Convergence of Particle Filter for Output Feedback Control

In the existing literature, convergence results for particle filters are given explicitly only for the case when the underlying dynamic model is a Markov process. When output feedback control is used, the evolution of the state process is no longer Markovian due to the dependence of inputs on the outputs. In this paper, it is shown that the random probability measures produced by the particle filter converge to the true prior and posterior measures in this nonMarkovian case. Firstly, it is proved that the recursive equations relating the prior and posterior measures continue to hold for output feedback control. These recursive equations are then used to show the required convergence of the random measures. Finally, the convergence is also illustrated using simulations on a nonlinear dynamical system.