H2H2 norm of linear time-periodic systems: A perturbation analysis

We consider a class of linear time-periodic systems in which the dynamical generator A(t)A(t) represents the sum of a stable time-invariant operator A0A0 and a small-amplitude zero-mean TT-periodic operator ϵAp(t)ϵAp(t). We employ a perturbation analysis to develop a computationally efficient method for determination of the H2H2 norm. Up to second order in the perturbation parameter ϵϵ we show that: (a) the H2H2 norm can be obtained from a conveniently coupled system of Lyapunov and Sylvester equations that are of the same dimension as A0A0; (b) there is no coupling between different harmonics of Ap(t)Ap(t) in the expression for the H2H2 norm. These two properties do not hold for arbitrary values of ϵϵ, and their derivation would not be possible if we tried to determine the H2H2 norm directly without resorting to perturbation analysis. Our method is well suited for identification of the values of period TT that lead to the largest increase/reduction of the H2H2 norm. Two examples are provided to motivate the developments and illustrate the procedure.