SIAM 1. CONTRO AND OPTIMIZATION Voi. 10.;N~&.~~1~2-2-28J. . . March 1992

@ 1992 Sodety Tor lndurnial and Applied Mafhcsads W2

A GAME THEORFTIC APPROACH TO S? CONTROL FOR TIMEVARYING SYSTEMS*

DAYID 1. N. LIMEBEER?, BRIAN D. 0. ANDERSON& PRAMOD P. KHARGONEKARB, AND MICHAEL GREENt

Abstract A representation formula for all controllen that satisfy an P-type constraint is derived for tim&varying systems. It is now known that a Cormnla based on two indefinite algebraic Rimti equations may be found for time-invariant systems over an infinite time support (see [J. C. Doyle el al., IEEE Tmns. Automat. Confro< AC-34 (19891, pp. 831-8471; [K. Glover and 1. C. Doyle, Systems Control Len, 11 (1988), pp. 167-1721; [K. Glover et al., SIAM I. Control Optim, 29 (1991), pp.283-3241; [M. Green et al., SIAM J. Contror Optim, 28 (1990), pp. 1350-13711; [D. J. N. Limebeer et al., in R o c IEEE wnf on Decision and

: , Control, Austin, TX, 19881; [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301-3241). < ,

In the t ime-va~na case. two indefinitk Riccati diSerential eauations are remired. A solution to the design . - . . problem exists if these equations have a solution on the optimization interval. The derivation of the representation formula illustrated in this paper makes explicit use of Linear quadratic differential game theory and extendsthe work in [I. C. ~ o ~ l e k h . . IEEE ~rak Automat. ControcAG34(1989),pp. 831:8471 and [G. Tadmor, Malh. Control Systems Signal Rmepsing, 3 (1990). pp.301-3241. The game theoretic approach is particularly simple, in that the background mathematics required for the sufficient conditions is little more than nandard arguments based on "completing the square."

Key words. @-optimal control, game theory, indefinite Riccati equations, four-block general distance problems, worst-case design

1. Introduction. Since the early 1980s, numerous techniques have been developed for solving T-control problems. In addition to their individual interest, further insights have been gained by studying the interplay between these various methodologies, and researchers have now acquired a good understanding of several aspects of the theory. Over the last two years, there has been a flurry of activity that has had a significant impact on both the accessibility of the theoretical ideas and the ease of computation. Once an T - n o r m bound has been decided, and provided that a solution exists, the computational burden associated with finding all %? wntroIlers is essentially the same as that required in solving a Linear quadratic Gaussian regulator problem [9], [lo], [ll], [13], [20], [21], [27]. T problems in which perfect information is assumed may be solved using a single Riccati equation with dimension equal to that of the problem [IS], 1191, [23], while the output-feedback problem requires the solution of two Riccati equations.

The "two Riccati equation" formula for all stabilizing wntrollers satisfying a closed loop F - n o r m constraint has been derived in a number of ways. A state-space solution via the "four-block" problem is reported in [11], [21], while an alternative approach reminiscent of classical linear quadratic theory may be found in 193. A solution based on J-spectral factorization theory is given in [2], [13], while the related notion of conjugation is used in [20]. An interesting by-product of this activity has

- *Received by the editors A ~ r i l 19. 1989: accevted for vublication (in revised form) December 13,1990. . . . t Department of Elemical Engineering, Imperial College, Exhibition Road, London, England. t De~artment of Svstem Engineering. Australian National Univenihi. Canberra. Australia. - -. . . 8 Department of Electrical Engineering and Computer Science, University of Michigan, ARn Arbor,

Michigan 48109-2122. This author was supposed in part by National Science Foundation grants ECS-8451519 and ECS-9096109, and in part by grants from Honeywell, Boeing, General Electric, Army Research Office, and United States Air Force Office of Scientific Research grants AFOSR-88-0020 and AFOSR-90-M3.

A QAME THEORETIC APPROACH TO TIME-VARYING DESIGN 263

been the discovery of a number of new interconnections. In [19], 1281, the authors note a connection between %?'-control and game theory. The interplay between indefinite factorization and game theory, probably first noted by Banker [3], has been rediscovered in the more general setting of %?' control [lo], [13], [221. The connection between risk-sensitive optimal control [lo], [30] and game theory, originally discovered by Jacobson in the perfect-information case 1171, has also received renewed interest in the wider setting of P control [7], [lo]. More recently still, Tadmor [27] has given results on the finite-horizon time-varying case using the maximum principle.

The aim of the present paper, which builds on the work given in [gland [27], is to give a solution to the finite-horizon time-varying %?'-control problems, which makes explicit use of the existing theory of linear quadratic games. This approach offers the advantage of introducing game theoretic intuition to the solution path, and also gives a game theoreticinterpretation to the change of variable introduced in [9]. We also show that the main infrastructure of a time-varying %?'-control theory may be developed using classical arguments based on completing the square. This simple solution is possible because of the linear quadratic (LQ) nature of the problem, and, as a final comment, we mention that we have been able to remove' all the removable assumptions in [27].

In this paper we will consider the finite-dimensional linear time-varying plant

which has a time-varying state-space realization

(l.lb) w (t), x(0) =O.

