In this paper, we will deal with positive discrete-time linear control systems in the state-space model, i.e., systems, whose states and inputs are no...

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Canonical forms for positive discrete-time linear control systems聻 Rafael Bru∗ , Sergio Romero, Elena Sánchez Dept. de Matemàtica Aplicada, ETSIA, Universitat Politècnica de València, Cami de Vera, s/n, 46022, València, Spain Received 25 July 1999; accepted 9 January 2000 Submitted by M. Neumann

Abstract In this paper, the properties of reachability, controllability and essential reachability of positive discrete-time linear control systems are studied. These properties are characterized in terms of the directed graph of the state matrix. From these characterizations canonical forms of those properties are deduced. © 2000 Published by Elsevier Science Inc. All rights reserved. Keywords: Positive control systems; Canonical forms; Controllability; Reachability

1. Introduction and preliminaries In this paper, we will deal with positive discrete-time linear control systems in the state-space model, i.e., systems, whose states and inputs are nonnegative. The nonnegativity condition yields a different treatment of these control systems based upon the theory of nonnegative matrices. The nonnegative systems appear in many different real situations such as in economics, biological, enviromental and chemical processes, among others. Many authors have studied different problems concerning positive systems. The invariant case has been studied by Ohta et al. [8], Coxson and Shapiro [5], van den Hof [10], Farina [7] and all references therein. Other researchers such as Bru and Hernández [2] and Bru et al. [3] deal with the positive periodic case. 聻 Supported by Spanish DGES grant number PB97-0334. ∗ Corresponding author. Tel.: +34 96 387 7660; fax: +34 96 387 7669.

E-mail address: [email protected] (R. Bru). 0024-3795/00/$ - see front matter 2000 Published by Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 0 4 4 - 6

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Recently, Valcher [9] has provided reachability and controllability criteria and studied the canonical forms of reachable and controllable positive discrete-time systems. The approach used in [9] is combinatorial and the main tool is the notion of deterministic path in the directed graph of a state matrix. The aim of our work consists mainly of obtaining characterizations of the reachability and controllability properties of positive invariant systems by partitioning the set of vertices of the directed graph of the state matrix in different kinds of deterministic paths (Section 2). These characterizations provide canonical forms yielding a refinement of those given in [9]. Such a refinement allows us to determine positive and zero entries of those canonical forms and so we give more detailed blocks in the structure of the canonical forms. These forms have the advantage of being the same for all cases considered in [9] (with or without zero columns). Moreover, we construct a canonical form for the essential reachability property. The structure of these forms is block upper triangular (in terms of the state matrix of its associated canonical form). This is done for standard reachability in Section 3, standard controllability in Section 4 and essential reachability and controllability in Section 5, in all cases for the multi-input case. Analogous results for periodic positive discrete-time linear systems are the subject of the technical report [4]. Given a general matrix A = [aij ], we write A > 0 if aij > 0 ∀i, j ; A > 0 if at least some entry aij > 0 and A 0 if aij > 0 ∀i, j . Moreover, we must bear in mind that an i-monomial vector is a (nonzero) multiple of the unit vector ei and that a monomial matrix has one and only one nonzero entry in each column and each row. We consider a positive invariant discrete-time linear control system given by x(k + 1) = F x(k) + Gu(k), Rn×n + ,

k ∈ N,

(1)

Rn×m + ,

G = [gij ] ∈ x(t) is the nonnegative state vector where F = [fij ] ∈ and u(t) is the nonnegative control or input vector. We denote that control system by (F, G) > 0. Note that if the initial state vector is nonnegative, that is x0 > 0, and the input vector u(k) is nonnegative for every k > 0, then the state vector x(k) is also nonnegative in any other instant k. We will study the structural properties of reachability and controllability for this kind of linear control systems. Usually (see [5]) a positive system (F, G) is said to be (a) reachable ( from 0) if for any nonnegative state xf , there exist k ∈ N and a nonnegative input sequence u(t) > 0, t = 0, . . . , k − 1, transferring the state of the system from the origin at t = 0 to xf at time t = k. (b) (completely) controllable if for any pair of nonnegative states x0 and xf there exist k ∈ N and a nonnegative input sequence u(t) > 0, t = 0, . . . , k − 1, transferring the state of the system from x0 at t = 0 to xf at time t = k. To study these structural properties, we introduce the reachability matrix in ksteps, defined as

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set of all nonnegative states xf which are reachable in k-steps, by means of a suitable sequence of nonnegative inputs u(0), . . . , u(k − 1). Then R∞ (F, G) =

∞ [

Rk (F, G),

n=1

is the set of all the reachable states in finite time. In [5] it was established that the pair (F, G) > 0 is reachable if and only if R∞ (F, G) = Rn+ , or equivalently, the reachability matrix

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α1 = α11 .. .

→

α12 .. .

→

... .. .

→

α1k1 .. .

(2)

αrkr .

