Semialgebraic Outer Approximations for Set-Valued Nonlinear Filtering

This paper addresses the set-valued filtering problem for discrete time-varying dynamical systems, whose process and measurement equations are polynomial functions of the system state. According to a set-membership framework, the process and measurement noises, as well as the initial state, are assumed to belong to bounded uncertainty regions, which are supposed to be generic semialgebraic sets described by polynomial inequalities. A sequential algorithm, based on sum-of-squares (SOS) representation of positive polynomials is proposed to compute a semialgebraic set described by an a-priori fixed number of polynomial constraints which is guaranteed to contain the true state of the system with certainty.