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H∞-control by state feedback: An iterative algorithm and characterization of high-gain occurence
Systems & Control Letters 12 (1989) 383-391 North-Holland
383
H -control by state feedback: An iterative algorithm and characterization of high-gain occurence Carsten S C H E R E R * Mathematisches Intsitut, Am Hublan~ D-8700 Wiirzbur& West Germany Received 29 December 1988 Revised 8 March 1989
Abstract: We present an iterative algorithm to compute the achievable H~-norm by state feedback for a standard regular problem and give a characterization when high-gain feedback is necessary to approach the optimal value. If there is no need for high-gain components to approximate the optimal value, we can reduce the problem to the computation of the H~-norm of a certain stable transfer matrix. Keywords: Hoo-optimization; high-gain feedback; Riccati equations; state feedback.
1. Introduction
Recently, Doyle, Glover, Khargonekar and Francis presented in [2] an algorithm for H~-control by state feedback which uses the fact that one can decide whether a prescribed number y is greater or smaller than the optimal value y, of the achievable Ho~-norm. This decision is made by solving an algebraic Riccati equation and looking for positive definiteness of the solution. In their paper [2], nothing is said about the behaviour of the suboptimal feedback matrices (which can be computed from the solution of this Riccati equation) if the parameter y approaches y,. Especially, the high-gain aspects of suboptimal feedback matrices (though dealing with regular problems) are completely neglected. In the present paper we will give a criterion for the case when high-gain feedbacks are necessary to approach the optimal value of the Ho~-norm and when the optimal value is really attained. Furthermore, we present an iterative algorithm which yields in each step a better upper bound approximation of the optimal value 7, together with suboptimal feedback controls. In each step we only have to solve a s t a n d a r d Riccati equation of optimal control and we can show convergence of the bounds to the optimal one. We denote by R and C the real and complex numbers where C is partitioned in the usual way into C - U C°U C + denoting the left half-plane, the imaginary axis and the right half-plane respectively. All matrices are to be considered as being real and of suitable dimension.
2. Assumptions and formulation of the problem
We assume our system to be given by =
+ Su + C.,
x ( O ) = O,
z = Cx + Du,
* Supported by Deutsche Forschungsgemeinschaft, KN 164/3-1.
0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
0a) (lb)
C. Scherer/H~-controlbystatefeedback
384
where all matrices are real and x is the state, u the control, v the external disturbance and z the controlled output. Furthermore we assume that the following regularity assumptions hold: • (A, B) is stabilizable and (C, A) is observable. • D T ( c D ) = (0 I). • G~O. These assumptions especially imply that the pair (C + DF, A + B F ) is observable for all matrices F of suitable dimension. We assume that the state of (1) is available for control and want to construct a compensator which stabilizes the system and reduces the L2[0, oo)-induced norm of the map from the disturbances v to the output z as far as possible. Since we are mainly interested in computing the least possible value of the norm and since this value does not change if using static state feedback instead of dynamic feedback laws (see [4,5]), we consider u=Fx
witho(A+BF)cC-
as admissible controls. Thus we are led to the optimization problem y,=
inf
]](C + D F ) ( s I - A
-BF)-IG[Io~
F admissible
where II H ( s ) II ~ is the H~-norm of the proper stable transfer matrix H(s). In this paper we wish • to compute y, (perhaps with the help of iteration); • to characterize when y, is attained and compute a corresponding F,; • to compute suboptimal feedback matrices when 3', is not attained; • to characterize the ease when high-gain feedbacks (i.e. 1]F ]l tends to infinity) are necessary to approach 3',. All these problems are investigated by considering the solutions of certain algebraic Riccati equations (ARE).
3. H~-optimization and Riccati equations The following main result of this section is partly proved in [2], but we want to include some extra results and give an independent elementary proof. Theorem 1. For the system (1) the following statements are equivalent: q admissible F: 3X>0:
]1(C + D F ) ( s I - a - B F ) - 1 a II ~ - Y;
(2)
GG T x A T + A X + - - - -,{2 BBT+XCTCX=O;
(3)
AT +eA+ I-7-
(4)
Every solution P > 0 of (4) is in fact positive definite and F = - BTp yields an admissible feedback law such that the inequality in (2) is satisfied. For this F we get equality in (2) if and only if
o A-BBTP+GaTP t/n C o
C Scherer/ H~-controlbystatefeedback
385
or equivalently a( A + P - 1 c T c ) C~C ° 4 ~ . The proof of this theorem proceeds by two lemmas which are interesting in their own right. The first one is a slight generalization of a result of [8], since we drop the assumption of controllability.
Lenuna 2. Suppose the matrix A is stable. The inequality GT( --i,M-- A X ) - ' Q ( i o d - A ) - ' G < r : I
(5)
holds for all ¢o ~ R iff there is a solution K -- K T of the algebraic Riccati equation ATK+/CA _ KGGT ./2 K _ Q = 0.
(6)
Proof. The frequency-domain inequality is of course necessary for the existence of a solution K = K T
of (6). For the converse we assume without loss of generality the pair (A, G) to be of the form A1,
A12
such that (A n, G , 0 is controllable. By partitioning Q and K accordingly, the Riccafi equation (6) can be split up into three equations
AT, K1, + K h A n
_
--al
G,,c .[2
,i
--n-
Qll = 0 ,
(7)
( .4T1-- gll GI'GK12+Kll.412+Kx2.422-Q12=O'./2 ------~I)
(8)
tc GnGT1 ATzK12 + AT2K22 + K2zAz2 + K2aA,2 - "'21 `/z "',2 - Q22 = 0.
