Sustainability-oriented applications of the half-derivative and of the half-integral in control system design.

The present paper deals with the applications of the half-derivative (derivative of order 1/2) and of the half-integral (integral of order 1/2) in control system design, with particular focus on sustainability. Fractional Calculus (FC) is the branch of mathematics which deals with derivatives and integrals whose order is non-integer. Nowadays FC is successfully applied in engineering applications, and in particular in control system design. The most widespread approach to apply FC to control systems is the so-called Fractional Order PID scheme (FOPID), which is a PID with non-integer order integrals and derivatives of the error. This scheme has two more control parameters, the orders of integration and derivation, thus improving the closed-loop behavior. An alternative approach to apply FC to control system is the Distributed Order PID (DOPID), in which the control output is the sum n differintegrators, with n odd and≥3, and with orders which are equally spaced in the interval [−1, +1]. For n=5, the differintegration orders are −1, −1/2, 0, +1/2, +1, adding to the PID the half-integral term (order −1/2) and the half-derivative term (order +1/2). The effectiveness of DOPID has been investigated in the scientific literature, and can lead to lower energy consumption with similar control readiness and error. Considering Multi-Input Multi-Output (MIMO) systems, and in particular robotic manipulators, a possible application of the half-derivative is the so-called KDHD impedance control. With respect to classical impedance control, KD, characterized by a stiffness and a damping term, a half-derivative damping term is added, proportional to the half-derivative of the end-effector error through the half-derivative damping matrix HD. Also in this case, additional control parameters, the elements of the matrix HD, allow a more efficient optimization of the closed-loop behavior. This kind of controllers represent an effective and nearly cost-free evolution of classical integer-order schemes, able to reduce energy consumption of mechatronic and automation devices. Another possible promising application of the half-derivative damping is the exploitation of renewable energies, for example for controlling the power take-off systems of Wave Energy Converters. In the paper, some possible future sustainability-oriented research directions involving the use of half-derivatives and half-integrals in control systems are outlined. [ABSTRACT FROM AUTHOR]