The Constrained Continuous Time Algebraic Riccati Equation

Copyright@ IFAC 12th Triennial World Congress. Sydney. Australia, 1993

THE CONSTRAINED CONTINUOUS TIME ALGEBRAIC RICCATI EQUATION V.lonescu* and M. WU** *Faculty of Automatic Contro~ Polytechnic Institute of Bucharest, 3 Emi/e Zola. 71272, Bucharest, Romanio **Malhematics Institute, University of Groningen, P.O. Box 8()(), 9700 AV Groningen, The Netherlands

Abstract. A genf'ralization of the well-known Riccati equation, suitable for singular linear quadratic control and the singular BOO-control problem, is studied. A necessary and sufficien t condition for the existence of the stabilizing solution is given in terms of a possibly sillgular matrix pencil. The result justifies a Schur-like algorithm for computing this stabilizing solution. Key Words. constrained continuous algebraic Riccati equation(CCTARE), extended Ha.miltonian pencil(EHP), proper deflating subspace, reducing subspaces, generalized eigenvalue

1.

In the sequel, we shall denote < V > for the subspace spanned by the columns of the matrix V. A(A, B) and A(A) will stand for the generalized spectrum of the matrix pencil ~A - B, respectively for the spectrum of the quadratic matrix A. Furthermore, A(A, B)I(x,y) will denote the spectrum of the matrix pencil with respect to the pair ofreducing subspaces (X, Y) . The reader should consult (van Dooren, 1983) for more details about these notions.

INTRODUCTION

The algebraic Riccati equation

and its stabilizing solution are already standard tools for linear-quadratic control theory, spectral factorization and, more recently, for BOO-control theory. In 1981, van Dooren (1981) gives an algorithm for computing the stabilizing solution of (1) that used the generalized eigenvalue decomposition of a suitably associated matrix pencil. It had the advantage of avoiding the inversion of the possibly ill-conditioned matrix R. A natural question would be what gives this algorithm when R is nonsingular. This question will be fully answered in the present paper.

2.

BASIC DEFINITIONS AND FACTS The objects we are interested in are given by the following two definitions.

=

Definition 1: Let A e Rnxn, B,L e Rnxm,Q QT e Rnxn, R RT e Rmxm. Then the following algebraic Riccati equation

=

Our main object will be a type of Riccati equation suitable for singular linear-quadratic control and spectral factorization of singular rational matrix functions. The term "constrained Riccati equation" is from (Chen and B.A.Francis, 1989) where it appeared in connection with a special case of the singular spectral factorization problem. For the discrete-time case, a similar object was studied by Silverman (1976) where the term "singular Riccati equation" was used.

6ubject to the conltraint (L +XB)(I - R+ R)

=0

(3)

=

(or, equivalently, (L + XB)IKerR 0), where R+ ,tand, for the Moore-Penro6e p6eudoinverle of R, will be called the constrained continuous-time algebraic Riccati equation (CCTARE).

The structure of the paper is the following. The basic tools and results needed are presented in Section 2. The main result is given in Section 3. It nicely generalizes a known result about the usual Riccati equation. Namely, the criterion "dichotomy+disconjugacy" for the existence of the stabilizing solution. The computer-implement able algorithm for computing the stabilizing solution of the constrained Riccati f'quation is presented is Section 4.

Definition 2: A 6ymmetric matru X ing (2) and (3) lJuch that

F

= _R+(L T + BTX)

e R nxn

6atilhl(4)

makelJ the pair (A+ BF, B(I - R+ R» IJtabilizable, will be called the stabilizing solution of the CCTARE.

809

Definition 4: Let'xA - B E R mxn be a matrix pencil. We call Cl proper deflating subspa.ce (to the right), a SUb6pace V C Rn of dimen6ion k for which there exillt kx a OO6ill matrix V =< V > and a matrix S E R " such that BV = AV S and AV i6 injective.

It is obvious that, if R is nonsingular, the classical algebraic Riccati equation and its stabilizing solution are retrieved from the above definitions. Remark 1: The existence of a 6ymmetric X llati8jying (!) and (3) i6 equivalent to the exi6tence of a pair (X X T E Rnxn,F. E Rmxn) 6ati6jying

=

We will ca.ll in the sequel, stable proper deflating subspace , a proper deflating subspace for which A(S) C C- . For the rest of the paper, we denote by k

The exi6tence of a stabilizing 60lution i6 equivalent to the exi6tence of a pair (X, F.) 6ati6jying (5) and 6uch that A + B F. i6 stable.

3. MAIN RESULT The following lemma can be readily proved, using the special structure of the EHP.

For the fir&t implication, one ha6 simply to take the Ilame X and F. = F+(I -R+ R)Fo with F given by (-4) and Fo &tabilizing for the pair (A+ BF, B(I - R+ R)). The conver6e easily follow6 by noticing that if X, F. satilljy (5), then F. = _R+(L T

+ BT X) + (I -

Lemma 1: If s is a finite generalized eigenvalue of the EHP, then -8 i6 also a generalized eigenvalue of the 6ame multiplicity. Furthermore, its 6et of right Kronecker indices is equal to its 6et of left Kronecker indices

R+ R)F•.

•

We are now in position to formulate our main result:

For the stabilizing solution of CCTARE, we have unicity:

Theorem 1: The CCTARE (2), (3) admit& a &tabilizing solution ifJ

(i) rankRp)('xM - N) = 2n

Proposition 1: If X = XT i6 a &tabilizing 60lution to CCTARE (!) and (3), then it i6 unique.

+ BF.)T(X -

X) + (X - X)(A

(iii) the first n lines of a basis matrix of a ndimensional stable proper deflating subspace form a nonsingular matrix. The above formulation needs some clarification concerning, especia.lly, the existence of the n-dimensional stable proper deflating subspace . The situation will be completely clarified by the following lemmas.

+ BF.) = o.

By the stability of A + BF. and A + BF., it follows that X - X = 0 and uniqueness is proved. •

Lemma 2: If the EHP satisfies (i) in Theorem 1, then it has exactly k generalized eigenvalues at 00.

Proposition 1 enables us to speak about the stabilizing solution of CCTARE. In the next section we shall give a result establishing a necessary and sufficient condition for the existence of this stabilizing solution. But first we need some extra definitions.

Proof: We show that M - ,XN has exactly k generalized eigenvalues at ,X = o. We make use of the identity

Definition 3: The matrix pencil ,XM - N with M, N E R(2n+m)x(2n+m) given by

M -,XN = [

,XM - N = I

=,x[ 00

0

I 0

~1 [

A -Q _LT

0

_AT _BT

~~ 1

+ k;

(ii) the EHP has no generalized eigenvalue6 on the imaginary axis;

Proof: Assume that X = XT is another stabilizing solution. According to Remark 1, let F. and F. be such that (X, F.) and (X, F.) sa.tisfy (5) and A+BF" A + BF, are stable. Using(5) for both pairs, some simple algebraic manipulations lead to (A

t:. = rankR.

(6)

x

-R

will be called the extended Hamiltonian pencil (EHP) a680ciated to CCTARE (!) (3).

~ 0

[ I-'xA ,XQ

1+ ,XAT

0

0

0

I

*

~1

X

'1«'~ 1[:

0

; }7)

I 0

where *'s stand for rational matrix functions with no poles in the origin, the particular expressions of which will play no role in the sequel and

In the sequel, we sha.ll be interested in the case when the EHP is, possibly, singular and an important part in our developments will be played by the eigenstructure of this pencil i.e. its generalized eigenvalues and the corresponding reducing subspa.ces. The theory of reducing subspa.ces is given in (van Dooren, 1983), but we sha.ll use another notion here which will simplify a lot our proofs. That is why we introduce

"R.('x)

=

R + 'x(LT(I - AA)-l B_BT(I + 'xAT)-l L_BT(I + AAT)-lQ(I - AA)-l B).

