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Systems & Control Letters 53 (2004) 407 – 414

www.elsevier.com/locate/sysconle

Positive N -periodic descriptor control systems Bego˜na Cant&o∗ , Carmen Coll, Elena S&anchez Departament de Matematica Aplicada, Universitat Politecnica de Valencia, 46071 Valencia, Spain Received 1 October 2002; received in revised form 11 August 2003; accepted 29 May 2004

Abstract This paper establishes the relationship between the positive N -periodic descriptor system in discrete-time and its associated invariant systems. Reachability and controllability properties of these kinds of systems are analyzed. Transmission of structural properties from the N -periodic system to their associated invariant systems is studied. Finally, some comments on the stability property are made. c 2004 Elsevier B.V. All rights reserved. Keywords: Descriptor control system; Reachability; Controllability; Stability; Positive periodic systems

1. Introduction Some types of systems associated with impulsive behavior called descriptor systems can appear in biological phenomena, in economics as in the Leontief dynamic model [25], in electrical [9] and in mechanical models [21], etc., where impulses are caused by the singular structure of the systems or by the inconsistent initial conditions. Descriptor systems are also called semi-state systems, di;erential-algebraic systems, singular systems or generalized state-space systems. In some descriptor systems, the state variables are concentrations, voltage or populations, and the trajectories generated by these states must be nonnegative, these models are called positive systems. Positive invariant standard systems have been discussed in the literature. Related results on reachability ∗

Supported by Spanish Grant DGI BFM2001-2783. Corresponding author. E-mail address: [email protected] (B. Cant&o).

and observability properties have been obtained in [7,12]. The relationship between positiveness and stability for invariant systems has been studied in [22] and the positive realization problem has been dealt with in [8,14,15]. Positive invariant descriptor systems have been recently studied in [5]. A survey of positive systems can be found in [19]. The analysis and design of digital, or discrete-time, models is interesting in practical engineering problems because some control laws are usually implemented using digital processors. Discretization and sample data of systems are common in control. In this way, a periodic discrete-time model (see [1,2]) is obtained by discretizing a continuous system. The technique used for that discretization requires the implementation of a multirate digital control using suitable sampling periods. Based on these previous works, we focus our attention in the positive N -periodic descriptor control systems. In particular, we study the N -periodic forward–backward control system (E(·), A(·), B(·))N

c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.05.017

408

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

given by E(k)x(k + 1) = A(k)x(k) + B(k)u(k);

(1)

where E(k) = diag[In1 ; N (k)]; B(k) = [B1T (k)B2T (k)]

A(k) = diag[A1 (k); In2 ];

T

with A1 (k + N ) = A1 (k) ∈ Rn1 ×n1 , N (k + N ) = N (k) ∈ Rn2 ×n2 , B1 (k + N ) = B1 (k) ∈ Rn1 ×m , B2 (k + N ) = B2 (k) ∈ Rn2 ×m , k ∈ Z+ , N ∈ Z+ . When N = 1, this system is an invariant system. If B(k) = 0, the system is called autonomous. System (1) can be separated in two parts: the dynamic part called forward subsystem, x1 (k + 1) = A1 (k)x1 (k) + B1 (k)u(k);

k ∈ Z+

(2)

k ∈ Z+ :

(3)

and the backward subsystem, N (k)x2 (k + 1) = x2 (k) + B2 (k)u(k);

This structure enables, see [13], systems (2)–(3) to be considered as a N -periodic realization of a N -periodic collection of nonproper rational matrices. We can provide some real applications where positive descriptor N -periodic systems can appear. One application is in the study of the well-known Leontief economic model, see [16,25]. In this case, the interrelationships among di;erent industrial sectors are described. When the capital and the demand are not constant through an annual cycle, for instance, they are variable depending on the season, then it appears an N -periodic Leontief model. This model is given by the descriptor system C(k)x(k + 1) = (I − P(k) + C(k))x(k) − D(k)u(k); where C(k + N ) = C(k) is the capital coeHcient matrix at time k, P(k + N ) = P(k), is the technological coeHcient matrix at k, D(k + N ) = D(k), is the demand coeHcient matrix (excluding investment) at time k, x(k) is the production level vector and u(k) is the demand level vector. Since the Leontief model is an economic model, C(k) ¿ 0, P(k) ¿ 0 and D(k) ¿ 0 that is, all the entries of these matrices are nonnegative. This kind of system is a descriptor system because the capital coeHcient matrix C(k) may be singular. The entries of the coeHcient capital matrix are the required capital per unit of production per sector. The

singularity of this matrix arises because no output from one sector is used in the production of some products. In the particular case that the model has two di;erent compartmental sectors at time k: the Jrst includes only the star products and the second the uninteresting products, the capital matrix has a nilpotent submatrix, because the uninteresting situation has to Jnish in a Jnite period of time. The system is N -periodic because the situation of star good can be changed into uninteresting good or new goods can be introduced in some season through annual cycle. We are considering N -periodic changes. The new N -periodic Leontief dynamic model, in this case is given by E1 (k) O x(k + 1) O N (k) In1 + E1 (k) O = x(k) O In2 + N (k) D1 (k) − u(k): D2 (k) Moreover, descriptor N -periodic systems can appear in the study of singularly perturbed systems, see [20,23,24]. This perturbed system, in the discrete-time case, can be given by x(k + 1) = A11 x(k) + A11 z(k) + B1 u(k); z(k + 1) = A21 x(k) + A22 z(k) + B2 u(k);

(4)

where x(k) and z(k) are n1 and n2 dimensional state vectors, respectively, u(k) is an m-dimensional control vector, and is a small positive parameter. This parameter tends to zero. System (4) can be considered as a descriptor system. There are several references on singularly perturbed systems with many applications to electrical and electronics, agricultural engineering, ecology and biology, see [23]. In particular, in [3] for modeling the dynamics of nitrogen absorption, distribution and translocation in citrus trees used a positive N -periodic compartmental system. The presence of some parameters such as small climatic changes or the existence of impurities in the soil generate a singular perturbation in the model. In this case, the singularly perturbed system will be a positive descriptor N -periodic system.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

The main aim of this work is to give some answers on the positive reachability and controllability properties of N -periodic forward–backward control systems (1). This problem has been discussed in [4,6] for positive N -periodic standard systems. Although, in the minimal realization problem the state spaces have minimal varying dimension at each time (see, for example [11,17] for the standard system and [13] for the forward–backward system), however, to study the transmission of structural properties from the N -periodic system to its associated invariant systems, it is not necessary to consider time-varying dimension for state spaces (see [6,18]). The work is structured as follows. In Section 2, we use the reachability cones to give a geometrical characterization of the positive reachability property, for system (1). In Section 3, we deJne the stability cones and we analyze the stability of the positive system using these cones. First, we study the solution of system (1). To achieve this, we deJne A1 (k; k0 ) = A1 (k − 1)A1 (k − 2) · · · A1 (k0 ), k ¿ k0 , A1 (k0 ; k0 ) = I , and N (k; k0 ) = N (k)N (k + 1) · · · N (k0 − 1), k ¡ k0 , N (k0 ; k0 ) = I . Note that, due to the periodicity of the system, we need only to consider the N -periodic matrices at time s, s = 0; 1; : : : ; N − 1. The matrices A1; s = A1 (s + N; s) and Ns = N (s; s + N ) are called forward monodromy and backward monodromy matrices, respectively. Assuming that the monodromy matrices, Ns , s = 0; 1; : : : ; N − 1, are nilpotent, there exists an integer h ∈ Z such that the general solution of systems (2)– (3) is given by the following expression: 0 In1 x2 (k) x1 (k) + x(k) = 0 I n2 =

