On the stability of T-S fuzzy control for non-linear systems

The main feature of a Takagi-Sugeno (T-S) fuzzy model is to express the local dynamics of each fuzzy implication (rule) by a linear system model. The overall fuzzy model of the system is achieved by fuzzy “blending” of the linear system models. Parallel or feedback connections of T-S fuzzy systems which preserve the properties of each system are possible [1]. Thus a simple and straightforward approach for the control of non-linear systems has emerged [2]. By representing the non-linear system by a T-S type fuzzy model, linear feedback control techniques can be utilized to design a linear controller for each local linear model. The overall controller is a fuzzy blending of each individual linear controller and therefore nonlinear but very simple to design. The closed loop fuzzy control system derived in this way is, in general, a system composed of rules with affine linear systems in their consequent parts. This design approach for the control of nonlinear systems does not have the local character of a linear control design for the linearized, around an operating point, non-linear system, nor the complication and involvement of feedback linearization controllers. Its appeal is, however, dependent on the stability issues involved. While stability conditions have been exploited for T-S fuzzy systems composed of rules with linear consequent parts, the stability of T-S fuzzy systems composed of rules with affine linear system in their consequent parts needs further investigation.