Periodic Invariant Subspaces in Control

Copyright (ji) IFAC Periodic Control Systems. Cemobbio-Como. Italy. 2001

PERIODIC INVARIANT SUBS PACES IN CONTROL Wen-Wei Lin·, Paul Van Dooren··, Quan-Fu Xu •••

• Department of Mathematics Tsing Hua University, Hsin-Chu Taiwan, R . O. C. [email protected] •• Centre for Engineering Systems and Applied Mechanics UniversitC catholique de Louvain B-1348 Louvain-la-Neuve, Belgium [email protected] ••• Department of Mathematics Tsing Hua University, Hsin-Chu Taiwan , R.O.C.

Abstract: In this paper we present several different characterizations of invariant subspaces of periodic eigenvalue problems. We analyze their equivalence and discuss their use in control theory. Copyright QP20011FAC Keywords: Periodic systems, discrete-time systems, canonical forms, eigenvectors, eigenvalues

1. MAIN RESULTS (2)

Consider the (homogeneous) linear time varying system:

kE N

(1)

is regular (i.e. det('x£ - A) periodic systems regular.

where N is the set of natural numbers, Xk is an n-dimensional vector of descriptor variables, Uk is an m-dimensional vector of input variables, the matrices Ek and Ak are n x n, and Bk is n x m. This system is said to be periodic with period K if Ek = E k + K , Ak = A k + K and Bk = Bk+K, for all kEN, and K is the smallest positive integer for which this holds. If we allow both the Ek and Ak matrices to be singular, then Xk may still uniquely be defined in the context of a two point boundary value problem, as e.g. in optimal control of periodic systems. It is shown in (Sreedhar and Van Dooren, 1999) that a necessary and sufficient condition for this is that the pencil

=I

0). We call such

We can always define a periodic similarity transformation of a regular periodic system (1.1) by multiplying it from the left by the invertible transformation Sk and substituting Xk by Xk ~ Tk-1Xk, where Sk and Tk are again K-periodic. Let us denote diagonal block transformations Sand T as follows:

Defining the periodic similarity transformation

19

f,

where X == [xl' xI' xl Y == [Yt Y[ Y3 are assumed to have full rank d. With

f

(3)

we can also rewrite this as follows:

one then obtains a new system kEN,

where

Eh ==

-M ](7)

(4)

AI

SkBk .

so that every d-dimensional eigenspace of the type (6) induces ad· K -dimensional eigenspace of the type (7), provided all individual matrices X k and Yk have full column rank . The following example shows that one must impose certain conditions on M, since this does not hold in general.

One important case of such a periodic similarity transformation is the so-called "Floquet transform" of a periodic system :

~1: AE2. .

[

-AI]

-AK AEK

=

[-~ AI

-M 5)

-M AI J

And let 1 w3

This reduces the homogeneous problem to a timeinvariant one, but the transformation may not always exist (Van Dooren and Sreedhar, 1994; Sreedhar and Van Dooren, 1997). Another important case is the Periodic Schur Form (Bojanczyk et al., 1992), where the transformation matrices Sk and Tk are constrained to be unitary, and the resulting matrices Ek and Ak can all be chosen upper triangular. This second form always exists and can be computed in a numerically stable manner (Bojanczyk et al., 1992).

= 3 and Ak = Ek = 12 . = [~~] with w -:j:. 1 and

Example 1 Consider K

= (1, l)T, M

= 1. Then we have

So (6) and (7) hold with Xl = Y1 = [1,w 2 1] X 2 = Y2 = [l,Wll] { X3 = Y = [l,wol]. 3

Both forms are closely linked to the concept of periodic invariant subspaces, which play a fundamental role in the solution of important control problem of periodic systems (Sreedhar and Van Dooren, 1994), (Sreedhar and Van Dooren, 1997), (Ferng et al., 1998) :

Although both the matrices

[~:l

• solution of periodic Riccati equations and their use in optimal and robust control, • solution of periodiC Lyapunov equations and their use in stability analysis, • solution of periodic Sylvester equations and their use in decoupling, • inverse eigenvalue problems and their use in pole placement.

and

ml

are of full column rank 2, all X k and Yk matrices are only of rank 1. 0 This example illustrates well that M cannot be arbitrary. The following theorem characterizes which matrix M will yield full column rank matrices X k and Yk · Theorem 1 Let Ak and Ek be nonsingular n x n matrices, for k = 1,2,"', K. Assume that X = [xl', ,XI] T is a full column rank matrix that generates a d-dimensional deflating subspace of A£ - A:

We now describe several characterizations of invariant subspaces of a regular periodic system. A first characterization uses the classical concept of deflating subspace, directly applied to the pencil (2) (we assume K = 3 for the sake of simplicity) :

XJ, ...

