Identifiability of Large Compartmental Models

IDENTIFIABILITY OF LARGE COMPARTMENTAL MODELS R. F. Brown and

J.

P Norton*

School of Electn·cal Engineen·ng, University of New South Wales, Kensington, N.S. W. 2033, Australia *Department of Electrical and Electronic Engineering, University of Birmingham, Birmingham, England

Abstract. Existing methods for testing the global identifiability of linear dynamic systems depend upon algebraic manipulation of nonlinear equations; this is a manual task not amenable to computerisation and one where the work effort tends to be prohibitive for systems with more than three states. A major breakthrough is reported based upon a novel variant of normal-mode analysis. The method begins with bilinear equations and reduces these by stages according to well-defined rules into subsets of independent linear equations. The effects on identifiability of changes in model structure are immediately apparent, which makes the procedure valuable for computer-aided design of optimum experimental conditions. The decomposition procedures are demonstrated via numerical examples, including application examples from chemical reaction kinetics and mineral ore processing. Keywords. Chemical reaction kinetics; compartmental modelling; dynamic response; identification; linear decomposition; mineral ore processing; normal-mode analysis; parameter estimation. INTRODUCTION

their superposed respective contributions may be separated.

Global identifiability is the property treated in this paper, and is defined as the ability to estimate uniquely under given ideal experimental conditions the free parameters in a given model structure. The practical issue of concern is to establish that a chosen experimental design is capable of uniquely identifying all free parameters under ideal conditions, namely where the model structure characterises exactly the system structure and the measured values of input and output signals are noisefree. Clearly, identifiability under these conditions is a prerequisite for consistent parameter estimates in the real-world situation. Global identifiability is vitally dependent upon the experimental conditions~ the most notable being

The resulm of this paper have general application to dynamic systems representable by finite sets of linear time-invariant state equations. However, the treatment deals specifically with compartmental models because these offer a realistic approach to the macromodelling of mass/energy flow in large-scale ill-defined systems, notably production processes (Hirata, 1979) and pharmacokinetic processes (Kusuoka and co-workers, 1978). Much literature now exists on the identifiability of linear compartmental models, and some of this is referred to by Godfrey, Jones and Brown (1979); Brown (1979) treats the identifiability of nonlinear compartmental models. However, the algorithms so far used in testing for identifiability (principally, the transfer function method for linear systems and the Pohjanpalo (1978) time domain method for nonlinear systems) are inapplicable to large systems because of the awkward sets of nonlinear equations involved. The reader may wish to convince himself of this by applying the transfer function method to some of the examples solved later by modal analysis. For reference, a sample example of the transfer function method follows.

(1) to which compartments test input flow rates can be applied, and in which compartments the state (mass or concentration) can be measured; (2) which known inputs and outputs are known absolutely, and which are known to within an unknown gain factor (as with non-calibrated measurement transducers); (3) which test inputs can be applied singly in separate experiments, and which need to be applied simultaneously in the one experiment. In the latter case, the waveforms of the simultaneously applied test inputs need to differ in order that

Example 1. In the three-compartment model of Fig. 1, a unit impulse function is applied at time 0 to compartment 1, setting up the initial state x 1 (0+) = 1, x (0+) = 0, x {0+) = O. 3 2

23

R. F. Brown and J. P. Norton

24

ent linear equations. Global identifiability results when the subset at each stage of reduction is unique, and when reduction proceeds until all elements of either the modal matrix or its inverse have been identified. The reduction process takes place according to welldefined rules and constitutes a linear decomposition process.

Assume that the transient responses Y1(t) = h x (t), with measurement gain hI unknown, and 1 1 Y2(t) = x (t) are measured exactly. 3 ~ From the network diagram of Fig. 1, derive the Laplace transformed responses P4(s s

3

2

+ Pss + P6)

+ PIs

2

( 1)

+ P2 s + P3

2.

The effects on identifiability of changes in model structure are immediately apparent, which makes the procedure valuable for computer-aided (interactive) design of optimum experimental conditions.

