NASA Contnctor Report 178240 

ICASE REPORT NO. 87-8 

ICASE 

RKH SPLICE iIPPROXI?LITIOYS FOR THE FEEDBACK 

OPERATOR 11; A LIXLIR HEREDITAR'1' CONTROL SYSTFI 

J. A. Reneke 

R. E. Fennel1 

Contract No. NAS1-1S107 

Febriiary 1987 

INSTITL'TE FOR C3MPUTER APPLICATIONS IN SCIENCE AN= ENGINEERING 

NASA Langley Research Center, Hampton, Virginia 23665 

Operated by the Universities Space Research Association 

National Aeronautics and 

Space Adminrs!ra!m 

LMQley R-rch Cerrtsr 

Hampton, Virginia 23665 
RKH Space Approximations for the Feedback 

Operator in a Linear Hereditary Control System* 

J. A. Reneke and R. E. Fennellt 

Department of Mathematical Sciences 

Clemson University 

Clemson S. C. 29634-1907 

Abstract 

Computational implementation of feedback control laws for linear hereditary systems requires the 

approximation of infinite dimensional feedback operators with finite dimensional operators. The 

dense subspaces of K-polygonal functions in reproducing kernel Hilbert spaces, RKH spaces, 

suggest finite dimensional approximations of the matrix representations of the control operators. A 

convergence theorem is developed for the approximations and the numerical implementation of the 

1 

I 

I 

I approximations is discussed. 

* This research was supported in part by the following grants AFOSR-84-02336 and 

t Partially supported by NASA Contract No. NAS 1-18 107 while the second author was in 

residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225. 

ONR-N00014-86-K-0693. 

~ ~~ 

i 
RKH Space Approximations for the Feedback 

Operator in a Linear Hereditary Control System 

1. Introduct ion. Linear hereditary systems are complicated by the fact that state space formulations 

require an infinite dimensional state space. For linear functional differential equations the usual 

choice of state space leads to a description by an abstract evolution equation and to approximations 

by semigroup techniques [2,5,8,9]. We have advocated in several papers an alternate approach, 

i.e., something other than ordinary differential equations in infinite dimensional spaces [lS, 161. 

Our approach is to describe system dynamics by operator equations defined on a reproducing 

kernel Hilbert (RKH) space. This approach, motivated by an integral equation or an input-output 

operator description of the system dynamics is general enough to include linear functional 

differential equations and Volterra integral equations. In this approach one gives up the familiar 

guide provided by f~te dimensional state space systems; but one gains the powerful methods 

associated with RKH spaces [l, 10, 13,181. In particular, matrix representations of continuous 

linear transformatran ' s aad the approximations suggested by the convenient dense subspaces of 

K-polygonal functions are available in RKH spaces. 

11 

.. 

In this paper we apply these methods to the problem of constructing suitable finite dimensional 

approximations to the feedback operator of a linear hereditary system. We are concerned with 

systems as diagrammed in Figure 1, where A denotes the system input-output operator, D denotes 

a feedback control operator, and f and h denote, respectively, the system input and response. One 

should think of the "physical plant" as being modelled by the input-output operator A. We assume 

A is linear and causal but is not necessarily time invariant or memoryless. 

In applications the feedback operator is determined by a control system designer to achieve 

some desired objective, i.e., stabililizing the system, tracking a desired trajectory, or obtaining a 

desired terminal condition for the system response. Such feedback operators may simply involve 

proportionaVintegraVderivative terms or more complex dynamic compensation. Synthesis methods 

for hereditary systems [3,4,8, 161 lead to feedback operators D which are linear and causal. 
D 

A 

Figure 1. Feedback Control System. 

In this paper we are not concerned with the synthesis of a feeback operator to achieve a desired 

objective. Motivated by problems of digital implementation of such control operators, we present 

methods to obtain finite dimensional (finite dimensional range) operator approximations to the 

system feedback operator. Digital implementation requires storage of a discrete representation of 

such approximations and rules for their evaluation. In Section 3 projection methods are used to 

obtain feedback operator approximations and show convergence of these and related system 

operator approximations as the sampling interval goes to zero. 

In order that these approximations satisfy the physical requirements of causality a simple 

"holdt of one sampling period is introduced in their definition. Such delays in digital control 

systems result from computational and data processing considerations which can become a 

significant factor in the control of high order systems [12]. Although one normally expects to use 

equally spaced samples, this is not a requirement for our methods. Convergence of related operator 

approximations has been considered in our previous effort [ 151. 

