IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL 2013 1013 Minimal Positive Realizations of Transfer Functions With Real Poles Luca Benvenuti Abstract—A standard result of linear-system theory states that a single–input–single–output (SISO) rational th–order transfer function always has a state–space realization of the same order. In some applica- tions, one is interested in having a realization with nonnegative entries (i.e., a positive system) and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order . In this technical note, necessary and sufﬁcient conditions for a third–order transfer function with distinct real poles to have a third-order positive realization are given: these conditions are ex- pressed in terms of lower bounds for the ﬁrst three samples of the impulse response and therefore are very easy to check. This result is an extension of a previous result for transfer functions with distinct real positive poles. Index Terms—Minimality, positive realization, positive systems. I. INTRODUCTION This technical note deals with the positive realization problem for discrete–time SISO systems, that is the problem of ﬁnding a state-space representation with nonnegative entries (called positive real- ization) realizing a given transfer function [7]. This problem goes back to the 1950s and was ﬁrst related to the identiﬁability problem for Hidden Markov Models (HMM) [10], then to the determination of in- ternal structures for compartmental systems [19] and later embedded in the more general framework of positive systems theory [18]. Recently, the problem appeared in the ﬁeld of signal processing, and in particular in the design and implementation of digital ﬁlters using technologies such as optical ﬁbers and charge coupled devices. In fact, whenever physical constraints imposed by the speciﬁc technology in use allow only nonnegative values for the state variables of the ﬁlter, then it can be implemented only by means of a positive realization. This is the case of Charge Routing Networks [2], [3], [14] in which the state variables are necessarily nonnegative since they represent quantities of electric charge, and the case of optical ﬁlters in which they represent inten- sity levels of light signals [6]. Finally, it was recently shown that the positive realization problem is essentially equivalent to the problem of ﬁnding the representation for a given phase-type distribution [11] and some ideas from positive realization theory were also used in [24] to establish new results in the realization problem for HMM. The problem of determining whether a positive realization of a given transfer function exists or not (existence problem) has been completely solved in [1], [12]. In this technical note the problem of ﬁnding the min- imal allowed order (minimality problem) for a positive realization of a given transfer function is considered. Such a problem is a key feature in many applications such as digital ﬁlter design using optical ﬁbers or charge-coupled devices. In fact, when designing a ﬁlter, one obviously wishes to reduce space occupation and power consumption. However, this is not the only situation in which minimality is important: one may Manuscript received March 30, 2012; revised July 17, 2012 and July 27, 2012; accepted August 03, 2012. Date of publication August 08, 2012; date of current version March 20, 2013. Recommended by Associate Editor A. Astolﬁ. The author is with the Dipartimento di Ingegneria Informatica, Automatica e Gestionale Sapienza Università di Roma, 00185 Roma, Italy (e-mail: luca. benvenuti@uniroma1.it). Digital Object Identiﬁer 10.1109/TAC.2012.2212612 think of the identiﬁcation problem where one wishes to obtain informa- tion of the system structure from data measurements. This considera- tion alone justiﬁes an effort in ﬁnding positive realization of minimal dimension. As shown in [4], the problem is quite intriguing, since the positivity constraint required on the system matrices, may “force” a given transfer function to have a minimal positive realization of order much greater than its degree. Though a general method to ﬁnd a minimal positive realization has not been found yet, several results appeared recently re- garding discrete–time transfer functions with different poles location [5], [16], [21], [23], [25]. In addition, lower and upper bounds for the order of a minimal positive realization were given in [15], [20]. In more detail, in [5], the case of third–order transfer functions with distinct pos- itive real poles was considered and necessary and sufﬁcient conditions for the transfer function to have a third–order positive realization were given. In this technical note, an extension of the above result is presented by considering also the case of nonpositive poles. In more details, the transfer functions are divided in three classes according to poles loca- tion. For two of such classes, necessary and sufﬁcient conditions for the transfer functions to have a third–order positive realization are pro- vided, while, for the remaining one, only sufﬁcient conditions are pre- sented. All the conditions are expressed in terms of lower bounds for the ﬁrst three samples of the impulse response and therefore are very easy to check. The technical note is organized as follows. In the next section some preliminary results are provided while Section III con- tains the main results of the technical note. Finally, some conclusions and open problems are drawn in Section IV. II. PRELIMINARY RESULTS In this technical note, we focus on strictly proper rational third–order transfer functions with distinct real poles (1) where , and provide conditions for these functions to have positive realizations of the same order. A trivial necessary condi- tion for a transfer function to have a positive realization of some ﬁnite dimension is that the impulse response has to be nonnegative for all , i.e., (2) If one restricts attention to third–order positive realizations, a neces- sary condition on is that the poles , , and have to be eigen- values of some 3 3 nonnegative matrix. Hence, by the Perron–Frobe- nius theorem [13], [22], one of the poles of maximum modulus, say , must be positive real, i.e., (3) Moreover, since the trace of a nonnegative matrix is nonnegative and it is also equal to the sum of the eigenvalues of the matrix, then the following condition has to hold: (4) The set deﬁned by conditions (3) and (4) is a polyhedral cone and its section with the plane is shown in Fig. 1. As proved in [17], conditions (3) and (4) are also sufﬁcient for the set to be the spectrum of a 3 3 nonnegative matrix. Consequently, without 0018-9286/$31.00 © 2012 IEEE