Digital implementation of fractional order PID controller and its application

A new discretization scheme is proposed for the design of a fractional order PID controller. In the design of a fractional order controller the interest is mainly focused on the s-domain, but there exists a difficult problem in the s-domain that needs to be solved, i.e. how to calculate fractional derivatives and integrals efficiently and quickly. Our scheme adopts the time domain that is well suited for Z-transform analysis and digital implementation. The main idea of the scheme is based on the definition of Grunwald-Letnicov fractional calculus. In this case some limited terms of the definition are taken so that it is much easier and faster to calculate fractional derivatives and integrals in the time domain or zdomain without loss much of the precision. Its effectiveness is illustrated by discretization of half-order fractional differential and integral operators compared with that of the analytical scheme. An example of designing fractional order digital controllers is included for illustration, in which different fractional order PED controllers are designed for the control of a nonlinear dynamic system containing one of the four different kinds of nonlinear blocks: saturation, deadzone, hysteresis, and relay.