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

We consider state and parameter estimation for a dynamical system having both time-varying and time-invariant parameters. It has been shown that the robustness of the Markov Chain Monte Carlo (MCMC) algorithm for estimating time-invariant parameters alongside nonlinear filters for state estimation provided more reliable estimates than the estimates obtained solely using nonlinear filters for combined state and parameter estimation. In a similar fashion, 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 to the MCMC algorithm. In a standard method, we augment the state vector to include the original states of the system and the subset of the parameters that are time-varying. Each time-varying parameter is perturbed by a white noise process, and we treat the strength of this artificial noise as an additional time-invariant parameter to be estimated by MCMC, circumventing the need for manual tuning. Conventionally, both time-varying and time-invariant parameters are appended in the state vector, and thus for the purpose of estimation, both are free to vary in time. However, allowing time-invariant system parameters to vary in time introduces artificial dynamics into the system, which we avoid by treating these time-invariant parameters as static and estimating them using MCMC. Furthermore, by estimating the time-invariant parameters by MCMC, the augmented state is smaller and the nonlinearity in the ensuing state space model will tend to be weaker than in the conventional approach. 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.