Time response computations of fractional order control systems and control applications

Fractional order calculus or differential equations has been known since classical calculations. However, fractional order mathematics has been the area of study for mathematicians only for a long time, due to its complicated structures. With the recent advances in computer technologies, it has been used in other fields as well. The main reason that fractional order differential equations are used widely in control engineering is its accuracy in modelling real world systems. In this thesis, two methods have been presented for computing time responses of fractional-order systems, which result very close values to analytical ones, using only frequency response data of a fractional order transfer function. With the help of these new methods, time response computation techniques have been developed for fractional order closed loop control systems with classical controllers like PI, PID, and Lag-Lead or with their fractional order structures. Moreover, methods for computing time response analyses of fractional-order control systems with time delay are presented. Effects of developed methods on the time response of systems with large time coefficient are analyzed and compared with Grünwald-Letnikov numerical method. Furthermore, new integer order approximate transfer functions have been calculated and tabulated using least-square curve fitting method for fractional order derivative operators. Using developed approximate transfer functions, circuit design for fractional order integrators has been created with active circuit components. In addition, computation methods of limit cycle frequency are presented for fractional order nonlinear control systems with ideal relay, relay with dead zone and relay with hysteresis. Finally, several interactive applications are developed using graphical based programming software LabVIEW and applications on pendulum control system have been done.