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Abstract
We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system , where the matrix coefficients
and
can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices.
1. Introduction
Linear periodic systems are frequently used mathematical models of periodic phenomena. A variety of such models is presented in the IFAC proceedings volume [Citation1]. The book [Citation2] provides fundamentals of modelling and analysis of both continuous and discrete-time linear periodic systems with emphasis on control theory. Modern algorithmic and computational issues related to linear periodic systems are discussed in papers [Citation3–5].
We are concerned with the general discrete-time linear periodic systems here, namely with the doubly infinite linear systems of the form
(1)
(1) where
is the set of integer numbers. The
complex matrices
and
are periodic in p with period
so that
and
for all integers k. The matrices
and
are allowed to be singular and
is a given complex N-vector. The doubly infinite vector sequences
and
are assumed to be elements of
, i.e.
and
. By
we denote the conjugate transpose of a matrix or vector V.
Let us first consider the time-invariant case where is the identity matrix and
is constant for all
, and the matrix A is discrete-time stable, that is, all its eigenvalues lie in the open unit disk
of the complex plane
. Then, the sensitivity of the stability of A to small perturbations can be characterized by the value
, where
denotes the minimum singular value of a matrix,
and
is the set of real numbers. The sensitivity can be alternatively characterized by the parameter
, where
is a positive definite solution of the discrete-time Lyapunov matrix equation
. Moreover, the magnitude
allows us to derive the decay estimate (see, e.g. [Citation6, chap. 9] or Remark 1.1)
(2)
(2) The interest of this estimate is not so much its sharpness (although it is very good for sufficiently large n), but rather the fact that it is easily implementable as it is based on standard Lyapunov matrix equations and eigenvalues of Hermitian positive-definite matrices. The literature on efficient algorithms dealing with these issues is indeed rich, see, e.g. [Citation7, Citation8]. As we shall see later in this section and throughout the paper, our goal is to extend this estimate to system (Equation1
(1)
(1) ). This will be achieved again by means of some Lyapunov matrix equations, but unlike the time-invariant case, here the literature is still in its infancy when it comes to implementation issues, see the discussion at the end of this section.
The concept of matrix stability is naturally extended to the concept of spectral dichotomy of matrices and regular matrix pencils, or operators as in [Citation9]. For example, a matrix A is said to possess the circular spectral dichotomy if it has no eigenvalues on the unit circle . A sensitivity theory of the circular dichotomy of matrices is given in [Citation10]. Its extension to regular matrix pencils is developed in [Citation11, Citation12]. The monograph [Citation6] presents a unified theory of the spectral dichotomy of matrices and regular matrix pencils.
The spectral dichotomy theory developed in [Citation6, Citation10–12] can be extended to discrete-time periodic systems. A standard way of extension is based on the so called lifting method [Citation13], which transforms the periodic system (Equation1
(1)
(1) ) to an augmented time-invariant system with matrix coefficients of size
, for example of the form
(3)
(3) where
,
and
Throughout this section we assume that the matrix
is nonsingular for all
(when this assumption fails, the system (Equation3
(3)
(3) ) may not be solvable, see Proposition 1.1). Consider the Fourier series
(4)
(4) with the matrix coefficients
, which we refer to as the Green matrices for the system (Equation3
(3)
(3) ). This denomination is used, for example, in [Citation10] for systems of type (Equation3
(3)
(3) ), where
is the identity matrix. It follows that the unique
-solution to (Equation3
(3)
(3) ) is represented in the convolutional form
(5)
(5) The matrix of the inverse operator determined by (Equation5
(5)
(5) ) is block Toeplitz and doubly infinite. The Toeplitz blocks are of size
.
Proposition 1.1
If is singular for some
, then system (Equation2
(2)
(2) ) has no solution
in
for some
.
Proof.
We may assume that the matrices and
are upper triangular and that
by using the QZ factorization of the matrix pencil
with eigenvalue reordering [Citation7]. Then the entries
and
satisfy the doubly infinite system
(6)
(6) When
, the right-hand side
must be 0 identically and this contradicts to existence of a solution for all sequences
. Now assume that
and
. Then the solution of (Equation6
(6)
(6) ) with
and
for all
has the components
for
and
for n<0. The sequence
belongs to
only if
, i.e.
must equal 0, which contradicts the assumption
.
Proposition 1.1 proves that the operator in
defined by (Equation2
(2)
(2) ) is bijective under the condition that
is nonsingular for all
. Since the inverse operator
is block convolutional as in (Equation5
(5)
(5) ),
as
. The matrix equation
yields the following identities for the matrix Fourier coefficients
:
(7)
(7) where
is the Kronecker symbol such that
and
for
. Similarly,
yields
.
The following theorem shows that the decay rate of as
can still be expressed analogously to (Equation2
(2)
(2) ), but the derivation is rather involved (see appendix for more detail).
Theorem 1.2
(8)
(8)
(9)
(9) where
and
are the orthogonal projectors onto the right deflating subspaces of the pencil
corresponding to the eigenvalues respectively inside and outside the unit circle, and
,
are Hermitian positive semidefinite matrices satisfying the discrete-time Lyapunov matrix equations
(10)
(10)
(11)
(11)
Remark 1.1
In the particular case when P = 1, and
for all p and A is discrete-time stable, then
,
,
if
,
if n<0,
,
, and the estimate (Equation8
(8)
(8) ) reduces to (Equation2
(2)
(2) ).
Given a sequence such that
whenever
for some positive integer
, the corresponding solution
to (Equation3
(3)
(3) ) decays as
. Owing to (Equation5
(5)
(5) ) the decay rate of
is bounded by the decay rate of
as
.
A drawback of estimates (Equation8(8)
(8) ) and (Equation9
(9)
(9) ) is that they involve the dense matrices
and
of large size
(which is the case, e.g. for small N and large P or for medium sizes N and P). This drawback is overcome by means of special periodic Lyapunov matrix equations of size
as in [Citation14]. However, this reference restricts the study to the case where
and nonsingular
and furthermore, the periodic Lyapunov matrix equations are constructed in a form which is not suitable for numerical computation. The present paper removes these restrictions and provides periodic Lyapunov matrix equations in the general case, where the matrices
and
can be singular. In addition, we give tighter decay estimates than those in [Citation14].
More specifically, our approach, where periodicity plays an essential role, is based on the periodic Schur decomposition of matrices with reordering and solutions of generalized periodic Sylvester systems, which allow us to block diagonalize periodic matrix sequences according to the spectrum parts of the monodromy matrix inside and outside the unit circle. The periodic Lyapunov matrix equations are then written for the diagonal blocks thus avoiding projected right-hand sides. The non-projected periodic Lyapunov equations that is without projected right-hand side have advantage that they admit unique solutions, which can be found by efficient algorithms of numerical linear algebra; see e.g. [Citation15]. These issues are developed in Sections 2, 3 and 4. Finally, in Sections 5 and 6, the desired decay estimates are written in terms of solutions to the non-projected periodic Lyapunov matrix equations.
The periodic Schur decomposition is a powerful tool for solving periodic Lyapunov matrix equations and periodic Riccati matrix equations; see e.g. [Citation3, Citation4, Citation13, Citation16] and references therein. Periodic Lyapunov matrix equations in the context of balanced model reduction for periodic linear descriptor systems are treated in [Citation17]. Note however that, so far, convergence failure issues in the periodic Schur decomposition limit its practical utility, but progress is being made; see [Citation18].
