1. Introduction
Induction machines play a crucial role in modern industry, ranging from simple applications such as driving fans or pumps to more precise usages such as conveyor belts or plastic injection molding [
1]. Conventional induction machine control methods such as Field-Oriented Control (FOC) [
2] and Direct Torque Control (DTC) [
3] can suffer from sensitivity to parameter and torque changes [
4], or torque and flux pulsations and control problems at low speeds [
5]. For high-performance drives, which require a fast dynamic response and disturbance rejection, these shortcomings must be improved, or new methods must be investigated. One direction of research is incorporating fuzzy logic into conventional control structures to improve the dynamics.
Research in [
6] shows an increasing trend of computational intelligence implementation in control applications. An overview of the recent literature proves the research interest in fuzzy logic in drive systems. The authors of [
7] investigate the implementation of FLC with a reduced computation burden. In [
8], the authors incorporate FLC into a classic DTC structure to improve the torque ripple of the high-performance drive. The authors of [
9] investigate FLC for low-speed induction machine operation. In [
10], the authors investigate the computation burden of FLC depending on the size of the fuzzy rules table. In [
11,
12], type 2 FLC is investigated to improve the DTC structure on five- and three-level inverters. In [
13,
14], the authors use FLC to improve the classic DTC in an induction machine drive with a two-level converter and dual-stator machine drive, respectively. The authors of [
15] improve the FOC structure for an induction machine by developing a fuzzy speed controller with an algorithm for automatic gain output adjustment. Tir et al. use fuzzy logic to improve the FOC structure for a single system in [
16] and a multi-machine drive system in [
17]. Bessaad et al. use a fuzzy system in [
18] for developing a correction regulator in a multi-machine system with a single inverter supply. The authors of [
19] develop a custom search algorithm for induction machine fuzzy-PI controller tuning. To improve efficiency, the authors of [
20,
21] use fuzzy logic for the online search of the minimum power losses of an induction machine drive and self-excited induction generator, respectively. In [
22], the authors use fuzzy logic to develop an expert system for induction machine fault diagnosis. The authors of [
23] use FLC to calculate a suitable voltage vector for a five-phase induction machine to reduce torque ripple. The authors of [
24] use fuzzy logic as a decision mechanism for weighing factor selection in predictive torque control. In [
25], the authors incorporate two FLCs as inputs to a feedback linearization algorithm to improve drive dynamics. Saghafinia et al. use FLC in [
26] as a speed controller in a sliding mode control structure. In [
27], the author presents a fuzzy-PI controller utilized for minimizing energy consumption while maximizing the performance of an induction machine drive. In [
28], Youb et al. develop a fuzzy-PI controller for a dual-star induction machine with online adaptation of proportional and integral gains.
There are a number of research papers showing the usage of fuzzy systems in parameter estimation. The authors of [
29] use fuzzy logic concepts for estimation purposes by building an observer using the Takagi–Sugeno model of an induction machine. Jabbour et al. use fuzzy logic in [
30] for online parameter estimation. In [
31], the authors utilize a type 1 and type 2 fuzzy controller for a model reference adaptive system and compare the results. The authors of [
32] build a Luenberger observer using a fuzzy logic system that outputs state variable estimates. In an older paper [
33], the authors optimize a fuzzy system for the purpose of estimating the rotor time constant. In [
34], Shukla et al. use fuzzy logic for the model reference speed adaptation of an induction machine that is controlled via DTC.
Fuzzy logic is also used in power and frequency control, as can be seen in papers such as [
35], where the authors use fuzzy logic to tune the proportional, integral, and derivative gains of a doubly fed induction generator (
DFIG) PID controller. In [
36], the authors also use fuzzy concepts to improve the DTC of a DFIG, while in [
37], FLC is used for active power control. In [
38], Dewangan et al. replace a classic PI controller with FLC to improve the performance of a wind-driven self-excited induction generator during fault and variable wind speed conditions. The authors of [
39] develop FLC for a six-phase induction generator that shows superiority over the classic controller in fault conditions.
