Identification and model (in)validation of switched ARX systems: A moment-based approach

Identification of switched systems has received considerable attention during the past few years. This chapter addresses the problem of identification and (in)validation of switched autoregressive models with external inputs from experimental data. In the first part of the chapter identification of switched linear systems from noisy measurements is considered. Given a finite number of measurements of input/output and a bound on the measurement noise, the objective is to identify a switching sequence and a set of linear submodels that are compatible with the a priori information, while minimizing the number of submodels. A deterministic approach based on convex optimization is proposed. Namely, by recasting the identification problem as polynomial optimization, we develop deterministic algorithms, in which the inherent sparse structure is exploited. A finite dimensional semi-definite problem is then given which is equivalent to the identification problem. Moreover, to address computational complexity issues, an equivalent rank minimization problem subject to deterministic linear matrix inequality constraints is provided, as efficient convex relaxations for rank minimization are available in the literature. Since the identification problem is generically NP-Hard, the majority of existing algorithms, including the one we proposed, are based on heuristics or relaxations. Therefore, it is crucial to check the validity of the identified models against additional experimental data. The second part of this chapter addresses the problem of model (in)validation for switched linear autoregressive exogenous systems with unknown switches. Necessary and sufficient conditions are obtained for a given model to be (in)validated by the experimental data. In principle, checking these conditions requires solving a sequence of convex optimization problems involving increasingly large matrices. However, as we show, if in the process of solving these problems either a positive solution is found or the so-called flat extension property holds, then the process terminates with a certificate that either the model has been invalidated or that the experimental data is indeed consistent with the model and a-priori information. By using duality, the proposed approach exploits the inherently sparse structure of the optimization problem to substantially reduce its computational complexity. The effectiveness of the proposed methods is illustrated using both academic examples and a non-trivial problem arising in computer vision: activity monitoring.