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

• Main Authors: Inés Tejado, Blas M. Vinagre, José Emilio Traver, Javier Prieto-Arranz, Cristina Nuevo-Gallardo
• Format: Article
• Language: English
• Published: MDPI AG 2019-06-01
• Series: Mathematics
• Subjects: fractional; control; PID; parameter; meaning
• Online Access: https://www.mdpi.com/2227-7390/7/6/530

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 <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>. 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, <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula>. Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> 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.