The fractional order proportional, integral, derivative and acceleration (PI ^λD ^µA) controller is an extension of the classical PIDA controller with real rather than integer integration action order λ and differentiation action order µ. Because the orders λ and µ are real numbers, they will provide more flexibility in the feedback control design for a large range of control systems. The Bode’s ideal transfer function is largely adopted function in fractional control systems because of its iso-damping property which is an essential robustness factor. In this paper an analytical design technique of a fractional order PI ^λD ^µA controller is presented to achieve a desired closed loop system whose transfer function is the Bode’s ideal function. In this design method, the values of the six parameters of the fractional order PI ^λD ^µA controllers are calculated using only the measured step response of the process to be controlled. Some simulation examples for different third order motor models are presented to illustrate the benefits, the effectiveness and the usefulness of the proposed fractional order PI ^λD ^µA controller tuning technique. The simulation results of the closed loop system obtained by the fractional order PI ^λD ^µA controller are compared to those obtained by the classical PIDA controller with different design methods found in the literature. The simulation results also show a significant improvement in the closed loop system performances and robustness using the proposed fractional order PI ^λD ^µA controller design.

This paper introduces a fractional-order PD approach (F-oPD) designed to control a large class of dynamical systems known as fractional-order chaotic systems (F-oCSs). The design process involves formulating an optimization problem to determine the parameters of the developed controller while satisfying the desired performance criteria. The stability of the control loop is initially assessed using the Lyapunov’s direct method and the latest stability assumptions for fractional-order systems. Additionally, an optimization algorithm inspired by the flight skills and foraging behavior of hummingbirds, known as the Artificial Hummingbird Algorithm (AHA), is employed as a tool for optimization. To evaluate the effectiveness of the proposed design approach, the fractional-order energy resources demand-supply (Fo-ERDS) hyperchaotic system is utilized as an illustrative example.