Representations and structural properties of periodic systems

Automatica 43 (2007) 1921–1931www.elsevier.com/locate/automatica

Representations and structural properties of periodic systems�

José Carlos Aleixoa,∗, Jan Willem Poldermanb, Paula Rochac

aDepartment of Mathematics, University of Beira Interior, 6201-001 Covilhã, PortugalbDepartment of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

cDepartment of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received 7 July 2006; received in revised form 7 January 2007; accepted 27 March 2007Available online 9 August 2007

Abstract

We consider periodic behavioral systems as introduced in [Kuijper, M., & Willems, J. C. (1997). A behavioral framework for periodicallytime-varying systems. In Proceedings of the 36th conference on decision & control (Vol. 3, pp. 2013–2016). San Diego, California, USA,10–12 December 1997] and analyze two main issues: behavioral representation, and controllability/autonomy. More concretely, we study theequivalence and the minimality of kernel representations, and introduce latent variable (and, in particular, image) representations. Moreover werelate the controllability of a periodic system with the controllability of an associated time-invariant system known as lifted system, and derivea controllability test. Further, we prove the existence of an autonomous/controllable decomposition similar to the time-invariant case. Finally,we introduce a new concept of free variables and inputs, which can be regarded as a generalization of the one adopted for time-invariantsystems, but appears to be more adequate for the periodic case.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Discrete-time systems; Time-varying systems; Difference equations; Behavior

1. Introduction

Linear models with periodically time-varying coefficientscan be found in numerous practical applications, such assatellite attitude control based on the periodicity of the earthmagnetic field, control of rotating machinery, or sampled-datasystems. An overview of the vast literature in the field of linearperiodic systems, in the classical input-state-output framework,can be found in the survey papers (Bittanti & Colaneri, 1996,2000) and the references therein.

In this paper periodic systems are studied within the behav-ioral framework. The behavioral theory of linear, time-invariantsystems has its roots in the mid-eighties when Willems startedhis pioneering work. The main paradigms are the emphasis on

� This paper was not presented at any IFAC meeting. This paper wasrecommended for pubication in revised form by the Associate Editor MariaElena Valcher under the direction of Editor Roberto Tempo.

∗ Corresponding author.E-mail addresses: [email protected] (J.C. Aleixo),

[email protected] (J.W. Polderman), [email protected](P. Rocha).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.03.013

the behavior, the set of possible trajectories of the system,rather than on its mathematical representations, and the absenceof an a prior partition of the signals into inputs and outputs.Nowadays, the behavioral approach has reached a fair level ofmaturity with branches in partial differential equations, discreteevent systems, computer science and coding theory. The reasonfor this diversity of applications is that all concepts are definedand studied in terms of the behavior and on a level of abstrac-tion that indeed enable translations in many different contexts.An example in that respect is the notion of controllability. Theclassical definition is stated for input-output systems in statespace form. The behavioral definition of controllability, how-ever, is more abstract and therefore more general. It can beapplied directly to systems of various natures, see Julius (2005)for a convincing account.

We present some results on the behavioral theory of linearperiodic systems. The paper builds on a framework developedby Kuijper and Willems, see Kuijper and Willems (1997) andthe references therein. An important tool is the lifting tech-nique that associates to each periodic behavior a time-invariantbehavior. The lifting map is the key ingredient in relating prop-erties of the periodic behavior to similar properties of the lifted,

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1922 J.C. Aleixo et al. / Automatica 43 (2007) 1921–1931

time-invariant, behavior. Using this map we obtain results aboutclassification of representations, controllability and autonomyof periodic behaviors, based on known results for the time-invariant case.

The paper is organized as follows. Section 2 contains thebasic material such as the precise definition of periodic be-havior, the lifting map, and a quick review of the results ob-tained in Kuijper and Willems (1997) that are relevant in ourcontext. The main new results in Section 3 are an analysis ofhow different kernel representations of the same periodic be-havior are related, and the characterization of minimal kernelrepresentations. Section 4 is concerned with latent variable andimage representations for periodic behaviors, we also exploitthe connections between these representations and kernel rep-resentations. Section 5 is dedicated to the study of controllableand autonomous periodic behaviors (some results concerningthe autonomous case have already been obtained in Kuijper &Willems, 1997). We prove that a periodic behavior is control-lable if and only if its lifted behavior is controllable; further-more a test for controllability is obtained. In case a periodicbehavior is neither controllable nor autonomous, we prove that,just like in the time-invariant case, it may be decomposed in acontrollable and an autonomous part. Finally, in Section 6 weintroduce the notion of free variable. A naive definition of freevariables would lead to the situation that a (nonzero) periodicbehavior without free variables is not necessarily autonomousand may even be controllable. In order to overcome this situa-tion we introduce a new concept of periodically free variable,which also allows us to define inputs and outputs for periodicsystems. Throughout the paper we provide examples to illus-trate the theory.

2. Periodic behavioral systems

In the behavioral framework, see Willems (1991), a dynami-cal system � is defined as a triple �= (T, W,B), with T ⊆ R

as the time set, W as the signal space and B ⊆ WT as the be-havior. Here we shall be concerned with the discrete-time case,that is, T = Z, assuming furthermore that the signal space isW = Rq , with q ∈ N. Let ��: (Rq)Z → (Rq)Z, defined by

(��w)(k) := w(k + �),

be the backward �-shift in case � ∈ Z+ or the forward �-shift in case � ∈ Z−. While the behavior B of a time-invariantsystem over Z is characterized by its invariance under the timeshift (and its inverse), which amounts to �B =B, P-periodicbehaviors are required to be invariant only w.r.t. the Pth powerof the shift, and its inverse.

Definition 1 (Kuijper & Willems, 1997). A system � is said tobe P-periodic, with P ∈ N, if its behavior B satisfies

�PB=B, (1)

but not �QB=B, for Q ∈ N smaller that P.

Note that (1) is equivalent to have �±P w ∈ B, ∀w ∈ B.

Example 2. The behaviorB ⊂ RZ defined by w(2k)=w(2k−1), k ∈ Z, describes a 2-periodic system since w ∈ B ⇒�±2w ∈ B and hence �2B = B. Note that �B = B so thatindeed B is 2-periodic.

