Frequency Shaped Linear Optimal Control with Transfer Function Riccati Equations

O PT IMAL CONTROL SYSTEMS---I

Copyright © IFAC 9t h Trie nn ia l World Co ngress Budapest, Hungary, 1984

FREQUENCY SHAPED LINEAR OPTIMAL CONTROL WITH TRANSFER FUNCTION RICCATI EQUATIONS l J.

B. Moore*, B. D. O. Anderson* and D. L. Mingori**

'Department of Systems Engineering, R esearch School of Phvsical Sciences, A ustratian National University, Canberra, ACT 260 I , A ustratia " M echanics allll Structures Department, Unj,'ersity of California, Los Angeles. CA 90024, USA

Abstract Standard linear optimal control theory is generalizeD using a spectral fact o rization approach to elucidate some effects of frequency shaped performance indices. The theory discusses robustness results which parallel those of standard optimal control design.Matrix transfer function Riccati equations emerge as a conceptual tool associated with the frequency shaped optimal designs. Keywords

1

Optimal control. procedures and robustness properties are derived from the spectral factorization formulations, rather than from the Riccati equations, although it is true that some of the computations or properties can be achieved working with augmented plants and augmented matrix Riccati equations.

INTRODUCTION

Matrix Riccati equations are a key theoretical and numerical- tool in optimal control pro blems associated with linear systems having state space descriptions and quadratic performance indices. For time invariant linear systems described by state space equations in terms of matrices ! A,B ) and quadratic contro l performance indices with parameters ! R=R' ) 0, 0=0' ) 0 ) , the associated steady-state Riccati equation is in continuous-time . P A

z

A'P - P BR-IB 'p + Q = 0

An important motivation for studying control and estimation problems characterized by polynomial or transfer function matrices ! A(s),B (s), Q(s), R(s») is to achieve good trades between performance for a nominal plant and robustness to variations from the nominal plant as discussed in many of the references.

(1.1 )

The optimal state feedback gain matrix is K = R-IBP

2

(1.2)

FREQUENCY DOMAIN 0PTIlfAL CONTROL

Plant and the optimal index is expressed in terms of P.

In Laplace transf o rm notation, the plant is defined by yes) = W(s ) u(s ) + W(s ) vI(s) (2.1) where all transf o rms are defined in terms of the bilateral Lapla ce transf o rm,

This paper studies linear systems having transfer function descriptions and quadratic indices with frequency dependent weightings, using a spectral factorization approach. In particular, we study linear continuous time systems des=fibed by transfer function matrices W(s) = A(s) B(s) and quadratic control performance indices with parameters ! R( s) , Q( s) ) , and in this way generalize in some dire c ti ons the ori&inal theory which is stated in terms of constant t A,B , Q,R ) t o theory in terms of ! A(s),B ( s),Q(s),R(s» ) . a Surprisingly perhaps matrix transfer function Riccati equations emerge as a conceptual too l associated with an interesting subclass of such problems. Thus in the case of square, full rank, and minimum phase plants, the o ptimal controller has a decomposit o n in terms of a Ri c cati equati on as P* (s)A(s) + A(s) * pes)

*

(2.2) We assume that v I (t) i s no t under our control, we shall require that (2.3) The plant transfer function W(s) is assumed to be real rational with a left c oprime matrix fraction description. W(s ) = A-I ( s)B(s ) Also,

W(s ) is strictly minimum phase, i.e. rank B(s) normal rank B(s) for

0.3a)

*

Re [s] ) 0

+ P (s2~(s)a.-I (s)B (s)P ( s) - Q(s) = 0 K(s) = R (s)B (s)P(s) (1.3b)

stress

that

in

the

paper,

(2.5a)

k~~ sW(s) is finite, normal rank of W(s)

The notation X*(s) = x'(-s ) is used throughout the paper where the prime denotes matrix transposition. *For s=j w with w real, and X(s) real rational, X (s) is the hermitian conjugate of X(s). We

(2.4 )

Performance Index. function as

*

with

rank

equal to the (2.5b)

Define a frequency domain cost

*

II J ~ [y (s) Q(s)y(s)+ u (s)R(s)u(s)] _'"

J =

computational

I

dljl

s-Jw (2.6)

1 Work supported in part b y research grant from Boeing Commercial Airplane Company and in part by the Aerospace Corporation.