Y 1 1c2 D21 ~~~1 L U J m e class of controls of interest is given by u =%y, where 83 is a linear time-varying (LTV) controller. Eliminating u and y gives rise to the closed loop operator z = O? w, where R = 13,,+Q,,%(I.-@,,R)~'~,, . Ourgoals are(1) togivenecessary and suf6cient conditions for the existence of LTV controllers such that /1z112< y l l~11~ for aU w $ 0 and a given y (11.112 denotes the norm on 22[0, TI), and (2) to characterize all such controllers when they exist. We assume that the matrices in (1.1) have entries that are continuous functions of time, that D,, has full column rank rn for all t E [0, TI, and that 9, has full row rank q for all t E [0, TI. The game theoretic natureof the problem is immediate: Roughly speaking, the w-player tries to maximize the energy in z, while the&ntroller or u-player simultaneously seeks tominimize it:

Section 2 has a large tutorial component and deals withthe time-varying problem in which perfect information is assumed. Section 2.1 establishes the necessaly conditions for the existence of a solutionto the %?'-control problemvia a conjugate point argument. We then show that XP-control problems with perfect information may be solved if and only if a solution to the Riccati differential equation (RDE) exists on [0, TI. Section 2.2 presents a representation formula for all solutions to the time-varying %?'-control problem in the perfect-information case. A brief review of some pertinent properties of adjoint systems is given in 8 2.3. Section 2.4 partially reconciles the more familiar time-invariant infinite-horizon P-control problem with the time-varying finite-horizon case. This reconciliation calls for a connection between the limiting solution of the RDE and its algebraic counterpart. . . .

264 LIMEBEER. ANDERSON, KHARGONEKAR AND GREEN . Section 3 contains the main results of the paper. We begin with an analysis of

problems in which both D,, and D2, are assumed to be square. A solution is found by calculating a plant inverse, and does not require the solution of any Riccati equations. It is interesting that the plant inverse has an observer structure that is indicative of the solution in the general &se. Following that, we treat problems in which either Dl, or D,, is square. Every problem of this type requires the solution of a single RDE. Finally, we treat the case in which neither D,, nor D,, is square, and we show that in these cases two RDEs are required. We give necessary and sufficient conditions for the existence of solutions and characterize all solutions (when they exist).

2. 8- control and linear quadratic differential games. Numerous variants of the linear quadratic differential game have been studied over the last twenty years. As an example of the extensive literature on the subject, we refer the interested reader to Basar and Olsder [4] and the references therein. The purpose of this section is to review those aspects of the existing game theory literature that are relevant to the solution of the time-varying 7P-control problem, and to study the connections between the two. We begin by considering a system with input vectors u(t) and w(t), dynamical description

and output,

We will assume that D(t) has full column rank, and that it has been scaled so that

Each of the matrices in (2.1) and (2.2) is assumed to have entries that are continuous functions of time. In the interests of notational compactness, this time dependence will not always be shown from now on.

With (2.1)-(2.3) given, the T-control problem is to find a linear causal wntrol u(t)= K(x(s), w(w( t), OCsC t, such that llFzYII = supw {llYmwl12: W C ~ ~ [ O , TI, 11 w 112 Z 1) < y for some given y > 0. The operator Fz, maps w to z when the control u(t)=K(~;;)isinplace.Ifz=F~w, then II.TmII<yifandonlyif

(2.4) J(6: w) = r (zrz- Jw'w) dt 5 -e 11 wI1:

for all w E S2[0, TI and some positive E.

In the language of game theory, the 7P-control problem is a leader-follower game and requires the control system designer to make the first move and select an admissible u(t) = K(., ., .) that minimizes J ( K , w). A control functional K(., ., .) is admissible if(1) u ( t ) ~ 9 ~ [ 0 , TI forevery w(t)~2'[0, T],and(2)x(O) =Oand w(t)=O+u(t)=O; we denote the set of admissible control functionals %. Aft'er the designer's choice of control strategy has been made public, we assume that nature is malicious and selects that w(t) E Z2[0, 71, which maximizes (2.4). The %?-control problem therefore has a solution if and only if

min max {J(K, w)}~-~llwll$ Kev W E ~ % O , T ]

is satisfied. In the notation of (2.5), the w-player maximizes J ( K , w) over all W E

5f2[0, TI affer the u-player has announced a choice of K c 45.

A GAME THEORETIC APPROACH TO TIME-VARYING 2" DDESIGN 265

Before studying the minimax problem (2.5) in detail, it is interesting to reconcile it with the classical linear quadratic optimal regulator. If we allow y to increase without bound, then it is necessarily the case that the energy in w ( t ) decreases to zero, rendering the w-player impotent. The game thus degenerates to the optimal regulator [ I ] as y is increased without bound.

Ow first result is based on a standard completion of squares argument [4, p. 2901, and shows that the %?-control problem has a solution if a particular RDE has a solution on the time interval [0, TI.

THEOREM 2.1. Suppose that the RDE

has a solulion on 10, TI. Then

result in

for any u (either open or closed loop) and w E Z2[O, TI in (2.1), noting that x(0) = 0 . With u = u*, we have

ProoJ S i (2.6) is assumed to have a solution on [0, TI with P ( T ) = 0 , we have, for any u and w, and x(0) = 0 ,

Substituting from (2.6) gives us

= Ilu-u*ll:-y211w- w*ll:

with w*, u* as in (2.7) and (2.8), which establishes (2.9). Suppose that 2 is the operator with realization

2 = ( A + B , K ) x + B , w , ( w - w * ) =-y-'B;Px+w,

266 LIMEBEER, ANDERSON. KHARGONEKAR, AND GREEN ,

which maps w to ( w - w * ) . Since 2 exists (and is given by i = (A+ B2K + y-2B1B:P)x+B,(w- w*), w = y - 2 ~ : ~ + ( w - w*)) , we can write

/ I ~ z w ~ l l ~ ~ ~ 2 I I ~ I I : = - y Z I I ~ - ~ * l l ~ = - y 2 1 1 ~ ~ I I ~ ~ - ~ j l ~ I I ~

for some positive K. Thus llFzwll < y, as required. O Remark 2.1 (Positive semidehiteness of P ( t ) ) . It is not hard to show that if (2.6)

with P ( T ) = 0 has a solution on [0, TI, then P ( t ) 2 0 for all t E [0, a. Consider system (2.1) with 0 5 1.d t S T, and with x(t,) =x , . If

T

(2.12) JtO(t& w ) := (z'z - Y ~ W ' W ) dt,

then we may complete the square as above to yield

J,..(t& w)->b~S. t , )x , ..... : . , . = Ilu-u*ll:-y211.w-w*II~, . . ' I

in which x, is regarded & ':an: arbitrary initial condition for the running period [ to , TI G [O, TI. Consequently,

(2.13) J,m(u*, 0 ) + y2/1w*I/:'=xbP(tO)x.

for every x.. Since Jte(u*, O)Z 0 forany x., it follows that P t t ) Z O for all t E 10, TI. Note that the game Riccati equation (2.6) and the corresponding linear quadratic

regulator Riccati equation "approach each other" as y increases. Since the linear quadratic regulator Riccati equation always has a solution, we expect (2.6) to have a solution if y is large enough. In the next section, we show that the existence of an admissible controller satisfying (2.5) is anecessary condition for (2.6) to have a solution on [o, TI .

