An integral equation approach to the singular values of a system with distributed output delays

Systems & Control Letters 20 (1993) 447-454 North-Holland

447

An integral equation approach to the singular values of a system with distributed output delays L. Pandolfi* Politecnico di Torino, Diparthnento di Matematica. Corso Duca degli Ahruzzi, 24, 10129 Torino. Italy

Received 27 June 1992 Revised 13 October 1992 Abstract: We present a state-space approach for the analysis of the Hankel singular values of a systemwith output delays and we derive asymptotic estimates for them. KeywortA': Input delays; singular values; integral equations.

1. Introduction The singular values and singular vectors o f a Hankel operator have a crucial role in H ~ control theory (the largest singular value being the 'Hankel norm' of the system) and in 'model reduction'. In particular, model reduction techniques allow the approximation of a given plant with finite-dimensional plants. These kinds of approximants give information, in particular, on the approximation of a normalized coprime description of the system. To find a normalized coprime factorization is a difficult task for nonrational transfer functions but Theorem 4.1 in [9] is an approximation result which shows that normalized coprime factorizations of approximating rational plants converge to a normalized coprime factorization of the nominal plant, with delays both in the graph and in the gap topology. In this paper we present a method for the analysis of the singular values of the Hankel operator of a system with distributed output delays which reduces the problem to the analysis of a set of integral equations. In particular, asymptotic estimates on the singular values will easily be derived (in Section 4) as a direct consequence of some known results for the eigenvalues of Fredholm integral equations. Several authors have studied the singular values of a Hankel operator which corresponds to systems with delays (see [4, 5, 9, 10] and references therein). The results in [4, 5-1, in particular, also cover the case of systems with distributed delays, although the main interest of the authors is with point delay systems. The main contribution of this paper is an approach to the Hankel singular values of systems with distributed delays described by

~ Y¢ = Ax + Bou(t), (S)

[y(t)=

C o x ( t ) + C x t ( ' ) + Ou,(').

(1)

Here x is an n-vector, u an m-vector and y a p-vector. The matrices A, B, Co are constant and have proper dimensions. The symbol xt(') denotes the function of 0 ' 0 - - . x(t + O) and analogously for u,(.). The

Correspondence to: L. Pandolfi, Politecnico di Torino, Dipartimento di Matematica, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy *Research supported by the Italian Ministero Della Ricerca Scientific E Tecnologica within the program of GNAFA-CNR.

0167-6911/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

operators

C and D are defined,

respectively,

0 C$(.,=

by 0

C(s)$~(s)ds,

Dti(.)

1 -h

=

D(s)$(s)ds.

(2)

s -h

Here @( .) and $( .) are, respectively, II- and m-vector functions and C( ‘), D( .) are matrices of proper dimensions. The matrices C( .), D( .) have square-integrable entries and also the input function u( .) is square-integrable. It is clear that system S defines a Hankel operator f in the sense of [8]. This is that bounded operator which to every input function u( .)EL’( - ~0) associates the function y( .)EL~(O. +x.),

s -r

v(r) =

mini-h,

-1)

D(.s)u(r + s)d.s + Co

eA(‘+‘)Bou(-r)dr

eA(r+c+r)

+ f

,. 0)

x B, u( - 5) dr ds,

where we assume condition. Assumption.

(3)

u(s) = 0 for s 2 0. The operator

The matrix

r is bounded

since we always

assume

the following

A is stable.

In this paper we make use of a state-space approach to study the singular values of the operator F. Our starting point is an extension of a result in [3] on the relations between the Hankel operator of a finitedimensional system and the solutions of the Lyapunov equations (see Section 2). Our main results, which show the relations between singular vectors and integral equations, are in Section 3; in Section 4 we present an asymptotic estimate for the singular values, based on known estimates for the eigenvalues of Fredholm operators.

2. State-space description

We present a state-space description for system S, i.e. we present a state-space system whose Hankel operator is the same as that of system S. This representation is inspired by the one used in [ 1 I]. The realization is I! = A.u + B,u, ;

V = A V + C(U)x(r) + D(U)u(t),

(4)

y(t) = C,.u(r) + V(r, 0). Here, we assume domd

that

V(t;)

belongs

= {4( .)E W’.2, 4(-h) A generates

Hence, the operator

bmM)(@ = and the operator

to Lz( -h,O)

= O),

(Ad4(N = - $

the semigroup

4(U - t)

for

o

otherwise,

and that A is the unbounded 4(O).

a(t) of the right shift.

