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

Mihailo Jovanovic, Makan Fardad

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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 A0 and a small-amplitude zero-mean T-periodic operator ε{lunate} Ap (t). We employ a perturbation analysis to develop a computationally efficient method for determination of the H2 norm. Up to second order in the perturbation parameter ε{lunate} we show that: (a) the H2 norm can be obtained from a conveniently coupled system of Lyapunov and Sylvester equations that are of the same dimension as A0; (b) there is no coupling between different harmonics of Ap (t) in the expression for the H2 norm. These two properties do not hold for arbitrary values of ε{lunate}, 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.

Original languageEnglish (US)
Pages (from-to)2090-2098
Number of pages9
Issue number8
StatePublished - Aug 2008


  • Distributed systems
  • Frequency responses
  • H norm
  • Linear time-periodic systems
  • Perturbation analysis


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