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

arXiv: https://arxiv.org/abs/2501.15849 · HTML: https://mlsanity.com/paper/2501.15849/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 44 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Builds on (most cited first, 21 of 21)

- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Data-Driven Model Predictive Control with Stability and Robustness Guarantees](https://mlsanity.com/paper/1906.04679/) (2019)
- [Stochastic data-driven model predictive control using Gaussian processes](https://mlsanity.com/paper/1908.01786/) (2019)
- [Distributionally Robust Chance Constrained Data-enabled Predictive Control](https://mlsanity.com/paper/2006.01702/) (2020)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [A trajectory-based framework for data-driven system analysis and control](https://mlsanity.com/paper/1903.10723/) (2019)
- [Robust and Kernelized Data-Enabled Predictive Control for Nonlinear Systems](https://mlsanity.com/paper/2206.01866/) (2022)
- [State space models vs. multi-step predictors in predictive control: Are state space models complicating safe data-driven designs?](https://mlsanity.com/paper/2203.15471/) (2022)
- [Nonlinear Data-Enabled Prediction and Control](https://mlsanity.com/paper/2101.03187/) (2021)
- [Basis functions nonlinear data-enabled predictive control: Consistent and computationally efficient formulations](https://mlsanity.com/paper/2311.05360/) (2023)
- [Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding](https://mlsanity.com/paper/2409.16389/) (2024)
- [Data-Driven Internal Model Control of Second-Order Discrete Volterra Systems](https://mlsanity.com/paper/2003.14158/) (2020)
- [Data-driven predictive control in a stochastic setting: a unified framework](https://mlsanity.com/paper/2203.10846/) (2022)
- [Data-based Control of Feedback Linearizable Systems](https://mlsanity.com/paper/2204.01148/) (2022)
- [Harnessing Uncertainty for a Separation Principle in Direct Data-Driven Predictive Control](https://mlsanity.com/paper/2312.14788/) (2023)
- [Data-Driven Optimal Control of Bilinear Systems](https://mlsanity.com/paper/2112.15510/) (2021)
- [Exploring the Links between the Fundamental Lemma and Kernel Regression](https://mlsanity.com/paper/2403.05368/) (2024)
- [Maximum Likelihood Estimation in Data-Driven Modeling and Control](https://mlsanity.com/paper/2011.00925/) (2020)
- [On the design of persistently exciting inputs for data-driven control of linear and nonlinear systems](https://mlsanity.com/paper/2303.08707/) (2023)
- [Robust and efficient data-driven predictive control](https://mlsanity.com/paper/2409.18867/) (2024)
- [Design of input for data-driven simulation with Hankel and Page matrices](https://mlsanity.com/paper/2109.04742/) (2021)

## Abstract (arXiv)

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.