Stability radius for structured perturbations and the algebraic Riccati equation

& Control North-Holland

Systems

Letters

8 (1986)

105

105-113

Stability radius for structured perturbations and the algebraic Riccati eauation or in state space form

D. HINRICHSEN ForsclItolgssclrwerpurlkt, Universitiit Bremen.

2800

Dvumische Brenletr

33,

i= 1-a2O

$vslet?~e. West Germutt~~

A.J. PRITCHARD Control Cooenrty

Received Revised

Theory Cenlre, Unicrersity CV4 7A L, Euglund

of Wunvick.

24 March 1986 1 July 1986

In this paper we generalize the notion of stability radius introduced in [l] to allow for structured perturbations. We then relate the stability radius to the existence of Hermitian solutions of an algebraic Riccati equation and give some applications of this result.

Abstrucf:

Kqtworrls:

Perturbations,

Riccati

equation,

Stability.

1. Introduction

In [l] we introduced the notions of real and complex stability radii of a stable matrix. Here we concentrate entirely on the complex stability radius but consider structured perturbations. If i = Ax is the nominal system we assume that the perturbed system can be represented in the form 1=(A+BDC)x

0)

where D is an unknown disturbance matrix and B, C are known scaling matrices defining the ‘structure’ of the perturbation. Throughout the paper AE@“~“, o(A)cC-, BE~“~“‘, CE Q)Px’?, D E Q=Yxp. The matrices B, C may reflect, for instance, the possibility that not all of the elements of A are subject to perturbations. Consider, for example, the linear oscillator E+ a,$+ az[ = 0 0167-6911/86/$3,50

8 1986. Elsevier

Science

Publishers

1 -a,

1 x*

If we assume that a,, a2 are uncertain we can account for this by taking B = [,“I, C = I,. Whereas if only a, is subject to perturbation we take B = [,“j, C-= [l 01. Formally, (1) may be interpreted as a closed loop system obtained by applying static linear output feedback (with unknown gain matrix D) to the system 1= Ax. However, we would like to emphasize that in this paper B, C do not represent input resp. output matrices but model the available knowledge about the structure and scale of uncertainty of the system parameters. Therefore, controllability and observability properties are not very natural assumptions for the triple (A, B, C) in our context. The entries of the perturbed matrix A + BDC depend affinely on the unknown entries d,, of D. But not every affine perturbation of the entries of A can be represented in this way. More structural possibilities would be covered by considering A + c BiDiCi. i-l

But a treatment of these more general perturbation structures is substantially more complicated and so we will not pursue this here. We proceed as follows. In Section 2, we introduce the stability radius of a stable matrix when subjected to perturbations of fixed structure (B, C). We then derive a formula for this ‘structured stability radius’ and characterize it by various conditions. In Section 3 we relate the stability radius to the existence of Hermitian solutions of a parametrized nonstandard algebraic Riccati equation. Solutions of general Riccati equations have been investigated by Brockett [3], Willems [2] and Popov [4]. However, their results are not directly applicable

B.V. (North-Holland)

D. Hinrichsen.

106

A.J.

Pritchurd

/ Stuhility

in our situation since we wish to avoid controllability assumptions. Fortunately, the main theorems remain valid without the assumption of controllability, if the system matrix A is stable. In Section 4 we study the parameter dependence of the stabilizing solution Pp of the algebraic Riccati equation and analyze in particular, how the stability radius and the spectrum of the closed loop system change with the parameter p. This question is in a sense the complement of the cheap control problem. In Section 5 we show how to determine a quadratic Liapunov function of maximal robustness for a linear system and describe how this function can be used to cope with nonlinear, time varying or even dynamic perturbations (i.e. perturbations ‘with memory’ resulting for example from neglected dynamics).

2. The structured stability radius first characterizations

definitions and

dim

for structured

perturhutions

Proof. First suppose that for some D E Cntx”, x E C”, x#O, wER, (A+BDC)x=iwx

(5)

then x = (iol-

A)-‘BDCx.

Since A is stable, Cx # 0. Set u = Cx, so u = G(io) Du.