287

J. B. Moore, B. D. O. Anderson and D. L. Mingori

288 Q(s) is real exists for all

rational,

Q(s)

Q*(s),

K( s) is proper

Q(jW)

K(s)W(s)

real W with Q(jw) ) 0 and

~!m

Q(s)

<~

J/2

R(s)

[~/2 (s»)*[~/2

Q(s)s[J/2 (s»)*[J /2 (s»)

Now

we

shall note a We require that

- 1/2

v (s) = R 1

further

restriction

(s)v ' (5)

on

Three other conditions on W(s) are: withS= {sO: nW(sO) n< ~,rank W(SO) = normal rank W(S)} d=dim

the controller has zero states at time t (2.15e) o The "external control" can be chosen freely except that we require v (t)sO for t
[nullspace

of

, sO=jWO' Wo real, any SoE:S) , linearly independent constanf a i - 1/2 W(SO) R (sO)ai=O for all So = jw o' Wo real, So E: S.

Suppose W(s), Q(s), R(s) and K(s) satisfy assumptions or inequalities (2.5), (2.8) (2.9) (2.11) and (2.15). Then K(s) satisfies the following natural generalization of the return difference equation of linear optimal control: W* (s)~(s)W~s) + R(s) =[I+W (s)K (s»)R(s)[I+K(s)W(s»)

(2.8)

1

where vi(t) is a time function zero outside the interval (to'O).

and

cancellation in Res (2.15d)

I.-.a 2.1

If

with R 1/2 (s), 1/2 (s) free of poles in Re [ s) ) O. and Q 12 (s) free of poles in Re [s) ) 0 and of constant rank in Re[s) > O. v (t). 1

pole-zero

Theory concerning the optimal selection of K(s) is developed in the following lemma. We defer for the moment the existence question.

be rational and satisfy (s»)

no

) 0

(2.7a)

R(s) is real rational, R(s) R*(s), R(jw) exists for all real w with R(jw) > 0 , and 111 R(s) is finite and nonsingular. (2.7b) Aiso introducing the stable, minimum phase spectral factors of R(s) and Q(s) . Let

~/2 and

has

(2.15c)

W(SO)R-tso) there exist such that

d

I + K(s)W(s) is strictly minimum phase, i.e [I+KW)-l has all poles in Re[s) < 0

(2 .16b )

Moreover, K( s) leads to an asymptotically stable closed-loop system and defines an optimal controller via (2.13) for t > O. The optimal control law is , v (s) = 0 (2.17) u(s) = -[K(s) y(s») 2

Proof

See Appendix.

(2.9)

3

CONSTRUCTION CF K( 8), SPECTRAL FACTORIZATION AND RIJIUSTNESS

With N( s), D( s) polynomial and right coprime, and with D(s) row proper, suppose that

Construction of K(s).

Q 1/2 (s) = N(s)D- 1 (s),

It proves convenient to following with quantities W(s) = D- 1 (S)W(S)R-

(2.10)

(2.16a)

work

for

much

1/ 2 (s)

Any right common divisor of N(s) B(s) and A(s) has all determinantal zeros in Re[s)
of

rational

u(s) = -K(s)y(s) + v(s) s o that the matrix is

transfer (2 .1 3)

closed loop system transfer function

WCL(s) = W(s)[I + K(s)W(s») - 1 [I + W(s)K(s»)-l W(s)

(2.14)

K(s) is a transfer matrix here which constrained to satisfy the following conditions

is

the

(3.1a)

K(s) = R 1/2 (s)K(s)D(s) for some right cop rime polynomial pair B ,A. The second assumption on W(s), Q(s), R(s) is as follows:

of

(3.1b)

Then the search for K(s) satisfying (2.1b) becomes one for K(s) satisfying

-* * W N NW

-* -*

--

+ I = (I + W K)(1 + KW) -

-1

(3.2)

- 1/2

Further, since W = D 1 WR condition (2.9) on the nullspace 5!.f WR- 12 carries over to a like c~ndition on W • Suppose to begi~ wit_h__ fhat W has full column rank. With NW = NBA , a right matrix fraction description of the strictly pr~per ~ with the greatest right divisor of NB and A possessing a Hurwitz determinant, see (2.11), it is well known that a polynomial C can be found, see e.g. (10) such that W*/NW+1 = [1+ (i\*)-1 C*)(1 + and

CA

CA

-1

det (A + proper

C)

is Hurwitz,

-1)

