H2 norm of linear time-periodic systems: a perturbation analysis

We consider a class of linear time-periodic systems in which dynamical generator A(t) represents a sum of a stable time-invariant operator A₀ and a small amplitude zero-mean T-periodic operator epsiA_P(t). We employ a perturbation analysis to develop a computationally efficient method for determination of the H ₂ norm. Up to a second order in perturbation parameter epsi we show that: a) the H₂ norm can be obtained from a conveniently coupled system of readily solvable Lyapunov and Sylvester equations; b) there is no coupling between different harmonics of A_p(t) in the expression for the H₂ norm. These two properties do not hold for arbitrary values of epsi, and their derivation would not be possible if we tried to determine the H₂ norm directly without resorting to perturbation analysis. Our method is well suited for identification of the values of period T that lead to the largest increase/reduction of the H₂ norm. Two examples are provided to motivate the developments and illustrate the procedure