In Section 4 we discuss the computational implementation of our approximation scheme. A 

matrix representation of the approximate feedback operator is presented. If the feedback operator is 

time invariant, it is shown that evaluation of the approximate feedback operator can be achieved 

through a simple convolution formula and computer storage requirements can be greatly reduced. 

We illustrate the effect of implementing our feedback operator approximations by considering two 

numerical examples in Section 5. These examples indicate a slight deterioration in the control 

2 
i 

system performance as the holding time increases and point to several interesting problems for 

future research. 

For finite dimensional state space systems, use of numerical integration rules and related 

approximation schemes to obtain digital filters is well understood [73. The significance of our 

results lies in the fact that we have provided an analysis of a much broader class of feedback 

operator approximations and a compatible scheme for their digital representation and 

implementation. Examples show that our methods reduce to standard approximations in the state 

space case. 

2. Preliminaries. We begin by presenting our abstract setting. One should refer to the text [ 191 by 

Willems for a basic discussion of the operator equation approach. For completeness, we present 

only necessary background information. Let r be a nonnegative number, S the number interval [-r, 

-), d a positive integer, X = Rd, e*, *> the usual inner product on X, and I 1 the corresponding 

inner product norm. Let G be the class of functions from S into X to which f belongs provided 1) f 

is continuous on [0, -) and 2) if r > 0 then f is continuous on [-r, 0) and f(0-) exists. Let {N,, x 

in S} be the family of pseudonorms defined on G by N,(f) = lub{ I f(t) I, -r I t I x}, f in G and x 

in S. 

Let B denote the class of linear transformations of G to which B belongs only in case 

1) [Bfl(x) = 0 for each fin G and x in [-r, 01 and 

2) for each compact subinterval [u, v] of [0, -) there is a number b such that 

I [BfJ(t) - [BfJ(s) I 5 b I', Nx(f)dx, for each fin G and subinterval [s, t] of [u, VI. 

The nondecreasing function bI is called a variation function for B on [u, VI. 

We will reserve C to denote the element of B defined by 

-r5tIO 

[cfl(t) = c" f(s) ds OIt 

3 
for fin G and tin S. The simple state space system given formally by h(t) = f(t) + a& h(x)dx, 

where a is a d x d matrix and f is in G, would be written as h = f + aCh. The general class of 

hereditary systems to be considered are of the form h = f + Bh with B in B. One should realize that 

our abstract setting is based upon an integral equation description of the system dynamics. 

We can obtain an equivalent class of models as follows. Let A denote the class of linear 

transformations of G to which A belongs only in case A - I is in B where I is the identity on G. If 

B is in B then I - B is a reversible map from G onto G and (I - B)-l is in A. Similarly, an element 

A of A is a reversible map from G onto G and I - A-' is in B. Thus we have equivalent system 

descriptions in terms of B = I - A-' or A = (I - B)-l [ 131. For the system as diagrammed in Figure 

1, if B =I - A-' andD is inB then h = A(f + Dh) = (I- B)-'(f + Dh). Thus h = f + Bh + Dh or h = 

(I - B - D)-'f. Here (I - B - D)-' exists since B + D belongs to B. 

Our analysis takes place in a reproducing kernel Hilbert space associated with the operators in 

A and B. Let k denote the increasing function defined on S by 

l+r+t if -rIt<O c 1+2r+t if OIt k(t) = 

for t in S. Let GH denote the subspace of functions in G which are Hellinger integrable with 

respect to k, i.e., f is in GH only in case for each compact subinterval [a, b] of S there is a number 

M such that 

p= 1 S 

for each partition {sp}i of [a, b]. Here df(sp-l, sp> = f(s ) - f(s ). The least such number M is 

denoted by ( 1 df 12 / dk. 

P P- 1 

Note that elements of G, are absolutely continuous on compact subintervals of [0, -31 and 

[-r, 0) if r > 0 [lo]. Furthermore, if each off and g is in G, and [a, b] is a compact subinterval of S 

then the integral ( <df, dg> / dk exists as a limit through refinement of partitions s of [a, b] of 

4 

, 
approximating sums of the form C, <df(sp-l, sp), dg(sp-l, sp)> I dk(~,~, sp). Let G, denote the 

-2 

-r subspace of GH to which f belongs only in case I df I I dk < - and Q, the inner product for 

Gm given by Q,(f, g) = cf(-r), g(-r)> + <df, dg> / dk. The space {G mm , Q } is a complete 

inner product space with norm N, and has a reproducing kernel K given by K(s, t)x = k(min(s, 

t))x for (s, t) in S x S and x in X. Thus K( , t)x belongs to G, for each t in S and x in X and 

<f(t), x> = QH(f, K( ,t)x) for each fin Goo, t in S and x in X. 

