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Structural identifiability of the parameters of a nonlinear batch reactor model
Structural Identifiability Batch Reactor Model
of the Parameters of a Nonlinear
MICHAEL J. CHAPPELL AND KEITH R. GODFREY Department of Engineering, University of Warwick, Coventry,
CV4 7AL, England
Received 20 March 1991; revised 17 August 1991
ABSTRACT The similarity transformation approach is used to analyze the structural identifiability of the parameters of a nonlinear model of microbial growth in a batch reactor in which only the concentration of microorganisms is measured. It is found that some of the model parameters are unidentifiable from this experiment, thus providing the first example of a real-life nonlinear model that turns out not to be globally identifiable. If it is possible to measure the initial concentration of growth-limiting substrate as well, all model parameters are globally identifiable.
1.
INTRODUCTION
In this paper, the structural identifiability of the parameters of a model of microbial growth in a batch reactor is considered. The model incorporates Michaelis-Menten-type nonlinearities and has previously been considered in this journal by Holmberg [5]. A comprehensive review of the kinetics of microbial growth processes is given by Button [ 11. Microbial growth in a batch reactor can be expressed as
(14 Pm=
‘= - Y(K,+s)
’
P-4
where x is the concentration of microorganisms; S, the concentration of growth-limiting substrate; pm, the maximum velocity of the reaction; K,, the Michaelis-Menten constant; Y, the yield coefficient; and K,, the decay rate coefficient. The experiment considered is one where the microorganisms are prepared in a mixture in which the concentration can be controlled, and the
MATHEMATICAL
BIOSCIENCES
108:241-251 (1992)
OElsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
241 0025-5564/92/$05.00
242
MICHAEL
substrate is also prepared controlled. Then, at time reactor; it is assumed that can reasonably be regarded t 2 0,
J. CHAPPELL
AND KEITH
R. GODFREY
in a mixture in which the concentration can be t = 0, the two mixtures are put into the batch this is a rapid procedure, so that the two inputs as impulsive. Thus the model equations are, for
i=$$-K,x+b,A(t), s
j=
-
pL,sx + bpqf),
Y(K,+s)
that is, the system can be considered as having zero initial conditions with impulsive concentration inputs of size b, and b,. In principle, the preliminary measurements (i.e., before the substances are placed in the reactor) allow b, and b, to be determined, but here it will be assumed, at least initially, that they are not known. With modern instrumentation, it should be possible to measure b, in almost all cases, but the measurement of b, depends on the exact nature of the growth-limiting substrate, and b, will be difficult to measure accurately in many cases. Holmberg [5] used the Taylor series approach [8] to analyze the structural identifiability of the parameters of this model on the assumption that both concentration of microorganisms x(t) and substrate concentration s(t) could be measured within the reactor (i.e., for t > 0). Both b, and b, are then globally identifiable from the values of x(t) and s(t) at t = O+, and by examining the expressions for the first and second derivatives of x(t) and s(t) at t = 0+ it is readily shown that the four parameters CL,,,, K,, K,, and Y are globally identifiable from the experiment. It is usually easy to measure x(t) within the reactor by means of, for example, optical density (absorbence) methods, but it is often difficult to obtain a measurement of s(t) within the reactor (i.e., for t > 0). In this paper, therefore, we consider identifiability of the parameters of the model of Equations (2a) and (2b) with measurement of x(t) only, that is, with observation y(t) = x(t) for t > 0. Thus the unknown parameter set is where x,, ( = bi) and sO ( = b2) are the initial {x,,,+,,P,,K,, Y,K,), concentrations of microorganisms and substrate, respectively. The Taylor series identifiability approach has been applied to this model and experiment [2]. Clearly y(t) at t = O+ is equal to x0, so that x,, ( = b,) is globally identifiable, reducing the number of unknowns to five. Thus at least five derivatives of y(t) are required to resolve the question of identifiability using this approach. More may be required because, as far as we are aware, there is no known upper bound to the number of derivatives required to obtain the maximum number of independent equations from this approach for the type of nonlinear system under consideration.
NONLINEAR
BATCH REACTOR
MODEL
243
Even with the help of a modem symbolic manipulation package (MATHEMATICA), the equations in the derivatives of y(t) at t = O+ fail to yield conclusive identifiability results. The situation is still not resolved if the additional assumption is made that se ( = b2) is known. The source of the problem is that three of the four remaining unknown parameters (pm, K,, and Kd) enter into the first derivative of the observation, while all four appear in the second and higher derivatives and in such a way that makes it extremely difficult to make progress toward a solution. In this paper, therefore, an alternative approach to structural identifiability analysis of nonlinear systems will be applied. This is the similarity transformation approach [3,10,11], which, as far as we are aware, is the only other approach to global identifiability analysis for systems with specified inputs. The basis of the approach has been described in previous papers in this journal [3,11], but it will be summarized briefly in the next section before proceeding with the particular application. 2.