20

since a circulant permutation of column and row blocks in (12) can bring any block X k to the first position. Observing (10) and using that Ek is nonsingular, it is obvious that Yk is also of full column rank d. This completes the proof. 0

If M has the property that >..K f /-L K whenever >.. and /-L are two distinct eigenvalues of M, Le.,

Ker(M - >..1) = Ker (MK - >..K 1),

Next , we want to link the system of equations (7) with one which only involves X k rather than X k and Yk . For this , we first rewrite (7) in a slight more general form, which we will retrieve also later on:

(9)

V>" E a(M) (the spectrum of M), then the X k and Yk matrices also have full column rank d, for k=I,2,· · ·,K. Proof. Expanding (8), yields

for k

or

= 1,2, · ··, K. Thus we have

Since the Ek matrices are nonsingular, we can define Sk = Ekl A k • Then (11) is equivalent to

where the regularity of )..[ - A implies the regularity of

provided the X k and Yk matrices are of full column rank d. Therefore , we have for all k:

or also

rank

----[Ek Ih]

=

d.

2d

And hence there exists a full column rank right null space, which we partition as follows:

Note that M is nonsingular, since the Ak matrices are also nonsingular.

(15)

We now show by contradiction that the matrices X k are full rank if (9) holds. Assume e.g. that X 1 does not have full column rank and that the columns of the matrix V span V := KerX 1 . Then it follows from (12) that X 2MV = S2XI V = o and hence MV C KerX2 . By similar arguments one shows that MkV C KerXk+l for k = 2,·· · ,K - 1 and finally, MKV C KerX 1 = V. So V is an invariant subspace of M K and it must contain an eigenvector v corresponding to some eigenvalue ~ of M K . Since the eigenvalues of M K are the K -th powers of those of M, there exists a value>.. E a(M) such that ~ = )...K. The assumption (9) then implies that (M - >..I)v = 0 and hence v is also an eigenvector of M. Since >.. f 0 we found a vector v which is in the kernel of all matrices X k and hence also of X . So we proved that rank(X) = d implies rank(Xd = d . The same proof also holds for all other matrices X k

Notice that for invertible matrices Ek, Ek this is like" swapping" factors since (15) implies --1-

-

--I

Ek Ak = AkEk . Multiplying the left equations of (13) by Ak and the right ones by Ek and equating corresponding terms yields finally:

(16) which does not involve Y k anymore. To go from (16) to (13) again, we rewrite (16) as

[EkXk

AkXk - d [ Ak

-Ek 21

1=

0,

which indicates that [EkXk left null space of [ given by [Ek such that

[EkXk

_At].

complex number). that satisfies ).3 = ~ (here ~ is an eigenvalue of M3 M2 Ml ) as an eigenvalue of M (see (Gantmacher, 1959)) . Thus, in particular, M can be chosen to satisfy the assumption of Theorem 1. That is, for each eigenvalue ~ of M3 Af2 Ml , we extract a fixed number). from the set of cubic roots of ~ as the only candidate for entering into the set of eigenvalues of M, even if there are multiple Jordan blocks corresponding to

AkXk- I] is in the

But a basis for that is

A k]. So there exists a matrix Y k

ih],

AkXk-d = Yk [Ek

(17)

which yields again (13) . From (14) it follows then that the Yk must be full rank or otherwise we would have dim (£V for

~.

With M satisfying the assumption of Theorem 1, all X k and Yk matrices in (7) are of full column rank d. Now we construct unitary matrices Qk and Zk such that

+ AV) < dim V,

V~Im [X, X,

xJ

which would mean ).£ - A is singular. Finally we point out that (17) can be expressed geometrically as the condition dim (Ek Vk

+ Ak Vk-d

and

= d = dim Vb

for Vk = Im (X k ), because rank [Ek

where the Rk and Sk matrices are square invertible. Putting these transformations in block form :

Ak] = d.

To go from (13) to (7) , we replace X k with X k and Yk with Yk in (13) :

. (18) we apply the block transformation to the cyclic pencil Q* ().£ - A) Z = ).t - .4,

Notice that the Ek and Ak matrices are nonsingular, since the E k , A k , X band Yk matrices are all full rank. Define -

Mk

- -1 = Ek A k,

k

for

and we find in the new coordinate system that

= 1, 2,3.

-Al] _ _ [AI X

And let M be a K-th root (K=3) of nonsingular matrix M3 M2 M1 . Then (18) induces (7) with

Xk

)'E3

= X k (Mk ··· MIM- k )

=Y

-M]

-M AI -M AI

,

with

{ Yk = YkEk (M k ·· . MIM- k ), for k = 1,2,3, since then This indicates that in this coordinate system the Ek and Ak matrices are upper block triangular:

EkXk = EkX k (M k ··· MIM- k ) =YkEk (Mk···MIM- k ) =Yk ; and

AkXk- l

= AkXk- l

(M k- l

.. .