→ → ... → αr = In what follows, we can assume that there are no repeated deterministic paths, if not, we eliminate them. The set of all vertices in these different deterministic paths given in (2) is denoted by A = {α11 , α12 , . . . , α1k1 , . . . , αr1 , αr2 , . . . , αrkr }. Note that the vertices of A determine the corresponding monomial vectors in

αr2

A1 = {αjm ∈ A|col

kj

αj

F = 0},

or equivalently k

A1 = {αjm ∈ A|(αj j , i) 6∈ C(F ) ∀i ∈ V }. The vertices of A which belong to deterministic paths that have access to some vertex of A1 are grouped in the set n o k k A2 = αjm ∈ A|∃α ∈ A1 : (αj j , α) ∈ C(F ), and (αj j , i) 6∈ C(F ) ∀i 6∈ A1 . By proceeding in this way, we introduce the sets n k Ah = αjm ∈ A|∃α ∈ Ah−1 : (αj j , α) ∈ C(F ), and o k (αj j , i) 6∈ C(F ) ∀i 6∈ A1 ∪ · · · ∪ Ah−1 , for h = 3, . . . , n − 1. Moreover, in order to achieve a partition of A, we define An = A − {A1 ∪ · · · ∪ An−1 }. Note that the vertices of An are in deterministic paths such that all of them are connected to some vertex included in V − {A1 ∪ · · · ∪ An−1 }. Next, we consider a second kind of deterministic paths starting from vertices β which are not associated with monomial vectors of G, and such that there exists a vertex k of An having access to β and access to some vertex of the set A1 ∪ · · · ∪ An−1 . That is, β ∈ V − A is such that ∃k ∈ An , ∃α ∈ A1 ∪ · · · ∪ An−1 : (k, β) ∈ C(F ), (k, α) ∈ C(F ) / β, (k, i) 6∈ C(F ). and ∀i 6∈ A1 ∪ · · · ∪ An−1 , i =

(3)

The set of all vertices in this second kind of deterministic paths p

β1 = β11 .. .

→

β12 .. .

→

··· .. .

→

β1 1 .. .

βs = βs1

→

βs2

→

···

→

βs s .

p

(4)

R. Bru et al. / Linear Algebra and its Applications 310 (2000) 49–71 p

53

p

is denoted by B = {β11 , β12 , . . . , β1 1 , . . . , βs1 , βs2 , . . . , βs s }. Additionally, we consider those deterministic paths starting from vertices γ ∈ V − {A ∪ B} such that there exists a column of G which can be written as col G = τ eγ + w, for some real number τ > 0, and for some / 0. real vector w > 0, with j ∈ A1 ∪ · · · ∪ An−1 if wj =

(5)

The set of all vertices in this third kind of deterministic paths q

γ1 = γ11 .. .

→

γ12 .. .

→

··· .. .

→

γ1 1 .. .

γt = γt1

→

γt2

→

···

→

γt t .

(6)

q

q

q

is denoted by C = {γ11 , γ12 , . . . , γ1 1 , . . . , γt1 , γt2 , . . . , γt t }. We will see in Proposition 2 that the deterministic paths given in (4) and (6) are closed deterministic paths, that is circuits, so the corresponding vertices correspond to monomial vectors in the reachability matrix. When studying essential reachability, we will consider circuits, whose communicating classes (blocks) have a spectral radius with bordering properties. If τ belongs to a circuit, by Cτ we denote the communicating class associated with this vertex (and hence with this circuit). We consider then the circuits whose vertices are not in A ∪ B ∪ C, and the set of their vertices H˜ = {τ ∈ V − {A ∪ B ∪ C}|∃ Cτ : j ∈ V − {A ∪ B ∪ C} ∀j ∈ Cτ } . Now, we shall only focus on those circuits having a unique vertex accesible from any vertex of An , that is, we shall consider vertices τ ∈ H˜ such that there exists k ∈ An with colk F = αeτ + w, for some real number α > 0, and some vector w ∈ Rn+ , where wj = 0 ∀j ∈ Cτ ,

(7)

or equivalently (k, τ ) ∈ C(F )

and (k, j ) 6∈ C(F ) ∀j ∈ Cτ ,

and with the three following conditions for any vertex k ∈ An connected to τ : (i) k does not have access to any vertex η ∈ V − Cτ such that from η there exist two outgoing arcs which reach two different vertices of Cτ . (ii) If η ∈ V − Cτ is accessible from k, and there exists only one arc from η to a vertex τ¯ of Cτ , then length(Pkη ) = length(Pτ τ¯ ) + q · length(Cτ ),

q ∈ Z+ ,

for all paths Pkη . ¯ 6 ρ(Cτ ). Moreover, (iii) For every communicating class C¯ accessible from k, ρ(C) ¯ ¯ = ρ(Cτ ), then Cτ is accessible from C. if ρ(C)

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The set of all vertices in these circuits

(8)

is denoted by D1 = {τ11 , τ12 , . . . , τ11 , . . . , τl1 , τl2 , . . . , τl l }. We will construct the last set D2 . Previously, we take the deterministic paths starting from vertices δ of H˜ − D1 such that there exists a column of G written as col G = αeδ + w, for some real number α > 0, and some w ∈ Rn+ with wj = 0

∀j ∈ Cδ ,

(9)

and with the additional condition (iv) If col G = αeδ + w, α > 0, w ∈ Rn+ , with wj = 0 ∀j ∈ Cδ , then any commu¯ < / 0 satisfies ρ(C) nicating class C¯ accessible from any vertex j such that wj = ρ(Cδ ). Then, the set of all vertices in these circuits

(10)

ν

is denoted by D2 = {δ11 , δ12 , . . . , δ1ν1 , . . . , δh1 , δh2 , . . . , δhh }. We illustrate the definitions given above by means of the following example. Example 1. Consider the pair (F, G) > 0 of order n = 11, where 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 1 0 0 2 1 F = 0 0 0 4 0 0 0 0 0 0 0 , 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

R. Bru et al. / Linear Algebra and its Applications 310 (2000) 49–71

0 1 0 0 0 G= 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 1 0

55

0 0 1 0 1 0 . 1 1 0 0 1

The directed graph associated with the matrix F is

Note that the monomial vectors of the matrix G correspond to the set of vertices {1, 2, 7, 8}. Moreover, G has the columns g1 = e10 + e8 and g2 = e11 + e8 + e7 + e 5 + e3 . We construct the deterministic paths generated by the vertices α1 = 1, α2 = 2, α3 = 7, α4 = 8. We have α1 = α11 = 1 α2 = α21 = 2

→

α22 = 4

→

α23 = 6

α3 = α31 = 7 α4 = α41 = 8. Then, A = {1, 2, 4, 6, 7, 8}. Now, we will construct the sets Aj , j = 0, 1, . . . , 11. The set of vertices of deterministic paths such that the corresponding last vertex has no outgoing arc is A1 = {8}. The set of vertices of deterministic paths such that from the last vertex all outgoing arcs reach vertices of the set A1 is A2 = ∅. Moreover, A3 = A4 = · · · = A10 = ∅. Then, A11 = {1, 2, 4, 6, 7}.

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Notice that for β1 = 3 there exists a vertex k = 1 ∈ A11 such that (1, 3) ∈ C(F ) and (1, 8) ∈ C(F ) with 8 ∈ A1 ∪ · · · ∪ A10 . So, we consider the deterministic path associated with the vertex β1 = 3 β1 = β11 = 3. Then, B = {3}. It is clear that for γ1 = 10, there exists a column in G, g1 = e10 + e8 with 8 ∈ A1 ∪ · · · ∪ A10. The deterministic path generated by γ1 = 10 is γ1 = γ11 = 10. Therefore, C = {10}. Note that V − {A ∪ B ∪ C} = {5, 9, 11}, so H˜ = {5, 9, 11} since C5 = C9 = {5, 9} and C11 = {11}. Observe that for τ1 = 5, there exists an arc (6, 5) ∈ C(F ) with k = 6 ∈ A11 and (6, 9) 6∈ C(F ), (6, 11) 6∈ C(F ). Note that the set of vertices accessible from k = 6 is {5, 9, 3, 8, 7, 2, 4, 6}. None of these vertices has two or more outgoing arcs to different vertices of C5 and only the vertex 7 has an outgoing arc (7, 9) to C9 (τ¯ = 9). Between 6 and 7, we have two possible paths: 1 :6→7 P67

2 or P67 :6→7→2→4→6→7

1 ) = 1 = length(P ) and length(P 2 ) = 5 = length(P )+2 length(C ). with length(P67 59 59 5 67 Moreover, the accessible communicating classes from 6 are: C¯ 1 = {2, 4, 6, 7}, ρ(C¯ 1 ) = 0.5,

C¯ 2 = {3},

ρ(C¯ 1 ) = 1/4,

C¯ 3 = {8},

ρ(C¯ 1 ) = 0

while ρ(C5 ) = 0.5. With these properties, we deduce that the conditions (i)–(iii) hold for τ1 = 5. Therefore, we construct the circuit generated by τ1 = 5 τ1 = τ11 = 5 → τ12 = 9. Then, D1 = {5, 9}. Finally, we construct the set D2 . Note that for δ1 = 11, there exists a column of G, g2 = e11 + w with w = e8 + e5 + e7 + e3 . Moreover, the accessible communicating classes from 8, 5, 7, 3 are C¯ 1 , C¯ 2 , C¯ 3 , C5 having each one of them spectral radius less than ρ(C11 ) = 1. Therefore, the corresponding circuit is δ1 = δ11 = 11. Then, D2 = {11}. Proposition 1. Let (F, G) > 0 be a positive system of order n such that [F |G] has a monomial submatrix of order n. For every i1 ∈ V − {A ∪ B ∪ C}, there exists a unique deterministic path containing i1 . Moreover, this deterministic path is a circuit and all its vertices are in the set V − {A ∪ B ∪ C}.

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Proof. Since i1 6∈ A, from the definition of the set A, the i1 -monomial vector is not in the matrix G, and since [F |G] has a monomial submatrix of order n then the i1 monomial vector is in the matrix F, i.e., there exists a unique vertex i2 such that the arc i2 → i1 , is in C(F ). If i2 = i1 , then the proof is completed. Otherwise, i2 6∈ A ∪ B ∪ C since i1 6∈ A ∪ B ∪ C and both vertices are in the same deterministic path. By reasoning in this way, we obtain a deterministic path il → · · · → ij → · · · → i2 → i1 . with ij 6∈ A ∪ B ∪ C ∀j ∈ {1, . . . , l}. Moreover, this deterministic path is closed since A is of finite order and the first vertex il always has a unique ingoing arc from any other vertex of C(F ). Finally, note that the constructed deterministic path is unique since the vertices i2 , . . . , il correspond to monomial vectors in the matrix F. From the construction of the sets A, B and C and the previous result, if a vertex of a deterministic path is in A, B, C or V − {A ∪ B ∪ C}, then all vertices of such deterministic path are in A, B, C or V − {A ∪ B ∪ C}, respectively. Then we can establish the following results. Proposition 2. Let (F, G) > 0 be a positive system of order n such that [F |G] has a monomial submatrix of order n. For every i1 ∈ B (i1 ∈ C) there exists a unique deterministic path containing i1 . Moreover, this deterministic path is a circuit and all its vertices are in B (C). Proof. If i1 ∈ B (i1 ∈ C), then i1 6∈ A, and hence we can apply a similar reasoning as in the proof of Proposition 1. Proposition 3. Let (F, G) > 0 be a positive system of order n such that [F |G] has a monomial submatrix of order n. For every i1 ∈ V such that (i1 , i) 6∈ C(F ) ∀i ∈ V , there exists a deterministic path containing i1 with its vertices in A1 . Proof. If (i1 , i) 6∈ C(F ) ∀i ∈ V , then the deterministic path ending in i1 is not a circuit. Thus, all vertices of that deterministic path must be in A. More precisely, by the construction of sets A1 , . . . , An , all these vertices must be in A1 .

3. Reachability property Consider a positive linear discrete-time system (F, G) given in (1). Note that the unit vector ei is reachable in a finite number of steps if and only if there exists an

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i-monomial vector in the reachability matrix. In [9] the set of vertices i such that there is an i-monomial vector in the reachability matrix is denoted by I (F, G). Thus, the positive system (F, G) is reachable if and only if I (F, G) = {1, . . . , n}. From the definitions of the sets A, B and C, and the propositions given in Section 2, we establish the following results in order to obtain a reachability canonical form, which has the structure of that given in [9], but with more detail. First, we relate the set I (F, G) to the sets A, B and C. Proposition 4. Assume that the matrix [F |G] has a monomial submatrix of order n. Then, I (F, G) = A ∪ B ∪ C. Proof. We will show that if the vertex i1 ∈ V is such that i1 6∈ A ∪ B ∪ C, the i1 -monomial vector is not in the reachability matrix

k

αj10 → αj20 → αjm0 = j → · · · → αj j0

0

with all vertices in A1 , and so they are different from i1 . Then, col kj 0 F = 0. αj 0

Moreover, we observe that

F ej = F eα m0 ∈ eα m0 +1 j0 j0 2 F ej ∈ F eα m0 +1 = eα m0 +2 j0 j0 .. . * + 0

F kj 0 −m +1 ej ∈

Fe

kj 0

αj 0

= 0,

and thus, F k ej is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1} and ∀j ∈ A1 . Now let us study the case in which j ∈ A2 . In this case j also belongs to a deterministic path in (2), such that the last vertex α is connected only to vertices of / 0. Thus, we find ourselves in the same A1 , i.e., colα F = v, such that j ∈ A1 if vj = situation as before, and we can conclude that F k ej is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1} and ∀j ∈ A2 .

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With the analogous line of reasoning, we deduce the same result for the other situations when the vertex j is in the sets Ah , h ∈ {3, . . . , n − 1}. Case 2: Let g = τ ei1 + w, where τ > 0, w ∈ Rn+ , wi1 = 0 and ∃j 6∈ A1 ∪ · · · ∪ / 0. We only need to prove that F k (τ ei1 + ej ) ∀k ∈ {1, . . . , n − An−1 such that wj = 1} is not an i1 -monomial vector ∀j 6∈ A1 ∪ · · · ∪ An−1 . We will study the following different possibilities: j ∈ B or j ∈ C or j ∈ An or j 6∈ A ∪ B ∪ C. First, note from Proposition 1 that since i1 6∈ A ∪ B ∪ C, the vertex i1 is in a circuit (11) having no vertices in A ∪ B ∪ C. If j ∈ B (j ∈ C), from Proposition 2 we know that j is in a circuit

with all vertices in B (C) and hence different from i1 . Therefore, we obtain

∀k ∈ {1, . . . , l}. F k (τ ei1 + ej ) ∈ hF k ei1 , F k ej i = eil+1−k , ejl0 −k+1

(12)

Thus, F k (τ ei1 + ej ) is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1}. From Proposition 1, if j 6∈ A ∪ B ∪ C, by the same line of reasoning and supposing that j is not in the same circuit as i1 given in (11), then F k (τ ei1 + ej ) is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1}. Otherwise, if j and i1 are in the same / i1 and so circuit, there is an index d ∈ {1, . . . , l} such that j = id = F k (τ ei1 + eid ) ∈ hF k ei2 , F k eid+1 i = heil−k+1 , eid−k i ∀k ∈ {1, . . . , n − 1}, which is not an i1 -monomial vector, since τ is positive because j 6∈ A. Finally, we consider the case j ∈ An . In this case the vertex j is in a deterministic path in (2), whose last vertex α is connected to some vertex included in V − {A1 ∪ · · · ∪ An−1 }. First, we suppose that from the last vertex α of the deterministic path there exists a unique outgoing arc, that is, there exists a unique vertex φ such that colα F = v / 0. Then, φ is in the same path of j. In this case we have a circuit, and thus, with vφ = the vector F k (τ ei1 + ej ) is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1}. Now, we suppose that from the last vertex α of the deterministic path there are two outgoing arcs. When one of the arcs is leading to a vertex of A1 ∪ · · · ∪ An−1 , we can prove that the other vertex is in a circuit not including the vertex i1 since i1 6∈ B. Thus, F k (τ ei1 + ej ) is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1}. Then, we consider that the outgoing arcs from the last vertex α are leading to vertices not included in A1 ∪ · · · ∪ An−1 . This study is reduced to the two following cases: / 0 and vid = / 0, where φ 6∈ A1 ∪ (a) colα F = v with only two entries nonzero, vφ = / id , and · · · ∪ An−1 , id is in the same circuit of i1 with φ = / 0 and vφ2 = / 0, where φ1 , φ2 6∈ (b) colα F = v with only two entries nonzero, vφ1 = / φ2 and they are not in the same circuit of i1 . A1 ∪ · · · ∪ An−1 , φ1 =

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Then, if φ, φ1 and φ2 6∈ An , reasoning as before and using Propositions 1 and 2, we obtain the desired result. Finally, if φ, φ1 and φ2 ∈ An , then we will be reiterating the process. Then, we prove that F k (τ ei1 + ej ) is not an i1 -monomial vector ∀k ∈ {1, . . . , n − 1} if j ∈ An . Conversely, suppose that the vertex i1 is in the set A ∪ B ∪ C. We only outline the case when i1 ∈ C (the other cases are similar). By definition of the set C, there / 0, exists a column g in G such that g = ρej + w, with α ∈ A1 ∪ · · · ∪ An−1 if wα = and vertices i1 and j are in the same circuit. Then, for some k, F k g is i1 -monomial since the vector sequence F k w, k ∈ Z, eventually becomes zero. Hence, i1 ∈ I (F, G). With this result, and Lemma 2 of [9], we can establish the following characterization of the reachability property of the positive system (F, G). Theorem 1. Assume (F, G) > 0. Then, (F, G) is reachable if and only if the matrix [F |G] has a monomial submatrix of order n and A ∪ B ∪ C = {1, . . . , n}. Now, we are going to construct a canonical form of reachability as follows. Suppose that (F, G) > 0 is reachable, then by Theorem 1, all vertices are in A ∪ B ∪ C. Next, we are going to construct some permutation matrices P and Q from the sets A = A1 ∪ · · · ∪ An , B and C in order to get the canonical form of the reachability property. We introduce A0 = A1 ∪ · · · ∪ An−1 , so that A = A0 ∪ An , and we take a partition of An = AnB ∪ AnR , where AnB is the set of vertices of An belonging to deterministic paths in (2) connected to vertices of set B, according to the definition of set B. We denote the number of deterministic paths in A, A0 , Ai , i = 1, . . . , n, AnB , AnR , B and C by r, r 0 , ri , i = 1, . . . , n, rnB , rnR , s, and t, respectively. Thus, r = r 0 + rn , r 0 = r1 + · · · + rn−1 , rn = rnB + rnR . Then, we relabel the vertices in decreasing order, from n to 1, considering the following order of the sets: AnR , AnB , An−1 , . . . , A1 , B and C, and in each set starting from the longest deterministic path and finishing with the shortest deterministic path. Define P as the permutation matrix associated with this relabelling, and let Q be the matrix which places the r-monomial vectors of G as the first r columns in the order that we will show within this proof. Then, the pair [P T F P |P T GQ] has the following structure: O D C O O O B O R D , O O A0 D D G (13) O O O An D B O O O O AnR

R. Bru et al. / Linear Algebra and its Applications 310 (2000) 49–71

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where C, B, A0 , AnB and AnR are the submatrices associated with the deterministic paths of the sets C, B, A0 , AnB and AnR , respectively. By construction of these sets we have C = diag [C(t), . . . , C(1)]

and B = diag [B(s), . . . , B(1)] ,

(14)

where C(i) and B(j ), ∀i = 1, . . . , t and ∀j = 1, . . . , s, are cyclic irreducible blocks, according to Proposition 2. Moreover, A1 D ··· D D O A2 · · · D D .. . . .. , 0 . .. .. .. (15) A = . . O O · · · An−2 D O O ··· O An−1 where

∗ .. D = [W, . . . , W] , with W = .

0 .. .

...

∗

0

...

0 .. , .

(16)

0

and ∗ denotes a nonnegative entry. By construction of the sets Ah , h = 2, . . . , n − 1, note that each block W of the blocks D in the superdiagonal of A0 in expression (15) has at least one positive entry. Moreover, Ai = diag [Ai (ri ), Ai (ri − 1), . . . , Ai (1)] , where

0 0 Ai (j ) = ... . ..

+ 0

0 .. . .. . 0 0

... . + .. . 0 .. .. . . . . 0 ...

0 .. . , 0 + 0

i = 1, . . . , n − 1,

j = 1, . . . , ri , i = 1, . . . , n − 1,

and + denotes a positive entry. In addition, AnB = diag AnB (s), AnB (s − 1), . . . , AnB (1) ,

(17)

(18)

where AnB (j ), ∀j = 1, . . . , s, has the same structure as the matrix given in (17). The relationship between set AnB and set B is expressed in the structure of the matrix R, with s diagonal blocks, 0 0 ... 0 .. .. .. . . R = diag [U, . . . , U] , where U = . (19) , 0 0 ... 0 + 0 ... 0

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with + denoting a positive entry. Moreover, the remaining relationships between the sets are given by the matrix D in (16). Note that these matrices are of appropriate dimensions in each case. The last diagonal block of P T F P , AnR , is a block matrix, where all off-diagonal blocks are W and the diagonal blocks are given by ∗ + 0 0 ... 0 ∗ 0 + 0 ... 0 .. .. .. . .. . . . . . 0 . (20) AnR (j ) = ∀j = 1, . . . , rR , .. .. ∗ 0 0 . . 0 . . .. .. .. .. . + . 0 ∗ 0 0 ... 0 0 with + denoting a positive entry and ∗ denoting a nonnegative entry. Finally, associated with each block of the submatrices A0 , AnB and AnR , we have a monomial vector in the matrix G. That is, r columns of G are multiples of distinct unit vectors. From the order established in the renumbering of the vertices, we can assume that the first column of G is a multiple of en , the second is a multiple of en−l , where l is the number of vertices of the longest deterministic path of AnR , and, in general, the ith-column, i = 1, . . . , r, is a multiple of en−li , where li is the sum of the number of vertices of all previous deterministic paths, according to the above order. Moreover, the following t columns of G have the structure considered in (5). This ordering of the columns of G defines the permutation matrix Q. Then, the pair (P T F P , P T GQ) is similar to the pair (F, G) and we have constructed a reachable canonical form. Moreover, it is easy to check that the reachability matrix of the similar pair (P T F P , P T GQ) given in (13) has a monomial submatrix of order n, and hence the pair is reachable. We summarize all the last results in the following theorem. Theorem 2. Assume (F, G) > 0. Then, (F, G) is reachable if and only if there exist permutation matrices P and Q such that the matrix [P T F P |P T GQ] has the structure given in (13), where the blocks are given in (14)–(20). The characterization of the reachability of the pair (F, G) > 0 in terms of directed graph C(F ) is given in the next result. Theorem 3 (Reachability in terms of C(F )). Let (F, G) be a positive discrete-time control system of order n. Consider the sets {β1 , . . . , βs } and {γ1 , . . . , γt } given in (3) and (5), respectively, and the set of vertices {α1 , . . . , αr } such that for each i ∈ {1, . . . , r}, there exists a column of G which is an αi -monomial vector. Then, (F, G) is reachable if and only if there exist in C(F ), r-deterministic paths generated from the vertices {α1 , . . . , αr }, and s + t closed deterministic paths (circuits)

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generated from the vertices {β1 , . . . , βs } and {γ1 , . . . , γt }, such that all these paths cover all the n vertices of C(F ). Example 2. Consider the control system (F, G) > 0, of order n = 10, given by 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 and 1 0 0 0 0 G= 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0

1 0 0 1 1 . 1 0 0 0 0

The following graph is associated with the matrix F.

