Structural Identifiability of Nonlinear Systems: Application to a Batch Reactor

Copyright © IFAC Identification and System Parameter Estimation. Budapest. Hungary 1991

STRUCTURAL IDENTIFIABILITY OF NONLINEAR SYSTEMS: APPLICATION TO A BATCH REACTOR M. J. Chappcll and K. R. Godfrcy Department of Engineering. University of Warwick. Coventry CV4 7AL. UK

~.

Only two known methods are currently available for analysing the structural identifiability of the parameters of a nonlinear system with specified input(s). These are the Taylor series and Similarity Transfonnation approaches. Application of either method can prove tedious. but the analysis can often be eased considerably by using a symbolic The two methods are illustrated in this paper by analysing the manipulation package. structural identifiability of the parameters of a nonlinear model of microbial growth in a batch reactor. The use of the symbolic manipulation package MATHEMATICA in the identifiability analysis is discussed. Keywords.

Biotechnology; Identifiability; Impulse response; Modelling; System identification.

INTRODUCTION There are several well-known methods available for analysing the structural identifiability of the parameters of linear. time invariant systems expressed in state space fonn (Godfrey and DiStefano. 1987). Structural identifiability is conccrned with whether the parameters could be identified from a particular It is input-output experiment if perfect data were available. also referred to as detenninistic identifiability and a priori identifiability and it should be emphasised that the problem is different from that of parameter estimation accuracy which is associated with the problems imposed by the limitations of the actual measurements.

TA YLOR SERIES IDENTIFIABILITY ANALYSIS Theory of the Method The basis of the appro~ch. (pohjanpalo. 1978) is that y(t.p) and Its successive time denvatlves are evaluated in tenns of the model parameters and initial conditions at a panicular time. usually t = 0+ :-

y( t .p)

ti

Here . we will consider structural identifiability of the parameters of nonlinear models of the fonn:

x(

t • p)

y( t .p)

y(o+.p) + y(1 )(O+.p) . t + + . .. + y(i)(O+.p)

(2)

i!+

f(x(t .p) .p) + u(t) g(x(t .p) .p) h(x(t.p).p)

(I)

diy

where y ( i ) (0+ • p) '" x(O.p)

Y(')(o+.p) . ~;

dtT

(0+. p) •

i = I. 2. 3 • . .. (3)

xo (p)

where t E [O.T]. x(t.p) ERn. y(t.p) E Rm and pEn C Rq (n being the parameter space). and restrict consideration to the Only two case where u(t) is a specified function of time . methods are currently available for the identifiability analysis of nonlinear systems with specified inputs. These are the Taylor series approach (pohjanpalo. 1978. 1982) and the Similarity Transfonnation approach (Vajda. Godfrey and Rabitz. 1989; Vajda and Rabitz. 1989; Chappell. Godfrey and Vajda. 1990). In general. it is difficult to predict which approach will involve the least effort for a particular problem (Chappell. Godfrey and Vajda. 1990) and it is important to have both approaches to hand. The application of either method for the global identifiability analysis of the parameters of nonlinear models with specified input can prove very tedious computationally and the use of symbolic manipulation packages can ease the algebra involved and so make the methods more accessible to the user. The use of the package SMP in this type of analysis has been described by the authors (Chappell and Godfrey. 1989. 1990; Chappell. Godfrey and Vajda. 1990). but in this paper. we will illustrate the use of the increaSingly widely used package MATHEMATICA (Wolfram. 1988). The model considered here is of microbial growth in a batch reactor. but before discussing the model in detail. the two identifiability methods. and the role of symbolic manipulation packages in applying the methods. will be considered.

