System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den

University of Groningen System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1996 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hof, J. M. V. D. (1996). System theory and system identification of compartmental systems s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 1 maximum. Download date: 7-2-218

15 Chapter 2 Compartmental Systems 2.1 Introduction Compartmental systems are mathematical systems that are frequently used in biology and mathematics. Also a subclass of the class of chemical processes can be modelled as compartmental systems. A compartmental system consists of several compartments with more or less homogeneous amounts of material. The compartments interact by processes of transportation and diusion. The dynamics of a compartmental system is derived from mass balance considerations. In this thesis linear compartmental systems consisting of inputs, states, and outputs are studied. The outputs of these systems are not the real outputs, i.e., material leaving the system, but the observations of the amount or concentration of material, for example in one or more compartments. The inputs, states, and outputs are positive, so these systems are in system theory called positive linear systems. In this chapter some properties of compartmental systems that are needed in this thesis are presented. For other properties of compartmental systems, see for example [2, 14, 55, 76, 77, 143]. For examples of positive systems that are not necessarily compartmental, see [123]. The outline of the chapter is as follows. In Section 2.2 continuous-time linear compartmental systems are considered and in Section 2.3 the discrete-time case is treated. A graphical representation of compartmental systems is presented in Section 2.4.

16 Chapter 2. Compartmental Systems Notation In this thesis the set R + =[;1) is called the set of positive real numbers and (; 1) the set of strictly positive real numbers. This terminology is used in [35, Section 2.2]. Let Z + = f1; 2; 3;:::g denote the set of the positive integers and NI = f; 1; 2;:::g the set of the natural numbers. Denote by R n + the set of n- tuples of the positive real numbers. Note that R n + is not a vector space over R because it does not admit an inverse with respect to addition. For n 2 Z + let Z n = f1; 2; 3;:::;ng and NI n = f; 1; 2;:::;ng. The set R km + of matrices over R + will be called the set of positive matrices of size k by m. For A 2 R km, A T denotes its transpose. For matrices A; B 2 R nm, write A B if a ij b ij for all i 2 Z n, j 2 Z m, and A>Bif A B and A 6= B. 2.2 Continuous-time compartmental systems A compartmental system is a system consisting of a nite number of subsystems, which are called compartments. Each compartment is kinetically homogeneous, i.e., any material entering the compartment is instantaneously mixed with the material of the compartment. Compartmental systems are dominated by the law of conservation of mass. They form also natural models for other areas of applications that are subject to conservation laws. Throughout this thesis it is assumed that the structure of the system is known, but that some parameters are unknown, for example the elements of F, I, and C, mentioned below. Consider an n-compartmental system. The behaviour of the ith compartment can be represented as in Figure 2.1. F ji I i q i F ij F i Figure 2.1: One compartment with possible ows. In this gure, q i denotes the amount of material considered in compartment i. The arrows represent the ows into and out of the compartment. The ow into compartment i from outside the system is denoted by I i and it is called the inow. Usually I i is a function of time or just a constant, but occasionally may be dependent onqas well. The symbols F ij and F ji represent the ow