(9)
Because
.i
,o (,..
1o1,)
we obtain from (5) the inequality
GT(--i~oI-- AT1)-'Qn(iwI-- A n ) - I G , I ~ `/21 for all ~0~ R and therefore (see [8], Theorem 5) there is a solution K,1 = K ~ to the ARE (7) with
G,I GT Thus, by stability of .4, the spectra of -(.411 - (GnG~,/`/2)Kn) and .422 are disjoint and (8) has, as a standard Sylvester equation, a solution Kx2. By defining K21 := K~, we infer in the same way the existence of a unique K22 = K ~ satisfying (9). []
C. Scherer/H~-controlbystatefeedback
386
L e m m a 3. Suppose the system (1) is influenced by some u(.), v(.) ~ L2o~[0, oo). I f there is a solution P >_0 of the A R E
ATp + PA + P ~
BBr
p + cTc=o
then for all T > 0 the following relation holds: ( T I I z(t)l[ 2 a t < ]¢2(T[[ v ( t ) H 2 dt + I T H B T p x ( t ) + u ( t ) l l 2 dt.
Jo
Jo
Jo
Proo|. The proof follows from a standard ' c o m p l e t i n g the squares' a r g u m e n t where one has to note z(t)Tz(t) = x(t)TCTCx(t) + u(t)Tu(t) and x(0) = 0 (see [6]). [] Proof of T h e o r e m 1. ( 2 ) = (3): F r o m L e m m a 2 we infer the existence of K = K T with
K + (C + DF)T ( c + D F )
( A + BF)TK + K( A + BF) = K-~ T
and thus, by stability of A + BF, we conclude K < O. With Z :~ - K this equation results in
ATz + ZA + Z GGT y2 Z + CTC + ( B T Z + F ) T ( B T Z + F ) - Z B B T Z = O. N o w Z x = 0 implies x T C T C x -~- 0, i.e. Cx = 0 and xTFTFx = xT(BTz + F)T(BTZ + F ) x and therefore we obtain Z A x = 0. Since (C, A) is observable, this implies x = 0. Thus Z is invertible and Z - 1 is a solution of the matrix inequality
A X + Xz~T + x c T c x
= 0,
i.e. Fx = 0
GG T + - -- BB T < O. y2
F r o m T h e o r e m 2.1 in [7] we obtain the existence of a matrix X > Z - 1 > 0 which solves the corresponding Riccati equation. (3) =, (4): Choose P :--- X -1, (4) ~ (2): F r o m the A R E in (4) we obtain
(ooT )
(A-BBTp)TP+P(A-BBTp)+P
7+BBT
p+cTc=o.
Since ( P ( G G T / y 2 + B B T ) P + c T c , (A - BBTp)) is an observable pair, we obtain from T h e o r e m 3.3 in [3] the stability of A - BBTp. L e m m a 3 with u = --BTpx and v ( . ) ~ L2[0, oo) leads to the inequality
II~(') IILqo,~) ~ 2 -< ~2 IIv(.) IIL2t0,~}. implies II(C + D F ) ( s I - A - B F ) - I G IJ ~ -< y for the feedback F . . . . BTp. Now we derive the remaining statements of Theorem 1: First, the nonsingularity of P follows from the chain of implications This
Px=O
~
xTCTCx=O
~
Cx=O
~
PAx=O
by the observability of (C, A). Secondly, it is well known that equality in (2) holds for F = - B T p matrix A -
BBTP
H=
GG T T2
--(cTc+~Te)
-(A-BBTp) T
if and only if the H a m i l t o n i a n
C. Scherer / Hoo-control by state feedback
387
has eigenvalues on the imaginary axis. (Note that (C + D F ) T ( c + D F ) = CTC + P B B T p holds true in our situation.) Now one easily checks
H
-v
=
i
e
o
I
t (1
°°T't
T
(
U-
as well as
(, 0
0
I
Thus the theorem is completely proved.
?
0
-(A+p-1cTc)
T
t
"
[]
Motivated by this theorem we denote by A R E ( g ) the algebraic Riccati equation A X + XA T + x c T c x
+ ttGG T - BB T = O.
Suppose now that A R E ( p ) has a symmetric solution. Since (A T, C T) is controllable, we deduce again from Theorem 2.1 in [7] the existence of a maximal symmetric solution X(g) of A R E ( g ) which has the additional property that the closed-loop matrix A ( g ) := A -I- X ( g ) c T c
has only eigenvalues in C + t.) C °. Furthermore, for v _
-4- vGG T - B B T = (it - g ) G G T <_<0
and thus, by maximality, the existence of X ( v ) with X ( v ) >__X(g). Therefore, X(#) is a nonincreasing map on its domain of definition. (Note that the results in [7] are stated for complex matrices but the given proofs carry over to the real case without change.) It is clear that we have thus reduced our original problem to the computation of g . defined by g . = s u p ( g I X ( g ) exists and X ( g ) > 0}. Once -¢. = 1 / g ~ .
is computed, it is easy to obtain suboptimal controls for any desired bound y > ~,..
4. Extreme solutions of ARE(p) The main idea of the paper consists of the following observation. Let us define F( Z ) :-= A Z -4- ZA T -4- z c T c z
and compute F(X+ Y) - F(X) = Y(A + xcTc) T + (A + xc'rc)y+
ycTcy.