The exponential decay of the entries of inverse of band matrices has been of some use in spline approximations as mentioned in [Citation19]. Decay patterns of matrix inverses have also attracted considerable interest in linear system preconditioning, low-rank approximation strategies such as hierarchical matrices, wavelets etc; see [Citation20, Citation21] and references therein. The exponential decay estimates of the present paper can be useful in the estimation of stability regions near periodic trajectories and in the Smith type methods [Citation22, Citation23] for periodic matrix equations; see e.g. the analysis in [Citation24] based on the decay bounds for the continuous periodic systems. The decay estimates also provide a quantitative convergence analysis of the so called doubling iterations for solving certain matrix equations [Citation25].
Part IV of [Citation26] contains other applications of exponential decay estimates and describes alternative decay estimates based on the Kreiss matrix theorem. However, the constants in the Kreiss matrix theorem are too expensive for computing [Citation27]. Moreover, these estimates depend on the matrix size.
2. Green's matrices and periodic Lyapunov equations
The doubly infinite system (Equation1(1)
(1) ) determines a bounded linear transform
in the space
of square-summable doubly infinite vector sequences. We assume that
is invertible and represent its bounded inverse
in the matrix form
(12)
(12) where we call the coefficient matrices
by Green's matrices for (Equation1
(1)
(1) ) in
. In the control theory literature, the Green matrices are usually called the fundamental matrices.
Formula (Equation5(5)
(5) ) determines the block structure of
:
(13)
(13) It follows from (Equation13
(13)
(13) ) that
is P-periodic:
(14)
(14) The structure (Equation13
(13)
(13) ) also implies that the positive semidefinite Hermitian matrices
(15)
(15) are the diagonal
blocks of the matrices
and
used in (Equation10
(10)
(10) ) and (Equation11
(11)
(11) ). The matrices
and
are periodic in p with period P on account of (Equation14
(14)
(14) ).
It follows from (Equation12(12)
(12) ) that Green's matrices satisfy the identity
(16)
(16) for all
. The Kronecker symbol
equals 1 if l = p or 0 otherwise. The identity (Equation16
(16)
(16) ) is simply the elementwise expression of equation
, where
stands for the identity operator in
. Inserting the equality
into (Equation12
(12)
(12) ) gives the elementwise expression of equation
:
(17)
(17) The matrices
and
are solutions to the following pair of periodic Lyapunov matrix equations
(18)
(18)
(19)
(19) These equations are direct implications of the identity (Equation17
(17)
(17) ). For instance, since
for
,
Equation (Equation19
(19)
(19) ) is derived analogously. We show in Section 4 that
and
are projectors for all
; see Corollary 4.2.
Derivations in Section 4 show that the solution of (Equation18
(18)
(18) ) and solution
of (Equation19
(19)
(19) ) are unique only under the additional conditions
(20)
(20)
(21)
(21) Note that computation of
and
from the systems of equations (Equation18
(18)
(18) ), (Equation20
(20)
(20) ) and (Equation19
(19)
(19) ), (Equation21
(21)
(21) ) is complicated by the fact that they are overdetermined. Below we get rid of the projected right-hand sides in (Equation18
(18)
(18) ), (Equation19
(19)
(19) ) and conditions (Equation20
(20)
(20) ), (Equation21
(21)
(21) ).
3. Periodic Schur decomposition and block diagonalization
By a slight abuse of notation we write the monodromy matrix formally as
(22)
(22) even if some matrices
are singular. Such usage can be justified for our purposes by small perturbation of the matrices
. A strict but more involved consideration is based on the lifting method. If some
are singular then the matrix (Equation22
(22)
(22) ) may have infinite eigenvalues and indefinite eigenvalues. The eigenvalue structure of M is easily revealed by means of the periodic Schur decomposition, which is an extension of the QZ factorization of linear matrix pencils to periodic matrix pencils; see [Citation28, Citation29] for more details.
Theorem 3.1
Periodic Schur decomposition
There exist unitary matrices ,
,
, and
such that all the transformed matrices
are upper triangular.