2.2. Continuous-time compartmental systems 17 from compartment j into compartment i and the ow from compartment i into compartment j, respectively. Finally, F i istheoutow to the environment from compartment i. The mass balance equations for every compartment can be written as _q i = X j6=i( F ji + F ij )+I i F i : (2.2.1) We assume that the ows F ij can be written as: F ij = f ij q j ; i =;:::;n; j =1;:::;n; i6=j; in which the functions f ij are called the fractional transfer coecients, see [77, Section 2.1]. In general, f ij are functions of q and time t. If f ij is independent of q, the system is a linear system. In this thesis it is assumed that f ij is also independent of the time t, i.e., the system is a time-invariant linear system. Using this, (2.2.1) can be written as _q = Fq+I; where q = q 1 q n T 2 R n +, F = (f ij ) 2 R nn, with f ij constants for i 6= j, f ii = (f i + P j6=i f ji), and I denotes the inow from outside the system. Since I i and q i, this system is easily seen to be a positive linear system, i.e., a system with positive input, state, and output, if for the output is taken y = Cq; y 2 R k ; C 2 R kn + ; where y denotes the vector of the observations. Note that the output is not the outow of the compartmental system. The outow, which is sometimes also called excretion, represents the ow of material leaving the system. The outputs of an experiment are measurements and usually dier from the material outows. On the other hand, the terms inow and input can be used interchangeably. Another property of compartmental systems is that the total ow out of a compartmentover any time interval cannot be larger than the amount that was initially present plus the amount that owed into the compartment during that interval. Together with the constraints on positive linear systems, this comes down to 1. f ij ; for all i; j 2 Z n ; i 6= j; 2. i=1 f ij ; for all j 2 Z n : The rst condition guarantees that positive solutions stay positive. The second condition stems from the inequality f j. A matrix F satisfying Conditions 1 and2above is said to be a compartmental matrix. Condition 2 states that all column sums of the matrix F 2 R nn are less than or equal to zero. Below some properties of compartmental matrices from the literature will be discussed that are needed in this thesis. References are [46, 48, 77, 136].

18 Chapter 2. Compartmental Systems Denition 2.2.1 A matrix A 2 R nn is said to be reducible if there exists a permutation matrix P 2 R nn such that PAP T U = ; Q R with U and R square matrices. The matrix A is said to be irreducible if A is not reducible. 2 Let F 2 R nn be a compartmental matrix. Then it follows from [11, Theorem 6.4.6] that (F ) f2cjre() < or=g. Here (F ) denotes the spectrum of F, and Re() means the real part of. Since a system _x = Fx is asymptotically stable if and only if (F ) f 2 C j Re() < g, a compartmental matrix is asymptotically stable if and only if =2 (F ). In the rest of this section compartmental matrices having zero as an eigenvalue are characterized. Proposition 2.2.2 (Adapted from [136, Theorem III].) Let F 2 R nn beanirreducible compartmental matrix. Then 2 (F ) if and only if P n i=1 f ij =for all j 2 Z n An important concept for the analysis of compartmental systems is that of a trap, dened below. Denition 2.2.3 Consider an n-compartmental system. A trap is a compartment or a set of compartments from which there are no transfers or ows to the environment nor to compartments that are not in that set. A trap is said to be simple if it does not strictly contain a trap. 2 In the physical literature traps are usually referred to as sinks. Let S be a linear compartmental system consisting of compartments C 1, C 2 ;:::;C n and let q j be the amount of material in C j. Let T S be a subsystem of S. Renumbering the compartments, assume T consists of the compartments C m ;C m+1 ;:::;C n, for m n. Let F 2 R nn be the compartmental matrix corresponding to S, consistent with this renumbering. Then T is a trap if and only if f ij =; for all (i; j) such that j = m; m +1;:::;n; i=;1;:::;m 1: The following two theorems are due to Fife [46]. (2.2.2) Theorem 2.2.4 The linear compartmental system S contains a trap if and only if one of the following conditions holds: 1. i=1 f ij =; for all j 2 Z n ; 2. there exists a permutation matrix P 2 R nn such that PFP T U = ; Q R with U, R square matrices and the sum of every column of R is zero.

