On a Non-homogeneous Singular Linear Discrete ... - Semantic Scholar

Circuits Syst Signal Process (2013) 32:1615–1635 DOI 10.1007/s00034-012-9541-8

On a Non-homogeneous Singular Linear Discrete Time System with a Singular Matrix Pencil Ioannis Dassios · Grigoris Kalogeropoulos

Received: 11 June 2012 / Revised: 5 December 2012 / Published online: 3 January 2013 © Springer Science+Business Media New York 2012

Abstract In this article, we study the initial value problem of a non-homogeneous singular linear discrete time system whose coefficients are either non-square constant matrices or square with an identically zero matrix pencil. By taking into consideration that the relevant pencil is singular, we provide necessary and sufficient conditions for existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally we provide some numerical examples based on a singular discrete time real dynamical system to justify our theory. Keywords Singular, linear, discrete time system · Non-homogeneous 1 Introduction Linear discrete time systems (or linear matrix difference equations), are systems in which the variables take their value at instantaneous time points. Discrete time systems differ from continuous-time ones in that their signals are in the form of sampled data. With the development of the digital computer, the discrete time system theory plays an important role in control theory. In real systems, the discrete time system often appears when it is the result of sampling the continuous-time system or when only discrete data are available for use. Discrete time systems have many applications in economics, physics, circuit theory, and other areas. For example in finance, there is the very famous Leontief model, see [2], or the very important Leslie population growth model and backward population projection, see also [2]. In physics the Host-parasitoid models, see [38]. Applications of absorbing Markov chains or the distribution of heat through a long rod or bar are other interesting applications I. Dassios () · G. Kalogeropoulos Department of Mathematics, University of Athens, Athens, Greece e-mail: [email protected]

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suggested in [38]. Thus many authors have studied discrete time systems and their applications, see [1–14, 16–34, 36–42]. In this article we study the initial value problem of a class of non-homogeneous generalized linear matrix difference equations whose coefficients are either nonsquare constant matrices or square with an identically zero matrix pencil. We consider the singular system F Yk+1 = GYk + Vk

(1)

Yk0

(2)

with known initial conditions where F ∈ M(r × m; F ), (i.e. the algebra of matrices with elements in the field F ) with Xk , Vk ∈ M(m × 1; F). For the sake of simplicity we set Mm = M(m × m; F) and Mrm = M(r × m; F ). The matrices F and G can be non-square (when r = m) or square (r = m) and F singular (det F = 0). Many authors use matrix pencil theory to study linear discrete time systems with constant matrices, see for instance [7–11, 16–19, 24–26]. The results of the paper apply to general non-square pencils and generalize various results regarded the literature which mainly are dealing with square and non-singular systems. A matrix pencil is a family of matrices sF − G, parametrized by a complex number s, see [15, 24, 26, 35]. When G is square and F = Im , where Im is the identity matrix, the zeros of the function det(sF − G) are the eigenvalues of G. Consequently, the problem of finding the non-trivial solutions of the equation sF X = GX is called the generalized eigenvalue problem. Although the generalized eigenvalue problem looks like a simple generalization of the usual eigenvalue problem it exhibits some important differences. First it is possible for F to be singular in which case the problem has infinite eigenvalues. To see this write the generalized eigenvalue problem in the reciprocal form F X = s −1 GX If F is singular with a null vector X, then F X = 0m,1 , so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue s −1 = 0; i.e., s = ∞. This case has been fully analyzed in [2–4, 7–11, 16–19, 25, 31–33]. A second non-trivial case is the determinant det(sF − G), when F , G are square matrices, to be identically zero, independent of s. Finally there is the case for both matrices F , G to be non-square (for r = m). In this article we will consider these last two cases. Actually the paper analyses pencils of the form sF − G in which F is not necessarily square or full rank. Explicit and easily testable conditions are derived for which the system has a unique solution. Another important characteristic of the general case considered here is that existence of solutions is not automatically satisfied. This is very important for many applications for which the model is significant only for certain range of its parameters. In this cases a careful interpretation of results or even a redesign of the system maybe needed.

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1617

Because of the singularity of the pencil sF − G, in order to solve and to study these type of systems there are in the literature two methods. The first method is by using the theory of the Drazin inverse, see [2–4, 31–33], and the second is by using matrix pencil theory and the Kronecker canonical form which is a generalization of the Weierstrass canonical form and the Jordan canonical form. The advantage of the second method is that it gives a better understanding of the structure of the system and more deep, elegant results. In this article we will present a theory based on the matrix pencil of the system and we will show how the invariants of the pencil are related with the solutions of singular systems with a singular matrix pencil.

