370 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002

Positive and Compartmental Systems

Luca Benvenuti and Lorenzo Farina

Abstract—When dealing with compartmental systems, an importantquestion is: given an experiment, i.e., an input–output sequence, andsupposing there is no error in the data, is the sequence compatible withthe compartmental assumption? If the process under analysis is linear,then the previous question is obviously equivalent to asking whether agiven transfer function is that of a compartmental system. In this note weprovide an answer to the latter question giving necessary and sufficientconditions for a transfer function to be that of a compartmental systemof some finite order (i.e., number of compartments). Another problemtackled in this note originates from the observation that in many cases onewants to determine the number of compartments involved in the process.In this note we report a step toward the solution of this fundamentalproblem by proving necessary and sufficient conditions for a given thirdorder transfer function with real poles to be that of a compartmentalsystem with three compartments.

Index Terms—Positive systems, minimal positive realization, compart-mental systems.

I. INTRODUCTION

Compartmental systems are effectively used in an impressivenumber of different applicative fields of mathematical modeling,ranging from tracer kinetics to distillation columns, from populationdynamics to water resources (see, for example, [1], [10], and [9]). Theyare composed of a finite number of subsystems, called compartments,interacting by exchanging material. Moreover, some type of massconservation condition holds for all transfers between compartmentsand, to/from the environment. Since the state variables represent theamount of material contained in each compartment, they are boundedto be nonnegative over time. Then, compartmental systems belong tothe class ofpositive systems(see, for example, [12], [7], and [16] fora general overview) which are characterized by the property that thestate and output variables remain nonnegative whatever the positiveinput sequence might be.

The process of assessing the compartmental nature of a system underanalysis involves a number of different professional skills depending onthe complexity of the problem. In most cases, however, a deep physicalknowledge is required. In fact, once the investigator is convinced thata compartmental model is reasonable for the process under study, thenhe tries to find experimental evidence of his assumption.

In this context, the first basic question is: given an experiment, i.e.,an input–output sequence, and supposing there is no error in the data,is the sequence compatible with the compartmental assumption?

If the process under analysis is linear, then the previous question isobviously equivalent to asking whether a given transfer function is thatof a compartmental system.

In this note we provide an answer to the latter question giving nec-essary and sufficient conditions for a transfer function to be that of acompartmental system of some finite order (i.e., number of compart-ments).

Another problem tackled in this note originates from the observationthat in many cases one wants to determine the minimum number ofcompartments involved in the process. In system’s theory, this problem

Manuscript received March 26, 2001; revised July 26, 2001. Recommendedby Associate Editor B. Chen.

The authors are with the Dipartimento di Informatica e Sistemistica,Università degli Studi di Roma “La Sapienza,” Rome, Italy (e-mail: luca.ben-venuti@uniroma1.it; lorenzo.farina@uniroma1.it).

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is known as the minimal realization problem and, when dealing withcompartmental systems, is known to be as a very difficult problem (see,for example [14], [15], and the references cited therein).

In this note, we report a step toward the solution of this fundamentalproblem by proving necessary and sufficient conditions for a giventhird order transfer function with real poles to be that of a compart-mental system with three compartments.

An outline of the note is as follows. In Section II, we give some pre-liminary definitions and known results on compartmental and positivesystems and in Section III, we state the main results of the note, that isTheorems 8 and 10.

II. BASICS OFCOMPARTMENTAL SYSTEMS

We will consider in the following continuous-time linear systems ofthe form:

_x(t) = Ax(t) + bu(t); y(t) = cTx(t) (1)

with A 2 N�N ; b; c 2 N . Such system is acompartmental system[1], [10], [9] provided that

bi � 0; ci � 0; (2)

aij � 0 for i 6= j (3)

aii +j 6=i

aji � 0 (4)

for i; j = 1; . . . ; N , and where theaij ’s are the entries ofA andbi;ci those ofb andc, respectively. The usual physical interpretation ofthe above constraints is that, a compartmental system consists of a fi-nite number of subsystems, called compartments, interacting by ex-changing material. Because the interactions between compartments aretransfers of material, some type of mass conservation condition holdsfor all transfers between compartments and, to/from the environment.The state variablesxi(t) represent the amount of resource present inthei-th compartment at timet; aij (i 6= j) is the rate constant inflowfrom thej-th to theith compartment,aii is the rate constant outflowfrom compartmentith (i.e., the sum of the outflow and of the losses ofresource in the compartment). Since the inputu(t) consists of materialinjection from the environment andb determines in which way this ma-terial is distributed among compartments, then at least one entry ofb

