Investigation of type 1 and type 2 fuzzy logic controllers performance: application of speed control of BLDC motor

Abstract

In this paper, a simulation study enhanced to model that the speed control of brushless direct current (BLDC) motors used in electric vehicles with intelligent control methods. The simulation study was prepared in Matlab/Simulink environment. The first control method is Type-1 fuzzy logic control (T1FLC), and the second control method is the Intermittent Type-2 fuzzy logic control (IT2FLC) model. Membership functions for different membership numbers have been created for both types of FLC models. These are 3×3, 5×5, 7×7. Control methods are prepared in Matlab/M-file environment. The model is defined as the input variable of the error, which is the difference between the reference speed and the motor speed, and the output variable of the Pulse Width Modulation (PWM) signal applied to the motor. The simulation study maintains the speed of the BLDC motor up to the reference speed with T1FLC and IT2FLC controllers, depending on the reference speed and applied load values. Depending on the number of different memberships, the effects of controller performances on the control of motor speed have been observed. The graphs and findings of the experiment are shown in the results and discussion section.

Keywords

Speed control fuzzy logic system BLDC motor

1 Introduction

BLDC motors are used in industrial systems, automobile industry, aviation systems, etc. in wide application areas. BLDC motors provide great virtue over direct current (DC) motors. The absence of friction and electrical losses due to the absence of brushes ensures low maintenance costs and long operating life. It also gives faster dynamic responses due to high performance and low inertia [1 –4]. Various methods have been used in speed control of BLDC motor[5, 6]. One of these methods is the fuzzy logic method. Today, T1FLC [7] has been successfully applied in many applications [8, 9]. However, with the emergence of IT2FLC, which takes into account the uncertainty in their design, the performance of both under similar degradation conditions can be evaluated comparatively.

IT2FLC’s ability to account for uncertainty in its designs is important in real applications. This is because there are many sources of uncertainty in problems encountered in real applications. These sources of uncertainty impressions the performance and accuracy of controllers used in applications. However, IT2FLC has disadvantages such as extra cost as it is difficult to calculate. It requires approximately twice as much computational throughput than T1FLCs. The degree of uncertainty that was taken into account when creating the IT2FLC, the cost issue can be ignored as the controller performance is significantly improved.

The purpose of this article is to analyze the motor performance results of a BLDC motor with T1FLC and IT2FLC controllers with different membership numbers.

The result of this article is to find the relationship between the controller performances by comparing different controller algorithms, to provide a better understanding of the performance of IT2FLCs in speed control for BLDC motors.

This study is planned as follows. The mathematical model and equations of the BLDC motor are presented in Chapter 2. Information on the historical development of the FLC is presented in Chapter 3. Design criteria for the T1FLC and IT2FLC controllers are presented in Chapter 4. In Chapter 5, the controllers prepared for the simulation study are presented. In Chapter 6, the Matlab/Simulink program prepared for the simulation study and the graphical results obtained are presented. Finally, interpretation of the results found and discussions about future work are presented in Chapter 7.

2 Mathematical model, equations and driver of BLDC motor

It is possible to model a BLDC motor whose mathematical equations are known. It has a motor model as PMSM (permanent magnet synchronous machine), which forms the basis of the BLDC motor in the Matlab/Simulink program. Different results can be obtained by changing the parameters of a BLDC motor or the controller, which saves time in simulation. To understand the mathematical equation of a BLDC motor, a DC motor mathematical equation must be observed [10].

Figure 1 shows a DC motor electrical equivalent circuit. When voltage is applied to the DC motor terminal, current flows through the DC motor windings and the rotor of the motor starts to rotate, producing BEMF (reverse electromagnetic force). DC motor mathematical expression is seen in Equation (1). $V = I . R + L . \frac{dI}{dt} + E$ (1)

Fig. 1

DC motor electrical equivalent circuit.

Here V denotes the voltage delivered to the motor terminals. ‘I’ is the current through its windings. ‘R’ is the resistance of the windings. ‘L’ is the inductance of the windings. ‘E’ is BEMF generated by the motor.

Fig. 2

BLDC motor electrical equivalent circuit.

