Closed-Loop Stackelberg Starategy via Output Feedback

Abstract

The purpose of this paper is to solve two-level closed-loop Stackelberg game problem for a linear time-invariant system described by the following state equation and output equation.
{x=Ax+B₁u₁+B₂u₂ y=Cx
The Stackelberg strategy for each decision maker u_i is generated via output feedback with constant feedback gain, that is,
u_i=-F_iy
The Stackelberg strategy via output feedback is effective for the problem with incomplete information with respect to state value. The cost functional associated with u_i is
J_i=E{(1/2)∫^∞₀[x^TQ_ix+u^T₁R_i1u₁+u^T₂R_i2u₂]dt}
that is, J_i is trace criteria which is interpreted as expected value of quadratic criteria. Assume the initial state of system is uniformly distributed over the unit sphere. We show that there exist optimal Stackelberg strategies to this problem.
In this paper, necessary conditions for a optimal Stackelberg strategy via output feedback are given in the infinite time problem. The necessary conditions are described by a set of coupled Riccati and Lyapunov algebraic matrix equations. In the special case that C^-1 exists, the necessary conditions are identical to ones derived by Medanic.