Physics-informed Gaussian Processes as Linear Model Predictive Controller with Constraint Satisfaction

TL;DR

The paper addresses constrained MPC for linear systems by embedding the dynamics in a physics-informed Gaussian Process (LODE-GP) within a Control-as-Inference framework. It introduces Matérn latent kernels to enhance expressivity, a cost-reweighting step to align with quadratic objectives, and a Hamiltonian Monte Carlo–based sampling scheme to enforce state/input constraints while preserving the ODE structure on a discretized grid. The authors prove that Matérn-based LODE-GPs are dense in the solution space and show how to obtain a closed-form, Gaussian posterior optimized for the quadratic cost, followed by constraint satisfaction via sampling from a truncated Gaussian. The approach is validated on a spring-mass system and a 2D integrator, demonstrating that constraint satisfaction is achieved and control performance is competitive, with Matérn kernels offering notable advantages over standard SE kernels for constrained scenarios.

Abstract

Model Predictive Control evolved as the state of the art paradigm for safety critical control tasks. Control-as-Inference approaches thereof model the constrained optimization problem as a probabilistic inference problem. The constraints have to be implemented into the inference model. A recently introduced physics-informed Gaussian Process method uses Control-as-Inference with a Gaussian likelihood for state constraint modeling, but lacks guarantees of open-loop constraint satisfaction. We mitigate the lack of guarantees via an additional sampling step using Hamiltonian Monte Carlo sampling in order to obtain safe rollouts of the open-loop dynamics which are then used to obtain an approximation of the truncated normal distribution which has full probability mass in the safe area. We provide formal guarantees of constraint satisfaction while maintaining the ODE structure of the Gaussian Process on a discretized grid. Moreover, we show that we are able to perform optimization of a quadratic cost function by closed form Gaussian Process computations only and introduce the Matérn kernel into the inference model.

Figures (2)

• Figure 1: Comparison of the outputs for the example given in Section \ref{['sec:exp:spring']}. The state $x_1$ is given in blue, the state $x_2$ in orange and the control $u$ in green. In the first row, we see GPSE (left), GPM (middle), GPMO (right), in the second row we see the samples generated via HMC for the respective constrained LODE-GP. The constraints are given via the dashed lines at $z_{\min}=-1$ and $z_{\max} = 1$. We see that none of the three approaches satisfies the constraints without sampling. While GPSE provides smooth paths with small oscillations, the GPM and GPMO paths are rougher and have no oscillations. For the sampled versions, GPSE still suffers from oscillations via its smoothness requirements. The GPM and GPMO paths are rougher and obtain more control advantage behavior.
• Figure 2: Comparison of the outputs for the integrator example in Section \ref{['sec:exp:int']}. The state $x_1$ is given in blue, the state $x_2$ in orange and the control $u$ in green. The constraint of $x_1$ is given via the dashed line in blue by $x_{\max} = 1$ and $x_{\min}=0$. The constraints of $x_2$ and $u$ are given via the dashed line in red. We observe a significantly better adapation to the constraints of GPM in comparison to GPSE. The smoothness of the se kernel prevents the control function from aligning the constraint boundaries more aggressively. All models using sampling satisfy the constraints.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• proof