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

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.

💡 Research Summary

This paper presents a robust control scheme for linear time‑invariant (LTI) systems that must operate under (i) time‑varying, a priori unknown cost functions, (ii) hard state and input constraints, and (iii) exogenous disturbances. The authors combine three well‑established ideas—online convex optimization (OCO), a reference governor (RG), and constraint tightening—to obtain a controller that is both computationally light and provably safe.

System model and problem statement
The plant is described by
xₜ₊₁ = A xₜ + B uₜ + B_w wₜ, yₜ = C₀ xₜ + D uₜ + D_w wₜ,
with full state measurement, compact convex constraint set 𝒴 for yₜ, and bounded disturbance set 𝒲. A stabilizing state‑feedback K is pre‑designed so that A_K = A + B K is Schur. The control input is written as uₜ = vₜ + K xₜ, where vₜ is a “virtual reference” that will be generated online. The steady‑state map S_K = (I – A_K)⁻¹ B links vₜ to the corresponding equilibrium of the disturbance‑free closed loop.

The performance index at each time t is a convex cost Lₜ(v, x) that is unknown before time t. The goal is to minimize the cumulative cost over a horizon while respecting the constraints for all t, despite the fact that only past cost functions L₀,…,Lₜ₋₁ are known at time t.

Algorithmic architecture

1. Online Gradient Descent (OGD) – The authors apply OGD to the static part of the cost, Lₛₜ(v) = Lₜ(v + K S_K v, S_K v). The update is
   rₜ₊₁ = Π_{𝒮_v}