Families of Solutions of Algebraic Riccati Equations
DDRAFT 1
Families of solutions of algebraic Riccati equations
Daniele Alpago, Augusto Ferrante
Abstract —We consider Homogeneous Algebraic Riccati Equa-tions in the general situation when the matrix of the dynamicscan be “mixed”. We show that in this case the equation may haveinfinitely many families of solutions. An analysis of these familiesis carried over and explicit formulas are derived. We also derivesufficient conditions under which the union of the families coversthe whole set of solutions.
I. I
NTRODUCTION
Since the seminal paper [17] of J. C. Willems where it wasfirst hinted that (under reasonable assumptions) once given areference solution of an Algebraic Riccati Equation (ARE), allthe others can be parametrized in terms of the solutions of anassociated Homogeneous Algebraic Riccati Equation (HARE),a huge amount of literature has been produced on this topic.In particular, [2], [16] developed the work of Willems estab-lishing what is referred to as the Willems-Coppel-Shaymanparametrization of the solutions of the ARE in terms ofthe invariant subspaces of a certain matrix. The discrete-time counterpart of this parametrization was established in[20]. Reducing the problem to the analysis of a HARE isalso one of the key ideas behind many theoretical results,see [1], [15], [18], [19] with control applications includingstochastic realization [12], [3], [10], spectral facorization [5],[4], and smoothing [7], [11]. The interest for this topiccontinues in recent literature [13], [14]. The advantage ofconsidering homogeneous ARE is that it is possible to obtaina geometric picture describing a family of solutions. Thisfamily of solutions is parametrized in terms of invariantsubspaces of a certain matrix. Under specific assumptions thisfamily is indeed the set of all the solutions of the HARE.In particular, this is true when the reference solution of theARE is stabilizing (or anti-stabilizing) so that the dynamicsof the associate HARE is stable (or anti-stable). It is possible,however, to generalize this property to a reference solution thatis unmixing i.e. such that the associated closed-loop matrixdoes not feature pairs of reciprocal eigenvalues. In fact, thisis the standing assumptions of most of the literature analysingthe set of solutions of HARE’s: the only exception to ourknowledge is in [8] where, however, only the ARE associatedto all-pass functions is considered.In this paper we consider the following general HARE Q = A (cid:62) QA − A (cid:62) QB ( R + B (cid:62) QB ) − B (cid:62) QA (1)where A ∈ R n × n , B ∈ R n × m and R = R (cid:62) ∈ R m × m andconsider the general case where A can be mixed so that itcan have pairs of eigenvalues λ and λ such that λ λ = 1 . D. Alpago, and A. Ferrante are with the Department ofInformation Engineering, University of Padova, Padova, Italy; e-mail: [email protected] (D. Alpago); [email protected] (A. Ferrante).
Our contribution is threefold: first we show that in general theHARE may have infinitely many families of solutions: eachfamily is associated to one non-singular solution of (1) and thesolutions in any fixed family are parametrized in terms of theinvariant subspaces of A . Also, we parametrize these familiesin terms of a linear equation. Second, we provide an explicitformula for the computation of the solutions of each family.This formula is very simple and it proves to be useful evenin the case when the “unmixing” assumption holds. Third,we derive sufficient conditions under which the union of thefamilies covers the the whole set of solutions of (1). Notation.
Given a matrix M , M (cid:62) denotes its transposeand M + its Moore-Penrose pseudo-inverse. The kernel of M is denoted by ker( M ) .II. S OLUTIONS OF HOMOGENEOUS A LGEBRAIC R ICCATI E QUATIONS
Given the HARE (1), we only consider symmetric solutionsso that when we say that Q is a solution of (1) we mean thatit is a symmetric matrix solving (1). Let I A be the set of A -invariant subspaces. The following well-known (see [20])classical result parametrizes the set of solutions of (1) in thecase when A is unmixed. Theorem 2.1:
Let ( A, B ) be a reachable pair and assumethat A is non-singular and that R = R (cid:62) > . If A is unmixedthen there is a bijective correspondence between the set ofsolutions of (1) and the set I A of A -invariant subspaces.Such correspondence is defined by the map assigning to eachsolution Q the A -invariant subspace ker( Q ) .In the following we relax the unmixing assumption. As afirst step we characterise the existence of invertible solutionsof (1) and parametrize the set of such solutions in terms of alinear equation. Consider the Stein (discrete-time Lyapunov)equation AP A (cid:62) − P = BR − B (cid:62) . (2) Lemma 2.1:
Assume that A and R are non-singular. Thereis a bijective correspondence between the set of non-singularsolutions of (1) and the set of non-singular solutions of (2).Such correspondence is defined by the map assigning to eachnon-singular solution P of (2) the matrix Q := P − which isa non-singular solution of (1). Proof.
