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  • POSITIVE LINEAR OBSERVERS FOR LINEAR COMPARTMENTALSYSTEMS

    J. M. VAN DEN HOFy

    SIAM J. CONTROL OPTIM. c 1998 Society for Industrial and Applied MathematicsVol. 36, No. 2, pp. 590{608, March 1998 007

    Abstract. Linear compartmental systems are mathematical systems that are frequently usedin biology and mathematics. The inputs, states, and outputs of such systems are positive, becausethey denote amounts or concentrations of material. For linear dynamic systems the observer problemhas been solved. The purpose of the observer problem is to determine a linear observer such thatthe state can be approximated. The dierence between the state and its estimate should convergeto zero. The interpretation in terms of a physical system requires that an estimate of the state bepositive, like the state itself. In this paper conditions on the system matrices are presented thatguarantee that there exists a positive linear observer such that both the error converges to zero andthe estimate is positive.

    Key words. compartmental systems, positive linear observers, asymptotic stability

    AMS subject classications. 93D20, 15A48

    PII. S036301299630611X

    1. Introduction. The purpose of this paper is to derive positive linear observersfor linear compartmental systems.

    Compartmental systems are mathematical systems that are frequently used inbiology and mathematics. In addition, a subclass of the class of chemical processescan be modeled as compartmental systems. A compartmental system consists ofseveral compartments with more or less homogeneous amounts of material. The com-partments interact by processes of transportation and diusion. The dynamics of acompartmental system are derived from mass balance considerations.

    In this paper linear compartmental systems consisting of inputs, states, and out-puts will be studied. The outputs of these systems are not the real outputs, i.e.,material leaving the system, but the observations of the amount or concentrations ofmaterial, for example, in one or more compartments. The inputs, states, and outputsare positive, so these systems are called positive linear systems in system theory. Asin linear system theory, the purpose is to determine a linear observer such that thestate x can be approximated by bx. The error, bx(t) x(t), should converge to zero.For positive linear systems, the observer provides an approximation of the positivestate. Therefore, the observer should be chosen in such a way that the approximationof the state, bx(t), is positive, like the state, x(t), itself.

    For linear systems the observer problem has been solved by Luenberger [11]. Seealso [10]. As far as we know, there is no literature on positive observers for positivelinear systems, in which the positivity of bx(t) is taken into account. It turns out thatthe existence of a positive linear observer satisfying the above conditions dependslargely on the structure of the system matrices, i.e., the zero/nonzero pattern. Somerelation can be found in the work of Sontag [13, 14].

    The outline of the paper is as follows. In section 2 the problem is posed. In sec-tion 3 continuous-time linear compartmental systems are considered, and in section 4the discrete-time case is treated. Concluding remarks are made in section 5.

    Received by the editors July 8, 1996; accepted for publication (in revised form) January 13, 1997.http://www.siam.org/journals/sicon/36-2/30611.htmlyCBS (Statistics Netherlands), P.O. Box 4000, NL 2270 JM Voorburg, the Netherlands (jhof@

    cbs.nl).

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  • POSITIVE OBSERVERS FOR LINEAR COMPARTMENTAL SYSTEMS 591

    2. Problem formulation. In this section some notation is introduced and theproblem is posed.

    The set R+ = [0;+1) is called the set of the positive real numbers. Let Z+ =f1; 2; : : :g denote the set of positive integers, Zn = f1; : : : ; ng, and NI = f0; 1; 2; : : :g.Denote by Rn+ the set of n-tuples of the positive real numbers. The set R

    nm+ will

    be called the set of positive matrices of size n by m. Note that Rn+ is not a vectorspace because it does not admit an inverse with respect to addition. For matricesA;B 2 Rnm, we will write A B if aij bij for all i 2 Zn, j 2 Zm, and A > Bif A B and A 6= B. A matrix A 2 Rnn is said to be a Metzler matrix if all itso-diagonal elements are in R+; see [9]. Metzler matrices can be characterized asfollows.

    PROPOSITION 2.1. A matrix A 2 Rnn is a Metzler matrix if and only if thereexists an 2 R+ such that (A+ I) 2 Rnn+ .

    DEFINITION 2.2. Consider a continuous-time linear dynamic system

    _x(t) = Ax(t) +Bu(t); x(t0) = x0;y(t) = Cx(t) +Du(t);(2.1)

    with x(t) 2 X Rn, u(t) 2 U Rm, y(t) 2 Y Rk, t 2 T = [t0;1). Equation (2.1)is said to represent a (continuous-time) positive linear system if for all x0 2 Rn+ andfor all u(t) 2 Rm+ , t 2 T , we have x(t) 2 Rn+ and y(t) 2 Rk+ for t 2 T ; in other words,X = Rn+, U = R

    m+ , and Y = R

    k+.

    The following proposition provides a characterization of continuous-time positivelinear systems.

