Design of a Robust Controller for Induction Motor Drive Systems Based on Extendable Fuzzy Theory

Abstract

In this paper, an extendable fuzzy robust speed controller suitable for induction motor drive systems was proposed. Firstly, the two-degrees-of-freedom (2DOF) robust control technology with feedforward control and disturbance elimination method was adopted. Upon parameter variation and load disturbance, the motor drive system could utilize a robust controller to generate compensation signals and reduce the impact on the controlling performance of the motor drive system. The magnitude of the compensation signal was adjusted via the weighting factor. However, should a fixed weighting factor be adopted, system instability might be generated easily when time delay and saturation of control force occur. Based on the above, the smart method of extendable fuzzy theory (EFT) was adopted in this paper to adjust adequate weighting factors, where the controlling performance of the induction motor drive system could be improved accordingly. Lastly, the simulation software Matlab/Simulink (R2023b version) was applied to simulate the utilization of the controlling method proposed for the induction motor drive system. The simulation results proved that the extendable fuzzy robust speed controller proposed provided better speed tracking and load regulation-controlling performance than the conventional robust controller.

1. Introduction

The design for a high-performance motor drive shall be equipped with numerous critical characters including fast speed tracking with zero overshoot, minimum speed drop and recovery time under load variation, zero steady-state error in speed tracking and load regulation, as well as the controlling performances not influenced by parameter variations. In terms of performance, the conventional proportional-plus-integral (P-I) controller [1] cannot execute tracking and load regulation at the same time, while the two-degrees-of-freedom (2DOF) controller integrating a feedforward controller and P-I controller [2] can improve the overall control performance. However, the parameters of the 2DOF controller are difficult to design and often obtainable with the trial-and-error method, which is time-consuming and inefficient. Therefore, the parameters of the 2DOF speed controller could be quantitatively designed by adopting the method proposed in [3] for the quantitative design of the controller. To solve the impact of parameter uncertainty on control performance, the robust control technology with disturbance elimination method could be introduced at the same time [4]. Upon variations in system parameters or load disturbance, the robust controller could generate compensation signals to eliminate the impact on motor-controlling performance and improve the robustness of the control system. The magnitude of compensation signals could be adjusted by the weighting factor w, which could offset the influence of the disturbance signal when w = 1 under ideal circumstances [4]. In practice, however, the greater the w, the greater the controlling force required. Thus, system instability could be induced easily. Therefore, the smart method of extendable fuzzy theory (EFT) was adopted in this paper [5]. Under different working conditions, such a method could automatically select suitable weighting factors and improve the controller performance of the induction motor drive system. The extendable fuzzy robust speed controller proposed was characterized by the speed difference between motor speed command and actual speed, as well as the variation rate for such speed difference, where a suitable weighting factor w was selected with EFT. Firstly, the range of two characteristic values, i.e., the speed difference and the variation rate for such a speed difference were divided into 20 categories for establishing the classical domain and neighborhood domain of the EFT model, where the weighting values of these two characteristics were set. The correlation degree between each speed difference and variation rate for such a speed difference in 20 categories could be derived after calculation. The weighting factor corresponding to the category with the maximum correlation degree was exported for adjusting the compensation signals of the controller. This method would provide good response performance in terms of speed tracking and load regulation for the induction motor drive system. In this paper, the simulation results were utilized to compare the extendable fuzzy robust speed controller proposed with the conventional robust controller on response performance in terms of speed tracking and load regulation. The main contribution of this paper is the use of an extendable fuzzy robust controller to automatically adjust the observer’s gains. This prevents the motor’s rotational speed from overshooting or system instability caused by excessive control force applied or controller output saturation.

2. The Field-Oriented Architecture of Induction Motor System

The core principle of field-oriented control (FOC) [6] was to convert the induction motor voltage and flux equation via coordinate conversion [7] from a three-phase rotating coordinate system to a two-axis synchronous coordinate system, which served to simplify the mathematical model of the induction motor [8,9] and reduce the complexity of the controller design further. Through feedback of parameters such as motor speed, voltage, and current, the system achieved the purpose of command tracking and steady-state error elimination with the speed controller. The space vector pulse width modulation (SVPWM) [10] technology was then applied to control the on-off function of the power semiconductor of the inverter [11].