(6)

If G = 0 this leads to a contradiction and so rc = co. If G f 0 then (6) implies ]I G(io) ]I ]I D II & 1 and thus 1 rc r IJ~ 11G(iw) I] . Suppose the maximum of w + ]I G(io) ]I occurs at w,,, say. The singular value decomposition of G(iw,) has the form ,ll G(iw,) = c sjuju,? j-l

Generalizing (2) of [l] we define the stability radius of the stable matrix A E Cnxn with respect to perturbations of a given structure (B, C) E q=“Xm x Q=PX”,by rc =r,(A;

=inf{

a(A+BDC)n6+#,@}

(2)

where I] D I] denotes the operator norm of the linear map D:CP+Q=” with respect to the Euclidian norms on Q:“‘, Q=Pand E+= {SEC;

II‘j IICr = II ‘j IIC”’ = ’ and

B, C)

[ID]];

where uj E Q=P, vi E Q=“I,

sl= ]]G(iwe)]] Let D = s;‘u~u:, C(iwcl-

x = (io,Z IIDII;

a(A+BDC)niR#$}.

(3)

x = (iw,l-

2.1. Proposition.

(io,l-

A)-‘BDu,

= ul, or

= ul.

Let

- A)-‘BDu,

A)-‘BDCx

or A)x i BDCx.

So that

A)-‘B.

a(A+BDC)d+#fl

Then

r, =

then G(iwo)Du,

then Cx = ui, hence x z 0 and

Note that m = p = n, B = C = I, corresponds to the unstructured case studied in [l]. .

G(s) = C(sI-

..a ks,.

If we set

ReskO}.

Clearly r,=inf{

zs,k

1 pig IIG(iw) II cc

and

ifG+O,

(4) ifG=O.

1 II DI’ = 11G(iw,)

1 I] = ff% ]I G(iw) ]I ’

• I

D. Hinrichsetr.

A.J.

Pritchurd

/ Stubi1it.v

As a consequence of the above characterization of the structured stability radius, we have

rudius

for structured

where uj(iw) E C J’, uj(iw) E Q=“I, 11

11CP= 11

Uj(ia)

r,(A;

B, C) =rc(TAT-‘;

TB, CT-‘),

(7)

This may seem surprising since we showed in [l] that the stability radius rc( A) = rc( A; I, I) can change substantially under similarity transformations on A. Note, however that in (7) the scaling matrices have also been transformed, whereas in [l] we considered r&TAT-‘; I, I) and not r,.(TAT-‘; T, T-l). A nother interpretation of the stability radius which will be useful later can be obtained by considering the convolution operator L2[0, co; cq

(Lu)(t)

--) L2[0, co; cq,

(8)

= JdC eAcrms)Bu(s) ds.

Note that G(s) is the Laplace transform of the weighting function. We have as an extension of a result in [5].

si(io)

= ]I G(io) ]I 2 s2(iw) 2 * . + 2 s,,(iw).

Then s,(iw), ui(iw), ut(iw) can be chosen to depend continuously on w and for every o such that G(iw) # 0 there exists D(iw)=

1 -ai(io)uT(iw), sl(iw)

satisfying G(iw)D(iw)ui(iw)

(9)

IILII= r,(A;l B, C). Proof. We denote the Fourier-Plancherel

trans-

form of u(t) by v^(iw). Then = G(io)o^(io)

for u E L*[O, co; C “‘1 and by Parseval’s Theorem

IILu Il&,.m;C~l= &/_m

IIG(io)u^(iw) 00

w ++ 11G(iw) 11 occurs at

CC(iw) = y,(io)D(iw)ui(iw), where

y,(iw) =

2.2. Proposition.

= ui(iw).

Suppose the maximum oo. Let

1

(z)(o)

1)~“1~ 1

Uj(i@>

and

TE Gl,,(C).

L:

107

perturbdons

E1/2 i 0

if wE [oc, We-i-e], otherwise.

Then 11Lo, lI:~~O,m;C~l= +-juocldo = $3 00 1 o,+e 1. II 0,ll;‘[O,m:C”‘]= G I w,, 11G(io) ]I * dw’ Hence

II& dw and 1

Hence

II-4 IIL*2[O,oo;C~] + 5

IILII s-.r;

as E 10. Thus I] L II = l/rc.

If rc = 00 equation (9) follows directly from (2) since G = 0 is equivalent to L = 0. Now let G $0 and suppose the singular value decomposition of G(iw) is

The stability radius also plays a role in the following optimization problem.

G(io) = csj(iw)uj(iw)uj*(iw). i

Minimize J( x0, u)

0

D. Hiwichsen.

108

A.J.

Pritclwrcl

/ .Wtbili~~~ rdius

subject to u E L2[0, 00; P], Z=Ax+Bu, x(0)=x0, y=cx.