Since the coefficients in the Taylor series expansion are unique. the problem reduces to one of detennining the number of solutions for the parameters in a set of algebraic equations which are. in general. nonlinear in the parameters. The parameter set is unidentifiable if the set of solutions is uncountable. is locally identifiable if the set of solutions is countable and is globally (uniquely) identifiable if there is a unique solution. When applying this approach to nonlinear systems. a problem anse.s m that. an upper bound on the number of equations reqUired for thiS set can often be difficult to detennine. For linear systems. it is known that there are at most 2n - I independent equations in this set (where. as noted above. n is the number of states) (Vajda. 1982). while for bilinear systems. the maximum number of independent equations is 22n - I For nonlinear systems of homogeneous (Vajda•. 1982). polynomial fonn. the corresponding maximum number is (q2n_I)!(q_I). where q is the degree of the polynomial (Vajda. 1987). It is clear that. as n increases. the number of equations reqUired for the latter two cases increases rapidly. For a general fonn of nonlinear system. no upper bound for the number of independent equations is yet known. as far as the authors are aware.

865

For a general main stagcs:-

nonlinear system .

the

approach

involves

I.

Successive differenti ation of y(t.p).

2.

Evaluation o f y(i)(o+ .p) by substitution o f quantities already known from y(O+.p) and lower deriv atives yCk)(O+.p). k .( i.

3.

Controllabilit y Rank Criterion (CRC)

three

Conside r. for vector fields
and
d
d
1<1" .
-d-X-
(x)


dx

(4)

where i1
A check on the independence of U1C equations in the successive derivatives and on what parameters. if any. can be identified from each coefficient.

er

,p i

It has The procedure is terminated at some appropriate stage. provided identifi ability results in many examplcs of practical interest (D rown . 1979: Chappell ,U1d Godfrcy. 1989; Chap[1cll . Godfrey and Vajda. 1990; Godfrey. 1983; Godfrey and DiStefano. 1987: Godfrey and Fitch. 1984; Holmherg. 19X2; Pohjanpalo. 1978. 1982; Waiter. 1982).

=

f (x) + uig (x) .

i = I. 2. 3 •.

(5)

Consider then the Lie Algebra f which has elements which can be reprcsented by finite linear combinations of elements of the form:

1
(6)

Note that g is in f . Let f (x) denote the space of vectors spanncd by the vector field s of f at x. The system of equations (I ) is said to satisfy the Controllability Rank Criterion (CRC) at x, if the dimension of (x,) is n (Hermann and Krcner. 1977).

Role of Sv mbolic Manipulation Packages The package which we currently use when performing thi s analysis. MATH EMATICA . Version 1.2 . can readily perform Stage I of the procedure described above since it has the capability to differentiate nonspecific functi ons . Not all of the symbolic manipulation packages presentl y available have such a facility. in which case. the differentiation would have to be performed by hand.

f

QbservabililY Rank Criterion CORC)

Stage 2 can in general be readily handled by almost all of the symbolic manipulation packages currently available. However. even when the series of ascending derivatives has been detemlincd. it can prove difficult to establish any identifiability properties of the unknown parameters due to the hi ghly nonlinear structure of the equations obtained. This is a problem that for certain examples cannot yet be solved. even WiUl the sophi sti cated solving algorithms available either sepa rately (Lecou rtie r and Raksan yi. 1987; R aksany i and co-workers. 1(85) or wiUlin cenain symbolic m ~mipulation packages such as the Groebner Bases library routine within REDUCE and MATHEMATICA. The independence check in Stage 3 can also be treatcd by s ymbolic manipulation packages. for example by applying a Jacobian rank test (Pohjanpalo. 1982). SIMILARITY TRANSFORMATION IDENTIFIABILITY ANALYSIS The Similarity Transformation (exhaustive modelling) approach for linear systems (Waiter and Lccounier. 198 1; Walter. 1982) has been extended to nonlinear systems (Vajda. Godfrcy and Rabitz. 1989). with the extension making use of the Local State Isomorphis m Theorem (Hermann and Krener. 1977; Isidori. 1985; Vajda and Rabitz. 1989). Before proceeding wiUl the identifi ability analysis using tlli s approach it is necessary to establish whether the system is controllable and observable (i.e. minimal). In several cases of practical intcrest. the minimality conditions for nonlinear systems c,m be established in the same manner as those for linear systems. Thi s is a consequence of Lemma I of Vajda, GoUfre y and Rabitz (1989) which states that if at a particul ar p* in n. the system described in equations (I) becomes linear. and if thi s linear system is contrOllable and observable. then so is Ihe corresponding nonlinear system (at x, = 0). for almost all p in n (i.e. except on a set of measure zero in n). However. if thi s result does not apply. then the controll ability and observability rank criteria described below necd to be sati sfied.

