Abstract
The application of Fractional Calculus to control mechatronic devices is a promising research area. The most common approach to Fractional-Order (FO) control design is the PIλDµ scheme, which adopts integrals and derivatives of non-integer order λ and µ. A different possible approach is to add FO terms to the PID control, instead of replacing integer order terms; for example, in the PDD1/2 scheme, the half-derivative term is added to the classical PD. In the present paper, by mainly focusing on the transitory behaviour, a comparison among PD, PDµ, and PDD1/2 control schemes is carried out, with reference to a real-world mechatronic implementation: a position-controlled rotor actuated by a DC brushless motor. While using a general non-dimensional approach, the three control schemes are first compared by continuous-time simulations, and tuning criteria are outlined. Afterwards, the effects of the discrete-time digital implementation of the controllers are investigated by both simulation and experimental tests. The results show how PDD1/2 is an effective and almost cost-free solution for improving the trajectory-tracking performance in position control of mechatronic devices, with limited computational burden and, consequently, easily implementable on most commercial motion control drives.
Highlights
The generalization of the concept of derivative and integral to non-integer order (Fractional Calculus) dates back to the beginning of the theory of differential calculus: there are notes by Leibnitz regarding the calculation of the half-derivative [1]
Besides the PIλ Dμ –PDμ scheme, in the scientific literature there are many other examples of extensions of control techniques based on Fractional Calculus; for instance, the performance of sliding mode control can be enhanced by using a FO disturbance observer [25] or by applying it to systems that are better approximated by FO models [26]
The PDD1/2 law is compared to the classical integer-order PD and the better-known FO PDμ in the control of a second-order linear system
Summary
The generalization of the concept of derivative and integral to non-integer order (Fractional Calculus) dates back to the beginning of the theory of differential calculus: there are notes by Leibnitz regarding the calculation of the half-derivative (that is the derivative of order 1/2) [1]. Besides the PIλ Dμ –PDμ scheme, in the scientific literature there are many other examples of extensions of control techniques based on Fractional Calculus; for instance, the performance of sliding mode control can be enhanced by using a FO disturbance observer [25] or by applying it to systems that are better approximated by FO models [26] Another possible approach to FO control is to add FO terms to the classical PID scheme, instead of replacing the integer order terms; for example, in the PDD1/2 scheme, the half-derivative term is added to the derivative term instead of replacing it as in the HPDμ [27,28,29,30,31,32]. FO controllers PDμ and PDD1/2 exhibit better performances, in both simulations and in experiments, than the classical PD scheme, with a limited increase of computational burden
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