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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 5, MAY 2004 651

A Tutorial on the Positive Realization ProblemLuca Benvenuti and Lorenzo Farina

Abstract—This paper is a tutorial on the positive realizationproblem, that is the problem of finding a positive state–space rep-resentation of a given transfer function and characterizing exis-tence and minimality of such representation. This problem goesback to the 1950s and was first related to the identifiability problemfor hidden Markov models, then to the determination of internalstructures for compartmental systems and later embedded in themore general framework of positive systems theory. Within thisframework, developing some ideas sprang in the 1960s, during the1980s, the positive realization problem was reformulated in termsof a geometric condition which was recently exploited as a tool forfinding the solution to the existence problem and providing par-tial answers to the minimality problem. In this paper, the readeris carried through the key ideas which have proved to be usefulin order to tackle this problem. In order to illustrate the main re-sults, contributions and open problems, several motivating exam-ples and a comprehensive bibliography on positive systems orga-nized by topics are provided.

Index Terms—Nonnegative matrices, polyhedral cones, positivesystems, realizations.

I. INTRODUCTION

MATHEMATICAL modeling is concerned with choosingthe relevant variables of the phenomenon at hand and

revealing the relationships among those. Positivity of the vari-ables often emerges as the immediate consequence of the natureof the phenomenon itself. A huge number of evidences are justbefore our eyes: Any variable representing any possible type ofresource measured by a quantity such as time [14], [35], moneyand goods [20], [21], buffer size and queues [31], data packetsflowing in a network [24], human, animal and plant populations[22], [25], concentration of any conceivable substance you maythink of [18], [19], [32] and also—if you have not conceivedthis—mRNAs, proteins and molecules [13], electric charge [7],[9], [15], and light intensity levels [10], [23]. Moreover, alsoprobabilities are positive quantities, so that one has to mentionin this list also hidden Markov models [6], [30] and phase-typedistributions models [28], [29].

In this paper, we focus on systems whose state variablesare positive (or at least nonnegative) in value for all times andconsider the class of models in which the relationships amongvariables are described by difference equations. Such systems,known as positive systems (see [1]–[5]), have the peculiarproperty that any nonnegative input and nonnegative initialstate generates a nonnegative state trajectory and output for alltimes.

Manuscript received December 18, 2002; revised June 10, 2003 andNovember 18, 2003. Recommended by Associate Editor F. M. Callier.

The authors are with the Dipartimento di Informatica e Sistemistica “A. Ru-berti,” Università degli Studi di Roma “La Sapienza,” 00184 Rome, Italy.

Digital Object Identifier 10.1109/TAC.2004.826715

For linear systems, positivity results in a specific sign pat-tern on the entries of the system’s matrices. In particular, a dis-crete-time system is described by nonnegative matrices whichhave been the subject of study in the mathematical sciences fromlong ago (see, for example, [82]–[91]). The main results regardthe characterization of the eigenvalue location and are due toPerron [90], Frobenius [85], [84], and Karpelevich [88].

The system theoretic approach to positive linear systems hasbeen initiated by Luenberger in his seminal work [5] during the1980s. From that time on, an impressive number of theoreticalcontributions to this field has appeared. Some of the topics are:reachability and controllability [55], [57], [59], [60], [66], [67],[71], [72], observability [41], realizability [92]–[128], stability[38]–[40], [44], [50], [51] and stabilization [64], [65], pole/zeropattern and pole assignment [45], [48], [49], [63], identification[96], [97], and two-dimensional (2-D) systems and behavioralapproach [73]–[80]. Moreover, there is an extensive literatureon nonlinear positive systems, also known as monotone cooper-ative systems [2], [33].

In this paper, we focus on the realizability problem fordiscrete-time single-input–single-output (SISO) positive linearsystems, that is, as described in Section II, the problem offinding a triple with nonnegative entries (calledpositive realization) realizing a given transfer function. Thisproblem goes back to the 1950s and was first related to theidentifiability problem for hidden Markov models (knownalso as functions of finite Markov chains) [12]. Moreover,during the 1960s, several publications appeared that provide anecessary and sufficient condition for the existence of a finitestochastic realization of a given distribution (see the referencesgiven in [115]). In the 1970s, such problem arose in the contextof the determination of internal structure for compartmentalsystems [109] and later was embedded in the more generalframework of positive systems theory [5]. More precisely [1],when a transfer function is given, the following issues appearto be fundamental: Find conditions on the transfer function forthe existence of a positive realization and provide an algorithmto construct such a realization (existence problem), determineits minimal allowed order (minimality problem) and find howall minimal positive realizations are related to each other(generation problem).

An important geometric interpretation of the positive realiza-tion problem (see [94] for the development of this idea in the lit-erature) was recently exploited as a tool for finding a completesolution to the existence problem [92], [101], [108] and for pro-viding partial answers to the minimality problem [120]–[128].Another geometrical interpretation is given in [16], [116], [125],and [126].

Moreover, in [115], a characterization of the positive realiza-tion problem is given in terms of positive factorizability of the

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652 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 5, MAY 2004

Hankel matrix. Finally, an extension of the results given in [92],[101], and [108] to the case of a positive realization having thematrix irreducible, strictly positive or primitive, is given in[103].