(3.3) is

strictly (3.4)

Accordingly, the task of finding K becomes one of solving KB = C, with B possessing full column rank. We can characterize all solutions of this equation in the following way. (This characterization will prove part of the properties of K, and in turn will allow a sharper characterization). Let U be a polynomial unimodular matrix such that (3 .5 )

+ [I-K(s)WCL(s») * R(S)[I- K(S)W

CL

(s»)

R(s) (2.15a)

WCL(s) has all poles in Re[s) < 0

(2.15b)

with

B1 square and nonsingular. Notice that is a &!eatest common right divisor of the co umns of B Then the set of all K satisfying

ifl

289

Frequency Shaped Linear Optimal Control

KB =

~

Wo (s) is minimum phase, but not necessarily strtctlY minimum phase; any jw axis zero must be simple.

is given by X] U- 1

K = [CB - 1 1

(3.6)

where X is an arbitrary rational matrix. If C ,B 1 are right coprime, all K _ necessarily have the (determi!!antal) zeros of BIas poles; otherwise, all K necessarily have as poles a subset of the zeros of B. Also, there obviously exist K with po\es confined to (a subset of) the zeros of B • 1 Now suppose that W does not have full rank. There exists by (1.9) an orthogonal constant P such that W P = [W 0] where W has full column rank. Premultiplying and postmultiplying (3.2) by p' and P yields

[r]

0] + I

N*N[W

We simply find

E

- -*-* [I+W K ] [I+KW]

in the same manner as

Properties of

=

[~* }*P] [I+P'K[W

0]]

K such that

-* * "'" WNNW+I

K

[I +

P

K:

r.e-a 3.1 Let K be determined via the above procedure, assuming (2.4), (2.5), (2.9) and (2.11) hold. Then there a~~ no unstable pole-zero can£~l!rtions in forming KW and KW. Moreover, (I+KW) and ) 0 ,

Passband robustness. The loop gain defined by the plant and controller is WOL(s) = K(s)W(s) (3.9) yields

m

+

[L( s)

+

~(:)] [.C(S) _

J

dimension Construct a M] such that

= 0

(3.14 )

-D(s)B(s)

Assuming for the moment that L(h) has full column rank then it is clear that all poles of HD must be zeros of B. With K =_Ho, K has poles which are zeros of B, and K = R- 1/2 K 0- 1 = R- 1;2 ii is proper, and we are done.

4

and

See Appendix.

(2.16)

+

have

1b

K and K

The spectral factorization return difference inequality

B(s) and C(s) Suppose that p x m and m x m with p ) m minimal left polynomial basis [L

The row degrees of [L M] are minimtfi and Jhe highest degree coefficient ma )x [L ) M(h] has full (row) rank. Now if L has full column rank, there exists, a proper H of least McMillan degree such that HoB = C .

We shall prove a number of desired properties of

Proof

It remains to be shown that we can find a proper K(s). For this purpose, let us employ a different constructive procedure for K(s) than that given earlier. This alternative procedure based on Forney (1975) constructs a K(s) of least McMillan degree which is proper, assuming there exists some proper solution. (Note that we were not in a position to present this procedure earlier, since to carry it through, we need to know that lim sKW exists and is finite, as described above.) s~

+

(3.7)

(3.8)

(I+KW)-l are free of poles in Re[s] WCL(s) has all poles in Re[s] < O.

See Appendix.

Properness of K(s):

t

K is found, and then set

I~ I

Proof

the

MATRIX RICCATI EQUATION FOR OPTIMAL CONTROL

In this section it is shown how a Riccati matrix transfer function equation c;an be associated with the previous results. The Riccati equation is perhaps also of independent interest, but this aspect remains to be explored. Associated with any K( s) satisfying the spectral factorizati'2,n (3.2), let us introduce a transfer function p( s) satisfying the linear matrix equation

-*

-

--

P (s)[A(s) + B(s)K(s)]

[I+WOL(-jW)], R(jW)[I+WOL(jw)] ) R(jw)

(3.10)

so that if R(s) = r(s)I for a scalar r(s), then

-*

-*

-*

-

+ [A (s) + K (s)B (s)]P(s)

(4.1)

-* K (s)K(s) + N* (s)N(s)

(3.11) Likewise, if we define

W (jw) = R(jw)w(jw) = R 1/2 (jw)W (jw)R - 1/2 (jw) OL OL (3.12) then [I+WOL(-jw)], [I+WOL(jw)] ) I

(3.13)

It is straightforward also' to see that these inequalities If,0ld with equality only for isolated w if Q 2 W has full column rank almost everywhere.

