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SIAM J. CONTROL AND OPTIMIZATION. Vol. 29, No. 6, pp. 1373-1393, November 1991

()

1991 Society for Industrial and Applied Mathematics OO6

H CONTROL WITH PRAMOD P.

KHARGONEKARt,

TRANSIENTS*

KRISHAN M. NAGPAL$, AND KAMESHWAR R.

POOLLA Abstract. In Ho (or uniformly optimal) control problems, it is usually assumed that the system initial conditions are zero. In this paper, an Ho-like control problem that incorporates uncertainty in initial conditions is formulated. This is done by defining a worst-case performance measure. Both finite and infinite horizon problems are considered. Necessary and sufficient conditions are derived for the existence of controllers that yield a closed-loop system for which the above-mentioned performance measure is less than a prespecified value. State-space formulae for the controllers are also presented.

Key words. Ho control theory, algebraic and differential Riccati equations, optimal control

AMS(MOS) subject

classifications. 93B50, 93C35, 93C05, 49A40

1. Introduction. Zames introduced the problem of H optimal control in his pioneering paper [18]. The essential idea was to design a controller to optimize the closed-loop system performance for the worst exogenous input. The expository book by Francis [6] presents a lucid account of the early developments in Ho control theory. A significant new development in H control theory in the last two years has been the introduction of state-space methods. This has led to a rather transparent solution to the standard problem of Ho control theory. See Doyle et al. [5], Khargonelmr [9], and the references cited there for the state-space approach to H control theory. The H norm of a system can be defined in many different but equivalent ways. However, it is always (at least, implicitly) assumed that the initial condition of the system is zero. Thus, in most of the H control theory literature, it is assumed that the plant initial conditions are zero. There are a few exceptions. For example, Nagpal and Khargonekar [12] have considered an H type of estimation problem with nonzero initial conditions. In this paper, our principal aim is to extend the basic ideas and the recent results from the state-space approach to H control theory taking initial conditions into account explicitly. We consider this as the key conceptual contribution of the present paper. In recent independent parallel work, Didinsky and Basar [4] consider a minimax design problem for discrete-time systems with nonzero initial states. However, their problem formulation, as well as the results, bear little resemblance to our work. In 2 we formulate an H-type optimal control problem that incorporates initial conditions. This is done by introducing a new performance measure that is essentially the worst-case norm of the regulated outputs over all exogenous signals and initial conditions. We define this performance measure for both finite time and infinite time Received by the editors March 5, 1990; accepted for publication (in revised form) September 10, 1990. This research was supported in part by National Science Foundation grants ECS-9096109 and ECS-8957461, U.S. Air Force Office of Scientific Research contract AFOSR-90-0053, U.S. Army Research Office grant DAAL03-90-G-0008, and grants from Honeywell, McDonnell-Douglas, and Rockwell International. The results of this paper were presented at the Dutch System and Control Theory Network summer school on Ho Control and Robust Stabilization, Schiermonnikoog, the Netherlands, June 1990, and at the 1990 IEEE Conference on Decision and Control, December 1990. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109-2122. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109-2122. Present address, Coordinated Sciences Laboratory, University of Illinois, Urbana, Illinois, 61801. Coordinated Sciences Laboratory, University of Illinois, Urbana, Illinois, 61801. 1373

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P.P. KHARGONEKAR, K. M. NAGPAL, AND K. R. POOLLA

