Maximizing the stability radius by output feedback In this section we investigate how the stability radius can be enhanced by dynamic output feedback. For this we consider the controlled system
&:
, x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D * (E x(t))b (t)#Bu(t), t3-, H H H H H
(36)
y(t)"Cx(t),
where (B, C)3*L"K;*N"L. As compensators we choose deterministic discrete-time systems & : x( (t#1)"A x( (t)#B y(t), ) ) ) u(t)"C x( (t)#D y(t), t3-, (37) ) ) where (A , B , C , D )3*L( "L( ;*L( "N;*K"L( ;*K"N. ) ) ) ) The resulting perturbed closed-loop system is , &N : x (t#1)"Ax (t)# Ax (t)w (t) G G G , # D * (E x (t))b (t), H H H H H x (0)"x 3*L>L( ,
(38)
where
x(t)
A#BD C BC ) ) , x( (t) B C A ) ) A 0 D "["H ], E "[E 0], A" G . H H H G 0 0 x (t)"
, A"
The nominal closed-loop system is , &L : x (t#1)"Ax (t)# Ax (t)w (t), G G G x (0)"x 3*L>L( .
(39)
Our aim is to determine conditions for the existence of a controller which stabilizes the nominal system (in the l sense) with a stability radius above a given threshold p'0. Following an LMI-approach we obtain the result below. The proof proceeds by applying Proposition 16 to the closed-loop system (38), for details see El Bouhtouri et al. (1997).
A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
Theorem 17. Given p'0, the following are equivalent. (i) There exists a stabilizing compensator & of dimension ) n( such that the nominal closed-loop system &L is l stable and rU @ (A, (A) , (D ,E ) )'p. * G GZ,M H H HZ,M (ii) There exist a3(0,R), and S, R3H>(*) with L S)R\Y0, rank(S!R\)4n( such that
p IlH !c DHSD Y0, j3N, H a H H H , , AHSA!S# j AHSA# aEHE O0 G G G H H H G H on ker C, , , R\! j AHSA! aEHE Y0, G G G H H H G H
(40)
(41) (42)
\ , , A R\! j AHSA! aEHE AH!RO0 G G G H H H G H on ker BH.
(43)
Remark 18. (i) The condition rank(S!R\)4n( is automatically satis"ed for n( 5n and the conditions (40)}(43) do not depend on n( . So if there exists a stabilizing compensator & of any dimension achieving r * 'p ) then there exists such a compensator of dimension 4n. (ii) Suppose we have found a, R, S for a given value of p, so that (ii) of the above theorem holds. Then stabilizing compensators of the form (37) which achieve r * 'p can be constructed by reversing the steps in the proof of the previous theorem, see El Bouhtouri et al. (1997). (iii) Applying Theorem 17 with n( "0 and S"R\ one can determine which stability radii can be achieved by static output feedback u(t)"D y(t). ) (iv) It follows from Theorem 17 that dynamic state feedback cannot increase the stability radius beyond those levels which can be achieved by static state feedback (where n( "0, p"n, C"I ). L (v) In the deterministic case, under some weak extra assumptions, the inequalities (41)}(43) can be formulated in terms of Riccati equations. We have shown that, in general, this cannot be done for discrete-time stochastic systems. A counterexample can be found in El Bouhtouri et al. (1997). If the structure operators D , E are chosen to H H be zero then the condition r * 'p can be neglected and Theorem 17 implies the following stabilizability conditions. Corollary 19. Given a system of the form , x(t#1)"Ax(t)# Ax(t)w (t)#Bu(t), y(t)"Cx(t) G G G (44)
1039
and a random sequence (w(t)) RZ Assumption 1. Then the following hold.
which
satisxes
(i) There exists a stabilizing compensator & of the form ) (37) for (44) if and only if there exist S, R3H>(*) with L S)R\Y0 such that , AHSA!S# j AHSAO0 on ker C, (45) G G G G \ , A R\! j AHSA AH!RO0 on ker BH G G G G , and R\! j AHSAY0. (46) G G G G (ii) There exists D 3*K"N such that the system ) , x(t#1)"(A#BD C)x(t)# Ax(t)w (t) ) G G G is l-stable if and only if there exists S3H>(*), SY0 L such that (45) and (46) hold with R\ replaced by S. (iii) There exists a stabilizing static state feedback control u"Fx for (44) if and only if (46) holds with R\ replaced by S.
In the special case of static state feedback, necessary and su$cient stabilizability conditions for continuous time systems with state-dependent noise (with C"I ) L have been derived in Willems and Willems (1976).
