Guaranteed cost structured control in infinite-horizon linear-quadratic cooperative differential games

TL;DR

This paper introduces feedback guaranteed cost structured control (GCSC) for infinite-horizon linear-quadratic cooperative differential games with output feedback, providing a practical approach to approximate Pareto optimal controls.

Contribution

It develops the concept of feedback GCSC, analyzes its properties, and offers conditions for its synthesis, addressing the difficulty of computing Pareto optimal controls.

Findings

Feedback GCSC ensures total team cost stays below a threshold.

Pareto optimal controls, if they exist, are within the class of feedback GCSCs.

Numerical examples demonstrate the approach's effectiveness in microgrid synchronization.

Abstract

In this paper, we consider infinite-horizon linear-quadratic cooperative differential games with output feedback information structure. We first demonstrate that, under output feedback information structure, computing Pareto optimal controls can be difficult even for simple low-dimensional differential games. To address this issue, this paper introduces the concept of feedback guaranteed cost structured control (GCSC). The feedback GCSC concept is inspired from suboptimal control. At a feedback GCSC, the total weighted team cost remains below a prescribed threshold while satisfying the structural constraints. We derive fundamental properties of the feedback GCSC and the admissible weight set, including their monotonicity properties. In particular, we show that if Pareto optimal controls exist, they belong to the class of feedback GCSCs. We also quantify the suboptimalty of Pareto optimal controls (if they exist) and the proposed GCSC with respect to output feedback optimal control. Furthermore, we provide the conditions for verification and the synthesis of a feedback GCSC. Finally, we illustrate the effectiveness of the proposed approach through numerical examples, including a case study on tracking synchronization in a microgrid.