horizons. For finite horizon problems we also allow for a penalty on the terminal state. This enables us to incorporate trade-offs between the norm of the controlled output and the size of the terminal state. In 3 we state the main results of this paper. Here we present necessary and sufficient conditions for the existence of a linear controller such that the above-mentioned performance measure of the closed-loop system is less than a prespecified number. These necessary and sufficient conditions are given in terms of existence and properties of solutions to certain algebraic and differential Riccati equations. In the event that these necessary conditions are met, we provide explicit formulae for controllers that yield the prespecified performance. Our results in this paper may be regarded as the H analogue of the nonstationary linear quadratic Gaussian (LQG) control theory results. The results for infinite time horizon problems are natural generalizations of the results of Doyle et al. [5] for the situation of nonzero initial states, while those for finite time horizon problems are natural generalizations of the results of Tadmor [15] and Limebeer et al. [10]. In 4 we give proofs of the main results. In this paper, we restrict our attention to finite-dimensional linear time-invariant plants. It should be noted that even though the plant is linear time-invariant, it is necessary to consider time-varying controllers since the natural (central) solutions even for linear time-invariant plants happen to be linear time-varying. This situation is analogous to the the finite horizon linear-quadratic regulator and Kalman filtering theory. Recall that for finite horizon linear-quadratic optimal control problems for linear time-invariant plants, the optimal controller turns out to be linear time-varying. Similarly, in the Kalman filtering problem, if the initial state covariance does not match the steady-state covariance, the Kalman filter is also linear time-varying. It is in this sense in which we regard our results in this paper as the Ho analogue of the nonstationary LQG control theory results. Results for finite horizon problems can be trivially extended to linear time-varying plants. Extensions to time-varying plants of the results for the infinite time horizon case are technically much more difficult but are possible along the lines of recent work by Tadmor [16] and Ravi, Nagpal, and Khargonekar [13]. These extensions are left for future research.

2. Worst-case H-type performance measures with nonzero initial conditions. Consider the finite-dimensional linear time-invariant system E"

(1)

-

dx Ax + Blw + B2u, dt z Clx + Dw + D2u, y C2x + D21w D22u.

Here x, w, u, z, and y denote, respectively, the state, exogenous input, control input, regulated output, and measured output. It is assumed that the initial state x(0) is possibly nonzero and unknown. The control problem that we wish to address is that of designing a controller that internally stabilizes the closed-loop system and reduces z uniformly for all w and x(0). More specifically, let Kbe a finite-dimensional linear (possibly time-varying) controller given by the system equations

d

(2)

d- F(t)(t) + G(t)y(t), u(t) H(t)(t) + J(t)y(t).

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H CONTROL WITH TRANSIENTS

Throughout this paper, we assume that the controllers are linear, finite-dimensional, and time-varying with continuous and bounded state-space realizations. (A time function f(t) is called bounded if and only if there exists M > 0 such that for all t _> 0 [If(t)l[ < M.) Let Ec denote the resulting closed-loop system. The closed-loop system is called well posed if and only if (I- JD22) -1 is bounded. The closed-loop system is called internally stable if and only if it is well posed and the unforced closedloop system (i.e., w 0,) with states x, is exponentially stable. In finite horizon problems, a controller is called admissible if and only if it yields a well-posed feedback system. In infinite horizon problems, a controller is called admissible if and only if it yields an internally stable feedback system. For a fixed time T > 0, a symmetric positive-semidefinite matrix S, and a symmetric positive-definite matrix R, define the worst-case closed-loop performace measure as

0, and the supremum is taken over all x(0) Xo e Rn, w e L2[0, T], I[wll+ In this definition, T is allowed to be c in which case S O. 0 and x’oRxo the supremum on the right-hand side is taken over all w E L2[0, c). Here IlfllT where (0)

The performance measure J(Ect, R, S, T) can be regarded as the induced norm of the linear operator generated by the closed-loop system Ect, which maps the pair (x(0), w) to (x(T), z). More explictly, consider the linear operator

F" Rn + L2[0, T]

+

Rn + L2[0, T]" (x(0), w)

(x(T), z).