-II I t) - t I 0,

Id on M 2 = R” x L2( - h, 0; UP’)defined

by

operator: (5)

L. Pando!li/Sinyular values of a system with distributed output delays

449

generates a C0-semigroup. If col(x(t), V(t, .))is the solution of equation (4) produced by a square-integrable input u(.) which is zero for t > 0 then the component V(t,. ) is given by D(z)u(z + t - O)dz +

V(t, O) = in(-h,O-t)

C(z)x(r + t - O)dz.

(7)

h

Consequently, the function 0 ~ V(t, O) is a continuous function and it is legitimate to calculate its value at 0 = 0: -t

v t t , o) =

vtstutt +

rnin(- h, -t)

=

in(- h. -t}

+

D(s)u(t + s)ds +

c(s)

h

C(s)

It +

ax(- t-s.O)

eA"+'~+'~Bu(-r)drds.

Comparing with (3) we see that the transformation from u(') to y('), y(t) = C o x ( t ) + V(t, 0), described by system (4) corresponds to the Hankel operator that we intend to study. It is to be noted that the system described by equation (4) is a system with bounded input operator but unbounded output operator. However, it is easy to see that the framework in [2] can be used: just define V = H = M 2 and W = ~" x C ( - h , O ) . Moreover, define the input operator B and the output operator C as follows. We define first the operators

~u=

E ol D(-)

l:cox+v,0,x

u,

V(')

,8,

and Bu =

~0+ ~

g(z)~u(T)dz,

C~b = cgg(t)q5

(9)

for all u(" )~L2(0, +oO) and all 4~(' ) ~M2- Then, it is easily seen from the definition of the semigroup $'(t) and from equation (7) that B is linear and continuous from L2(0, 7"; R m) to W for each T > 0, and C is linear and continuous from V to L2(0, T; R p) for each T. Moreover, the operator F is given equivalently by F = CB. By definition, the singular values of F are the nonzero eigenvalues of F * F = B * C * C B . These numbers coincide with the nonzero eigenvalues of PQ, P = BB*, Q = C*C. We shall see in this paper that the calculation of the operators P, Q is quite easy and leads to a set of Fredholm integral equations for the evaluation of the singular values. This is important from the point of view of systems theory because good numerical methods exist for the eigenvalues of Fredholm equations and many results are known which relate the asymptotic behavior of the eigenvalues to the regularity of the kernel of the equation. Now we can see that the operator Q can be computed by its definition, while the operator P is most easily computed from a related Lyapunov equation. First we calculate the operator P = BB*, which transforms dom d * to dom d . It is convenient to note that the operator P is the unique solution to the following Lyapunov equation: ~PX

+ P,~*X = -~*X

VXEdomd*.

(10)

Here the operator ~¢ is defined in equation (6) and ~ is defined in equation (8). It is easily seen that the adjoint of the operator ~¢ has the following domain:

{Exol ~t)

' ~()~w~'2'

{(o) = o

}

450

L. Pando(]i /Singular values t~la system with distributed output delays

and s ¢ * [ ~.('x°)l=

-A*xo + f° C*(s)¢(s)ds (ll) ~ ~(" )

M o reo ve r,

The solution P to the Lyapunov equation (10) should transform the domain of d * to the domain of d and it should be continuous in the norm of L z ( - h, 0). This suggests the following block form for the operator P:

=

P

h P*(0)yo +

.

(13)

J_hn(O,s) (s)ds

A simple substitution of this expression in the Lyapunov equation (10) shows that the matrices Po, P('), 17(',') must solve the following equations:

APo + PoA*

=

(14)

-BOB*,

15(0) = AP(s) + PoC*(s) + BoD*(s),

(15)

17o + lls = D(O)D*(s) + C(O)P(s) + P*(O)C*(s),

(16)

P(-h) = 0,

(17)

17(0, - h ) = 0.

