Discrete-Time Linear Periodic Systems: The Realization Problem

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

DISCRETE-TIME LINEAR PERIODIC SYSTEMS: THE REALIZATION PROBLEMl P. Colanerl* and S. Longhl*· ·Diparl~fIlo

di Elettronica ed In!ormazione, Polilecnico di Milano, P.za L. do Vinci. 32. 20133, Milano,llaly "Diparl~nlo di Eltllronica ed AuJomalica. Universila di Ancona. via Breece Bianche. 60131 Ancona,llaly

Abstract. This paper deals with the minimal realization problem for discrete-time linear periodic systems. Two different time-invariant reformulations are introduced and compar.ed in terms of their frequency domain behaviour. The knowledge of the structural properties of a discrete-time periodic system is exploited in order to get a deep insight on the various and articulated aspects of minimal, quasi minimal and uniform realization of a given periodic impulse response pattern. Keywords. Discrete time systems, Linear systems, Periodic systems, Realization problem, Time-varying systems. terms of its structural properties, is not taken into account. In particular, in (Sanchez et al., 1992) In the last years an increasing attention has been the main results are uncorrectly stated, since the devoted to the analysis and control of discrete-time reachable and observable part of a periodic realizalinear periodic systems, and algebraic as well as ge- tion has in general a time-varying dimension. A ometric techniques have been used for the solution discussion on this point is provided in the remark of both analysis and design problems . Besides the ending Section 3. reasons related to the wide spreading diffusion of The present paper aims at filling up this deficiency. digital schemes for control purposes, the researches Necessary and sufficient conditions for the solvabilon this topic have been also spurred by the possi- ity of the periodic realization problem are given in bility of enhancing the properties of linear time in- terms of the existence of a minimal (reachable and variant plants with the use of periodic controllers observable at any time-instant) periodic realization, (see, e.g., Khargonekar et ai, 1985). Following which is generally described by periodic difference this line, an important technical tool consists in equations whose matrices have time-varying dimenusing the natural lifting isomorphism between pe- sions. The possibility of computing a minimal or riodic systems and time-invariant ones (see, e.g., "quasi" minimal (reachable and observable at least Meyer and Burrus, 1975) which proved useful for in one time-instant) uniform (fixed-dimensional) remany purposes ranging from the definition of ze- alization is also investigated. The method adopded ros (see Bolzern et al., 1986; Grasselli and Longhi, herein brings to an algorithm which can be consid1988) to the tracking and optimal H2/ Hoo prob- ered an extension to periodic systems of the welllems (see, e.g., Colaneri, 1991; Dahleh et al., 1992). known one proposed by Ho and Kalman (1966). The nice feature about the use of the lifting isomor- The proposed algorithm is based on two input-outphism is that one can exploit the theory of time- put equivalent time-invariant reformulations which invariant systems for the control of periodic ones, can be associated with a periodic system. Moreover, provided that the results achieved are easily rein- main importance here deserves the notions related terpreted in a periodic framework. More precisely, to the structural properties of a periodic system, the design procedure of a periodic controller, carried which, in discrete-time, split in a number of differout in the context of time-invariant systems, must ent characterizations due to the possible system non incorporate the constraineds on the periodic real- reversibility. For an overview of the strucural propizability of the achieved time-invariant controller. erties see Dittanti (1986) and Grasselli and Longhi As pointed out in various papers (see, e.g., Meyer, (1991). 1990) the structure of the time-invariant controller must reflect an obvious causality condition of the The paper is organized as follows. In Section 2, the periodic input-output map . From the discussions so-called lifted and cyclic time-invariant reformulaabove it turns out that a main role, in the design tions of a given linear discrete-time periodic system of periodic controllers, is played by the issue of de- are introduced and their relationships are pointed termining a periodic state-space model from input- out in terms of a frequency domain approach. The output maps (periodic realization problem). Un- strucural properties of the periodic system are also fortunately, only partial results are now available interpretated at the light of these shift-invariant deon this topic (Sanchez et al., 1992; Lin and King, scriptions. The arguments discussed herein are suc1992). In fact in the quoted papers, the very pe- cessive ly used in Section 3 for deriving the main reculiar structure of discrete-time periodic systems in sults on the realization of a given impulse response pattern (transfer function). An illustrative example ends the paper. lThis work was supported by Ministero Universita

