System identification with input uncertainties an EM kernel-based approach

Abstract: Many classical problems in system identification, such as the classical predictionerror method and regularized system identification, identification of Hammersteinand cascaded systems, blind system identification, as well as errors-in-variablesproblems and estimation with missing data, can be seen as particular instancesof the general problem of the identification of systems with limited information.In this thesis, we introduce a framework for the identification of linear dynamicalsystems subject to inputs that are not perfectly known. We present the class ofuncertain-input models—that is, linear systems subject to inputs about which onlylimited information is available. Using the Gaussian-process framework, we modelthe uncertain input as the realization of a Gaussian process. Similarly, we model theimpulse response of the linear system as the realization of a Gaussian process. Usingthe mean and covariance functions of the Gaussian processes, we can incorporateprior information about the system in the model. Interpreting the Gaussian processmodels as prior distributions of the unknowns, we can find the minimum mean-square-error estimates of the input and of the impulse response of the system. Theseestimates depend on some parameters, called hyperparameters, that need to beestimated from the available data. Using an empirical Bayes approach, we estimatethe hyperparameters from the marginal likelihood of the data. The maximizationof the marginal likelihood is carried out using an iterative scheme based on theExpectation-Maximization method. Depending on the assumptions made on themodels of the input and of the system, the standard E-step may not be available inclosed form. In this case, the E-step is replaced with a Markov Chain Monte Carlointegration scheme based on the Gibbs sampler. After showing how to estimate thesystem and the hyperparameters, we show how to specialize the general uncertain-input model to particular structures and how to modify the general estimationmethod to account for these particular structures. In the last chapter, we show inwhat sense the aforementioned classical system identification problems can be seenas uncertain-input model identification problems; we show the effectiveness of theframework in dealing with these classical problems in several numerical examples.