Automatica 38 (2002) 861–868www.elsevier.com/locate/automatica

Brief Paper

Identi!cation of positive linear systems with Poisson outputtransformation�

Alberto De Santis, Lorenzo Farina ∗

Dipartimento di Informatica e Sistemistica, “A. Ruberti”, Via Eudossiana 18, 00184 Roma, Italy

Received 11 August 2000; received in revised form 8 March 2001; accepted 19 November 2001

Positive systems are systems in which the input=state=output variables are always positive since theyrepresent quantities. We propose an identi'cation procedure for a class of positive linear systems.

Abstract

We study the identi!cation problem for third-order linear time invariant positive systems in experiments where the output is a Poissonprocess. The problem well-posedness is investigated when the input–output model is described by a sum of real exponentials. A maximumlikelihood procedure is then proposed and the admissible set for the unknown parameters is characterized. The novelty of the approachconsists in solving a constrained maximum likelihood problem to estimate residues and eigenvalues based on theoretical results onminimality of positive realizations recently obtained in the literature. Numerical results are also provided. ? 2002 Elsevier Science Ltd.All rights reserved.

Keywords: Positive systems; Positive realization; Identi!cation algorithms; Poisson processes; Pharmacokinetics; Network tra7c analysis

1. Introduction

In this paper we deal with the identi!cation problem forlinear time invariant (LTI) positive systems whose impulseresponse h(t) is described by a sum of real exponentials, sothat the parameters one wishes to estimate are the residuesand exponents, i.e.

h(t) =n∑i=1

Rie�it ; t¿ 0: (1)

Such a problem has been studied by a large number of au-thors (see Anderson, 1983 and references therein, for a de-tailed discussion on this important topic) since the aboverepresentation is suitable in many diverse !elds of applica-tion such as medicine, biology, econometrics, telecommu-nications and industrial plants.

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor AntonioVicino under the direction of Editor Torsten Soederstroem.

∗ Corresponding author. Tel.: +39-06-44585690; fax: +39-06-44585367.

E-mail addresses: desantis@dis.uniroma1.it (A. De Santis),lorenzo.farina@uniroma1.it (L. Farina).

In the literature relevant to this problem a priori in-formation is often exploited at diAerent levels of gener-ality: within the state space models the zero pattern (i.e.the zero entries) of the system matrices is supposed tobe known so that only the remaining parameters needto be estimated, thus obtaining an identi!ed model of!xed structure among the many available of the same or-der n. For input–output models such as (1), the residuesRi are often assumed to be positive since robust nu-merical procedures are available when !tting real datato a sum of exponentials which, otherwise, is recog-nized to be an ill-posed problem (see Anderson, 1983,p. 225–226).We are interested in studying the case in which the in-

formation available on the process is positivity (Farina &Rinaldi, 2000). De!ning the positive orthant Rn+ as the setof x∈Rn such that xi¿ 0, we have the following.

De�nition 1. A linear system described by an input=state=output representation is said to be a positive linear system iffor any input function with values in Rp+ and for any initialstate vector in Rn+; the state and the output vectors remainsin Rn+ and Rq+; respectively.

0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0005 -1098(01)00277 -1

862 A. De Santis, L. Farina / Automatica 38 (2002) 861–868

Note that, while the notion of positivity may be appliedto a very general setup (see for example Nieuwenhuis, 1998for the behavioral approach), in this study we shall restrictourselves to single input–single output linear time invariant(SISO LTI) systems which, anyway, represent a very largeclass of applications.The positivity property stems directly from the intrinsic

physical nature of the variables involved in the process andtherefore this information is easily available. It is worth not-ing that according to the above de!nition, a positive systemhas a positive impulse response (i.e. h(t) nonnegative forany t¿ 0), but this does not necessarily imply positivity ofthe residues. On the contrary, as it will be clear in the sequel,positivity of the impulse response function is not su7cientto guarantee positivity of the system, unless h(t) is strictlypositive for t ¿ 0 (see Farina, 1996).Identi!cation of positive systems whose input–output rep-

