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A construction method for positive realizations with an order bound
Systems & Control Letters 61 (2012) 759–765
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A construction method for positive realizations with an order bound Kyungsup Kim Department of Computer engineering, Chungnam National University, Yuseong-gu, Daejeon, Republic of Korea
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Article history: Received 21 October 2011 Received in revised form 15 March 2012 Accepted 18 April 2012 Available online 23 May 2012
This paper presents an efficient construction method to address the positive realization problem with an order bound for primitive transfer functions with multiple real or complex conjugate pairs of poles using polyhedral cone concepts. Through a down-sampling step, the non-primitive transfer function is decomposed into a sum of primitive transfer functions. A construction method for the positive realization of a primitive transfer function with multiple poles is proposed by using a regular polygon in the complex plane. A sufficient condition for the existence of the positive realization with an order bound is analyzed. Numerical examples are provided to demonstrate the efficiency of the proposed method for obtaining positive realizations. © 2012 Elsevier B.V. All rights reserved.
Keywords: Positive realization Positive linear system Discrete time linear systems Impulse response sequence
1. Introduction This paper discusses the realization problem of a class of linearinvariant system, in which state variables, input and output are restricted to be non-negative to reflect physical constraints. The non-negative constraints can be encountered in many applications in engineering, medicine and economics. The positive systems can be applied to applications such as compartmental models in pharmacokinetics [1], charge-routing networks [2], fiber-optical filters [3] and TCP-like congestion control problems [4]. A positive linear system denotes a linear dynamical system in which the input u, the state x and the output y are non-negative real values. In this work, positive linear systems are considered in discrete time. Assume that a rational transfer function of a given positive system exists with McMillan degree n, expressed as H (z ) =
b n −1 z n −1 + · · · + b 0 zn
+ an − 1
x n −1
+ · · · + a0
=
hk z −k .
k≥0
Then, the impulse response sequence {hk } of H (z ) is non-negative (i.e., hk ≥ 0 for all k ≥ 0). Thus, the linear system can be represented by: x(k + 1) = Ax(k) + bu(k) y(k) = cx(k), where A ∈ Rn×n , b ∈ Rn , c T ∈ Rn and c T is a matrix transpose of c. The rational transfer function H (z ) has a state-space realization
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in the form H (z ) = c (zI − A)−1 b and the triple (A, b, c ) denotes a state-space realization of H (z ). A transfer function H (z ) with a realization (A+ , b+ , c+ ) has a positive realization if a triple (A+ , b+ , c+ ) exists such that A+ ∈ Rn+×n , b+ ∈ Rn+ , and c+T ∈ Rn+ , where R+ denotes the set of non-negative real numbers. A linear system with a positive realization (A+ , b+ , c+ ) clearly induces a positive linear system. Conversely, the positive realization problem is to find a positive realization (A+ , b+ , c+ ) that represents a given positive linear system. The problems associated with the existence and the minimality of a positive realization have been studied widely in the past few decades, but a complete answer is not yet available. The positive constraint may enforce realizations of larger dimensions than the McMillan degree [5,6]. The minimality problem of positive realizations has been considered in a number of papers [7,8]. The lower-bound problems of positive realizations have already been discussed in [7,9,10]. The construction of a positive realization of a transfer function with simple complex poles was discussed in [11]. The construction of a positive realization of a transfer function with non-negative, real, multiple poles was considered in [12]. A straightforward algorithm to construct a positive realization for a transfer function with multiple poles (complex or negative) was described in [9]. In this paper, we will consider positive realizations with tighter bounds for transfer functions with possibly multiple real or complex poles. We extend earlier results on positive realizations from Benvenuti et al. [11] and Nagy et al. [9], and we provide a simple and unified solution to the positive realization problem of the primitive transfer function with tighter upper bounds and multiple poles. We demonstrate the proposed construction
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K. Kim / Systems & Control Letters 61 (2012) 759–765
method for positive linear systems with multiple complex or real poles using two examples. The paper is organized as follows. In Section 2, preliminary results for positive linear systems are introduced. A construct method for a positive realization of a primitive positive linear system and a sufficient condition for the efficacy of the proposed algorithm are presented in Section 3. In Section 4, two examples are provided to demonstrate the efficiency of the proposed algorithm. Finally, the conclusions are presented in Section 5. 2. Preliminary results related to positive linear systems The notation used throughout the paper is introduced first. X = cone(X ) denotes the smallest convex cone of a set X , which consists of all finite, non-negative, linear combinations of elements of the set X . The dual cone, X∗ , of a cone X is defined by X∗ = {y|xT y ≥ 0, ∀x ∈ X}. A convex cone X is said to be a polyhedral cone if it is spanned by a finite set of vectors X = {x1 , . . . , xm } with xi ∈ Rn ; i.e., X = cone({x1 , . . . , xm }), and X is called a polyhedral generator of X. X can denote the matrix with columns xi ∈ X , i.e., X = x1 x2 · · · xm , as well, if it is not ambiguous. An extreme point of a convex cone is one that is not a proper positive linear combination of any two points of the set. A finite set X is said to be a frame of the polyhedral cone X if the points of X are extreme points in X and X spans X [13]. Consider the necessary and sufficient conditions for the existence of a positive realization introduced in [14,15,11]. We refer to the result in [11], without a detailed proof, in the next theorem. Theorem 2.1. Let H (z ) be a strictly proper rational transfer function with a (minimal) realization (A, b, c ). There exists a positive realization (A+ , b+ , c+ ) and a generator matrix K with K = cone(K ) such that 1. AK ⊂ K 2. K ⊂ O where an observability cone O is defined by O = {x|cAk x ≥ 0, k = 0, 1, . . .} 3. b ∈ K . Furthermore, there is a positive realization {A+ , b+ , c+ } satisfying FK = KA+ , b = Kb+ , and c+ = cK . The above theorem provides an interpretation of the positive realization problem. Given any (minimal) realization of a transfer function, a polyhedral cone K satisfying the above three conditions corresponds to any positive realization. The number of edges of the cone K equals the dimension of the positive realization [11]. The construction method of the positive realization is related to the construction of the polyhedral cone generator K that satisfies the above three conditions. Let H[M ] (z ) denote a transfer function corresponding to the ∞ −j shifted sequence {hk }k≥M +1 i.e., H[M ] (z ) = . The j=1 hj+M z transfer function H[M ] (z ) is called the M-shifted transfer function of H (z ). We refer to the result in [5]. −j Lemma 2.1 ([5]). Let H (z ) = be a strictly proper rational j =1 hj z transfer function with a non-negative impulse response hk for k ≥ 1 and H (z ) = H c (z ) + z −M H[M ] (z ) where H c (z ) is a transfer function with a finite impulse response sequence. Assume that the M-shifted transfer function H[M ] (z ) admits a positive realization (A2 , b2 , c2 ) with dimension K . Then, H (z ) admits a positive realization of dimension K + M.
∞
Proof. The transfer function H c (z ) with a non-negative finite impulse response has an M-dimensional positive realization
(A1 , b1 , c1 ) as follows: 0 1 0 ··· 0 0 1 · · · . .. A1 = . .. 0 0 0 · · · 0
0
c1 = hM
···
0
hM −1
···
0 0
0
0 . b1 = .. 0
.. . , 1 0
1
h2
h1 .
The positive realization of the transfer matrix H[M ] (z ) with dimension K is given by (A2 , b2 , c2 ). The following combination of the two realizations yields a non-negative realization (A, b, c ) of H (z ) as follows:
A1 A= b2 0
0 , A2
b=
b1 , 0
c = c1
c2 .
Therefore, H (z ) admits a positive realization of dimension K + M. If H (z ) has a realization (A, b, c ), then the M-shifted transfer function H[M ] (z ) is given by H[M ] (z ) = cAM (zI − A)−1 b, and has a realization (A, b, cAM ). A necessary condition for the existence of a positive realization is that one of the dominant poles of H (z ) (i.e., the poles with maximum modulus) is non-negative and real [14]. The distinct pole set {λi } of H (z ) is ordered by |λj | ≤ |λi | ≤ 1 for j > i. Let λ0 be the dominant pole with λ0 > 0. The transfer function H (z ) is called primitive if λ0 is a unique dominant pole. The transfer function H (z ) has a positive realization if and only if β H (α z ) has a positive realization for α > 0 and β > 0 [5]. Therefore, we can set λ0 = 1 without any loss of generality by choosing values of α and β . If H (z ) is not primitive, a necessary condition for the existence of a positive realization is that the dominant poles of H (z ) be cyclic p (z p = λ0 for some p) [14]. It is known that if the dominant poles are not cyclic, then there is no positive realization of H (z ) [16]. If the dominant poles are cyclic with index p, then the necessary and sufficient condition for a positive realization is that all the downsampled transfer functions H(j) (z ) = c (zI − Ap )−1 Aj b for 0 ≤ j < p must be positive realizable [16]. If some of the functions H(j) (z ) are not primitive, then the down-sampling step is applied recursively to these functions, until they become primitive. After recursively applying the down-sampling step, H (z ) is decomposed into the following form: H (z ) =
z νs H(s) (z µs ),
(1)
s
where µs and νs are integers with 0 ≤ νs < µs and all the functions H(s) (z ) are primitive [16]. It is known that if every primitive transfer function H(s) (z ) is realizable, then H(s) (z µs ) and the original transfer function H (z ) can be positive realizable [16]. Therefore, it is sufficient to find a positive realization of only the primitive transfer functions. Using a partial fraction expansion, a primitive positive transfer function H (z ) is given by H (z ) =
n0 k=1
nj τ βk(0) βk(j) + , (z − λ0 )k (z − λj )k j=1 k=1
(2)
where nk is the maximum number of the multiplicity of λk . From the result in [16], if a primitive transfer function H (z ) has a nonnegative impulse response sequence, then the leading coefficient βn(00 ) of the first term in Eq. (2) is necessarily positive. We divide H (z ) into H (z ) = H c (z ) + z −M H[M ] (z )
(3)
K. Kim / Systems & Control Letters 61 (2012) 759–765
for sufficiently large M > 0. For sufficiently large M > 0, the Mshifted transfer function H[M ] (z ) can be decomposed into H[M ] (z ) = F[M ] (z ) + G[M ] (z )
(4)
and F[M ] (z ) and G[M ] (z ) have non-negative impulse response sequences. The transfer function F[M ] (z ) is given by n0
F[M ] (z ) =
k=1
(0)
βk (z − λ0 )k
(5) (0)
with the leading coefficient βn0 > 0 and βk and G[M ] (z ) is given by
≥ 0 for 1 ≤ k < n0 ,
j τ τ τ β0 βi(j) + = Gj (z ) z − λ0 (z − λj )i j=1 i=1 j=1
n
for some sufficiently large β0 > 0, where Gj (z ) = (j)
βi
nj
i=1 (z −λj )i .