2 Singular Matrix Pencils: Mathematical Background and Notation In this section we will give the mathematical background and the notation that is used throughout the paper Definition 2.1 Given F, G ∈ Mrm and an indeterminate s ∈ F, the matrix pencil sF − G is called regular when r = m and det(sF − G) = 0. In any other case, the pencil will be called singular. In this article, we consider the case that the pencil is singular. The class of sF − G is characterized by a uniquely defined element, known as complex Kronecker canonical form, sFK − QK , see [15, 24, 26, 35], specified by the complete set of invariants of the singular matrix pencil sF − G. These invariants are the elementary divisors (e.d.) and the minimal indices (m.i.). The set of e.d. is obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over field F . In the case where sF − G is regular, we have e.d. of the following type: • e.d. of the type (s − aj )pj , are called finite elementary divisors (f.e.d.), where aj is a finite eigenvalue of algebraic multiplicity pj • e.d. of the type sˆ q = s1q are called infinite elementary divisors (i.e.d.), where q the algebraic multiplicity of the infinite eigenvalues. Unlike the case of the regular pencils where the pencil is characterized only from the e.d. the characterization of a singular matrix pencil apart from the set of the determinantal divisors requires the definition of additional sets of invariants, the minimal indices. The distinguishing feature of a singular pencil sF − G is that either r = m or r = m and sF − G ≡ 0. Let Nr , Nl be right, left null space of a matrix, respectively. Then the equations (sF − G)U (s) = 0m,1

(3)

V T (s)(sF − G) = 01,m

(4)

where ()T is the transpose tensor, have solutions in U (s), V (s), which are vectors in the rational vector spaces Nr (sF − G) and Nl (sF − G), respectively. The binary vectors U (s) and V T (s) express dependence relationships among the columns or rows of

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sF − G, respectively. U (s), V (s) are polynomial vectors. Let d = dim Nr (sF − G) and t = Nl (sF − G). It is known, see [15, 24, 26, 35], that Nr (sF − G) and Nl (sF − G) as rational vector spaces are spanned by minimal polynomial bases of minimal degrees 1 = 2 = · · · = g = 0 < g+1 ≤ · · · ≤ d

(5)

ζ1 = ζ2 = · · · = ζh = 0 < ζh+1 ≤ · · · ≤ ζt

(6)

and

respectively. The set of minimal indices 1 , 2 , . . . , d and ζ1 , ζ2 , . . . , ζt are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i) of sF − G, respectively. To sum up, as already stated in the case of a singular matrix pencil we have invariants of the following type: • e.d. of the type (s − aj )pj , finite elementary divisors (f.e.d.) • e.d. of the type sˆ q = s1q , infinite elementary divisors (i.e.d.) • m.c.i. of the type 1 = 2 = · · · = g = 0 < g+1 ≤ · · · ≤ d , column minimal indices • m.r.i. of the type ζ1 = ζ2 = · · · = ζh = 0 < ζh+1 ≤ · · · ≤ ζt , row minimal indices. Definition 2.2 Let B1 , B2 , . . . , Bn be elements of Mrm . The direct sum of them denoted by B1 ⊕ B2 ⊕ · · · ⊕ Bn is the blockdiag[B1 B2 . . . Bn ]. The existence of a complete set of invariants for singular pencils implies the existence of canonical form, known as Kronecker canonical form, see [15, 24, 26, 35], defined by sFK − QK := sIp − Jp ⊕ sHq − Iq ⊕ sF − G ⊕ sFζ − Gζ ⊕ 0h,g

(7)

where sIp − Jp is uniquely defined by the set of f.e.d. (s − a1 )p1 , . . . , (s − aν )pν ,

ν 

pj = p

(8)

j =1

of sF − G and has the form sIp − Jp := sIp1 − Jp1 (a1 ) ⊕ · · · ⊕ sIpν − Jpν (aν )

(9)

The q blocks of the second uniquely defined block sHq − Iq correspond to the i.e.d. sˆ q1 , . . . , sˆ qσ ,

σ 

qj = q

(10)

j =1

of sF − G and has the form sHq − Iq := sHq1 − Iq1 ⊕ · · · ⊕ sHqσ − Iqσ

(11)

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1619

Thus, Hq is a nilpotent element of Mn with index q˜ = max{qj : j = 1, 2, . . . , σ }, where q˜

Hq = 0q,q and Ipj , Jpj (aj ), Hqj are defined as ⎡ Ipj

⎢ ⎢ ⎢ ⎢ Jpj (aj ) = ⎢ ⎢ ⎢ ⎣ ⎡

0 0

...

⎢ ⎢0 1 ⎢ =⎢. . ⎢ .. .. ⎣ 0 0 ⎡

Hqj

1 0

⎥ 0 0⎥ ⎥ ∈ Mpj .. .. ⎥ . .⎥ ⎦ 0 1

... ..

⎤

.

...

⎤

aj

1

...

0

0

0 .. .

aj .. .

... .

0 .. .

0 .. .

0

0

...

aj

⎥ ⎥ ⎥ ⎥ ⎥ ∈ Mpj ⎥ ⎥ 1⎦

0

0

...

0

aj

0 1

..

...

0 0

⎤

⎢0 0 ⎢ ⎢ ⎢ = ⎢ ... ... ⎢ ⎢ ⎣0 0

...