SIMILARITY TRANSFORMATION STRUCTURAL IDENTIFIABILITY
The approach considers structural nonlinear models of the form
APPROACH ANALYSIS identifiability
TO of the parameters
of
~(t,P)=f(x(f,P),P)+u(t)g(x(t,P),P), J+,P) = +(t,P)J+ x(0, P) = X0(P) 7
(3)
where tE[O,T], x(t, ~)ER”, y(t, p@Rm, u(t)EU[O, T], and peQ c Rq, Q being the feasible parameter space. The approach is an extension of the exhaustive modeling approach for linear systems [12, 131, with the extension making use of the local state isomorphism theorem [4, 6, lo]. Before proceeding with the identifiability analysis using this approach we need to establish whether the system is controllable and observable (i.e., minimal) in the sense described by Hermann and Krener [4]. In several cases of practical interest, the minimality conditions for nonlinear systems can be established in the same manner as those for linear systems. This is a consequence of Lemma 1 in [ 111, which states that if at a particular p* in Q, the system described in Equations (3) becomes linear, and if this linear system is controllable and observable, then so is the corresponding nonlinear system (at x0 = 0), for almost all p in B (i.e., except on a set of measure zero in Q). However, if this result does not apply, then the controllability and observability rank criteria described below must be satisfied.
244
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
CONTROLLABILITY
Consider,
RANK CRITERION
(CRC)
for vector fields (a’ and (o*, the Lie bracket [(o’ , (p*] defined
by
where 8 cp’/ a x denotes the Jacobian matrix of (oi, i = 1,2. For a piecewise constant control ui, the vector field is (oi is defined by pi=f(x)+Uig(x),
i=1,2,3
.
,...
(5)
Consider then the Lie algebra f , which has elements that can be represented by finite linear combinations of elements of the form
Note that g is in f . Let f(x) denote the space of vectors spanned by the vector fields of f at x. The system of Equations (3) is said to satisfy the controllability rank criterion (CRC) at x0 if the dimension of f (x,) is n [4]. OBSERVABILITY
RANK CRITERION
(ORC)
Consider a system with a single output; the output function h(x) in Equations (3) is assumed to be continuously differentiable in x. The Lie derivative of h along the vector field cpi is defined by
&i(h)(X) where dh(x)
= dh(x).cp’(x),
denotes the gradient vector field
ah(x)
dh(x) = [
Consider the space of combinations of elements
a X”
I
l-forms dg of the form
d(L,i(L,i-I(...
ah(x)
-J-y,“‘,--
whose
&(h(x)))...
I .
elements
)))
are
finite
linear
(9)
Note that dh is in dg . Let da(x) denote the space of vectors obtained by evaluating the elements of dg at x. The system of Equations (3) is said to satisfy the observability rank criterion (ORC) at x0 if the dimension of dg ( x0) is n [4]. The concept extends readily to multi-output systems.
NONLINEAR
BATCH REACTOR
MODEL
Finally, the system of Equations (3) is said to be locally (minimal) at x0 if it satisfies both the CRC and the ORC at x0. STRUCTURAL
245
reduced
IDENTIFIABILITY
The identifiability of the system of Equations (3) is first studied at some parameter value p in Q in the experiments specified by (x,,, U[O, T]), where the initial state x,, is well defined after selecting p and is generally parameter-dependent; note that, for the moment, u( t)~ U[O, T], the set of bounded and measurable functions defined on the time interval [0, T]. The notation p - p denotes that the parameter values p, p in Q are indistinguishable in the experiments (x0, U[O, T]). The system is globally identifiable at p E Q if d - p implies that j = p, and it is locally identifiable if there exists an open neighborhood W of p in s2 such that d - p for p in W implies that d = p. The method for testing for the identifiability of the unknown parameters is given in Theorem 1, the proof of which is given in [ 111. THEOREM
I
Assume that the system of Equations (3) is locally reduced at x,,(p) for almost all p in Q. Consider the parameter values of p, p in $2, an open neighborhood V of x0(p) in R”, and an analytical mapping X: V -+ R” defined on V G R” such that
ax(x)
(i) Rank 7
for all 2’EV
= n
(ii) Wxo(P)) = x0(P)
(iii) f(x(% g(h(Z),
(10) (11)
ax(n) P)=
an
UW
f(n,P)
p)= %$Qg(R,
h(X(% PI= 42, i%
P)
(12b) WC)
for all 2 in V. Then there exists T > 0 such that the system of Equations (3) is globally identifiable at p in the experiments (x,(p), U(0, TJ) if and only if conditions (i), (ii), and (iii) imply p = p. For a globally identifiable model, the only value fi in Q satisfying Equations (lo)-(12) is j5 = p, for which A(Z) = 2, that is, the identity mapping on R”. If the model is not globally identifiable to every solution @ # p of Equations (lo)-(12) there corresponds a local isomorphism X(Z) # 2. If the system is not locally identifiable at p, then every open neighborhood of p contains a point j such that j5 - p; thus the number of solutions is infinite. From Corollary 1 of [ 111, the above result can be extended to systems
246
MICHAEL
with specified input if all the following (a) vector (b) (c) is, the 3.