1

Introduction

Ricerca Scientifica Tecnologica

321

2

Notations and preliminary results

Let denote by x(m,p , T) be the class of proper rational matrices W( z ) = [Wij(Z) E (CPxmi,j = 1, ···, T,] with Wij(ex:» = 0, i < j , Consider the T-periodic discrete-time system ~ de- i,j=l, . .. ,T. scribed by By the structure of matrix E£ it results that the transfer matrix WE(z) E x(m, p, T) so that the x(t + 1) = A(t)x(t) + B(t)u(t) (2 .1) class X(p, m, T) characterizes the transfer matrice of yet) = C(t)x(t) + D(t)u(t) (2.2) lifted reformulations of periodic systems. Moreover, Definition 2.2.

where t E Z, x(t) E IRn is the state , u(t) E IR is the input, yet) E lRP is the output and AO , BO , C( .), DO are periodic matrices of period T . Denote also by <1>(t, to) the transition matrix associated with AO.

in (Colaneri and Longhi, 1992) it is shown th at the transfer matrice of the lifted reformulation can be related to the impulse responses of the T-periodic system. A different time-invariant reformulation is now introduced.

In the sequal it will be usefull to introduce th e block · th e r10II OWIng . . . partitIOn 0f a t ma 'rIX P E ".,aRxbR \1J In way :

[v~(t)' vq(t)' ... v~(t)']' , y~(t) := [w~(t)' wq(t)' . .. w~(t)'l', x~(t) := [~~(t)' ~q(t)' ... ~~(t)']', wh ere vf(t) := u(t) , wf(t) := yet), ~f(t) := x(t),

m

Definition 2.3. For h E [0 , T - 1], let u~(t) :=

for t = i-I + h + jT, j E Z+, i = 1, . . . , T, and consider th e tim e-invariant system ~~ defin ed by

p = [Pij E lR a X b, i = 1, .. . ,R, j = 1, .. . , 5] (2 .3) Analogously, the column block partition of a matrix

F~x~(t)

+ C~u~(t) Il~x~(t) + E~u~(t)

Q E(CaxbS will be denoted as : Q = [Qj E IRaxb , j

= 1, ... ,5] .

(2.4)

(2 .8) (2. 9)

wh ere

The analysis ofT-periodic systems can be performed by making reference to a lifting isomorphism between these systems and time-invariant ones. In the sequel, two particular time-invariant reformula- Fh _ ctions of ~ are introduced and their properties and relationships are analized .

[ Alh) 0 0

A(Tf h)]

0 0

A(h + I)

0 0 0

0

A(T-2+h)

Definition 2.l. For h E [0, T - 1], let u1(k) :=

[u'(kT+h) u'(kT+h+1) . .. u'(kT+h+T-1)]', yl(k) := [y'(kT + h) y'(kT + h + 1) . .. y'(kT + Gh _ ch + T - 1)]', xi(k) := x(kT + h) and consider the time-invariant system ~i described by:

0 0

B(h+l)

0 0 0

0

0

B(T-2+h)

= diag{ C(h) , E~ = diag{ D(h ), H~

+ Cl ul(k) (2.5) H£xl(k) + E£ul(k) (2.6)

FE x l(k)

B(T_~>+h)]

[ BCh0 )

C(h + I) ,

C(T-I+h ) },

D(h + I),

D(T-I+h)} .