We construct the deterministic paths generated by α1 = 1, α2 = 2, α3 = 3, α4 = 4, α5 = 5, α6 = 9, which are

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1→2 2 3 4 5 9 → 10. The unrepeated deterministic paths are α11 = 1 → α12 = 2 α21 = 3 α31 = 4 α41 = 5 α51 = 9 → α52 = 10. Note that A = {1, 2, 3, 4, 5, 9, 10}. Following the steps of Example 1, we construct the deterministic paths in B β11 = 8 → β12 = 7. Then, s = 1, p1 = 2 and B = {8, 7}. And finally, we have the deterministic path in C given by γ11 = 6. Therefore, t = 1, q1 = 1 and C = {6}. Since A ∪ B ∪ C = {1, . . . , 10} and the pair (F, G) has a monomial submatrix of order 10, then the pair (F, G) is reachable and its canonical form [P T F P |P T GQ] is given by 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 C

B

A1

A2 A3 A4 A10B

G

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4. Complete controllability Now we shall study the complete controllability property of a positive discretetime control system. By the relationship between this property and the reachability property (see [5]), we can establish the following characterization in terms of the sets A = A1 ∪ · · · ∪ An , B, and C. Theorem 4. Let (F, G) be a positive discrete-time control system. Then, (F, G) is completely controllable if and only if A1 ∪ A2 ∪ · · · ∪ An−1 = {1, . . . , n}. Proof. (H⇒): Suppose the pair (F, G) is completely controllable, then we know (see [5]) that the pair (F, G) is reachable and the matrix F is nilpotent. Then, by Theorem 1, the matrix [F |G] has a monomial submatrix of order n and A ∪ B ∪ C = {1, . . . , n}. To prove that A1 ∪ A2 ∪ · · · ∪ An−1 = {1, . . . , n}, we suppose, by contradiction, that there exists a vertex j ∈ B or j ∈ C or j ∈ An . In all these cases, by Proposition 2 and the construction of the set An , one can find a circuit in C(F ), and thus, F is not nilpotent. (⇐H): We have to show that the pair (F, G) is reachable and the matrix F is nilpotent. To see reachability, from Theorem 1, we have to prove that the pair (F, G) has a monomial submatrix of order n. That is true by the construction of the sets {Ah , h = 1, . . . , n − 1} and the condition A1 ∪ A2 ∪ · · · ∪ An−1 = {1, . . . , n}. In this case, the obtained canonical form is [P T F P |P T GQ] = [A0 |G] where the matrix A0 , given in (15), is strictly upper triangular. Thus, the matrix F is nilpotent. From the proof of Theorem 4, we can deduce the complete controllability canonical form. Theorem 5. Assume (F, G) > 0. Then, (F, G) is completely controllable if and only if there exists permutation matrices P and Q such that the matrix [P T F P |P T GQ] = [A0 , G], where the matrix A0 is given in (15). In the following result, we characterize the complete controllability of the pair (F, G) > 0 in terms of the directed graph C(F ). We recall that a matrix A is nilpotent if and only if there exists a permutation matrix P such that P T AP is strictly upper triangular, equivalently the vertices of C(A) can be relabelled 1, 2, . . . , n, in such a way that each arc (i, j ) satisfies j < i. Theorem 6. Let (F, G) be a positive discrete-time control system and let {α1 , . . . , αr } be the set of vertices such that, for each αi with i ∈ {1, . . . , r}, there exists a column of G which is an αi -monomial. Then (F, G) is completely controllable if and only if there exists in C(F ) r-deterministic paths generated from the vertices {α1 , . . . , αr }, such that these paths cover all vertices in C(F ) and the vertices of C(F ) can be relabelled in such a way that each arc (i, j ) satisfies j < i.

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Example 3. Consider the control system (F, G) > 0, with n = 5, given by 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 F = 0 0 0 1 1 and G = 0 1 0 0 . 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 The graph associated with the matrix F is

From the above graph, we have A1 = {1, 2}, A2 = {3}, A3 = {4}, A4 = {5}. Since A1 ∪ · · · ∪ A4 = {1, 2, 3, 4, 5}, then the pair (F, G) is completely controllable and its canonical form is given by 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 T T [P F P |P GQ] = 0 0 0 1 1 0 0 1 0 . 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 Theorem 7. Let (F, G) be a positive discrete-time control system. Then (F, G) is completely controllable if and only if (F, G) is reachable and for each column of G, there exists a vertex k ∈ {1, . . . , n} such that F k g = 0.

