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Regret of $H_\infty$ Preview Controllers

Jietian Liu, Peter Seiler

TL;DR

This paper analyzes preview control for discrete-time LTI systems under $H_\infty$ and regret-optimal criteria, introducing a non-causal baseline with full disturbance knowledge. By augmenting the plant with preview information, it constructs $p$-step preview controllers and shows that, as $p$ increases, the closed-loop $H_\infty$ performance converges to the non-causal bound and the minimal regret converges to zero, corroborated by a numerical example. A separate $H_2$ result establishes convergence of the preview cost to the non-causal cost for deterministic disturbances, which then implies similar convergence for the $H_\infty$ case. Overall, the work demonstrates that longer disturbance preview enables causal controllers to asymptotically emulate non-causal performance in both $H_\infty$ and regret frameworks.

Abstract

This paper studies preview control in both the $H_\infty$ and regret-optimal settings. The plant is modeled as a discrete-time, linear time-invariant system subject to external disturbances. The performance baseline is the optimal non-causal controller that has full knowledge of the disturbance sequence. We first review the construction of the $H_\infty$ preview controller with $p$-steps of disturbance preview. We then show that the closed-loop $H_\infty$ performance of this preview controller converges as $p\to \infty$ to the performance of the optimal non-causal controller. Furthermore, we prove that the optimal regret of the preview controller converges to zero. These results demonstrate that increasing preview length allows controllers to asymptotically achieve non-causal performance in both the $H_\infty$ and regret frameworks. A numerical example illustrates the theoretical results.

Regret of $H_\infty$ Preview Controllers

TL;DR

This paper analyzes preview control for discrete-time LTI systems under and regret-optimal criteria, introducing a non-causal baseline with full disturbance knowledge. By augmenting the plant with preview information, it constructs -step preview controllers and shows that, as increases, the closed-loop performance converges to the non-causal bound and the minimal regret converges to zero, corroborated by a numerical example. A separate result establishes convergence of the preview cost to the non-causal cost for deterministic disturbances, which then implies similar convergence for the case. Overall, the work demonstrates that longer disturbance preview enables causal controllers to asymptotically emulate non-causal performance in both and regret frameworks.

Abstract

This paper studies preview control in both the and regret-optimal settings. The plant is modeled as a discrete-time, linear time-invariant system subject to external disturbances. The performance baseline is the optimal non-causal controller that has full knowledge of the disturbance sequence. We first review the construction of the preview controller with -steps of disturbance preview. We then show that the closed-loop performance of this preview controller converges as to the performance of the optimal non-causal controller. Furthermore, we prove that the optimal regret of the preview controller converges to zero. These results demonstrate that increasing preview length allows controllers to asymptotically achieve non-causal performance in both the and regret frameworks. A numerical example illustrates the theoretical results.
Paper Structure (8 sections, 6 theorems, 43 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 43 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. BACKGROUND: $H_\infty$ CONTROL WITH PREVIEW
  4. Full Information $H_\infty$ Synthesis
  5. $H_\infty$ Control With Preview
  6. Main Results
  7. Example
  8. CONCLUSIONS

Key Result

Theorem 1

Let $(A,B_u,B_d,Q,R)$ be given and assume: (i) $Q\succeq 0$ and $R \succ 0$, (ii) $\left(A, B_u\right)$ stabilizable, (iii) $A$ is nonsingular, and (iv) $(A, Q)$ has no unobservable modes on the unit circle. Then:

Figures (2)

  • Figure 1: Closed-loop $H_\infty$ norm of the $p$-step preview controller versus preview length $p$. The closed-loop $H_\infty$ norm with the optimal non-causal controller is also shown.
  • Figure 2: Optimal additive regret $\gamma_{R,p}$ versus preview length $p$.

Theorems & Definitions (10)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • proof