In1

−

I n2

k+h−1

N (k; j)B2 (j)u(j);

(6)

where H (s) = diag[In1 ; O] and Hi (s) = diag[O; In2 ] N (s; s + i)B2 (s + i), i = 0; : : : ; h − 1. Note that, the integer h ∈ Z in the trajectory of the system depends on the indices of backward monodromy matrices. So in the next result, we give the relationship among the nilpotent indices of these matrices, and using this result we obtain that h = qN , where q = m&ax{ind(Ns ); ∀s = 0; 1; : : : ; N − 1}, with ind(Ns ) denoting the nilpotence index of Ns . Proposition 1. Let qt and qs be the nilpotence index of the backward monodromy matrices Nt and Ns , respectively. Then |qs − qt | 6 1, ∀s; t = 0; 1 : : : ; N − 1. Proof. Without loss of generality, we prove the result for the index qo and qs of the matrices N0 and Ns , respectively. It suHces to prove that qo − 1 6 qs 6 qo + 1: For that, we prove that if N0qo = 0 then (a) Nsqo +1 = 0

and

(b) Nsqo −2 = 0:

(a) By deJnition we have Nsqo +1 = (

N (s; s

+ N ))qo +1 =

N (s; s

+ (qo + 1)N )

= N (s) · · · N (N − 1)N (N ) · · · N (qo N + N − 1)N (qo N + N ) · · · N (qo N + N + s − 2)N (qo N +N +s−1) = N (s) · · · N (N − 1)

N (0; qo N )N ((qo

+ 1)N )

· · · N ((qo + 1)N + s − 1): As N0q0 = 0, then Nsqo +1 = 0. (b) We suppose that Nsqo −2 = 0, and we develop qo −1 N0 ,

j=s

0

X0 (s) = Im[H (s); H0 (s); : : : ; Hh−1 (s)];

= N (s) · · · N (N − 1)N0q0 N ((qo + 1)N )

k−1 + A1 (k; j + 1)B1 (j)u(j)

And the set of initial conditions X0 (s), s = 0; 1; : : : ; N − 1, for system (1) is

· · · N ((qo + 1)N + s − 1)

A1 (k; s)x1 (s)

0

409

k ¿ s:

N0qo −1 = (

j=k

(5)

qo −1 N (0; N ))

=

N (0; (qo

= N (0) · · · N (s − 1)N (s)

− 1)N )

410

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

· · · N (qo N − 2N + s − 1)N (qo N − 2N + s) · · · N (qo N − N − 1) = N (0) · · · N (s − 1)

N (s; (q0

− 2)N + s)

Now, we consider the N -periodic descriptor system (1) with nonnegative restrictions.

N ((qo − 2)N + s) · · · N ((qo − 1)N − 1) = N (0) · · · N (s − 1)Nsq0 −2 N ((qo − 2)N + s) · · · N ((qo − 1)N − 1): As Nsq0 −2 =0 then N0qo −1 =0. That is, ind(N0 ) 6 qo −1, but we have that ind(N0 ) = q0 then we arrive at a contradiction.

For each s ∈ Z, system (1) has associated a forward–backward invariant linear system, (Es ; As ; Bs ), see [13], given by x1; s (k + 1) x1; s (k) Es = As + Bs us (k); (7) x2; s (k + 1) x2; s (k) with x1; s (k)=x1 (s+kN ), x2; s (k)=x2 (s+kN ), us (k)= col[u(s + kN ); : : : ; u(s + kN + N − 1)], and I n1 O A1; s O Es = ; As = ; O Ns O In 2 B1; s Bs = ; B2; s where N (s; s

+ N ) ∈ Rn2 ×n2 ;

B1; s = row[A1 (s + N; s + j + 1) −1 ×B1 (s + j)]Nj=0 ∈ Rn1 ×mN ;

B2; s = row[

N (s; s

−1 + j)B2 (s + j)]Nj=0 ∈ Rn2 ×mN :

For each s=0; 1; : : : ; N −1, the set of initial conditions X0; s at time k = 0, for system (7), (Es ; As ; Bs ), is the subspace X0; s = Im[Hs ; H0; s ; : : : ; Hh−1; s ];

Denition 1. System (1) is a positive N -periodic descriptor system when for each s = 0; 1; : : : ; N − 1, for each nonnegative initial state x(s) = x0; s ∈ X0 (s) ∩ Rn+ and for each nonnegative control sequence u(k) ¿ 0, k ¿ s, the state trajectory belongs to Rn+ . A characterization of positive descriptor systems is given in [10] by the following proposition. Proposition 3. Consider an N -periodic forward –backward system (E(·), A(·), B(·))N and q = m&ax{ind(Ns ); s = 0; 1; : : : ; N − 1}. The system is positive, if and only if, A1 (k) ¿ 0, B1 (k) ¿ 0, N (k; j)B2 (j) 6 0, k ∈ Z, and j = k; k + 1; : : : ; k + qN − 1. Note that when N = 1, a forward–backward invariant system (E; A; B) is positive, if and only if, A1 ¿ 0; B1 ¿ 0, and N i B2 6 0, i = 0; 1; : : : ; q − 1 (see [5]). The positiveness property of the N -periodic system is transmitted to its associated invariant systems in the following proposition. Proposition 4. If an N -periodic forward–backward system (E(·), A(·), B(·))N is positive then its associated invariant systems (Es ; As ; Bs ) are positive for all s = 0; 1; : : : ; N − 1.

A1; s = A1 (s + N; s) ∈ Rn1 ×n1 ; Ns =

Proposition 2. Consider an N -periodic forward– backward system (E(·), A(·), B(·))N . For each s = 0; 1; : : : ; N − 1, X0; s = X0 (s).

(8)

Hi; s = diag[O; In2 ]Nsi B2; s ,

where Hs = diag[In1 ; O] and i = 0; : : : ; qs − 1. Next, we record a technical result given in (see [10]).