Define the inner product on the domain of F to be

XlRX2 nt- (wl, W2)L2[O,T] and the inner product on the co-domain as :=

+

1,

where L[0,T] is the usual inner product in L2[0, T]. These inner products induce corresponding (semi)norms in the domain and the range of F. Then the performance measure J is the induced operator norm of F. A similar interpretation can be given for the infinite horizon case. We can now state the control problems considered in this paper. Given a real number > 0: (i) Infinite Horizon Problem: Does there exist, and if so find, an internally stabilizing bounded linear time-varying controller such that

J(Ect, R, O, ) < "? (ii) Finite Horizon Problem: Does there exist, and if so find, time-varying controller such that

J(Ect, R, S, T)
_ R2 then J(Ea, RI, S,T) _ $2, then J(Ea, R, St, T) >_ J(Ea, R, $2, T). (c) If T >_ T2, then J(Ea, R, 0, T) >_ J(a, R, 0, T2). (d) Suppose that Ea is time-invariant and asymptotically stable, then lim

p T-- o

where

Tzw

:=

J(Ea, pI, 0, T)= [ITzw[[ := sup{(Tzw(S)): Re(s) _> 0},

C(sI A)-B denotes the system transfer function and, moreover,

(6) Thus, J(Ecl, R, S, T) is a generalization of the more familiar concept of the H norm of a system accommodating the possibility of nonzero initial conditions. The weighting matrix R is a measure of the relative importance of the uncertainty in initial conditions vis-k-vis the uncertainty in the exogenous signals w. A "smaller" choice of R reflects greater uncertainty in the initial condition. This connection is best illuminated by observing that as the smallest eigenvalue )min(R) of R approaches oc, the unit ball in Rn (R) L2 defined by

Bt$L2 := { (Xo, w) e Rn @ L2

xoRxo + II vll

}

tends to (0, BL2), where BL. := {w We now describe how the performance measure J can be computed for a given system. These results may be viewed as natural generalizations of existing work on the computation of the H norm of a linear time-invariant system (see, for example, Anderson [1], Willems [17], and noyd, Salakrishnan, and Kabamba [2]). THEOREM 1.2. Consider the linear (possibly time-varying) system Ea as in (4) above. Let R, S be given symmetric matrices such that S is positive semidefinite and R is positive definite. Then the following are equivalent:

(a) J(Ea, R, S, T) < /. (b) There exists a symmetric

(7) -/(t) and P(O)

(c) (8) ((t)

matrix function

A’(t)P(t) + P(t)A(t) +

2P(t)B(t)B’(t)P(t) + C’(t)C(t),P(T)

< "2R. There exists a symmetric matrix function

A(t)Q(t) + Q(t)A’(t) +

P(t), t e [0, T] such that S

-

Q(t) > O, t e [0, T] such that

Q(t)C’(t)C(t)Q(t) + B(t)B’(t), Q(O)

R

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and 72Q-1(T)

> S.

THEOREM 1.3. Let Fa in (4) be a linear time-invariant, asymptotically stable system. Let R be a given symmetric positive-definite matrix. Then the following are equivalent:

(a) J(Ea, R, O, oc) < 7. (b) There exists a symmetric

matrix P such that

A’P + PA + (1/72)PBB’P + C’C,

0

(A + (1/72)BB’P) is asymptotically stable, and P < 72R. (c) There exists a symmetric Q(t) that satisfies the Riccati differential eqution (8) .for all t >_ 0 and is such that the autonomous system O(t) [A + (1/72)q(t)C’C]q(t) is exponentially stable. Moreover, limt_ Q(t) exists and equals Q, where Q is

_

the unique symmetric matrix with the following properties" 1

AQ + QA’ + -QoC CQ + BB’

(9)

O,

(1/72)QC’C is asymptotically stable. Proofs of Theorems 2.2 and 2.3. Equivalence of (a) and (b). We will first prove that (a) (b) in Theorem 2.3. From Lemma 2.1, it follows that g(a,R, O, oc) IIC(sI- A)-IBII. It follows from Anderson [1], Willems [17], and Soyd, Balakrishnan, and Kabamba [2] that IIC(sI- A)-IBII < 7, if and only if there exists a and A +