Consider a system with a single output; the output function hex) in eq uat ions (I) is ass umed to be continuously diff~ rentiable in x. The Lie derivative of h along the vector field is defined by:

er

(7)

where dh(x) denotes the gradient vector fi eld

dh(x) =

[d~~~)

... ....

d~~:) 1

(8)

Cons ider the· space of I -forms d(j- whose elements are iinear combinations of elements of the form :

finite

(9)

Note that dh is in d!f. Let d§-(x) denote the space of vectors obtained by evaluating the elemenL~ of d
P.P

p

The system IS globally Identifiable at p E ~2 if P _ P impli es that = p and It is locally ldentifiable If there exists an open neighbourhood W of p in n such that p for p in W implies Ulat = p.

p

p

p_

The method for testing for the identifiability of the unknown parameters is given in Theorem I below. the proof of which is given in Vajda. Godfrey and Rabitz (1989).

866

Theorem I

The Model and The Experiment

Assume that the system of equations (I) is locally reduced ,~ Xo (p) for almost all p in n. Consider the parameter values p,p in n, an open neighbourhood Y of Xo (p) in Rn and an analytical mapping A. : Y .... Rn defined on Y C; Rn such tJlat:

Microbial growtil in a batch reactor can be expressed as ~(s)

x

(BA)

x - Kd x

1

Y~(s)

aA.(x) (i)

rank

(ii)

A.(xo (p»

(iii)

f(A.(X) ,p)

for all x E V

n

ax

(10)

where

concentration of microorg,misms concentration of growth limiting substrate Y yield coefficient Kd = decay rate coefficient and fl(S) = overall velocity of the reaction, which is assumed to be cl' Michaelis-Menten form, with

(11)

xo(p) aA.eX) ax

(l3B)

x

x s

f(x,p)

(12A)

g(x,p)

(l2B)

~(S)

(12C)

where flm Ks and

dA.(X) g(A.(X) ,p)

ait

h(A.eX) ,p)

h(x,p)

for all x in Y. Then there exists T > 0 such tJlat the system of equations (1) is globally identifiable at p in the experiments (xo (p),U[O,TJ) if and only if conditions (i), (ii) and (iii) imply p = p.

p

(a)

g(x,p)

b(p) ;

(b)

f(O,p)

0;

(c)

to systems

with

~m

s x (15A)

Ks + s

spe~ified

~ s x Y(Ks + s)

i.e. the input is multiplied by a constant veetor b.

+ b, o( t)

(15B)

i.e. tile conditions b , . The (b, ,b,)T

i.e. with no input, the system stays at rest.

Tx, where T is an n x n constant nonsingular matrix; i.e. the equivalence linear and time transformations are invariant.

The Role of Symbolic Manipulation

maximum velocity Michaelis-Menten constant

The experiment which is considered is where the microorganisms are prepared in a mixture in which the concentration can be both measured and controlled, while the substratc is also prepared in a mixture in which the concentration can also be both measured and controlled. Then, at time t = 0, the two mixtures arc put into the batch reactor; it is assumed that this is a rapid procedure, so that the two inputs can reasonably be regarded as impulsive. Thus the model equations are, for t '" 0,

For a globally identifiable model, the only value p _ in n satisfying equations (10) to (12) is p = p, for which A.(x) = X, i.e. the identity mapping on Rn If the model is not globally identifiable, to every solution = p of ~quatio~ls (10) to (12), there corresponds a local isomorphism A.(x) *- x. The system of equations (1) is not locally identifiable at p if the number of solutions to equations (10) to (12) is uncountable. The above result can be extended input if the following hold:-