Proof. Since the N -periodic forward–backward system (E(·), A(·), B(·))N is positive from Proposition 3, A1 (k) ¿ 0, and B1 (k) ¿ 0, k ∈ Z. Then, by construction of A1; s and B1; s given in (7), we have A1; s ¿ 0; B1; s ¿ 0, s ∈ Z and thus the associated forward invariant linear subsystem (In1 ; A1; s ; B1; s ) is positive. Now, we consider the backward invariant linear subsystem (Ns ; In2 ; B2; s ). By Proposition 3, N (k; j)B2 (j) 6 0, k ∈ Z, j = k; k + 1; : : : ; k + qN − 1 and by Proposition 2, we have Nsi B2; s 6 0, for all i = 0; 1; : : : ; q − 1 and s ∈ Z. Thus, the associated invariant systems (Es ; As ; Bs ) are positive.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

In the following example we show that a forward –backward N -periodic system (E(·); A(·); B(·))N is not positive although its associated invariant systems (Es ; As ; Bs ) are positive, s = 0; 1; : : : ; N − 1. Example 1. We consider a 2-periodic forward–backward system, (E(·); A(·); B(·))2 with E(0) = E(1) = I3 , −1 0 −2 −1 −1 A1 (0) = 0 6 0; −3 −1 −2 1 2 −3 0 A1 (1) = −5 −2 ; −2 −3 −1 and the matrices B1 (k), B2 (k) and N (k) such that B1 (k) ¿ 0 and N (k; j)B2 (j) 6 0, k ∈ Z, j = k; k + 1; : : : ; k + qN − 1, with q = max{ind(Ns ); s = 0; 1; : : : ; N − 1}. We observe that this system is not positive because of A1 (0) and A1 (1) are not nonnegative matrices. If we consider the invariant associated systems we have 8 1 2 A1; 0 = A1 (2; 0) = A1 (1)A1 (0) = 5 2 12 5 4 9 and

3

A1; 1 = A1 (3; 1) = A1 (2)A1 (1) = 7 6

4 5 2

5

1 : 11

Nsi B2; s

Moreover, B1; s ¿ 0, 6 0, i = 1; : : : ; q − 1. Thus, we have the invariant associated systems (Es ; As ; Bs ) are positive. 2. Structural properties In this section we study the reachability and controllability properties of forward–backward positive systems. We want to learn if there exists a nonnegative control sequence transferring the system from a nonnegative initial state to a nonnegative Jnal state. First, we give the following deJnition.

411

Denition 2. Consider the system (E(·); A(·); B(·))N ¿ 0 and the set of admissible initial conditions X0 (s), for any s = 0; 1; : : : N − 1. (i) The state x ∈ Rn+ is positively reachable at time s, if there exists a time k ∈ Z+ and a nonnegative control sequence u(j) ¿ 0, j = s; s + 1; : : : ; k + s + qN − 1, transferring the state of the system from the origin at time s, x(s) = 0, to x in k steps. The system is positively reachable at time s if every x ∈ Rn+ is positively reachable at time s. The system is positively reachable if it is positively reachable at time s, for all s = 0; 1; : : : ; N − 1. (ii) The state xf ∈ Rn+ is positively controllable at time s for all s = 0; 1; : : : ; N − 1, if any Jnal state xf ∈ Rn+ can be reachable from any initial state x(s) ∈ X0 (s) by a Jnite control sequence. The system is positively reachable at time s if every xf ∈ Rn+ is positively reachable at time s, for all s=0; 1; : : : ; N −1. (iii) In this last case, when xf = 0, the system is called positively null-controllable. To study these properties, it is essential to introduce reachability cones of N -periodic forward– backward systems. For that, given the N -periodic forward–backward system (1), (E(·); A(·), B(·))N , the following matrices are used in the reachability property when we consider all reachable states in kN steps. (i) For each s, the set of all reachable states at time s in kN steps of the forward subsystem (In1 , A1 (·), B1 (·))N , is the cone RfkN (In1 ; A1 (·); B1 (·); s) which is generated by the column of the reachable matrix RfkN (In1 ; A1 (·); B1 (·); s) =row[A1 (s; s − j + 1)B1 (s − j)]kN j=1 :

(9)

(ii) If we observe the construction of the solution to the N -periodic system, we see that the part corresponding to the backward subsystem has qN terms. Then, we deJne the cone Rb (N (·); In2 ; B2 (·); s) where Rb (N (·); In2 ; B2 (·); s) = row["N (s; s + j − 1)B2 (s + j − 1)]qN j=1 ;

(10)

is the reachability matrix associated with the backward subsystem (N (·), In2 , B2 (·))N at time s. Hence, we deJne the reachability matrix at time s; of the N -periodic forward–backward system in kN

412

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

steps as the block diagonal matrix RkN (E(·); A(·); B(·); s) = diag[RfkN (In1 ; A1 (·); B1 (·); s); Rb (N (·); In2 ; B2 (·); s)]:

(11)

Thus, we can establish the following result. Proposition 5. The system (E(·); A(·), B(·))N positively reachable at time s, if and only if ∞

RkN (In1 ; A1 (·); B1 (·); s) = Rn+ ;

or equivalently

k=1

RfkN (In1 ; A1 (·); B1 (·); s) = Rn+1

and

j=s

is

k=1

∞

then there exist k ∈ Z and a control sequence u(j) ¿ 0, j = s; : : : ; k + s + qN − 1 such that x(k + s) = 0, that is, I n1 A1 (k + s; s)x1 (s) 0= 0 k+s−1 + A1 (k + s; j + 1)B1 (j)u(j)

Rb (N (·); In2 ; B2 (·); s) = Rn+2 :

If the system is positively null-controllable, we can give the following result. Proposition 6. Consider the N -periodic forward– backward system (E(·); A(·), B(·))N ¿ 0. The system is positively null-controllable, if and only if, A1; s is a nilpotent matrix, for all s ∈ Z. Proof. If we consider x(s) ∈ X0 (s), u(j) = 0, for all j = s; : : : ; s + k + qN − 1, the solution of the system is given by I n1 x(k + s) = A1 (k + s; s)x1 (s); k ¿ 0: 0 Now, we suppose that A1; s are nilpotent matrices, s ∈ Z and % = max{ind(A1; s )}. If k ¿ %N , we have A1 (k + s; s) =A1 (k + s − 1) · · · A1 (%N + s)A1 (s + %N; s) =A1 (k + s − 1) · · · A1 (%N + s)A1; s % = 0: Thus, x(k + s) = 0, and the system is positively null-controllable. Conversely, if the positive system (E(·); A(·); B(·))N is positively null-controllable, for any x(s) ∈ X0 (s),

−

0 In2

k+s+qN −1

N (k+s; j)B2 (j)u(j);

k¿s:

j=k

Since the system is positive, the three terms of the above expression are positive or zero. In this case, each term is zero because the Jnal state is zero. In particular, choosing the canonical vectors as initial state, x1 (s) = ei , we have In1 0= A1 (k + s; s)x1 (s) 0 In 1 = A1 (k + s; s)ei : 0 Thus, A1 (k + s; s) = 0. For each s, we take %s ∈ Z, such that %s ¿ k, then A%1;s s = A1 (%s N + s; s) = 0. Then A1; s is a nilpotent matrix for all s ∈ Z. The relationship between reachability matrices of the N -periodic system and its associated invariant systems is given in the following result. The proofs are rather technical. Proposition 7. The reachability matrices given in (9)–(11) satisfy the following relations, for all s ∈ Z: 1. RfkN (In1 ; A1 (·); B1 (·); s) = Rfk (In1 ; A1s ; B1s ), k ∈ Z, 2. Rb (N (·); In2 ; B2 (·); s) = Rb (Ns ; In2 ; B2s ), 3. RkN (E(·); A(·); B(·); s) = Rk (Es ; As ; Bs ), where Rfk (In1 ; A1s ; B1s ), Rb (Ns ; In2 ; B2s ), and Rk (Es ; As ; Bs ) are the reachability matrices of the associated invariant systems. Thus, we have the following relation between the structural properties of the N -periodic system and its associated invariant systems.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