2.1. The necessary conditions. The aim of this section is to show that if an adrnis- sible feedback control exists that solves the F-control problem, then (2.6) has a solution on [0, TI. Our proof uses conjugate point arguments that are reminiscent of those found in 1251, [26] . To begin, we introduce the two-point boundary value problem

andits associated transitionmatrix, whichis generatedhy the linear differentialequation

with @(T, T ) =I. Next, we note that

+ C'C. Thus, if @ z ( t , T ) exists on 1 E [0, TI, the RDE (2.6) has a solution on [0, TI given by P( t )=@z , ( z , T)@;:(t, TI.

The existence of @;i( t , T) is equivalent to a conjugate point condition, and this observation forms the basis of the necessity proof.

A ~ A M E THEORETIC APPROACH TO TIME-VARYING z.?" DESIGN 267

DEFINITION 2.1. Suppose that to < t,. Then to and t, are conjugate points if it is possible to find a nontrivial solution to (2.14) such that x(t.) = p ( t , ) =O. We note also that p(t,) # 0 and x(t,) $0, since otherwise (2.14) would only have the trivial solution x ' p'](t) - 0.

LEMMA 2.2. The matrix @,,(to, t f ) defined by (2.15) is singular if and only i f i, and, t, are conjugate points.

Proof: Suppose that to and t, are conjugate points. Since

~ ( $ 1 [;::;I = [:: : r l ( t o l .)[,(,)I2 and since t, and l, are conjugate points, we obtain

Hence @,,(t,,, t f ) is singular, as x ( t f ) is nonzero. If, on the other hand, @,,(to, i f ) is singular, there exists a vector g ZO such that

@,,(to, +)g=O. By considering the final value problem with x ( t f ) = g and p(tf)=O, we see that x(t.) =0, which establishes that to and t, are conjugate. O

THEOREM 2.3. Consider the linear system (2.1). If there exists a closed loop control B E Cg such that

(2.17) l l ~m l l < 7,

then @,,(t, T ) is nonsingular for all t e[O, TI . Consequently, the RDE (2.6), with boundary condition P ( T ) = 0, has a solution on [0, T I .

Proof: We suppose for contradiction that t* E [O, TI is the largest time such that @,,(t*, T ) is singular. Since @,,(T, T ) = I, t'< T by continuity. It follows from Lemma 2.2 that t* and Tare conjugate points, giving x( t*) = p ( T ) =O. Next, we suppose that

Since (2.14) has a nontrivial solution, p(t)+O t E [t*, TI. Furthermore, if B : p ( t ) =O, then (under this assumption) (2.14) becomes the two-point boundary problem for the LQ-regulator; this can have no conjugate points [ I ] . Thus C ( t ) PO. Next, we see that J ( K , a)> J,.($ G), since $ ( t ) = O for t < t*. Let tS(t) be the function generated by x(t*)=O, I? and #( t ) . Then it follows that

T

(2.18a) J,*(E, 3) = J,-(G, $) 2 y;? j,. { ? l ~ t ~ + u*u - 7 2 f i t ~ ] dt,

subject to

(2.18b) ?=A<+B,a+B,y ? ( f L ) = O .

The tilde is used to distinguish the state trajectory associated with (2.14) from that associated with the minimization problem in (2.18). The initial condition ?(tL) = O is a consequence of (if x(0) =0, (ii) LC - 0 for all t E [0, t*), and (ii) K E (e. The minimiz- ation on the right-hand side of (2.18a) subject to (2.18b), with u ( . ) being sought in open-loop form is almost a standard LQ control problem and is solved as follows: Fonn

H(~,?,A, u )=; (?~c~G+u~u- y 2 ~ 1 a ) + ~ ( & + ~ l # + ~ 2 ~ ) . . .

Then

268 LIMEBEER, ANDERSON, KHARGONEKAR, AND GREEN '

and

giving

Using %= Y - ~ B : ~ , and subtracting (2.19) from (2.14), gives us

Since there are no conjugate points associated with the LQ boundary problem in (2.20), i ( t ) x ( t ) and A ( t ) - p ( t ) for all t ~ [ t \ T ] . Consequently,

which contradicts (2.17). 0 Remark 2.2. In our later work, we will need to consider problems in which the

output is given by

rather than (2.2). This change in output has the effect of introducing a cross-coupling term between u and x in the functional J (K , w). This may be removed by the change of variable u = I? - D ' C x In the case of an output given by (2.21), the Riccati equation (2.6) becomes

and the corresponding minimizing feedback control is given by u*(t)= - ( D I C + B : P ) ( t ) x ( t ) , rather than (2.7).

2.2. A representation formula for all solutions. In Theorem 2.1 we found one feedback control that solves the time-varying P-con t ro l problem. In the infinite- horizon case, it is well known that there are usually many feedback controls that result in 11TZwll < y. The aim of this section is to show how to construct all linear full- information (access to w and x) feedback controls such that llTmll < y. Our analysis is based on the output equation (2.21), rather than (2.2).

