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IDENTIFIABILITY
OF PARAMETRIC MODELS
by
E. WALTER
Ecole Supérieure d'Electricité
Gif-sur-Yvette, France
PERGAMON PRESS
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Copyright © 1987 Pergamon Books Ltd.
All Rights Reserved. No part of this publication may be reproduced,
stored in a retrieval system or transmitted in any form or by any means:
electronic, electrostatic, magnetic tape, mechanical, photocopying,
recording or otherwise, without permission in writing from the
publishers.
First edition 1987
Library of Congress Cataloging-in-Publication Data
Walter, Eric.
Identiflability of parametric models.
Edited, updated, and expanded papers from the
7th IFAC/IFORS Symposium on Identification and
System Parameter Estimation held in July 1985 in
York, England.
Includes bibliographies and indexes.
1. System identification—Congresses.
2. Parameter estimation—Congresses. I. IFAC/IFORS
Symposium on Identification and System Parameter
Estimation (7th: 1985: York, North Yorkshire)
II. Title.
QA402.W365 1987 003 87-2324
British Library Cataloguing in Publication Data
Walter, E.
Identifiability of parametric models
1. System theory
I. Title
003 Q295
ISBN 0-08-034929-3
In order to make this volume available as economically and as rapidly
as possible the authors' typescripts have been reproduced in their
original forms. This method unfortunately has its typographical
limitations but it is hoped that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
FOREWORD
Why is identiflability receiving so much attention? There suggests simplifying assumptions for removing the
are two main reasons in my opinion. The serious one is ambiguity. Chapter 5, by Lecourtier, Lamnabhi-
that it is of great practical importance. Those involved in Laggarigue and Walter, describes in a tutorial way
estimating parameters from measurements would of various methods, based on recent results on Volterra
course like to know whether they stand any chance of and generating power series approaches, that can be
succeeding. Whenever a model is not uniquely used to test nonlinear models for structural identifiability.
identifiable, there are several values of the parameters Linear time-varying models and bilinear models
that correspond to exactly the same input-output are treated as special cases.
behavior; and the very meaning of an attempt to
estimate them is questionable, although we shall see in Chapter 6, by Chavent, is concerned with infinite-
the sequel that there may still be some hope. A less dimensional models, such as those described by
serious (but very important!) reason is that it is a partial differential equations. It investigates the relations
fascinating subject. Many researchers who planned to between identifiability and the well-posedness of the
give it only a passing thought have found themselves estimation problem.
trapped in a long-term project. Part of the attraction
comes from the fact that the basic problem is very simple Chapter 7, by Lecourtier and Raksanyi, describes the
to explain, part from the fact that no one can claim to facilities offered by computer algebra for performing
have solved it definitively. the algebraic manipulations required for testing a model
for structural controllability, observability, identifiability or
The idea of this volume took shape at York during the distinguishability. It also presents interesting conjectures
7th IFAC/IFORS Symposium on Identification and on the relations between the global injectivity of an
System Parameter Estimation in July 1985. Many of application and the global properties of its Jacobian.
those working in the field of identifiability were present,
and they shared the feeling that the subject was now Chapters 8 to 11 are devoted to the relations
mature enough for a coordinated presentation. Papers between identifiability« and parameter
given at York form the backbone of the book. They have uncertainty. Chapter 8, by Cobelli and Toffolo,
been edited, updated and significantly expanded. One describes methods that can be used to estimate
paper has been written especially for the occasion. All parameters even when the model proves to be
the authors were aware of the subjects that were to be structurally unidentifiable. Chapter 9, by Hadaegh and
treated by the other contributors . They were asked to Bekey, investigates the consequences of the fact that the
take advantage of the availability of the conference model is only an approximation to the real system under
preprints and send each other suggestions for study. Chapter 10, by Happel, Walter and Lahanier,
improvement and for maximizing complementarity. shows by a realistic chemical example how it may be
possible to obtain quantitative information on the
parameters of interest even when there are several
Chapter 1, by Godfrey and DiStefano, is a tutorial that competing model structures that are neither uniquely
recalls the basic methods for structural identifiability identifiable nor distinguishable. Finally Chapter 11, by
testing. In addition it provides tools that can be used to Walter and Pronzato, is devoted to robust experiment
obtain bounds on the possible values of unidentifiable design, viewed as the maximization of some measure of
parameters. identifiability.