Remark 3. In order to specify that a system �, or the corre-sponding behavior B, is described by certain equations we usethe notation � ∼ or B ∼ followed by these equations. For in-stance, in the previous example,

B ∼ w(2k) = w(2k − 1), k ∈ Z.

A widely common approach when dealing with periodic sys-tems is to relate them with some suitable time-invariant onesas, for instance, the lifted and the twisted systems, see Bittantiand Colaneri (2000), Khargonekar, Poolla, and Tannenbaum(1985), Kuijper and Willems (1997), Yamamoto (1994). Here,following Kuijper and Willems (1997), we consider the linearmap LP : (Rq)Z → (RPq)Z, defined by

(LP w)(k) :=⎡⎢⎣

w(Pk)

...

w(Pk + P − 1)

⎤⎥⎦ , P ∈ N,

and associate with a P-periodic system �=(Z, Rq,B) the liftedsystem, �L

P = (Z, RPq, LPB), with behavior

LPB= LP (B) := {w ∈ (RPq)Z|w = LP w, w ∈ B}.

Note that, in this definition of the lifted trajectory, the firstcomponent is w(Pk) rather than w(Pk + 1) as adopted inKhargonekar et al. (1985), Kuijper and Willems (1997).

As proved in Kuijper and Willems (1997), the lifting mapLp satisfies the properties presented in the next propositions.

Proposition 4.

(i) LP �P = �LP ;(ii) LP is a homeomorphism; consequently LP is a closed map.

This allows to relate a P-periodic system with the corre-sponding lifted system, yielding the following result.

Proposition 5.

(i) � is P-periodic if and only if P is the smallest positiveinteger for which �L

P is time-invariant;(ii) B is linear if and only if LPB is linear;

(iii) B is closed if and only if LPB is closed.1

In the sequel, for simplicity of notation, we shall drop thesubscript P in LP and �L

P .

1 Here, we say that a subspace of (Rr )Z is closed if it is closed in thetopology of pointwise convergence.

J.C. Aleixo et al. / Automatica 43 (2007) 1921–1931 1923

3. Kernel representations

Following the behavioral spirit, Definition 1 of a P-periodicsystem is not given in terms of equations representing the sys-tem. It has been shown in Kuijper and Willems (1997), Willems(1991) thatB is a �P -invariant linear closed subspace of (Rq)Z

if and only if

B ∼ (Rt (�, �−1)w)(P k + t) = 0,

t = 0, . . . , P − 1, k ∈ Z, (2)

where each Rt(�, �−1) ∈ Rgt×q [�, �−1] is a Laurent-polynomial matrix in the indeterminate �. Notice that theLaurent-polynomial matrices Rt need not have the same num-ber of rows (in fact we could even have some gt equal to zero,meaning that the corresponding matrix Rt would be void andno restrictions were imposed at the time instants Pk + t).

Example 6. Consider the 2-periodic system � with behaviorB ⊂ (R2)Z defined by

B ∼ (Rt (�, �−1)w)(2k + t) = 0, t = 0, 1, k ∈ Z,

with

R0(�, �−1) = [�2 − � �3] ∈ R1×2[�, �−1],

R1(�, �−1) =[

1 − � �3 − �

2�2 � − �2

]∈ R2×2[�, �−1].

This definition leads to the periodically time-varying differenceequations(

[�2 − � �3][

w1

w2

])(2k) = 0,

([1 − � �3 − �

2�2 � − �2

] [w1

w2

])(2k + 1) = 0,

that is,

w1(2k + 2) − w1(2k + 1) + w2(2k + 3) = 0,

w1(2k + 1) − w1(2k + 2) + w2(2k + 4) − w2(2k + 2) = 0,

2w1(2k + 3) + w2(2k + 2) − w2(2k + 3) = 0.

Observe now that, since

(Rt (�, �−1)w)(P k + t) = ((�tRt (�, �−1))w)(P k),

Eq. (2) can also be written as

(R(�, �−1)w)(P k) = 0, k ∈ Z, (3)

where

R(�, �−1) :=

⎡⎢⎢⎢⎢⎣

R0(�, �−1)

�R1(�, �−1)

...

�P−1RP−1(�, �−1)

⎤⎥⎥⎥⎥⎦ ∈ Rg×q [�, �−1], (4)

with g := ∑P−1t=0 gt . Analogously to the time-invariant case,

although with some abuse of language, we refer to (3) as a

P-periodic kernel representation (P-PKR) and to the matrixR(�, �−1) as a P-PKR matrix, or just representation matrix, ofthe corresponding behavior.

Remark 7. Note that by considering the P-PKR matrix R, weare ignoring the partition that is initially given by the matricesR0, . . . , RP−1. Indeed, this partition is irrelevant, as can beseen in Example 6. In this example P = 2, R0 has one rowand R1 has two rows and the final description consists of threedifference equations, which could as well be obtained by takingadequately defined R0 and R1 matrices with two and one row,respectively.

Since, for a given P, any integer i has a unique representationi = j +niP with ni ∈ Z and j ∈ {0, . . . , P − 1}, the followingdecomposition of R(�, �−1)

R(�, �−1) =∑i∈Z

Ci�i =

P−1∑j=0

∑i,i=j+niP

Ci�j�niP

=P−1∑j=0

�j∑ni∈Z

Cj+niP �niP

︸︷︷︸=:RL

j (�P,�−P

)

= RL(�P , �−P )�P,q(�) (5)

with

�P,q(�) := [Iq �Iq · · · �P−1Iq ]T (6)

and

RL(�, �−1) = [RL0 (�, �−1) RL

1 (�, �−1) · · · RLP−1(�, �−1)],

(7)

is unique.It follows from the decomposition (5) and the definition

of the lifted trajectory Lw associated to w, that (3) can bewritten as

(RL(�, �−1)(Lw))(k) = 0, k ∈ Z. (8)

Taking into account that this reasoning can be reversed, weobtain the following result.

Lemma 8 (Kuijper & Willems, 1997). A P-periodic behaviorB ⊂ (Rq)Z is given by a P-PKR if and only if the associatedlifted behavior LB is given by a kernel representation. More-over if (3) is a P-PKR of B, that is,

B ∼ (R(�, �−1)w)(P k) = 0, k ∈ Z,

then (8) is a kernel representation of LB, that is,

LB ∼ (RL(�, �−1)(Lw))(k) = 0, k ∈ Z,

or simply LB = ker RL(�, �−1), where RL(�, �−1) ∈Rg×Pq [�, �−1], g = ∑P−1

t=0 gt , is defined as in (7).


