On the convexity of H∞ Riccati solutions and its applications

The authors consider the two-Riccati-equation solution to a standard H^∞ control problem, which can be used to characterize all possible stabilizing optimal or suboptimal H^∞ controllers if the optimal H^∞ norm (or γ), an upper bound of a suboptimal H^∞ norm is given. Some eigen properties of these H^∞ Riccati solutions are revealed. The most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of γ on the domain of interest. Based on these properties, a quadratically convergent algorithm is developed to compute the optimal H^∞ norm