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postgraduate thesis: Domain of attraction in hybrid systems
Title  Domain of attraction in hybrid systems 

Authors  
Issue Date  2015 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Luk, C. [陸傳傑]. (2015). Domain of attraction in hybrid systems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5610962 
Abstract  Domain of Attraction (DoA) is a set of initial conditions for which the system converges to the equilibrium point. In fact, it is a key problem in control engineering to guarantee stability within a workspace and avoid system failures. Classical applications include pendulum systems, tunnel diode circuits, massspring systems, negativeresistance oscillators and more recently, these have been found in other fields such as biology and ecology.
This thesis firstly addresses the estimation of the DoA for a class of hybrid nonlinear systems in both discrete and continuoustime. The state space is partitioned into several regions which are described by polynomial inequalities, and the union of all the regions is a complete cover of the state space. The system dynamics are defined on each region independently from the others by polynomial functions. The problem of computing the largest sublevel set of a Lyapunov function included in the DoA is considered. An approach is proposed for addressing this problem based on linear matrix inequalities (LMIs), which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region.
Secondly, a sufficient and necessary condition is firstly provided for establishing optimality of the found lower bound. This requires to solve linear algebra operations in typical cases. The problem of looking for variable Lyapunov functions that maximize the estimate of the DoA is also considered, describing several strategies where the proposed approach can be readily adopted.
Thirdly, the computation of static nonlinear output feedback controllers for increasing the DoA of an equilibrium point of continuous hybrid nonlinear polynomial systems is addressed. A dynamical system where the state space is partitioned into possibly overlapping regions, and where the vector field is defined independently among the regions by polynomial functions, will be considered. The computation of static nonlinear output feedback controller that increase the estimate of the DoA provided by a polynomial Lyapunov function is addressed. The controller can be common or vary among the regions that partition the state space. A strategy is proposed which provides guaranteed estimates of the increased DoA controllers and the controllers required to achieve them. Moreover, this strategy can be readily exploited with optimality test and variable Lyapunov functions through the use of approaches described. 
Degree  Doctor of Philosophy 
Subject  Mathematical optimization Systems engineering 
Dept/Program  Electrical and Electronic Engineering 
Persistent Identifier  http://hdl.handle.net/10722/221182 
HKU Library Item ID  b5610962 
DC Field  Value  Language 

dc.contributor.author  Luk, Chuenkit   
dc.contributor.author  陸傳傑   
dc.date.accessioned  20151104T23:11:55Z   
dc.date.available  20151104T23:11:55Z   
dc.date.issued  2015   
dc.identifier.citation  Luk, C. [陸傳傑]. (2015). Domain of attraction in hybrid systems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5610962   
dc.identifier.uri  http://hdl.handle.net/10722/221182   
dc.description.abstract  Domain of Attraction (DoA) is a set of initial conditions for which the system converges to the equilibrium point. In fact, it is a key problem in control engineering to guarantee stability within a workspace and avoid system failures. Classical applications include pendulum systems, tunnel diode circuits, massspring systems, negativeresistance oscillators and more recently, these have been found in other fields such as biology and ecology. This thesis firstly addresses the estimation of the DoA for a class of hybrid nonlinear systems in both discrete and continuoustime. The state space is partitioned into several regions which are described by polynomial inequalities, and the union of all the regions is a complete cover of the state space. The system dynamics are defined on each region independently from the others by polynomial functions. The problem of computing the largest sublevel set of a Lyapunov function included in the DoA is considered. An approach is proposed for addressing this problem based on linear matrix inequalities (LMIs), which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region. Secondly, a sufficient and necessary condition is firstly provided for establishing optimality of the found lower bound. This requires to solve linear algebra operations in typical cases. The problem of looking for variable Lyapunov functions that maximize the estimate of the DoA is also considered, describing several strategies where the proposed approach can be readily adopted. Thirdly, the computation of static nonlinear output feedback controllers for increasing the DoA of an equilibrium point of continuous hybrid nonlinear polynomial systems is addressed. A dynamical system where the state space is partitioned into possibly overlapping regions, and where the vector field is defined independently among the regions by polynomial functions, will be considered. The computation of static nonlinear output feedback controller that increase the estimate of the DoA provided by a polynomial Lyapunov function is addressed. The controller can be common or vary among the regions that partition the state space. A strategy is proposed which provides guaranteed estimates of the increased DoA controllers and the controllers required to achieve them. Moreover, this strategy can be readily exploited with optimality test and variable Lyapunov functions through the use of approaches described.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject.lcsh  Mathematical optimization   
dc.subject.lcsh  Systems engineering   
dc.title  Domain of attraction in hybrid systems   
dc.type  PG_Thesis   
dc.identifier.hkul  b5610962   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Electrical and Electronic Engineering   
dc.description.nature  published_or_final_version   