Projected Generalized Discrete-Time Periodic Lyapunov ... - CiteSeerX

Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems Eric King-Wah Chu∗

Hung-Yuan Fan†

Wen-Wei Lin‡

Abstract From the necessary and sufficient conditions for complete reachability and observability of periodic time-varying descriptor systems, the symmetric positive semi-definite reachability/observability Gramians are defined. These Gramians can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and an illustrative numerical example is given. As an application of our results, the balanced realization of periodic descriptor systems is discussed.

Key words. periodic systems, descriptor systems, reachability and observability Gramians, Hankel singular values, balanced realization.

1

Introduction

In the second-half of the last century, the development of systems and control theory, together with the achievements of digital control and signal processing, has set the stage for renewed interest in the study of periodic systems, both in continuous and discrete time; see, e.g., [29, 54, 44, 12, 52, 14] and the survey papers [3, 4]. This has been amplified by specific application demands in the aerospace realm [21, 30, 20], computer control of industrial processes [5] and communication systems [43, 12, 42, 53]. The number of contributions on linear time-varying discrete-time periodic systems has been increasing in recent times; see, e.g., [15, 19, 22, 45, 47, 49] and the references therein. This increasing interest in periodic systems has also been motivated by the large variety of processes that can be modelled through linear discrete-time periodic systems (e.g., multirate sampled-data systems, chemical processes, periodically time-varying filters and networks, and seasonal phenomena [2, 3, 7, 16, 28, 33, 51]). We consider here a periodic time-varying descriptor system of the form Ek xk+1 = Ak xk + Bk uk ,

yk = Ck xk ,

k ∈ Z,

(1.1)

where the matrices Ek , Ak ∈ Rn×n , Bk ∈ Rn×m , Ck ∈ Rp×n are periodic with period K ≥ 1, i.e., Ek = Ek+K , Ak = Ak+K , Bk = Bk+K , Ck = Ck+K , and the matrices Ek are allowed to be singular for all k. Recently, this class of periodic descriptor systems (1.1) is discussed and studied extensively in the problem of solvability and conditionability [36], the computation of H∞ -norm and system zeros [35, 50], and the compensating and regularization problems for periodic descriptor systems [9, 23]. ∗ School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia ([email protected]). † Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan ([email protected]). ‡ Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan ([email protected]).

1

It is well known that the dynamics of the discrete-time periodic descriptor system (1.1) depend K−1 critically on the regularity and the eigenstructure of the periodic matrix pairs {(Ek , Ak )}k=0 which satisfy the homogeneous systems of (1.1): Ek xk+1 = Ak xk ,

k ∈ Z.

(1.2)

K−1 The set of matrix pairs {(Ek , Ak )}K−1 / 0, where k=0 is said to be regular when det[C((αk , βk )k=0 )] ≡



α0 E0 −β1 A1    C((αk , βk )K−1 ) ≡ k=0    0

···

0 α1 E1 .. .

..

0

.

..

. 0

..

. −βK−1 AK−1

−β0 A0 0 .. . 0 αK−1 EK−1

    ,   

(1.3)

in which αk , βk are complex variables for k = 0, . . . , K − 1. Note that we are considering the regularity of the set of matrix pairs {(Ek , Ak )}K−1 k=0 , rather than the regularity of the individual matrix pairs (Ek , Ak ). Definition 1.1. [26] Let {(Ek , Ak )}K−1 k=0 be n × n a regular set of matrix pairs. If there exist complex numbers α0 , · · · , αK−1 , β0 , · · · , βK−1 which satisfy ! K−1 K−1 Y Y K−1 det[C((αk , βk )k=0 )] = 0 , αk , βk ≡ (πα , πβ ) 6= (0, 0) (1.4) k=0

k=0

then (πα , πβ ) is an eigenvalue pair of {(Ek , Ak )}K−1 k=0 . Note that if (πα , πβ ) is an eigenvalue pair of {(Ek , Ak )}K−1 k=0 , then (πα , πβ ) and (τ πα , τ πβ ) represent the same eigenvalue for any nonzero τ . If πβ 6= 0, then λ = πα /πβ is a finite eigenvalue; otherwise (πα , 0) represents an infinite eigenvalue.
The spectrum, or the set of all eigenvalue pairs, K−1 of {(Ek , Ak )}K−1 k=0 is denoted by σ {(Ek , Ak )}k=0 . We shall assume throughout the paper that K−1 the set of periodic matrix pairs {(Ek , Ak )}k=0 is regular, and also use the notation σ(M ) to denote the spectrum of a square matrix M . It is easily seen that the determinant of C((αk , βk )K−1 k=0 ) is a homogeneous polynomial in πα and πβ of degree n of the form n X ck παk πβn−k , (1.5) k=0

where c0 , · · · , cn are complex numbers uniquely determined by {(Ek , Ak )}K−1 k=0 . For the regular K−1 set of matrix pairs {(Ek , Ak )}k=0 , at least one of the ck ’s is nonzero, and hence we see from K−1 Definition 1.1 that there are exact n eigenvalue pairs (counting multiplicity) for {(Ek , Ak )}k=0 . It was shown in [36] that the solvability of (1.2) is equivalent to the condition that the pencil   αE0 0 ··· 0 −βA0 −βA1 αE1 0     ..  . . . .  . . .  (1.6) αE − βA :=     . . .. ..  0  0 0 −βAK−1 αEK−1 is regular i.e. det(αE − βA) ≡ / 0. From (1.5) it is easy to check that
 K K σ {(Ek , Ak )}K−1 = (α , β ) | det(αE − βA) = 0 . k=0 Hence, from (1.7), the solvability of (1.2) is equivalent to the regularity of {(Ek , Ak )}K−1 k=0 . 2

(1.7)

For discrete-time descriptor systems, the concepts of reachability and observability Gramians, causal and noncausal Hankel singular values, and balanced realization are well-established [1, 41]. Moreover, numerical methods are proposed in [37, 11] to solve the projected generalized Lyapunov equations for continuous-time descriptor systems. However, to our best knowledge, similar results have not been developed for periodic descriptor systems. In summary, there are two main contributions in this paper. First, with the aid of the fundamental matrices Ψi,j defined as in (2.6), the reachability/observability Gramians and their corresponding projected generalized discrete-time periodic Lyapunov equations (GDPLE) are derived in terms of the original system matrices Ek , Ak , Bk and Ck , k = 0, 1, . . . , K − 1, respectively. These fundamental matrices play an important role here and are not natural extension of those defined for the descriptor system with period K = 1 [37, 41]. Second, in Sections 6 and 7, Hankel singular values and balanced realization are discussed, for the first time, for completely reachable and observable periodic descriptor systems. These concepts are likely to be crucial in the model reduction problem of periodic descriptor systems. This paper is organized as follows. Section 2 contains some notations and definitions, as well as some preliminary results. In Section 3 the necessary and sufficient conditions for complete reachability and observability of periodic descriptor systems, respectively, are quoted from [11]. With these equivalent conditions, the periodic reachability and observability Gramians, which satisfy some generalized periodic Lyapunov equations, are developed in Section 4. In Section 5 we propose a numerical method for solving these equations under the assumption of pd-stability. A numerical example is given to illustrate its feasibility and reliability. The concept of Hankel singular values is generalized for periodic descriptor systems in Section 6. The problem of balanced realization for the completely reachable and completely observable periodic descriptor systems is discussed in Section 7.

2

Preliminaries

For period K = 1 and a regular matrix pair (E, A), it is well known that the discrete-time descriptor system (E, A, B, C) is asymptotically stable if and only if all finite eigenvalues of (E, A) lie inside the unit circle [13, 38, 39]. Similarly, the asymptotic stability of the periodic descriptor system (1.1) can be characterized in terms of the spectrum of the periodic matrix pairs {(Ek , Ak )}K−1 k=0 . Definition 2.1. Let {(Ek , Ak )}K−1 k=0 be n × n a regular set of matrix pairs. The periodic descriptor system (1.1) is asymptotically stable if and only if all finite eigenvalues of the periodic matrix pairs K−1 K−1 {(Ek , Ak )}k=0 lie inside unit circle. The periodic matrix pairs {(Ek , Ak )}k=0 are called pd-stable K−1 if the set of periodic matrix pairs {(Ek , Ak )}k=0 is regular and all their finite eigenvalues lie inside the unit circle. In a similar fashion to the Kronecker canonical form for a regular matrix pair, we can transform a regular set of periodic matrix pairs into a periodic Kronecker canonical form [23] (see also [34] for the history of the canonical form). K−1 Lemma 2.1. Suppose that the set of periodic matrix pairs {(Ek , Ak )}k=0 in systems (1.1) is regular. Then for k = 0, . . . , K − 1, there exist nonsingular matrices Xk and Yk such that   f   I 0 Ak 0 , Xk Ak Yk = , (2.1) Xk Ek Yk+1 = 0 Ekb 0 I

where YK ≡ Y0 , Afk+K−1 Afk+K−2 · · · Afk ≡ Jk is an n1 × n1 Jordan matrix corresponding to the b b finite eigenvalues, Ekb Ek+1 · · · Ek+K−1 ≡ Nk is an n2 × n2 nilpotent Jordan matrix corresponding to the infinite eigenvalues, and n = n1 + n2 . Remark. If νk is the nilpotency of the Nk for k = 0, 1, . . . , K − 1, then these K values are K−1 defined as the indices [23] of a regular set of periodic matrix pairs {(Ek , Ak )}k=0 . Hence we define the index of the periodic descriptor system (1.1) as ν ≡ max{ν0 , ν1 , · · · , νK−1 }. We say that the 3

periodic descriptor system (1.1) is of index at most 1 if ν ≤ 1, i.e., Ek are all nonsingular or Nk = 0 for all k. For each k ∈ Z, we let " xk = Yk