Figure 2 shows a BLDC motor electrical equivalent circuit. The phase inductances and resistances of the motor are considered equal. The mathematical equation of the engine is seen in Equation (2) in matrix form. $\begin{matrix} [\begin{matrix} V_{an} \\ V_{bn} \\ V_{cn} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + \frac{d}{dt} \times \\ [\begin{matrix} L_{aa} & L_{ab} & L_{ac} \\ L_{ba} & L_{bb} & L_{cb} \\ L_{ca} & L_{cb} & L_{cc} \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + [\begin{matrix} E_{a} \\ E_{b} \\ E_{c} \end{matrix}], \end{matrix}$ (2) Where, V_an, V_bn and V_cn is the voltage difference between the motor terminal and the midpoint of V_n. V_n, V_an, V_bn and V_cn are calculated as in Equations (3)–(6). Rs is the phase resistance, I_a, I_b and I_c are phase currents, L_aa, L_bb and L_cc are phase inductances, L_ab, L_ac, L_ba, L_bc, L_ca and L_cb are mutual inductances, E_a, E_b and E_c are BEMF: $V_{n} = (V_{a} + V_{b} + V_{c} - E_{a} - E_{b} - E_{c}) / 3,$ (3) $V_{an} = V_{a} - V_{b}$ (4) $V_{bn} = V_{b} - V_{n}$ (5) $V_{cn} = V_{c} - V_{n}$ (6) Equation (7) gives the phase inductances. In Equation (8), it gives the mutual phase inductances as M. Mutual inductance can be defined as the flow of current in a winding wound on the core, affecting other windings on the same core: $L = L_{aa} = L_{bb} = L_{cc}$ (7) $M = L_{ab} = L_{ac} = L_{ba} = L_{bc} = L_{ca} = L_{cb}$ (8) Equation (9) is obtained by writing Equations (7) and (8) in the appropriate places in Equation (2). $\begin{matrix} [\begin{matrix} V_{an} \\ V_{bn} \\ V_{cn} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + \frac{d}{dt} \times \\ [\begin{matrix} L & M & M \\ M & L & M \\ M & M & L \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + [\begin{matrix} E_{a} \\ E_{b} \\ E_{c} \end{matrix}], \end{matrix}$ (9) The sum of the current coming into the node is zero as seen in Equation (10). Mathematical expressions are given in Equations (10) and (11). Equation (12) is obtained by writing Equation (11) in the appropriate places in Equation (9): $0 = I_{a} + I_{b} + I_{c}$ (10) $M . I_{a} = - M . I_{b} - M . I_{c}$ (11) $\begin{matrix} [\begin{matrix} V_{an} \\ V_{bn} \\ V_{cn} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + \frac{d}{dt} \times \\ [\begin{matrix} L - M & 0 & 0 \\ 0 & L - M & 0 \\ 0 & 0 & L - M \end{matrix}] \times [\begin{matrix} I_{a} \\ I_{b} \\ I_{c} \end{matrix}] + [\begin{matrix} E_{a} \\ E_{b} \\ E_{c} \end{matrix}], \end{matrix}$ (12)

Since the BLDC motor has no brushes, the speed control takes place by electronic commutation. For this, a special electronic control circuit is needed. A three-phase inverter circuit synchronized with the rotor position is used to control the current polarity. Switches such as MOSFET or IGBT are used in the inverter circuit. The rotor position inside the motor can be detected using the Hall Effect sensor fixed in the stator. The drive can be controlled in sensored or sensorless mode. Sensorless mode can reduce the cost and dimensions of the inverter circuit but impairs its performance. Hall Effect sensor is used in this article. The schematic representation of the sample inverter circuit for the BLDC motor is shown in Fig. 3.

Fig. 3

Equivalent circuit of BLDC motor driver.

