Analysis Of Stability And Study On Control For Dynamic Systems With Respect To Partial State Variables

As is well known, the stability of system is the most essential and the most important property, which must be considered in system analysis and control system design. Without stability, there is no need to talk about any other property. After the foundations of movement stability theory established by Lyapunov in 1892, the theory has already been accepted by the whole world and numerous realms. With the development of science and technology, partial stability, namely concerning stability of parts states variables of system, was produced based on Lyapunov stability theory. Since 1957, the importance of partial stability has been gradually recognized and this theory has also been accepted universally later on, and the related researches have been developed greatly. First, the partial stability theory is a kind of thought and method. Many phenomena in nature, many problems in social economy, and many techniques in actual engineering conceal many dynamic rules, which can be described by the different system. Owing to its complicated structure and numerous factors, it is becoming very difficult to process some problems. Sometimes one will have to clear up dynamic behavior of the part states of the system before dynamic characteristic of the whole system can be obtained. Secondly, partial stability theory is also an actual demand. In many actual problems, people are only interested in part state variables of systems, or because of the technical difficulty, other state variables of the systems cannot be controlled or measured, this will oblige people to study partial stability property of system. Again, partial stability is also the true reflection of objective thing. In many problems concerning science and technology, the stability property of each state variable for a system may be different: some variables are unstable, while others are stable or asymptotically stable; or some variables are stable, while others are asymptotically stable or exponentially stable. Therefore, studying partial stability for systems has its theoretical importance and practical values. Last, partial stability theory and application study for systems are a modern topic. The theory itself is not perfect; some theory results have not yet been built up; many existed results can be also further expanded; and there have been few results for partial stabilization of systems. This thesis will continue the study on such aspect. As is mentioned above, this thesis will mainly investigate the partial stability, partial stabilizations and applications of partial stability with respect to part states of some differential system, delay system and stochastic linear system. Some correlative conditions on various partial stability of the correspondent system will be given. The relationship among partial solutions of differential systems, Lyapunov functions and various partial stabilities between delay systems and certainty systems will be obtained. Concrete contents are as follows. In this thesis, the relation among various partial stabilities and partial attractions and class K function and class L function has been investigated. Some equivalent relations have been derived. Based on these equivalent relations, the converse theorems of various partial stabilities have been studied and some necessary conditions for Lyapunov functions constructed by partial solutions of systems have been obtained. For applications of these converse theorems of partial stability, under certain conditions, partial uniform stability, partial asymptotic stability and partial exponential stability of nonlinear differential systems and partial exponential stability of a class of delay system are discussed and partial stability property of nonlinear system under continuous disturbance has been further researched. For the differential dynamic system, some related criteria of partial equiasymptotical stability and partial exponential stability have been built up. Under some reasonable assumption, partial asymptotic stability and partial stability under continuous disturbance are discussed and a series of sufficient conditions are obtained. These results generalize and improve the corresponding ones existed. At the same time, globally partial stability for a class of nonlinear time-varying system is also studied by using Lyapunov's function method for autonomous systems with separated variables, and the system studied and these results obtained here have generalized the related models and the corresponding conclusions existed. Design method of partial stabilization controller of null solution for time-varying system is given via using the property of the M -matrix and some of these stabilizing controllers are only related to partial state variables. The problem of partial stability is investigated for several types of time-varying systems and delay systems. Based on Cauchy matrix solution and intercept Cauchy matrix solution of linear system and method of variation of parameters, various partial stabilities, including partial stability, partial uniform stability, partial asymptotic stability and partial exponential stability, are investigated for differential system under linear disturbance of part variables and nonlinear disturbance, and some sufficient and necessary conditions are obtained as well as some sufficient conditions. Making further use of this method, the partial uniformly asymptotic stability and partial exponential stability for several delay systems includingsingle-delay systems, multi-delay systems and full-delay systems are discussed by accepting the time-delay term in nonlinear system as the disturbance of linear system, these results obtained here are both independent and dependent on the magnitudes of the delays. Finally, some numerical examples are given to verify the effectiveness of the results. Based on aggregation-decomposition methods of large-scale system, the partial stability and design method of partial stabilization controllers are studied. Partial asymptotic stability and partial exponential stability of the large-scale systems are studied via making use of both scalar and vector Lyapunov function methods, the property of matrix and relevant inequality, where the null solution of the corresponding isolated sub-systems are partially stable or unstable. A series of sufficient conditions have been obtained under the condition that the related terms satisfy certain conditions and the main results extend the corresponding results existed. Meanwhile, the design method of two partial stabilization controllers is studied by using of these methods. First a nonlinear feedback controller for control large-scale systems is studied, where the corresponding isolated subsystems is unstable. Then a linear feedback controller of part state variables for control large-scale systems is also considered via the partial stability of the corresponding isolated subsystems with linear feedback control terms of part state variables in the presence of unknown stability of the original isolated subsystems for the control large-scale systems. In both cases some sufficient conditions for the partial stabilization of control large-scale systems are obtained by matrix properties and inequality analysis technique. These results generalize and improve the corresponding ones existed. The new concept of partial relative stability is put forward, and the theorem of partial relative stability for the two systems of different dimensions is built up by using Lyapunov function method. These concepts and theorems include the relevant concepts and the relevant theorems of Lyapunov stability and partial stability, respectively. Then based on the theory of partial relative stability built up, partial synchronization of angular motions of two solid bodies with bidirectional couple of linear feedback for part state variables and unidirectional couple of nonlinear feedback for part state variables are discussed. Finally, partial synchronization with respect to part states for the two different chaotic systems with unidirectional couple is also investigated via adaptive control, and an adaptive controller is designed to control the part states of the response system synchronizing that of the driving system. The virtual result has proved the feasibility of the method applied. The problem of strong stability with respect to partial variables for Ito? stochastic linear system is discussed. Left interceptive Cauchy matrix solutions and right interceptive Cauchy matrix solutions were introduced by using Cauchy matrix solutions and interceptive Cauchymatrix solutions, the almost sure strong stability with respect to partial variables in probability for Ito? stochastic linear system were studied and some equivalent criteria for different kinds of almost sure strong stability of the system were obtained, which only depended on the boundedness of the left interceptive Cauchy matrices and asymptotic properties of the right interceptive Cauchy matrices. Some equivalent relations of various strong stabilities with respect to partial variables in probability are also obtained via utilizing these matrices and the monotony and continuity of measure. These results popularize some existed ones in relevant literatures.