J. CHAPPELL
conditions
AND KEITH
R. GODFREY
hold:
g(x, p) = b(p); that is, the input is multiplied by a constant b. f(0, p) = 0; that is, with no input, the system stays at rest. X( 2) = TZ, where T is an n x n constant nonsingular matrix; that equivalence transformations are linear and time-invariant.
APPLICATION MODEL
OF THE APPROACH
TO THE BATCH REACTOR
It is noted that the system of Equations (2a) and (2b) is of the form of Equations (3) with g = b = (b,, b,)* and u(t) = s(t). In order to apply Theorem 1, it is necessary to show first that the system of Equations (2a) and (2b) with observation y = x is locally reduced. The system becomes linear if pm = 0, but this linear system is not observable, so it is necessary to check that the CRC and ORC are satisfied. To check the CRC, initially we have the vector b = (b,, b,)T# (O,O)T. Then consider the vector field cp’ = f and the Lie bracket PCLmS --K, K,+s
[f,b]= -
which, when evaluated
LLms Y(K,+s)
-
at (0,O) yields
[fJ](O)=(&h&
(14
Since K,b, # 0 generically, this implies that b and [f,b](O) are linearly independent so that dim f(0) = 2 and the CRC is satisfied. To check the ORC, initially we have the vector h = [ 1, 01. Then consider the Lie derivative L#z)=(l
O).f=+&K,x, s
(15)
so that
which when evaluated at (0,O) yields d(L,(h))(O) = (- K,,O), which is not linearly independent to (l,O). Denoting F = d(LJh)), we then
NONLINEAR
BATCH REACTOR
247
MODEL
consider the Lie derivative L&2)=(1
O).F=~-K,,
(17)
s
so that
which when evaluated at (0,O) yields d( LF( h))(O) = (0, pm /KS). Since pm /K, # 0 generically, this is linearly independent to (l,O), so that dim dg(0) = 2 and the ORC is satisfied. The system is thus locally reduced at (O,O), and we can proceed with the identifiability analysis using Theorem 1. IDENTIFIABILITY
ANALYSIS
As with the Taylor series analysis, we start by assuming that the inputs b, and b, are unknown. Thus the unknown parameter vector is p = (b,, b,, pm, K,, Y, KJT. As before, y = x for t > 0. The state isomorphism being sought is denoted by h= (X,, &)r with arguments (2, QT. Initially, it is assumed that u( t)~U[0, T]. From Equation (12c), we have, for the system of Equations (2a) and (2b) 3
[l
o]
[;I=11
0]
[;I,
which implies that A,=.%,
(20)
so that
ah,
an=’ From Equation
and
ah
x=0.
(21)
(12b), we have
which implies that b, = &,
(23)
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
248
and
Using the fact that A, = 2, the first equation
of (12a) gives
(25) This equation is true for all 2, so that 5? cancels out; hence a& /a 5? = 0, giving, from Equation (24),
ax,
b,
==z, Substituting
in the second equation
From Equations
Multiplying
(26)
of (12a),
(25) and (27), with 2 cancelled
out,
through by Z?S + S,
~(~)p,s-K,(~,+s)=~,S-R,(R,+s).
(29)
2
By comparing
coefficients
and since, generically,
we have from the constant term
kS is nonzero.
K, = Ed.
(30)
From the term in S, b,/Y=&,/i?