=

symmteric matrix P such that

0

AP + PA +

PBBP

+ CC,

and (A + (1/72)BB’P) is asymptotically stable. To complete the necessity it remains to be shown that P < 72R. Suppose that this is not the case. Then there exists a nonzero x0 such that Xo(P- 72R)x0 >_ 0. Straightforward algebra reveals that for system (4),

d(x’Px) dt

72w, w x’C’Cx

7w

1B’Px 7

7w

_

B’Px

(1/72)B’Px, x(O) xo. Using stability of (A / (1/72)BB’P) and integrating the above equation from 0 to oc along the trajectory of Ea, we obtain

Now set w

72xoRxo / ")’211wll2 -Ilzll 2 72xoRxo xoPxo

O,

J(E, R, 0, c) < 7. The proof of (b) =v (a) of Theorem 2.3 can be readily completed by reversing the steps in the above argument to establish that the existence of P with the requisite properties is sufficient for J(Ea, R, O, oc) < 7. Next we provide a brief sketch of the proof of equivalence of parts (a) and (b) of Theorem 2.2. Suppose J(Ea, R,S,T) < 7. It follows that the cost in the optimal control problem which contradicts

inf

wEL2[O,t]

I1 11

I

j (llCxIl + x’ (T)Sx(T))

}

subject to the system Ea with x0 0 is nonnegative. The existence of P(t) satisfying the Riccati differential equation of Theorem 2.2 now follows from standard arguments

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P.P. KHARGONEKAR, K. M. NAGPAL, AND K. R. POOLLA

in classical linear-quadratic control theory (see, e.g., Brockett [3], Limebeer et al. [10]). The rest of the proof is analogous to that of Theorem 2.3. Equivalence of (a) and (c). We next prove the existence of Q(t) with the desired properties if J(Ea, R, S, T) < (respectively, g(Ea, R, 0, oo) < "7). Since Hoo analysis Riccati equations are seldom written in this form, our proof of this part will be in much greater detail than the one involving P(t) or P. The following arguments also appear implicitly in [12] and [13]. The following Hamiltonian system plays an important role in establishing the existence of Q(t)

’

Let the transition matrix of this system be given by

(11)

(I)21 (t 0) (I)22 (t 0)

x(t)

x(0)

T e [0, T] (for the infinite horizon case T e [0, OO)), we will use rr to denote the projection operator defined as (rrf)(t) f(t) when t _< 0 for t > T. Suboptimality of J implies that for some T and (rrf)(t) 0, j2(Ea, R,S,T) < /2(1-2) (respectively, J2(Ea, R,O, oc) < "2(1-52)). Now for any T E (0, T] (respectively, T e (0, OO)), this implies that

In the following, for any

(2)

x’oRxo + I1

1[

1

-

> (x’oRxo +

11 (C )11

for all x(0) e Rn and w e L2[0, T] for system (4). From observation (12) we will now show that [011 (t, 0)R + (I’12 (t, 0)] and [O2x (t, 0)R + O22(t, 0)] are both nonsingular for all t [0, T] (respectively, t e [0, oo)). First, suppose, contrary to what we want to prove, that for some T e (0, T] (respectively, e (0, oo)), [(I,(T, 0)R + O12(T 0)] is 0. Thus 0 such that [OI(T, 0)R + (I’2(T, 0)]x0 singular. Then there exists x0 for system (10) with x(0) x0 and p(0) Rxo, p(’) 0. Choosing w(t) B’p(t) for t e [0, T], systems (4) and (10) have the same trajectory for x. Differentiating the product x’(t)p(t) along the trajectory of (10), we obtain

-

(13) Integrating

-

d(x’P)dt

w’ w

x’C’ Cx.

