Positive and compartmental systems

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<ul><li><p>370 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002</p><p>Positive and Compartmental Systems</p><p>Luca Benvenuti and Lorenzo Farina</p><p>AbstractWhen dealing with compartmental systems, an importantquestion is: given an experiment, i.e., an inputoutput 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.</p><p>Index TermsPositive systems, minimal positive realization, compart-mental systems.</p><p>I. INTRODUCTION</p><p>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.</p><p>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.</p><p>In this context, the first basic question is: given an experiment, i.e.,an inputoutput sequence, and supposing there is no error in the data,is the sequence compatible with the compartmental assumption?</p><p>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.</p><p>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).</p><p>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 systems theory, this problem</p><p>Manuscript received March 26, 2001; revised July 26, 2001. Recommendedby Associate Editor B. Chen.</p><p>The authors are with the Dipartimento di Informatica e Sistemistica,Universit degli Studi di Roma La Sapienza, Rome, Italy (e-mail: luca.ben-venuti@uniroma1.it; lorenzo.farina@uniroma1.it).</p><p>Publisher Item Identifier S 0018-9286(02)02082-2.</p><p>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).</p><p>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.</p><p>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.</p><p>II. BASICS OFCOMPARTMENTAL SYSTEMS</p><p>We will consider in the following continuous-time linear systems ofthe form:</p><p>_x(t) = Ax(t) + bu(t); y(t) = cTx(t) (1)</p><p>with A 2 NN ; b; c 2 N . Such system is acompartmental system[1], [10], [9] provided that</p><p>bi 0; ci 0; (2)</p><p>aij 0 for i 6= j (3)</p><p>aii +j 6=i</p><p>aji 0 (4)</p><p>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 ofbis 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.</p><p>Lemma 1: Given an asymptotically stable matrixA 2 NN , thenA 2 M if and only if there exists a positive diagonal matrixT suchthatTAT1 2 C.</p><p>Proof: We obviously prove only thatA 2 M implies that thereexists a positive diagonal matrixT such thatTAT1 2 C. Considerthe matrix</p><p>T = diagf(1 . . . 1)A1g := diagf(t1 . . . tN )g:</p><p>00189286/02$17.00 2002 IEEE</p></li><li><p>IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002 371</p><p>Since asymptotic stability ofA implies nonnegativity of the entries ofA1 (see [5]) and that there are no zero column, then the valuestiare positive. Then</p><p>TAT1 =</p><p>a11 a12t</p><p>t a1N</p><p>t</p><p>t</p><p>a21t</p><p>ta22 a2N</p><p>t</p><p>t</p><p>.... . .</p><p>aN1t</p><p>taN2</p><p>t</p><p>taNN</p><p>belongs to the setC if, for every index i, we have aii +</p><p>j 6=iaji(tj=ti) 0, that is</p><p>(1 . . . 1)(TAT1) = (t1 . . . tn)AT1</p><p>= (1 . . . 1)A1AT1</p><p>= 1</p><p>t1. . .</p><p>1</p><p>tn&lt; 01N :</p><p>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].</p><p>Definition 2: Let fA; b; cT g and f ~A;~b; ~cT g; A; ~A 2 NN ;b; c;~b; ~c 2 N , two realizations of a given system. Then they arediagonally equivalent if there exists a positive-diagonal matrixTsatisfying ~A = TAT1;~b = Tb; c = T ~c.</p><p>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:</p><p>Theorem 3: A positive asymptotically stable system is diagonallyequivalent to a compartmental system.</p><p>A. Basics of Positive Systems</p><p>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 systems 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</p><p>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)?</p><p>ii) what is the minimal dimension over all realizations?Recently the question i) has been solved in [2], [6], and [11] where</p><p>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]).</p><p>Theorem 4: Let fF; g; hT g be any minimal realization of dimen-sion n of a discrete-time systemD . Then,D has a positive re-alization of dimensionN n if and only if there exist matricesA+ 2</p><p>NN+ ; b+; c+ 2</p><p>N+ ; P 2</p><p>nN such that</p><p>FP = PA+ g = Pb+ cT+ = h</p><p>TP:</p><p>The corresponding positive realization ofD is fA+; b+; cT+g.</p><p>Theorem 5: Let fF; g; hT g be any minimal realization of a con-tinuous-time systemC . Then,C has a positive realization of di-mensionN if and only if there exist a value 2 and matricesA+ 2</p><p>NN+ ; b+; c+ 2</p><p>N+ ; P 2</p><p>nN such that</p><p>(F + I)P = PA+; g = Pb+; cT+ = h</p><p>TP:</p><p>The corresponding positive realization ofC is fA+ I; b+; cT+g.We state now two lemmas (whose proof is based on the previous</p><p>two 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.</p><p>Lemma 6: Let fF; g; hT g be any minimal realization of a contin-uous-time systemC . 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 .</p><p>Roughly speaking, the above lemma says that shifting the real partof the spectrum of the system leaves unchanged its property of beingpositively realizable.</p><p>Lemma 7: Let fF; g; hT g be any minimal realization of a contin-uous-time systemC . Then,C has a positive realization of dimen-sionN if and only if there exists a value &gt; 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 .</p><p>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.</p><p>III. COMPARTMENTAL REALIZATIONS</p><p>As for positive systems, the relevant problem in this note is theso-calledcompartmental realization problem: Suppose a prescribedrationalH(s) is given, then</p><p>i) is there a compartmental systemfA; b; cT g of some finite di-mension such thatH(s) = cT (sI A)1b?</p><p>ii) what is the minimal dimension over all realizations?In the next paragraph we provide the answer to question i) while</p><p>a specific result related to question ii) for the case of compartmentalsystems with real eigenvalues, will be then given.</p><p>A. Existence Conditions</p><p>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, frominputoutput 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 inputoutput 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</p></li><li><p>372 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002</p><p>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:</p><p>Theorem 8: A given asymptotically stable transfer functionH(s)is that of a compartmental system if and only if the following hold:</p><p>1) the impulse response functionh(t) is such that1 h(t) &gt; 0 foreveryt &gt; 0 andh(0) 0;</p><p>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</p><p>H(s). To prove condition 1. consider the matrix~A = A+ +I whichis nonnegative for a sufficiently large value of. Then, we can write</p><p>h(t) = cT+eA tb+ = c</p><p>T+e</p><p>( ~AI)tb+</p><p>= etcT+e~Atb+ = e</p><p>t</p><p>1</p><p>k=0</p><p>cT+( ~At)k</p><p>k!b+:</p><p>Sinceh(t) is not identically zero, then there is at least one coefficientcT+ ~A</p><p>kb+ &gt; 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...</p></li></ul>