It may be shown that the norm N, restricted to PTGH T > 0 is equivalent to the Sobolev norm 

T 

(jo { I f(t) 12 + I f'(t) 12) dt)ln. However use of the norm N- or equivalently the kernel K results 

in simplified representations and approximations of continuous linear operators. 

One can show [ 151 that an element B of B maps G into GH and hence an element A of A 

maps GH onto GH. We will be concerned with the restrictions of elements of A and B to GH but 

will not introduce any special notation for these restrictions. 

Let {P,, x in S} denote the family of projections on G given by 

-rItlx 

xlt [Pxfl(t) = 

for each fin G and (x, t) in S x S. It is important to note that operators B in B are causal, i.e., 

PTBPT for each T 2 0. 

To illustrate the properties of the kernel K let us compute the adjoint in {G-, Q,} of PTC, 

T > 0. Iff is in G-, tis in S, andx is inX then 

5 
T 

= lo <df, dCK( ,t)x> / dk 

T 

= jo <df, K( ,t)x> 

= <f(T), K(T, t)x> 

- <f(O), K(0, t)x> - <f, dK( ,t)x> . 

If -r ,< t I O then <[PTC)*fl(t),x> = <f(T) - f(O), k(t)x>. If O I t ,< T then <[(P&)*fJ(t),x> = 

<f(T), k(t)x> - <f(O), k(O)x> - <[Cfl(t), x>. If T I t then <[(p,C)*fl(t), x> = <f(T), k(T)x> - 

<f(O), k(O)x> - <[CfJ(T), x>. 

Thus 

This expression will be used in the following section. 

3. Matrix representations and the convergence theorem. One of the many advantages of an RKH 

space setting for our analysis is the tractability of the dense linear subspace defined in terms of the 

kernel K, the K-polygonal functions [lo]. Consider the family of projections {n,} from G into 

G, with t = {t }n and increasing sequence in S defined by PO 

f(t0) -r 5 s I to 

(1/dk(tp-17 tp)){dk(s7 tp>f(tp-l) + dk(tp-17 S)f(tp)} tp-l tp 

t(tn) tn < s 

rntflcs, = 

for each fin G and s in S. Elements of ntG are called K-polygons. One can show that for each 

T > -r, the union of the subspaces n,G,, {t }" a partition of [-r, TI is dense in PTGH with 

PO 

respect to N,. 

Suppose that D is an element of B, {t }n is a partition of some interval [-r, TI, and PO 

c = mesh(t) = max (tp - tp-l), p = 1, 2, ..., n. When T > 0 we assume that 0 is the partition. Let Dt 

6 
denote the linear transformation of G defined by 

To -r I u I c - r 

for each fin G. At first it is natural to choose npn, to approximate D [ 151 but QDn, does not 

satisfy the necessary causality requirement. Here D, is causal and we show D, approximates D. 

Thinking of D as a feedback operator and nJ as sampled data, DS represents an approximation to 

Df obtained from a translation of nPn,,f. 

For linear functional differential equations the basic feedback operator arising from linear 

quadratic regulator design [3,4, 6, 8 3 has the form W(t) = -K,(t)h(t) - (-k2(t, z)h(z)dz where 

K, and K, denote matrix valued functions. Since our approach is based upon an integral equation 

description of the system dynamics the feedback operator D would be of the form Iph](t) = 

t 

- Io {Kl(z)h(z) + I' K,(u, z)h(u)du} dz. With appropriate boundedness assumptions on K, 7-r 

and K,, D belongs to B. 

To further illustrate the possibilities for the feedback operator D, we consider a finite 

7 

dimensional state space example. Robust control system design methods for a plant xp' = ap xp + 

13 u , z = y x lead to a compensator x,' = a,x, + Rcz, y = ycxc + 6,z and interconnection 

up = r,, - y where rref denotes a reference signal. Integration yields xp = xp(0) + apCxp + RpCup 

or xp = (I - a, C)-l(xP(O) + RpCup), x, = (I - a, C)-'R,Cz, and y = {y,(I - acC)-lB,C + 6,}yPxp. 

