STOCHASTIC H 1 Introduction

12 downloads 0 Views 359KB Size Report
University of Warwick. Coventry CV4 7AL. U.K. ajp@maths.warwick.ac.uk ...... would require that the kernel of Ccl must contain the smallest Acl-invariantĀ ...
STOCHASTIC H1 D. Hinrichsen Institut fur Dynamische Systeme Universitat Bremen D-28334 Bremen F.R.G. [email protected]

A. J. Pritchard Mathematics Institute University of Warwick Coventry CV4 7AL U.K. [email protected]

Abstract

We consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations. Our objective is to develop an H 1 type theory for such systems. We prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations. This enables us to obtain necessary and sucient conditions for the existence of a stabilizing compensator which keeps the e ect of the perturbations on the to be controlled output below a given threshhold > 0. In the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we get coupled nonlinear matrix inequalities instead. The connection between H 1 theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases.

Keywords:

stochastic systems, state dependent noise, H 1 control, bounded real lemma, matrix inequalities

1 Introduction

The objective of this paper is to develop a general H 1-type theory for the disturbance attenuation of stochastic systems by dynamic output feedback. We consider systems  described by Ito stochastic di erential equations of the form

dx(t) = Ax(t)dt + A0 x(t)dw1(t) + B0v(t)dw2(t) + B1 v(t)dt + B2u(t)dt z(t) = C1x(t) + D11v(t) + D12 u(t) y(t) = C2x(t) + D21v(t);

(1)

where wi; i = 1; 2 are zero mean scalar Wiener processes, not necessarily independent. In applications such models are often obtained by linearization and then x(t); z(t) and u(t) represent deviations from desired xed values of the state, the output and the control (for instance, in a tracking problem, see [30]). We view v as an unknown nite energy stochastic disturbance which adversely a ects the to be controlled output z (whose desired value is represented by 0). The 1

disturbing e ect is to be ameliorated via control action u based on dynamic feedback from the measured output y. A feedback controller K : y 7! u has to be chosen in such a way that the closed loop system cl is stabilized. The e ect of the disturbances on the to be controlled output z of cl is then described by the perturbation operator L cl : v 7! z of cl which (for zero initial state) maps nite energy disturbance signals v into the corresponding nite energy output signals z of the closed loop system. The size of this linear operator is measured by the induced norm. The larger this norm is, the larger is the e ect of the unknown disturbance v on the to be controlled output z in the worst case. The problem is to determine, whether or not for each

> 0 there exists a stabilizing controller K achieving kL cl k < . Moreover we want to know how such controllers, if they exist, can be constructed. In the deterministic case the norm kL cl k is given by the H 1-norm of the associated rational transfer matrix and so the theory dealing with the above problem is known as H 1-control theory. In the present stochastic context the term H 1-control may be a misnomer, but we use it nevertheless to refer in a succinct and suggestive way to the above disturbance attenuation problem. System (1) may be regarded as a perturbed version of the stochastic system

x_ (t) = (A + A0 w_ 1(t))x(t) + B2u(t) z(t) = C1x(t) + D12 u(t); y(t) = C2x(t):

(2)

representing a linear time-invariant system with multiplicative white noise. Such systems are widely considered in the stochastic literature, especially in stochastic stability analysis, see [10, ch. 6], [1, ch. 11], [4, ch. 11]. By adding unknown disturbances to this equation we lay the ground for an analysis of robust stability of these systems, see Section 5. The disturbance of the state equation in our model (1) is composed of two parts, B1 v(t)dt and B0v(t)dw2(t). Although v is in general a stochastic vector we view the rst term as the deterministic and the second as the stochastic component of the disturbance. To motivate this terminology let v =  z where  is an unknown matrix and assume that the two Wiener processes in (1) are equal: w1 = w2 = w. Then the state equation in (1) reads

dx(t) = (A + B1 C1)x(t)dt + (A0 + B0 C1 )x(t)dw(t) + B2 u(t)dt: (3) So the deterministic disturbance term B1 C1 represents a perturbation of A, i.e. of the deterministic parameters, and the stochastic disturbance term B0C1 represents a perturbation of

the stochastic parameters of the system. The presence of both types of disturbances in (1) is essential to obtain a full generalisation of the H 1-control problem to the stochastic context. As special cases it contains on the one hand (A0 = 0; B0 = 0) the general deterministic H 1-control problem (without regularity assumptions), as stated e.g. in [8], [9], [17], [25], [26]. On the other hand it also includes the \purely stochastic" case where B1 = 0, see [16]. It may seem odd that we use the same disturbance vector v in both the deterministic and the stochastic disturbance terms. But this is in fact more general since distinct "disturbance vectors # v0 and v1 can be accounted for by setting B0 = [B00 0]; B1 = [0 B11] and v = vv0 . Similarly, it 1 would simplify the situation if we assumed that the two Wiener processes w1 and w2 are independent. But we avoid this assumption in order to derive formulae which are equally applicable to the case where e.g. w1 = w2 = w, see (3). 2

In order to keep the notational burden as low as possible we do not deal with more general models and more general multi-perturbation structures where the single stochastic terms in (1) are replaced by sums of similar terms. We make some comments about the extension of our results to such systems in Section 6. The main complication in the H 1-control problem studied here is due to the presence of both deterministic and stochastic perturbations terms in (1). In the case of purely stochastic disturbances the problem has been solved in [16]. The key result on which the general solution will be based is the Stochastic Bounded Real Lemma which will be proved in the next section. This result states necessary and sucient conditions for a given stochastic system to be stable with kL k < . It is of independent interest since it allows one to determine kL k which measures the in uence of the disturbances in the worst case scenario. The centrepiece of our conditions is no longer a Riccati type equation or the corresponding matrix inequality (as in the deterministic case), but a rational matrix equation which appears to be new. For the associated matrix inequality we could not nd existence results in the literature. Our proof proceeds | as in the theory of algebraic Riccati equations | via the study of a nite time optimization problem which due to the structure of our disturbance model has a number of subtleties. While Section 2 deals with a problem of system analysis (under which conditions has the input output operator L of a stable stochastic system a norm kL k < ?) the synthesis problem of H 1-control will be treated in Section 3. Here we follow the LMI approach developed for deterministic systems, see [8], [17]. The idea is to apply the Stochastic Bounded Real Lemma to the compensated system cl in such a way that the matrices of compensator parameters which achieve stability and kL cl k < are characterized by a linear matrix inequality. Then applying the projection lemma [8] we are able to obtain necessary and sucient conditions in terms of the given data. The result is a characterisation in terms of a pair of matrix inequalities. But in contrast to the deterministic case the two matrix inequalities are coupled and nonlinear. Specializing to the case A0 = B0 = 0, however, the inequalities are linear and decoupled, and we regain the deterministic results as given in [8]. In Section 4 we deal with the so-called regular case. The matrix inequalities | although still coupled and nonlinear | are greatly simpli ed via the regularity assumptions and it is possible to derive explicit formulae for full order suboptimal compensators. In the deterministic context this was the case for which the H 1-control problem was rst resolved via a pair of Riccati equations [5]. In the stochastic context, we do not obtain an analogous pair of rational matrix equations. In fact it has been shown for the special case A0 = B1 = 0 that, in general, it is not possible to replace the two inequalities by equalities, see [16]. Therefore, even for regular data, matrix inequalities seem to be an indispensable tool of an H 1 type optimal control theory in the stochastic context. For the deterministic case A0 = B0 = 0, however, our conditions reduce to the well-known Riccati inequalities from which the results in [5] follow via standard theorems about the relationship between Riccati inequalities and equations. In Section 5 we consider stability radii for a nominal system where all direct input output couplings are zero (a singular problem). If (2) is stable, u = 0 and v = z, the associated stability radius is the maximum  such that all the perturbed systems with kk <  are stable. We will use the results of Section 2 to obtain a lower bound for the radius. In the deterministic case there is a close relationship between the singular H 1-control problem and the problem of maximizing the stability radius of a given system by state or dynamic output feedback, see [13]. We will use this relationship to show how the radius may be enhanced by feedback. However, we do not determine precise formulae for the stability radius or for the supremal stability radius achievable 3

by feedback. The general problem of characterizing and maximizing stability radii of stochastic systems of the form (1) is still open. In some concluding remarks (Section 6) we will comment on further open problems and possible extensions of our work. The stabilization of stochastic systems with multiplicative noise has been studied since the late sixties, particularly in the context of linear quadratic optimal control, see e.g. [20], [27], [29]. The subject of robust stabilization is of more recent vintage. An early reference is [28] where the problem is considered in an almost disturbance decoupling framework. More recently a number of papers have been published which deal with robust stability and robust stabilization problems in the spirit of H 1-control or the stability radius approach. In [6] El Ghaoui describes how the maximization of an estimate for the stability radius by state feedback can be formulated | for general multi noise structures | as a convex optimization problem over linear matrix inequalities. In [22] the norm of the perturbation operator of a time-varying stable linear system with state dependent noise (B0 = 0; B1 = 0) is related to a parametrized Riccati equation. Formulae for stability radii and supremal stability radii of stochastic systems have been obtained for various special cases. The rst formula was derived for the case A0 = 0; B1 = 0 in [2]. Morozan [21] extended this formula to the case where B1 = 0 and the nominal system contains a sum of white noise terms. The maximization of stability radii via state feedback was rst considered in [13]. A full H 1 and stability radius theory for stochastic multi-perturbations of a deterministic system was developed in [16]. However, all these characterizations of stability radii only apply to the special case of purely stochastic parameter disturbances (B1 = 0). First results concerning stability radii subjected to simultaneous deterministic and stochastic parameter perturbations have been presented in [15]. These are improved in the present paper. However, our main intention is to develop a counterpart of H 1-control for linear stochastic systems. The conceptual di erence between H 1-control theory and the theory of stability radii, which has been blurred by the fact that they yield similar results for deterministic time-invariant linear systems, stands out more clearly in the stochastic context.