So (6) induces a block triangular periodic Schur decomposition, provided M satisfies the assumption of Theorem 1.

MIM-(k-l»)

= YkAk (M k- 1 • • . M1M-(k-l») = Yk ( EkMk) (M k- l

.. .

We thus closed the following set of equivalence relations for pencils with invertible matrices E k , A k .

Ml M - (k - l»)

= YkEk (Mk ·· . MIM- k ) . M

RI

=YkM, which yield (7). Moreover, in the extraction of M such that M3 = M3 M2 Ml , we can take any

22

)'El

-AI]

[Xl] [Yl] X Y =

(AI - M) 2 Y3 with M satisfying the assumption of Theorem 1, X k and Y k matrices having full column rank d.

[

-A2 )'E2

- A3 )'E3

2

X3

(3kAkxk - 1 = akEkxk , for k = 1, 2,·· · , K (20)

with (n:= laj,n~l (3j) f:. (0, 0) , we say that (xdf=1is an eigenvector sequence of ((Ak, Ek))f=1 with eigenvalue < n~1 aj , n:=1(3j >. 0

ACKNOWLEDGMENTS where EkAk = AkEk , for k = 1, 2,3. R4 dim (Ek Vk + Ak Vk - I) = d = dim Vk , where Vk = Im(Xt} = Im k) , for k = 1,2, 3. R5 There exist a block triangular periodic Schur decomposition with leading d x d blocks.

This paper presents research supported by NSF contract CCR-97-96315 and by the Belgian Programme on Inter-university Poles of Attraction , initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with its authors.

(X

We can also show that relations (R3) , (R4) , and (R5) are still valid for Ek, Ak arbitrary for as long as >..£ - A is regular. In this case (RI) and (R2) only apply to the invariant subspaces with finite nonzero reduced spectrum. Example 2 The pencil (n

= 1, K

REFERENCES A. Bojanczyk, G . Golub , a nd P . Van Dooren, "The periodic Schur decomposition . Algorithms and applications", in Proceedings of the SPIE Conferen ce, San Diego , Vo!. 1770, July 1992, pp. 31-42. W . R. Ferng, W.-W. Lin , and C.-S. Wang , "On computing the stable deflating subspaces of cyclic symplectic matrix pairs" , submitted for publication , 1998. F . R. Gantmacher, Th e Th eory of MatT'ices Vo!. I , Chelsea, New York , 1959. J . Sreedhar, P. Van Dooren, "A Schur approach for solving some periodic matrix equations," in Systems and N etworks : Mathematical The ory and Applications, pp. 339-362 , Eds. U. Helmke, R. Mennicken , J. Saurer, Mathematical Research, Vol 77, Akademie Verlag, Berlin 1994. J. Sreedhar , P. Van Dooren, "Periodic descriptor systems : solvability and conditionability", IEEE Trans. Aut. Contr., pp. 310-313, February 1999. J . Sreedhar, P. Van Dooren, "Forward/backward decomposition of periodic descriptor systems and two point boundary value problems", in Proceedings of the European Control Conf., paper FR-A-L7, 1997. P. Van Dooren and J . Sreedhar, "When is a periodic discrete-time system equivalent to a time-invariant one ?", Linear Algebra B Applications, Vo!. 212/213, pp. 131-152, 1994.

= 3)

100] - [000] >.. 010 100 [ 001

010

is clearly regular with triple eigenvalue O. It is already in triangular periodic Schur form , but (RI) with M = 0 (d = 1) has and So (RI) is not equivalent to (R5) in the case, neither is (R2). 0 Finally, this also allows to give a new definition of eigenvalue/eigenvector pairs for periodic pencils. Proofs of the validity of these definition are given in the full paper. Definition 1 Let ((A k , Ed):=1 be regular periodic n x n matrix pairs. If there exist complex numbers al ," ' , aK and /31," ' , (3K such that

(nf=1 aj, n:=l (3j) f:.

(0,0) and

(19)

is singular, then we say

< nf=l aj , nf=l /3j > is

an eigenvalue of (A k , Ek)f:l ' The set of eigenvalues of (Ak, Ek):=1 is denoted as a (Ab Ek):=l' 0 Definition 2 Let (Ab Ek):=l be regular periodic n x n matrix pairs. If there exist complex numbers al , ' " ,aK,(3I,'" ,(3K, and nonzero vectors Xl, • . . ,XK such that

23