# xfk }n1 , xbk }n2

"

# Bkf }n1 X k Bk = , Bkb }n2

 Ck Yk = Ckf n1

 Ckb , n2

(2.2)

and by using Lemma 2.1 we can decompose the original system (1.1) into forward and backward periodic subsystems, respectively: xfk+1 = Afk xfk + Bkf uk ,

ykf = Ckf xfk ,

(2.3)

Ekb xbk+1

ykb

(2.4)

=

xbk

+

Bkb uk ,

=

Ckb xbk ,

with yk = ykf + ykb , k ∈ Z. Notice that the state transition matrix of the forward subsystem (2.3) equals Φf (i, j) = f Ai−1 Afi−1 · · · Afj when i > j with Φf (i, i) := In1 . The state transition matrix of the backward b b subsystem (2.4) is Φb (i, j) = Eib Ei+1 · · · Ej−1 when i < j with Φb (i, i) := In2 . The state transition n1 ×n1 matrix over one period Φf (τ + K, τ ) ∈ R is called the monodromy matrix of the forward subsystem (2.3) at time τ . It is well known that its eigenvalues, called the characteristic multipliers, are independent of τ [46, 27]. For k = 0, 1, . . . , K − 1, the n × n matrices     In1 0 −1 0 −1 In1 Pr (k) = Yk Y , Pl (k) = Xk Xk , (2.5) 0 0 k 0 0 are respectively the spectral projections onto the kth right and left deflating subspaces of the periodic matrix pairs {(Ek , Ak )}K−1 k=0 corresponding to the finite eigenvalues. Moreover, the fundamental matrices Ψi,j (i, j ∈ Z) of the periodic descriptor system (1.1) are defined by    Φf (i, j + 1) 0   Y Xj , if i > j,  i 0 0   Ψi,j = (2.6)  0 0   Yi Xj , if i ≤ j. 0 −Φb (i, j) These matrices play an essential role for the periodic discrete-time descriptor system (1.1). For the discrete-time descriptor system with period K = 1, these fundamental matrices coincide with the coefficient matrices of the Laurent expansion of the generalized resolvent (λE − A)−1 at infinity [25, 41]. Some of the results in this paper can be developed through the application of the results of Stykel [39], after “lifting” the periodic system into a descriptor system of higher dimension [6]. We shall not following this path in this paper.

3

Complete reachability and observability

In this Section we shall give a characterization of complete reachability and observability for the periodic discrete-time descriptor systems (1.1). Proofs can be found in [11] and are omitted. Definition 3.1. (i) The periodic descriptor system (1.1) is reachable at time t if for any state x ¯ ∈ Rn , there exist two integers s, ` with s < t < ` and a set of control inputs {ui }`i=s which carry xs = 0 into xt = x ¯. The periodic descriptor system (1.1) is called completely reachable if it is reachable at all time t. (ii) The forward subsystem (2.3) is reachable at time t if for any state ξ¯1 ∈ Rn1 , there exists f f ¯ an integer s with s < t and a set of control inputs {ui }t−1 i=s which carry xs = 0 into xt = ξ1 . The periodic subsystem (2.3) is called completely reachable if it is reachable at all time t. 4

(iii) The backward subsystem (2.4) is reachable at time t if for any state ξ¯2 ∈ Rn2 , there exists an integer ` with ` > t and a set of control inputs {ui }`i=t such that xbt = ξ¯2 . The periodic subsystem (2.4) is completely reachable if it is reachable at all time t. Remark. It is easily seen from Definition 3.1 that the periodic discrete-time descriptor system (1.1) is completely reachable if and only if both its forward and backward subsystems are completely reachable. Theorem 3.1 (Forward Reachability). The following statements are equivalent. (a) The forward subsystem (2.3) is completely reachable. (b) For t = 0, 1, 2, · · · , K − 1, the matrices i h f f f Rf (t) ≡ Bt−1 , Aft−1 Bt−2 , · · · , Φf (t, t − n1 K + 1)Bt−n 1K have full row rank . (c) For t = 0, 1, 2, · · · , K − 1, and i h f f f f , Btf ≡ Bt−1 , · · · , Φf (t, t − K + 1)Bt−K , Aft−1 Aft−2 Bt−3 , Aft−1 Bt−2 the matrices  f  Bt , Φf (t, t − K)Btf , (Φf (t, t − K))2 Btf , · · · , (Φf (t, t − K))n1 −1 Btf have full row rank. (d) For

K−1 Q

αi ∈ σ(Φf (K, 0)), the matrix

i=0



α0 I

  −Af1   U f (α0 , · · · , αK−1 ) ≡  0   .  .. 0

0 α1 I −Af2 .. . ···

··· .. . .. . .. . 0

0

−Af0

..

.

0 .. .

..

.

0

−AfK−1

B0f

 B1f ..

       

. ..

.

αK−1 I

f BK−1

has full row rank. (e) For t = 0, 1, 2, · · · , K − 1, f y T Φf (t + K, t) = λy T and y T Φf (t, j)Bj−1 = 0 for j = t − K + 1, · · · , t − 1, t

imply y = 0. Theorem 3.2 (Backward Reachability). The following statements are equivalent. (a) The backward subsystem (2.4) is completely reachable. (b) For t = 0, 1, 2, · · · , K − 1, the matrices   b b , · · · , Φb (t, t + νK − 1)Bt+νK−1 Rb (t) ≡ Btb , Etb Bt+1 have full row rank. (c) For t = 0, 1, 2, · · · , K − 1, and   b b b b · · · Et+K−2 Bt+K−1 , · · · , Etb Et+1 Btb ≡ Btb , Etb Bt+1 ,   the matrices Nt , Btb have full row rank. 5

(d) The pair (Eb , Bb ) is reachable, where  0 E0b  0 0 E1b   .. .. .. .. . . . . Eb ≡    . . . Eb  0 0 K−2 b EK−1 0 · · · · · · 0





B0b

       and Bb ≡       

 B1b ..

    . (3.1)   

. ..

. b BK−1

Definition 3.2. (i) The periodic descriptor system (1.1) is observable at time t if there exist two integers s, ` with s < t < ` such that any state at time t can be determined from knowledge of {yi }`i=s and {ui }`i=s . The periodic descriptor system (1.1) is called completely observable if it is observable at all time t. (ii) The forward subsystem (2.3) is observable at time t if there exists an integer ` with ` > t such that any state at time t can be determined from knowledge of {yi }`i=t and {ui }`i=t . The periodic subsystem (2.3) is called completely observable if it is observable at all time t. (iii) The backward subsystem (2.4) is observable at time t if there exists an integer s with s < t such that any state at time t can be determined from knowledge of {yi }ti=s and {ui }ti=s . The periodic subsystem (2.4) is completely observable if it is observable at all time t. Remark. It is easily seen from Definition 3.2 that the periodic discrete-time descriptor system (1.1) is completely observable if and only if both its forward and backward subsystems are completely observable. Theorem 3.3 (Forward Observability). The following statements are equivalent. (a) The forward subsystem (2.3) is completely observable. (b) For t = 0, 1, 2, · · · , K − 1, the matrices 

Ctf f Ct+1 Aft f f Ct+2 At+1 Aft .. .



      f   O (t) ≡       f Ct+n Φ (t + n K − 1, t) f 1 1 K−1 have full column rank. (c) For t = 0, 1, 2, · · · , K − 1, and h iT f f Ctf ≡ (Ctf )T , (Aft )T (Ct+1 , )T , · · · , Φf (t + K − 1, t)T (Ct+K−1 )T the matrices



Ctf f Ct Φf (t + K, t) f Ct (Φf (t + K, t))2 .. .



              f n1 −1 Ct (Φf (t + K, t)) have full row rank.