(2)

s

3

+ PIs

2

+ P2 s + P3

where PI to Ps are eight known (measured) parameters and are nonlinearly related to the eight unknowns hI' k 12 , k 02 ' k , k 31 , k 32 , k 03 : 23

P3

PI

k 21 + k 31 + Ps

(3)

P2

(k 21 + k 31 )Ps + P6 - k 21 k 12

(4)

(k Z1 + k 31 )P6

k12[k21(k03 + k Z3 )+ kZ3k31J (5)

P4

P6

hI

(6)

+ k 12 + k 32 + k + k (7) 03 23 (k 02 + k 1Z + k 32 )k + (k 02 + k 12 )k 23 (s) 03

Ps

k

P7

k 31 k 21 k 32 + k 31 (k 02 + k 12 + k 32 )

Ps

02

(9)

~:

Solve equations 3 to 10 algebraically. This step is necessary to distinguish between the following three possible outcomes:

(2)

(3)

the eight equations are not independent, in wtich case there is an infinite number of solutions and the system is not completely identifiable; elimination of seven of the eight unknowns leaves a quadratic, cubic, etc. equation in the remaining unknown, in which case there is a finite number of solutions and the system is locally identifiable; the solution is unique, in which case the system is globally identifiable.

The reader is left to verify that the solution is unique in this instance. Recently, Norton (1979) has devised a novel variant of normal-mode analysis (applicable to Zinear compartmental models) for testing identifiability. His basic method sets up two sets of bilinear equations, one in terms of the (unknown) elements of the modal matrix and its inverse, and the other in terms of the constraint relations imposed by prior knowledge, for example, that certain rate coefficients are known nonzero or known zero. In this paper, refinements of this method are presented which make it attractive for large-systems analysis. The new contributions in this paper are: 1.

BASIC THEORY

(10)

A FORTRAN G computer program exists to execute step 1 (Bossi and co-workers, 1979).

(1)

In the remainder of the paper, the basic theory is first developed, next the concept of linear decomposition is introduced at the simplest level via Example 2, and then a problem-dependent approach via a series of numerical examples is presented. Thereby, questions of mathematical rigour are avoided, and the examples can serve the reader as standard problems against which to test any subsequently developed computer programs. Examples 3 and 4 illustrate compartmental structures where the decomposition process needs to become hierarchical. Example 5 illustrates sparsity considerations.

The bilinear equations are reduced by stages into successive subsets of independ-

The mass balance equations for a linear ncompartment model with states x , ... ,x and 1 n inputs glu1, ... ,gnun may be written n

dx.

1

I

dt

j=l

0,

a .. x. + gi u. , x. (0 1 1 1J J

i

(11)

1, ... ,n

The compartmental structure imposes the linear constraints k

a

ij

ij

~

0

i

(12)

=1 j

1

a.. ~ 0 , j = 1 , · · · ,n ( 13 ) i= 1 1J where k .. is the rate coefficient for the mass fl5~ rate to compartment i from compartment j, and k . is the rate coefficient for OJ the mass flow rate to the environment 0 from compartment j. k J' O

-

In equa1ions 11, let the coefficient matrix [a .. J - A have n distinct eigenvalues 1J

AI' ... ,An' forming a diagonal matrix 1\.

Let

the modal matrix (whose col~s are the respective eigenvectors) be M = [r TJ where r T i

i'

is the ith row of M, and let the inverse modal matrix M-I be N ~ [n J where n. is the j

r.

T

1

n. J

,

1, i

=

j

0, otherwise r.

1

T

1\ n.J

J

Then

jth colunm of N.

a .. for all i, j 1J

(14) (15)

Identifiability of Large Compartmental Models Attention is restricted to the two commonest test input signals, namely, on a normalized basis, the unit impulse function u. = o(t) and

r

niT

T n 1\ 1 T n /\ 1

1

the unit step function u = u(t). The coeffi icients gi in equations 11 allow for the various input gains (which are zero in the case of non-existent inputs). For an impulse-function input to compartment j and no inputs to all other compartments, the solution of equations 11 is

T At

x. = (r.

e

0 0

i = 1, ... ,n (16) For a step function input to compartment k and no inputs to all other compartments, the solution of equations 11 is 1

