NASA ContractorReport 178082

I_E REPORT NO. 86-16 NASA-CR-178082 19860015696

ICASE

SHIFTING THE CLOSED-LOOP SPECTRUM IN THE OPTIMAL

LINEAR QUADRATIC REGULATOR PROBLEM FOR HEREDITARY SYSTEMS

J. S. Gibson

and

I. G. Rosen

Contract Nos. NASI-17070 and NASI-18107

March 1986

INSTITUTE FOR CO_UTER APPLICATIONS IN SCIENCE AND ENGINEERING

NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association

LIBRARYCOPY

MAY2 0 1986

National Aeronautics anti

Space Administration I-AHGLEY RESEARCH CENIER

I _l'_ey R_l'_-'_h _l'ltei' LIBRARY,NASA

Hampton. Virginia 23665 ItAIVI_TO_N. VIRGI , BIP_

SHIFTING THE CLOSED-LOOP SPECTRUM IN

THE OPTIMAL LINEAR QUADRATIC REGULATOR

PROBELM FOR HEREDITARY SYSTEMS (t)

by

J.S. Gibson

Department of Mechanical, Aerospace

and Nuclear Engineering

University of California, Los Angeles

Los Angeles, CA 90024

and

I.G. Rosen

Department of Mathematics

University of Southern California

Los Angeles, CA 90089

ABSTRACT

In the optimal linear quadratic regulator problem for finite dimensional

systems, the method known as an a-shift can be used to produce a closed-loop system

whose spectrum lies to the left of some specified vertical line; that is, a closedloop system with a prescribed degree of stability. This paper treats the extension

of the a-shift to hereditary systems. As in finite dimensions, the shift can be

accomplished by adding a times the identity to the open-loop semigroup generator and

then solving an optimal regulator problem. However, this approach does not work

with a new approximation scheme for hereditary control problems recently developed

by Kappel and Salamon. Since this scheme is among the best to date for the

numerical solution of the linear regulator problem for hereditary systems, an

alternative method for shifting the closed-loop spectrum is needed. An a-shift

technique that can be used with the Kappel-Salamon approximation scheme is

developed. Both the continuous-time and dlscrete-time problems are considered. A

numerical example which demonstrates the feasibility of the method is included.

t_,_This research was supported in part by The Air Force Office of Scientific

Research under contract No. AFOSR-84-0309.

(**)This research was supported in part by the The Air Force Office of Scientific

Research under contract No. AFOSR-84-0393.

(t)Part of this research was carried out while the authors were visiting scientists

at the Institute for Computer Applications in Science and Engineering (ICASE), NASA

Langley Research Center, Hampton, VA, which is operated under NASA contracts

No. NASI-17070 and No. NASI-18107.

-o05/6_o7d:

I. INTRODUCTION

In this paper we consider the problem of computing optimal linear state

feedback control laws for linear hereditary systems which yield a resulting

optimal state trajectory that exhibits a prescribed degree of stability. This

problem is sometimes referred to as the linear quadratic regulator problem

with e-shift, u being the desired degree of stability, since it involves a

linear state constraint, the minimization of a quadratic payoff functional and

the shifting of the closed-loop spectrum to the left of the line Re z = -u in

the complex z-plane. A solution is a control law of the form

(I.I) u (t) =-K(x (t), xt) , t > 0

where K is a linear function of the optimal trajectory x*(t) and its past

history x t at time t, x = x*(u*) is the solution to the underlying hereditary

system, u minimizes a performance index which is quadratic in the state and

the control and x satisfies a uniform exponential bound of the form

(1.2) [x*(t) I < Me -_t t > 0

These ideas will be made precise in the subsequent Section 2.

In finite dimensions, i.e. when the state is given by a linear ordinary

differential equation of the form

(1.3) x(t) = Ax(t) + Bu(t), t ) 0,

the linear quadratic regulator problem with u-shift and its solution are well

known (see [i], [2]). The matrix A is simply replaced by the matrix
--2-- \

A + =I and the resulting standard linear quadratic regulator problem (the

shifted problem) is solved in the usual fashion.

A hereditary system, on the other hand, is infinite dimensional. Instead of the matrix A being replaced by A + _I, it is the infinitesimal

generator A of the solution semigroup which is replaced by A + el. The

solution of the resulting shifted regulator problem requires the use of some

form of finite dimensional approximation. One is tempted to take the general

approach which by now has become standard in the control of infinite

dimensional or distributed systems. That is, approximate the unshifted system

using one of the currently available schemes for the regulator problem for

hereditary systems and then apply the standard finite dimensional theory and

techniques to the finite dimensional approximating systems to obtain

approximations to the shifted system. However, as we discovered, this

approach may not work.

The linear spline-based approximation scheme for the linear quadratic

regulator problem for hereditary systems recently developed by KapDel and

Salamon in [14] has been shown to be, in many respects, one of the most

attractive approximation methods currently available for this class of

problems. However, the finite dimensional approximating systems possess

eigenvalues which do not converge to eigenvalues of the original underlying

hereditary system. Those eigenvalues are stable and hence do not cause

difficulties when the unshifted problems are solved. However, they are

extremely difficult, if not impossible to shift. Consequently, even if the

poles of the hereditary system which are to the right of the line Re z = -e in

the complex z-plane can be shifted (i.e. the finite dimensional subspace

spanned by the eigenvectors corresponding to the eigenvalues with real part

greater than or equal to -_ is controllable), when the s-shift is
--3--

applied to the approximating systems the solution of the resulting finite

dimensional regulator problems fail.

We have found a relatively simple and straight forward way to overcome

this difficulty. It involves a modification of the infinite dimensional

shifted system which permits the extraneous eigenvalues introduced by the

approximation to remain stable while the true eigenvalues of the hereditary

system are forced to the left of the line Re z = -e.

In the paper we have treated both the continuous-time and the discretetime or sampled problems. The Kappel-Salamon approximation and the e-shift

are discussed in Section 3. In Section 4 we provide an example with numerical

results.
-4-

2. THE OPTIMAL LINEAR QUADRATIC REGULATOR PROBLEM FOR HEREDITARY SYSTEMS WITH

A PRESCRIBED DEGREE OF STABILITY

We consider linearhereditarycontrol systems of the form

(2.1) x(t) = Lx t + B0u(t), t • 0,

(2.2) x(0) = n, x0 = _ ,

where x(t) _ Rn , u _ L2(0,tf;R m) for each tf with 0 < tf < _ ,

B0 E i(Rm,Rn), n _ Rn , _ _ L2(-r,0;Rn) and for each t • 0,

x t _ L2(-r,0;R n) denotes the past history of the state x on the interval

[t-r, t]. That is, xt(8) = x(t+0), -r ( 8 ( 0. The linear transformation L

is assumed to be of the form

(2.3) L, = Z Ai_(-r i) + f0rA(O )*(O )dO_ i=0

with Ai E i(Rn,Rn),i = 0,I,2,...,_,(_ < _), A _ L2(-r,0; /(Rn,Rn))and

0 = r0 < rI < r2 < r9 = r.