References Doyle, J. (1982). Analysis of feedback systems with structured uncertainties. Proceedings of IEE, 129, 242}250. El Bouhtouri, A., & Pritchard, A. J. (1992). Stability radii of linear systems with respect to stochastic perturbations. Systems and Control Letters, 19, 29}33. El Bouhtouri, A., Hinrichsen, D., & Pritchard, A.J. (1997). Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations. IDS-Report Nr. 420, UniversitaK t Bremen. Curtain, R.F. (1972). Stability of stochastic dynamical systems. Lecture Notes in Mathematics, vol. 294. Berlin: Springer. Hinrichsen, D., & Pritchard, A.J. (1990). Real and complex stability radii: A survey. In: Hinrichsen, D., & Ma rtensson B. (Eds.), Control of uncertain systems, Progress in system and control theory, vol. 6. (pp. 119}162). Basel: BirkhaK user. Hinrichsen, D., & Pritchard, A. J. (1996). Stability radii of systems with stochastic uncertainty and their optimization by output feedback. SIAM Journal Control, 34, 1972}1998. Hinrichsen, D., & Son, N. K. (1991). Stability radii of linear discrete * time systems and symplectic pencils. International Journal of Robust and Nonlinear Control, 1, 79}97. Ibrahim, R. A. (1985). Parametric Random Vibration.. Letchworth, England: Research Studies Press. Kozin, J. (1969). A survey of stability of stochastic systems. Automatica, 5, 95}112. Kubrusly, C. S. (1986). On discrete stochastic bilinear systems stability. Journal of Mathematical Analysis and Applications, 113, 36}58. Kushner, H. (1967). Stochastic stability and control. New York: Academic Press.
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Morozan, T. (1983). Stabilization of some stochastic discrete-time control systems. Stochastic Analysis and Applications, 1, 89}116. Morozan, T. (1997). Stability radii of some discrete-time systems with independent random parameters. Stochastic Analysis and Applications, 15, 375}386. Polyak, B.T. (1977). Convergence and convergence rate of iterative stochastic algorithms. II. The linear case. Automation and Remote Control. (4), 537}542. Willems, J. C., & Blankenship, G. L. (1971). Frequency domain stability criteria for stochastic systems. IEEE Transactions on Automatic Control, AC-16, 292}299. Willems, J. J., & Willems, J. C. (1976). Feedback stabilizability of stochastic systems with state and control dependent noise. Automatica, 12, 277}283. Abdelmoula El Bouhtouri was born in Marrakech, Morocco in 1960. He was educated at the University Mohammed V, Faculty of Sciences, Rabat where he received the Licence de matheH matiques appliqueH es in 1984, Le certi"cat d'eH tudes approfondies in systems theory and control in 1985, Le Doctorat de troisieH me cycle on the stabilizability of linear distributed parameter stochastic systems in 1989 and Le Doctorat d'etat on the robustness of stability of linear uncertain stochastic systems in 1994. His present position is Directeur de recherche au laboratoire d'ingeH niere matheH matique, University Chouaib Doukkali, El Jaddida, Morocco. His research interests include stochastic linear systems, H control, robustness of stability of jump processes and linear stochastic distributed parameter systems. Dr. A. El Bouhtouri is a member of the American Mathematical Society and an a$liate member of IFAC.
Diederich Hinrichsen was born in NuK rnberg, Germany, in 1939. He studied mathematics and German literature in Hamburg and Paris, passed the Staatsexamen at the University of Hamburg in 1965 and received the degree of Dr. rer. nat. from the University of Erlangen in 1966. He held positions at the Universities of Hamburg, Havana and Bielefeld. Since 1973 he has been Professor of Mathematics at the University of Bremen where he served as Director of the Institute of Dynamical Systems from its foundation in 1978 until 1990. His current research interests are in analysis and control of uncertain dynamical systems.
Anthony J. Pritchard was born in Llwyny-pia, Wales in 1937. He graduated in mathematics from Kings College, London in 1958, obtained a DIC from Imperial College, London in 1959 and a Ph.D. from Warwick University in 1969. He has been at the University of Warwick since 1968, "rst as Lecturer, then Reader in 1974, Director of the Control Theory Centre in 1975 and has been a Professor in the Mathematics Department since 1984. He is a Fellow of the Institute of Mathematics and Its Applications and a Chartered Mathematician. His main research interest is in"nite dimensional systems theory, although recently he has worked on robustness problems. He has published a number of books including Inxnite Dimensional Linear Systems Theory with Ruth Curtain and Sensors and Controls in the Analysis of Distributed Systems with Abdelhaq El Jai.
