In industrial control applications, the plants are often represented by integer-order models and controlled by proportional-integral-derivative controllers. The control scheme for permanent magnet synchronous motors (PMSM) includes two nested loops, each employing a PI controller: the inner loop is dedicated to control the current, the outer loop is devoted to control the angular speed. The optimum modulus and symmetrical optimum criteria are widely accepted techniques to tune the two PI controllers. However, to obtain improvements, one may think to apply non-integer order controllers. To this aim, if one uses a fractional-order PI (FOPI) controller in the inner loop, the consequent inner feedback system of non-integer order becomes a non-integer order plant in the outer loop. Then, a FOPI controller should be more effective to control a real example of non-integer order plant. This paper proposes an appropriate design approach to obtain performance and robustness specifications by FOPI controllers in both loops. The approach provides analytical formulas to determine the parameters of the controllers, which are characterized by stability, minimum-phase and interlacing properties. The simulation of a real PMSM shows the effectiveness of the approach and could help to increase the confidence in non-integer order controllers.
In this work, we prove a generalization of the Gronwall type inequality. This relation can be used in the qualitative analysis of the solutions to fractional differential equations with the ψ -fractional derivatives.
The paper is an attempt of an application of the fractional order derivative in modeling of power system elements. The electrical part of the generating unit contains, first of all, the power generator equipped with an excitation system. Three other components may be identified, when the electromachine excitation system is considered. This type of excitation uses an AC electric machine as an exciting device. The mathematical model of high frequency AC exciter with additional regulator, being one of three possible submodels of electromagnetic excitation system model, was chosen intentionally and used as a simulation platform. The presented model in its simplicity includes all elements that characterise far more advanced and extended models, for example power generators. It contains gain factors and time constants as well as saturation components. Another important factor is that this particular model operates only using positive signals developed by an additional regulator. The paper presents the method and exemplary results of parameter estimation of the fractional model of the high frequency AC exciter with an additional regulator. To preserve full reliability of the computations, true waveforms measured in a power plant were used as input and output signals of the model. The advantages of applying fractional order calculus were verified by comparing measured and computed model output waveforms. Both integer and fractional order models were used in computations. The aspect of filtering the recorded measurement signals is also presented in the paper.
On the basis of simple two-component nonlinear incommensurate fractional-order systems with positive and negative feedbacks, some general properties of fractional auto-oscillation systems are established. By linear stability analysis and numerical simulation, it is shown that fractional derivative orders and ratio between them can substantially change the stability conditions of the system and lead to appearing of complex oscillations and attractors, which cannot be found in their integer counterparts.
