## 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 ε{lunate} 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{lunate} 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{lunate}, 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.

Original language | English (US) |
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Pages (from-to) | 2090-2098 |

Number of pages | 9 |

Journal | Automatica |

Volume | 44 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2008 |

## Keywords

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

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