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Gaussian Process-Based Prediction and Control of Hammerstein-Wiener Systems

Mingzhou Yin, Matthias A. Müller

TL;DR

This work tackles data-driven prediction and control of Hammerstein-Wiener systems with unknown input/output nonlinearities. It introduces an implicit Gaussian process predictor by imposing GP priors on the nonlinearities and treating the linear part as hyperparameters learned under a stable-spline prior, yielding explicit multi-step predictions. Key contributions include the implicit GP model design with kernel-based nonlinearities, a joint MAP/marginal likelihood hyperparameter learning scheme, and a receding-horizon controller with chance-constraint guarantees. Empirical results on simulated Hammerstein-Wiener examples show superior prediction accuracy and constraint-aware control compared with a black-box GP and linear predictors, highlighting the method's practical impact for data-driven nonlinear systems.

Abstract

This work investigates data-driven prediction and control of Hammerstein-Wiener systems using physics-informed Gaussian process models. Data-driven prediction algorithms have been developed for structured nonlinear systems based on Willems' fundamental lemma. However, existing frameworks cannot treat output nonlinearities and require a dictionary of basis functions for Hammerstein systems. In this work, an implicit predictor structure is considered, leveraging the multi-step-ahead ARX structure for the linear part of the model. This implicit function is learned by Gaussian process regression with kernel functions designed from Gaussian process priors for the nonlinearities. The linear model parameters are estimated as hyperparameters by assuming a stable spline hyperprior. The implicit Gaussian process model provides explicit output prediction by optimizing selected optimality criteria. The 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 Gaussian process models.

Gaussian Process-Based Prediction and Control of Hammerstein-Wiener Systems

TL;DR

This work tackles data-driven prediction and control of Hammerstein-Wiener systems with unknown input/output nonlinearities. It introduces an implicit Gaussian process predictor by imposing GP priors on the nonlinearities and treating the linear part as hyperparameters learned under a stable-spline prior, yielding explicit multi-step predictions. Key contributions include the implicit GP model design with kernel-based nonlinearities, a joint MAP/marginal likelihood hyperparameter learning scheme, and a receding-horizon controller with chance-constraint guarantees. Empirical results on simulated Hammerstein-Wiener examples show superior prediction accuracy and constraint-aware control compared with a black-box GP and linear predictors, highlighting the method's practical impact for data-driven nonlinear systems.

Abstract

This work investigates data-driven prediction and control of Hammerstein-Wiener systems using physics-informed Gaussian process models. Data-driven prediction algorithms have been developed for structured nonlinear systems based on Willems' fundamental lemma. However, existing frameworks cannot treat output nonlinearities and require a dictionary of basis functions for Hammerstein systems. In this work, an implicit predictor structure is considered, leveraging the multi-step-ahead ARX structure for the linear part of the model. This implicit function is learned by Gaussian process regression with kernel functions designed from Gaussian process priors for the nonlinearities. The linear model parameters are estimated as hyperparameters by assuming a stable spline hyperprior. The implicit Gaussian process model provides explicit output prediction by optimizing selected optimality criteria. The 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 Gaussian process models.
Paper Structure (11 sections, 3 theorems, 51 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 3 theorems, 51 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement and Background
  3. Data-Driven Prediction as Linear Systems
  4. Gaussian Process Data-Driven Prediction
  5. Gaussian Process Regression
  6. Kernel Design and Posterior Prediction
  7. Hyperparameter Tuning
  8. Recovering Nonlinearities
  9. Data-Driven Predictive Control with Implicit Gaussian Process Model
  10. Numerical Examples
  11. Conclusions

Key Result

Lemma 1

Suppose $\left(\mathbf{u}^d,\mathbf{y}^d\right):=\left(u_k^d,y_k^d\right)_{k=1}^N$ is a noise-free trajectory of the Hammerstein-Wiener system eq:sys. Let $\bar{\mathbf{u}}^d := \Psi(\mathbf{u}^d)$, $\bar{\mathbf{y}}^d := \Phi(\mathbf{y}^d)$. If $\bar{\mathbf{u}}^d$ is persistently exciting of order

Figures (5)

  • Figure 1: Block diagram of Hammerstein-Wiener systems.
  • Figure 2: Comparison of data-driven prediction with different predictors.
  • Figure 3: Estimation of input and output nonlinear functions $\psi(u)$ and $\phi(y)$.
  • Figure 4: Closed-loop output trajectories with different data-driven predictive controllers.
  • Figure 5: Prediction accuracy and error bounds of the proposed control algorithm.

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • Corollary 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Lemma 2