Identification of a Duffing oscillator using particle Gibbs with ancestor sampling

T J Rogers,A Lindholm,K Worden,E J Cross,T B Schön

doi:10.1088/1742-6596/1264/1/012051

T J Rogers, A Lindholm + Show 3 more

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https://doi.org/10.1088/1742-6596/1264/1/012051

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Abstract

The Duffing oscillator remains a key benchmark in nonlinear systems analysis and poses interesting challenges in nonlinear structural identification. The use of particle methods or sequential Monte Carlo (SMC) is becoming a more common approach for tackling these nonlinear dynamical systems, within structural dynamics and beyond. This paper demonstrates the use of a tailored SMC algorithm within a Markov Chain Monte Carlo (MCMC) scheme to allow inference over the latent states and parameters of the Duffing oscillator in a Bayesian manner. This approach to system identification offers a statistically more rigorous treatment of the problem than the common state-augmentation methods where the parameters of the model are included as additional latent states. It is shown how recent advances in particle MCMC methods, namely the particle Gibbs with ancestor sampling (PG-AS) algorithm is capable of performing efficient Bayesian inference, even in cases where little is known about the system parameters a priori. The advantage of this Bayesian approach is the quantification of uncertainty, not only in the system parameters but also in the states of the model (displacement and velocity) even in the presence of measurement noise.

Highlights

The Duffing equation presented 100 years ago by George Duffing [1] remains one of the most studied equations in nonlinear dynamics
This paper has presented the use of Particle Gibbs with Ancestor Sampling and Particle Rejuvenation as a viable technique for Bayesian estimation of nonlinear dynamical systems encountered in structural dynamics
It has been shown that the methodology is capable of recovering the smoothing distributions of the states and the distributions of the model parameters with low error in relation to the maximum a posteriori estimates and that the known true values and trajectories are within the probability mass of the estimated distributions

Summary

Introduction

The Duffing equation presented 100 years ago by George Duffing [1] remains one of the most studied equations in nonlinear dynamics. The contribution of this paper is to simultaneously estimate the smoothing distributions of the states (displacement and velocity of the oscillator) alongside the parameters of the Duffing oscillator This is achieved by use of a particle Gibbs scheme with ancestor sampling and particle. The procedure for Particle Gibbs with Ancestor Sampling is shown, the use of a Particle Rejuvenation scheme is shown for handling degenerate models The application of this methodology to the identification of a nonlinear dynamic system, namely the Duffing oscillator, is presented in Section 4 where the smoothing distributions of the states and the distributions of the model parameters are recovered. For matrices the same notation is adopted but with commas separating dimensions and following column major order This is again approximated using importance sampling such that the (unnormalised) importance weights of the filtering density are given by, wti gθ (yt | xt, ut) wti−1fθ The procedure for running a bootstrap particle filter is shown in Algorithm 1

9: Weighting

Particle Gibbs with Ancestor Sampling

Discussion