(31)
NONLINEAR BATCH REACTOR MODEL
From Equations
249
(27) and (31),
so that
(32) But aX2 /a$=
b2 /“b2, so that, equating terms in S-*, Pm = Pm.
There is then no term in s”; equating
(33)
constant terms,
(34) Thus, from this experiment, the individual parameters b,, pm, and K, are globally identifiable and so are the ratios b, / Y and b, /KS, but the individual values of b, , K,, and Y are unidentifiable. If the additional assumption is made that b, is known, then K, and Y are also globally identifiable. The transformations h in both cases are linear, so that condition (c) of Section 2 as well as conditions (a) and (b) apply for these models and experiments. Hence the results also apply to the specified form of input, that is, with impulsive input of microorganism and substrate. 4.
CONCLUSIONS
The case with b, unknown is of particular interest because it is the first example of a real-life nonlinear model that turns out not to be globally identifiable. The source of the unidentifiability is that 2 cancels out in Equations (25) and (27), thus reducing the number of coefficients that can be equated. The first generation of unidentifiable nonlinear models consisted of examples that were trivial mathematically, for example, with parametric parts that were not connected to the outputs. The second generation, which included the bilinear models of Examples 6 and 8 of [7] and the polynomial model of Example 1 of [l l] (considered earlier in [9]), was no longer trivial, but the examples were still purely artificial exercises. The example in the present paper could be regarded as the start of the third generation of unidentifiable nonlinear models, because the model is widely used in practice and was considered earlier in identifiability studies [5].
250
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
The experiment with s(t) unknown has been considered precisely because in many cases it is difficult to obtain a measurement of substrate concentration within a reactor. In some cases (depending on the nature of the substrate), it is possible to determine b, from a preliminary measurement (i.e., before the substances are placed in the reactor), and if b, is known, the parameters pm, K,, K,, and Y are globally identifiable. If only x is observed (but with b, = so known), Taylor series identifiability analysis proves very complicated [2], and it does not appear to be possible to achieve a conclusive result even with the help of a symbolic manipulation package. In contrast, the similarity transformation analysis has proved to be straightforward, once the controllability and observability rank criteria have been checked. This is by no means always the case, and in general it is far from easy to predict which approach will prove the easier for a particular example [3].
The work described in this paper is financed by grant GR /F 6381.7 from the U.K. Science and Engineering Research Council. We are grateful to Dr. David Hodgson of the Department of Biological Sciences at the University of Warwick for advice on the problems of measuring microbial growth in batch reactors.
REFERENCES 1
D. K. Button, Kinetics of nutrient-limited transport and microbial growth, Microbio. Rev. 49:210-291 (1985). 2 M. J. Chappell and K. R. Godfrey, Structural identifiability of nonlinear systems: application to a batch reactor model, 9th IFACIIFORS Symposium on Zdentifcation and System Parameter Estimation, Budapest, 8-12 July 1991. 3 M. J. Chappell, K. R. Godfrey, and S. Vajda, Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods, Math. Biosci. 102:41-73 (1990). 4 R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Autom. Control AC-22:728-X0 (1977). 5 A. Holmberg, On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities, Math. Biosci. 62:23-43 (1982). 6 A. Isidori, Nonlinear Control Systems: An Introduction, Springer-Verlag, New York, 1985. 7 Y. Lecourtier, F. Lamnabhi-Lagarrigue, and E. Walter, Volterra and generating power series approaches to identifiability testing, in Identifiability of Parametric Models, E. Walter, Ed., Pergamon, Oxford, 1987, Chapter 5, pp. 50-66. 8 H. Pohjanpalo, System identifiability based on the power series expansion of the solution, Math Biosci. 41:21-33 (1978).
NONLINEAR BATCH REACTOR MODEL 9
10 11 12 13
251
S. Vajda, Identifiability of polynomial systems: structural and numerical aspects, in Identifiability of Parametric Models, E. Walter, Ed., Pergamon, Oxford, 1987, Chapter 4, pp. 42-49. S. Vajda and H. Rabitz, State isomorphism approach to global identifiability of nonlinear systems, IEEE Trans. Autom. Control AC-34:220-223 (1989). S. Vajda, K. R. Godfrey, and H. Rabitz, Similarity transformation approach to structural identifiability of nonlinear models, Math. Biosci. 93:217-248 (1989). E. Walter, Identifiability of State Space Models (Lect. Notes Biomath. Vol. 46), Springer-Verlag, New York, 1982. E. Walter and Y. Lecourtier, Unidentifiable compartmental models: what to do? Math. Biosci. 56:1-25 (1981).
