Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov-like functions
Xiuliang Zheng
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
College of Science, Hebei North University, Zhangjiakou, China
Search for more papers by this authorCorresponding Author
Zhikun She
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Correspondence
Zhikun She, State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing 100191, China.
Email: [email protected]
Search for more papers by this authorQuanyi Liang
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Search for more papers by this authorMeilun Li
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Search for more papers by this authorXiuliang Zheng
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
College of Science, Hebei North University, Zhangjiakou, China
Search for more papers by this authorCorresponding Author
Zhikun She
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Correspondence
Zhikun She, State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing 100191, China.
Email: [email protected]
Search for more papers by this authorQuanyi Liang
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Search for more papers by this authorMeilun Li
State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics Informatics and Behavioral Semantics, School of Mathematics and Systems Science, Beihang University, Beijing, China
Search for more papers by this authorSummary
Domain of attraction plays an important role in control systems analysis, which is usually estimated by sublevel sets of Lyapunov functions. In this paper, based on the concept of common Lyapunov-like functions, we propose an iteration method for estimating domains of attraction for a class of switched systems, where the state space is divided into several regions, each region is described by polynomial inequalities, and any region has no intersection among with each other. Starting with an initial inner estimate of domain of attraction, we first present a theoretical framework for obtaining a larger inner estimate by iteratively computing common Lyapunov-like functions. Then, for obtaining a required initial inner estimate of domain of attraction, we propose a higher-order truncation and linear semidefinite programming–based method for computing a common Lyapunov function. Successively, the theoretical framework is under-approximatively realized by using S-procedure and sum-of-squares programming, associated with a coordinatewise iteration idea. Finally, we implement our method and test it on some examples with comparisons. These computation and comparison results show the advantages of our method.
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