In this tutorial paper, we review, by means of several exam-ples, the development of the key ideas leading to the completesolution of the existence problem and giving new insight intothe minimality and generation problem. The main difference ofthis work w.r.t. the tutorials [1], [3] is that only some sufficientconditions for the existence problem are there considered; more-over, only few proofs are there sketched as well examples. Fi-nally, note that, all results herein presented have a correspondingcontinuous-time formulation which can be easily derived [92],[95], [101], [114], [125]. In this paper, we will draw particularlyfrom [61], [92], [101], [110], [111], [120], and [121].

The paper is organized as follows. In Section II, the positiverealization problem is formally stated together with some basicdefinitions and notations. Section III contains well known basicresults on the spectrum of nonnegative matrices useful in thesubsequent sections. Sections IV and VI are collections of ex-amples devoted to introduce and illustrate the key issues relatedto the existence and minimality problem, respectively. The com-plete solution of the existence problem is given in Section Vand some preliminary results on minimality are illustrated inSection VII. The problem of finding how different minimal pos-itive realization are related each other is briefly discussed inSection VIII and open problems and directions are addressedin Section IX.

The bibliographic section does not contain only referencesstrictly related to the positive realization problem, but to themore general area of positive systems theory and applications.In particular, the references have been grouped by topicsas follows: Basics of positive systems theory (the absolutebeginner may start with [3] and [5]), applications in differentfields (economy, biology, electronics, ), state-space issues(stability, invariance, ), control of positive systems andthat of general systems with a positive control (reachability,controllability, ), multidimensional positive systems and thebehavioral approach (asymptotic behavior, realization, ),basics of nonnegative matrix theory (the interested reader maybegin with [89]), and positive realization problem (existenceand minimality).

II. PROBLEM FORMULATION

In this section, we provide the formulation of the positive re-alization problem together with some basic definitions. More-over, the geometric reformulation in terms of invariant cones,mentioned in the Introduction, is given at the end of this section.

We begin with some definitions: denotes the set of non-negative real numbers; given a matrix , denotes its spec-trum and , with , is the size of the largest blockcontaining in the Jordan canonical form of . A matrix (or avector) is said to be nonnegative if all its entries are nonneg-ative and at least one is positive (so to avoid the trivial case ofan all-zero matrix). Any eigenvalue of a nonnegative matrix

such that will be called

a dominant eigenvalue of and will be called the spec-tral radius of . Given a transfer function ,1 any pole ofmaximal modulus will be called a dominant pole of , andits modulus will be denoted by . Moreoverdenotes the maximal modulus of the poles apart from the domi-nant ones, if any. Any pole of modulus will be calleda subdominant pole of .

Definition 1: A discrete-time SISO linear system of the form

(1)

is said to be a positive system provided that for any nonnegativeinput sequence and nonnegative initial state , the state tra-jectory and the output are always nonnegative.

The following theorem characterizes positive systems interms of the sign pattern of the system’s matrices.

Theorem 1: [3] A system of the form (1) is a positive systemif and only if , , and are nonnegative.

From Definition 1 immediately follows that the impulse re-sponse of a positive system is nonnegative for alltimes . Hence, in the sequel we will consider onlysystems with a nonnegative impulse response. We are now readyto give a formal statement of the positive realization problem,that is the realization problem for positive systems [1], [93],[111]: Given a strictly proper rational transfer functionwith nonnegative impulse response , the tripleis said to be a positive realization if

with , , nonnegative. The positiverealization problem consists of providing answers to the fol-lowing questions.

• The existence problem: Is there a positive realizationof some finite dimension and how may

it be found?• The minimality problem: What is the minimal value for

?• The generation problem: How can we generate all possible

positive minimal realizations?It is worth noting that when no specific sign pattern is re-

quired for the system’s matrices, the above problems have awell known solution: existence is always guaranteed, the min-imal order of a realization equals the order of the transfer func-tion and all minimal realizations can be generated by using anyinvertible change of coordinates. It will be clear in the sequelhow positivity dramatically changes this situation leading to anintriguing scenario where the solutions are far from trivial. It isplain that, as stated in [94], we consider the positive realizationproblem solved only when an algorithm for checking the exis-tence of solutions or allowing their construction is given.

The geometrical interpretation of the positive realizationproblem provided in the sequel, requires some basic definitionsfrom cone theory. A set is said to be a cone providedthat for all ; if contains an open ball of

, then is said to be solid; if , thenis said to be pointed. A cone which is closed, convex, solid

1Without loss of generality, we consider only strictly proper transfer func-tions of finite order in which the numerator and denominator are coprime poly-nomials.

BENVENUTI AND FARINA: TUTORIAL ON THE POSITIVE REALIZATION PROBLEM 653

and pointed will be called a proper cone. A cone is said tobe polyhedral if it is expressible as the intersection of a finitefamily of closed half-spaces. The notationindicates the closed convex cone consisting of all nonnegativelinear combinations of vectors , with possiblyinfinite. The following is a reformulation presented in [110]of a theorem which gives a geometrical interpretation of thepositive realization problem. The development of this approachin the literature is given in [94].

Theorem 2: Let be a strictly proper rational transferfunction of order and let , with and ,

be a minimal (i.e. jointly reachable and observable)realization of . Then, has a positive realization if andonly if there exists a polyhedral proper cone such that

1) , i.e. is -invariant;2) ;3)

where is called theobservability cone.