0 0⎥ ⎥ ⎥ .. .. ⎥ ∈ M qj . .⎥ ⎥ ⎥ 0 1⎦

0 0

...

0 0

... ..

.

(12)

For algorithms about the computations of the Jordan matrices see [15, 24, 26, 35]. For the rest of the diagonal blocks of FK and GK , sF − G and sFζ − Gζ , the matrices F , G are defined as F = blockdiag{Lg+1 , Lg+2 , . . . , Ld }

(13)

. where L = [I .. 0,1 ], for  = g+1 , . . . , d G = blockdiag{L¯ g+1 , L¯ g+2 , . . . , L¯ d }

(14)

. where L¯  = [0,1 .. I ], for  = g+1 , . . . , d . The matrices Fζ , Gζ are defined as Fζ = blockdiag{Lζh+1 , Lζh+2 , . . . , Lζt } where Lζ =



Iζ 01,ζ



(15)

, for ζ = ζh+1 , . . . , ζt Gζ = blockdiag{L¯ ζh+1 , L¯ ζh+2 , . . . , L¯ ζt }

(16)

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0

, for ζ = ζh+1 , . . . , ζt . For algorithms about the computations of where L¯ ζ = I1,ζ ζ these matrices see [15, 24, 26, 35].

3 Main Results Following the above given analysis, there exist non-singular matrices P , Q with P ∈ Mr , Q ∈ Mm , such that P F Q = FK (17)

P GQ = GK Let

Q = Qp

Qq

Q

Qζ

Qg



(18)

where Qp ∈ Mmp , Qq ∈ Mmq , Q ∈ Mm , Qζ ∈ Mmζ and Qg ∈ Mmg and let ⎡

P1

⎤

⎢P ⎥ ⎢ 2⎥ ⎢ ⎥ ⎥ P =⎢ ⎢ P3 ⎥ ⎢ ⎥ ⎣ P4 ⎦

(19)

P5 where P1 ∈ Mpr , P2 ∈ Mqr , P3 ∈ Mr , P4 ∈ Mζ r P5 ∈ Mgr Lemma 3.1 System (1) is divided into five subsystems: p

p

(20)

Hq Zk+1 = Zk + P2 Vk

q

(21)

 = G Zk + P3 Vk F Zk+1

(22)

Zk+1 = Jp Zk + P1 Vk the subsystem q

the subsystem

the subsystem ζ

ζ

Fζ Zk+1 = Gζ Zk + P4 Vk

(23)

and the subsystem g

g

0h,g · Zk+1 = 0h,g · Zk + P5 Vk

(24)

Proof Consider the transformation Yk = QZk

(25)

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1621

Substituting the previous expression into (1) we obtain F QZk+1 = GQZk + Vk whereby, multiplying by P and using (17), we arrive at FK Zk+1 = GK Zk + P Vk Moreover, we can write Zk as

⎡

(26)

p⎤

Zk

⎢ q⎥ ⎢ Zk ⎥ ⎥ ⎢ ⎥ ⎢ Zk = ⎢ Zk ⎥ ⎥ ⎢ ⎢ Zζ ⎥ ⎣ k⎦ g Zk where Zpk ∈ Mp1 , Zqk ∈ Mq1 , Zk ∈ Mp1 , Zζk ∈ Mq1 and Zgk ∈ Mp1 , Zqk ∈ Mq1 . Taking into account the above expressions, we arrive easily at the subsystems (20), (21), (22), (23), and (24).  Solving the system (1) is equivalent to solving subsystems (20), (21), (22), (23), and (24). Proposition 3.1 The subsystem (20) is a regular type system and its solution is given from, see [7, 21, 27–29, 36–39] p Zk

p = Jpk−k0 Zk0

+

k−1 

Jpk−i−1 P1 Vi ,

k ≥ k0

(27)

i=k0

Proposition 3.2 The subsystem (21) is a singular type system but its solution is very ease to compute, see [7–11, 16–19, 25] q

Zk = −

q ∗ −1

Hqi P2 Vk+i

(28)

i=0

Proposition 3.3 The subsystem (22) has infinite solutions and can be taken arbitrary Zk = Ck,1 Proof Let

⎡



Vk g+1

(29) ⎤

⎢ g+2 ⎥ ⎢ Vk ⎥ ⎥ ⎢ P3 Vk = ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ 

Vk d

(30)

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If we set

⎡ Zk



Zkg+1

⎤

⎢ g+2 ⎥ ⎢ Zk ⎥ ⎢ ⎥ =⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ 

Zkd By using (13), (14) and (30), the system (22) can be written as ⎡ g+1 ⎤ Zk+1 ⎢ g+2 ⎥ ⎢ Zk+1 ⎥ ⎥ ⎢ blockdiag{Lg+1 , . . . , Ld } ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ 

d Zk+1 ⎡ g+1 ⎤ ⎡ g+1 ⎤ Vk Zk ⎢ g+2 ⎥ ⎢ g+2 ⎥ ⎢ Zk ⎥ ⎢ Vk ⎥ ⎥ ⎢ ⎥ ⎢ = blockdiag{L¯ g+1 , . . . , L¯ d } ⎢ . ⎥ + ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦

Zkd

(31)

Vkd

Then for the non-zero blocks a typical equation from (37) can be written as i Li Zk+1 = L¯ i Zki + Vk i ,







i = g + 1, g + 2, . . . , d

(32)

or I i

.. .

i = 0i ,1 0i ,1 Zk+1

 ..  . Ii Zki + Vk i

or ⎡

1 0 ... ⎢ ⎢0 1 ... ⎢ ⎢. . ⎢ .. .. . . . ⎣ 0 0 ...