If X(#I) and X ( g : ) are defined, the f o l l o ~ n g holds for A X : = X(g2) - X(gl): F(X(ga))+glGGT-BBT-=O, =
i(gl)
F(X(g2))+g2GGT-BBT=O
AX-.I-AXA(gl)T..jt-AXCTCAX-4 - (g2-gl)GaT=o.
(10)
388
C Scherer/ Hoo-control by state feedback
We observe that we can compute X(g2) from X ( g l ) by solving (10) for A X via
/ ( g 2 ) ~- X ( g l ) "+"AS. N o w we can describe what we want to call extreme solutions. First of all, we compute X ( 0 ) > 0 which exists, because ARE(0) is a standard Riccati equation of optimal control and which has the additional property that A(0) has no purely imaginary eigenvalues. In the second step, we compute the maximal value g,, for which a symmetric solution of A R E ( g ) exists at all. This is equivalent to finding the maximal g,, such that
A(O)Y+ YA(O) T + y c T c y + g a g T = 0
(11)
still has a symmetric solution for/~ = g,,. This can be decided in a straightforward manner because - A ( 0 ) is stable and according to L e m m a 2, a solution exists iff the inequality II C(sI + A(O))-IG II oo < - 1 / f f f holds. Thus we have to compute the H ~ - n o r m g,~ of the transfer matrix
C(sI + A(0))-'G which is clearly positive since (C, A(0)) is observable and G does not vanish. In this way we obtain g,, .'= 1 / y ~ < oo and compute X(g,,) with the help of the maximal solution of (11) for g = ttm. It is now clear that we call • X(O) for g = O, • X(g,.) for the maximal possible value g,., extreme solutions of A R E ( g ) because the optimum g , lies in the interval [0, g,.] and because we can construct (sub)optimal controls with the help of the solutions X ( g ) with the property x(O) > X ( g ) > X(g.,).
5. An iterative algorithm Suppose that the extreme solutions of A R E ( g ) are computed and the assumption X ( g ~ ) ~' 0 holds. Now we are in the position to explain how the iteration proceeds. For this reason we suppose that g , and X ( g , ) > 0 are already computed (starting with g0 = 0). Then we search at t, such that
x(g.)
+
t(x(g=)
-
x(g.))
>
0
holds for all t ~ (0, t , ) and that the left-hand side becomes singular for t = t o. Because of the equivalences
x ( g . ) + t.( x ( g ~ ) - X ( g . ) ) >_ 0 ~, I + t a X ( g , ) - l / 2 ( X(gm) •.
Lxtn
x(~.l-'/2(x(g.)
-
X ( g , ) ) X ( g , ) -1/z > 0
- x ( g m ) ) X ( g . ) - ' / 2 >__0,
we define for X ( g ) > 0 the function
t(g) ,=
1
II I
-
X(g)-'/2 x(t~.,)x(~) - ' / 2
II
~ (0, 1]
and take t, = t ( g . ) . With this value we get the new parameter from the convex combination
g . + l = (1 - t . ) g . + t.gm = g . + t.(gm - g . ) ~ ( g . , gm]. Since t,2 <_ t,, we obtain for A X = X(gm) - X ( g , ) the inequality
A O , . ) ( t . a x ) + (t. a x ) a ( g . )
T + (t. a x ) C c ( t °
a x ) + t.(~,m - ~ . ) c G T < 0.
C Scherer/ Hoo-controlby statefeedback
389
Thus we conclude that the maximal solution Y of the corresponding A R E satisfies Y ~ t. (X(It,.) - X(It.)) and we obtain, by the definition of t., the inequality
X(~.+ 0 = X(~.) + :+_ O. We could arrive at two possible situations: X(it.+l ) is singular. In this case we have reached the optimal value # , = # . + 1 : Indeed, for # . < # < It,+1 we can determine t ~ (0, tn) with It = # . + t(it," - Its) and therefore X ( # . ) + t ( X ( # . ) X(it.)) > O, i.e. X ( # ) > O. • X ( # . + I ) > O. Then it is clear how to iterate. F r o m these considerations it follows that X ( / z . ) > 0 implies t o - 1 and thus p~ = # . , i.e. p , = ~ . . In case of X ( I t . ) ~ 0 we obtain It, < I t . and the algorithm either stops with #~+1 = # , or yields an infinite sequence (P,)n. In the latter case, ( # . ) . is increasing and bounded b y It,", i.e. convergent to a limit/2 ~ (0, # . ] . Besides this we obtain in each step suboptimal feedback matrices - B T X ( I t . ) -1 for the bound y = 1 / p ~ . Note that ( X ( p . ) ) , is nonincreasing, bounded from below by 0 and thus converges too, where the limit must be x(#). Now we want to show ~ = # , by assuming ~ < # , . By definition of It,, X(/~) must be positive definite and therefore t(/~) > 0 well defined. As an intermediate step we show that t(it) is a nonincreasing function for increasing # in the domain of definition of t(p); this can be seen as follows: It >_ ~ ~
x(it) ~ x(~)
[ 1 - ( t ( ~ ) - e)] X(#) < [ 1 - ( t ( # ) - e)] X(v) =
x(~)
+ (t(~) - ,)(x(~.)
- x(~))
~ x(~) + (t(~) - ~)(x(~,")
- x(~))
for all e ~ (0, t(it)). Since the left-hand side is positive definite for these values of e, we obtain t ( # ) < t(v) as desired. By t(#.) = #.+l-it. it is clear that ( t ( # . ) ) . converges to 0 for n ~ oo and that there exists some n o with 0 < t(#.o) < t(g). But this impfies #.o > t~, contrary to the monotone convergence of ~. to g.