resentation is given by model (1), when the only availableinformation is positivity, seems to provide a very generalframework since neither a !xed structure is imposed norpositivity of the residues is assumed, allowing for a more re-alistic description of data. However, any identi!cation pro-cess strictly depends on the experiment performed, i.e. howdata are obtained, so that we shall consider the case in whichthe output sequence of data is a Poisson process. The lat-ter is representative of a wide class of real processes (suchas in the dynamics of tracers in the human body, (see Ka-jiya, Kodama, & Abe, 1984), and for networks tra7c mod-els (see Ryden, 1994; San-qi Li & Chia-lin Hwang, 1997))and, moreover, it has the important property of being con-sistent with the positivity assumptions, since the output is asequence of positive numbers. Recently, the theory of LTIpositive systems has received many contributions in clas-sical system’s theory problems such as realization, mini-mality, structural properties and so forth (Anderson, 1997),nevertheless there is a lack of research in the identi!cationarea (apart from van den Hof, 1996). In particular, neces-sary and su7cient conditions for the existence of at leastone positive realization of some !nite order are available(Anderson, Deistler, Farina, & Benvenuti, 1996; Kitano &Maeda, 1998). On the other hand, minimality is still an openproblem (see van den Hof, 1996 for an interesting character-ization of minimality); indeed the previous conditions resultto be only necessary. However, for third-order systems withreal poles, a set of necessary and su7cient conditions for theexistence of a positive realization involving the parametersin model (1) can be found in Benvenuti, Farina, Anderson,and De Bruyne (2000). For this reason we shall focus onthis latter case in order to fruitfully exploit these results indesigning optimal identi!cation procedures. Moreover, it isworth noting that third-order systems represent a signi!cantsubset of applications as one can see from the related lit-erature (see, for example, Anderson, 1983; Jacquez, 1985;Kajiya, Kodama, & Abe, 1984; Walter & Contreras, 1999).The paper is organized as follows: in Section 2 we brieKy

recall some properties and results concerning positive sys-

tems, while the problem formulation is given in Section 3and the analysis of its well posedness is there also provided.In Section 4 a constrained optimization problem is formu-lated and a numerical procedure is provided, in order tosolve the maximum likelihood estimation problem. Simula-tion results are provided in Section 5.

2. Some results on positive systems

We will make use of the following de!nitions:

De�nition 2. A matrix A∈Rn×n is said to be nonnegativeprovided that its entries are nonnegative; is said to be a Met-zler matrix provided that its oA-diagonal entries are nonneg-ative. A vector x∈Rn is said to be nonnegative providedthat x∈Rn+.

De�nition 3. A function f(t)∈Rm; t¿ 0 is said to be pos-itive provided that it maps R+ into Rm+.

In this paper we will consider SISO LTI system, namelya triple (A; b; cT) where A∈Rn×n and b; cT ∈Rn.The following results state necessary and su7cient con-

ditions for a given triple to represent positive systems eitherdiscrete-time or continuous-time. The proof can be found inFarina and Rinaldi (2000).

Theorem 4. A discrete-time SISO LTI system (A; b; cT) ispositive if and only if A; b and cT are nonnegative.

Theorem 5. A continuous-time SISOLTI system (A; b; cT)is positive if and only if A is a Metzler matrix; b and cT

are nonnegative vectors.

Remark. We would like to point out that the de!nitionswhich can be found in the relevant literature may bemisleading; since some authors like to refer to positive sys-tems as nonnegative systems. However; in the following; wemaintain the de!nitions given by Luenberger in (Chapter 6;Luenberger; 1979) as already stated.