⎤

⎡

i ,1 zk+1

⎤

⎤

⎡

⎡

zki ,1

⎤ ⎡

vki ,1

⎤

0 0 ⎢  ,2 ⎥ ⎢  ,2 ⎥ i ⎢ ⎥ ⎥ ⎢ zki ⎥ ⎥ ⎢ vk ⎥ 0 0⎥⎢ ⎢ ⎥ ⎥ ⎥⎢ . . ⎢ ⎢ ⎥ ⎥ .. .. ⎥ ⎢ .. ⎥ + ⎢ .. ⎥ ⎥ ⎥ ⎢ ⎥ . . ⎦⎢ ⎢ zi ,i ⎥ ⎢ v i ,i ⎥ ⎣ k ⎦ ⎣ k ⎦ 0 1 i ,i +1 i ,i +1 zk vk

0 1 ... 0 0 ⎢ i ,2 ⎥ zk+1 ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢0 0 ... 0 0⎥⎢ ⎥⎢ . ⎥ ⎥=⎢ . .. .. .. .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . . ... . .⎥ ⎣ ⎦⎢  , ⎢ zi i ⎥ ⎣ k+1 ⎦ 0 0 ... 1 0 i ,i +1 zk+1

or i ,1 zk+1 = zki ,2 + vki ,1 i ,2 zk+1 = zki ,3 + vki ,2

.. .  ,

 , +1

i i i i = zk+1 zk+1

 ,i

+ vk i

(33)

Circuits Syst Signal Process (2013) 32:1615–1635

1623

The system (33) is of a regular type system of difference equations with i equations and i + 1 unknowns. It is clear from the above analysis that in every one of the d − g subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrary Zk = Ck,1



Proposition 3.4 For the subsystem (23) we assume that ⎡ ζ ⎤ V h+1 ⎢ kζ ⎥ ⎢ V h+2 ⎥ ⎢ k ⎥ ⎥ P4 Vk = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎦ ⎣

(34)

ζ

Vk t and

⎡

ζ

Vk j

ζ ,0

vk j

⎢ ζ ,1 ⎢v j ⎢ k ⎢ ζ ,2 ⎢ j = ⎢ vk ⎢ ⎢ . ⎢ .. ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

j = h + 1, h + 2, . . . , t

(35)

ζ ,ζj

vk j

Then the subsystem (23) has a solution if and only if ζj 

ζ ,ρ

j vk+ρ = 0,

j = h + 1, h + 2, . . . , t

(36)

ρ=0

If (36) holds, the unique solution of the system (23) is given from ⎡ ζ ⎤ Z h+1 ⎢ kζ ⎥ ⎢ Z h+2 ⎥ ⎢ k ⎥ ζ ⎥ Zk = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎦ ⎣ ζ

Zkt where ⎡

ζ ,1

ζ ,2

ζ ,ζ

j j j −vkj − vk+1 − · · · − vk+ζ j −1

⎤

⎥ ⎢ ζ ,2 ⎢ −v j − v ζj ,3 − · · · − v ζj ,ζj ⎥ ⎢ k k+ζj −2 ⎥ k+1 ζ ⎥, Zkj = ⎢ ⎥ ⎢ . .. ⎥ ⎢ ⎦ ⎣ ζ ,ζj

−vkj

j = h + 1, h + 2, . . . , t

(37)

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For ζj 

ζ ,ρ

j vk+ρ = 0,

j = h + 1, h + 2, . . . , t

(38)

ρ=0

the subsystem (23) has no solutions. Proof From (15), (16), (34) and (37) the subsystem (23) can be written as ⎡

ζ

h+1 Zk+1

⎤

⎢ ζ ⎥ ⎢ Z h+2 ⎥ ⎢ k+1 ⎥ ⎥ blockdiag{Lζh+1 , . . . , Lζt } ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎦ ⎣ ζt Zk+1 ⎡ ζ ⎤ ⎡ ζ ⎤ V h+1 Z h+1 ⎢ kζ ⎥ ⎢ kζ ⎥ ⎢ Z h+2 ⎥ ⎢ V h+2 ⎥ ⎢ k ⎥ ⎢ k ⎥ ⎥ ⎢ ⎥ = blockdiag{L¯ ζh+1 , . . . , L¯ ζt } ⎢ ⎢ .. ⎥ + ⎢ .. ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎦ ⎣ ⎦ ⎣ ζ