3 History of FLC

Lotfi A. Zadeh first expressed the concept of Fuzzy Logic in 1965 [11], who suggested the substructures of fuzzy sets. This innovation allowed to immence progress in control systems, which were evolved by analogy with the decision-making process of humans. After this innovation in 1975, linguistic synthesis control rules based on expert operators expressed by Ebrahim Mamdani were created and used fuzzy inference system to control a steam engine and boiler [12]. He created the infrastructures of the currently used fuzzy interface system, which includes fuzzing and defuzzifying the full input variables to obtain the absolute output variables. Mamdani’s FLC system has gained wide acceptance and is still used in today’s applications. Ten years after Mamdani’s invention, a new FLC system has been proposed that should be linear/constant except for the output membership function expressed by Takagi-Sugeno and work similarly to Mamdani’s method [13]. The proposed method, unlike the Mamdani type, is compact and calculatingly efficient as it uses fixed/linear output membership functions that can generate an offline table, which increases the online computation capabilities of the fuzzy system. However, creating an offline table takes time and its accuracy can degrade system performance. Mamdani and Takagi-Sugeno methods are of great interest in various applications [14]. In the last four decades, FLC based on Mamdani and Takagi-Sugeno fuzzy inference systems has found wide use in various industrial applications such as motor drives [15], power converters [16], power systems [17, 18]. According to FLC experts, the above-mentioned FLC theories are considered type-1 fuzzy sets with strictly selected two-dimensional MFs. In addition, another type of set, called a type 2 fuzzy set, uses three-dimensional fuzzy membership functions [19]. This method was first introduced by Zadeh [20] as a supplement to the T1 fuzzy sets introduced earlier. Such fuzzy sets can be applied to systems with faulty MFs and manageable ones. In recent studies, T2 fuzzy sets have attracted great interest in motor driver applications due to their adaptive nature [21]. Despite the advantageous properties of T2 fuzzy sets, they require extra high computational costs compared to T1 fuzzy sets. Because of this, they are not widely used in cost-sensitive real-time applications and high-capacity processors are needed [22, 23].

In short, the developments experienced in FLC systems have enabled great developments in control system applications. Its success in fuzzy systems applications has made it a leading controller in various industrial applications as it can compensate for system uncertainties, disturbance effects and parameter deviations. With the improvement of different fuzzy systems, the Mamdani T2FLC system is preferred in many applications due to its performance accuracy and stability, although it is a design challenge compared to Takagi-Sugeno and T1FLC systems.

4 T1FLC and T2FLC

This section explains the basic concepts of Fuzzy Logic, T1FLC and T2FLC.

4.1 T1FLC

FLC is a member of the artificial intelligence controller family used to control processes. Based on the Mamdani fuzzy system [24], FLC consists of three stages: preprocessing, main processing and postprocessing. Precise inputs are transformed into fuzzy variables in the preprocessing stage. MFs designed to generate the fuzzy output are in the main processing stage and fuzzy values are calculated according to the membership degree. After main processing, the fuzzy expressions are clarified and converted back to strictly numerical values for use as control signals. However, the design of the rule base and membership functions differ depending on the characteristics of the system to be controlled. The common form of the FLC processing steps is shown in the block diagram in Fig. 4. The following sections examine FLC design and simplification approaches. There are 3 different categories of FLC systems. The first fuzzy system developed by Lotfi A. Zadeh [20] is Takagi-Sugeno (TS). The TS system proposed by TS [13] and Mamdani or the turbidity system proposed by Ebrahim Mamdani is the first FLC [12]. The first fuzzy system is a combination of fuzzy rules that relate its inputs and outputs. Fuzzy sets are the inputs and outputs of a pure fuzzy system. Neverthless, the inputs and outputs of co-engineered systems are exact values, making it difficult to implement a fuzzy system that works well in such applications. A TS system with real or specific inputs and outputs is proposed to overcome the limitations of fuzzy systems. It is possible to obtain TS outputs with a simple mathematical equation. Neverthless, as the TS system consists of mathematical formula outputs, it may not naturally represent human knowledge, and this method has limitations that do not allow the application of various principles. Therefore, the TS system is the most important factor limiting the acceptability of fuzzy systems. To solve the problems related to the first fuzzy and TS systems discovered, a defuzzifier to convert the fuzzy set outputs to full value outputs and a fuzzifier to convert the full value inputs to the fuzzy set inputs is recommended. The differences distinguishing FLC variants from each other are highlighted in [25]. Mamdani type FLCs are most commonly used in engineering applications and are most useful for hardware application.

Fig. 4

Mamdani T1FLC block diagram.

It proposes defining expressions known as MF with mathematical equations and applying membership degrees accordingly. These MFs are linked to linguistic expressions, and decision-making conditions can be specified by special operations. The mathematical form of fuzzy sets is expressed by Equation (13). $A = {(x, μ_A (x)) | x \in X}$ (13) Where μA (x) : X → [0, 1] it is the degree of membership of x to the set A. Mamdani [26] successfully implemented a Fuzzy Logic based controller, in other words FLC, in 1974 and improved the results obtained with its easy controller design compared to conventional controllers.

Variations of FLCs are available today for many control applications [7 , 28] and they achieve better results than conventional controllers.

