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.
Highlights
This work proposes a Bayesian framework to address joint input-state estimation tasks, with application to structural dynamics
This rigourous uncertainty quantification allows a fuller understanding of the behaviour of a system and can be invaluable in safety-critical applications. Recovering these distributions allows futher analysis of the system to be made under uncertainty—an example being fatigue damage accural calculations; doing so opens up the possibility of risk-based asset management. To achieve this aim a Gaussian process latent force model is combined with a nonlinear state-space representation of the system and joint inference is performed over the states, unknown inputs and hyperparameters of the Gaussian process by means of particle Gibbs [1]
In order to do this, the Duffing oscillator is chosen as the nonlinear system of interest, since it is a conceptually simple system which is still capable of exhibiting many interesting behaviours
Summary
In the Duffing oscillator used as a case study in this paper, this means the distributions over the displacement, velocity and forcing time histories at every point in time given a measured acceleration signal and the parameters of that system This rigourous uncertainty quantification allows a fuller understanding of the behaviour of a system and can be invaluable in safety-critical applications. As with almost any analysis in structural dynamics, the challenge increases significantly when moving to a nonlinear setting Despite this difficulty, attention is turning to nonlinear systems; Lei et al [11] present an approach based on an unscented Kalman filter for the task of input-state estimation.
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