It is worth noting that, once has been found,2

where the columns of are the extremal vectors (see [82]) of ,a positive realization can be obtained by solving

(2)

Hence, the number of extremal vectors of the cone equalsthe dimension of the positive realization. This fact amounts tosaying that polyhedrality of corresponds to a finite dimensionof the positive realization. Moreover, the multiple-input–mul-tiple-output (MIMO) case with inputs and outputs can beeasily handled by replacing conditions 2) and 3) of the afore-mentioned theorem by

2) , ;3) , ;

where is the -th column [row] of and.

Another interesting geometric interpretation of the positiverealization problem can be found in [16], [116], [125], and[126]. This interpretation refers to the impulse responseinstead of the transfer function and relies on the conceptof shift invariance of a cone.

Theorem 3: [16] Consider a nonnegative impulse response. Then, has a positive realization3 if and

only if there exists a polyhedral cone such that

1) cone ;2) is -shift invariant;

where a cone is said to be -shift invariant ifimplies .

Note that, the conditions of Theorem 2 are given in terms of-invariance of a cone , where is such that

is a minimal realization of while those of Theorem 3 aregiven only in terms of the impulse response and of shift in-

2Obviously, the notation cone(K) indicates the cone generated by thecolumns of the matrix K .

3A nonnegative impulse response H of a MIMO system is said to have apositive realization if there exists a nonnegative triple fA ;B ;C g such thatH = C A B .

variance of a cone . The main advantage of this last ap-proach is that it provides a theoretical framework for the positiverealization problem which may give new insights and new direc-tions for the solution of those problems which are currently un-solved, such as minimality and generation. In fact, this approachleads naturally to the concept of positive system rank [16] whichallows the reformulation of the minimality problem in terms ofpositive factorization (see [116]). However, the evaluation of thepositive system rank involves an infinite test so that, at the mo-ment, such result is mainly theoretical.

In this paper, we will refer to the approach considered inTheorem 2.

III. EIGENVALUE LOCATIONS FOR POSITIVE SYSTEMS

In this section, we present those results related to nonnegativematrices which can be used, in view of Theorem 1, to charac-terize the eigenvalues location for positive systems.

We first state the celebrated Perron–Frobenius Theorem[85]–[87], [89], [90] in a suitable reformulation.

Theorem 4: [Perron–Frobenius] The dominant eigenvaluesof a nonnegative matrix of dimension are all the roots of

for some (possibly more than one) values of. In particular, one of the dominant eigenvalues is

positive real, i.e., . Moreover,for any dominant eigenvalue .

The following theorem (whose statement presented later isdue to Ito [84] and in view of [89, Th. 1.2, p. 168]) is the cele-brated Karpelevich Theorem [88], [89] that completely charac-terizes the regions of points in the complex plane that areeigenvalues of nonnegative matrices with spectral radius

. In fact, and the following holds.Theorem 5: [Karpelevich] The region is symmetric rel-

ative to the real axis, is included in the disc , and inter-sects the circle at points , where and run overthe relatively prime integers satisfying . Theboundary of consists of these points and of curvilinear arcsconnecting them in circular order. Let the endpoints of an arc be

and . Each of these arcs is givenby the following parametric equation:

where the real parameter runs over the interval anddenotes the nearest integer to .

For the sake of illustration, the regions and are de-picted in Fig. 1.

We end this section with two technical lemmas which will beused in the sequel.

Lemma 6: [92] Let be a strictly proper rational transferfunction and let be a positive realization of .If , then there exist another positive real-ization of lesser dimension of such that

.Lemma 7: Let be a strictly proper rational transfer

function with nonnegative impulse response ( ,). Then, is a pole of and has maximal

multiplicity among all the dominant poles.


Fig. 1. Karpelevich regions � (dark gray) and � (dark and light gray).

IV. EXISTENCE: PROLOGUE VIA EXAMPLES

This section is devoted to introducing the key ideas and pit-falls regarding the existence problem for positive systems bymeans of several examples. First note that since

, then it is immediate to see bydirect substitution that nonnegativity of the impulse response

is equivalent to the following cone condition:

which defines the reachability cone . In view of the ge-ometrical interpretation of the positive realization problem,i.e., Theorem 2, one needs to find a polyhedral proper conesatisfying conditions 1)–3). By construction and from Lemma7 (see [114]), the reachability cone fulfills these conditionsapart from polyhedrality, so that it is worth investigatingconditions for polyhedrality of . Let’s explore this possibilityby considering the following example.

Example 1: Consider the system with transfer function

Its impulse response

is clearly nonnegative for all . Consider then the minimal real-ization in Jordan canonical form

The reachability cone , as shown on the left-hand side ofFig. 2, is polyhedral with five extremal vectors and, in fact

since can be expressed as a nonnegative linear combinationof vectors , , , , and . For example

A positive realization of order 5 can then be found by solving(2) with thus obtaining

The previous example shows that the reachability cone canplay the role of the cone in Theorem 2 being polyhedral. Isit always the case? That is, can one always use the reachabilitycone in order to find a positive realization? The next two exam-ples illustrate this point.




The reachability cone , as the picture in the middle of Fig. 2shows, is not polyhedral. In fact, for every one has

This implies that the vector

belongs to the boundary of but not to . Accordingly, theclosure of is given by

In particular, in this example

is -invariant and is polyhedral with three extremal vectors. Apositive realization of order 3 can then be found by solving (2)with thus obtaining

The previous example makes clear that the reachability conecan be nonpolyhedral but its closure may be such. Hence,

a positive realization can be found by using the cone insteadof . Unfortunately, this is not always the case as shown in thenext example.