Positive systems feature a number of peculiar properties,the most relevant one being undoubtedly the impulse re-sponse function positivity. Signi!cantly enough, the latterproperty does not guarantee positivity of the system. In-deed, one can formulate the positive realization problem fordiscrete-time [continuous-time] systems () as follows:

2.1. The positive realization problem

Given a discrete-time positive impulse response functionh(k), the triple (A; b; cT) is said to be a positive realizationif h(k)= cTAk−1b, k=1; 2; : : : with A; b; cT nonnegative. Onthe other hand, the triple (A; b; cT) is a positive realization of

A. De Santis, L. Farina / Automatica 38 (2002) 861–868 863

a continuous-time positive impulse response function h(t)if h(t) = cTeAtb with A Metzler, and b; cT nonnegative.Some questions arise:

• (The existence problem) Is there a positive realization{A; b; cT} of some !nite dimension N¿ n and how it maybe found?

• (The minimality problem) What is a minimal value forN?

• (The generation problem) How can we generate all pos-sible minimal positive realizations?

In (Anderson, Deistler, Farina, & Benvenuti, 1996; Fa-rina, 1996) the existence problem has been solved and aconstructing procedure for such realizations is also giventhere. Recently, in Forster and Nagy (2000) further resultsare obtained using an interesting diAerent approach.It is known, for transfer functions of degree 1 or 2, that

positivity of the impulse response is a necessary and su7-cient condition for the existence of a positive realization ofsome dimension N . Moreover, in those two cases, the min-imal dimension of a positive realization coincides with thedegree of the transfer function (Ohta, Maeda, & Kodama,1984). On the other hand, the situation for the case of sys-tems of order n¿ 2 is totally diAerent, as discussed recentlyin Benvenuti and Farina (1999).Actually, a necessary and su7cient condition for the ex-

istence of a minimal positive realization for the case ofdiscrete-time third-order systems is provided in Benvenutiet al. (2000) which reads as follows.

Theorem 6. Let the impulse response of a discrete-timeSISO LTI system be

h(k) = r1�k−11 + r2�k−1

2 + r3�k−13 ; k¿ 1 (2)

with nonzero residues r1; r2 and r3 and distinct positivereal eigenvalues �1¿�2¿�3. Then; h(k) has a third-orderpositive realization if and only if the following conditionshold:

(1) r1¿ 0(2) r1 + r2 + r3¿ 0(3) (�1 − P�)r1 + (�2 − P�)r2 + (�3 − P�)r3¿ 0(4) (�1 − �)2r1 + (�2 − �)2r2 + (�3 − �)2r3¿ 0 for all �

such that P�6 �6 �3 where

P�=max

{�1 + �2 + �3 − 2

√�

3; 0

}

with �:=(�2 − �3)2 + (�1 − �2)(�1 − �3):

In order to obtain the corresponding result for continuous-time systems, as shown in Farina (1997) it su7ces to applythe previous theorem to the Euler discrete-time approxima-tion of the given system. The proof is straightforward andis omitted for the sake of brevity.

Theorem 7. Let the impulse response of a continuous-timeSISO LTI system be

h(t) = r1e�1t + r2e�2t + r3e�3t t¿ 0 (3)

with nonzero residues r1; r2 and r3 and distinct real eigen-values �1¿�2¿�3. Then; h(t) has a third-order positiverealization if and only if the following conditions hold:

(1) r1¿ 0(2) r1 + r2 + r3¿ 0(3) (�1 + P�)r1 + (�2 + P�)r2 + (�3 + P�)r3¿ 0(4) (�1 + �)2r1 + (�2 + �)2r2 + (�3 + �)2r3¿ 0 for all �

such that −�36 �6 P� where

P�:=− �1 + �2 + �3 − 2√�

3:

We conclude this section with some remarks. We wouldlike to stress that, despite we are dealing with the seeminglysimple case of third-order transfer functions with distinctreal poles, because of the minimality property, the problemof !nding a minimal realization, is all but trivial. Indeed, afamily of third-order transfer functions, found in Benvenutiand Farina (1999), features the following property: for anyinteger N¿ 3, the corresponding member of the family ad-mits a minimal positive realization of dimension not smallerthan N . This shows the need for conditions of Theorems 6and 7, taking also into account that the problem of !nding anonnegative matrix with a given spectrum, is currently un-solved (see Boyle & Handelman, 1991 for details).