2.1. Dynamic Equation of FOC

The purpose of FOC was to realize the control of the induction motor per an excited DC motor [12], which could control the magnetic field and torque independently. The dynamic equations of the squirrel cage induction motor under a synchronous rotational reference frame can be expressed as Equations (1)–(4), respectively [8,9].

$$v_{ds}^e=R_s{i}_{ds}^e+L_{\sigma }{\displaystyle \frac{d{i}_{ds}^e}{dt}}-{\omega }_e{L}_{\sigma }{i}_{qs}^e+{\displaystyle \frac{L_m}{L_r}}{\displaystyle \frac{d{\varphi }_{dr}^e}{dt}}-{\displaystyle \frac{{\omega }_e{L}_m}{L_r}}{\varphi }_{qr}^e$$

(1)

$$v_{qs}^e={\omega }_e{L}_{\sigma }{i}_{ds}^e+R_s{i}_{qs}^e+L_{\sigma }{\displaystyle \frac{d{i}_{qs}^e}{dt}}+{\displaystyle \frac{{\omega }_e{L}_m}{L_r}}{\varphi }_{dr}^e+{\displaystyle \frac{L_m}{L_r}}{\displaystyle \frac{d{\varphi }_{qr}^e}{dt}}$$

(2)

$$0=-R_r{L}_m{i}_{ds}^e+R_r{\varphi }_{dr}^e+L_r{\displaystyle \frac{d{\varphi }_{dr}^e}{dt}}-({\omega }_e-{\omega }_r)L_r{\varphi }_{qr}^e$$

(3)

$$0=-R_r{L}_m{i}_{qs}^e+({\omega }_e-{\omega }_r)L_r{\varphi }_{dr}^e+R_r{\varphi }_{qr}^e+L_r{\displaystyle \frac{d{\varphi }_{qr}^e}{dt}}$$

(4)

Equations (1)–(4) are the voltage equations for the stator and rotor of the induction motor, where $R_s$, $R_r$, $L_s$, $L_r$, $L_m$, ${\omega }_e$, and ${\omega }_r$ are stator resistance, rotor resistance, stator inductance, rotor inductance, mutual inductance of motor winding, synchronous angular velocity, and electrical angular velocity of rotor, respectively, while $i_{ds}^e$ and $i_{qs}^e$ are the axial current d and q of stator with synchronous coordinates, ${\varphi }_{dr}^e$ and ${\varphi }_{qr}^e$ are the axial flux d and q of the rotor with synchronous coordinates, and $L_{\sigma }\triangleq L_s-L_m^2/L_r$ is the leakage inductance.

The electromagnetic torque and mechanical equation for motor can be expressed as Equation (5).

$$T_e={\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m}{L_r}}(i_{qs}^e{\varphi }_{dr}^e-i_{ds}^e{\varphi }_{qr}^e)=T_L+J_m{\displaystyle \frac{d{\omega }_{rm}}{dt}}+B_m{\omega }_{rm}$$

(5)

In Equation (5), $T_L$ is the load torque, $J_m$ is the moment of inertia for the motor, and $B_m$ is the viscosity coefficient for the motor. The relationship formula electrical angular velocity ${\omega }_r$ and mechanical angular velocity ${\omega }_{rm}$ were ${\omega }_{rm}=(2/P){\omega }_r$,while $P$ is the motor pole number.

Should axis d under synchronous rotation coordinates be aligned with the rotor flux links, all the rotor fluxes would fall on axis d, i.e., ${\varphi }_{dr}^e={\varphi }_r$, flux ${\varphi }_{qr}^e$ of axis q would then be zero. Thus, Equations (1)–(4) can be modified to Equations (6)–(9) [9].

$${\displaystyle \frac{d{i}_{ds}^e}{dt}}=(-{\displaystyle \frac{R_s}{L_{\sigma }}}-{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r}})i_{ds}^e+{\omega }_e{i}_{qs}^e+{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r{L}_m}}{\varphi }_r+{\displaystyle \frac{v_{ds}^e}{L_{\sigma }}}$$

