Back to Basics: Meaning of the Parameters of Fractional Order PID Controllers

Abstract

The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order λ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with λ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, μ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of μ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes.

Low-Power GPS-Disciplined Oscillator Module for Distributed Wireless Sensor Nodes

“…This closed-loop negative feedback can be implemented very simply with a proportional controller (cf. [19]). Due to the extended period of time it takes to make a single frequency measurement, anywhere from a few seconds to several minutes depending on the required resolution, and because the duty cycle of the GPS power needs to be as low as possible, the proportional gain of the controller can be calibrated to accurately steer the oscillator to the desired frequency in a single adjustment.…”

Design of an Optimized Fractional High Order Differential Feedback Controller for Load Frequency Control of a Multi-Area Multi-Source Power System With Nonlinearity

“…There are several linear and non-linear control techniques employed in LFC problem of power systems such as classical integral (I), proportional-integral (PI), and proportionalintegral-derivative (PID) controllers [3]- [13], PID controller with derivative filter (PIDF) [14], two degree of freedom PID controller (2-DOF PID) [15], [16], PID plus second order derivative controller (PID+DD) [17], PD-PID cascade controller [18], fractional order PID controller (FOPID) [19]- [24], fuzzy fractional order PI and PD controller (FFOPI-FOPD) [25], fuzzy logic based PID (FPID) and FOPID controllers with derivative filter (FFOPIDF) [26], fuzzy PID with filter and fractional order integer controller (FPIDN-FOI) [27], fuzzy tuned PI (FPI) and PID controllers [28], [29], fuzzy tuned fractional order integerderivative controller (FFOID) [30], tilt integral-derivative controller with derivative filter (TIDF) [31], neuro-fuzzy hybrid intelligent PI controller [32], linear active disturbance rejection control (LADRC) [33], LADRC controller with two anti-GDB schemes [34], and etc. Since, the intelligent and modern control techniques generally require long computational complexities like learning process, expert knowledge and inference mechanism, PID controller and its expanded versions are highly popular for LFC problem because of its two main advantages of simplicity and efficiency [35]. However, dynamic responses of area frequency and tie-line power of LFC system with classical I/PI/PID controllers have large oscillations and longer settling time under consideration of physical limitations, system uncertainties, and change in loading conditions [27].…”

Variation of fraction in FOPID controller for vibration control of Euler–Bernoulli beam

“…Furthermore, PID controllers being simple in nature, are ideal to analyze the behavior of its complicated fractional order form against the classical controller form. The extension of PID controllers from the classical to fractional order translates into more robust control design of the system and better flexible in tuning the frequency response and the settling time of systems [6]. The tuning methods for fractional order controllers are reported in [7][8][9].…”