Brief paper: H2 norm of linear time-periodic systems: A perturbation analysis

Authors:
Mihailo R. Jovanović;Makan Fardad
Affiliations:
Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, United States;Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, United States
Venue:
Automatica (Journal of IFAC)
Year:
2008

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Abstract

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 @eA"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 @e 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 @e, 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.