(13) from 0 to and noting that p(T)

X’onXo + II

-w[I

0, we get

which is a clear contradiction to (12) since x(0) 0. Nonsingularity of xo [O2(t, 0)R + 22(t, 0)] is shown similarly. (If [O2(T, 0)R + O2(T, 0)] is singular for some T, then with an appropriately chosen x(0) 0, X(T) is zero and we arrive at a similar contradiction.) Next, setting Q(t)"= [O2(t, O)R+222(t, 0)][11(t, 0)R+ O2(t, 0)] -1, straightforward differentiation shows that Q(t) satisfies the desired Riccati differential equation. Note that Q(t) is nonsingular for all t > 0 since [O21 (t, 0)R + 22(t, 0)] is also nonsingular. Since Q(t) is nonsingular for all t > 0, Q(0) > 0, and Q(t) is a continuous function of t, it now follows that Q(t) > 0 for all t > 0.

Hoo CONTROL WITH TRANSIENTS

1379

For the following proposition, we will need the Lyapunov differential equation

(t)

(14)

AC + CA’ + BB’,

C(O)

R -1.

Since A is a constant stable matrix, C(t) is bounded. Let p >_ supt>0 [[C(t)ll, where the norm of a matrix is defined as the largest singular value. Before proceeding with the remainder of the proof, we state a straightforward result from quadratic optimization theory which will prove very useful in some remaining parts of the proof. (The following result can be easily proven by standard completion of squares.) PROPOSITION 1.4. Consider the system given by (4) (in the infinite horizon case A is also stable), and let R > 0 be a given positive-definite matrix. Then for any e [0, T] (respectively, T e [0, OC) for the infinite horizon case)

-

a

inf

z (o) e P,., ,w e

2 2, z: [O,-r] II--wll + x’(O)Rx(O)- -ll-.zll x(-)

x.

x.Q --1 (T)X.,

()

(ll-.wll 2 + x’(O)Rx(O) X(T) ,wez:2 [0,r]

(16)(0)ertnf

X)

1

X’C-()X

llxll

2

where p is defined as above. If J(Ea, R, S, T) < 7, then, inf

( I1-oll = + x’(O)Rx(O)- II’Tzll x(T)=xT)--2XTSXT 1

x(O)EIn,wEE2[O,T]

2

>0

for all XT E Rn. Invoking the above proposition, it now follows that ,2Q-I(T) > S. For part (c) of the infinite horizon case (Theorem 2.3), it still remains to be shown that Q(t) is bounded, stabilizing, and asymptotically converges to the stabilizing solution of the corresponding algebraic Riccati equation. We first show that there exists a positive number u < oc such that Q(t) < uI for all t > 0. Based on Proposition 2.1, now consider the following series of inequalities:

from

(12)

The above bound, which is independent of T and true for all x E Rn, shows that Q-(T) is bounded below for all T > 0 or that Q(T) is bounded. Now we show that the time-varying system generated by A + (1/’2)Q(t)C’C is exponentially stable. The Hamiltonian system associated with the optimization problem inf

(0)ert,o2 [0,]

II.wll 2 + x’(O)Rx(O) ll--zll 2" x(-) x-

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P.P. KHARGONEKAI, K. M. NAGPAL, AND K. R. POOLLA

which has a solution for all written as

T

>

0 and

x

E

Rn when J(Ea,R, 0, c)
P>0. (ii) There exists a symmetric bounded matrix function Q(t) > 0 for all t >_ 0 such that

1 ((t) AQ(t)+Q(t)A’ +[-CIC-Q(t) [C2C2]Q(t)+BB, Q(O)

R-

(34) and the

unforced linear time-varying system

(35)

i5(t)

[A-Q(t)(CC.-- CC1)] p(t)

is exponentially stable.

