Hybrid system identification with faulty measurements and its application to activity analysis

This paper addresses the problem of set membership identification of a class of discrete-time affine hybrid systems, switched affine models, in the presence of sensor failures. Given a finite collection of input/output measurements and a bound on the number of subsystems, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information. Contrary to existing work, here we allow for instantaneous failures in the measurement sensors at unknown times. These failures lead to corrupted input/output data, that if used in the identification process would result in substantial identification errors. The main result of the paper shows that, exploiting the fact that these failures are infrequent, combined with an algebraic-geometric argument, allows for recasting the problem into an optimization form where the objective is to simultaneously minimize the rank of a matrix and the number of nonzero rows of a second one. While in principle this is a challenging, non-convex problem, exploiting recent results on convex relaxations of rank and block-sparsity leads to an efficient, semi-definite optimization based identification algorithm. Finally, these results are illustrated using both simulations and a practical example that arises in computer vision where the aim is to analyze the activity of a person in the presence of sensor failures.