Fig. 2. Reachability coneR considered in example 1 (left), in example 2 (center), and in examples 3 and 5 (right).


which, in this case, is a positive realization. Nevertheless, thereachability cone and its closure as well, depicted at the righthand side of Fig. 2, are nonpolyhedral, i.e., they have an infinitenumber of extremal vectors. Hence, neither the reachabilitycone nor its closure can function as the cone in Theorem2, while, in this case, the positive orthant can.

The examples considered so far show that one cannot con-sider only the reachability cone (or its closure) when searchingfor the cone of Theorem 2. Consequently, on one hand, onehas to find conditions assuring polyhedrality of the reachabilitycone (or of its closure), while on the other hand it is necessaryto develop more general methods in order to find such a cone

. However, it is worth noting that even if the impulse responseis nonnegative, a cone satisfying the conditions of Theorem2 may not exist at all, that is nonnegativity of the impulse re-sponse alone is not a sufficient condition for a transfer functionto have a positive realization. The next example illustrates thissituation.


whose poles are 1 and . Its impulse response


If is an irrational number, then neither the reachability conenor its closure and, indeed, any other -invariant proper cone,

can be polyhedral. Hence, a positive realization of finite orderdoes not exist.

To show this, suppose there exists an invariant polyhedralproper cone and consider any of its extremal vectors . Sincethe third component of remains unchanged under and the

first two components are rotated by an angle in theplane, then it is easily seen that, as goes to infinity, the cone

is an ice-cream cone (see [82, p. 2]), thus contradicting the poly-hedrality hypothesis. This conclusion can also be derived byusing the Perron–Frobenius theorem and Lemma 6. In fact, fromLemma 6 it follows that, without loss of generality, one can con-sider a positive realization of with spectralradius which, in this case, equals 1. Hence,from the Perron–Frobenius theorem, the dominant eigenvaluesof would be roots of for some . This isobviously not the case since, when is an irrational number,then there is no integer such that holds. Finally,observe that the previous conclusion does hold independently ofthe specific value for the input and output vectors and , sinceit relies only on the Perron–Frobenius theorem.

This last example reveals that the location of the dominantpoles plays an important role in the positive realization problem.In fact, in view of Lemma 6, the Perron–Frobenius theorem im-poses a specific poles pattern on the dominant poles ofand defines a new necessary condition together with nonnega-tivity of the impulse response. Are those conditions sufficient?Section V provides a negative answer to this intriguing questionthus revealing that the situation is far more complicated than onemay expect.

V. EXISTENCE: THE EPILOGUE VIA THEOREMS

In this section we will provide necessary and sufficient condi-tions for the existence of a positive realization of a given transferfunction with nonnegative impulse response. In particular, webegin the section by giving necessary and sufficient conditionsfor the reachability cone (or its closure) to be polyhedral(Theorems 8–10). As mentioned before, polyhedrality of , orof its closure, together with nonnegativity of the impulse re-sponse allow to construct a positive realization by using(or ) in Theorem 2. These results provide a complete answerto the questions raised by Examples 1–3. Detailed proofs ofTheorems 9 and 10 are not given here since they are straight-forward consequences of [61, Ths. 1 and 2].4 In fact, roughlyspeaking, these last theorems provide conditions for polyhe-drality in terms of the spectrum of a minimal realization. Hence,

4Detailed proofs of these theorems can be found in [53].


in view of Lemma 7, [61, Ths. 1 and 2] can be reformulatedin terms of the poles of and this is done in Theorems 9and 10.

The remaining part of the section deals with necessary andsufficient conditions for a transfer function to have a positiverealization when polyhedrality of , or of its closure, may notbe present. Moreover, the constructive procedure to obtain thecone is also sketched.

The following theorem considers the simple case ofwith all-zero poles.

Theorem 8: Let be a strictly proper rational transferfunction of order with nonnegative impulse response( , ) and . Then the reach-ability cone of any minimal realization ofis and is polyhedral withextremal vectors. Moreover, has a positive realization oforder .

Sketch of proof: Condition implies thatis a nilpotent matrix, so that for every

. Moreover, it is immediate to see that the Markov canonicalrealization (see [3, p. 83]) is positive.

In view of the aforementioned theorem, we will consider inthe sequel without loss of generality. This case isconsidered in what follows where denotes the leastcommon multiple between the numbers and .

Theorem 9: Let be a strictly proper rational transferfunction of order with nonnegative impulse response ( ,

) and . Then the reachability coneof any minimal realization of is polyhedral if and only ifthe poles of satisfy the following conditions.

1) The dominant poles are simple.2) The dominant poles are among the th roots of

for some positive integer . Moreover, taking the minimalvalue of , no nonzero non dominant pole can havean argument which is an integer multiple of .

Furthermore, has a positive realization of some finiteorder where is the number of extremal vectors of .

Applying the aforementioned theorem to Examples 1–3 oneobtains , so that polyhedrality is assured provided thatthere are no positive real poles apart from the dominant one.Hence, the reachability cone is polyhedral only in Example 1.To deal with the situation illustrated by Examples 2 and 3, onecan resort to the following theorem.


) and . If the reachability coneof any minimal realization of is nonpolyhedral, then itsclosure is polyhedral if and only if the poles of satisfyeither of the following.

1a) The dominant and subdominant poles are simple.2a) The dominant poles are among the th roots of

for some positive integer and the subdom-inant poles are among the th roots offor some positive integer .

