A Gaussian process latent force model for joint input-state estimation in linear structural systems

Abstract

The problem of combined state and input estimation of linear structural systems based on measured responses and a priori knowledge of structural model is considered. A novel methodology using Gaussian process latent force models is proposed to tackle the problem in a stochastic setting. Gaussian process latent force models (GPLFMs) are hybrid models that combine differential equations representing a physical system with data-driven non-parametric Gaussian process models. In this work, the unknown input forces acting on a structure are modelled as Gaussian processes with some chosen covariance functions which are combined with the mechanistic differential equation representing the structure to construct a GPLFM. The GPLFM is then conveniently formulated as an augmented stochastic state-space model with additional states representing the latent force components, and the joint input and state inference of the resulting model is implemented using Kalman filter. The augmented state-space model of GPLFM is shown as a generalization of the class of input-augmented state-space models, is proven observable, and is robust compared to conventional augmented formulations in terms of numerical stability. The hyperparameters governing the covariance functions are estimated using maximum likelihood optimization based on the observed data, thus overcoming the need for manual tuning of the hyperparameters by trial-and-error. To assess the performance of the proposed GPLFM method, several cases of state and input estimation are demonstrated using numerical simulations on a 10-dof shear building and a 76-storey ASCE benchmark office tower. Results obtained indicate the superior performance of the proposed approach over conventional Kalman filter based approaches.

Bayesian Joint Input-State Estimation for Nonlinear Systems

“…One of the most useful aspects of this approach is that the whole system remains a linear Gaussian state-space model which can be solved with the Kalman filter and RTS smoother to recover the filtering and smoothing distributions exactly. It is this form of the model that has been exploited previously in structural dynamics [5,6,10].…”

“…Some examples include, Naets et al [2] who use a Kalman filter approach for force identification, the Dual Kalman Filter approach of Azam et al [3], and Maes et al [4] who show the benefit of a smoothing approach to the problem at hand. Nayek et al [5] adopt a Gaussian process latent force approach to the joint input-state problem, in a similar manner to the work of Rogers et al [6] who also include parameter estimation.…”

Stochastic Oblique Impact on Composite Laminates: A Concise Review and Characterization of the Essence of Hybrid Machine Learning Algorithms

“…In this work, we are only interested in supervised learning techniques. Popular supervised learning techniques include Gaussian process or Kriging [19][20][21][22], Polynomial chaos expansion (PCE) [23][24][25], analysis-of-variance decomposition [26][27][28][29], Polynomial chaos based Kriging (PC-Kriging) [30][31][32][33] etc. In this work, we review three machine learning techniques in the context of stochastic low-velocity impact analysis.…”

On the application of Gaussian process latent force models for joint input-state-parameter estimation: With a view to Bayesian operational identification

“…The authors of this paper have previously introduced the use of this model for joint input-state-parameter estimation for structural dynamical systems in [20], where the method is demonstrated on a single-degree-of-freedom system and the mass of the system is considered known. Independently of this, Nayek et al [21] have recently shown the usefulness of this method in the case of input-state estimation only, i.e. the parameters of the system are considered known.…”