Example 9. Refer to Example 6. By definition the matrixR(�, �−1) is given by

[R0(�, �−1)

�R1(�, �−1)

]=

⎡⎢⎣

�2 − � �3

� − �2 �4 − �2

2�3 �2 − �3

⎤⎥⎦ .

By decomposing R as in (5), we obtain

R(�, �−1) =⎡⎢⎣

�2 0

−�2 �4 − �2

0 �2

⎤⎥⎦ + �

⎡⎢⎣

−1 �2

1 0

2�2 −�2

⎤⎥⎦

= [RL0 (�2, �−2) RL

1 (�2, �−2)]�2,2(�).

Therefore the matrix RL is given by

RL(�, �−1) = [RL0 (�, �−1) RL

1 (�, �−1)]

=⎡⎢⎣

� 0

−� �2 − �

0 �

∣∣∣∣∣∣∣−1 �

1 0

2� −�

⎤⎥⎦ .

For time-invariant behaviors the following result is standardand a proof, for the continuous time-setting, may be found inPolderman and Willems (1998).

Theorem 10. Consider the two time-invariant behaviors H =ker H(�, �−1) and H′ = ker H ′(�, �−1). Then H ⊆ H′ if andonly if there exists a Laurent-polynomial matrix V (�, �−1) suchthat H ′(�, �−1) = V (�, �−1)H(�, �−1).

Recalling the established relation between a P-periodicbehavior and its associated lifted-behavior, and invoking The-orem 10, we can immediately conclude that for P-periodicbehaviors B and B′, B ⊂ B′ if and only if any matricesRL(�, �−1) and R

′L(�, �−1), that represent the correspond-ing lifted behaviors LB and LB′, respectively, are relatedby R

′L(�, �−1) = V (�, �−1)RL(�, �−1), for some Laurent-polynomial matrix V (�, �−1). This constitutes an indirectcharacterization of behavior inclusion for the periodic case.However, our next result provides a more direct condition,since it is stated in terms of the P-PKR matrices themselves.

Theorem 11. Let B,B′ ⊂ (Rq)Z be two P-periodic behav-iors given by the P-PKR matrices R(�, �−1) and R′(�, �−1),respectively. Then B ⊂ B′ if and only if there exists a Laurent-polynomial matrix V (�, �−1) such that

R′(�, �−1) = V (�P , �−P )R(�, �−1). (9)

Proof. Assume that B ⊂ B′. By Theorem 10, the matricesRL(�, �−1) and R

′L(�, �−1), that represent the correspondinglifted behaviors, are related by

R′L(�, �−1) = V (�, �−1)RL(�, �−1), (10)

for some Laurent-polynomial matrix V (�, �−1). Note nowthat (10) may also be written as R

′L(�P , �−P )=

V (�P , �−P )RL(�P , �−P ), immediately yielding (9) after rightmultiplication by �P,q(�). Due to the uniqueness of decompo-sition (5), the previous reasoning can be entirely reversed. �

Corollary 12. LetB,B′ ⊂ (Rq)Z be two P-periodic behaviorsgiven by the P-PKR matrices R(�, �−1) and R′(�, �−1), respec-tively, possessing the same number, g, of rows. Then B = B′if and only if there exists a unimodular matrix U(�, �−1) suchthat

R′(�, �−1) = U(�P , �−P )R(�, �−1). (11)

Proof. By (Polderman & Willerms, 1998, Theorem 3.6.2),the two corresponding lifted behaviors, LB and LB′, arethe same (and therefore the same happens with the P-periodic behaviors B and B′) if and only if there existsa unimodular matrix U(�, �−1) ∈ Rg×g[�, �−1] such thatR

′L(�, �−1) = U(�, �−1)RL(�, �−1). Due to the uniqueness ofdecomposition (5), this yields (11). �

In the behavioral study of time-invariant systems one im-portant issue is the question of minimality of a given kernelrepresentation.

Definition 13. A P-PKR (matrix) R ∈ Rg×q [�, �−1] of a P-periodic system � = (Z, Rq,B) is said to be minimal if forany other representation R′ ∈ Rg′×q [�, �−1] of �, there holdsg�g′.

This definition is analogous to the one given for the time-invariant case, according to which a representation is minimalif it has a minimum number of equations. However, contrary towhat happens for the time-invariant case, see Polderman andWillems (1998), a minimal P-PKR matrix may not have fullrow rank.

It is easy to check that a P-PKR, R(�, �−1), of a P-periodicsystem � is minimal if and only if the same happens for the cor-responding representation RL(�, �−1) of the associated (time-invariant) lifted system �L. Thus R(�, �−1) is minimal if andonly if RL(�, �−1) is of full row rank over R[�, �−1]. The nextlemma (that can be shown based on the uniqueness of the de-composition (5)) translates this property in terms of the matrixR(�, �−1) itself.

Lemma 14. Let P ∈ N and R(�, �−1) ∈ Rg×q [�, �−1]. Con-sider the corresponding matrix RL(�, �−1) ∈ Rg×Pq [�, �−1]given by (5) and (7). Then, the following conditions are equiv-alent:

(i) RL(�, �−1) has full row rank over R[�, �−1];(ii) R(�, �−1) has full row rank over R[�P , �−P ].2

This result, together with the previous considerations, yieldsthe following characterization of minimality.

2 I.e., if r(�P , �−P ) ∈ R1×g[�P , �−P ] is such thatr(�P , �−P )R(�, �−1) = 0, then r(�P , �−P ) = 0.


Theorem 15. Let R(�, �−1) ∈ Rg×q [�, �−1] be the P-PKRmatrix of a P-periodic system �. Then R(�, �−1) is a minimalP-PKR if and only if it has full row rank over R[�P , �−P ].

Example 16. The representation R(�, �−1) ∈ R3×2[�, �−1] ofExample 6 is minimal. Indeed, although it is clearly not fullrow rank over R[�, �−1], it can be shown that it has full rowrank over R[�2, �−2].