3a) Taking the minimal values , of and re-spectively, then no nonzero nonsubdominant pole can

have an argument which is an integer multiple of ,with .

Or

1b) has multiplicity 2.2b) The dominant poles are among the th roots of

for some positive integer . Moreover,taking the minimal value of , no nonzero nondominant pole can have an argument which is aninteger multiple of .

Furthermore, has a positive realization of some finiteorder where is the number of extremal vectors of .

As an example of application of the aforementioned theorem,consider Examples 2 and 3 where case a) applies with

. Then polyhedrality of the closure of is assuredprovided that there are no positive real eigenvalues apart fromthe dominant and the subdominant ones. Therefore, the closureof the reachability cone is polyhedral only in Example 2.

As a final comment to the previous theorems of this section,we note that the observability cone shares the same poly-hedrality properties of the reachability cone since a dualityproperty holds, as shown in [110]. Moreover, polyhedrality of

can be interpreted as the property of finite-time reachability.Finally, when using for finding a positive realization, the re-sulting realization has the positive orthant as the reachabilitycone [122]. This property of positive systems is called positivereachability and has been studied by several authors (see [56],[57], [59], [60], [66], [67], and [71]).

When the easily testable conditions of the previous theoremsdo not hold, then one cannot use the reachability cone in order tofind a positive realization, so that this scenario calls for a moregeneral methodology for finding the cone satisfying the con-ditions of Theorem 2. As discussed at the end of the previoussection, besides nonnegativity of the impulse response, the dom-inant pole pattern must be consistent with the Perron–Frobeniustheorem for a given transfer function to have a positiverealization, that is the dominant poles must have an argumentsuch that is a rational number.

We begin our discussion by considering the case in which thedominant pole is unique in order to check whether nonnegativityof the impulse response is a sufficient condition when the dom-inant pole pattern does not play any role. The answer is affirma-tive and is provided by the following result. Moreover, the proofis constructive, so that a methodology for finding a positive re-alization is also briefly illustrated. Finally, the following resultis the “building block” for the general case, as it will clearly ap-pear in the sequel.

Theorem 11: [92] Let be a strictly proper rationaltransfer function of order with nonnegative impulse response( , ) and . If has a unique(possibly multiple) dominant pole, then has a positiverealization of some finite order .

The proof of the previous theorem relies on the property thatwhenever nonpolyhedrality of the reachability cone occurs,then its extremal vectors accumulate on the vector


Roughly speaking, nonpolyhedrality shows up in the fact thatcone has infinitely many extremal vectors approaching .Then, the key idea of the constructive proof of the previoustheorem is

1) surround with an appropriate cone;2) add the vector to such a cone;3) add appropriate vectors so that the overall cone is in-

variant under .To be more precise, consider any polyhedral proper cone

strictly containing and contained in .Such cone surely exists because is strictly contained in theobservability cone . In fact, since the realization is observable,then for any . Consider then the cone

which is proper, polyhedral, contained in and containingby construction. Hence, only invariance is required in order tofulfill all the conditions of Theorem 2. To obtain invariance, onecan take the cone

(3)

which is by construction invariant, is contained5 in and main-tains polyhedrality since all the extremal vectors accumulate on

that is strictly contained in the interior of .For the sake of illustration, such methodology is applied to

the system considered in Example 3 even if, in this case, theJordan canonical form is already a positive realization.

Example 5: Consider the system as in Example 3 for whichthe reachability cone and its closure are not polyhedral cones, asshown in Fig. 2 on the right. Then, chooseas follows:

It easy to verify that with this choice and .The cone is depicted in white on the left-hand side of Fig. 3together with the cone . One has

since, in this case, the vectors , , and are con-tained in . The cone is depicted on the right-hand side ofFig. 3. It is worth noting that, in this case, the cone is invariantbut in general this is not the case; nevertheless this methodologyalways produces a cone that is invariant. A positive realiza-tion can then be found using (2) thus obtaining

5Note that the observability cone O is A-invariant by definition.

Fig. 3. The reachability cone R and the cone Q depicted in white (left) andthe final cone K (right).

When the dominant pole is not unique but the dominant polepattern is consistent with the Perron–Frobenius theorem, thenone can try to extend the previous result. First of all, considerthe case of simple dominant poles and note that there is a finiteset of vectors such that in the limit as ,

cycles through them. Moreover, is equal toas defined in condition 2) of Theorem 9 and

(4)

In order to extend Theorem 11, one has to consider that inthis case, even if the realization is minimal and conse-quently observable, one may have for some but not all’s. Hence, in order to extend in a straightforward manner the

proof of Theorem 11 to the case of nonuniqueness of the domi-nant pole, one needs the additional assumption that

(5)

which is equivalent to condition 2) of the following theorem asit will be clear in the sequel.

Theorem 12: [92] Let be a strictly proper rationaltransfer function of order with nonnegative impulse response( , ) and . If the poles ofsatisfy the following conditions:

1) the dominant poles are simple and are among the -rootsof for some positive integer ;

2) ;then has a positive realization of some finite order

.It is plain that the proof of the previous theorem goes in the

same way of that of Theorem 11 by choosing an appropriatepolyhedral cone contained in and strictly containing all thevectors . The case , i.e., when condition 2) of theprevious theorem does not hold, implies that the corresponding

belongs to the boundary of the observability cone . Hence,it is not possible anymore to find a polyhedral cone containedin and strictly containing all the vectors . Because condi-tions ensure that the associated cone , definedby (3), is polyhedral and condition ensures ,then the reasoning used to prove Theorems 11 and 12 cannot be“relaxed” to consider the case .