Chapters 2 and 3 are devoted to linear time- I sincerely believe that the book as it stands gives a
invariant models. Chapter 2, by Delforge, d'Angio more comprehensive presentation of identifiability than
and Audoly, presents methods that are especially could have been written by any of the authors alone. I
interesting when dealing with large-scale models and hope you will share this view.
proposes conjectures on global identifiability deserving
further consideration. Chapter 3, by Chapman and
Godfrey, addresses the problems of initial model Eric Walter
selection and of generating the set of all models having
exactly the same input-output behavior.
Chapters 4 and 5 deal with nonlinear models.
Chapter 4 , by Vajda, gives a finite algebraic condition
for the structural identifiability of the parameters of a
class of polynomial models. It also addresses the
detection of dependences among the parameters that
can result from the effect of measurement noise even
when the model considered is structurally identifiable.
When such dependences occur, the proposed method
TUTORIAL
Chapter 1
IDENTIFIABILITY OF MODEL PARAMETERS
K. R. Godfrey* and J. J. DiStefano, III**
* Department of Engineering, University of Warwick, Coventry CV4 7AL, UK
**Biocybernetics Laboratory and Departments of Computer Science and
Medicine, University of California, Los Angeles, CA 90024, USA
Abstract. The problem is whether the parameters of a model can be identified
(uniquely or with more than one distinct solution) from a specified input-output
experiment. If perfect data are assumed, and the models are linear and time-invariant,
there are several approaches for identifiability analysis, alternatively referred to as
structural, deterministic or a nriori identifiability analysis, and five such approaches
are described. Only one, based on the Taylor series expansion of the observations, is
directly applicable to nonlinear or time-varying models.
When a model is unidentifiable from a proposed experiment, physically based constraints
on the model often provide a means of computing finite bounds for the parameters
(interval identifiability). This is illustrated for the class of linear, time-
invariant models in which the system matrix is of comoartmental form. Under certain
conditions, model parameter intervals are sufficiently small for all practical purposes,
and they are thus called quasi identifiable.
The identifiability question in the presence of real, noisy data, often referred to in
the literature as numerical or a posteriori identifiability, is classified and treated
as a separate problem. In the context of parameter estimation, the numerical identi
fiability problem is the same as the well known and long studied problem of parameter
estimation accuracy or nrecision, given that the parameters are known to be structurally
identifiable. Thus, the identifiability question arises as an issue separate from para
meter estimation in the classical or general sense only in the context of specifically
structured models; only in this topological sense is it new or different.
Keywords. Identification; Laplace transforms; Linear systems; Modelling;
Multi vari able systems; Nonlinear systems; Parameter estimation; State-space methods;
Time-domain analysis.
1. INTRODUCTION Identifiability analysis was put on a formal basis
by Bellman and Aström (1970), although specific
models were considered a good deal earlier than
The notion of identifiability is fundamentally a this, e.g., Skinner et al (1959). Bellman and
problem in uniqueness of solutions for specific Aström describe this as structural identifiability,
attributes of certain classes of mathematical and the term a priori identifiability also
models. The identifiability problem usually has has been used auite widely, on the grounds
meaning in the context of unknown model para that the analysis can (and should) be done before
meters, although it occasionally has had other a proposed experiment is carried out. The term
meanings. The usual question is whether or not deterministic identifiability also has been pro
it is possible to find a unique solution for each posed, in an attempt to circumvent certain inad
of the unknown parameters of the model, from data equacies of the adjective "structural" in cases
collected in experiments performed on the real where identifiability properties are dependent on
system. It is clearly a critical aspect of the the form of the input, normally considered to be
modelling process, especially when the parameters an independent variable in a dynamic system model.
are analogs of physical attributes of interest These issues are treated anew in Section 3.1.
and the model is needed to quantify them. Identi
fiability analysis is normally used to determine The question of identifiability in the presence
the extent to which a particular model is suitable of noisy data has come to be known as the
for reduction of data from a specific parameter numerical or a posteriori identifiability problem.
estimation experiment. As a consequence, it is In the context of model parameter estimation, this
also of significant value in experiment design. is none other than the classical problem of para
meter estimation accuracy (precision) in disguise.
