Improvements to the linear differential inclusion approach to stability analysis of linear systems with saturated linear feedback

Ellipsoids, as level sets of quadratic Lyapunov functions, and the convex hull of ellipsoids, as a level set of a certain composite quadratic Lyapunov function, have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. By expressing...

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Published in	Automatica (Oxford) Vol. 49; no. 3; pp. 821 - 828
Main Authors	Li, Yuanlong, Lin, Zongli
Format	Journal Article
Language	English
Published	Kidlington Elsevier Ltd 01.03.2013 Elsevier
Subjects	Actuator saturation Applied sciences Attraction Composite Lyapunov function Computer science; control theory; systems Control system analysis Control theory. Systems Domain of attraction Ellipsoids Exact sciences and technology Feedback Hulls Hulls (structures) Law Linear systems Lyapunov functions Set invariance System theory Domain of attraction Composite Lyapunov function Actuator saturation Set invariance Saturation Feedback regulation Linear condition Invariant set Dynamical system Conservative system Convex hull Convex function Differential inclusion Ellipsoid Actuator Lyapunov function Quadratic function
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ISSN	0005-1098 1873-2836
DOI	10.1016/j.automatica.2012.12.002

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Abstract	Ellipsoids, as level sets of quadratic Lyapunov functions, and the convex hull of ellipsoids, as a level set of a certain composite quadratic Lyapunov function, have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. By expressing the saturated linear feedback law on the convex hull of a group of linear feedback laws, which in turn expresses the linear system under this saturated linear feedback in a linear differential inclusion, conditions have been established under which an ellipsoid or the convex hull of a group of ellipsoids are contractively invariant sets and are thus estimates of the domain of attraction. These conditions are usually less conservative for single input systems than for multiple input systems. In this paper, we consider multiple input systems and establish conditions for contractive invariance of the convex hull of ellipsoids that are less conservative than the existing conditions.
AbstractList	Ellipsoids, as level sets of quadratic Lyapunov functions, and the convex hull of ellipsoids, as a level set of a certain composite quadratic Lyapunov function, have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. By expressing the saturated linear feedback law on the convex hull of a group of linear feedback laws, which in turn expresses the linear system under this saturated linear feedback in a linear differential inclusion, conditions have been established under which an ellipsoid or the convex hull of a group of ellipsoids are contractively invariant sets and are thus estimates of the domain of attraction. These conditions are usually less conservative for single input systems than for multiple input systems. In this paper, we consider multiple input systems and establish conditions for contractive invariance of the convex hull of ellipsoids that are less conservative than the existing conditions.
Author	Li, Yuanlong Lin, Zongli
Author_xml	– sequence: 1 givenname: Yuanlong surname: Li fullname: Li, Yuanlong email: liyuanlong0301@163.com organization: Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing of Ministry of Education, Shanghai 200240, China – sequence: 2 givenname: Zongli surname: Lin fullname: Lin, Zongli email: zl5y@virginia.edu organization: Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing of Ministry of Education, Shanghai 200240, China
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Keywords	Domain of attraction Composite Lyapunov function Actuator saturation Set invariance Saturation Feedback regulation Linear condition Invariant set Dynamical system Conservative system Convex hull Convex function Differential inclusion Ellipsoid Actuator Lyapunov function Quadratic function
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Snippet	Ellipsoids, as level sets of quadratic Lyapunov functions, and the convex hull of ellipsoids, as a level set of a certain composite quadratic Lyapunov...
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SubjectTerms	Actuator saturation Applied sciences Attraction Composite Lyapunov function Computer science; control theory; systems Control system analysis Control theory. Systems Domain of attraction Ellipsoids Exact sciences and technology Feedback Hulls Hulls (structures) Law Linear systems Lyapunov functions Set invariance System theory
Title	Improvements to the linear differential inclusion approach to stability analysis of linear systems with saturated linear feedback
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