5. Essential reachability When not all nonnegative states can be reached in finite time, one considers the essential reachability property. A positive system (F, G) is said to be essentially reachable (see [5]) if all positive states are asymptotically reachable. From the construction of the reachability cones, one can say that (F, G) is essentially reachable if and only if R∞ (F, G) = Rn+ . That is, the states not reachable in a finite number of steps are limits of states which are reachable in a finite number of steps. In [9] it is established that a positive system (F, G) is essentially reachable if and only if for every vertex i 6∈ I (F, G) the following facts hold: (a) i belongs to some communicating class Ci of C(F ) consisting either of a single vertex i or of a circuit, (b) there exists some column vector gτi in G such that for every positive integer t > n the block of the components of F t gτi corresponding to Ci , denoted by

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blockCi (F t gτi ) (as we said in Section 1), constitutes a monomial vector. Moreover, for each communicating class C¯ = / Ci such that blockC¯ (F t gτi ) > 0 for + ¯ 6 ρ(Ci ). In addition, if ρ(C) ¯ = ρ(Ci ), then C¯ has some t ∈ Z we have ρ(C) access to Ci . We will establish conditions equivalent to those conditions (a) and (b) of the above result, and we will construct an essential reachability canonical form. First, we shall show the following result. Proposition 5. Assume (F, G) > 0. Then, for each vertex i 6∈ I (F, G) the condition (a) holds if and only if the matrix [F |G] has a monomial submatrix of order n. Proof. (H⇒): We will show that the matrix [F |G] includes, among its columns, all i-mononial vectors, i = 1, . . . , n. If i ∈ I (F, G), then we know that i-monomial vector is in [F |G] (see [9]). And if i 6∈ I (F, G) then, by hypothesis, we know that i is in some communicating class which consists of either the single vertex i or of hi vertices connected by a single circuit, i.e.

Note that in both cases the i-monomial vector appears in F. (⇐H): Since the pair [F |G] has a monomial submatrix of order n, from Proposition 4 we have I (F, G) = A ∪ B ∪ C. Then, by Proposition 1, for all i 6∈ I (F, G) = A ∪ B ∪ C, there exists a closed deterministic path containing the vertex i which is the circuit that we are looking for. Moreover, we have the following characterization of the condition (b) in terms of the subsets of V = {1, . . . , n}. Proposition 6. Assume that (F, G) > 0 is such that [F |G] has a monomial submatrix. Then, the condition (b) holds if and only if A ∪ B ∪ C ∪ D1 ∪ D2 = {1, . . . , n}. Proof. Since [F |G] has a monomial submatrix of order n, by Proposition 4 we have I (F, G) = A ∪ B ∪ C, and moreover, by Proposition 5, condition (a) holds for all i 6∈ I (F, G). (H⇒): We have to prove that D1 ∪ D2 = {1, . . . , n} − I (F, G). Suppose i 6∈ I (F, G) and i 6∈ D1 ∪ D2 . Then, by Proposition 1, there exists a circuit,

with id 6∈ I (F, G) ∀ d = 1, . . . , p. Moreover, by definition of sets D1 and D2 , id 6∈ D1 ∪ D2 ∀d = 1, . . . , p.

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Denoting by Ci the communicating class associated with the above circuit, and since id 6∈ D2 , then none of the columns of G have the special form indicated in the definition of set D2 . We can distinguish three cases: Case 1: If g = col G with at least two nonzero entries in the blockCi (g), then the blockCi (F t g) is not monomial for t > n, in contradiction with condition (b). Case 2: Suppose g = col G with only one nonzero entry in the blockCi (g) and with some nonzero entry outside this block such that the corresponding vertex has ¯ = ρ(Ci ) ¯ > ρ(Ci ). Then, by condition (b), ρ(C) access to a class C¯ satisfying ρ(C) and so, the class C¯ has access to the class Ci . By the construction of the sets A, B and C, and since the matrix [F |G] has a monomial submatrix of order n, we see that all vertices of the class C¯ are in An , since the vertices in B, C, A1 , . . . , An−1 and V − (A ∪ B ∪ C) are only connected to vertices in such sets. Thus, let k be the last vertex of a deterministic path of the set An which is in the class C and such that from k there exists one outgoing arc reaching some vertex ij , for some j ∈ {1, . . . , p} (note that from k there cannot be two or more outgoing arcs leading to the class Ci since the blockCi (F t g) is a monomial vector for some t ∈ Z+ by condition (b)). Earlier, we noted that if id 6∈ D1 ∪ D2 then one at least, among conditions (i)– (iii) of the construction of the set D1 does not hold. In this case, condition (iii) holds, thus we can suppose that either condition (i) or condition (ii) is not satified. If (i) does not hold, then there exists an index t ∈ {1, . . . , n} such that blockCi (F t g) is not a monomial vector, which is not possible. If condition (ii) is not fulfilled, then there exists a vertex η ∈ V − Ci , accessible from k, such that from η there exists a unique outgoing arc to a vertex ij 0 of Ci , and / length(Pij ij 0 ) + q · length(Ci ), q ∈ Z+ , length(Pdη ) = for some path Pdη . The difference between the lengths allows us to assert that blockCi (F t g) is not a monomial vector for some t ∈ Z+ , in contradiction with the hypothesis. Case 3: Finally, assume that g = col G has all zero entries in the blockCi (g). If a / 0 and h has access to a unique id does not exist, then vertex h ∈ An such that gh = there does not exist any integer t such that the blockCi (F t g) is a monomial vector, in contradiction with the hypothesis. If such h exists, then by the same reasoning, we find that id ∈ D1 , which is in contradiction with our assumption. (⇐H): We have to prove that condition (b) holds for the vertices in D1 and D2 . If i ∈ D2 then i is in a circuit (communicating class Ci ) of the type given in (10),

By construction of the set D2 , this path is associated with a column of G, g = αeδ 1 + w with α > 0 and w ∈ Rn+ such that wj = 0 ∀j ∈ Ci and satisfying condition (iv). Then such a column g satisfies the condition (b).