Zkt

(39)

ζ

Vk t

Then for the non-zero blocks, a typical equation based on (39) can be written as j Lζj Zk+1 = L¯ ζj Zkj + Vk j ,

ζ

ζ

ζ

j = h + 1, h + 2, . . . , t

or ⎡

I ζj

⎤

⎡

01,ζj

⎤

ζj ⎥ ζj ⎥ ζj ⎢ ⎢ ⎣ · · · ⎦ Zk+1 = ⎣ · · · ⎦ Zk + Vk 01,ζj I ζj

or ⎡

1 0

⎢0 1 ⎢ ⎢ ⎢ .. .. ⎢. . ⎢ ⎢ ⎣0 0 0 0

⎡ ⎤ 0 . . . 0 ⎡ ζj ,1 ⎤ zk+1 ⎢ ⎥ ⎥ ⎢1 ... 0⎥⎢ ζj ,2 ⎥ ⎥⎢ z ⎢ ⎥ ⎢ k+1 .. ⎥ ⎢ ⎥ ⎢ .. ⎥ ... . ⎥⎢ . ⎥ = ⎢ ⎢. . ⎢ ⎢ ⎥⎣ . ⎥ ⎦ ⎣0 ... 1⎦ ζj ,ζj z k+1 0 ... 0

0 ... 0 ... .. . ... 0 ... 0 ...

⎡ ζj ,0 ⎤ ⎤ vk 0 ⎡ ζj ,1 ⎤ ⎢ ζ ,1 ⎥ zk j ⎥ ⎥ ⎢ 0⎥ ⎢ vk ⎥ ⎥⎢ ζj ,2 ⎥ ⎢ ⎥⎢ ⎥ z ⎥ ζj ,2 ⎥ .. ⎥ ⎢ ⎢ k ⎥+⎢ v ⎢ ⎥ . ⎥⎢ . ⎥ ⎢ k ⎥ ⎥ ⎥ ⎢ .. ⎥ ⎥ ⎦ ⎢ ⎢ ... ⎥ 0⎦⎣ ⎣ ⎦ ζj ,ζj zk ζj ,ζj 1 vk

(40)

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1625

or ζ ,1

ζ ,0

ζ ,2

ζ ,1

j zk+1 = vk j

ζ ,1

j zk+1 = zkj + vkj

.. .

(41) ζ ,ζj −1

ζ ,ζ

j j = zkj zk+1

ζ ,ζj

0 = zkj

ζ ,ζj −1

+ vk j ζ ,ζj

+ vk j

We have a system of ζj + 1 difference equations and ζj unknowns. Starting from the last equation we get the solutions ζ ,ζj

zkj

ζ ,ζj −1

zkj

ζ ,ζj −2

zkj

ζ ,ζj

= −vkj

ζ ,ζj −1

= −vkj

ζ ,ζj −2

= −vkj

ζ ,ζ

j j − vk+1

ζ ,ζ −1

j j − vk+1

ζ ,ζ

j j − vk+2

.. . ζ ,1

zkj

ζ ,1

ζ ,2

ζ ,ζ

j j j = −vkj − vk+1 − · · · − vk+ζ j −1

In order to solve the system we used the last ζj equations. By applying these results in the first equation we get ζ ,0

vk j

ζ ,1

ζ ,2

ζ ,ζ

j j j j = −vk+1 − vk+2 − · · · − vk+ζ j

or ζj 

ζ ,ρ

j vk+ρ =0

ρ=0

which is a necessary and sufficient condition for the system (23) to have a solution. If (36) does not hold, then the system has no solution. If (36) holds, then the system (23) has the unique solution (37).  Corollary 3.1 If the system (1) has a zero input vector, that is Vk =0m,1 or if P4 Vk =0,1 , then the solution of the system (23) is unique and it is the zero solution ζ

Zk = 0t−h,1

(42)

Proposition 3.5 If P5 Vk is a non-zero vector then the system (24) has no solution. If P5 Vk is the zero vector, then the subsystem (24) has an infinite number of solutions

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that can be taken arbitrary g

Zk = Ck,2

(43)

We can state the following Theorem. Theorem 3.1 Assume the non-homogeneous singular discrete time system (1), with known initial conditions (2) and a singular matrix pencil sF − G. By using the transformation (25) and the matrices P , Q, as described from (18), (19), respectively, the system can be divided into five subsystems, (20), (21), (22), (23), and (24). Let ζ V h+1 ⎢ kζ ⎥ ⎢ V h+2 ⎥ ⎢ k ⎥

P4 Vk = ⎢ ⎢ ⎢ ⎣

⎡

⎤

⎡

.. .