Letting u = RpCup, we obtain u = RPCr,, - D% with D = RpC{yC(I - a,C)-'B,C + 6,) yp. Here D 

belongs to B and our approximation metohds apply. 

PP PP 

Theorem 1. The operator D, is in B . Furthermore, any variation function bk of D is also a 

variation function of D,. 
Proof. If bk is a variation function of D, f is in G, and 0 I tpl I u - c I v - c I t then P 

.. 

P" 

i=p+l 

and we are through. 

We are concerned in this paper with the problem of substituting the finite dimensional 

approximation D,, i.e., D, has finite dimensional range, for a given feedback operator D, see Figure 
c 

1. We begin by studing the convergence of nets D,, t a partition of [-r, TI. In an application the 

feedback operator is chosen to achieve some desired objective, i.e., the plant h = f + Bh + Dh, or h 

= (I - B - D)-'f with B = I - A-' behaves satisfactorily. Since D, represents a digital implementation 

of D, it is important to also consider convergence of the nets (I - B - DJ1 to the system 

imputfoutput operator (I - B - D)-'. In later sections we take up a computational implementation of 

these ideas and apply then to two examples. 

A function L from S x S into the continuous linear transformations of X is called the matrix 

representation of D provided <[Dfl(t), x> = Q,(f, L( , t)x) for each fin G,, tin S, and x in X. A 

fundamental property of RKH spaces is that every continuous linear transformation between such 

spaces has a matrix representation. A problem is that D maps GH into GH rather than G, into G,. 

For T in S, PTD is continuous from G, into G, with respect to N, and so PTD has a matrix 

representation. Note that <pTDfl(t), x> = Q,(PTDPTf, K( , t)x) ='Qm(PTf, (PTD)*K(, t)x), for 

-r I t I T, fin GH, and x in X. Let L(s, t) x = [(PTD)*K( , t)x](s) for -r I s, t I T and x in X. To 

obtain L without having to compute (PTD)* note that, for 1 I i, j I d and -r I s, t I T, < ei, L(s, 

t)ej > = Q,(PTDK( , s)ei, K( , t)ej) = <[DK( , s)e,](t), ej>, where ei and ej are standard basis 

elements of X = Rd. In light of this equation we speak of the matrix representation of D rather than 

of PTD. 

Note that L(s, t) = 0 for -r I t I 0. For L causality means that given -r I t I s, <x, L(s, t)y> = 

<[DK( ,s)x](t),y> = <[DK( ,t)x](t),y> = <x, L(t, t)y> for each x and y in X, i.e., L(s, t) = L(t, t). 

Hence L( ,t)x is in G, for each t in S and x in X. 

For each positive number T, let N, denote the operator pseudonorm defined on the linear 

operators F of GH which are continuous with respect to NT by N,(F) = SUP{NT(F,)/N,(PT~) I g in 

GH, PTg # 0). Assume for the rest of the paper that T is a fixed positive number. We will discuss 

convergence of D, and the control systems {(I - €3 - D,)-'}, t a partition of [-r, TI, using N, and 

the matrix representations L and M of D and D,, respectively. 

9 
Theorem 2. The operator PTD is the limit through refinement of the net of finite dimensional 

operators {D,}, t a partition of [-r, T], with respect to the operator pseudonorm NHT. 

Lemma. If t is a partition of [-r, TI, u is in the range of t, and x is in X then N,(L( ,u)x - 

M( , u + c)x)~ = N,(L( ,u)x)~ - C, I dL( , U) x 12/dk. 

Proof. If y is in X, v is in S, and c = mesh(t) then 

Lemma. If [u, v] is a subinterval of [0, T] and x is in X then Nm(L(, v)x - L( , u)x) I 

Nm(PTD) I x I dk(u, v)ll2. Also, for each partition t of [-r, T], Nw(M( , v)x - M( , u)x) I 

Nm(PTD) I I &(u, 

Proof. Note that N,(L( , v)x-L( , u)x) = Nm((PTD)*(K(, v) - K( , u))x) I N,(PTD) I x 1. 

Wu,v)ln and N,(M( , v)x - M( , u)x) = Nw(n,(PTD)*n,(K( , v - c) - K( , u - c))x) I 

N,PTD) 1 I *(u, v)1/2* 

10 
! 

