Using model classes in system identification for robust response predictions

Abstract eng:
The goal of system identification for dynamic systems is to use experimental data to improve mathematical models of the input-output behavior of a dynamic system so that they make more accurate predictions of the system response to a prescribed excitation. A common approach is to take a parameterized model of the system and then use system data to estimate the value of the uncertain model parameters. This parameter estimation must be done with great care, however, because it is unrealistic to expect any model to be an exact representation of the system behavior and so one cannot expect true parameter values, and often the estimation gives non-unique results. Furthermore, in order to make more robust predictions, the uncertainties in modelling the system, as well as in modelling the future system input, should be explicitly treated; in particular, because of the approximate nature of any system model, one should explicitly treat the uncertain prediction error (the difference between the outputs or state of the real system and those of the system model). Probability logic with Bayesian updating has emerged in recent years as a rigorous approach to address these points. This paper first present an overview of probability logic, which provides a multi-valued propositional logic for plausible reasoning and which gives a meaning for the "probability of a model" as a measure of the relative plausibility of the model within a proposed set of models; this plausibility is quantified by a probability distribution for the model parameter values which is conditional on specified information. A summary is then given of the application of probability logic to quantifying modelling uncertainty for robust predictive analysis of systems. Instead of using system data to estimate the model parameters, Bayes’ Theorem is used to update the relative plausibility of each model in a model class, which is a parameterized set of predictive input-output probability models for the system behavior together with a prior probability distribution over this set that expresses the initial plausibility of each model. Then to produce robust predictive analyses, the entire model class is used with the probabilistic predictions of each model being weighted by its posterior probability, in accordance with the Total Probability Theorem. An additional level of robustness can be performed by combining the robust predictions of each model class in a set of candidate model classes for the system, where each contribution is weighted by the posterior probability of the corresponding model class. These robust analyses involve integrals over high-dimensional parameter spaces that usually cannot be evaluated analytically. Useful computational tools for these evaluations are Laplace's method of asymptotic approximation and Markov Chain Monte Carlo methods.

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