⎥ ⎥ ⎥ ⎦

ζ

and Vk j

ζ

Vk t s

ζ ,0

vk j

⎢ ζ ,1 ⎢v j ⎢ k ⎢ ζ ,2 ⎢ j = ⎢ vk ⎢ ⎢ . ⎢ .. ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

j = h + 1, h + 2, . . . , t

ζ ,ζj

vk j

Then

ζj ζj ,ρ 1. If ρ=0 vk+ρ = 0, j = h + 1, h + 2, . . . , t or if P5 Vk is a non-zero vector then the system (1) has no solutions. ζj ζj ,ρ vk+ρ = 0, j = h + 1, h + 2, . . . , t and P5 Vk is a zero vector then the 2. If ρ=0 solution is unique if and only if the c.m.i. are zero dim Nr (sF − G) = 0

(44)

and Yk0 ∈ colspan Qp − Qq

q ∗ −1

ζ

Hqi P2 Vk0 +i + Qζ Zk0

(45)

i=0

The unique solution is then given by the formula Yk = Qp

p Jpk−k0 Zk0

+

k−1  i=k0

Jpk−i−1 P1 Vi

− Qq

q ∗ −1

ζ

Hqi P2 Vk+i + Qζ Zk

(46)

i=0

ζ

where Zk is given from (37). In any other case the system has infinite solutions.  ζj ζj ,ρ vk+ρ = 0, j = h + 1, h + 2, . . . , t, then from Proposition 3.4 the Proof 1. If ρ=0 subsystem (23) has no solution. If P5 Vk is a non-zero vector then from Proposition 3.5 the subsystems (24) has no solution. Consequently the system (1) has no solutions. ζj ζj ,ρ 2. We assume now that ρ=0 vk+ρ = 0, j = h + 1, h + 2, . . . , t and P5 Vk is a zero vector. Then from the transformation (25) and the solutions (27), (28), (29), (37) and

Circuits Syst Signal Process (2013) 32:1615–1635

1627

(43) of the subsystems (20), (21), (22), (23), and (24), respectively, we have ⎡ Yk = QZk = Qp

Qq

Q

⎢ ⎢

⎢ ⎢ Qg ⎢ ⎢ ⎢ ⎣

Qζ

k−k0

Jp

⎤  p k−i−1 P V Zk0 + k−1 1 i i=k0 Jp ⎥ q∗ −1 i ⎥ − i=0 Hq P2 Vk+i ⎥ ⎥ ⎥ Ck,1 ⎥ ⎥ ζ Zk ⎦ Ck,2

or Yk = Qp

p Jpk−k0 Zk0

+

k−1 

Jpk−i−1 P1 Vi

i=k0

− Qq

q ∗ −1

ζ

Hqi P2 Vk+i + Q Ck,1 + Qζ Zk + Qg Ck,2

i=0

Since Ck,1 and Ck,2 can be taken arbitrary, it is clear that the non-homogeneous singular linear discrete time system (1) for every suitable defined initial condition has an infinite number of solutions. It is clear that the existence of c.m.i. is the reason that the subsystems (22) and consequently (24) exist. These systems as proved in Propositions 3.3 and 3.5 have infinite solutions. Thus a necessary condition for the system to have unique solution is not to have any c.m.i. which is equal to dim Nr (sF − G) = 0 In this case the Kronecker canonical form of the pencil sF − G has the following form: sFK − QK := sIp − Jp ⊕ sHq − Iq ⊕ sFζ − Gζ

(47)

and then the system (1) is divided into the three subsystems (20), (21), (23) with solutions (27), (28), (37), respectively. Thus ⎡ Yk = QZk = Qp

Qq

⎢ Qζ ⎢ ⎣

k−k0

Jp

⎤  p k−i−1 P V Zk0 + k−1 1 i i=k0 Jp ⎥ q∗ −1 i ⎥ − i=0 Hq P2 Vk+i ⎦ ζ

Zk or Yk = Qp

p Jpk−k0 Zk0

+

k−1  i=k0

Jpk−i−1 P1 Vi

− Qq

q ∗ −1 i=0

ζ

Hqi P2 Vk+i + Qζ Zk

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Circuits Syst Signal Process (2013) 32:1615–1635

The solution that exists if and only if p

Yk0 = Qp Zk0 − Qq

q ∗ −1

ζ

Hqi P2 Vk0 +i + Qζ Zk0

i=0

or Yk0 ∈ colspan Qp − Qq

q ∗ −1

ζ

Hqi P2 Vk0 +i + Qζ Zk0



i=0

Theorem 3.2 Consider the system (1), with known initial conditions (2), a singular matrix pencil sF − G and a zero input vector, Vk = 0m,1 . Then its solution is unique if and only if the c.m.i. are zero dim Nr (sF − G) = 0

(48)

Yk0 ∈ colspan Qp

(49)

and

The unique solution is then given from the formula p

Yk = Qp Jpk−k0 Zk0

(50)

p

p

where Zk0 is the unique solution of the algebraic system Yk0 = Qp Zk0 . In any other case the system has infinite solutions. Proof First we consider that the system has non-zero c.m.i and non-zero r.m.i. By using transformation (25), when the input vector Vk is a zero vector then from (27), (28), (29), (37), and (43) the solutions of the subsystems (20), (21), (22), (23), and (24), respectively, are ⎡ p ⎤ ⎡ k−k p ⎤ Zk Jp 0 Zk0 ⎥ ⎢ q⎥ ⎢ ⎢ Zk ⎥ ⎢ 0q,1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (51) Zk = ⎢ Zk ⎥ = ⎢ Ck,1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Zδ ⎥ ⎢ 0 ⎣ k ⎦ ⎣ t−h,1 ⎦ g Zk Ck,2 Then

