FOPID controller optimization based on SIWPSO-RBFNN algorithm for fractional-order time delay systems

In this study, the stochastic inertia weight particle swarm optimization (SIWPSO) algorithm and radial basis function neural network (RBFNN) methods were used to identify the optimal controller gain for the fractional order proportional integral derivative (FOPID) controller of time-delay systems; furthermore, a graphic approach was used to plot 3D stability regions in the $$k_p $$kp, $$k_i $$ki, and $$k_d$$kd parameter space. This paper presents an intelligent SIWPSO-RBF algorithm for identifying the optimal solution for a FOPID control system. To explain how to use the SIWPSO-RBFNN method, this paper presents two cases describing how the proposed algorithm can be useful in FOPID-type controllers with two fractional-order time-delay systems. Furthermore, the proposed algorithm can be used in two desired procedures if the system transfer functions are known. The first procedure involves identifying the optimal $$k_p$$kp and $$k_i$$ki gains while $$k_d $$kd varies and the parameters $$\lambda $$ź and $$\mu $$μ are known. The second procedure involves identifying the optimal $$k_p $$kp, $$k_i $$ki and $$k_d$$kd gains while $$\lambda $$ź and $$\mu $$μ vary. Finally, several simulations of the proposed algorithm verified the effectiveness of a FOPID controller regarding fractional-order with time-delay systems.