1

-

(iii) The function (1 p((1//2)Q(t)P)) > 0 for all t >_ 0 and is bounded. (b) Moreover, if Q(t) with above properties exists for all t >_ O, then limt__. Q(t) exists and equals Q, where Qo is the unique symmetric matrix with the following properties:

(36)

AQ

+ QocA’ Q CC2

-CC

Q + B1B

O,

A Qo [CC2 (1//2)CC] is asymptotically stable, and Q >_ O. (c) If the conditions above are met, then one controller that achieves J(Ecl, R, O, c) < / is given

by

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P.P. KHARGONEKAR, K. M. NAGPAL, AND K. R. POOLLA

d&(t)

A+

B1BP &(t) +

I

Q(t)P

Q(t)C[y(t)- C2&(t)] + B2u(t),

=o

(37) u(t)

It should be noted that as in Doyle et al. [5], the necessary and sufficient conditions for the existence of controllers in the output-feedback problems are two decoupled Riccati equations and a "spectral radius" condition. From the formulae for the controllers, we note that in contrast to the statefeedback problems, the initial condition weighting matrix R plays an important role in the controller formulae in the output-feedback problems. An intuitive explanation for this phenomenon is as follows. In the state-feedback problem, since the entire state is available for feedback, there is no uncertainty in the initial state as far as the controller is concerned. The only issue in the state-feedback case is whether the desired performance bound is achievable. This is verified by checking the inequalities in the infinite-horizon P(0) < "2R in the finite-horizon case and 0 < P < case. On the other hand, in the output-feedback case, the controller does not have complete knowledge of the initial state. Consequently, the controller gains depend on the relative weighting between the uncertainty in the initial state vis-a-vis that in the exogenous input w to satisfy the desired performance bound for all w and x(0). Also, the controller gains change with time reflecting the relative importance of the information contained in measurements as compared to the prior information on the initial state. Finally, we would like to note that in the infinite time horizon case, as t the controller approaches the central controller for an associated standard H problem. This is intuitively appealing since assuming that the controller is internally the effect of nonzero initial states should disappear and the stabilizing, as t controller should be required only to attenuate the effect of exogenous signals w on z. This is exactly how the controller given in Theorem 2.4 behaves. For sufficiently large R, the steady-state central controller itself has sufficient robustness to account for uncertainty due to unknown initial condition and we need not necessarily require a linear time-varying controller to achieve the desired performance. This is formally stated in the following result. COROLLARY 2.5. Let the conditions of Theorem 3.4 be satisfied and R be such that R < Q. Then the linear time-invariant controller of the form given in Theorem 3.4(c) with Q replacing Q(t) achieves J(Ecl, R, 0, ) < "),.

’R

,

,

-

,

1. 3. Proofs. Without loss of generality, we will assume that Proof of Theorem 3.2. Suppose that both x, w are available as measurements to the controller and suppose that there exists a controller such that J(Ecl, R, 0, ) < 1. Then it follows from Lemma 2.1 that for the system E with x0 0, we have

sup inf 0 such that P satisfies (28) and (A + BBP- B2BP) is asymptotically stable. It remains to be shown that P < R. We will prove this by contradicition. 0 such that Xo(P- R)xo >_ O. Now in the Suppose that there exists x0 system E, set x(0)

x0 and

w(t)

BPe(A+(B1B1-B2B2)P)tx 0

Using the fact that

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A / (B1B

B2B)P is asymptotically stable, it follows that w belongs to L2[0, c).

For this particular w, we claim that sup {

II (t)ll -IIz(t)lI }

-xoPxo.

Indeed, the (unique) optimal input u for the above optimal control problem and the corresponding state trajectory are

u(t)

-B2Pe (A+(B1B-B2B)P)txo,

t(t)

e (A+(BI

2(t)

B-B2B)P)(t-’)xO.

This can be shown, for example, by noting that for any w E L2[0, c), the optimal u denoted by fi is obtained from the following two-point boundary value problem

Px

+

5(t)

x(0)

Xo,

lim p(t)

O,

Bp(t).

