Second and third order digital integrators (DIs) have been optimized first using Particle Swarm Optimization (PSO) with minimized error fitness function obtained by registering mean, median, and standard deviation values in different random iterations. Later indirect discretization using Continued Fraction Expansion (CFE) has been used to ascertain a better fitting of proposed integer order optimized DIs into their corresponding fractional counterparts by utilizing their refined properties, now restored in them due to PSO algorithm. Simulation results for the comparisons of the frequency responses of proposed 2nd and 3rd order optimized DIs and proposed discretized mathematical models of half integrators based on them, with their respective existing operators, have been presented. Proposed integer order PSO optimized integrators as well as fractional order integrators (FOIs) have been observed to outperform the existing recently published operators in their respective domains reasonably well in complete range of Nyquist frequency. 1. Introduction In fractional calculus (FC) initially the development of mathematical models of fractional order integrators (FOIs) and fractional order differentiators (FODs) [1, 2] started presumably for searching different generalizing approaches for switching from integer order to the fractional order domain. The frequency response of ideal fractional order differ-integrator is where and gives angular frequency in radians. Variable “ ” defines the order of fractional order operators and its value lies in between 0 and 1. Due to its exotic nature FC has strengthened its grasp over different research areas such as statistical modeling, mechanical system analysis [3], control [4], automated control [5], instrumentation [6], signal processing [7], and radio engineering and image processing [8, 9]. At present time, FC is also extensively spreading its niche in different design methods for improving fractional order controllers by fractional integral and derivative functions [10, 11]. An important design method which has undoubtedly helped in sharpening up the focus of an efficient formulation of new blocks of fractional operators is “linear interpolation” method [12–16]. This most vastly followed tri-step method involves the following steps. (1) First step deals with linear interpolation of two existing digital integrators. (2) In second step, a new fractional order integrator is formulated by discretization of integer order integrator developed in first step by anyone of the two existing discretization (direct and indirect)
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