Soft-constrained output feedback guaranteed cost equilibria in infinite-horizon uncertain linear-quadratic differential games

TL;DR

This paper introduces a new class of guaranteed cost equilibria for infinite-horizon uncertain linear-quadratic differential games, expanding the set of feasible strategies and providing computational methods for their synthesis.

Contribution

It proposes the concept of soft-constrained output feedback guaranteed cost equilibrium (SCOGCE), broadening the scope of equilibrium strategies in uncertain differential games.

Findings

Sufficient conditions for SCOGCE existence are derived from coupled bi-linear matrix inequalities.

Semi-definite programming relaxations enable iterative algorithms for strategy synthesis.

Numerical examples demonstrate the effectiveness of SCOGCE controllers.

Abstract

In this paper, we study infinite-horizon linear-quadratic uncertain differential games with an output feedback information structure. We assume linear time-invariant nominal dynamics influenced by deterministic external disturbances, and players' risk preferences are expressed by a soft-constrained quadratic cost criterion over an infinite horizon. We demonstrate that the conditions available in the literature for the existence of a soft-constrained output feedback Nash equilibrium (SCONE) are too stringent to satisfy, even in low-dimensional games. To address this issue, using ideas from suboptimal control, we introduce the concept of a soft-constrained output feedback guaranteed cost equilibrium (SCOGCE). At an SCOGCE, the players' worst-case costs are upper-bounded by a specified cost profile while maintaining an equilibrium property. We show that SCOGCE strategies form a larger class of equilibrium strategies; that is, whenever an SCONE exists, it is also an SCOGCE. We demonstrate that sufficient conditions for the existence of SCOGCE are related to the solvability of a set of coupled bi-linear matrix inequalities. Using semi-definite programming relaxations, we provide linear matrix inequality-based iterative algorithms for the synthesis of SCOGCE strategies. Finally, we illustrate the performance of SCOGCE controllers with numerical examples.