2 A stochastic version of the Bounded Real Lemma The main tool that we will use in our analysis of the stochastic disturbance attenuation problem is an extension of the Bounded Real Lemma to stochastic systems. This result is of independent interest and in fact we regard it as the main result of this paper. In order to describe it we consider the following system

dx(t) = Ax(t)dt + A0x(t)dw1 (t) + B0v(t)dw2(t) + Bv(t)dt z(t) = Cx(t) + Dv(t);

(4)

where (A; A0 ; B0; B; C; D) 2 K nn  K nn  K n`  K n`  K qn  K q` ;

K

= R or C :

(5)

w1; w2 are zero mean real scalar Wiener processes on a probability space ( ; F ; ) relative to an increasing family (Ft)t2R+ of -algebras Ft  F . We assume that

E ((wi(t) ? wi(s))(wj (t) ? wj (s))) = qij (t ? s); i; j = 1; 2; t; s 2 R + ; t > s: 4

So " Q =# (qij ) is the incremental covariance matrix of the two-dimensional Wiener process w1(t) . w2(t) In (4) the input process v(t) is viewed as a stochastic disturbance and the output process z(t) as vector of the to be controlled variables. The system equation contains multiplicative state and input dependent noise terms which may be interpreted as white noise parameter perturbations of the matrices A and B :

dx(t) = (A + A0w_ 1 (t))x(t)dt + (B + B0 w_ 2(t))v(t)dt In this paper we provide all spaces K k ; k  1 with the usual inner product h; i and the corresponding 2-norm k  k. Let L2 ( ; K k ) denote the space of square-integrable K k -valued functions (modulo equivalence) on the probability space ( ; F ; ). For any 0 < T  1 we write [ 0; T ] for the closure of the open interval (0; T ) in R and denote by L2w ([0; T ]; L2 ( ; K k )) the space of nonanticipative stochastic processes y() = (y(t))t2[0; T ] with respect to (Ft )t2[0; T ] (see e.g. [7]) satisfying ! Z Z T T 2 2 ky()kL2w = E ky(t)k dt = E (ky(t)k2)dt < 1 : (6) 0

0

For arbitrary 0 < T < 1 and (v; x0 ) 2 L2w ([0; T ]; L2 ( ; K ` ))  K n , there exists a unique solution x() = x(; v; x0 ) 2 L2w ([0; T ]; L2( ; K n )) of (4) with x(0) = x0 [18], i.e. x() is a continuous nonanticipative stochastic process satisfying the Ito integral equation Z t

"

Z t

#

x(t) = x0 + 0 (Ax(s) + Bv(s))ds + 0 [A0x(s) B0 v(s)]d ww1 ((ss)) ; t 2 [ 0; T ]: 2

(7)

Moreover x() has bounded second moments on [ 0; T ]. De nition 2.1 The system (4) is called internally stable if there exists a constant c > 0 such that Z 1 E kx(t)k2 dt  ckx0 k2; x0 2 K n ; 0 0 where x() = x( ; 0; x ) is the free trajectory of (7) starting at x0 (i.e. v = 0).

An equivalent condition (see [4]) is that there exist constants M  1; ! > 0, such that

Ekx(t; 0; x0)k2  Me?!t kx0 k2 for all x0 2 K n ; t  0: Let Hn(K ) denote the set of Hermitian matrices in K nn . It is known [4] that (4) is stable in the above sense if and only if there exists P 2 Hn(K ); P  0 such that

PA + A P + q11 A0PA0 = In:

(8)

It is easily seen that in this stability criterion the identity matrix (on the RHS of (8)) may be replaced by any other positive de nite matrix Q0 2 Hn(K ). The following de nition generalizes the concept of nite gain L2 stability from deterministic input output systems to stochastic systems of the form (4). 5

De nition 2.2 The system (4) is said to be externally stable or L2 input-output stable if, for every v() 2 L2w (R + ; L2 ( ; K ` )), z() = Cx(; v; 0) + Dv() 2 L2w (R + ; L2( ; K q )) and there exists a constant  0 such that kz()kL2w (R+;L2( ;K q ))  kv()kL2w (R+;L2( ;K ` )) ; v 2 L2w (R + ; L2 ( ; K ` )): (9) De nition 2.3 Suppose that (4) is externally stable. The operator L : L2w (R + ; L2 ( ; K ` )) ! L2w (R + ; L2 ( ; K q )); de ned by

v() 2 L2w (R + ; L2( ; K ` )) (10) is called the perturbation operator of (4). Its norm is de ned as the minimal  0 such that (9) (L v)(t) = Cx(t; v; 0) + Dv(t); t  0;

is satis ed, i.e.

kCx(; v; 0) + Dv()kL2w (R+;L2 ( ;Kq )) : (11) kv()kL2w(R+ ;L2( ;K` )) v2L2w (R+ ;L ( ;K ` ));v6=0 kL k is a measure of the worst e ect the stochastic disturbance v() may have on the to be controlled output z() of the system. Therefore it is important to nd a way of determining the norm kL k. The Stochastic Bounded Real Lemma which we will derive in this section provides a method for computing kL k. kL k =

sup 2

We proceed by associating a nite time quadratic cost functional with the problem,

Z T h i i 2 2 2 E kv(t)k ? kz(t)k dt = 0 E 2 kv(t)k2 ? kCx(t) + Dv(t)k2 dt (12) where x() = x(; v; x0) denotes the solution of (4) with x(0) = x0 and v() 2 L2w = L2 ([ 0; T ]; L2( ; K ` )), and z() = z(; v; x0 ) is the corresponding output. We will see that the Z

T JT 2 (x0 ; v) = 0

h

w

problem of minimizing this functional will lead us to a solution of the supremum problem on 2 0

the RHS of (11). Formally the problem of minimizing JT (x ; v) has the form of an optimal control problem and so in our development in this section we will refer to the disturbance v as a \control". Our rst step is to show that an internally stable system (4) is also externally stable. For every P 2 Hn(K ), we set "

#

 A0PA0 ? C C PB + q12 A0 PB0 ? C D : M (P ) = PAB+ PA+Pq+Bq11 PA

2 I` + q22 B0PB0 ? DD 12 0 0 ? D C Lemma 2.4 Suppose P () : [ 0; T ] 7! Hn(K ) is continuously di erentiable, T > 0. Then

(13)

JT 2 (x0; v) = hx0 ; P (0)x0i ? Ehx*" (T ); P (T#)x(T )i " #+! Z T x ( t ) x ( t ) 0 2 K n ; v () 2 L2 (14) dt; x + E hx(t); P_ (t)x(t)i + v(t) ; M (P (t)) v(t) w 0 where M (P ) is de ned by (13) and x() = x(; v; x0 ).

6

Proof: Let x0 2 K n ; v() 2 L2w and let x() = x(; v; x0) denote the corresponding solution of (4). Then the vector function '(s) = (Ax(s) + Bv(s)) and the n  2 matrix function (s) =

[A0x(s) B0 v(s)] satisfy the conditions of Ito's Lemma, see [4, 4.5] and, by (7) " # Z t Z t w ( s ) 1 0 x(t) = x + 0 '(s)ds + 0 (s) d w (s) ; t 2 [ 0; T ]: 2 Applying Ito's formula to F (t; x(t)) = hx(t); P (t)x(t)i and taking expectations we obtain, for every T > 0, * " #+ Z T Z T w ( s ) 1 Ehx(T ); P (T )x(T )i ? hx0; P (0)x0i = E 0 hx(t); P_ (t)x(t)idt + E 0 2 Re P (t)x(t); (t) d w2(s) +E

Z T

0

2 RehP (t)x(t); Ax(t) + Bv(t)idt + E

Z T

0

tr fP (t)[A0x(t) B0 v(t)] Q [A0x(t) B0v(t)] g dt

where tr denotes the trace. We rst prove (14) under the condition that v() 2 L2w is bounded, i.e. 9c > 0 : kv(t; !)kK`  c; (t; !) 2 [0; T ]  : Applying an estimate for the moments of x(), see [19, p. 81, Corollary 6], there exist constants c0; c1 > 0 such that Zt 0 2 0 2 (15) Ekx(t; v; x )kKn  c0 kx kKn + c1 E 0 kv(s)k2K` ds: Hence Ekx(t; v; x0)k2Kn is bounded on [0; T ] and therefore " #+ Z T* Z T Z T w ( s ) 1 E 0 P (t)x(t); (t) d w2(s) = E 0 hP (t)x(t); A0x(t)idw1(t)+E 0 hP (t)x(t); B0v(t)idw2(t) = 0: Now ) ("  A # x ( t )  0 tr fP (t)[A0 x(t) B0 v(t)] Q [A0x(t) B0v(t)] g = tr v(t)B  P (t) [A0x(t) B0 v(t)] Q = =

0 #" #)     A0 P (t)A0x(t) x(t) A0 P (t)B0v(t) q11 q12 tr vx((tt)) B  P (t)A0 x(t) v (t) B  P (t)B0 v (t) q12 q22 0 0 *" # " #" #+   x(t) ; q11 A0P (t)A0 q12 A0 P (t)B0 x(t) : v(t) q12 B P (t)A0 q22 B  P (t)B0 v(t) ("

0

0

Hence JT 2 (x0; v() + Ehx(T ); P (T )x(T )i ? hx0 ; P (0)x0i Z T hx(t); P_ (t)x(t)i + 2 kv(t)k2 ? kCx(t) + Dv(t)k2 + hP (t)x(t); Ax(t) + Bv(t)i =E 0

*"

# "

#"

#+ )

A0 P (t)A0 q12 A0P (t)B0 x(t) +hAx(t) + Bv(t); P (t)x(t)i + xv((tt)) ; qq11B dt v(t) 12 0 P (t)A0 q22 B0 P (t)B0 *" # " Z T(  P (t) ? C  C P (t)B ? C  D # " x(t) #+ x ( t ) P ( t ) A + A _ =E hx(t); P (t)x(t)i + v(t) ; B  P (t) ? DC

2 I ? D D v(t) 0 *" # " #" #+ ) A0 P (t)A0 q12 A0 P (t)B0 x(t) + xv((tt)) ; qq11B dt v(t) 12 0 P (t)A0 q22 B0 P (t)B0 *" # " #+ ) Z T (D E x ( t ) x ( t ) _ dt: = E x(t); P (t)x(t) + v(t) ; M (P (t)) v(t) 0 7

This proves (14) for all bounded v() 2 L2w and x0 2 K n . Now consider the linear maps L2w  K n ! L2w ([ 0; T ]; L2( ; K n )); (v; x0) 7! x(; v; x0) L2w  K n ! L2 ( ; K n ); (v; x0) 7! x(T; v; x0 ) where we endow L2w  K n with the norm k(v; x0)k = (kx0 k2Kn + kv()k2L2w )1=2 . By the estimate (15) these are bounded linear operators. As a consequence, for any xed x0 2 K n , the LHS and the RHS of (12) depend continuously on v 2 L2w . They coincide on the linear subspace L2b of bounded v 2 L2w which is dense in L2w . Therefore (12) holds for all v 2 L2w ; x0 2 K n . Proposition 2.5 Suppose (4) is internally stable. Then (4) is externally stable. Moreover there exist > 0 and P 2 Hn(K ); P  0 such that "  P + q11 A PA0 ? C  C PB + q12 A PB0 ? C  D # PA + A 0 0 M (P ) = (16) B P + q12 B0PA0 ? DC 2I` + q22 B0 PB0 ? DD  0: and, for each pair ( ; P ) 2 (0; 1)  Hn (K ) satisfying (16) and P  0, we have kL k < . Proof: Since (4) is internally stable there exists P 2 Hn(K ); P  0 such that PA + AP + q11 A0PA0 ? C C  0: (17) "

#

11 M12 (n+`)(n+`) we have the well known For any Hermitian block matrix M = M M21 M22 2 K

de niteness criterion

M  0 , (M22  0 and M11 ? M12 M22?1 M21  0): (18) Applying this criterion to M (P ) (with the above P ) we see that, for suciently large, M (P )  0. Hence there exists a pair ( ; P ) 2 (0; 1)  Hn(K ) satisfying (16) and P  0. Now assume that ( ; P ) 2 (0; 1)  Hn(K ) is any pair satisfying (16) and P  0. Choose " > 0 suciently small such that M (P )  "2I . Then, setting P (t) = P and x0 = 0 in (14), we obtain for all v() 2 L2w (R + ; L2 ( ; K ` )) and all T > 0, since P  0. It

Z T h Z T i 2

2 2 2 2 JT (0; v) = 0 E kv(t)k ? kz(t)k dt  " 0 Ekv(t)k2dt follows that z() = z(; v; 0) 2 L2w (R + ; L2( ; K q )) and

kL vk2L2w (R+;L2( ;K q )) =

Z 1

0

Ekz(t)k2 dt  ( 2 ? "2 )

Z1

0

(19)