    PROPOSITION 2.3. A continuous-time linear dynamic system of the form (2.1) isa positive linear system if and only if

    B 2 Rnm+ ; C 2 Rkn+ ; D 2 Rkm+ ; and A is a Metzler matrix.Proof. Suppose rst u(t) = 0 for all t 2 T . For i 2 Zn, xi(t) 0 if and only if

    _xi 0 whenever xi = 0 and xj 0 for all j 6= i. This is equivalent to aij 0 forall j 6= i. Moreover, y(t) = Cx(t) 0 for x(t) 0 if and only if C 2 Rkn+ . Nowsuppose u(t) 6= 0. For i 2 Zn, xi(t) 0 if and only if _xi 0 whenever xj = 0 forall j 2 Zn. This is equivalent to bir 0 for r 2 Zm. Furthermore, if x(t) = 0, theny(t) = Du(t) 0 if and only if D 2 Rkm+ .

    For discrete time, the denition of a positive linear system is presented below.DEFINITION 2.4. Consider a discrete-time linear dynamic system

    x(t+ 1) = Ax(t) +Bu(t); x(0) = x0;y(t) = Cx(t) +Du(t);(2.2)

    with x 2 X Rn, u 2 U Rm, y 2 Y Rk, t 2 T = NI . Equation (2.2) is saidto represent a (discrete-time) positive linear system if for all x0 2 Rn+ and for allu(t) 2 Rm+ , t 2 T , we have x(t) 2 Rn+ and y(t) 2 Rk+ for t 2 T ; in other words,X = Rn+, U = R

    m+ , and Y = R

    k+.

    A characterization of discrete-time positive linear systems is as follows.PROPOSITION 2.5. A discrete-time linear system of the form (2.2) is a positive

    linear system if and only if

    A 2 Rnn+ ; B 2 Rnm+ ; C 2 Rkn+ ; D 2 Rkm+ :The positive linear observer problem is as follows. A positive linear observer for

    a positive linear system is a positive linear system described by the equations

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  • 592 J. M. VAN DEN HOF

    _bx(t) = Hbx(t) +Ky(t) + Eu(t); bx(t0) = bx0;bx(t+ 1) = Hbx(t) +Ky(t) + Eu(t); bx(t0) = bx0;for the continuous-time case and the discrete-time case, respectively, which yields anestimate bx(t) of the state x(t) at time t 2 T of system (2.1), (2.2), respectively. As inlinear system theory, the observer has to satisfy the following two conditions:

    1. bx(t0) = x(t0) implies bx(t) = x(t) for all t t0 and for all input functionsu(t), t t0;

    2. bx(t) should converge to x(t) for t!1, for all input functions u(t), t t0.For linear systems the problem of nding an observer satisfying 1 and 2 has beencompletely solved [11]. The solution is

    _bx(t) = (AKC)bx(t) +Ky(t) +Bu(t);bx(t+ 1) = (AKC)bx(t) +Ky(t) +Bu(t);respectively, with K 2 Rnk such that A KC is asymptotically stable; i.e., for thecontinuous-time case, (A KC) f 2 C j Re() < 0g, and for the discrete-timecase, (A KC) f 2 C j jj < 1g. Here (A) denotes the spectrum of A. Thenecessary and sucient conditions for the existence of a matrix K 2 Rnk such thatA KC is asymptotically stable depend on the matrices A and C; i.e., the pair(A;C) should be detectable. Equivalent conditions for detectability can be foundin, for example, [3, pp. 259 and 293], respectively. The interpretation in terms of aphysical system requires that an estimate bx(t) be, like x(t), positive. So a positivelinear observer for a positive linear system should also satisfy the following condition:

    3. bx(t) 2 Rn+, for all t t0, if bx(t0) 2 Rn+, y(t) 2 Rk+, and u(t) 2 Rm+ for allt t0.

    This third condition is satised if and only if K 2 Rnk+ and, for the continuous-timecase, AKC is a Metzler matrix, or for the discrete-time case, AKC 2 Rnn+ . Thisfollows from Propositions 2.3 and 2.5, respectively. Now detectability of (A;C) denedin [3] cannot be used, because then it may be possible that K =2 Rnk+ . Of course,detectability is a necessary condition but is not sucient. Therefore, new necessaryand sucient conditions on A and C have to be found. The problem considered inthis paper is stated below.

    Problem 2.6.Continuous time. Formulate necessary and sucient conditions on a Metzler

    matrix A 2 Rnn and a positive matrix C 2 Rkn+ such that there exists aK 2 Rnk+ , K 6= 0, with

    1. AKC a Metzler matrix;2. (AKC) f 2 C j Re() < 0g.

    Discrete time. Formulate necessary and sucient conditions on positive ma-trices A 2 Rnn+ and C 2 Rkn+ such that there exists a K 2 Rnk+ , K 6= 0,with

    1. AKC 2 Rnn+ ;2. (AKC) f 2 C j jj < 1g.

    These problems will be solved for linear compartmental systems, which form asubclass of positive line

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