Particle Observers for Contracting Dynamical Systems

Abstract

In the present paper we consider a class of partially observed dynamical systems. As in the Rao-Blackwellized particle filter (RBPF) paradigm (see e.g., Doucet et al. 2000), we assume the state x can be broken into two sets of variables x=(z,r) and has the property that conditionally on z the system’s dynamics possess geometrical contraction properties, or is amenable to such a system by using a nonlinear observer whose dynamics possess contraction properties. Inspired by the RBPF we propose to use particles to approximate the r variable and to use a simple copy of the dynamics (or an observer) to estimate the rest of the state. This has the benefits of 1- reducing the computational burden (a particle filter would sample the variable x also), which is akin to the interest of the RBPF, 2- coming with some indication of stability stemming from contraction (actual proofs of stability seem difficult), and 3- the obtained filter is well suited to systems where the dynamics of x conditionally on z is precisely known and the dynamics governing the evolution of z is quite uncertain.