Here, we proceed by stages to show that (4.1) leads to the matrix transfer function Riccati equation as in (1.3). Le-.. 4.1

Let the conditions specified in the hypothesis of Lemma 2.1 hold and let K(s) be defined as described in the Lemma statement. Let N(s), W(s) and K(s) be as defined in (2.11) and (3.1) and let W( s) have a left coprime mat rix fraction description A -1(s) iHs). Then there exists a rational P(s) such that (4.1) holds. Proof

A Minimum Response.

Phase

Property

of

the

Open-Loop

We assert:

r.e-a

3.2

For the optimal defigns of section 2 2(s) = WOL(s)[I+WOL(s)]is positive real, and

As background, let uS observe that if K( s) has a right coprime matrix fraction description 1 (s), K(s) = ~(s) then (4.1) implies and is implied by

Ra-

-* -* (s)[A(s)KO(s) - KO(s) P + B(S)KN(S)] -*

-*

-*

-*

-

+ [~(S)A (s) + ~(S)B (s)]P(s)~(s)

J. B. Moore, B. D. O. Anderson and D. L. Mingori

Z90

(4.Z) yes) - WCL(s) [vI(s) + vZ(s)] which causes us to look at the polynomial equation u(s) - -K(s)WCL(s)vI(s) + lI-K(s)WCL(s)]vZ(s)

G*(s)[X(s)Ro(s) + ~(s)RN(s)] + [R;(s)X*(s) + K~(S)B*(s)]G(s)

Thus,

Q I/z (s)y(s) _ Q I/z (s)WCL(s)v (s) I + Q I/z (s)WCL(s)vZ(s)

Any solution of (4.3)(whether or not polynomial) of (4.1) by defines a solution pes) - G(S)KD-I(s) • We do however wish to isolate solutions of (4.1) with special properties.

R I/z (s)u(s) - -R l/z (s)K(s)WCL(s)v I (s)

Let U(s) be a nonmodular matrix such that

After some algebra one can show

8(s)

U(s)

=

[8~ (s) )

+ R I/z (s) [I-K(s)WCL(s)]vZ(s)

y*(s)Q(s)y(s) + u*(s)R(s)u(s)

(4.4)

=

BI(~) square. r!n~, ~I (s) _ _

with

Since

B(s)

* *R(I-K WCL)]v Z + VI* [W *CL 0 WCL- WCLK * * * + V [W Q W - (I-K W ) R(I-K WCL)]v I z CL CL CL * * * + Vz [WCLO WCL+ (I-K WCL ) R K WCL]v Z

has full column

is ~~n!!..ingular. Repla~e A,B and P by UA, UB and U P . Now wi~1 P any solution of (4.1) obtained as G(s) KD (s) with polynomial G, define 0]

MZ = [

8~-*

P

* WCL + WCLK * *R K WCL]v VI* [WCLQ I

Using (Z.15), this equation may be written as

(4.5a)

MI -strictly

J

- -I proper part of M ] [B I I P = P -~ (A -tBK) I

0]

(4.5b) (4.5c) Thus the cost function (Z.6) becomes

Le..a 4.Z. W!~h _ th!, above definition, PI satisfies (4.1), B P WCL is strictly proper.! :>ith !l! poles at the eterminantal zeros of (A Kn + B KN) . If any other solution P of (4.1) obtainable as G( s) Kn-I (s) _ for a pO!.ln~mi':.l G ~~2 is used to derive a PI ,than B PI WCL = B PIW CL

f:~[y*OY

=

J

J

co * * RK]WCLv j s=jWdW f_~ VI* WCL[Q+K I

_

Proof

See Appendix.

Reaarlr.s Now let us turn to connecting

K(s) and pes)

+ u*R u] jS=jWdW

+

f:~v ;(RK

+

f:~

f~v;

f:LRK

WCLv Z] jS=jWdW +

WCLVZ]*VljS=jWdW

v; R vzjs=jWdW [RK WCLVZ]js=jWdW

=

0

Le..a 4.3

5!t

Under the hypotheses L<:.mma ~.I, any solution pes) of (4.1) with B (s)p (s)WCL(s) strictly proper satisfies

-

-

-*

-

-

K(s)W(s) = B (s)P(s)W(s) Proof

(4.6)

See Appendix.