6

(d) For

K−1 Q

αi ∈ σ(Φf (K, 0)), the matrix

i=0



α0 I

··· .. . .. . .. . 0

0

  −Af 0    0   .  ..   f 0 V (α0 , · · · , αK−1 ) ≡    Cf  0       

α1 I −Af1 .. . ···

−AfK−1

0 ..

        0   αK−1 I            0 .. .

.

..

. −AfK−2

C1f ..



. f CK−2

f CK−1

has full column rank. (e) For t = 0, 1, 2, · · · , K − 1, Φf (t + K, t)x = λx and Cif Φf (i, t)x = 0 for i = t, t + 1, · · · , t + K − 1 imply x = 0. Theorem 3.4 (Backward Observability). The following statements are equivalent. (a) The backward subsystem (2.4) is completely observable. (b) For t = 0, 1, 2, · · · , K − 1, the matrices 

Ctb b b Ct−1 Et−1 b b b Ct−2 Et−2 Et−1 .. .



      O (t) ≡       b Ct−νK+1 Φb (t − νK + 1, t) b

have full column rank . (c) For t = 0, 1, 2, · · · , K − 1, and  T b b b )T )T (Ct−1 )T , · · · , Φb (t − K + 1, t)T (Ct−K+1 , Ctb ≡ (Ctb )T , (Et−1 the matrices



Ctb Ctb Nt Ctb Nt2 .. .



            b ν−1 Ct Nt have full column rank b (d) The pair (Eb , Cb ) is observable, where Eb is defined in (3.1) and Cb ≡ diag(C0b , C1b , · · · , CK−1 ).

7

4

Periodic reachability and observability Gramians

It is well known that Gramians play an important role in many applications, such as the model reduction problem [17, 31, 55]. In this Section, the concepts of reachability and observability Gramians are generalized for periodic discrete-time descriptor systems (1.1). Consider the causal and noncausal reachability matrices given by   R+ (t) ≡ Ψt,t−1 Bt−1 , Ψt,t−2 Bt−2 , · · · , Ψt,i Bi , · · · (t = 0, 1, · · · , K − 1) and   R− (t) ≡ Ψt,t Bt , Ψt,t+1 Bt+1 , · · · , Ψt,t+νK−1 Bt+νK−1

(t = 0, 1, · · · , K − 1),

respectively, with Ψi,j (i, j ∈ Z) as defined in (2.6). Definition 4.1 (Reachability Gramians). Suppose that the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable. (i) The causal reachability Gramians of the periodic descriptor system (1.1) are defined by T

Gcr k ≡ R+ (k)R+ (k) =

k−1 X

Ψk,i Bi BiT ΨTk,i ,

k = 0, 1, . . . , K − 1.

i=−∞

(ii) The noncausal reachability Gramians of the periodic descriptor system (1.1) are defined by T

Gnr k ≡ R− (k)R− (k) =

k+νK−1 X

Ψk,i Bi BiT ΨTk,i ,

k = 0, 1, . . . , K − 1.

i=k

(iii) The reachability Gramians of the periodic descriptor system (1.1) are defined via nr Grk ≡ Gcr k + Gk ,

k = 0, 1, . . . , K − 1.

The causal and noncausal observability matrices are respectively defined by T  T , · · · , Ψi,t−1 CiT , · · · O+ (t) ≡ ΨTt,t−1 CtT , ΨTt+1,t−1 Ct+1

(t = 0, 1, · · · , K − 1)

and   T T T , · · · , ΨTt−1,t−1 Ct−1 , ΨTt−νK+1,t−1 Ct−νK+1 O− (t) ≡ ΨTt−νK,t−1 Ct−νK

(t = 0, 1, · · · , K − 1).

K−1 Definition 4.2 (Observability Gramians). Suppose that the periodic matrix pairs {(Ek , Ak )}k=0 are pd-stable. (i) The causal observability Gramians of the periodic descriptor system (1.1) are defined by

T

Gco k ≡ O+ (k) O+ (k) =

∞ X

ΨTi,k−1 CiT Ci Ψi,k−1 ,

k = 0, 1, . . . , K − 1.

i=k

(ii) The noncausal observability Gramians of the periodic descriptor system (1.1) are defined by T

Gno k ≡ O− (k) O− (k) =

k−1 X

ΨTi,k−1 CiT Ci Ψi,k−1 ,

k = 0, 1, . . . , K − 1.

i=k−νK

(iii) The observability Gramians of the periodic descriptor system (1.1) are defined by no Gok ≡ Gco k + Gk ,

k = 0, 1, . . . , K − 1.

8

co Remarks. (i) The infinite series appeared in the definition of Gramians Gcr k and Gk converge K−1 because of the pd-stability of the periodic matrix pairs {(Ek , Ak )}k=0 . nr co no (ii) The Gramians Gcr k , Gk , Gk and Gk are n × n symmetric positive semi-definite matrices for all k. (iii) Definitions 4.1 and 4.2 are natural generalizations of the Gramians defined for descriptor systems with period K = 1; see, e.g., [1, 41].

The following theorem indicates that these Gramians of the periodic descriptor system (1.1) satisfy some projected generalized discrete-time periodic Lyapunov equations with special righthand sides. Theorem 4.1. Consider the periodic discrete-time descriptor system (1.1), where the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable. K−1 nr K−1 (i) The causal and noncausal reachability Gramians {Gcr k }k=0 and {Gk }k=0 are the unique symmetric positive semi-definite solutions of the projected GDPLE T

T cr T T Ek Gcr k+1 Ek − Ak Gk Ak = Pl (k)Bk Bk Pl (k) , T

cr Gcr k = Pr (k)Gk Pr (k) ,

and

k = 0, 1, 2, . . . , K − 1,

T nr T T T Ek Gnr k+1 Ek − Ak Gk Ak = −(I − Pl (k))Bk Bk (I − Pl (k)) , Pr (k)Gnr k = 0, 1, 2, . . . , K − 1, k = 0,

(4.1)

(4.2)

nr cr nr respectively, where Gcr K ≡ G0 and GK ≡ G0 . K−1 no K−1 (ii) The causal and noncausal observability Gramians {Gco k }k=0 and {Gk }k=0 are the unique symmetric positive semi-definite solutions of the projected GDPLE T T co T T Ek−1 Gco k Ek−1 − Ak Gk+1 Ak = Pr (k) Ck Ck Pr (k), T

co Gco k = Pl (k − 1) Gk Pl (k − 1),

and

k = 0, 1, . . . , K − 1,

T T no T T Ek−1 Gno k Ek−1 − Ak Gk+1 Ak = −(I − Pr (k)) Ck Ck (I − Pr (k)), k = 0, 1, 2, . . . , K − 1, Gno k Pl (k − 1) = 0,

(4.3)

(4.4)

no co no respectively, where Gco K ≡ G0 , GK ≡ G0 , E−1 ≡ EK−1 and Pl (−1) ≡ Pl (K − 1). K−1 (iii) The reachability and observability Gramians {Grk }k=0 and {Gok }K−1 k=0 are the unique symmetric positive semi-definite solutions of the projected GDPLE T

Ek Grk+1 EkT − Ak Grk ATk = Pl (k)Bk BkT Pl (k) − (I − Pl (k))Bk BkT (I − Pl (k))T , T

Pr (k)Grk = Grk Pr (k) ,

k = 0, 1, 2, . . . , K − 1,

(4.5)

and T

T Ek−1 Gok Ek−1 − ATk Gok+1 Ak = Pr (k) CkT Ck Pr (k) − (I − Pr (k))T CkT Ck (I − Pr (k)), T

Pl (k − 1) Gok = Gok Pl (k − 1),

k = 0, 1, 2, . . . , K − 1,

(4.6)

respectively, where GrK ≡ Gr0 , GoK ≡ Go0 , E−1 ≡ EK−1 and Pl (−1) ≡ Pl (K − 1). Proof. We shall verify only (4.1) here and the other cases can be shown similarly. Rewrite (4.1) into an enlarged Lyapunov equation EGE T − AGAT = BB T , where E = diag(E0 , E1 , · · · , EK−1 ),

B = diag(Pl (0)B0 , Pl (1)B1 , · · · , Pl (K − 1)BK−1 ), 9

(4.7)



A0

A1  A= 

..

   , 

.

 cr G1   G= 

 Gcr 2 ..

. Gcr 0

AK−1

  . 