System ~~ will be referred to as the cyclic reformulation at time h of L

where FE := <1>(T+h, h), ci := [(Cih ... (Ci)T] , with (Ci)i := (T + h, i + h)B(i - 1 + h), i = 1" " , T, Hi := [(Hi)~ . .. (Hi)T]" with (Hi)i := C(i-1+h)(i-1+h,h) , i = 1, "' ,T, E£ := [(Ei)ij E lRPxm , i,j = 1,· ·· ,T], with (Ei)ij:= if i < j, (Ei)jj := D(i - 1 + h) , (Ei)ij := C(i 1 + h)(i - 1 + h,j + h)B(j - 1 + h), if i > j, i, j = 1"" ,T. System ~1 will be referred to as the lifted reformulation at time h of~.

=

It is important to note that whe n ~?(h) x(h) , system ~~ actually ge n e rates~f(t) x(t) and wf(t) = yet), t = i-I + h + jT, with i = 1, . .. , T and j E Z+. If course , with the cyclic reformulati on ~~ can be asso ciated a transfer matrix, henceforth

=

°

denoted by

(2.10) It is worth noticing that system ~t is equivalent to the original T-periodic system ~ In the sense that In the next result the relationships between sysxi(O) = x(h) reproduces yet) for t 2 hand x(kT + tems ~1 and ~~ in terms of their transfer matrices h) for k E Z+ (the set of non-negative integers). WE( z ) and W~(z) are prec isely stated . The time instant h can be consid ered as the initial time of T-rate sampling for the state of system ~ . Lemma 2.1. For any integer h, it results Of course, with the lifted reformulation ~i can be diag{1p, z-l/p," " z-T+I Ip} W£( zT) associated a transfer matrix , henceforth denoted by W~( z )

=

diag{lm, z lm, ···,zT- I/ m where In denotes the identity matrix of dimension n. The dependence of WE(z) with respect to the initial sampling time h is explained in (Grasselli and Longhi , 1988).

}.

(2.11)

In the sequel the relationships between ~i and ~~ will be further develop ed in terms of the Hankel matrices a
322

lifted reformulation and the cyclic reformulation, respectively.

It is easy to see that:

(Mb(rT))i = I pT , (Nb(rT))i = ImT, i

From the Markov parameters Jf(i) define now the Hankel matrix (r) of order r of the lifted reformulation (2.5), (2.6), as

si

J£(+ T) 1) J£(T

1

= jT + 1,

Vj E Z+,

(2.24)

(Mb(rT))i+iT = (Mb(rT))i' (Nb(rT))i+iT = (Nb(rT))i, i = I, . .. , T - 1, Vj E Z+ . (2 .25)

J£(2; - 1) . (2.12) Analogously, let denote by S~(r) the Hankel matrix of order r of the ciclic reformulation (2.8), (2.9), i.e.

J~(T)

J~(T+1)

1

In the following result, whose proof is reported in (Colaneri and Longhi, 1992), the relaltionships between the two Hankel matrices (2.12), (2.13) is precisely stated .

Lemma 2.3. For any integer h and any positive integer r, it results

J~(2~-l)

(2 .13) Moreover, for any integer r define the following matrices:

Ma(rT) := [(Ma(rT))ii E IRprTxpT, It will be of much importance in the solution of the i = 1, ... , T,j = 1, ... , rT] (2.14) realization problem to precisely define the relationships between the structural properties of ~ and Na(rT) := [(Na(rT))ii E IRmTxmrT, those of ~1 and ~~ . In the sequel such relationi= 1, .. . ,rT,j= 1, . . . ,T] (2 .15)

ships are precisely recalled for reachability (observability), controllability (reconstructibility) .

where

(Ma(rT)ii (Na(rT)ii

:=

:=

[((Ma(rT))ij ).q E IRPx p , S = 1, ... ,rT,q = 1, .. . ,T] (2.16)

[((Na(rT))ii).q E IRmxm , s=I, ... ,T,q=I, . .. ,rT] (2.17)

Lemma 2.7. (Bittanti, 1986; Grasselli and Longhi, 1991c) ~ is reachable (observable) at time t ijJ system ~i is reachable (observable).

