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Robust Regret Control with Uncertainty-Dependent Baseline

Jietian Liu, Peter Seiler

TL;DR

The paper addresses robust regret control for a discrete-time LTI plant with parametric uncertainty by introducing an uncertainty-dependent noncausal baseline $K^{nc}_{d,\Delta}$ and reducing the regret constraint to a robust $H_\infty$-type condition on an augmented plant. It then develops a practical synthesis pipeline using spectral factorization, a linear approximation of $F_{\Delta}^{-1}$ to obtain a linear fractional transformation, and standard $\mu$-synthesis (DK-iteration) on an augmented plant to yield a causal controller. The key contributions are the formulation of an uncertainty-dependent regret objective, the linearization-based reduction to robust synthesis, and the demonstration on a scalar example showing improved regret guarantees compared to fixed-baseline designs. This approach reduces conservatism in regret guarantees under model uncertainty and provides a tractable path to implementable robust regret controllers in discrete-time settings.

Abstract

This paper proposes a robust regret control framework in which the performance baseline adapts to the realization of system uncertainty. The plant is modeled as a discrete-time, uncertain linear time-invariant system with real-parametric uncertainty. The performance baseline is the optimal non-causal controller constructed with full knowledge of the disturbance and the specific realization of the uncertain plant. We show that a controller achieves robust additive regret relative to this baseline if and only if it satisfies a related, robust $H_\infty$ performance condition on a modified plant. One technical issue is that the modified plant can, in general, have a complicated nonlinear dependence on the uncertainty. We use a linear approximation step so that the robust additive regret condition can be recast as a standard $μ$-synthesis problem. A numerical example is used to demonstrate the proposed approach.

Robust Regret Control with Uncertainty-Dependent Baseline

TL;DR

The paper addresses robust regret control for a discrete-time LTI plant with parametric uncertainty by introducing an uncertainty-dependent noncausal baseline and reducing the regret constraint to a robust -type condition on an augmented plant. It then develops a practical synthesis pipeline using spectral factorization, a linear approximation of to obtain a linear fractional transformation, and standard -synthesis (DK-iteration) on an augmented plant to yield a causal controller. The key contributions are the formulation of an uncertainty-dependent regret objective, the linearization-based reduction to robust synthesis, and the demonstration on a scalar example showing improved regret guarantees compared to fixed-baseline designs. This approach reduces conservatism in regret guarantees under model uncertainty and provides a tractable path to implementable robust regret controllers in discrete-time settings.

Abstract

This paper proposes a robust regret control framework in which the performance baseline adapts to the realization of system uncertainty. The plant is modeled as a discrete-time, uncertain linear time-invariant system with real-parametric uncertainty. The performance baseline is the optimal non-causal controller constructed with full knowledge of the disturbance and the specific realization of the uncertain plant. We show that a controller achieves robust additive regret relative to this baseline if and only if it satisfies a related, robust performance condition on a modified plant. One technical issue is that the modified plant can, in general, have a complicated nonlinear dependence on the uncertainty. We use a linear approximation step so that the robust additive regret condition can be recast as a standard -synthesis problem. A numerical example is used to demonstrate the proposed approach.
Paper Structure (12 sections, 2 theorems, 23 equations, 4 figures)

This paper contains 12 sections, 2 theorems, 23 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notation
  4. Problem Formulation
  5. Main Results
  6. Uncertainty-Dependent Non-causal Controller
  7. Output Feedback Control Design
  8. Numerical Approximation
  9. Example
  10. Problem Data
  11. Performance Analysis
  12. Conclusions

Key Result

Theorem 1

Let $(A,B_u,C_e,D_{eu})$ be given and define $Q:=C_e^\top C_e$, $S:=C_e^\top D_{eu}$, and $R:=D_{eu}^\top D_{eu}$. Assume: (i) $R \succ 0$, (ii) $(A,B_u)$ is stabilizable, (iii) $A-B_uR^{-1}S^\top$ is nonsingular, and (iv) $\left[ \right]$ has full column rank for all $\theta \in [0,2\pi]$. Then:

Figures (4)

  • Figure 1: Feedback interconnection $CL(P,K,\Delta)$ for robust synthesis.
  • Figure 2: Approximate representation of $CL(P,K,\Delta)F_\Delta^{-1}$ (Left). This serial interconnection is in the standard form for robust synthesis with model $\hat{P}$ and uncertainty $\hat{\Delta}$ (Right).
  • Figure 3: Additive regret using nominal baseline versus uncertainty $\Delta$.
  • Figure 4: Additive regret using uncertainty-dependent baseline versus uncertainty $\Delta$.

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Lemma 1