Bounded complexity ℓ℞ filters for switched systems

This paper considers the worst-case estimation problem in the presence of unknown but bounded noise for piecewise linear switched systems. Contrary to stochastic approaches, the goal here is to confine the estimation error within a bounded set. Previous work dealing with the problem has shown that the complexity of estimators based upon the idea of constructing the state consistency set (e.g. the set of all states consistent with the a-priori information and experimental data) cannot be bounded a-priori, and can, in principle, continuously increase with time. To avoid this difficulty in this paper we propose a class of bounded complexity filters, based upon the idea of confining r-length error sequences (rather than states) to hyperrectangles. The main result of the paper shows that this approach leads to computationally tractable filters, that only require the on-line solution of a bounded complexity convex optimization problem. Moreover, as we show in the paper, these filters are (worst-case) optimal when operating in a simplified, restricted information scenario.