3. Problem formulation and well-posedness

In order to give a precise formulation of the identi!-cation problem for SISO LTI third-order positive systemsdescribed by Eq. (1), we need to study in some detail theexperiment which generates the data. We actually refer tothe class of applications in which the output generated by adiscrete impulse is a sequence of data {zk} having a Poissondistribution. This is for instance the case of kinetics of drugsin the human body when the amount of tracer is put intothe circulatory system (see for example, Kajiya, Kodama,& Abe, 1984). According to this physical mechanism, themeasured data zk are integer values provided by a radiationdetector (Geiger counter); it is known that such data followa Poisson distribution so that, the probability that the mea-sured data zk ; for any k¿ 1, are equal to an integer valuenk is given by

P(zk = nk) =�nkknk !

e−�k k¿ 1; (4)

which is indeed a Poisson law with count rate �k given by

�k =1�

∫ k�(k−1)�

h(t) dt: (5)

864 A. De Santis, L. Farina / Automatica 38 (2002) 861–868

The count rate �k is a positive number when h, given bymodel (1), is the impulse response of a positive system andtherefore is a positive function.

3.1. Problem formulation

Given a sample of data {zk}; k=1; : : : ; Pk, !nd a functionh(t); 06 t6 Pk�, given by model (3), such that (4) and (5)hold.Such a problem is well posed if existence and unique-

ness of a solution can be proved; moreover, the solutionh(t); t ∈ [0; T ] must be robust w.r.t. data perturbations. Tothis purpose we de!ne the admissible set of functions rep-resented by (3).

De�nition 8. Let E(T ) be the set of functions given by (3)de!ned over [0; T ] and such that for any h∈E(T ) we have

(i) |ri|6R; i = 1; 2; 3.(ii) �m6 �i6 �M ; i = 1; 2; 3.

Note that functions in E(T ) are equibounded, i.e. h(t)¡3Re�MT, t ∈T ; moreover they are also equicontinuous sincethe derivatives are equibounded, i.e. |h′(t)|¡ 3R|�M |e�MT,t ∈T . Then, according to the Ascoli–ArzelRa theorem (seeSimmons, 1963), E(T ) is a compact set in the uniform topol-ogy given by the sup norm.Now, we can de!ne the linear operator

S : E(T ) → R Pk+; T = Pk�;

where

z = Sh; zk =1�

∫ k�(k−1)�

h(t) dt k = 1; : : : ; Pk: (6)

It is easy to see that S is a bounded operator. We show nextthat S has a bounded inverse.

Proposition 9. Operator S; de'ned by (6); is one to oneand onto.

Proof. To this aim we need only to show that operator S hasa trivial kernel; i.e. Sz = 0 implies z = 0. Let h1; h2 ∈E(T )be two distinct impulse responses yielding the same outputz ∈R Pk

+. Then; we can write

0 = S(h1 − h2)that is; for any k¿ 1;

0 =1�

∫ k�(k−1)�

(h1(t)− h2(t)) dt: (7)

Now; provided T is chosen large enough; functions in E(T )have a monotone long-term behavior; therefore there existsa k06 Pk such that; from Eq. (7) follows that h1(t)=h2(t) for(k−1)�6 t6 k�; for Pk¿ k¿ k0. Since functions in E(T )are analytic; we obtain h1(t)=h2(t) for any t6T = Pk�.

According to Proposition (9) the operator S has an inverseS−1, and this accounts for the existence and uniqueness ofthe solution of the identi!cation problem. Nevertheless, theissue of robustness of the solution w.r.t. data perturbationasks for continuity of S−1 over R Pk

+. To this aim we state thefollowing.

Theorem 10. Operator S−1 :R Pk+ → E(T ) is continuous.