(14)

Packa~cs

An important role for symbolic manipulation packages is in determirting the necessary controllability and observability conditions for application of Theorem I. As the state space dimension increases, these calculations become tedious and Symbolic manipulation packages can prove very lengthy. helpful since the appropriate Lie brackets and Lie derivatives (or, if the system can be reduced to a linear model, as discussed above, the appropriate controllabi lity and observability matrices) can be generated readily and evaluated at appropriate points. In the identifiability analysis itself (using Theorem 1), symbolic manipulation packages can help with functional differentiation for the lacobian matrix aA.(x)!ax, with substitution of known quantities and with comparison of coefficienLs in polynomial equations; see, for example, the procedure in Examples 4 and 5 of Yajda, Godfrey and Rabitz (1989). STRUCTURAL IDENTIFIABILITY OF THE PARAMETERS OF A MODEL OF MICROBIAL GROWTH

system can be considered as having zero initial with impulsive ~...1!l!Jl.Qn inpUL~ of size h, and system is of the form of equations (1) with g = b = and u(t) o(t). In principle, the preliminary measuremenL~ (i.e. before the substances are placed in the reactor) allow b, and b, to be determined, but in the identifiability analysis, we will assume, at least initially, that they are not known. Holmberg (1982) considered the structural identi fiability of the parameters of this model on the assumption that both concentration of microorganisms x(t) and substrate concentration set) could be measured within the reactor (i.e. for t > 0). She used the Taylor series approach, from which it is reasonably straightfOlward to show that the four parameters ~m' Ks ' Kd ,md 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 set) for t > O. In this example, therefore, we will consider identifiability of the model of equations (l5A) and (15B) with measurement of x(t) only, i.e . with y = x, for t > O.

Taylor Series Identifiability Analysis On the assumption that the two mixtures arc put into the batch reactor rapidly, the behaviour for t '" 0+ is described by ~m

x

The example considered is of a model of microbial growth in a batch reactor; the model incorporates Michaelis-Menten type nonlinearities. A comprehensive review of the kinetics of microbial growth processes is given by Bulton (1985).

S x

~m

s y

867

s x

Y(Ks + s) x

(16A)

Xo

Ks + s

So

b,

(l6B) (16C)

The first four Taylor series MATHEMATICA are as follows:y,

x,

Y,

(~

y,

coefficients

obtained

Substituting for ~m from equation (21) and for Y from equation (23) into the equation for y, and then factorising the resulting expression yields

within (17)

s, x, / (Ks + s,»

Y,

(18)

- Kd x,

(x,(KdK~Y + 3KdK~Yso - 2KdK~~m Ys, + 3KdKsYs: (24)

+ ~Ys: - Ks~s,x,»

This equation contains the two unknowns for Kd gives

(19)

/ (Y(Ks + s,)').

Kct

and

Ks:

solving

y,

(25) Substituting for ~m from equation (21), for Y from equation (23) and for Kd from equation (25) into the equation for y., expanding and then faclOrising the resulting expression yields

y.

(20)

In these equations and equations (21) 10 (26), the i'th derivative of yet) at t = 0+ has been denoted by Yi for ease of nOl
s, Yo

+

y,

+

So

(21)

+

Substituting this expression for ~ into equation (19) and then factorising the resulting expression yields

Ys, (Ks + s, )y,. This equation contains the three unknowns is linear in Y. Solving for Y gives Y

(22)

Kct ,

Y and

Ks

and

(-y: + y,y,».