Proof of the theorem. If {sp}: is a partition of [-r, TJ, {t,}: is a refinement of s, c = mesh(t), 

I 

, 

f is in G,, x is in X, and u is in [s~-~, sp] for some positive integer p then 

i t’ is a refinement of s such that if t refines t’, p = 1, 2, ..., n and i = 1,2, ..., d then 

N,&( , sp-l))ei)2 - C, I dL( , sp-l)ei l2 / dk < (~/2)~/d. If t refines t’ and 1 x I = 1 then 

I < [Dflb) - [Dtflw, x> I 

I 2 I xi I I <[Dfl(u) - [D,fl(u), ei> I 

i= 1 

d 

I I xi I Nm(PTf) E /d“ 

i= 1 

i.e., NHT (D - D, ) I E. 

Theorem 3. If B is in B then (I - B - D)-’ is the limit through refinement of the net {(I - D - D,)-l}, 

t a partition of [-r, TJ, with respect to the operator pseudonorm N,. 

Lemma. If bk is a variation function for D then bebTk is a variation function for (I - D)-’ - I on 

[-r, TI. 

11 
Proof. If h = f + Dh then by the Gronwall inequality N,(h) I ebuN,(f) for each u in [0, TI. 

Thus for each subinterval [u, v] of [0, TI 

I [(I - D)-'fl(v) - f(v) - [(I - D)-'fl(u) + fW I 

V 

I b j, ebsNs(f)dk(s) 

V 

I bebTju Ns(f) dk(s) . 

Proof of the theorem. Assume b,k and b,k are variation functions for B and D, respectively. 

Then NT((1- B - Dt)-') I 1 + bebT(k(T) - k(O)), where b = b, + b,. Hence for each fin G,. 

NT((1- B - D,)-'f - (I - B - D)-'f) 

I NT((1- B - Dt)-l)N,(D - Dt)Nm(PT(I - B - D)-')N,(PTf) 

I { 1 + bebT(k(T) - k(O))}N,(D - Dt)Nm(PT(I - B - D)-') Nm(PTf), 

.i.e., Nm((1- B - DJ1 - (I - B - D)-') I { 1 + bebT(k(T) - k(0))}Nm(PT(I - B - D)-')N,(D - Dt). 

Thus (I - B - D)-' is the limit with respect to the operator pseudonorm N, of the net {(I - B - 

DJ-'}, t a partition of [0, A. 

Thus NT(1- B - D,)-'f - (I -B - D)-'f) converges to zero with respect to refinements for each f 

in G, Le., the solution of h = f + Bh + Dth converges uniformly on [-r, TI to the solution of h = f 

+ Bh + Dh. 

4. Computational imFlementation. In this section we elaborate on the discrete structure of the 

approximation Dt which facilitiates a digital implementation. Recall that L and M denote the matrix 

representations of D and D, respectively. In this paper we are not concerned with computing L(s, t) 

or M(s, t) for various values of s and t in S but with the effect of using the approximations D, in 

12 
place of D. We would expect that most Ds met in practice would have representations in terms of 

familiar integrals and we can obtain the various values of L and M using some existing numerical 

quadrature code. 

In addition, the extensive literature [2,3,8,9] on numerical approximations for hereditary 

systems is available for evaluations of those L and M arising from linear quadratic regulator 

designs. 

If {t }" is an increasing sequence in S, t I s I tp, and f is in G then [n,Dn,fJ(s) = PO P-1 

(dk(s, $)[Dn&XtP-,> + W$+ s)[D~,fl(tP))~dk(tp-,, tp). Thus to evaluate npn, we need to 

know [Dn,fl($,) for p = 0, 1,2, ..., n. Let ft denote the (n + l)d vector given by ft = col(f(t,), 

f(tl), ..., f(Q) and L, the (n + l)d x (n + l)d block matrix whose (p, q) block is given by L,(p, q)ij 

= cL(tp, tq)ej, ei> for 0 I p, q I n and 1 I i, j I d. Define K, in an analogous way. 

We have ntf = K( , t )c where c = [(K,)-' f,l(q) for q = 0, 1,2, ..., n. Hence 

99 P 

1 Dn,f = C" DK( , t )C = Z:a Utq, )*cq and, for P = 0, 1, ... , n [Dntfl(tP) = C(L)*@J fJ(p>. q=o 99 

Let M, denote the (n + l)d x (n +l)d block matrix defined by 

Then [Dfl(tp) = [M,f,l(p). 

The matrices (LJ*(K,)-' and M, denote discrete representations of the operators npn, and D,: 

respectively. Let us look closer at the structure of these matrices. 