Theorem 8. Consider the N -periodic forward–backward system (E(·), A(·), B(·))N ¿ 0. (a) The system is positively reachable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively reachable for any s ∈ Z. (b) The system is positively controllable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively controllable for any s ∈ Z. (c) The system is positively null-controllable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively null-controllable for any s ∈ Z. Proof. Parts (a) and (b) are consequence of Proposition 7 and part (c) follows from Proposition 6. 3. On stability

Next, we deJne the stability cones associated with the positive N -periodic system as S(E(·); A(·); s) = {x ∈ Rn+1 : [In1 − A1 (s)]x ¿ 0}; for all s ∈ Z. And, we deJne the stability cone associated with the corresponding invariant system as S(Es ; As ) = {x ∈ Rn+1 : [In1 − A1; s ]x ¿ 0}; s = 0; 1; : : : ; N − 1: A condition for the stability of the positive N periodic forward–backward system is given in the following result. Theorem 9. Consider the positive N -periodic forward–backward system (12), (E(·); A(·))N . If there

N −1 exists a vector e ∈ s=0 int(S(E(·); A(·); s)), then the system is asymptotically stable. Proof. If A1 (s) ¿ 0 and there exists a vector

N −1 e ∈ s=0 int(S(E(·); A(·); s)), that is, there exists e ¿ 0 satisfying

In this section we study the stability property associated with a positive N -periodic system. Consider the autonomous positive N -periodic forward–backward system

then, we have that

E(k)x(k + 1) = A(k)x(k):

e ¿ A1 (N − 1)e ¿ A1 (N − 1)A1 (N − 2)e ¿

(12)

We shall analyze this property and its relationship with the stability of its associated positive invariant systems by constructing some stability cones. First, we recall the deJnition of asymptotic stability. The positive N -periodic system (12) is asymptotically stable, if its free response satisJes lim x(k) = 0:

k→∞

Using the fact that the monodromy matrices Ns ; s ∈ Z, are nilpotent, we have that the positive N -periodic system (E(·); A(·))N is asymptotically stable, if and only if, its positive N -periodic forward subsystem (In1 ; A1 (·)) is asymptotically stable, that is, if A1; s is a stable matrix, ∀' ∈ ((A1; s )|'| ¡ 1. The stability of the N -periodic system (12), (E(·); A(·))N , is transmitted to its associated invariant systems (Es ; As ), for all s ∈ Z, and reciprocally. This is due to the fact that the relationship between the state of the N -periodic system and the state of sth invariant system is given by xs (k) = x(s + kN ), for all k ¿ s, s ∈ Z.

413

e ¿ A1 (0)e; e ¿ A1 (1)e; : : : ; e ¿ A1 (N − 1)e;

· · · ¿ A1 (N; 0)e = A1; 0 e: Thus, e ∈ int(S(E0; A0 )). Using the periodicity of the matrices we obtain e ∈ S(Es ; As ), for all s = 0; 1; : : : ; N − 1, that is [In1 − A1; s ]e ¿ 0. Consider the Perron root * ¿ 0 of A1; s and its associated eigenvector v ¿ 0, we have that, e ¿ A1; s e; vT e ¿ *vT e; (1 − *)vT e ¿ 0: Thus, * ¡ 1, and A1; s is a stable matrix. Hence, the N -periodic system is asymptotically stable. 4. Conclusions In this paper positive N -periodic forward–backward systems are considered. The structural properties of this kind of system, the positive reachability and controllability properties, are discussed. Using a

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geometrical approach, a relationship between the structural properties of the system and its associated invariant systems is given. In this way, the reachability cones are constructed. Finally, some results about the stability property are given. More speciJcally, a geometrical condition is stabilized to assure the asymptotic stability of the N -periodic system. References [1] P. Albertos, Block multirate input–output model for sampled-data control systems, IEEE Trans. Automat. Control 35 (9) (1990) 1085–1088. [2] M. Bidani, M. Djemai, A multirate digital control via a discrete-time observer for non-linear singularly perturbed continuous-time systems, Internat. J. Control 75 (8) (2002) 591–613. [3] R. Bru, R. Cant&o, B. Ricarte, Modelling nitrogen dynamics in citrus trees, Math. Comput. Modelling 38 (2003) 975–987. [4] R. Bru, C. Coll, V. Hern&andez, E. S&anchez, Geometrical conditions for the reachability and realizability of positive periodic discrete systems, Linear Algebra Appl. 256 (1997) 109–124. [5] R. Bru, C. Coll, E. S&anchez, Structural properties of positive linear systems time-invariant di;erence-algebraic equations, Linear Algebra Appl. 349 (2002) 1–10. [6] R. Bru, V. Hern&andez, Structural properties of discrete-time linear positive periodic systems, Linear Algebra Appl. 121 (1989) 171–183. [7] R. Bru, S. Romero, E. S&anchez, Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl. 310 (2000) 49–71. [8] L. Caccetta, V.G. Rumchev, Reachable discrete-time positive systems with minimal dimension control sets, Dynamics Continuous, Discrete Impulsive Systems 4 (1998) 539–552. [9] S.L. Campbell, Singular Systems of Di;erential Equations, Pitman, London, 1980. [10] B. Cant&o, C. Coll, E. S&anchez, On feedbacks for positive discrete-time singular systems, in: Preprints 15th IFAC World Congress, 2002, pp. 1–6.

[11] P. Colaneri, S. Longhi, The realization problem for linear periodic systems, Automatica 31 (1995) 775–779. [12] P.G. Coxon, H. Shapiro, Positive input reachability and controllability of positive systems, Linear Algebra Appl. 94 (1987) 35–53. [13] V. Estruch, V. Hern&andez, E. S&anchez, C. Coll, Forward– backward periodic realizations of nonproper rational matrices, J. Math. Systems, Estimation Control 8 (2) (1998) 155–175. [14] L. Farina, On the existence of a positive realization, Systems Control Lett. 28 (1996) 219–226. [15] L. Farina, L. Benvenuti, Positive realizations of linear systems, Systems Control Lett. 26 (1995) 1–9. [16] G. Gandolfo, Economic Dynamics: Methods and Models, Elsevier, New York, 1985. [17] I. Gohberg, M.A. Kaashoek, J. Kos, ClassiJcation of linear di;erence equations under periodic or kinematic similarity, SIAM J. Matrix Anal. Appl. 21 (1999) 481–507. [18] V. Hern&andez, A. Urbano, Pole-assignment problem for discrete-time linear periodic systems, Internat. J. Control 46 (2) (1987) 687–697. [19] T. Kaczorek, Positive 1D and 2D Systems, Springer, London, 2002. [20] P. Kokotovic, H.K. Khalil, J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, SIAM, Philadelphia, 1999. [21] P.C. MRuller, Linear mechanical descriptor systems: identiJcation, analysis and design, in: Preprints of IFAC Conference on Control of Independent Systems, Belfort, France, 1997, pp. 501–506. [22] S. Muratori, S. Rinaldi, Excitability, stability, and sign of equilibria in positive linear systems, Systems Control Lett. 16 (1991) 59–63. [23] D.S. Naidu, Singular perturbations and time scales in control theory and applications: an overview, Dynamics Continuous, Discrete Impulsive Systems, Ser. B 9 (2002) 233–278. [24] G. Obinata, B.D.G. Anderson, Model Reduction for Control System Design, Springer, Berlin, 2001. [25] M.S. Silva, T.P. de Lima, Looking for nonnegative solutions of a Leontief dynamic model, Linear Algebra Appl. 364 (2003) 281–316.

