Physics-informed Gaussian Processes for Model Predictive Control of Nonlinear Systems

TL;DR

We address model predictive control for nonlinear dynamical systems by leveraging a physics-informed Gaussian Process (GP) that encodes linear ODE dynamics in its kernel (LODE-GP). The nonlinear system is handled via local linearization around an asymptotically stable equilibrium, enabling an MPC scheme that treats control as an inference problem conditioned on setpoints and constraints. The approach yields open-loop stability and finite-time convergence to the equilibrium through an endpoint constraint, demonstrated on a nonlinear two-tank system. The method offers data-efficient control with uncertainty quantification, integrating hard and soft constraints as datapoints and providing a path toward global extension through gain scheduling and bounded likelihoods.

Abstract

Recently, a novel linear model predictive control algorithm based on a physics-informed Gaussian Process has been introduced, whose realizations strictly follow a system of underlying linear ordinary differential equations with constant coefficients. The control task is formulated as an inference problem by conditioning the Gaussian process prior on the setpoints and incorporating pointwise soft-constraints as further virtual setpoints. We apply this method to systems of nonlinear differential equations, obtaining a local approximation through the linearization around an equilibrium point. In the case of an asymptotically stable equilibrium point convergence is given through the Bayesian inference schema of the Gaussian Process. Results for this are demonstrated in a numerical example.

Figures (5)

• Figure 1: (Left) A gp prior with zero mean and se covariance function. (Right) The same gp, but conditioned on datapoints (black asterisk). The blue line is its mean and the blue area is two times its standard deviation ($2\sigma$).
• Figure 2: Nonlinear water tank system. The tanks are connected by the valve $V_{12}$. Water can be pumped into the first tank with the control input $u_1(t)$ and is drained from the second tank with the valve $V_{2R}$.
• Figure 3: Model (A): The reference is present in the gp prior mean. Note the different time span.
• Figure 4: Model (B): The reference is added to the training data as endpoint constraint at $t=100$ s.
• Figure 5: Model (C): The reference is incorporated as soft constraints.

Theorems & Definitions (2)

• definition 1
• definition 2