4. Latent variable and image representations

In addition to kernel representations, which only involve thesystem variables, it is sometimes useful to also consider de-scriptions with other auxiliary variables. This is what happens,for instance, when models are derived from first principles, orwhen one is interested in obtaining simplified descriptions, suchas state space models. In the behavioral setting, those auxiliaryvariables are usually called latent variables as opposed to thesystem variables which are said to be manifest.

Definition 17. A system � = (Z, Rq,B) is said to have a P-periodic latent variable representation (P-PLVR) if its behav-ior consists of all the trajectories w that satisfy, together withsome latent variable trajectory v ∈ (R�)Z, the following set ofequations:

(Rt (�, �−1)w)(P k + t) = (Mt(�, �−1)v)(P k + t),

t = 0, . . . , P − 1, k ∈ Z, (12)

where Rt(�, �−1) and Mt(�, �−1) are Laurent-polynomialmatrices of suitable dimensions. In other words, B = {w ∈(Rq)Z: ∃v ∈ (R�)Z s.t. (12) holds}.

Remark 18. Note that if B has a P-PLVR, B is a P-periodicbehavior.

Example 19. Consider a system � described by the discretestate space model{

(�x)(k) = A(k)x(k) + B(k)u(k),

y(k) = C(k)x(k) + D(k)u(k),k ∈ Z, (13)

where A(k), B(k), C(k), D(k) vary with period P ∈ N. Lettingw := [u y]T and v := x, we may rewrite (13) as

(Rk(�, �−1)w)(k) = (Mk(�, �−1)v)(k), k ∈ Z, (14)

by taking

Rk(�, �−1) :=[

B(k) 0

−D(k) I

], Mk(�, �−1) :=

[�I − A(k)

C(k)

].

Since the periodicity of matrices A, B, C and D leads to

Rk+P (�, �−1) = Rk(�, �−1), Mk+P (�, �−1) = Mk(�, �−1),

we may write down (14) in the form (12).

Note that Eqs. (12) can be written as

(R(�, �−1)w)(P k) = (M(�, �−1)v)(P k), k ∈ Z, (15)

where R ∈ Rg×q [�, �−1], g := ∑P−1t=0 gt , is defined as in (4)

and M is defined in a similar way. We denote the latent variablerepresentation (15) by (R, M). By decomposing matrices R andM as

R(�, �−1) = RL(�P , �−P )�P,q(�),

M(�, �−1) = ML(�P , �−P )�P,�(�), (16)

and recalling the definition of lifted trajectories, it is straight-forward to see that (15) is equivalent to

(RL(�, �−1)w)(k) = (ML(�, �−1)v)(k), k ∈ Z,

where w=Lw and v =Lv. This reasoning allows to obtain thefollowing result.

Lemma 20. A P-periodic behavior B ⊂ (Rq)Z is given bya P-PLVR if and only if the associated lifted behavior LB isgiven by a latent variable representation (LVR). Moreover if(R, M) is a P-PLVR of B, then (RL, ML) is a LVR of LB.

Remark 21. Note that not every LVR, Rw=Mv, for the time-invariant behavior LB corresponds to a P-PLVR for B, sincethe number of components of the latent variable v must be amultiple of P. Nevertheless, if a LVR (R, M) of LB is given,one can always introduce “extra” components in v (and accord-ingly zero columns in M) in order to achieve the aforemen-tioned requirement on the cardinality of the components of v.

Theorem 22 (Latent variable elimination). Let �=(Z, Rq,B)

be a P-periodic system. Then B has a P-PLVR (R, M) ifand only if it has a P-PKR R∗. Given (R, M), R∗ may beobtained as

R∗(�, �−1) := H(�P , �−P )R(�, �−1), (17)

where H is a minimal left annihilator (MLA)—see Zerz(2000)—of the matrix ML(�, �−1), in the decomposition (16).

Proof. (⇒): Using the relationship with the lifted system(Lemma 20), and taking into account the results on latentvariable elimination for time-invariant systems (see Polderman& Willems, 1998), we obtain the following equivalences,with k ∈ Z,

B ∼ (R(�, �−1)w)(P k) = (M(�, �−1)v)(P k)

⇔ LB ∼ (RL(�, �−1)w)(k) = (ML(�, �−1)v)(k)

⇔ LB ∼ (H(�, �−1)RL(�, �−1)w)(k) = 0

⇔ B ∼ (R∗(�, �−1)w)(P k) = 0,

where as usual w = Lw, v = Lv, H(�, �−1) is a MLA ofML(�, �−1), and R∗ is given by (17).

(⇐): This implication is obvious. Since every P-PKR is alsoa P-PLVR with M = 0. �

A special case of latent variable representation occurs whenRt = Iq , t = 0, . . . , P − 1.


Definition 23. A system � = (Z, Rq,B) is said to have a P-periodic image representation (P-PIR) if its behavior can bewritten as

B= {w ∈ (Rq)Z: ∃v ∈ (R�)Z s.t.

w(Pk + t) = (Mt(�, �−1)v)(P k + t), t = 0, . . . , P − 1},where Mt(�, �−1) ∈ Rq×�[�, �−1], t = 0, . . . , P − 1. In thiscase the matrix M of (15) is said to be a P-PIR matrix, of B.

Remark 24. Analogously to the latent variable representationcase if B has a P-PIR, then is P-periodic.

As a consequence of Lemma 20, it turns out that a behaviorB has a P-PIR if and only if LB has an image representation.The next example illustrates how to obtain such a representationfor the periodic behavior, given a representation of its liftedbehavior, and puts into evidence the issue with the cardinalityof the latent variables mentioned in Remark 21.

Example 25. Consider the 2-periodic behavior B of Example2, with P-PKR R(�, �−1) = 1 − �−1. Since

R(�, �−1) = 1 − �−1 = [1 − �−2][

1

�

],

its associated lifted behavior LB is described by the kernelrepresentation(

[1 − �−1][

w1

w2

])(k) = 0, k ∈ Z.

It is also possible to describe this lifted behavior in terms of animage representation, namely

LB= im

[1

�

].

In order to achieve the decomposition in Eq. (16), we considerthat LB= imML(�, �−1), with ML(�, �−1) ∈ R2×2�[�, �−1],given by

ML(�, �−1) =[

1 0

� 0

].