Since the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable, the matrix pencil λE − A is regular and all its generalized eigenvalues lie inside the unit circle. Then the Lyapunov equation (4.7) has a unique solution and hence the uniqueness of solutions of the projected GDPLE (4.1) is guaranteed. On the other hand, it can be shown that the causal reachability Gramians Gcr k , k = 0, 1, . . . , K − 1, satisfy the projected GDPLE (4.1). Indeed, simple calculation gives that T cr T Ek Gcr k+1 Ek − Ak Gk Ak

= Ek

k X

! Ψk+1,i Bi BiT ΨTk+1,i

EkT − Ak

i=−∞

k−1 X

! Ψk,i Bi BiT ΨTk,i

ATk

i=−∞

   ! k T X Φf (k + 1, i + 1) 0 0 Φ (k + 1, i + 1) f T = Ek Yk+1 Xi Bi BiT XiT Yk+1 EkT 0 0 0 0 i=−∞  ! k−1 T X Φf (k, i + 1) 0 0 T T Φf (k, i + 1) −Ak Yk Xi Bi Bi Xi YkT ATk 0 0 0 0 i=−∞  Pk Φf (k + 1, i + 1)Bif (Bif )T Φf (k + 1, i + 1)T 0 −1 −∞ = Xk Xk−T 0 0 Pk−1  f f T T 0 −∞ Φf (k + 1, i + 1)Bi (Bi ) Φf (k + 1, i + 1) −Xk−1 Xk−T 0 0  f f  Bk (Bk )T 0 = Xk−1 Xk−T = Pl (k)Bk BkT Pl (k)T , 0 0 and T Pr (k)Gcr k Pr (k) !     k−1 In1 0 −1 X 0 T −T In1 T T = Yk Y Ψk,i Bi Bi Ψk,i Yk Y 0 0 k 0 0 k i=−∞ Pk−1  Φf (k, i + 1)Bif (Bif )T Φf (k, i + 1)T 0 T i=−∞ = Yk Y = Gcr k , 0 0 k K−1 for k = 0, 1, . . . , K − 1. Therefore, the causal reachability Gramians {Gcr k }k=0 are the unique symmetric positive semi-definite solutions of the projected GDPLE (4.1).

The following theorem shows that complete reachability/observability of the periodic descriptor system (1.1) can be characterized via the reachability/observability Gramians. Theorem 4.2. Consider the periodic discrete-time descriptor system (1.1). Assume that the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable. (i) The periodic descriptor system (1.1) is completely reachable if and only if the reachability Gramians Grk are positive definite for k = 0, 1, 2, . . . , K − 1. (ii) The periodic descriptor system (1.1) is completely observable if and only if the observability Gramians Gok are positive definite for k = 0, 1, 2, . . . , K − 1. Proof. Here we shall only prove statement (i) and statement (ii) can be verified similarly. For k = 0, 1, . . . , K − 1, pre-multiply (4.5) by Xk and post-multiply (4.5) by XkT , it follows that  f f T  0 b r Y T E T X T − Xk Ak Yk G b r Y T AT X T = Bk (Bk ) Xk Ek Yk+1 G , (4.8) k+1 k+1 k k k k k k 0 −Bkb (Bkb )T 10

b r ≡ Y −1 Gr Y −T . where G k k k k From Definition 4.1 it is easily seen, for k = 0, 1, . . . , K − 1, that " # cr b G 0 k,1 b rk = Y −1 Grk Y −T = G k k b nr , 0 G

(4.9)

k,2

with b cr ≡ G k,1

k−1 X

Φf (k, i + 1)Bif (Bif )T Φf (k, i + 1)T ,

b nr ≡ G k,2

k+νK−1 X

i=−∞

Φb (k, i)Bib (Bib )T Φb (k, i)T .

i=k

Then by (2.1) and (4.9), equations (4.8) are decomposed into two periodic Lyapunov equations, for k = 0, 1, 2, . . . , K − 1: f b cr f T f f T b cr G k+1,1 − Ak Gk,1 (Ak ) = Bk (Bk ) ,

(4.10)

b b nr b T b b T b nr G k,2 − Ek Gk+1,2 (Ek ) = Bk (Bk ) .

(4.11)

Rewrite (4.10) and (4.11) to two enlarged Lyapunov equations: Gcr − Af Gcr ATf = Bf BfT , Gnr −

Eb Gnr EbT

=

(4.12)

Bb BbT ,

(4.13)

b cr b nr b nr b nr b cr b cr , · · · , G where Gcr = diag(G K−1,1 , G0,1 ), Gnr = diag(G0,2 , G1,2 , · · · , GK−1,2 ), Eb and Bb as dek,1 fined in (3.1), and Af0

 Af  1 Af =  

..

 f B0   Bf =  

   , 

. AfK−1

 B1f ..

  . 

. f BK−1

Since the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable and the matrix Eb is nilpotent with index ν, the pairs (Af , Bf ) and (Eb , Bb ) are reachable if and only if the solutions Gcr and Gnr of Lyapunov equations (4.12), (4.13) are symmetric positive definite. Equivalently, followed from (4.9), all reachability Gramians Grk (k = 0, 1, . . . , K − 1) are symmetric positive definite. Moreover, from Theorems 3.1–3.2 and the Remark following Definition 3.1, we know that the periodic descriptor system (1.1) is completely reachable if and only if the pairs (Af , Bf ) and (Eb , Bb ) are reachable. This completes the proof of statement (i).

5

Numerical solutions of projected GDPLEs

In this Section, a numerical method is proposed for the symmetric positive semi-definite solutions of the projected generalized discrete-time periodic Lyapunov equations (4.1) and (4.3), for pd-stable {(Ek , Ak )}K−1 k=0 . We first consider the numerical solutions of the GDPLE (4.3).

GDPLE for observability Gramians Gco k As {(Ek , Ak )}K−1 k=0 are pd-stable, there exist orthogonal matrices Vk and Uk , with UK ≡ U0 and for k = 0, 1, . . . , K − 1, such that     Ek,1 Ek,3 Ak,1 Ak,3 T T Vk Ek Uk+1 = , Vk Ak Uk = (5.1) 0 Ek,2 0 Ak,2 are upper triangular except V0T A0 U0 is quasi-upper triangular [8, 19]. The matrices Ek,1 and Ak,2 are nonsingular, and Ek,2 Ek+1,2 · · · Ek+K−1,2 are nilpotent for k = 0, 1, . . . , K − 1. All finite 11

eigenvalues of the periodic matrix pairs {(Ek,1 , Ak,1 )}K−1 k=0 lie inside the unit circle and the spectrum of the periodic matrix pairs {(Ek,2 , Ak,2 )}K−1 contains only infinite eigenvalues, with k=0

K−1 σ {(Ek,1 , Ak,1 )}K−1 = ∅. (5.2) k=0 ∩ σ {(Ek,2 , Ak,2 )}k=0 Computationally, these matrix decompositions can be accomplished via the periodic QZ algorithm (PQZ) with reordering strategies. Notice that       I Zk Ek,1 Ek,3 I −Wk+1 Ek,1 0 = , (5.3) 0 I 0 Ek,2 0 I 0 Ek,2       I Zk Ak,1 Ak,3 I −Wk Ak,1 0 = , (5.4) 0 I 0 Ak,2 0 I 0 Ak,2 if the matrices Zk and Wk , with WK ≡ W0 and for k = 0, 1, . . . , K − 1, satisfy the generalized periodic Sylvester equations Ek,1 Wk+1 − Zk Ek,2 = Ek,3 , (5.5) Ak,1 Wk − Zk Ak,2 = Ak,3 . From condition (5.2), the generalized periodic Sylvester equations (5.5) have unique solutions Zk and Wk . Therefore, the nonsingular matrices Xk , Yk in (2.1) satisfy     I Zk I −Wk Xk = VkT , Yk = Uk , 0 I 0 I and the right and left spectral projections Pr (k), Pl (k) are given as     I Zk I Wk Pl (k) = Vk VkT , Pr (k) = Uk UkT . 0 0 0 0

(5.6)

Let, for k = 0, 1, . . . , K − 1, " T Vk−1 Gco k Vk−1

=

Gco k,1

Gco k,3

T (Gco k,3 )

Gco k,2

# ,

Ck Uk =



Ck,1 ,

Ck,2



.

