Pareto optimal generalized $H_2$-control and vibroprotection problems

We consider a novel multi-objective control problem where the criteria are generalized $H_2$-norms of transfer matrices of individual channels from the disturbance input to various objective outputs. We obtain necessary conditions for Pareto optimality. We show that synthesis of Pareto optimal controls can be done in terms of linear matrix inequalities based on optimizing Germeier's convolution, which also turns out to be the generalized $H_2$-norm of the closed-loop system with output composed of the objective outputs multiplied by scalars. As applications we consider multi-objective problems of vibration isolation and oscillation suppression with new types of criteria.