Moreover, 17(0,s) = 17"(s, O) since the operator P is self-adjoin(. These equations can easily be solved. The first one is a finite-dimensional Lyapunov equation. We assumed that the matrix A is stable so that the solution is Po = J'o ~ eA'Bo B(~e A*'dt. Moreover, efficient algebraic methods exist for its solution. The second equation is an ordinary differential equation with a Cauchy condition: P(O)= ~°heA(°-s){PoC*(s)+ BD*(s)} ds. If F(O, s)= D(O)D*(s) + C(O)P(s) + P*(O)C*(s) then the solution of the third equation is given by

17(0,s) =

{ f~ F(O - s + r,r)dr, h f ~ h F ( r ' s - O + r)dr'

- h < s < O < O, (18)

- h <_O <_s <_O.

So, we calculated explicitly a representation for the operator P, derived from the relative Lyapunov equation. Instead, it is easier to derive a representation for the operator Q directly from its definition. We saw that Q = C*C. We note that the semigroup generated by d is 8(0

[xo] [eA'xo1 ~(.) = [_v(t,')['

where

v(t,O)={~ ( O - t )

for - h <_O - t <_O ;~ C(s)eA(s+ t-°~ X 0 ds. otherwise + axl~0-t~. -h~

Hence, c

Xo

= CoeAtx ° + = ~IXo

"[- ( ~ 2 ~ ( ' )

o

for 0 < t < h otherwise A¢- ~ 3 X o "

+

.fS C(s) e A(s + o d s Xo axl-,.-n~ (19)

L. Pando!fi/Singular values of a system with distributed output delays

451

It is an elementary calculation to compute the adjoints of the operators L~'i, i = 1, 2, 3, and their compositions:

Q

I~O I -'~?'~1 "]'-'~?'~3 -'~-'~'~1 + ~'~"~3'~?'~2 "]- " ~ 2 1 I ) = ,,~"~'al + '~C~7~3 ,,~,,~2

XO 1 4(')

-Qoxo + f°hO(s)¢(s)ds(20)

Q*(O)xo + 4(0) We see that Q(.) is the solution to

Q.(s) = - A * Q ( s ) -

C*(s),

(21)

Q(O)= C~.

The derivation of the equation for Qo is more difficult and can be found in the appendix, where it is proved that

Qo =

eA"Q(-h)Q*(-h)eA' dt +

Q(-t)Q*(-t)dt.

(22)

h

We note that the improper integral can be calculated with purely algebraic methods, since it is the solution of the finite-dimensional Lyapunov equation VA + A* V = - Q ( - h ) Q * ( - h). We recall now that a # 0 is a singular value if a2e%(PQ), i.e. if there exist nonzero V, W which solve

P V = aW,

Q W = aV.

(23)

with V, W nonzero elements of R" x L 2 ( - h , 0 ; •"). We shall see in the next section that these abstract equations lead to a 'concrete' integral equation.

3. The integral equation for the singular values In this section and the next, we write v('

'

w("

'

The block forms of P and Q show that equations (23) are equivalent to

av(O) = Q*(O)wo + w(O),

(24)

~vo= Qowo + ~o Q(s)w(s)ds, 3- h

(25)

Po,:o + fo P(s)vts)ds, 3- h

(26)

aw(O) = P*(O)vo + ~t, 17(0, s) v(s)ds. 3- h

(27)

,,wo

=

Now we can simplify the previous set of equations. We substitute the expressions for Vo and v(. ) into Eqs. (27) and (26) and obtain that the number a is a singular value for F if and only if there exists a nonzero solution to

O.,s,ds}wo+;

L. Pandolfi/Sinqular values qf a system with distributed output delays

452

Hence, the calculation of the singular values has been reduced to the calculation of the eigenvalues of an operator acting on ~ " × L 2 ( - h , 0 ; R"), whose structure is fairly simple. Moreover, let H(~r) = [(72I - PoQo -~°-hP(S)Q*(s)ds] 1 (if the inverse exists). Then we have

a2w(O) =

{

P*(O)Qo +

;

t ;o

hIl(O's)Q*(s)ds H(a) -h [PoQ(s) + P(s)] w(s) ds (30)

f,O

+|

3- h

EP*(o)Q(s)+.(O,s)lw(s)ds

and w ( - h ) = 0, and hence the following theorem. Theorem 3.1. The number tr is a singular value ofF if and only !f there exists a nonzero solution to Eqs. (28) and (29). With the possible exception of a.finite number of them, the singular values are those numbers tr such that equation (30) admits a nonnull solution w(" ). We note that any solution to equation (30) satisfies w ( - h ) = 0.

