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^N contains 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.