=

Because of (i), it follows that rankRp) "R.(A) k and "R.(A) has no zeroes at A = o. Hence, by a result

810

diag(Li(.~),i

in (McMillan, 1952), there exist X(A), Y(A) rational functions regular at 0, such that

X(A)'R(A)Y(A) 1

=

=

,} ·

L;(»

~

[

= l...m -1 A

>

k) where -1

o m-I<

L

o

o

n, =n-p

i=1

o

We choose now arbitrarily m - k sets Ai E C- of n, complex numbers each (Ai 0 if ni O)and form the monic polynomials of degree ni with Ai the set of its roots and let fi = Lt? ... f~;-11 be the row-vector of coefficients in increasing order of these polynomials. Then

=

,or

X(A)A'R(A)Y(A)

=

,}

=

o

=

o

o o relation which shows, by the same cited result, that the order of the zero at A = 0 in A'R(A) is k. By (7), this is also the order of the eigenvalue i the origin of

M-AN .

where A(Ai) = Ai . Using these one can easily obtain a nonsingular S such that

•

= [diag(AI

Lemma 3: If the EHP 6atilljie6 (i) and (ii), then it ha6 an n-dimensional stable proper deflating subspace.

from Z I S11 [I

Im -

Q·(AM - N)Z = [ Hiu -

Q·MV=

that

= [ Mi'i

k

[

~[

£111 ~ Nu

o o

In-

1

O

diag(A~ 1

1

N11

= A(M, N)lx.)I U Q"(NV - MVS) = 0

diag(A;)]' A(S)

=

=

AM11 - N11

o

o ] Q.

U~_1 Ai, so S is stable and hence NV MVS . As MV is injective by the invertibility of M11 , V is the basis matrix of an n dimensional stable proper deflating subspace and the lemma is proved. •

=

with A(Mu, Nu) A(M, N)lx .)I and AMu-Nuleft invertible. Now, from AMu - Nu, the regular part can be extracted i.e. there exist Tu, Su nonsingular matrices such that

[

P]

=

AM12 - N12 AM22 - N22

TU(AM11 - NU )S11

k)

But A(M11 ,N11 ) = A(M,N)lx.)I does not contain A = 00, hence £111 is invertible. Denoting S

=

Nl~

-

which is clearly invertible, we have

=

Y =< Ql >, X =< ZI > such

= 1, m

Ai; i

s] and denoting

p

q"NV with

n; -

where P E R(n-p)x(m-k) . Taking then V to be the injective matrix obtained by taking the first n columns

Proof: Let p be the number of stable generalized eigenvalues ofthe EHP (p ~ 0). By Lemmas 1 and 2 it follows that the sum of right Kronecker indices is n-p and there are exactly m-k right Kronecker blocks (the same situation on the left). According to Theorem 2.4 in (van Dooren, 1983), there exists a unique pair of reducing subspaces (X, Y) such that A(M, N)lx .)I consists of the p stable generalized eigenvalues. According to the theory in the same cited reference, dimY n,dimX n + m - k and there exist Q and Z, (2n + m) X (2n + m) orthogonal matrices

=

=

(AMr - Nr)S

AMr

-

Remark 2: Condition (i) i6 a dichotomy condition, while condition (iii) i6 a disconjugacy condition. So Theorem 1 a6sert6 the "dichotomy+disconjugacy" criterion for the existence of the stabilizing solution of the CCTARE and when R is nonsingular, the wellknown result for the u6ual Riccati equation is retrieved. •

0 ] Nr

where AM11 - N11 is a p x p regular pencil and AMr - N r contains the right null structure of the initial pencil, hence it has n - p + m - k columns and n - p rows. We can also assume that AMr - N r

= 811

4. CONCLUSIONS A necessary and sufficient condition for the existence of the stabilizing solution of the constra.ined Riccati equation was given. The discrete counterpart of this result is a.lso ava.ilable and will be reported elsewhere. Applications of the constra.ined Riccati equation to spectra.l fa.ctorization of singular rationa.l matrices and to the singular H oo -control problem are under current investigation. Some results in these directions will a.lso be soon reported .