⎡ Yk = QZk = Qp

p

Qq

Q

Qζ

k−k0

Jp

⎢ ⎢ 0q,1

⎢ ⎢ Qg ⎢ Ck,1 ⎢ ⎢ 0 ⎣ t−h,1 Ck,2

Yk = Qp Jpk−k0 Zk0 + Q Ck,1 + Qg Ck,2

p

Zk0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Circuits Syst Signal Process (2013) 32:1615–1635

1629

Since Ck,1 and Ck,2 can be taken arbitrary, it is clear that the general singular discrete time system for every suitable defined initial condition has an infinite number of solutions. As in Theorem 3.1, it is clear that the existence of c.m.i. is the reason that the systems (22) and consequently (24) exist. These systems as shown in Propositions 3.3 and 3.5 have always infinite solutions. Thus a necessary condition for the system to have unique solution is not to have any c.m.i. which is equal to dim Nr (sF − G) = 0 In this case the Kronecker canonical form of the pencil sF − G has the following form: sFK − QK := sIp − Jp ⊕ sHq − Iq ⊕ sFζ − Gζ

(52)

and then the system (1) is divided into the three subsystems (20), (21), (23) with solutions (27), (28), (37), respectively. Thus ⎡ Yk = QZk = Qp

Qq

⎢ Qζ ⎣

k−k0

Jp

p

Zk0

0q,1

⎤ ⎥ ⎦

0t−h,1 or p

Yk = Qp Jpk−k0 Zk0 The solution that exists if and only if p

Yk0 = Qp Zk0 or Yk0 ∈ colspan Qp In this case the system has the unique solution p

Yk = Qp Jpk−k0 Zk0



4 Numerical Example In [2] we find a description of the Leontief model. The Leontief model is a dynamic model of a multisector economy. It is constructed as follows. Suppose the economy is divided into n sectors. Let Yk have ith component the output in the kth time period of sector i. Let dk have ith component the final demand (demand in the kth time period excluding investment demand). Let fij be the amount of commodity i that sector j must have to produce one unit of commodity j . Let fij be elements of the matrix F . Let aij be the proportion of commodity j that gets transferred to commodity i in the kth time period. Let aij be elements of the matrix A. A is called the Leontief input-out matrix or the matrix of flow coefficients. The Leontief model then says that amount of commodity i is equal to amount of commodity i input by all sectors

1630

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plus amount needed for production of the next output Yk+1 plus amount used to meet noninvestment demand (consumption). In matrix form Yk = AYk + F (Yk+1 − Yk ) + dk

(53)

or if G = A − B − I and Vk = −dk F Yk+1 = GYk + Vk

(54)

4.1 Example 1 Consider the discrete time system (1) with a zero input vector Vk and let ⎡

2 1 1 0 0 0 0

⎤

⎥ 3 1 1 0 0 0⎥ ⎥ 1 2 1 0 0 0⎥ ⎥ ⎥ 1 1 1 0 0 0⎥ ⎥, ⎥ 0 0 0 0 0 0⎥ ⎥ 0 0 0 1 0 0⎥ ⎦ 0 1 0 0 0 0 1

⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢ F =⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎣

⎡

1 1 1 0 0 0 1

⎤

⎥ 3 2 2 0 1 1⎥ ⎥ 2 3 2 0 0 0⎥ ⎥ ⎥ 2 2 2 0 0 0⎥ ⎥ ⎥ 0 0 0 1 0 0⎥ ⎥ 0 0 0 0 0 0⎥ ⎦ 0 0 0 0 0 1 0

⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢ G=⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎣

Then det[sF − G] ≡ 0. The invariants of the pencil are, s − 2, s − 1 the finite elementary divisors, 1 =0, 2 =2 the c.m.i and ζ1 =0, ζ2 =1 are the r.m.i. Then from Theorem 3.2 the solutions of the system are infinite. 4.2 Example 2 Consider the matrix differential equation (5) and let ⎡

1

⎢ ⎢0 ⎢ ⎢1 ⎢ F =⎢ ⎢0 ⎢ ⎢ ⎣1 0

1

1

1

1

1

0

1

1

1

1

1

0

0

1

0

0

1

1

1

⎤

⎥ 1⎥ ⎥ 1⎥ ⎥ ⎥, 1⎥ ⎥ ⎥ 0⎦ 1

⎡

1

⎢ ⎢0 ⎢ ⎢1 ⎢ G=⎢ ⎢0 ⎢ ⎢ ⎣0 1

2

2

1

2

2

0

2

2

2

2

3

1

0

0

0

0

1

0

2

⎤

⎥ 2⎥ ⎥ 3⎥ ⎥ ⎥ 3⎥ ⎥ ⎥ 0⎦ 0

The matrices F, G are non-square. Thus the matrix pencil sF − G is singular with invariants, s − 2, s − 1 the finite elementary divisors, the infinite elementary divisors