(11)

We consider (10) as a parametrized optimization problem. If p I 0 this is the usual linear quadratic regulator problem. In the case where p > 0 the state penalty is negative and we would expect the resulting feedback law to deteriorate the stability of A and this will in fact come out in the analysis (Proposition 4.1). The next proposition characterizes the set of parameter values p for which the optimal value starting at zero is nonnegative. 2.3. Proposition. rrGL’[O

inf ,mC”‘].

for sirrtclu,.erl

perIrrco’l)ulio~ls

literature there are two basic approaches to solving general algebraic Riccati equations (see [2,3,4]). One is via the associated linear quadratic optimisation problem [2] and the other proceeds by direct analysis. Optimization techniques can only be used if the minimum costs are finite. To this end we have: 3.1. Proposition. Suppose a( A) C Q:-, rc < 03, rhen for alf p E (- 00, rz),

inf J(xa, u)i
(13)

Proof. Since a( A) c Q=_ we need only consider the case p E (0, rc) and show

inf J(x,, L~EL*[O.cu:C”‘]

J(0, u) = 0

xaEQZ=“.

0) > -co,

XOEC”.

We have

eplrz

ol-pG*(iw)G(iw)

20

foralf

J(O, 0) =lm[ Ilu(r> Il&--r~ll(~u)(~)

WEIR. (12)

+(rd-~)~wlI(~~)(~)ll~~ 0

Proof. We have

II&] dt dt.

Hence

y(t) = Cp”(+‘)Bu(s)

ds= (tu)(r).

J(0, u)>(r,2-

Hence

~)&=ll(Ld(d

II& dt.

But

J(Ovu>= IIlJIl:‘~o.m:c”‘]-P IIJ!JIlL2’[0.m;Cq. For p I 0, (12) is obvious and for p > 0 the first equivalence follows from Proposition 2.2. The second equivalence is a consequence of Parseval’s Theorem which implies J(0, u) = +-/_” (I-

do.

0

3: Characterization of r,(A; B, C) via a parametrized algebraic Riccati equation

In this section we consider the parametrized algebraic Riccati equation A*P+PA-pC*C-PBB*P=O.

-2p Re om(C eA’xo, (L)(t)) /

dt

+J(O, 0) and since for OL> 0, 2ab s a2/a + ab2, we have

(O(iw), p~*(i~)G(i~))8(iti))c~,

J(‘% 9 u) = -p / oml( C eA’xp II& dr

Jbo 3 u)a

-p(l+~)~mlJCe”‘x~ll~~dt

f ($2 - P - ~)~mll(Lo)(~) II& dt. For p < r,j we may choose a! sufficiently that

J(x,, u) 1 -p 1 + (

(ARE, >

We will say that a solution P of (ARE,) is a stabilizing solution if a(A - BB*P) C @-. In the

:I/,

small so

*UC eA’x, ll~n dt

from which the result follows since a( A) Cl

c

Q3-.

D. I-liwichsetr,

A .J. Pritclwrd

/ Stuhility

Using this proposition it is in fact possible although by no means trivial - to show that there exists a unique stabilizing solution P, of (ARE,) for p E (- 00, rz), without any controllability assumption. Moreover 5(t) = - B*P,x(t) minimizes the performance index (10) subject to the dynamics (11) and Jb,,

for structured

109

perturhutions

is not equivalent to inf J(x,,

u) > -CO

VEL

(even though (A, 8) is controllable). Moreover (ARE,lc)-may have a solution P with a(,4 BB*P) c C _ although the associated optimal control problem (lo), (11) does not have a solution.

C) = (x0, P~x,)c,L

However, we tried without optimization approach to which is of special interest optimization techniques lustrated by the following tive example.

success to extend the the case where p = rc, to us. The reason why fail for p = rc is ilsimple but representa-

,t= -x+0

,

x(0) =x0

and u)=/“[Iu(t)I’-Ix(t)I’]

dt.

0

It is easy to see that r, = 1. (ARE,.;) has the form -2p-l-p’=Oandissolvedbyp=-1. On the other hand,

=

dw

a(A-BB*Pr&

cE=_.

ForaNpE(-m,

$1,

Pp=P,,*.

Proof. Choose T E Gl,,(C) such that

TB=

4 10 I’

CT-‘=

[C,

C,]

and (A,, B,) is controllable. Note that

= --&-(2Rex,J(iw)+

1x01*]

1

dw.