1

T

xi = r i

J

-1

1\

n.)g., u. = o(t), J

J

At

(e

- I)nk~' uk i = 1, ... ,n

u(t), (17)

0

l

0

0

rZ

o

0

0

nZ

k

0

0

n

k Z1

r T 1 T r 1 /\ T r3 T r 3 /\

0

25

0

0

0

0

0

0

3

J

01

-(r

h1r11n11e

AZt + h1r1ZnZ1e + h1r13n31e

YZ(t)

=

A1t r 31 n 11 e

Azt + r 3Z n Z1 e

A t 3

(18)

At + r 33n e 3 31 (19)

Clearly, the eigenvalues AI' A ' A are measurZ 3 able exactly, as also the coefficients h1r1ini1' r 3in , i = 1,Z,3. The unknown i1 measurement gain hI is immediately given by hI

Y1(O+)

=

(ZO)

Arbitrarily define r1

T

n =

0

0 T r1 T r 1 1\ T r3

(ZZ)

T

T

T r Z An z

k

T Z n3

0

T r z I\n 3

k

r TAn

k Z3

r

z

3

T T OZ -(r 1 + r 3 )An z

03

-(r T + r TA ) n 1 3 3

(Z3)

Stage Z : Examining (ZZ), no solution for r Z or for n Z is possible by setting rate coefficients to zero. But a linear solution for n is possible by setting k = 0: 3 13 -1 T r1

1\

Then hI' r 1 ' r 3 ' n 1 ,I\are known. Stage 1: Using (14) and (15), list the equations which are linear in the unknowns r T Z ,nZ' n 3 :

3

Z,

(ZZ) and (Z3) contain 7 equations not containing rate coefficients, whereas the unknown vectors T contain 9 unknowns. r Z ' nZ' n 3 Therefore, it is necessary to impose at least two constraints on the values of the rate coefficients in order to obtain sufficient independent equations. To simplify the presentation, consider only zero-value constraint:.s .

(Zl)

[l,l,lJ

l\n 1

0

Z,

Example Z. In the maximally connected threecompartment model of Fig. Z, a unit impulse function is applied at time 0 to compartment 1, setting up the initial state x (0+) = 1, 1 + + xZ(O ) = 0, x (0 ) = O. Assume that the 3 transient responses Y1(t) = h x (t), with 1 1 measurement gain hI unknown, and YZ(t) = x (t) 3 are measured exactly. From (16),

=

3

Next list the equations which are bilinear in the unknowns r T n n :

LINEAR DECOMPOSITION AT LEVEL 1

Y1(t)

+r

o

0

where I is the identity matrix.

A1t

T~

T

1

n

3

T r 1 /\ r3

r0 0

, k

13

0

(Z4)

T

With k 13 = 0 and n 3 known, list the equations which are linear in the unknowns r T , n Z : Z

R. F. Brown and J. P. Norton

26 T 1 T n 1 /\ T 1 n T" n3 n

0 0

[::j

-.

0

[::]

k

01

0

k 21

0

0

n T/\ 3

0

k

y\

0

k

n

3

03

) n -(r 1T+r 3TA

simultaneously. However, the above global identifiability test gives only sufficient conditions, and can exclude many practicable designs. In the next Section, it is shown that simple rules can be applied which, under certain conditions, reduce some of the bilinear equations at each stage of the above process (equations 23 at the first stage and equations 26 at the second stage) into linear equations. Example 3 illustrates this.

1

) n -(r T+r T/\ 3 3 1

23

In the following numerical examples, the convention is followed that the stated numeric values have been truncated to 4 significant figures; values stated with less than 4 significant figures are exact.