(i) System

(2.18)

(ii) System ~ is reachable (observable) at any time ijJ system ~i is reachable (observable) for any t, or equivalently system ~~ is reachable (observable) for an arbitrary t.

((Na(rT))ij).q := s = T if j = 1, s = j - 1 if j > 1, Im , q = (2(i].l) + I)T - i + I; (2.19)

(iii) System ~ is controllable (reconstructible) at any time ijJ, for an arbitrary t, system ~L or equivalently system ~~, is controllable (reconstructible) .

with

((Ma(rT))ii).q :=

{

0,

{~~'

s=),q=z; otherwise;

otherwise.

and the symbol (a) denotes the integer part of a. Moreover, let

(Mb(rT)h := IpT, i-I

(Mb(rT))i:=

[JP

Ip(T_l)

o

'

]

• _ I -

2,3, . .. (2.22)

In order to consistently introduce the concepts of minimality of a realization from impulsive responses of a periodic systems, it is convenient to define Tperiodic equations of the type

(Nb(rT))1 := ImT, (Nb(rT))i:=

[L

Im(~-I) ]

As well known the dimension of the reachable and observable subspaces have T-periodic time-varying dimension, whereas the dimensions of the controllable and reconstructible subspaces do not change with time . It turns out that the Kalman canonical decomposition of a discrete-time periodic system into the reachable (observable) and unreachable (unobservable) parts can be performed only if time-varying dimensional part of systems are allowed. Moreover, when dealing with the properties of controllability and recontructibility, the Kalman canonical decomposition brings to subsystems with time-invariant dimensions.

i-I

,

i

= 2,3, .. . (.2 .23)

z(t

+ 1) y(t)

323

=

A(t)z(t) + B(t)u(t) C(t)z(t) + V(t)u(t)

(2.27) (2.28)

where t E Z, z(t) E IRn(/) , with net) T-periodic in- Definition 3.1. tegerfuntion, u( t) E IR m, y( t) E IRP, and AO, BO, (i) Given a transfer matrix W(z) E
zi(k + 1) yi(k)

(ii) A T-periodic realization of W(z) E
F£zi(k) + 9iui(k) (2 .29) Hizi(k) + E£ui(k) (2 .30)

where zi (k) := z( kT + h) and matrices .1'£ ,92, Hi, E£ are defined as the matrices FE ,Gi , H£, E2

1).

AO,

in srstem (2.5), (2.6) by substituting EO, CO, with AC) , BO, CC), 'DO, respectively. With equations (2 .27) and (2 .28) can be associated also the cyclic reformulation at time h, denoted by I:~,

DO

(iii) A T-periodic realization of W(z) E
in a straighforward way. Notice that I:i and I:~ are time-invariant systems (with constant dimensions). Definition 2.4. The pair (AC) , BC)) is reachable (control/able) at time t if I:t is reachable (controllable). Analoguously, the pair (CC) ,AC)) is observable (reconstructible) at time t if I:~ is observable (reconstructible) .

(iv) A T-periodic realization of W(z) E
= nh(z,AC),BO,C(-), VC» where A(t) = 2(t + I)A(t)2- 1 (t), 8(t) = 2(1+1)B(t), C(t) = C(t)2-1(t)

