Selection of PID controller design plane for time-delay systems using genetic algorithm

Abstract

The design of a Proportional-Integral-Derivative (PID) controller with proportional, integral, and derivative, gain, k p , k i , and k d , respectively, for a time-delay system, is quite common, particularly in the k i - k d plane, for a fixed k p or in the k p - k i plane, for a fixed k d . These design methods have been widely reported in the literature, however, the process of investigating the effects of using any of these design planes on system performance has not been given serious attention hence the need for this study. The stability region in the k i - k d and k p - k i design plane for a fixed value of k p and k d respectively were determined. For every determined stability region, the optimum value of controller gains in the plane was determined using a genetic algorithm (GA) with the integral of time multiplied by absolute error (ITAE) used as the objective function. The optimum value of the fixed gains was graphically determined by plotting the minimum of ITAE (Min-ITAE) for each stability region against the fixed gains. The overall optimum controller gains are the fixed gain that gives minimum of Min-ITAE (Min (Min-ITAE)) and the gains that resulted in Min-ITAE that yielded the Min (Min-ITAE). Using the determined overall optimum controller gains, the system closed-loop step response was plotted for the two design planes and the time domain performance measures (TDPMs) were determined. Based on TDPMs obtained for examples 1, 2, and 3, the k i -k d design plane yielded a faster response while the k p - k i design plane yielded a response that closely tracks the input irrespective of the system type and order. The study will enable control system designers to select the design plane that will give the best system performance right from the start of controller design without involving trial and error once the system transfer function and design specifications are known.