Theorem
4 (Iterative algorithm). Suppose that the extreme solutions X(0) and X(it,") of A R E ( # ) available for the standard system (1). (a) For X ( # m ) > 0 the equality # . = It., holds and there is no need for iteration. (b) In case of X(#,") 7" 0 consider the following iteration: Set t% = O. Suppose It. and X ( # . ) > 0 are already given. I f X ( # . ) is singular then STOP. Otherwise define ].t. + 1 :'-~ I t . +
are
It ," - it .
fl I- x(It.)-l/2 x(It.) x(It.) -'/2 If
and compute X(#.+a) = X ( I t . ) + Y with the maximal solution Y of A(~.)Y+
Y A ( ~ . ) T + y c T c Y + (It.+1 - I t . ) a ~ ~ = 0.
Equivalently one obtains X ( # . + 1) = X(it,") + Y with the maximal solution of the familiar Riccati equation of optimal control (with fixed matrices) A(~.)Y+
rA(~.)
~ + rC~C~ - (it. - It.+,)6e"
= O.
C Scherer/ Hoo-controlby statefeedback
390
Then X ( g . + l ) is positive semidefinite and the algorithm is thus well defined. • In case of X ( g . ) > 0 in each step, the sequence (1/~fff~.). converges to y, and X ( g . ) converges to X ( g , ) >_ 0. • Otherwise there exists an index n o such that X(/xn0 ) is singular and then the equality g.0 = g* holds.
Remarks. (a) We could of course prevent the matrices X ( g . ) in the iteration from becoming perhaps singular but this would make our algorithm more poorly convergent. (b) Note that we do not run into numerical difficulties when applying the algorithm because the considered solutions of A R E ( g . ) do not grow when /t approaches g , . The high-gain character of feedbacks appears only in inverting these solutions to obtain the proposed controls - B T x ( g . ) - ~ . (c) Exactly in case of X(gm) > 0 we do not need iteration to obtain g , . (d) It is possible to replace g,. and X(gm) in the iteration by any other value 0 < t~ < g m such that X(~) ~ 0.
6. The high-gain aspects Having presented our algorithm, we are aware of the cases arising in the discussed optimization problem. In case X(/xm) > 0 we conclude that the optimum ,/, = 1 / ~ is attained for F, = - B T x ( g . , ) - 1. On the other hand suppose that the optimum y, is achieved for F,. Then the inequality X ( g , ) > 0 must hold. If/~, < g,. then g , could be increased by one step of our algorithm in Theorem 4, contradicting the optimality of/~,. Therefore, 7, is attained iff ~¢, = 1/#~-~ or equivalently X(gm) > 0. Suppose now that the optimal value 7, is not attained. Then we consider for some sequence 7.---' 7, (n ---, ~ ) suboptimal feedbacks F. with ll(C + D F ~ ) ( s I - A - B F . ) - a G I[ oo < Y~We claim that (F.) n must satisfy II f~ II ---' ~ for n ~ oo. Otherwise there exists a subsequence ( F . . ) . which is convergent to some F,. From the admissibility we obtain o ( A + B F , ) c C - u C O and [(C + DF,)(i~oI - A - B F , ) - I G ] * ( C + DF,)(i~oI - A - B F , ) - I G < 7,
(12)
for all ~0 ~ R. Therefore, the matrix on the left-hand side has no poles on the imaginary axis and, by observability of ( C + D F , , A + BF,), we conclude the same for
(st- A
- 8F,)-'G.
Without loss of generality we now assume that the system Y¢= ( A + B F , ) x + Bw + Gv, z = ( C + D F , ) x + Dw
(which results from (1) with u = F , x + w ) is given in the controllability canonical form
f¢= z = (C.
h 2 2 } x + ~B21 W+
U,
G2)x + Dw,
where (Axl, G u ) is controllable and therefore A u is stable. N o w (A2z, Bzl ) is of course stabilizable and we can compute F12 such that A22 + B21F12 is stable, Since the control w = F x with F = (0 1712) is admissible and does not change the closed loop transfer matrix from v to z, the inequality (12) still holds for F. + F instead of F.. Therefore we conclude that the optimal value is achieved, being a contradiction.
C Scherer / Hoo-control by statefeedback
391
T h e o r e m 5. The optimal value %,, is a t t a i n e d i f and only i f
x(
m) > o
and we can choose F, = -BTX(~m)
-1
as an optimal control f o r %,, = 1 / ~ . Otherwise it is necessary to use h i g h - g a i n f e e d b a c k to approach %,,, where suboptimal controls can be computed with the algorithm gioen in Theorem 4.
References [1] T. Ando, Matrix Quadratic Equations (Sapporo, 1988). [2] J. Doyle, K. Glover, P. Khargonekar and B. Francis, State-space solutions to standard H~ and H 2 control problems, submitted to IEEE Trans. Automat. Control (1988). [3] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their Loo-error bounds, Internat. J. Control 39 (1984) 1115-1193. [4] P.P. Khargonekar, I.R. Petersen and M.A. Rotea, H~o-optimal control with state feedback, IEEE Trans. Automat. Control 33 (1988) 786-788. [5] P.P. Khargonekar, I.R. Petersen and K. Zhou, Robust stabilization of uncertain systems and H~-optimal control (University of Minnesota, 1987). [6] E.F. Magdrou, Values and strategies for infinite time linear quadratic games, IEEE Trans. Automat. Control 21 (1976) 447-550. [7] A.C.M. Ran and R. Vrengdenhil, Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems, Linear Algebra Appl. 99 (1988) 63-83. [8] J.C. Willems, Least-squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control 21 (1971) 319-338.
