Control Of Several Classes Of Nonlinear Systems Using Partition Of Unity Method

Control of nonlinear systems has long been a main and difficult issue in control area.The most used by now and the uppermost method is Lyapunov direct method, which is to analyze and justify system stability by introducing a Lyapunov function. Yet due to nonlinear systems' diversity and complexity,there are no uniform effective methods in choosing Lyapunov functions.All the existing exact methods in nonlinear system control,are only applicable to a certain kind of nonlinear systems.Nonlinear system control using approximation-based method,such as fuzzy control,neural network control,etc,though has gained plentiful theoretic and actual achievements,there are still many theoretic and practical problems leaving unsolved.In the 70s of last century,differential geometry method was introduced into control theory of nonlinear systems,which makes nonlinear system theory gain a big boost. Partition of unity is an important concept in differential geometry.Its combination has the capability of approximating any continuous functions in a compact region of Eu-clidian space within any specified precision.In this dissertation,by taking advantage of this character to approximate nonlinear systems,then using Lyapunov stability theory, etc,combined with linear matrix inequality technology,we studied the control problems of several classes of nonlinear systems.The main contents can be concluded as:●Stability and stabilizability problemsIn chapter 2,for a class of time-delay systems with uncertainties,we analyzed its robust asymptotical stability problem by using Lyapunov-Krasovskii method,then presented a sufficient condition for the time-delay systems to be robust asymptotically stable in terms of LMI.In chapter 3,we firstly made use of partition of unity method to approximate the nonlinear functions of a class of nonlinear time-delay systems,so as to transform the original system into an equivalent system,then by using Lyapunov-Krasovskii method,combined with LMI technology,we presented a sufficient condition for the existence of a stabilization controller which can make the original system robust asymptotically stable.●H_∞control problemsIn chapter 4,we also used partition of unity method to approximate the nonlinear functions of a class of nonlinear systems,then analyzed the approximation precision by using continuous modulus of continuous functions.For a given reference model,by using Lyapunov stability theory,we studied the H_∞control problem.In chapter 5,by using partition of unity method and Lyapunov stability theory,we analyzed system stability and disturbance-restrained problem of a class of nonlinear systems with disturbance when attenuating at a specified decay rate,presented a sufficient condition for the existence of a state feedback Hoo controller in terms of LMI.●Guaranteed cost control problemsIn chapter 6,we firstly made use of partition of unity method to transform a class of nonlinear systems into an equivalent system,then for a H₂ quadratic performance index,by using Lyapunov stability theory,combined with LMI technology,we analyzed guaranteed cost control problem of the nonlinear system.In chapter 7,by using partition of unity method,we firstly transform a class of nonlinear time-delay systems into an equivalent system with local linear structure,which is easier to be dealt with.Then for a H₂ quadratic performance index,by using Lyapunov-Krasovskii method,combined with LMI technology,we analyzed guaranteed cost control problem of the original system.In chapter 8,by using partition of unity method and Lyapunov stability theory,we analyzed the stability problem of a class of nonlinear systems with uncertainties,presented an adaptive guaranteed cost controller and proposed a sufficient condition for the existence of the controller in terms of LMI.The adaptive law can be obtained by solving the LMIs.●State observer design problemIn chapter 9,for a class of nonlinear systems with uncertainties whose bound is unknown,we first took advantage of partition of unity method to approximate the unknown bound function of the uncertainties.The approximation error was adjusted by the adaptive law of parameters.By using Lyapunov stability theory,we studied the stability problem of the error system between the original nonlinear system and the system under the control of an adaptive state observer.Finally we drew a conclusion on the whole work of this dissertation,pointed out some existent problems and the future orientation of research.