Standardargumentsyield the existenceof a unique solution x( ; n,_,u) to

(2.1), (2.2) which is absolutelycontinuouswith x( ; q,_,u) € L2(0,tf;Rn)

for any tf, 0 < tf < m, which satisfies(2.1) for almost every t • 0 and which

depends continuouslyon 0,$, and u.

A one parameterfamily of solutionoperatorsfor the homogeneoussystem

correspondingto (2.1), (2.2) can be defined by

(2.4) T(t)(O,¢) = (x(t;n,¢,0), xt(n ,@,O)).
-5-

If we let Z = Rn x L2(-r,0;Rn) together with the usual inner product

(2.5) <(_,O),(_,X)>z = _T + f0rO(0)T(0)d0,_

then the family of operators, { T(t) : t ) O} forms a CO semigroup of bounded

linear operators on the Hilbert space Z. The infinitesimal generator is given

by

Dom(A) __{(_,_) _ Z : _ _ HI(-r,o;Rn), _ = _(0)}

(2.6) A(_(O),_) = (L_,D_) .

If we define the operator B: R m . Z by

(2.7) Bu = (Bou,0)

then an equivalence exists between solutions to (2.1), (2.2) and mild or

generalized solutions to the abstract evolution equation

(2.8) z(t) =Az(t) + Bu(t), t ) 0

with initial condition

(2.9) z(O) = (n,@) •

That is, z(t) = (x(t;n,_u), xt(n,_,u)), where
-6-

(2.10) z(t) = T(t)(q,_) + f0t T(t-s) Bu(s)ds, t _ 0 .

In the subsequent discussion, when the solution to the system (2.8), (2.9) is

referred to, it should be understood to imply the mild solution given by

(2.10).

2.1 THE CONTINIJOUS-TIME PROBLEM

The control problem which is of interest to us here is the infinite time

horizon linear quadratic regulator (LQR) problem given by

Find u s L2(0,=;Rm ) which minimizes the performance index

(2.11) _I(u) = f x(t)TQ0x(t) + u(t)TRu(t)dt 0

where x is the solution to (2.1), (2.2) corresponding to u.

The matrix Q0 s i (Rn,R n) is assumed to be nonnegative, symmetric and the

matrix R € /(Rm,R m) is assumed to be positive definite, symmetric.

Defining the nonnegatlve, symmetric operator Q : Z . z by

(2.12) Q(_,_) = (Q0_,0),

we treat the equivalent LQR problem given by

(PI) Find u g L2(0,=*;Rm) which minimizes

(2.13) J(u) = f <Qz(t),z(t)> Z + u(t)TRu(t)dt 0
-7-

where z is the solution to (2.8), (2.9) correspondin_ to u.

We summarize the results from [7] concerning the solution of problem

(PI). An admissible control for the initial state z(O) = (n,_) € Z is a

u g L2(O,_;Rm) for which J(u) < _. Under the assumptions

function

(AI) for each initial state, z(0) = (n,_) € Z there exists an admissible

control,

and

(BI) the operators L, B 0 and QO are such that any admissible control u

drlves the state z(t), t > 0 to zero, asymptotically as t . _ ,

there exists a unique nonnegative, self-adjoint solution P _ i(z,z) to the

Riccati algebraic equation

(2.14) A*P + PA- PBR -I B* P+ Q = O.

The unique solution u € L2(O,_;Rm ) to problem (PI) is given in feedback form

by

* -I * *

(2.15) u (t) = -R B Pz (t), t ) 0

and

(2.16) 3(u ) = < P(n,(F) , (n ,€)>z .

The optimal trajectory, z , is given by
-8-

(2.17) z (t)=S(t)(q,_)

where IS(t) : t ) 0} is the CO semlgroup generated by A- BR -I B* P. The

semlgroup (S(t) : t _ 0} is uniformly exponentially stable, i.e. there exist

positive constants M and _ for which

(2.18) Is(t)[< Me -_t , t > 0 .

The operator A* is given by

Dom(A*) = {(E,_) € Z : D_ _ L2(-r,0;Rn), _ absolutely

continuous on [-r,0] except at the points -rl,...,-rg_ 1

where _((-rl)+)-_((-rl) -) = A_E, I < i < v - I

and _(-r) = _E} ,

(2.19) A*(E,_) = (ATE + _(0), ATE - D_)

and is the infinitesimal generator of the CO semigroup {T (t) : t _ 0}.

We have P Z c Dom (A*).

The operator P can be represented by a matrix of operators

220 Ip°°10 11°I)

where p00 € i(Rn,R n) is a nonnegative symmetric matrix,
-9-

p10 € L2 (-r,0; /(Rn,Rn)), p01 = p10* with

0

(2.21) p01_ = f pl0(o)T_(o)dO, _ € L2(-r,0;Rn),

-r

and -FII g i(L2(-r,0;Rn),L2(-r,0;Rn)) is nonnegative and self-adjoint. We

have

0

* 0 * pl * (2.22) u (t) = - p x (t) -f (O)xt(O)dO , t _ 0

-r

0 -I T p00 1 -I T )T where p = R B0 and p (O) = R B 0 pI0(o , -r _ O _ 0.

We note that Assumption (AI) is satisfied if the unstable subspace of L

(which is finite dimensional, see [9], [23]) is controllable. Assumption (BI)

is certainly satisfied if Q0 is positive definite.

The Continuous-Time Problem with a-Shift

An a-shlft, we recall, is a technique which is used in conjuctlon with

the standard LQR theory to obtain an optimal feedback control which yields not

just an asymptotically stable closed-loop system, but rather, one which

exhibits a prescribed degree of stability, that is, one for which the state

-_t

z(t) decays at least as fast as e , i.e.

(2.23) [z(t)l Z _ M e'_tlz(0)l Z

where M is a positive constant and _ > 0 is the desired degree of stability.

For the hereditary systems of interest to us here, this is completely
-I0-

equivalent to requiring that the eigenvalues of the closed-loop system have

real part less than -e. A discussion of this problm and its solution for

finite dimensional systems first appeared in [I] and can also be found in [2].

One approach to solving this problem involves the inclusion of the

2ct

multlplicatlve factor e under the integral sign in the performance index

given in (2.13). However, making the change of variables

A

(2.24) z(t) = eetz(t)

A

(2.25) u(t) = eCtu(t)

it is easily seen that if one solves the modifiedLQR problem

^_

(PI) Find u £ L2(0,=;Rm) which minimizes

oo

A

(2.26) J (u) = f <Qz(t),z(t)> Z + u(t)TRu(t)dt

where z is the (mild) solution to the abstract evolution system

d A A A

(2.27) _ z(t) = ( A+ cl)z(t) +Bu(t), t ) 0

A

(2.28) z(0) = (n,¢)

and applies

-_t ^,

(2.29) ue*(t)- = e u (t), t ) 0

to the original control system (2.8), (2.9), the resulting optimal trajectory,
-IIz , will satisfy (2.23).

