Discrete-time H ∞ algebraic Riccati equation and parametrization of all H ∞ filters

This paper is concerned with the algebraic Riccati equations (AREs) related to the H ∞ filtering problem. A necessary and sufficient condition for the H ∞ problem to be solvable is that the H ∞ ARE has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a stabilizing solution is a monotonically non-increasing convex function of the prescribed H ∞ norm bound γ. This property of the H ∞ ARE is very important for the analysis of the performance of the H ∞ filter. In this paper, the size of the set of all H ∞ filters is considered on the basis of the monotonicity of the above Riccati solution. It turns out that, under a certain condition, the degree of freedom of the H ∞ filter reduces a1 the optimal H ∞ norm bound. These results provide a guideline for selecting the value of γ Some numerical examples are included.