New results in state estimation and regulation

hutomatica, Vol. 17, No. 5, pp. 745-748, 1981 Printed in Great Britain

0005-1098/81/050745-04 $02.00/0 PergamonPressLtd © 1981InternationalFederationof AutomaticControl

Brief Paper New Results in State Estimation and Regulation* VLADIMIR

KU(~ERAt

Key Words---Linear optimal estimator; linear optimal regulator; spectral factorization; matrix polynomial equations.

Abstract--An alternative technique to design linear state estimators and regulators is presented. This technique is based on transfer matrix considerations. The optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices, This approach provides further insight, displays the duality of estimation and control nicely, and bridges the state-space and frequency-domain techniques. The resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form.

where K=P1HTR -1

(5)

and P~ is the stabilizing solution of the algebraic Riccati equation FP ~+ P~FT - p~HTR- 1HP~ + Q = 0

(6)

that is, F - K H is a stability matrix. The optimal control law u = - Lx is linear too, and

1. Introduction LINEAR state estimators and regulators have obtained much attention in the control literature. In fact, these two problems have become standard and indispensable part of modern control theory and have found numerous practical applications. Using the standard notation, the (infinite-time, stationary) state estimation problem is defined as follows. Given a process y modelled by an observable system Yc=Fx +w y=Hx+v

(1)

starting in the infinite past, where w and v are two independent zero-mean Gaussian white random processes with intensities Q > 0 and R > 0 , respectively. Find the current estimate ~ of x, generated from y by an asymptotically stable system, such that the expectation J1 = E f r ( x - x ) ( x - x ) r f

(2)

is minimized for arbitrary vector f. The dual problem of (infinite-horizon, time-invariant) state regulation can be phrased as follows. Given a controllable system ~=Fx+Gu, plus the measure of performance

x(to)=x o

(3)

o0 J 2 = S (xrQx+uTRu)dt (4) '° with the weighting matrices Q > 0 and R > 0 . Find the control law, relating u to x, which makes the resultant system asymptotically stable and minimizes ,J2 for every initial state x o. It is well-known (Kwakeernaak and Sivan, 1972) that the optimal estimator is linear and given by "~=F~+K(y-H:~)

L=R-1G'rp 2

(7)

where P2 is the stabilizing solution of the algebraic Riccati equation FTP2 + PzF - P2 GR- 1GTp2 + Q = 0 (8) that is, F - GL is a stability matrix. The purpose of this paper is to present a new approach to the solution for the gain matrices K and L. This approach is based on transfer function considerations and makes use of polynomial algebra. Throughout this decade we have witnessed the growing presence of algebra and the comeback of transfer matrix methods in system and control theory. They have proved useful in obtaining further insight (Rosenbrock, 1970) as well as in facilitating some design procedures (Ku~era, 1979b). The principal idea here is to use suitable matrix fractions to describe the system; the calculation of the optimal regulator or estimator gain is then reduced to spectral factorization and the solution of a simple equation in polynomial matrices. 2. Preliminaries The mathematical objects studied in this paper are real rational or polynomial matrices in the complex variable s. Given a polynomial matrix P(s), then d e g P denotes the degree of P(s), i.e. the highest degree occurring among its polynomial entries. Similarly one writes deg~i P and deg~j P for the degrees of the ith row and jth column of P, respectively. The P(s) is said to be row reduced if its highest row-degreecoefficient matrix Ph, has full row rank and it is called column reduced if its highest column-degree-coefficient matrix Phc has full column rank. Any rational matrix R(s) can be written in the form of matrix fractions R(s)=A~(s)B~(s) = B2 (s)A21(s). ........................................