383
H -control by state feedback: An iterative algorithm and characterization of high-gain occurence Carsten S C H E R E R * Mathematisches Intsitut, Am Hublan~ D-8700 Wiirzbur& West Germany Received 29 December 1988 Revised 8 March 1989
Abstract: We present an iterative algorithm to compute the achievable H~-norm by state feedback for a standard regular problem and give a characterization when high-gain feedback is necessary to approach the optimal value. If there is no need for high-gain components to approximate the optimal value, we can reduce the problem to the computation of the H~-norm of a certain stable transfer matrix. Keywords: Hoo-optimization; high-gain feedback; Riccati equations; state feedback.
1. Introduction
Recently, Doyle, Glover, Khargonekar and Francis presented in [2] an algorithm for H~-control by state feedback which uses the fact that one can decide whether a prescribed number y is greater or smaller than the optimal value y, of the achievable Ho~-norm. This decision is made by solving an algebraic Riccati equation and looking for positive definiteness of the solution. In their paper [2], nothing is said about the behaviour of the suboptimal feedback matrices (which can be computed from the solution of this Riccati equation) if the parameter y approaches y,. Especially, the high-gain aspects of suboptimal feedback matrices (though dealing with regular problems) are completely neglected. In the present paper we will give a criterion for the case when high-gain feedbacks are necessary to approach the optimal value of the Ho~-norm and when the optimal value is really attained. Furthermore, we present an iterative algorithm which yields in each step a better upper bound approximation of the optimal value 7, together with suboptimal feedback controls. In each step we only have to solve a s t a n d a r d Riccati equation of optimal control and we can show convergence of the bounds to the optimal one. We denote by R and C the real and complex numbers where C is partitioned in the usual way into C - U C°U C + denoting the left half-plane, the imaginary axis and the right half-plane respectively. All matrices are to be considered as being real and of suitable dimension.
2. Assumptions and formulation of the problem
We assume our system to be given by =
+ Su + C.,
x ( O ) = O,
z = Cx + Du,
* Supported by Deutsche Forschungsgemeinschaft, KN 164/3-1.
0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
0a) (lb)
C. Scherer/H~-controlbystatefeedback
384
where all matrices are real and x is the state, u the control, v the external disturbance and z the controlled output. Furthermore we assume that the following regularity assumptions hold: • (A, B) is stabilizable and (C, A) is observable. • D T ( c D ) = (0 I). • G~O. These assumptions especially imply that the pair (C + DF, A + B F ) is observable for all matrices F of suitable dimension. We assume that the state of (1) is available for control and want to construct a compensator which stabilizes the system and reduces the L2[0, oo)-induced norm of the map from the disturbances v to the output z as far as possible. Since we are mainly interested in computing the least possible value of the norm and since this value does not change if using static state feedback instead of dynamic feedback laws (see [4,5]), we consider u=Fx
witho(A+BF)cC-
as admissible controls. Thus we are led to the optimization problem y,=
inf
]](C + D F ) ( s I - A
-BF)-IG[Io~
F admissible
where II H ( s ) II ~ is the H~-norm of the proper stable transfer matrix H(s). In this paper we wish • to compute y, (perhaps with the help of iteration); • to characterize when y, is attained and compute a corresponding F,; • to compute suboptimal feedback matrices when 3', is not attained; • to characterize the ease when high-gain feedbacks (i.e. 1]F ]l tends to infinity) are necessary to approach 3',. All these problems are investigated by considering the solutions of certain algebraic Riccati equations (ARE).
3. H~-optimization and Riccati equations The following main result of this section is partly proved in [2], but we want to include some extra results and give an independent elementary proof. Theorem 1. For the system (1) the following statements are equivalent: q admissible F: 3X>0:
]1(C + D F ) ( s I - a - B F ) - 1 a II ~ - Y;
(2)
GG T x A T + A X + - - - -,{2 BBT+XCTCX=O;
(3)
AT +eA+ I-7-
(4)
Every solution P > 0 of (4) is in fact positive definite and F = - BTp yields an admissible feedback law such that the inequality in (2) is satisfied. For this F we get equality in (2) if and only if
o A-BBTP+GaTP t/n C o
C Scherer/ H~-controlbystatefeedback
385
or equivalently a( A + P - 1 c T c ) C~C ° 4 ~ . The proof of this theorem proceeds by two lemmas which are interesting in their own right. The first one is a slight generalization of a result of [8], since we drop the assumption of controllability.
Lenuna 2. Suppose the matrix A is stable. The inequality GT( --i,M-- A X ) - ' Q ( i o d - A ) - ' G < r : I
(5)
holds for all ¢o ~ R iff there is a solution K -- K T of the algebraic Riccati equation ATK+/CA _ KGGT ./2 K _ Q = 0.
(6)
Proof. The frequency-domain inequality is of course necessary for the existence of a solution K = K T
of (6). For the converse we assume without loss of generality the pair (A, G) to be of the form A1,
A12
such that (A n, G , 0 is controllable. By partitioning Q and K accordingly, the Riccafi equation (6) can be split up into three equations
AT, K1, + K h A n
_
--al
G,,c .[2
,i
--n-
Qll = 0 ,
(7)
( .4T1-- gll GI'GK12+Kll.412+Kx2.422-Q12=O'./2 ------~I)
(8)
tc GnGT1 ATzK12 + AT2K22 + K2zAz2 + K2aA,2 - "'21 `/z "',2 - Q22 = 0.
