OPTIMAL TUNING OF IMPLEMENTABLE FRACTIONAL ORDER PI CONTROLLERS BASED ON ISE PERFORMANCE INDEX

Benefiting from the great potential of the mathematical tool of fractional calculus, fractional order proportional integral (PI) controllers have been proposed as the next generations of popular PI controllers. On the other hand, the integral performance indices such as the integral square error (ISE), integral absolute error (IAE), integral time square error (ITSE), and integral time absolute error (ITAE) are commonly used in the design and evaluation of the performance of practical control systems. Considering these points, the present paper deals with the optimal tuning of the free parameters of fractional order PI controllers, to be used in control of first order plus dead time (FOPDT) processes in a unity negative control structure, on the basis of ISE performance index. Using the approach of ``tuning based on the implementable form of the controller'' instead of the approach of ``tuning based on the ideal form of the controller'' causes that no incompatibility is seen between the ideal behavior of the controller and the behavior of the implementable controller. Also, to avoid approximation error in the calculation of ISE based cost functions, algebraic relations have been used for analytically finding the values of ISE in the under study time delay control system. In fact, the main contribution of the paper is to propose simple tuning rules for implementable fractional order PI controllers with the aim of achieving an optimal control system in the viewpoint of the ISE performance criterion. These rules have been obtained by iteratively using the steepest descent (gradient descent) optimization algorithm with the constraint of optimizing the step size in each iteration. The proposed rules for tuning of the free parameters of implementable fractional order PI controllers yield in a control system whose performance is better than the control system with an optimal PI controller. This point has been successfully confirmed by some numerical and experimental examples.