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```AD-A17J 748 SOLUTION TO THE ALGEBRAIC RICCATI EQUATION FOR

PARABOLIC SVSTEHS(U) JOHNS HOPKINS UNIV BALTIHORE HD
DEPT OF ELECTRICAL ENGINEERIN. . L R RIDDLE ET AL.
UNCLASSIFIED 22 OCT 86 JHU/EE-86/21 NB0814-85-K-B255 F/G 12/1

•■■m r-

Solution to the Algebraic
Riccati Equation for Parabolic Systems *

Laurence R. Riddle and Howard L. Weinert

Department of Electrical Engineering
The Johns Hopkins University
Baltimore, MD 21218

ABSTRACT

This paper presents an analytical solution to the operator alge¬
braic Riccati equation (ARE) for selfadjoint parabolic systems.
The solution to the operator ARE is important in the design of the
steady-state, on-line filter for estimating the system’s states. This
analytical solution is derived by considering the operator analog of
Potter’s method of using the Hamiltonian system’s eigenvectors
and eigenvalues to solve a finite-dimensional ARE. As an example
of using this analytical solution, the steady-state filtering error
covariance for the 2-D heat equation is studied.

This work supported by the Office of Naval Research under Contract N00014-85-K-

V. W-'\ .1 i.1 A -.1 ,V„Tf A’■J\k

1. Introduction

In this paper, we derive an analytical solution to the operator algebraic
Riccati equation (ARE) on a Hilbert space H:

AP+PA'+Q =PRP (1.1)

when —A is a strongly positive (coercive) selfadjoint operator that generates a

continuous semigroup, Q is positive-definite, bounded and commutes with A,
and R is bounded and nonnegative-definite. We also assume that the domain
of A is dense in H, and is contained in the domain of A , and that the positive
square root of Q commutes with A. In the context of distributed parameter
filtering and control, one needs a solution to Eq. (1.1) that is bounded, nonnega¬
tive, selfadjoint and that maps the domain of A into itself. Gibson [3] has
shown that under the above assumptions, Eq. (1.1) has a unique solution with
these properties.

An example where one would solve Eq.(l.l) with the above assumptions on
A , Q and R is the design of a steady-state filter to estimate a process governed
by the heat equation:

u(x,t) — V u(x,t)+e(x,t ) , x G H C R 3 , <>0 (1.2)

dt

- u(x,t) = 0 , x G dCl

dn

d

where denotes the outward normal derivative, and the observations are
dn

Vj{t) = f Cj(x)u(x,t)dx+Wj(t) , j = 1, • • • ,N
n

In Eq (1.2), e(x,J) is a random process that is white in space and time with

constant intensity Q, w-{t) is a scalar white noise process with intensity R~ ,
and A — V 2 . Several applications using this model to describe the tempera¬
ture distribution of heated metals have been reported [l],[9]. Numerical exam¬
ples of calculating the steady-state filtering error covariance P for this model
are given in [ll].

2. Solving the ARE

In this section we will solve Eq. (l.l) by using the operator analog to the
Potter method [10] of solving the ARE in finite dimensions. The Potter method
of solving the ARE is summarized as follows: let the Hamiltonian matrix L be

defined as

A Q

R -A

and diagonalize L via

where

LM = MA

A = diag(\, ■ ■ ■ ,\ n ) , * • • \ > 0

M =

■M n M 12

M 2 i A/gj

Then

p = m 11 m 2 " 1

is a nonnegative symmetric solution to the ARE , assuming A/ 21 is invertible.

To justify the formal solution given by Eq. (2.3) we need to show that P is
bounded, nonnegative, selfadjoint, and maps the domain of A into itself. Since
—A > 0 it follows that A +R has a unique positive square root that is selfad¬
joint and that the domain of A 2 is dense in the domain of (A 2 -kR)*[7, p. 281].
Therefore —A +(A +R ) is strongly positive and selfadjoint with dense domain:

D{-A+{A 2 +Rf) =D{-A) nD({A 2 +R)*)CZD{A 2 )CD(A) (2.4)

where D( ) denotes domain; so that P x as defined in Eq. (2.2) is bounded [ 4, p.

209 ]. Furthermore, P l is selfadjoint and nonnegative because (—A+(A 2 +R) k )

is [ 7, p. 272]. We will prove that P x maps the domain of A into the domain of

..1 VlV

The range of F, is contained in the domain of A by Eq. (2.4), and so it is
sufficient to show that the domain of P { contains the domain of A. Since P x is
bounded and necessarily closed (being selfadjoint), it follows [7, p. 269] that the
domain of is H, and hence contains the domain of A.

Since we have assumed that the operator S commutes with A, if x G D(A)
then Sx £ D (A), hence P in Eq. (2.3) also maps D(A) into D(A). We have
thus shown that the operator defined in Eq. (2.3) is the correct solution to Eq.
(1.1) needed for filtering and control applications.

3. Numerical Considerations and Example

The analytical approach to solving the operator ARE presented in this
chapter provides a computationally faster way of implementing the optimal
gain for filtering and control applications. In our approach, finite-dimensional
approximations to the operators in Eq (1.1) are done after the analytical solu-
tion has been obtained, resulting in a computational complexity of 0(6n ),
where n is the size of the matrix that approximates the operator A , whereas in
other techniques [2] one approximates Eq (l.l) by a matrix ARE and then solves
this equation using algorithms developed for the finite-dimensional case which
require approximately 0(75n ) operations [8]. We remark that a similar
approach (with the same computational complexity 0(6n ) ) to the finite¬
dimensional ARE has been considered [5],[6].

Another important difference between the analytical approach used in this
chapter and those using a high order finite-dimensional approximation to Eq.

