Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors

Abstract

In this article we address the problem of finding feedback Nash equilibria for linear quadratic differential games defined on descriptor systems. First, we decouple the dynamic and algebraic parts of a descriptor system using canonical projectors. We discuss the effects of feedback on the behavior of the descriptor system. We derive necessary and sufficient conditions for the existence of the feedback Nash equilibria for index 1 descriptor systems and show that there exist many informationally non-unique equilibria corresponding to a single solution of the game. Further, for descriptor systems with index greater than 1, we give a regularization based approach and discuss the associated drawbacks.

Disturbance attenuation problem using a differential game approach for feedback linear quadratic descriptor systems

“…We will characterize the set of FBSP solutions for the game (1,7) using the reduced ordinary differential game described by the dynamical system (10) with the cost function (11). The general index case was studied in [25], where the theory of projector chains is used to decouple algebraic and differential parts of the descriptor system, and then the usual theory of ordinary differential games is applied to derive both necessary and sufficient conditions for the existence of feedback Nash equilibria for linear quadratic differential games. Furthermore, the open-loop version of such a game has been studied in [22], while the hard constrained version for such games can be found in [26] and [30].…”

Feedback saddle point equilibria for soft-constrained zero-sum linear quadratic descriptor differential game

“…and there exists a nonnegative definite solution to (26), say K + 1γ , with the further property thatF…”

“…If γ < γ ∞ , the upper value is infinite, and (26) has no real solution which also satisfies property 4.…”

“…All of them solved the game by converting it to a reduced ordinary game. The general index case was studied in [26], where the theory of projector chains is used to decouple algebraic and differential parts of the descriptor system, and then the usual theory of ordinary differential games is applied to derive both necessary and sufficient conditions for the existence of feedback Nash equilibria for linear quadratic differential games. Moreover, the open-loop framework of soft-constrained descriptor differential game was studied in [20], also with its application in robust optimal control design.…”

Feedback Properties of Descriptor Systems Using Matrix Projectors and Applications to Descriptor Differential Games

“…The matrices N i in (2.3b) are given in terms of canonical projectors; see equations (27) and (28b) of [35] for details. This decomposition is unique for a regular pair (E, A).…”