(9)
Because
.i
,o (,..
1o1,)
we obtain from (5) the inequality
GT(--i~oI-- AT1)-'Qn(iwI-- A n ) - I G , I ~ `/21 for all ~0~ R and therefore (see [8], Theorem 5) there is a solution K,1 = K ~ to the ARE (7) with
G,I GT Thus, by stability of .4, the spectra of -(.411 - (GnG~,/`/2)Kn) and .422 are disjoint and (8) has, as a standard Sylvester equation, a solution Kx2. By defining K21 := K~, we infer in the same way the existence of a unique K22 = K ~ satisfying (9). []
C. Scherer/H~-controlbystatefeedback
386
L e m m a 3. Suppose the system (1) is influenced by some u(.), v(.) ~ L2o~[0, oo). I f there is a solution P >_0 of the A R E
ATp + PA + P ~
BBr
p + cTc=o
then for all T > 0 the following relation holds: ( T I I z(t)l[ 2 a t < ]¢2(T[[ v ( t ) H 2 dt + I T H B T p x ( t ) + u ( t ) l l 2 dt.
Jo
Jo
Jo
Proo|. The proof follows from a standard ' c o m p l e t i n g the squares' a r g u m e n t where one has to note z(t)Tz(t) = x(t)TCTCx(t) + u(t)Tu(t) and x(0) = 0 (see [6]). [] Proof of T h e o r e m 1. ( 2 ) = (3): F r o m L e m m a 2 we infer the existence of K = K T with
K + (C + DF)T ( c + D F )
( A + BF)TK + K( A + BF) = K-~ T
and thus, by stability of A + BF, we conclude K < O. With Z :~ - K this equation results in
ATz + ZA + Z GGT y2 Z + CTC + ( B T Z + F ) T ( B T Z + F ) - Z B B T Z = O. N o w Z x = 0 implies x T C T C x -~- 0, i.e. Cx = 0 and xTFTFx = xT(BTz + F)T(BTZ + F ) x and therefore we obtain Z A x = 0. Since (C, A) is observable, this implies x = 0. Thus Z is invertible and Z - 1 is a solution of the matrix inequality
A X + Xz~T + x c T c x
= 0,
i.e. Fx = 0
GG T + - -- BB T < O. y2
F r o m T h e o r e m 2.1 in [7] we obtain the existence of a matrix X > Z - 1 > 0 which solves the corresponding Riccati equation. (3) =, (4): Choose P :--- X -1, (4) ~ (2): F r o m the A R E in (4) we obtain
(ooT )
(A-BBTp)TP+P(A-BBTp)+P
7+BBT
p+cTc=o.
Since ( P ( G G T / y 2 + B B T ) P + c T c , (A - BBTp)) is an observable pair, we obtain from T h e o r e m 3.3 in [3] the stability of A - BBTp. L e m m a 3 with u = --BTpx and v ( . ) ~ L2[0, oo) leads to the inequality
II~(') IILqo,~) ~ 2 -< ~2 IIv(.) IIL2t0,~}. implies II(C + D F ) ( s I - A - B F ) - I G IJ ~ -< y for the feedback F . . . . BTp. Now we derive the remaining statements of Theorem 1: First, the nonsingularity of P follows from the chain of implications This
Px=O
~
xTCTCx=O
~
Cx=O
~
PAx=O
by the observability of (C, A). Secondly, it is well known that equality in (2) holds for F = - B T p matrix A -
BBTP
H=
GG T T2
--(cTc+~Te)
-(A-BBTp) T
if and only if the H a m i l t o n i a n
C. Scherer / Hoo-control by state feedback
387
has eigenvalues on the imaginary axis. (Note that (C + D F ) T ( c + D F ) = CTC + P B B T p holds true in our situation.) Now one easily checks
H
-v
=
i
e
o
I
t (1
°°T't
T
(
U-
as well as
(, 0
0
I
Thus the theorem is completely proved.
?
0
-(A+p-1cTc)
T
t
"
[]
Motivated by this theorem we denote by A R E ( g ) the algebraic Riccati equation A X + XA T + x c T c x
+ ttGG T - BB T = O.
Suppose now that A R E ( p ) has a symmetric solution. Since (A T, C T) is controllable, we deduce again from Theorem 2.1 in [7] the existence of a maximal symmetric solution X(g) of A R E ( g ) which has the additional property that the closed-loop matrix A ( g ) := A -I- X ( g ) c T c
has only eigenvalues in C + t.) C °. Furthermore, for v _
-4- vGG T - B B T = (it - g ) G G T <_<0
and thus, by maximality, the existence of X ( v ) with X ( v ) >__X(g). Therefore, X(#) is a nonincreasing map on its domain of definition. (Note that the results in [7] are stated for complex matrices but the given proofs carry over to the real case without change.) It is clear that we have thus reduced our original problem to the computation of g . defined by g . = s u p ( g I X ( g ) exists and X ( g ) > 0}. Once -¢. = 1 / g ~ .
is computed, it is easy to obtain suboptimal controls for any desired bound y > ~,..
4. Extreme solutions of ARE(p) The main idea of the paper consists of the following observation. Let us define F( Z ) :-= A Z -4- ZA T -4- z c T c z
and compute F(X+ Y) - F(X) = Y(A + xcTc) T + (A + xc'rc)y+
ycTcy.
If X(#I) and X ( g : ) are defined, the f o l l o ~ n g holds for A X : = X(g2) - X(gl): F(X(ga))+glGGT-BBT-=O, =
i(gl)
F(X(g2))+g2GGT-BBT=O
AX-.I-AXA(gl)T..jt-AXCTCAX-4 - (g2-gl)GaT=o.