THEOREM 2.4. Suppose that

A 'GAME THEORETIC APPROACH TO TIME-VARYING P DESIGN 269

is given. Then there exists a control g E '% such that llYzwll < y if and only if

has a solution on [0, TI with P ( T ) = 0. In this case, all closed-loop systems .Tm generated by controls of the form

where 2, and 0, m e arbitIary cnusal linear operators such that

IlYmll <r are also generated by

where

and y-'U is an arbitrary linear causal strictly contractive operator on S2[0 , TI ( i .b ,

I l~ l l<Y) . Equivalently, w r y 2, and 2, may be parametrized by

Equations (2.22), (2.26), and (2.27)may be represented diagrammatically, as in Fig. 1. Proof: The first part, that the existence of a controller such that ilYswll < y is

necessary and sufficient for the existence of P ( t ) is just Theorems 2.1 and 2.3 for the cross-coupled game.

Now suppose that the RDE (2.23) with P(T)=O has a solution on [0, TI. We need to show (1) that there exists a causal I2 in (2.26) corresponding to every control of the form given in (2.24), and ( 2 ) that /lTmll < y if and only if y-'U is strictly contractive.

Substituting (2.24) and (2.27b) into the dynamical equation (2.22a) gives us

~ = ( A + Y - ~ B , B : P - B2(l?,.+ y - 2 2 2 ~ ; ~ ) ) ~ + ( B l - B Z 2 Z ) ( ~ - w * ) , x(O)=O,

FIG. 1. All solutions to the P-controlproblem wirh perfect information,

270 LIMEBEER, ANDERSON, KHARGONEKAR, AND G R E E N . i

which shows that x depends causally on (w - w*), and so x(t) = &(w- w*), in which

. .. ives us E, is a linear causal operator. Substituting (2.27) into (2.24) g'

= [ - ( Z , - F , - Y - ~ ~ ~ B ; P ) ~ ? ~ - ~ ~ I ( W - w*)

= U(w-w*)

for some linear causal U. This establishes the existence of the causal mapping in (2.26). The proof is completed by noting that y-'U is strictly contractive as follows:

* 2 - 2 Ilzll:-~~Ilwll:=IIu-u 112 Y IIw-w*II:

= IIU(w-w*)Il:-y211w-w*II:

= IIU3wIlI:- r2113wli:

< 0 for all w # O E Z2[0, TI- y-'U is strictly contractive.

In the above, 2 is the causal and causally invertible operator linking w and (w - w*)., The invertibiiity of 3 was established in the proof of Theorem 2.1. 17

At this point it is interesting to reexamine Fig. 1. By reviewing (2.26), wesee that r = (w - w*), and that u = ( v t. u*) drives B, with v = Ur. If w = w*, there is no signal into the U parameter, and the corresponding control is given by u* (irrespective of U). If w # w*, we do not have to use the control u*. Thus for the purpose of solving. the P-control problem, the u-player only has to play well enough to ensure that J(K,w)<Oforag ivenw#O~~~[O,T] . ' :

2.3. Adjoint systems. In 5 3, where w e derive a representation formula for all solutions to the time-varying P problem with output feedback, we willrequire an elementary property of adjoint systems [S], 1141.

Let 2: X + Y be a linear operator between two Hibert spaces X and Y. Then 2*: Y -t X, the adjoint of a, is the unique linear operator such that, for all z E Y and W E X, (z, ZW) =((2*z, W) [14, Thm. 2, p. 391. Note also that /I211 = ll2*II. Now suppose that 2 is described by the state-space equations

or, equivalently,

Thus

=I: ($2) .-JOT pf(Ax-k Bw) dti- if(Cx+ Dw) dt. r .

A GAME THEORETIC APPROACH TO TIME-VARYING Z? DESIGN 27 1

Integrating the first tern by parts gives us

= (O*Z, w),

where the adjoint S* is a linear system with realization

(2.3 la) -$(t) =A'(t)p(t)+C'(t)z(t), p(T) =0,

(2.3 1b) q(t) = BC(t)p.(t)+D'(t)z(t).

In the following, we will require the solution of a dual LQ game system, which is obtained by applying Theorem 2.4 to its associated adjoint system.

2.4; The asymptotic properties of the game equation. This section represents a digression from the main stream of the time-varying finite-horizon P- control problem, and its purpose is to make some connections with the more familiar time-invariant, infinite interval results [9]. In particular, we show that in the limit as T + co, the solution of (2.6) approaches the smallest nonnegative solution of an algebraic Riccati equation (if such a solution exists). We also show that this smallest nonnegative solution is stabilizing in the sense alluded to in [9]. We begin by connecting the properties of the solution of the RDE (2.6) as T + m (when the various coefficient matrices are assumed to be constant) to the solution of the algebraic Riccati equation

(2.32) O = PA+ A'P- P(B2B;- y-2BlB:)P+ C'C.

In doing this, we will supposeThat each of the matrices in (2.1) and (2.2) is time-invariant and that (A, B,, C) is stabilizable and observable '(it is not hard to remove the observabiiity assumption, but we will not address this issue here).

If (2.6) is solved backward from the terminal condition P(T) = Q, then we will refer to the solution as P(t, T, Q). If a limiting solution to (2.6) exists, we will call it P(t) = limtt, P(t, T, Q). The first result of this se.ction is standard and shows that P(t) is nonincreasing i f P(T) =O.

LEMMA 2.5. If (2.6) has a solution on 10, TI with P(T)=O, then P(t) is non- increasing (in the sense of semidefinite matrices).

Proof: Differentiating (2.6) gives us

- ? = P ( A - ( B ~ B : - ~ - ~ B , B I ) P ) + ( A - ( B ~ B : - ~ - ~ B , B ; ) P ) ~ P , &TI=--clc, which has solution [8, Thm. 1, p. 591 &t)= -Q(t, T)'CfCQ(t, T), where @(t, T) is the transition matrix associated with (A-(B2B;- y-'B,B;)P)'. It is evident that P ( ~ ) s o for all t, so P(t) is nonincreasing. 0

If (2.32) has at least one nonnegative solution, then we will show that the smallest of these solutions, Y say, is an upper bound for P( t ) provided that Y Z P(T).

