Bayesian identification of LPV Box-Jenkins models

In this paper, we introduce a nonparametric approach in a Bayesian setting to efficiently estimate, both in the stochastic and computational sense, linear parameter-varying (LPV) input-output models under general noise conditions of Box-Jenkins (BJ) type. The approach is based on the estimation of the one-step-ahead predictor model of general LPV-BJ structures, where the sub-predictors associated with the input and output signals are captured as asymptotically stable infinite impulse response models (IIRs). These IIR sub-predictors are identified in a completely nonparametric sense, where not only the coefficients are estimated as functions, but also the whole time evolution of the impulse response is estimated as a function. In this Bayesian setting, the one-step-ahead predictor is modelled as 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. The unknown hyperparameters that parameterize the kernel are tuned using the empirical Bayes approach, i.e., optimization of the marginal likelihood with respect to available data. It is also shown that, in case the predictor has a finite order, i.e., the true system has an ARX noise structure, our approach is able to recover the underlying structural dependencies. The performance of the identification method is demonstrated on LPV-ARX and LPV-BJ simulation examples by means of a Monte Carlo study.