4.1.1 T1FLC design

A general closed-loop control system is accustomed examine the components of the FLC. In closed-loop control systems, the operator pays the most attention to the error (E), which is the result of the comparison between the reference control target and the real control output [29]. It usually uses system error (E) and error variation (dE) as input variables for FLCs. Fuzzy systems work by imitating human reasoning that is fuzzy or ambiguous. Thus, the FLC partially relates its variables to membership sets or degrees of membership. Besides, the changing range of a variable is between 0 and 1 instead of 0 or 1 as in conventional logic. Another feature of fuzzy variables is the linguistic variables assigned as natural language words used to describe the values of these linguistic variables. These words are explained by membership functions (MFs) [20]. Many different MF shapes are used in fuzzy systems such as Triangle, Trapezoid, Gaussian and Sigmoid MFs. Triangular MFs are most widely used due to their simplicity and computational efficiency.

As shown in Fig. 4, FLC enables linguistic input variables to be transformed into fuzzy sets with appropriate MFs by fuzzification. These MFs are usually chosen based on a thorough understanding of the physical behavior of the system to be controlled. The fuzzy rule base is expressed in the (IF-THEN) model to reveal the relationship between inputs and output variables in linguistic terms. With an FLC interface, the contribution of each fuzzy rule in the rule base is taken into account to find the overall value of the fuzzy control output. A precise result is obtained by applying defuzzification to the fuzzy expression obtained from the FLC interface.

The fuzzy rule is often expressed as (IF-THEN) because of its simplicity, common use, and computational ease. The number of fuzzy rules is directly related to the total number of linguistic expressions in MFs in the system. Standard 9, 25 or 49 rule base structure can be used for closed loop speed control of DC motor drives using two input and one output FLC [29, 30]. Depending on the number of different rules for FLC, different controllers are designed. But there are two commonly used methods to create fuzzy rules. The Heuristic Method (HM) introduced by Mamdani using control systems and human behavior modeling is the first method. Deterministic Method proposed by Takagi-Sugeno is fuzzy modeling and other method called self-learning fuzzy controller [13].

HM is generally preferred by an expert FLC designer or engineer because it requires engineering skills and system operating experience to create a fuzzy rule base [31]. For closed-loop speed control of a BLDC motor driver, rules and HM in IF-THEN format can be created as follows:

Rule1: IF E is A1 and dE is B1, THEN dU is C1↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

Rulen: IF E is An and dE is Bn, THEN dU is Cn

The n specified here refers to the number of the most recently created rule. E and dE represent error and variation of error, respectively. dU stands for the output of the FLC controller. An and Bn are linguistic expressions for the input variables and Cn is the linguistic output expression for the output variable.

It is possible to design the rules of the FLC based on expert human knowledge without the need for mathematical calculation of the system. In speed control of a BLDC motor, it is possible to establish FLC rules using a fixed reference speed signal. It is possible to select the appropriate ones to decrease or increase the BLDC motor speed to follow the reference speed among the rules. Creating rules in this way is accepted and practiced by many researchers [14 , 32].

There are several methods for building a rule base with HM [33]. The most preferred technique is the Phase Plane Trajectory (PPT) technique. The rules created by this technique are based on a closed-loop trajectory in the phase plane. FFT technique has been preferred in many areas [30, 34] and to create fuzzy rules in control systems. In this study, an FLC based on the PPT method was designed. The PPT method is preferred to fill the gap between time-response and rules. It is possible to easily create a rule base by following the general structure of the system. In this study, considering a rule base of 49 for BLDC motor drivers, the regions and orbital route [35] of the rules table based on the PPT technique are shown in Fig. 5. Zone 1 is the zone where it is responsible for the stability of the system. Zones 2 and 4 are the zone where the system is responsible for its sensitivity. There are five zones in total, with Zones 3 and 5 rarely activated by the system.

Fig. 5

Regions in the rule table and the changing route of the rule base.

Another of the core elements of FLC are MFs. Two MFs are used at the inputs and one MF at the output of the FLC. However, it is possible to use MFs of different shapes and sizes in FLC. As shown in Fig. 6, 3 different sizes of triangular MFs, 3×3, 5×5, and 7×7, are used in this article.

Fig. 6

Different triangular MFs, (a) 3×3, (b) 5×5, (c) 7×7.

In summary, it is possible to apply different approaches to create rule tables in fuzzy systems. Nevertheless, depending on the type of Mamdani FLC, heuristics based on PPT in particular are the most widely used and most suitable for hardware application.