of the Parameters of a Nonlinear
MICHAEL J. CHAPPELL AND KEITH R. GODFREY Department of Engineering, University of Warwick, Coventry,
CV4 7AL, England
Received 20 March 1991; revised 17 August 1991
ABSTRACT The similarity transformation approach is used to analyze the structural identifiability of the parameters of a nonlinear model of microbial growth in a batch reactor in which only the concentration of microorganisms is measured. It is found that some of the model parameters are unidentifiable from this experiment, thus providing the first example of a real-life nonlinear model that turns out not to be globally identifiable. If it is possible to measure the initial concentration of growth-limiting substrate as well, all model parameters are globally identifiable.
1.
INTRODUCTION
In this paper, the structural identifiability of the parameters of a model of microbial growth in a batch reactor is considered. The model incorporates Michaelis-Menten-type nonlinearities and has previously been considered in this journal by Holmberg [5]. A comprehensive review of the kinetics of microbial growth processes is given by Button [ 11. Microbial growth in a batch reactor can be expressed as
(14 Pm=
‘= - Y(K,+s)
’
P-4
where x is the concentration of microorganisms; S, the concentration of growth-limiting substrate; pm, the maximum velocity of the reaction; K,, the Michaelis-Menten constant; Y, the yield coefficient; and K,, the decay rate coefficient. The experiment considered is one where the microorganisms are prepared in a mixture in which the concentration can be controlled, and the
MATHEMATICAL
BIOSCIENCES
108:241-251 (1992)
OElsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
241 0025-5564/92/$05.00
242
MICHAEL
substrate is also prepared controlled. Then, at time reactor; it is assumed that can reasonably be regarded t 2 0,
J. CHAPPELL
AND KEITH
R. GODFREY
in a mixture in which the concentration can be t = 0, the two mixtures are put into the batch this is a rapid procedure, so that the two inputs as impulsive. Thus the model equations are, for
i=$$-K,x+b,A(t), s
j=
-
pL,sx + bpqf),
Y(K,+s)
that is, the system can be considered as having zero initial conditions with impulsive concentration inputs of size b, and b,. In principle, the preliminary measurements (i.e., before the substances are placed in the reactor) allow b, and b, to be determined, but here it will be assumed, at least initially, that they are not known. With modern instrumentation, it should be possible to measure b, in almost all cases, but the measurement of b, depends on the exact nature of the growth-limiting substrate, and b, will be difficult to measure accurately in many cases. Holmberg [5] used the Taylor series approach [8] to analyze the structural identifiability of the parameters of this model on the assumption that both concentration of microorganisms x(t) and substrate concentration s(t) could be measured within the reactor (i.e., for t > 0). Both b, and b, are then globally identifiable from the values of x(t) and s(t) at t = O+, and by examining the expressions for the first and second derivatives of x(t) and s(t) at t = 0+ it is readily shown that the four parameters CL,,,, K,, K,, and Y are globally identifiable from the experiment. It is usually easy to measure x(t) within the reactor by means of, for example, optical density (absorbence) methods, but it is often difficult to obtain a measurement of s(t) within the reactor (i.e., for t > 0). In this paper, therefore, we consider identifiability of the parameters of the model of Equations (2a) and (2b) with measurement of x(t) only, that is, with observation y(t) = x(t) for t > 0. Thus the unknown parameter set is where x,, ( = bi) and sO ( = b2) are the initial {x,,,+,,P,,K,, Y,K,), concentrations of microorganisms and substrate, respectively. The Taylor series identifiability approach has been applied to this model and experiment [2]. Clearly y(t) at t = O+ is equal to x0, so that x,, ( = b,) is globally identifiable, reducing the number of unknowns to five. Thus at least five derivatives of y(t) are required to resolve the question of identifiability using this approach. More may be required because, as far as we are aware, there is no known upper bound to the number of derivatives required to obtain the maximum number of independent equations from this approach for the type of nonlinear system under consideration.
NONLINEAR
BATCH REACTOR
MODEL
243
Even with the help of a modem symbolic manipulation package (MATHEMATICA), the equations in the derivatives of y(t) at t = O+ fail to yield conclusive identifiability results. The situation is still not resolved if the additional assumption is made that se ( = b2) is known. The source of the problem is that three of the four remaining unknown parameters (pm, K,, and Kd) enter into the first derivative of the observation, while all four appear in the second and higher derivatives and in such a way that makes it extremely difficult to make progress toward a solution. In this paper, therefore, an alternative approach to structural identifiability analysis of nonlinear systems will be applied. This is the similarity transformation approach [3,10,11], which, as far as we are aware, is the only other approach to global identifiability analysis for systems with specified inputs. The basis of the approach has been described in previous papers in this journal [3,11], but it will be summarized briefly in the next section before proceeding with the particular application. 2.