(6) β0 z −λ0
+
Therefore, for sufficiently large values of M and β0 ,
we can obtain the transfer functions H (z ), F[M ] (z ) and G[M ] (z ) with non-negative impulse response sequences as given in Eqs. (3), (5) and (6), based on step 4 in [16]. A positive realization of F[M ] (z ) is obtained easily in [16]. Therefore, using Lemma 2.1, we need to find only a positive realization method for Gi (z ) for all i. c
3. Explicit positive realization algorithm We will consider the construction realization of Gi (z ) in this section. The given in Eq. (6) has a minimal canonical
(Aj , bj , cj ) such that Ai = 1 ⊕ J (λj ), bi =
O
cˆ = cˆ2n
···
cˆ1
λj 0 . J (λj ) , .. 0 (nj )
0 1
.. .
··· ··· .. .
0 0
0 0
··· ···
1
0 cˆj = βj
λj
(nj −1)
βj
···
method of a positive transfer function Gi (z ) Jordan-form realization 1 eni
and ci = β0
cˆj
0
0 0
0 . en , .. 0
.. . , 1
λj
βj(1)
defined by the block matrix A =
O
···
C
O
0
.. .
.. . ˆb = 0 1
(8)
0
for a proper number w , where a matrix C = C (x, y) ,
x y
−y x
yields
Proof. The transfer function H (z ) with the partial fraction expansion in Eq. (7) has a canonical Jordan form realization (A, b, c ) such that
¯ j ), A = J (λj ) ⊕ J (λ
b=
en , en
c = βn
β1
β¯ n
βn−1
···
1
A O
O B
with proper zero matrices
Lemma 3.1. Assume that a real, rational transfer function H (z ) with multiple complex conjugate poles is decomposed into
βk β¯ k + , (z − λ1 )k (z − λ¯ 1 )k
(7)
where the pole λ1 and the coefficients βi are complex and β¯ i is defined as the conjugate of βi . Then, H (z ) has a real Jordan-form realization (Jˆ(x, y, w), bˆ , cˆ ) whereby Jˆ(x, y, w) ∈ R2n×2n , bˆ ∈ R2n and cˆ T ∈ R2n are given by
β¯ 1 .
···
Additionally, there exists a permutation matrix P such that a block Jordan-form realization is given by
··· ··· ··· .. .
E O O A1 = P −1 AP =
I E O
O I E
O
O
O
...
β¯ n
β1
···
¯ . Let H = where E = λ ⊕ λ
O O O ,
0
.. .
.. . b1 = 0 , 1
··· E ¯β1 , 1 1
−i i
1
. Using a block-diagonal matrix
n
P1 = k=1 H as a similarity transformation, we obtain a new realvalued realization (A2 , b2 , c3 ) such that C O O A2 = P1−1 A1 P1 =
I C O
O I C
··· ··· ··· .. .
O O O ,
O
O
O
···
C
c2 = β˜ 2n
O, and the k-th element of the basis vector ek is 1 and the rest (i) of elements 0. We note that the poles λi and the coefficients βj could be complex for j ≥ 1 because complex poles or zeros of a rational transfer function with real coefficients in the numerator and denominator polynomials could exist as complex conjugate pairs. Hence, we consider the real-valued realization of a transfer function with multiple complex poles.
k=1
O O O ,
wC
...
for 1 ≤ j ≤ r and A0 = λ0 = 1.A block diagonal matrix A ⊕ B is
n
C
··· ··· ··· .. .
O
C O
H (z ) =
wC
C
O ˆJ (x, y, w) , O . ..
c1 = βn
where
761
a representation of a complex number x + iy and cˆk for all k are real. (0)
G[M ] (z ) =
β˜ 2n−1
···
0
.. .
.. . b2 = 0 , 1 0
β˜ 1 ,
where the entries of c2 are defined by β˜ 2k = 2Re(βk ) and β˜ 2k−1 = 2Im(βk ) for each k. Finally, we obtain the real block Jordan-form matrix (Jˆ(x, y, w), bˆ , cˆ ) such that Jˆ(x, y, w) = P3−1 A2 P3 in Eq. (8), bˆ is the same as b2 , and cˆ = c2 P3 has real-valued n components, k−1 where P3 is defined by P3 = D1 D2 with D1 = , and k=1 (w C ) n −n +1 0 D2 = (w C ) . We assume that B = I for any square k=1 matrix B. Let Pm (m ≥ 1) denote the set of points in the complex plane that lie in the interior of a regular polygon, with m edges, which has one vertex at point 1 and includes point 0. For m ≥ 2, the polygon Pm is defined by a subset in R2 through the following inequalities:
Pm = (x, y)|r cos
(2k + 1)π m
π − ϕ ≤ cos , m
for all k with 0 ≤ k ≤ m, where x = r cos ϕ and y = r sin ϕ . For m ≥ 3, the polygon Pm was introduced in [11]. The polygon Pm can be extended to the case where m = 1 and m = 2 without lose of generality. For m = 2, P2 is defined by P2 = {(x, 0)||x| ≤ 1}, because there are two vertices that have one vertex at point 1 and π cos m = 0. For m = 1, P1 is defined by P1 = {(x, 0)|0 ≤ x ≤ 1}, because it has at least vertices at point 1 and 0.
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K. Kim / Systems & Control Letters 61 (2012) 759–765
k > 1, A(λ1 , w)Vk has only the 1-st, k-th and k + 1-th row block components with non-zero values; the others are zeros, as follows:
Theorem 3.1. Assume that a transfer function G1 (z ) =
n βk β¯ k β0 + + (z − 1) k=1 (z − λ1 )k (z − λ¯ 1 )k
(9) 1
has a non-negative impulse response sequence for a sufficiently large ¯ 1 } ⊂ Pm value of β0 > 0, where λ1 = r cos θ + ir sin θ has {λ1 , λ for some m ≥ 3 and n ≥ 1. Let us define rm as
A(λ1 , w)Vk =
k
rm = max{ˆr |(ˆr cos θ , rˆ sin θ ) ∈ Pm , rˆ > 0}
k+1
for the given angle θ . Assume that for the given λ1 and the given m, conditions of w and {βk } for 1 ≤ k ≤ n are given by 0
1−
r
rm
rm
(10)
r
β0 . (2w r )k−n
|βk | ≤
(11)
Then, there exists a positive realization (A+ , b+ , c+ ) of the transfer function G1 (z ) that has the order mn.
¯ 1 } where Proof. The transfer function G1 (z ) has a pole set {1, λ1 , λ λ1 = r cos θ + ir sin θ ∈ Pm and has a positive impulse response sequence. Using the result of Lemma 3.1, we obtain a real block Jordan-form realization (A(λ1 , w), b, c ) of the transfer function H (z ) in Eq. (9) as follows:
A(λ1 , w) = P −1 AP = 1 ⊕ Jˆ(x, y, w),
b=
1 bˆ
cˆ ,
c= 1
O
I O Z = . .. O
O O I
.. .
··· ··· ··· .. .
O O O
O O O ,
O
···
I
O
.. .
ˆ
K
O . K = .. ,
.. .
O O
where I is a 2 × 2 identity matrix, O is a 2 × 2 zero matrix and a 2 × m matrix Kˆ is defined as
1
cos
Kˆ = 0
sin
2π
cos
m 2π
sin
m
4π m 4π m
···
cos
···
sin
2π (m − 1)
m . 2π (m − 1) m
A cone generator matrix V ∈ R(2n+1)×mn is defined as
V = V1
V2 · · · Vn ,
where Vk is defined as Vk =
Z
eT k−1
K
for 1 ≤ k ≤ n and e represents
an m × 1 vector with all of its entries equal to 1. The columns of matrix V represent the edges of a finite generated cone V in R2n+1 (i.e., cone(V ) = V ) and are positive independent. We shall prove that V is A(λ1 , w)-invariant(i.e., A(λ1 , w)V ⊂ V ). Let Vk be the cone generated from the column vectors of Vk (i.e.,Vk = cone(Vk )). When λ1 ∈ Pm , we can easily see that A(λ1 , w)V1 ∈ V1 . For
eT
.. .
O wC Kˆ C Kˆ O .. .
.
(12)
If A(λ1 , w)Vk ∈ cone{Vk−1 , Vk } for every k > 1 is satisfied, then V is sufficiently A(λ1 , w)-invariant. Choose {Wk−1 , Wk } for each k such that A(λ1 , w)Vk = α1 Wk−1 + α2 Wk , where all αj ≥ 0, α1 + α2 = 1 and the entries in the first row of Wk ∈ Vk are equal to 1. Let a 2 × m matrix Wk,k+1 be defined by the k + 1-th row component of Wk as shown in Eq. (12). Then, from Eq. (12), we can see that α1 Wk−1,k = w C Kˆ and α2 Wk,k+1 = C Kˆ . For the direction of the angle θ , the maximum norm of the column vectors of Wk,k+1 is less than rm for any k, and the norm of the column vectors of C Kˆ r ≤ α2 and rrm ≤ α1 . A is r. We can derive two inequalities, w rm sufficient condition for the existence of a feasible solution (α1 , α2 ) r that satisfies w ≤ α2 , rr ≤ α1 , and α1 +α2 = 1 can be obtained as r m
m
0 < w ≤ (1 − rr ) rrm . Therefore, we can show that Vk is A(λ1 , w)m invariant because A(λ1 , w)Vk ∈ cone{Vk−1 , Vk }. We can easily verify that the second condition is satisfied because b ∈ Vn . Finally, we will prove that c = β0 cˆ ∈ V ∗ . The 2k and the 2k − 1 entries of cˆ are given by cˆ2k cˆ2k−1 =
β˜ 2k−1 (w C )k−n from cˆ = D1 D2 c2 in Lemma 3.1. An inequality cVk ≥ 0 for each k ≥ 1 leads to the sufficient condition
where x = r cos θ , y = r sin θ and (Jˆ(x, y, w), bˆ , cˆ ) is defined in Eq. (8). We can now demonstrate that there exists an A(λ1 , w)invariant cone V with mn edges satisfying Theorem 2.1 (i.e., AV ⊂ V ). We generalize the concept of the cone generator introduced in [11] for the case with multiple complex poles. To formulate a cone generator, a block shift matrix Z ∈ R2n×2n and a matrix K ∈ R2n×m are defined as:
β˜ 2k
in Eq. (11) as follows:
β0 e + β˜ 2k
β˜ 2k−1 (w C )k−n Kˆ ≥ 0.
The l2 norm of each column vector of C k−n Kˆ equals r k−n . To satisfy the above inequality, we impose a sufficient condition such that
|βk | ≤
β0 (2w r )k−n
for sufficiently large β0 . W can show that cVk ≥ 0. Therefore, we obtain a positive realization (A+ , b+ , c+ ) with ×nm T nm order nm such that A+ ∈ Rnm , b+ ∈ Rnm + + and c+ ∈ R+ satisfy A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV , respectively, using a suitable linear programming tool. The transfer function Gk (z ) in Eq. (9) is derived from the shifted transfer function H[M ] (z ) in Eq. (4). The shifted transfer function H[M ] (z ) has the realization (A, b, cAM ). The norm of cˆ (proportional to r M ) decreases as M increases. Therefore, we can impose a tradeoff between the condition (11) and the shifted dimension M. Our approach provides a more general and unified solution to the problem of finding a positive realization of a transfer function with multiple complex poles or multiple real poles and has a tighter upper bound on the dimension than that of [9]. Our approach additionally covers the known results for the positive realization of the transfer function G1 (z ) with negative or positive real multiple poles [9,12]. First, it is worth noting that the positive realization of the transfer function G1 (z ) with negative real multiple poles in Theorem 2 of [9] can be explained as a special case of Theorem 3.1. Theorem 3.2. Assume that the transfer function G1 (z ) =
n βk β0 + z−1 ( z − λ1 ) k k=1
K. Kim / Systems & Control Letters 61 (2012) 759–765
has a non-negative impulse response sequence where n ≥ 1 is an integer, the real pole satisfies −1 < λ1 < 0 (i.e., λ1 ∈ P2 ). Assume that for the given λ1 and m = 2, conditions of w and {βk } for 1 ≤ k ≤ n are given by 0
1−r
(13)
r
β0 |βk | ≤ (w r )k−n
(14)
for r = |λ1 |. Then, G1 (z ) has a positive realization of dimension 2n. Proof. The pole λ1 < 0 is included in P2 . We have m = 2. The transfer function G1 (z ) has a Jordan-form realization (A(λ1 , w), b, c ) given by 1
0 . b= .. , 1
A(λ1 , w) = 1 ⊕ J (x, 0, w),
0
βn
c= 1
β1
0···
0 ,
where x = r and y = 0. A 2 × 2 matrix Kˆ is given by
Kˆ =
cos 0
cos
sin 0
sin
2π
2 = 1 0 2π
−1 0
.
2
(2n+1)×2n
Thus, K ∈ R and V ∈ R(2n+1)×2n are defined in the same manner as in Theorem 3.1. We note that V equals an augmented matrix that can be obtained by inserting a zero in every odd k-th row such that k ≥ 3, and it yields a result equivalent to that of [9]. Because we have rm = 1, the condition (13) is obtained similar to the way Eq. (10) was. The remaining derivations are the same as those of Theorem 3.1. We obtain a positive realization (A+ , b+ , c+ ) with dimension nm such that A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV using linear programming. Next, we show the cases of the transfer functions with nonnegative multiple poles in our framework as well. The result is the same as that shown in [12,9]. The resulting dimension is observed to be minimal in this case. Theorem 3.3. Let λ1 be a non-negative number with 0 ≤ λ1 < 1 (i.e. λ1 ∈ P1 ). Assume that a transfer function G1 (z ) =
n βk β0 + z−1 ( z − λ1 )k k=1
Proof. By definition, the pole λ1 > 0 is included in P1 . We have m = 1. The transfer function G1 (z ) has a minimal Jordan-form realization (A, b, c ) such that A ∈ R2n+1×2n+1 , b and c are defined as: 1
0 . b= .. , 1 0
c= 1
βn
0···
β1
0 ,
where x = r and y = 0. A 2 × 1 matrix Kˆ is defined as Kˆ =
cos 0 1 = . sin 0 0
K ∈ R2n+1 is defined in a similar manner as in Theorem 3.1. Vk is
defined by Vk =
eT
Z k−1 K
for 1 ≤ k ≤ n. In this definition, we note
that {Vk } cannot cover the vector e1 induced by the zero in P1 . In addition, a vector V0 is defined as follows: V0 = e1 ∈ R2n+1 . An augmented cone generator matrix V ∈ R(2n+1)×mn is defined by
V = V0
V1
V2 · · · Vn .
The remaining derivations are similar to those for Theorems 3.1 and 3.2. We obtain a positive realization (A+ , b+ , c+ ) with dimension n + 1 by solving a suitable linear programming such that A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV . The dimension n + 1 is equal to the minimal realization dimension of G1 (z ). We summarize the construction process of the positive realization of a given transfer function H (z ) as follows: 1. If H (z ) is not primitive, then it is decomposed into the sum of primitive transfer functions as in Eq. (1) using a down-sampling step recursively. If the primitive transfer functions have positive realizations, then the original transfer function is also positive realizable as well. 2. Each induced primitive transfer function H (z ) can be divided into the form H (z ) = H c (z )+ z −M H[M ] (z ) for a sufficiently large M. Using a partial fraction τ expansion, H[M ] (z ) is rearranged into H[M ] (z ) = F[M ] (z ) + j=1 Gj (z ) as in Eq. (6). 3. With Theorems 3.1–3.3, we can compute positive realizations of the primitive positive systems Gk (z ). Let N0 denote the maximal order of the multiple dominant poles and N0 equal to the order of the positive realization of F[M ] (z ). Let N1 denote the sum of the orders of the (possibly multiple) non-negative poles in P1 except for a dominant pole, and let N2 be the number of (possibly multiple) negative real poles in P2 except for a dominant pole. Let Nm (m ≥ 3) denote the total number of complex conjugate pairs of (possibly multiple) complex poles of the transfer functions in the form Gk (z ), belonging to the following region: m−1
Pm −
Pj ,
j =1
except for a dominant pole. Using Theorem 3.1, we can show that the upper bound order of the positive realization is mNm . 4. Finally, we obtain a positive realization of the primitive transfer function H (z ) by using Lemma 2.1 and the upper bound of the positive realization of H (z ) is given by
has a positive linear system where n ≥ 1 is an integer. Assume that for the given λ1 and m = 1, conditions of w and {βk } for 1 ≤ k ≤ n are given by Eqs. (13) and (14). Then, G1 (z ) has a positive realization of dimension n + 1.
A(λ1 , w) = 1 ⊕ J (x, 0, w),
763
N = M + N0 +
1 + N1 +
jNj
.
(15)
j ≥2
Remark 3.1. The proposed construction method can cover the known positive realization results for the case of transfer functions with exclusively simple (complex) poles, multiple positive poles and multiple negative poles in the Refs. [7,11,12]. The case of a transfer function with multiple complex poles remains open. The proposed method additionally works well for the case of a transfer function with multiple complex poles. The proposed upper bound N in Eq. (15) of the positive realization is lower than that of [9] if λk lies in Pm for m ≥ 3. We provide a tighter upper bound for the case of a transfer function with multiple complex poles. The explicit sufficient condition for the existence of a positive realization is additionally established in the previous theorems. We remark that the proposed method provides a unified and more general method
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K. Kim / Systems & Control Letters 61 (2012) 759–765
for the positive realization of primitive transfer functions with a tighter upper bound than the previous ones. The notion of IPR (Internally positive realization) has been introduced in [17] as a generalization of realization of non-positive filters by means of the difference of positive filters [11]. In IPRs, the difference of the outputs of two positive systems is replaced by a more general output transformation applied to the state of a single positive system. We note that the positive N-representation of IPR in completely different issue is closely related to our proposed method in the point of using a regular polygon containing complex eigenvalues.
leads to a positive realization (A1+ , b1+ , c1+ ) with order 2N2 = 6 using Theorem 3.2. Similarly, the transfer function G2 (z ) leads to a positive realization (A2+ , b2+ , c2+ ) with order 5N5 = 10 using Theorem 3.1. From Eq. (15), the total upper bound N = 16 is less than the total upper bound 22 in the Ref. [9]. Example 4.2. Consider a primitive transfer function with multiple conjugate complex poles, multiple dominant poles and a nonnegative impulse response sequence such that H (z ) =
z − 0.99
−
+
0.1 1000(z + 0.9)3
100(z − 1)
(50z 2 − 70z + 29)
−
50z 2 − 70z + 20
(50z 2 − 70z + 29)2
.
The transfer function H (z ) has the pole λ0 = 0.99, the pole λ1 = −0.9 that is of order 3 and the complex conjugate pair of poles λ2 = 0.7 + 0.3i and λ¯ 2 that are of order 2. In this case, the shifting of the transfer function is not required. We have H c (z ) = 0 and F (z ) = 0. The transfer function H (z ) is properly divided into two terms H (z ) = G1 (z ) + G2 (z ), where the positive transfer function G1 (z ) has the pole λ0 = 0.99, the pole λ1 = −0.9 that is of order 3 and the positive transfer function G2 (z ) has the pole λ0 = 0.99 ¯2 and the complex conjugate pair of poles λ2 = 0.7 + 0.3i and λ that are of order 2. Because λ1 is included in P2 , set m = 2. We get rm = 1 and r = 0.9. The maximum of w is computed as w1 = 1/9 by using Eq. (13). We can verify that the condition (14) with respect to {βk } is satisfied. The transfer function G1 (z ) is determined by the real Jordan-form realization (A1 , b1 , c1 ) where A1 = 0.99 ⊕ A(λ1 , w1 ) with the augmented matrix A(λ1 , w1 ) given by
−0.9 0 0 A(λ1 , w1 ) = 0
0 0
0 −0.9 0 0 0 0
−0.1 0
−0.9 0 0 0
0 −0.1 0 −0.9 0 0
0 0 −0.1 0 −0.9 0
0 0 0 −0.1 0 −0.9
with w1 = 1/9 and c1 = 5.00 0.02 0 0 0 0 0 . We use an augmented matrix A(λ1 , w1 ) for the purpose of the unified and general approach. It can be reduced to a 3 × 3 Jordan-form matrix by removing the unobservable part. Because λ2 is included in P5 , we set m = 5. We get rm = 0.8296 and r = 0.7616. The maximum of w is computed as w2 = 0.0894 by using Eq. (10) in Theorem 3.1. We can verify that the condition (11) with respect to {βk } is satisfied. The transfer function G2 (z ) is determined by the real Jordan form realization (A2 , b2 , c2 ), where A2 = 0.99 ⊕ A(λ2 , w2 ) with
0.7 −0.3 A(λ2 , w2 ) = 0 0
β1 β¯ 1 , + (z − λ1 )3 (z − λ¯ 1 )3
0.3 0.7 0 0
0.0626 −0.0268 0.7 −0.3
0.0268 0.0626 0.3 0.7
w2 = 0.0894 and c2 = 5 −0.2701 0.1158 −2 −2 . We have the order N2 = 3 for G1 (z ) and the order N5 = 2 for G2 (z ) because λ1 ∈ P2 and λ2 ∈ P5 . We obtain a positive realization (A+ , b+ , c+ ) such that A(λ1 , w1 )V = VA1+ , b1 = Vb1+ and c1+ = c1 V using linear programming. The transfer function G1 (z )
β0 (z −1)
β
β
β¯
and G(z ) = (z −01) + (z −λ1 )3 + (z −λ¯1 )3 for a given 1 1 β0 = 8. There is a sufficiently large M such that the M-shifted transfer functions F[M ] (z ) of F (z ) have positive realizations, and the M-shifted transfer functions G[M ](z ) of G(z ) have non-negative impulse response sequences and are positive realizable. H (z ) is decomposed into H (z ) = H c (z ) + z −M (F[M ] (z ) + G[M ] (z )). The order of the positive realization of F[M ] (z ) is N0 = 3. Now, we consider a positive realization of G[M ] (z ). Because λ1 is included in P4 , set m = 4. We get rm = 0.7115 and r = 0.6403. The maximum of w is computed as w1 = 0.1111 by using Eq. (10) in Theorem 3.1. We can verify that the condition (11) with respect to {βk } is satisfied. Therefore, G[M ] has a realization (A1 , b1 , c1 ) such that A1 = 1 ⊕ A(λ1 , w1 ) is a 7 × 7 matrix with w1 = 0.1111 and c1 = 8. −7.938 5.022 −3.528 2.232 −0.588 0.372 . We have M = 4 and N4 = 3. We can obtain a positive realization (A1+ , b1+ , c1+ ) with order 4N4 = 12 such that A(λ1 , w1 )V = VA1+ , b1 = Vb1+ and c1+ = c1 V using Theorem 3.1. Finally, we can obtain the positive realization of H (z ) with the total order bound N = 18 using Eq. (15). 4
Example 4.1. For the purpose of a comparison, we consider the positive transfer function introduced in [9]: 10
+
where λ1 = 0.4 + 0.5i and β1 = 0.1 + i0.1. The transfer function is divided into H (z ) = F (z ) + G(z ) where F (z ) =
4. Numerical examples
H (z ) =
4
(z − 1)3
(z −1)3
−
5. Conclusion We presented a novel construction method for positive realizations of discrete positive linear systems with multiple complex poles. The transfer function can be decomposed into a primitive transfer function with a simple dominant pole and other poles and a transfer function with only multiple dominant poles. We extended and modified earlier positive realization results of Benvenuti et al. [11] and Nagy et al. [9] and provided a simple, more general and unified solution to the positive realization problem for transfer functions with multiple complex or real poles. We provided a novel construction method for positive realizations, resulting in easier computations and a tighter dimensional bound. We note that the internally positive realization method in [17] has a more general framework. It is worthwhile to extend our method for internally positive realization as an open problem. Acknowledgments The author wishes to thank the reviewers for constructive comments and suggestions. References [1] R.F. Brown, Compartmental system analysis: state of the art, Biomedical Engineering, IEEE Transactions on 27 (1980) 1–11. [2] A. Gersho, B. Gopinath, Charge-routing networks, Circuits and Systems, IEEE Transactions on 26 (1979) 81–92. [3] L. Benvenuti, L. Farina, The design of fiber-optic filters, Lightwave Technology, Journal of 19 (2001) 1366–1375. [4] R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results, Networking, IEEE/ACM Transactions on 14 (2006) 616–629. [5] C. Hadjicostis, Bounds on the size of minimal nonnegative realizations for discrete-time LTI systems, Systems & Control Letters 37 (1999) 39–43.
K. Kim / Systems & Control Letters 61 (2012) 759–765 [6] L. Benvenuti, L. Farina, An example of how positivity may force realizations of ‘large’ dimension, Systems & Control Letters 36 (1999) 261–266. [7] L. Benvenuti, L. Farina, B. Anderson, F.D. Bruyne, Minimal positive realizations of transfer functions with positive real poles, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 47 (2000) 1370–1377. [8] L. Benvenuti, L. Farina, A note on minimality of positive realizations, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 45 (1998) 676–677. [9] B. Nagy, M. Matolcsi, M. Szilvasi, Order bound for the realization of a combination of positive filters, Automatic Control, IEEE Transactions on 52 (2007) 724–729. [10] W. Czaja, P. Jamingc, M. Matolcsi, An efficient algorithm for positive realizations, Systems & Control Letters 57 (2008) 436–441. [11] L. Benvenuti, L. Farina, B. Anderson, Filtering through combination of positive filters, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 46 (1999) 1431–1440.
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[12] B. Nagy, M. Matolcsi, Minimal positive realizations of transfer functions with nonnegative multiple poles, Automatic Control, IEEE Transactions on 50 (2005) 1447–1450. [13] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994. [14] B. Anderson, M. Deistler, L. Farina, L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 43 (1996) 134–142. [15] L. Farina, On the existence of a positive realization, Systems & Control Letters 28 (1996) 219–226. [16] B. Nagy, M. Matolcsi, Algorithm for positive realization of transfer functions, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 50 (2003) 699–702. [17] F. Cacace, A. Germani, C. Manes, R. Setola, A new approach to the internally positive representation of linear MIMO systems, Automatic Control, IEEE Transactions on 57 (2012) 119–134.
Contents lists available at SciVerse ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
A construction method for positive realizations with an order bound Kyungsup Kim Department of Computer engineering, Chungnam National University, Yuseong-gu, Daejeon, Republic of Korea
article
abstract
info
Article history: Received 21 October 2011 Received in revised form 15 March 2012 Accepted 18 April 2012 Available online 23 May 2012
This paper presents an efficient construction method to address the positive realization problem with an order bound for primitive transfer functions with multiple real or complex conjugate pairs of poles using polyhedral cone concepts. Through a down-sampling step, the non-primitive transfer function is decomposed into a sum of primitive transfer functions. A construction method for the positive realization of a primitive transfer function with multiple poles is proposed by using a regular polygon in the complex plane. A sufficient condition for the existence of the positive realization with an order bound is analyzed. Numerical examples are provided to demonstrate the efficiency of the proposed method for obtaining positive realizations. © 2012 Elsevier B.V. All rights reserved.
Keywords: Positive realization Positive linear system Discrete time linear systems Impulse response sequence
1. Introduction This paper discusses the realization problem of a class of linearinvariant system, in which state variables, input and output are restricted to be non-negative to reflect physical constraints. The non-negative constraints can be encountered in many applications in engineering, medicine and economics. The positive systems can be applied to applications such as compartmental models in pharmacokinetics [1], charge-routing networks [2], fiber-optical filters [3] and TCP-like congestion control problems [4]. A positive linear system denotes a linear dynamical system in which the input u, the state x and the output y are non-negative real values. In this work, positive linear systems are considered in discrete time. Assume that a rational transfer function of a given positive system exists with McMillan degree n, expressed as H (z ) =
b n −1 z n −1 + · · · + b 0 zn
+ an − 1
x n −1
+ · · · + a0
=
hk z −k .
k≥0
Then, the impulse response sequence {hk } of H (z ) is non-negative (i.e., hk ≥ 0 for all k ≥ 0). Thus, the linear system can be represented by: x(k + 1) = Ax(k) + bu(k) y(k) = cx(k), where A ∈ Rn×n , b ∈ Rn , c T ∈ Rn and c T is a matrix transpose of c. The rational transfer function H (z ) has a state-space realization
E-mail address: [email protected]. 0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.04.009
in the form H (z ) = c (zI − A)−1 b and the triple (A, b, c ) denotes a state-space realization of H (z ). A transfer function H (z ) with a realization (A+ , b+ , c+ ) has a positive realization if a triple (A+ , b+ , c+ ) exists such that A+ ∈ Rn+×n , b+ ∈ Rn+ , and c+T ∈ Rn+ , where R+ denotes the set of non-negative real numbers. A linear system with a positive realization (A+ , b+ , c+ ) clearly induces a positive linear system. Conversely, the positive realization problem is to find a positive realization (A+ , b+ , c+ ) that represents a given positive linear system. The problems associated with the existence and the minimality of a positive realization have been studied widely in the past few decades, but a complete answer is not yet available. The positive constraint may enforce realizations of larger dimensions than the McMillan degree [5,6]. The minimality problem of positive realizations has been considered in a number of papers [7,8]. The lower-bound problems of positive realizations have already been discussed in [7,9,10]. The construction of a positive realization of a transfer function with simple complex poles was discussed in [11]. The construction of a positive realization of a transfer function with non-negative, real, multiple poles was considered in [12]. A straightforward algorithm to construct a positive realization for a transfer function with multiple poles (complex or negative) was described in [9]. In this paper, we will consider positive realizations with tighter bounds for transfer functions with possibly multiple real or complex poles. We extend earlier results on positive realizations from Benvenuti et al. [11] and Nagy et al. [9], and we provide a simple and unified solution to the positive realization problem of the primitive transfer function with tighter upper bounds and multiple poles. We demonstrate the proposed construction
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K. Kim / Systems & Control Letters 61 (2012) 759–765
method for positive linear systems with multiple complex or real poles using two examples. The paper is organized as follows. In Section 2, preliminary results for positive linear systems are introduced. A construct method for a positive realization of a primitive positive linear system and a sufficient condition for the efficacy of the proposed algorithm are presented in Section 3. In Section 4, two examples are provided to demonstrate the efficiency of the proposed algorithm. Finally, the conclusions are presented in Section 5. 2. Preliminary results related to positive linear systems The notation used throughout the paper is introduced first. X = cone(X ) denotes the smallest convex cone of a set X , which consists of all finite, non-negative, linear combinations of elements of the set X . The dual cone, X∗ , of a cone X is defined by X∗ = {y|xT y ≥ 0, ∀x ∈ X}. A convex cone X is said to be a polyhedral cone if it is spanned by a finite set of vectors X = {x1 , . . . , xm } with xi ∈ Rn ; i.e., X = cone({x1 , . . . , xm }), and X is called a polyhedral generator of X. X can denote the matrix with columns xi ∈ X , i.e., X = x1 x2 · · · xm , as well, if it is not ambiguous. An extreme point of a convex cone is one that is not a proper positive linear combination of any two points of the set. A finite set X is said to be a frame of the polyhedral cone X if the points of X are extreme points in X and X spans X [13]. Consider the necessary and sufficient conditions for the existence of a positive realization introduced in [14,15,11]. We refer to the result in [11], without a detailed proof, in the next theorem. Theorem 2.1. Let H (z ) be a strictly proper rational transfer function with a (minimal) realization (A, b, c ). There exists a positive realization (A+ , b+ , c+ ) and a generator matrix K with K = cone(K ) such that 1. AK ⊂ K 2. K ⊂ O where an observability cone O is defined by O = {x|cAk x ≥ 0, k = 0, 1, . . .} 3. b ∈ K . Furthermore, there is a positive realization {A+ , b+ , c+ } satisfying FK = KA+ , b = Kb+ , and c+ = cK . The above theorem provides an interpretation of the positive realization problem. Given any (minimal) realization of a transfer function, a polyhedral cone K satisfying the above three conditions corresponds to any positive realization. The number of edges of the cone K equals the dimension of the positive realization [11]. The construction method of the positive realization is related to the construction of the polyhedral cone generator K that satisfies the above three conditions. Let H[M ] (z ) denote a transfer function corresponding to the ∞ −j shifted sequence {hk }k≥M +1 i.e., H[M ] (z ) = . The j=1 hj+M z transfer function H[M ] (z ) is called the M-shifted transfer function of H (z ). We refer to the result in [5]. −j Lemma 2.1 ([5]). Let H (z ) = be a strictly proper rational j =1 hj z transfer function with a non-negative impulse response hk for k ≥ 1 and H (z ) = H c (z ) + z −M H[M ] (z ) where H c (z ) is a transfer function with a finite impulse response sequence. Assume that the M-shifted transfer function H[M ] (z ) admits a positive realization (A2 , b2 , c2 ) with dimension K . Then, H (z ) admits a positive realization of dimension K + M.
∞
Proof. The transfer function H c (z ) with a non-negative finite impulse response has an M-dimensional positive realization
(A1 , b1 , c1 ) as follows: 0 1 0 ··· 0 0 1 · · · . .. A1 = . .. 0 0 0 · · · 0
0
c1 = hM
···
0
hM −1
···
0 0
0
0 . b1 = .. 0
.. . , 1 0
1
h2
h1 .
The positive realization of the transfer matrix H[M ] (z ) with dimension K is given by (A2 , b2 , c2 ). The following combination of the two realizations yields a non-negative realization (A, b, c ) of H (z ) as follows:
A1 A= b2 0
0 , A2
b=
b1 , 0
c = c1
c2 .
Therefore, H (z ) admits a positive realization of dimension K + M. If H (z ) has a realization (A, b, c ), then the M-shifted transfer function H[M ] (z ) is given by H[M ] (z ) = cAM (zI − A)−1 b, and has a realization (A, b, cAM ). A necessary condition for the existence of a positive realization is that one of the dominant poles of H (z ) (i.e., the poles with maximum modulus) is non-negative and real [14]. The distinct pole set {λi } of H (z ) is ordered by |λj | ≤ |λi | ≤ 1 for j > i. Let λ0 be the dominant pole with λ0 > 0. The transfer function H (z ) is called primitive if λ0 is a unique dominant pole. The transfer function H (z ) has a positive realization if and only if β H (α z ) has a positive realization for α > 0 and β > 0 [5]. Therefore, we can set λ0 = 1 without any loss of generality by choosing values of α and β . If H (z ) is not primitive, a necessary condition for the existence of a positive realization is that the dominant poles of H (z ) be cyclic p (z p = λ0 for some p) [14]. It is known that if the dominant poles are not cyclic, then there is no positive realization of H (z ) [16]. If the dominant poles are cyclic with index p, then the necessary and sufficient condition for a positive realization is that all the downsampled transfer functions H(j) (z ) = c (zI − Ap )−1 Aj b for 0 ≤ j < p must be positive realizable [16]. If some of the functions H(j) (z ) are not primitive, then the down-sampling step is applied recursively to these functions, until they become primitive. After recursively applying the down-sampling step, H (z ) is decomposed into the following form: H (z ) =
z νs H(s) (z µs ),
(1)
s
where µs and νs are integers with 0 ≤ νs < µs and all the functions H(s) (z ) are primitive [16]. It is known that if every primitive transfer function H(s) (z ) is realizable, then H(s) (z µs ) and the original transfer function H (z ) can be positive realizable [16]. Therefore, it is sufficient to find a positive realization of only the primitive transfer functions. Using a partial fraction expansion, a primitive positive transfer function H (z ) is given by H (z ) =
n0 k=1
nj τ βk(0) βk(j) + , (z − λ0 )k (z − λj )k j=1 k=1
(2)
where nk is the maximum number of the multiplicity of λk . From the result in [16], if a primitive transfer function H (z ) has a nonnegative impulse response sequence, then the leading coefficient βn(00 ) of the first term in Eq. (2) is necessarily positive. We divide H (z ) into H (z ) = H c (z ) + z −M H[M ] (z )
(3)
K. Kim / Systems & Control Letters 61 (2012) 759–765
for sufficiently large M > 0. For sufficiently large M > 0, the Mshifted transfer function H[M ] (z ) can be decomposed into H[M ] (z ) = F[M ] (z ) + G[M ] (z )
(4)
and F[M ] (z ) and G[M ] (z ) have non-negative impulse response sequences. The transfer function F[M ] (z ) is given by n0
F[M ] (z ) =
k=1
(0)
βk (z − λ0 )k
(5) (0)
with the leading coefficient βn0 > 0 and βk and G[M ] (z ) is given by
≥ 0 for 1 ≤ k < n0 ,
j τ τ τ β0 βi(j) + = Gj (z ) z − λ0 (z − λj )i j=1 i=1 j=1
n
for some sufficiently large β0 > 0, where Gj (z ) = (j)
βi
nj
i=1 (z −λj )i .
