Systems & Control Letters 36 (1999) 261–266
An example of how positivity may force realizations of ‘large’ dimension Luca Benvenuti a; ∗ , Lorenzo Farina b aDipartimento
bDipartimento
di Ingegneria Elettrica, UniversitÂa degli Studi di L’Aquila, Poggio di Roio, 67140 L’Aquila, Italy di Informatica e Sistemistica, UniversitÂa degli Studi di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy Received 10 November 1997; received in revised form 17 June 1998; accepted 2 October 1998
Abstract A standard result of linear system theory states that an SISO proper rational transfer function of degree n always has a realization of dimension n. In some applications one is interested in having a realization with nonnegative entries and it is known that, when the dominant poles display a speci c pattern, forcing nonnegativity leads to a system which is not jointly reachable and observable. In this paper, we show that the minimal dimension of a positive realization may be ‘large’ even in the case of a single dominant pole. More precisely, we provide a family of transfer functions, each of which is of degree n = 3, such that for any integer N ¿3 the corresponding member of the family admits a minimal positive realization of state c 1999 Elsevier Science B.V. All rights reserved. space dimension not smaller than N . Keywords: Positive realizations; Positive systems; Minimality; Discrete-time systems
1. Introduction 1.1. Positive systems The work presented in this paper was motivated by a research interest of the authors in the area of positive linear systems (see, for general overviews, [5,14,19]). Positive systems are, by de nition, systems whose variables take only nonnegative values. From a general point of view, they should be considered as very particular. From a practical point of view, however, such systems are anything but particular since positive systems are often encountered in applications. In fact, positive systems are, for instance, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems (memories, warehouses, : : :), promo∗
Corresponding author. E-mail:
[email protected].
tional systems, compartmental systems (frequently used when modelling transport and accumulation phenomena of substances in human body), water and atmospheric pollution models, stochastic models where state variables must be nonnegative since they represent probabilities and many other models commonly used in economy and sociology. Just to cite the most popular, consider the Leontie model used by economists for predicting productions and prices [12], the Leslie model used to study age-structured population dynamics [13], the hidden Markov models [18] mainly adopted for speech recognition, the compartmental models [10] commonly encountered in pharmacokinetics and radio-nuclide tracer dynamics and the birth and death processes, relevant to the analysis of queueing systems [20]. More recently, it has been developed into a MOS-based technology for discrete-time ltering, the so-called charge routing networks, in which the state variables are
c 1999 Elsevier Science B.V. All rights reserved. 0167-6911/99/$ – see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 9 8 - X
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positive since they represent quantity of electrical charge [3, 8]. 1.2. The positive realization problem In recent times a number of issues regarding positive systems has been addressed (such as, for example, positive orthant reachability [22]) but, for a number of reasons, the so-called positive realization problem has been the most studied (see, for example, the references cited in [2, 7, 15]). In the following, we will restrict ourself to consider only the SISO discrete-time case. The formulation of this problem is as follows. The positive realization problem for discrete-time systems [1, 16]. Given a strictly proper rational transT fer function H (z), the triple {A; P b; c } is said to be a positive realization if H (z) = k¿1 cT Ak−1 bz −k with A; b; cT nonnegative (i.e. with nonnegative entries). The positive realization problem consists of providing answers to the questions: • (The existence problem) Is there a positive realization {A; b; cT } of some nite dimension N and how may it be found? • (The minimality problem) What is the minimal value of N ? • (The generation problem) How can we generate all possible positive realizations? In Refs. [2,7,11] the existence problem has been completely solved and a means of constructing such realizations is also given there. In this paper, we shall not consider the interesting question of characterizing the relationship between equivalent realizations, and we shall concentrate on what we have termed the minimality problem (see, for example, [4, 17, 21]). It is known that, for transfer functions of degree 1 or 2, nonnegativity of the