Proof. Let us consider a sequence {zn}∈R Pk+ converging to

an element z. Then; let {hn}∈E(T ) be the correspondingsequence of impulse responses obtained as hn=S−1zn. Now;by compactness; there exists a subsequence hnk convergingto a limit h∈E(T ). Let znk be the subsequence of zn cor-responding to hnk . Since zn converges to z; any convergingsubsequence converges to z as well and therefore we obtainh= S−1z; i.e. S−1 is continuous.

Concluding, Proposition 9 and Theorem 10 enable us toensure well-posedness of the identi!cation problem for func-tions in E(T ). This result is the appropriate starting point tosetup, in the following section, an identi!cation procedureaiming to estimate the parameters of a third-order positiverealization thus exploiting the a priori information on thepositivity of the process generating the data.

4. The proposed identi�cation procedure

Suppose one has Pk = N independent observationsn1; : : : ; nN with Poisson distribution as prescribed by therelationship (4). Then, the joint probability distributioncorresponding to the data vector is

L(#) =N∏i=1

P(zi = ni) =�n11 · : : : · �nNNn1! · : : : · nN ! e

(−∑Ni=1 �i)

#= ( r1 r2 r3 �1 �2 �3 )T; (8)

where, according to (5), �i is given by

�i =1�

3∑j=1

rj�je�jti(1− e−�j�) ti = i�:

Function L(#) is called likelihood function and the set ofparameters (residues and exponents) giving the maximumvalue for L(#) characterizes the impulse response whichmore likely gave rise to the observed data.Then, according to Theorem 7 and De!nition 8, we

can formulate the following constrained optimizationproblem P:

max#∈DL(#);

where D is de!ned by #∈R6 which satis!es the followingconstraints:

P1: |#i|6R; i = 1; 2; 3; �m6 #i6 �M ; i = 4; 5; 6;

A. De Santis, L. Farina / Automatica 38 (2002) 861–868 865

P2: #6¡#5¡#4;P3: #1¿ 0;P4: #1 + #2 + #3¿ 0;P5: (#4 + P�)#1 + (#5 + P�)#2 + (#6 + P�)#3¿ 0, where

P�:=− #4 + #5 + #6 − 2√&

3with

&:=(#5 − #6)2 + (#4 − #5)(#4 − #6);P6: (#4 + �)2#1 + (#5 + �)2#2 + (#6 + �)2#3¿ 0 for all �

such that

−#66 �6 P�;

where R; �m and �M are known constants.Constraints P2–P6 represent the necessary and su7cient

conditions of Theorem 7 for the existence of a minimalthird-order positive realization of the unknown impulse re-sponse h. On the other hand, constraint P1 implies thath∈E(T ), T ¿ 0 so that the identi!cation problem is wellposed.Note that, because of constraint P6, problem P is a

semi-in!nite programming problem. In order to use a morestandard technique we will work out P6 to transform theoriginal problem into an equivalent one with a !nite numberof constraints.

Proposition 11. The admissible set D can be characterizedas follows:

P1: |#i|6R; i = 1; 2; 3; �m6 #i6 �M ; i = 4; 5; 6;P2: #6¡#5¡#4;P3: #1¿ 0;P4: #1 + #2 + #3¿ 0;

PP5: if − (#6 − #4)2(#5 − #6)2 6

#2#16#6 − #4#5 − #6 then (#5 − #6)2#2#3

+ (#5 − #4)2#2#1 + (#6 − #4)2#3#1¿ 0 and (#4 + P�)#1+ (#5 + P�)#2 + (#6 + P�)#3¿ 0.