2.3. Discrete-time compartmental systems 19 Theorem 2.2.5 The linear compartmental system S contains a trap if and only if 2 (F ). In response of Fife [46], Foster and Jacquez [48] derived the following result. See also Theorems 1 and 2 together with their proofs in [77]. Theorem 2.2.6 Let S be a compartmental system with system matrix F. 1. Zero is an eigenvalue of F of multiplicity m 2 Z + if and only if S contains m simple traps. 2. Assume zero is an eigenvalue of F of multiplicity m 2 Z +. Then there exists a partition of S into a disjoint union of subsystems S = S 1 [ S 2 [[S p ; such that S i receives no input from S i+1 ;:::;S p, i = 1;:::;p 1, and S p m+1 ;:::;S p are traps. Relative to this partition the system matrix is given by PFP T = F, e with ef = B @ F 11.... F p m;1 F p m;p m F p m+1;1 F p m+1;p m F p m+1;p m+1..... F p1 F p;p m F pp where F ii is irreducible for all i 2 Z p and zero is an eigenvalue of F ii of multiplicity 1 for i = p m +1;:::;p, and the sum of every column of F ii, i = p m +1;:::;p,iszero. An additional consequence of this theorem is that if zero is an eigenvalue of a compartmental matrix of (algebraic) multiplicity m, the geometric multiplicity is also m, so there are always m independent eigenvectors for the eigenvalue zero. 1 C A ; 2.3 Discrete-time compartmental systems In this section discrete-time compartmental systems will be considered. For that purpose it is assumed that transfer of material occurs at discrete times t 1 ;t 2 ;:::; or a continuous-time system is sampled at discrete times, in which case the state at time t k has been changed into the state at time t k+1. What happens in between will not be considered explicitly. So this can also be seen as if a transfer has occurred at time t k+1. The discrete times will be assumed to be equally spaced, to obtain a time-invariant system. Let this space be the unit time, so t k+1 = t k +1.

2 Chapter 2. Compartmental Systems Let q i (t) be the amount of material in the ith compartment at time t. The amount transferred from the jth to the ith compartment between time t and time t +1 is G ij (t). This transferred material will be assumed to be linear dependent onq j, i.e., G ij (t) =g ij q j (t). The state at time t + 1 will be given by q i (t +1)= X j6=i g ij q j (t)+i i (t)+g ii q i (t); where g ii q i (t) is the amount of material that was in compartment i at time t and is still (or again) in compartment i at time t +1. This amount g ii q i (t) is equal to q i (t) minus the amount that left compartment j: X n g ii q i (t) =q i (t) g oi q i (t) g ji q i (t) = 1 g oi g ji qi (t): Hence dene g ii =1 g oi n X j=1;j6=i g ji : j=1;j6=i j=1;j6=i The total outow of a compartment at time t +1 cannot be larger than the amount that was present at time t, if the inow from outside is assumed to be zero. Together with the constraints on positive linear systems (in discretetime), this comes down to 1. g ij ; for all i; j 2 Z n ; 2. i=1 g ij 1; for all j 2 Z n : A matrix G satisfying Conditions 1 and 2 above is said to be a compartmental matrix (in the discrete-time case). Condition 2 states that all column sums of G =(g ij ) 2 R nn + are less than or equal to one. Below properties of compartmental matrices in discrete-time will be discussed, analogous to the continuous-time case. In the rest of this section, G refers to a discrete-time compartmental matrix, whereas F refers to a continuous-time compartmental matrix. Let G 2 R nn + be a compartmental matrix. Then (G) f2cjjj1g, since the sum of every column of G is less than or equal to one, see [97, Section 6.2], or [11, Chapter 2]. Because a system x(t +1) = Gx(t) is asymptotically stable if and only if (G) f 2 C j jj < 1g, it follows from the Perron-Frobenius Theorem, see [97], that a compartmental system is asymptotically stable if and only if the spectral radius (G) 6= 1, which is equivalent to 1 =2(G). Analogously to the continuous-time case, compartmental matrices having spectral radius one are characterized. Proposition 2.3.1 Let G 2 P R nn + be an irreducible compartmental matrix. Then n (G) =1if and only if i=1 g ij =1for all j 2 Z n.