If we set $,,(io) =

In order to derive a comprehensive result including the case p = rc we will now analyze (ARE,) directly following Brockett’s development in [3] and extend this theorem to not necessarily controllable systems. 3.3. Theorem. Suppose a( A) c Q=-, r, < co, p E (- 00, r,?). Then there exists a unique stabilizing solution Pp of (ARE,). Moreouer when p = rc there exists a unique solution Pr; of (ARE,;) having the property

3.2. Example. Consider the scalar equation

Jb,,

radius

C(iwI-A)-lB=Cl(iwl-Al)-lBI,

o E Iw. Multiplying (ARE,) on the left by T-l* and on the right by T-’ and setting

T-‘*pT-’ = (un312

i0

gives

otherwise,

then for large n, 27rJ( x0, u,,) - fa’ - TT1x0 12 - 2c&2

Re x0.

Choosing a such that OLRe xe > 0 we see that u,,) 3 -co as n -+ co. This shows that

J(x,,

min J(0, u) = 0 LIEL’

-

P,B,B:P, P3B,B:P,

P,B,BfP2 P3B,B:P2

I

=‘*

.

D. Hitrrichsen,

110

A.J.

Pritchard

/ Stuhili(v

so P,A, + A,*P, - pC$Z, - P,B,B:P, P2A3 + (A, - B,B:P,*)*P,

= -P,A,

= 0,

(144

+ pC;LC,,

radius

for structured

perturhutions

Theorem 3.3 and Proposition 3.4 show that rc can be characterized as the maximal parameter value p for which (ARE,) has an Hermitian solution.

W) P3(A, - B,B:P,)

i- ATP, = -A;P,

4. Parameter dependence of the solution Pp

+ pC2Cl, (14c)

+ P3B, B,*P, .

In this section we examine the map

p +BPp on

(- 00,$1.

P4A, + ATP., = - P3A, - A;P, + pC,*C, (144

Since (A,, B,) is controllable there is a unique Hermitian solution PIP of (14a) with the property

4.1. Proposition. Suppose u(A) c Q:_ and set A,, = A- BB*P,. 9 (a) The maps p c) P,, p * A,, are differentiable on (- 00, r,‘) and continuous on (- 00, $1.

a(A,-BIB,*PI,)cCof i:

if pE(-oo,

~~~~~~~$

22~~

- Pp, is a solution

r,.?) and

u( A, - B,B:( PI),;)

c E-

A,*,P+PA,,-(p-pl)c*c-PBB*P=O

(see [3]). Replacing PI by PIP in the Liapunov equations (14b), (14~) and using the fact that a(A,) c Q=- it follows (see [6]) that the solutions Pzp, P3p are uniquely determined by (14b), (14~). Moreover P3p = P$,. Substituting on the right hand side of (14d) we see that P+, is uniquely determined and P4*p= P+. Finally note

05)

satisfying u ( API -BB*(P,-P,,))c@_.

(d) If p -< r,?‘, then r,?( A,; B, C) = $(A;

B, C) - p.

06)’

Proof. Assume p1 < r,?, then u(A ,) c @-. Let P,, be the solution of (ARE,), p I r,-j< then

qp-5,)

+ P-P,,)A,,

-(p-pl)C*C-(P-P,,)BB*(P-P,,)=o. 07)

As a converse to this theorem we have: 3.4. Proposition. Suppose a( A) = Q=_. If there exists a solution of (ARE,) with P = P*, then necessarily p I rz. Proof. If P = P* satisfies (ARE,), (A-iwl)*P+P(A-iwI)-pC*C-PBB*P =o,

OER.

so Os(B*P(A-ill)-‘B-I)* .(B*P(A-~J)-‘B-I)

= I-

pG*(iw)G(iw),

Thus p I rc by (12).

w E NJ!. IJ

Hence Pp - Pp, = -Jgme”2[(p

- pl)C*C

+ ( Pp - P,,,) BB*( P,, - Pi,)] e%’ dt

10

then

ifpkp,.

(18)

So Pp is decreasing on (-co, $1 and bounded below by P,.;. It follows that Pp tends to a limit F as p T rc. Since 1; satisfies (ARE,z) and a(A ‘BB*k) c c-, we musi have j”= Pr; by the uniqueness statement in Theorem 3.3. This proves the continuity in (a). It then follows from (18) that the map p r) P, is differentiable on (- 00, rz) and in fact

dPPdp --

/oD”eA%*C

eApr dt I 0.

09

D. Hinrichsen,

A.J.

Pritcburd

/ Stuhili!y

So we have proved (a), (b). Part (c) follows from (17) and the continuity of the map p * Pp on (- co, r,?]. It remains to prove (d). Since P,,, - Pp, is an Hermitian solution of (15) by Proposition 3.4 we must have p2-p1

-:(Ap,;

B, C)

for any p2 I r&,4;

B, C) 5 rz(A,;

p&A;

B, C).

B, C)=rc(A,;

B, C) + p,

II c” = rc II Cx II cp-

D = -B*P,zCx(Cx)*/IICxl&,

then I] D II = r, and B, C)+f--E

(A + BDC)x

B, C)-p+$e)C*C

-PBB*P=O

with the property that a( A, - BB*P) C c-. But then it is easy to see that P,, + p is an Hermitian solution of A*P+PA-(&A;

(A-BB*Prg)x=iwx

Now let

for some E > 0, and p < &A; B, C). Then by Theorem 3.3 there exists an Hermitian solution P of A,*P + PA, -($(A;

111

may be considered as a converse of the cheap control approach. (e) By (16) there exists x E Q=“, o E R such that

II B*Prp

Now suppose that &A;

perturbations

and it is easy to show that

B, C). Hence

&A;

ruclius for structured

B, C)+$)C*C

-PBB*P=O

satisfying a(A-BB*(P,+P))=cr(A,-BB*+@-.

This contradicts Proposition 3.4 and

= Ax - BB*P,.gx = iwx.

Note that by (16), r&A,; B, C) + co as .p + - cc. This might suggest that the feedback “p.= - B*P,x will arbitrarily increase the stabihty radius. However B, C do not represent input and output operators so in general this control cannot be implemented. If the uncertain system (1) is controlled via a known input matrix B E aBnxA Z=(A+BDC)x+h

one can formulate a state space version of the robustness optimization problem (cf. [9,10]) as follows: Find a (minimal norm) state feedback matrix FE ~t?IXrl which maximizes rc (A + BF, B, C). A similar formulation can be given for static or dynamic output feedback.

SO

r;( A; B, C) = rc( A,,; B, C) + p,

5. Liapunov functions for maxinial robustness

p c&A;

One major source of difficulty in robust stability analysis is due to the fact that the set of (nicely) stable systems is not convex. However, the set of systems i = Ax for .which a given positive definite quadratic form V(x) = (x, Px) is a Liapunov function turns out to be convex. Based on this observation Kiendl, [ll] has developed a method for checking the stability of parameter dependent matrices of the form

B, C).

It is clear that (16) holds when p = rc since B, C) =O. 0 r&Q; 4.2. Remark. (a) More general monotonicity results can be found in Wimmer [7]. (b) Employing the implicit function theorem as in Delchamps [8] one can show, in fact, that if it exists the stabilizing solution is an analytic function of A, B, C and p. (c) It is easy to show that if (A, C) is observable then P,, < 0 (resp. Pp > 0) if 0 < p I rz (resp. p -c 0).

(d) As p --) - 00 one obtains the so called cheap control problem. Thus the analysis of A,, as p ---)rz

