# FINDING A COMMON QUADRATIC LYAPUNOV FUNCTION USING CONICAL HULLS

### Rianto Adhy Sasongko, J. C. Allwright

#### Abstract

Consider a set of linear time-invariant continuous-time systems that is a convex hull with vertices formed by a given set of systems. The problem of finding a common Lyapunov function v, specified in terms of a symmetric positive definite matrix, for the convex hull of systems is tackled by searching for a symmetric positive definite (PD) matrix P which causes dv/dt to be negative definite for each vertex system. The approach involves an extension of an existing method for solving optimization problems for positive semidefinite (PSD) matrices that is based on a representation of the cone of PSD matrices as a conical hull. The condition that the derivative of the Lyapunov function for each vertex system is negative definite is converted naturally into the condition that the matrix P belongs to the interior of the intersection of several conical hulls: one for each vertex system to ensure dv/dt for it is negative definite. The determination of a P in the intersection is viewed as the solution of a quadratic programme on the product space of the cones. Then the existing theory and algorithms for conical hull problems are adapted to the solution of the quadratic programme. The numerical results suggest that the proposed algorithm is faster than the projective method used in MATLAB for small problems. Effort is being devoted to improve it for larger problem.

#### References

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#### Paper Citation

#### in Harvard Style

Adhy Sasongko R. and C. Allwright J. (2006). **FINDING A COMMON QUADRATIC LYAPUNOV FUNCTION USING CONICAL HULLS** . In *Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-972-8865-59-7, pages 113-118. DOI: 10.5220/0001218001130118

#### in Bibtex Style

@conference{icinco06,

author={Rianto Adhy Sasongko and J. C. Allwright},

title={FINDING A COMMON QUADRATIC LYAPUNOV FUNCTION USING CONICAL HULLS},

booktitle={Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2006},

pages={113-118},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001218001130118},

isbn={978-972-8865-59-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - FINDING A COMMON QUADRATIC LYAPUNOV FUNCTION USING CONICAL HULLS

SN - 978-972-8865-59-7

AU - Adhy Sasongko R.

AU - C. Allwright J.

PY - 2006

SP - 113

EP - 118

DO - 10.5220/0001218001130118