2.3. Discrete-time compartmental systems 21 Proof. This follows from [11, Theorem 2.2.35]. 2 A trap in an n-compartmental system is dened in the same way as for continuous-time systems, see Denition 2.2.3. As in the continuous-time case, let C 1 ;C 2 ;:::;C n be the compartments of a linear compartmental system S. After renumbering, let T S consist of the compartments C m ;C m+1 ;:::;C n, for m n. Then T is a trap if and only if g ij =; for all (i; j) such that j = m; m +1;:::;n; i=;1;:::;m 1; (2.3.1) where G = (g ij ) 2 R nn + is the compartmental matrix corresponding to S. Consider F = G I 2 R nn. Since 1. f ij = g ij ; for i; j 2 Z n ; i 6= j; 2. j=1 f ji = g ii 1+ j=1;j6=i g ji = j=1 g ji 1 ; F is a continuous-time compartmental matrix. Assume F is the system matrix for a continuous-time compartmental system S F and let T F S F consist of the last n m + 1 compartments e Cm ;:::; e Cn. Proposition 2.3.2 Consider T S and T F S F dened above. Then T is a (simple) trap if and only if T F is a (simple) trap. Proof. The subsystem T is a trap if and only if (2.3.1) holds, which is equivalent to g ij =; for all (i; j) such that j = m; m +1;:::;n; i=1;2;:::;m 1; and P g jj =1 n i=1;i6=j g ij; for all j = m; m +1;:::;n; since g j =. Because f ij = g ij for i 6= j and f jj = g jj to f ij =; for all (i; j) such that j = m; m +1;:::;n; i=1;2;:::;m 1; and P n f jj = for all j = m; m +1;:::;n; i=1;i6=j f ij; P n 9 >= >; (2.3.2) 1, (2.3.2) is equivalent which is, because f j = i=1 f ij, equivalent to (2.2.2), i.e., T F is a trap. In the same way itcan be proved that T is a simple trap if and only if T F is a simple trap. 2 Using Proposition 2.3.2, the following theorems, analogous to Theorems 2.2.4, 2.2.5, and 2.2.6, can be proved. Theorem 2.3.3 The linear compartmental system S contains a trap if and only if one of the following conditions holds. 9 >= >;

22 Chapter 2. Compartmental Systems 1. i=1 g ij =1; for all j 2 Z n ; 2. there exists a permutation matrix P 2 R nn such that PGP T U1 = ; R 1 Q 1 with U 1, R 1 square matrices and the sum of every column of R 1 is one. Proof. Since i=1 f ij = n X i=1 g ij 1 and PFP T = P(G I)P T = PGP T I = U =: ; Q R U1 I Q 1 R 1 I in which the sum of every column of R = R 1 I is equal to the sum of every column of R 1 minus one, it follows that the conditions stated in the theorem are equivalent to the conditions stated in Theorem 2.2.4. The theorem now follows using Proposition 2.3.2. 2 Theorem 2.3.4 The linear compartmental system S contains a trap if and only if 1 2 (G). Proof. The following statements are equivalent: (i) 2 (F ); (ii) det(f ) = ; (iii) det(g I) = ; (iv) 1 2 (G); (v) (G) = 1. The last equivalence relation follows from the Perron-Frobenius Theorem. With Theorem 2.2.5 and Proposition 2.3.2, the theorem is proved. 2 Theorem 2.3.5 Let S be a compartmental system with system matrix G. 1. One is an eigenvalue of G of multiplicity m 2 Z + if and only if S contains m simple traps. 2. Assume one is an eigenvalue of G of multiplicity m 2 Z +. Then there exists a partition of S into a disjoint union of subsystems S = S 1 [ S 2 [[S p ; such that S i receives no input from S i+1 ;:::;S p, i = 1;:::;p 1, and S p m+1 ;:::;S p are traps. Relative to this partition the system matrix is