4. Asymptotic estimates for the singular values We already know that F is a compact operator so that the sequence or, of its singular values is finite or converges to zero (of course, the sequence (rr is finite when F has finite rank, i.e. when the system possesses a finite-dimensional realization). In this section we present some more precise estimates on the singular values, expressed in terms of the regularity of the matrices C('), D(.) which appear in the system. More precisely, the regularity properties of the matrix//(., .) imply asymptotic estimates for the singular values at. In order to present these estimates, we need some definitions and results which can be found in [6, l]. If A is any operator then 21(A) denotes the sequence of its eigenvalues ordered so that 2i(A) > 2i + ~(A), while s~(A) denotes the sequence of the singular values of A ordered so that s~(A)> s~+ ~(A). Hence, looking at equations (28) and (29) we see that the sequence {(r~} of the singular values of F is given by the square roots of the eigenvalues Ai(A), where A is an operator on M 2 defined by the equations (28) and (29). These equations are the best starting points for the estimates that we are going to present. Now, let A be a compact operator. We say that it belongs to the Von Neumann class Sfp. ~ when k~/PSk(A) is bounded. If this is the case then the sequence kl/Pl2k(A)l is bounded and

kl/Pl2k(A)l < C(p)supkl/Ptrk(A) (cf. [1, Lemma 1.4]). Now, let us write the operator A defined by equations (28) and (29) in the following form:

-Rowo + f°hR(s)w(s)ds + K(O)wo+ f° P*(O)O(s)w(s)ds

[i: ]El El A 1 w°

,ll(O,s)w(s)ds

w(')

w°

+A2

w(')

'

(31)

where Ro = PoQo + ~°-h P(S)Q*(s)ds, R(s) = [PoQ(s) + P(s)] and/~(0) = P*(O)Qo + ~°h H(O, s) Q*(s)ds. The operator A1 has finite rank, at most n 2, n being the dimension of the vector x. The operator A2 is compact. The following relation holds for the singular values of the sum of the operators A~ and Az [8, Corollary 1.5]: let us fix two natural numbers m, r. Then

Sm+r 1(A1 + A 2 ) < s , , ( A 1 ) + sr(A2).

(32)

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453

In our special case, the operator A1 has only finitely many singular values (n 2 at most) so that s,~+,(Al + A2) _< st(A2).

(33)

Hence, we must only give an estimate for the numbers srCA2). From [1, Proposition 2.1] we know the following result. Lemma 4.1. Let us assume that the functions s ~ H (0, s) belongs to W ~"2 and that the function 0 ~ fl I1 (0, .)II w~.2 is square-integrable. Then, the sequence {s~(A2)} belongs to 6t~+ 1/2 .... Furthermore, r ~+ t/Zs,(A2) --* O. Consequently, if the assumption of the previous theorem holds, then there exists some number H such that s~+,(Al + A2) < sr(A2) < H/r~/2+'; furthermore, r ~+ 1/2s~+, ~ 0 for r ~ + oo. Hence, we can state the following theorem.

Theorem 4.2. I f the assumptions of Lemma 4.1 hold then the sequence {a,} of the Hankel singular value of the system S belongs to oc~'(~/2~+~1/41.~.

More precise estimates can be given: if the assumption of Lemma 4.1 holds then there exists a constant C (which does not depend on H(.,.)) such that r~+l/2sv+r(Al+A2)~r~+l/EsrCA2)<_ c(SO-h IIH(O,')ll2"2dO) ;/2. Consequently, r ~/2+ 1/40v+r _~ N///CCf°h 11H(O, .)ll~,, 2d0) ;/4. In fact, r ~/2÷ l/4av+r ~ 0 for v ~ + ~ . In particular, let us assume that the matrix functions C ( ' ) , D ( - ) which appear in the representation of the system S belong to WE2; then s---, 17(0, s) belongs to W 1'2 and 0--, I117(0,')II w,2 is square-integrable. In this case the sequence {tr, } of the singular values of system S is such that the sequence {r3/4o'r} tends to zero. Finally, let us consider a special example, as suggested by one referee: let us choose A to be an exponentially stable matrix, Bo = 0, Co = 0 and CCs) = O. Instead, D(') is differentiable. In this case we get the 'distributed delay line' examined in [7]. As we assumed that D(') is differentiable, then (0, s ) ~ 17(0, s) belongs to W 1'2 over the square I - h , 0 ] x I - h , 0 ] (cf. equation 08)). Hence, the sequence {a,} of the singular values is at least of the order 1/r 3/4. This is a general estimate, which holds for any differentiable weighting functions D(- ). In special cases a direct analysis can give more stringent results: for example, let us assume that n = 1 and that D(s) - 1. In this case the singular values {a,} can be explicitly computed and a, = h/(rrt + rt/2) (cf. [4] and [7]). The sequence {tr,} is of the first order and not simply o(1/r3/4).