Therefore the original 2-periodic behavior has a P-PIR matrixM given by

M(�, �−1) =[

1 0

�2 0

]�2,1(�) =

[1 0

�2 0

] [1

�

]=

[1

�2

],

that is, the 2-periodic behavior B allows the P-PIR[w(2k)

w(2k + 1)

]= (M(�, �−1)v)(2k), k ∈ Z.

The following result particularizes Theorem 22 to the P-PIRcase.

Corollary 26. Let �=(Z, Rq,B) be a P-periodic system. ThenB has a P-PIR M only if it has a P-PKR R. Given M, R

can be obtained as R(�, �−1) = H(�P , �−P )�P,q(�), where His a MLA of ML.

Taking Theorem 11 into account we can conclude that twoP-PIR M and M ′ describe the same behavior if and only ifthere exist two Laurent-polynomial matrices V (�, �−1) andV ′(�, �−1) such that the following relations hold:

H ′(�P , �−P )�P,q(�) = V (�P , �−P )H(�P , �−P )�P,q(�),

H(�P , �−P )�P,q(�) = V ′(�P , �−P )H ′(�P , �−P )�P,q(�),

where H ′ and H are MLAs of M′L and ML, respectively, as in

Theorem 22. Due to the uniqueness of decomposition (5) thisleads to

H ′(�, �−1) = V (�, �−1)H(�, �−1),

H(�, �−1) = V ′(�, �−1)H ′(�, �−1),

meaning that the sets of annihilators of ML and M′L coincide,

or equivalently, that ML and M′L are related by ML = M

′LG′and M

′L=MLG, where G′ and G are suitable rational matrices.Hence, M = ML�P,� = M

′LG′�P,� and M ′ = M′L�P,�′ =

MLG�P,�′ .

Theorem 27. Let M and M ′ be two P-PIR describing, respec-tively, the behaviorsB andB′. ThenB=B′ if and only if thereexist two rational matrices G and G′ such that

M(�, �−1) = M′L(�P , �−P )G′(�P , �−P )�P,�(�)

and

M ′(�, �−1) = ML(�P , �−P )G(�P , �−P )�P,�′(�),

where M′L and ML are given as in (16).

Since P-PIR are a particular type of P-PLVR, it is naturalto expect that they represent a more restrict class of systems.As we shall see, in the next section, P-periodic systems with aP-PIR are precisely those which are controllable.

5. Controllability and autonomy

In this section we first focus our attention on the behavioralcontrollability characterization of P-periodic systems. A keytool to achieve this is the relationship between the behavioralcontrollability of a P-periodic system and its associated liftedsystem. We start by recalling the definition of behavioral con-trollability.

Definition 28. A system � is said to be behaviorally control-lable if for all w1, w2 ∈ B and k0 ∈ Z, there exist w ∈ B andk1 �0 such that the following holds:

w(k) ={

w1(k), k�k0,

w2(k), k > k0 + k1.

By means of Theorem 29 below, it is possible to link thecontrollability of a P-periodic system with the controllabilityof its associated lifted system.


Theorem 29. A P-periodic system � is behaviorally control-lable if and only if the associated lifted system �L is behav-iorally controllable.

Proof. (⇒): Assume that � is controllable. Let w1, w2 ∈ LB

and k0 ∈ Z. By construction there exist w1, w2 ∈ B such thatLwi = wi , i = 1, 2.

Take k0 := P k0 +P . Then, by the controllability of �, thereexists k1 ∈ Z+ and a trajectory w ∈ B satisfying: w(k)=w1(k)

for k�k0 and w(k)=w2(k) for k > k0+k1. Take k1=�k1/P �+1.Then, the trajectory w := Lw ∈ LB coincides with w1forinstants k� k0 and with w2 for instants k > k0 + k1, showingthat LB is controllable.

(⇐): Assume now that �L is controllable. Let w1, w2 ∈ Band k0 ∈ Z. By construction there exist w1, w2 ∈ LB such thatLwi = wi , i = 1, 2.

Define k0=�k0/P �−1. Since LB is controllable, there existsk1 ∈ Z+ (which can clearly always taken to be not less than 1)and a trajectory w ∈ LB such that w(k)= w1(k) for k� k0 andw(k) = w2(k) for k > k0 + k1. Take k1 := P (k1 − 1) + 1�0,and let w := L−1(w) ∈ B. Then, w(k) = w1(k) for k�k0and w(k) = w2(k) for k > k0 + k1, which proves that B iscontrollable. �

In Kuijper and Willems (1997) an analogous result ofTheorem 29 has been obtained for the alternative (time-invariant) twisted system.

The behavioral controllability characterization of time-invariant systems, given in (Willems, 1991, Theorem V.2),together with Theorem 29, allows to conclude the following.

Proposition 30. Let � = (Z, Rq,B) be a P-periodic systemwith a P-PKR matrix R as in (4). Then � is controllable if andonly if the corresponding matrix RL (see (5) and (7)) is suchthat RL(�, �−1) has constant rank over C\{0}.

Furthermore if in addition the matrix RL(�, �−1) ∈Rg×Pq [�, �−1] has full row rank, then the previous conditionof RL(�, �−1) having constant rank over C\{0} is equivalentto say that RL(�, �−1) is left-prime, i.e, all its left divisors areunimodular matrices in Rg×g[�, �−1]. It turns out that, due tothe uniqueness of the decomposition (5), the left-primenessof RL(�, �−1) can be related with the following primenessproperty for R(�, �−1).

Definition 31. A Laurent-polynomial matrix R(�, �−1) ∈Rg×q [�, �−1] with full row rank over R[�P , �−P ] is said to beleft-prime over R[�P , �−P ], or simply P-left-prime, if when-ever it is factored as R(�, �−1) = D(�P , �−P )R(�, �−1), withD(�P , �−P ) ∈ Rg×g[�P , �−P ], then the factor D(�P , �−P )

and, equivalently D(�, �−1), is unimodular (over R[�P , �−P ]and R[�, �−1], respectively).

Lemma 32. Let P ∈ N and R(�, �−1) ∈ Rg×q [�, �−1]have full row rank over R[�P , �−P ]. Consider the asso-ciated matrix RL(�, �−1) ∈ Rg×Pq [�, �−1] according tothe decomposition (5). Then, the following conditions are

equivalent:

(i) RL(�, �−1) is left-prime;(ii) R(�, �−1) is P-left-prime.