SIMILARITY TRANSFORMATION STRUCTURAL IDENTIFIABILITY
The approach considers structural nonlinear models of the form
APPROACH ANALYSIS identifiability
TO of the parameters
of
~(t,P)=f(x(f,P),P)+u(t)g(x(t,P),P), J+,P) = +(t,P)J+ x(0, P) = X0(P) 7
(3)
where tE[O,T], x(t, ~)ER”, y(t, p@Rm, u(t)EU[O, T], and peQ c Rq, Q being the feasible parameter space. The approach is an extension of the exhaustive modeling approach for linear systems [12, 131, with the extension making use of the local state isomorphism theorem [4, 6, lo]. Before proceeding with the identifiability analysis using this approach we need to establish whether the system is controllable and observable (i.e., minimal) in the sense described by Hermann and Krener [4]. In several cases of practical interest, the minimality conditions for nonlinear systems can be established in the same manner as those for linear systems. This is a consequence of Lemma 1 in [ 111, which states that if at a particular p* in Q, the system described in Equations (3) becomes linear, and if this linear system is controllable and observable, then so is the corresponding nonlinear system (at x0 = 0), for almost all p in B (i.e., except on a set of measure zero in Q). However, if this result does not apply, then the controllability and observability rank criteria described below must be satisfied.
244
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
CONTROLLABILITY
Consider,
RANK CRITERION
(CRC)
for vector fields (a’ and (o*, the Lie bracket [(o’ , (p*] defined
by
where 8 cp’/ a x denotes the Jacobian matrix of (oi, i = 1,2. For a piecewise constant control ui, the vector field is (oi is defined by pi=f(x)+Uig(x),
i=1,2,3
.
,...
(5)
Consider then the Lie algebra f , which has elements that can be represented by finite linear combinations of elements of the form
Note that g is in f . Let f(x) denote the space of vectors spanned by the vector fields of f at x. The system of Equations (3) is said to satisfy the controllability rank criterion (CRC) at x0 if the dimension of f (x,) is n [4]. OBSERVABILITY
RANK CRITERION
(ORC)
Consider a system with a single output; the output function h(x) in Equations (3) is assumed to be continuously differentiable in x. The Lie derivative of h along the vector field cpi is defined by
&i(h)(X) where dh(x)
= dh(x).cp’(x),
denotes the gradient vector field
ah(x)
dh(x) = [
Consider the space of combinations of elements
a X”
I
l-forms dg of the form
d(L,i(L,i-I(...
ah(x)
-J-y,“‘,--
whose
&(h(x)))...
I .
elements
)))
are
finite
linear
(9)
Note that dh is in dg . Let da(x) denote the space of vectors obtained by evaluating the elements of dg at x. The system of Equations (3) is said to satisfy the observability rank criterion (ORC) at x0 if the dimension of dg ( x0) is n [4]. The concept extends readily to multi-output systems.
NONLINEAR
BATCH REACTOR
MODEL
Finally, the system of Equations (3) is said to be locally (minimal) at x0 if it satisfies both the CRC and the ORC at x0. STRUCTURAL
245
reduced
IDENTIFIABILITY
The identifiability of the system of Equations (3) is first studied at some parameter value p in Q in the experiments specified by (x,,, U[O, T]), where the initial state x,, is well defined after selecting p and is generally parameter-dependent; note that, for the moment, u( t)~ U[O, T], the set of bounded and measurable functions defined on the time interval [0, T]. The notation p - p denotes that the parameter values p, p in Q are indistinguishable in the experiments (x0, U[O, T]). The system is globally identifiable at p E Q if d - p implies that j = p, and it is locally identifiable if there exists an open neighborhood W of p in s2 such that d - p for p in W implies that d = p. The method for testing for the identifiability of the unknown parameters is given in Theorem 1, the proof of which is given in [ 111. THEOREM
I
Assume that the system of Equations (3) is locally reduced at x,,(p) for almost all p in Q. Consider the parameter values of p, p in $2, an open neighborhood V of x0(p) in R”, and an analytical mapping X: V -+ R” defined on V G R” such that
ax(x)
(i) Rank 7
for all 2’EV
= n
(ii) Wxo(P)) = x0(P)
(iii) f(x(% g(h(Z),
(10) (11)
ax(n) P)=
an
UW
f(n,P)
p)= %$Qg(R,
h(X(% PI= 42, i%
P)
(12b) WC)
for all 2 in V. Then there exists T > 0 such that the system of Equations (3) is globally identifiable at p in the experiments (x,(p), U(0, TJ) if and only if conditions (i), (ii), and (iii) imply p = p. For a globally identifiable model, the only value fi in Q satisfying Equations (lo)-(12) is j5 = p, for which A(Z) = 2, that is, the identity mapping on R”. If the model is not globally identifiable to every solution @ # p of Equations (lo)-(12) there corresponds a local isomorphism X(Z) # 2. If the system is not locally identifiable at p, then every open neighborhood of p contains a point j such that j5 - p; thus the number of solutions is infinite. From Corollary 1 of [ 111, the above result can be extended to systems
246
MICHAEL
with specified input if all the following (a) vector (b) (c) is, the 3.