4.2 T2 FLC

T2FLC was proposed as an extension of T1FLC, based on the phrase “Words can mean different things to different people” [34]. T2FLC is expressed as an extension of fuzzy systems first proposed by “Zadeh”. In the T2 fuzzy set, the membership degree is a normal decimal number that can vary in the range [0,1] for each point. Such sets are used when the membership functions are uncertain. The uncertainties can be in the form of MFs or the uncertainty of some membership parameters. For example, a Gaussian function with an indeterminate mean has a definite shape and indeterminate parameters.

T1 fuzzy systems have at least four sources of uncertainty:

The meanings of the words are used in the premise and the conclusion of the rules may be ambiguous. In other words, vocabulary has different concepts for different people.

They can create a graph of the findings, especially when result sets are constructed from a group of experts that are not fully agreed upon.

The inputs of the fuzzy system are likely to be electrically noisy and therefore may not clearly belong to an MF.

There is a possibility that the information and data used to set the parameters of a fuzzy system may be corrupt.

All of the above uncertainties arise with a fuzzy system and indeterminate MF. T1 fuzzy sets cannot be modeled directly on unpredictability because the MFs are not completely fuzzy. Since T2 fuzzy sets have fully fuzzy MFs; can model the above unpredictability. For that reason, modeling the unpredictability with the help of the controller will also reduce the effects of the uncertainties.

Interval Type-2 Membership functions (IT2MF) are used to express the unpredictability in their design. It consists of two MFs, upper and lower. The region between the upper and lower T1MF is called the Footprint of Uncertainty (FOU) [36]. The mathematical expression of IT2MF is expressed by Equation (14). $\tilde{A} = {((x, u), 1) | \forall x \in X, \forall u \in J_{x} \subseteq [0, 1]}$ (14)

A schematic representation of an IT2MF is shown in Fig. 7. As can be seen from Fig. 7, the administration of Mamdani FLC with IT2FLC is very similar to the administration of T1FLC. The difference between the two controllers is that the inference block is used with two separate fuzzy sets for the upper and lower fuzzy sets, and a new fuzzy set is used together in a block called type reduction. The block diagram of IT2FLC is shown in Fig. 8.

Fig. 7

IT2 MF.

Fig. 8

Mamdani IT2FLC.

Some samples for IT2 FLC have been extensively studied due to their specificity [37 –39]. Various variations of IT2 FLC have been reported and presented recently [38 –40].

5 Experimental studies

In this section, it starts with the experimental setup, which is first prepared to better understand the experiments performed. Secondly, the results of the input and output membership functions obtained by designing each controller in Matlab/M-file environment, simulation study and experiment results designed in Matlab/Simulink environment are expressed. Finally, based on these results, the results of T1FLC and IT2FLC are given comparatively.

5.1 Creation of experimental setup and controllers

The block diagram of the system designed to find the T1FLC and IT2FLC behaviors for the experimental setup is shown in Fig. 9.

For this study, FLC with two inputs and one output was used as error and error variation. Input and output MFs for T1FLC and IT2FLC were created with the results obtained with the code created through the Matlab M-file program. First, a controller consisting of 3 input and output membership functions was created. It consists of 9 rules and the rule table is presented in Table 1. Secondly, a controller consisting of 5 input and output membership functions was created. It consists of 25 rules and the rule table is shown in Table 2. Thirdly, a controller consisting of 7 input and output membership functions was created. It consists of 49 rules and the rule table is shown in Table 3. The created controller structures are shown in Figs. 10–12 as input and output membership functions, respectively.

Fig. 9

Block diagram of the experimental setup created.

Fig. 10

3×3 MF (a) T1FLC Input MF, (b) T1FLC Output MF, (c) IT2FLC Input MF, (d) IT2FLC Output MF.

