### 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) |
---|---|

Pages (from-to) | 2090-2098 |

Number of pages | 9 |

Journal | Automatica |

Volume | 44 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering

### Cite this

*Automatica*,

*44*(8), 2090-2098. https://doi.org/10.1016/j.automatica.2007.12.015

**H2 norm of linear time-periodic systems : A perturbation analysis.** / Jovanović, Mihailo R.; Fardad, Makan.

Research output: Contribution to journal › Article

*Automatica*, vol. 44, no. 8, pp. 2090-2098. https://doi.org/10.1016/j.automatica.2007.12.015

}

TY - JOUR

T1 - H2 norm of linear time-periodic systems

T2 - A perturbation analysis

AU - Jovanović, Mihailo R.

AU - Fardad, Makan

PY - 2008/8

Y1 - 2008/8

N2 - 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.

AB - 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.

KW - Distributed systems

KW - Frequency responses

KW - H norm

KW - Linear time-periodic systems

KW - Perturbation analysis

UR - http://www.scopus.com/inward/record.url?scp=47249137465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47249137465&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2007.12.015

DO - 10.1016/j.automatica.2007.12.015

M3 - Article

VL - 44

SP - 2090

EP - 2098

JO - Automatica

JF - Automatica

SN - 0005-1098

IS - 8

ER -