0

0 0

, k I2

0

0

0

~k32

LINEAR DECOMPOSITION AT LEVEL 2 (25)

Note that knowledge of n has converted three 3 of the bilinear equations in (23) into linear equations. Next list the equations which are bilinear in the unknowns r T , n : 2 2

3. In the three-compartment model 3, a unit impulse function is applied 0 to compartment 1, setting up the state xl (0+) = 1, x (0+) = 0, x (a) = O. 2 3 The following transient responses are measured: Example of Fig. at time initial

y1

hI xl 0.8922e-0.8225t+0.4247e-3.321t +1.683e-5.855t

(26) Stage 3: Examining (25), no solution for n 2 is possible by setting rate coefficients to zero. But a solution for r is possible: 2

(29)

x

2 0.09360e-0.8225t+0.2086e-3.321t

Case 1

(27)

r 21 n 11 e

A1 t

Arbitrarily set Case 2

+r

22

_0.3023e-5.855t A2 t At 3 n 21 e +r 23n 31 e

Al

-0.8225

A 2 A 3

-3.321

(30)

(31)

-5.855

The unknown measurement gain hI is immediately given by

o

+

hI = Y1(0)

o (28) This terminates the solution since knowledge of r T completes the calculation of the modal 2 matrix. Therefore, the compartmental structures of Cases 1 and 2 are globally identifiable under the given experimental conditions. Case 1 will be seen to be identical to that treated by the transfer function method in Example 1. The above approach is extremely simple to program, is interactive, and rapidly enables global identifiability to be tested under various possible experimental conditions: in particular, in regard to the choice of input/ output locations, whether input/output gains are measured, whether signals are impulse or step functions, and whether signals are applied singly in separate experiments or

Define r 2 Then n

1

T

~

(32)

3

[1, 1, 1J

(33)

[0.09360,0.2086,-0.3023J [3.177,0.6783,-1.855J

T

(34) (35)

Stage List those of equations 14 and 15 which are linear in the unknowns T r 3 ' n2 , n 3 :

Identifiability of Large Compartmental Models

~

n1

T

n 1Tf\

0

0

0

0 0

0

0

:0

0

r T 1 r T 2 r T 2

i

io I i 1

n

where

I

0

10 i

3

1_

n

r T /\ n

3

o

(38) give the linear dependence equation T T TA a r 1 + b r 2 + r2/\ = 0 a

-r T/\n = -1 1 2

b

-r T/\ n

(39)

2

Substituting the known values for a, r T

r 2T n 2

T

1

'

and/\ into (39) yields the linear indepen-

(45)

10.05892J 0.6152

2

(37)

3

o

2

The unknown a is immediately given by

whence n

Examining (36), no direct solution for r T , 3 However, the singular n 2 or n 3 is possible. simultaneous equations

r

(44)

0.1628

0.5a

o

2

1

J

(36)

T

n

=4

1:°.02946-1 0.3076 a

o

2

/\ r 2

(/\+ bI)-l a n

2

o

T

T 2

(43) gives the three linear equations

o

r 3 n2

where

-n

Substituting the known values of b, n ,/\ into 1

Next list the equations which are bilinear in the unknowns r T 3 ' n2, n3 :

3

-n T/\ r 1 2

=

b

0

i

r

a

2

n L

I

10

r3 i

T r1 r T 2

,0

I

27

(46)

0.3257

'-

Stage 2: List the equations which are linear in the remaining unknowns r T n . 3 ' 3· T 0 0 n1 3 -(r T+r ) n 1 0 n3 2 1 n T 0 0 2

-[r

n1Tf\

n T/\ 2

T/\

0

0

0

T r1

0

0

r2

0

r T/\ 2

T

0 L

O

.J

(47)

The solution for r 3 is possible:

dent equation (40)

-1.539

o

1.443

(48) _J

(41)

-4 is not soluble for n since the matrix to be 2 inverted is singular (equation 38). Further singular simultaneous equations can be written incorporating the first two bilinear equations in (37). The singular simultaneous equations T '

n1

n2

This terminates the solution since knowledge of r T completes the calculation of the modal 3 matrix. Linear decomposition at level 2 is seen to require the examination of singular simultaneous equations via linear dependence equations. LINEAR DECOMPOSITION AT LEVEL 3 Example 4.