Definition 2.5. The pair (A(-), BC» is reachable and Vet) = 'D(t), where 2(t) is an arbitray Tperiodic nonsingular matrix with dimension net) (control/able) at aI/ times if I:~ is reachable (con- equal to the dimension of the vector z(t) in the trol/able) at an arbitrary integer t. Analoguously, equation (2.27), (2 .28) ; in other words the transthe pair (C(-),AO) is observable (reconstructible) fer function nh(z,AC),BC),CC),'DO) does not deat all times if I:~ is observable (reconstructible) at pend on the choice of the coordinates used for dean arbitrary integer t. scribing equations (2 .27), (2 .28). It is worth mentioning that the above definitions are In the main result below, whose proof is reported consistent, in the sense that the classical system- in (Colaneri and Longhi, 1992), the existence of theoretic interpretation of the structural properties minimal, quasi miniminal and uniform T-periodic are preserved . For example, according to Definition realizations is dealt with by resorting to the time2.4, if the pair (AC), B( ·» is controllable then, for invariant realization algorithm of Ho (Ho and Kalmeach initial state zi(O) = z(h) ofI:2 there exists an an, 1966) . Preliminarly, denote by Wi(z) the following rational matrix: input sequence u2c) which drives the state z2(k) = z(kT + h) to zero in a finite time. This fact leads ° z-lI~i] to the same property for the state of the T-periodic W'(z) := Z I pi I.(T-il] W(Z) [ I~(T-il equation (2.27) . i = 0, . .. , T - 1. (3 .2)

.

3

Solution of the realization problem

Hl(zIn(h) -

= .1'£)-19£ + EE

(3.1)

°

°

,

Note that, W(z) = WO(z) . Theorem 3.1. Given a transfer matrix W(z) E
In this section the problem of realizability of transfer matrices into T-periodic systems is dealt with . It will be shown that in order to have a "minimal" realization, one has to consider T-p eriodic equations described by matrices with time-varying dimensions. Specific definitions and results for "quasi minimal" and "uniform realizations" in terms of (constant-dimensional) T-periodic systems are also provided. First of all, define

nh(Z, A(·), BO,Cc), 'DO)

[°

(i) there exists a T-periodic minimal realization A(-), B(-),C(-), 'DC) if and only ifW(z) E X(P, m,T); (ii) there exists a T-periodic uniform minimal realization A(-), BO, CC), 'D(-) if and only if W( z ) E x(p, rn, T), and the Hankel matrices SiC) associated with Wi(z) has a constant rank for i = 0, 1, ... , T - 1. The sufficiency proof of Theorem 3.1 (i) provides a computational scheme for the construction of a T-periodic minimal realization .

so that nh(z, A( ·), BO, Cn, VC)) represents the tra- Remark 3.1. Let be given a transfer matrix W( z) E o,sfer matrix associated with the lifted reformulation X(P, rn, T) . A T-periodic minimal realization (.4(-), of the time-vaying dimensional T-periodic equaBC), CC),'DC) can be found by the following steps : tions (2.27), (2.28).

E2

324

Step 1 Compute v = degree of the monic least common multiple of all denominators of the entries of W(z). Let p = v+ 1. Step 2 Compute matrices Si(p), i = 0, .. . , T 1, which are the Hankel matrices associated with rational matrices Wi(z), i = 0, ... , T - 1, defined by (3.2). Denote by TJi the rank of Si(p), for i =

0, ... , T - 1.

= 1(0). [h 01 if b < a; h if b = a; E

where b

La := { [lOa]

if b > a.

Step 6 Compute the T-periodic minimal realization according to the following expressions:

Step 3 Compute S(pT) and S(pT) where S(pT) is the Hankel matrix of order pT assotiated with the A(i + kT) := R(i + I)F R(i)', B(i + kT) := rational matrix W(z) defined by R(i+l)CR(i)', i=0, ... ,T-2, VkEZ, 00

W(z) =

L

J(q)z-q := diag{Ip, z-l I p," "

A(T - 1 + kT) := R(O) F R(T - I)', B(T - 1 + kT) := R(O) CR(T - I)' Vk E Z,

q=O

z-T+l Ip}W(zT)diag{Im, zIm , . .. , zT-l Im},

C(i+kT) := R(i) H R(i)', i = 0, . . . , T-l, Vk E Z, and S(pT) is defined by

1(3)

S(pT) := [

1(p; + 1)

V(i+kT) := R(i) E R(i)' , i = 0, ... , T-2, Vk E Z,

1)]