Brief Paper
Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations夽 A. El Bouhtouri , D. Hinrichsen, A. J. Pritchard * Universite& Chouaib Doukkali, Faculte& des Sciences, BP 20 El Jadida, Morocco Institut fu( r Dynamische Systeme, Universita( t Bremen, D-28334 Bremen, Federal Republic of Germany Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Received 10 February 1998; revised 11 March 1999; received in "nal form 10 November 1999
Abstract We consider stochastic discrete-time systems with multiplicative noise which are controlled by dynamic output feedback and subjected to blockdiagonal stochastic parameter perturbations. Stability radii for these systems are characterized via scaling techniques and it is shown that for real data, the real and the complex stability radii coincide. In a second part of the paper we investigate the problem of maximizing the stability radii by dynamic output feedback. Necessary and su$cient conditions are derived for the existence of a stabilizing compensator which ensures that the stability radius is above a prespeci"ed level. These conditions consist of parametrized matrix inequalities and a coupling condition. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Stability radius; Discrete-time stochastic systems; State-dependent noise; k-analysis; Scaling; Linear matrix inequalities
1. Introduction Consider a perturbed system of the form: , & : x(t#1)"Ax(t)# D * (E x(t))b (t), t3-, (1) H H H H H where (b (t)) - , j3N"+1,2, N, are N independent real H RZ random processes. M We view & as a stochastic perturbation of the nom inal discrete-time linear system & : x(t#1)"Ax(t). The family (D , E ) of matrix pairs describes the structure H H HZ,M of these perturbations whilst * , j3N are unknown "nite H M operators * are gain nonlinearities. If the disturbance H linear, then & describes a linear system with noisy parameters. Models of this type play an important role in applications where dynamical systems operate in a
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor F. Jabbari under the direction of Editor R. Tempo. The "rst author was supported by a grant from the German Academic Exchange Service DAAD. The hospitality of the University of Bremen is gratefully acknowledged. * Corresponding author. Tel.: #44-1203-523-128; fax: 44-1203-524182. E-mail address: [email protected] (A. J. Pritchard).
random environment. A survey of theoretical and experimental results on systems with parametric random vibrations is given in Ibrahim (1985). This book contains many references and discusses a number of mechanical examples and real-life control problems. Since the late 1960s there has been a good deal of research on the stability of systems with multiplicative noise, see Kushner (1967), Curtain (1972), Kubrusly (1986) and Kozin (1969) for an early survey. Most of these papers consider the continuous-time case. Frequency domain stability criteria for both continuous- and discrete-time parameter excited systems are derived in Willems and Blankenship (1971). Discrete-time systems with stochastic parameters also play a role in the convergence analysis of iterative stochastic algorithms, see Polyak (1977). In this paper we will study problems of robust stability and robust stabilization for discrete-time systems with blockdiagonal random parameter perturbations (N51 arbitrary) which, in the linear case, take the form A A# , D * E b (t) (see (1)). Thus the perturbaH H H H H tions considered in this paper are a (nonlinear) stochastic version of the blockdiagonal parameter perturbations which are studied in deterministic k-analysis. The largest o50 which guarantees mean square stability of all the perturbed systems & , ""* ""(o, is called the stability H radius of & .
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 1 3 - 3
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For deterministic systems with deterministic parameter perturbations the concept of stability radius was introduced in 1986; for a survey see Hinrichsen and Pritchard (1990), and for further development of the discrete-time case, see Hinrichsen and Son (1991). In the special case where all the E are equal the stability radius H of a continuous-time system with respect to stochastic perturbations was characterized in El Bouhtouri and Pritchard (1992). A discrete-time version of this result can be found in Morozan (1997). In the present paper we overcome the restrictive assumption that all the E are H equal, which essentially reduces the mathematical analysis to the single perturbation case (N"1). Thus, in a stochastic framework, this paper makes the step from `unstructureda to `structureda perturbations (k-analysis) and from H-theory to k-synthesis. The study of genuine multiperturbations requires the use of scaling techniques (i.e. multiplying E , by a positive H real a and D by a\, see Doyle (1982), Hinrichsen and H H H Pritchard (1990)). In deterministic k-analysis, scaling techniques have been used to obtain bounds for the k-function which are computationally feasible. But if there are more than three blocks * these bounds will H not, in general, be tight (Doyle, 1982). In the deterministic case the problem of determining the stability radius for arbitrary block diagonal perturbation structures is still far from being solved. This is in contrast with the stochastic case where we will show that a complete characterization of the stability radius can be obtained via scaling techniques. This is the main contribution of the paper. It solves the stability radius problem of stochastic k-analysis for discrete-time systems. For the continuous time case, see Hinrichsen and Pritchard (1996). In the second part of the paper we examine the problem of optimizing the stability radius by dynamic output feedback, which, in our framework, is a problem of ksynthesis.