The above claim now follows by directly verifying that p(t) -Pe (A+(B l-)P)txo. (Similar optimization plays an important role in Tadmor [15].) Using this observation and the fact that J(Ect, R, 0, c) < 1, it follows that 0

< 4Ro + I111 -I111 _< 4R0 + up{llll -IIzll }

’o(R- P)xo 0 such that P satisfies (28) and (A + B2BP) is of asymptotically stable, then using detectability (A, C), standard Lyapunov techniques and a completion of squares argument together with P < R can be employed to show that the control law (27) is internally stabilizing and achieves the desired performance bound. Proof of Theorem 3.1. Suppose that both x, w are available to the controller and suppose that there exists a controller such that J(Ect, R, S, T) < 1. Now consider the system E with x0 0. Then it follows that

BIBP-

sup inf

Ilzll + x(T)’Sx(T)


_ 2(x’oRxo + II ,vll 2 + II ,(C2x)ll 2)

x(0) e R’ and v e L2[0, T) for system (48). From Nagpal and Khargonekar [12, Tams. 3, 4], the above inequality implies (a)-(ii) for Theorem 3.3 and (a)-(ii) and (b) for Theorem 3.4.

for all

To obtain conditions (a)-(iii) of Theorems 3.3 and 3.4, we use the "separation" from Khargonekar [9]. This "separation" idea is roughly as follows 1) We separate the time interval of the original problem into two subintervals; 2) during the idea

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first part, w is chosen so that y 0 (thus during this interval u 0); and 3) in the second interval, one chooses the "worst" w in an appropriate sense (which will soon become clear). Such a choice of w together with the inequality implied by the existence of a controller that achieves J(EcI, R,S,T) < 1 (respectively, J(Ecl,R, 0, oc) < 1 for the infinite horizon case), would lead us to conclude conditions (iii) of the two theorems. Fix T e (0, T] (respectively, e (0, oc)) and x e Rn. Let w e W(-) such that v(t) D2lBQ-l(t)x(t) for t e [0, T]. With this choice of X(T) and v(t), it is easily seen by completion of squares that for system (48),

-

.

x’Q-()x x(O)’Rx(O)/ II:ll / II(c)ll x(O)’Rx(O)/ IIwll -IIzll

(5o)

-

We now consider the finite and the infinite horizon cases separately. Finite horizon case. Fix 6 (0, T] and x 6 R ’ and let w W(T) be chosen as in the last paragraph. For t (-,T], let w(t) BP(t)x(t). Now integrating (38) from

(51)

-

to T, we get

(I r)wll =

(I 7rr)zll = x(T)’Sx(T) -x’rP(z’)x -[1(I- rr)(U + BPx)[I 2.

Combining 0

(50)and (51)we obtain

P. Hence p(Q(T)P) < 1 for all T e [0, OC). Now limt_ Q(T) Q. From Theorem 3 of Doyle et al. [5], p(QP) < 1. Thus, (1- p(Q(r)P)) -1 exists for all T [0, oc), is continuous, and has a limit as T OC. Therefore it is bounded on [0, oc). Hence condition (iii) of the theorem follows. This completes the proof.

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4. Conclusions. In this paper we have formulated and solved an H-like control problem, where, in addition to the exogenous signals in the state and measurement equations, we must also account for uncertainty in the initial condition. This was done by defining a suitable worst-case performance measure. It is hoped that these results may allow us to design control strategies that are robust to both exogenous signals and nonzero initial states. It was shown by Mustafa and Glover [11] that the central solutions to H control problems have the additional property of maximizing the entropy of the closed-loop transfer matrix. It is clear that the controllers obtained in the present paper are analogous to the central solutions to the H control problem. The question arises then as to what is the analogue of entropy maximization property in the present context. This problem does not seem to have immediate answers since entropy is defined in terms of the closed-loop transfer function matrix evaluated along the imaginary axis. This definition has no obvious extension to our context where the controllers are time-varying. We leave a full investigation of this issue for future research. Finally, we have formulated our problem in a purely deterministic context. A stochastic formulation of this problem should be an interesting undertaking. Acknowledgment. The authors are grateful to R. Ravi for many helpful discussions regarding the content of this paper. REFERENCES

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