Detailed first-principles-based models often exhibit dynamics that is occurring at multiple time and length scales. Additionally, these models tend to include a large number of parameters, some of which may be estimated where selecting the set of parameters to be estimated is an important task. Identifiability of a set of parameters is determined by the effect that changes in the parameter values have on the output. This effect is represented by the sensitivity vectors. Several methods for parameter selection based on sensitivity vectors have been proposed in the literature. These include, but are not limited to, a collinearity index method,¹ a column pivoting method,² an extension of the relative gain array,³ a Gram-Schmidt orthogonalization method,⁴ a recursive approach based upon principal component analysis,⁵ and a combination of Hankel singular values and singular value decomposition.⁶ A systematic approach for parameter selection is based on optimality criteria computed from the Fisher information matrix as the inverse of Fisher information matrix provides a lower bound for the covariance matrix of parameter estimators. A subset of identifiable parameters can be selected based upon optimizing some experimental criteria such as the D-optimality or the modified E-optimality criterion of the Fisher information matrix.^7,8

Most of the parameter selection approaches mentioned can be formulated as a combinatorial optimization problem, however, solving these optimization problems is nontrivial. For systems with only few parameters an exhaustive search⁷ can be used. However, the total number of possible combination of parameters is too large to be enumerated even for systems with only a few dozen parameters. Stochastic search techniques such as genetic algorithms⁹ can provide a solution for larger systems, however, convergence of these algorithms is not guaranteed. Another approach is to use a sequential approach where one parameter is selected at a time, e.g., the orthogonalization method.^2,4 The main disadvantage of these types of approaches is that some combinations of parameters may give better results, but could be excluded because of parameters selected at earlier steps. Techniques used for the orthogonalization method include the householder transformation² and the Gram-Schmidt procedure⁴. Both of these procedures can be regarded as an approach to maximize the D-optimality criterion one parameter at a time.⁹

The methods mentioned above focus on searching the space of possible parameter sets, a procedure that is strongly affected by the number of parameters of the model. This work presents a different approach for parameter selection. The number of parameters to be considered is first reduced by determining several groups of parameters where the parameters within a group are pair-wise indistinguishable, i.e., they cannot be estimated together. It is then possible to only consider one parameter per group for the parameter set selection procedure. This technique significantly reduces the combinatorial problem resulting from a large number of parameters and enables solution of the parameter set selection problem using existing approaches.

The main contribution of this work consists of developing the analytic and numerical methods for sorting parameters into groups for models consisting of nonlinear differential equations. It will be shown that parameters in an analytical pair-wise indistinguishable set can be re-parameterized by a new parameter. A procedure for carrying out this step numerically will also be presented.

References

(1)        Brun, R.; Reichert, P.; Kunsch, H. R.; Practical identifiability analysis of large environmental simulation models. Water Resour. Res. 2001, 37, 1015.

(2)        Velez-Reyes, M.; Verghese, G. C.; Subset selection in identification, and application to speed and parameter estimation for induction machines. Proceedings of the 4th IEEE Conference on Control Applications. 1995, 991.

(3)        Sandink, C. A.; McAuley, K. B.; McLellan, P. J. Selection of parameters for updating in on-line models. Ind. Eng. Chem. Res. 2001, 40, 3936.

(4)        Yao, K. Z.; Shaw, B. M.; Kou, B.; McAuley, K. B.; Bacon, D. W. Modeling ethylene/butene copolymerization with multi-site catalysts: Parameter estimability and experimental design. Polym. React. Eng. 2003, 11, 563.

(5)        Li, R. J.; Henson, M. A.; Kurtz, M. J. Selection of model parameters for off-line parameter estimation. IEEE Trans. Control Syst. Technol. 2004, 12, 402.

(6)        Sun, C. L.; Hahn, J. Parameter reduction for stable dynamical systems based on Hankel singular values and sensitivity analysis. Chem. Eng. Sci. 2006, 61, 5393.

(7)        Weijers, S. R.; Vanrolleghem, P. A. A procedure for selecting best identifiable parameters in calibrating activated sludge model no.1 to full-scale plant data. Water Sci. Technol. 1997, 36, 69.

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

(9)        Chu, Y.; Hahn, J. Parameter Set Selection for Estimation for Nonlinear Dynamic Systems. AICHE J. 2007, 53, 2858.