Ekv(t)k2dt

for all v() 2 L2w (R + ; L2 ( ; K ` )). This concludes the proof. Remark 2.6 (i) By setting C = In and D = 0, we see that x(; v; 0) 2 L2w (R + ; L2( ; K n )) for v() 2 L2w (R + ; L2( ; K ` )) and since x(t; v; x0) = x(t; 0; x0)+x(t; v; 0) we conclude that x(; v; x0) 2 L2w (R + ; L2( ; K n )) for all (v(); x0) 2 L2w  K n . (ii) Suppose that M (P )  0 for some > 0 and P 2 Hn (K ) with P  0 (as in Proposition 2.5). Then there exists  > 0 such that P  ?2 I and by (14) Z 1

2 0 Ekv(t)k2dt  JT 2 (x0 ; v)  hx0 ; Px0i ? Ehx(T ); Px(T )i  hx0 ; Px0i + 2 Ekx(T )k2; T > 0 It follows that, Ekx(t; v; x0)k2 is bounded in t 2 R + , for all (v(); x0) 2 L2w  K n . 8

Corollary 2.7 Suppose that (16) holds for some pair ( ; P ) 2 (0; 1) Hn(K ) with P  0. Then (4) is internally stable and kL k < . Proof: Suppose that (16) holds. Since (16) implies (17), system (4) is internally stable. Hence kL k < follows from the previous proposition. We will now show the converse of Corollary 2.7, i.e. we will prove the following characterization of the norm of the perturbation operator which can be viewed as a stochastic version of the Bounded Real Lemma. Theorem 2.8 (Stochastic Bounded Real Lemma) For any set of data (5) and any positive real number the following statements are equivalent: (i) The system (4) is internally stable and kL k < , (ii) there exists P 2 Hn (K ) such that (16) is satis ed. It remains to prove that (i) implies (ii). In order to do this we need a number of lemmata. Using the notation

H 2 (P ) = 2I` + q22 B0 PB0 ? DD 2 H`(K ); K (P ) = PB + q12 A0PB0 ? C  D 2 K n` : we can write M (P ) de ned by (13) in the following way "  P + q11 A PA0 ? C  C K (P ) # PA + A 0 (20) M (P ) = K (P ) H 2 (P )  0:

Lemma 2.9 Suppose F () 2 C ([ 0; T ]; K `n ) and PF 2 () satis es the linear di erential matrix

equation X_ (t) + X (t)(A + BF (t)) + (A + BF (t))X (t) + q11 A0 X (t)A0 + q22F (t) B0X (t)B0 F (t) +q12A0 X (t)B0F (t) + q12 F (t) B0X (t)A0 + 2F (t) F (t) ? (C + DF (t))(C + DF (t)) = 0 (21) 2

with PF (T ) = 0. Then if v() 2 L2w ([ 0; T ]; L2( ; K ` )), we have Z

T JT 2 (x0 ; v + FxF ) = hx0; PF 2 (0)x0 i + 0

h

D

Ei

E hv; NxF i + hNxF ; vi + v; H 2 (PF 2 )v dt; (22)

where xF () = xF (; v(); x0) = x(; F ()xF () + v(); x0 ) is the solution of

dxF (t) = (A+BF (t))xF (t)dt+A0xF (t)dw1(t)+B0 F (t)xF (t)dw2(t)+B0v(t)dw2(t)+Bv(t)dt (23) 2

2

with xF (0) = x0 and N (t) = K (PF (t)) + H 2 (PF (t))F (t). In particular, if v = 0 then

JT 2 (x0 ; FxF ) = hx0 ; PF 2 (0)x0 i:

(24)

Proof: The left hand side of (21) can be written " # " #" # X (t)A + A X (t) + q11 A0X (t)A0 ? C  C X (t)B + q12 A0 X (t)B0 ? C D I _X (t) + I F (t) B  X (t) + q12 B0X (t)A0 ? DC

2I + q22 B0 X (t)B0 ? DD F (t) 9

2

Hence PF (t) satis es

"

#

X_ (t) + [I F (t)]M (X (t)) F I(t) = 0; X (T ) = 0:

(25)

2

Therefore, applying Lemma 2.4 with P () = PF () and F ()xF () + v() for v() we obtain that 2 JT (x0 ; FxF + v) is equal to

" # " #) Z T(D E 2 2 2 x ( t ) x ( t )

F F 0 0 _ hx ; PF (0)x i + E 0 xF (t); PF (t)xF (t) + F (t)xF (t) + v(t) M (PF (t)) F (t)xF (t) + v(t) dt Z Th D Ei 2 2 = hx0 ; PF (0)x0i + E h v (t); N (t)xF (t)i + hN (t)xF (t); v(t)i + v(t); H 2 (PF (t))v(t) dt 0

Hence (22) holds. Setting v = 0 in (22) we obtain (24).

Lemma 2.10 Suppose (4) is internally stable and kL k < . Then there exists c > 0 such that JT 2 (x0 ; v)  ?ckx0 k2;

x0 2 K n ; v() 2 L2w ([ 0; T ]; L2( ; K ` )); T > 0

(26)

Proof: Denote by XT (t) the solution of (21) with F (t)  0 and nal value XT (T ) = 0, i.e. XT (t) solves:

X_ (t) + X (t)A + A X (t) + q11 A0 X (t)A0 ? C C = 0; X (T ) = 0: By time-invariance XT (t) = XT ?t (0). By linearity we have x(t; v; x0) = x(t; 0; x0 ) + x(t; v; 0). Applying (22) with F (t)  0 we get Z T hD E D Ei 2 2 0

0 0 JT (x ; v) ? JT (0; v) = hx ; XT (0)x i + E 0 v(t); NT (t)x(t; 0; x0 ) + NT (t)x(t; 0; x0); v(t) dt

where NT (t) = K (XT (t)) . Let 0 < "2 < 2 ? kL k2 . Then

JT 2 (0; v)  2 kvk2L2w ([0;1];L2( ;K ` )) ? k(L v )k2L2w ([0;1];L2( ;K ` ))  "2kvk2L2w ([0;1];L2( ;K ` )) = "2kvk2L2w ([ 0;T ];L2( ;K` )) ; where v denotes the extension of v from [ 0; T ] to R + by 0. Hence Z Th

D

E D

Ei

JT 2 (x0 ; v) hx0 ; XT (0)x0 i + 0 E "2hv(t); v(t)i + v(t); NT (t)x(t; 0; x0 ) + NT (t)x(t; 0; x0 ); v(t) dt = hx0 ; XT (0)x0i +

 hx0 ; XT (0)x0 i ?

Z T

0

Z T

0







E

"v(t) + "?1NT (t)x(t; 0; x0)

2 ?

"?1NT (t)x(t; 0; x0 )

2 dt



E

"?1NT (t)x(t; 0; x0 )

2 dt:

Since (4) is stable there exists c0 > 0 such that Z 1

0

Ekx(t; 0; x0)k2dt  c0 kx0 k2: 10

(27)

Hence by (24) there exist constants c1 ; c2 > 0 independent on T such that 2

0  hx0; XT (t)x0 i = hx0 ; XT ?t(0)x0i = JT ?t(x0 ; 0)  ? and

Z 1

0

EkCx(s; 0; x0)k2ds  ?c1 kx0 k2

kNT (t)k = kXT (t)B + q12 A0 XT (t)B0 ? C  Dk  c2 ; t 2 [ 0; T ]; T > 0:

Thus by (27)

JT 2 (x0 ; v)  ?c1 kx0 k2 ? c22"?2c0 kx0k2 ;

This concludes the proof.

T > 0:

Lemma 2.11 Suppose (4) is2 internally stable, kL k < , F () 2 C ([ 0; T ]; K `n ); T > 0 and 2 PF () satis es (21) with PF (T ) = 0. Then

2 I ? DD  0 and H 2 (PF 2 (t))  ( 2 ? kL k2 )I` ;

t 2 [ 0; T ]:

(28)

Proof: We will rst prove that H 2 (P2F 2 (t))2  0. Suppose this is false and there exists t^ 2 [ 0; T ]; u 2 K ` ; kuk = 1 such that hu; H (PF (t^))ui  ? for some  > 0. Assume t^ < T . Then for  > 0 suciently small D

E

u; H 2 (PF 2 (t))u  ?=2; t 2 [ t^; t^ + ]  [ 0; T ]:

De ne

8
0 suciently small, the integrand becomes negative since E xF (t) is continuous 2 2

and E xF (t^) = 0. This yields a contradiction whence H (PF (t))  0. If t^ = T , a similar proof applies replacing the interval [ t^; t^ + ] by [ T ? ; T ]. Now let " be any positive number such that kL k2 < 2 ? "2. Applying the2 previous step with

~ (t) of (21) (with ~

~ = ( 2 ? "2)1=2 instead of

we obtain for the corresponding solution P F 2 instead2 of ): H ~2 (PF ~ (t))  0. For any t0 2 [ 0; T ), de ne Ft0 (t) = F (t + t0); t 2 [ 0; T ? t0]. Let PF ~t0 (t) be the solution of (21) with replaced by ~ and F replaced by Ft0 on the interval 2 [ 0; T ? t0 ] such that PF ~t0 (T ? t0) = 0. Then

PF ~t20 (t) = PF ~2 (t + t0); 11

t 2 [ 0; T ? t0]:

Hence by (24), for any t0 2 [ 0; T ), x0 2 K n ,

hx0 ; PF ~2 (t0)x0 i = hx0 ; PF ~t20 (0)x0i = JT ~2?t0 (x0 ; Ft0 xFt0 )  JT 2?t0 (x0 ; Ft0 xFt0 ) = hx0 ; PF 2 (t0 )x0i 2

2

2

and so H 2 ?"2 (PF (t0 ))  H 2?"2 (PF ~ (t0 ))  0, i.e. H 2 (PF (t))  "2I for all t 2 [ 0; T ] (by continuity). Since this holds for arbitrary "2 < 2 ? kL k2 , (28) follows and

2I ? DD = H 2 (PF 2 (T ))  0: This completes the proof. We will now study the matrix di erential equation X_ + XA + A X + q11 A0XA0 ? C  C ? K (X )H 2 (X )?1K (X ) = 0;

X (T ) = 0:

(29)

The function

f (X ) = XA + A X + q11 A0 XA0 ? C C ? K (X )H 2 (X )?1K (X ) is continuously di erentiable on its domain of de nition Df = fX 2 Hn(K ) ; det(H 2 (X )) 6= 0g in the real vector space Hn(K ). For every T > 0 there exists a (unique) solution of (29) backwards in time on a maximal interval (t?(T ); T ]. The following proposition shows, in particular, that t?(T ) < 0 for all T > 0. Proposition 2.12 Suppose (4) is internally stable and kL k < . Then (29) has a unique solution PT () on [ 0; T ] for every T > 0. Moreover, the feedback control vT (t) = FT (t)xFT (t);

FT (t) = ?H 2 (PT (t))?1K (PT (t));

(30)

where xFT () satis es

dxFT (t) = (A + BFT (t))xFT (t)dt + A0 xFT (t)dw1(t) + B0FT (t)xFT (t)dw2(t); xFT (0) = x0; 2

minimizes JT (x0 ; v) and the optimal cost is

2 (x0 ; v ) = Dx0 ; P (0)x0 E : min J T v2L2 T w

(31)

Proof: Since (29) is time-invariant we have PT (t) = PT ?t (0); t 2 (t?(T ); T ] and t? (T ?  ) = t?(T ) ? ;  2 R : (32) Let T~ = inf fT  0 ; t? (T )  0g. Then T~ > 0, and t?(T~) = 0 if T~ < 1. For every T < T~ we have t? (T ) < 0 and PT () is continuously di erentiable on [0; T ]. Setting F (t) = FT (t); t 2 [0; T ] in (25) we get from (20) and (30)

" # " # " # "  PT + q11 A PT A0 ? C  C K (PT ) # " I # P A + A I I I T 0 _ _ PT + F M (PT ) F = PT + F K (PT ) H 2 (PT ) FT T T T = P_T + PT A + A PT + q11 A0PT A0 ? C C ? K (PT )H 2 (PT )?1K (PT ) = 0

12

Hence PT () satis es (25), or equivalently (21), with F (t) = FT (t) on [ 0; T ] for all T < T~, i.e.

