# Learning dynamical systems using particle filters

**Thomas Schön
**

Division of Systems and Control

Department of Information Technology

Uppsala University

### Abstract:

The particle filter/smoother provide computational solutions to the nonlinear state filtering/smoothing problem in nonlinear/non-Gaussian state space models. The particle filter and the particle smoother and the most popular members of the Sequential Monte Carlo (SMC) family of methods. The particle filter is introduced and it is then used to solve two type of problems. The first problem type is that of state estimation in various industrial sensor fusion and positioning applications, including navigation of fighter aircraft and indoor positioning. The second problem type is that of learning nonlinear dynamical models based on measured data (commonly referred to as the nonlinear system identification problem).

The particle filters/smoothers open up for nonlinear system identification (both maximum likelihood and Bayesian solutions) in a systematic way. As we will see it is not a matter of directly applying the SMC algorithms, but there are several ways in which they enter as a natural part of the solution. The use of SMC for nonlinear system identification is a relatively recent development and the aim here is to first provide a brief overview of how SMC can be used in solving challenging nonlinear system identification problems by sketching both maximum likelihood and Bayesian solutions. The Bayesian solution consists of a systematic combination of SMC and Markov chain Monte Carlo (MCMC) samplers, where SMC is used to construct the high-dimensional proposal density for the MCMC sampler. The first results emerged in 2010 and since then we have witnessed a steadily increasing activity within this area. We focus on our new sampler "Particle Gibbs with ancestor sampling" and show its use in computing the posterior distribution for a general Wiener model (i.e. identifying a Bayesian Wiener model).