Optimizing Model Predictive Control Performance and Stability via Constrained Bayesian Optimization of Neural Cost Functions

Core Concepts

This work explores a Bayesian optimization approach to learn the cost function parameters of a model predictive controller, optimizing closed-loop performance while ensuring stability through Lyapunov-based constraints.

Abstract

This paper proposes a framework for learning the cost function parameters of a model predictive controller (MPC) using constrained Bayesian optimization. The key aspects are: Parameterization of the MPC stage cost function using a feedforward neural network, providing flexibility to account for model-plant mismatch. Bayesian optimization is employed to systematically learn the neural network parameters that optimize the closed-loop performance, as measured by a high-level cost function. Lyapunov-based constraints are incorporated into the Bayesian optimization to ensure stability of the learned MPC controller. Specifically, the optimal value function of the MPC is used as a Lyapunov function candidate, and constraints are imposed to guarantee positive definiteness and decrease along the closed-loop trajectory. The effectiveness of the proposed approach is demonstrated in simulation on a double pendulum system, where the learned MPC controller outperforms the nominal controller in terms of transient behavior while provably ensuring stability.

Stats

The authors consider a double pendulum system with the following dynamics: ¨ψ1 = (m2l1 ˙ψ2 1s21c21 + m2gs2c21 + m2l2 ˙ψ2 2s21 - (m1+m2)gs1)/(m1+m2)l1 - m2l1c2 21 + u ¨ψ2 = (-m2l2 ˙ψ2 2s21c21 + (m1+m2)(gs1c21 - l1 ˙ψ2 1s21 - gs2))/(l2/l1)(m1+m2)l1 - m2l1c2 21 The control objective is to bring the pendulum to the upright position (π, π, 0, 0) and stabilize it there.

Quotes

"Doing so offers a high degree of freedom and, thus, the opportunity for efficient and global optimization towards the desired and optimal closed-loop behavior." "We extend this framework by stability constraints on the learned controller parameters, exploiting the optimal value function of the underlying MPC as a Lyapunov candidate."

Key Insights Distilled From

Stability-informed Bayesian Optimization for MPC Cost Function Learning

by Sebastian Hi... at arxiv.org 04-19-2024

https://arxiv.org/pdf/2404.12187.pdf

Deeper Inquiries

How can the proposed approach be extended to handle state and input constraints more explicitly during the learning process

To handle state and input constraints more explicitly during the learning process, the proposed approach can be extended by incorporating barrier functions or penalty terms in the cost function. By penalizing violations of constraints in the cost function, the learning algorithm can be guided to prioritize parameterizations that adhere to the constraints. This can be achieved by modifying the cost function to include terms that penalize deviations from the desired state and input constraints. Additionally, the constraints can be enforced probabilistically by incorporating them as soft constraints in the Bayesian optimization framework. This approach allows for a balance between optimizing the closed-loop performance and ensuring constraint satisfaction during the learning process.

What are the potential challenges in scaling the Bayesian optimization to higher-dimensional parameter spaces, and how can hybrid approaches with reinforcement learning help address them

Scaling Bayesian optimization to higher-dimensional parameter spaces poses challenges such as increased computational complexity and the curse of dimensionality. In higher-dimensional spaces, the search for optimal parameters becomes more challenging due to the larger search space and increased computational resources required. Hybrid approaches with reinforcement learning can help address these challenges by leveraging the strengths of both methods. Reinforcement learning can assist in exploring the parameter space more efficiently by guiding the search towards promising regions based on past experiences. By combining Bayesian optimization for global optimization and reinforcement learning for local exploration, the hybrid approach can improve the efficiency and effectiveness of parameter learning in higher-dimensional spaces.

Can the framework be adapted to learn the prediction model and the cost function simultaneously, further improving the closed-loop performance

The framework can be adapted to learn the prediction model and the cost function simultaneously by integrating both tasks into a unified learning objective. This can be achieved by jointly optimizing the parameters of the prediction model and the cost function using a multi-objective optimization approach. By formulating a composite objective function that considers both the accuracy of the prediction model and the performance of the cost function, the learning algorithm can simultaneously improve the model's predictive capabilities and the closed-loop performance. This integrated approach can lead to a more holistic optimization process, where the prediction model and cost function are jointly optimized to enhance the overall closed-loop performance.