746

Brief Paper

3. State estimation A transfer matrix approach will now be applied to solve the state estimator problem. After Laplace transformation, the process equations (1) are

(s1-F)x(s)=w(s) y(s)=Hx(s)+v(s) and let

H(sI-F)-I=A~1(s)BI(s)

(9)

where A~(s), B~(s) are left coprime polynomial matrices with Aj (s) row reduced. Clearly degr~ B~
(10)

for all i. The observability assumption implies that H and sl - F are right coprime; hence s l - F and A~ share the same non-unit invariant polynomials. Theorem 1. The estimator problem is solvable if and only if D 1(s), any greatest common left divisor of A l(s)R and B a(s)Q, has det D t (s)4:0 on Re s = 0. The optimal estimator gain, if it exists, is unique and given by

K=Y1X~ ~

(11)

where Xa and Y~ is the constant solution of the equation

A l(s)X1 + B I(s)Yj = C t(s)

(12)

and Cl(s ) is a polynomial matrix with det Cl(s)4:0 in Res_>0 and satisfying -

AI(S)RAT(-s)+B~(s)QB~(-s)=C~(s)C~(-s ).

up to a right orthogonal multiplier [see Youla t1961)]. Thus (14) yields

[AI is) + Bj (s}K]RI"2U = C 1(s)

(15)

where U is an unknown orthogonal matrix. Then the claim in (11) and (12) is immediate. The existence of K hinges on the existence of Cl(s ). A factor C a (s) with det C1 (s) 4:0 in Re s > 0 can always be found to satisfy (13), and detCl(s)4:0 on R e s = 0 if and only if detOl(s), where Ol(s ) is any greatest common left divisor of AI(s)R and Bits)Q, does not vanish on R e s = 0 . The uniqueness of K is an immediate consequence of (15). [] The matrices A l(s ) and Bits) being left coprime, equation (12) has a polynomial solution X(s), Y(s) for any C~(s). Any other solution X~, Y~ is related to X(s), Y(s) by

X(s)=X 1- H T t s ) Y(s) = Y~ + (sl -F)T(s) where T(s) is a polynomial matrix. Dividing s l - F into Y(s) we can always make Yt constant; then (10) entails that X~ is also constant and satisfies AtnTX ~=C~hr- This constant solution is obviously unique. Moreover, in view of (10), Al(s ) row reduced makes C~(s) also row reduced; hence X~ is invertible. 4. State regulation The transfer matrix approach will now be applied to solve the dual problem of state regulation. To this effect, write the system equation (3) in the form

(sl-F)x(s)=Gu(s)+x°

(13) ~,,ld let

This result can be proved by expressing J~ as a complex integral of the spectral density of f r ( ~ _ x ) and then using frequency-domain arguments (Ku~era, 1981). Our objective here, however, is to relate the state-space and frequencydomain techniques and thus the following proof is preferred, Proof Add and subtract sP from equation (6) to obtain

(sI--F)p+p(--sI--FT)=Q-PHTR

1Hp.

Then premultiply the result by H ( s l - F ) ~ and postmultiply by ( - s l - F ' r ) - ~HT. Using (5) and (9) we get

AI(s)RKTBT(_s)+BI(s)KRAT(_s) = B1 (s)QB~(- s ) - B 1 (s}KRKTBTI(- s). Adding A~(s) RArI(-s) to both sides above, we finally obtain the fundamental identity

I s / - F ) - J G = Bz(s)A ~ ~(s)

(16}

~here Az(s), Bz(s) are right coprime polynomial matrices with A2ls) column reduced. Clearly

deg~j B 2 < deg~j A 2

(17)

for all j. The controllability assumption implies that s l - F and G are left coprime; hence s l - F and A2(s) share the same non-unit invariant polynomials. Theorem 2. The regulator problem is solvable if and only if Dz(s), any greatest common right divisor of RA2(s) and QBz(s ), has detDz(s)@0 on R e s = 0 . The optimal control law, if it exists, is unique and given by

L=X2 ~Y2

(18)

where X 2 and Y2 is the constant solution of the equation

[A, (s) + B 1(s)K]R[AT( - s) + KTBT( -- s)] =AI(s)RAT(-s)+B~(s)QBT(-s). The next step is to relate the matrices s I - F + K H AI(s)+B~(s)K. Using (9)

(14) and

X2A2(s)+ YzB2(s)=Cz(s )

1191

and C2(s) is a polynomial matrix with det C2(s):p0 in R e s t 0 and satisfying

A,(s)H~B,(s)(sI-F).