If the model is deterministic and the data are We address the noise-free problem as one that must
noise-free, the problem is generally a nonlinear be resolved, ascertaining identifiability under
algebraic one and, unfortunately, solution of this ideal circumstances, prior to attempting the
algebraic problem, as we see below, is generally usually difficult problem of parameter estimation
both nontrivial and nonunique for all but the with real data. The latter issue is addressed in
simplest of models. In the presence of real, noisy Section 5.
data, the problem is compounded. Nevertheless,
structural identifiability conditions for the noise-
free case are minimal, necessary conditions for
achieving a successful estimation of model para
meters of interest from real input/output data.
2
2. BASIC CONCEPTS AND IDENTIFIABILITY 2.2 The Laplace Transform or Transfer
ANALYSIS FOR NOISE-FREE LINEAR TIME- Function Approach
INVARIANT MODELS
It is convenient to employ the Laplace transform
2.1 Basic Concepts and Linear Models of equations (6)-(8) for further analysis of the
identifiability properties of linear constant
Before turning to formal definitions, we review the coefficient models. We assume xQ(£) = 0^ here,
basic concept and models, for linear deterministic for convenience, but hasten to emphasise that the
systems, and present several examples to illustrate approach applies for any initial or other boundary
some of the subtleties of the identifiability con conditions. With x0 = °
cept,
Y(s,£) = C(£)[sI-A(p)]"'B()U(s) (9)
The following simple first-order model is adapted £
from Bellman and AstrSm (1970):
The identifiability properties are established by
x(t) = - x(t) + p u(t) , (1) examining the expressions in the powers of s in
Pl 2
the numerators and denominators of the measured
X(0) = 0 (2) outputs, together with any other information
available about £ .
y(t) = p x(t) . (3)
3
A parameter p. is uniquely (globally) identi
The model has three unknown parameters: p, , p« » fiable from a let of measurements y|<(t,£) ,
k = 1, 2 ... q if it can be evaluated uniqely
P . For any known u , the explicit solution
3 from the equations for y (t,£) , plus any other
k
of equations (l)-(3) is:
information about p. ; it is locally identifi
- (t-i) able from y.(t,£) if it has a countable number
Pl
y(t) = P P [ u(i)dx (4)
2 3 (> 1) of solutions or nonuniquely identifiable
if this number > 1 , while it is unidentifiable
from y (t,£) if it has an infinite number of
If the input is an impulse u(t) = 6(t) , k
solutions. If all p. of a model are identifi
y(t) = p p e Pl . (5) able, the model is said to be identifiable. The
2 3
definitions are easily extended for subsets of
It is well known that a semi logarithmic plot of the parameter vector £ = [p, p p ] .
2 p
the data, represented as y(t) for this model,
Also, it is often of interest to know which
yields the coefficient A = p p and exponent
2 3 combinations of individually unidentifiable
λ = p, of this model. Thus only p, and the parameters are identifiable (uniquely or other
wise). We formalise and extend these definitions
product P2P3 can be determined and not p2 or in Section 3.1, following presentation of several
P , i.e. the model is unidentifiable. This is examples that illustrate specific problems.
3
also clear from equation (4) for any known u(t) .
Identifiability analyses of linear, single-input,
If p or p were known, or if a unique
2 3 single-output (SISO) models may be performed
functional relationship between p and p were directly from the impulse responses. These
2 3
generally have the form
known, all parameters could be uniquely determined
from y(t) , and we could say the model (or model
parameters) is (are) uniquely (globally)
y(t,£) = Σ A.(£) exnCX. (£)]t (10)
identifiable,
k=l k k
This result generalises quite easily for linear
The n coefficients A. and n exponents λ.
constant coefficient systems. A system of n
first-order linear ordinary differential equations, are typically determined by fitting this model
depending on a set of P unknown constant para output function to the data in a least squares
sense and the 2n relationships among the A, ,
meters p ,p ,..., Pp , may be written conven
1 2
iently in vector-matrix form as λ. and p. determine the extent to which the
p. are identifiable.
x(t, ) = A(£)x(t,£) + B(£)u(t) (6)
£
Taking the Laplace transform of equation (10) for
with
the case with n = 2 and distinct eigenvalues,
X(0",£) - ^(P) (7)
Z(t,£) » C(£)x(t,£) . (8) Y(s,£) s-λ1 as Λ-2x0
(A,+A)s - (A, λ+Αλ-|)
Suppressing the arguments, the n state variables 2 2 2
of the model are denoted by the vector —
s -(x +x )s + λ λ
x_ = [x, x .... x ] , with initial conditions 1 2 1 2
2
given by (7); £ = [u-j u2 uf] is the ~8-2, s + 82 (11)
vector of r known inputs, ^s [y, y?····^
s + ou s + dp
is the vector of m output (measurement) variables
If there are no common factors in these numerator
of the model and A , B and C are constant
and denominator polynomials, it is clear that all
matrices of appropriate dimension, each consisting
of the 2n a and 8 coefficients (often called
Qf some or all of the unknown parameters moment invariants) can be determined from the
£= [p p ... pp] . The superscript T repre experiment (data), just as all of the 2n co
1 2
efficients A. and λ, of the sum of expo-
sents the vector transpose operation.