Clearly, Q is a non-singular solution of (1) if and onlyif Q − = A − [ Q − Q B ( R + B (cid:62) Q B ) − B (cid:62) Q ] − A −(cid:62) which, in view of the Sherman-Morrison-Woodbury matrixinversion formula, is equivalent to Q − = A − [ Q − + BR − B (cid:62) ] A −(cid:62) . a r X i v : . [ m a t h . O C ] F e b RAFT 2
The latter is clearly equivalent to Q − being a non-singularsolution of (2). (cid:3) Corollary 2.1:
Let ( A, B ) be a reachable pair. Then, allthe solutions of equation (2) are invertible so that given anysolution P = P (cid:62) , the set P of all non-singular solutions of(2) is parametrized by P = { P ∆ := P + ∆ : ∆ = ∆ (cid:62) solves A ∆ A (cid:62) = ∆ } . Proof.
Let B R := BR − / . Since ( A, B ) is reachable, ( A, B R ) is reachable as well so that in view of [6, Lemma3.1], all the solutions of equation (2) are invertible. (cid:3) The following result shows that even when we drop all theassumptions, the kernel of any solution of (1) is still an A -invariant subspace. Lemma 2.2:
Let Q be a solution of (1). Then ker( Q ) ∈ I A . Proof.
Let Q be fixed. We write (1) (which is now an identity)as Q = LQ ( A − εI ) + εLQ with L := A (cid:62) − A (cid:62) QB ( R + B (cid:62) QB ) − B (cid:62) and ε being a constant such that both A − εI and I − εL are non-singular. This may be rewritten as ( I − εL ) Q ( A − εI ) − = LQ , so that we immediately see that if v ∈ ker( Q ) then Q ( A − εI ) − v = 0 , and hence ker( Q ) is ( A − εI ) − -invariant. Thus ker( Q ) is also ( A − εI ) -invariantand, eventually, A -invariant. (cid:3) To any invertible solution P ∆ of (2) we can associate afamily of solutions of (1). This family is parametrized withrespect to the set I A of A -invariant subspaces and in termsof the matrix Π S that orthogonally projects into S . In formalterms, we have Theorem 2.2:
Let ( A, B ) be a reachable pair and assumethat A is non-singular and that R = R (cid:62) > . Let P ∆ bea solution of (2); let S ∈ I A and Π S be the orthogonalprojector into S . Then Q = [( I − Π S ) P ∆ ( I − Π S )] + (3)is a solution of (1). Thus, for each given P ∆ ∈ P , Q ∆ := { Q = ( I − Π S ) P ∆ ( I − Π S ) : S ∈ I A } defines a family of solutions of (1) parametrized in I A . Proof.
Let S be a matrix whose columns form an orthonormalbasis for S and S ⊥ be a matrix whose columns form anorthonormal basis for S ⊥ . Consider a change of basis inducedby the orthogonal matrix T := [ S ⊥ | S ] . Clearly, Π S = SS (cid:62) and ¯ A := T − AT has the form ¯ A = (cid:2) A A A (cid:3) . We partition ¯ B := T − B conformably as ¯ B = (cid:2) B B (cid:3) . Finally we set ¯Π S := T (cid:62) Π S T = [ I ] and ¯ P ∆ := T (cid:62) P ∆ T which we partitionconformably as ¯ P ∆ = (cid:104) P P P (cid:62) P (cid:105) . Notice that since ( A, B ) is reachable, ( ¯ A, ¯ B ) and hence ( A , B ) are also reachable.Taking into account that P ∆ is by assumption a solution of(2), by a change of basis we immediately get ¯ A ¯ P ∆ ¯ A (cid:62) − ¯ P ∆ = ¯ BR − ¯ B (cid:62) . (4)which, by employing the partitions just defined reads: (cid:2) A A A (cid:3)(cid:104) P P P (cid:62) P (cid:105)(cid:104) A (cid:62) A (cid:62) A (cid:62) (cid:105) − (cid:104) P P P (cid:62) P (cid:105) = (cid:2) B B (cid:3) R − [ B (cid:62) B (cid:62) ] . (5) The upper-left block of this equation provides the followingreduced-order Stein Equation A P A (cid:62) − P = B R − B (cid:62) . (6)Since ( A , B ) is reachable, P is non-singular [6, Lemma3.1] so that, in view of Lemma 2.1, we have P − = A (cid:62) P − A − A (cid:62) P − B ( R + B (cid:62) P − B ) − B (cid:62) P − A . (7)By direct inspection, we can check that this implies that ¯ Q := (cid:104) P −
00 0 (cid:105) is a solution of the ARE ¯ Q = ¯ A (cid:62) ¯ Q ¯ A − ¯ A (cid:62) ¯ Q ¯ B ( R + ¯ B (cid:62) ¯ Q ¯ B ) − ¯ B (cid:62) ¯ Q ¯ A. (8)Therefore, Q := T ¯ QT (cid:62) is a solution of (1). It remains only to show that Q =[( I − Π S ) P ∆ ( I − Π S )] + , or equivalently that Q + = ( I − Π S ) P ∆ ( I − Π S ) . Since T is orthogonal, we have T (cid:62) Q + T =( T (cid:62) QT ) + so that it is sufficient to show that ( T (cid:62) QT ) + = T (cid:62) ( I − Π S ) P ∆ ( I − Π S ) T (9)The left-hand side of (9) is ( T (cid:62) QT ) + = ¯ Q + = (cid:20) P −
00 0 (cid:21) + = (cid:20) P
00 0 (cid:21) (10)The right-hand side of (9) is T (cid:62) ( I − Π S ) P ∆ ( I − Π S ) T = T (cid:62) ( I − Π S ) T T (cid:62) P ∆ T T (cid:62) ( I − Π S ) T = (cid:20) I