(26)

This expression is quadratic in the only remaining unknown Solving this expression would therefore yield parameter Ks. two values for Ks which have varying degrees of feasibility (Ks must be real and positive) depending on the values of the known quantities s,' Yo' y" y" y, and y.. Thus no conclusive identifiability results can be obtained for the parameter set (~m' Y, Kd' Ksl from this observation using the Taylor series approach. As noted above, there is no known upper bound on the number of derivatives to be examined (as far as the authors arc aware), but the expression for y, is cubic in Ks ' so it is still not possible to obtain conclusive identifiability results. The analysis is also inconclusive if a different ordering for the substitution of calculated parameter expressions is applied (for example, initially solving for Kct and then continuing to solve for the other three parameters). Similarity Transformation Approach In order to apply Theorem I, it is necessary to show first that the system of equations (l5A) and (l5B) with observation y = x is locally reduced. The system becomes linear if ~ = 0, but this linear system is not observable, so that 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 '" (0,0)T Then consider the vector field
(KdKsY: + 2KdKsY:y, + KsY,Y:) / (Kss,y: + s:y: (23)

868

Ilms

Ks Ilm x

Ks + s

(Ks + S)'

[f, b]

b,

and Ilms

Ks Ilm x

Ks + s

(Ks + s)'

(37)

which implies that b l

b,

+

~m A,

which, when evaluated at (0,0) yields

x

(28)

Ks

+

(39)

s

This equation is true for all x, so that x cancels out; hcnce aA, lax = 0, giving, from equation (38),

Since Kdbl l' 0 generically, this implies that b and [f,b](O) are linearly independent, so that dim PO) = 2 and the CRC is satisfied.

(40)

To check the ORC, initially we have the vector h Then consider the Lie derivative Lf(h)

so that

(1,0) . f

Substituting in the second equation of (12A), ~m

(29)

Ilms

Ks Ilm x

Ks + s

(Ks + s)'

1 (30)

so that

0,

d(LF(h) )

[

Ks

~m

(31)

1

b,

5,

~m

s

5, Y Ks

+

~m s x

Y (Ks

+

Y

(41)

s) x cancelled

out,

(42)

S

Multiplying through by Ks + s,

b,

Ilms Ks + s - Kd

(I,O).F

x

A,)

From equations (39) and (41), with

b,

which when evaluated at (0,0) yields d(Lj-(h))(O) = (-Kct,O) which is not linearly independent to (1,0). Denoting F = d(Lr(h» we then consider the Lie derivative

LF(h)

A,

Y(Ks + d(Lf(h) )

Y

5, Y

~mS

By comparing coefficients we have from the constant tenn (32)

(Ks + s)'

which, when evaluated at (0,0) yields d(LF(h»(O) = (O,~/Ks). Since ~m/Ks l' 0 generically, this is linearly independent to (1,0), so that dim d~(O) = 2 and the ORC is satisfied.

and since, generically, Ks is nonzero,

The system is thus locally reduced at (0,0) and we can proceed with the identifiability analysis using Theorem 1.

From the tenn in s,

Identifiability Analysis

Y

As with the Taylor series analysis we will start by assuming that the inputs b l and b, are unknown. Thus the unknown parameter vector is p = [b l , b" ~, Kd Y, Ks]T As before, y = x, for t > O. The state isomorphism being sought is denoted by A = (AI' A,)T with arguments (x, s)T Initially, it is assumed that u(t) E U[O,T].

From equations (41) and (45),

(44)

5,

b,

(45)

so that

From equation (12C), we have, for the system of equations (15),

[I, 0]

[ >, ]

[I, 0]

!I.

which implies that

so that

Al

(38)

Ilm s x

Ks + A,

[f,b](O)

aA, as

x, the first equation of (I2A) gives

Using the fact that Al (27)

5,

[ X]

(33)

But

x

(34)

o.

and

(46)

~m

=

aA,

b,

as

5,

,so that, equating tenns in s',

~m

(47)

There is then no tenn in s; equating constant tenns, (35)

b,

5,

From equation (l2B), we have

[: I [

o

I[

51

5,

(36)

(48)

Thus, from this experiment, the individual parameters bl , ~ and Kd are globally identifiable and so are the ratios b, /Y and b,/Ks' but the individual values of b" Ks and Y are unidentifiable.

