Discrete-time algebraic Riccati equation arising in H∞ filtering problem

A necessary and sufficient condition for the H_∞ filtering problem to be solvable is that the H_∞ algebraic Riccati equation (ARE) has a positive semi-definite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a solution is a monotonically non-increasing convex function of the prescribed H_∞ norm bound γ. Moreover, in this paper, the degree of freedom contained in the H_∞ filter is investigated based upon this property of the Riccati solution. It turns out that the degree of freedom reduces to zero as γ tends to the optimal value in a certain case. These results provide a guideline for the design of an H_∞ filter