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

Abstract

In this paper the feedback saddle point equilibria of soft-constrained zero-sum linear quadratic differential games for descriptor systems that have index one will be studied for a finite and infinite planning horizon. Both necessary and sufficient conditions for the existence of a feedback saddle point equilibrium are considered

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

“…In this section we recall some theorems of feedback zero-sum linear quadratic softconstrained differential games for index one descriptor systems from [21] that will be used as a framework in designing robust optimal controls. 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).…”

“…Following the procedure in [21], yields that ( ) * * , u w is a Feedback Saddle Point (FBSP) solution for the zero-sum differential game (1) with cost function (7) if and only if ( ) …”

“…Following this same procedure, the robust control design problem for descriptor systems can also be translated into a regular differential game. Hence, by merging results obtained in [1] and [21], we will solve the robust optimal control problem for descriptor systems here. Further, [20] solved this problem for an open-loop information framework.…”

“…Section 2 introduces the precise problem formulation of designing a disturbance attenuation problem by designing a robust optimal control using the differential game approach and translates the problem for a descriptor system into a regular differential game. Section 3 recalls from [21] the feedback zero-sum linear quadratic soft-constrained descriptor differential game for both a finite planning horizon and an infinite planning horizon. Section 4 presents the main result for the problem.…”

“…[1] used this approach to find robust controllers for regular systems (nonsingular systems). In [21] the feedback zero-sum linear quadratic soft-constrained descriptor differential game for index one systems was solved by transforming it into a regular differential game. Following this same procedure, the robust control design problem for descriptor systems can also be translated into a regular differential game.…”