On risk-coherent input design and Bayesian methods for nonlinear system identification

Abstract: System identification deals with the estimation of mathematical models from experimental data. As mathematical models are built for specific purposes, ensuring that the estimated model represents the system with sufficient accuracy is a relevant aspect in system identification. Factors affecting the accuracy of the estimated model include the experimental data, the manner in which the estimation method accounts for prior knowledge about the system, and the uncertainties arising when designing the experiment and initializing the search of the estimation method.As the accuracy of the estimated model depends on factors that can be affected by the user, it is of importance to guarantee that the user decisions are optimal. Hence, it is of interest to explore how to optimally perform an experiment in the system, how to account for prior knowledge about the system and how to deal with uncertainties that can potentially degrade the model accuracy.This thesis is divided into three topics. The first contribution concerns an input design framework for the identification of nonlinear dynamical models. The method designs an input as a realization of a stationary Markov process. As the true system description is uncertain, the resulting optimization problem takes the uncertainty on the true value of the parameters into account. The stationary distribution of the Markov process is designed over a prescribed set of marginal cumulative distribution functions associated with stationary processes. By restricting the input alphabet to be a finite set, the parametrization of the feasible set can be done using graph theoretical tools. Based on the graph theoretical framework, the problem formulation turns out to be convex in the decision variables. The method is then illustrated by an application to model estimation of systems with quantized measurements.The second contribution of this thesis is on Bayesian techniques for input design and estimation of dynamical models. In regards of input design, we explore the application of Bayesian optimization methods to input design for identification of nonlinear dynamical models. By imposing a Gaussian process prior over the scalar cost function of the Fisher information matrix, the method iteratively computes the predictive posterior distribution based on samples of the feasible set. To drive the exploration of this set, a user defined acquisition function computes at every iteration the sample for updating the predictive posterior distribution. In this sense, the method tries to explore the feasible space only on those regions where an improvement in the cost function is expected. Regarding the estimation of dynamical models, this thesis discusses a Bayesian framework to account for prior information about the model parameters when estimating linear time-invariant dynamical models. Specifically, we discuss how to encode information about the model complexity by a prior distribution over the Hankel singular values of the model. Given the prior distribution and the likelihood function, the posterior distribution is approximated by the use of a Metropolis-Hastings sampler. Finally, the existence of the posterior distribution and the correctness of the Metropolis-Hastings sampler is analyzed and established.As the last contribution of this thesis, we study the problem of uncertainty in system identification, with special focus in input design. By adopting a risk theoretical perspective, we show how the uncertainty can be handled in the problems arising in input design. In particular, we introduce the notion of coherent measure of risk and its use in the input design formulation to account for the uncertainty on the true system description. The discussion also introduces the conditional value at risk, which is a risk coherent measure accounting for the mean behavior of the cost function on the undesired cases. The use of risk coherent measures is also employed in application oriented input design, where the input is designed to achieve a prescribed performance in the intended model application.