The article deals with the application of fractional order desired forms to extend the range of possible fractional controllers adjustment in the synthesis of electromechanical systems loops and to provide better quality of transition processes as compared with integer order controllers. Experimental tests of the fractional order controllers, synthesized by proposed approaches in the control system of frequency converter-induction motor speed, have confirmed the effectiveness in terms of the expansion of adjusting capabilities of PI ^γ D ^μ controller as compared with the classical PID-controller.

The approximation of a fractional order PI^λD^μ-controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of $s^{\pm \alpha }$ components of the transfer functions of PI^λD^μ-controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of D^μ (α = μ = 0.5) and I^λ (α = λ = −0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PI^λD^μ-controller transfer function by the method of chain fractions with different valuesof λ and μ. For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for α = ±0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the PI^λD^μ-controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of PI^λD^μ-controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.

Devising a procedure for the synthesis of electromechanical systems with cascade-enabled fractional-order controllers and their study

Devising a procedure for the parametric synthesis of fractional order controllers and their implementation in the FC–IM system

The article presents the creation of characteristic polynomials on the basis of fractional powers j of dynamic systems and problems related to the determination of the stability intervals of such systems.