(1.1), is that the implementation of Eq. (2.3) calls for approximations of A and

2 2
A separately, whereas in approximating Eq. (1.1) one is implicitly using X ,

where X is a finite-dimensional approximation to A, as an approximation to A .

The fact that X may not be a good approximation to A suggests that our

approach may be more accurate.

The steady-state filtering error covariance for estimating the temperature
profile of a heated square aluminum slab was calculated using the results
obtained in the previous section. These calculations were based on the follow¬
ing model for the variation in temperature u (x,y,t) above an assumed known
ambient temperature T Q :

d 2

— u(x,y,t) — <*V u(a:,y,t)+e(a:,y,t) 0 < x , y < L
dt

with boundary conditions

«(0 ,y,t) — u(L,y,t) — u(x,0,t) = u(x,L,t) = 0

where

■—5 2 /

L — Imeter , a = 5 X 10 meter /second
and the observations are

y{t) = u{.5,.5,t)+w{t )

The input e(x,y,t) is assumed to be white in space and time with intensity

Q — 1 (degreeC) /{meter ) second
and the observation noise w(t) is white with intensity

R~ X = 10 ~ 2 {degreeC) 2 /second

In Eq. (2.3) ,a 10 X 10 finite-difference approximation to the operators A and

A was used. Figure 1 shows a cross-section of the variances of the tempera¬
ture variations u(x,.5,t) and the filtering error It is evident from Fig¬

ure 1 that the unobserved temperatures more that 10 cm away from the sensor
are not being estimated very effectively. This result places some doubt on the
possibility of estimating, with few sensors, the unobserved temperature distribu¬
tion of heated metals in the presence of spatially white process noise.

References

[1] M. B. Ajinkya, M. Kohne, M. F. Mader, and W. H. Ray, ‘ The experimen¬
tal implementation of a distributed parameter filter,’ Automatica, vol 11,
pp. 571-577, 1975.

[2] H.T. Banks and K. Kunisch, ‘The linear regulator problem for parabolic
systems,’ SIAM J. Contr. Opt. ,vol. 22, pp 684-695, 1984.

[3] J.S. Gibson, ‘The Riccati integral equations for optimal control problems
on Hilbert spaces,’ SIAM J. Contr. Opt. vol. 17, pp 537-565, 1979.

[4] K. E. Gustafson, Introduction to partial differential equations and Hilbert
space methods, Wiley, New York, 1980.

[5] M. Hamidi, ‘A method of optimal actuator and sensor placement’, Proc.
Fourth IFAC Symp. Contr. Dist. Parm. Syst., Los Angeles, 1986.

[6] F. Incertis, ‘ An extension on a new formulation of the algebraic Riccati
equation problem’, IEEE Trans. Auto. Contr., vol AC-28, pp. 235-237, 1983.

[7] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New

York, 1966.

[8] A.J. Laub, ‘A Schur method for solving algebraic Riccati equations,’ IEEE
Trans. Auto. Contr., vol. AC-24, pp. 913-922,1979.

[9] G. K. Lausterer, W. H. Ray and H. R. Martens, ‘The real time application
of distributed parameter state estimation theory to a two dimensional
heated ingot,’ pp. 451-467 of Proc. Second IF AC Symp. Contr. Dist. Parm.
Syst., S. P. Banks, A. J. Pritchard Eds., Peragamon Press, Oxford,1978.

[10] J.E. Potter, ‘Matrix quadratic solutions,’ SIAM J. Appl. Math., vol. 14, pp
496-501, 1966.

[11] K. Watanabe, M. Iwasaki, ‘A fast computational approach in optimal dis¬
tributed parameter state estimation,’ J. Dynamical Systems, Measurement
and Contr., vol. 105, pp. 1-10, 1983.

location

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ELEMENT NO.

1 1 TITLE (Include Security Clarification) SOLUTION TO THE ALGEBRAIC
RICCATI EQUATION FOR PARABOLIC SYSTEMS (UNCLASSIFIED)

12. PERSONAL AUTHOR(S)

RIDDLE, L.R. AND WEINERT, H.L.

134. TYPE OF REPORT

INTERIM

PROJECT

WORK UNIT

NO.

NO.

NO-

NR661-019

rrrr«

mMmmsMM

14. OATE OF REPORT (Yr . Mo.. Day) 15. PAGE COUNT

OCTOBER 22, 1986 9

COSATl COOES

18. SUBJECT TERMS (Continue on rwuerte if neceuary and identify by block number)

OPERATOR RICCATI EQUATION, PARABOLIC SYSTEMS, DISTRIBUTED
PARAMETER FILTERING AND CONTROL.

19. ABSTRACT iContinue on reverte if necerary and identify by block number /

This paper presents an analytical solution to the operator algebraic Riccati equation
(ARE) for selfadjoint parabolic systems. The solution to the operator ARE is important
in the design of the steady-state, on-line filter for estimating the system's states.

This analytical solution is derived by considering the operator analog of Potter's method
of using the Hamiltonian system's eigenvectors and eigenvalues to solve a finitei-dimension-
al ARE. As an example of using this analytical solution, the steady-state filtering error
covariance for the 2-D heat equation is studied.

20 OISTRIBUTION/AVAILABILITY OP ABSTRACT
UNCLASSIPIEO/UNLlMlTED £2 SAME AS RPT Z OTIC USERS □

22*. NAME OP RESPONSIBLE INDIVIDUAL

DR. NEIL L. GERR

DD FORM 1473, 83 APR EDITION OF 1 JAN

21 ABSTRACT SECURITY CLASSIFICATION

UNCLASSIFIED

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(202)696-4321

EOITION OF 1 JAN 73 iS OBSOLETE.

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