is positive; moreover, since the outputy(t) measures the material con-tained in some compartment(s), then at least one entry ofc is positive.It is worth noting that compartmental systems belong to a broader classof systems calledpositive systems[12], [7]. In fact, a continuous-timesystems of the form (1) is a positive system provided that only con-straints (2) and (3) hold. In order to establish a link between the classesof compartmental and positive systems, we first show that the class ofasymptotically stable matrices satisfying (3) is similar to the class ofasymptotically stable matrices satisfying (3) and (4). To this end, letus define the setM of matrices for which (3) holds, and the setC ofmatrices for which (3) and (4) hold. A matrixT is said to bepositive di-agonalprovided that it is diagonal with positive diagonal entries. Then,we have the following lemma.

Lemma 1: Given an asymptotically stable matrixA 2 N�N , thenA 2 M if and only if there exists a positive diagonal matrixT suchthatTAT�1 2 C.

Proof: We obviously prove only thatA 2 M implies that thereexists a positive diagonal matrixT such thatTAT�1 2 C. Considerthe matrix

T = diagf�(1 . . . 1)A�1g := diagf(t1 . . . tN )g:

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Since asymptotic stability ofA implies nonnegativity of the entries of�A�1 (see [5]) and that there are no zero column, then the valuestiare positive. Then

TAT�1 =

a11 a12t

t� � � a1N

t

t

a21t

ta22 a2N

t

t

.... . .

aN1t

taN2

t

taNN

belongs to the setC if, for every index i, we have aii +

j 6=iaji(tj=ti) � 0, that is

(1 . . . 1)(TAT�1) = (t1 . . . tn)AT�1

= �(1 . . . 1)A�1AT�1

= �1

t1. . .

1

tn< 01�N :

Observe that the proof of the previous theorem provides an explicitform for the equivalence transformation matrixT . This matrix is pos-itive diagonal so that we consider the following definition of equiva-lence between realizations, coming from [13].

Definition 2: Let fA; b; cT g and f ~A;~b; ~cT g; A; ~A 2 N�N ;b; c;~b; ~c 2 N , two realizations of a given system. Then they arediagonally equivalent if there exists a positive-diagonal matrixTsatisfying ~A = TAT�1;~b = Tb; c = T ~c.

When considering an asymptotically stable system, using the pre-vious definition, then Lemma 1 can be fruitfully employed to establishthe link between positive and compartmental systems:

Theorem 3: A positive asymptotically stable system is diagonallyequivalent to a compartmental system.

A. Basics of Positive Systems

Positive systems are characterized by the specific property that thestate and output variables remain nonnegative whatever the positiveinput sequence might be [12], [7]. These systems are quite commonin applications where input, output and state variables represent posi-tive quantities such as populations, consumption of goods, densities ofchemical species and so on. The fundamental result of positive systemstheory is that a discrete-time positive linear system is characterized bynonnegativity of the system’s matricesfA; b; cT g, i.e., each of the en-tries ofA; b andcT are nonnegative real numbers. On the other hand,a continuous-time positive system is characterized by nonnegativity ofb; cT and of the off-diagonal entries ofA (see [12], [7] for details).The relevant problem in this note is the so-calledpositive realizationproblem[2], [6], [11]: Suppose a prescribed rationalH(s) [H(z)] isgiven, then

i) is there a positive systemfA; b; cT g of some finite dimensionsuch thatH(p) = cT (pI � A)�1b (wherep = z for the dis-crete-time case andp = s for the continuous-time one)?

ii) what is the minimal dimension over all realizations?Recently the question i) has been solved in [2], [6], and [11] where

conditions for the existence of a positive realization are given in termsof pole locations of the given transfer function with nonnegative im-pulse response and in [3] a partial answer to the question ii), i.e., to theminimality problem, has been presented. In the following, we restate,in a suitable form to our purposes, some known results on the positiverealization problem for the discrete and continuous-time case (see [2]and [13]).