One can show that (KJ1 is symmetric block tridiagonal. For p = 1,2, ..., n. the p-th block in 

the first subdiagonal is -(l/dk(tp-l, tp))I. The first main diagonal block is (lk(to) + l/dk(to, t,))I, 

the last main diagonal block is (l/dk(tn-l, t,))I, and, for p = 2, ..., n the p-th main diagonal block is 

(l/dk(tP_2, tp-J + l/dk(tp-l, tp))I. If0 I p < q 5 n then [&J*(K,)-'l@, q) = (L,(P, p))*{[(K,)-'l(q -17 

q) + [(KJ'](q, q) + [(K,)-'](q + l), q)} = 0, i.e., (L,)*(K,)-' is block lower triangular. 

Clearly M, is also block lower triangular. 

13 
If D is a time invariant operator, i.e., @3g](t + b) = @3fl(t) with g(t) = f(t - b) for each f in (I - 

Po)G,, b 2 0, t 2 0 then the structure of the matrices simplifies further. In this case L(v, w) - L(u, 

w) =L(v+ b, w t b) -L(a+ b), w + b) for0 Iu Iv Iw and b20 [15]. Let tp = - r +pc, p= 0, 

..., n with r = p’c. It follows that 

(LJ*Wt)-’(P, q) = 0 OIqIpSp’ 

= {L,(p’+l, p’+l)* - Lt(p’, p’+l)}/(r+c) p = q = p’ + 1 

= {L,(p’+l, p’+l)* - L,(p’, p’+l)*}/c 

= {L,(p’+l, PI* - L,(p’, p>*}/(r+c> 

p=q>p’+ 1 

- {L,(p’+l, p-1)” - L,(p’, p-l)*}/c p>q=p’+ 1 

= {L,(p’+l, p’+l+j)* - L,(p’, p’+l+j)*}/c p = q + j > q > p’ 

- {L,(p’+l, p’+j)* - L,(p’, p ‘+j)*}/c . 

Thus the storage requirements for (L,>*KJ’ and M, are significantly reduced. One need only 

store (L,)*(K,)-’(p,q) for p = p’ < q I n, p = p’ + 1 I q S n, 0 I q I p’ < p. Here [D,fl(tp) = 0 for 

PIP’+ 1 andforp’+ 1 <pIn 

where aj = ((Lt)*(KJ1(n, n-j) for j = 0, ..., n - (p’+l). This expression simplifies further if r = 0, 

which would be the case if D is of the form [Dfl(t) = If0 g(t - s)f(s) ds for example. If r = 0 then 

I 

p =Oand 

for p’ + 1 < p I n. Thus in the time invariant case evaluation of [Dtfl(tp) involves a convolution 

sum and, consequently, computation time can be reduced through use of an appropriate fast 

algorithm. 

14 
In the examples to follow, we solve the system h = ntFf + (B + D,),. h with t' a refinement oft 

to obtain approximate solutions of the system h = f + Bh + Dth. 

5. Numerical examples. In order to simulate h = f + Bh + D$ with t = {t }" a partition of [-r, TI 

one should choose, depending on the nature of B, a differential equation solver, see for example 

[12]. On the interval [-r, 01 we have h = f. Let tp# = 0 < tp#+l. On the interval [0, tp,+l] we know 

f and DP and so can solve for h. Proceeding in this manner one steps across the interval [0, TI. 

PO 

The following examples, were chosen to illustrate the effect of implementing the 

approximations D, in a feedback control system. Because the examples are simple we chose to 

base our calculations on the corollary to Theorem 3. 

Example 1. Consider the time invariant system h = f + Bh + Dh with Ph](t) = -4 If, (t-z)h(t) dz + 

t t Io h(z-.25) dt and [Dh(t) = -4 Io h(z) dz for t 2 0. Here the feedback operator D stabilizes the 

system h = f + Bh. The effect of this stabilizing feedback is illustrated in Figure 2. In Figure 2 h 

denotes the approximate solution of the system with D = 0 and h,, h,, h,, denote approximate 

solutions of the feedback system with c = tp - tpl = 1/128, 1/32, 1/8 respectively. It is interesting 

to note that stability is maintained for relatively large values of the "holding time" c. Here we 

obtain N2(fi-hp) I .031036, .061612, .12045, .22959, .41532 for c = p/256, p = 2, 4, 8, 16, 32 

respectively, where fi denotes the solution of h = f + Bh + Dh. 

