The proportional-integral-derivative (PID) controllers have experienced series of structural modifications and improvements. Example of such modifications are set-point weighting and fractional ordering. While the former is to achieve two-degree-of-freedom (2DOF) ability of set-point tracking and disturbance rejection, the latter is to ensure smooth control action. Therefore, this paper reviews various forms of PID controllers and provides a comparative analysis of 2DOF PID and 2DOF fractional order PID (FOPID) controllers. The paper also discusses the conversion of one PID form to another. For the comparative analysis of the various controllers, a class of unstable systems are considered. Simulation result shows that in most cases the conversion from one form to another does not significantly affect the performance of the system. It is also observed that the 2DOF controllers (2DOF PID and 2DOF FOPID) improved significantly the performance of the ordinary PID controllers.

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.