PF T2 (t) = PT (t);

t 2 [ 0; T ]:

Moreover, with this choice of F (t),

N (t) = K (PT (t)) + H 2 (PT (t))FT (t) = 0 and so Lemma 2.9 implies that Z

E D Ei T hD JT 2 (x0 ; v + FT x) = x0 ; PT (0)x0 + 0 E v(t); H 2 (PT (t))v(t) dt:

But by Lemma 2.11

H 2 (PT (t)) = H 2 (PF T2 (t))  ( 2 ? kL k2 )I`  0; t 2 [ 0; T ]:

(33)

2 Hence the control vT (t) = FT (t)x(t) minimizes JT (x0 ; v) and the optimal costs are given by (31), for all T < T~. As a consequence we obtain

hx0 ; PT ( )x0 i = hx0; PT ? (0)x0i = JT 2? (x0 ; vT ? )  JT 2? (x0 ; 0)  0;  2 [ 0; T ]: On the other hand

hx0; PT ( )x0 i = JT 2? (x0 ; vT ? )  ?ckx0 k2 ; x0 2 K n ;  2 [ 0; T ]; for all T < T~ by Lemma 2.10. Hence ~ t 2 [0; T ]; T < T:

? cIn  PT (t)  0;

(34)

Now suppose T~ < 1 so that t?(T~) = 0. Then ?cI  PT~ (t)  0 for all t 2 (0; T~] and hence the solution PT~ (t) of (29) (with T = T~) cannot escape to 1 as t # 0. It follows that there exists a is a limit point of PT~ (t) boundary point P 0 2 Hn(K ); det(H 2 (P 0)) = 0 of the 2domain Df which 2

as t # 0. But this contradicts the fact that by (33) H (PT~ (t)) = H (PT~?t(0))  ( 2 ? kL k2 )I` for all t 2 (0; T~). Thus T~ = 1 and the proposition is proved. Now we examine what happens as T ! 1. Lemma 2.13 Suppose (4) is internally stable and kL k < . Then PT (t) decreases as T increases, for each t 2 [0; T ] .

Proof: Suppose T 0 > T; t 2 [0; T ] and x0 2 K n . Let vT ?t be optimal for x0 on [ 0; T ? t], and 0 set v( ) = vT ?t ( ) for  2 [ 0; T ? t] and v( ) = 0 for  2 (T ? t; T ? t]. Then hx0 ; PT (t)x0 i JT ?t (x0 ; v) = JT ?t (x0; vT ?t ) ? 0

0

Z T ?t 0

T ?t

13

Ekz(s)k2 ds  JT ?t(x0 ; vT ?t ) = hx0; PT (t)x0 i:

We are now in a position to prove Theorem 2.8. Proof of Theorem 2.8: By Corollary 2.7 it only remains to prove that (i) implies (ii). Assume (i), i.e. (4) is internally stable and kL k < . Using (34) it follows from Lemma 2.13 that PT (t) converges as T ! 1 for any t  0. But PT (t) = PT ?t(0), so the limit limT !1 PT (t) = limT !1 PT (0) = P is constant. It follows from (34) and (33) that P satis es

P  0 and H 2 (P )  0

(35)

and is a solution of the Rational Matrix Equation

PA + A P + q11A0 PA0 ? C C ? K (P )H 2 (P )?1K (P ) = 0: (36) " # " # C D Now replace C by C = I and D by D = 0 in De nition 2.3 to obtain the perturbation operator L  for the modi ed data. Then kL  k < for suciently small  > 0 and so applying the above result to the modi ed data we nd that there exists P 2 Hn(K ); P  0 satisfying P A + A P + q11 A0 P A0 ? C C ? 2 I ? K (P )H 2 (P )?1K (P ) = 0; H 2 (P )  0: (37) By stability P  0, and the above equation implies P A + A P + q11 A0P A0 ? C  C ? K (P )H 2 (P )?1K (P )  0; H 2 (P )  0: Applying the de niteness criterion (18) we get that M (P )  0, and (ii) is proved. 2 For later use we add the following consequence of Theorem 2.8. Corollary 2.14 For any set of data (5) the following conditions are equivalent: (a) the system (4) is internally stable and kL k < , (b) there exist  > 0 and P  0 satisfying (37). (c) there exists P 2 Hn (K ); P  0 such that 3

2

PA + A P + q11A0 PA0 PB + q12 A0PB0 C  6 B P + q12 B0PA0 2I + q22 B0 PB0 D 75  0 4 C D I

(38)

Proof: In the above proof we have shown that (a) implies (b) and (b) implies condition (ii) of Theorem 2.8 (hence (a)). So it remains to show the equivalence of condition (ii) and (c). This follows from the equality 2

32

32

I 0 ?C  7 6 PA + AP + q11 A0PA0 PB + q12 A0PB0 C  7 6 I 0 6 B P + q12 B0PA0 2I + q22 B0 PB0 D 5 4 0 I 4 0 I ?D  5 4 ?C ?D C D I 0 0 I 3 2 PA + A P + q11A0 PA0 ? C C PB + q12 A0PB0 ? C D 0 7 " M (P ) 6

2 I + q22 B0PB0 ? DD 0 5 = = 4 B  P + q12 B0 PA0 ? DC 0 0 0 I The following scalar example illustrates the above results. 14

0 0

I 0

I

3 7 5

#

Example 2.15 Consider a system of the form (4), with n = 1, K = R and D = 0: dx(t) = ax(t)dt + a0 x(t)dw1(t) + b0 v(t)dw2(t) + bv(t)dt; y(t) = cx(t)

(39)

where a; a0; b; b0 ; c 2 R and w1(t); w2(t) are Wiener processes as before. (16) is equivalent to 2pa + q11a20 p ? c2 ? (b + q12 a0b0 )2 p2=( 2 + q22 b20 p) > 0; 2 + q22 b20 p > 0

(40)

Suppose a = ?1; a0 = b = b0 = c = 1 and assume rst that w1 = w2 = w, q11 = q12 = q22 = 1. The inequalities (40) become

?p ? 1 ? 4p2=( 2 + p) > 0; 2 + p > 0: and these in turn are equivalent to 0 > 5p2 + (1 + 2 )p + 2 and 2 + p > 0:

(41)

The rst inequality holds if and only if

p

p

(9 + 80 < 2 or 2 < 9 ? 80) and 10p < ?(1 + 2 ) + ( 4 ? 18 2 + 1)1=2 : So

10( 2 + p) < ( 4 ? 18 2 + 1)1=2 + 9 2 ? 1: p Hence thepconstraint 2 + p >p 0 requires 2 > 1=9 > 9 ? 80 which excludes the alternative

2 < 9 ? 80. Therefore 9 + 80 < 2 is a pnecessary and sucient condition for (41) to have a joint negative solution p. Thus kL k2 = 9 + 80. We will now analyze what happens if the incremental covariance matrix of the system is changed. Let q11 = 1; q12 = q22 = 0 so that the stochastic perturbation term v(t)dw2(t) is absent from (39). In this case the inequalities (40) reduce to

?p ? 1 ? p2= 2 > 0: This inequality has a negative solution p if and only if 2 > 4. Hence kL k = 2.

3 Resolution of the general disturbance attenuation problem

We will study the H 1 type disturbance attenuation problem for stochastic systems of the form :

dx(t) = Ax(t)dt + A0x(t)dw1 (t) + B0 v(t)dw2(t) + B1v(t)dt + B2 u(t)dt z(t) = C1x(t) + D11 v(t) + D12 u(t) y(t) = C2x(t) + D21 v(t);

where (A; A0 ; B0; B1 ; B2) 2 (C1; C2; D11; D12; D21) 2

 K nn  K n`  K n`  K nm K qn  K pn  K ql  K qm  K p` :

K nn

15

(42)

and w1; w2 are as in the previous section. There are two vector valued input variables u; v and two vector valued output variables y; z. v represents an unknown stochastic disturbance signal, u the control, z the vector of the to be controlled variables and y the measurements. As compensator we choose { as usual in H 1-theory { a nite dimensional time-invariant deterministic linear system which is driven by the measurement process y() of  and produces the (random) control values u(t): K : dx^(t) = AK x^(t)dt + BK y(t)dt;

u(t) = CK x^(t) + DK y(t) (43) where (AK ; BK ; CK ; DK ) 2 K n^ n^  K n^ p  K mn^  K mp and the dimension n^  0 is arbitrary. If n^ = 0 the state equation of K vanishes and we obtain u(t) = DK y(t), i.e. a static linear output feedback control law. For arbitrary n^  0, let " # A B K K MK = C D K K The resulting closed loop system is = Acl x(t)dt + A0clx(t)dw1(t) + Bcl0 v(t)dw2(t) + Bclv(t)dt cl : dxz((tt)) = (44) Ccl x(t) + Dcl v(t); where " # " # " # " # x A + B D C B C A 0 B 2 K 2 2 K 0 0 0 0 x = x^ ; Acl = (45) BK C2 AK ; Acl = 0 0 ; Bcl = 0 " # B + B D D 1 2 K 21 Bcl = ; Ccl = [C1 + D12 DK C2 ; D12 CK ]; Dcl = D11 + D12 DK D21 : B D K 21

Suppose (44) is internally stable in the sense of the previous section and the linear operator L cl

: L2w (R + ; L2 ( ; K ` )) ! L2w (R + ; L2( ; K q ));

is de ned by

t  0; v() 2 L2w (R + ; L2 ( ; K ` )): (46) where x(t; v; x0) is the solution of (44) with x(0) = x0 , for every v() 2 L2w (R + ; L2 ( ; K ` ). L cl describes the e ect of the disturbance signal v() on the to be controlled output vector z() of the closed loop system. Given > 0, our aim is to determine whether or not there is a compensator (43) which stabilizes system (42) internally and achieves kL cl k < . Such controllers will be called suboptimal of level . In case of existence we want to know how these compensators MK (L cl v)(t) = Cclx(t; v; 0) + Dcl v(t);

can be constructed. Remark 3.1 In the linear quadratic Gaussian control problem additive noise terms are present in the basic model and one may ask why we have excluded them here. Consider a model with additive white noise in the state and the measurement equations:

dx(t) = Ax(t)dt + A0 x(t)dw1(t) + B0 v(t)dw2(t) + B1v(t)dt + B2 u(t)dt + E1 dw3(t) z(t) = C1 x(t) + D11 v(t) + D12 u(t) dy(t) = C2 x(t)dt + D21 v(t)dt + E2dw3(t); 16

where (E1; E2) 2 K nr  K pr and w3 is a vector of r scalar Wiener processes. Suppose the compensator has the form

dx^(t) = AK x^(t)dt + BK dy(t);

u(t) = CK x^(t):

