Data-Driven Prediction and Control of Hammerstein-Wiener Systems with Implicit Gaussian Processes

TL;DR

This paper introduces a novel Gaussian process-based approach for data-driven prediction and control of Hammerstein-Wiener systems, effectively handling output nonlinearities and leveraging structured kernel functions for improved performance.

Contribution

It develops an implicit GP model with structured kernels and virtual derivative points, enabling accurate prediction and control of nonlinear systems with theoretical guarantees.

Findings

Outperforms black-box GP models in prediction accuracy.

Provides explicit output prediction with optimized criteria.

Ensures chance constraint satisfaction in control applications.

Abstract

This work investigates data-driven prediction and control of Hammerstein-Wiener systems using physics-informed Gaussian process (GP) models that encode the block-oriented model structure. Data-driven prediction algorithms have been developed for structured nonlinear systems based on Willems' fundamental lemma. However, existing frameworks do not apply to output nonlinearities in Wiener systems and rely on a finite-dimensional dictionary of basis functions for Hammerstein systems. In this work, an implicit predictor structure is considered, leveraging the linearity for the dynamical part of the model. This implicit function is learned by GP regression, utilizing carefully designed structured kernel functions from linear model parameters and GP priors for the nonlinearities. Virtual derivative points are added to the regression by expectation propagation to encode monotonicity information of the nonlinearities. The linear model parameters are estimated as hyperparameters by assuming a stable spline hyperprior. The implicit GP model provides explicit output prediction by optimizing selected optimality criteria. The implicit model is also applied to receding horizon control with the expected control cost and chance constraint satisfaction guarantee. Numerical results demonstrate that the proposed prediction and control algorithms are superior to black-box GP models without model structure knowledge.