(6) β0 z −λ0
+
Therefore, for sufficiently large values of M and β0 ,
we can obtain the transfer functions H (z ), F[M ] (z ) and G[M ] (z ) with non-negative impulse response sequences as given in Eqs. (3), (5) and (6), based on step 4 in [16]. A positive realization of F[M ] (z ) is obtained easily in [16]. Therefore, using Lemma 2.1, we need to find only a positive realization method for Gi (z ) for all i. c
3. Explicit positive realization algorithm We will consider the construction realization of Gi (z ) in this section. The given in Eq. (6) has a minimal canonical
(Aj , bj , cj ) such that Ai = 1 ⊕ J (λj ), bi =
O
cˆ = cˆ2n
···
cˆ1
λj 0 . J (λj ) , .. 0 (nj )
0 1
.. .
··· ··· .. .
0 0
0 0
··· ···
1
0 cˆj = βj
λj
(nj −1)
βj
···
method of a positive transfer function Gi (z ) Jordan-form realization 1 eni
and ci = β0
cˆj
0
0 0
0 . en , .. 0
.. . , 1
λj
βj(1)
defined by the block matrix A =
O
···
C
O
0
.. .
.. . ˆb = 0 1
(8)
0
for a proper number w , where a matrix C = C (x, y) ,
x y
−y x
yields
Proof. The transfer function H (z ) with the partial fraction expansion in Eq. (7) has a canonical Jordan form realization (A, b, c ) such that
¯ j ), A = J (λj ) ⊕ J (λ
b=
en , en
c = βn
β1
β¯ n
βn−1
···
1
A O
O B
with proper zero matrices
Lemma 3.1. Assume that a real, rational transfer function H (z ) with multiple complex conjugate poles is decomposed into
βk β¯ k + , (z − λ1 )k (z − λ¯ 1 )k
(7)
where the pole λ1 and the coefficients βi are complex and β¯ i is defined as the conjugate of βi . Then, H (z ) has a real Jordan-form realization (Jˆ(x, y, w), bˆ , cˆ ) whereby Jˆ(x, y, w) ∈ R2n×2n , bˆ ∈ R2n and cˆ T ∈ R2n are given by
β¯ 1 .
···
Additionally, there exists a permutation matrix P such that a block Jordan-form realization is given by
··· ··· ··· .. .
E O O A1 = P −1 AP =
I E O
O I E
O
O
O
...
β¯ n
β1
···
¯ . Let H = where E = λ ⊕ λ
O O O ,
0
.. .
.. . b1 = 0 , 1
··· E ¯β1 , 1 1
−i i
1
. Using a block-diagonal matrix
n
P1 = k=1 H as a similarity transformation, we obtain a new realvalued realization (A2 , b2 , c3 ) such that C O O A2 = P1−1 A1 P1 =
I C O
O I C
··· ··· ··· .. .
O O O ,
O
O
O
···
C
c2 = β˜ 2n
O, and the k-th element of the basis vector ek is 1 and the rest (i) of elements 0. We note that the poles λi and the coefficients βj could be complex for j ≥ 1 because complex poles or zeros of a rational transfer function with real coefficients in the numerator and denominator polynomials could exist as complex conjugate pairs. Hence, we consider the real-valued realization of a transfer function with multiple complex poles.
k=1
O O O ,
wC
...
for 1 ≤ j ≤ r and A0 = λ0 = 1.A block diagonal matrix A ⊕ B is
n
C
··· ··· ··· .. .
O
C O
H (z ) =
wC
C
O ˆJ (x, y, w) , O . ..
c1 = βn
where
761
a representation of a complex number x + iy and cˆk for all k are real. (0)
G[M ] (z ) =
β˜ 2n−1
···
0
.. .
.. . b2 = 0 , 1 0
β˜ 1 ,
where the entries of c2 are defined by β˜ 2k = 2Re(βk ) and β˜ 2k−1 = 2Im(βk ) for each k. Finally, we obtain the real block Jordan-form matrix (Jˆ(x, y, w), bˆ , cˆ ) such that Jˆ(x, y, w) = P3−1 A2 P3 in Eq. (8), bˆ is the same as b2 , and cˆ = c2 P3 has real-valued n components, k−1 where P3 is defined by P3 = D1 D2 with D1 = , and k=1 (w C ) n −n +1 0 D2 = (w C ) . We assume that B = I for any square k=1 matrix B. Let Pm (m ≥ 1) denote the set of points in the complex plane that lie in the interior of a regular polygon, with m edges, which has one vertex at point 1 and includes point 0. For m ≥ 2, the polygon Pm is defined by a subset in R2 through the following inequalities:
Pm = (x, y)|r cos
(2k + 1)π m
π − ϕ ≤ cos , m
for all k with 0 ≤ k ≤ m, where x = r cos ϕ and y = r sin ϕ . For m ≥ 3, the polygon Pm was introduced in [11]. The polygon Pm can be extended to the case where m = 1 and m = 2 without lose of generality. For m = 2, P2 is defined by P2 = {(x, 0)||x| ≤ 1}, because there are two vertices that have one vertex at point 1 and π cos m = 0. For m = 1, P1 is defined by P1 = {(x, 0)|0 ≤ x ≤ 1}, because it has at least vertices at point 1 and 0.
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K. Kim / Systems & Control Letters 61 (2012) 759–765
k > 1, A(λ1 , w)Vk has only the 1-st, k-th and k + 1-th row block components with non-zero values; the others are zeros, as follows:
Theorem 3.1. Assume that a transfer function G1 (z ) =
n βk β¯ k β0 + + (z − 1) k=1 (z − λ1 )k (z − λ¯ 1 )k
(9) 1
has a non-negative impulse response sequence for a sufficiently large ¯ 1 } ⊂ Pm value of β0 > 0, where λ1 = r cos θ + ir sin θ has {λ1 , λ for some m ≥ 3 and n ≥ 1. Let us define rm as
A(λ1 , w)Vk =
k
rm = max{ˆr |(ˆr cos θ , rˆ sin θ ) ∈ Pm , rˆ > 0}
k+1
for the given angle θ . Assume that for the given λ1 and the given m, conditions of w and {βk } for 1 ≤ k ≤ n are given by 0
1−
r
rm
rm
(10)
r
β0 . (2w r )k−n
|βk | ≤
(11)
Then, there exists a positive realization (A+ , b+ , c+ ) of the transfer function G1 (z ) that has the order mn.
¯ 1 } where Proof. The transfer function G1 (z ) has a pole set {1, λ1 , λ λ1 = r cos θ + ir sin θ ∈ Pm and has a positive impulse response sequence. Using the result of Lemma 3.1, we obtain a real block Jordan-form realization (A(λ1 , w), b, c ) of the transfer function H (z ) in Eq. (9) as follows:
A(λ1 , w) = P −1 AP = 1 ⊕ Jˆ(x, y, w),
b=
1 bˆ
cˆ ,
c= 1
O
I O Z = . .. O
O O I
.. .
··· ··· ··· .. .
O O O
O O O ,
O
···
I
O
.. .
ˆ
K
O . K = .. ,
.. .
O O
where I is a 2 × 2 identity matrix, O is a 2 × 2 zero matrix and a 2 × m matrix Kˆ is defined as
1
cos
Kˆ = 0
sin
2π
cos
m 2π
sin
m
4π m 4π m
···
cos
···
sin
2π (m − 1)
m . 2π (m − 1) m
A cone generator matrix V ∈ R(2n+1)×mn is defined as
V = V1
V2 · · · Vn ,
where Vk is defined as Vk =
Z
eT k−1
K
for 1 ≤ k ≤ n and e represents
an m × 1 vector with all of its entries equal to 1. The columns of matrix V represent the edges of a finite generated cone V in R2n+1 (i.e., cone(V ) = V ) and are positive independent. We shall prove that V is A(λ1 , w)-invariant(i.e., A(λ1 , w)V ⊂ V ). Let Vk be the cone generated from the column vectors of Vk (i.e.,Vk = cone(Vk )). When λ1 ∈ Pm , we can easily see that A(λ1 , w)V1 ∈ V1 . For
eT
.. .