Strictly speaking (2.27), (2.28) is not a hereditary system. However,

the results outlined above concerning the solution of problem (PI) in closedloop form are in fact derived from a more general abstract theory (see [6]).

A

This more general theory can be applied directly to problem (PI). Under

assumptions (At) and (BI) (with z, u, and j replaced by

A A A A

z, u and J respectively) the unique solution to problem (PI) is given in

state feedback form by

"* --1 * " "*

(2.30) u -- -R B P z (t), t _ 0

where P s L(Z,Z) is the unique nonnegative self-adjoint solution to the

Riccati algebraic equation (2.14) with A and A* replaced by A + _I and

, ^

A + =I respectively. The operator p can be represented by a matrix of

operators analogous to the one given in (2.20). From (2.24) and (2.29) we

obtain

-1B* p *

(2.31) us•(t) = -R z (t), t _ 0 •

It then follows that

(2.32) z (t) = Se(t)(rl,¢), t > 0

with

A

-et

(2.33) IS=(t)I _ M e , t _ 0
-12-

_A

where {S (t) : t ) 0} is the C O semigroup generated by A- BR -I B P •

Since the shifted system (2.27) is not a hereditary system, it would seem

that to obtain the estimate (2.33) from the general theory presented in [6],

the coerclvlty of Q would be required. (In the case of a hereditary system,

assumption (BI) is sufficient.) However, as will become clear in the next

section, (2.27) is in fact related to a hereditary system through a bounded

similarity transformation. Consequenly, assumption (BI) is sufficient to

obtain the uniform exponential bound (2.33) for the shifted system as well.

The controllability of the finite dimensional generalized eigenspaces

corresponding to the eigenvalues of A with real part greater than or equal to

-e is sufficent to conclude that assumption (AI) holds for the shifted system.

2.2 THE DISCRETE-TIME PROBLEM

The discrete-time or sampled analog of problem (PI) is given bY

(P2) Find u = {Uk} k=0 s £2 (0'=;Rm) which minimizes

T

(2.34) J(u) = _ <Qzj,zj> Z + ujRuj where z = {Zk}k= 0 satisfies the recurrence j=O

(2.35) Zk+ I = Tz k + BUk, k = 0,1,2,...,

with

(2.36) z0 = (n,@).

The operators T g L(Z,Z) and B g i(Rm;Z) are defined by
-13-

T

(2.37) T = T(_) and B = f0 T(s) Bds,

respectively, where T denotes the length of the sampling interval.

The characterization of the solution to the dlscrete-time LQR problem in

state feedback form for infinite dimensional systems is treated in [18] and

[24]. The application of the general theory to problems involving hereditary

systems is discussed in [8]. The results are completely analogous to those

given above for the continuous-time problem. We briefly summarize them here.

An admissible control sequence u _ £2(0,_;R m) for the initial condition

z0 = (n,_) E Z is one for which J(u) < _. If the assumptions

(A2) for each initial condition z0 = (B,_) € Z there exists an admissible

control

and

(B2) the operators L, B0 and Q0 are such that if u is an admissible

control for the initial condition z0 = (n,_) then the state

z = {Zk}_= 0 given by (2.35) satisfies lim IZklz = 0

hold, then there exists a unique solution to problem (P2) which is given in

linear state feedback form by

(2.38) u k = -Fz k , k = 0,1,2,...

where
-14-

(2.39) F = R-IB*PT,

(2.40) R = R + B PB

and P is the unique nonnegative self-adjoint solution to the Riccati algebraic

equation

(2.41) P = T*(P - PB(R + B*PB)-IB*P)T + Q .

The minimum value of the performance index can be computed from

(2.42) J(u ) = <P(n,_),(n,_)> Z

and the optimal trajectory z satisfies

* sk(n (2.43) zk = ,_), k = 0,1,2,...

where S s i(Z,Z) is given by

(2.44) S = T-BF .

We also have the following result.

Theorem 2.1 If assumptions (A2) and (B2) hold then the operator S has

spectral radius less than i and there exist positive constants M and p with

p < 1 for which
-15-

(2.45) ISk] _ M0k, k = 0,I,2,...

Proof

Since {T (t) : t > 0} is the solutionsemiKroupfor the hereditarysystem

(2.8) and the operatorBF is of finite rank, the operatorsSk = (T - BF)k are

compact for all k sufficientlylarge. It follows therefore(see [5],Chapter

VII, Section 4, Theorem 6) that the spectrumnof S, a(S), containsat most a

countablenumber of pointswith no accumulationpoints in the complexplane

except possibly_ = 0. The non-zeroelementsin o(S) are in the point

spectrumof S; that is, they are eigenvaluesof S.

Now suppose_ £ a(S), _ # 0 and S(_,_) = l(_,_) with (_,_)# 0. If

# 0, then for z0 = (_,_) the optimal trajectoryis

* = sk lk (2.46) zk (_,_)= (_,_), k = 0,1,2,....

Consequently assumption (B2) implies _ < I. If _ = 0 and

(0,_) _ N(F), the null space of F, then z0 = (0,_) implies

, , k(0 kF (2.47) u k = -Fz k = -FS ,_) = -_, (0,_), k = 0,1,2,...

Since u E £2(0,=;Rm), (2.47) implies _ < I. Finally, if (0,_) _ N(F), then

(2.48) _(0,_) = S(0,_) = T(0,_) = (x(T;0,_,0), xT(0,_,0))

which implies _ = 0 and consequently that _ _ o(S).
-16-

Therefore,we concludethat the spectralradius of S is less than 1 and

that Iskl( pk for all k sufficientlylarge for some positivep < I. The

uniform exponentialbound (2.45)immediatelyfollows.

The operator F _ i(Z,R m) can be represented by a matrix of operators,

(F0,F I) where F0 E i(Rn,R m) can be represented by an mxn matrix f0 and

F 1 _ L(L2(-r,0;Rn),R m) can be represented by a square integrable mxn matrix

valued function fl defined on the interval [-r,O]. We have

(2.49) u*k = -fOx*(kr) - f_0rfI(8)x_ (O)dO, k = 0,I,2,...

with (x (kr), Xk_) = Zk, k = 0,1,2,....

