(368a) Parameter Set Selection for Estimation of Nonlinear Dynamic Systems

Mathematical modeling and
analysis plays an important role in the study of the complex dynamic systems.
Parameter estimation forms an essentially component of deriving mathematical
models. However, accurate estimation of parameters in a complex system can be
challenging as models can contain hundreds or even thousands of parameters
while at the same time experimental data used for parameter estimation may be
sparse and noisy. Due to these reasons it is usually not possible to estimate
the values of all the parameters accurately from the experimental data.

Parameter selection is often
based on an optimality criterion^1-3 involving the Fisher information
matrix. A combination of the D-optimality and the modified E-optimality
criteria has been used to determine identifiable parameters of models for
biological wastewater treatment processes.^4,5 The Fisher information
matrix should be far from singularity to maximize an optimality criterion. This
situation occurs when the norm of the sensitivity vectors is likely to be
reasonably large and the angles between the sensitivity vectors are not small,
either. Following these two rules, several parameter-selection techniques have
been developed in the past based on the sensitivity vectors which indirectly
maximize a criterion involving the Fisher information matrix, e.g., an
orthogonal method has been presented for the analysis of an ethylene-butane
copolymerization reactor model⁶ and a recursive approach based upon
principal component analysis (PCA) has been applied to a polymerization reactor
model.⁷

The drawback to these procedures
is that they are based on sensitivity vectors and the calculation of
sensitivity vectors for nonlinear systems is changing with the nominal value of
the parameters. At the same time, the nominal values of the parameters are not
exactly known as it is the purpose of parameter estimation to determine their
values. For dynamic systems the sensitivity vector is also dependent on the
inputs and the initial conditions. It is the aim of this work to present a
parameter set selection technique for dynamic systems described by nonlinear
autonomous differential equations which will take the effect of uncertainties
of the parameter values and initial states as well as changes of the inputs
into account.

The parameter selection problem
can be formulated as an optimization problem

max E[f(x,y,z)|x,y],

where the objective function f
is some kind of optimality criterion of the Fisher information matrix. The
objective functions depends upon three types of variables: x belonging
to {0,1}^M denotes which parameters are selected; y
belonging to R^Ncontains adjustable experimental
conditions; z belonging to R^P includes uncertain
factors. These uncertain factors are regarded as random variables according to
some distribution. x, y are varied to maximize the conditioned
expectation, i.e. to improve the average performance over the uncertainty range
of z. To determine the optimal parameter set for estimation it is
required that the optimal experimental conditions to estimate the parameters
are obtained while the effects of the uncertain factors are taken into account.
However, for complex systems the dimension of the variable space can be large
and solution of the optimization problem typically requires repeated simulation
of the model, both of which results in a large computational burden for solving
the problem. Furthermore, it is often preferred in practice to not just look at
one optimal set of parameters but instead to focus on several sets of
parameters that could be used depending upon changes in operating conditions or
limitations in the amount of available data.

In this work, a genetic
algorithm (GA) is used to determine a collection of (sub)optimal parameter sets
based upon a chosen nominal value of the parameters. In next step sensitivity
analysis of the criterion function is performed to identify the key factors
which have large effects on the accuracy of the estimation. Then a stochastic
approximation method is applied to search the optimal experimental settings for
a selected parameter set. The outcome of this procedure is a collection of
parameter sets which can be used for parameter estimation, additional
information about of how the estimation accuracy is affected by the uncertain
factors and the optimal experimental settings to estimate them. The developed
techniques is illustrated in systems.

Literature cited

1.             
Silvey SD. Optimal design: an
introduction to the theory for parameter estimation. London: Chapman and Hall,
1980.

2.             
Walter E, Pronzato L. Qualitative
and quantitative experiment design for phenomenological models - A survey.
Automatica. 1990; 26 (2): 195-213.

3.             
Kiefer J. Optimum experimental
designs. Journal of the Royal Statistical Society Series B - Statistical
Methodology. 1959; 21 (2): 272-319.

4.             
Weijers SR, Vanrolleghem PA. A
procedure for selecting best identifiable parameters in calibrating activated
sludge model no.1 to full-scale plant data. Water  Science and Technology.
1997; 36 (5): 69-79.

5.             
Brun R, Kuhni M, Siegrist H, Gujer
W, Reichert P. Practical identifiability of ASM2d parameters - systematic
selection and tuning of parameter subsets. Water Research. 2002; 36 (16):
4113-4127.

6.             
Yao KZ, Shaw BM, Kou B, McAuley KB, Bacon DW. Modeling ethylene/butene
copolymerization with multi-site catalysts: Parameter estimability and
experimental design. Polymer Reaction Engineering. 2003; 11 (3): 563-588.

7.             
Li RJ, Henson MA, Kurtz MJ.
Selection of model parameters for off-line parameter estimation. IEEE
Transactions on control systems technology. 2004; 12 (3): 402-412.