LEMMA 2.6. Suppose that P(t, T, Q) exists [0, TI and that Y = Y'ZO satisfies (2.32) with Y 2 Q. Then Y Z P( t ) for all t E [0, TI.

272 LIMEBEER ANDERSON, KHARGONEKAR, AND GREEN r

Proof: Completing the square using (2.32) gives us T

J w ) = [(u+ B;Yx)'(u + B; Y x ) - y2(w - Y-'B; Yx)'(w - f 2 B ; Y x ) ] dl

+xr( t )Yx ( t ) - x l (T )Yx(T)

for any u and w. In the same way,

J,(u, W ) = [(u+B:Px)'(u+ B;PX) - Y Z ( ~ - y - Z ~ : P X ) ' ( ~ - Y-~B:PX)] d7 ItT by invoking (2.6). Subtracting and setting C:=-B$Yx and G:= y-2B:PX yields

x ' ( t ) [ Y - P ( t ) ] x ( t ) = x ' ( T ) [ Y - P ( T ) ] x ( T ) + y2 ( G - y - 2 B : ~ ~ ) ' ( G - y - 2 ~ : Y ~ . ) d~

i J , T ( ~ + B i f i f i c + B : f i ) d r i - o

for all x( t ) . O We remark that a variation on the above argument, in conjunction with Lemma

2.5, will establish the following result. COROLLARY 2.7. Assume that Y = Y'Z 0 satisfies (2.32) with Y 2 Q 2 0. Then

P(t, T, Q) exists for all t 5 T and Y 2 P(t, T, Q ) Z P(i, T, 0 ) Z 0. If P ( t ) exists for some Q, it is easy to argue that it satisfies (2.6). Suppose that

t S T I C T. Then P(t, T, Q ) = P(t , T I , P(T1, T, Q)) , and thus

P(t)=lim P(t, T, Q)= lim P(t, T I , P (T l , T, Q)). Tim Tt-

For any fixed T,, the solution P(t, T I , 6) depends continuously on 0 and therefore

(2.33) p( t )=P(f , T I , lim P(T, , T, Q))=P(t , T I , P(T,)), 7tm

which shows that P(t) is a solution to (2.6) for all t. Since the system and output matrices have been assumed to be time-invariant, the value of the game J,(K*, w*) = x1( t )P( t )x( t ) is invariant under time translations. Consequently, P( t ) = I' is constant and therefore satisfies the algebraic equation (2.32).

If a nonnegative stabilizing (i.e., (A- (B2Bi- y-2BlB:)P) asymptotically stable) solution to (2.32) exists, it is the smallest of the nonnegative solutions.

THEOREM 2.8. Suppose that [A, C ] is obsemable and that X 2 0 and Y Z 0 satisfy

with A - DX asymptotically stable. Then Y > X. Proof: Due to the observability of [A, C ] , every solution to (2.32) is nonsingular.

We can therefore write X ( A - DX)X-' = -(A'+ C'CX-I), which shows that A'+ C'CX-' is completely unstable. Now

Subtracting gives us A(X-1- Y - I ) + (X-' - Y-')Af+X-'C'CX-I- y - ' C ' C T 1 = 0

+(A+x-~c*c)(x-~ - Y-')+ (x- 'Y-~)(A+x-~c 'c) '

= (X- I - Y - ' )C 'c (X- ' - T I ) .

-A OAME THEOW APPROACH TO TIME-VARYSNG DESIGN 273

Since (A+X-'C'C) is completely unstable, we must have (X-'-Y-')zO and SOXSY. 0

Remark 2.3. Theorem 2.8 remains valid when the obse~abii ty assumption on [A, C] is weakened to one of detectability. Suppose that

with [A,,, C,] observable, then argue as before on the smatter system [A,,, C,].

3. Solution to the time-varying fl-control problem: Main results. Now that the background game theory is in place, we arein a position to solve the general time-varying %?-control problem. As mentioned in the Introduction, the plant is described by

(3.la) 2(t) =A( t )x ( f )+B , ( t )w( t )+B2( t )u ( t ) , x(0) =0,

(3.lb) z(t)= c,(t)x(t)+D12(t)~(f),

(3:lc) . ~(f)=Cz(t)x(f)+Dz~(t)w(t) ,

in which we assume that all the matriceshave entries that are continuous functions of time, and that D,, and D2, have full column and row rank, respectively, for all t e [0, TI. Under this assumption, we may scale the problem so that D',,D1, = I and D2,D;, = I. By a corollary of Doleial's theorem [29, Cor. 3, p. 701, we know that there exist continuous extensions D, and fiL to DI2 and Dz,, respectively, such that IDl2 DL] and [D:, 5:] are orthogonal for all t e [0, T]. Note that by generalizing the loop-shifting transformations in [Ill, [24] to the time-varying case, there is no loss of generality in the assumption that D,, and D, are zero. Details of the scding and loop-shifting transformations required in the time-varying case may be found in the Appendix.

3.1. Problems of the first kind. We begin with a preliini'nary result in which we assume that D12 and D2, are square; we call such problems problems of the first kind. Under this assumption, scaling arguments allow us to assume, without loss of generality, that Dl+ I and D2, s I. As we will now show, finite-horizon YP problems of the first kind may be solved by inverting the plant, and the resulting controllers have a remarkably simple observer structiue.

THEOREM 3.1. C 0 ~ i d e r the generalizedphnt described by

(3.2a) i ( t ) = ~ ( t ) x ( t ) + ~ ' ( t )w(t ) + ~ ~ ( t ) u ( f ) , ~ ( 0 ) =o, (3.2b) z(t) = c,(t)x(t) + 14th

(3.2~) ~ ( t ) = c2(tlx(tj+ ~ ( t ) .

