Generalized H^∞ control theory

Abstract

The present H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> theory has some constraints in application, e.g. it can not deal with the servo problem. This is due to the superfluous requirement for the internal stability of the weighted feedback system In this paper, we alleviate the stability of the feedback system of G and K to admit pure imaginary poles, only assuring the internal stability of the feedback loop of G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> and K. After such generalization of H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> control problem, the servo problem is naturally incorporated into the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> synthesis. The solvability condition and the structure of solution are similar to those of the standard H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> control problem. The difference lies in the requirement for the solution of Riccati equation. Here instead of stabilizing solution, a solution of Riccati equation called quasi-stabilizing solution is used. Further, we reveal that all invariant zeros of G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> and G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> in the closed left half plane are hidden modes of the generalized H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> feedback system for almost all H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> controllers.