3
nentials solution can be determined from the function matrix (14), the additional observation
y(t,£) data. is of the form
Identiflability has to be qualified as being for
almost all parameter values. For example, partic Y (s) = (19)
2 T
ular (isolated) combinations of parameter values S +a,S+a0
or particular input functions which give rise to
pole-zero cancellations in Y(s,£) do not invali
so that the extra equation, in addition to
date the general analysis.
equations (17) and (18), is
The following model is used to illustrate many of
the concepts and problems introduced in this and c2bla21 = K2 (20)
the next several sections:
which only gives a , if c^b, is known.
2
*1 an ai2 xl bl 0 Ul Experiment 3: Dual-Input Single-Output, u-, f- 0 ,
+ b (12) Up t β » y? no^ measured. if the two test-
J
.X2. a21 a22. X2. 0 2 U2. inputs are applied at different times, so that
the two responses are completely distinguishable,
then the extra equation is
"c, 0 V
= (13) clb2a12 = K3 (2i ;
y2. 0 c2 X2. where K~ is a known constant. In this case,
a-,« could be found only if c-^ were known a
It is easy to show that the transfer function
priori. If, on the other hand, the perturbations
matrix for this model is:
are applied simultaneously, then from equations
(14) and (15),
H(s) Ξ C(sI-A)"'B
sb U (s)-a b U (s) a b U (s)
[Y^sJ/U^s) Y^sJ/Ugis) 1 1 22 1 1 + 12 2 2
Y-|tsj - C| —2 \ίά)
s -(a-|-|+a22^s+a'|la22~a12a21
Y (s)/U (s) Y (s)/U (s)
2 1 2 2
We now distinguish between two different input
types and two different corresponding identifi-
clbl(s"a22) clb2a12 ability results. If the two input waveforms are
1
(14) the same, the numerator gives c-.b, and
c2bla21 c b (s-a )
2 2 11
(-c,b,a +C|ba.|) , providing neither a nor
22 2 2 22
where A(S) = s -(a-.-.+a22)s+a-i-ιa22""aT2a21 * ^15^ Ί2 individually. But, suppose the input wave
forms were different, e.g., with u-.(t) a
We now consider several individual experiments in
turn. unit impulse and u (t) a unit step, so that
2
Experiment 1: SISO Case, u, f 0 , u^ = 0 and b,s -b-,a22S+b2a-|2
only y-, is measured. From equations (14) and ΥΊ(5) .(23)
sEs -(an+a22)s+alla22"a12a21]
(15),
S+3, Now the numerator gives c-.b^ and a?2 and, if
Y^S) = K, -y Ms) (16) c-jbp were known, a-^ ·
S +α-| S+αρ
Remark 1 : The example illustrated in Experiment
where KΊ, ' pl ' al and a0 are assumed known 3 is important because it illustrates that, for
linear systems with more than one input, identi-
and, from equation (14)
flability results may depend on whether the in
puts are applied simultaneously or separatesly.
cibi = S (17A) And, if they are applied simultaneously, the
result also can depend on the shape of the input
a22 51 (17B) waveforms. Thus, for multiple inputs, it is
essential to examine the Laplace transform of the
while from equation (15),
observations (available to the experimenter)
rather than just the individual entries in the
all = "al + β1 (ISA) transfer function matrix.
a12a21 = ^ι(αι"^ι)_α2 ' (18B) Remark 2: All of the above examples have illus-
trated either unique identi fiabi1ity or uni denti-
Thus from this experiment a^, a and tne Pro fi ability results. One way in which our two-state
22
ducts c-.bn and a-|oa2i are uniquely identifiable, example can give a locally identifiable (nonunique)
result is when the denominator of the output trans
but the parameters a,« sind a?1 are unidenti form has intrinsic parameters as its roots, in
fiable individually. It is of interest to see which case they are indistinguishable. Consider
the following example.