2. Preliminaries Consider discrete-time systems of the form , & : x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D * (E x(t))b (t), t3-, H H H H H
probability space (), F, k) with Ew (t)"Eb (t)"0, E(w (s)w (t))"j d d , G H G H G GH QR E(b (s)b (t))"c d d , E(w (s)b (t))"0, G H H GH QR G H for all s, t3-, i, j3N. Here d , d are Kronecker symbols. The assumption GH QR that the w's and b's are mutually independent excludes the interesting case where w "b which would allow us H H to study the e!ect of disturbing the di!usion parameters A A#D * E . If Assumption 1 is not satis"ed it is H H H H H still possible to obtain a lower bound for the stability radius, but, as yet no complete characterization is available. However, our results can be applied to arbitrary deterministic discrete-time systems with random parameter disturbances as discussed in the introduction (A"0, i3N). We denote by F the sigma algebra G R F "p(+b (s), w (s); 04s4t, i, j3N,), R H G then b (t), w (t), i, j3N are independent of F . For H G norm is taken on *OH ,R\ each j3N the Euclidean *lH . We D (*) the vector space of all measurable denote by H "nite gain maps * : *OH P*lH with * (0)"0 and set H H D(*)" ,D (*). By de"nition each * 3D (*) H H H satis"es ""* (y)""*lH ""* "" " : sup H (R, j3N. (3) H ""y""*OH M W$ For any given family * 3D (*), j3N, let * denote the H H combined (blockdiagonal) perturbation , *" * : *OP*l, H where l"l #2#l , q"q #2#q , ,
and
""*"""max ""* "". (4) H HZ,M Without loss of generality we assume the initial state of (2), x(0)"x3*L is deterministic. For every *3D(*) there exists a unique solution of (2) with x(0)"x3*L, which we denote by x ( ) , x). Note that x (t, x) is F measurable for all t3- (we set F "+, ),). R\ \ De5nition 2. The zero equilibrium state of (2) is said to be l-stable if there exists a constant c'0 such that
(2)
where (A, A, D , E )3*L"L;*L"L;*L"lH ;*OH "L, G H H i, j3N, *"1 or " are given. We suppose Assumption 1. The w (t), b (t), t3-, i, j3N are sequences G H of real independent random variables on a complete
E(""x (t, x)"")4c""x"". R The nominal system is , x(t#1)"Ax(t)# Ax(t)w (t), t3-, G G G x(0)"x3*L.
(5)
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If '(t, s) is the fundamental matrix associated with (5),
3. The input}output operator
'(t, t)"I , '(t, s)"¸(t!1)¸(t!2)2¸(s), L s, t3-; t5s#1,
Let ¸(), *I) be the space of square integrable *I-valued functions on the probability space (), F,k) with norm "" ) "". For any positive ¹4R, we set - "+t3-; t4¹, (hence - "-) and de"ne the 2 space l (- ; *I) of k-dimensional nonanticipatory U @ 2 square integrable stochastic processes on - as the space 2 of all sequences y( ) )"(y(t)) -2 3l(- ; ¸(), *I)) such RZ 2 that y(t)3¸(), *I) is F -measurable for all t3- . So R\ 2 every y( ) )3l (- ; *I) is adapted to the shifted seU @ 2 quence of p-algebras (F ) -2 . The space l (- ; *I) U @ 2 R\ RZ is provided with the l-norm:
where , ¸(t)"A# Aw (t), G G G then '(t, s) is F -measurable for all s, t3-, t5s and R\ the solution of (5) is x(t, x)"'(t, 0)x. We have the following result; see Morozan (1983,1997). Proposition 3. The following are equivalent. (i) The nominal system (5) is l-stable. (ii) There exist constants M51, u'0, such that E ""'(t, s)""4Me\SR\Q, s, t3-, t5s. (iii) There exists P3H (*), PY0 such that L , AHPA!P# j AHPAO0. (6) G G G G Moreover if E3*O"L and (i) holds, then there exists a unique solution P)0 in H (*) of the discrete-time L Liapunov equation , AHPA!P# j AHPA#EHE"0, G G G G given by P" E['(t, 0)HEHE'(t, 0)]. R
(7)
(8)
De5nition 4. The stability radius of the system (5) with respect to the multiperturbation structure (D , E , b ( ) )) is de"ned to be H H H HZ,M r* "r@ U (A, (A) , (D , E ) ) * G GZ,M H H HZ,M , "inf ""*""; *" * 3D(*) and H G
(2) is not l-stable . Remark 5. (i) A stability radius with respect to linear perturbation can be de"ned by restricting the disturbance operators * : *OH P*lH to be linear. The characH terization of this radius is an open problem for stochastic systems. (ii) If the data A, A, D , E , i, j3N are real, two stabilG H H ity radii can be obtained depending on whether one chooses *"1 (real perturbations), or *"" (complex perturbations). In the deterministic framework, the real and the complex stability radii are in general distinct. We will show that the two stability radii are equal in our stochastic framework.