Then the closed loop system is

dx(t) = Acl x(t)dt + A0cl x(t)dw1(t) + Bcl0 v(t)dw2(t) + Bcl v(t)dt + Ecldw3(t) z(t) = Cclx(t) + Dclv(t); " # " # x E 1 0 0 where x = x^ , Acl ; Acl ; Bcl ; Bcl ; Ccl; Dcl are as in (45) with DK = 0 and Ecl = B E . K 2 In order that the H 1 problem makes sense, the map from v to z (with initial state x(0) = 0)

must be linear or at least map the zero input onto the zero output. For the preceding closed loop system this will only be the case if the additive noise term w3 is completely decoupled from z. In the case where the di usion term is absent in the nominal system equation (A0 = 0) this would require that the kernel of Ccl must contain the smallest Acl -invariant subspace generated by the range of Ecl . Thus the presence of additive white noise would impose an additional very restrictive condition on the controllers. In fact it is our opinion that adding a speci c white noise term is not really appropriate in an H 1 type disturbance attenuation problem. In this framework measurement and state disturbances are modelled by the unknown random process v (so that e.g. measurement noise is represented by the term D21v(t) in the second output equation). We will show that the above disturbance attenuation problem can be solved via the resolution of matrix inequalities. Our approach follows the one developed by Gahinet for the deterministic case [8]. The key tool which makes this possible is the stochastic version of the Bounded Real Lemma derived in the previous section. From deterministic H 1 control theory we will need the following, so called, Projection Lemma. A proof of this lemma can be found in [8]. Lemma 3.2 (Projection Lemma) Suppose N 2 K `m ; M 2 K nm and H 2 Hm (K ). Then the linear matrix inequality H + N  X M + M  XN  0 has a solution X 2 K n` if and only if H is positive de nite on ker N and on ker M . To simplify the presentation the following notations will be used.

A0 =

"

A

0

#

B0 =

"

#

B1 ; C 0 = [C ; 0 ] ; D0 = [0 ; D ] 1 qn^ qn^ 12 12 0

0 0n^n^ ; n^ ` " # " # " # 0 B 0 I 0 2 n ^ n ^  ` I I 0 B = I 0 ; C = C 0 ; D21 = D : n^ 2 21 Then the closed loop matrices can be written as

Acl = A0 + B I MK C I ; Bcl = B 0 + B I MK D210 ; Ccl = C 0 + D120 MK C I ; Dcl = D11 + D120 MK D210 ; A0cl; Bcl0 as in (45):

(47)

In order to save space we will not write out the upper triangle of large Hermitian matrices but use a ? notation. 17

Theorem 3.3 For any system of the form (42) and > 0 the following conditions are equivalent: (i) There exists a compensator (43) of dimension n^ such that the resulting closed loop system (44) is internally stable and kL cl k < . (ii) There exists a Pcl 2 Hn+^n(K ), Pcl  0, such that the matrix Pcl = Pcl is positive de nite on ker U and Pcl = Pcl is positive de nite on ker V , where 2

Pcl =

6 4 2

Pcl = and

6 4

3

(A0) Pcl + Pcl A0 + q11 (A0cl) PclA0cl ? ? (B 0)Pcl + q12 (Bcl0 )PclA0cl

2I` + q22 (Bcl0 )Pcl Bcl0 ? 75 ;

P ?1

0 0

cl

3

C0

2

P ?1 cl

0 0

D11

3

Iq

I` 0 75 Pcl 64 0 I` 0 75 ; 0 0 Iq 0 Iq

0 0

U = [(B I ); 0(^n+m)` ; (D120 )] ; V = [C I ; D210 ; 0(^n+p)q ]:

(48)

Proof: Applying Corollary 2.14 with A = Acl ; A0 = A0cl ; etc. we see that (i) is equivalent to the existence of Pcl  0 such that 2 6 4

(Acl) Pcl + Pcl Acl + q11 (A0cl )Pcl A0cl ? ?  0 2 0  0  0

I` + q22 (Bcl) PclBcl ? (Bcl ) Pcl + q12 (Bcl) PclAcl

Ccl

Dcl

Iq

3 7 5

 0:

Substituting for Acl; Bcl0 etc. the expressions in (47), we obtain that this is equivalent to 2 







3

A0 + BI MK C I Pcl + Pcl A0 +B I MK C I + q11 (A0cl ) Pcl A0cl ? ? 7 6 6 0 ) M  (B I ) Pcl + q12 (B 0 ) Pcl A0 (B 0 ) + (D21

2 I` + q22 (Bcl0 ) Pcl Bcl0 ? 75  0: 4 K cl cl D11 + D120 MK D210 Iq C 0 + D120 MK C I Or, separating the data and the design parameters, 2

3

3

2

(C I ) i h 7 6 6 I 0 Pcl + 4 0`(^n+`) 5 MK [C ; D21 ; 0(^n+p)q ] + 4 (D210 ) 75 MK (B I )Pcl ; 0(^n+`)` ; (D120 )  0: 0q(^n+p) D120

PclB I

That is

(49) Pcl + UPcl MK V + V  MK UPcl  0; where UPcl = [(B I )Pcl ; 0(^n+m)` ; (D120 )] and V is de ned as in (48). Applying the Projection Lemma we conclude that (i) is equivalent to Pcl being positive de nite on ker V and ker UPcl . To complete the proof, note that 3

2

Pcl 0 0 7 6 UPcl = U 4 0 I` 0 5 : 0 0 Iq 18

The characterization in the above theorem is awkward since it involves both Pcl and its inverse. However a simpler form can be obtained by partitioning Pcl. To achieve this the following lemma will be useful, see Lemma 7.5 in [23]. Lemma 3.4 Let n; n^  1. Suppose P 2 Hn+^n(K ) and its inverse P ?1 are partitioned as follows "

and P  0, then

#

"

#

P = NS NQ ; P ?1 = MR M T ; R; S 2 Hn(K );

(50)

S  R?1  0 and rank [R?1 ? S ]  n^:

(51) Conversely, if R; S 2 Hn(K ) are given such that (51) is satis ed then there exists P 2 Hn+^n (K ), P  0 such that P and its inverse can be partitioned as in (50) (with suitable N; Q; M; T ). Theorem 3.5 For any system of the form (42) and > 0 the following conditions are equivalent: (i) There exists a stabilizing compensator (43) of dimension n^ such that kL cl k < . (ii) There exists (R; S ) 2 Hn(K )  Hn(K ) such that S  R?1  0; rank (R?1 ? S )  n^ and  (S ) = 2I` + q22 B0 SB0  0; (52) "

#

"

"

#

#

AR + RA + q11 RA0SA0 R RC1 ? B1 + q12 RA0 SB0  (S )?1 B1 + q12 RA0SB0   0

D11 C1 R I` D11  ] (53) on ker [B2 D12

and

#

"

"

#"

#

SA + AS + q11 A0 SA0 SB1 + q12 A0SB0 ? C1 C1   0 on ker [C D ]: (54) 2 21 D11 D11 B1 S + q12 B0SA0  (S ) Proof: By Theorem 3.3 (i) is equivalent to the existence of Pcl 2 Hn+^n(K ), Pcl  0, such that the matrix Pcl is positive de nite on ker U and Pcl is positive de nite on ker V . If we partition # # " " R M S N ? 1 Pcl = N  Q ; Pcl = M  T ; R; S 2 Hn(K ); we obtain from the preceding lemma that S  R?1  0 and rank [R?1 ? S ]  n^: Let us rst consider the condition that Pcl is positive de nite on ker V . Since by de nition (48) " # 0 I 0 0 n ^ n ^  q V= C 0 D 0 ; 2 21 pq ker V can be represented as 2 V1 0 3 7 6 ker V = Im 664 V0 00 775 2 0 Iq 19

"

#

where VV1 is a basis matrix for ker [C2; D21 ]. Partitioning Pcl accordingly, a straight forward 2 calculation yields

Now

2

SA + AS + q11 A0 SA0 ? ? 6  N A 0 ? Pcl = 664 B  S + q B SA B1 N  (S ) 12 0 0 1 C1 0 D11

?3 ? 777 : ?5 Iq

SA + A S + q11 A0SA0 AN SB1 + q12 A0 SB0 C1 3 2 V1 0 N A 0 N  B1 0 777 666 0 0 V1 0 V2 0 666 B1N  (S ) D11 5 4 V2 0 0 0 0 Iq 4 B1S + q12 B0 SA0 C1 0 D11# Iq " 0# Iq " #" 2  3  S + q11 A SA0 SB1 + q12 A SB0 SA + A V C 1 1 0 0     [V1 V2 ] D 77 6 [V1 V2 ] B1 S + q12 B0SA0 " #  (S ) V2 6 11 7 6 = 6 7 V 4 5 [C1 D11] V1 Iq "

#

2

3 7 7 7 5

2

Using (18) it follows therefore that Pcl is positive de nite on ker V if and only if # " # " SA + A S + q11 A0SA0 SB1 + q12 A0SB0 ? C1 [C D ]  0 on ker [C D ]; 1 11 2 21 B1 S + q12 B0 SA0  (S ) D11 i.e. if and only if (54) is satis ed. The condition that the matrix Pcl is positive de nite on ker U can be analyzed in a similar way. Since by de nition (48) # " 0 I 0 0 n ^ n ^  ` U = B 0 0 D : ker U can be represented by

"

2

2

m`

12

U1 0

3

7 6 ker U = Im 664 00 I0 775 ; ` U2 0

#

where UU1 is a basis for ker [B2 D12 ]. Partitioning Pcl accordingly, we obtain by a straight2 forward calculation that 2

AR+RA+q11 RA0SA0 R ? ? 6 (AM + q RA SA M ) q M  A SA M ? 11 0 0 Pcl = 664 (B + q11 RA0 SB0 )  q B SA M  1 12 0 0 12 0 0

(S ) C1 R C1 M D11

Now "

2

? ? ? Iq

AR+RA+q11 RA0 SA0R ? ? 6       U1 0 0 U2 66 (AM + q11 RA0SA0 M ) q11 M A0 SA0M ? q12 B0 SA0M  (S ) 0 0 I` 0 4 (B1 + q12RA0 SB0) C1 R C1 M D11 #

20

3 7 7 7 5

? ? ? Iq

32 76 76 76 54

U1 0

0 0 0 I` U2 0

3 7 7 7 5

2

=

6 [U1 U2 ] 6 6 6 4

"

AR+RA+q11 RA0SA0 R RC1 C1 R I#q " # "  B1+q12 RA0 SB0 U1 D11 U2

#"

U1 U2

#

[U 

U ]

"

1 2

B1+q12 RA0SB0 D11  (S )

# 3 7 7 7: 7 5

Hence, using again (18), Pcl is positive de nite on ker U if and only if  (S )  0 and (53) hold. Altogether we see that (i) implies (ii). Conversely, suppose (R; S ) 2 Hn(K )  Hn(K ), R  0, S  0 satisfy the conditions in (ii). Applying Lemma 3.4 we obtain that there exist N; M 2 K n` ; Q; T 2 H`(K ) such that "

#

"

#

Pcl := NS NQ  0; Pcl?1 = MR M T :