~:[RK This

WCLVZ]*VljS=jWdW

follows

from Parseval's

that L -I[R I/z VI]

vi(t)

theorem and noting is zero for t

>

0,

while 5

CONCLUSIONS

The paper has built a bridge from the state space to a transfer function approach for optimal control system design. Once the bridge is built, it is not surprising that many of the insights and results of the state space approach to optimal design apply to the frequency domain approach. The details associated with the corresponding results for the discrete time case are not presented in this paper, but should be obtainable by similar methods.

ilz K WCL v Z = ilz KW( I+KW) -Iv z

=

ilz [1-( I+KW) -1]v Z

is zero for t < O. (Note that the stability of and (I+KW)-I is relevant here.) Hence J -

+

co * * * f_~ VI WCL[Q+K R K]WCLvljS=jWdW

f:~

v; R vzjs:jWdW

Now only Vz is at our disposal to minimize J in this expression. Clearly, the minimization is accomplished by setting

APPENDIX

Proof of Le..a Z.1 Solve (Z.I), (Z.13) and (Z.14) for yes) and u(s) in terms of VI (s) and vZ(s).

J/Z

which means that the optimal u(s) is given by u(s) = -K(s) yes)

291

Frequency Shaped Linear Optimal Control

subscript

In terms of the + and introduced in Section 2,

notation

The second term in this expression is due to the initial conditions on the (now dynamic) feedback control Law. Note that if K is constant, [K y_ (s) J+ is zero. Proof of Le8aa 3.1 Suppose

first

Equation strictly rank in Re[sJ ~ poles of Re[sJ <

(2. 5a) implies that Wand so WR- 1/2 is minimum phase, and so B has constant Re[sJ ~ 0.. Hence K has no ~les in O. In forming the product KW, the K are cancelled, and_these are all in 0.. No poles of Ware cancelled

that

W(s)

has

full

column

When W does not have full rank, a simple modification of the above argument shows that, as before, in forming KW ,there is no unstable pole - zero cancellation. ~ext det (I+KW) has a Hurwitz numerator and KW is strictly proper. This is because det (A + "2) is Hurwitz, which ensures that the numerator of I + KW has full rank

in the

Re(s) ~ 0

or

that

(I+KW)-1

is

KW = CA -1

Now let us relate these results back to K and W. First, (3.2) implies (2.16a). Next, since

=

I+KW

I + il2 KW R- 1/2

= il2

(I+KW)R - 1/2

we have (I+KW) -1 = il2 (I+KW) -1 R- 1/2 and so, (s)J

[using the properties of

JI2 (s)

and R- 1/2

(I+KW)-1 has all poles in Re[sJ < O.

J/

Third,

J

because KW = R- 1/2 KW 2 and I2 (00) is finite and nonsingular since R has this property, KW has no unstable pole zero cancellation and is strictly proper because KW has this property. Finally, because [I+KWJ- 1 is stable and KW has nO_fnstab1e pole-zero cancellations, WCL = W[ I+KWJ is stable, i.e. has all its poles in Re(s) < O.

Then the

Preliminary result for Eroof of Lemma 4.1 1 . - . A.l

Let Y,Z be polynomial matrices such that det Y is not identically 7.lro and has all zeros in Re[sJ < 0, and Z - Z Then there exists a polynomial X with

rank.

unless CA -1 has a cancellation, i.e. C,A have a right common divisor. By (3.4) such a divisor would necessarily have a Hurwitz determinant, and therefore there can be no unstable _Eo1e-zero cancellation in forming the product KW.

free of -poles in Re[sJ ~ O. Also, which is strictly proper by (3.4).

real and nonsingular almost everywhere. earlier argument can be applied.)

*

* XY+YX-Z Proof

Write, using a partial fraction expansion, Y-* Zy- 1 _ V+ + V + U

where U is polynomial, V+ is strictly proper with poles in Re[sJ < 0 and V_ is strictly p~oper with pl/les in Re[sJ > O. Obviously V+ - V_ and U a U. Now

and since the left side has poles only in Re[sJ < the right side has poles only in Re [sJ > 0, both Si~ys are polynomial. Let V+Y = N i.e. V+ - NY for some polynomial N. Then

0,

Y-*Z y- 1

Y-*N* + N y- 1 + U * * * NY+YN+YUY

Z

- (N

* + 1/2 Y*U)Y

Rellarlt [14 J.