whether a, and a could be identified by
2 21
either observing an additional output or by stimu Experiment 4: SISO, u1 t 0 u2-= ° J?l not
lating an additional input.
measured; prior knowledge that a,2 = 0 From
equations (14) and (15),
Experiment 2: Single-Input Dual-Output, u-j _£_0_»
Up = 0 , y·. and y measured. From the transfer
2
4
and similarly from equation (27B),
c2bla21
Y (s) « . ϋ (5) (24)
2 Ί
(s-an)(s-a IV x2(0+) = a2lDlD . (29B)
and we see that intrinsically unknown parameters
Differentiating equation (27A),
a,, and a cannot be distinguished from the
22
denominator, i.e. a-,, and a are locally
22 x (0+) = a (0+) + a x (0+)
1 llXl 12 2
identifiable, with two solutions, from this experi
ment. For a system with three or more states,
there are many other possibilities for non-unique = (a^ + a12a2l)biD · (30A)
ness; see Section 2.7 for an example.
Differentiating equation (27B),
Remark 3: The Laplace transform method is con-
ceptually simple and derivation of the equations
x (0+) = a (0+) + a x (0+)
relating observations to system parameters is 2 2lXl 22 2
straightforward. Unfortunately, these equations
are not linear so that it is difficult to see = a2l(an + a22)blD * (30B)
whether there are multiple solutions or whether
redundancy exists. It is not clear how to modify
Further differentiation of equation (27A) gives
the model structure, input and observed variables
to achieve identi fi ability for a configuration re
sulting in unidentiflability and it is necessary x"(0+) = βΊ1χΊ(0+) + a12'x2(0+)
to re-work for each trial modification. No con
sistent simple structure carries over from one case
to another, so that it is difficult to generalise [all(all+a12a2V τ+ αa1i2oαa2o1nαa1-1M +
conclusions drawn from specific cases.
+ a12a21a22]blD ' (31)
Other examples of the Laplace transform approach
are given by Skinner et al (1959), Di Stefano et al
For Experiment 1, with only y. observed,
(1975), Milanese and Molino (1975), Cobelli et al
(1979b), Norton (1982) and Godfrey (1983), Chapter6. information is obtained only from equations (28A),
(29A), (30A) and (31), from which we see that, at
2.3 Taylor Series Expansion of the Observations the successive stages of differentiation, cLlbü,l ·
iInn ath Tiasy laoprp rsoearcihe,s tahbeo uotu tputt =w a0v+e f,o rmtsh ea rsuec cexepsasnidveed all ' a12a21 ancl a22 may be identified. Vlith
terms of the expansion being expressed as functions observation restricted to y-, , further differ
of the model unknowns (Pohjanpalo, (1978)). Spec
entiation does not yield a, and a-i individ
ifically, for an observation y.(t) , 2 2
ually, as the reader is invited to confirm.
2
y^t) =y .(o+) +ty .(o+) +7yy (o+) +... (25) For Experiment 2, in which y also is observed,
n n 1 2
equation (29B) gives cb-.a·. uniquely, which
2 2
Successive derivatives are, in principle, measurable
if the product cb·. is known, gives a , and
and contain information about the parameters to be 2 21
identified. If the Laplace transform of equation then a-,2 » uniquely. Thus, as exnected, the
(25) is taken,
result is the same as for the Laplace transform
analysis.
Υ·(*) = ^ ·(0+) +^y (0+) + ^.(OV... (26)
Ί Ί i
Remark: This example has illustrated that the
method suffers from the same drawbacks as the
from which it may be seen that the test is equiva
Laplace transform approach as described in Remark
lent to expanding the Laplace transform of the ob
3 of Section 2.2. the method has not been used
servation vector in a power series in s~^ .
much for linear, time-invariant systems. It has
the decided advantage, however, that it is
To illustrate the approach, consider again Experi
applicable to nonlinear and time-varying systems,
ments 1 and 2 above, with only input 1 applied
for which the Laplace transform approach is not.