Proof. Consider the following variable transformation:

t2 =#2#1; t3 =

#3#1; d= #4 + �;

Pd= #4 + P�; t5 = #5 − #4; t6 = #6 − #4 (9)

then constraints P2;P4;P5;P6 become

P′2: t6¡t5¡ 0;

P′4: 1 + t2 + t3¿ 0;

P′5: Pd+ (t5 + Pd)t2 + (t6 + Pd)t3¿ 0 where

Pd:=− t5 + t6 − 2√(t5 − t6)2 + t5t63

;

P′6: d

2 + (t5 + d)2t2 + (t6 + d)2t3¿ 0 for all d such that−t66d6 Pd;

which de!ne the transformed admissible set D′. We nowdescribe the subset of the boundary of D′ given by

1 + t2 + t3 = 0; (10)

Pd+ (t5 + Pd)t2 + (t6 + Pd)t3 = 0; (11)

d2 + (t5 + d)2t2 + (t6 + d)2t3 = 0; −t66d6 Pd: (12)

In particular; Eq. (12) represents a family of straight linesin the plane (t2; t3) given (t5; t6) and considering d as aparameter whose values range in the interval [− t6; Pd]. Then;it is known that the parameter can be eliminated by solvingthe following system of equations with respect to d:

d2 + (t5 + d)2t2 + (t6 + d)2t3 = 0;

d+ (t5 + d)t22 + (t6 + d)t23 = 0

and obtaining the equation of the envelope

(t5 − t6)2t2t3 + t25 t2 + t26 t3 = 0

−t66d=− t5t2 + t6t31 + t2 + t3

6 Pd; 1 + t2 + t3¿ 0; (13)

which is easily recognized to be an hyperbole. Moreover;condition on d in (13) select a segment of the curve betweenthe vertical asymptote; obtained for d = −t6 with abscissat2a=−t26 =(t5− t6)2; and the point P ≡ (t2P; t3P) which solves(13) for d= Pd and (11)

t2P =Pd

t5 + Pd

t6t5 − t6 ¡ 0;

t3P =−Pd

t6 + Pd

t5t5 − t6 ¿ 0:

It can be easily checked that solutions (t2; t3) of (13);with t2a6 t26 t2P; actually imply that 1 + t2 + t3¿ 0 andPd+ (t5 + Pd)t2 + (t6 + Pd)t3¿ 0; so that (13) determines theboundary of D′ in this interval.For t2¿ t2P only two boundary Eqs. (10) and (11) remain

which share the solution Q ≡ (t2Q; t3Q) given by

t2Q =t6t5 − t6 ¿t2P;

t3Q =− t5t5 − t6 :

We can !nally note that for t2P6 t2¡t2Q one has thatsolutions (t2; t3) of (11), verify 1 + t2 + t3¿ 0; while fort2¿ t2Q equation 1 + t2 + t3 = 0 determines Pd+ (t5 + Pd)t2+ (t6 + Pd)t3¿ 0. The diagram in the Fig. 1 depicts theshape of the admissible set in the (t2; t3) plane as obtainedin the previous analysis. Finally, constraints P′

4–P′6 turn to

the following form:

P′4. 1 + t2 + t3¿ 0,

P′′5 . if t2a ¡ t2¡t2Q then

(t5 − t6)2t2t3 + t25 t2 + t26 t3¿ 0 and

Pd+ (t5 + Pd)t2 + (t6 + Pd)t3¿ 0

so that, recalling substitutions (9), the Proposition remainsproved.

866 A. De Santis, L. Farina / Automatica 38 (2002) 861–868

Fig. 1. The transformed admissible set D′ in the (t2; t3) plane.

Because of constraintsP1, setD is bounded; nevertheless,due to P2 and P3 it is not closed. In this case Weierstra*theorem does not apply, thus missing a su7cient conditionfor the existence of an optimal solution. To overcome thisdi7culty, while retaining a third-order model with distincteigenvalues, we change constraints P2 and P3 as follows:

P2. #6 + +16 #56 #4 − +2,P3. #1 − +3¿ 0;

where +1, +2 and +3 are positive numbers whatever small.Now D is closed as well and Weierstra* theorem applies;obviously would the optimal solution be found on the partof the boundary of D arti!cially introduced, it should bedisregarded as dependent on the arbitrary constants chosen,meaning that the model (1) need to be reformulated. Finally,we note that the constrained optimization problemP is gen-erally nonconvex so that it is well known that it might behard to !nd an e7cient numerical algorithm; global conver-gent algorithms should be enforced with trust regions meth-ods to deal with the initialization problem.

