Theory and implementation of weighted distributed order integrator

In this paper, the weighted distributed order integrator is discussed in both time and frequency domains. The physical backgrounds and the construction of a distributed order element is illustrated by means of the fractional order capacitors and some other real world phenomena. The impulse response and the asymptotic properties of weighted distributed order integrators are derived in the whole time domain, which show that its impulse response is bounded by an exponential function (lower bound) and a power law function with time varying index (upper bound). The impulse response invariant discretization technique is applied to the digital implementation of the discussed operators. A number of figures are provided to validate the concepts.