### Abstract

We consider the problem of finding a common quadratic Lyapunov function to demonstrate stability of a family of matrices which incorporate design freedoms. Generically, this can be viewed as picking a family of controller (or observer) gains so that the family of closed-loop system matrices admit a common Lyapunov function. We provide several conditions, necessary and sufficient, for various structures of matrix families. Families of matrices containing a subset of diagonal matrices invariant under the design freedoms is also considered since this is a case that occurs in many applications. Conditions for uniform solvability of the Lyapunov equations are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.

Original language | English (US) |
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Title of host publication | Proceedings of the American Control Conference |

Pages | 3896-3901 |

Number of pages | 6 |

Volume | 5 |

State | Published - 2004 |

Event | Proceedings of the 2004 American Control Conference (AAC) - Boston, MA, United States Duration: Jun 30 2004 → Jul 2 2004 |

### Other

Other | Proceedings of the 2004 American Control Conference (AAC) |
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Country | United States |

City | Boston, MA |

Period | 6/30/04 → 7/2/04 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*Proceedings of the American Control Conference*(Vol. 5, pp. 3896-3901)

**Conditions for uniform solvability of parameter-dependent Lyapunov equations with applications.** / Krishnamurthy, P.; Khorrami, Farshad.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the American Control Conference.*vol. 5, pp. 3896-3901, Proceedings of the 2004 American Control Conference (AAC), Boston, MA, United States, 6/30/04.

}

TY - GEN

T1 - Conditions for uniform solvability of parameter-dependent Lyapunov equations with applications

AU - Krishnamurthy, P.

AU - Khorrami, Farshad

PY - 2004

Y1 - 2004

N2 - We consider the problem of finding a common quadratic Lyapunov function to demonstrate stability of a family of matrices which incorporate design freedoms. Generically, this can be viewed as picking a family of controller (or observer) gains so that the family of closed-loop system matrices admit a common Lyapunov function. We provide several conditions, necessary and sufficient, for various structures of matrix families. Families of matrices containing a subset of diagonal matrices invariant under the design freedoms is also considered since this is a case that occurs in many applications. Conditions for uniform solvability of the Lyapunov equations are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.

AB - We consider the problem of finding a common quadratic Lyapunov function to demonstrate stability of a family of matrices which incorporate design freedoms. Generically, this can be viewed as picking a family of controller (or observer) gains so that the family of closed-loop system matrices admit a common Lyapunov function. We provide several conditions, necessary and sufficient, for various structures of matrix families. Families of matrices containing a subset of diagonal matrices invariant under the design freedoms is also considered since this is a case that occurs in many applications. Conditions for uniform solvability of the Lyapunov equations are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.

UR - http://www.scopus.com/inward/record.url?scp=8744282564&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=8744282564&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:8744282564

VL - 5

SP - 3896

EP - 3901

BT - Proceedings of the American Control Conference

ER -