Example 2. Let [Bh](t) = -4((t-z)h(z) dz - (h(z-.25) dz. For a given disturbance f, we wish to 

choose a control u to drive the response of the system h = f + Bh + u to zero over a terminal 

interval [1.25, 1.51. Optimization techniques [14] can be used to obtain such a control in the form 

u = Dh + g, i.e., u is the sum of a feedback term and an open loop term. Here D = A-'C2 with A-l 

= I - B and g depends on the known disturbance f and known parameters. The effects of 

16 
More specific knowledge of the feedback operator leads to further simplifications. For 

example, if [Dh](t) = Jf, lch(z)dz, which is the case for a state space system h'(t) = ah(t) + Rv(t) 

with feedback v(t) = K h(t), then [D$](c) = 0 and for p > 1 [D,h](pc) = K (c/2){h(0) + 2h(c) + ... + 

2h((p-2)c) + h((p-l)c), Le., the approximation scheme reduces to the trapezoid rule. 

Implementation of this approximation yields h'(t) = a h(t) + RK {h((p-1)c) + h(pc)}/2 for pc < t I 

(p+l)c, i.e., a simple "first order hold# approximation for the feedback operator. 

Before illustrating the effect of implementing the approximation D, in a feedback control 

system, we present a corollary to Theorem 3 which provides a method to obtain approximate 

solutions to the system h = f + Dh with D in B. 

For t = {t }" a partition of [-r, TI, if h = n$ + Dth then h is in ntGH and for 0 I p I n 

PO 

h(tJ = f(tp> + [Dthl(tp) or h,(p) = f,(p) + [M$J(p). Thus h, = ft + Mtht or h, = ((I - D,)-ll&f), = 

(I(n+l)d - M$'fr 

Corollarv to Theorem 3. Iff is in GH, h = (I - D)-' f, and E is a positive number then there is a 

partition s of [0, TI such that if {tp}," refines s and 0 S p S n then I h(tp) - [(I(n+l)d - MJ-'fJ(p) I < E. 

Proof. Using Theorem 3 with B = 0, there is a partition s of [0, TI such that if g}: refines 

s then Nm((I - D)-' - (I - DJ1) is smaller than either 1 or (1/2) &/(No0(f) + 1) and N,(f - n,f) < 1/2 

&/(Nm(I - D)-') + 1). Thus 

NT((I - D)-lf - (I - D,)-'ntf) 

I NHT((I - D)-l - (I - D$')N,(f) 

+ Nm((I - D,)-')Nw(f - ntf) 

S 1/2 E + Nm((I - D)-l + l)Nw(f - nS, 

<E. 

Since ((I - DJ'n,f), = - MJf, we are through. 

15 
I 

implementing this control and the approximations u = Dth + g with f(t) = 1 + t for -.25 I t I 1.5 are 

illustrated in Figure 3. Here h denotes the approximate solution of the system with D = 0 and h,, 

h,,, and h,, denote approximate solutions with c = tp - tp-l = 1/128, 1/16. 1/8 respectively. Note 

that the desired terminal condition holds when c = 1/128 and there is a slight deterioration in the 

control system performance for larger values of the "holding time" c. For the example we obtain 

N,.,(h - hp) I .0032262, .0072059, .015469, .032368, .067435, .14337 for c = p/256 and 

p = 2,4, 8, 16,32 respectively, where Fi denotes the solution of h = f + Bh + Dh + g. 

6. Concluding Remarks. We have presented a theory for the approximation of feedback control 

operators which arise in the analysis of linear hereditary systems. The reproducing kernel Hilbert 

space setting introduced facilitates the analysis and digital representation of such approximations. 

Advances in on board computational capabilities indicate that storage of our discrete representations 

of a feedback operator is feasible. Furthermore, in the design of feedback controllers for infinite 

dimensional systems one need not be restricted to the search for simple feedback gains as in the 

finite dimensional case. 

Although we have not addressed control design issues in this paper, such problems have been 

discussed using our abstract setting within a number of contexts [ 11,161. We are presently 

investigating the robustness of control designs based upon an input/output description of the 

systems. The results and examples presented in this paper indicate that such designs should have 

good robustness properties. 

A unified discussion of our approach to system analysis, approximation, and control will 

appear in the lecture notes, "Structured Hereditary Systems" [ 171. 

