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 ) represents the sum of a stable time-invariant operator A 0 and a small-amplitude zero-mean T -periodic operator ϵ A p ( t ) . We employ a perturbation analysis to develop a computationally efficient method for determination of the H 2 norm. Up to second order in the perturbation parameter ϵ we show that: (a) the H 2 norm can be obtained from a conveniently coupled system of Lyapunov and Sylvester equations that are of the same dimension as A 0 ; (b) there is no coupling between different harmonics of A p ( t ) in the expression for the H 2 norm. These two properties do not hold for arbitrary values of ϵ , and their derivation would not be possible if we tried to determine the H 2 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 H 2 norm. Two examples are provided to motivate the developments and illustrate the procedure.