Appendix The derivation of Qo

We recall that Qo = £~'* 56'1 + £-a*.Laa. It is easy to compute the adjoints of the operators £#i and to see that the matrix Qo is the sum of So~'ea*tC*CoeA'dt and of the three integrals

11 = =

I3 =

fo f: f2,f+: ea*tC*

_

ax(-t, -h) C ( z ) e A ( t + ° d z dt,

eA*U+"C*(z)Co eAtdtdz,

fo f: ax(-t. -hi e A*u+o C • (r) dr .lmax(-t, ro -h) C(s) e Ats +t) ds dr.

L. Pando!~i/Singular values of a system with distributed output delays

454

From (21 ),

t f

.o

C(z)eAO + ~ dz

=

Q*(-t)-

Co eat,

r

o C(z)eA" +~ldr = I Q * ( - h ) _ CoeAh} eA"-ht" h

We change the order of integration in the first integral 11 and find that 11 =

C(z)eA"+'dzdt +

e A 'C*

C(z)e A~'+

t =

h

f2 eA*'C*[Q*(-t)-Coe

A ' ] d t + fh + " ~~A*tt~*'n*t ,~o~ ,- h)-

C°eAh]eA(' h)dt"

Interchanging the order of integration we see that lz =

Q ( - t ) C o e A ' dt -

eA*'C* CoeA' dt +

eA*"-h~Q(--h)CoeA' dt --

eA*'C*CoeA' dt.

We proceed analogously on the third integral and find that 13 =

f:

[Q(-t)-

eA*'Ct' l [ Q * ( - t ) -

[Q*(--h)--CoeAh}e

Coemldt +

f/

e A * l ' - h ) [ Q ( - - h ) - eAhC~l

~" h~dt.

W e a d d t h e i n t e g r a l s t o get e x p r e s s i o n (22) for t h e m a t r i x Qo.

Acknowledgment The author thanks an unknown referee for useful comments and observations.

References [I] M.Sh. Birman and M.Z. Solomyak, Estimates of singular numbers of integral operators, Russian Math. Surveys 32(1977) 15 89. [2] R.F. Curtain, Sufficient conditions for infinite-rank Hankel operators to be nuclear, IMA J. Math. Control In]brm. 2 (1985) 171 182. [3] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L'-error bounds, lnternat. J. Control 39 (1984) 1115 1193. [4] K. Glover, J. Lain and J.R. Partington, Rational approximation of a class of infinite-dimensional systems I: singular values of Hankel operators, Math. Control Signals Systems 3 11990) 325-344. [5] K. Glover, J. Lam and J.R. Partington, Rational approximation of a class of infinite-dimensional systems I1: optimal convergence rates of L, approximants, Math. Control Signals Systems 4 (1991) 233 246. [6] M.G. Krein, Topics in Differential and Integral Equations and Operator Theory, Operator Theory, Advances and Applications, Vol. 7 (Birkh~user, Basel, 1983). [7] L. Pandolfi, The Hankel singular values of a distributed delay line A Fredholm equation approach, in: Proc. IOth Internat. Con]: on Analysis and Optimization ~?]Systems, Antibes {1992) xx xx. [8] J.R. Partington, An Introduction to Hankel Operators [London Math. Sac., London, 1988L [9] J.R. Partington and K. Glarer, Robust stabilization of delay systems by approximation ofcoprime factors, Systems Control Lett. 14 [1990) 325 331. [10] G. Tadmor, An interpolation problem associated with H • -optimal design in systems with distributed input lags, Systems Control Lett. 8 ii987) 313 319. [11] R.B. Vinter and R.H. Kwong, The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach, SIAM J. Control Optim. 19 (1981) 139 153.