impulse response is a necessary and sucient condition for the existence of a positive realization. Moreover, in those two cases, the minimal dimension of a positive realization coincides with the degree of the transfer function [16]. On the other hand, the situation for the case of transfer functions of degree n¿2 is totally di erent. In fact, it is known [1,9,16] that, when the dominant poles display a speci c pattern, one can nd a positive realization of minimal dimension which is not jointly reachable and observable. In this paper we show that the minimal dimension of a positive realization may be ‘large’ even in the case of a single dominant pole. More precisely, we provide a family of transfer functions with positive real poles, each of which is of McMillan
degree n = 3; such that for any integer N ¿3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than N . 2. Some preliminaries We give next a quick listing of basic de nitions (taken from [5]) and results which will be needed in the sequel. A set K is said to be a cone provided K ⊆ K for all ¿0; if K contains an open ball of R n then K is said to be solid, if K ∩ {−K} = {0} then K is said to be pointed. A cone which is closed, convex, solid and pointed will be called a proper cone. A cone K is said to be polyhedral if it is expressible as the intersection of a nite family of closed half-spaces. cone(v1 ; : : : ; vM ) denotes the polyhedral closed convex cone consisting of all nite nonnegative linear combinations of vectors v1 ; : : : ; vM : (A) and pA () denote the spectrum and the characteristic polynomial of a matrix A; respectively. We will also make use of the following. Theorem 1 (Ohta et al. [16]). Let H (z) be a strictly proper rational transfer function of McMillan degree n and let {F; g; hT }, with F ∈ R n×n and g; h ∈ R n ; be a minimal (i.e. jointly reachable and observable) realization of H (z). Then, H (z) has a positive realization, if and only if there exists a polyhedral proper cone K such that 1. FK ⊂ K, i.e. K is F-invariant; 2. K ⊂ O; 3. g ∈ K; where O = {x ∈ R n | hT F k x¿0; k = 0; 1; : : :} is called the observability cone. Moreover, a positive realization {A; b; cT } is obtained by solving FK = KA;
g = Kb;
cT = hT K;
where K is such that K = cone(K): The above theorem provides a geometrical interpretation of the positive realization problem: given any realization of a transfer function, then to any positive realization corresponds an invariant cone K satisfying conditions 1–3 and vice versa. Since the number of edges of the cone K equals the dimension of the
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positive realization, then the minimality problem can be equivalently stated as the problem of nding the invariant polyhedral proper cone K with the minimal number of edges contained in the observability cone O and containing the vector g. 3. The possible need for nonminimality To show that the minimality problem for positive linear systems is inherently di erent from that of ordinary linear systems we shall make use of the following two examples. Example 1 (Anderson [1]). Consider the transfer function H (z) =
1 1 1 : − + z − 0:8 z + 0:8 z + 0:5
Clearly, there exists a third-order realization of H (z). What about a positive realization? Suppose there is one. Then, the nonnegative dynamic matrix A would have the eigenvalues ± 0:8 and −0:5 but, as a consequence of the Frobenius theorem (see, for example, [6]), the spectrum of A must remain unchanged under a rotation of radians. Since +0:5 is not an eigenvalue of A, then we arrive at a contradiction, i.e. there is no third order positive realization of H (z). However, the following 0 0:25 0 0 1 1 0 0 0 0 ; A= b= 0 0:39 0 0 ; 0:8 0 0 0:8 0 0 1 1:1 c= 0 ; 2 is a fourth-order positive realization of H (z). Example 2. Consider the following family of transfer functions: H (z; q) =
1 ; (z − 1)(z 2q + 1)
q¿1:
By exploiting the rotational symmetry of the dominant poles, we can prove that for any integer q¿1; the state space dimension N of any minimal positive realization of the corresponding member of the family equals 2q+1 while H (z; q) is of McMillan degree n = 2q +1. To see
Fig. 1. Poles pattern of H (z; q) for q = 1 and q = 2. q
this, note that since e=2 and 1 are dominant poles of H (z; q) (see Fig.1 for the cases q = 1 and q = 2), then from the Frobenius theorem, the spectrum of the dynamic matrix of any positive realization must remain unchanged under a rotation of =2q radians. This implies that all the 2q+1 th roots of unity must belong to the spectrum of the dynamic matrix of any positive realization. Consequently, the minimal dimension for a positive realization of H (z; q) is not smaller than 2q+1 . The following positive realization of H (z; q) 2q+1
z
0 1 0 A= 0 .. . 0
}|
0 0 1
::: :::
0 .. .
.
.. .
0 0 0 .. .
:::
1 0
0 1
0
..