The diagonal entries of ,
provide spectral information about system (Equation1
(1)
(1) ). By the aid of Theorem 3.1 the monodromy matrix (Equation20
(20)
(20) ) reads
(23)
(23) The eigenvalues
of M are the diagonal entries of
, namely
When
, we say that
is indefinite. From now on, we restrict ourselves to the regular case, when the monodromy matrix M has no indefinite eigenvalues. If
(
), then
is a finite (infinite) eigenvalue of M.
We also assume below that the monodromy matrix M in (Equation22(22)
(22) ) has no eigenvalues on the unit circle
. Otherwise, it is easy to construct a solution
of (Equation1
(1)
(1) ) such that the solution
and the corresponding right-hand side
.
It is possible to select an arbitrary order of eigenvalues in the periodic Schur decomposition of Theorem 3.1; see [Citation30]. Let the eigenvalues be ordered such that , …,
lie in the open unit disk, i.e.
for
, and
, …,
lie outside the closed unit disk, so that
and
for
.
Let us block partition the transformed matrices subject to the above ordering
(24)
(24) The matrices
and
are then simultaneously block-diagonalized as
(25)
(25)
(26)
(26) where the blocks
and
satisfy the generalized P-periodic Sylvester system (an analog of the generalized Sylvester matrix equations; see, e.g. [Citation31])
(27)
(27)
(28)
(28) Since
and
are nonsingular due to the ordering, the Sylvester matrix equations can be rewritten in the form
and
. After subsequent elimination of
we obtain the periodic Lyapunov matrix system with respect to the set of matrices
,
,
(29)
(29) where
, and
.
The system of P equations (Equation29(29)
(29) ) can be further reduced to a single nonsymmetric discrete-time Lyapunov matrix equation
(30)
(30) where
and
Equation (Equation30
(30)
(30) ) has a unique solution because both matrices
and
are discrete-time stable; see, e.g. [Citation6]. This leads to the following
Theorem 3.2
Periodic block diagonalization with eigenvalue ordering
Assume that the monodromy matrix M has no indefinite eigenvalues and has no eigenvalues on the unit circle. Then, there exist nonsingular matrices ,
,
, and
such that
where all the transformed matrices
,
,
, are upper triangular and block diagonal. The diagonal blocks
,
,
,
of the transformed matrices
are such that the eigenvalues of the monodromy matrices
and
lie inside the open unit disk for all integer p.
Proof.
The matrices
provide the desired block diagonalization; see (Equation25
(25)
(25) ), (Equation26
(26)
(26) ).
4. Block diagonal representations
Recall that the matrix sequences ,
,
,
are P-periodic in p. The Green matrices
are also periodic as
for all
. If necessary, the matrices
,
,
,
are extended P-periodically for all
.
The transformed Green matrices
(31)
(31) satisfy the analogue of (Equation7
(7)
(7) ), namely
(32)
(32) and also the similar equation
.
Theorem 4.1
The transformed Green matrices are block diagonal
(33)
(33) with the blocks
(34)
(34)
(35)
(35)
Proof.
It is easy to verify that matrices (Equation33(33)
(33) ) with the blocks (Equation34
(34)
(34) ) and (Equation35
(35)
(35) ) satisfy Equation (Equation32
(32)
(32) ). Moreover,
(36)
(36) Convergence in (Equation36
(36)
(36) ) is linear. Therefore, the operator (Equation12
(12)
(12) ) constructed from the matrices (Equation33
(33)
(33) ) is bounded in
and coincides with
owing to uniqueness of the inverse operator in
.
Corollary 4.2
The identities and
are valid and
are projectors.
Let us denote the upper triangular factor of the Cholesky factorization of by
and that of
by
.
Proposition 4.3
(37)
(37)
(38)
(38)
Proof.
If , then
. If
, then
.
The transformed matrices
(39)
(39) are block diagonal
(40)
(40) with the diagonal blocks
(41)
(41) The P-periodic matrix sequences
and
satisfy the periodic discrete-time Lyapunov matrix systems
(42)
(42)
(43)
(43) To estimate the norms in (Equation37
(37)
(37) ) and (Equation38
(38)
(38) ) we introduce the following matrices:
(44)
(44)
(45)
(45)
(46)
(46)
(47)
(47) Note that the monodromy matrices
and
are discrete-time stable. We also obtain
if
and
if
.