Now de ne Pcl , Pcl as in Theorem 3.3. We have just proved that (53) and (54) imply that the matrix Pcl is positive de nite on ker U and Pcl is positive de nite on ker V . But this is equivalent to (i). Let opt be the optimal value of our H 1 control problem, i.e.

opt = inf f  0; 9 compensator (43) s.t. (44) is internally stable and kL cl k < g: (55) By the previous theorem opt is the in mum of all  0 for which there exist (R; S ) 2 Hn(K )  Hn(K ) such that (52), (53) and (54) are satis ed. Now the condition rk(R?1 ? S )  n^ is automatically satis ed for n^  n and the conditions (53), (54) do not depend on n^. Therefore we obtain, as a consequence of the previous theorem, that for every > opt there exists a stabilizing controller (43) of dimension  n such that kL cl k < . Remark 3.6 (i) In the deterministic case (A0 = 0; B0 = 0) we have  (S ) = 2 I and so (53) and (54) become # " AR + RA ? B1B1 = 2 RC1 ? B1 D11 = 2  0; on ker [B  D ] (56) 2 12 C1R ? D11 B1= 2 I ? D11 D11 = 2 " # SA + A S ? C1 C1 SB1 ? C1D11  0; on ker [C D ] 2 21 B1S ? D11 C1 2I ? D11 D11

These together with

S  R?1  0;

(57)

rank (R?1 ? S )  n^; are precisely the LMI solvability conditions for the suboptimal H 1 synthesis problem as stated in [8]. (ii) Suppose R, S satisfying the conditions in part (ii) of the previous theorem have been found for a given value of . Then stabilizing compensators which achieve kL cl k < can be constructed just as for deterministic systems [8]. First one constructs Pcl, followed by Pcl and Pcl and then one solves (49) for MK . An explicit construction will be given in the next section for the regular case. (iii) Comparing (53), (54) with (56), (57) we see that the presence of multiplicative state and control dependent noise leads to nonlinear instead of linear matrix inequalities and to a onesided coupling of the \controller" and the \observer" matrix inequalities. Note, however, that the \observer" inequality (54) is still linear and independent of (53) so that it can be considered separately. Its solutions then have to be fed into the controller inequality (53) which is a quadratic matrix inequality in R. 21

We can replace the inequalities (53), (54) by inequalities on the whole space at the sake of introducing scalar parameters. Namely (53) is equivalent to the existence of > 0 such that

"

# ! ! " # AR+RA+q11RA0 SA0R RC1 ? B1+q12RA0SB0  (S )?1 B1+q12RA0SB0 + 2 B2 [B  D ]  0:

C1 R I D11 D11 D12 2 12

(58)

And (54) is equivalent to the existence of > 0 such that "

#

"

SA + A S + q11 A0 SA0 SB1 + q12 A0 SB0 ? C1 B1 S + q12 B0 SA0  (S ) D11

#"

C1 D11

#

+ 2

"

#

C2 [C D ]  0: 2 21 D21

(59) Further we can replace the above inequalities with lower dimensional ones. In fact, applying (18), (58) is equivalent to I + 2 D12D12 ? D11 (S )?1D11  0 and

AR+RA+q11 RA0SA0 R ? (B1+q12 RA0SB0 ) (S )?1(B1+q12 RA0 SB0) + 2B2B2 ?[RC1 ? (B1+q12RA0 SB0) (S )?1D11 + 2B2D12 ][I + 2D12 D12 ? D11  (S )?1D11 ]?1 [RC1 ? (B1+q12RA0 SB0) (S )?1D11 + 2B2D12 ]  0: (60) Similarly we obtain that (59) is equivalent to  (S ) + 2D21 D21 ? D11 D11  0 and SA + AS + q11 A0 SA0 ? C1C1 + 2C2C2 ? [SB1 + q12A0 SB0 ? C1 D11 + 2C2 D21 ] [ (S ) + 2 D21 D21 ? D11 D11 ]?1 [SB1 + q12 A0 SB0 ? C1D11 + 2C2D21 ]  0: (61) Thus

Corollary 3.7 For any system of the form (42) and any > 0 the following conditions are equivalent: (i) There exists a stabilizing compensator (43) of dimension n^ such that kL cl k < . (ii) There exist (R; S ) 2 Hn(K )  Hn(K ), > 0; > 0, such that

S  R?1  0; rk(R?1 ? S )  n^;  (S ) = 2 I + q22 B0SB0  0; I + 2D12 D12 ? D11  (S )?1D11  0;  (S ) + 2D21 D21 ? D11 D11  0 and (60), (61) hold. We conclude this section with a brief discussion of the state feedback case where C2 = In and D21 = 0. Then (54) is equivalent to

 (S ) ? D11 D11 = 2 I + q22 B0SB0 ? D11 D11  0 The following corollary determines what can be achieved by static state feedback:

u(t) = Fx(t); F 2 K mn

(62)

Corollary 3.8 There exists a stabilizing static state feedback controller (62) such that kL cl k < if and only if there exists R 2 Hn(K ), R  0, satisfying  (R?1) ? D11 D11 = 2 I + q22 B0R?1 B0 ? D11 D11  0 (63) 22

and "

#

"

#

"

#

AR+RA+ q11RA0 R?1 A0R RC1 ? B1+q12RA0 R?1 B0  (R?1)?1 B1+q12 RA0 R?1B0  0

D11 C1 R I D11   on ker [B2 D12 ]: (64) Proof: In the static state feedback case we have n^ = 0 and hence (52) implies S = R?1. But with S = R?1 (53) and (54) are equivalent to (64) and (63), respectively. Thus the statement follows from Theorem 3.5.

An interesting question is whether lower levels of can be achieved by employing dynamic state feedback. This has been answered in the negative for the deterministic case A0 = 0; B0 = 0 (see e.g. [26]) and for the special stochastic case A0 = 0; B1 = 0, see Proposition 5.6 in [16]. The following corollary generalizes these results to the general stochastic case where the nominal system's noise w1(t) and the perturbation noise w2(t) are independent. Corollary 3.9 For any system of the form (42) with q12 = 0 and any > 0 the following conditions are equivalent: (i) There exists a stabilizing static state feedback controller (62) such that kL cl k < . (ii) There exists a stabilizing dynamic state feedback controller of dimension n^  0 (i.e. (43) with y(t) = x(t)) such that kL cl k < .

Proof: Only the implication (ii) ) (i) needs to be proved. Assume (ii), then by Theorem 3.5 there exist (R; S ) 2 Hn(K )  Hn(K ), such that S  R?1  0 and the following inequalities are satis ed: " # # " # " AR + RA + q11RA0 SA0R RC1 ? B1  (S )?1 B1  0 on ker [B  D ] 2 12 D11 D11  (S ) ? D11 D11 = 2 I + q22 B0SB0 ? D11 D11  0 But since S  R?1  0 it follows that 0   (S )   (R?1 ) and q11 RA0 SA0R  q11 RA0R?1 A0R. Hence the previous two inequalities hold with S replaced by R?1 . There-

C1R

I

fore (i) follows from Corollary 3.8.

4 The regular case In this section we consider the so-called regular case and make the usual assumptions [5]:

D11 = 0; D12 D12 = I; D21D21 = I; D12 C1 = 0; D21 B1 = 0:

(65)

Then (60) becomes

AR + RA + q11 RA0SA0 R ?(B1 + q12 RA0 SB0) (S )?1(B1 + q12 RA0 SB0) + 2B2B2 ?(RC1 + 2B2D12 )(I + 2D12 D12 )?1 (RC1 + 2B2 D12 )  0: By (65)

(I + 2D12 D12 )?1 D12 = (1 + 2 )?1D12 ; (I + 2D12 D12 )?1 C1 = C1 23

and hence

RC1(I + 2D12 D12 )?1(RC1 + 2B2D12 ) = RC1 C1R:

Therefore (60) is equivalent to

AR + RA + q11 RA0 SA0R? (B1 + q12 RA0 SB0) (S )?1(B1 + q12 RA0 SB0) + 2B2B2 ?RC1 C1R ? 4B2 D12 (I + 2D12 D12 )?1D12 B2  0: Now D12 (I + 2D12D12 )?1D12 = (1 + 2)?1 I , so (60) is equivalent to

AR + RA + q11 RA0SA0 R ? (B1 + q12 RA0 SB0) (S )?1(B1 + q12 RA0 SB0) + 2(1 + 2)?1B2 B2 ? RC1C1 R  0:

(66)

(61) becomes

SA+ A S + q11 A0SA0 ? C1 C1 + 2C2C2 ?(SB1 + q12 A0SB0 + 2C2D21 )( (S ) + 2D21 D21 )?1(SB1 + q12 A0 SB0 + 2C2 D21 )  0: Suppose in addition that D21B0 = 0, then after similar calculation to the ones above we obtain the equivalent inequality

SA + A S + q11 A0SA0 ? C1C1 + 2 2 ( 2 + 2)?1 C2C2 ?(SB1 + q12 A0 SB0) (S )?1(SB1 + q12 A0SB0 )  0: (67) If (66), (67) are satis ed for some given ; > 0 they are also satis ed for all larger values. Taking limits as ! 1; ! 1 we see that (66) and (67) are equivalent, respectively, to AR + RA + q11RA0 SA0R + B2 B2 ? RC1C1 R ? (B1 + q12 RA0SB0) (S )?1 (B1 + q12 RA0SB0)  0 SA + A S + q11A0 SA0 ? C1C1 + 2C2C2 ? (SB1 + q12A0SB0 ) (S )?1 (SB1 + q12A0 SB0)  0: (68) Setting R?1 = P , the rst inequality is equivalent to PA+A P +q11A0 SA0 +PB2B2P ?C1 C1 ?(PB1 +q12 A0SB0 ) (S )?1(PB1 +q12 A0 SB0)  0: (69) Altogether we have derived the following consequence of Corollary 3.7. Proposition 4.1 Suppose the regularity conditions (65) and D21 B0 = 0. Then the following statements are equivalent: (i) There exists a stabilizing compensator (43) of dimension n^ such that kL cl k < . (ii) There exist P; S 2 Hn (K ) such that S  P  0; rank (P ? S )  n^ ; 2 I + q22 B0SB0  0 and (68), (69) hold. We now show how to explicitly calculate a compensator in the special case that n^ = n. Suppose that condition (ii) of the previous proposition is satis ed with n^ = n. Then there exist P; S 2 Hn(K ) such that S  P  0, 2 I + q22 B0SB0  0 and S  0; P  0 where S = SA + A S + q11 A0 SA0 ? C1 C1 + 2 C2 C2 ? (SB1 + q12 A0 SB0 ) (S )?1 (SB1 + q12 A0 SB0 ) ; P = PA + A P + q11 A0 SA0 + PB2 B2 P ? C1 C1 ? (PB1 + q12 A0 SB0 ) (S )?1 (PB1 + q12 A0 SB0 ) : 24

De ne then

BK = 2(P ? S )?1C2 ; CK = B2 P; DK = 0; "

#

"

#

(70) "

#

 Acl = 2(P ? SA)?1 C C B2AB2 P ; A0cl = A00 00 ; Bcl0 = B00 ; 2 2 # K " Bcl = 2(P ? SB)1?1 C D ; Ccl = [C1 D12 B2 P ]; Dcl = 0: 2 21 So condition (ii) of Theorem 2.8 is equivalent to the existence of Pcl  0 such that  := PclAcl + Acl Pcl + q11 A0cl PclA0cl ? Ccl Ccl ?(PclBcl + q12 A0cl Pcl Bcl0 )( 2I + q22 Bcl0  PclBcl0 )?1(Pcl Bcl + q12A0cl  PclBcl0 )  0: (71) " # S N Choosing Pcl = N  Q , with N = ?Q = (P ? S ) then Pcl  0 and partitioning  = " #