In this paper, the authors present an optimization procedure for the fuzzy speed controller that enhances the speed tracking and torque response of the induction machine, which is controlled using the PCC algorithm. In recent years, a few papers have dealt with this problem in the induction machine drive field. In [
40], George et al. use an optimization procedure to optimize the speed control of an electric vehicle. Similarly, the authors of [
41] use an optimization approach to optimize frequency control in multi-area interconnected power systems. Both papers offer several criteria for objective function value calculation. In [
37], the authors use the integral time squared error to calculate the objective function value for the particle swarm optimization algorithm, which is used to optimize DFIG power control. The authors of [
42] optimize FLC for an induction machine and use the mean average error as a criterion to optimize seven membership functions for inputs and output while using the Mamdani-type defuzzification process. In this paper, the authors use five membership functions for inputs, and a Takagi–Sugeno-type defuzzification process. Several criteria for objective function value calculation are investigated and results show improvements upon the classic speed controller. The paper is organized as follows: In
Section 2, the dynamic model of the induction machine is presented and a control overview is given. In
Section 3, the fuzzy logic speed controller that is being optimized in this paper is presented and each part of the controller is described. The problem statement and optimization procedure (optimizer, controller parameters, and the choice of objective function) are presented in
Section 4. In
Section 5, results for different optimization approaches are displayed and discussed. Control system performance is shown in
Section 6, while the comparison with several model predictive methods is carried out in
Section 7. In
Section 8, the authors discuss caveats and future research possibilities.
2. Induction Machine Model and Control Overview
The dynamic model of the induction machine is written in the stationary
-reference frame and it is shown by Equations (1)–(4), where
J is the inertia constant of the machine,
is the rotor shaft speed,
p is the number of pole pairs,
is electromechanical torque, and
is load torque.
Equation (5) displays electrical quantities, where
represents the stator voltage vector,
and
represent the stator and rotor current vector, while
and
describe stator and rotor flux linkages.
Electrical parameters are described by Equation (6), where
and
are the stator and rotor resistance matrices, while
represents the rotation matrix.
Additionally, the relationship between fluxes and currents is described by Equations (7) and (8), but for a deeper understanding of the induction machine model, the reader is referred to [
43].
Model predictive control has attracted research interest in recent years, as seen in papers such as [
44,
45,
46]. This being the case, the control method that is being modified in this paper is PCC for an induction machine, which falls into the family of model predictive control structures. In the following text, it is explained how it works.
Firstly, a discrete state space model for computing the current prediction
is formed by selecting stator currents
and rotor fluxes
as state variables. Equation (9) represents the final expression to calculate the current predictions, where
represents the leakage inductance factor,
represents the rotor time constant, and
represents the discretization sampling time.
Equation (10) represents the cost function that is used to derive the control law by minimizing the squared error between reference currents
,
and current predictions. By inserting (9) into (10) and solving (11), the optimal voltage vector
can be obtained to minimize the cost function and drive the machine in the desired state.
Equation (12) represents the solution of (11), which is used as a control law.
Reference current value
is calculated from the desired rotor flux of the machine, while reference current
is generated by the FLC acting as a speed controller.
Figure 1 represents the final topology of the control structure, and more details about the PCC structure can be found in [
47].
5. Optimization Results
In this section, the optimization results, obtained by criteria from Equations (22), (25), and (26), are presented.
Figure 7 shows the speed and torque responses that are obtained by using only criteria from Equation (22). It can be seen that the dynamics of the drive system have almost no impact on the speed response; in other words, the speed response has only around 0.2 rpm maximum tracking error and a fast settling time, but when observing the torque response, an unacceptable overshoot of around 20 Nm is observed. The reason for the large overshoot is the fact that the speed error is the only criterion for the optimization. To keep it at its lowest value, large control action is required at every instance when the speed diverges from the reference value. Since there is no criterion that would limit or penalize the torque, optimized parameters allow this kind of behavior. An attempt is made to remedy this problem with a multi-objective optimization approach. The first objective is selected as
from Equation (22) and the second as overshoot from Equation (25).