J. CHAPPELL
conditions
AND KEITH
R. GODFREY
hold:
g(x, p) = b(p); that is, the input is multiplied by a constant b. f(0, p) = 0; that is, with no input, the system stays at rest. X( 2) = TZ, where T is an n x n constant nonsingular matrix; that equivalence transformations are linear and time-invariant.
APPLICATION MODEL
OF THE APPROACH
TO THE BATCH REACTOR
It is noted that the system of Equations (2a) and (2b) is of the form of Equations (3) with g = b = (b,, b,)* and u(t) = s(t). In order to apply Theorem 1, it is necessary to show first that the system of Equations (2a) and (2b) with observation y = x is locally reduced. The system becomes linear if pm = 0, but this linear system is not observable, so it is necessary to check that the CRC and ORC are satisfied. To check the CRC, initially we have the vector b = (b,, b,)T# (O,O)T. Then consider the vector field cp’ = f and the Lie bracket PCLmS --K, K,+s
[f,b]= -
which, when evaluated
LLms Y(K,+s)
-
at (0,O) yields
[fJ](O)=(&h&
(14
Since K,b, # 0 generically, this implies that b and [f,b](O) are linearly independent so that dim f(0) = 2 and the CRC is satisfied. To check the ORC, initially we have the vector h = [ 1, 01. Then consider the Lie derivative L#z)=(l
O).f=+&K,x, s
(15)
so that
which when evaluated at (0,O) yields d(L,(h))(O) = (- K,,O), which is not linearly independent to (l,O). Denoting F = d(LJh)), we then
NONLINEAR
BATCH REACTOR
247
MODEL
consider the Lie derivative L&2)=(1
O).F=~-K,,
(17)
s
so that
which when evaluated at (0,O) yields d( LF( h))(O) = (0, pm /KS). Since pm /K, # 0 generically, this is linearly independent to (l,O), so that dim dg(0) = 2 and the ORC is satisfied. The system is thus locally reduced at (O,O), and we can proceed with the identifiability analysis using Theorem 1. IDENTIFIABILITY
ANALYSIS
As with the Taylor series analysis, we start by assuming that the inputs b, and b, are unknown. Thus the unknown parameter vector is p = (b,, b,, pm, K,, Y, KJT. As before, y = x for t > 0. The state isomorphism being sought is denoted by h= (X,, &)r with arguments (2, QT. Initially, it is assumed that u( t)~U[0, T]. From Equation (12c), we have, for the system of Equations (2a) and (2b) 3
[l
o]
[;I=11
0]
[;I,
which implies that A,=.%,
(20)
so that
ah,
an=’ From Equation
and
ah
x=0.
(21)
(12b), we have
which implies that b, = &,
(23)
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
248
and
Using the fact that A, = 2, the first equation
of (12a) gives
(25) This equation is true for all 2, so that 5? cancels out; hence a& /a 5? = 0, giving, from Equation (24),
ax,
b,
==z, Substituting
in the second equation
From Equations
Multiplying
(26)
of (12a),
(25) and (27), with 2 cancelled
out,
through by Z?S + S,
~(~)p,s-K,(~,+s)=~,S-R,(R,+s).
(29)
2
By comparing
coefficients
and since, generically,
we have from the constant term
kS is nonzero.
K, = Ed.
(30)
From the term in S, b,/Y=&,/i?
(31)
NONLINEAR BATCH REACTOR MODEL
From Equations
249
(27) and (31),
so that
(32) But aX2 /a$=
b2 /“b2, so that, equating terms in S-*, Pm = Pm.