The matrices and
satisfy the periodic discrete-time Lyapunov matrix systems
(48)
(48)
(49)
(49) which are uniquely solvable. Moreover, all coefficient matrices
,
,
and
in systems (Equation48
(48)
(48) ) and (Equation49
(49)
(49) ) are upper triangular. Therefore, (Equation48
(48)
(48) ) and (Equation49
(49)
(49) ) are triangular linear systems when writing them with the Kronecker product of matrices.
5. Decay estimates I
Theorem 5.1
The following decay estimates hold (50)
(50)
(51)
(51) where the product over an empty set of indices is equal to 1.
Proof.
The bound (Equation50(50)
(50) ) is valid for l = p as a consequence of the inequality
. Further we need the variables
for
such that
when l>p and
. If l>p, then owing to (Equation48
(48)
(48) ) and
we obtain the estimates
Corollary 4.2 and (Equation44
(44)
(44) )–(Equation46
(46)
(46) ) imply that
. Since
we arrive at the inequality
which yields (Equation50
(50)
(50) ). The estimate (Equation51
(51)
(51) ) is derived similarly using (Equation47
(47)
(47) ), (Equation49
(49)
(49) ), the variables
for
satisfying
when l<p−1 and
, the identities
and
Remark 5.1
Let and
be the orthogonal projectors corresponding to the spectral projector pair
and
. It can be shown that for all
6. Decay estimates II
In this section we propose decay estimates which are based on the P-periodic sequences of the discrete-time stable monodromy matrices and
from Theorem 3.2. The discrete-time Lyapunov matrix equations
and
have unique solutions
(52)
(52) and (Equation2
(2)
(2) ) yields the following estimates for all integer
:
(53)
(53)
Theorem 6.1
For all integer
(54)
(54)
(55)
(55)
Proof.
It is easy to see that for
and
for
. The estimates (Equation53
(53)
(53) ) yield the desired estimates.
Alternatively, one can apply the decay estimates from [Citation32] formulated in the following
Theorem 6.2
If A is a discrete-time stable matrix, then
(56)
(56) where
.
Corollary 6.3
Determine the parameters
For all integer
,
(57)
(57) For all integer
,
(58)
(58)
Example
This simple illustrative example shows that the decay estimates of Theorem 6.1 can be better than the decay estimates of Theorem 5.1.
Let P = 2 and α be a very large real number, say, . The coefficients of a stable 2-periodic linear system are
,
,
. The monodromy matrices are
and
. Green's matrices
for p = 0 are
,
,
, and
for l<0.
Solution to the system (Equation42(42)
(42) ) is
. Since
and
, the bound (Equation50
(50)
(50) ) gives the weak estimate
On the other hand, from (Equation52
(52)
(52) ) we obtain
, so
, and (Equation54
(54)
(54) ) yields the tight estimate
We also have
. Corollary 6.3 provides the weaker estimate
for k>1.
When the matrices and
, p = 1, 2, are interchanged, we easily obtain an illustrative example with singular
. The Green matrices
for p = 0 are
,
,
, and
for
.
Solution to the system (Equation43(43)
(43) ) is
,
. Since
and
, the bound (Equation51
(51)
(51) ) gives the estimate
On the other hand, from (Equation54
(54)
(54) ) we obtain
, so
, and (Equation54
(54)
(54) ) yields the estimate
Acknowledgments
The authors thank the anonymous referees for their criticisms and suggestions that helped to improve the presentation of this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
References
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Appendix
Proof of Theorem 1.2
The proof requires some intermediate results which we give in the form of lemmas in order to facilitate the reading.