In order to gain insight into the case , note that iffor some , then using (4), one has that there exists a

value such that

(6)

In other words, there exists a subsequence of the im-pulse response for which (6) holds so that condition 2) ofTheorem 12 rules out this case. This lead us to try to tacklethis situation by changing the focus on the properties of subse-quences. Therefore, consider the following procedure whichcan be used in order to decompose the impulse response ofa given transfer function in appropriate subsequences.

Procedurebeginif has a unique dominant

pole or has at least one of the domi-nant poles with an argument suchthat is not a rational number

then let and.

otherwise let the minimal positiveinteger for which the dominant polesof are among the th rootsof . Hence, decompose theimpulse response in thesubsequences

with corresponding transfer functions

where .end.

To decompose , let , and performthe procedure , starting with . Then, repeat ititeratively for each new transfer function produced by the pro-cedure itself until only ’s are produced.

The transfer functions with obtainedat the end of this decomposition procedure will have, by con-struction, either a unique dominant pole or at least a domi-nant pole having an argument such that is not a rationalnumber. Moreover, as shown in [101], is finite.

The following example illustrates such decomposition proce-dure.


whose impulse response

is nonnegative for every . Note moreover that

so that Theorem 12 does not apply.The dominant poles of are and. Since , then the procedure produces the

two subsequences and whosetransforms are

First note that has one unique dominant pole. Ifis not a rational number, say , then has domi-nant poles (which are not dominant poles of ) with argu-ment such that is not a rational number. In this case,by performing the decomposition procedures andthen , the decomposition ends with and

, .Otherwise, that is if is a rational number, say ,

then has three dominant poles, namely ,and . Since , then the proce-

dure decomposes the subsequence inthree more subsequences

with , i.e.

whose -transforms are

The dominant poles of the previous transfer functions areunique. Hence, the decomposition procedure stops with

and

We are now ready to state the necessary and sufficient condi-tions for a transfer function to have a positive realization.Note that, as shown in [101], also the case of multiple poles isencompassed.


Theorem 13: [101] Let be a strictly proper rationaltransfer function of order with nonnegative impulse response( , ) and . Then has a pos-itive realization of some finite order if and only if all the

’s, , obtained by iteratively applying thedecomposition procedure , have a unique (possibly multiple)dominant pole.

A simple procedure to obtain a positive realization from thepositive realizations of the ’s is not reported here for thesake of brevity and can be found in [101] and [108]. The pre-vious theorem states that, in general, nonnegativity of the im-pulse response together with a specific pole pattern for the dom-inant poles, as required by the Perron–Frobenius theorem, arenecessary but not sufficient conditions for a transfer function

to have a positive realization. For example, the transferfunction considered in Example 6 has dominant poles consis-tent with the Perron–Frobenius theorem but, from Theorem 13,has not a positive realization in the case is not a rationalnumber.

An example considering a transfer function for which theconditions of Theorems 8–12 are not fulfilled but having a pos-itive realization, that is conditions of Theorem 13 are satisfied,is Example 6 in the case .

In order to check positive realizability, one can also resort tothe following corollary.

Corollary 14: [101] Let be a strictly proper rationaltransfer function of order with nonnegative impulse response( , ) and . Then has apositive realization of some finite order if every polehas the property that is a root of unity.

VI. MINIMALITY: A PROLOGUE VIA EXAMPLES

This section deals with the minimality problem for positivesystems as defined in Section II. Hereafter, it will be shownthat the minimal dimension of a positive realization of a giventransfer function may be much “larger” than its order andhow this can be related to different mechanisms. First notethat the poles of a given transfer function are a subset of theeigenvalues of any of its positive realizations. Then, the polesmust be a subset of eigenvalues consistent with a nonnegativematrix. Basing on this consideration, the next example con-siders the mechanism related to the rotational symmetry of thedominant eigenvalues of a nonnegative matrix required by thePerron–Frobenius theorem.


Its impulse response is clearlynonnegative for all . Note that since , and 1 are the dom-inant poles of , then from the Perron–Frobenius theorem,all the 4th roots of unity must belong to the spectrum of the ma-trix of any positive realization. Consequently, the minimal

dimension for a positive realization of is not smaller than4 so that the following realization:

is minimal as a positive linear system. A more general case isthe following one parameter family of transfer functions of order

It has been shown (see [120]) that for any , the minimal pos-itive realization of is of dimension .Hence, the difference between thedimension of any minimal positive realization and the order ofthe transfer function goes to infinity as increases.