(u« = 0) and let the input be impulsive, u,(t) =
Applications to some nonlinear models are given
D.ó(t) with D known. This input can be incor in Section 3.
porated as an initial condition, so that the
equations may be written 2.4 Markov parameter matrix approach
Grewal and Glover (1976) described a technique
X](t) = θ χ (ΐ) + a x (t) , t > 0 (27A) for testing whether two sets of parameter values
η Ί 12 2
can give the same observed responses for all ad
missible forms of excitation. First, the Markov
x (t) = a^x^t) + a x (t) , t > 0 (27B)
2 22 2 parameter matrix
with initial conditions x-.(0 ) = b,D (28A) G = [(CB)T(CAB)T(CA2B)T (CA2n_1B)T] (32)
x2(0+) = 0 . (28B) (where n is the model order) is formed. It is
possible to use G itself for a global identifi-
ability test by determining whether
From equation (27A),
X](0+) = a^x^O4") + a x (0+) G(£) = G(£')**£=£'
12 2
but this again suffers from the same drawbacks as
= a b D (29A)
11 ] the Laplace transform approach. What Grewal and
Glover (1976) did was to find the rank of the
5
Jacobian of G with respect to the unknown model system described by eqrua tions (6), (7) and (8)
parameters. If the rank is equal to the number
of parameters, the system is identifiable, but not
necessarily uniquely. The rank test is also applic x(t) = eAtx(Cf ) + eA(t~x)BuJT)dT (33)
able to Jacobians from other approaches, for example,
the equations resulting from the Laplace transform
or Taylor series approaches. and in particular the form for a zero initial
state experiment. Then parameter sets £ and £
Taking our two-state example, are indistinguishable if and only if
a l C(p)Ak(p)B(£) = C(£)Ak(£)B(£) (34)
Ml 12
*21 a22 for k = 0,1, It is well known that we
can restrict consideration to the first 2n
equations of the form (34). The upper bound
all+a12a21 a12(an+a225 k = (2n-l) is readily arrived at through the use
of Tether's continuation lemma (Tether, 1970),
i21(all+a22) a22+a12a21 wMh ichs ucsht attheast that, given matrices M,' , ML9 and
3
all+2alla12a21+a22a12a21 rank M-. = rank(M. NL) = rank/» (35A)
2 2 2
(all+a22)a21+alla22a21+a12a21 there exists, at most, one matrix M, for which
2 2 2
(a +a )a +a a a +a a Ί "2 \
11 22 12 11 22 12 12 21 rank! MJ = rank Mi · (35B)
M3
a22+2a22a12a21+alla12a21
If we consider the matrices
If only u, is applied and only y, is observed,
CB CAB .... 0Αη"ΊΒ
CAB CA2B .... CAnB
c1b1 Ml "
clVll CA""1 3 CAnB .... CA2n_2B
G(£)
c1bl(all+a12a2l' CAnB
3
c1b1(a11+2a11a12a21+a22a12a21) CAn+1B
CA2n"V
where the parameter vector £= [c,b, a.,, a a a««]
12 21
It is clear that — cannot possibly be of rank 5, M = [CAnB CAn+1B .... CA2"1"^]
3
d£
so let us consider the case where there is prior M = [CA2nB]
knowledge that a, = α(=)= 0) . Then 4
2
£= Ccb a a a ] ,
1 1 n 21 22
we know that the rank conditions (35A) and (35B)
follow from the Cayley-Hamilton theorem and so
from Tether's lemma, CA2nB is uniquely defined.
Hence if equations (34) hold for k = 0,1,.. .(2n-l),
8G Ml clbl then they also hold for any k and no new inform
3£ ation is forthcoming from examining higher powers
(all+a12a21) clbl2all of A . For our two-state example, with only
3 2 state 1 perturbed and observed, the (1 1) ele-
(α +2a aa +a aa ) c b (3a-j +2aa ] 4 4 2
Ί 1 11 21 22 21 ] ] 1 21 ment of A is a,, + 3a,,a a, + 2a-iia22a12a21 +
12 2
0 0 aα2o2oα^1i2oα^2o1i τ + ^(α1ai2α?2a1?/- |) ™s oι "t,nαaυt α1a2i o and a0a 21
0 0 still do not appear individually.
clbla 0 Also, if we consider a unit impulse input
b.u.(t) = b.6(t) , with x(0") = 0 , then
cb(2a a+a a)
1 1 11 22 clblaa21
x(t) = eAt
and it is readily seen that rank 4 provided bjij
9£
clbl a and a 21 are not zero. where δ_. is a vector zero except for a 1
Further examples are given by Grewal and Glover row j Then
(1976) and by Carson, Cobelli and Finkelstein (1983,
Section 7.5). y.(t) - c. eAt
bj
Remark 1 : This method has the advantage that it 2 t
= c.b. + c.Ab.t + c.A b. π + ...
is computationally convenient and amenable to com
puter implementation.