Table 1

3×3 flc rule base

e∖de	N	Z	P
N	NB	NM	Z
Z	NM	Z	PM
P	Z	PM	PB

Table 2

5×5 flc rule base

e∖de	NB	NS	ZE	PS	PB
NB	NB	NB	NB	NS	ZE
NS	NB	NB	NS	ZE	PS
ZE	ZE	NS	ZE	PS	PB
PS	NS	ZE	PS	PB	PB
PB	ZE	PS	PB	PB	PB

Table 3

7×7 flc rule base

e∖de	NB	NM	NS	ZE	PS	PM	PB
NB	NB	NB	NB	NB	NM	NS	ZE
NM	NB	NB	NB	NM	NS	ZE	PS
NS	NB	NB	NM	NS	ZE	PS	PM
ZE	NB	NM	NS	ZE	PS	PM	PB
PS	NM	NS	ZE	PS	PM	PB	PB
PM	NS	ZE	PS	PM	PB	PB	PB
PB	ZE	PS	PM	PB	PB	PB	PB

Fig.11

5×5 MF (a) T1FLC Input MF, (b) T1FLC Output MF, (c) IT2FLC Input MF, (d) IT2FLC Output MF.

Fig. 12

7×7 MF (a) T1FLC Input MF, (b) T1FLC Output MF, (c) IT2FLC Input MF, (d) IT2FLC Output MF.

6 Simulation study and experiment results

To prove the effect of the designed FLC types, it was tested and evaluated by providing speed control of the BLDC motor. The parameters of the BLDC motor tested during the simulation period are given in Table 4. Motor response was investigated at various reference speed values using T1FLC and T2FLC. The scaling factors for both controllers were kept constant. Rule bases with 3 different membership degrees were created for both controllers. The model of the FLC system with an output and error versus error variation as input variable is shown in Fig. 13. The step rate response comparisons of T1FLC and IT2FLC, created with the 9, 25 and 49 rules designed as standard, were made to show their effects on speed performance depending on the FLC type.

Fig. 13

FLC structure used in experimental studies.

Table 4

BLDC motor parameters

Motor Parameters	Values
Number of poles	8
Number of phases	3
Stator Resistance	0.0095 Ω
Stator Inductance	0.0144 10^-3H
Rated speed of motor	3000 RPM
Rated torque of motor	0.095 N.m
Rotor inertia of motor	0.00019 kg.m²

In order to carry out experimental studies, a simulation program was designed over the Matlab/Simulink program. The created simulation program is shown in Fig. 14. The created programs were applied to the simulation setup shown in Fig. 14, respectively.

Fig. 14

Simulation program created in Matlab/Simulink program.

BLDC motor parameters used during the experiments are given in Table 4. In order to simulate variable road conditions, the load applied to the engine is applied at different values depending on time. The variable load graph designed for application to a BLDC motor is shown in Fig. 15.

The reference speed value required for the speed control of the motor was created as a step function in the simulation program. Three different reference values, 1000, 2000 and 2500 rpm, were created. The reference speed values created for 1000 rpm are shown in Fig. 16.

Fig. 15

Load value created in the simulation program.

Fig. 16

Reference speed value created in the simulation program for 1000 rpm.

The test results for T1FLC and IT2FLC formed with 3×3 MF, 5×5 MF and 7×7 MF are shown in Figs. 17–19, respectively. The numerical results obtained from the graphics are given in Table 5 comparatively.

Fig. 17

Speed-time graph of BLDC motor for 3×3 MF. (a) 1000 rpm, (b) 2000 rpm, (c) 2500 rpm.

Fig. 18

Speed-time graph of BLDC motor for 5×5 MF. (a) 1000 rpm, (b) 2000 rpm, (c) 2500 rpm.

Fig. 19

Speed-time graph of BLDC motor for 7×7 MF. (a) 1000 rpm, (b) 2000 rpm, (c) 2500 rpm.

Table 5

Comparison of controller performance results

	Types of controllers and MF’s
		3×3 MF		5×5 MF		7×7 MF
Characteristics of Dynamic Response	RPM	T1FLC	IT2 FLC	T1FLC	IT2 FLC	T1FLC	IT2 FLC
Rising Time (Tr) Sec	1000	0,04	0,021	0,033	0,026	0,028	0,027
	2000	0,04	0,022	0,033	0,031	0,030	0,026
	2500	0,04	0,023	0,031	0,028	0,030	0,026
Maximum Overshoot (Rpm)	1000	150	128	40	2	53	2
	2000	330	188	131	4	93	1
	2500	486	216	169	7	122	3
Setting Time (Ts) Sec	1000	0,12	0,070	0,10	0,067	0,078	0,033
	2000	0,10	0,073	0,067	0,048	0,070	0,032
	2500	0,15	0,086	0,088	0,05	0,082	0,036