11 12 we obtain from (71) 12 22 11 = SA + A S + q11 A0 SA0 + 2 2 C2 C2 ? C1 C1 ?(SB1 + q12A0 SB0 + 2C2D21 ) (S )?1 (SB1 + q12A0 SB0 + 2 C2D21 ) ;

12 = SB2 B2P + NAK + A N ? 2 C2 C2 ? (SB1 + q12 A0 SB0 + 2 C2 D21 ) (S )?1 (NB1 ? 2 C2D21 ) ; 22 = NB2 B2 P + PB2 B2 N ? NAK ? AK N ? PB2 B2 P ? (NB1 ? 2 C2 D21 ) (S )?1 (NB1 ? 2 C2 D21 ) :

Now 11 = S , and 12 simpli es to 12 = SB2B2 P + NAK + AN ? (SB1 + q12 A0 SB0) (S )?1B1 N: Thus choosing h i (72) AK = ?N ?1 SB2 B2P + AN ? (SB1 + q12 A0 SB0) (S )?1B1N + S ; we get 12 = ?S . Finally 22 simpli es to 22 = NB2 B2P + PB2B2 N ? NAK ? AK N ? PB2B2 P ? 2 C2C2 ? NB1  (S )?1B1N: " #  ?  S S Substituting for AK , we get 22 = P + S . Hence  = ?  +  and since S  S P S 0; P  0 it follows that   0, i.e. the above Pcl satis es (71). We conclude from Theorem 2.8 that the closed loop system is stable and kL cl k < . Remark 4.2 (i) Using (68) and (69) one can show that AK = A ? BK C2 + B2CK ? B1  (S )?1(B1 P + q12 B0 SA0) ? N ?1 P : The rst three terms are familiar from pole-placement based deterministic dynamic output feedback stabilization. If B0 = 0 the fourth term is ?2B1 B1P and this is familiar from deterministic H 1 control. In this case the last term (N ?1P ) is zero for the so called \central controller". (ii) We have shown [16] that in a stochastic setting it is not, in general, possible to replace both inequalities (68) and (69) by equalities in Proposition 4.1. 25

5 Stability radii In this section we turn to a singular control problem and discuss the application of our general results to stability radii. First we show how the results of Section 2 can be used to derive a lower bound for stability radii of stochastic systems. Then we use the results of Section 3 to show how to increase the radii via feedback. The corresponding H 1 type control problem is singular because all three feedthrough matrices D11 ; D12 ; D21 are zero in this case. For the analysis problem we adopt the notation of Section 2 and for the synthesis problem that of Section 3. Suppose that a stable linear stochastic model

dx(t) = Ax(t)dt + A0x(t)dw1 (t)

(73)

dx(t) = (A + B C )x(t)dt + A0x(t)dw1 (t) + B0Cx(t)dw2 (t)

(74)

0 : is perturbed to  : where

(A; A0; B0; B; C ) 2 K nn  K nn  K n`  K n`  K qn : The Wiener processes wi; i = 1; 2 are as in Section 2 and  2 D(K ) = K `q represents an unknown disturbance matrix. We view the term B C in (74) as a parameter perturbation of the nominal system matrix A and B0 Cx(t)dw2(t) as a stochastic perturbation (multiplicative noise). If w1 = w2 = w we can interpret B0 C as a parameter perturbation of A0 and write (74) in the following way:

x_ (t) = (A + B C )x(t) + (A0 + B0C )x(t)dw(t)

(75)

The fact that the same  is used in " both # perturbations is not really a restriction since if we set  0 B0 = [B00 0]; B = [0 B 1 ] and  =  , we obtain 1

x_ (t) = (A + B 11 C )x(t) + (A0 + B00 0 C )x(t)dw(t): We will take this up again in Example 5.8. The size of each  2 D(K ) is measured by its operator norm (with respect to the Euclidean norms on K q ; K ` ). Our aim is to determine which bounds  > 0 on the size of the perturbations ensure the stability of the perturbed system (74). The maximum  for which all the perturbed systems (74) with kk <  are stable is called the stability radius of (73). De nition 5.1 The stability radius of the stochastic system (73) with respect to perturbations as in (74) is

rKw = rKw (A; A0; B; B0 ; C ) = inf fkk;  2 D(K ); (74) is not stableg:

(76)

In particular, rKw = 1 if there does not exist  2 D(K ) such that (74) is not stable. The stability radius is a quantitative index of robust stability of the system 0 (73) under perturbations of the form 0 ;  (74). Since robust stability is a basic requirement for every control system with uncertain parameters it is of considerable interest to have computable formulae or good estimates for the stability radii of a given system. 26

To connect the stability radius problem with the analysis in Section 2 observe that the perturbed system (74) is identical with the closed loop system obtained from dx(t) = Ax(t)dt + A0x(t)dw1 (t) + B0v(t)dw2(t) + Bv(t)dt (77) z(t) = Cx(t); by setting v(t) = z(t). The open loop gain of this closed loop system is given by kL k where L : v () 7! Cx(; v; 0) is the perturbation operator associated with (77) (see De nition 2.3). It is therefore reasonable to expect that the norm of the perturbation operator plays a crucial role in the determination of the stability radius. Remark 5.2 (i) If the data A; A0; B; B0 ; C are real, two stability radii are obtained depending on whether one chooses K = C (complex perturbations) or K = R (real perturbations) in (76). In a deterministic framework the real and the complex stability radii are, in general, distinct, see [14]. The complex stability radius is equal to kL k?1 [12], whereas the real stability radius is characterized via second order singular values [24]. (ii) A stability radius with respect to time-varying and/or nonlinear perturbations can be de ned via (76) by extending the perturbation class D(K ) appropriately. For example let Dtn(K ) denote the set of all Lebesgue measurable  : R +  K q 7! K ` which are Lipschitz bounded and linearly bounded in y, that is for all T > 0 there exists L=L(T) such that k(t; y) ? (t; y^)k  Lky ? y^k for all y; y^ 2 K q ; t 2 [ 0; T ] and there exists K > 0 such that k(t; y)k  K kyk for all t 2 R + ; y 2 K q : (78) The size of  2 Dtn (K ) is measured by the smallest K for which (78) holds. The stability radius of (73) with respect to time-varying nonlinear perturbations of the form dx(t) = (Ax(t) + B (t; Cx(t)))dt + A0 x(t)dw1(t) + B0(t; Cx(t))dw2 (t) (79) is then de ned by (76) with (74) replaced by (79) and D(K ) replaced by Dtn(K ). Here the nonlinear system (79) is said to be stable (recall De nition 2.1) if the solutions x (; x0) of (79) satisfy Z 1 Ekx(t; x0 )k2dt  c kx0 k2; x0 2 K n 0 for some suitable constant c. In the deterministic context (A0 = 0; B0 = 0) it is known [14] that the complex stability radius is not changed by such an extension of the perturbation class, whereas the real stability radius is. In the special case where the nominal model (73) is deterministic and the perturbations are purely stochastic, i.e. A0 = 0; B = 0, it has been shown in [2] that the real and the complex stability radii coincide and are equal to the inverse of the norm of the perturbation operator (kL k?1 ) if nonlinear disturbances are considered. In this case it is also possible to analyze the e ect of blockdiagonal perturbations where the single stochastic PN perturbation term B0 Cx(t)dw2 (t) is replaced by a sum of the form i=1 B0i i C ix(t)dw2i (t). In the deterministic context the analysis of blockdiagonal perturbations is the object of -analysis. In [16] it was shown that in the case of purely stochastic perturbations of a deterministic system the real and complex radii coincide and although they are not equal to kL k?1 , they are equal to the inverse of the norm of a suitably scaled perturbation operator. Analogous results are not available for deterministic blockdiagonal perturbations. 27

Proposition 5.3 Suppose that (73) is stable and  2 Dtn(K ) is a time-varying nonlinearity satisfying kk = supfk(t; y)k=kyk ; t  0; y 2 K q ; y 6= 0g < kL k?1 where L is the perturbation operator associated with the data (A; A0 ; B0 ; B; C; 0) (see De nition 2.3). Then the perturbed system (79) is stable. In particular,

rKw (A; A0; B; B0 ; C )  kL k?1

(80)

Proof: Since  is Lipschitz bounded and linearly bounded, for every x0 2 K n ; T > 0, there exists a unique solution x() = x(; x0) 2 L2w ([0; T ]; L2( ; K n )) of (79) satisfying x (0) = x0 with bounded second moments [19]. x() is a continuous nonanticipative stochastic process on R+

satisfying the Ito integral equation

" # Z t Z t w ( s ) 1 0 x (t) = x + 0 (Ax(s) + B (s; Cx(s)))ds + 0 [A0 x (s) B0(s; Cx(s))] d w (s) ; t  0: 2 So x() satis es (7) with v() = v() = (; Cx()) 2 L2w ([0; T ]; L2 ( ; K ` )) for every T > 0. Since kk < kL k?1 there exists > kL k such that kk < 1. Applying Theorem 2.8 to (74) there are  > 0 and P = P   0 satifying M (P )  2I . By Lemma 2.4 we obtain, for every x0 2 K n and T > 0, Z

T JT 2 (x0 ; v) = hx0; Px0i ? Ehx(T ); Px(T )i + 0 E

*"

#

"

x (t) ; M (P ) x(t) v (t) v(t)

#+!

dt:

Substituting (; Cx()) for v() and making use of the de nition (12) and the inequality M (P )  2I we obtain Z Th i 0 0 hx ; Px i ? Ehx(T ); Px(T )i  0 2 Ek(t; Cx(t)k2 ? EkCx(t)k2 ? 2 Ekx(t)k2 dt: Now 2Ek(t; Cx(t))k2  2 kk2EkCx(t)k2  EkCx(t)k2 and ?P  I for some  > 0.

Hence

Z

T Ekx(T )k2  ?Ehx (T ); Px(T )i  kP k kx0k2 ? 0 2Ekx(t)k2dt;

and it follows that i.e. (79) is stable.

