Solution to the general mixed H2/H∞ control problem - necessary conditions for optimality

A synthesis method is developed to design an output feedback controller to minimize the two-norm of one transfer function while ensuring the infinity-norm of another is held below a chosen level. This is known as the general mixed H₂/H_∞ optimization problem. The solution minimizes the actual two-norm, rather than an upper bound to it, and therefore is not conservative. Seven coupled nonlinear matrix equations are derived which represent the necessary conditions such a solution must satisfy. By analyzing the case where the desired controller has the same order as the plant, it is found that the solution lies on the boundary of the infinity-norm constraint whenever the two objectives are competing. In this case, the mixed controller requires the neutrally stabilizing solution to a Riccati equation and solution of a Lyapunov equation which has no unique solution.