The singular $H_\infty$ control problem with dynamic measurement feedback

This paper is concerned with the $H_\infty$ problem with measurement feedback. The problem is to find a dynamic feedback from the measured output to the control input such that the closed loop system has an $H_\infty$ norm strictly less than some a priori given bound $\gamma$ and such that the closed loop system is internally stable. Necessary and sufficient conditions are given under which such a feedback exists. The only assumptions we have to make is that there are no invariant zeros on the imaginary axis for two subsystems. Contrary to recent publications no assumptions are made on the direct feed through matrices of the plant. It turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability, i.e. the problem in which we can make the $H_\infty$ norm arbitrarily small. Keywords: Quadratic matrix inequality, Riccati equation, Almost disturbance decoupling, Measurement feedback, Internal stability.