With

Z-I,

Proof of

r-

4.1

the

+ Y* (N * + 1/2 UY) above

result

appears

in

The fact that (I+KW) is strictly minimum phase while . KW has no unstable pole-zero cancellations means that the same is true of (I+KW) and KW, and so X(s)KO(s) + B(s)KN(s) has all determinantal zeros in Re[sJ < O. This guarantees by Lemma A.l that we can find a polynomial G( s) satisfying (4.3) and thus a pes) satisfying (4.1). Proof of

r-

4.2

It is easily verified that if* X is a rational matrix for which X + X = 0, then G - X(AKO+BK ) is a (not necessarily polynomial) N solution of t4.3) i f and only i f G is a solution of (4.3). Hence P - X(X + B K) is a rational solution of (4.1) if and only if is a rational solution. In light of (4. 5c), to show that P * is a solution of (4.1), we can show that 1 M2 + M2 = 0 •

P

Now (4.3) implies that

Proof of Lemaa 3.2 Z(s) is asymptotically stable from the results of the last section. Moreover for all s = jw, (3.2) implies trivially that

[I+~L(s)J[I+~OL(s)J ~ I.

This

inequality

G[AK

O

+ BK J- 1 + [K~ X* + ~ B*J- 1 G* N

can

be reorganized as .J

*_

[I+WOL(s)][Z(s) + Z (s)J[I+WOL(s)J ~ 0 for s=jw

*

from which it is clear that Z(s) + Z (s) all s=j w. Hence Z(s) is positive real.

~

0 for

The zeros of WOL(s) are zeros of Z(s). Because Z(s) is positive real, its jw-axis zeros must be simple and all other zeros must be in Re[sJ< O. (To see this, we note that i f Z(s) is nonsingu1ar almost everywhere, Z-I(s) exists and is positive real. The poles of z-1 (s) which are the zeros of Z(s), are all in Re[sJ < 0 or are simple on the jw -axis. If Z(s) is not nonsingu1ar almost everywhere, there exists a full row rank constant T such that Z(s) = T'ZI (s)T with ZI positive

Now it is not hard to verify, that KN [AKO + BKN J- 1 B a n d N[AKO + BKN -1 B must be strictly proper. Hence

[B~ +

is

OJ G [ARO+ BR NJ-

[B"/

O][K~

strictly

X* +

~

proper.

1

[~1]

1 B*J- G*

Hence

[~IJ

if

yes)

=

M (S) 1

J. B. Moore, B. D. O. Anderson and D. L. Mingori

292

strictly proper part of Ml (S), then Using (4.5b), we see then that

Y*(s) + yes) - O.

*

-

M2 (s)+M2 (S) ~ 0, and so PI Next, observe that

~

M -[M - strictly proper part of M ) l l l

-*

-

-

Further, the poles of B PI WCL(s) are evidently the poles of the strictly proper part of Ml , which are, by (4.5a) a subset of the poles of the strictly pro~er part of P (A + BR) = G (AKD + BRN) - . These lie in Re[s) < 0 as already observed. the

strictly ' proper part of Ml is unique t"!? ~
-* -

B

-

PI WCL

is unique.

Proof of Le..a 4.3 PE~-

and Post-multiplication of (4.1) W (s), W(S) respectively, application of spectral factorization (3.2) written as

by the

W*R*KW + w*R*+ KW and substitution of the relationship A- I B ~ W yields W*N*NW

W* (p*B-R~ (I+RW) + (I+w*R*) (B*p-R)W = 0 Pre-multiplication by fI+W*R*)-l and multiplication by (I+KW)then yields

-*

-*

-

post-

-*

WCL(s)[P (s)B(s)-K (s»)

-*

-

-

-

+ [B (s)P(S)-K(S»)WCL(s) = 0 Regard this equation as expressing z* + Z = 0 where Z = S*PW

IlEFEllEHCES

satisfies (4.1).

- strictly proper part of Ml

Next,

holds.

CL

- RWCL

-*-By the Lemma hypothesis and Lemma 4.2, B PW is st rictly proper and has all poles in Re [s fL< 0, while KWCL has the same properties by the results of Section 2. Hence Z = 0, whence (4.6)

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