Study On Dissipativity And Invariance Principles Of Switched Nonlinear Systems

Switched systems are an important and special class of hybrid systems,which have drawn considerable attention in the last decades since a large class of practical systems can be modeled as switched systems and there exist many systems that cannot be effectively controlled by a single controller,but the control goals can be achieved by controllers switching.However,due to the complexity arising from interaction between the continuous dynamics and discrete dynamics,the dynamic behavior of switched systems becomes very complicated and this makes the design of switched systems very difficult.Thus,a lot of problems deserve investigation.On the other hand,the dissipativity property and invariance principles have been proven to be powerful and effective tools for the analysis and design of non-switched systems and the systematical theories have been established.For switched systems,however,the studies on dissipativity and invariance principles are just at the early stage.Many problems are far from being understood and thus deserve investigation.This dissertation studies the dissipativity property,invariance principles and a few synthesis issues including the stabilization under asynchronous switching,H?control and output synchronization based on the dissipativity property and the invariance principle developed for switched systems.The main contributions are as follows.1.The stabilization issue of switched nonlinear systems in which the controllers are switched asynchronously with the switching of system modes is investigated.For given average dwell time,given passivity rate and admissible switching delay,the method of the controllers design are given to achieve stabilization under asynchronous switching for switched nonlinear systems consisting of passive subsystems and non-passive subsystems by using an average dwell time approach.2.The H? control problem of switched nonlinear systems is addressed using an average dwell time approach and by designing a state-dependent switching law respectively.The first part gives the method of the controllers design to guarantee the solvability of the problem for switched nonlinear systems consisting of passive subsystems and non-passive subsystems by using an average dwell time approach.The second part focuses on the control problem for a class of switched nonlinear systems based on passivity.The solvability conditions of the control problem are obtained by designing a state-dependent switching law and state feedback controller of each subsystem.In addition,if all subsystems are not passive,we make a partition of the state space and design controllers for subsystems such that each subsystem has the passivity property on the associated region,and then give the method of the controllers design to guarantee the solvability conditions of the control problem.3.The output synchronization problem for discrete-time dynamical networks with identical nodes is considered.Firstly,if each node of a network is geometrically incrementally dissipative,the entire network can be viewed as a geometrically dissipative nonlinear system by choosing a particular input-output pair.Then,based on the geometrical dissipativity property,we consider two cases:output synchronization under arbitrary topology and switching topology,respectively.For the first case,we establish several criteria of output synchronization under arbitrary switching between a set of communication topologies by employing a common Lyapunov function.For the other case,we give the design method of a switching signal to achieve output synchronization even if all subnetworks are not synchronous.4.The asymptotical behavior problem for discrete-time switched nonlinear systems and continuous-time switched nonlinear systems is addressed,respectively.For discrete-time switched nonlinear systems,three invariance principles is developed based on common Lyapunov function,multiple Lyapunov functions and multiple weak Lyapunov functions,respectively,which allows the first differences of multiple weak Lyapunov functions to be positive on some set.It is shown that the solution of the system is attracted to the largest weakly invariant set in some specific region.For continuous-time switched nonlinear systems,an invariance principle is proposed based on Lyapunov-like functions.5.The generalized output synchronization problem is studied for discrete dynamical networks with non-identical nodes.Three cases is considered:generalized output synchronization for fixed topology,generalized output synchronization under arbitrary switching topology and generalized output synchronization for designing switching topologies,respectively.For the first case,two criteria for generalized output synchronization are established by employing the geometrical dissipativity.For the second case,the generalized output synchronization under arbitrary switching topology is given based on the geometrical dissipativity and an invariance principle.For the final case,based on the geometrical dissipativity and an invariance principle,the generalized output synchronization is obtained for discrete-time dynamical networks with non-identical nodes by an appropriate switching among several communication topologies.6.The problem of the generalized L2-gain property for switched systems is investigated by using multiple storage functions.The generalized L2-gain property characterized by a positive function of the state describes the non-uniform gain property for switched systems.For interconnected switched systems,a small-gain theorem using generalized L2-gain is obtained,which generalizes the well-known small-gain theorem.Moreover,if each subsystem of a switched system does not admit a generalized L2-gain,by choosing an appropriate switching law,the overall system can be made to satisfy a generalized L2-gain condition.The conclusions and perspectives are presented in the end of the thesis.