## 1 Introduction
In [32], a sliding mode backstepping control method is proposed, which involves the use of a fractional-order command filter, a fuzzy logic system approximator, and a grey wolf and weighted whale optimization algorithm for multi-input multi-output nonlinear dynamic systems. Instead of replacing the IO terms as in the PI\({}^{\lambda}\)D\({}^{\mu}\) scheme, an alternative way to apply FC to control systems is to add FO terms to the PID scheme. This approach was introduced in 2009 by [PERSON] et al. [33] with the PDD\({}^{1/2}\) scheme, in which a half-derivative term is added to the classical PD controller. In [34] the dynamic behavior of the PDD\({}^{1/2}\) control in combination with a purely inertial system is discussed, adopting a nondimensional approach. In [35] the PDD\({}^{1/2}\) scheme is applied in simulation to position control of a nonlinear multi-input multi-output plant (a Parallel Kinematics Machine). The effectiveness of the PDD\({}^{1/2}\) scheme has been experimentally validated in the position control of a micrometric linear axis [36] and of a rotor [37]. In [38] the PD, PD\({}^{\mu}\) and PDD\({}^{1/2}\) controls of a purely inertial system are compared by simulation, and the results indicate that the two FO schemes have similar performance, but the PDD\({}^{1/2}\) is characterized by a slightly better readiness and a slightly higher overshoot. In [39] the comparison among PD, PD\({}^{\mu}\) and PDD\({}^{1/2}\) is validated by experimental tests, highlighting the benefits of the proposed control approach in real working conditions, not limited to the classical step response. The proposed PDD\({}^{1/2}\) scheme did not include an integral action, as the focus of the research was the optimization of the transient behavior of the system. Compared to the PD performances, with the same control effort, the introduction of the half-derivative term reduces the transient tracking error, also in case of complex MIMO nonlinear mechanical systems, with possible applications in position, force or impedance control [40] of serial and parallel robots. In 2017, a similar approach has been proposed by [PERSON] et al. and named Distributed Order PID (DOPID) [41], and then applied to the control of permanent magnet synchronous motor drives [42; 43]. In the DOPID\({}_{\rm N}\), the control action is given by the linear combination of an odd number \(n\) (with \(n\geq 3\)) of differintegrators of equally spaced orders ranging from \(-1\) to \(+1\). Accordingly, for \(n=3\), the \(\text{DOPID}_{3}\) corresponds to the classical PID, since the three differintegration orders are \(-1\), \(0\), \(+1\). For \(n=5\), the \(\text{DOPID}_{5}\) orders are \(-1\), \(-1/2\), \(0\), \(+1/2\), \(+1\), etc. The \(\text{PDD}^{1/2}\) scheme is a subcase of \(\text{DOPID}_{5}\) with null integral gains. In the present paper the integral terms are added to the \(\text{PDD}^{1/2}\) controller, to obtain a more general control scheme, capable of providing the required accuracy also in a steady state. Therefore, three control schemes are compared: the classical integer-order PID, the fractional-order \(\text{PI}^{3}\text{D}^{\mu}\) and the \(\text{P}\text{II}^{1/2}\text{DD}^{1/2}\), which corresponds to \(\text{DOPID}_{5}\), while in [41; 42; 43] the \(\text{DOPID}_{7}\) is mainly considered.