Tuning PID and PI ^λ D ^δ controllers using particle swarm optimization algorithm via El-Khazali's approach

The major goal of this paper is to tune PID and PI^λD^δ controllers using Particle Swarm Optimization (PSO) algorithm via El-Khazali's approach. This is the first time that such a method is employed for meeting this problem. El-Khazali's approach is, usually, used to approximate fractional-order Laplacian operators of order α; s^±α, 0 < α ≤ 1, by finite-order rational transfer functions. The significance of this approach lies in developing an algorithm that depends only on α, which enables one to synthesize both fractional-order inductors and capacitors. To illustrate the proposed design method, an objective function is presented as well as the results of using the PI^λD^δ controller are compared with the results of using the conventional PID controller to minimize several error functions; ITAE, IAE, ISE and ITSE. These results of such comparisons that are related to step response specifications, allow us to see the effectiveness of the best controller.