The rotational symmetry of the dominant spectrum of a non-negative matrix, due to the specific dominant poles pattern, isnot the only reason for the dimension of any minimal positiverealization to be greater than the order of the transfer function.This is illustrated by the following example in which the domi-nant eigenvalue is unique so that no symmetry of the dominantspectrum is required by the Perron–Frobenius theorem:



is clearly nonnegative for all . Since the poles of are 1,and, as one can easily check from Fig. 1, lie outside ,

then the matrix of any minimal positive realization must beof dimension greater than 3. Therefore, the following fourth-order positive realization

is minimal as a positive system.This last mechanism, related to a specific poles pattern,

is—again—not the only reason for the dimension of anyminimal positive realization to be greater than the order of thetransfer function even when the dominant eigenvalue is unique.This should be not surprising since positivity of the systemimplies restrictions not only on the dynamic matrix but on theinput and output vectors also. The next example is based on thegeometrical point of view given by Theorem 2.




is such that , , , and for

Then, it is nonnegative for all . Since the impulse response ofthe system is such that , then for any minimalrealization we have

(7)

Suppose then that there exists a third-order positive realizationof . Hence, the cone , where

is the similarity transformation between and, has three extremal vectors and satisfies the conditions

of Theorem 2. From conditions 1) and 3), it follows thatfor . Moreover, in view of (7) the following hold:

for . Consequently, the three vectors , andlie on different extremal vectors of the observability cone .Then, from condition 2) of Theorem 2, i.e., , the vectors, and are necessarily extremal vectors of so that

is the polyhedral closed convex cone consisting of all finitenonnegative linear combinations of vectors , and , i.e.,

Since , then is not -invariant, thus arriving at acontradiction. Therefore the following fourth-order positive re-alization:

(8)

is minimal as a positive system. This example has been gener-alized in [127] using a digraph approach. In particular, a familyof transfer functions with four fixed real simple polesis there considered and it is shown that for any , the orderof any minimal positive realization of the fourth order transferfunction is not less than . This is quite surprisingsince the value can be chosen arbitrarily large.

The examples considered in this section reveal that the loca-tion of the poles plays an important role in the minimal dimen-sion of a positive realization. In fact, the Perron–Frobenius the-orem imposes a specific pattern on the dominant poles ofand the Karpelevich regions on the remaining poles. Moreoverthe geometric characterization of the realization problem givenby Theorem 2 and the digraph approach lead to a different mech-anism enforcing the order of a positive realization to be greaterthan the order of the transfer function . The next sectionprovides partial answers to the minimality problem that, differ-ently from the realization problem, is still an essentially openproblem.

VII. MINIMALITY: A PARTIAL EPILOGUE VIA THEOREMS

In this section, some results on the minimality problem arepresented. First, a lower bound to the minimal order of a positiverealization is given basing on the Karpelevich theorem. Then,necessary and sufficient conditions are given for a third-ordertransfer function with distinct real positive poles and nonnega-tive impulse response to have a (minimal) positive realization oforder three.

As examples 7 and 8 make clear, since the set of theeigenvalues of any positive realization of a given transferfunction contains the poles of the transfer function itself, thenthe poles pattern must be consistent with the Karpelevich andPerron–Frobenius theorems. This is formally stated in thefollowing theorem.


) and . Let , with , bethe poles of , then the minimal order of a positive realiza-tion of is not less than where is the minimalvalue such that

Other interesting lower and upper bounds for the order of aminimal positive realization can be found in [124] and [127].

On the other hand, as Example 9 shows, the minimal order ofa positive realization may be much larger than the lower boundgiven by Theorem 15. This may happen, as in Example 9, whenall the poles of are real and the Karpelevich theorem doesnot play any role in the determination of the minimal order ofa positive realization. In order to gain partial insight into thepositive minimality problem, we state hereafter necessary andsufficient conditions for a third order transfer functions withdistinct positive real poles to have a third-order (minimal) pos-itive realization.

Theorem 16: [121] Let

be a third-order transfer function (i.e., , , ) withdistinct positive real poles . Then,has a third-order positive realization if and only if the followingconditions hold.

1) .2) .3) .4) for all

such that where

with .It is worth noting that the a priori knowledge about nonneg-

ativity of the impulse response is not required by the previoustheorem so that there is no need to check such condition onbefore applying the theorem.

Conditions for a positive realization to be minimal have beengiven for a special class of positive systems such as the tree-


compartmental systems considered in [109] and the positivereachable systems in [122].

Moreover, as previously stated, a reformulation of the mini-mality of a positive realization in terms of the factorization ofthe Hankel matrix into two nonnegative matrices (called posi-tive factorization) can be found in [16], [115], [116], and [126].However, in this context, a procedure to evaluate the minimalorder of a positive realization is not available so far.

Finally, in [123], the connections between the McMillan de-gree of a given transfer function, the size of any minimalpositive realization and the minimum of the number of ex-tremal vectors of cones satisfying the conditions of Theorem 2,are discussed. It is there shown that, in general and ex-amples are given, for the MIMO case, for which this inequalityis strict.

VIII. GENERATION PROBLEM

This section deals with the generation problem, that is theproblem of finding how all the minimal positive realizations ofa given transfer function are related one to the other. To the bestof our knowledge there are few results on this problem [111],[126]; to have a glimpse on the difficulties encountered whentackling it, hereafter we provide a very simple example showingthat a change of coordinates consisting of positive linear combi-nations of the state variables of a positive system, may destroythe positivity of the representation in the new coordinates.

Example 10: Consider the positive system described by thetriple with nonnegative entries

Consider then the change of coordinates ,so that the system representation becomes

It is straightforward to notice that positivity is generally lost,since it may be well the case that ,

or .In general, positivity is certainly maintained only when the

change of coordinates reduces to a simple positive rescaling andreordering (i.e., when using a nonnegative generalized permu-tation matrix6 ) of the state variables, as stated by the followingtheorem.

Theorem 17: Given a change of coordinateswith full rank, then every positive system

of dimension is transformed into a positivesystem if and only if is anonnegative generalized permutation matrix.

