Prediction-error identification of LPV systems: A nonparametric Gaussian regression approach

Abstract

In this paper, a Bayesian nonparametric approach is introduced to estimate multi-input multi-output (MIMO) linear parameter-varying (LPV) models under the general noise model structure of Box–Jenkins (BJ) type. The approach is based on the estimation of the one-step-ahead predictor of general LPV-BJ structures. Parts of the predictors associated with the input and output signals are modeled as asymptotically stable infinite impulse response (IIR) models. Then, these IIR models are identified in a completely nonparametric sense: not only the coefficients are estimated as functions, but also the whole time evolution of the impulse response w.r.t. the scheduling signal of the LPV system. In this Bayesian setting, the estimate of the one-step-ahead predictor is a realization from a zero-mean Gaussian random field, where the covariance function is a multidimensional Gaussian kernel that encodes both the possible structural dependencies and the stability of the predictor. Two different kernel formulations are presented for the LPV setting, namely a diagonal (DI) like and tuned/correlated (TC) like kernels, where the TC-like kernel is able to describe the correlation between coefficient functions associated with different time indices. The unknown hyperparameters that parameterize the DI or TC kernel are tuned by maximizing the marginal likelihood w.r.t. the observed data. Moreover, we provide a nonparametric realization scheme to recover the original process and noise IIRs from the identified one-step-ahead predictor. The performance of the presented identification approach is tested on a MIMO LPV-BJ simulation example by means of an extensive Monte-Carlo study.