Computationally Efficient System Level Tube-MPC for Uncertain Systems

TL;DR

This work addresses robust constrained control for linear systems with both additive disturbances and model uncertainties by introducing a filter-based system-level tube-MPC (SLTMPC) with online disturbance-set optimization and an asynchronous computation scheme. It combines a secondary process that designs online error-dynamics tubes and a primary process that optimizes a nominal trajectory using a convex fusion of stored tubes, yielding rigorous closed-loop guarantees (recursive feasibility and ISS) under memory updates. The key contributions are a new terminal controller design with an online terminal set, an asynchronous architecture that decouples tube and nominal trajectory optimization to reduce computation, and thorough numerical validation on a double integrator and a VTOL model demonstrating improved feasibility and significant speedups. The approach has practical impact for robust, real-time controller synthesis in uncertain, constrained systems where online adaptivity of uncertainty descriptions and computational load are critical.

Abstract

Tube-based model predictive control (MPC) is one of the principal robust control techniques for constrained linear systems affected by additive disturbances. While tube-based methods with online-computed tubes have been successfully applied to systems with additive disturbances, their application to systems affected by additional model uncertainties is challenging. This paper proposes a tube-based MPC method - named filter-based system level tube-MPC (SLTMPC) - which overapproximates both types of uncertainties with an online optimized disturbance set, while simultaneously computing the tube controller online. For the first time, we provide rigorous closed-loop guarantees for receding horizon control of such a MPC method. These guarantees are obtained by virtue of a new terminal controller design and an online optimized terminal set. To reduce the computational complexity of the proposed method, we additionally introduce an asynchronous computation scheme that separates the optimization of the tube controller and the nominal trajectory. Finally, we provide a comprehensive numerical evaluation of the proposed methods to demonstrate their effectiveness.

Figures (6)

• Figure 1: Visualization of the asynchronous computation scheme for filter-based SLTMPC: the secondary process computes error system responses $\bm{\Phi}^\mathbf{e}\, \bm{\Phi}^{\bm{\nu}}$, disturbance filter $\bm{\Sigma}, \, \bm{\Xi}$, and terminal set scaling $\alpha$, which are then passed to the primary process and stored in memory $\mathtt{M}$.
• Figure 2: Comparison of \ref{['SLTMPC:rec-feas']} to Chen2023 and other standard tube-based MPC methods Bujarbaruah2021Bujarbaruah2022Langson2004Lorenzen2019Kohler2019Lu2019: Region of attraction (RoA) relative to the maximal RCI set in percent, where we vary the uncertainty in the dynamics matrix $A$, i.e. $\epsilon_A$, (left) and the additive disturbance, i.e. $\sigma_w$, (right). The other uncertainty parameters are fixed to $\epsilon_B = 0.1$, $\sigma_w = 0.1$ and $\epsilon_A = 0.1$, $\epsilon_B = 0.1$, respectively, and the horizon is $N=\{5, \, 10\}$.
• Figure 3: RoA relative to the maximal RCI set in percent for \ref{['SLTMPC:rec-feas']} and Chen2023, where we vary $\sigma_w$ and the choice of $\bar{\mathcal{W}}$, but keep $\epsilon_A = \epsilon_B = 0.1$ and $N=5$ fixed.
• Figure 4: Visualization of the experimental setup for the VTOL vehicle.
• Figure 5: Closed-loop trajectories of length $25$ for Chen2023 and \ref{['SLTMPC:rec-feas']}. The average costs are $2143.7$ for Chen2023 and $1951.1$ for \ref{['SLTMPC:rec-feas']}.

Theorems & Definitions (8)

• Remark 5
• Proposition 5
• Corollary 6
• Remark 6
• Definition 4: Support function
• Lemma 7: Lemma 1 in Sieber2023
• Lemma 8
• Lemma 9