The Discrete-TimeProblem wlth c-Shlft

The shiftedproblem in dlscrete-tlmeinvolves the finding of an optimal

control us for which the resulting optimal trajectory z satisfies

* M_k,

(2.50) [z ,k[ ( k = 0,1,2,...

where c, the prescribed degree of stability is a positive number less than I. The

modified discrete-time problem (analogous to problem (PI)) takes the form

^ ^,

(P2) Find u g £2(0,=;R m) which minimizes

(2.51) J(u) = _ <Qzj,zj>z + u.mu. where z = { satisfiesthe recurrence j=0 3 3 0

^ 1 ^ + 1 u

(2.52) Zk+l = --c Tz k _ B k' k = 0,1,2,...
-17-

(2.53) z0 = (n,_) •

Assumptions analogous to (A2) and (B2) yield

^, ^^,

(2.54) u k = - Fz k , k = 0,1,2,...

^

where F is given by (2.39) - (2.41) with T and B replaced by

1 B respectively. It follows that _T and

(2.55) u ,k -Fz ,k , k = 0,1,2,...

with the optimal trajectory given by

* k

(2.56) z, k = S (n,_), k = 0,1,2,...

where S s i(Z,Z) is defined by

^

(2.57) S = T - BF .

The operator S has spectral radius less than 1 and is uniformly exponentially

bounded;

(2.58) [sk[ < M_ k, k = 0,1,2,..

^

where M is a positive constant which does not depend on k.
-18-

3. APPROXIMATION

A

The infinite dimensionality of problems (PI) (or (PI)) and (P2)

A

(or (P2)) necessitates the use of some form of finite dimensional

approximation to solve them. The standard approach involves the use of finite

element (Rayleigh-Ritz, Galerkin, etc.) techniques to discretize the state

equations (2.8) (or (2.27)) and (2.35) (or (2.52)). A sequence of finite

dimensional LQR problems result, each of which can be solved in linear state

feedback form using standard techniques and readily available software. The

averaging , or AVE scheme, which uses piecewise constant elements with finite

differencing, and its application to the continuous time problem is carefully

studied in [7]. A linear spline based Galerkin method is treated in [4].

More recently, methods using piecewise linear elements [21] and Legendre

polynomials [13] and a method based upon Lanczos' T-method for partial

differential equations which also uses Legendre polynomials [12] have yielded

promising results.

While AVE yields strong L2 convergence of the approximating feedback

kernals, the observed rate of convergence is relatively slow. The spline

based scheme, by some measures, offers superior performance. However, it

appears that only weak L 2 convergence of the approximating feedback kernals

can be obtained. Kappel and Salamon [14] have developed a new linear spline

based method which performs at the level of the original spline scheme and

which seems to yield strong L2 convergence of the approximating functional

feedback gains. It is this approximation scheme which is the focus of our

discussions below.
-19-

3.1 AN APPROXIMATION SCHEME FOR LINEAR HEREDITARY SYSTEMS

We briefly outline the details of the formulation of the Kappel-Salamon

scheme. Fundamental to their approach (and unlike the standard Galerkin

approach) is the choosing of the approximating spaces so that they are not

subspaces of either Dom(A) or Dom(A*). Herein lies the key to obtaining

strong L 2 convergence of the approximating feedback kernals.

For each N = 1,2,... let

k,j_ Ark

(3.1) eN - -rk_ 1 - j --_--, k = 1,2,...,_, j = 0,1,2,.. .,N

where

(3.2) Ar k = rk - rk_1, k = 1,2,...,_ .

Let {_'J N}j=0 denote the usual linear B-spline elements with respect to the

k,N ,0k,0. mesh {8N ,... N _ on the interval [-rk, -rk_l), k = 1,2,...N and extended

to be zero elsewhere on the interval [-r,0]. That is for each k = 1,2,...,N

^k,1. k,1 ^k,0.

N (8 - _N ) 8 _ [8N _N ) ' ,

_k,0(8 ) =

N

0 elsewhere

-N (8 k,j-I k,j-I

eN ), e t0 ,JeN j

_k,j(e) = N (8 - 8 k'j+l k,j+l k,j]

(3.3) "N _ N )' 8 € [8N - ,8N

0 elsewhere
-20-

j = 1,2,...,N-I and

-N ^ k,N-I °^ k,N ^ k,N-I

- UN , UN

_ k,N.^ (o ) e € N , ]

N Iv) =

0 elsewhere .

Defining

(3.4) eN0 = (In,0) and eNk'j = (0, #Nk,j in )

in Rnxn x L2(-r,0;R nxn) where In denotes the nxn identity matrix, we let

N

0 k,j

(3.5) ZN = {(_,_N) € Z: ($,_N) = eN a0 + _ _ eN ak,j, a0, ak,j € Rn}. k=! j=0

The collection {eO, ek'j} is a basis for the KN = n((N+l)_ + i) dimensional

)r

subspace of Z, ZN and a = (a0, al,0,...,a_, N is referred to as the

k,j_

coordinate vector with respect to the basis {eO, eN 2 for the element

(_,_N) g Z N. Defining

0 1,0 _,N), (3.6) EN = (eN,eN ,...,e N

we have ($,_N) = ENaLet PN: Z . ZN denote the orthogonal projection of Z onto ZN. It is

immediately clear that pN(_,_) = (_,_N_) where _N is the orthogonal projection

of L2(-r,0;R n) onto span{_'Jl n} .
-21-

Noting that ZN _ Dom(A), approximations AN: ZN . ZN to the operator A are

defined by first extending A to all of ZN. for (_,_N) E ZN, define

(3.7) A($,_N ) = (_(_,_N),D(_,¢N))

where

0

(3.8) _(_,_N) = A0_ + Y Ak_N(-r k) + f A(e)_N(e)de k=l -r

and

(3.9) D(_,_N) = D+_N + 60(_ - lim_¢N(e))+ [ _k(¢N(-rk) - lim _¢N(e))

8.0 k=l 9.-r k

+

with 6i the Dirac delta impulse centered at -ri, i = 0,i,2,...9-I and D _N the

derivative from the right of _N "

Let M N € i(R _, R_) be given by

T

(3.I0) MN = <EN,EN>Z

6k,+ k,- n

and define N ' _N g i(R ,ZN) by

Akg+ E k,+

(3.11) 6N _ = NYN _, k = 1,2,...,_,

k_ m k_

(3.12) _N _ = ENYN -_' k = 0,I,2,...,9-I ,

where
-22-

(3.13) YNk,+ = _I(0 '¢N1,0(-rk)"'''¢N_,N (-rk))T e9In,

(3.14) yk,- = MI_I(o,lim -_N1,0(8),...,lim ___I'N(B))TQ91

8+_rk e+_rk n

and _ denotes the Kroneckerproduct. The approximatingoperators

AN : zN . zN and their adjoints are given by

o + 0,-

(3.15) AN(_,_ N) = (L(_,_N),_ND _N ) + 6N (_ - llm_ _N(8))

8+0

v-i

+ Z 6Nk'-(_N(_rk) - lim _N(8))

k=l 8+-rk

and

(3.16) AN($,_ * N) = ( lim_ _N(O) + A 0_, _ N (AT$ - D +_N )) O+0

v-I

+ Z 6k'+N (Ak_ + lim _ _N(8) -_N(-rk))

k=l 8.-rk

+ 6_9,+ N (AT_ - _N(-r))

respectively.