0 0
{ 1 0 0 ; 0 0
1 0 0 b = . ; .. 0 0
cT = (0 : : : 0 1 : : : 1) | {z } | {z } 2q
2q
is therefore minimal as a positive linear system. For the sake of illustration consider the case q = 1. Then H (z; 1) = and
1 (z − 1)(z 2 + 1)
0 0 1 0 A= 0 1 0 0 0 0 c= 1 : 1
0 0 0 1
1 0 ; 0 0
1 0 b= 0 ; 0
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Fig. 2. Impulse response of the system in Example 3.
Since the poles of H (z; 1) are 1; i and −i, then the dynamic matrix of any minimal positive realization must include also the eigenvalue −1; consequently, the above realization is minimal as a positive system. The case q = 1 is also studied in [9] where the minimality of the realization is proved via primes in the positive matrices. As it would appear in the sequel, this rotational symmetry of the spectrum of a nonnegative matrix, due to the speci c dominant poles pattern, is not the only cause for the need of nonminimality in the positive realization problem, as is shown by the following example. Example 3. Consider the transfer function H (z) =
25 75 1 − + ; z − 1 z − 0:4 z − 0:2
which has a fourth-order positive 0 0√ 0 1 63+4 26 0√ 85√ A= 63−4 26 22−4 26 0 85 85√ 22+4 26 0 0 85 0 6 0 0 b= c= 0 ; 0 ; 1 51
realization 1 0 ; 0 0
whose nonnegative impulse response has the fourth and the fth element of the sequence equal to zero as depicted in Fig. 2. Although the dominant eigenvalue of A is unique, so that no symmetry of the spectrum is required by the
Fig. 3. The observability cone of the system in Example 3.
Frobenius theorem, we claim that there is no third order positive realization of H (z). Suppose there is one. Then, by taking any minimal (i.e. jointly reachable and observable) realization {F; g; hT } of H (z), there exists a cone K with three edges satisfying the conditions of Theorem 1. From conditions 1 and 3, it follows that F k g ∈ K for k = 0; 1; : : : : Moreover, since in this case hT F 2 g = hT F 3 q = 0; then the following hold: hT (g)¿0; hT F(g)¿0; hT F 2 (g) = 0; hT F 3 (g) = 0; hT F m (g)¿0; m¿4;
hT (Fg)¿0; hT F(Fg) = 0; hT F 2 (Fg) = 0; hT F m (Fg)¿0; m¿3;
hT (F 2 g) = 0; hT F(F 2 g) = 0; hT F m (F 2 g)¿0; m¿2; Consequently, the rst three vectors of the free evolution emanating from g lie on di erent edges of the observability cone as depicted in Fig. 3. Then, from condition 2 of Theorem 1, i.e. K ⊂ O, the vectors g; Fg; F 2 g are necessarily edges of K. Since F 3 g 6∈ cone(g; Fg; F 2 g) = K; then K is not F-invariant, thus arriving at a contradiction. To state the main result of the paper we need the following lemma: Lemma 2. Given a reachable pair {F; g} with F ∈ R n×n ; g ∈ R n ; n¿2 and F having only distinct
L. Benvenuti, L. Farina / Systems & Control Letters 36 (1999) 261–266
and positive real eigenvalues, then
Moreover,
F k g 6∈ cone(g; Fg; : : : ; F k−1 g);
hN −1 = 1 − 25 · (0:4)2 + 75 · (0:2)2 = 0;
∀k¿1:
hN = 1 − 25 · (0:4)3 + 75 · (0:2)3 = 0;
Proof. We will prove the lemma by contradiction. Suppose then F k g ∈ cone(g; Fg; : : : ; F k−1 g); i.e.
and, for k¿N , we have
F k g = 0 g + 1 Fg + · · · + k−1 F k−1 g;
hk = 1 − 25 · (0:4)3−N +k + 75 · (0:2)3−N +k
i ¿0:
(1)
If k¡n; from reachability of the pair {F; g}, we arrive at a contradiction. Suppose then k¿n. From Eq. (1), it follows that we can write (2)
FP = PA+ with P = [g Fg : : : 0 ::: 1 A+ = 0 . . . . .. 0
:::
F k−1 g] and 0 .. . 0
0 .. . .. .