Then (i) the set ofaU linear causal output-feedback control laws such that \I %z,\\ < 7 is paramehized by

(3.3a) i ( t ) = (A-BIC, -B ,C , ) ( t ) f ( t )+B, ( t ) y ( t ) +B2(t)u(t), 2(O) =O,

(3.3b) u(t) = u(t)- C,(t)x^(t),

(3.3~) r(t) = y(t) - Cz(t)x^(t),

(33d) u(t) = (lIr)(t),

in which 11 is a causal strictly contraetiue operator on Z2[0, TI, and (ii) there exists a linear causal controller such that the closed-loop mappingfrom w foz is identically zero.

274 LIMEBEER, ANDERSON, KHARGONEKAR, AND GREEN 2

. . Proof: Eliminating y(t) and u( t ) from (3.2) and,(3.3) gives us. ~

and

Consequently,

A- B2Cl ' BI Cz X - X 0

and

[;I=[" 0 "I[ C2 x-2 * ]+[o I 0 I ] [ " ] . v

The second row of (3.4) implies that the obse~ations enor e ( t ) = ( x - i ) ( t ) is identically zero, thereby establishing the observer property.'. In addition, it is clear. that [a(t) = [:](t) for all r E [0, TI, which shows that the controller is plant inverting.

For the first part, we note that if W is any strictly contractive mapping from w(t) to z(t), then W will he generated by setting U = W in (3.3d). Setting U=O proves the second part. D

. The controller given in Theorem 3.1 is presented diagrammatically in Fig. 2, and is seen to have an o b s e ~ e r structure with observer gain matrix -B, and state estimate feedback Cl .

. 3.2.Problems of the second kind. In this section we show that all problems in which either Dl, or D2, is square, which we will refer to as problems of the second kind, have controllers that may be characterized by the solution. of a single RDE. Simple plant inversion is no longer possible, since this requires that both Dl, and D2, be square. Nevertheless, a solution is still possible and involves the game theory of P 2. If D,, is assumed to be we can replace (3 .1~) with . .

(3.ld) ~ ( t ) = C,(t)x(t)+ w ( t )

by an appropriate scaling operation. .. . . .

. .

FIG. 2. neplant inwrse as an observer. . . . . .

. ..

:'.Since we are dialing with a finitefuoeproblem, the question of stability does not arise.

A GAME THEORETIC APPROACH TO TIME-VARYING DESIGN 275

We now consider the possibility of obtaining a controller via a combination of an observer for x, and a linear feedback law as computed in 8 2. Since the controllers that we are considering do not have access to w, the observer can only be driven by u and y. Suppose that

(3.5) i=.42+B2u+K(CZx^-y), f ( O ) = O .

Subtracting (3.1) from (3.5) and substituting (3.ld) gives us

(3.6) ( x ^ - ~ ) = ~ ( 2 - x ) + K ( C , x ^ - C ~ x - w ) - B , w = ( A + K C , ) ( x ^ - x ) - ( K + B , ) w .

Setting K =-El as before, and e := 2 -x gives us

(3.7) ;=(A-B,C,)e, e(O)=O

+ e(t)=O and so x^(t)= ~ ( t ) . Again, the stability of A- BICz is not an issue for finite terminal times.

With the required observer established, we may now find a11 linear closed-loop controls K E F2 such that (IYz,l(<y as if full state information were available. As before,

(3.8) J ( U , W ) = IoT (Z'Z - y 2 ~ r ~ ) dl.

Using the results of 8 2, weintroduce the Ricciti equation -2' . r -&= (A-B2D[zCl)'X,+X,(A- BzD;2C1) -X,(BzBI-y BlB1)Xw

(3.9) + C:DLD:Cl

with terminal condition X,(T) =0, and define

(3.10) u* = -(D;zC,+ B:X,)x = -F& (3.11) w* = y-ZB:X,x,

which gives us

(3.12) J(-F& w ) s - E ~ / w ~ ~ :

for some E > 0. By combining the observer given in (3.5) with K = -B,, the equilibrium strategies

(3.10), (3.11), and the characte~ization of all solutions in 5 2.2, we obtain the controller confignration illustrated in Fig. 3 and the main result of this section.

THEOREM 3.2. Given

(3.134 . . i ( t ) = ~ ( t ) x ( t ) + ~ , ( t ) ~ ( t ) + ~ ~ ( t ) ~ ( t ) , ~ ( 0 ) =0,

(3.13b) 2 0 ) = c l ( t ) x ( t ) + D , k t ) ~ ( t ) ,

(3.13~) Y ( t ) = Cz(t)x(t)+ wft),

then (i) an output-feedback controller exists such that j.Tm-,/j < y ifand on& ifthe R D E (3.9) has a soIution on [O, TI with teminalconditionX,(T) = 0, and (ii) every closed-loop operator Ym with ( 1 Ys,)l < y, corresponding to an outputfeedback control u =By, is generated by

(3.144 j ( t ) = ( ~ - ~ , ~ , - B , ~ , ) ( t ) x ^ ( t ) + B,(t)y(t)+B,(t)u(t), i ( 0 ) =0,,

(3.14b) u(t)=u(t)-F,(t)i(t),

(3.14~) r(t)= y ( t ) -(c,+ y-2B:X,)(t)x^(t),

(3.14d) v ( t ) = (Ur(t) ,

in which y-lU is acausal, strictly contractive operator on Z2[0, TI.

LIMEBEER. ANDERSON, KHARGONEKAR, A N D GREEN a

. . Rc. 3. AN eontroff~rsfor problems ofthe second kind.

Proof: (i) Suppose that a control of the form u =%y exists such that llTmll < y. Then setting u = W[C2 I][:] in Theorem 2.3 proves the "only if" part. To prove tbe "if" part, we suppose that a solution to (3.9) exists. We then invoke the observer property developed in (3.5) to (3.7) and the completing-the-square argument given in Theorem 2.1.