3.2 THE APPROXIMATE SOLUTION OF THE REQULATOR PROBLEMS

The Continuous-Time Problem

(3.17) SN = pN s IZN = BIZN

and
-23-

(3.18) QN : PNQIz N = QIZN

and assume that problem (PI) with A,B, Q and (n,_) replaced by nN, B N, QN

and pN(n,_) satisfies assumptions analogous to (AI) and (BI). (Under certain

conditions, if the original system satisfies (AI) and (BI) so too will the

approximating systems if N is sufficiently large, see [14].) The

approximating solutions to problem (P1) are then given in feedback form by

* -I * *

(3.19) UN(t) = -R BN PN PNZN(t)' t _ 0

where PN is the unique nonnegative self-adjoint solution to the Riccati

algebraic equation

* -I *

(3.20) AN PN + PN AN + PN BN R BN PN + QN = 0

and zN is given by

(3.21) ZN(t) =sN(t)(_,_), t > 0

where {sN(t) : t > 0} is the CO semigroup with infinitesimal generator

A BR -1 *

- B N PNPN •

In practice, the approximating feedback gains are computed by solving the

KN dimensional matrix Riccati algebraic equation

(3.22) [AN]THN + HN[AN] + HN[BN]R-I[BN]THN + QN = 0
-24-

where brackets denote the operator's matrix representation with respect to the

basis {e , eNk,j__ and the matrix _N is given by

(3.23) IIN = MN[PN].

Then, if we write

1,0

PN

(3.24) [FN] = PN

0 pk,J

where PN and °N ' k = 1,2,...,_, j = 0,1,2,...,N are n×n matrices, we have

* 0 * 0 1 *

(3.25) UN(t) = PN XN(t) -f-r pN(8 )(XN)t(8 )dB, t _ 0

with

N

(3.26) PN0 = R -IBT0 _0'N , pNI(8) = _ _ R -IBT0 PNk,j #Nk,J(8), -r _ 8 _ 0

_-1 j=o

and (_.(t),(XN) t)_ = ZN(t),_ _ t > 0
-25-

The Discrete-Time Problem

For the dlscrete-tlmeproblem,we let

(3.27) TN = IN(T) and BN = fO TN(t) BNdt

where BN is given by (3.17)and {TN(t) : t ) 0} is the CO semiKroupwith

infinitesimalgeneratorAN. The approximatingsolutionsto problem (P2) are

then given in feedbackform by

(3.28) UN, k = - FNPNZN, k, k = 0,I,2,...

where

(3.29) FN = RN BNPNT N ,

(3.30) RN = R + BNPNBN,

PN is the unique, nonnegative, self-adjoint solution to the Riccati alKebraic

equation

(3.31) PN = TN* (PN _ PNBN(R + BNPNBN * )-iBN* PN)TN + QN

and zN is given by

(3.32) ZN,k = (sN)k(n,_), k = 0,I,2,...
-26-

with

(3.33) SN= T - BFNP N.

The approximating feedback kernals can be computed from

(3.34) [FN] = [RN ]-I[BN]TFN[TN ] ,

and

(3.35) [RN] = R + [BN]TFN[BN ]

where

(3.36) [TN] = exp ([AN]T) , [BN] = fO exp([AN]t)[BN]dt

and rN is the unique nonnetative symmetric solution to the KN dimensional

matrix Riccati algebraic equation

(3.37) rN = [TN]T(rN -FN[BN](R + [BN]TFN[BN])-I[BN]TPN[TN ] + [QN]

If we set

(3.38) [FN]M I = (F_ , FN1,0 ,...,F 9,N N .)

0 k,j

where FN and FN , k = 1,2,...,_, j = 0,1,2,...,N are mxn matrices, we have
-27-

* 0 * 0 1 *

= XN(kT ) -_ (0) (0)d0 k = 0 1 2, (3.39) UN, k -fN -r fN (XN)k_ ' ' ' "'"

with

v N

_k,j _k,J(0 ) -r ( 0 ( 0

(3.40) fN0 = FN0 ' fNI(0 ) = l l _"N "N

k=l j=0

and (XN(k_)' (XN)k_) = ZN,k' k = 0,1,2, ....

3.3 CONVERGENCE

The Continuous-Time Problem

Elementary approximation properties of spline functions and the TrotterKato Theorem on the approximation of semigroups (stability together with

consistency imply convergence, see [15], [20]) can be used to argue that

(3.41) TN(t) + T(t) and TN(t) + T (t), t • 0

strongly on Z as N . _ , uniformly in t for t in bounded sub-intervals, where

{TN(t) : t • 0} and {Y (t) : t • 0} are the CO semigroups with infinitesimal

generators AN and A respectively. Observing (numerically) that IPNI is

bounded in N, it follows (see [7], Theorem 6.7) that PN converges weakly to P

0 0 Rmxn 1 1

as N . _ and consequently that PN + p in and PN . _ weakly in

; Rmxn) L2(-r 0 as N . = . To obtain strong convergence of PN to P and strong

1 I, L2 convergence of PN to P the only known result (see [7], Theorem 6.9)

requires the existence of positive constants M and m, independent of N, for

which
-28-

(3.42) [SN(t) [ < Me -rot, t • 0, N • 1

where { SN(t) : t • 0} is the CO semigroup with infinitesimal generator

-- *

AN BNR-I BN PNp N • While all numerical results indicate that strong

convergence of the approximating feedback kernals holds, analysis in [14] and

numerical studies point to the fact that a uniform exponential bound of the

form (3.42) can not be obtained for the Kappel-Salamon scheme. Indeed, both

the open and closed-loop approximating systems yield a sequence of extraneous

eigenvalues (i.e. ones which do not appear to be converging to an element of

the spectrum of the original open or closed-loop herediatary system)

{iN }:=I for which Re I N + 0- and Im lN + + _ (or - _) as N + _

We note that the N-independent uniform exponential bound (3.42) which is

sufficient for strong (in fact trace norm, see [7]) convergence of PN to P has

been shown to hold for the AVE scheme in [22] and for the Legendre-tau method

in [11].