Figure 8 represents the optimization result in the form of a Pareto front, where the
x-axis represents the
value and the
y-axis represents the total overshoot value.
Depending on the application, a range of solutions are available to choose from, and they are represented by the circles in the graph. The green hexagon in the figure is the area of the Pareto front that the optimizer selected a solution from, and
Figure 9 represents the system response produced by its parameters.
It can be seen from
Figure 9 that the overshoot in the torque response is greatly reduced. To obtain the unique solution that offers the best system response, criteria from Equation (26) are investigated.
Figure 10 represents the speed and torque responses of the criteria from Equation (
26) with the weighing factor of value 10. It can be seen that the torque overshoot is greatly reduced, while preserving a good speed tracking response.
To summarize, nine different optimization procedures are conducted for nine different types of objective functions.
Table 3 shows relevant metrics for each optimization procedure.
It can be seen from the results that single-objective optimizations that utilize objective functions
and
produce the best results regarding torque overshoot, while keeping the speed tracking error below 1 rpm. Even though they do not differ significantly, the
criterion produces slightly better results, which is why it was chosen to be further investigated.
Figure 11 shows the optimized membership functions and
Table 4 shows the optimized gain and output level function coefficient values for the
criterion.
It should be noted that coefficients , and have opposite values to their positive counterparts.
7. Discussion
The optimization procedures conducted in the study show improvements in the speed tracking response of the induction machine drive.
Figure 9 shows the improvement when using multi-objective optimization over single-objective optimization, whose results are represented by
Figure 7. The reason for the improvement is that torque overshoot penalization is added via a second objective function, and the result is a Pareto front that offers a range of solutions to chose from, based on the application.
Figure 10 shows further improvements: single-objective optimization is used and torque overshoot can be arbitrarily penalized using a weighing factor, which results in the smallest amount of overshoot and good speed tracking behavior. A comparison between the proposed method and PCC method that utilizes a classic PI speed controller is shown in
Figure 13. In
Figure 13a, it can be seen that the speed response is greatly improved, with a much smaller value of maximum speed tracking error but similar settling time. Torque responses are filtered to better represent overshoots, and as can be seen from the same figure, the optimized fuzzy speed controller produces less torque overshoot than the classic PI controller. The reason for the better response is the fact that the fuzzy speed controller acts on the speed tracking error derivative along with the speed tracking error. The speed tracking error derivative can be understood as a form of torque estimation which increases control action in the instances when the torque is changing. The classic PI controller does not have this advantage, since it only acts on the speed tracking error.
Figure 13b shows the unfiltered torque response for both methods. It can be seen that the chattering produced by both methods is in the same range. To further confirm the effectiveness of the method, two more comparisons with different predictive control methods are conducted.
Figure 14 shows a comparison of the proposed method with Finite Control Set-Predictive Current Control (FCS-PCC), while
Figure 15 represents a comparison of the proposed method with Finite Control Set-Predictive Torque Control (FCS-PTC). To gain an understanding of both methods, the reader is referred to [
48,
49]. It can be seen from
Figure 14a and
Figure 15a that the speed tracking error and filtered torque responses are similar to the original comparison: the proposed method has less torque overshoot and superior speed tracking.
Table 5 represents relevant numerical values for each method.
Figure 14b and
Figure 15b show unfiltered torque responses. It can be concluded that finite control set methods produce a larger amount of chattering compared to the proposed method, which means that the proposed method produces less stress on the rotor shaft during operation.
In future research, alternative inputs to the FLC will be investigated, since the speed derivative has several drawbacks: it can be computationally unstable and it can be a cause of high control action. Estimated load torque can be explored as an alternative input to the FLC. This could provide more stable input to the controller, which would produce a more stable output with less control action and potentially less torque overshoot.