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Stability Analysis And Control Design Of Multiple-Equilibrium Switched Systems

Posted on:2012-02-20Degree:DoctorType:Dissertation
Country:ChinaCandidate:R W GuoFull Text:PDF
GTID:1118330371450987Subject:Control theory and control engineering
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In the past two decades, the switched system has attracted great attention in control systems community due to their significance in both theory and applications. The re-search of switched systems has drawn a considerable attention and a lot of nice results have obtained for switched systems. For the stability analysis problem, one impor-tant stability analysis tool for switched systems is Common Lyapunov Function (CLF) method. But, for a given switched system, it is very difficult to verify that whether a common Lyapunov function for such system exists or not. Another powerful stability analysis tool for switched systems is Multiple Lyapunov Functions (MLF) method. It was also shown that the switched system is stable if switching among stable subsystems is slow on the average, in other words, the dwell time is sufficiently large. Therefore, how to find the minimum dwell time (MDT) is very important. However, the present results on finding the MDT for the nonlinear switched systems are elementary and have some limitations. Accordingly, it is important and very urgent to solving this problem.It is noted that up to now, most of the existing literature on stability issues of switched systems assumes that all subsystems share a common equilibrium (typically the origin) and hence the stability of such systems is actually the stability of the com-mon equilibrium. However, in many cases, we are interested not only in a system's local behavior but also in its behavior in a larger region around the origin. Further-more, for many practical switched systems, not only does each subsystem have several equilibria, but also different subsystems have different equilibrium points. In practical switched systems, the most encountered case is that each subsystem has a unique equi-librium which is different from those of other subsystems. To facilitate the depiction, a switched system with each subsystem having different unique equilibrium is called a Multi-Equilibrium (ME) switched system in this thesis. Such a system is a kind of generation of ordinary switched systems, and it reduced to an ordinary switched system when all the equilibria become one point, i.e., the common unique equilibrium. The ME switched system is a natural representation of dynamic systems and describes a larger class of systems than the ordinary switched system. Obviously, research on the ME switched system is of great significance not only in the theoretical development of switched systems, but also in applications of many practical analysis and synthesis problems. It is noted that the ME switched system is very difficult to study, because such a kind of switched system is much more complicated than ordinary switched sys-tems, and also many existing analysis methods of ordinary switched systems, such as the well-known CLF and MLF methods, are no longer applicable to such a kind of sys-tem. Accordingly, works on the analysis and synthesis of the ME switched systems are important and very challenging.This thesis mainly investigates the region stability and control design of ME switched systems. Firstly, noticing that the ordinary switched systems are special ME switched systems, we investigate the stability of the ordinary switched systems, propose some new results for the stability of such systems, and obtain the minimum dwell time method for investigating the stability for such systems. By the MDT method, the input-to-state stability and practical stability for the ordinary switched systems are investigated. In comparison with the previous results, the obtained results not only are simple, but also have some advantages over others in some cases. Secondly, by using the common Lya-punov function-like method, multiple Lyapunov function-like method and the MDT method, some sufficient conditions for investigating the region stability for the ME switched system are proposed, and the corresponding estimates of region of conver-gence are obtained. For some special systems, the regions of convergence is obtained by investigating the dynamics of the ME switched systems. Using the componentwise ultimate bound method, the relatively exact estimations of the region of convergence of the ME switched systems under arbitrary switching are obtained. In comparison with the previous results, the obtained estimations of the region of convergence are more tighter, which reduce the conservative property of the estimation in some ways. Thirdly, we design controllers which can guarantee the controlled ME switched linear systems region stable under arbitrary paths by extending the previous results for ordi-nary switched linear systems, and obtain the corresponding estimate of the region of convergence. On the other hand, the region stabilization, robust region stabilization and adaptive region stabilization of the ME switched nonlinear systems are proposed applying the Hamilton function method. In the end, conclusions and future researches are obtained. The main contents of this thesis are composed of the following five parts:The first part investigates the stability analysis, input-to-state stability, practical stability of the switched systems. Based on the 2-norm technique, the MDT method for the nonlinear switched systems is obtained, by which the stability, input-to-state stability and practical stability of the nonlinear (stochastic) switched systems are then investigated. In comparison with the previous results, the obtained results in this part have some advantages over those in some cases. Study of examples with simulations verify the correctness and effectiveness of the obtained results in this part.The second part studies the region stability of the ME switched linear systems. Through investigating the dynamical behavior of the ME switched system, both com-mon Lyapunov-like function method and multiple Lyapunov-like function method are established for the region stability, based on which several new approaches to the re-gion stability analysis are then obtained. It is shown that the main results obtained in this note not only guarantee the region stability of the ME switched linear system under arbitrary switching, but also provide several new methods to determine the correspond-ing estimates of the region of convergence. Then, using the componentwise ultimate bound method, the relatively exact estimations of the region of convergence of the ME switched systems are obtained. In comparison with the previous results, the obtained es-timations of the region of convergence are more tighter, which reduce the conservative property of the estimation in some ways. Finally, illustrative examples with numerical simulations are studied by using the results obtained in this part. The study of exam-ples shows that our analysis methods work very well in analyzing the region stability of some classes of ME switched linear systems.The third part considers the region stability of the ME switched nonlinear sys-tems. Through investigating the dynamical behavior of the ME switched system, both common Lyapunov function method and multiple Lyapunov function method are es-tablished for the region stability, based on which several new approaches to the region stability analysis are then obtained. It is shown that the main results obtained in this note not only guarantee the region stability of the ME switched nonlinear system un-der arbitrary switching, but also provide several new methods to determine the corre-sponding estimates of the region of convergence. Illustrative examples with numerical simulations are studied by using the results obtained in this part. The study of exam-ples shows that our analysis methods work very well in analyzing the region stability of some classes of ME switched nonlinear systems.The forth part investigates the control designs of a class of ME linear systems in two cases. Firstly, we consider the case that the control is not bounded. In this case, we design controller such that the close loop systems have CLF, and give several sufficient conditions to obtain the above results. Secondly, we consider the case that the control is bounded. In this case, we find the MDT for the ME switched linear systems. Illustrative examples with numerical simulations verify the correctness and the effectiveness of the obtained results in this part.The fifth part investigates the region stabilization, adaptive region stabilization and robust adaptive region stabilization of a class of ME switched nonlinear systems by the Hamilton function method. Firstly, we express the ME switched nonlinear systems as ME switched Hamilton systems. Then, we design controllers for region stabilization, adaptive region stabilization and robust adaptive region stabilization of such systems, and obtain the corresponding estimates of the regions of convergence. Finally, illustra-tive examples with numerical simulations verify the correctness and the effectiveness of the obtained results in this part.Innovations of the thesis mainly include the following five aspects:●It is first that a class of novel switched systems are proposed, i.e., the ME switched system. Moreover, the region stability and control design problems of such systems are investigated. Based on the common Lyapunov function-like method and multiple Lyapunov function-like method, some sufficient conditions for the region sta-bility of such systems are obtained, and the corresponding estimates of the regions of convergence for such systems are also obtained. Especially, for some low dimensional (one dimensional or two dimensional) systems, the regions of convergence are obtained by analysis method. Then, by the Hamilton function method, the controllers for region stability, adaptive region stability and robust adaptive region stability of the nonlinear ME switched systems are designed, and the corresponding estimates of the regions of convergence of such systems are obtained.●It is first that the stability of a class of switched linear systems under arbitrary switching paths is investigated by the 2-norm method. These results can unify the proofs of the existing results studied case by case in literature, such as, if Ai is symmetric, or pairwise commutative, i.e., AiAj= AjAi,(?)i,j∈∮, or normal, i.e., AiTAi= AiAiT, i∈∮then the switched linear system x= Aix is asymptotically stable under arbitrary switching.●It is first that the minimum dwell time method for investigating the nonlinear switched systems is obtained by the 2-norm method. Especially, for all linear time-invariance switched systems, the minimum dwell time can be obtained by this method. In comparison with the previous results, the obtained results not only have extensive applications, but also have some advantages over others in some cases.●It is first that the stability of nonlinear switched system with unstable subsystems is investigated by the minimum dwell time method.●It is first that the input-to-state stability of nonlinear switched (stochastic) sys-tems is studied by the minimum dwell time method. In comparison with the previous results, the results obtained in this paper not only are simple, but also have some advan-tages over others in some cases.●It is first that the practical stability of linear switched systems is studied by the minimum dwell time method. In comparison with the previous results, the results obtained in this paper not only are simple, but also have some advantages over others in some cases.
Keywords/Search Tags:Nonlinear Hamiltonian Descriptor Systems, Multiple -Equilibria Switched System, Hamilton Function Method, 2-norm, Region Stability, Region of Convergence, Estimate of Region of Convergence, Minimum Dwell Time Method, Input-to-State Stability
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