Singular Infinite-Horizon $H_{\infty }$ Problem

Abstract

In this chapter, we consider a system consisting of an uncertain controlled linear time-invariant differential equation and a linear time-invariant output algebraic equation. For this system, an infinite-horizon $H_{\infty }$ problem is studied in the case where the rank of the coefficients’ matrix for the control in the output equation is smaller than the Euclidean dimension of this control. In this case, the solvability conditions, based on the game-theoretic matrix Riccati algebraic equation, are not applicable to the solution of the considered $H_{\infty }$ problem meaning its singularity. To solve this $H_{\infty }$ problem, a regularization method is proposed. Namely, the original problem is replaced approximately with a regular infinite-horizon $H_{\infty }$ problem depending on a small positive parameter. Thus, the first-order solvability conditions are applicable to this new problem. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix algebraic equation, arising in these conditions, yields a controller solving the original singular $H_{\infty }$ problem. Properties of this controller are studied.