Stability And Stabilization Of Fractional Order Systems Based On Takagi-Sugeno Fuzzy Model

Over the last few decades, with the development of fractional calculus theory, theory of fractional order control systems now have become a new hot subject. The research of stability analysis and control problems is one of the most fundamental tasks for these fractional order dynamic systems, and more and more results have been given. The research of stability analysis and control problems about nonlinear fractional order system is still difficult problem in present study. In control theory, T-S fuzzy model has been recognized as an effective method for approximating nonlinear dynamic systems. In this dissertation, the integer order T-S fuzzy model was extended to fractional order T-S fuzzy system for deal with the fractional order nonlinear systems. Based on theory of fractional calculus, T-S fuzzy systems theory, fractional stability theory, combined with existing research results, we proposed the stability and stabilization conditions of the fractional order T-S fuzzy systems. The conditions are obtained in terms of linear matrix inequalities. Then, some numerical examples and fractional order chaotic systems are given to show the effectiveness of our results.1. The problem of robust stability and stabilization of fractional order systems uncertain T-S fuzzy model with 1 < α < 2 are studied. A sufficient and necessary condition of asymptotical stability for fractional order uncertain T-S fuzzy model is given, and a parallel distributed compensate fuzzy controller is designed to asymptotically stabilize the model.The results are obtained in terms of linear matrix inequalities. Finally, a numerical example and fractional order Van der Pol system are given to show the effectiveness of our results.2. The problem of robust stability and stabilization of fractional order systems uncertain T-S fuzzy model with 0 < α < 1 are studied. The fractional order uncertain T-S fuzzy system is converted into the corresponding integer order T-S fuzzy systems with uncertainties.A sufficient condition of asymptotical stability for fractional order uncertain T–S fuzzy model is given by researching the asymptotical stability of integer order T–S fuzzy model.And then a parallel distributed compensating fuzzy controller is designed to asymptotically stabilize the model. The sufficient conditions are formulated in the format of linear matrix inequalities. The fractional order T–S fuzzy models of chaotic systems, which have complex nonlinearity, are developed as test bed. The effectiveness of the approach is tested on fractional order R ¨ossler system and fractional order uncertain Lorenz system.3. The problem of Mittag-Leffler stability and stabilization of fractional order systems based on uncertain T-S fuzzy model are studied. By using the fractional order Lyapunov direct method, the Mittag-Leffler stability of the corresponding fractional order systems can be get.In this dissertation, formula which can be applied to fractional order Lyapunov direct method is given. Then, based on Mittag-Leffler theory and fractional Lyapunov direct method, some sufficient conditions of stability and stabilization are given in the terms of LMI format.Finally, a numerical example and fractional order uncertain Lorenz system are presented to show the effectiveness of the proposed approach.4. The stabilization of fractional order uncertain T-S fuzzy model with constant delay is studied. Based on linear matrix inequality theory and parallel distributed compensate, fuzzy controller is designed to asymptotically stabilize the model, and a sufficient condition of robust stabilization is given in the format of linear matrix inequalities. Fractional order uncertain Chen system with time-delayed is introduced to demonstrate the effectiveness of the proposed approach.5. The state feedback stabilization problem for the fractional order Takagi-Sugeno fuzzy interconnected systems with multiple time delays is studied. Based on linear matrix inequality theory, a parallel distributed compensate controller based on fractional order Takagi-Sugeno fuzzy model is designed to asymptotical stabilize the fractional order interconnected systems, and a sufficient condition is given in the format of linear matrix inequalities. Finally,a numerical example and interconnected fractional order chaotic systems are introduced to demonstrate the effectiveness of the proposed theoretical results.