Z 1

0

T > 0;

Ekx(t)k2dt  kP k kx0k2=2;

We illustrate the above result by considering the same scalar stochastic system as in Example 2.15. We will see that in this simple example the estimate (80) is tight. Example 5.4 Consider dx(t) = ?x(t)dt + x(t)dw1 (t) + v(t)dw2(t) + v(t)dt; z(t) = x(t) (81) and assume rst that q11 = 1; q12 = q22 = 0 so that the stochastic perturbation term v(t)dw2(t) is absent from (81). We have shown in Example 2.15 that kL k = 2 in this case. The corresponding perturbed equation (74) takes the form dx(t) = ?(1 ? )x(t)dt + x(t)dw1(t): 28

By (8) this stochastic equation is stable if and only if there exists p < 0 such that ?(2 ?  ?  )p + p > 0; i.e. ? (2 ?  ?  ) + 1 < 0: Hence  = 1=2 is a destabilizing real disturbance and by Proposition 5.3 there is no smaller disturbance  2 C which destabilizes. So rRw = rCw = kL k?1 = 1=2. Now supposep w1 = w2 = w and q11 = q12 = q22 = 1. We have shown in Example 2.15 that kL k2 = 9 + 80. The perturbed model takes the form dx(t) = ?(1 ? )x(t)dt + (1 + )x(t)dw(t): By (8) this stochastic equation is stable if and only if there exists p < 0 such that ?(2 ?  ?  ) p + j1 + j2 p > 0 i.e. ? (2 ?  ?  ) + j1 + j2 < 0: p C violating this A short calculation shows that p = 5 ? 2 is the smallest q disturbance q 2 p p p w w ? 1 condition. Hence rR = rC = 5 ? 2. But kL k = 1= 9 + 80 = 9 ? 80 = 5 ? 2. Therefore in this case we again have rRw = rCw = kL k?1 . As was to be expected the presence of the stochastic disturbance term x(t)dw(t) e ectively decreases the stability radius of the system. This is not necessarily so if there is more than one disturbance parameter, i.e. maxf`; qg > 1. We now turn to the synthesis problem. The perturbed closed loop equation obtained by setting v = z in (44) is dx(t) = (Acl + Bcl Ccl)x(t))dt + (A0cldw1(t) + Bcl0 Ccl dw2(t))x(t): (82) As an immediate corollary of Proposition 5.3 and Theorem 3.3 we have Corollary 5.5 Let opt be de ned by (55). Then for any > opt, there exists a stabilizing compensator (43) such that the corresponding closed loop system has a stability radius rKw (Acl ; A0cl; Bcl; Bcl0 ; Ccl) > ?1. In the following examples we indicate how the synthesis problem may be solved under simplifying assumptions. We rst consider two system classes (4) for which the estimate (80) is tight. Example 5.6 : A0 = 0; B0 = 0.{ For this deterministic case we have shown [12] that rCw = kL cl k?1, but in general rRw > rCw [14]. (60) and (61) are equivalent to R = AR + RA ? RC1 C1R ? B1 B1= 2 + 2B2B2  0: (83) S = SA + AS ? C1C1 ? SB1 B1S= 2 + 2C2C2  0: (84) Let ropt = sup frCw (Acl ; 0; Bcl; 0; Ccl) ; 9 compensator MK s. t. Acl is stableg: ?1 . For any > 0, ?1 < r , using the same procedure as that in Section Then we have ropt = opt opt 4, it is easy to verify that provided there exist R; S 2 Hn(K ); ; > 0 satisfying S  R?1  0 and (83), (84), the following compensator of order n achieves rCw > ?1: BK = 2(R?1 ? S )?1C2 ; CK = 2B2 R?1; DK = 0; AK = A ? BK C2 + B2CK ? B1 B1R?1 = 2 ? (I ? RS )?1R R?1 : where R is de ned by (83). 29

Example 5.7 : A0 = 0; B1 = 0. For this case we have shown that if nonlinear perturbations  are allowed then rRw = rCw = kL cl k?1 [16]. (53) and (54) are equivalent to AR + RA ? RC1 C1R  0 on ker B2; (85) SA + A S ? C1C1  0 on ker C2;  (S ) = 2 I + q22 B0SB0  0: (86) Let

ropt = sup frCw (Acl ; 0; 0; Bcl0 ; Ccl) ; 9a compensator MK s. t. Acl is stableg: ?1 . And for any > 0, ?1 < r , we can use (85) and (86) together Then again we have ropt = opt opt with (52) to obtain a compensator which achieves rCw > ?1. In fact in [16] we have shown that (by scaling) it is possible to explicitly construct such compensators for more general perturbation structures where B0 C1x(t)dw2 (t) is replaced by a sum of the form PNi=1 B0i iC1i x(t)dw2i (t), and the w2i are independent Wiener processes.

Finally we consider an example where we do not know whether (80) is tight or not. 0 1 0 n`1 ; B 1 2 K n(`?`1 ) . For Example 1 " # 5.8 : A0 = 0; B0 = [ B0 0]; B1 = [ 0 B1 ], where B0 2 K v v = v0 , (42) has the form 1

dx(t) = Ax(t)dt + B00v0 (t)dw2(t) + B11 v1(t)dt + B2 u(t)dt z(t) = C1x(t) y(t) = C2x(t): Moreover if

"

then the perturbed equation is

#

"

#

v0 = z = 0 C x v1 1 1

dx(t) = (A + B111 C1)x(t)dt + B000 C1x(t)dw2 (t) + +B2u(t)dt: Because of the presence of both stochastic and deterministic perturbations we do not know whether it is equal to kL cl k?1 or whether this is just a lower bound. Since B1B0 = 0 we have B1 ( 2 I + q22 B0 SB0)?1 B1 = B1 B1= 2 and hence (60) and (61) are equivalent to R = AR + RA ? RC1 C1R ? B1 B1= 2 + 2B2B2  0; (87) S = SA + AS ? C1C1 ? SB1 B1S= 2 + 2C2C2  0: (88) For > 0, provided there exist R; S 2 Hn(K ); ; > 0 satisfying S  R?1  0; 2I + q22 B0SB0  0 and (87), (88), the same compensator as that given in Example 5.6 achieves rCw > ?1 .

30

6 Concluding remarks

We have posed and solved an H 1-type problem were both stochastic and deterministic perturbations are present. In opening up this eld, which appears to be fruitful, we think that it would be interesting to pursue research in the following problems.  Since our theory includes both the cases where w1, w2 are independent and w1 = w2 , we think we have laid the foundation for considering the stochastic multiperturbation H 1 problem

dx(t) =

N N N X X X i i i i i Ax(t)dt + A x(t)dw (t) + B v (t)dw (t) + B i vi(t)dt + B

z(t) = C1x(t) + y(t) = C2x(t) +

i=1 N X i=1 N X i=1

0

i=1

D11i vi(t) + D12u(t)

0

i=1

1

2 u(t)dt

(89)

D21i vi(t);

where wi are independent Wiener processes.  In Proposition 5.3 we have obtained the estimate rKw (A; A0; B; B0 ; C )  kL k?1 for the stochastic stability radii. It would be interesting to know under what conditions equality holds. To do this it is necessary to construct a destabilizing perturbation with norm as close as we like to kL k?1 .  A stability radius can be associated with the above multiperturbation problem (89) in a number of di erent ways. For example if all the D's are zero and vi = i z = iC1x, we get the so called full block case, whereas if vi = i zi = iC1i x we get the blockdiagonal case. In both cases kL cl k?1 will be a lower bound for the radii and for full block perturbations it may well be tight. But we cannot expect this for blockdiagonal perturbations since in a deterministic setting this is a  problem. The estimate can be improved by scaling B0i 7! iB0i ; B1i 7! iB1i ; C1i 7! i?1C1i ; i > 0 which does not change the radius but does change the corresponding kL cl k. And we have shown [16] that for suitable i this estimate is tight for purely stochastic multiperturbations of a deterministic system. It would be interesting to know under what conditions this is true when deterministic perturbations are also present.

References [1] L. Arnold. Stochastic Di erential Equations: Theory and Applications. J. Wiley, New York, 1974. [2] A. El Bouhtouri and A. J. Pritchard. Stability radii of linear systems with respect to stochastic perturbations. Systems & Control Letters 19: 29{33, 1992. [3] A. El Bouhtouri and A. J. Pritchard. A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations. Systems & Control Letters 21: 475{484, 1993. 31

[4] G. Da Prato and J. Zabczyk. Stochastic Equations in In nite Dimensions. Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1992. [5] J. Doyle, K. Glover, P. P. Khargonekar, and B. Francis. State space solutions to standard H2 and H 1 control problems. IEEE Transactions on Automatic Control, AC-34: 831-847, 1989. [6] L. El Ghaoui. State-feedback control of systems with multiplicative noise via linear matrix inequalities. Systems and Control Letters 24: 223-228, 1995. [7] A. Friedman. Stochastic Di erential Equations and Applications. Probability and Mathematical Statistics 28, Academic Press, 1975. [8] P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H 1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994. [9] M. Green and D. J. N. Limebeer. Linear Robust Control. Prentice-Hall, Englewood Cli s, 1995. [10] R. Z. Has'minskii. Stochastic stability of di erential equations. Sijtho & Noordho , Alphen aan den Rijn, 1980 (translation of the Russian edition, Moscow, Nauka 1969). [11] U. G. Haussmann. Optimal stationary control with state and control dependent noise. SIAM J. Control and Optimization 9:184-198, 1971. [12] D. Hinrichsen and A. J. Pritchard. Stability radius for structured perturbations and the algebraic Riccati equation. Systems & Control Letters 8: 105{113, 1986. [13] D. Hinrichsen and A. J. Pritchard. Riccati equation approach to maximizing the complex stability radius by state feedback. Int. J. Control 52: 769-794, 1990. [14] D. Hinrichsen and A. J. Pritchard. Real and complex stability radii: a survey. In D. Hinrichsen and B. Martensson, editors, Control of Uncertain Systems, volume 6 of Progress in System and Control Theory, pages 119{162, Basel, 1990. Birkhauser. [15] D. Hinrichsen and A. J. Pritchard. Stability margins for systems with deterministic and stochastic uncertainty. Proc. 33rd IEEE Conf. Decision and Control, Florida 1994. [16] D. Hinrichsen and A. J. Pritchard. Stability radii of systems with stochastic uncertainty and their optimization by output feedback. SIAM J. Control and Optimization 34: 1972-1998, 1996. [17] T. Iwasaki and R. E. Skelton. All controllers for the general H 1 control problem: LMI existence conditions and state space formulas. Automatica 30: 1307-1317, 1994. [18] N. V. Krylov. Introduction to the Theory of Di usion Processes. Translations of Mathematical Monographs 142. AMS Providence, Rhode Island 1995 [19] N. V. Krylov. Controlled Di usion Processes. Springer Verlag 1980. [20] P. J. McLane. Optimal Stochastic control of linear systems with state and control-dependent disturbances. IEEE Transactions on Automatic Control, AC-16: 292-299, 1971. 32

[21] T. Morozan. Stability radii for some stochastic di erential equations. Stochastics and Stochastics Reports 54: 281-291, 1995. [22] T. Morozan. Parametrized Riccati equations associated to input-output operators for timevarying stochastic di erential equations with state-dependent noise. Institutul de Matematica al Academiei Romane, Preprint No. 37, Bucarest 1995. [23] A. Packard, P. Zhou, P. Pandey, G. Becker. A collection of robust control problems leading to LMIs. Proc. 30th IEEE Conf. Decision and Control, 1245-1250, Brighton 1991. [24] L. Qiu, B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, and J. C. Doyle. On the real structured stability radius. Proc. 12th IFAC World Congress. 71{78, IFAC 1993. [25] C. Scherer. H1 optimization without assumptions on nite or in nite zeros. SIAM J. Control and Optimization 30: 123-142, 1992. [26] A. A. Stoorvogel. The H1 Control Problem. Prentice-Hall, New York, 1992. [27] J. L. Willems and J. C. Willems. Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica 12: 277-283, 1976. [28] J. Willems and J. C. Willems. Robust stabilization of uncertain systems. SIAM J. Control and Optimization 21: 352{374, 1983. [29] W. M. Wonham. Optimal stationary control of a linear system with state dependent noise. SIAM J. Control and Optimization 5: 486-500, 1967. [30] W. M. Wonham. Random di erential equations in control theory. Probabilistic Methods in Applied Mathematics, vol. 2, A.T. Bharucha-Reid, ed., Academic Press, New York, 1970.

33