The Discrete-Time Problem

For the discrete-time problem (see [8], Theorem 3.12) IPNI bounded in N

0 f0 Rmxn 1 fl

implies PN + P weakly, FN . F strongly, fN + in and fN + weakly in

L2(-r,0;R mxn) as N . = The existence of positive constants M and P which do

not depend on N, with O < I and for which

(3.43) [S_I < MP k , k = 0,I,2,..., N • I,

where

(3.44) SN = TN - BNFNP N
-29-

is sufficient (see [8], Theorem 3.10) to conclude strong convergence of PN to

P, uniform norm convergence of FN to F and strong L2 convergence of

1 fl fN to as N . _ . Although numerical studies indicate that the stronger

modes of convergence hold, the approximating open and closed-loop discretetime systems constructed using the Kappel-Salamon scheme yield a sequence of

extraneous eigenvalues {_N}N=I with I_NI . I as N . _ .

3.4 THE APPROXIMATE SOLUTION OF THE REGULATOR PROBLEMS WITH u-SHIFT

One obvious approach for approximating the solutions to the LQR problems

A

with e-shift is to replace the operators A,B and Q in problem (PI) with the

operators AN, B N and QN or the operators T, B and Q in problem (P2) with TN,

BN and QN and then solve the finite dimensional shifted problems with states

given either by

d A A A

(3.45) _ ZN(t) = (A N + _l)ZN(t) + BNU(t) , t _ 0

or

^ 1 ^ I ^

(3.46) ZN, k+l = --a TN ZN, k + _ BNU k , k = 0,1,2, ....

However, if the Kappel-Salamon scheme outlined above is used, this approach

will not work. Indeed, our numerical studies indicate that as a result of the

extraneous eigenvalues introduced by the approximation scheme, for N

and a sufficiently large the resulting finite dimensional systems are, at

best, marginally stabilizable. The solutions to the matrix Riccati algebraic

equations begin to deteriorate. Eventually the eigenvalue-eigenvector or

Schur vector methods used to solve them fail completely. We
-30-

observe this independently of the stabilizability of the original underlying

hereditary system. Since only the poles of the original hereditary system are

to be shifted, this situation can be remedied by observing that the

operator A+ _I is related to the intinitesimal generator for a hereditary

system (semigroup) through a bounded similarity transformation.

For T a real number, define Uy _ L (Z,Z) by

To

(3.47) uy(_,_)--(_,e _) ,

where the function eY'_evaluated at 8 is eY8_(8), r < 8 _ 0. Then

(3.48) U$1(_,kb)= ($,e-T'_b).

The Continuous-Time Problem

A A A

For the continuous-timeproblem,set w(t) = U z(t). Then w satisfies

d A A ^

(3.49) _-_ w(t) = A w(t) + Bu(t), t _ 0

A

(3.50) w(0) ffi(n, ee'#)

where for T g R, AT: Dom(A)c Z + Z is given by

(3.51) AT(_(0), _) = UT(A+TI) u_l(_(0), £b) = (Lye, D_b)

with
-31-

Yr i

(3.52) Ly* = (A0 + yl)*(0) + [ e Ai,(-r i) + fO r e-YOA(O),(O)d8 .

i=1

7

U I is self-adjoint and U_IQ Uyl- = Q for all y € R, the Kappel-Salamon

Since

A

scheme can be applied to problem (PI) with A + aI replaced

A

with Aa and z replaced with w . The extraneous eigenvalues will now remain

stable and therefore cause no problems when the approximating feedback laws

are computed. The approximating solutions to the continuous-time problem

with e-shift are given by

* -I * ^ * -I * *

(3.53) UaN(t) = -R BN PaNPNZaN(t) = -R B N PeNPNUeZaN(t), t _ 0

where PaN is the solutionto the Riccati algebraicequation(3.20)with AN

and AN replacedby AaN and AaN respectivelyand ZaN is given by

(3.54) ZaN(t) = S (t)(n,_), t ) 0

where { S_(t) : t _ 0} is the C O semigroup with infinitesimal _enerator

A - BR-I B*

N PaNPN" We have

0 * 0 1 )eae *

= - PaN XeN(t) - f-r PaN(e (XaN)t(O)d8, t ) 0

0 1

where pON and pi N are obtained from PaN in the same manner that PN and PN

are obtained from PN (as in (3.2.6)) and (XeN(t), (x N)t ) = ZaN(t), t ) 0 .
-32-

The Discrete-Time Problem

A

For the discrete-tlmeproblem,we set wk = U_zk. Then

A A

(3.56) Wk+1 = T_wk + BaUk, k = 0,I,2,...

(3.57) ^w0 = (n,e )

where

(3.58) T=--FI UBTUsI= T8(r),

{Ts(t) : t ) 0} the CO semlgroupwhose infinitesimalgeneratoris AS,

(3.59) Be = _I f_0 e-St TB(t) B dt

and 8 = -(ine)/T. The approximating solutions to the dlscrete-tlme problem

with =-shlft are given by

, ^ , *

(3.60) UaN,k = -FaNPNZaN,k = -FaNPNUsZaN,k , k = 0,1,2,...

where FeN is computed according to (3.29)-(3.31) with FN, _, BN, TN and PN

replaced by FeN , RN, BaN , TeN and PeN respectively and ZaN is given by

* N k(n

(3.61) ZaN,k = (Se) ,_), k = 0,1,2,...

with
-33-

A

(3.62) SN = T

a - BFaNPN"

Finally, we have

(3.63) UaN,k * = -faNX_N ^0 * (kr) - f0rflN(e)(XeN)kr * (O)dO

= _fOaN X_N(k_) * -f O-r f1N(8)eSe(XaN)k_(O)d8 * ' k = 0,I,2,.""

where fO fl 0 1

aN and aN are obtained from F N as were fN and fN from FN in (3.40)

and (XaN(kr) , (XaN)kr) = ZaN,k , k = 0,1,2, ....
-34-

4. AN E_LE AND NDMERICAL RESULTS

We consider the second order linear harmonic oscillator with delayed

damping given by

o.

(4.1) y(t) + y(t-l) + y(t) = u(t), t _ 0 .

We take the continuous-time performance index to be

(4.2) J(u) = / y(t) 2 + y(t) 2 + u(t)2dt .

0

Setting x(t) = (y(t), y(t)) T, we rewrite (4.1) as a first order system;

For this example we have n = 2, m = I, r = I, _ = I, A _ 0,

A01? C00 -I0)' B0 = (?)' Q0 = (0 I) '

and R = i.

