Fractional-order integrator

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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A fractional-order integrator or just simply fractional integrator is an integrator device that calculates the fractional-order integral or derivative (usually called a differintegral) of an input. Differentiation or integration is a real or complex parameter. The fractional integrator is useful in fractional-order control where the history of the system under control is important to the control system output.^{[citation needed]}

Some industrial controllers use fractional-order PID controllers (FOPIDs), which have exceeded the performance of standard ones, to the extent that standard ones are sometimes considered as a special case of FOPIDs. Fractional-order integrators and differentiators are the main component of FOPIDs.^[1]