Feedback Particle Filter

Abstract

A new formulation of the particle filter for nonlinear filtering is presented, based on concepts from optimal control, and from the mean-field game theory. The optimal control is chosen so that the posterior distribution of a particle matches as closely as possible the posterior distribution of the true state given the observations. This is achieved by introducing a cost function, defined by the Kullback-Leibler (K-L) divergence between the actual posterior, and the posterior of any particle.The optimal control input is characterized by a certain Euler-Lagrange (E-L) equation, and is shown to admit an innovation error-based feedback structure. For diffusions with continuous observations, the value of the optimal control solution is ideal. The two posteriors match exactly, provided they are initialized with identical priors. The feedback particle filter is defined by a family of stochastic systems, each evolving under this optimal control law.A numerical algorithm is introduced and implemented in two general examples, and a neuroscience application involving coupled oscillators. Some preliminary numerical comparisons between the feedback particle filter and the bootstrap particle filter are described.

Leader-based versus soft control of multi-agent swarms

“…(1)), the Kalman-Bucy filter is well known to be optimal. For nonlinear problems, a recent filtering mechanism uses feedback particles (i.e., agents) to empirically form the distribution P[X(t) = x | Z(t) = z] [9]. The Feedback Particle Filter (FPF) lets a swarm of N particles follow the same dynamic f as the noisy input signal Y (t) (g, however, is assumed to be linear).…”

A coupled oscillators-based control architecture for locomotory gaits

“…The methodology of this paper is a synthesis of several papers from our research group at the University of Illinois: the feedback particle filtering methodology for diffusion appears in [30], [29], [28]; its application to the problem of phase estimation is described in [24], [26]; a feedback particle filter-based approach to optimal control of partially observed diffusions is the subject of [21], [25]. In the present paper, we bring together these research threads to propose a CPG architecture for control of locomotion.…”

Control with rhythms: A CPG architecture for pumping a swing

“…The considerations of this section are based on [18]. Feedback Particle Filter (FPF) is a simulation-based algorithm to approximate the posterior distribution for Itô diffusions,…”

“…The optimal control is shown to be a function of certain averages taken over the population, {Θ i t : 1 ≤ i ≤ N}. The methodology of this paper is a synthesis of several papers from our research group at the University of Illinois: the feedback particle filtering methodology for diffusion appears in [17], [16], [18]; its application to the problem of phase estimation is described in [13], [14]; a feedback particle filter-based approach to optimal control of partially observed diffusions is the subject of [11]. In the present paper, we bring together these research threads to propose a CPG architecture for optimal control of rhythmic behavior.…”