""y( ) )""lU @ -2 _
"E ""y(t)"" " E+""y(t)"",. (9) RZ-2 RZ-2 We associate with (2) the stochastic control system *I
, , x(t#1)"Ax(t)# Ax(t)w (t)# D v (t)b (t), G G H H H G H (10) x(0)"x3*L, z(t)"Ex(t), where E"[E?, E?,2, E? ]?3*O"L. Note that the per , turbed system (2) is obtained from (10) by the nonlinear output feedback v (t)"* (z (t))"* (E x(t)), j3N. For H H H H H M arbitrary (x, v)3*L;l (-; *l) there exists a unique U @ solution of (10) in l (- ; *L) for all ¹3-, which we U @ 2 denote by x(t, x, v). Lemma 6. Suppose P3H (*) satisxes the Liapunov L equation (7), then for (x, v)3*L;l (- ; *l) and U @ 2 x( ) )"x( ) , x, v), z( ) )"Ex( ) ), we have 2 E(""z(t)"")#E1x(¹#1), Px(¹#1)2 R 2 , "1x, Px2# c E1DHPD v (t), v (t)2. H H H H H R H
(11)
The lemma is proved by direct calculation. A detailed proof can be found in El Bouhtouri, Hinrichsen and Pritchard (1997). Applying Proposition 3 and Lemma 6 with E"I and ¹PR one obtains L Corollary 7. Suppose the nominal system (5) is l-stable. If (x, v)3*L;l (-; *l), then there exists c'0, such that U @ E(""x(t)"")4c[""x""#""v""lU @ - *l ]. _ R Lemma 6 and Corollary 7 lead to the following lemma. Lemma 8. Suppose the nominal system (5) is l-stable and P)0 in H (*) satisxes the Liapunov equation (7), then for L
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(x, v)3*L;l (-; *l), we have U @
and vice versa. The control system associated with (17) (compare (2) and (10)) is
E(""z(t)"")"1x, Px2 R , # c E1DHPD v (t), v (t)2. H H H H H R H
(12)
De5nition 9. The system (10) is said to be externally l-stable if z( ) )"Ex( ) , 0, v)3l (-; *O) for every U @ v3l (-; *l), and there exist c50 such that U @ ""z""lU @ - *O " E(""z(t)"")4c""v""lU @ - *l . _ _ R
(13)
, x?(t#1)"Ax?(t)# Ax?(t)w (t) G G G , # D?v?(t)b (t), z?(t)"E(a)x?(t), H H H H where
v? (t) E? x? (t) v?(t)" $ , E(a)" $ , x?(t)" $ . v? (t) ,
E?, ,
(18)
(19)
x? (t) ,
By Corollary 7 if the nominal system (5) is l-stable then (10) is externally l-stable.
Setting *?" , *?H the scaled system (17) is obtained H H from (18) by the nonlinear feedback v?(t)"*?(E(a)x?(t)). Hence if x( ) )"x ( ) , x) is any solution of (2) or, equiva lently of (17), then
De5nition 10. Suppose (10) is externally l-stable. The operator +:l (-; *l)Pl (-; *O) de"ned by U @ U @
x (t, x)"x?(t, x , v?), where v?(t)"*?(E(a)x (t, x)).
R\ , (+v)(t)"Ex(t,0, v)"E '(t, s#1)D v (s)b (s), H H H Q H t3-, v3l (-;*l) U @
(14)
is called the input}output operator of the system (10). Its norm is de"ned as the minimal c50 such that (13) is satis"ed. Corollary 11. Suppose the nominal system (5) is l-stable and P)0 in H (*) satisxes the Liapunov equation (7), then L the input}output operator + has operator norm ""+"""max c ""DHPD "". H H H HZ,M
(15)
Theorem 12. Suppose the nominal system (5) is l-stable. If, for some p'0, there exist a3(0,#R), and P(a)3H>(*) satisfying L , , AHP(a)A!P(a)# j AHP(a)A# aEHE "0, G G G H H H G H (21)
Suppose that a3(0,R),, a"(a ,a ,2, a ), then the , nonlinear perturbed system (2) remains unchanged if we replace D , E , * with D?H , E?H , *?H , where H H H H H H *?H "a * (a\ ) ), H H H H
Although the solutions of (2) and (17) are the same and do not depend on a, the input}output operator of the scaled system, +? does change with a. We use this added freedom to characterize r* . First we obtain a lower bound.
