Optimal Control of Continuous-Time Symmetric Systems with Unknown Dynamics and Noisy Measurements

Core Concepts

An iterative learning algorithm is presented for continuous-time linear-quadratic optimal control problems with unknown dynamics, globally convergent to the optimal solution, unbiased under noisy measurements, and computationally efficient.

Abstract

The content discusses an iterative learning algorithm for optimal control in continuous-time symmetric systems with unknown dynamics. It covers the background of linear-quadratic regulation problems, state-of-the-art methods, convergence conditions, measurement noise considerations, and extension to infinite-horizon problems. The algorithm's key features include global convergence, unbiasedness under noise, and low computational complexity.

Introduction

Linear-quadratic regulation (LQR) problem aims to minimize a quadratic cost subject to system dynamics.
Direct approach focuses on solving optimal control without knowing the system model.

State-of-the-Art

Kleinman’s algorithm sets a foundation for solving LQR problems without system model access.
Model-free algorithms emerged from Kleinman’s algorithm for LQR problems.

Symmetric Systems

Definition of external symmetry in systems based on input-output relations.
Completely symmetric systems are internally and externally symmetric.

Main Results

Algorithm presented solves optimal control problem without prior model knowledge.
Convergence analysis shows the algorithm's effectiveness in reaching the optimal solution.

Extension to Infinite-Horizon Problems

Algorithm adapted for infinite-horizon problems with state feedback gain derivation.
Theoretical analysis ensures convergence and reliability under noisy measurements.

Stats

It is shown that limk→+∞ ∥uk − u⋆∥2,tf = 0 holds from Lemma 1.
Condition (45) ensures limk→+∞ ∥uk − u⋆∥∞,tf = 0 as per Lemma 3.

Quotes

Key Insights Distilled From

Optimal control of continuous-time symmetric systems with unknown dynamics and noisy measurements

by Hamed Taghav... at arxiv.org 03-21-2024

https://arxiv.org/pdf/2403.13605.pdf

Deeper Inquiries

How does the proposed algorithm compare to traditional dynamic programming methods

The proposed algorithm differs from traditional dynamic programming methods in several key aspects. Firstly, the algorithm presented in the context is model-free, meaning it does not require prior knowledge of the system dynamics. This contrasts with traditional dynamic programming approaches that rely on a known system model to derive optimal control strategies. Additionally, the iterative learning algorithm converges to the optimal solution by observing input-output data and updating the control signal directly without assuming a specific feedback structure. In contrast, traditional dynamic programming methods often involve solving complex optimization problems based on known system models and cost functions.

What implications does measurement noise have on real-world implementation of this algorithm

Measurement noise can have significant implications for real-world implementation of this algorithm. In scenarios where noisy measurements are present, such as in practical control systems or experimental setups, there is a risk of biasing the update rule used in Algorithm 1. The presence of measurement noise can lead to deviations between expected outcomes and actual results obtained during iterations of the algorithm. However, Theorem 2 shows that under Assumption 2 and proper conditions on step size α, estimation errors due to measurement noise are unbiased with bounded variance.

How can this iterative learning approach be applied to other fields beyond control systems

This iterative learning approach can be applied beyond control systems to various fields where optimization problems need to be solved based on observed input-output data rather than explicit models. For example:

Finance: It could be utilized for portfolio optimization or trading strategies by iteratively learning from market data.
Healthcare: The method could help optimize treatment plans or resource allocation based on patient outcomes.
Marketing: Marketers could use this approach for optimizing advertising campaigns or customer targeting strategies using observed response data.
Manufacturing: It could assist in optimizing production processes or supply chain management by learning from operational data.
By adapting this iterative learning framework to different domains and problem settings, it offers a flexible and versatile tool for solving optimization problems without relying on detailed mathematical models upfront.