(5.7)

Substituting (5.1), (5.6) and (5.7) into the projected GDPLE (4.3), for k = 0, 1, . . . , K − 1, we have T T co T Ek−1,1 Gco k,1 Ek−1,1 − Ak,1 Gk+1,1 Ak,1 = Ck,1 Ck,1 , T Ek−1,1 Gco k,1 Ek−1,3 T Ek−1,3 Gco k,1 Ek−1,3 T co Ak,3 Gk+1,1 Ak,3 −

(5.8)

T T co T co T + Ek−1,1 Gco k,3 Ek−1,2 − Ak,1 Gk+1,1 Ak,3 − Ak,1 Gk+1,3 Ak,2 = Ck,1 Ck,1 Wk , (5.9) T T co T T co + Ek−1,3 Gco k,3 Ek−1,2 + Ek−1,2 (Gk,3 ) Ek−1,3 + Ek−1,2 Gk,2 Ek−1,2 − T co T T co T T ATk,3 Gco k+1,3 Ak,2 − Ak,2 (Gk+1,3 ) Ak,3 − Ak,2 Gk+1,2 Ak,2 = Wk Ck,1 Ck,1 Wk .

(5.10) Again from the pd-stability of {(Ek,1 , Ak,1 )}K−1 k=0 , the generalized discrete-time periodic Lyapunov equations (5.8) have unique symmetric positive semi-definite solutions Gco k,1 . Furthermore, it follows from (5.5) that (5.9) can be rearranged as T co T co co Ek−1,1 (Gco k,3 − Gk,1 Zk−1 )Ek−1,2 − Ak,1 (Gk+1,3 − Gk+1,1 Zk )Ak,2 = 0.

(5.11)

Again, from (5.2), we deduce that co Gco k,3 = Gk,1 Zk−1 ,

k = 0, 1, . . . , K − 1.

(5.12)

From (5.5), (5.8) and (5.12), (5.10) can be rewritten as T T co T co T co Ek−1,2 (Gco k,2 − Zk−1 Gk,1 Zk−1 )Ek−1,2 − Ak,2 (Gk+1,2 − Zk Gk+1,1 Zk )Ak,2 = 0.

12

(5.13)

Now, since the periodic matrix pairs {(Ek,2 , Ak,2 )}K−1 k=0 have only infinite eigenvalues, we then have T co Gco k,2 = Zk−1 Gk,1 Zk−1 ,

k = 0, 1, . . . , K − 1.

Therefore, the solutions of the projected GDPLE (4.3) have the form   Gco Gco T k,1 k,1 Zk−1 Gco = V Vk−1 , k = 0, 1, . . . , K − 1, k−1 T T k Zk−1 Gco Zk−1 Gco k,1 k,1 Zk−1

(5.14)

(5.15)

where the matrices Gco k,1 are the unique symmetric positive semi-definite solutions of the generalized periodic Lyapunov equations (5.8). Moreover, from (5.6) and (5.15) they also satisfy Pl (k − co 1)T Gco k Pl (k − 1) = Gk . In many applications it is necessary to have the Cholesky factors of the solutions of the Lyapunov equations rather the solutions itself [24]. In particular, these full-ranked factors are useful for computing numerically the Hankel singular values (see next Section). If Lk,1 denotes a Cholesky co T factor of each matrix Gco k,1 , i.e., Gk,1 = Lk,1 Lk,1 , then we compute the QR factorization  Lk,1 = Qk,L

Tk,L 0

 ,

where Qk,L is orthogonal and Tk,L has full row rank, for k = 0, 1, . . . , K − 1. The full-ranked factorizations of the solutions Gco k , for k = 0, 1, . . . , K − 1, are given by # " LTk,1   T co Lk,1 , Lk,1 Zk−1 Vk−1 Gk = Vk−1 T T Zk−1 Lk,1 " # T Tk,L   T Tk,L , Tk,L Zk−1 Vk−1 = Vk−1 T T Zk−1 Tk,L

where Lk ≡



≡ LTk Lk ,  T Tk,L Zk−1 Vk−1 has full row-rank.

Tk,L ,

GDPLE for reachability Gramians Gcr k Similarly for the projected GDPLE (4.1), for k = 0, 1, . . . , K − 1, we let " #   Gcr Gcr Bk,1 k,1 k,3 T cr T Uk Gk Uk = , V B = . k k T Bk,2 (Gcr Gcr k,3 ) k,2

(5.16)

Substituting (5.1), (5.6) and (5.16) into the projected GDPLE (4.1), we then have T cr T cr T cr T T cr T Ek,1 Gcr k+1,1 Ek,1 − Ak,1 Gk,1 Ak,1 = −Ek,1 Gk+1,3 Ek,3 − Ek,3 (Gk+1,3 ) Ek,1 − Ek,3 Gk+1,2 Ek,3 T cr T T cr T + Ak,1 Gcr k,3 Ak,3 + Ak,3 (Gk,3 ) Ak,1 + Ak,3 Gk,2 Ak,3

+ (Bk,1 + Zk Bk,2 )(Bk,1 + Zk Bk,2 )T , T Ek,1 Gcr k+1,3 Ek,2 T Ek,2 Gcr k+1,2 Ek,2

− −

T Ak,1 Gcr k,3 Ak,2 T Ak,2 Gcr k,2 Ak,2

=

T −Ek,3 Gcr k+1,2 Ek,2

= 0,

+

T Ak,3 Gcr k,2 Ak,2 ,

k = 0, 1, . . . , K − 1.

(5.17) (5.18) (5.19)

Since the periodic matrix pairs {(Ek,2 , Ak,2 )}K−1 k=0 have only infinite eigenvalues, it follows from (5.19) that Gcr k = 0, 1, . . . , K − 1. (5.20) k,2 = 0, Furthermore, (5.18) can be simplified to T cr T Ek,1 Gcr k+1,3 Ek,2 − Ak,1 Gk,3 Ak,2 = 0.

13

(5.21)

Then from (5.2), we have Gcr k,3 = 0,

k = 0, 1, . . . , K − 1.

(5.22)

From (5.20) and (5.22), (5.17) can be rewritten as T cr T T Ek,1 Gcr k+1,1 Ek,1 − Ak,1 Gk,1 Ak,1 = (Bk,1 + Zk Bk,2 )(Bk,1 + Zk Bk,2 ) .

Therefore, the solutions of the projected GDPLE (4.1) have the form  cr  Gk,1 0 Gcr = U UkT , k = 0, 1, . . . , K − 1, k k 0 0

(5.23)

(5.24)

where the matrices Gcr k,1 are the unique symmetric positive semi-definite solutions of the generalized periodic Lyapunov equations (5.23). Moreover, from (5.6) and (5.24) they also satisfy T cr Pr (k)Gcr k Pr (k) = Gk . cr T If Rk,1 denotes a Cholesky factor of each matrix Gcr k,1 , i.e., Gk,1 = Rk,1 Rk,1 , then we compute the QR factorization  T  Tk,R T Rk,1 = Qk,R , 0 where Qk,R is orthogonal and Tk,R has full column-rank. The full-ranked factorizations of the solutions Gcr k are given by " # R  T  k,1 Rk,1 , 0 UkT Gcr k = Uk 0 " # Tk,R  T  Tk,R , 0 UkT = Uk 0 ≡ Rk RkT , where RkT ≡



T Tk,R , 0



UkT has full row-rank for k = 0, 1, . . . , K − 1.

Algorithm GDPLEs We now summarize the main steps for computing the full-ranked Cholesky factors of the causal Gramians, via the solution of the GDPLEs (4.1) and (4.3). For simplicity in Algorithm 5.1, we shall ignore the obvious qualification for k, i.e., k = 0, 1, · · · , K − 1. Algorithm 5.1 (GDPLEs) K−1 Input: System matrices (Ek , Ak , Bk , Ck ), with {(Ek , Ak )}k=0 being pd-stable.

Output: Full-ranked Cholesky factors Rk and Lk (k = 0, 1, . . . , K − 1), where T co T Gcr k = Rk Rk and Gk = Lk Lk . Step 1. Use the PQZ algorithm [8, 19] to compute orthogonal matrices Vk and Uk , with UK ≡ U0 , such that     Ek,1 Ek,3 Ak,1 Ak,3 T T Vk Ek Uk+1 = , Vk Ak Uk = 0 Ek,2 0 Ak,2 are upper triangular except V0T A0 U0 is quasi-upper triangular. The matrices Ek,1 and Ak,2 are nonsingular, and Ek,2 Ek+1,2 · · · Ek+K−1,2 are nilpotent. Step 2. Use the Cyclic Schur and Hessenberg-Schur methods [10] to compute the solutions of the generalized periodic Sylvester equations Ek,1 Wk+1 − Zk Ek,2 = Ek,3 , Ak,1 Wk − Zk Ak,2 = Ak,3 . 14

Step 3. Compute the matrices   Bk,1 VkT Bk = , Bk,2

Ck Uk =



Ck,1 , Ck,2



.