5. Simulation results

As stated in the introduction, we considered the case inwhich the output sequence of data is a sample of a Poissonprocess. As a matter of fact, this case is representative ofa wide class of real processes, including the dynamics oftracers in the human body, (see Kajiya, Kodama, & Abe,1984). For this reason, in the following we shall consideran application of our result to the case of a compartmentalsystem.As discussed in Walter and Contreras (1999), in param-

eters identi!cation of a compartmental model it is often un-clear how many compartments one has to include in themodel. While a minimum number of terms can be inferredby simply observing the exponential decays in the data, theupper bound is hard to de!ne due to possible high-order dy-

Fig. 2. The generated experimental data (ragged line) and the systemimpulse response (smooth line).

namics. In this respect, our result allows the identi!cationof the dominant dynamics within the assumed class of sys-tems, i.e. within the class of third-order positive systems.For the sake of illustration, consider then the impulse

response given by

h(t) = 10− 30e−0:3t + 100e−0:6t + 2e−t − 5e−2t ;

which is positive for any t¿ 0. The last two terms of h(t)have been introduced to model unwanted high-order dynam-ics. Then, by choosing T = 25 and �= 0:25, we generate asample of 100 Poisson distributed experimental data pointsaccording to (4) and (5) (the data are shown in Fig. 2). Thenwe considered the likelihood function L(#) de!ned in (8)and solved problem P subject to constraints P1 ÷P4 andPP5. Using the MatLab? optimization subroutine fmincon forconstrained programs, problem P is solved optimizing L(#)by a quasi-Newton sequential quadratic programming tech-nique (SQP). With an initial parameters estimate

#0 = ( 10 −35 90 −0:001 −0:6 −0:9 )

we obtained the following constrained maximum likelihoodestimate:

#con = (10:25 −25:87 106:55 −0:0026 −0:2905 −0:6473);

which well agrees with the !rst three terms of the true h(t).The presence of a null eigenvalue can be easily checked byobserving a constant steady-state behavior of the data; onthe other hand the value of the corresponding residue canbe reliably estimated by the sample mean.Then we considered the same problem and solved the cor-

responding unconstrained program by using the MatLab?

function fminunc which makes use of a trust-region tech-nique. With the same initial point #0 the following uncon-strained maximum was obtained:

#unc = ( 8:7996 14:537 65:346 0:0015 −0:8237 −0:8235 ):

A. De Santis, L. Farina / Automatica 38 (2002) 861–868 867

Fig. 3. The identi!ed impulse responses in the two cases considered(constrained and unconstrained).

Fig. 4. The generated experimental data (ragged line) and the systemimpulse response (smooth line).

The optimum is still inside the feasible set but the nega-tive residue was lost and the eigenvalues are signi!cantlydiAerent from the true ones. Nevertheless, as shown inFig. 3, the estimated impulse responses do not look verydiAerent. This is not surprising, since the problem of !ttingdata with sum of exponentials is well known to be ill posed(see, for example, Anderson, 1983) so that almost identicalimpulse responses can be generated by very diAerent valuesof residues and eigenvalues.In order to obtain a more signi!cant experiment to test the

positivity constraints eAectiveness, we considered the caseof a system with impulse response starting at zero for t=0,with the following true parameters:

#= ( 50 −150 100 −0:6 −0:3 0 ):

The Poisson distributed data were generated according tothe already mentioned mechanism (see Fig. 4), and the

likelihood function L(#) de!ned in (8) was again consid-ered. With an initial parameters estimate

#0 = ( 48 −148 98 −0:7 −0:4 −0:01 )

solving both the constrained and the unconstrained prob-lems, we obtained

#con = ( 47:41 − 149:2 101:8 −0:4 −0:26 0 )

and

#unc = ( 48:2 −147:7 99:13 −0:38 −0:27 0:0014 ):

We note that, despite the initial condition #0 was quite closeto the true parameters values, #unc is outside the feasibleset; actually constraint P4 is violated, that is, the estimatedimpulse response becomes negative near t = 0.Finally, it is worth noting that the inadequacy of the

unconstrained solution was indeed observed in many othertrials performed using diAerent initial conditions, thus show-ing the necessity of including the positivity constraints inthe identi!cation process.

