Bayesian Joint Input-State Estimation for Nonlinear Systems

Abstract

This work suggests a solution for joint input-state estimation for nonlinear systems. The task is to recover the internal states of a nonlinear oscillator, the displacement and velocity of the system, and the unmeasured external forces applied. To do this, a Gaussian process latent force model is developed for nonlinear systems. The model places a Gaussian process prior over the unknown input forces for the system, converts this into a state-space form and then augments the nonlinear system with these additional hidden states. To perform inference over this nonlinear state-space model a particle Gibbs approach is used combining a “Particle Gibbs with Ancestor Sampling” Markov kernel for the states and a Metropolis-Hastings update for the hyperparameters of the Gaussian process. This approach is shown to be effective in a numerical case study on a Duffing oscillator where the internal states and the unknown forcing are recovered, each with a normalised mean-squared error less than 0.5%. It is also shown how this Bayesian approach allows uncertainty quantification of the estimates of the states and inputs which can be invaluable in further engineering analyses.

Nonlinear Gaussian Process Latent Force Models for Input Estimation in Hysteretic Systems

“…This paper has sought to understand the performance of a nonlinear input-state estimation methodology proposed in [1] when the nonlinear system contains a hysteretic nonlinearity. A Bouc-Wen system was chosen as a typical example of such a nonlinear dynamic model.…”

“…It should be noted at this point, that this formulation gives the state-space form for a single output GP; however, it is trivial to extend this to a number of independent GPs acting on the different degrees of freedom in equation (1).…”

“…The algorithm for inferring the smoothing distribution using an MCMC approach incorporating the CPF with ancestor sample will now be given, a more thorough review and comparison with an alternative method can be found in [12]. It should also be noted that, should the (hyper)parameters of the system need to be inferred this can be done as part of a full Particle Gibbs with Ancestor Sampling [13] scheme; an example of this approach in the context of a GP latent force model can be found in [1].…”

“…Hysteresis, informally, introduces a memory effect into the system which can greatly complicate identification, including input-state estimation. The contribution of this work is to extend previous work on nonlinear input-state estimation [1] to the case of a hysteretic nonlinearity in order to demonstrate the general and powerful approach of nonlinear Gaussian process latent force modelling.…”

Nonparametric nonlinear restoring force and excitation identification with Legendre polynomial model and data fusion

“…33 A novel hybrid output-only structural damage identification method is presented without requiring the estimation of the input excitation. 34 Rogers et al 35 proposed a solution for joint input-state estimation for nonlinear systems. A novel framework to accurately estimate nonlinear structural model parameters and unknown external inputs using sparse sensor networks is proposed and numerically validated with a realistic three-dimensional nonlinear steel frame subjected to unknown seismic ground motion.…”