T Step 4. Compute the Cholesky factors Rk,1 and Lk,1 of the solutions Gcr k,1 = Rk,1 Rk,1 co T and Gk,1 = Lk,1 Lk,1 of the generalized discrete-time periodic Lyapunov equations T cr T T Ek,1 Gcr k+1,1 Ek,1 − Ak,1 Gk,1 Ak,1 = (Bk,1 + Zk Bk,2 )(Bk,1 + Zk Bk,2 ) , T T co T Ek−1,1 Gco k,1 Ek−1,1 − Ak,1 Gk+1,1 Ak,1 = Ck,1 Ck,1 .

Step 5. Compute the QR factorizations  T  Tk,R T Rk,1 = Qk,R , 0

 Lk,1 = Qk,L

Tk,L 0

 .

Step 6. Compute the full-ranked Cholesky factors     T Tk,R Rk = Uk , Lk = Tk,L , Tk,L Zk−1 Vk−1 . 0 Remark. One can extend the techniques in [32], for the numerical solution of the generalized Lyapunov equations, to solve the generalized discrete-time periodic Lyapunov equations given in Step 4. A thorough error analysis and practical implementation details for the algorithm extended from [32] are still under investigation.

A numerical example We shall illustrate the feasibility and reliability of the proposed algorithm with an example. All computations were performed in MATLAB/version 6.5 on a PC with an Intel Pentium-III processor at 866 MHz, with 768 MB RAM, using IEEE double-precision floating-point arithmetic. The machine precision is approximately 2.22 × 10−16 . ek of the projected generalized discrete-time periodic Lyapunov For approximate solutions X equations (4.1) and (4.3), we compute the relative residuals defined by γkcr =

ek+1 E T − Ak X ek AT − Pl (k)Bk B T Pl (k)T k2 kEk X k k k , e kX k k2

γkco =

T T T e ek Ek−1 − AT X kEk−1 X k k+1 Ak − Pr (k) Ck Ck Pr (k)k2 . ek k2 kX

Example 1. We consider a periodic discrete-time descriptor system (1.1) with n = 10, m = 2, p = 3 and period K = 3. For k = 0, 1, 2, we have   0 0 0 0 0 0 0 0 0 0 0 1 0 c1 s1 0 0 0 0 0   0 0 1 −s1 c1 0 0 0 0 0   0 c1 s1 1 0 c2 s2 0 0 0   0 −s1 c1 0 1 −s2 c2 0 0 0 (0)  , Ek =  0 0 c2 s2 1 0 c3 s3 0 0  0 0 0 −s2 c2 0 1 −s3 c3 0   0 0 0 0 0 c3 s3 1 0 0   0 0 0 0 0 −s3 c3 0 1 0 0 0 0 0 0 0 0 0 0 0 15

(0)

Ak = diag(1.01, A01 , A02 , A03 , A04 , 1.001), θk := 2πk/K,   4 −1 3 5 0 −2 0 8 1 0 T Bk = , 1 1 s1 + 1 −2 1 0 0 −3 0 1   0 0 1 0 0 1 0 0 0 0 0 0 1 , Ck = 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0.05 + c1 0 0 where c1 = cos(θk ), s1 = sin(θk ),

A01 A03

c2 = 0.2c1 , c3 = 0.6c1 , s2 = 0.2s1 , s3 = 0.6s1 ,     r1 cos(π/3) r1 sin(π/3) r2 cos(7π/5) r2 sin(7π/5) = , A02 = , −r1 sin(π/3) r1 cos(π/3) −r2 sin(7π/5) r2 cos(7π/5)     r3 cos(π/4) r3 sin(π/4) r4 cos(π/10) r4 sin(π/10) = , A04 = , −r3 sin(π/4) r3 cos(π/4) −r4 sin(π/10) r4 cos(π/10)

and r1 = 0.5,

r2 = 0.05,

r3 = −0.02,

r4 = 0.12.

√ We define a Householder transformation V = I − 2uu with u = [ 1, 1, · · · , 1, 1 ]T / 10 ∈ R10 , and the K-periodic system matrices (Ek , Ak , Bk , Ck ) are given by T

(0)

Ek ≡ V T Ek V,

(0)

Ak ≡ V T Ak V,

k = 0, 1, 2.

The computed open-loop spectrum of the periodic matrix pairs {(Ek , Ak )}K−1 k=0 consists of two infinite eigenvalues and four pairs of complex conjugate finite eigenvalues lying inside the unit circle. Thus, the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable with n1 = 8 and n2 = 2. Accurate numerical results were produced by the proposed algorithm, as shown in Table 1. k 0 1 2

kGcr k k2 8.30 × 104 7.11 × 103 5.82 × 102

γkcr 2.17 × 10−16 3.11 × 10−16 6.73 × 10−16

kGco k k2 1.14 × 103 9.70 × 100 9.74 × 101

γkco 1.39 × 10−16 4.17 × 10−15 9.18 × 10−15

Table 1: Norms and relative residuals of causal Gramians

6

Hankel singular values

Similar to standard state space systems [17] and continuous-time descriptor systems [37, 40], the controllability and observability Gramians can be used to define Hankel singular values for the periodic descriptor systems (1.1), which are of great importance in the model reduction problem via the balanced truncation method. For the discrete-time descriptor systems, the causal and noncausal Hankel singular values are defined via the nonnegative eigenvalues of the matrices Gdcc E T Gdco E and Gdnc AT Gdno A. Here Gdcc , Gdnc , Gdco and Gdno denote the causal/noncausal reachability Gramians and the causal/noncausal observability Gramians, respectively [41]. c Lemma 6.1. Let the periodic matrix pairs {(Ek , Ak )}K−1 k=0 be pd-stable. Then the matrices Hk ≡ cr T co nc nr T no Gk Ek−1 Gk Ek−1 and Hk ≡ Gk Ak Gk+1 Ak , k = 0, 1, 2, . . . , K − 1, have real and nonnegative eigenvalues.

Proof. From Definitions 4.1, 4.2 and (2.6) and for k = 0, 1, 2, . . . , K − 1, we have  cr co  b G b G 0 −1 k,1 k,1 Y , Hck = Yk 0 0 k 16

where b cr G k,1 ≡

k−1 X

Φf (k, i +

1)Bif (Bif )T Φf (k, i

T

+ 1) ,

i=−∞

b co G k,1 ≡

∞ X

Φf (i, k)(Cif )T Cif Φf (i, k).

i=k

b cr and G b co are symmetric positive semi-definite, it follows that Hc Since the n1 × n1 matrices G k,1 k,1 k have real and nonnegative eigenvalues. Similarly, it can be shown that Hnc k also share the same property. Notice that, in the proof of Lemma 6.1, the matrices Hck and Hnc k have at least n2 and n1 zero eigenvalues, respectively. Hence, we have the following definition of Hankel singular values for the periodic descriptor system (1.1). K−1 Definition 6.1. Suppose that the periodic matrix pairs {(Ek , Ak )}k=0 are pd-stable and let n1 , n2 be the dimensions of the periodic deflating subspaces of {(Ek , Ak )}K−1 k=0 corresponding respectively to the finite and infinite eigenvalues. (i) For k = 0, 1, . . . , K − 1, the square roots of the largest n1 eigenvalues of the matrices Hck , denoted by ζk,j , are called the causal Hankel singular values of the periodic descriptor system (1.1). (ii) For k = 0, 1, . . . , K − 1, the square roots of the largest n2 eigenvalues of the matrices Hnc k , denoted by θk,j , are called the noncausal Hankel singular values of the periodic descriptor system (1.1).

Remarks. (i) When K = 1, the causal and noncausal Hankel singular values defined in Definition 6.1 coincide with those for discrete-time descriptor systems (see [41] and references therein). For Ek = I, the causal Hankel singular values are the classical Hankel singular values of linear periodic discrete-time systems [48]. (ii) As in the case of descriptor systems, the causal and noncausal Hankel singular values of the periodic descriptor system (1.1) are invariant under system equivalence transformations. From Theorem 4.2 and Lemma 6.1 we obtain the following result. Corollary 6.2. Consider the periodic discrete-time descriptor system (1.1), where the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable. The following statements are equivalent. (a) The periodic descriptor system (1.1) is completely reachable and completely observable. (b) For k = 0, 1, 2, . . . , K − 1, we have co c rank(Gcr k ) = rank(Gk ) = rank(Hk ) = n1 , no nc rank(Gnr k ) = rank(Gk ) = rank(Hk ) = n2 .

