Fast Particle Filters and Their Applications to Adaptive Control in Change-Point ARX Models and Robotics

The Kalman filter has provided an efficient and elegant solution to control problems in linear stochastic systems. For nonlinear stochastic systems, control problems become much more difficult and a large part of the literature resorts to linear approximations so that an "extended Kalman filter" or a "mixture of Kalman filters" can be used in place of the Kalman filter for linear systems. Since these linear approximations are local expansions around the estimated states, they may perform poorly when the true state differs substantially from its estimate. Substantial progress was made in the past decade for the filtering problem with the development of particle filters. This development offers promise for solving some long-standing control problems which we consider in this chapter. As noted by Ljung & Gunnarsson (1990), a parameterized description of a dynamic system that is convenient for identification is to specify the model's prediction of the output yt as a function of the parameter vector and past inputs and outputs us and ys, respectively, for s < t. When the function is linear in , this yields the regression model , which includes as a special case the ARX model (autoregressive model with exogenous inputs) that is widely used in control and signal processing. Here the regressor vector is