Bilinear Data-Driven Min-Max MPC: Designing Rational Controllers via Sum-of-squares Optimization

TL;DR

This work tackles control of unknown discrete-time bilinear systems from noisy input-state data by formulating a data-driven min-max MPC using sum-of-squares optimization. It defines a data-consistent model set \Sigma via SOS-based set-membership constraints and derives a receding-horizon SOS program that computes a rational controller $u(x)=\frac{1}{d(x)}K(x)x$ while minimizing an upper bound $\gamma$ on the worst-case cost. Theoretical guarantees establish recursive feasibility, robust stability to a robust positively invariant (RPI) set, and satisfaction of input/state constraints for all models in \Sigma. A numerical example on a zone-temperature process demonstrates feasibility and trade-offs between constraint tightening and computational load. Overall, the approach advances direct data-driven control for bilinear systems by avoiding explicit identification and leveraging discrete measurements only.

Abstract

We propose a data-driven min-max model predictive control (MPC) scheme to control unknown discrete-time bilinear systems. Based on a sequence of noisy input-state data, we state a set-membership representation for the unknown system dynamics. Then, we derive a sum-of-squares (SOS) program that minimizes an upper bound on the worst-case cost over all bilinear systems consistent with the data. As a crucial technical ingredient, the SOS program involves a rational controller parameterization to improve feasibility and tractability. We prove that the resulting data-driven MPC scheme ensures closed-loop stability and constraint satisfaction for the unknown bilinear system. We demonstrate the practicality of the proposed scheme in a numerical example.

Figures (2)

• Figure 1: Closed-loop state and input trajectories of the proposed bilinear data-driven min-max MPC scheme with different input constraints.
• Figure 2: Closed-loop cost and computation time with different values of $\alpha$.

Theorems & Definitions (8)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Remark 2