A new kernel-based approach to hybrid system identification

All the approaches for hybrid system identification appeared in the literature assume that model complexity is known. Popular models are e.g. piecewise ARX with a priori fixed orders. Furthermore, the developed numerical procedures have been tested only on simple systems, e.g. composed of ARX subsystems of order 1 or at most 2. This represents a major drawback for real applications. This paper proposes a new regularized technique for identification of piecewise affine systems, namely the hybrid stable spline algorithm (HSS). HSS exploits the recently introduced stable spline kernel to model the submodels impulse responses as zero-mean Gaussian processes, including information on submodels predictor stability. The algorithm consists of a two-step procedure. First, exploiting the Bayesian interpretation of regularization, the problem of classifying and distributing the data to the subsystems is cast as marginal likelihood optimization. We show how an approximated optimization can be efficiently performed by a Markov chain Monte Carlo scheme. Then, the stable spline algorithm is used to reconstruct each subsystem. Numerical experiments on real and simulated data are included to test the new procedure. They show that HSS not only solves all the most popular benchmark problems proposed in the literature without having exact information on ARX subsystems order, but can also identify more complex (high-order) piecewise affine systems. MATLAB code implementing the approach, called Hybrid Stable Spline Toolbox, is also made available.