A~(-s)RA2(s)+B~(-s)QB2(s)=C~(-s)C2(s).

Add BI(s)KH to both sides of the above equation and rearrange to give

Proof This is a dual of the proof of Theorem 1. l h e matrices A2(s) and Bz(s) being right coprime, equation 119) has a polynomial solution X(s), Y(s) for any C2{s). Any other solution X2, Y2 is related to X(s), Y(s) by

HI~I--F4-KH1-1 = rA. t~i+ s , t s w q - ~s, is).

(20)

Brief Paper

5. Discussion

747

T++masou+to++ma++r I

J [;1

example, let

and regulator problems advanced in this paper has several interesting features. The design procedure consists of three steps: the calculation of an appropriate matrix fraction, the spectral factorization of a positive matrix polynomial, and the solution of a linear equation in polynomial matrices, We shall describe these steps and give approximate operation counts for the calculation of the regulator gain matrix L; the results for K are clearly analogous. Given the system

F=

G=

a 0 ...

"1 -a._

Then Az{s )=sn+a._

matricesFandGofsize, say, n×nandnxmthenthematrix

fraction decomposition defined in (16) can be found by elementary row operations (Sain, 1975). This construction requires approximately 4n a operations to obtain A2(s) and B2
I-QFr HXR-~H] [ QF GR 'G' 1 •

Fa

with zero real part. In addition to providing further insight, displaying the duality of estimation and control nicely, and bridging the state-space and frequency-domain techniques, this approach is computationally attractive. The approximate total amount of 30n3 operations is to be compared with the work count associated with the solution of nth order algebraic Riccati

'

isn

'

1+... d-ao,

[il s I

B2(s)= "and if C2~,}__(n s .s . . . ~ . _ (.n _. i s - 1 _ ] _ . . . 4 _ C o

we simply have

Uco

c, ,

L=m---ao...-a,_ ~ . I_C, c,,

J

(21)

The design procedure then consists of spectral factorization only; the first and third steps are completely obviated thus leaving the operation count by an order of magnitude lower. Indeed, some 20n 2 operations will produce L in (21) provided the efficient spectral factorization algorithm for polynomials (Vostr~,, 1976; Ku~era, 1979a) is applied. 6.

Conclusions

A new method has been presented for the design of linear state estimators and regulators, as an alternative to the standard state space techniques. The method is based on transfer matrix considerations and consists of three steps: the calculation of an appropriate matrix fraction, the spectral factorization of a positive matrix polynomial, and the solution of a linear equation in polynomial matrices. It can be interpreted as the assignment of desired dynamics to the optimal system, the dynamics being specified by spectral factorization. The resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form.

AcknowledgementsThe author wishes to thank the reviewers for suggesting a shorter proof of Theorem 1. R~lerences

Kailath, T. (1980). Linear Systems. Prentice Hall, Englewood Cliffs. Kazanjian. N.N.(1977).Bauer-typefactorizationofpositive matrices and the theory of matrix polynomials orthogonal on the unit circle. Ph.D. thesis, Polytechnic Institute of New York, Farmingdale. Kui~era. V. (1979a). Design of steady-state minimum variance controllers. Automatica,15, 411. Ku~:era. V. (1979b). Discrete Linear Control: The Polynomial Equation Approach.JohnWiley, Chichester. Ku~era. V. (1981). Discrete linear regulator revisited. Kyhernetika17, 62. Kwakernaak+ H. and R. Sivan (1972) Linear OptimulControl Systems.John Wiley. New York.

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Brief P a p e r

Vostr~,, Z. (1976). New algorithm for polynomial spectral factorization with quadratic convergence. Kybernetika 12, 248.

Wolovich, W. A. (1974). Linear Muhivariable Syster~. Springer, New York. Youla, D. C. (1961). On the factorization of rational matrices. IRE Trans ln]ormation Theory IT-7, 172.