This work formalizes a deterministic trigonometric feature Gaussian process (DTF-GP), a finite-dimensional kernel approximation based on discretized trigonometric features that reduces GP regression to Bayesian linear regression in feature space, and derives a high-probability uniform uncertainty bound for the proposed DTF-GP.
Abstract
Learning-based Model Predictive Control (MPC) using Gaussian processes (GPs) is an effective approach for safe control in the presence of model mismatch. High-probability safety guarantees typically require uncertainty bounds that hold uniformly over the entire state--input domain, but existing bounds are available only for full GP regression. Since exact GP inference scales poorly with the number of data points, its deployment is impractical in large-data regimes. We close this gap by developing a scalable GP framework that admits the derivation of uniform uncertainty bounds. We formalize a deterministic trigonometric feature Gaussian process (DTF-GP), a finite-dimensional kernel approximation based on discretized trigonometric features that reduces GP regression to Bayesian linear regression in feature space. We derive a high-probability uniform uncertainty bound for the proposed DTF-GP and provide its closed-form solution for the squared-exponential kernel case. Finally, we integrate the DTF-GP into a learning-based MPC scheme and demonstrate that it provides high-probability safety guarantees and exploration performance comparable to a full GP while improving computational efficiency in large-data regimes.
We propose a novel Stochastic Nonlinear Model Predictive Control (SNMPC) framework for nonlinear systems with additive noise. Building on recent advances in nonlinear uncertainty propagation, we show that the state distribution of the system can be tractably approximated over time by Gaussian mixture distributions, with formal error bounds in Wasserstein distance. This representation yields closed-form expressions for expected costs and chance constraints, which become exact for affine constraints and exact up to a constant for quadratic costs. Consequently, the resulting control problem can be solved efficiently via nonlinear programming, while providing formal open-loop guarantees of correctness and asymptotic optimality. Experiments on a set of benchmarks demonstrate that the proposed approach compares favorably with existing methods in nonlinear settings with multi-modal disturbances, where standard approaches lead to poorly scaled solutions and unsafe or overly conservative control actions.
Konstantinos Prattis, Luca Laurenti, A. Dabiri· 0 citations
Gaussian processes (GPs) are powerful, nonparametric models, widely recognized as universal function approximators due to their ability to provide robust probabilistic predictions alongside quantified uncertainty estimates. This has allowed for the modeling of complex processes in chemical engineering with applications to process optimization, prediction, and control. However, the practical adoption of standard GPs is severely constrained by their computational training complexity O(n3) . This bottleneck is particularly problematic in modern high-throughput environments, such as those associated with Industry 4.0, big data applications, and advanced manufacturing. Addressing this demand requires the implementation of highly efficient online learning and model updating strategies. This paper specifically looks at deterministic nonlinear Model Predictive Control (MPC) and proposes a framework for efficiently identifying a highly performant Gaussian process approximation model and re-identification metric, alongside stability and feasibility analyses. The effectiveness of these techniques in a real-time system is demonstrated through a cascaded tanks experiment.
Michael W. Fouts, David S. Mebane, Fernando V. Lima· Industrial & Engineering Che...· 0 citations
Intermittent state measurements pose fundamental challenges to model predictive control of constrained nonlinear systems because prediction uncertainty grows during feedback outages and measurement-triggered resets disrupt nominal state propagation, potentially compromising closed-loop stability and recursive feasibility. This paper develops a Koopman-based stochastic MPC framework with probabilistically truncated soft constraints. Specifically, a Lipschitz-constrained deep Koopman model provides a linear latent predictor, enabling computationally efficient online optimization. The intermittent measurement process is modeled as a two-mode discrete-time Markov chain, yielding a unified Markov jump error model for open-loop propagation and measurement-triggered resets. Under numerically verifiable sufficient conditions, the prediction error is shown to be mean-square ultimately bounded, and an explicit uniform second-moment bound is obtained. A distribution-free probabilistic error radius is then constructed for a prescribed confidence level and used to truncate dropout-dependent constraint tightening. An exact-penalty soft-constraint mechanism accommodates reset-induced jumps and prolonged dropouts. Under the stated terminal compatibility and bounded-disturbance conditions, recursive feasibility and mean-square ultimate boundedness of the closed-loop regulation error are established. Numerical simulations on a visual-servoing tracking task corroborate these theoretical results and demonstrate effective tracking under stochastic measurement unavailability.
Guanzhi Liu, Tong Wu, Lixian Zhang et al.· 0 citations
The extended state observer is widely used for states and disturbances estimation in uncertain systems. However, the traditional extended state observer typically assumes that disturbances eventually converge to constants, making it difficult to guarantee theoretical convergence for disturbances with non-zero derivatives. To address this issue, we model the lumped disturbance as a stationary Gaussian process via a Matérn covariance kernel, reformulated as a stochastic differential equation within an extended system. The Gaussian kernel extended state observer is implemented through a Kalman filter, enabling accurate disturbance estimation. Stability is rigorously proven using a Lyapunov function for Itô processes, establishing mean-square exponential practical stability under detectability and stabilizability conditions. Comparative numerical simulations on a permanent magnet synchronous motor model demonstrate that the Gaussian kernel extended state observer largely outperforms the conventional solution in disturbance estimation accuracy and exhibits strong robustness to non-Gaussian noise. It yields smaller observer errors across most states, thereby offering a theoretically rigorous and practical framework for disturbance estimation and compensation in control systems with stationary stochastic disturbances.
Liangda Hu, Yinghua Jin· Transactions of the Institut...· 0 citations
A RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations is proposed and robust satisfaction of the original nonlinear constraints is proved by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state.