We computed the optimal feedback gains for the shifted and unshifted,

continuous and discrete-time control problems on an IBM PC personal computer

using the Kappel-Salamon approximation scheme and the a-shift technique

outlined in the previous section. The matrix Riccati algebraic equations

(3.22) and (3.37) were solved using either a standard eigenvalue/eigenvector

(see [16]) or Schur vector (see [17], [19]) decomposition of the Hamiltonian

matrix. For the discrete-time problem, matrix exponentials were computed

using an eigenvalue/eigenvector decomposition.
-35-

A Scheme to Compute Eigenvalues of Linear Hereditary Systems

In order to evaluate the performance of the method, we had to be able to

compute approximations to the closed-loop elgenvalues of contlnuous-time and

sampled hereditary systems. To do this we used a spline-based scheme

developed in [I0]. In the case of the contlnuous-tlme problem, the closedloop system is a homogeneous hereditary system. Let A N = A-BR -I B_ P NPN

denote the infinitesimal generator of the closed-loop semigroup

{sN(t) : t _ 0} and let {ZM} denote a sequence of finite dimensional splinebased subspaces of Z which are contained in Dom(A). Let qM : Z + ZM denote

the orthogonal projection of Z onto ZM with respect to the inner product

(4.4) <<(K,_), (K,X)>> z = <AN(_,_),AN(K,X)>z .

An approximation to the spectrum of AN is obtained by computing the

i

eigenvalues of the matrix representation of the inverse of the operators

(4.5) qM( AN)-1

IzM •

Spectral convergence is argued in [I0] using the theory of collectively

compact families of operators (see [3]).

The approach outlined above is used to obtain approximations TM and BM to

the discrete-time open-loop state transition operator T and input operator

A

B. The feedback gains F NPN are projected (with respect to the standard Z

product) onto ZM to obtain the operators F_N " The eigenvalues of the

inner

operator TM B_FM - aN are taken to be an approximation to the closed-loop

spectrum of the discrete-time system.
-36-

Numerleal Findings

The eigenvalues in the examples which follow were computed using the

method we have described above with quintic splines and M taken large enough

to declare a sufficient number of the eigenvalues converged. Typically,

taking M = 30, which results in a 70 dimensional eigenvalue problem, sufficed

to yield 21 converged eigenvalues. The resulting matrix eigenvalue problems

were solved using IMSL routines EIGRF (the QR method for the standard

eigenvalue problem) or EIGZF (the QZ method for the generalized problem).

These computations were performed in double precision on the IBM 3081 at the

University of Southern California.

The eigenvalues of A N, the Nth Kappel-Salamon approximation to A, with

real part greater than -3.5 (ordered by decreasing real part) are given in

Tables 4.2 and 4.3 for various values of N. The first nine "true" continuoustime open-loop eigenvalues (eigenvalues of the operator A) can be found in

Table 4.1. Upon careful inspection of Tables 4.1, 4.2 and 4.3, one can easily

discern the true eigenvalues of the hereditary system emerging and observe the

behavior of the extraneous, artifactual eigenvalues which was described in the

previous section as N increases.

For the present example, we used the schemes described in Section 3 to

^ ^i

compute the continuous-time feedback gains, P2N and PeN for _ = 0, 2.0 and

discrete-time gains f2N and flN for e = 1.0, .98 and .975.

3.0, and the

As _ is increased, larger values of N are necessary to ensure that the

approximating optimal feedback laws have essentially converged. The results

presented below were computed with N = I0. The scalar gains PaN and N are

given in Table 4.4 and 4.6 respectively. The kernals or functional gains

(PAN)2 and ( N)2 (where ( )j, J = 1,2 denotes the jth component) are plotted

in Figures 4.5 and 4.7. Note that the initial conditions
-37-

(4.6) n = (0,0)T, (€)i arbitrary, (€)2 = 0

for the system (4.3) yield x(t) = 0, t _ 0 and consequently that the optimal

control is u(t) = 0, t _ 0 . This will also be true for the corresponding

"a-shifted" systems. It therefore immediately follows from this observation

and the basic structure of the finite dimensional approximating systems that

the true and approximating, continuous and discrete-time optimal control laws

do not feedback displacement history; that is

(4.7) (PaN)I = (Pa)l = ( N)I = ( )I = 0

for all N.

The resulting closed-loop eigenvalues for the continuous-time systems are

plotted in Figure 4.8 and are tabulated for the discrete-time systems in Table

4.9. In the discrete-time example, the length of the sampling interval T was

taken to be .01.
-38-

i I

i

1,2 .0219 ± 1.6020i

3 -.7384

4.5 -2.0469 ± 7.58201

6,7 -2.6484 ± 13.94771

8,9 -3.0179 ± 20.2719i

TABLE 4.1: EIGENVALUES OF A

N=4 N=6 N=8

1,2 .0220 ± 1.6017i .0219 ± 1.6019i .0219 ± 1.6019i

-.5833 ± 12.77561

-.6503 ± 12.7819i

3 -.7384 -.7384 -.7384

-1.2679 ± 6.1063i -.9824 ± 9.21361

-1.5000 ± 5.80951 -1.1564 ± 9.11741

-3.4286 ± 2.96921

4,5 -2.0876 ± 7.0492i -2.1251 ± 7.4637i

-2.8600 ± 6.31731 -2.1392 ± 9.99171

-2.8084 ± I0.04311

TABLE 4.2: EIGENVALUES OF A N
-39-

N = I0 N = 20 N = 30

1,2 .0219 ± 1.6019i .0219 ± 1.6020i .0219 ± 1.6020i

-.4513 ± 16.3731i - .1227 ± 34.0989i - .0575 ± 51.5897i

-.4681 ± 16.4023i - .1265 ± 34.0977i - .0576 ± 51.5885i

- .4868 ± 32.5168i - .2258 ± 50.4897i

- .5013 ± 32.5050i - .2264 ± 50.4946i

- .4869 ± 48.6959i

- .4938 ± 48.7064i

3 - .7384 - .7384 - .7384

-1.3706 ± 13.7131i -1.0176 ± 30.0843i - .8459 ± 46.3080i

-1.5800 ± 13.8084i -i.0330 ± 30.0323i - .8707 ± 46.3184i

-1.6310 ± 26.7096i -1.2170 ± 43.3964i

-1.7094 ± 26.8311i -1.2651 ± 43.3778i

-1.7362 ± 40.0062i

-1.7657 ± 39.9081i

4,5 -2.0765 ± 7.5469i -2.0483 ± 7.5804i -2.0471 ± 7.5817i

-2.9685 ± i0.3102i -2.4723 ± 23.1169i -2.2485 ± 36.2831i

-2.3628 ± 36.4404i

-2.4687 ± 32.3647i

6,7 -2.5840 ± 13.9956i -2.6381 ± 13.9567i

-2.7313 ± 19.6455i -2.7968 ± 32.2934i

-2.8700 ± 23.2028i

-3.3004 ± 19.0864i

8,9 -2.9769 ± 20.1502i

-3.3039 ± 27.2171i

-3.3823 ± 28.1149i

TABLE 4.3: EIGENVALUES OF A

N
-40-

_ = 0.0 a = 2.0 _ : 3.0

IpOl0)i .4142 31.8304 129.1725

IpOl0)2 1.4291 10.8678 21.6132

TABLE 4.4: SCALAR GAINS - CONTINUOUS-TIME

FIGURE 4.5: FUNCTIONAL GAINS - CONTINUOUS-TIME
-41-

a = l.0 a = .98 a = .975

"'0 (fa1O )1 .4041 30.9821 63.8527

"'0 (fa1O )2 1.4215 10.5734 14.9272

TABLE 4.6: SCALAR GAINS - DISCRETE-TIME

0::;)

w,

- ....