When the speed graphs obtained during the simulation process are examined, it is seen that the T1FLC created with 3×3 MF exceeds 150 rpm over the 1000 rpm reference value and reaches the reference value after 120 ms. IT2FLC created with 3×3 MF, on the other hand, exceeded the 1000 rpm reference value by 128 rpm and reached the reference value after 70 ms. It is seen that the T1FLC created with 3×3 MF exceeds 330 rpm over the 2000 rpm reference value and reaches the reference value after 10 ms. IT2FLC created with 3×3 MF, on the other hand, exceeded the 2000 rpm reference value by 188 rpm and reached the reference value after 73 ms. It is seen that T1FLC created with 3×3 MF exceeds 486 rpm over 2500 rpm reference value and reaches reference value after 150 ms. It is seen that IT2FLC created with 3×3 MF exceeds 216 rpm over the 2500 rpm reference value and reaches the reference value after 86 ms.

It is seen that T1FLC created with 5×5 MF exceeds 40 rpm over 1000 rpm reference value and reaches reference value after 100 ms. IT2FLC created with 5×5 MF, on the other hand, exceeded the 1000 rpm reference value by 2 rpm and reached the reference value after 67 ms. It is seen that the T1FLC created with 5×5 MF exceeded 131 rpm over the 2000 rpm reference value and reached the reference value after 67 ms. IT2FLC created with 5×5 MF, on the other hand, exceeded the 2000 rpm reference value by 4 rpm and reached the reference value after 48 ms. It is seen that the T1FLC created with 5×5 MF exceeded 169 rpm over the 2500 rpm reference value and reached the reference value after 88 ms. IT2FLC created with 5×5 MF, on the other hand, exceeded the 2500 rpm reference value by 7 rpm and reached the reference value after 50 ms.

It is seen that T1FLC created with 7×7 MF exceeds 53 rpm over 1000 rpm reference value and reaches reference value after 78 ms. IT2FLC created with 7×7 MF, on the other hand, exceeded the 1000 rpm reference value by 2 rpm and reached the reference value after 33 ms. It is seen that the T1FLC created with 7×7 MF exceeded 93 rpm over the 2000 rpm reference value and reached the reference value after 70 ms. IT2FLC created with 7×7 MF, on the other hand, exceeded the 2000 rpm reference value by 1 rpm and reached the reference value after 32 ms. It is seen that the T1FLC created with 7×7 MF exceeded 122 rpm over the 2500 rpm reference value and reached the reference value after 82 ms. IT2FLC created with 7×7 MF, on the other hand, exceeded the 2500 rpm reference value by 3 rpm and reached the reference value after 36 ms. All experimental results found are given in Table 5 comparatively.

7 Conclusions and argument

This article explored the design and operation of different T1FLC variations for speed control of BLDC motor drives. Its use has been increasing rapidly since the discovery of T1FLC in the 1970 s. It is used in many applications, including high performance motor drives. T1FLC’s ability to account for many uncertainties is one of its key advantages. The fact that it can easily control uncertain and very complex systems has made it the preferred controller in applications. In BLDC motor drives, the T1FLC is first used as a speed controller where the motor speed is compared to the reference speed. In BLDC motor drives, the T1FLC is first used as a speed controller where the motor speed is compared to the reference speed. The resulting fault is applied as input to T1FLC and output control signal is generated. FLC has different applications. Besides, the Mamdani T1FLC variant is the most widely used FLC. However, in some cases, T1FLC is insufficient in calculating uncertainties. In these cases, IT2FLC is used, which calculates the FOU value. A rule base was created using the step response of the second-order BLDC motor driver system and the PPT technique. This rule is in three different ways; It is designed to consist of 9, 25 and 49 rules. Controllers created with three different rule bases for T1FLC and IT2FLC were designed and their performances were examined. Performance evaluation was carried out in the simulation program designed in Matlab/Simuklink environment. With the results found, performance evaluation was carried out depending on the number of rules and controller type.

Finally, it was seen that the controller created with 49 rules improved the driver performance. It has been observed that IT2FLC reaches the reference value faster than T1FLC and does not overshoot over the reference value. In this way, both the driver performance is increased and the absence of sudden speed fluctuations protects the driver and motor health. The system has been provided to work more efficiently. Increasing the number of rules and MFs had a positive impact on supervisor performance. It was observed in the results that IT2FLC, which was created with the same number of rules and MF, worked more stable than T1FLC. In the next work, an optimization algorithm can be proposed to further improve the performance of IT2FLC by changing the FOU value.

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