An LMI-based fuzzy control system design with application to nonlinear magnetic bearings

This dissertation proposes an LMI-based systematic design methodology for the fuzzy control of a class of nonlinear systems with multi-objective requirements (i.e., stability, disturbance rejection, and transient response) in a unified framework. The framework is based on the Takagi-Sugeno (TS) fuzzy model and Parallel Distributed Compensation (PDC), which form a fuzzy gain scheduling scheme. The following two issues of a synthesis of a multi-objective fuzzy control system are considered. First, the synthesis of an LMI-based stable fuzzy control system with pole-placement constraint is presented. The requirements of stability and pole-placement region are formulated based on the Lyapunov direct method. By recasting these constraints into LMIs, we formulate an LMI feasibility problem for the design of the fuzzy state feedback control system that guarantees stability and satisfies desired transient responses. Second, the synthesis of an LMI-based fuzzy control system with guaranteed {dollar}Hsbinfty{dollar} performance is presented in detail. Combining pole-placement requirements, the {dollar}Hsbinfty{dollar} fuzzy control synthesis problem is converted to the minimization of a linear objective under LMI constraints. It should be noted that while most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. To demonstrate the usefulness and effectiveness of the proposed design methodology, it is applied to a nonlinear Active Magnetic Bearing (AMB) system concerning the issues of rotor position control with guaranteed stability and desired transient performance, and rotor vibration control with disturbance rejection and desired transient performance. For each issue of control of AMB, each fuzzy state feedback controller is designed by the proposed LMI-based multi-objective design scheme. Experimentation and simulation results show that the proposed LMI-based design methodology for each issue yields better performances than those of a linear local controller or single objective controller.