Regret Optimal Control for Uncertain Stochastic Systems: A Scenario Optimization Approach

Core Concepts

Designing control policies that minimize regret robustly over randomly sampled system parameters can be achieved through semidefinite programming, providing strong probabilistic out-of-sample regret guarantees.

Abstract

The content discusses regret optimal control for uncertain stochastic systems using scenario optimization. It introduces a competitive framework focusing on minimizing regret relative to a clairvoyant optimal policy. The method proposed involves sampling uncertainty instances and solving a semidefinite program to compute the policy with robust regret minimization. The approach extends to include safety constraints with high probability, showcasing improved closed-loop performance across various system dynamics.
I. INTRODUCTION

Regret minimization in control systems.
Competitive framework for designing efficient control laws.
Importance of minimizing loss relative to an optimal policy.
II. PROBLEM STATEMENT AND PRELIMINARIES

Description of uncertain linear time-varying dynamical systems.
Formulation of robust regret minimization problem.
Linear disturbance feedback policy for causality enforcement.
III. MAIN RESULTS

Solution to robust regret minimization problem based on scenario optimization.
Semidefinite programming approach for computing the policy.
Strong probabilistic out-of-sample regret guarantees demonstrated through numerical simulations.
IV. NUMERICAL RESULTS

Validation of theoretical results through numerical experiments.
Comparison between exact and approximate solutions in terms of performance guarantees and computation times.
Illustration of improved closed-loop performance using regret minimization approach.
V. CONCLUSION

Novel method for convex synthesis of robust control policies with provable regret and safety guarantees.
Potential applications in adapting to heterogeneous dynamics and disturbances.

Stats

"Research supported by the Swiss National Science Foundation (SNSF) under the NCCR Automation (grant agreement 51NF40 80545)."
"Mass m = 1 kg, spring constant k = 1 N m−1, damping constant c = 1 N m−1."
"Sampling time Ts = 1 s."
"Uniform distribution: δk ∼ U[−0.2,0.2] and δc ∼ U[−0.2,0.2]."

Quotes

"We prove that this policy optimization problem can be solved through semidefinite programming."
"Our method naturally extends to include satisfaction of safety constraints with high probability."

Key Insights Distilled From

Regret Optimal Control for Uncertain Stochastic Systems

by Andr... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2304.14835.pdf

Deeper Inquiries

How does the proposed scenario optimization approach compare to traditional worst-case oriented synthesis methods?

The proposed scenario optimization approach differs from traditional worst-case oriented synthesis methods in several key aspects. Firstly, while worst-case methods aim to find a single control policy that performs well under all possible scenarios, the scenario optimization approach focuses on minimizing regret over a finite set of randomly sampled system parameters. This allows for more flexibility and adaptability in handling uncertainties compared to rigid worst-case strategies.
Secondly, the scenario optimization method leverages sampling techniques to approximate the optimal policy against an unknown clairvoyant benchmark policy. In contrast, traditional worst-case approaches often rely on analytical expressions or norm inequalities to derive upper bounds on performance without explicitly considering uncertainty realizations.
Furthermore, by using semidefinite programming and probabilistic guarantees, the scenario optimization framework provides strong out-of-sample regret guarantees even in the face of uncertain dynamics. This contrasts with worst-case methods that may struggle with computational complexity when evaluating performance across all possible scenarios.
Overall, the scenario optimization approach offers a more nuanced and probabilistic way of designing control policies that can adapt effectively to uncertain and complex systems compared to traditional worst-case oriented synthesis methods.

How can competitive ratio be applied beyond control systems into other domains?

The concept of competitive ratio, which measures how well an algorithm performs relative to an optimal benchmark strategy in hindsight, can be applied beyond control systems into various other domains where decision-making processes are involved.
In finance and investment management, competitive ratio analysis can help evaluate trading algorithms or portfolio management strategies by comparing their performance against an idealized optimal strategy given perfect information about market conditions. This allows investors to assess how well their decisions fare relative to what could have been achieved with complete foresight.
In healthcare operations and resource allocation, competitive ratio metrics can be used to optimize patient scheduling procedures or hospital bed management protocols by quantifying how efficiently different scheduling algorithms perform compared to an oracle-like scheduler who knows future patient arrivals perfectly.
In transportation logistics and supply chain management, competitive ratio analysis can aid in optimizing route planning algorithms or inventory management strategies by measuring their effectiveness relative to a clairvoyant planner who has full knowledge of demand patterns and traffic conditions beforehand.
By applying the concept of competitive ratio outside control systems contexts into these diverse domains mentioned above (finance/investment management, healthcare operations/resource allocation), organizations can make data-driven decisions based on insights gained from comparing algorithmic performance against theoretically optimal benchmarks.