p IlH !c DHP(a)D )0, j3N, H a H H M H
4. Characterization of r*
D?H "a\D , H H H
(20)
E?H "a E . (16) H H H
More precisely, every solution of the system (2) is also a solution of the scaled system , &? ? : x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D?H *?H (E?H x(t))b (t), H H H H H
t3-
(17)
(22)
then r* 5p. Proof. Suppose P(a) satis"es (21), (22) and let ""*""(p. We prove that (2) is l-stable. Let x(t)"x (t, x) be the solution of (2). By (20), x(t)"x (t, x) coincides with the state trajectory of the scaled system (18) generated by the input v?(t)"*?(E(a)x (t, x)). By Lemma 6 (with P re placed by P(a), E by E(a), etc.) for ¹'0, the corresponding output signal z?(t) of (18) satis"es 2 E(""z?(t)"")#E1x?(¹#1), P(a)x?(¹#1)2 R 2 , "1x, P(a)x2# c E1D?HP(a)D v?(t), v?(t)2. H H H H H R H
A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
Then P(a))3H>(*) is the unique solution of the L equation
But 2 2 ""v?(t)""4 E+""*?""""z?(t)"", R R 2 4""*?"" E""z?(t)"", R and since P(a))0, we have
, AHPA!P# j AHPA# aEHE "0. G G G I I I G IZ) Moreover
2 E(""z?(t)"")41x, P(a)x2 R c 2 , #max H ""DHP(a)D "" E""v?(t)"" H H a H H HZ,M R H 2 41x, P(a)x2#p\""*?"" E""z?(t)"". R Now ""*?H """""* "", so ""*?""(p and c " : p\""*?""(1. H H Thus 2 E""z?(t)""4(1!c)\1x, P(a)x2, ¹3-. R Therefore z?( ) )3l (-; *O) and there exists c '0, U @ such that ""z?""lU @ - *O 4c ""x"". Hence v?3l (-;*l) U @ and ""v?""lU @ - *l 4""*?"" ""z?""lU @ - *O 4pc ""x"". So by _ Corollary 7 there exists a c'0, such that E(""x (t, x )"")4c(""x""#""v?""lU @ - *l ) _ R 4c(1#pc )""x"". This shows that (2) is l-stable and concludes the proof. 䊐 It follows immediately from Theorem 12 that
\ c max H DHP(a)D , (23) r* 5 sup H H a H ?Z , HZ,M where P(a) is the unique solution of (21). By constructing a destabilizing perturbation whose norm is equal to the expression on the RHS of (23), we will show that equality holds. To do this we use a result on a minimax problem for quadratic forms which was proved in Hinrichsen and Pritchard (1996). Suppose the nominal system (5) is lstable. Let H "c E(DH'(t, 0)HEHE '(t, 0)D ), k, j3N. (24) IH H H I I H R For any KLN let (0,R)) denote the set of all families M reals and for any a)"(a ) 3(0,R)), (a ) of positive I IZ) I IZ) set P(a))" aE('(t, 0)HEHE '(t, 0)). I I I IZ) R
1037
(25)
c a H DHP(a))D " I H , j3N. H H IH a a H IZ) H Consider the minimax problem
(26)
(27)
, a G H Minimize max GH a HZ,M G H with respect to a3(0,R),. Let k( be the minimal value
, a I H . k( " inf max (28) IH a ?Z , HZ,M I H The next lemma follows directly from Theorem 2.4 in Hinrichsen and Pritchard (1996). Lemma 13. There exist a subset KLN and for every d'0, a vector a3(0,R),, such that
, a I H 4k( #d, j3N, IH a I H a I H "k( , j3K. IH a IZ) H
(29)
The following characterization of r* is the main result of the "rst part of this paper. Theorem 14. Suppose the nominal system (5) is l-stable, then (A, (A) , (D , E ) ) r@ U * G GZ,M H H HZ,M \ c " sup max H DHP(a)D , H H a H ?Z , HZ,M where P(a) is the unique solution of (21).
(30)
Proof. Let k( be de"ned by (28). If k( "0 then r* "R by (27) and Theorem 12, and so (30) is satis"ed. Now assume k( '0. Let p(a) be the largest p for which (21) and (22) have joint solution P(a)3H>(*). Then L sup p(a)"k( \. By Lemma 13 and (27), there ?Z >, exists KLN and a)3(0,R)) satisfying a c I k( " H " H DHP(a))D , j3K. (31) IH H a a H H IZ) H Let p( "k( \, then there exists v 3*lH such that ""v """1 H H and
a v, I H v H IH H a IZ) H c " v , H DHP(a))D v "k( , j3K. H a H H H H
(32)
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A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
De"ne * ( ) )3D (*) by H H * (z )"p( ""z ""v H H H H
if j3K, (33)
* (z )"0 if j3N!K, z 3*OH . H H H M Then ""* """p( , j3K and so ""*"""p( . We will show that H for this *, (2) cannot be stable. Assume the contrary then for arbitrary x3*L, the solution x ( ) , x) of (17) satis"es E(""x (t, x)"")(R. Now R *?H (z )"a * (a\z )"p( ""z ""v , z 3*lH , j3K. H H H H H H H H De"ne E(a)) analogously to E(a) in (19) and let z?)(t)"E(a))x (t, x). Then , x(t#1)"Ax(t)# Ax(t)w (t) G G G #p( D?H v ""z?H (t)""b (t). H H H H HZ) Applying Lemma 8 with P(a)) instead of P, E(a)) instead of E, D?H instead of D , j3K, D "0, j3N!K and H H H p( v ""z?H ( ) )"" instead of v ( ) ), we have by (32) M H H H E(""z?)(t)"")"1x, P(a))x2 R p( # c 1v , DHP(a))D v 2E(""z?H (t)"") H a H H H H H R HZ) H "1x, P(a))x2 # E(""z?)(t)""), x3*L. R So P(a))"0 which contradicts the fact that k( '0. Hence the system (2) is not stable for this *. Note that the (nonlinear) destabilizing disturbance constructed in the previous proof is real when the data A, (A) , (D , E ) are real. Thus G GZ,M H H HZ,M Corollary 15. If the data (A, A) , (D , E ) are real G GZ,M H H HZ, then the complex and the real stability radii of (2)M coincide, r" "r1 . In the next section we will use the following characterization of the stability radius in terms of strict linear matrix inequalities. It is proved from Theorem 14 by a perturbation argument, for details see El Bouhtouri et al. (1997). Proposition 16. The nominal system (5) is l-stable and r* 'p if and only if there exist a3(0,R), and
P3H>(*), PY0 such that L , , AHPA!P# j AHPA# aEHE O0, G G G H H H G H p IlH !c DHPD Y0, j3N. H a H H H
(34) (35)
5. Maximizing the stability radius by output feedback In this section we investigate how the stability radius can be enhanced by dynamic output feedback. For this we consider the controlled system
&:
, x(t#1)"Ax(t)# Ax(t)w (t) G G G , # D * (E x(t))b (t)#Bu(t), t3-, H H H H H
(36)
y(t)"Cx(t),
where (B, C)3*L"K;*N"L. As compensators we choose deterministic discrete-time systems & : x( (t#1)"A x( (t)#B y(t), ) ) ) u(t)"C x( (t)#D y(t), t3-, (37) ) ) where (A , B , C , D )3*L( "L( ;*L( "N;*K"L( ;*K"N. ) ) ) ) The resulting perturbed closed-loop system is , &N : x (t#1)"Ax (t)# Ax (t)w (t) G G G , # D * (E x (t))b (t), H H H H H x (0)"x 3*L>L( ,
(38)
where
x(t)
A#BD C BC ) ) , x( (t) B C A ) ) A 0 D "["H ], E "[E 0], A" G . H H H G 0 0 x (t)"
, A"
The nominal closed-loop system is , &L : x (t#1)"Ax (t)# Ax (t)w (t), G G G x (0)"x 3*L>L( .
(39)
Our aim is to determine conditions for the existence of a controller which stabilizes the nominal system (in the l sense) with a stability radius above a given threshold p'0. Following an LMI-approach we obtain the result below. The proof proceeds by applying Proposition 16 to the closed-loop system (38), for details see El Bouhtouri et al. (1997).
A.E. Bouhtouri et al. / Automatica 36 (2000) 1033}1040
Theorem 17. Given p'0, the following are equivalent. (i) There exists a stabilizing compensator & of dimension ) n( such that the nominal closed-loop system &L is l stable and rU @ (A, (A) , (D ,E ) )'p. * G GZ,M H H HZ,M (ii) There exist a3(0,R), and S, R3H>(*) with L S)R\Y0, rank(S!R\)4n( such that
p IlH !c DHSD Y0, j3N, H a H H H , , AHSA!S# j AHSA# aEHE O0 G G G H H H G H on ker C, , , R\! j AHSA! aEHE Y0, G G G H H H G H
(40)
(41) (42)
\ , , A R\! j AHSA! aEHE AH!RO0 G G G H H H G H on ker BH.
(43)
Remark 18. (i) The condition rank(S!R\)4n( is automatically satis"ed for n( 5n and the conditions (40)}(43) do not depend on n( . So if there exists a stabilizing compensator & of any dimension achieving r * 'p ) then there exists such a compensator of dimension 4n. (ii) Suppose we have found a, R, S for a given value of p, so that (ii) of the above theorem holds. Then stabilizing compensators of the form (37) which achieve r * 'p can be constructed by reversing the steps in the proof of the previous theorem, see El Bouhtouri et al. (1997). (iii) Applying Theorem 17 with n( "0 and S"R\ one can determine which stability radii can be achieved by static output feedback u(t)"D y(t). ) (iv) It follows from Theorem 17 that dynamic state feedback cannot increase the stability radius beyond those levels which can be achieved by static state feedback (where n( "0, p"n, C"I ). L (v) In the deterministic case, under some weak extra assumptions, the inequalities (41)}(43) can be formulated in terms of Riccati equations. We have shown that, in general, this cannot be done for discrete-time stochastic systems. A counterexample can be found in El Bouhtouri et al. (1997). If the structure operators D , E are chosen to H H be zero then the condition r * 'p can be neglected and Theorem 17 implies the following stabilizability conditions. Corollary 19. Given a system of the form , x(t#1)"Ax(t)# Ax(t)w (t)#Bu(t), y(t)"Cx(t) G G G (44)
1039
and a random sequence (w(t)) RZ Assumption 1. Then the following hold.
which
satisxes
(i) There exists a stabilizing compensator & of the form ) (37) for (44) if and only if there exist S, R3H>(*) with L S)R\Y0 such that , AHSA!S# j AHSAO0 on ker C, (45) G G G G \ , A R\! j AHSA AH!RO0 on ker BH G G G G , and R\! j AHSAY0. (46) G G G G (ii) There exists D 3*K"N such that the system ) , x(t#1)"(A#BD C)x(t)# Ax(t)w (t) ) G G G is l-stable if and only if there exists S3H>(*), SY0 L such that (45) and (46) hold with R\ replaced by S. (iii) There exists a stabilizing static state feedback control u"Fx for (44) if and only if (46) holds with R\ replaced by S.