17 
References 

I 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

N. Aronzajn, Theory of reproducing kernels, Trans Amer. Math. SOC., 68( 1950), 337-404. 

H. T. Banks and J. A. Bums, Hereditary control problems: numerical methods based on 

averaging approximations, SIAM J. Control Opt. 16( 1978), 169-208. 

H. T. Banks and A. Manitius, Application of abstract variational theory to hereditary systems - A survey, IEEE Trans. Auto. Cont., AC-19(1974), 524-533. 

H. T. Banks, I. G. Rosen and K. Ito, A spline based technique for computing Raccati 

operators and feedback controls in regulator problems for delay equations, SIAM J. Sci. Stat. 

Comput. 5(1984), 830-855. 

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, 

Springer-Verlag, Berlin and New York 1978. 

M. C. Delfour, Linear optimal control of systems with state and control variable delays, 

Automatica, 20( 1984), 69-77. 

G. F. Franklin and J. D. Powell, Digital Control of Dynamic Systems, Addison-Wesley 

Publishing Co., Reading Mass., 1980. 

J. S. Gibson, Linear quadratic optimal control of hereditary systems: infinite dimensional 

Riccati equations and numerical approximations, SIAM J. Control Opt. 21 (1983), 95-139. 

K. Kunisch, Approximation schemes for linear quadratic optimal control problems associated 

with delay equations, SIAM J. Control Opt. 20(1982), 506-540. 

10. J. S. Mac Nerney, Hellinger integrals in inner product spaces, J. Elisha Mitchell Sci. SOC., 

76( 1960), 252-273. 

11. R. B. Minton and J. A. Reneke, An RKH space approach to state feedback control for a class 

of linear stochastic systems, SIAM J. Control Opt., 24(1986), 700-714. 

12. T. Mita, Optimal digital feedback control systems counting computation time of control laws, 

IEEE Trans. Auto. Contr. AC-30(1985), 542-548. 

13. R. E. Moore, Computational Functional Analysis, Ellis Honwood Limited, Chester, England, 

1985. 

14. Neves, K. W. Automatic integration of functional differential equations: an approach, ACM 

Trans. Math. Software, 1(1975), 357-368. 

15. Reneke, J. A. and R. E. Fennell, An indentification problem for hereditary systems, J. 

Applicable Anal., 11(1981), 167-183. 

16. Reneke, J. A. and R. E. Fennell, Decentralized control strategies for hereditary systems, 

17. Reneke, J. A. and R. E. Fennell and R. B. Minton, Structured Hereditary Systems, to 

Proceedings IEEE Conference and Control, Dec. 1985. 1479-1483. 

appear, Marcel Dekker Inc., Series on Pure and Applied Mathematics. 

18 
18. Weinert, H. L., Editor, Reproducing Kernel Hilbert Spaces - Applications in Statistical Signal 

Processing, Hutchenson Ross Publishing Co., Strougsburg, PA., 1982. 

19. J. C. Willems, The Analysis of Feedback Systems, The MI" Press, Cambridge, Mass., 

1971. 

19 
I I I I I I I 1 

I 
I I I I I I 

In 0 

N ON 0 Ln 

0 I 

u") 0 Ln 

k In 

Ln 0 N 

0 

I 

0 0 d 0 - 9 - 

21 
Standard Bibliographic Page 

. Report No. NASA CR-178240 

ICASE Report No. 87-8 

2. Government Accession NO. 

. Author(s) 

J. A. Reneke and R. E. Fennel1 

1. P rformjng Org 'zatio Name and A dr fnstituteyor tomputer 61piYfications in Science 

and Engineering 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

5. Supplementary Notes 

Langley Technical Monitor: 

J. C. South 

Final Report 

3. Recipient's Catalog No. 

5. Report Date 

February 1987 

6. Performing Organization Code 

8. Performing Organization Report No. 

87-8 

10. Work Unit No. 

11. Contract or Grant No. 

NASl-18107 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

505-90-21-01 

Submitted to Internat. J. Control 

6. Abstract 

Computational implementation of feedback control laws for linear hereditary 

systems requires the approximation of infinite dimensional feedback operators 

with finite dimensional operators. The dense subspaces of K-polygonal functions 

in reproducing kernel Hilbert spaces, RKH spaces, suggest finite dimensional 

approximations of the matrix representations of the control operators. A 

convergence theorem is developed for the approximations and the numerical 

implementation of the approximations is discussed. 

L7. Key Words (Suggested by Authors(s)) 

hereditary systems, operator 

approximations, matrix representations 

~ 

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66 - Systems Analysis 

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