Lihat lebih banyak...
Systems & Control Letters 53 (2004) 407 – 414

www.elsevier.com/locate/sysconle

Positive N -periodic descriptor control systems Bego˜na Cant&o∗ , Carmen Coll, Elena S&anchez Departament de Matematica Aplicada, Universitat Politecnica de Valencia, 46071 Valencia, Spain Received 1 October 2002; received in revised form 11 August 2003; accepted 29 May 2004

Abstract This paper establishes the relationship between the positive N -periodic descriptor system in discrete-time and its associated invariant systems. Reachability and controllability properties of these kinds of systems are analyzed. Transmission of structural properties from the N -periodic system to their associated invariant systems is studied. Finally, some comments on the stability property are made. c 2004 Elsevier B.V. All rights reserved. Keywords: Descriptor control system; Reachability; Controllability; Stability; Positive periodic systems

1. Introduction Some types of systems associated with impulsive behavior called descriptor systems can appear in biological phenomena, in economics as in the Leontief dynamic model [25], in electrical [9] and in mechanical models [21], etc., where impulses are caused by the singular structure of the systems or by the inconsistent initial conditions. Descriptor systems are also called semi-state systems, di;erential-algebraic systems, singular systems or generalized state-space systems. In some descriptor systems, the state variables are concentrations, voltage or populations, and the trajectories generated by these states must be nonnegative, these models are called positive systems. Positive invariant standard systems have been discussed in the literature. Related results on reachability ∗

Supported by Spanish Grant DGI BFM2001-2783. Corresponding author. E-mail address: [email protected] (B. Cant&o).

and observability properties have been obtained in [7,12]. The relationship between positiveness and stability for invariant systems has been studied in [22] and the positive realization problem has been dealt with in [8,14,15]. Positive invariant descriptor systems have been recently studied in [5]. A survey of positive systems can be found in [19]. The analysis and design of digital, or discrete-time, models is interesting in practical engineering problems because some control laws are usually implemented using digital processors. Discretization and sample data of systems are common in control. In this way, a periodic discrete-time model (see [1,2]) is obtained by discretizing a continuous system. The technique used for that discretization requires the implementation of a multirate digital control using suitable sampling periods. Based on these previous works, we focus our attention in the positive N -periodic descriptor control systems. In particular, we study the N -periodic forward–backward control system (E(·), A(·), B(·))N

c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.05.017

408

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

given by E(k)x(k + 1) = A(k)x(k) + B(k)u(k);

(1)

where E(k) = diag[In1 ; N (k)]; B(k) = [B1T (k)B2T (k)]

A(k) = diag[A1 (k); In2 ];

T

with A1 (k + N ) = A1 (k) ∈ Rn1 ×n1 , N (k + N ) = N (k) ∈ Rn2 ×n2 , B1 (k + N ) = B1 (k) ∈ Rn1 ×m , B2 (k + N ) = B2 (k) ∈ Rn2 ×m , k ∈ Z+ , N ∈ Z+ . When N = 1, this system is an invariant system. If B(k) = 0, the system is called autonomous. System (1) can be separated in two parts: the dynamic part called forward subsystem, x1 (k + 1) = A1 (k)x1 (k) + B1 (k)u(k);

k ∈ Z+

(2)

k ∈ Z+ :

(3)

and the backward subsystem, N (k)x2 (k + 1) = x2 (k) + B2 (k)u(k);

This structure enables, see [13], systems (2)–(3) to be considered as a N -periodic realization of a N -periodic collection of nonproper rational matrices. We can provide some real applications where positive descriptor N -periodic systems can appear. One application is in the study of the well-known Leontief economic model, see [16,25]. In this case, the interrelationships among di;erent industrial sectors are described. When the capital and the demand are not constant through an annual cycle, for instance, they are variable depending on the season, then it appears an N -periodic Leontief model. This model is given by the descriptor system C(k)x(k + 1) = (I − P(k) + C(k))x(k) − D(k)u(k); where C(k + N ) = C(k) is the capital coeHcient matrix at time k, P(k + N ) = P(k), is the technological coeHcient matrix at k, D(k + N ) = D(k), is the demand coeHcient matrix (excluding investment) at time k, x(k) is the production level vector and u(k) is the demand level vector. Since the Leontief model is an economic model, C(k) ¿ 0, P(k) ¿ 0 and D(k) ¿ 0 that is, all the entries of these matrices are nonnegative. This kind of system is a descriptor system because the capital coeHcient matrix C(k) may be singular. The entries of the coeHcient capital matrix are the required capital per unit of production per sector. The

singularity of this matrix arises because no output from one sector is used in the production of some products. In the particular case that the model has two di;erent compartmental sectors at time k: the Jrst includes only the star products and the second the uninteresting products, the capital matrix has a nilpotent submatrix, because the uninteresting situation has to Jnish in a Jnite period of time. The system is N -periodic because the situation of star good can be changed into uninteresting good or new goods can be introduced in some season through annual cycle. We are considering N -periodic changes. The new N -periodic Leontief dynamic model, in this case is given by E1 (k) O x(k + 1) O N (k) In1 + E1 (k) O = x(k) O In2 + N (k) D1 (k) − u(k): D2 (k) Moreover, descriptor N -periodic systems can appear in the study of singularly perturbed systems, see [20,23,24]. This perturbed system, in the discrete-time case, can be given by x(k + 1) = A11 x(k) + A11 z(k) + B1 u(k); z(k + 1) = A21 x(k) + A22 z(k) + B2 u(k);

(4)

where x(k) and z(k) are n1 and n2 dimensional state vectors, respectively, u(k) is an m-dimensional control vector, and is a small positive parameter. This parameter tends to zero. System (4) can be considered as a descriptor system. There are several references on singularly perturbed systems with many applications to electrical and electronics, agricultural engineering, ecology and biology, see [23]. In particular, in [3] for modeling the dynamics of nitrogen absorption, distribution and translocation in citrus trees used a positive N -periodic compartmental system. The presence of some parameters such as small climatic changes or the existence of impurities in the soil generate a singular perturbation in the model. In this case, the singularly perturbed system will be a positive descriptor N -periodic system.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

