## 1 Introduction  
Fractional Calculus (FC) is the generalization of the concepts of derivative and integral from integer to non-integer order [1]. The origin of FC dates back even to the seventeenth century: as a matter of fact, it was discussed by [PERSON]\({}^{\prime}\)[PERSON], [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON]. The birth and the historical development of FC are outlined in [2]. After a long period of being forgotten, in the last few decades there has been a renewed research interest in FC, also due to the discovered relationship with the chaos theory. Some real systems can be better modelled by Fractional Order (FO) equations than by Integer Order (IO) ones, in particular in the case of multi-scale problems, with wide dimensional or time scales, as found, for example, in viscoelasticity problems [3, 4]. FC is used to provide more accurate models in mechanics [5], physics [6] and biology [7]. Recently, FC has been used to model the evolution of the COVID-19 pandemic [8, 9]. Nowadays FC is not only used as modelling tool, but also in engineering applications, and in particular in the field of control system design. Most control algorithms are based on IO derivatives and integrals of the error; the extension to FO derivatives and integrals introduces additional parameters which can be tuned to improve the closed-loop system performance. The most widespread approach to apply FC to control system design is the well-known PI\({}^{\lambda}\)D\({}^{\mu}\) scheme, which adopts integral and derivative terms of non-integer orders \(\lambda\) and \(\mu\)[10]. Design techniques, optimization tools and practical applications of PI\({}^{\lambda}\)D\({}^{\mu}\) controllers are widely discussed in the scientific literature. Some approaches are derived from classical tuning criteria: in [11] tuning is obtained by modified Ziegler-Nichols and Astrom-Hagglund methods, while in [12] the classical isodamping condition is generalized for the PI\({}^{\lambda}\)D\({}^{\mu}\) controller. Other approaches are based on numerical optimization techniques, for example Artificial Bee Colony algorithms [13], Particle Swarm Optimization [14] or optimal shaping of the Bode plot to achieve robustness [15]. In [16] a population-based optimization approach named the Sine-Cosine Algorithm is applied to PI\({}^{\lambda}\)D\({}^{\mu}\) generation control in wind farms. In [17] an optimized PI\({}^{\lambda}\)D\({}^{\mu}\) controller is compared to the Linear Quadratic Gaussian and \(H\infty\) controllers. The robustness to parametric uncertainties and the rejection of external disturbances is considered in [18; 19]. The application of PI\({}^{\lambda}\)D\({}^{\mu}\) can greatly improve the performance in the transient behavior for motion control applications, for example in case of mechatronic devices actuated by DC motors. In [20] PI\({}^{\lambda}\)D\({}^{\mu}\) is used for speed control of a buck converter-fed DC motor. In [21] an analog implementation exploiting the Operational Transconductance Amplifier is used for controlling a DC motor. In [22] PI\({}^{\lambda}\)D\({}^{\mu}\) is applied to speed control of a chopper-fed DC motor drive. Chaotic Atom Search Optimization [23] and Flower Pollination Algorithms [24] are proposed to tune the PI\({}^{\lambda}\)D\({}^{\mu}\) parameters for DC motor speed control. In [25] a methodology for the quantitative robustness evaluation of PI\({}^{\lambda}\)D\({}^{\mu}\) controllers employed in DC motors is proposed. In [26] an inertial load elastically connected to a DC motor is studied, comparing IO and FO controllers. Other works are related to the application of PI\({}^{\lambda}\)D\({}^{\mu}\) to different actuators, for example synchronous motors [27], linear motors [28; 29], linear positioning systems [30]. Besides the PI\({}^{\lambda}\)D\({}^{\mu}\) scheme, FC can be profitably applied to control system design in different ways, for example for enhancing the performance of sliding mode control by applying a FO disturbance observer [31].