(10)
388
C Scherer/ Hoo-control by state feedback
We observe that we can compute X(g2) from X ( g l ) by solving (10) for A X via
/ ( g 2 ) ~- X ( g l ) "+"AS. N o w we can describe what we want to call extreme solutions. First of all, we compute X ( 0 ) > 0 which exists, because ARE(0) is a standard Riccati equation of optimal control and which has the additional property that A(0) has no purely imaginary eigenvalues. In the second step, we compute the maximal value g,, for which a symmetric solution of A R E ( g ) exists at all. This is equivalent to finding the maximal g,, such that
A(O)Y+ YA(O) T + y c T c y + g a g T = 0
(11)
still has a symmetric solution for/~ = g,,. This can be decided in a straightforward manner because - A ( 0 ) is stable and according to L e m m a 2, a solution exists iff the inequality II C(sI + A(O))-IG II oo < - 1 / f f f holds. Thus we have to compute the H ~ - n o r m g,~ of the transfer matrix
C(sI + A(0))-'G which is clearly positive since (C, A(0)) is observable and G does not vanish. In this way we obtain g,, .'= 1 / y ~ < oo and compute X(g,,) with the help of the maximal solution of (11) for g = ttm. It is now clear that we call • X(O) for g = O, • X(g,.) for the maximal possible value g,., extreme solutions of A R E ( g ) because the optimum g , lies in the interval [0, g,.] and because we can construct (sub)optimal controls with the help of the solutions X ( g ) with the property x(O) > X ( g ) > X(g.,).
5. An iterative algorithm Suppose that the extreme solutions of A R E ( g ) are computed and the assumption X ( g ~ ) ~' 0 holds. Now we are in the position to explain how the iteration proceeds. For this reason we suppose that g , and X ( g , ) > 0 are already computed (starting with g0 = 0). Then we search at t, such that
x(g.)
+
t(x(g=)
-
x(g.))
>
0
holds for all t ~ (0, t , ) and that the left-hand side becomes singular for t = t o. Because of the equivalences
x ( g . ) + t.( x ( g ~ ) - X ( g . ) ) >_ 0 ~, I + t a X ( g , ) - l / 2 ( X(gm) •.
Lxtn
x(~.l-'/2(x(g.)
-
X ( g , ) ) X ( g , ) -1/z > 0
- x ( g m ) ) X ( g . ) - ' / 2 >__0,
we define for X ( g ) > 0 the function
t(g) ,=
1
II I
-
X(g)-'/2 x(t~.,)x(~) - ' / 2
II
~ (0, 1]
and take t, = t ( g . ) . With this value we get the new parameter from the convex combination
g . + l = (1 - t . ) g . + t.gm = g . + t.(gm - g . ) ~ ( g . , gm]. Since t,2 <_ t,, we obtain for A X = X(gm) - X ( g , ) the inequality
A O , . ) ( t . a x ) + (t. a x ) a ( g . )
T + (t. a x ) C c ( t °
a x ) + t.(~,m - ~ . ) c G T < 0.
C Scherer/ Hoo-controlby statefeedback
389
Thus we conclude that the maximal solution Y of the corresponding A R E satisfies Y ~ t. (X(It,.) - X(It.)) and we obtain, by the definition of t., the inequality
X(~.+ 0 = X(~.) + :+_ O. We could arrive at two possible situations: X(it.+l ) is singular. In this case we have reached the optimal value # , = # . + 1 : Indeed, for # . < # < It,+1 we can determine t ~ (0, tn) with It = # . + t(it," - Its) and therefore X ( # . ) + t ( X ( # . ) X(it.)) > O, i.e. X ( # ) > O. • X ( # . + I ) > O. Then it is clear how to iterate. F r o m these considerations it follows that X ( / z . ) > 0 implies t o - 1 and thus p~ = # . , i.e. p , = ~ . . In case of X ( I t . ) ~ 0 we obtain It, < I t . and the algorithm either stops with #~+1 = # , or yields an infinite sequence (P,)n. In the latter case, ( # . ) . is increasing and bounded b y It,", i.e. convergent to a limit/2 ~ (0, # . ] . Besides this we obtain in each step suboptimal feedback matrices - B T X ( I t . ) -1 for the bound y = 1 / p ~ . Note that ( X ( p . ) ) , is nonincreasing, bounded from below by 0 and thus converges too, where the limit must be x(#). Now we want to show ~ = # , by assuming ~ < # , . By definition of It,, X(/~) must be positive definite and therefore t(/~) > 0 well defined. As an intermediate step we show that t(it) is a nonincreasing function for increasing # in the domain of definition of t(p); this can be seen as follows: It >_ ~ ~
x(it) ~ x(~)
[ 1 - ( t ( ~ ) - e)] X(#) < [ 1 - ( t ( # ) - e)] X(v) =
x(~)
+ (t(~) - ,)(x(~.)
- x(~))
~ x(~) + (t(~) - ~)(x(~,")
- x(~))
for all e ~ (0, t(it)). Since the left-hand side is positive definite for these values of e, we obtain t ( # ) < t(v) as desired. By t(#.) = #.+l-it. it is clear that ( t ( # . ) ) . converges to 0 for n ~ oo and that there exists some n o with 0 < t(#.o) < t(g). But this impfies #.o > t~, contrary to the monotone convergence of ~. to g.