Theorem 4: Let fF; g; hT g be any minimal realization of dimen-sion n of a discrete-time system�D . Then,�D has a positive re-alization of dimensionN � n if and only if there exist matricesA+ 2 N�N

+ ; b+; c+ 2 N+ ; P 2 n�N such that

FP = PA+ g = Pb+ cT+ = hTP:

The corresponding positive realization of�D is fA+; b+; cT+g.

Theorem 5: Let fF; g; hT g be any minimal realization of a con-tinuous-time system�C . Then,�C has a positive realization of di-mensionN if and only if there exist a value� 2 and matricesA+ 2 N�N

+ ; b+; c+ 2 N+ ; P 2 n�N such that

(F + �I)P = PA+; g = Pb+; cT+ = hTP:

The corresponding positive realization of�C is fA+ � �I; b+; cT+g.

We state now two lemmas (whose proof is based on the previoustwo theorems and is omitted for the sake of brevity) which are the fun-damental devices which will be used in the next sections to prove themain results of the note.

Lemma 6: Let fF; g; hT g be any minimal realization of a contin-uous-time system�C . Then,�C has a positive realization of dimen-sionN if and only if the continuous-time system~�C described by thetriple fF + aI; g; hT g has a positive realization of dimensionN foranya 2 .

Roughly speaking, the above lemma says that shifting the real partof the spectrum of the system leaves unchanged its property of beingpositively realizable.

Lemma 7: Let fF; g; hT g be any minimal realization of a contin-uous-time system�C . Then,�C has a positive realization of dimen-sionN if and only if there exists a value�� > 0 such that the dis-crete-time system~�D described by the triplef((F + �I)=�); g; hT ghas a positive realization of dimensionN for any� � ��.

This last lemma establishes the equivalence w.r.t. the property ofa system to be positively realizable between a given continuous-timesystem and an appropriate family of discrete-time systems.

III. COMPARTMENTAL REALIZATIONS

As for positive systems, the relevant problem in this note is theso-calledcompartmental realization problem: Suppose a prescribedrationalH(s) is given, then

i) is there a compartmental systemfA; b; cT g of some finite di-mension such thatH(s) = cT (sI � A)�1b?

ii) what is the minimal dimension over all realizations?In the next paragraph we provide the answer to question i) while

a specific result related to question ii) for the case of compartmentalsystems with real eigenvalues, will be then given.

A. Existence Conditions

In most approaches to the study of identifiability of the parame-ters of a compartmental model, the size is assumed known. Yet, inmany biomedical applications, it is unclear whether to include certaincompartments in the model so that one of the most important prob-lems in the analysis of compartmental systems is the determinationof internal structures, in particular the number of compartments, frominput–output observations. The first main result of this note consistsof the characterization of compartmental systems’ transfer functions.This result allows to recognize whether a given transfer function isthat of a compartmental system of some finite order and may be worthusing in the determination of the number of compartments involvedby input-output observations. Consider, for the sake of illustration, onehas performed an experiment (i.e., an input–output time series has beencollected) on a linear system which isa priori known to be a com-partmental system, but its order is unknown. A typical identificationsession consists of estimating the parameters of a family of systemsof increasing order until some estimation index (say, the output meansquare error) reach a “satisfying” value. When the family of systemsunder consideration is the family oflinear systems, then the problemof actually estimating the parameters is an ill-posed problem, that is,the output may be close to the true output, whereas the parameters arestill quite different than the true ones. On the other hand, restricting the

372 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002

family of linear systems to that ofcompartmental and linearsystems,may reduce parameter sensitivity during the estimation process. Thefollowing theorem gives the conditions for a transfer functionH(s) tobe that of a compartmental system:

Theorem 8: A given asymptotically stable transfer functionH(s)is that of a compartmental system if and only if the following hold:

1) the impulse response functionh(t) is such that1 h(t) > 0 foreveryt > 0 andh(0) � 0;

2) the pole� ofH(s)with maximal real part is negative and unique.Proof: (Necessity) Let fA+; b+; cT+g be a positive realization of