Part (u) is a consequence of Theorem 2.4 on substituting f ( t ) for x ( t ) . A solution for problems of the second kind in which Dl,, rather than DZl, is

square is now given without proof. This particular result is needed in the next section in solutions to probiems of the third kind and is solved via the dual game associated with the operator [?TI @$;I (see (1.1) for a definition).

THEOREM 3.3. Given

then (i) a feedback controller exists such that ll.Twll < y if and only if the RDE

f',= (A- BlD:,C2) Y,+ Y,(A- B1Di1C2)'- Y,(C$C,- V-2C;C,) Y, (3.16) + B,B:~~,B: has a solution on [0, T ] with Y,(O)=O, and (ii) eoery closed-1oop.operator Tz, with I(g;u /( < y, corresponding to an ourput feedback control u =By, is generated by

in which y-Ill is a causal, strictly contractive operator on S2[0, T ] and H,= Dz1B:+C2Y,.

Proof: Apply Theorem 3.2 to theadjoint system associated with(3.15). . . 0 . ..:

A CAME THEORETIC APPROACH m TIME-VARYING P DESIGN 277

3.3. Problems of the third kind. In this section we treat the case in which neither D2, nor D,, is square. We call such problems problems of the third kind. In the case where Dzl has fewer rows than columns, the observer analysis in § 3.2 breaks down because the equation WD21(t) = -Bl(t) need not he soluble for %. What is required is a norn-preserving transformation to a new problem for which state reconstruction is possible. Thedesired transformation, which was first suggested in [9] in the time- invariant case, is immediate from identity (2.9). Suppose that

(3.18) r = w- y-2B:X,x,

Then (3.18), (3.19), and the completion-of-squares argument resulting in (2.9) gives us

(3.20) lor ( - yzWw) dt

or, what is the same,

(3.21) l l ~ l l ~ - ~ ~ l l ~ l l ~ = l l ~ ! l ~ - ~ ' ~ l r I l ~

for any r and v. Substituting (3.18) and (3.19) into (3.1) yields

(3.22a) ~ = ( A + y - Z B , B : X , ) x + B l r + B 2 y x(O)=O,

(3.22~) =(C,+y-2~1B;X,)~+D21r.

LEMMA 3.4. Suppose that the RDE (3.9) with X,(T) = 0 has a solution on [0, TI. Suppose also that 8 is any controller, and that the control law u = Wy is applied to the systems given in (3.1) and (3.22)'. Then JIFz,IJ < y if and only 5 JjFwJJ < y.

ProoJ First, note that the relationship between r and w in(3.18) is causally invertible. The result now follows directly from (3.21). O

Lemma 3.4 shows that the tasks of designing controllers for the systems in (3.1) and (3.22) are interchangeable. The key feature of (3.22) is that Dl, is square, and thus this system description is a problem of the second kind, which makes state reconstruction possible along the lines of O 3.2. If we define

then it is immediate that

and that

A direct application of the results of 8 2.3 (see (2.31)) shows that a state-space model for the adjoint operator associated with (3.22) is given by

(3.26a) -p"= (A+ y-2B,B;X,)'p+ FLU, +(CZ+ y-2D21B:Xi,)'u2, p(T)=$J,

(3.26b) ' y,= B ~ P + D L ~ U Z ,

(3.26~) y2 = Bkp+ u, . . .

. . . . . . .

. .. . . 'For the purposes of this result, 3 need not be linear. ' . . . .

278 LIMEBEER, ANDERSON, KHARGONEKAR, AND GREEN :,

If we now substitute (3.26) into (3.14), using (3.9) to (3.1.1), we obtain a representation formula for R* and thus also for $3. This calculation forms the basis of the next result.

THEOREM 3.5. Given

then (i) a causal outputrfeedback control of the form u = By exists such that JIFa.,jl < y if and only ifthe RDEs (3.9) and (3.30a) (given below) haw solutions on [0, TI, and (ii) every closed-loop operator .Tz, with il.Tm 11 < y, corresponding to an output feedback control u = By, is generated by

in which y-'U is a causal, strictly contractive operator on g2[0, TI and

(3.29~) [B,, Bk2]:= [-B,D;, -z,(c,+ y - Z ~ z , ~ ; ~ , ) ' I -B,- Y-~Z,FLI,

with

in which

(3.30b) A~; .A- B,D;,c,+~-~B,~~B,B:x,. Proof: (i) Suppose that a control given by u = By exists such that 11 Fs, 11 < y. Then

u=%[C2 D2,][:] together with Theorem 2.3 implies that (3.9) has a solution on [0, TI. Since X,(t) exists on [0, TI, we may apply the control u,= %*y2 to (3.26) to obtain lly,l12< yllull12 for all u, $0 by Lemma 3.4 and (3.26). We may now use u2=W*[B; I ] [ : ] together with the "only if" part of Theorem 3.3 to establish the existence of Z,(t) on [0, TI. Sufficiency is immediate from Lemma 3.4 and Theorem 3.3.

(ii) The set of ail closed-loop operators corresponding to (3.27) and u =Ry is the same as the set of all closed-loop operators corresponding to (3.26) and u2= R*y2 by Lemma 3.4. Since (3.26) is a problem of the second kind, we may invoke Theorem 3.3 to complete the proof. U

In the last part of this section, we replace Z, with Y,, which is the solution to the RDE introduced in Theorem 3.3. As we will show, Y, is the dual of X,, and there is a close connection between X,, Y,, and 2,. Before supplying this connection, we give a more general version of the connection between a Riccati equation and the Hamiltonian matrix associated with the two-point boundary value problem.