Now we can proceed to

Proo/:Theorem 1 "only if" Suppose the CCTARE (2)(3) has a stabilizing solution X. Using Remark 1, let F. such that (X, F.) satisfies (5) and A + BF. is stable. Then, making use of (5), it is easy to check that

[ -~

0 I 0

-[

>.I - (A

Fr I

1(~M

+ BF.)

F.

),,1 + (A

0 0

i ~1 J + ~

- N) [

0 I 0

0 BF.)T BT

=

-B

The proof of Theorem 1 leads to a computer-implementable a.lgorithm for computing the stabilizing solution of a CCTARE very similar to the a.lgorithm for computing the stabilizing solution of a usual a.lgebra.ic Riccati equation devised in van Dooren (1981) . The ma.in difference consists in the precaution one has to take when working with possibly singular pencils. We will not discuss here the computationa.l issues, but it is worth mentioning that there are some very recent progresses in numerica.l methods for singular pencils as (Kagstrom , 1986) and (Beelen and van Dooren, 1988) which may be very useful for computing a stable proper deflating subspace . Of course, the problem of determining the stabilizing solution of CCTARE may be very ill-conditioned like the problem of determining the eigenstruct ure of a singular pencil, as is pointed out in the above mentioned references.

8)

The last relation shows that rankR(~)()"M - N) = 2n + rankR 2n + k and that there are no genera.lized eigenva.lues on the imaginary axis, as A + BF. is stable. It rema.ins to prove (iii). Let

=

V=

Then NV

MV

[i 1 = MV(A + BF.) and =[

~ j.

5. Hence V is the basis matrix of a n-dimensional stable proper deflating subspa.ce which obviously satisfies (iii) .

Beelen, T. and P van Dooren (1988) . 'An improved a.lgorithm for the computation of Kronecker's Canonica.l form of a singular pencil'. Linear Algebra and Its Applications 105, 9-65 . Chen, T . and B.A.Francis (1989). 'Spectra.l and innerouter factorization of rationa.l matrices'. SIAM J. Matrix Annal. Appl. 10(1), 1-17. Kagstrom, B. (1986) . 'RGSVD-An a.lgorithm for computing the Kronecker structure and reducing subspa.ces of singular A - >'B pencils' . SIAM J. Sci. Stat. Comp. 7(1), 185-211. McMillan, B. (1952) . 'Introduction to forma.l rea.lizability theory I & II' . Bell Syst. Tech . J. 31 , 121135, 541-600. Silverman, L. M. (1976) . Discrete Riccati equations: a.lternative a.lgorithms, asymptotic properties and system theoretic interpretations. In C .T.Leondes (Ed .). ' Control and Dynamic Systems: Advances in theory and applications' . Vol. 12. Academic Press. van Dooren, P. (1981) . ' A genera.lized eigenva.lue approach for solving Riccati equations'. SIAM J. Sci. Stat. Comp. 2(2), 121-135. van Dooren, P. (1983) . Reducing subspaces : definitions, properties and algorithms. Vol. 973 of Lecture Notes in Mathematics. Springer Verlag: Berlin, Heidelberg, New York. pp. 58-73.

"if" Suppose (i),(ii),(iii) are satisfied. Let

}n }n

E R(2n+m)xn

(9)

}m be the basis matrix of the stable proper deflating subnonsingular and let S be the spa.ce of >'M - N with n x n stable matrix such that

"'1

NV= MVS.

(10)

If we denote X ~ V2 Y;-l , F ~ V3 V1V1S~-1, then relation (10) becomes A+BF. -Q _ATX -LF. _LT - BT X - RF.

=

1

a.nd S ~

S

=

XS

=

O.

REFERENCES

(11)

Multiplying the third equation to the left by F,! and adding it to the second, it follows that X is the solution of a Lyapunov equation with symmetric free term hence it is symmetric. Introducing S from the first equation into the second, one may easily check th&t the pa.ir (X, F.) satisfies (5). From the first equation, A+BF. is stable, so, conforma.lly to Remark 1, X is the stabilizing solution of the CCTARE and the proof is complete. •

812