(36)
Remark 2: It is instructive at this point to
consider the general time domain solution of the This makes it clear that, for a unit impulse
6
input, exactly the same parameter combinations equations (12) and (13). These equations are
occur as successive derivatives of y.(t) at known as the input-output equations and similar
+ expressions can be obtained for other forms of
t = 0 in the Taylor series approach and as
input. For example, if u.(t) had been a unit
successive elements of c.A b. , k = 0, 1,
step (again with x_(0") =0) , then
Remark 3: Some identi fi ability approaches have
examined properties of the transition matrix Yi(t) = c.rjA'V*-^J (4i :
At
e , for example that described in Chapter 4 of
Walter (1982). Generally these techniques prove Prior knowledge of any elements of A can be
rather complicated algebraically and are not easy incorporated through the prior knowledge equations
to apply for n > 2 . A more systematic time
domain approach, using a modal expansion is des
cribed in Section 2.5. TJ i^i 1,2...n 1,2,. .n (42)
As noted above, the objective of the method is to
2.5 Modal Matrix Approach
reduce the bilinear matrix equations to linear
The methods presented in Sections 2.2 and 2.3 re equations by using the input-output and prior
sult in equations nonlinear in the unknown para knowledge equations and so to solve for
meters, so it is difficult to see whether redund
r. and n_. . Let us now consider the four experi
ancy exists. There are general methods for solving
these equations, some of which are presented by ments detailed in Section 2.2. For all the two-
Lecourtier and Raksanyi in this book, but the state examples, the modal matrix equations are:
problem is that sometimes the computations get so
complex that no result is obtained. Another method,
based on the modal matrix and its inverse, results r-iJi! (43A)
in bilinear equations. Many (often all) of these
equations are then reduced to linear equations by (43B)
the incorporation of information about outputs and
prior knowledge of the elements of A (Norton,
1980a; Norton, Brown and Godfrey, 1980). (43C)
The eigenvalues of A A. and, up to a sealing (43D)
factor, the eigenvectors Ü1 are defined by
Experiment 1 : SIS0 case,u 6(t) 0 ,
Am. = A.m. 1,2,... n . (37) 0 Since both modes appear in x and
*2-
T
the scaling of eigenvectors is arbitrary, r.
The eigenvectors are collected together as columns
can be taken as [1 1] . The input-output
of the modal matrix M :-
equation is
AM = ΜΛ
y^t) = c]r|eAV1b1 t > 0 (44)
where Λ is a diagonal matrix having \. as
principal diagonal element i . The eigenvalues This gives the proportions of ru from the ratio
are assumed distinct; systems such as distillation
of the normal modes and the scaling is then given
columns which have simple structure but repeated
by equation (43A). In the absence of prior know
eigenvalues are probably best treated by a special
ledge, the only other equations in the remaining
ised analysis exploiting their particular structures.
unknowns r? and ru are (43B) and (43C) which
The modal matrix equations
*s
are linear and (43D) which is bilinear. The pro
duct c,b, is obtained uniquely from equations
1 , i = j (38A)
(43A) and (44). Note that equation (43A) is
= 0 , i + j (38B) redundant if there is prior knowledge of c-.b, .
Prior knowledge of a,« (/ 0) would give
where r. is row i of M and n. is column j
1
of N Ξ Μ"1 are bilinear in the unknown elements _r.iAIb a12 (45)
of jr. and n. .
Assuming distinct eigenvalues, Λ ru is independ
The system response is given by equation (33). If
we consider an impulse input b.u.(t) = b.fi(t) ent of ru so with r.i known, equations (43B)
with x_(0") = 0_ , and (45) give ru ; r? is then 0Dtainecl from
equations (43C) and (43D).
x(t) = eAtb.fi. = MeAt Nb.fi. (39)
In some cases, prior knowledge can result in two
equations bilinear in the unknowns, but these can
where <5_. is a vector zero except for a 1
be rearranged to give a singularity equation in
Hence one of the unknown rows or columns (Norton,
1980a; Norton, Brown and Godfrey, 1980); an
ni eA V,· example will be given in Experiment 4.
(t) t > 0
Xi
Experiment 2: One Input, Two Outputs, u-, (t) =
and
y^tj-c^je^bj t > 0 (40) 6(t) , Ug(t) = 0 . The available equations
are (43A) to (43D) and (44), from which r^ and
for square, diagonal B and C matrices, as in ru are known, and