A GAME THEORETIC APPROACH TO TIME-VARYING DESIGN 279

LEMMA 3.6. The RDE

hasasolutionon [0, TI $and only ifthere existsanxsuch that the boundary valueproblem

has a solution on [0, TI with P,(t) nonsingular for all t E [0, TI and P2(0)Pl(O)-' = M. In this case, P ( t ) = P2(t)Pl(t)-' is a solution to (3.31) and

Proof: Suppose that the RDE has a solution P(t) . Then (3.33) is immediate, and the result follows. . .

Conversely, with P = P~P;' we have from (3.32),

THEOREM 3.7. Suppose that (3.9) has a solution X,(t) on [0, TI. Then (3.30) has a solution Z,(t) on [O, 77 ij and only if (3.16) has a solution Y,(t) on [O, TI and p(X,Y,)(t) i y2 for all t E [0, n. Furthermore, Z,(t) = Y,(I - y-2X,Y,)-'(t).

Proof: A straightforward calculation using (3.9) shows that

where H, and HZ are the Hamiltonian matrices from the boundary value problems associated with the RDEs for Y, and Z,, respectively.

Suppose that Z, exists. SinceX,;Z,--20, it follows that ( I + y-2X,Z,)(t) is nonsingularforall t E [0, TI. Also, Z,(I + y-2X,Z,)-1(~)= 0. Using (3.30a) and (334), we see that

So Y,=Z,(I+ y-2Xdm)- ' is a solution to (3.16) by Lemma 3.6. Furthermore,

Conversely, if Y, exists and p(X,Y,) < -y2, then I - y2XWY, is nonsingular on [O, TI, Y,(I- y-2X,Y,)-'(0) =0, and from (3.16) we have that

Thus Z,= Y,(I - y-2XwY,)-' is a solution to (3.30a). 0

280 LIMEBEER ANDERSON. K H A R G O N E K A R , AND GREEN ::

Substituting Zm= Ym(I - y-2XmY,)-' = ( I - y-2Yzm)-'Y, into Theorem 3.5 gives our final result.

THEOREM 3.8. Given

then (i) an ourput feedback u = W y exists such that IIYmII < y if and only if (a) the Riccati equation (3.9) has a solution X, on [0, TI, (b) the Riccati equation (3.16) has asolution Y, on 10, a, and (c) p(X,Y,) < yZ for all t E [ 0 , 0 ; and (ii) every closed-loop operator Tz, with llTzwll c y, corresponding to an output feedback control u = By, is generated by

in which y-'U is a causal, strictly contractive operator on 32[0, TI and

. . Proof: This follows from Theorems 3.5 and 3.7. 0 As a check, we note that (3.38) and (3.39) reduce to those in [9], [ll], [131, and

[Zl] in the time-invariant case. In the infinitehorizon case, we mustestablish an internal stability property; this has already been done in the case of time-invariant problems [91, [Ill , [131, C211.

Appendix. Suppose that the design problem is described by the equation

in. which each of the matrices has entries that are continuous functions of time. We will also assume that Dl, and Dzl have, respectively, full column and row rank for all t E [0, TI. The controller measures y and generates the control u via u = B y and has the task of minimizing the worst-case energy gain between z and w.

It is the aim of this appendix to establish that, without loss of generality, we may assume that

A GAME THEORETIC APPROACH TO TIME-VARYING %? DESIGN 281

If one prefers, it is possible to have (i) and (ii) replaced by

(A.2)' (i)' DL=EL,

(A.3)' (ii)' D2, = [Iq Oj-q],

which are clearly special cases of (i) and @). To begin, we consider the. closed-loop configuration in Fig. A.l, which contains

the direct feedthrough matrix from ( k l ) , the controller, and four scaling matrices. S, and S, will be chosen to enforce (A.2) and (A.3), while S, and S, ensure that (k2) ' and ( k 3 ) ' are satisfied (should we wish to include them).

Suppose that

(A.6) D:zD,z= N'N

and

(A.7) DZ1D;, = MM'

are the Cholesb factorizations. Then the entries of M and N are continuous, since they are expressible in terms of the entries of Dl, and D,, [12, p. 881. It is easy to check that Sl = M-' and S, = N-' will achieve the desired orthogonalization of D2, and D,,, and that these scaling matrices will not destroy the continuity properties assumed for the original problem. It is evident from Fig. A.l that we would design the controller I? for the scaled problem, and then back substitute through Sl and S2 for K. Next, we introduce the orthogonal extensions DL and 6,. such that [Dl , Dl] and ID:, fi:] areprthogonal. It is a consequence of Doleial's theorem that the extensions DL and DL exist and are continuous [29, Cor. 3, p. 701. Setting

('4.8) ' s l = [ D ~ z DL], S4=tDh f i ~ l in Fig. A.1 leads to a direct feedthrough matrix of the form

Dill1 0 1 1 1 2 Im (A.9) [D,- :;;; %].

The partitioning of D,, into DllV i, j = l , 2 is induced by (A.2)' and (k3)'. Since S, and S, are orthogonal and thus norm preserving, these scale factors do not change the set of admissible controllers and therefore the controller representation formula.

FIG. A.1. Scaling the D-matrix

282 LIMEBEER, ANDERSON, KHARGONEKAR, AND GREEN c.

We are now in a position to invoke a loop-shifting argument that eliminates both D,, and D2,, and leads to a signscant simplification in the main analysis [ll], [24]. Defining

(-4.11) Qm=-(D1111+ DIIIZ(Y;I -D~IuDIIz~)-'D~IuDII~I),

(A.12) F m = ( I + Q J A Z ) - ~ Q ~ , allows us to mimic the arguments in [ll],[24], which remove D,, and D,,. In conclusion, we mention that the Cholesky factors in the Julia operator of [ I l l always exist and have continuous entries. U

Addendum. After the completion of the k t version of this paper, we became aware of [ 5 ] and [28]. Reference [28j discusses the connections between Z? control and game theory, while [ S ] applies discrete game theory to the state-feedback e- control problem.

. ..

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