6. Conclusions

In this paper we have studied the identi!cation problemfor positive systems whose impulse response is described bya sum of real exponentials, referring to the class of applica-tions in which the output generated by a unit step input isa sequence of data having a Poisson distribution. The nov-elty of the approach we proposed consists in solving a con-strained maximum likelihood problem to estimate residuesand eigenvalues. Based on theoretical results on minimal-ity of positive realizations recently obtained in Benvenuti etal. (2000), the constraints we considered guarantee the ex-istence of a minimal (third-order) positive realization. We!rst established identi!ability results for a class of impulseresponses, thus providing a well-posed setup for the estima-tion process. Then, we considered a maximum likelihoodestimation procedure to identify the parameters of a mini-mal positive realization. The resulting optimization problemis a semi-in!nite program, i.e. it has an in!nite number ofconstraints. To overcome this di7culty, we have proved thatsuch a program is equivalent to another one having !nitenumber of constraints, also including a logical condition.This allowed us to implement standard constrained optimiza-tion algorithms. It’s worth emphasizing that our procedureallows to !nd a solution of the identi!cation problem withinthe assumed class of systems, i.e. where some residues arealso allowed to be negative, the actual constraint being noth-ing but the existence of a minimal positive realization of thesystem under study. Indeed, simulations have shown thatunconstrained optimization, while being less accurate, mayeven result in a nonpositive estimated impulse response.

868 A. De Santis, L. Farina / Automatica 38 (2002) 861–868

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Alberto De Santis was born in Rome, Italy,in 1958. He received the Laurea degreein Electrical Engineering in 1984 and thepostgraduate diploma in system and con-trol theory in 1986, both at University “LaSapienza”, Rome, Italy. From 1987 to 1992he was a researcher with the Istituto di Anal-isi dei Sistemi e Informatica of CNR, Rome.He then joined the faculty of engineering,dept. di Informatica e Sistemistica, Univer-sity “La Sapienza”, where he is currentlyassociate professor. In 1991 he spent one

year at the Henry Samueli School of Engineering and Applied Science,University of California at Los Angeles. His research interests are in sys-tem identi!cation and signal processing, control of distributed parametermechanical systems, nonlinear stochastic control.

Lorenzo Farina was born in Rome, Italy, onOctober 3, 1963. He received the “Laurea”degree in Electrical Engineering (summacum laude) and the Ph.D. degree in sys-tems engineering from the University ofRome, “La Sapienza”, Rome, Italy, in 1992and 1997, respectively. He was a Scien-ti!c Consultant at the Interdepartmental Re-search Centre for Environmental Systemsand Information Analysis, at the Politecnicodi Milano, Milan, Italy, in 1993. He wasthe Project Coordinator at Tecnobiomedica

S.p.A., in the !eld of remote monitoring of patients with heart diseases, in1995, and held a visiting position at the Research School of InformationSciences and Engineering, the Australian National University in 1997.Since 1996 he has been with the Department of Computer and SystemsScience, the University of Rome “La Sapienza”, where he is currentlyAssociate Professor of Modelling and Simulation. He is co-author ofthe book “Positive Linear Systems: Theory and Applications” with S.Rinaldi, Series on Pure and Applied Mathematics, Wiley-Interscience,New York, 2000. He is co-recipient of the IEEE Circuits and Systems“Guillemin-Cauer Award” for 2001.