### Abstract

This technical note develops linear matrix inequality (LMI) conditions to test whether an uncertain linear system is exponentially stable with a given decay rate α. These new α- exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time-invariant system and a 'troublesome' perturbation. The perturbation can contain uncertain parameters, time delays, or nonlinearities. This technical note presents two key contributions. First, α- exponential stability of the uncertain LTI system is shown to be equivalent to (internal) linear stability of a related scaled system. This enables derivation of α- exponential stability tests from linear stability tests using integral quadratic constraints (IQCs). This connection requires IQCs to be constructed for a scaled perturbation operator. The second contribution is a list of IQCs derived for the scaled perturbation using the detailed structure of the original perturbation. Finally, connections between the proposed approach and related work are discussed.

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

Article number | 7393522 |

Pages (from-to) | 3631-3637 |

Number of pages | 7 |

Journal | IEEE Transactions on Automatic Control |

Volume | 61 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2016 |

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### Keywords

- Exponential convergence rate
- integral quadratic constraint
- robustness

### Cite this

**Exponential Decay Rate Conditions for Uncertain Linear Systems Using Integral Quadratic Constraints.** / Hu, Bin; Seiler Jr, Peter J.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 61, no. 11, 7393522, pp. 3631-3637. https://doi.org/10.1109/TAC.2016.2521781

}

TY - JOUR

T1 - Exponential Decay Rate Conditions for Uncertain Linear Systems Using Integral Quadratic Constraints

AU - Hu, Bin

AU - Seiler Jr, Peter J

PY - 2016/11/1

Y1 - 2016/11/1

N2 - This technical note develops linear matrix inequality (LMI) conditions to test whether an uncertain linear system is exponentially stable with a given decay rate α. These new α- exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time-invariant system and a 'troublesome' perturbation. The perturbation can contain uncertain parameters, time delays, or nonlinearities. This technical note presents two key contributions. First, α- exponential stability of the uncertain LTI system is shown to be equivalent to (internal) linear stability of a related scaled system. This enables derivation of α- exponential stability tests from linear stability tests using integral quadratic constraints (IQCs). This connection requires IQCs to be constructed for a scaled perturbation operator. The second contribution is a list of IQCs derived for the scaled perturbation using the detailed structure of the original perturbation. Finally, connections between the proposed approach and related work are discussed.

AB - This technical note develops linear matrix inequality (LMI) conditions to test whether an uncertain linear system is exponentially stable with a given decay rate α. These new α- exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time-invariant system and a 'troublesome' perturbation. The perturbation can contain uncertain parameters, time delays, or nonlinearities. This technical note presents two key contributions. First, α- exponential stability of the uncertain LTI system is shown to be equivalent to (internal) linear stability of a related scaled system. This enables derivation of α- exponential stability tests from linear stability tests using integral quadratic constraints (IQCs). This connection requires IQCs to be constructed for a scaled perturbation operator. The second contribution is a list of IQCs derived for the scaled perturbation using the detailed structure of the original perturbation. Finally, connections between the proposed approach and related work are discussed.

KW - Exponential convergence rate

KW - integral quadratic constraint

KW - robustness

UR - http://www.scopus.com/inward/record.url?scp=84994728070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994728070&partnerID=8YFLogxK

U2 - 10.1109/TAC.2016.2521781

DO - 10.1109/TAC.2016.2521781

M3 - Article

VL - 61

SP - 3631

EP - 3637

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 11

M1 - 7393522

ER -