O wC Kˆ C Kˆ O .. .
.
(12)
If A(λ1 , w)Vk ∈ cone{Vk−1 , Vk } for every k > 1 is satisfied, then V is sufficiently A(λ1 , w)-invariant. Choose {Wk−1 , Wk } for each k such that A(λ1 , w)Vk = α1 Wk−1 + α2 Wk , where all αj ≥ 0, α1 + α2 = 1 and the entries in the first row of Wk ∈ Vk are equal to 1. Let a 2 × m matrix Wk,k+1 be defined by the k + 1-th row component of Wk as shown in Eq. (12). Then, from Eq. (12), we can see that α1 Wk−1,k = w C Kˆ and α2 Wk,k+1 = C Kˆ . For the direction of the angle θ , the maximum norm of the column vectors of Wk,k+1 is less than rm for any k, and the norm of the column vectors of C Kˆ r ≤ α2 and rrm ≤ α1 . A is r. We can derive two inequalities, w rm sufficient condition for the existence of a feasible solution (α1 , α2 ) r that satisfies w ≤ α2 , rr ≤ α1 , and α1 +α2 = 1 can be obtained as r m
m
0 < w ≤ (1 − rr ) rrm . Therefore, we can show that Vk is A(λ1 , w)m invariant because A(λ1 , w)Vk ∈ cone{Vk−1 , Vk }. We can easily verify that the second condition is satisfied because b ∈ Vn . Finally, we will prove that c = β0 cˆ ∈ V ∗ . The 2k and the 2k − 1 entries of cˆ are given by cˆ2k cˆ2k−1 =
β˜ 2k−1 (w C )k−n from cˆ = D1 D2 c2 in Lemma 3.1. An inequality cVk ≥ 0 for each k ≥ 1 leads to the sufficient condition
where x = r cos θ , y = r sin θ and (Jˆ(x, y, w), bˆ , cˆ ) is defined in Eq. (8). We can now demonstrate that there exists an A(λ1 , w)invariant cone V with mn edges satisfying Theorem 2.1 (i.e., AV ⊂ V ). We generalize the concept of the cone generator introduced in [11] for the case with multiple complex poles. To formulate a cone generator, a block shift matrix Z ∈ R2n×2n and a matrix K ∈ R2n×m are defined as:
β˜ 2k
in Eq. (11) as follows:
β0 e + β˜ 2k
β˜ 2k−1 (w C )k−n Kˆ ≥ 0.
The l2 norm of each column vector of C k−n Kˆ equals r k−n . To satisfy the above inequality, we impose a sufficient condition such that
|βk | ≤
β0 (2w r )k−n
for sufficiently large β0 . W can show that cVk ≥ 0. Therefore, we obtain a positive realization (A+ , b+ , c+ ) with ×nm T nm order nm such that A+ ∈ Rnm , b+ ∈ Rnm + + and c+ ∈ R+ satisfy A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV , respectively, using a suitable linear programming tool. The transfer function Gk (z ) in Eq. (9) is derived from the shifted transfer function H[M ] (z ) in Eq. (4). The shifted transfer function H[M ] (z ) has the realization (A, b, cAM ). The norm of cˆ (proportional to r M ) decreases as M increases. Therefore, we can impose a tradeoff between the condition (11) and the shifted dimension M. Our approach provides a more general and unified solution to the problem of finding a positive realization of a transfer function with multiple complex poles or multiple real poles and has a tighter upper bound on the dimension than that of [9]. Our approach additionally covers the known results for the positive realization of the transfer function G1 (z ) with negative or positive real multiple poles [9,12]. First, it is worth noting that the positive realization of the transfer function G1 (z ) with negative real multiple poles in Theorem 2 of [9] can be explained as a special case of Theorem 3.1. Theorem 3.2. Assume that the transfer function G1 (z ) =
n βk β0 + z−1 ( z − λ1 ) k k=1
K. Kim / Systems & Control Letters 61 (2012) 759–765
has a non-negative impulse response sequence where n ≥ 1 is an integer, the real pole satisfies −1 < λ1 < 0 (i.e., λ1 ∈ P2 ). Assume that for the given λ1 and m = 2, conditions of w and {βk } for 1 ≤ k ≤ n are given by 0
1−r
(13)
r
β0 |βk | ≤ (w r )k−n
(14)
for r = |λ1 |. Then, G1 (z ) has a positive realization of dimension 2n. Proof. The pole λ1 < 0 is included in P2 . We have m = 2. The transfer function G1 (z ) has a Jordan-form realization (A(λ1 , w), b, c ) given by 1
0 . b= .. , 1
A(λ1 , w) = 1 ⊕ J (x, 0, w),
0
βn
c= 1
β1
0···
0 ,
where x = r and y = 0. A 2 × 2 matrix Kˆ is given by
Kˆ =
cos 0
cos
sin 0
sin
2π
2 = 1 0 2π
−1 0
.
2
(2n+1)×2n
Thus, K ∈ R and V ∈ R(2n+1)×2n are defined in the same manner as in Theorem 3.1. We note that V equals an augmented matrix that can be obtained by inserting a zero in every odd k-th row such that k ≥ 3, and it yields a result equivalent to that of [9]. Because we have rm = 1, the condition (13) is obtained similar to the way Eq. (10) was. The remaining derivations are the same as those of Theorem 3.1. We obtain a positive realization (A+ , b+ , c+ ) with dimension nm such that A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV using linear programming. Next, we show the cases of the transfer functions with nonnegative multiple poles in our framework as well. The result is the same as that shown in [12,9]. The resulting dimension is observed to be minimal in this case. Theorem 3.3. Let λ1 be a non-negative number with 0 ≤ λ1 < 1 (i.e. λ1 ∈ P1 ). Assume that a transfer function G1 (z ) =
n βk β0 + z−1 ( z − λ1 )k k=1
Proof. By definition, the pole λ1 > 0 is included in P1 . We have m = 1. The transfer function G1 (z ) has a minimal Jordan-form realization (A, b, c ) such that A ∈ R2n+1×2n+1 , b and c are defined as: 1
0 . b= .. , 1 0
c= 1
βn
0···
β1
0 ,
where x = r and y = 0. A 2 × 1 matrix Kˆ is defined as Kˆ =
cos 0 1 = . sin 0 0
K ∈ R2n+1 is defined in a similar manner as in Theorem 3.1. Vk is
defined by Vk =
eT
Z k−1 K
for 1 ≤ k ≤ n. In this definition, we note
that {Vk } cannot cover the vector e1 induced by the zero in P1 . In addition, a vector V0 is defined as follows: V0 = e1 ∈ R2n+1 . An augmented cone generator matrix V ∈ R(2n+1)×mn is defined by
V = V0
V1
V2 · · · Vn .
The remaining derivations are similar to those for Theorems 3.1 and 3.2. We obtain a positive realization (A+ , b+ , c+ ) with dimension n + 1 by solving a suitable linear programming such that A(λ1 , w)V = VA+ , b = Vb+ and c+ = cV . The dimension n + 1 is equal to the minimal realization dimension of G1 (z ). We summarize the construction process of the positive realization of a given transfer function H (z ) as follows: 1. If H (z ) is not primitive, then it is decomposed into the sum of primitive transfer functions as in Eq. (1) using a down-sampling step recursively. If the primitive transfer functions have positive realizations, then the original transfer function is also positive realizable as well. 2. Each induced primitive transfer function H (z ) can be divided into the form H (z ) = H c (z )+ z −M H[M ] (z ) for a sufficiently large M. Using a partial fraction τ expansion, H[M ] (z ) is rearranged into H[M ] (z ) = F[M ] (z ) + j=1 Gj (z ) as in Eq. (6). 3. With Theorems 3.1–3.3, we can compute positive realizations of the primitive positive systems Gk (z ). Let N0 denote the maximal order of the multiple dominant poles and N0 equal to the order of the positive realization of F[M ] (z ). Let N1 denote the sum of the orders of the (possibly multiple) non-negative poles in P1 except for a dominant pole, and let N2 be the number of (possibly multiple) negative real poles in P2 except for a dominant pole. Let Nm (m ≥ 3) denote the total number of complex conjugate pairs of (possibly multiple) complex poles of the transfer functions in the form Gk (z ), belonging to the following region: m−1
Pm −
Pj ,
j =1
except for a dominant pole. Using Theorem 3.1, we can show that the upper bound order of the positive realization is mNm . 4. Finally, we obtain a positive realization of the primitive transfer function H (z ) by using Lemma 2.1 and the upper bound of the positive realization of H (z ) is given by
has a positive linear system where n ≥ 1 is an integer. Assume that for the given λ1 and m = 1, conditions of w and {βk } for 1 ≤ k ≤ n are given by Eqs. (13) and (14). Then, G1 (z ) has a positive realization of dimension n + 1.
A(λ1 , w) = 1 ⊕ J (x, 0, w),
763
N = M + N0 +
1 + N1 +
jNj
.