The main aim of this work is to give some answers on the positive reachability and controllability properties of N -periodic forward–backward control systems (1). This problem has been discussed in [4,6] for positive N -periodic standard systems. Although, in the minimal realization problem the state spaces have minimal varying dimension at each time (see, for example [11,17] for the standard system and [13] for the forward–backward system), however, to study the transmission of structural properties from the N -periodic system to its associated invariant systems, it is not necessary to consider time-varying dimension for state spaces (see [6,18]). The work is structured as follows. In Section 2, we use the reachability cones to give a geometrical characterization of the positive reachability property, for system (1). In Section 3, we deJne the stability cones and we analyze the stability of the positive system using these cones. First, we study the solution of system (1). To achieve this, we deJne A1 (k; k0 ) = A1 (k − 1)A1 (k − 2) · · · A1 (k0 ), k ¿ k0 , A1 (k0 ; k0 ) = I , and N (k; k0 ) = N (k)N (k + 1) · · · N (k0 − 1), k ¡ k0 , N (k0 ; k0 ) = I . Note that, due to the periodicity of the system, we need only to consider the N -periodic matrices at time s, s = 0; 1; : : : ; N − 1. The matrices A1; s = A1 (s + N; s) and Ns = N (s; s + N ) are called forward monodromy and backward monodromy matrices, respectively. Assuming that the monodromy matrices, Ns , s = 0; 1; : : : ; N − 1, are nilpotent, there exists an integer h ∈ Z such that the general solution of systems (2)– (3) is given by the following expression: 0 In1 x2 (k) x1 (k) + x(k) = 0 I n2 =

In1

−

I n2

k+h−1

N (k; j)B2 (j)u(j);

(6)

where H (s) = diag[In1 ; O] and Hi (s) = diag[O; In2 ] N (s; s + i)B2 (s + i), i = 0; : : : ; h − 1. Note that, the integer h ∈ Z in the trajectory of the system depends on the indices of backward monodromy matrices. So in the next result, we give the relationship among the nilpotent indices of these matrices, and using this result we obtain that h = qN , where q = m&ax{ind(Ns ); ∀s = 0; 1; : : : ; N − 1}, with ind(Ns ) denoting the nilpotence index of Ns . Proposition 1. Let qt and qs be the nilpotence index of the backward monodromy matrices Nt and Ns , respectively. Then |qs − qt | 6 1, ∀s; t = 0; 1 : : : ; N − 1. Proof. Without loss of generality, we prove the result for the index qo and qs of the matrices N0 and Ns , respectively. It suHces to prove that qo − 1 6 qs 6 qo + 1: For that, we prove that if N0qo = 0 then (a) Nsqo +1 = 0

and

(b) Nsqo −2 = 0:

(a) By deJnition we have Nsqo +1 = (

N (s; s

+ N ))qo +1 =

N (s; s

+ (qo + 1)N )

= N (s) · · · N (N − 1)N (N ) · · · N (qo N + N − 1)N (qo N + N ) · · · N (qo N + N + s − 2)N (qo N +N +s−1) = N (s) · · · N (N − 1)

N (0; qo N )N ((qo

+ 1)N )

· · · N ((qo + 1)N + s − 1): As N0q0 = 0, then Nsqo +1 = 0. (b) We suppose that Nsqo −2 = 0, and we develop qo −1 N0 ,

j=s

0

X0 (s) = Im[H (s); H0 (s); : : : ; Hh−1 (s)];

= N (s) · · · N (N − 1)N0q0 N ((qo + 1)N )

k−1 + A1 (k; j + 1)B1 (j)u(j)

And the set of initial conditions X0 (s), s = 0; 1; : : : ; N − 1, for system (1) is

· · · N ((qo + 1)N + s − 1)

A1 (k; s)x1 (s)

0

409

k ¿ s:

N0qo −1 = (

j=k

(5)

qo −1 N (0; N ))

=

N (0; (qo

= N (0) · · · N (s − 1)N (s)

− 1)N )

410

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

· · · N (qo N − 2N + s − 1)N (qo N − 2N + s) · · · N (qo N − N − 1) = N (0) · · · N (s − 1)

N (s; (q0

− 2)N + s)

Now, we consider the N -periodic descriptor system (1) with nonnegative restrictions.

N ((qo − 2)N + s) · · · N ((qo − 1)N − 1) = N (0) · · · N (s − 1)Nsq0 −2 N ((qo − 2)N + s) · · · N ((qo − 1)N − 1): As Nsq0 −2 =0 then N0qo −1 =0. That is, ind(N0 ) 6 qo −1, but we have that ind(N0 ) = q0 then we arrive at a contradiction.

For each s ∈ Z, system (1) has associated a forward–backward invariant linear system, (Es ; As ; Bs ), see [13], given by x1; s (k + 1) x1; s (k) Es = As + Bs us (k); (7) x2; s (k + 1) x2; s (k) with x1; s (k)=x1 (s+kN ), x2; s (k)=x2 (s+kN ), us (k)= col[u(s + kN ); : : : ; u(s + kN + N − 1)], and I n1 O A1; s O Es = ; As = ; O Ns O In 2 B1; s Bs = ; B2; s where N (s; s

+ N ) ∈ Rn2 ×n2 ;

B1; s = row[A1 (s + N; s + j + 1) −1 ×B1 (s + j)]Nj=0 ∈ Rn1 ×mN ;

B2; s = row[

N (s; s

−1 + j)B2 (s + j)]Nj=0 ∈ Rn2 ×mN :

For each s=0; 1; : : : ; N −1, the set of initial conditions X0; s at time k = 0, for system (7), (Es ; As ; Bs ), is the subspace X0; s = Im[Hs ; H0; s ; : : : ; Hh−1; s ];

Denition 1. System (1) is a positive N -periodic descriptor system when for each s = 0; 1; : : : ; N − 1, for each nonnegative initial state x(s) = x0; s ∈ X0 (s) ∩ Rn+ and for each nonnegative control sequence u(k) ¿ 0, k ¿ s, the state trajectory belongs to Rn+ . A characterization of positive descriptor systems is given in [10] by the following proposition. Proposition 3. Consider an N -periodic forward –backward system (E(·), A(·), B(·))N and q = m&ax{ind(Ns ); s = 0; 1; : : : ; N − 1}. The system is positive, if and only if, A1 (k) ¿ 0, B1 (k) ¿ 0, N (k; j)B2 (j) 6 0, k ∈ Z, and j = k; k + 1; : : : ; k + qN − 1. Note that when N = 1, a forward–backward invariant system (E; A; B) is positive, if and only if, A1 ¿ 0; B1 ¿ 0, and N i B2 6 0, i = 0; 1; : : : ; q − 1 (see [5]). The positiveness property of the N -periodic system is transmitted to its associated invariant systems in the following proposition. Proposition 4. If an N -periodic forward–backward system (E(·), A(·), B(·))N is positive then its associated invariant systems (Es ; As ; Bs ) are positive for all s = 0; 1; : : : ; N − 1.