00 0 (cid:21) (cid:20) P P P (cid:62) P (cid:21) (cid:20) I
00 0 (cid:21) = (cid:20) P
00 0 (cid:21) which together with (10) proves (9). (cid:3)
Remark 1:
Notice that it is immediate to compute the kernelof Q given by (3): ker( Q ) = ker([( I − Π S ) P ∆ ( I − Π S )] + ) = S . Therefore, when A is unmixed and hence (2) has exactlyone solution P , in view of Theorem 2.1, the only family Q := { Q = ( I − Π S ) P ( I − Π S ) : S ∈ I A } (11)provides the set of all the solutions of (1). Even in this case,Theorem 2.2 is an important improvement with respect toTheorem 2.1 because of the explicit parametrization (11) thatallows to compute Q = [( I − Π S ) P ( I − Π S )] + as a functionof the corresponding A -invariant subspace S . Remark 2:
Assume that (2) has non-singular solutions. Itis natural to ask whether or not the union of the solutionsdescribed in Theorem 2.2 covers the whole set of the solutionsof (1). Somehow surprisingly, the answer is negative as provenby the following counter-example. Let A := (cid:34)
00 1 0 2 (cid:35) and B = I . By direct computation, we easily see that Q := − − is a non-singular solution of (1).Therefore, our assumptions are satisfied. On the other hand,we easily see that also Q := (cid:34) −
313 913
913 1213 (cid:35) is a solution of
RAFT 3 (1). We now show, however, that such a solution is somehow spurious in the sense that it does not have the form in the right-hand side of (3). In fact, Q +1 = (cid:34) − (cid:35) so that if Q hadthe form of the right-hand side of (3), then the correspondingmatrix P ∆ should have the form P ∆ = (cid:34) − p p p p p p p p p p p p (cid:35) forsuitable values of the entries p ij . It is, however, easy to checkby direct inspection that with such a P ∆ the entry in position (2 , of the matrix AP ∆ A (cid:62) − P ∆ − BB (cid:62) is equal to forevery choice of the parameters p ij . Therefore, P ∆ cannot bea solution of (2).The following result provides a sufficient condition ensuringthat the union of the solutions described in Theorem 2.2 coversthe whole set of the solutions of (1). Theorem 2.3:
Let ( A, B ) be a reachable pair and assumethat A is non-singular and that R = R (cid:62) > . Moreover,assume that (2) admits solutions. If A has at most one pair ofreciprocal eigenvalues and these eigenvalues (when present)have algebraic multiplicity equal to then each solution of(1) is given by (3) for a suitable solution P ∆ of (2) and asuitable S ∈ I A . Proof.
Let Q be a solution of (1). In view of Lemma 2.2, weknow that ker( Q ) is A -invariant. Then, since we can performthe change of basis described in the proof of Theorem 2.2, wecan assume, without loss of generality, that A = (cid:2) A A A (cid:3) , B = (cid:2) B B (cid:3) and Q = (cid:2) Q
00 0 (cid:3) with Q being non-singular.By direct inspection, we can check that Q is a non-singularsolution of the reduced-order ARE (7) so that P := Q − isa solution of the reduced-order Stein equation (6). We onlyneed to show that P can be “extended” to a solution of (2),i.e. that there exist matrices X and X = X (cid:62) of suitabledimensions such that P := (cid:104) P X X (cid:62) X (cid:105) solves (2). If A isunmixed, equation (6) admits a unique solution; moreover, forany solution P of (2) it is easy to check that the upper-leftblock of P must satisfy (6) so that it is necessarily equal to P . Consider now the case when A is not unmixed. In thiscase, since A has at most one pair of reciprocal eigenvalueswhich are both simple, not only is A unmixed but we alsohave σ ( A ) ∩ σ ( A − ) = ∅ . (12)Now we consider (2) with P := (cid:104) P X X (cid:62) X (cid:105) as an equation inthe unknowns X and X = X (cid:62) . If we write this equationblock by block, for the upper-left block we get equation(6) which is an identity, for the upper-right block we get A P A (cid:62) + A X A (cid:62) − X = B R − B (cid:62) which admits asolution X because of (12). Let ¯ X be such a solution andconsider the upper-right block that now reads A P A (cid:62) + A ¯ X (cid:62) A (cid:62) + A ¯ X A (cid:62) + A X A (cid:62) − X = B R − B (cid:62) which admits a solution X because A is unmixed. (cid:3) R EFERENCES[1] D. J. Clements, and H. K. Wimmer. Existence and uniqueness of un-mixed solutions of the discrete-time algebraic Riccati equation,
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