This leads to the following direct characterization of control-lability.

Theorem 33. A P-periodic system � = (Z, Rq,B), with P-PKR, is controllable if and only if its minimal representationmatrices R(�, �−1) are P-left-prime.

Unfortunately, it may not be easy to (directly) verify the P-left-primeness of a Laurent-polynomial matrix. This also ap-plies, in fact, to full row rankness over R[�P , �−P ]. The studyof these and analogous properties is currently under investiga-tion.

Theorem 34. Let �=(Z, Rq,B) be a P-periodic system. ThenB has a P-PIR M if and only if B is controllable.

Proof. Due to the one-to-one correspondence between theimage representations of B and LB we have the followingscheme:

B controllable ⇔ LB controllable�

B has a P -PIR ⇔ LB has an IR �

Example 35. Consider the 2-periodic behavior of Examples 2and 25, with P-PKR matrix R(�, �−1)= 1 − �−1. According toTheorem 33, B is controllable, since R(�, �−1) is 2-left-prime,as the polynomial 1−�−1 has no nontrivial Laurent-polynomialfactors in �2. In fact, as we have seen in Example 25, B allowsa P-PIR.

Remark 36. This example shows that, contrary to time-invariant systems, a periodic behavior may be controllablewithout having free variables, i.e., without having variableswhose values may be freely assigned on the whole time axis.

As introduced, controllability, roughly speaking, means thatevery trajectory w1 ∈ B can be driven to another arbitrary tra-jectory w2 ∈ B. At the opposite end of the spectrum standsautonomy, which, roughly speaking, is the impossibility of con-necting a system trajectory with another different one, mean-ing that every trajectory in an autonomous behavior is uniquelydetermined by its past.

Definition 37. The P-periodic system � = (Z, Rq,B) is saidto be autonomous if for all k0 ∈ Z and all w1, w2 ∈ Bw1(k) = w2(k) for k < k0 ⇒ w1 = w2.

Similar to what happens with controllability, the autonomyof B and of LB are one-to-one related.

Theorem 38 (Kuijper & Willems, 1997). Let � = (Z, Rq,B)

be a P-periodic system. Then B is autonomous if and only ifLB is autonomous.


Taking into account the characterization of autonomy fortime-invariant behaviors given in Polderman and Willems(1998), the following result is trivially obtained.

Corollary 39 (Kuijper & Willems, 1997). Let � = (Z, Rq,B)

be a P-periodic system given by a P-PKR matrix R. Then Bis autonomous if and only if the corresponding representationmatrix of the associated lifted system, RL, has full columnrank (fcr).

It is well known that every linear time-invariant behavior B,which is a closed subspace of (Rq)Z, can be decomposed as thedirect sum of an autonomous sub-behavior with a controllablesub-behavior, more concretely the following result holds true.

Theorem 40 (Polderman & Willems, 1998). Let �=(Z, Rq,B)

be linear, time-invariant and such that B is a closed subspaceof (Rq)Z. Then:

(i) there exists an autonomous sub-behavior, Ba, of B, and acontrollable sub-behavior,Bc, ofB, such thatB=Ba⊕Bc;

(ii) if Ba, autonomous, and Bc1,B

c2, controllable, are all sub-

behaviors of B such that B=Ba ⊕Bc1 =Ba ⊕Bc

2, thenBc

1 =Bc2.

Remark 41. The controllable behaviorBc is defined byBc :=Bcp where Bcp is the set of trajectories of B with compactsupport and its closure is taken w.r.t. the topology of pointwiseconvergence.

We now prove a similar result for periodic behaviors.

Theorem 42. Let � = (Z, Rq,B) be P-periodic. Then thereexist Ba,Bc ⊂ B such that

(i) Ba is autonomous;(ii) Bc is controllable;

(iii) B=Ba ⊕Bc.

Proof. Define B := LB. Then, there exist sub-behaviors Ba

and Bc such that B=Ba ⊕Bc. As stated in Proposition 4, L is ahomeomorphism and therefore its inverse L−1 is well defined.Let’s use it to define Ba := L−1Baand Bc := L−1Bc. Notethat, see Theorem 38,Ba is autonomous andBc is controllable(due to Theorem 29). Since L is a homeomorphism, we thenhave that

Ba ∩Bc = L−1Ba ∩ L−1Bc = L−1(Ba ∩ Bc)

= L−1({0}) = {0}.Now, take w ∈ B. Let w := L(w) and take wa and wc to besuch that w = wa + wc, wa ∈ Ba, wc ∈ Bc. Define wa :=L−1(wa), wc := L−1(wc). Then, w = L−1(w) = L−1(wa +wc) = L−1(wa) + L−1(wc) = wa + wc. �

Example 43. Let �=(Z, R2,B) be the 2-periodic system withrepresentation matrix

R(�, �−1) = [�2 − 1 �4 − 1] = RL(�2, �−2)�2,2(�),

with RL(�, �−1) = [� − 1 �2 − 1| 0 0]. It can be shownthat the time-invariant lifted behavior LB, represented byRL(�, �−1), has an autonomous/controllable decompositionLB=(LB)a⊕(LB)c, where the autonomous behavior (LB)a isrepresented by

RLa (�, �−1) =

⎡⎢⎢⎢⎣

� − 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤⎥⎥⎥⎦

and the controllable behavior (LB)c has a representation

RLc (�, �−1) = [1 � + 1| 0 0].

We refer the reader to Polderman and Willems (1998) for furtherdetails on how to obtain such a decomposition for time-invariantsystems. This implies that B can be decomposed as B=Ba ⊕Bc, with Ba = L−1((LB)a) represented by

Ra(�, �−1) = RLa (�2, �−2)�2,2(�) =

⎡⎢⎢⎢⎣

�2 − 1 0

0 1

� 0

0 �

⎤⎥⎥⎥⎦

and Bc = L−1((LB)c) represented by

Rc(�, �−1) = RLc (�2, �−2)�2,2(�) = [1 �2 + 1].

6. Free variables and P-periodic inputs

Recall now that given a behavior B ⊂ (Rq)Z, a componentwi of the system variable w is said to be free if for all � ∈ RZ

there exists a trajectory w∗ ∈ B such that w∗i (k)=�(k), k ∈ Z.

