Approximation of Transfer Function With a Characteristic Equation in a Fractional Order Binomial Form

The approximation of one of the desired fractional order transfer functions characterized by the absence of zeros and the expression of a fractional characteristic polynomial in the form of a fractional order binomial are considered. For the purpose of approximation, it is proposed to first implement the approximation of a fractional characteristic polynomial and then write the approximated transfer function as the expression inverse to the approximated characteristic polynomial. The approximation of the characteristic polynomial of the fractional order is carried out by means of a chain fraction. The inverse Laplace transformation is applied for a comparative analysis of the dynamic properties of systems described by the fractional order desired transfer function and the systems described by approximated transfer functions. Consequently, the expression of the system reference transition function described by the desired fractional order transfer function is obtained. It is shown that for such approximation it is sufficient to use expressions of integer order, where the highest degree of the Laplace operator does not exceed two (second-order approximation). It is revealed that the adequacy of such systems frequency characteristics depends on the approximation order. It is necessary to choose higher order of approximation in order to expand the range of frequencies for which the frequency characteristics of the approximated transfer function will coincide with the frequency characteristics of the desired fractional order transfer function.