(469g) Parameter Reduction for Nonlinear Models Based on Hankel Singular Values and Sensitivity Analysis

Model reduction is an essential step for many important tasks like controller and observer design or state and parameter estimation. Most of the existing techniques on model reduction for these applications focus on reduction of the number of states [1, 2, 3], whereas relatively few techniques are available for reduction of the number of parameters in these models. However, as models often contain a large number of process parameters and since there is always an uncertainty range associated with the values of parameters, it is crucial to systematically investigate techniques that reduce the parameters in the model.

A novel reduction technique for model parameters is presented in this paper. This novel methodology reduces the set of parameters to be considered for a model via a technique derived from balanced model reduction [4] and further reduces this set of parameters via sensitivity analysis [5].

Reduction of the set of parameters is performed by comparing Hankel singular values [6] of the original system to systems where only the parameters are used as inputs. Each parameter is varied within its uncertainty range and a set of Hankel singular values can be derived for each individual parameter. It is then possible to compare the sum of the Hankel singular values for variations in each individual parameter and determine an error bound for reducing this parameter. Based upon the error bounds it is possible to determine a set of parameters to be retained.

The set of parameters can be further reduced by parameter sensitivity analysis and principle component analysis. Those parameters (or combinations of parameters) whose variation can result in a large system response are determined to be relatively important for the system behavior. In order to take the interdependence between parameters into account, a sensitivity correlation matrix can be obtained based on the system responses. It is shown that for linear systems, this correlation matrix can be easily derived from the observability gramian. The eigenvalues of the correlation matrix can then be used to determine the relative importance of parameters.

Each of these two techniques has its advantages and its drawbacks: parameter reduction via comparison of the Hankel singular values will retain the physical meaning of the parameters, while sensitivity analysis of the correlation matrix can result in a significantly more reduced subset of the parameters to be included in the model.

Considering the advantages and drawbacks of these two basic methods, a two-step procedure is developed that combines aspects of both. In a first step, the effect of the parameters on the outputs is screened via comparison of the Hankel singular values. Those parameters with small Hankel singular values are reduced. Sensitivity analysis is then performed on the reduced system, as it can further reduce parameters by taking correlations between them into account.

An example of continuous stirred tank reactor model [7] is applied to illustrate this technique. Monte Carlo simulation method is implemented to characterize the performance of the reduced system.

Keywords: Model Reduction, Parameter Reduction, Hankel Singular Values, Sensitivity Analysis

References:

[1] S. Lall, J.E. Marsden, S. Glavaski, Empirical model reduction of controlled nonlinear systems, 14th IFAC World Congress, Beijing, 1999.

[2] J. Hahn, T.F. Edgar, W. Marquardt, Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems, Journal of Process Control 13 (2003), 115-127.

[3] C. Sun, J. Hahn, Reduction of differential-algebraic equation systems via projections and system identification, Journal of Process Control 15 (2005), 639-650.

[4] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Transactions on Automatic Control 26 (1) (1981), 17-32.

[5] A. Saltelli, K. Chan, E. M. Scott, Sensitivity Analysis, Wiley Series in Probability and Statistics, Chichester, New York, Wiley, 2000

[6] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, Englewood Cliffs, New Jersey, 1996.

[7] Michael A. Henson, Dale E. Seborg, Input-output linearization of general nonlinear processes, AIChE Journal, 36 (11), 1990, 1753-1757