A1; s = A1 (s + N; s) ∈ Rn1 ×n1 ; Ns =

Proposition 2. Consider an N -periodic forward– backward system (E(·), A(·), B(·))N . For each s = 0; 1; : : : ; N − 1, X0; s = X0 (s).

(8)

Hi; s = diag[O; In2 ]Nsi B2; s ,

where Hs = diag[In1 ; O] and i = 0; : : : ; qs − 1. Next, we record a technical result given in (see [10]).

Proof. Since the N -periodic forward–backward system (E(·), A(·), B(·))N is positive from Proposition 3, A1 (k) ¿ 0, and B1 (k) ¿ 0, k ∈ Z. Then, by construction of A1; s and B1; s given in (7), we have A1; s ¿ 0; B1; s ¿ 0, s ∈ Z and thus the associated forward invariant linear subsystem (In1 ; A1; s ; B1; s ) is positive. Now, we consider the backward invariant linear subsystem (Ns ; In2 ; B2; s ). By Proposition 3, N (k; j)B2 (j) 6 0, k ∈ Z, j = k; k + 1; : : : ; k + qN − 1 and by Proposition 2, we have Nsi B2; s 6 0, for all i = 0; 1; : : : ; q − 1 and s ∈ Z. Thus, the associated invariant systems (Es ; As ; Bs ) are positive.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

In the following example we show that a forward –backward N -periodic system (E(·); A(·); B(·))N is not positive although its associated invariant systems (Es ; As ; Bs ) are positive, s = 0; 1; : : : ; N − 1. Example 1. We consider a 2-periodic forward–backward system, (E(·); A(·); B(·))2 with E(0) = E(1) = I3 , −1 0 −2 −1 −1 A1 (0) = 0 6 0; −3 −1 −2 1 2 −3 0 A1 (1) = −5 −2 ; −2 −3 −1 and the matrices B1 (k), B2 (k) and N (k) such that B1 (k) ¿ 0 and N (k; j)B2 (j) 6 0, k ∈ Z, j = k; k + 1; : : : ; k + qN − 1, with q = max{ind(Ns ); s = 0; 1; : : : ; N − 1}. We observe that this system is not positive because of A1 (0) and A1 (1) are not nonnegative matrices. If we consider the invariant associated systems we have 8 1 2 A1; 0 = A1 (2; 0) = A1 (1)A1 (0) = 5 2 12 5 4 9 and

3

A1; 1 = A1 (3; 1) = A1 (2)A1 (1) = 7 6

4 5 2

5

1 : 11

Nsi B2; s

Moreover, B1; s ¿ 0, 6 0, i = 1; : : : ; q − 1. Thus, we have the invariant associated systems (Es ; As ; Bs ) are positive. 2. Structural properties In this section we study the reachability and controllability properties of forward–backward positive systems. We want to learn if there exists a nonnegative control sequence transferring the system from a nonnegative initial state to a nonnegative Jnal state. First, we give the following deJnition.

411

Denition 2. Consider the system (E(·); A(·); B(·))N ¿ 0 and the set of admissible initial conditions X0 (s), for any s = 0; 1; : : : N − 1. (i) The state x ∈ Rn+ is positively reachable at time s, if there exists a time k ∈ Z+ and a nonnegative control sequence u(j) ¿ 0, j = s; s + 1; : : : ; k + s + qN − 1, transferring the state of the system from the origin at time s, x(s) = 0, to x in k steps. The system is positively reachable at time s if every x ∈ Rn+ is positively reachable at time s. The system is positively reachable if it is positively reachable at time s, for all s = 0; 1; : : : ; N − 1. (ii) The state xf ∈ Rn+ is positively controllable at time s for all s = 0; 1; : : : ; N − 1, if any Jnal state xf ∈ Rn+ can be reachable from any initial state x(s) ∈ X0 (s) by a Jnite control sequence. The system is positively reachable at time s if every xf ∈ Rn+ is positively reachable at time s, for all s=0; 1; : : : ; N −1. (iii) In this last case, when xf = 0, the system is called positively null-controllable. To study these properties, it is essential to introduce reachability cones of N -periodic forward– backward systems. For that, given the N -periodic forward–backward system (1), (E(·); A(·), B(·))N , the following matrices are used in the reachability property when we consider all reachable states in kN steps. (i) For each s, the set of all reachable states at time s in kN steps of the forward subsystem (In1 , A1 (·), B1 (·))N , is the cone RfkN (In1 ; A1 (·); B1 (·); s) which is generated by the column of the reachable matrix RfkN (In1 ; A1 (·); B1 (·); s) =row[A1 (s; s − j + 1)B1 (s − j)]kN j=1 :

(9)

(ii) If we observe the construction of the solution to the N -periodic system, we see that the part corresponding to the backward subsystem has qN terms. Then, we deJne the cone Rb (N (·); In2 ; B2 (·); s) where Rb (N (·); In2 ; B2 (·); s) = row["N (s; s + j − 1)B2 (s + j − 1)]qN j=1 ;

(10)

is the reachability matrix associated with the backward subsystem (N (·), In2 , B2 (·))N at time s. Hence, we deJne the reachability matrix at time s; of the N -periodic forward–backward system in kN

412

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

steps as the block diagonal matrix RkN (E(·); A(·); B(·); s) = diag[RfkN (In1 ; A1 (·); B1 (·); s); Rb (N (·); In2 ; B2 (·); s)]:

(11)

Thus, we can establish the following result. Proposition 5. The system (E(·); A(·), B(·))N positively reachable at time s, if and only if ∞

RkN (In1 ; A1 (·); B1 (·); s) = Rn+ ;

or equivalently

k=1

RfkN (In1 ; A1 (·); B1 (·); s) = Rn+1

and

j=s

is

k=1

∞

then there exist k ∈ Z and a control sequence u(j) ¿ 0, j = s; : : : ; k + s + qN − 1 such that x(k + s) = 0, that is, I n1 A1 (k + s; s)x1 (s) 0= 0 k+s−1 + A1 (k + s; j + 1)B1 (j)u(j)

Rb (N (·); In2 ; B2 (·); s) = Rn+2 :

If the system is positively null-controllable, we can give the following result. Proposition 6. Consider the N -periodic forward– backward system (E(·); A(·), B(·))N ¿ 0. The system is positively null-controllable, if and only if, A1; s is a nilpotent matrix, for all s ∈ Z. Proof. If we consider x(s) ∈ X0 (s), u(j) = 0, for all j = s; : : : ; s + k + qN − 1, the solution of the system is given by I n1 x(k + s) = A1 (k + s; s)x1 (s); k ¿ 0: 0 Now, we suppose that A1; s are nilpotent matrices, s ∈ Z and % = max{ind(A1; s )}. If k ¿ %N , we have A1 (k + s; s) =A1 (k + s − 1) · · · A1 (%N + s)A1 (s + %N; s) =A1 (k + s − 1) · · · A1 (%N + s)A1; s % = 0: Thus, x(k + s) = 0, and the system is positively null-controllable. Conversely, if the positive system (E(·); A(·); B(·))N is positively null-controllable, for any x(s) ∈ X0 (s),