~(pT + J(pT + 2)

1(3) 1( 4)

J(2)

1(pT + 2)

where, for i

1(2pT)

= O,j =f i + 1, Rj = Im,j = i + 1, R(i):= [Rj E IRPx p, j = 1, .. . , TJ, Rj = 0, j =f i + 1, R j = lp, j = i + 1. Rj

the following expressions:

U(p):=

:= :=

diag{Uo(p), U1(p)"", UT-l(p)} diag{Vo(p), V1(p)"", VT-1(p)}

[~~~~n,

V(p) := [V1(p)

The above quadruplet is reachable and observable at all times and its lifted reformulation at time 0 has a transfer matrix coincident with W(z), in other words nO(z,A(-),B(-),C(-), V(·» = W(z). b-

V2(p)] ,

It should be apparent that a minimal realization of W(z) E Xp ,m,T cannot in general be uniform, since the reachable and observable part of a Tperiodic system may have time-varying dimensions. On the contrary, a quasi minimal realization can be selected to have constant dimensions, of order fj = maXiE[O,T-l] rankSi(p), where p is defined at the step 1 of the above remark, by adding suitable unreachable and unobservable dynamics to the minimal realization . This point is clarified in the following result, whose proof is reported in (Colaneri and Longhi, 1992) .

with

U1(p) := diag{[ 1'10

U2(p):= diag{[O

0], [I'll 0], .. . , [I'IT_l 0 j},

I ppT-'1o] , [0

" ', [0

1,

R(i) := [Rj E IR'IiX'Ij-l, j = 1, . .. , Tj, Rj = O,j =f i+ I,Rj = I'Ii,j = i+ 1, R(i) := [Rj E IR mxm , j = 1, . . . , TJ,

Step 4 Compute matrices Ma(pT), Mb(pT), Na(pT) andNb(pT), according to (2.14), (2.20), (2 .15), (2.21), and matrices U(p), V(p), U(p) V(p) according to

U(p) V(p)

= 0, ... ,T -

IppT-'11] '

I pPT-'1T_l]},

Corollary 3.1. Let be given a transfer matrix W(z)

v 2(P) :=

diag{

E
[ImpT-'1o 0 ], [I 0 ] , ... , mpT-'11

(i) there exists aT-periodic minimal realization of W( z );

[ImpT~'1T_l] }.

(ii) there exists a uniform T -periodic quasi mini-

mal realization of W(z) . Step 5 Compute F, C, H, E according to

F

= L~pTl U(p) U(p) Ma (pT) Mb(pT) S(pT) Nb(pT) Na(pT) V(p) V(p) U;pT

°

1 ,

C = L;pTl U(p) U(p) Ma (pT) Mb(pT)S(pT) L;'//', H

= L~~Tl S(pT) Nb(pT) Na(pT)V(p) V(p) L';PT

Suppose that there exists a T-periodic minimal realization A(i) E IR'Ii+IX'li, B(i) E IR'Ii+l xm , C(i) E IRPxm. h v lE , l -, .. . , T - 1,were IRPX'li an d 'T"I(') TJi = rankSi(p), i = O,I, ... ,T-l, 7JT:= TJo and st) are the Hankel matrices of Wi(z), i = 0,1, .. . , T-l. Now, denote with fj := maXiE[O,T-l]TJi, and define T-periodic matrices A(i) E IR'ifX'if, B(i) E

1 ,

325

IRtl xm , C(i) E IRP Xi), and D(i) E IRPxm, following way:

In

the

4

Concluding remarks

In the paper the problem of realizability of a transfer matrix into a T-periodic system has been addressed. The solution has been derived by proper B(i) [ B~i)] , Vi E Z (3.4) definitions of minimal, quasi minimal and uniform T-periodic realization, along with a deep insight in (3.5) the characterization of the structural properties of C(i) [ C(i) 0], Vi E Z D(i) D(i), V E Z, (3.6) periodic systems. Necessary and sufficient conditions for the existence of such realizations have been carried out and an algorithm for their effective comwhere the zero blocks can vanish in some time instant. The T-periodic quadruplet AC), BO, CO, putation has been provided . The algorithm is easily D(·) defines a T-periodic system which is completely implemented in a high level language (e.g., MATLAB too!), and does not require a complex comreachable and observable in t, where t is such that putational burden, since the calculations only con1ft = r;. This realization is obviously quasi minimal sist in elementary column and row operations in the and uniform. matrices appearing in the algorithm. The proposed As an illustractive example consider procedure has been tested in an illustrative example. diag{A(i),O}, Vi E Z

A(i)

1/ z2] I/z

(3 .3)

.

References

It is easy to see that W(z) E X(I, 1,2), then, by Theorem 3.1 (i), there exits a quadruplet (AO, BO, CO, D( .)) which is a 2-periodic realization of W(z) . Following the algorithm provided in Remark 3.1, It turn out that 11 = 2, p = 3. Then, the 2-periodic minimal realization of W(z) takes the form:

A(O) = [0

1],

B(O) = 1, C(O) = (1

0], D(O) = 0,

[6]'

(3.7)

B(I)=[~],

(3.8)

A(I) =

Bittanti S. (1986). Deterministic and stochastic linear periodic systems. Time Series and Lin ear Systems, S.Bittanti Ed., Springer Verlag, 141. Bolzern P., P.Colaneri and R.Scattolini (1986) . Zeros of discrete-time linear periodic systems. IEEE Trans. Aut. Contr., AC-31, 1057. Colaneri P. (1991) . Output stabilization via pole-placement of discrete-time linear periodic systems. IEEE Aut . Contr., AC-36, 739. Colaneri, P. aIlS S. Longhi (1992b). The realization problem for lin ear periodic systems. Dipartimento di Eletlronica ed Automatica, Universitd di Ancona, R-92-01 .

(3 .9) (3.10)

C(I) = 1, D(I) = O.

Dahleh, M.A., P.G . Voulgaris and L.S . Valavani (1992). Optimal and robust controllers for periodic and multirate systems. IEEE Trans. Autom. Control, vo\. AC-37, no.1, pp 90-99. Grasselli O.M. and S.Longhi (1988b). Zeros and poles of linear periodic discrete-time systems. Circuits

It is easily seen that the above quadruplet is reachable and observable at all times and its lifted reformulation at time 0 has a transfer matrix coincident with W(z), in other words nO(z,AO, BO, CO, D( ·)) = W(z). By (3.3), (3.4) , (3.5) and (3.6), an uniform complete controllable and reconstructible 2-periodic realization is obtained as

[~

6]' B(O) = [6] ,

A(O) =

C(O) = [1

D(O)

0],

= 0,

A(I)

= [6 ~]

,(3.11)

[~],

(3.12)

B(1) =

C(I) = [1 D(I) = O.

Systems Signal Process., 7,361-380.

Grasselli O.M. and S.Longhi (1991). The geometric approach for linear periodic discrete-time systems, Linear Algebra and its Applications, 158, 27-60. Ho , B.L. and R.E. I
0], (3.13) (3.14)

Remark 3.2 The above example serves also as a counterexample for two results in (Sanchez et al., 1992) . In particular, in the quoted paper, part (b) of Theorem 4.1 is incorrectely stated. In fact, the example shows that does not exist a completely reachable and observable constant dimension realization having order less than or equal to 2. As a consequence, also the last statement of Theorem 5.2 in the quoted paper turns out to be technically not sound. Actually, the rank of the Henkel matrix S1(r) defined by (2.12) of the lifted reformulation associated with (3 .11) (3 .14) depends on h (the initial sampling time) . In fact the conclusion can be finally be drawn that the completely reachable and observable realization, for this example, has timevarying dimensions, see equations (3.7) - (3 .10). I:::. 326