In other words, this component wi it is not restricted by thesystem laws.

The properties of controllability and autonomy, in the time-invariant case, are related to the existence or the absence, re-spectively, of free variables. Indeed, a nontrivial time-invariantcontrollable behavior must have free variables while the ab-sence of free variables is equivalent to autonomy, (Polderman& Willems, 1998; Willems, 1991). As the next examples show,this no longer holds in the P-periodic case.

Example 44. Consider again the 2-periodic behavior ofExamples 2, 25 and 35. As shown in Example 35,B is control-lable. However B has no free variables, since the values of w

on each odd time instant and its consecutive one must coincide.

Example 45. Let B ⊂ R be the 2-periodic behavior de-scribed by w(2k) = 0, k ∈ Z. Clearly the only system vari-able w is not free, since it is required to be zero on eventime instants. However, B is not autonomous. Indeed fixingthe values of w(k) for k�0 does not yield a unique trajec-tory, since the values of w(2k + 1), k�0 can still be chosenfreely. Thus the absence of free variables does not implyautonomy.


These two examples suggest that a more sophisticated notionof free variable should be considered in the P-periodic case.

Definition 46. Let B ⊂ (Rq)Z be a P-periodic behavior in qvariables. The ith system variable wi , i ∈ {1, . . . , q}, is saidto be P-periodically free with offset t or t-P-periodically free,for t = 0, . . . , P − 1, if wi(P k + t), k ∈ Z, is not restrictedby the behavior. More precisely, if for all � ∈ RZ, there existsw∗ ∈ B such that its ith-component satisfies

w∗i (P k + t) = �(k), k ∈ Z.

Moreover, wi is said to be just P-periodically free if it is P-periodically free with offset t, for some t = 0, . . . , P − 1.

Note that, regarding time-invariance as 1-periodicity, Defini-tion 46 yields to the usual definition of free variable for time-invariant behaviors.

The following lemma is a direct consequence of Definition 46.

Lemma 47. Given a P-periodic behavior B ⊂ (Rq)Z, the ithsystem variable wi , i ∈ {1, . . . , q}, is t-P-periodically free (inB), for t = 0, . . . , P − 1, if and only if (Lw)tq+i is free in LB.

It is now easy to conclude that a controllable P-periodicbehavior must have P-periodically free variables.

Example 48. Recall Example 44. As we have seen there, w isnot free. However, w is 2-periodically free. To see this recallthat the associated lifted behavior LB is described by

([1 − �−1]

[w1

w2

])(k) = 0, k ∈ Z,

or, equivalently, w1(k)=w2(k−1), k ∈ Z, showing that eitherw1 or w2 are free in LB. Thus w is 2-periodically free since itis 2-periodically free with offsets t = 0 or t = 1.

Analogously, for autonomy, the following characterization interms of P-periodically free variables holds.

Theorem 49. Let �=(Z, Rq,B) be a P-periodic system. ThenB is autonomous if and only if B has no P-periodically freevariables.

Proof. Recall that, see Theorem 38, B is autonomous if andonly if LB is autonomous. In turn the lifted system is au-tonomous if and only if LB has no free variables which, byLemma 47, is equivalent to the autonomy of B. �

Example 50. Recall Example 45. As we have seen there, al-though B is not autonomous, the system variable w is not free.Notice, however, that w is 2-periodically free since, in this case,we have

R(�, �−1) = 1 = [1 0][

1

�

],

which leads to RL(�, �−1) = [1 0]. Therefore the associatedlifted behavior LB is described by(

RL(�, �−1)

[w1

w2

])(k) = 0 ⇔ w1(k) = 0, k ∈ Z.

Thus w2 is free and w is 2-periodically free since it is 2-periodically free with offset t = 1.

Remark 51. The absence of P-periodically free variables im-plies the absence of free variables inB which, in turn, is equiv-alent to a fcr condition on R (this can be shown by similararguments as used in Polderman & Willems, 1998 for the time-invariant case). Thus the fcr condition on R is only a necessarycondition for autonomy, but not a sufficient one (as illustratedin Example 45, where R(�, �−1) = 1). This can also be seen(using Corollary 39) from the fact that RL fcr implies R fcr,but not vice versa.

In the time-invariant case, an input is defined as a maximallyfree set of system variables, i.e., as a set of variables whichare simultaneously free and which, once fixed, leave no extrafree variables in the system. When defining simultaneously freecomponents of a trajectory, in a P-periodic behavior, one has totake into account that such components may be P-periodicallyfree with different offsets. This is illustrated by the followingexample.

Example 52. Let B ⊂ (R2)Z be the 3-periodic behavior givenby the equations

w2(3k) = w2(3k + 1) = w1(3k + 2) = 0, k ∈ Z.

Clearly the values of w1(3k), w1(3k + 1) and w2(3k + 2), (k ∈Z), are free, i.e., w1 is 3-periodically free with offsets 0 and 1,and w2 is 3-periodically free with offset 2. Note further, thatnone of the variables is free at all the possible offsets t =0, 1, 2,i.e., there is no variable whose values can be freely assigned atall time instants.

Given a P-periodic behavior B with variable w, a choice(possibly repeated) of components of w, (wi1 · · · wim)T,ir ∈ {1, . . . , q} for r = 1, . . . , m, is said to be (t1, . . . , tm)-P-periodically free, tr ∈ {0, . . . , P − 1} for r = 1, . . . , m, if forall �r ∈ RZ, there exists w∗ ∈ B such that its ir th componentsatisfies

w∗ir(P k + tr ) = �r (k), k ∈ Z.

Note that (wi1 · · · wim)T is (t1, . . . , tm)-P-periodically free ifand only if

u = ((�P,q(�)w)t1q+i1 , . . . , (�P,q(�)w)tmq+im)

is a free set of variables of �P,q(�)w, with �P,q(�) defined asin (6).

Definition 53. Given a P-periodic behavior B ⊂ (Rq)Z

with variable w = (w1 · · · wq)T, a choice of componentsu = ((�P,q(�)w)�1 , . . . , (�P,q(�)w)�m

) of �P,q(�)w is said


to be a P-periodic input of B if u is a maximally free set ofvariables of �P,q(�)w in the following sense:

(i) u is free, i.e., ∀� ∈ (Rm)Z ∃w∗ ∈ B s.t.

u∗(P k)

= ((�P,q(�)w∗)�1 , . . . , (�P,q(�)w∗)�m)(P k)

= �(k), k ∈ Z;

(ii) The set {(�P,q(�)w)(P k), w ∈ B|u(Pk)= 0} has no freevariables.

