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Online convex optimization for robust control of constrained dynamical systems

Marko Nonhoff, Emiliano Dall'Anese, Matthias A. Müller

TL;DR

The paper addresses controlling constrained linear systems with time-varying, unknown costs and disturbances while strictly satisfying state and input constraints. It blends online convex optimization with robust MPC-style constraint tightening to track optimal steady states and guarantee feasibility. A dynamic regret bound linear in cost variation and disturbance magnitude is established, with a practical autonomous-vehicle tracking case study validating the approach. The work offers a computationally efficient controller capable of adapting to changing objectives while maintaining safety constraints.

Abstract

This article investigates the problem of controlling linear time-invariant systems subject to time-varying and a priori unknown cost functions, state and input constraints, and exogenous disturbances. We combine the online convex optimization framework with tools from robust model predictive control to propose an algorithm that is able to guarantee robust constraint satisfaction. The performance of the closed loop emerging from application of our framework is studied in terms of its dynamic regret, which is proven to be bounded linearly by the variation of the cost functions and the magnitude of the disturbances. We corroborate our theoretical findings and illustrate implementational aspects of the proposed algorithm by a numerical case study on a tracking control problem of an autonomous vehicle.

Online convex optimization for robust control of constrained dynamical systems

TL;DR

The paper addresses controlling constrained linear systems with time-varying, unknown costs and disturbances while strictly satisfying state and input constraints. It blends online convex optimization with robust MPC-style constraint tightening to track optimal steady states and guarantee feasibility. A dynamic regret bound linear in cost variation and disturbance magnitude is established, with a practical autonomous-vehicle tracking case study validating the approach. The work offers a computationally efficient controller capable of adapting to changing objectives while maintaining safety constraints.

Abstract

This article investigates the problem of controlling linear time-invariant systems subject to time-varying and a priori unknown cost functions, state and input constraints, and exogenous disturbances. We combine the online convex optimization framework with tools from robust model predictive control to propose an algorithm that is able to guarantee robust constraint satisfaction. The performance of the closed loop emerging from application of our framework is studied in terms of its dynamic regret, which is proven to be bounded linearly by the variation of the cost functions and the magnitude of the disturbances. We corroborate our theoretical findings and illustrate implementational aspects of the proposed algorithm by a numerical case study on a tracking control problem of an autonomous vehicle.
Paper Structure (9 sections, 4 theorems, 67 equations, 6 figures, 1 algorithm)

This paper contains 9 sections, 4 theorems, 67 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting
  3. Algorithm
  4. Theoretical Results
  5. Numerical Case Study
  6. Conclusion
  7. Proof of Lemma \ref{['lem:feasibility']}
  8. Proof of Lemma \ref{['lem:mincontr']}
  9. Proof of Theorem \ref{['thm']}

Key Result

Lemma 1

Suppose Assumptions ass:noise, ass:system, and ass:init are satisfied. Let $\mu\geq\mu^*$ and $c_g\geq \left\lVert S_c^\top\left(S_cS_c^\top\right)^{-1}\right\rVert$. It holds that

Figures (6)

  • Figure 1: Schematic illustration of Algorithm \ref{['alg']}: The closed-loop cost function $L_{K,t-1}$ is visualized via its sublevel sets (dotted) together with the (tightened) constraint sets and the tightened steady-state manifold $\bar{\mathcal{S}}$ (red, dotted). Note that the constraints are tightened the farther Algorithm \ref{['alg']} predicts into the future. First, Algorithm \ref{['alg']} predicts the system state $\mu$ times state ahead $\hat{x}^{\mu}_t$. Then, it applies OGD evaluated at $\hat{x}^{\mu}_t$ in [S2] (blue, dashed) to obtain an estimate of the optimal steady state $\hat{\theta}_t$. Next, an additional input sequence $g_t$ is computed that steers the closed-loop system to the estimate $\hat{\theta}_t$. Note that the additional input sequence $g_t$ may violate the constraints. Therefore, a scaling $\beta_t$ (gray, dashed) is applied to ensure that the trajectory emerging from application of the predicted input sequence $\hat{u}^{\mu}_t$robustly satisfies the constraints. However, due to the scaling $\beta_t$, the predicted input sequence steers the system to a vicinity of the steady state $\hat{x}^s_t$ instead of to the estimate $\hat{\theta}_t$. Finally, the first part of the predicted input sequence is applied to system \ref{['eq:sys']}.
  • Figure 2: Schematic illustration of the scenario considered in the simulation. The border of the road is indicated by the thick black lines, the two lanes are illustrated by the dashed line. The position of the controlled car relative to the slower vehicle is shown in blue (for the optimization based solution of \ref{['algo:additional_input']}) and yellow (for the explicit solution of \ref{['algo:additional_input']}). The slower vehicle is indicated by the black rectangle.
  • Figure 3: Lateral position of the controlled car in closed loop for two variants of Algorithm \ref{['alg']} (optimization based solution of \ref{['algo:additional_input']} (blue) and the explicit solution of \ref{['algo:additional_input']} (yellow)) and reference position (green).
  • Figure 4: Velocity of the controlled car for two variants of Algorithm \ref{['alg']} (optimization based solution of \ref{['algo:additional_input']} (blue) and the explicit solution of \ref{['algo:additional_input']} (yellow)), reference velocity (green) and constraint (red).
  • Figure 5: Steering angle of the controlled car in closed loop for two variants of Algorithm \ref{['alg']} (optimization based solution of \ref{['algo:additional_input']} (blue) and the explicit solution of \ref{['algo:additional_input']}) and constraints (red).
  • ...and 1 more figures

Theorems & Definitions (10)

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