Theorem
4 (Iterative algorithm). Suppose that the extreme solutions X(0) and X(it,") of A R E ( # ) available for the standard system (1). (a) For X ( # m ) > 0 the equality # . = It., holds and there is no need for iteration. (b) In case of X(#,") 7" 0 consider the following iteration: Set t% = O. Suppose It. and X ( # . ) > 0 are already given. I f X ( # . ) is singular then STOP. Otherwise define ].t. + 1 :'-~ I t . +
are
It ," - it .
fl I- x(It.)-l/2 x(It.) x(It.) -'/2 If
and compute X(#.+a) = X ( I t . ) + Y with the maximal solution Y of A(~.)Y+
Y A ( ~ . ) T + y c T c Y + (It.+1 - I t . ) a ~ ~ = 0.
Equivalently one obtains X ( # . + 1) = X(it,") + Y with the maximal solution of the familiar Riccati equation of optimal control (with fixed matrices) A(~.)Y+
rA(~.)
~ + rC~C~ - (it. - It.+,)6e"
= O.
C Scherer/ Hoo-controlby statefeedback
390
Then X ( g . + l ) is positive semidefinite and the algorithm is thus well defined. • In case of X ( g . ) > 0 in each step, the sequence (1/~fff~.). converges to y, and X ( g . ) converges to X ( g , ) >_ 0. • Otherwise there exists an index n o such that X(/xn0 ) is singular and then the equality g.0 = g* holds.
Remarks. (a) We could of course prevent the matrices X ( g . ) in the iteration from becoming perhaps singular but this would make our algorithm more poorly convergent. (b) Note that we do not run into numerical difficulties when applying the algorithm because the considered solutions of A R E ( g . ) do not grow when /t approaches g , . The high-gain character of feedbacks appears only in inverting these solutions to obtain the proposed controls - B T x ( g . ) - ~ . (c) Exactly in case of X(gm) > 0 we do not need iteration to obtain g , . (d) It is possible to replace g,. and X(gm) in the iteration by any other value 0 < t~ < g m such that X(~) ~ 0.
6. The high-gain aspects Having presented our algorithm, we are aware of the cases arising in the discussed optimization problem. In case X(/xm) > 0 we conclude that the optimum ,/, = 1 / ~ is attained for F, = - B T x ( g . , ) - 1. On the other hand suppose that the optimum y, is achieved for F,. Then the inequality X ( g , ) > 0 must hold. If/~, < g,. then g , could be increased by one step of our algorithm in Theorem 4, contradicting the optimality of/~,. Therefore, 7, is attained iff ~¢, = 1/#~-~ or equivalently X(gm) > 0. Suppose now that the optimal value 7, is not attained. Then we consider for some sequence 7.---' 7, (n ---, ~ ) suboptimal feedbacks F. with ll(C + D F ~ ) ( s I - A - B F . ) - a G I[ oo < Y~We claim that (F.) n must satisfy II f~ II ---' ~ for n ~ oo. Otherwise there exists a subsequence ( F . . ) . which is convergent to some F,. From the admissibility we obtain o ( A + B F , ) c C - u C O and [(C + DF,)(i~oI - A - B F , ) - I G ] * ( C + DF,)(i~oI - A - B F , ) - I G < 7,
(12)
for all ~0 ~ R. Therefore, the matrix on the left-hand side has no poles on the imaginary axis and, by observability of ( C + D F , , A + BF,), we conclude the same for
(st- A
- 8F,)-'G.
Without loss of generality we now assume that the system Y¢= ( A + B F , ) x + Bw + Gv, z = ( C + D F , ) x + Dw
(which results from (1) with u = F , x + w ) is given in the controllability canonical form
f¢= z = (C.
h 2 2 } x + ~B21 W+
U,
G2)x + Dw,
where (Axl, G u ) is controllable and therefore A u is stable. N o w (A2z, Bzl ) is of course stabilizable and we can compute F12 such that A22 + B21F12 is stable, Since the control w = F x with F = (0 1712) is admissible and does not change the closed loop transfer matrix from v to z, the inequality (12) still holds for F. + F instead of F.. Therefore we conclude that the optimal value is achieved, being a contradiction.
C Scherer / Hoo-control by statefeedback
391
T h e o r e m 5. The optimal value %,, is a t t a i n e d i f and only i f
x(
m) > o
and we can choose F, = -BTX(~m)
-1
as an optimal control f o r %,, = 1 / ~ . Otherwise it is necessary to use h i g h - g a i n f e e d b a c k to approach %,,, where suboptimal controls can be computed with the algorithm gioen in Theorem 4.
References [1] T. Ando, Matrix Quadratic Equations (Sapporo, 1988). [2] J. Doyle, K. Glover, P. Khargonekar and B. Francis, State-space solutions to standard H~ and H 2 control problems, submitted to IEEE Trans. Automat. Control (1988). [3] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their Loo-error bounds, Internat. J. Control 39 (1984) 1115-1193. [4] P.P. Khargonekar, I.R. Petersen and M.A. Rotea, H~o-optimal control with state feedback, IEEE Trans. Automat. Control 33 (1988) 786-788. [5] P.P. Khargonekar, I.R. Petersen and K. Zhou, Robust stabilization of uncertain systems and H~-optimal control (University of Minnesota, 1987). [6] E.F. Magdrou, Values and strategies for infinite time linear quadratic games, IEEE Trans. Automat. Control 21 (1976) 447-550. [7] A.C.M. Ran and R. Vrengdenhil, Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems, Linear Algebra Appl. 99 (1988) 63-83. [8] J.C. Willems, Least-squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control 21 (1971) 319-338.