Optimal continuous approximation of basic fractional elements: Theory and applications

In the last two decades, a boom of fractional calculus applications started in many technical areas including automation and process control. The generalization of integrals and derivatives to arbitrary real order (FO - Fractional Order) simplifies solution of many problems especially in frequency domain. Unfortunately, switching into time domain is always quite difficult due to the necessity to approximate fractional elements by integer-order ones. For this purpose, often a high order zero/pole transfer function is employed. This paper extends the authors' previous work and summarizes the results of numerical optimization of zero/pole positions for two important fractional elements: fractional integro-differential operator and fractional pole. The optimization is done on a limited frequency band up to four decades. The quadratic difference between the frequency response of ideal FO element and its zero/pole approximation was taken as an optimality criterion. It is shown, that the optimization decreases markedly the criterion value compared to traditional methods. The paper main results are provided in a form of analytical functions parametrizing the zero/pole positions dependent on element order. Additionally, prospective applications of presented fractional elements are discussed from both controller synthesis and process modeling point of view.