869

Retuming now lO the precise experiment considered in the main pan of the Taylor se ries analysis, i.c. WiUl b, known, it is seen that the remaining parameters b" ~m ' Kd ' Y and Ks arc globally identifiablc. The transformation A = I, so that condition (C) , as well as conditions (a) and (b) apply for Uli s Hence the result also applies to the model and experiment specified form of input, i.e. with impulsive input of mi croo rganism ;md substrate.

Chappell, MJ . and K.R. Godfrey (1990). Structural identifiability of a model developed for optimal tumour targeting by antibodies using the Similarity Transformation approach. UniversilY of Warwick Control and Instrument Systems Cen[re, Repon No. 7. Chappell, MJ., K.R. Goctrrey and S. Vajda ([990). Global identifiability of the parameters of nonlinear systems with Math. Biosci. specified inputs: a comparison of methods. .l.QZ, 41-73.

The case with h, unknown is interesting, because it provides a funher example of an unidentifiable nonlinear model, in addition lO Exanlple I of Vajda, Godfrey and Rabitz (1989) . The source of the unidentifiahility in the present example is th at it cimcels out in bOUl equations (39) and (41), so reducing the number of coefficients which can be equated.

Godfrcy, K.R. Application,

(1983). Companmen[al Models Academic Press, New York.

and

[he ir

Godfrey, K.R. and W .R. Fitch (1984). The deterministic identifiability of nonlinear pharmacokinetic models. .!n!... Pharmacokin Biopharm .. 12, 177-191.

CONCLUSIONS The Taylor series identifiability analysis of the model of equations (15) if both x and s arc observed proves reasonably straightforward (Holmberg, 1982) and it is necessary to go to onl y the first and second dcrivatives of x and s to show that the parameters ~m' Ks ' Y and Kd arc globall y idelllili able. Similar analysis if only x is observed (but wi th \J kno wn) proves ve ry complicated and, as seen above, it does not appear to be possible to achieve a conclusive resu lt using this approach, even with the help of a symbolic manipulation package. In contrast, the Similarity Transformation idemi li ahi lit y analysis for the case where only x is observed has proved very straightforward, once Ule controllability and obsc.vability rank This is by no means al ways the criteria hi\.ve been checked. case and in general it is far from easy to pred ict in advance which approach will prove the easier fo r a panicular example (Chappell , Godfrey and Vajda, 1990). In mallY cases , symbolic manipulation packages signifi calll ly ease the tedious algebra involved in eiUler approach ilnd they open the door" to many structural identifiability prohlems that would have appeared inconccivable by hand. The pack;lge used for the analysis in this paper, MATHEMATICA. does have one shoneoming for identifiability JI,~·j 'i ~is ;n 'Jut there is currcntly no library routine available f0. calcuiat j ~g 3acobian matrices, which arc used in the local identifiability test of Pohjanpalo However, MATHEMATICA is capable of factori sing (1982). lengthy algebraic expressions (as seen in Ule Taylor series identifiability analysis above), which is an area where some other symbolic manipulation packages arc found w,mting.

ACKNOWLEDGEMENTS The work described in thi s paper is financed by Grant GR/F 6381.7 from the U.K. Science and Engineering Research Council. The authors are gratefUL lO Dr David Hodgson of Ule Depanment of Biological Sciences at Ule University of Warwick for advice on the problems of measuring microbial grow th in batch reactors. REFERENCES Brown, R.F. ([979). The identifiability of nonlinear 'i[h IFAC Svmposium on companmental models. Identification and System Parameter Es[imation. Darm stadt Germany. Paper M 11.5 . Bulton , D.K. (1985). microbial growth.

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Waiter, E. (1982). Identifiability of State Space Models. Lccu[re No[es in BiomaLhematics. Vol 46, Sp.ingcr, New York. Waiter, E. and Y. Lccounier companmental models : what 1-25 .

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(1981). Unidentifiable do? MiIlh Biosci 56,

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870