A(p)

=A,+p,A,

+ .a. +p,A,

where A, E Rnx”, k = 1,. . . , r, and p = (P 1,. . . , p,) is a real parameter vector in a multidimensional interval asp I b. This shows that, given a nominal system f = Ax it may be of interest to construct a Liapunov function V(x) of

D. Hinrichsen.

112

A.J.

Pritclwd

/ Stahili[v

maximal robustness. By this we mean that V is a Liapunov function for all matrices 2 in a ball with radius p around A, where p is the largest possible radius for which such a (quadratic) Liapunov function can be found. If we assume structured perturbations as in (1) it is clear that a joint Liapunov function can exist for the set {A + BDC; 11D 112 < p} only if p 5 r&,4; B, C). The following proposition shows that the converse is true.

rudiru

for smcrtrred

Then if x0 E G we have by (22) that for all t 2 0. Suppose Y(t%

xl))

=Yl(h

x0)

+Yz(t,

P,)’

,

xc) E G

y(t,

x0)

where y,(t,

x0)

E (ker

u2(t,

x0>

E ker

Then since Pp is a solution of (ARE,) y2( t, x0) E ker C. Thus X,EG

5.1. Proposition. Suppose a(A) C Q=--) p E (0, rc], and P, is the solution of (ARE,) with u( A BB*P,) c d=-. Then V(x) = -(x, P,x) is a Liapunov function for all linear systems z?= (A + BDC)x with II D II 2 -z p.

perrurbutions

-

yl(t,xo)~G,

and so there exists Msuch But Y(tv

x0>

=Yl(t* =

x0>

eA’x

Pp.

we see that

120, that 11yl(t,

+Y2(tt

x0)

11 =&f

4))

0

We do not prove this proposition since it is an immediate corollary of Proposition 5.2 where we analyze time varying and nonlinear perturbations. Consider

Since a(A) C C-, there exist positive numbers M, o such that

1=Ax+BiV(Cx,

IleA’Il lMe-w’,

t),

x(0)=x0,

(20)

where N : R P x Iw + + Iwn is continuously differentiable, and. N(0, t) = 0, t 2 0, so that 0 is an equilibrium state of (20).

+

/0

‘eA(‘-‘)BN(Cy(s,

120.

so Ilub.

x0> II ~Me-w’IIxoII

w(‘-s)IlN(Cy,(s,

+lPlIMjbe-

5.2. Proposition. Suppose u(A) c Q)-, p E (0, rcz] and there exists d = d(p) > 0 such that

and hence

-(x,

II Y(L x0> II s M e-“‘II x0 II

*

P,x)
t>ll&sPIICXll&,

120.

(21)

Then {xEC”, -(x, Ppx)
Ppx), then

-2 Re(B*P,,x,

N( Cx, t))cm, .

II WY(b

Denote the’solution G= {@I”:

-(x,

Ppx) < d.

(22)

of (20) by y(t, x0) and let P,x) cd}.

*

y(t,xo)+S

ast+co,

tker

C,

so for any E > 0, there exists T such that for all t 2 T, II CY(C ~0) II
IINCX> m.~].

5 0 whenever -(x1

II C II ’ -;-U’.

Thus the orbit through x0, 0(x,), is relatively compact. We may now use La Salle’s theorem [12] to conclude that

Gn v-l(O)

Thus f(x)

xc,), 3) II ds

where S is the largest invariant set in Gn p-l(O). But

= - 11B*P,x + N( Cx, t) II 2

-[PIIwI&-

+ 11B II Mfl-

X,EG

3(x> = -P II Cx II& - II B*P,x II 2

x0), s) ds.

x0)3 t) < GE.

Now y(t, x0) = eA(‘-T)x(T) +

sT

‘eA(f-S)BN(Cy(s,

x0),

S)

ds

D. Hinrichsen,

A.J.

Prirchurd

/ Stuhili
x0> II sMe-“(‘-T)Ilx(T)II + z [l - e-“(‘-T)]

113

References x0)+

(0)

asr-,co.

[l]

0

[2]

Proposition satisfies

II

perrurburions

11B 116~.

Thus y(t,

for structured

Hence if 11D II -C rC we have 11y(a) 11Lz < co for all x,, E C’l and this in turn implies that the origin of (23) is asymptotically stable.

and so

llv(t,

rudius

&NY,

5.1 shows in particular

that, if N [3]

r> ly=o
[4]

then the origin is an asymptotically stable equilibrium point of (20). Finally let us note that a stable system i = Ax can even tolerate dynamic structured perturbations as long as the gain of the perturbation operator is smaller than rc. In fact, consider i(t)

=/lx(t)

Y(t)

= cx(t),

+B(Dy)(t),

x(0) =x0,

[5] [6] [7]

[8]

(23) [9]

where D: L’[O, co; C”] + L2[0, co; C,‘] [lo]

is any (nonlinear, finite gain

time varying)

operator

with [ll]

II D II = inf{ Y E R; II DY II L’ I

Y

II Y II L2

for all y E L2 } .

[12]

We suppose a(A) E C _. By Proposition 2.2,

Ilv(*> II,z~ IIC eA~xollL~+~IIDIl

lI~(-)ll~~. (24)

D. Hinrichsen and A.J. Pritchard, Stability radii of linear systems, SJatents und Control Left. 7 (1986) l-10. J.C. Willems. Least squares stationary optimal control and the algebraic Riccati equation, IEEE Truns. Auromut. Control 16 (6) (1971) 621-634. R. Brockett, Finite Dimensionul Linear Qsrems (Wiley, New York, 1970). V.M. Popov, Hyperstuhili