In the special case of static state feedback, necessary and su$cient stabilizability conditions for continuous time systems with state-dependent noise (with C"I ) L have been derived in Willems and Willems (1976).
References Doyle, J. (1982). Analysis of feedback systems with structured uncertainties. Proceedings of IEE, 129, 242}250. El Bouhtouri, A., & Pritchard, A. J. (1992). Stability radii of linear systems with respect to stochastic perturbations. Systems and Control Letters, 19, 29}33. El Bouhtouri, A., Hinrichsen, D., & Pritchard, A.J. (1997). Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations. IDS-Report Nr. 420, UniversitaK t Bremen. Curtain, R.F. (1972). Stability of stochastic dynamical systems. Lecture Notes in Mathematics, vol. 294. Berlin: Springer. Hinrichsen, D., & Pritchard, A.J. (1990). Real and complex stability radii: A survey. In: Hinrichsen, D., & Ma rtensson B. (Eds.), Control of uncertain systems, Progress in system and control theory, vol. 6. (pp. 119}162). Basel: BirkhaK user. Hinrichsen, D., & Pritchard, A. J. (1996). Stability radii of systems with stochastic uncertainty and their optimization by output feedback. SIAM Journal Control, 34, 1972}1998. Hinrichsen, D., & Son, N. K. (1991). Stability radii of linear discrete * time systems and symplectic pencils. International Journal of Robust and Nonlinear Control, 1, 79}97. Ibrahim, R. A. (1985). Parametric Random Vibration.. Letchworth, England: Research Studies Press. Kozin, J. (1969). A survey of stability of stochastic systems. Automatica, 5, 95}112. Kubrusly, C. S. (1986). On discrete stochastic bilinear systems stability. Journal of Mathematical Analysis and Applications, 113, 36}58. Kushner, H. (1967). Stochastic stability and control. New York: Academic Press.
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Morozan, T. (1983). Stabilization of some stochastic discrete-time control systems. Stochastic Analysis and Applications, 1, 89}116. Morozan, T. (1997). Stability radii of some discrete-time systems with independent random parameters. Stochastic Analysis and Applications, 15, 375}386. Polyak, B.T. (1977). Convergence and convergence rate of iterative stochastic algorithms. II. The linear case. Automation and Remote Control. (4), 537}542. Willems, J. C., & Blankenship, G. L. (1971). Frequency domain stability criteria for stochastic systems. IEEE Transactions on Automatic Control, AC-16, 292}299. Willems, J. J., & Willems, J. C. (1976). Feedback stabilizability of stochastic systems with state and control dependent noise. Automatica, 12, 277}283. Abdelmoula El Bouhtouri was born in Marrakech, Morocco in 1960. He was educated at the University Mohammed V, Faculty of Sciences, Rabat where he received the Licence de matheH matiques appliqueH es in 1984, Le certi"cat d'eH tudes approfondies in systems theory and control in 1985, Le Doctorat de troisieH me cycle on the stabilizability of linear distributed parameter stochastic systems in 1989 and Le Doctorat d'etat on the robustness of stability of linear uncertain stochastic systems in 1994. His present position is Directeur de recherche au laboratoire d'ingeH niere matheH matique, University Chouaib Doukkali, El Jaddida, Morocco. His research interests include stochastic linear systems, H control, robustness of stability of jump processes and linear stochastic distributed parameter systems. Dr. A. El Bouhtouri is a member of the American Mathematical Society and an a$liate member of IFAC.
Diederich Hinrichsen was born in NuK rnberg, Germany, in 1939. He studied mathematics and German literature in Hamburg and Paris, passed the Staatsexamen at the University of Hamburg in 1965 and received the degree of Dr. rer. nat. from the University of Erlangen in 1966. He held positions at the Universities of Hamburg, Havana and Bielefeld. Since 1973 he has been Professor of Mathematics at the University of Bremen where he served as Director of the Institute of Dynamical Systems from its foundation in 1978 until 1990. His current research interests are in analysis and control of uncertain dynamical systems.
Anthony J. Pritchard was born in Llwyny-pia, Wales in 1937. He graduated in mathematics from Kings College, London in 1958, obtained a DIC from Imperial College, London in 1959 and a Ph.D. from Warwick University in 1969. He has been at the University of Warwick since 1968, "rst as Lecturer, then Reader in 1974, Director of the Control Theory Centre in 1975 and has been a Professor in the Mathematics Department since 1984. He is a Fellow of the Institute of Mathematics and Its Applications and a Chartered Mathematician. His main research interest is in"nite dimensional systems theory, although recently he has worked on robustness problems. He has published a number of books including Inxnite Dimensional Linear Systems Theory with Ruth Curtain and Sensors and Controls in the Analysis of Distributed Systems with Abdelhaq El Jai.