Model (in) validation of switched ARX systems with unknown switches and its application to activity monitoring

Identification of switched linear systems has received considerable attention during the past few years. Since the problem is generically NP-Hard, the majority of existing algorithms are based on heuristics or relaxations. Therefore, it is crucial to check the validity of the identified models against additional experimental data. This paper addresses the problem of model (in)validation for multi-input multi-output switched affine autoregressive exogenous systems with unknown switches. Our main result provides necessary and sufficient conditions 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 in the paper, 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 method is illustrated using both academic examples and a non-trivial problem arising in computer vision: activity monitoring.