(c) The causal and noncausal Hankel singular values of (1.1) are nonzero. For pd-stable {(Ek , Ak )}K−1 k=0 , the causal and noncausal reachability and observability Gramians are symmetric and positive semi-definite. Thus, there exist full-ranked factorizations T Gcr k = Rk R k , nr e e Gk = Rk RkT ,

T Gco k = Lk Lk , co e ek , Gk = LTk L

(6.1)

˜ k and L ˜ T are of full column-rank. The connections between the where the matrices Rk , LTk , R k causal/noncausal Hankel singular values and the singular values of the matrices Lk Ek−1 Rk and e k+1 Ak R ek are considered in the following Lemma. L Lemma 6.3. For the periodic descriptor system (1.1), where the periodic matrix pairs {(Ek , Ak )}K−1 k=0 are pd-stable. Suppose that the causal and noncausal Gramians of (1.1) have the full-ranked factorizations defined as in (6.1). Then for k = 0, 1, 2, . . . , K − 1, the nonzero causal Hankel singular values are the nonzero singular values of the matrices Lk Ek−1 Rk , while the nonzero noncausal e k+1 Ak R ek . Hankel singular values are the nonzero singular values of the matrices L 17

Proof. Notice that for k = 0, 1, . . . , K − 1, we have 2 T T ζk,j = λj (Rk RkT Ek−1 LTk Lk Ek−1 ) = λj (RkT Ek−1 LTk Lk Ek−1 Rk ) = σj2 (Lk Ek−1 Rk ), 2 ek R ekT ATk L e Tk+1 L e k+1 Ak ) = λj (R ekT ATk L e Tk+1 L e k+1 Ak R ek ) = σj2 (L e k+1 Ak R ek ), θk,j = λj (R

where λj (·) and σj (·) denote, respectively, the eigenvalues and singular values of the corresponding matrices.

7

Balanced realization

It is well known [17] that for any minimal realization (A, B, C) of a stable continuous-time or discrete-time system, there exists a transformation such that the controllability and observability Gramians for the transformed realization equal to some diagonal matrix. Such a realization is called a(n) (internally) balanced realization. Recently, the issues of balanced realization and model reduction via the balanced truncation method are discussed for continuous-time descriptor systems [37, 40] and asymptotically stable linear discrete-time periodic systems [47, 48]. In this Section the problem of balanced realization is generalized for periodic descriptor systems. We shall assume K−1 that the periodic descriptor system (1.1) is completely reachable/observable with {(Ek , Ak )}k=0 being pd-stable. Definition 7.1. A realization (Ek , Ak , Bk , Ck ) of the periodic descriptor system (1.1) is called balanced if     Dk,1 0 0 0 co nr no Gcr = G = and G = G = , k k k k+1 0 0 0 Dk,2 where Dk,1 and Dk,2 are diagonal matrices for k = 0, 1, . . . , K − 1. We shall show that for a realization (Ek , Ak , Bk , Ck ) of the periodic descriptor system (1.1), there exist nonsingular periodic matrices Sk and Tk (k = 0, 1, . . . , K − 1) with TK ≡ T0 , such that the transformed realization bk , A bk , B bk , C bk ) ≡ (S T Ek Tk+1 , S T Ak Tk , S T Bk , Ck Tk ) (E k k k

(7.1)

is balanced. Consider the full-ranked factorizations (6.1) of the causal/noncausal reachability/observability Gramians. For k = 0, 1, · · · , K − 1, let Lk Ek−1 Rk = Uk Σk VkT ,

e k+1 Ak R ek = U ek Θk VekT , L

(7.2)

e k+1 Ak R ek . Here Uk , Vk , U ek , Vek are be the singular value decompositions [18] of Lk Ek−1 Rk and L orthogonal, and Σk and Θk are diagonal and nonsingular. From Corollary 6.2 and Lemma 6.3, we have Σk = diag(ζk,1 , · · · , ζk,n1 ) > 0 and Θk = diag(θk,1 , · · · , θk,n2 ) > 0. Furthermore, it is easily seen from Theorem 4.1 and (2.5) that cr T T co Gcr Gco k = Pr (k)Gk Pr (k) , k = Pl (k − 1) Gk Pl (k − 1), Pr (k)Gnr Gno k = 0, k Pl (k − 1) = 0, Ek−1 Pr (k) = Pl (k − 1)Ek−1 , Ak Pr (k) = Pl (k)Ak . nr no cr co nr cr co Simple calculations then yield Gno k Ek−1 Gk = Gk Ek−1 Gk = Gk+1 Ak Gk = Gk+1 Ak Gk = 0. Hence, for k = 0, 1, . . . , K − 1, we have

e k Ek−1 Rk = Lk Ek−1 R ek = L e k+1 Ak Rk = Lk+1 Ak R ek = 0. L Now for k = 0, 1, . . . , K − 1, consider the n × n matrices h i h e T e −1/2 , Sˇk = Ek Rk+1 Vk+1 Σ−1/2 , Sk = LTk+1 Uk+1 Σ−1/2 k+1 , Lk+1 Uk Θk k+1 18

(7.3)

ek Vek Θ−1/2 Ak R k

i

,

It follows from (7.2) and (7.3) that " −1/2 −1/2 T Σk+1 Uk+1 Lk+1 Ek Rk+1 Vk+1 Σk+1 T ˇ Sk Sk = −1/2 e T e −1/2 Θk U k Lk+1 Ek Rk+1 Vk+1 Σk+1

−1/2 T ek Vek Θ−1/2 Σk+1 Uk+1 Lk+1 Ak R k −1/2 e T e ek Vek Θ−1/2 Uk Lk+1 Ak R k

# = In ,

Θk

i.e., the matrices Sk and Sˇk are nonsingular and Sk−1 = SˇkT . Similarly, it can be shown that the matrices h i h i eT U ek Θ−1/2 ek Vek Θ−1/2 , Tˇk = E T LT Uk Σ−1/2 , AT L Tk = Rk Vk Σ−1/2 , R k−1 k k k+1 k k k k are also nonsingular and Tk−1 = TˇkT . Therefore, with the transformation matrices Sk and Tk defined above and (7.3), the causal reachability and observability Gramians of the transformed periodic descriptor system (7.1) become −1 cr −T b cr ˇ G = TˇkT Gcr k ≡ T k Gk T k k Tk " −1/2

Σk

=

−1/2

Θk

 =

Σk 0

T UkT Lk Ek−1 Rk RkT Ek−1 LTk Uk Σk

−1/2

Σk

−1/2 T ekT L e k+1 Ak Rk RkT Ek−1 U LTk Uk Σk

Θk

0 0

= =

Σk 0

−1/2

T VkT RkT Ek−1 LTk Lk Ek−1 Rk Vk Σk

0 0

−1/2

ekT L e k+1 Ak Rk RkT ATk L e Tk+1 U ek Θ−1/2 U k

−1/2

T ek−1 Vek−1 Θ VkT RkT Ek−1 LTk Lk Ak−1 R k−1

,

Σk

−1/2 −1/2 T T ek−1 ATk−1 LTk Lk Ek−1 Rk Vk Σk Θk−1 Vek−1 R



e Tk+1 U ek Θ−1/2 UkT Lk Ek−1 Rk RkT ATk L k



and −1 co −T cr ˇ b co ˇT G k ≡ Sk−1 Gk Sk−1 = Sk−1 Gk Sk−1 " −1/2 Σk

#

−1/2

−1/2

#

−1/2 T T ek−1 ek−1 Vek−1 Θ−1/2 Θk−1 Vek−1 R ATk−1 LTk Lk Ak−1 R k−1

 .

On the other hand, one can also show that the noncausal reachability and observability Gramians of the transformed periodic descriptor system (7.1) satisfy   −T b nr ≡ T −1 Gnr T −T = 0 0 b no , k = 0, 1, . . . , K − 1. G = Sk−1 Gno ≡G k k k+1 Sk k+1 k k 0 Θk Consequently, Sk and Tk (k = 0, 1, . . . , K − 1) are the desired balancing transformations such that the realization (7.1) is balanced. In summary, we have the following theorem. Theorem 7.1. For completely reachable and completely observable periodic discrete-time descriptor system (1.1) with {(Ek , Ak )}K−1 k=0 being pd-stable, there exist nonsingular periodic matrices Sk and Tk (k = 0, 1, . . . , K − 1) with TK ≡ T0 such that the transformed realization (7.1) is balanced. Remark. As in the cases of standard state space systems [17, 31] and descriptor systems [37, 40], the balancing transformation matrices for periodic descriptor system (1.1) are not unique. Indeed, if {(Sk , Tk )}K−1 k=0 denotes a set of balancing transformation pairs for the periodic descriptor system (1.1), then for any diagonal matrix D with diagonal entries ±1, the set of matrix pairs {(Sk D, Tk D)}K−1 k=0 are also the balancing transformation matrices for the periodic descriptor system (1.1).