Continuous pyrolysis of benzene

i

TJI

r3

o

(42)

Ln 2 T/\ are the only useful ones, and give the linear dependence equation an1T+bn2T+n2T/\

0

(43)

Chemical reaction kinetics can give rise to high-order sparsely-connected compartmental models; an example of particular importance in the chemical industry is polymerisation kinetics (Ray, 1977; Amrehn, 1977). Consider here a small-scale example relating to the continuous pyrolysis of benzene. The kinetics are nonlinear but give rise to the following (linear) tracer kinetics (see Brown,

28

R. F. Brown and J. P. Norton

1979a, for background): -

-

n -

1-

T 1

0

0

0

T 1

r

,-

xl

-(2k1x1+k3x2)x1+k2x4x2+ 3 k 4x 4x 3

0

0

0

r

x2

22k1x1x1-(k2x4+k3x1)x2+ 3 k 4x 4 x 3

n/A u1T/\ 0

0

n

0

n

(49) The corresponding model is shown in Fig. 4 for numeric values k 1=k 2=k =k = 1 for the 3 4 reaction rates and = 1 = 2 = 1 5 1 '2 '4 · for the steady state. The rate coefficients in the compartmental model are seen from (49) to be interdependent, and satisfy the constraint relations

x

x

x

(50) (51) A unit impulse function is applied at time 0 to compartment 1, setting up the initial states x (0+) = 1, x (0+) = 0, x (0+) = O. 2 1 3 The following transient responses are measured: Y1

0

r T 1

0

h 2x 2

0

0

h 2r 2

0

0

0

r T 1

0

0

0

h 2r

0

0

0

0

2

-r 1Tf\n 1

3-.1

0 h2

0

0

T

0

2

T T (2h r 1 -h 2 r 2 2

o

J

L

(59) Next list the equations which are bilinear in the unknowns T T T r 2 ' r 3 ' n 2 , n 3 , h 2 (with h 2 r 2 known):

o o

o o o

1

0.6779 + 0.2370e-2.881t_0.9150e-5.118t ~ A1t A2 t A3 t + h2r22n21e + h2r23n31e = h2r21n11e (53) Arbitrarily

set

Al

0

A 2

-2.881

A 3

(60)

Examining (59), no direct solution for r T ' 2 T However, the r , n 2 , n 3 or h 2 is possible. 3 singular simultaneous equations

"I

-T r

h

-5.855

(54)

The unknown measurement gain hI is immediately given by

1

2

r

T T

(2h 2 r have zero

1

T

~

(56) T

o

-1.236

3.236

(57)

1= 0

(63)

(58)

Stage 1: List the equations which are linear in the unknowns r T ' r T ' n , n 2 3 2 3 T (with h r known): 2 2

I

-5.763h +9.326 -10. 23h -6. 32~ 2 2 (62)

whence [3.636, 3.236, -1.236J

(61)

namely,

(55)

= [1, 1, 1J

[0.1864, 0.07326, 0.7402J

- h 2r 2

o

3

T

determinant~

det 3.636 r1

n

2

1

+

Y 1(0 ) = 10

Define

T

3

o

hIxl 1.864 + 0.7326e-2.881t+ 7.402e-5.118t A3 t ~ A1t A2 t = h r 11 n 11 e + h1r12n21e + h1r13n31e (52)

Y2

n

-;

r 2- I '0

Further singular simultaneous equations can be written incorporating the first six bilinear equations in (60). The singular simultaneous equations r

o o

r

o

r

T

2 T 2 T

2

+ r3 T>!\n 1 +

+

r3T )A.n z r3T >A.n 3

(64)

are the only useful ones, and give the linear dependence equation

Identifiability of Large Compartmental Models (r

T T + r ZT + r 3 1

l\=

0

(65)

Since Al = 0, (65) contains only two nontrivial equations. Combining these with the equation r Tn = 0 gives 3 1

-A (r + r 22 ) 2 12

transfer function method (after tedious algebra). It is not possible in the limited space available to present the large numbers of equations to be considered. Suffice it to say that solution at levels 1, 2 or 3 of linear decomposition is aided by use of inferred equations and the exploitation of the sparsity of the modal matrix. This constitutes linear decomposition at level 4. For the compartmental structure of Fig. 5, the modal matrix M and the inverse modal matrix N have a common incidence matrix, which is shown in Fig. 6 for the following arbitrary ordering of the measured eigenvalues:

-A (r 13 + r 23 ) 3

(66)

This terminates the solution since knowledge . of r T completes the calculat10n of the modal 3 matrix. Linear decomposition at level 3 is seen to require the examination of singular simultaneous equations via zero detenminant conditions. LINEAR DECOMPOSITION AT LEVEL 4 Example 5.

