Physics-informed Gaussian Processes as Linear Model Predictive Controller (2412.04502v1)

Abstract: We introduce a novel algorithm for controlling linear time invariant systems in a tracking problem. The controller is based on a Gaussian Process (GP) whose realizations satisfy a system of linear ordinary differential equations with constant coefficients. Control inputs for tracking are determined by conditioning the prior GP on the setpoints, i.e. control as inference. The resulting Model Predictive Control scheme incorporates pointwise soft constraints by introducing virtual setpoints to the posterior Gaussian process. We show theoretically that our controller satisfies asymptotical stability for the optimal control problem by leveraging general results from Bayesian inference and demonstrate this result in a numerical example.

• The paper integrates Gaussian Processes (specifically Linear Ordinary Differential Equation Gaussian Processes) with Model Predictive Control for linear systems, framing control synthesis and dynamics under a unified framework called Control as Inference.
• Theoretical analysis establishes asymptotic stability for the proposed controller using Bayesian principles and demonstrates optimality of the output control function within a Reproducing Kernel Hilbert Space.
• The approach handles dynamic constraints flexibly using soft constraints within the GP model and is validated on a spring-mass-damper system, showing effective regulation and constraint adherence.

Physics-Informed Gaussian Processes as Linear Model Predictive Controller

This paper presents a novel integration of Gaussian Processes (GPs) with Model Predictive Control (MPC), focusing specifically on linear systems defined by ordinary differential equations (ODEs). The authors, Tebbe, Besginow, and Lange-Hegermann, propose an advancement in model predictive control through the introduction of the Linear Ordinary Differential Equation Gaussian Processes (LODE-GPs), in which the synergy of GPs and MPC substantiates a unified model for control synthesis and system dynamics. This innovative approach leverages the predictive distribution of GPs to achieve both system control and adherence to dynamic constraints through posterior inference, forming a strategy aptly described as Control as Inference (CAI).

The methodology expands upon classical tracking MPC, which typically separates the predictive model and control strategy, by employing GPs to simultaneously manage the synthesis of control laws, thereby incorporating the control inputs and system dynamics within a single framework. This integration is expected to enhance the control strategy, enabling smooth transition control functions and satisfying control constraints more flexibly through the nuanced use of GPs.

Key Contributions

1. Unified Control Framework: The paper elucidates the union of dynamics and control law synthesis under a GP framework. The control problem thus reduces to a simple GP posterior inference problem with a strong connection to Bayesian inference principles.
2. Stability and Optimality: Asymptotic stability of the controller is theoretically established by utilizing fundamental Bayesian principles, ensuring that the posterior GP aligns with the marginalization principles as time extends to infinity. Additionally, the output control function is constrained and optimized within the Reproducing Kernel Hilbert Space (RKHS), ensuring optimal solutions relative to the chosen covariance kernel.
3. Handling Constraints: Unlike traditional methods that employ hard constraints, the use of soft constraints within GPs offers a level of flexibility and robustness in handling dynamic restrictions, effectively incorporating these constraints within the GP’s probabilistic model instead of purely on the optimization scale.
4. Practical Implementation and Results: Utilizing a spring-mass-damper system as a test case, the paper demonstrates the controller's effectiveness at regulating an unstable system. The conformity to desired states is achieved with minimal constraint violations and enhanced control precision, as evidenced by computational results showing reduced control error within small ranges.

Implications and Future Directions

This research is pivotal in showcasing the usefulness of physics-informed Gaussian processes in model-based predictive control, marking significant potentials for industries requiring precise and adaptive control solutions such as robotics, autonomous systems, and automated process control. The theoretical foundation laid out forms a path for further exploration in discrete versions of the LODE-GP model and application in areas where measurement noise and data sparsity present substantial challenges.

Future developments could extend into exploring non-linear extension versions of LODE-GPs for complex systems, deep integration with reinforcement learning paradigms for adaptive modeling over time, and adapting this approach for real-time applications in resource-constrained environments where model inference speed is critical.

Conclusion

The presented controller leverages GPs for system control and stability through the innovative use of LODE-GPs, offering a robust alternative to traditional MPC implementations. The formalism of Bayesian inference in controlling linear systems, with the comprehensive consideration of system dynamics and constraints, signifies a substantial forward step in applied control theory. As these methods gain traction, their impact across varying control systems will likely catalyze considerable advancements in both theoretical and practical domains of AI-driven control systems.