Robust Control of Constrained Linear Systems using Online Convex Optimization and a Reference Governor

TL;DR

This work addresses robust control of constrained linear systems under time-varying and unknown costs and exogenous disturbances. It integrates online gradient descent for reference generation with a reference governor and a constraint-tightening step based on a robust invariant set and a maximal output admissible set, ensuring recursive feasibility and robust constraint satisfaction. Theoretical results yield a dynamic regret bound that scales linearly with cost variation and disturbance magnitude, confirming performance guarantees under disturbance influence. A numerical example with a mobile robot illustrates effective tracking while maintaining constraint satisfaction despite measurement noise, highlighting practical applicability.

Abstract

This article develops a control method for linear time-invariant systems subject to time-varying and a priori unknown cost functions, that satisfies state and input constraints, and is robust to exogenous disturbances. To this end, we combine the online convex optimization framework with a reference governor and a constraint tightening approach. The proposed framework guarantees recursive feasibility and robust constraint satisfaction. Its closed-loop performance is studied in terms of its dynamic regret, which is bounded linearly by the variation of the cost functions and the magnitude of the disturbances. The proposed method is illustrated by a numerical case study of a tracking control problem.

Figures (4)

• Figure 1: Block diagram of the robust OCO-RG scheme.
• Figure 2: Closed-loop position $p_t$ (red) and reference position $p_t^{\text{ref}}$ (green) together with the projections of the sets $\mathcal{Y}$, $S_K\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{S}\mkern-1.5mu}\mkern 1.5mu_v$, and $\mathcal{O}_\infty^\lambda$ (centered at $0$) onto the $(p^x,p^y)$-plane (black)
• Figure 3: Reference governor parameter $\alpha_t$ (blue; compare \ref{['eq:RG_opt']})
• Figure 4: Closed-loop velocity $\nu^x_t$ (in direction of $p^x$; blue) together with the corresponding velocity constraints (red)

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof