로컬 안정화 접근법을 통한 비선형 마르코비안 점프 시스템의 제어

Abstract

Over the past decades, the development of control technologies has witnessed rapid growth in a number of research studies on hybrid systems that consist of both continuous and discrete dynamic behavior. As a special class of stochastic hybrid systems, Markov Jump Systems (MJSs), in which discrete dynamics behavior are governed by Markov processes, have received much attention from control societies due to their great abilities to represent linear dynamic systems subjected to random abrupt variations such as component fault or failures, sudden environmental changes, and packet-loss phenomemnon in communication. In reality, the MJS model has been successfully utilized in a variety of practical applications such as network control systems, power systems, and communication systems. Thanks to Takagi-Sugeno (T-S) fuzzy model which can represent a large class of nonlinear systems by means of an averaged sum of linear models, the study on nonlinear MJSs have been intensively carried out in the framework Markovian jump fuzzy systems (MJFSs) regarding to stability and stabilization control problems, the filtering problem, the output-feedback control problems.

Recently, the control synthesis-based linear matrix inequality (LMI) approach has been well-rooted in formulating control problems of MJFSs. These works, however, have brought conservative results associated with the feasibility of controller synthesis and performance. To be specific, relaxation techniques, by which LMI-based conditions are obtained, have not been appropriately considered in the existing studies on MJFSs. It is worth noting that the stabilization for MJFSs has been formulated in terms of nonconvex or parameterized matrix inequalities. Therefore, the absence of effective relaxation techniques possibly leads to conservative results and computational burden in the LMI-based conditions. Further, the previous studies of MJFSs related to nonquadratic Lyapunov functions have been lacks of theoretical contributions on local stabilization approach.

This dissertation aims at providing control synthesis methods for stabilization and dissipative control of continuous- and discrete-time nonlinear MJSs with incomplete transition descriptions. Goals of this works can be highlighted as follows: i) By transforming the nonlinear MJSs into MJFSs, our works investigates the global and local stabilization approach for control of MJFSs. With consideration to nonhomogeneous Markov process, our study can cover more realistic phenomena such as time-varying and unknown transition descriptions. ii) Improving existed results of MJFSs with respect to feasibility of controller synthesis and performance enhancement by using nonquadratic stochastic (fuzzy-mode-dependent) Lyapunov functions. Based on the nonquadratic Lyapunov functions, the global and local stabilization approaches are employed to obtain less conservative results in LMI-based stabilization conditions. iii) New effective relaxation techniques that can deal with multiparameterized linear matrix inequalities are proposed to relax the derived stabilization conditions in terms of LMIs. These techniques have the capability of systematically applying to wide-range controller-based LMIs design problems of MJFSs.