Circuits Syst Signal Process (2013) 32:1615–1635

1631

are of degree 1 and ζ1 =0, ζ2 =1 are the r.m.i. There exist non-singular matrices ⎡

1

−1 0 0 0 0

⎤

⎥ 0⎥ ⎥ 1 0 0 0⎥ ⎥ ⎥ 0 0 1 0⎥ ⎥ ⎥ 0 0 0 1⎦ 0 1 0 0

⎢ 1 ⎢ 0 ⎢ ⎢ −1 0 ⎢ P =⎢ ⎢ 0 0 ⎢ ⎢ 0 ⎣ 0 0 −1

0

0

0

and ⎡

0

⎢ 1 ⎢ ⎢ Q=⎢ ⎢ 0 ⎢ ⎣ 1

0

1

1

−1 −1

0

−1

0

−1 −1

−1 0

−1

1

2

⎤

0 ⎥ ⎥ ⎥ 1 ⎥ ⎥ ⎥ 1 ⎦

0

−1

1

such that ⎡

1

⎢ ⎢0 ⎢ ⎢0 ⎢ P F Q = FK = ⎢ ⎢0 ⎢ ⎢ ⎣0 0 ⎡

1

⎢ ⎢0 ⎢ ⎢0 ⎢ P GQ = GK = ⎢ ⎢0 ⎢ ⎢ ⎣0 0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 1⎦ 0

0

0

0

0

2

0

0

0

1

0

0

0

0

0

0

1

0

0

0

with ⎡

0

⎢ 1 ⎢ ⎢ Qp = ⎢ ⎢ 0 ⎢ ⎣ 1

0

⎤

1⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎦

−1 0

0

⎤

0

⎤

⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎦ 1

(55)

1632

Circuits Syst Signal Process (2013) 32:1615–1635

Let the initial values of the system be ⎡

0

⎤

⎢ −1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ Y0 = ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎣ 1 ⎦ −1 Then Y0 ∈ colspan Qp and from Theorem 3.2 the solution of the system is ⎡

0

0

⎤

 1⎥ ⎥ ⎥ 1 0 p ⎥ 0⎥ Z0 k ⎥ 0 2 0⎦

⎢ 1 ⎢ ⎢ Yk = ⎢ ⎢ 0 ⎢ ⎣ 1

−1 0 p

and by calculating Z0 we get ⎡

0

0

⎤

1⎥ ⎥ ⎥ p 0⎥ ⎥ Z0 ⎥ 0⎦

⎢ 1 ⎢ ⎢ Y0 = ⎢ ⎢ 0 ⎢ ⎣ 1

−1 0 or  p Z0

=

1



−2

and the solution of the system is ⎡

0

⎤

⎢ 1 − 2k+1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Yk = ⎢ 0 ⎥ ⎢ ⎥ ⎣ ⎦ 1 −1

(56)

Circuits Syst Signal Process (2013) 32:1615–1635

1633

Next assume the initial conditions ⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Y0 = ⎢ ⎢0⎥ ⎢ ⎥ ⎣1⎦ 1 Then Y0 ∈ / colspan Qp and the solution for every k ≥ 0 is ⎡

0

⎢ 1 ⎢ ⎢ Yk = ⎢ ⎢ 0 ⎢ ⎣ 1 −1

0

⎤

2k ⎥ ⎥ ⎥ 0⎥ ⎥C ⎥ 0⎦ 0

c

where C = c12 is constant and the dimension of the solution vector space of the system is 2. From the results of the above examples we see that by using the explicit and easily testable conditions that are derived from the Theorem 3.2 we see that the existence of unique solution is not automatically satisfied. This is very important because in cases where the solution is not unique a redesign of the system maybe needed.

5 Conclusions In this article, we study a class of linear generalized discrete time systems whose coefficients are either non-square constant matrices or square matrices with an identically zero matrix pencil. Actually the results of the paper apply to general non-square pencils and generalize various results regarded the literature which mainly are dealing with square and non-singular systems. By taking into consideration that the relevant pencil of the system (1) is singular, we decompose the non-homogeneous linear matrix difference equation into five subsystems. Afterwards, we provide necessary and sufficient conditions for existence and uniqueness of solutions. Explicit and easily testable conditions are derived for which the system has a unique solution. Surprisingly this can include cases in which the matrix pencil sF − G is singular. More analytically we analyze the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Another important characteristic of the general case considered here is that existence of solutions is not automatically satisfied. This is very important for many

1634

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applications for which the model is significant only for certain range of its parameters. In this cases a careful interpretation of results or even a redesign of the system maybe needed. Finally we provide some numerical examples based on a singular discrete time real dynamical system, to justify our theory. Acknowledgements

We are very grateful to the anonymous referees.

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