H(s). To prove condition 1. consider the matrix~A = A+ +�I whichis nonnegative for a sufficiently large value of�. Then, we can write

h(t) = cT+eA tb+ = cT+e

( ~A��I)tb+

= e��tcT+e~Atb+ = e��t

1

k=0

cT+( ~At)k

k!b+:

Sinceh(t) is not identically zero, then there is at least one coefficientcT+ ~Akb+ > 0. From this, Condition 1) immediately follows. To proveCondition 2), first note that since the system is asymptotically stable,then the pole� of H(s) with maximal real part has negative real part.Moreover, as shown in [12],A+ has a real eigenvalue at�(A+)2 andno other eigenvalue has a real part equal to�(A+), that is it is unique.Furthermore, the associated right and left eigenvectorsu+ andv+ arenonnegative. We shall now show that eithercT+u+ 6= 0, that is theeigenvalue is observed, or unobservable states can be removed withoutdestroying the nonnegativity of the realization following the same lineof [2].

Suppose then thatcT+u+ = 0. Without loss of generality, reorder theentries of the state vector so that

cT+ = cT+ 0 0 uT+ = 0 0 uT+

with c+ andu+ positive. Let

A+ =

A+ A+ A+

A+ A+ A+

A+ A+ A+

:

BecauseA+u+ = �(A+)u+, the zeros inu+ force A+ = 0;A+ = 0. However, now the zero blocks inc+ andA+ mean thatan unobservable part is displayed, and a lower dimension but still non-negative realization is provided by

A+ A+

A+ A+;

b+b+

;c+0

T

:

Similarly, if vT+A+ = �(A+)vT+ andvT+b+ = 0, we can immediately

eliminate certain uncontrollable blocks and still retain the nonnega-tivity of the realization. Starting with an arbitrary nonnegative realiza-tion, then we can reduce it by eliminating uncontrollable and/or unob-servable blocks, and retaining the nonnegativity, until the real domi-nant eigenvalue is controllable and observable, i.e., we can consider,without loss of generality, a positive realizationfA+; b+; c

T+g of H(s)

such that�(A+) = �. Then,� is real negative and unique, that is con-dition 2).

(Sufficiency) Since the system is asymptotically stable, then by virtueof Theorem 3, without loss of generality we will consider positive sys-tems in place of compartmental ones. For the sake of simplicity, con-sider then the transfer function~H(s) = H(s + �) whose pole withmaximal real part is zero. From Lemma 6, the systemH(s) has a posi-tive realization if and only if~H(s) has one. Let thenfF; g; hT g be any

1In order to avoid trivial cases, we will assume thath(t) is not identicallyzero.

2We will denote by�(M) the real part of the maximal real part eigenvalue ofM .

minimal realization of~H(s). Consider the Cauchy–Euler discrete-timeapproximation

xk+1 � xk�

= Fxk + guk

yk = hTxk

whose impulse response is~hd(k;�) = �hT (�F + I)k�1g. Thetheory of numerical approximation to the solution of differential equa-tions ensures that, for any time intervalT > 0 and any� > 0, thereexists a positive integer such thatj~h(k�) � ~hd(k;�)j < � fork = 1; 2; . . . ; � 1, where� = T= (see [4], [8]). From Condi-tion 1), it follows that also the impulse response~h(t) = L�1[ ~H(s)] =e��th(t) satisfies Condition 1). Moreover, since the pole with maximalreal part of ~H(s) is zero, thenlimt!1

~h(t) > 0. From this followsthat, also when consideringT ! 1, there exists a sufficiently smallvalue� such that for any� � � one has3 j~h(k�) � ~hd(k; �)j < �for k = 1; 2; . . . with ~hd(k; �) > 0 for k = 1; 2; . . .. Consequently,�hd(k; �) = hT (�F + I)k�1g > 0 for k = 1; 2; . . . Then, as proved in[2], �hd(k; �) has a (discrete-time) positive realization for any� � �,so that, in view of Lemma 7 with� = (1=�) and�� = (1=�), the triplefF; g; hT g has a (continuous-time) positive realization.