Sketch of Proof: The if part is obvious so that we consideronly the only if part. Since must be nonnegative for anynonnegative , then is a nonnegative matrix. Analogously,

is also a nonnegative matrix when considering nonnega-tivity of . Hence, from [89, Lemma 1.1], is a general-ized permutation matrix.

6A nonnegative generalized permutation matrix is a permutation matrix inwhich the 1’s are replaced by positive real numbers.

IX. OPEN PROBLEMS AND NEW DIRECTIONS

As it is clear from the issues so far discussed, there is a con-siderable number of open issues related to the minimality andgeneration problems for positive systems. For example, it is notclear what kind of mathematical “instruments” should be usedto effectively tackle these problems. In fact, the geometrical ap-proach (i.e., that of considered in Theorem 2) has proved to bea fundamental tool for determining the existence of a positiverealization. By contrast, such approach, has led so far to the de-termination of necessary and sufficient conditions for the exis-tence of a minimal positive realization for the third order caseonly. Nevertheless, several different promising reformulationsof the positive realization problem have been proposed in theliterature and are mainly related to the positive factorization ap-proach and to the concept of positive system rank proposed byPicci et al. in [116] and by van den Hof in [126], to the factor-ization of the transfer function in functions as proposedby Maeda et al. in [109] or to the Tarski–Seidenberg theory asproposed by Anderson in [1].

An important issue related to minimality of positive systemsis the study of “hidden” eigenvalues, i.e., of the eigenvalueswhich are not poles and that possibly one has to add in order toobtain a minimal positive realization. A full characterization ofthis property may lead to a deeper and valuable insight into theproblem. For instance, in Examples 7 and 8, the eigenvalue ofthe minimal order matrix which is not a pole of is oneof the dominant eigenvalues. Consequently, its value is relatedto the Perron–Frobenius theorem; on the other hand the situa-tion is in general far more complicated as Example 9 clearly hasshown.

Moreover, in the case of SISO systems, it is worth inves-tigating whether the inequality between the sizeof any minimal positive realization and the minimum of thenumber of extremal vectors of cones satisfying the conditionsof Theorem 2, reduces to .

Further, among all possible minimal realizations, it would bevery interesting to define a canonical minimal positive realiza-tion. Some preliminary results are provided in [121] for the caseof third order transfer functions with distinct positive real poles,and by Commault in [2].

Another important issue related to minimality [1] is the de-termination of “tight” lower and upper bounds to the minimalorder of a positive realization. Some results are presented in[124] and [127]. Moreover, it would be very interesting [1] todetermine directly from the system’s parameters (say, residuesand eigenvalues), the minimum number of samples of the im-pulse response to be checked in order to infer nonnegativity ofthe whole impulse response. In [112], an upper estimate of thisnumber is provided.

Finally, it would be very useful for applications [1] to approx-imate a positive realization by a lower dimension one.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their sug-gestions which helped to substantially improve the quality andreadability of this paper. Moreover, they would like to expresstheir gratitude to B. D. O. Anderson for giving them support andnew ideas which encouraged the pursuit of the study of positivesystems.


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Luca Benvenuti was born in Rome, Italy, in 1966.He received the “Laurea” degree in electricalengineering (summa cum laude) and the Ph.D.degree in systems engineering from the Universityof Rome “La Sapienza,” Rome, Italy, in 1992 and1995, respectively.

He was a Visiting Graduate Student at the Univer-sity of California, Berkeley, in 1995. In 1997, he wasScientific Consultant for Magneti Marelli, Bologna,Italy. From 1997 to 1999, he had a postdoctoral po-sition at the Department of Electrical Engineering,

University of L’Aquila, L’Aquila, Italy. From 1997 to 2000, he was a Scien-tific Consultant with the Project on Advanced Research on Architectures andDesign of Electronic Systems (PARADES), a European Group of Economic In-terest supported by Cadence Design Systems, Magneti Marelli, and STMicro-electronics. He is currently an Assistant Professor at the Department of Com-puter and System Science, University of Rome “La Sapienza,” Rome, Italy.

Dr. Benvenuti was Co-Recipient of the IEEE Circuits and Systems“Guillemin-Cauer Award” in 2001. He was Co-Chairman of the First Mul-tidisciplinary International Symposium on Positive Systems: Theory andApplications (POSTA03), held in Rome, 2003.

Lorenzo Farina was born in Rome, Italy, in 1963.He received the “Laurea” degree in electricalengineering (summa cum laude) and the Ph.D.degree in systems engineering from the Universityof Rome, “La Sapienza,” Rome, Italy, in 1992 and1997, respectively.

He was a Scientific Consultant with the Inter-departmental Research Centre for EnvironmentalSystems and Information Analysis, the Politecnicodi Milano, Milan, Italy, in 1993. He was the ProjectCoordinator at Tecnobiomedica S.p.A., Rome, Italy,

in the field of remote monitoring of patients with heart diseases, in 1995, andheld a visiting position at the Research School of Information Sciences andEngineering, the Australian National University, Canberra, in 1997. Since1996 he has been with the Department of Computer and Systems Science,the University of Rome “La Sapienza,” Rome, Italy, where he is currentlyan Associate Professor. He is coauthor of the book Positive Linear Systems:Theory and Applications (New York: Wiley, 2000).

Dr. Farina was Co-Recipient of the IEEE Circuits and Systems“Guillemin-Cauer Award” in 2001. He was a Co-Chairman of the FirstMultidisciplinary International Symposium on Positive Systems: Theory andApplications (POSTA03), held in Rome, Italy, 2003.

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