8

Concluding remarks

In Theorem 4.1, the reachability/observability Gramians are shown to satisfy some projected GDPLE which can be computed numerically by applying the PQZ algorithm with reordering strategies. We have developed these important concepts of reachability/observability Gramians, Hankel singular values and balanced realization for periodic discrete-time descriptor systems, based on the necessary and sufficient conditions for complete reachability and observability. These are useful in the model reduction problem via the balanced truncation method. A numerical example is given to illustrate the feasibility and reliability of the proposed algorithm in Section 5. 19

References [1] D. J. Bender, Lyapunov-like equations and reachability/observability gramians for descriptor systems, IEEE Trans. Auto. Control, 32 (1987), pp. 343–348. [2] M. C. Berg, N. Amit, and J. D. Powell, Multirate digital control system design, IEEE Trans. Auto. Control, 33 (1988), pp. 1139–1150. [3] S. Bittanti, Deterministic and stochastic linear periodic systems, in Time Series and Linear Systems, S. Bittanti, ed., Springer Verlag, New York, 1986, pp. 141–182. [4] S. Bittanti and P. Colaneri, Analysis of discrete-time linear periodic systems, in Control and Dynamic Systems, C. T. Leondes, ed., vol. 78, Academic Press, New York, 1996. [5] S. Bittanti and P. Colaneri, Periodic control, in Wiley Encyclopedia of Electrical and Electronic Engineering, J. G. Webster, ed., vol. 16, Wiley, New York, 1999, pp. 59–74. [6] S. Bittanti and P. Colaneri, Invariant representations of discrete-time periodic systems, Automatica, 36 (2000), pp. 1777–1793. [7] S. Bittanti, P. Colaneri, and G. D. Nicolao, The difference periodic Riccati equation for the periodic prediction problem, IEEE Trans. Auto. Control, 33 (1988), pp. 706–712. [8] A. Bojanczyk, G. H. Golub, and P. Van Dooren, The periodic Schur decomposition. Algorithms and applications, in Proc. SPIE Conference, vol. 1770, San Diego, 1992, pp. 31–42. [9] R. Bru, C. Coll, and N. Thome, Compensating periodic descriptor systems, Sys. Contr. Lett., 43 (2001), pp. 133–139. [10] R. Byers and N. Rhee, Cyclic Schur and Hessenberg Schur numerical methods for solving periodic Lyapunov and Sylvester equations, Technical Report, Dept. of Mathematics, Univ. of Missouri at Kansas City, 1995. [11] E. K.-W. Chu, H.-Y. Fan, and W.-W. Lin, Reachability and observability of periodic descriptor systems, preprint, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2005. [12] R. E. Crochiere and L. R. Rabiner, Mutirate Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1993. [13] L. Dai, Singular Control Systems, Springer-Verlag Berlin, Heidelberg, 1989. [14] A. Feuer and G. C. Goodwin, Sampling in Digital Signal Processing and Control, Birkhauser, New York, 1996. [15] D. S. Flamm, A new shift-invariant representation of perioidc linear systems, Syst. Contr. Lett., 17 (1991), pp. 9–14. [16] B. Francis and T. T. Georgiou, Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Auto. Control, 33 (1988), pp. 820–832. [17] K. Glover, All optimal Hankel norm approximations of linear multivariable systems and their L∞ error bounds, Int. J. Control, 39 (1984), pp. 1115–1193. [18] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996. [19] J. J. Hench and A. J. Laub, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Auto. Control, 39 (1994), pp. 1197–1210. [20] R. W. Isniewski and M. Blanke, Fully magnetic attitude control for spacecraft subject to gravity gradient, Automatica, 35 (1999), pp. 1201–1214. 20

[21] W. Johnson, Helicopter Theory, Princeton University Press, Princeton, NJ, 1996. [22] M. Kono, Eigenvalue assignment in linear discrete-time system, Int. J. Control, 32 (1980), pp. 149–158. [23] Y.-C. Kuo, W.-W. Lin, and S.-F. Xu, Regularization of linear discrete-time perioidc descriptor systems by derivative and proportional state feedback. To appear in SIAM J. Matrix Anal. Appl., 2004. [24] A. J. Laub, M. T. Heath, C. C. Paige, and R. C. Ward, Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms, IEEE Trans. Auto. Control, 32 (1987), pp. 115–122. [25] F. L. Lewis, Fundamental, reachability, and observability matrices for discrete descriptor systems, IEEE Trans. Auto. Control, 30 (1985), pp. 502–505. [26] W.-W. Lin and J.-G. Sun, Perturbation analysis for eigenproblem of periodic matrix pairs, Lin. Alg. Appl., 337 (2001), pp. 157–187. [27] W.-W. Lin and J.-G. Sun, Perturbation analysis of the periodic discrete-time algebraic Riccati equation, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 411–438. [28] M.-L. Liou and Y.-L. Kuo, Exact analysis of switched capacitor circuits with arbitrary inputs, IEEE Trans. Circuits and Systems, 26 (1979), pp. 213–223. [29] A. Marzollo, Periodic Optimization, Springer Verlag, Berlin, 1972. [30] R. McKillip, Periodic model following controller for the control-configured helicopter, Journal of the American Helicopter Society, 36 (1991), pp. 4–12. [31] B. C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Auto. Control, 26 (1981), pp. 17–32. [32] T. Penzl, Numerical solution of generalized Lyapunov equations, Adv. Comput. Math., 8 (1998), pp. 33–48. [33] J. A. Richards, Analysis of Periodically Time-Varying Systems, Springer-Verlag, Berlin, 1983. [34] V. V. Sergeichuk, Computation of canonical matrices for chains and cycles of linear mappings, Lin. Alg. Appl., 376 (2004), pp. 235–263. [35] J. Sreedhar and P. Van Dooren, Forward/backward decomposition of periodic descriptor systems and two point boundary value problems, in European Control Conf., 1997. [36] J. Sreedhar and P. Van Dooren, Periodic descriptor systems: solvability and conditionability, IEEE Trans. Auto. Control, 44 (1999), pp. 310–313. [37] T. Stykel, Model reduction of descriptor systems, Technical Report 720-2001, Institut f¨ ur Mathematik, TU Berlin, D-10263 Berlin, Germany, 2001. [38] T. Stykel, Analysis and numerical solution of generalized Lyapunov equations, PhD Dissertation, Institut f¨ ur Mathematik, Tecnische Universit¨at Berlin, Berlin, 2002. [39] T. Stykel, Stability and inertia theorems for generalized Lyapunov equations, Lin. Alg. Appl., 355 (2002), pp. 297–314. [40] T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Technical Report 04-2003, Institut f¨ ur Mathematik, TU Berlin, D-10263 Berlin, Germany, 2003. [41] T. Stykel, Input-output invariants for descriptor systems, preprint PIMS-03-1, Pacific Institute for the Mathematical Sciences, Canada, 2003. 21

[42] L. Tong, G. Xu, and T. Kailath, Blind identification and equalization based on second-order statistics: A time domain approach, IEEE Trans. Information Theory, 40 (1994), pp. 340– 349. [43] P. P. Vaidyanathan, Multirate digital filters, filter banks, polyphase networks, and applications: A tutorial, in Proc. IEEE, vol. 78, 1990, pp. 56–93. [44] P. P. Vaidyanathan, Mutirate Systems and Filter-Banks, Prentice-Hall, Englewood Cliffs, NJ, 1993. [45] P. Van Dooren and J. Sreedhar, When is a periodic discrete-time system equivalent to a time invariant one?, Lin. Alg. Appl., 212/213 (1994), pp. 131–151. [46] A. Varga, Periodic Lyapunov equations: some applications and new algorithms, Int. J. Control, 67 (1997), pp. 69–87. [47] A. Varga, Balancing related methods for minimal realization of periodic systems, Sys. Contr. Lett., 36 (1999), pp. 339–349. [48] A. Varga, Balanced truncation model reduction of periodic systems, in Proc. of IEEE Conference on Decision and Control, Sydney, Australia, 2000. [49] A. Varga, Robust and minimum norm pole assignment with periodic state feedback, IEEE Trans. Auto. Control, 45 (2000), pp. 1017–1022. [50] A. Varga and P. Van Dooren, Computing the zeros of periodic descriptor systems, Sys. Contr. Lett., 50 (2003), pp. 371–381. [51] J. Vlach, K. Singhai, and M. Vlach, Computer oriented formulation of equations and analysis of switched-capacitor networks, IEEE Trans. Circuits and Systems, 31 (1984), pp. 735– 765. [52] e. W. A. Gardner, Cyclostationarity in Communications and Signal Processing, IEEE, New York, 1994. [53] J. Xin, H. Kagiwada, A. Sano, H. Tsuj, and S. Yoshimoto, Regularization approach for detection of cyclostationary signals in antenna array processing, in IFAC Symposium on System Identification, vol. 2, 1997, pp. 529–534. [54] V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Wiley, New York, 1975. [55] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1996.

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