B. The Third-Order Case

The second main result of this note consists of the characterization ofthird-order compartmental systems’ transfer functions with real poles.Analogously to the previous case, suppose one has performed an ex-periment on a linear system which isa priori known to be a third-ordercompartmental system with real poles, but still no information is givenon the internal structure of the system. Again, restricting the family oflinear systems to that ofcompartmentalsystems, may reduce parametersensitivity during the estimation process. The main result of this sec-tion relies on the next theorem, which has been recently proved in [2]and is here reported for the sake of readability. This theorem providesnecessary and sufficient conditions for a third-order transfer functionwith positive real poles to be that of a positive system of the same order,in the discrete-time domain.

Theorem 9 [3]: Let H(z) be a third order transfer function withdistinct positive real poles�1 > �2 > �3 and letfF; g; hT g be anyminimal realization ofH(z). Then,H(z) has a third-order positiverealization if and only if the following conditions hold:

1) hT (F � �2I)(F � �3I)g > 0;2) hT g � 0;3) hT (F � ��I)g � 0;4) hT (F � �I)2g � 0 for all � such that�� � � � �3;

where

�� = max� � 2 (�2 � �3)2 + (�1 � �2)(�1 � �3)

3; 0

with � = �1 + �2 + �3.On the basis of the previous result, we can state now the condition

for a transfer function to be that of a compartmental system with threecompartments.

Theorem 10: LetH(s)be a transfer function with three distinct-realpoles0 > �1 > �2 > �3 and letfF; g; hT g be any minimal realiza-tion ofH(s). Then,H(s) has a realization with three compartments ifand only if the following conditions hold:

1) hT (F � �2I)(F � �3I)g > 0;2) hT g � 0;3) hT (F + ��I)g � 0;4) hT (F + �I)2g � 0 for all � such that��3 � � � ��;

3At this point it worth recalling that the absence of purely imaginary eigen-values ofF ensures that�F + I has ultimately eigenvalues within the unitcircle, while the dominant eigenvalue ofF maps to 1.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002 373

where

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

3with � = �1 + �2 + �3.

Proof: From Lemma 7,H(s) has a third order positive realiza-tion if and only if there exists a value�� > 0 such that the discrete-timesystemf((F + �I)=�); g; hT g has a third-order positive realizationfor any� � ��, i.e., if and only if the conditions of Theorem 9 holdby replacingF with ((F + �I)=�) and�i with ((�i + �)=�). Aftersimple manipulations, one obtains thatH(s) has a third-order positiverealization if and only if there exists a value�� > 0 such that

1) hT (F � �2I)(F � �3I)g � 0;2) hT g � 0;3) hT (F + �(1 � ��)I)g � 0;4) hT (F + �(1 � �)I)2g � 0;

for all � such that�� � � � ((�3 + �)=�) with

�� = max3�+ � � 2

p�

3�; 0

and for any� � ��, with

� = (�2 � �3)2 + (�1 � �2)(�1 � �3):

Moreover, since� � �� > 0, there also exists an~� > 0 such that forany� � ~�, the above Conditions 1–4) hold with

�� =3�+ � � 2

p�

3�> 0

Finally, setting

�(1� ��) = � 1� 3�+ � � 2p�

3�

= �� � 2p�

3:= ��

condition�� � � � ((�3 + �)=�) reduces to��3 � �(1 � �) � ��.Then, the theorem remains proved letting� := �(1� �).

REFERENCES

[1] D. H. Anderson, “Compartmental modeling and tracer kinetics,”LectureNotes in Biomathematics, vol. 50, 1983.

[2] B. D. O. Anderson, M. Deistler, L. Farina, and L. Benvenuti, “Nonneg-ative realization of a linear system with nonnegative impulse response,”IEEE Trans. Circuits and Syst. I, vol. 43, pp. 134–142, 1996.

[3] L. Benvenuti, L. Farina, B. D. O. Anderson, and F. De Bruyne, “Minimaldiscrete-time positive realizations of transfer functions with positive realpoles,”IEEE Trans. Circuits and Syst. I, vol. 47, 2000.

[4] A. Berman, R. J. Plemmons, and R. J. Stern,Nonnegative Matrices inDynamic Systems. New York: Wiley, 1989.

[5] A. Berman and R. J. Plemmons,Nonnegative Matrices in the Mathe-matical Sciences, Classics in Applied Mathematics. Philadelphia, PA:SIAM, 1994.