−

0 In2

k+s+qN −1

N (k+s; j)B2 (j)u(j);

k¿s:

j=k

Since the system is positive, the three terms of the above expression are positive or zero. In this case, each term is zero because the Jnal state is zero. In particular, choosing the canonical vectors as initial state, x1 (s) = ei , we have In1 0= A1 (k + s; s)x1 (s) 0 In 1 = A1 (k + s; s)ei : 0 Thus, A1 (k + s; s) = 0. For each s, we take %s ∈ Z, such that %s ¿ k, then A%1;s s = A1 (%s N + s; s) = 0. Then A1; s is a nilpotent matrix for all s ∈ Z. The relationship between reachability matrices of the N -periodic system and its associated invariant systems is given in the following result. The proofs are rather technical. Proposition 7. The reachability matrices given in (9)–(11) satisfy the following relations, for all s ∈ Z: 1. RfkN (In1 ; A1 (·); B1 (·); s) = Rfk (In1 ; A1s ; B1s ), k ∈ Z, 2. Rb (N (·); In2 ; B2 (·); s) = Rb (Ns ; In2 ; B2s ), 3. RkN (E(·); A(·); B(·); s) = Rk (Es ; As ; Bs ), where Rfk (In1 ; A1s ; B1s ), Rb (Ns ; In2 ; B2s ), and Rk (Es ; As ; Bs ) are the reachability matrices of the associated invariant systems. Thus, we have the following relation between the structural properties of the N -periodic system and its associated invariant systems.

B. Cant&o et al. / Systems & Control Letters 53 (2004) 407 – 414

Theorem 8. Consider the N -periodic forward–backward system (E(·), A(·), B(·))N ¿ 0. (a) The system is positively reachable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively reachable for any s ∈ Z. (b) The system is positively controllable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively controllable for any s ∈ Z. (c) The system is positively null-controllable, if and only if, the invariant associated systems (Es ; As ; Bs ) are positively null-controllable for any s ∈ Z. Proof. Parts (a) and (b) are consequence of Proposition 7 and part (c) follows from Proposition 6. 3. On stability

Next, we deJne the stability cones associated with the positive N -periodic system as S(E(·); A(·); s) = {x ∈ Rn+1 : [In1 − A1 (s)]x ¿ 0}; for all s ∈ Z. And, we deJne the stability cone associated with the corresponding invariant system as S(Es ; As ) = {x ∈ Rn+1 : [In1 − A1; s ]x ¿ 0}; s = 0; 1; : : : ; N − 1: A condition for the stability of the positive N periodic forward–backward system is given in the following result. Theorem 9. Consider the positive N -periodic forward–backward system (12), (E(·); A(·))N . If there

N −1 exists a vector e ∈ s=0 int(S(E(·); A(·); s)), then the system is asymptotically stable. Proof. If A1 (s) ¿ 0 and there exists a vector

N −1 e ∈ s=0 int(S(E(·); A(·); s)), that is, there exists e ¿ 0 satisfying

In this section we study the stability property associated with a positive N -periodic system. Consider the autonomous positive N -periodic forward–backward system

then, we have that

E(k)x(k + 1) = A(k)x(k):

e ¿ A1 (N − 1)e ¿ A1 (N − 1)A1 (N − 2)e ¿

(12)

We shall analyze this property and its relationship with the stability of its associated positive invariant systems by constructing some stability cones. First, we recall the deJnition of asymptotic stability. The positive N -periodic system (12) is asymptotically stable, if its free response satisJes lim x(k) = 0:

k→∞

Using the fact that the monodromy matrices Ns ; s ∈ Z, are nilpotent, we have that the positive N -periodic system (E(·); A(·))N is asymptotically stable, if and only if, its positive N -periodic forward subsystem (In1 ; A1 (·)) is asymptotically stable, that is, if A1; s is a stable matrix, ∀' ∈ ((A1; s )|'| ¡ 1. The stability of the N -periodic system (12), (E(·); A(·))N , is transmitted to its associated invariant systems (Es ; As ), for all s ∈ Z, and reciprocally. This is due to the fact that the relationship between the state of the N -periodic system and the state of sth invariant system is given by xs (k) = x(s + kN ), for all k ¿ s, s ∈ Z.

413

e ¿ A1 (0)e; e ¿ A1 (1)e; : : : ; e ¿ A1 (N − 1)e;

· · · ¿ A1 (N; 0)e = A1; 0 e: Thus, e ∈ int(S(E0; A0 )). Using the periodicity of the matrices we obtain e ∈ S(Es ; As ), for all s = 0; 1; : : : ; N − 1, that is [In1 − A1; s ]e ¿ 0. Consider the Perron root * ¿ 0 of A1; s and its associated eigenvector v ¿ 0, we have that, e ¿ A1; s e; vT e ¿ *vT e; (1 − *)vT e ¿ 0: Thus, * ¡ 1, and A1; s is a stable matrix. Hence, the N -periodic system is asymptotically stable. 4. Conclusions In this paper positive N -periodic forward–backward systems are considered. The structural properties of this kind of system, the positive reachability and controllability properties, are discussed. Using a

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geometrical approach, a relationship between the structural properties of the system and its associated invariant systems is given. In this way, the reachability cones are constructed. Finally, some results about the stability property are given. More speciJcally, a geometrical condition is stabilized to assure the asymptotic stability of the N -periodic system. References [1] P. Albertos, Block multirate input–output model for sampled-data control systems, IEEE Trans. Automat. Control 35 (9) (1990) 1085–1088. [2] M. Bidani, M. Djemai, A multirate digital control via a discrete-time observer for non-linear singularly perturbed continuous-time systems, Internat. J. Control 75 (8) (2002) 591–613. [3] R. Bru, R. Cant&o, B. Ricarte, Modelling nitrogen dynamics in citrus trees, Math. Comput. Modelling 38 (2003) 975–987. [4] R. Bru, C. Coll, V. Hern&andez, E. S&anchez, Geometrical conditions for the reachability and realizability of positive periodic discrete systems, Linear Algebra Appl. 256 (1997) 109–124. [5] R. Bru, C. Coll, E. S&anchez, Structural properties of positive linear systems time-invariant di;erence-algebraic equations, Linear Algebra Appl. 349 (2002) 1–10. [6] R. Bru, V. Hern&andez, Structural properties of discrete-time linear positive periodic systems, Linear Algebra Appl. 121 (1989) 171–183. [7] R. Bru, S. Romero, E. S&anchez, Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl. 310 (2000) 49–71. [8] L. Caccetta, V.G. Rumchev, Reachable discrete-time positive systems with minimal dimension control sets, Dynamics Continuous, Discrete Impulsive Systems 4 (1998) 539–552. [9] S.L. Campbell, Singular Systems of Di;erential Equations, Pitman, London, 1980. [10] B. Cant&o, C. Coll, E. S&anchez, On feedbacks for positive discrete-time singular systems, in: Preprints 15th IFAC World Congress, 2002, pp. 1–6.

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