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If i ∈ D1 , then i is in a circuit (communicating class Ci ) of the type given in (8). Moreover, by condition (7) there exists a vertex in An having access to this path. Then, the monomial column in the matrix G associated with this vertex satisfies the condition (b). From the above propositions, we can establish the following result. Theorem 8. Assume (F, G) > 0. Then, (F, G) is essentially reachable if and only if [F |G] has a monomial submatrix of order n and A ∪ B ∪ C ∪ D1 ∪ D2 = {1, . . . , n}. Next, we give the essential reachability canonical form. For that, we construct some permutation matrices P and Q from the sets A, B, C, D1 and D2 , which constitute a partition of the set V = {1, . . . , n}, following the steps of the proof of Theorem 2 but adding the sets D1 and D2 . We keep the notation corresponding to the sets A = A0 ∪ An , B and C, but we take a finer partition of An = AnB ∪ AnD1 ∪ AnR , where AnB is defined as in the proof of Theorem 2 and AnD1 is set of the vertices of An in deterministic paths associated with vertices of set D1 , according to the definition of condition (7). Then, we denote the number of deterministic paths in AnD1 , D1 and D2 by rnD1 , l and h, respectively. Thus, rn = rnB + rnD1 + rnR . Then, we relabel in decreasing order the vertices, from n to 1, considering the sets ordered in the following way: AnR , AnD1 , AnB , An−1 , . . . , A1 , D1 , B, C and D2 , and in each set starting from the longest deterministic path and finishing with the shortest deterministic path. Define P as the permutation matrix associated with this relabelling, and let Q be the matrix reordering the columns of G in the order used in Theorem 2, that is, the first columns of G are monomial vectors, the following t columns of G have the structure considered in (3) and in addition, the following h columns have the form given in (9). Then, the pair [P T F P |P T GQ] has the following structure O O D2 O O O D D O O O C O O D D O O B O O R D D O O O D1 O O R D G (21) , O O O O A0 D D D O O O O O AnB D D O O O O O O An D D1

O

O

O

O

O

O

D

AnR

0

where C, B, A , D, AnB , and R are the matrices given in (14)–(16), (18) and (19), respectively. Moreover, AnR , is a block matrix where all off-diagonal blocks are W and the blocks in the diagonal have the structure given in (20). In addition, D1 = diag[D1 (l), . . . , D1 (1)]

and D2 = diag [D2 (h), . . . , D2 (1)] , (22)

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where D1 (i) and D2 (j ), ∀i = 1, . . . , l and ∀j = 1, . . . , h, are cyclic irreducible blocks. Further h i (23) AnD1 = diag AnD1 (l), . . . , AnD1 (1) , where AnD1 (j ), j = 1, . . . , l, has the same structure as the matrices given in (17). Note that if there exists some nonzero entry in the matrix D, connecting AnD1 with AnR , then there exists restrictions given by the definition of set D1 on the nonzero entries in the column block D associated with AnR . Moreover, it is possible to check that the pair (P T F P , P T GQ) given in (21) is essentially reachable. Therefore, we summarize the above discusion in the following theorem. Theorem 9. Assume (F, G) > 0. Then, (F, G) is essentially reachable if and only if there exists permutation matrices P and Q such that [P T F P | P T GQ] has the structure given in (21), where the blocks are given in (14)–(23). Example 4. Consider the control system (F, G) > 0 given in Example 1. We note that A ∪ B ∪ C ∪ D1 ∪ D2 = {1, . . . , 11}. Then, the pair (F, G) has a monomial submatrix of order 11. Therefore, the pair (F, G) is essentially reachable and its canonical form is given by

P T F P |P T GQ 1 0 0 0 0 = 0 0 0 0 0 0

0 1 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0

0 0 0

0 0 1

0 0 1

0 0 0

0 0 0

0 0 0

1 2

1 2

0 0

0 0 0

0 0 0 0

0 0 1 0

0 1 1 0

0 0 0 0

0 0 0 0

1 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0

1 4

0

0 0 1

0

0

0

0

0

0 0 0

1 4

1 4

D2 C B

D1

A0 AnB

0 0 1

AnD1

0 0

0

AnR

0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 0

1 0 1 0 1 1 0 0 0 0 1

G

Finally, since the essential reachability property together with the stability property is equivalent to the essential controllability property, the canonical form of this last property coincides with that of the former property adding that the spectral radius of F is less than 1.

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Acknowledgement We are grateful to Elena Valcher for her helpful comments and to an anonymous referee for constructive comments which have significantly improved this paper.

References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [2] R. Bru, V. Hernández, Structural properties of discrete-time linear positive periodic systems, Linear Algebra Appl. 121 (1989) 171–183. [3] R. Bru, C. Coll, V. Hernández, E. Sánchez, Geometrical conditions for the reachability and realizability of positive periodic discrete systems, Linear Algebra Appl. 256 (1997) 109–124. [4] R. Bru, S. Romero, E. Sánchez, On Reachability and controllability of periodic positive discrete-time linear system, Tech. Report 30, Univ. Politècnica de València, València, Spain, 1998. [5] P.G. Coxson, H. Shapiro, Positive input reachability and controllability of positive systems, Linear Algebra Appl. 94 (1987) 35–53. [6] P.G. Coxson, I.C. Larson, H. Schneider, Monomial patterns in the sequence Ak b, Linear Algebra Appl. 94 (1987) 89–101. [7] L. Farina, Necessary conditions for positive realizability of continuous-time linear systems, Syst. Cont. Lett. 25 (1995) 121–124. [8] Y. Ohta, H. Maeda, S. Kodama, Reachability observability and realizability of continous-time positive systems, SIAM J. Cont. Optim. 22 (1984) 171–180. [9] E. Valcher, Controllability and reachability criteria for discrete time positive system, Int. J. Cont. 65 (3) (1996) 511–536. [10] J.M. van den Hof, Realization of positive linear systems, Linear Algebra Appl. 256 (1997) 287–308.

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