[6] L. Farina, “On the existence of a positive realization,”Syst. Control Lett.,vol. 28, pp. 219–226, 1996.

[7] L. Farina and S. Rinaldi,Positive Linear Systems: Theory and Applica-tions, Pure and Applied Mathematics, Series of Text, Monographs andTracts. New York: Wiley, 2000.

[8] J. H. Hubbard and B. H. West,Differential Equations: A Dynamical Sys-tems Approach, Texts in Applied Mathematics. New York: Springer-Verlag, 1995.

[9] J. A. Jacquez,Compartmental Analysis in Biology and Medicine. NewYork: Elsevier, 1972.

[10] F. Kajiya, S. Kodama, and H. Abe,Compartmental Analysis—MedicalApplications and Theoretical Background: Karger, 1984.

[11] T. Kitano and H. Maeda, “Positive realization of discrete-time system bygeometric approach,”IEEE Trans. Circuits Syst. I, vol. 45, pp. 308–311,1998.

[12] D. G. Luenberger, “Positive linear systems,” inIntroduction to DynamicSystems. New York: Wiley, 1979, ch. 6.

[13] H. Maeda and S. Kodama, “Positive realization of difference equation,”IEEE Trans. Circuits Syst., vol. CAS-28, pp. 39–47, 1981.

[14] H. Maeda, S. Kodama, and F. Kajiya, “Compartmental system analysis:Realization of a class of linear systems with physical constraints,”IEEETrans. Circuits Syst., vol. CAS-24, pp. 8–14, 1977.

[15] Y. Ohta, H. Maeda, and S. Kodama, “Reachability, observability andrealizability of continuous-time positive systems,”SIAM J. ControlOptim., vol. 22, pp. 171–180, 1984.

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Active Sensing Policies for Stochastic Systems

Shuo Liu and L. E. Holloway

Abstract—In systems with sensing cost, anactive sensing policyis neededto determine when to collect sensing observations. This note presents anactive sensing policy for systems with additive and parametric white noise.The policy uses an open-loop estimator between sensings and a Kalmanfilter when observations are requested. We present two active sensing poli-cies. The goal of the first policy is to maintain the uncertainty (variance) ofthe state estimate below a given threshold. Sufficient conditions are pre-sented that guarantee that this goal is achievable and will be met. Thesecond policy senses when needed to distinguish discrete state regions forcontrol. Sufficient conditions are presented that show within any specifiedprobability, the control under the active sensing will be identical to the con-trol under conventional sensing. Experiments demonstrate that sensing andsensing communications can be significantly reduced with active sensingpolicies, while still meeting control objectives.

Index Terms—Active sensing, Kalman filtering, observers, stochastic sys-tems.

I. INTRODUCTION

System and control theory typically considers systems for whichsensing is available either continuously or at each discrete time step.In many systems, however, this assumption of regular sensing may notbe reflective of practice. In this note, we present a framework for sys-tems which haveactive sensing, where the time instances of sensingsare determined dynamically. The framework is shown in Fig. 1. A stateestimator operates in open loop during periods without sensing andwith feedback when sensing is available. The estimator outputs an ex-pected state value and an estimated state covariance matrixPk. Theseare input into one or moreactive sensing policies. These policies eval-uate the state and the covariance estimate and then determine, based ondifferent criteria, when to request system sensing.

Nonperiodic sampling has been considered in the past. Troch pre-sented a method of optimal choice of nonperiodic sampling instantsover a time interval in order to reduce propagation of measurement er-rors and to provide sensing for control at predetermined discrete controltimes [1]. De La Sen showed that proper selection of nonperiodic sam-pling instances improved identifiability of a system’s state transitionmatrix and improved model matching [2].

Manuscript received April 4, 2000; revised July 25, 2001. Recom-mended by Associate Editor Q. Zhang. This work was supported in partby USARO GRANT DAAH04-96-1-0399, National Science FoundationGrants ECS-9807106 and ECS-0115694, and the Center for Robotics andManufacturing Systems at the University of Kentucky.

The authors are with the Department of Electrical and Computer Engineeringand Center for Manufacturing Systems, University of Kentucky, Lexington,Kentucky 40506 USA (e-mail: holloway@engr.uky.edu).

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