2.4. Graphical representations of compartmental systems 23 given by PGP T = e G, with eg = B @ G 11.... G p m;1 G p m;p m G p m+1;1 G p m+1;p m G p m+1;p m+1..... G p1 G p;p m G pp where G ii is irreducible for all i 2 Z p and one is an eigenvalue of G ii of multiplicity 1 for i = p m +1;:::;p, and the sum of every column of G ii, i = p m +1;:::;p, is one. Proof. 1. The following statements are equivalent one is an eigenvalue of G of multiplicity m 2 Z + ; det(g I) =( 1) m p() with p(1) 6= ; det(f I) = m p 1 () with p 1 () = p(1) 6= ; zero is an eigenvalue of F of multiplicity m 2 Z +. The equivalence between the second and the third statement follows from det(f I) = det(g I I) = det(g ( +1)I)=((+1) 1) m p( +1)= m p 1 () with p 1 () =p(+1). Now statement 1 follows from Proposition 2.3.2 and statement 1 of Theorem 2.2.6. 2. Consider the following statements. a. one is an eigenvalue of G of multiplicity m 2 Z + ; b. zero is an eigenvalue of F of multiplicity m 2 Z + ; c. statement 2 in Theorem 2.2.6; d. statement 2 in Theorem 2.3.5. From 1 it follows that (a, b), and Theorem 2.2.6 provides (b ) c). Noting that PGP T = PFP T +I, G ij = F ij for i 6= j, and G ii = F ii + I, where the sum of every column of G ii is equal to the sum of every column of F ii plus 1, the implication (c ) d) follows from the statements of the proof of part 1 for m =1. This completes the proof of part 2. 2 1 C A ; 2.4 Graphical representations of compartmental systems A compartmental system can also be represented by a directed graph, see for example [34] or [76, Chapter 3]. Every compartment is represented by a vertex and there is a directed arc from q j to q i if and only if f ij >. To incorporate outows (f j > ) and inows into this graphical representation,

24 Chapter 2. Compartmental Systems the following vertices and arcs can be added. The environment is treated as a single compartment q, so all outows go to this compartment. Then there is a directed arc from P q j to q if and only if f j >. Note that in the continuoustime case f j = n P i=1 f ij and in the discrete-time case f j =1 n i=1 f ij. Every inow can be treated as a separate source point, or all inows can be lumped into one single source point. The choice between these two possibilities, or even a combination of the two, depends on the nature of the inows and on the problem. In any case, there is an arc between a source point andavertex q j if I j 6=. Example 2.4.1 Consider the compartmental system with input I 2 R 4 and system matrix F 2 R 44, F = B @ f 21 f 12 f 21 f 12 f 32 f 23 f 32 f 23 f 43 f 34 f 43 f 34 f 4 1 C A ; I = B @ I 1 1 C A : f21 f32 f43 I1 f4 I q1 q2 q3 q4 q f12 f23 f34 Figure 2.2: Example of catenary system. Figure 2.2 represents its graph. An n-compartmental system of such a form is called a catenary system. In a catenary system the n compartments are arranged in a linear array such that every compartment exchanges only with its immediate neighbours. This particular example, in which there is only inow into the rst compartment and only outow from the last compartment, is a special case. The system matrix for a catenary system has nonzero entries only on the main diagonal and on the rst super-diagonal and the rst sub-diagonal. 2 Example 2.4.2 Consider the compartmental system with system matrix F = B @ 1 f 41 f 14 f 42 f 24 C f 43 f 34 A 2 R44 ; f 41 f 42 f 43 f 44 with f 44 = f 14 f 24 f 34 f 4. The corresponding graph is Figure 2.3. An n-compartmental system of this form, with a central compartment, is called a mammillary system. Its central compartment is called the mother compartment. There exists only exchange between the mother compartment and

2.4. Graphical representations of compartmental systems 25 q2 f24 f42 f43 f14 q3 q1 f34 q4 f41 f4 q Figure 2.3: Example of mammillary system. another compartment, not directly between the other compartments. The excretion and the inow can in principle be in any compartment. In Chapter 8 examples of mammillary systems will be treated.