1 0
0 1
0
1 .. : . .. . k−1
From Eq. (2) it follows that, P being full rank by hypothesis, (A+ ) ⊃ (F). Since pA+ () has only nonpositive coecients, from the Descartes’ rule of signs it has, at most, one positive root thus contradicting the hypothesis. The main result of the paper consists of the following example: Example 4. Consider the following family of transfer functions: H (z; N ) =
0:44−N 1 − 25 · z−1 z − 0:4 0:24−N ; N ¿4 +75 · z − 0:2
For any N ¿4; the impulse response sequence hk of the member of the family corresponding to N , is nonnegative with hN −1 = hN = 0. In fact, if 16k6N − 2, we can write hk = 1 − 25 · (0:4)3−N +k + 75 · (0:2)3−N +k 5N −3−k + 75 · 5N −3−k 2N −3−k 1 N −1−k =1+5 3 − N −3−k ¿0: 2 = 1 − 25 ·
265
23−N +k 1 + 75 · 3−N +k 53−N +k 5 51−N +k − 23−N +k + 3 51−N +k (22 + 1)1−N +k − 23−N +k + 3 51−N +k 2−2N +2k − 23−N +k + 3 2 51−N +k 2−N +k k−N (2 − 2) + 3 2 ¿0: 1−N +k 5
= 1 − 25 · = = ¿ =
Then from Theorem 4.1 in Ref. [2], the transfer function of the member of the family corresponding to N is positively realizable. Hence, there exists a cone K satisfying the conditions of Theorem 1. Taking any minimal realization {F; g; hT } of H (z; N ), with F ∈ Rn×n and g; h ∈ Rn ; then hT F N −2 g = hT F N −1 g = 0; so that the following hold: hT F m (F j g)¿0; 06m¡N − 2 − j; hT F N −2−j (F j g) = 0; hT F N −1−j (F j g) = 0;
j = 0; 1; : : : ; N − 2
hT F m (F j g)¿0; m¿N − 1 − j; Moreover, from conditions 1 and 3 of Theorem 1, it follows that F k g ∈ K for k = 0; 1; : : : : Consequently, the rst N − 1 vectors of the free evolution emanating from g lie on di erent edges of the observability cone and, from condition 2, i.e. K ⊂ O, the vectors g; Fg; F 2 g; : : : ; F N −2 g are necessarily edges of K. Finally, since from Lemma 2, F N −1 g 6∈ cone (g; Fg; F 2 g; : : : ; F N −2 g), the minimal order of any positive realization is not smaller than N . This is quite surprising since, in spite of the fact that we are dealing with the seemingly simple case of a third order transfer function with distinct positive real poles, the minimal positive realization may possibly have a ‘large’ state-space dimension.
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Acknowledgements This work was partially supported by MURST (Italian Ministry for Scienti c and Technological Research) under the National Project ‘Teoria dei Sistemi e del Controllo’, and ASI (Italian Space Agency) under Grant ASI-ARS221. References [1] B.D.O. Anderson, New developments in the theory of positive systems, in: Byrnes, Datta, Gilliam, Martin (Eds.), Systems and Control in the Twenty-First Century, Birkhauser, Boston, 1997. [2] B.D.O. Anderson, M. Deistler, L. Farina, L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, IEEE Trans. Circuits Systems-I: Fundamental Theory Appl. 43 (1996) 134 –142. [3] L. Benvenuti, L. Farina, On the class of lters attainable with charge routing networks, IEEE Trans. Circuits and SystemsII: Analog and Digital Signal Processing 43 (1996) 618 – 621. [4] L. Benvenuti, L. Farina, A note on minimality of positive realizations, IEEE Trans. Circuits and Systems-I: Fundamental Theory Appl., to appear. [5] A. Berman, M. Neumann, R.J. Stern, Nonnegative Matrices in Dynamic Systems, Wiley, New York, 1989. [6] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 2nd ed., SIAM, Philadelphia, 1994. [7] L. Farina, On the existence of a positive realization, Systems Control Lett. 28 (1996) 219 – 226. [8] A. Gersho, B. Gopinath, Charge-routing networks, IEEE Trans. Circuits and Systems CAS-26 (1979) 81– 92. [9] J.M. van den Hof, Realization of positive linear systems, Linear Algebra Appl. 256 (1997) 287–308.
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