Lemma A.1
The Hermitian positive semidefinite matrices
(A1)
(A1) satisfy the discrete-time Lyapunov matrix equations
(A2)
(A2)
(A3)
(A3)
Proof.
The proof is based on Equation (Equation7(7)
(7) ). For
,
, and so
Similarly, for
,
, and so
Lemma A.2
The matrices
and
are the spectral projectors onto the right deflating subspaces of the pencil
corresponding to the eigenvalues respectively inside and outside the unit circle.
The orthogonal projectors
and
corresponding to
and
satisfy
(A4)
(A4)
The matrices
satisfy the identities
(A5)
(A5)
(A6)
(A6)
Proof.
The spectral projectors onto the right deflating subspace of the pencil
corresponding to the eigenvalues inside the unit circle is given by (see, e.g. [Citation33, chap. 1] or [Citation6, chap. 10])
where γ is any closed contour enclosing the eigenvalues in the open unit disk and excluding the other eigenvalues. Letting γ be the unit circle
,
leads to
.
On the other hand, it follows from (Equation4
(4)
(4) ) that
. Therefore
, and equation (Equation7
(7)
(7) ) gives
.
The orthogonal projectors associated with
and
are given by
and
. From
we deduce that
and hence
. Similarly, we have
.
In fact,
for any projector
onto the same subspace as
, i.e. when
. Indeed, for each vector x,
. Analogously,
for any projector
onto the same subspace as
.
Let
and
be closed contours enclosing the eigenvalues in the open unit disk and excluding the other eigenvalues and suppose that
is inside
. Then
where we have used the Cauchy formulas
The other equalities in (EquationA5
(A5)
(A5) ) and (EquationA6
(A6)
(A6) ) can be derived similarly.
Proof of Theorem 1.2.
Note first that the equalities for
and
for n<0 imply the equations
(A7)
(A7)
(A8)
(A8) which should be added to the Lyapunov Equations (EquationA2
(A2)
(A2) ) and (EquationA3
(A3)
(A3) ) in order to provide uniqueness of their solutions
and
.
Now, owing to (EquationA2(A2)
(A2) ), the equalities
hold if
. In view of (EquationA4
(A4)
(A4) ),
. As a consequence of (EquationA7
(A7)
(A7) )
and
Hence
and using again (EquationA7
(A7)
(A7) ) leads to the estimate
Since
, we can derive the estimate
(A9)
(A9) The estimate (Equation8
(8)
(8) ) of Theorem 1.2 then follows from the inequality
. The estimate (Equation9
(9)
(9) ) is derived analogously.
Comments
The Lyapunov equations (EquationA2
(A2)
(A2) ) and (EquationA3
(A3)
(A3) ) are often called generalized Lyapunov equations [Citation6, chap. 10]) because the matrices
and
are projectors. The generalized Lyapunov equations were first introduced by Godunov for continuous-time linear systems in [Citation34]. Since the equations from [Citation34] have nonunique solutions, Bulgakov supplemented them by additional equations in [Citation35] in order to get a matrix system, which has a unique solution. Generalized Lyapunov equations with unique solution in the general discrete-time case, including singular
, were derived by Malyshev in [Citation11, Citation12]. Generalized Lyapunov equations for descriptor continuous-time case were introduced by Stykel in [Citation36].
To our best knowledge, the earliest publication which contains decay estimates of the form (Equation2
(2)
(2) ) for the continuous-time Lyapunov matrix equations, is [Citation9, proof of Theorem 5.1]. Decay estimates for discrete-time linear systems
with an arbitrary matrix
first appeared in [Citation11, Citation12]. Decay estimates for descriptor continuous-time linear systems are derived in [Citation36].
In previous publications, the denominators in (Equation8
(8)
(8) ) and (Equation9
(9)
(9) ) are
instead of
and
instead of
. The variant in (Equation8
(8)
(8) ) and (Equation9
(9)
(9) ) gives tighter estimates, which, as far as we know, are not available in the literature.