Control Optimization of Electromechanical Systems by Fractional-Integral Controllers

The object of research in the work is electromechanical systems, a characteristic feature of which is the presence of significant power dependence in the mathematical description. Because of this, problems arise when choosing the structure and parameters of controllers. In particular, in a DC motor with series excitation, a switched reluctance motor and electromagnetic retarders, saturation of the magnetic system in static and dynamic modes can occur. The apparatus of fractional-integral calculus used in the work allows us to describe such nonlinear objects with high accuracy by linear transfer functions of fractional order. So, when approximating the anchor circuit of a DC motor with series excitation by a fractional transfer function, the smallest standard error was obtained. The combination of a conventional PID controller with fractional integral components of the order of 0.35 and 1.35 ensured the best quality of the transient process - the current reaches the set value as quickly as possible without overshoot. Secondly, the switched reluctance motor, in the model of which it is necessary to take into account the power dependences, is described by the aperiodic function of the order of 0.7 when describing the transient processes of the speed during a voltage jump. From the family of controllers studied, the traditional PI controller with additional fractional-integral components of the order of 0.7 and 1.7 ensured the astaticism of the speed loop of the order of 1.7 and the smallest overshoot. Thirdly, the electromagnetic retarders of the driving wheels of a car, used to tune the internal combustion engine, are also most accurately described after testing by the fractional transfer function. Using the PIDIγIµ controller, which ensured closed loop astaticism of the order of 1.63, stabilization of the rotation speed of two wheels without out-of-phase oscillations and the accurate development of a triangular tachogram were achieved. Thus, thanks to the apparatus of fractional-integral calculus, a more accurate identification of object parameters is provided, the mathematical description is reduced to linear transfer functions of fractional order. And in closed systems it is possible to ensure astaticism of fractional order 1.3–1.7 and to achieve a better quality of transient processes than using classical methods.


Introduction
The beginning of the development of fractional calculus is considered to be 1695, when Leibniz in a letter to Francois L'Hospital discussed the differentiation of the order 1/2 [1]. More than 300 years have passed since then, mathematicians have conducted numerous studies in this field. But a new surge of interest in it is noticeable in recent decades. This is primarily due to the fact that differential equations with fractional order made it possible to describe some physical processes with greater accuracy than integer ones [2,3]. Also, the computing capabilities of modern single-chip processors already correspond to the complexity of calculating fractional integrals and derivatives in real time. Therefore, ways of applying this mathematical apparatus in technical systems, in particular in acoustics, electronics, and thermodynamics, have opened [4,5]. In controlled systems, PI γ D m controllers are used that can improve the quality of transients in comparison with classical integer PID controllers, especially in nonlinear systems [6,7].
The aim of this research is to study the possibility of using the apparatus of fractional integral calculus in electromechanics to optimize transient and steady-state processes. DC motor with series excitation, switched reluctance motor and an induction brake are considered as objects of research.

Methods of research
The study of each of the listed objects was carried out according to the generally accepted methodology. Based on the testing of objects by supplying standard signals, identification of model parameters was carried out with the subsequent synthesis of optimal controllers and verification of closed systems. However, at the identification stage, in addition to the generally accepted ones, models are considered that include fractional differential equations. This choice is due to the fact that the apparatus of fractional integral numbering is based on power functions, and in the considered electromechanical objects the magnetization ISSN 2664-9969 curve is also close to the power function. It is precisely this which suggests that controllers with integral and differential components of non-integer order can provide the best dynamic and static indicators of systems.

Research results and discussion
Transients significantly differ from solutions of differential equations of the first or second order in DC motor with series excitation at a current in the armature circuit above the rated value [8].
The smallest standard error is provided when using the following mathematical model: where for the studied motor with a power of 450 W received K = 0.193 -gain coefficient; m = 0.35 -indicator of the fractional differential equation; а 1 = 0.0062, а 0 = 0.127parameters equivalent to the time constants in degrees 1+m and m, respectively. Accordingly, to ensure optimum tuning with a given integer or fractional orders of astatism, D m I γ I-, PI γ I-, PIDI γ I m controllers are required.
In the course of experimental studies, the best quality of the transient process was obtained with the PIDI γ I m controller ( Fig. 1, a). In this case, the current reaches the reference as quickly as possible and without overshoot.
It turned out that such settings are effective even with reference signals half the maximum, as well as with stepwise change of reference ( Fig. 1, b, c). It can be seen that the quality indicators of the system remain unchanged and correspond to the desired settings.
Another electric machine, in the description of which it is necessary to take into account power dependencies, is a switched reluctance motor. To identify a closed speed loop, a fractionally aperiodic transfer function of the following form was also chosen [9]: The approximation results are illustrated in Fig. 2. As can be seen, at different applied voltages, the nature of the transition process changes, but the use of the fractional transfer function (2) provides the smallest deviation. And this also allows for the synthesis of controllers to abandon the complex motor model described by a system of nonlinear differential equations.
Accordingly, the adjustment of the closed loop is greatly simplified -the modular optimum provides the II γ controller, and the fractional order of astaticism 1.7 is achieved using the PII γ I m controller. Fig. 3 shows the simulation results of transients in the speed control loop of a switched reluctance motor. In Fig. 3, a, the graphs of transients during the jump of the reference are shown, in Fig. 3, b -with a step change in the reference signal. In all cases, a system with a fractional order of astaticism has the best dynamic characteristics.
A similar approach was used for the synthesis of the electromagnetic retarders control system of the driving wheels of a car on a stand for tuning and measur-ing the power (torque) of an internal combustion engine (ICE) [10].
According to the experimental data obtained, the control object is approximated with an error of about 1 % by the following transfer functions: Using the transfer function (3), the coefficients for the PID controller were found, and according to (4), two types of fractional-integral controllers were synthesized: D γ I m I, which provides tuning to the modular optimum, and PIDI γ I m , which allows obtaining the fractional astaticism order 1.63.
When conducting experimental studies, transient graphs were obtained with stabilization of the speed of the semiaxles of the car (Fig. 4, a). The PID controller provided the best performance (2.55 s), but with the greatest overshoot δ 1 = 27.6 %. With the D γ I m I controller, the overshoot was δ 2 = 16.3 %, and the duration of the transient process was 4.42 s. The smallest overshoot was obtained with the PIDI γ I m controller -δ 3 = 3.3 % for a duration of 3.8 s.
An important step in checking the results of engine settings is to measure the power and maximum torque when forming a triangular tachogram.
The results of such a test with the fastest (PID) and most accurate (PIDI γ I m ) controllers are shown in Fig. 4     Both controllers provide stabilization of speed in the entire range of power measurement, but less overshoot and oscillation at the beginning of acceleration (the initial sections of transient processes with various controllers are compared in the upper left in Fig. 4, b). This allowed to make the final decision on choosing a control system block diagram in favor of the PIDI γ I m controller.

Conclusions
Thus, the use of the fractional integral calculus for three types of nonlinear electromechanical objects, in the ISSN 2664-9969 description of which there are power functions, allowed us to obtain the following: 1. A more accurate identification of the parameters at which the smallest discrepancy between the calculated and experimental data is achieved when testing open systems.
2. A simplified mathematical description due to the use of fractional-order linear transfer functions in models.
3. The best dynamic and static indicators, especially when using fractional-integral controllers, providing a fractional (1.5-1.7) order of closed loop astaticism.