There is then no term in s”; equating
(33)
constant terms,
(34) Thus, from this experiment, the individual parameters b,, pm, and K, are globally identifiable and so are the ratios b, / Y and b, /KS, but the individual values of b, , K,, and Y are unidentifiable. If the additional assumption is made that b, is known, then K, and Y are also globally identifiable. The transformations h in both cases are linear, so that condition (c) of Section 2 as well as conditions (a) and (b) apply for these models and experiments. Hence the results also apply to the specified form of input, that is, with impulsive input of microorganism and substrate. 4.
CONCLUSIONS
The case with b, unknown is of particular interest because it is the first example of a real-life nonlinear model that turns out not to be globally identifiable. The source of the unidentifiability is that 2 cancels out in Equations (25) and (27), thus reducing the number of coefficients that can be equated. The first generation of unidentifiable nonlinear models consisted of examples that were trivial mathematically, for example, with parametric parts that were not connected to the outputs. The second generation, which included the bilinear models of Examples 6 and 8 of [7] and the polynomial model of Example 1 of [l l] (considered earlier in [9]), was no longer trivial, but the examples were still purely artificial exercises. The example in the present paper could be regarded as the start of the third generation of unidentifiable nonlinear models, because the model is widely used in practice and was considered earlier in identifiability studies [5].
250
MICHAEL J. CHAPPELL AND KEITH R. GODFREY
The experiment with s(t) unknown has been considered precisely because in many cases it is difficult to obtain a measurement of substrate concentration within a reactor. In some cases (depending on the nature of the substrate), it is possible to determine b, from a preliminary measurement (i.e., before the substances are placed in the reactor), and if b, is known, the parameters pm, K,, K,, and Y are globally identifiable. If only x is observed (but with b, = so known), Taylor series identifiability analysis proves very complicated [2], and it does not appear to be possible to achieve a conclusive result even with the help of a symbolic manipulation package. In contrast, the similarity transformation analysis has proved to be straightforward, once the controllability and observability rank criteria have been checked. This is by no means always the case, and in general it is far from easy to predict which approach will prove the easier for a particular example [3].
The work described in this paper is financed by grant GR /F 6381.7 from the U.K. Science and Engineering Research Council. We are grateful to Dr. David Hodgson of the Department of Biological Sciences at the University of Warwick for advice on the problems of measuring microbial growth in batch reactors.
REFERENCES 1
D. K. Button, Kinetics of nutrient-limited transport and microbial growth, Microbio. Rev. 49:210-291 (1985). 2 M. J. Chappell and K. R. Godfrey, Structural identifiability of nonlinear systems: application to a batch reactor model, 9th IFACIIFORS Symposium on Zdentifcation and System Parameter Estimation, Budapest, 8-12 July 1991. 3 M. J. Chappell, K. R. Godfrey, and S. Vajda, Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods, Math. Biosci. 102:41-73 (1990). 4 R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Autom. Control AC-22:728-X0 (1977). 5 A. Holmberg, On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities, Math. Biosci. 62:23-43 (1982). 6 A. Isidori, Nonlinear Control Systems: An Introduction, Springer-Verlag, New York, 1985. 7 Y. Lecourtier, F. Lamnabhi-Lagarrigue, and E. Walter, Volterra and generating power series approaches to identifiability testing, in Identifiability of Parametric Models, E. Walter, Ed., Pergamon, Oxford, 1987, Chapter 5, pp. 50-66. 8 H. Pohjanpalo, System identifiability based on the power series expansion of the solution, Math Biosci. 41:21-33 (1978).
NONLINEAR BATCH REACTOR MODEL 9
10 11 12 13
251
S. Vajda, Identifiability of polynomial systems: structural and numerical aspects, in Identifiability of Parametric Models, E. Walter, Ed., Pergamon, Oxford, 1987, Chapter 4, pp. 42-49. S. Vajda and H. Rabitz, State isomorphism approach to global identifiability of nonlinear systems, IEEE Trans. Autom. Control AC-34:220-223 (1989). S. Vajda, K. R. Godfrey, and H. Rabitz, Similarity transformation approach to structural identifiability of nonlinear models, Math. Biosci. 93:217-248 (1989). E. Walter, Identifiability of State Space Models (Lect. Notes Biomath. Vol. 46), Springer-Verlag, New York, 1982. E. Walter and Y. Lecourtier, Unidentifiable compartmental models: what to do? Math. Biosci. 56:1-25 (1981).