Local Stability Analysis And Controller Design Of Continuous-time T-S Fuzzy Systems

The research of fuzzy control theory has always been a hotspot in the control field,especial-ly,the stability analysis and controller design of complex nonlinear systems based on T-S fuzzy models.Scholars are mostly committed to improving existing analysis methods and perfecting the present control theory,and research results are yielded fruitfully.However,among all these literatures,most of the results are focusing on global stability analysis of T-S fuzzy systems,and there are few studies on local stability analysis and control synthesis of T-S fuzzy systems.Due to the limitations of actual systems,global stability conditions of many T-S fuzzy systems are demanding or conservative.However,global stability conditions can not be satisfied and local stability may occur for the actual systems.Therefore,it is practical to study the local stability of systems.In general,the local stability conditions which are easy to acquire and make the systems more suitable are weaker than the global stability ones,and also provide a new stability criterion of the T-S fuzzy system.It should be pointed out that not only the local stability of the T-S fuzzy systems is stable at the equilibrium point,but is stable in the region containing the equilibrium point.Based on the shortcomings and difficulties of existing literatures,this paper focuses on the local stability analysis and controller design of continuous-time T-S fuzzy systems.The improved local stability analysis and new control synthesis methods are proposed by applying the generalised fuzzy Lyapunov function and the linear matrix inequality(LMI)method.The main works are as follows:(1)For a class of continuous-time T-S fuzzy systems,the stability analysis of the system un-der the bounded derivative of the membership function is studied,and the more general stability conditions of the system are given.Firstly,the extended fuzzy Lyapunov function is designed,and the sufficient stability conditions of open-loop continuous-time T-S fuzzy systems are given by using the homogeneous polynomial method and the augmented multi-index matrix method.Secondly,the non-quadratic Lyapunov functions and homogeneous polynomials non-parallel control law with arbitrary dimensions are designed.The stability conditions of the closed-loop continuous-time T-S fuzzy system are obtained.At the same time,it is verified that the proposed stability conditions include previous results as special cases,and the conditional conservation decreases with the increase of the matrix function dimension.(2)For a class of continuous-time T-S fuzzy control systems,the local controller design problem of the system with unknown input of membership function derivative is studied,and a new local state feedback controller design method is proposed.Firstly,the homogeneous polynomial non-parallel control law is designed to transform the constraint of the membership function derivative into a linear matrix inequality.Secondly,under the conditions of the obtained linear matrix inequality,the local stability conditions based on the extended non-quadratic fuzzy Lyapunov function are given.The constraints of the Lyapunov level set in a given compact set convert into a linear matrix inequality condition by using the Lagrange multiplier method,which is more reasonable than the existing results.(3)For a class of continuous-time T-S fuzzy control systems with bounded disturbances,the problem of system disturbance energy bound is studied,and a local controller design method for calculating disturbance energy bounds is given.Firstly,considering the case of the distur-bance amplitude bounded,the homogeneous polynomial non-parallel control law is designed to transform the constraint of the membership function derivative into a linear matrix inequality.Secondly,considering the bounded energy of the disturbance,under the conditions of the ob-tained linear matrix inequality,the local stability conditions based on the extended non-quadratic Lyapunov function are given,and two local regions(one smaller and one larger)are obtained such that all trajectories start in smaller areas and still fall in larger areas.Finally,the main works of this paper are summarized,and future works for the continuous T-S fuzzy system are prospected.