Bayesian inference in non-Markovian state-space models with applications   to fractional order systems
  Battery impedance spectroscopy models are given by fractional order (FO) differential equations. In the discrete-time domain, they give rise to state-space models where the latent process is not Markovian. Parameter estimation for these models is therefore challenging, especially for non-commensurate FO models. In this paper, we propose a Bayesian approach to identify the parameters of generic FO systems. The computational challenge is tackled with particle Markov chain Monte Carlo methods, with an implementation specifically designed for the non-Markovian setting. The approach is then applied to estimate the parameters of a battery non-commensurate FO equivalent circuit model. Extensive simulations are provided to study the practical identifiability of model parameters and their sensitivity to the choice of prior distributions, the number of observations, the magnitude of the input signal and the measurement noise. 
Pierre E. Jacob, S.M.Mahdi Alavi, Adam Mahdi, Stephen J. Payne, and   David A. Howey
math.OC stat.CO