:.:

l:=;:t

..,...

10

,::a

10

I

C"

I

a = 1.000 --....,

- -- ------..

100.0 -80.0 -.:.0.0 -40.0 -20.0 0.0

~-r--...----,---r---.---..----r---,----...-----r---, I

FIGURE 4.7: FUNCTIONAL GAINS - DISCRETE-TIME
-42-

[] 60.0

• OPEN-LOOP

[] A a:O.O

0 e:2.0

[] [] a: 3.0 45.0

[]

[]

30.0

[]

[]

[] ® ]5.o

[]l 0 : : ,_'_ i

-5.0 -3.0 -2.0 -1.0 1.0

-6.0 -4.0

[] @ ,15.0

[]

[]

-30.0

[]

[]

[] -45.0

[]

[] 60.0

FIGURE 4.8: OPEN AND CLOSED-LOOP SPECTRUM - CONTINUOUS TIME;

EIGENVALUES OF A AND A-BR -I B* ^

10 P_10
-43-

OPEN-LOOP = = 1.0 _ = .98 _ = .975

MAG ARG MAG ARG MAG ARG MAG ARG

.9926 0 .9919 0 .9693 0 .9630 0

1.0002 ± .0160 .9938 ± .0170 .9594 ± .0177 .9484 ± .0211

.9797 ± .0758 .9796 ± .0759 .9787 ± .0758 .9702 ± .0757

.9739 ± .1395 .9738 ± .1395 .9738 ± .1395 .9737 ± .1395

.9703 ± .2027 .9702 ± .2027 .9703 ± .2027 .9703 ± .2027

.9677 ± .2658 .9677 ± .2658 .9677 ± .2658 .9678 ± .2657

.9656 ± .3288 .9656 ± .3288 .9656 ± .3288 .9657 ± .3288

.9639 ± .3918 .9639 ± .3918 .9639 ± .3918 .9639 ± .3918

.9625 ± .4547 .9625 ± .4547 .9625 ± .4538 .9625 ± .4546

.9612 ± .5176 .9612 ± .5176 .9613 ± .5176 .9613 ± .5175

.9599 ± .5806 .9599 ± .5806 .9600 ± .5806 .9602 ± .5805

TABLE 4.9: OPEN AND CLOSED-LOOP SPECTRUM - DISCRETE-TIME;

EIGENVALUES OF T AND T - BFal 0
-44-

REFERENCES

[I] B.D.O. Anderson and J.B. Moore, Linear system optimization with

prescribed degree of stability, Proc. IEEE, 116 (1969), 2083-2087.

[2] B.D.O. Anderson and J.B. Moore, Linear Optimal Control, Prentice

Hall, Englewood Cliffs, N.J., 1971.

[3] P.M. Anselone, Collecltively Compact Operator Approximation Theory

and Applications to Integral Equations, Prentice Hall, Englewood

Cliffs, N.J., 1971.

[4] H.T. Banks, I.G. Rosen and K. Ito, A spline based technique for

computing Riccati operators and feedback controls in regulator

problems for delay equations, SIAM J. Scl. Stat. Comput., 5 (1984),

830-855.

[5] N. Dunford and J.T. Schwartz, Linear operators, Part I: General

Theory, Wiley-lntersclence, New York, 1957.

[6] J.S. Gibson, The Riccati integral equations for optimal control

problems in Hilbert spaces, SIAM J. Control and Opt. 17 (1979),

537-565.

[7] J.S. Gibson, Linear quadratic optimal control of hereditary

differential systems: Infinite dimensional Riccati equations and

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

[8] J.S. Gibson and I.G. Rosen, Numerical approximation for the infinitedimensional dlscrete-tlme optimal llnear-quadratlc regulator problem,

ICASE Report No. 86-15, Institute for Computer Applications in

Science and Engineering, NASA Langley Research Center, Hampton, VA,

1986. (Available as NASA CR-178081.)

[9] J. Hale, Theory of Functional Differential Equations, Springer

Verlag, New York, 1977.

[I0] K. Ito, On the approximation of eigenvalues associated with

functional differential equations, J. Diff. Equ., 60 (1985), 285-300.

[II] K. Ito, Legendre-tau approximation for functional differential

equations, Part III: Eigenvalue approximations and uniform

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1. Report No. NASA CR-178082 12. Government Accession No. 3. Recipient's Catalog No.

ICASE Report No. 86-16

4. Title and Subtitle 5. Report Date

SHIFTING THE CLOSED-LOOP SPECTRUM IN THE March 1986

OPTIMAL LINEAR QUADRATIC REGULATOR PROBLEM FOR 6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

J. S. Gibson and I. G. Rosen 86-16

9. Performing Organization Name and Address 10. Work Unit No.

Institute for Computer Applications in Science

and Engineering 11. Contract or Grant No.

Mail Stop 132C, NASA Langley Research Center NASI-17070, NASI-18107

Hamoton VA 23665-5225 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address

Contractor Report

National Aeronautics and Space Administration 14. Sponsoring Agency Code

Washington. D.C. 20546 _rnr ... 1 0... n 1

15. Supplementary Notes

..,v.., .., ... ___

Langley Technical Monitor: Submitted to IEEE Trans. Auto. J.

C. South Control

Final Report

16. Abstract

In the optimal linear quadratic regulator problem for finite dimensional

systems, the method known as an a-shift can be used to produce a closedloop system whose spectrum lies to the left of some specified vertical line;

that is, a closed-loop system with a prescribed degree of stability. This

paper treats the extension of the a-shift to hereditary systems. As in

finite dimensions, the shift can be accomplished by adding a times the

identity to the open-loop semigroup generator and then solving an optimal

regulator problem. However, this approach does not work with a new

approximation scheme for hereditary control problems recently developed by

Kappel and Salamon. Since this scheme is among the best to date for the

numerical solution of the linear regulator problem for hereditary systems, an

alternative method for shifting the closed-loop spectrum is needed. An

a-shift technique that can be used with the Kappel-Salamon approximation

scheme is developed. Both the continuous-time and discrete-time problems are

considered. A numerical example which demonstrates the feasibility of the

method is included.

17. Key Words (Suggested by Authors(s» 18. Distribution Statement

linear quadratic regulator, 64 - Numerical Analysis

a-shift, hereditary system, 67 - Theoretical Mathematics

approximation

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Unclassified Unclassified 47 A03

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