PID controller design for fractional-order systems with time delays

Classical proper PID controllers are designed for linear time invariant plants whose transfer functions are rational functions of s α , where 0 < α < 1 , and s is the Laplace transform variable. Effect of input–output time delay on the range of allowable controller parameters is investigated. The allowable PID controller parameters are determined from a small gain type of argument used earlier for finite dimensional plants.