Robust Bayesian state and parameter estimation framework for stochastic dynamical systems with combined time-varying and time-invariant parameters

Abstract

We consider the joint state and parameter estimation for dynamical systems having both time-varying and time-invariant parameters. For systems with time-invariant parameters, it has previously been demonstrated that although nonlinear filters may be used for concurrent state and parameter estimation, employing the Markov Chain Monte Carlo (MCMC) algorithm for estimating time-invariant parameters with nested nonlinear filters for state estimation provides more reliable estimates. Following established methods for parameter estimation using filters, we augment the state vector to include the original system states in addition to the subset of the parameters that are known to vary in time. Conventionally, both time-varying and time-invariant parameters would be appended in the state vector, and thus for the purpose of estimation, both would free to vary in time. However, allowing parameters that are known to be time-invariant to change in time introduces non-physical dynamics into the original system. To avoid inducing artificial dynamics, the time-invariant parameters are estimated using MCMC. Furthermore, by estimating the time-invariant parameters by MCMC, the dimensionality of the augmented state is smaller and the nonlinearity in the ensuing state space model will tend to be weaker compared to the conventional approach. The time-varying parameters are perturbed by random noise processes, and the amplitude of the artificial noise is estimated alongside the time-invariant system parameters using MCMC. Estimating the noise strength in this procedure circumvents the need for manual tuning. We illustrate the above-described approach for a simple dynamical system in which some model parameters are time-varying, while the remaining parameters are time-invariant. We adopt the extended Kalman filter (EKF) for state estimation and the estimation of the time-varying system parameters, but reserve the task of estimating time-invariant parameters, including the model noise strength and the artificial noise strength associated with the time-varying parameter to the MCMC algorithm.