Controller Design for Fractional-Order Systems

In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives has found wide application in the control of dynamical systems, when the controlled system or/and the controller is described by a set of fractional order differential equations. In the present work a fractional order system has been represented by a higher integer order system, which is further approximated by second order plus time delay (SOPTD) model. The approximation to a SOPTD model is carried out by the minimization of the two norm of the actual and approximated system. Further, the effectiveness of a fractional order controller in meeting a set of frequency domain specifications is determined based on the frequency response of an integer order PID and a fractional order PID (FOPID) controller, designed for the approximated SOPTD model. The advent of fuzzy logic has led to greater flexibility in designing controllers for systems with time varying and nonlinear characteristics by exploiting the system observations in a linguistic manner. In this regard, a fractional order fuzzy PID controller has been developed based on the minimization different optimal control based integral performance indices. The indices have been minimized using genetic algorithms. Simulation results show that the fuzzy fractional order PID controller is able to outperform the classical PID, fuzzy PID and FOPID controllers