Copper-ore grinding process

In Fig. 5 is shown a compartmental model of a copper-ore grinding process (full documentation of the physical process is given by Kortela and Niemi, 1978). Compartments 1, 2, 3 simulate the ball mill, compartments 4 and 5 the autogenous mill and compartment 6 the classifier. Practicable inputs are a step-function input to compartment 1 and an impulse-function input to compartment 4. The only accurately measurable variable is the copper concentration at the classifier overflow. The numeric values in Fig. 5 correspond to recycle coefficients k = 0.8 1 for the ball mill and k = 0.72 for the auto2 genous mill. Conservation of mass flow rate of the water carrier for the crushed ore imposes the following known constraints on the rate coefficients:

-0.6311

A 2

-2

AS

-7.702

A 3

-1.5

A 6

-10

Define

r6

T b.

= [1, 1, 1, 1, 1, 1J

where y = y(l) if the unit step function ~~ly is applied (to compartment 1), and y = y( if the unit impulse function only is applied (to compartment 4). Since the response y(l) to a unit step function applied to compartment 1 is measured in compartment 6, which is 3 compartments removed from yompartment 1, the first 3 derivatives of y(l evaluated at time 0+ are zero:

; (1)

(0+)

= -; ( 1 )

(0+)

The above example turns out to be globally identifiable, as can be verified by the

(71)

Substituting (71) into (69), and applying the zero-nonzero configuration of Fig. 6, 6 1 A.t ~ A.t Y = L h 6A.- (e 1 1) + h e 1 n. i=l 1 - nil i=4 6 14 ~ y(1) + y(2) (72)

o=

(69)

(70)

Permutation of the ordering of the eigenvalues gives corresponding row-column permutations of the incidence matrix. Let us choose the incidence matrix of Fig. 6. At the same time, let us use the ordering of the eigenvalues given in equations 70, but recognise that there is ambiguity in the ordering of the eigenvalues, and that if this is not resolved identifiability will be at best LocaL.

(68)

_ 5.626e-0.6311t_ 2.995e-7.702t + 2.25ge- 10t

+ eAt n 4 J

A 4

;(1)(0+)

9.1S3e- t + 1. 118e- 2t _ 4.90ge- 1 . St

1

-1

o

h 6x 6

h r TrA-1(At 6 6 ~\ e -I)n

Al

(67)

Consider a unit step function and a unit impulse function to be simultaneously applied to respective compartments 1 and 4, giving rise to a measured transient response

Y

29

o

(73)

Substituting measured values from (69), using (72), into (73) enables solution" for the remaining unknowns:

30

R. F. Brown and J. P. Norton

1

-\-r .02"5"' 4

t1.530~

-0.6311 -7.702 -10

! 0.3983

1-

59.32

2.580

100J

JI

'4.014

0.03029

_-0.01865 (74)

In turn, ....,

r-

-1

-1

--,

r-

-,

h 6n 44 i :h 6 (A 4 n41+n44)-A4

J h6(A~ln61+n64)-A~lh6n61J ~.258

h 6n 64

(75)

h 6n 1 and h 6n are now separately known. At 4 this point, the ambiguity in the ordering of the eigenvalues can be resolved by noting that the steady-state response is 6 -1 y(oo) = 1 = h 6 Ai nil (76) i=l

I

The identity in (76) is satisfied only for the ordering of the eigenvalues given in (70). The remainder of the analysis is straighforward but lengthy and proceeds at level 2. Full details are reported elsewhere (Brown, 1979b). Two fine points are worth noting:

linear decomposition; thereby questions of mathematical rigour have been avoided, in particular, whether a symbolic set of simultaneous equations can be inverted, a question which becomes trivial when numerical values are substituted. Indeed, it is recommended that the initial implementation of the above method be based upon numerical simulation. That is, numerical values of rate constants are (randomly) chosen, the modal matrix M and its inverse N are calculated, the measurement data are calculated, and then an attempt is made to reconstruct M (or N) via computeraided application of the identifiability test. Logic faults, whether originating in the user or in the program, become self-checking. Later, if demand warranted, the program could be implemented at the symbolic level to avoid matrix inversions. REFERENCES Amrehn, A. (1977). Computer control in the polymerization industry. Automatica, 11, 533-545. Bossi, A. and co-workers (1979). A method of writing symbolically the transfer matrix of a compartmental model. Math. Biosci., 43, 187-198. Brown, R.F. (1979a). The identifiability of nonlinear compartmental models. 5th IFAC Symposium on Identification and System Parameter Estimation, Darmstadt, F.R. Germany.

(1) The constraint k = 0 is not required 64 in order to establish identifiability. This is apparent from (75) which makes h n known and hence also h k 6 4 6 64 h r TI\n (which latter is only inciden6 6 4 tally constrained to be zero).

Brown, R.F. (1979b). Identifiability of a copper-ore grinding process. EE Report, Univ. N.S.lv.

(2) In the omitted level-2 analysis, the constraint k + k = 1.5 k 21 is not 43 63 used in the derivation of global identifiability.

Godfrey, K.R., R.P. Jones and R.F. Brown (1979). The identifiability of linear compartmental models. 5th IFAC Symposium on Identification and System Parameter Estimation, Darmstadt, F.R. Germany.

In a parameter estimation scheme the two abovementioned zero constraints may either be imposed to improve the accuracy of estimation or relaxed to act as a cross-check on the validity of the model.

Hirata, H. (1979). The stability of composite production systems based on a mass-energy flow model. IEEE Trans. Syst., Man, Cybern., SMC-9, 296-300.

CONCLUSIONS A variant of modal analysis developed by Norton for use in the testing of identifiability has been shown to be amenable to linear decomposition procedures, and thereby to have applicability to large-scale systems. At the simplest level of linear decomposition (level 1), the method provides sufficient conditions for global identifiability. Successively higher levels of linear decomposition successively broaden the class of globally identifiable compartmental processes (where process implies here input/output signals plus compartmental structure) which are amenable to the method. A series of numerical examples has been used to demonstrate the successive levels of

Kortela, U.K.J. and Niemi, T.J. (1978). Modelling and prediction of copper concentration of a grinding process. Automatica, ~, 547-556. Kusuoka, H. and co-workers (1978). Optimal control of drug administration. IEEE Cybernetics and Society: Proc. Int. Conf. on Cybernetics and Society, Tokyo-Kyoto, Japan, 1, 63-68. Norton, J.P. (1979). Normal-mode analysis of structural identifiability of linear compartmental systems. Submitted to Math. Biosci. Pohjanpalo, H. (1978). System identifiability based on the power series expansion of the solution. Math. Biosci., ~, 21-33.

Identifiability of Large Compartmental Models

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Ray, W.R. and Laurence, R.L. (1977). Polymerization reaction engineering. In: Lapidus, L. and Amundson, N.R. (Eds.) (1977). Chemical Reactor Theory: A Prentice-Rall, New Jersey, Review. ~ 532-582.

Fig. 1.

Compartmental model of Example 1. Measurements Yl = h 1x , Y2 = x 3 · 1

Fig. 3.

Compartmental model of Example 3. Measurements Yl = h1x , yZ = x • 1 Z

benzene

Fig. 2.

Maximally-connected compartmental model of Example 2. Measurements Yl = h 1x 1 , Y2 = x 3 ·

diphenyl

triphenyl Fig. 4.

Tracer model of the continuous pyrolysis of benzene (Example 4). Measurements Yl = ~lxl' yz = h x · 2 2

R. F. Brown and J. P. Norton

32

1- -

BALL MILL

-

-

-

-

-

-

-

I

I

U (t) I I

Fig. 5.

(Examp le 5). Compa rtmenta l model of a copper -ore crushin g proces s Measur ement y = h 6x 6 •

x

0 0 0 0 0 X X 0 0 0 0 X X X 0 0 0 X X X X X 0 X X X X X 0 X X X X X X

Fig. 6.

Inciden ce matrix for M and N in Exampl e 5.