(15)
j ≥2
Remark 3.1. The proposed construction method can cover the known positive realization results for the case of transfer functions with exclusively simple (complex) poles, multiple positive poles and multiple negative poles in the Refs. [7,11,12]. The case of a transfer function with multiple complex poles remains open. The proposed method additionally works well for the case of a transfer function with multiple complex poles. The proposed upper bound N in Eq. (15) of the positive realization is lower than that of [9] if λk lies in Pm for m ≥ 3. We provide a tighter upper bound for the case of a transfer function with multiple complex poles. The explicit sufficient condition for the existence of a positive realization is additionally established in the previous theorems. We remark that the proposed method provides a unified and more general method
764
K. Kim / Systems & Control Letters 61 (2012) 759–765
for the positive realization of primitive transfer functions with a tighter upper bound than the previous ones. The notion of IPR (Internally positive realization) has been introduced in [17] as a generalization of realization of non-positive filters by means of the difference of positive filters [11]. In IPRs, the difference of the outputs of two positive systems is replaced by a more general output transformation applied to the state of a single positive system. We note that the positive N-representation of IPR in completely different issue is closely related to our proposed method in the point of using a regular polygon containing complex eigenvalues.
leads to a positive realization (A1+ , b1+ , c1+ ) with order 2N2 = 6 using Theorem 3.2. Similarly, the transfer function G2 (z ) leads to a positive realization (A2+ , b2+ , c2+ ) with order 5N5 = 10 using Theorem 3.1. From Eq. (15), the total upper bound N = 16 is less than the total upper bound 22 in the Ref. [9]. Example 4.2. Consider a primitive transfer function with multiple conjugate complex poles, multiple dominant poles and a nonnegative impulse response sequence such that H (z ) =
z − 0.99
−
+
0.1 1000(z + 0.9)3
100(z − 1)
(50z 2 − 70z + 29)
−
50z 2 − 70z + 20
(50z 2 − 70z + 29)2
.
The transfer function H (z ) has the pole λ0 = 0.99, the pole λ1 = −0.9 that is of order 3 and the complex conjugate pair of poles λ2 = 0.7 + 0.3i and λ¯ 2 that are of order 2. In this case, the shifting of the transfer function is not required. We have H c (z ) = 0 and F (z ) = 0. The transfer function H (z ) is properly divided into two terms H (z ) = G1 (z ) + G2 (z ), where the positive transfer function G1 (z ) has the pole λ0 = 0.99, the pole λ1 = −0.9 that is of order 3 and the positive transfer function G2 (z ) has the pole λ0 = 0.99 ¯2 and the complex conjugate pair of poles λ2 = 0.7 + 0.3i and λ that are of order 2. Because λ1 is included in P2 , set m = 2. We get rm = 1 and r = 0.9. The maximum of w is computed as w1 = 1/9 by using Eq. (13). We can verify that the condition (14) with respect to {βk } is satisfied. The transfer function G1 (z ) is determined by the real Jordan-form realization (A1 , b1 , c1 ) where A1 = 0.99 ⊕ A(λ1 , w1 ) with the augmented matrix A(λ1 , w1 ) given by
−0.9 0 0 A(λ1 , w1 ) = 0
0 0
0 −0.9 0 0 0 0
−0.1 0
−0.9 0 0 0
0 −0.1 0 −0.9 0 0
0 0 −0.1 0 −0.9 0
0 0 0 −0.1 0 −0.9
with w1 = 1/9 and c1 = 5.00 0.02 0 0 0 0 0 . We use an augmented matrix A(λ1 , w1 ) for the purpose of the unified and general approach. It can be reduced to a 3 × 3 Jordan-form matrix by removing the unobservable part. Because λ2 is included in P5 , we set m = 5. We get rm = 0.8296 and r = 0.7616. The maximum of w is computed as w2 = 0.0894 by using Eq. (10) in Theorem 3.1. We can verify that the condition (11) with respect to {βk } is satisfied. The transfer function G2 (z ) is determined by the real Jordan form realization (A2 , b2 , c2 ), where A2 = 0.99 ⊕ A(λ2 , w2 ) with
0.7 −0.3 A(λ2 , w2 ) = 0 0
β1 β¯ 1 , + (z − λ1 )3 (z − λ¯ 1 )3
0.3 0.7 0 0
0.0626 −0.0268 0.7 −0.3
0.0268 0.0626 0.3 0.7
w2 = 0.0894 and c2 = 5 −0.2701 0.1158 −2 −2 . We have the order N2 = 3 for G1 (z ) and the order N5 = 2 for G2 (z ) because λ1 ∈ P2 and λ2 ∈ P5 . We obtain a positive realization (A+ , b+ , c+ ) such that A(λ1 , w1 )V = VA1+ , b1 = Vb1+ and c1+ = c1 V using linear programming. The transfer function G1 (z )
β0 (z −1)
β
β
β¯
and G(z ) = (z −01) + (z −λ1 )3 + (z −λ¯1 )3 for a given 1 1 β0 = 8. There is a sufficiently large M such that the M-shifted transfer functions F[M ] (z ) of F (z ) have positive realizations, and the M-shifted transfer functions G[M ](z ) of G(z ) have non-negative impulse response sequences and are positive realizable. H (z ) is decomposed into H (z ) = H c (z ) + z −M (F[M ] (z ) + G[M ] (z )). The order of the positive realization of F[M ] (z ) is N0 = 3. Now, we consider a positive realization of G[M ] (z ). Because λ1 is included in P4 , set m = 4. We get rm = 0.7115 and r = 0.6403. The maximum of w is computed as w1 = 0.1111 by using Eq. (10) in Theorem 3.1. We can verify that the condition (11) with respect to {βk } is satisfied. Therefore, G[M ] has a realization (A1 , b1 , c1 ) such that A1 = 1 ⊕ A(λ1 , w1 ) is a 7 × 7 matrix with w1 = 0.1111 and c1 = 8. −7.938 5.022 −3.528 2.232 −0.588 0.372 . We have M = 4 and N4 = 3. We can obtain a positive realization (A1+ , b1+ , c1+ ) with order 4N4 = 12 such that A(λ1 , w1 )V = VA1+ , b1 = Vb1+ and c1+ = c1 V using Theorem 3.1. Finally, we can obtain the positive realization of H (z ) with the total order bound N = 18 using Eq. (15). 4
Example 4.1. For the purpose of a comparison, we consider the positive transfer function introduced in [9]: 10
+
where λ1 = 0.4 + 0.5i and β1 = 0.1 + i0.1. The transfer function is divided into H (z ) = F (z ) + G(z ) where F (z ) =
4. Numerical examples
H (z ) =
4
(z − 1)3
(z −1)3
−
5. Conclusion We presented a novel construction method for positive realizations of discrete positive linear systems with multiple complex poles. The transfer function can be decomposed into a primitive transfer function with a simple dominant pole and other poles and a transfer function with only multiple dominant poles. We extended and modified earlier positive realization results of Benvenuti et al. [11] and Nagy et al. [9] and provided a simple, more general and unified solution to the positive realization problem for transfer functions with multiple complex or real poles. We provided a novel construction method for positive realizations, resulting in easier computations and a tighter dimensional bound. We note that the internally positive realization method in [17] has a more general framework. It is worthwhile to extend our method for internally positive realization as an open problem. Acknowledgments The author wishes to thank the reviewers for constructive comments and suggestions. References [1] R.F. Brown, Compartmental system analysis: state of the art, Biomedical Engineering, IEEE Transactions on 27 (1980) 1–11. [2] A. Gersho, B. Gopinath, Charge-routing networks, Circuits and Systems, IEEE Transactions on 26 (1979) 81–92. [3] L. Benvenuti, L. Farina, The design of fiber-optic filters, Lightwave Technology, Journal of 19 (2001) 1366–1375. [4] R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results, Networking, IEEE/ACM Transactions on 14 (2006) 616–629. [5] C. Hadjicostis, Bounds on the size of minimal nonnegative realizations for discrete-time LTI systems, Systems & Control Letters 37 (1999) 39–43.
K. Kim / Systems & Control Letters 61 (2012) 759–765 [6] L. Benvenuti, L. Farina, An example of how positivity may force realizations of ‘large’ dimension, Systems & Control Letters 36 (1999) 261–266. [7] L. Benvenuti, L. Farina, B. Anderson, F.D. Bruyne, Minimal positive realizations of transfer functions with positive real poles, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 47 (2000) 1370–1377. [8] L. Benvenuti, L. Farina, A note on minimality of positive realizations, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 45 (1998) 676–677. [9] B. Nagy, M. Matolcsi, M. Szilvasi, Order bound for the realization of a combination of positive filters, Automatic Control, IEEE Transactions on 52 (2007) 724–729. [10] W. Czaja, P. Jamingc, M. Matolcsi, An efficient algorithm for positive realizations, Systems & Control Letters 57 (2008) 436–441. [11] L. Benvenuti, L. Farina, B. Anderson, Filtering through combination of positive filters, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 46 (1999) 1431–1440.
765
[12] B. Nagy, M. Matolcsi, Minimal positive realizations of transfer functions with nonnegative multiple poles, Automatic Control, IEEE Transactions on 50 (2005) 1447–1450. [13] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994. [14] B. Anderson, M. Deistler, L. Farina, L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 43 (1996) 134–142. [15] L. Farina, On the existence of a positive realization, Systems & Control Letters 28 (1996) 219–226. [16] B. Nagy, M. Matolcsi, Algorithm for positive realization of transfer functions, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 50 (2003) 699–702. [17] F. Cacace, A. Germani, C. Manes, R. Setola, A new approach to the internally positive representation of linear MIMO systems, Automatic Control, IEEE Transactions on 57 (2012) 119–134.