In this case the remaining components, y, of �P,q(�)w are saidto constitute a P-periodic output of B. Finally, an input/outputstructure forB is defined as a partition (u, y) of the componentsof �P,q(�)w, such that u is an input and y is an output.

Remark 54. The fact that

u = ((�P,q(�)w)�1 , . . . , (�P,q(�)w)�m)

is a P-periodic input means that the values of

wir (P k + tr ), ir ∈ {1, . . . , q}, r = 1, . . . , m,

where lr = trq + ir , may be freely assigned.

Noticing that (�P,q(�)w)(P k)= (Lw)(k), k ∈ Z, leads eas-ily to the following result.

Proposition 55. u = ((�P,q(�)w)�1 , . . . , (�P,q(�)w)�m) is a

P-periodic input of B if and only if

u = ((Lw)�1 , . . . , (Lw)�m)

is an input of the time-invariant behavior LB.

Given this established relationship between the P-periodicallyfree variables, of a P-periodic behavior, and the free variablesof its associated lifted system, it is now possible to define in-put/output structures in the periodic case based on the availableresults for time-invariant systems. This leads to the followingresult.

Theorem 56. Every P-periodic behavior B admits an in-put/output structure.

Example 57. Consider the 2-periodic behavior B of Example9. The lifted system can be represented as

P(�, �−1)

⎡⎢⎣

(Lw)1

(Lw)2

(Lw)3

⎤⎥⎦ = Q(�, �−1)[(Lw)4],

where P(�, �−1) and Q(�, �−1) consist of the first threecolumns and minus the last column of RL(�, �−1), respec-tively. Since det P(�, �−1) = 0, u := Lw4 is an input in LB

and, consequently, u= (�2,4(�)w)4 is a 2-periodic input forB.

7. Conclusions

In this paper we have pursued with the work on periodicsystems started by Kuijper and Willems in order to completethe results obtained in Kuijper and Willems (1997), giving ananswer to some basic open questions that arise within the be-havioral framework. In this context, we have characterized theequivalence and the minimality of kernel-like representationsfor periodic behaviors, and we have introduced and studied la-tent variable representations and, in particular, image represen-tations. Further, we have investigated the controllability andautonomy of a periodic behavior B, based on the characteri-zation of such properties for the corresponding lifted behaviorLB. Motivated by the time-invariant case, where controllabil-ity and autonomy are strictly related to the presence or absenceof free variables, we have analyzed this issue for the periodiccase and have obtained a new notion of freeness which is ade-quate for both cases. This also allowed us to define new notionof inputs and outputs for periodic systems.

In this line of research there are still many open questions left.Future work includes the study of topics such as observabilityand stability, as well as partitions and controller design forperiodic behaviors.

Acknowledgements

This research was partially supported by the European Com-munity Marie Curie Fellowship in the framework of the Con-trol Training Site programme (Grant no. HPMT-GH-01-00278-98) during the first author’s stay at the Department of AppliedMathematics, University of Twente, The Netherlands, as wellas by the Fundação para a Ciência e Tecnologia (FCT) throughthe Unidade de Investigação Matemática e Aplicações (UIMA),University of Aveiro, Portugal.

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José Carlos Aleixo was born in Covilhã, Por-tugal, in 1969. He received the Licenciatura inElectrical and Computer Engineering from In-stituto Superior Técnico, Lisbon, Portugal, in1993, and the M.S. degree in Mathematics fromUniversity of Beira Interior, Covilhã, Portugal,in 2001. Currently he is a Mathematics Ph.D.student at University of Aveiro, Aveiro, Portu-gal, under the supervision of P. Rocha. He hasreceived a Marie Curie grant for a three monthstay at the University of Twente, Enschede, TheNetherlands, under the supervision of J.W.

Polderman, and, currently, he is a FCT-Portuguese Foundation for Scienceand Technology fellow.J.C. Aleixo currently holds a Teaching Assistant position in the Departmentof Mathematics, University of Beira Interior, Covilhã, Portugal. He is alsoaffiliated with the Mathematical Systems Theory Group, in the research unitof Mathematics and Applications, at the University of Aveiro. His researchinterests include behavioral periodic systems.

Jan Willem Polderman was born in Kruinin-gen, The Netherlands. He received the M.Sc.degree in mathematics from the Universityof Groningen, Groningen, The Netherlands in1983, and the Ph.D. degree in mathematics, alsofrom the University of Groningen, in 1987. Thetitle of his dissertation was “Adaptive Controland Identification: Conflict or Conflux?” From1983 to 1987 he was a Research Assistantat the Center for Mathematics and Computer

Science in Amsterdam, where he worked under the supervision of J.H. vanSchuppen and J.C. Willems. Since 1987 he has been with the Department ofApplied Mathematics of the University of Twente, Enschede, The Netherlands,first as a Lecturer, and since 2003, as an Associate Professor. He has madecontributions to adaptive control, the behavioral apporach to systems theory,coding theory, and hybrid systems. With I.M.Y. he wrote the book AdaptiveControl: An Introduction (Basel, Switzerland: Birkhäuser 1996). With J.C.Willems he published the book Mathematical Systems Theory: A BehavioralApproach (Berlin, Germany: Springer Verlag, 1998).

Paula Rocha graduated in Applied Mathematicsin 1983 at the University of Oporto, Portugal.In 1990, she received a Ph.D. degree in Math-ematics and Natural Sciences from the Univer-sity of Groningen, The Netherlands, after hav-ing worked 4 years as a Ph.D. student in theSystems and Control Group of the Departmentof Mathematics of that university.From 1990 till the end of 1992, she was anassistant professor at the Faculty of Techni-cal Mathematics and Informatics of the DelftUniversity of Technology, The Netherlands.

Since 1993, she has been with the Department of Mathematics of the Uni-versity of Aveiro, Portugal, where she presently holds a Professor position.Her research interests are in the area of mathematical systems theory andinclude the behavioral approach to one (1D) and multi-dimensional (nD)systems.

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