### Abstract

This technical note considers the stability of a feedback connection of a known linear, time-invariant system and a perturbation. The input/output behavior of the perturbation is described by an integral quadratic constraint (IQC). IQC stability theorems can be formulated in the frequency domain or with a time-domain dissipation inequality. The two approaches are connected by a non-unique factorization of the frequency domain IQC multiplier. The factorization must satisfy two properties for the dissipation inequality to be valid. First, the factorization must ensure the time-domain IQC holds for all finite times. Second, the factorization must ensure that a related matrix inequality, when feasible, has a positive semidefinite solution. This technical note shows that a class of frequency domain IQC multipliers has a factorization satisfying these two properties. Thus the dissipation inequality test, with an appropriate factorization, can be used with no additional conservatism.

Original language | English (US) |
---|---|

Article number | 6915700 |

Pages (from-to) | 1704-1709 |

Number of pages | 6 |

Journal | IEEE Transactions on Automatic Control |

Volume | 60 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2015 |

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### Keywords

- Integral quadratic constraint (IQC)
- linear time-invariant (LTI)

### Cite this

**Stability analysis with dissipation inequalities and integral quadratic constraints.** / Seiler, Peter.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 60, no. 6, 6915700, pp. 1704-1709. https://doi.org/10.1109/TAC.2014.2361004

}

TY - JOUR

T1 - Stability analysis with dissipation inequalities and integral quadratic constraints

AU - Seiler, Peter

PY - 2015/6/1

Y1 - 2015/6/1

N2 - This technical note considers the stability of a feedback connection of a known linear, time-invariant system and a perturbation. The input/output behavior of the perturbation is described by an integral quadratic constraint (IQC). IQC stability theorems can be formulated in the frequency domain or with a time-domain dissipation inequality. The two approaches are connected by a non-unique factorization of the frequency domain IQC multiplier. The factorization must satisfy two properties for the dissipation inequality to be valid. First, the factorization must ensure the time-domain IQC holds for all finite times. Second, the factorization must ensure that a related matrix inequality, when feasible, has a positive semidefinite solution. This technical note shows that a class of frequency domain IQC multipliers has a factorization satisfying these two properties. Thus the dissipation inequality test, with an appropriate factorization, can be used with no additional conservatism.

AB - This technical note considers the stability of a feedback connection of a known linear, time-invariant system and a perturbation. The input/output behavior of the perturbation is described by an integral quadratic constraint (IQC). IQC stability theorems can be formulated in the frequency domain or with a time-domain dissipation inequality. The two approaches are connected by a non-unique factorization of the frequency domain IQC multiplier. The factorization must satisfy two properties for the dissipation inequality to be valid. First, the factorization must ensure the time-domain IQC holds for all finite times. Second, the factorization must ensure that a related matrix inequality, when feasible, has a positive semidefinite solution. This technical note shows that a class of frequency domain IQC multipliers has a factorization satisfying these two properties. Thus the dissipation inequality test, with an appropriate factorization, can be used with no additional conservatism.

KW - Integral quadratic constraint (IQC)

KW - linear time-invariant (LTI)

UR - http://www.scopus.com/inward/record.url?scp=84930657546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930657546&partnerID=8YFLogxK

U2 - 10.1109/TAC.2014.2361004

DO - 10.1109/TAC.2014.2361004

M3 - Article

AN - SCOPUS:84930657546

VL - 60

SP - 1704

EP - 1709

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 6

M1 - 6915700

ER -