(6)

$${\displaystyle \frac{d{i}_{qs}^e}{dt}}=-{\displaystyle \frac{R_s}{L_{\sigma }}}{i}_{qs}^e-{\omega }_e{i}_{ds}^e-{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r{L}_m}}{\omega }_e{\varphi }_r+{\displaystyle \frac{v_{qs}^e}{L_{\sigma }}}$$

(7)

$${\displaystyle \frac{d{\varphi }_r}{dt}}=-{\displaystyle \frac{R_r}{L_r}}{\varphi }_r+R_r{\displaystyle \frac{L_m}{L_r}}{i}_{ds}^e$$

(8)

$$-R_r{\displaystyle \frac{L_m}{L_r}}{i}_{qs}^e+{\omega }_{sl}{\varphi }_r=0$$

(9)

In Equations (6)–(9) of ${\tau }_r=L_r/L_m$, stator leakage inductance is $L_{\sigma }=\sigma L_s$, the leakage inductance coefficient is $\sigma =1-L_m^2/L_s{L}_r$, and the slip speed is ${\omega }_{sl}={\omega }_e-{\omega }_r$. Since ${\varphi }_{qr}^e=0$, with substitution into the electromagnetic torque Equation (5), it could be modified to Equation (10). Thus, the stator current command for axis q can be derived per Equation (11).

$$T_e={\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m}{L_r}}(i_{qs}^e{\varphi }_{dr}^e)$$

(10)

$${i_{qs}^e}^{*}={\displaystyle \frac{4}{3P}}{\displaystyle \frac{L_r}{L_m{{\varphi }_{dr}^e}^{*}}}{T_e}^{*}$$

(11)

From the relationship between torque ${T_e}^{*}$ and current ${i_{qs}^e}^{*}$ in Equation (11), the torque constant $k_t$ can be derived per Equation (12) [8,9].

$$k_t={\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m{{\varphi }_{dr}^e}^{*}}{L_r}}$$

(12)

The flux command is the nominal flux for the motor, but since the actual motor rotor flux cannot not be measured directly, the values are derived from estimation. The estimated flux ${\widehat{\varphi }}_{dr}^e$ can be derived by conducting the Laplace transform in Equation (8), which is shown as Equation (13) [8,9].

$${\widehat{\varphi }}_{dr}^e={\displaystyle \frac{L_m{i}_{ds}^e}{\frac{L_r}{R_r}s+1}}$$

(13)

Among them, ${\widehat{\varphi }}_{dr}^e$ is the estimated flux, and s is the Laplace operator.

Equations (6) and (7) include the ordinary differential equation (ODE) and nonlinear coupling for axial current d and q. After the Laplace transform [13] on ODE, the transfer function for the current of axis d and q can be derived. Among them, the command for axial current q is in Equation (11). After introducing the P-I controller for current control, the systematic block diagram for the control loop of axial current d and q can be constructed. Regarding the control loop for rotor flux, the error from subtracting rotor flux and rotor flux command estimated with Equation (13) can be utilized to achieve flux control by forming control loop through the P-I controller. In the speed control loop, the motor mechanical equation in Equation (5) was acquired via Laplace transform $1/(J_{m}s+B_m)$, where the speed control could be realized through the P-I controller. Summing up the above, the block diagram of an induction motor with FOC consisting of current, flux, and speed control loops is shown in Figure 1 [8].

2.2. Sensorless FOC System for Induction Motors

Figure 1 displays the fundamental architecture of FOC for the induction motor drive system [8]. However, the actual system still requires coordinate conversion, feedforward compensation of decoupling [14], and pulse width modulation (PWM) of the inverter. Moreover, since the sensorless direct FOC strategy was adopted in this paper, the present flux and speed should be acquired via the flux estimator [15] and speed estimator [16]. Summing up the above, the complete architecture of sensorless FOC for the induction motor drive system is shown in Figure 2 [8].

3. The Design of Robust 2DOF Controller with EFT Proposed

If the speed loop controller in FOC only adopted the use of a P-I controller, accommodating both speed tracking and load regulation performance would be difficult. Therefore, the 2DOF controller architecture was adopted in this paper [17]. The control architecture integrated a feedforward controller and robust control technology to offset interference through compensation signals. The weight factor was then dynamically adjusted with EFT to determine the magnitude of the compensation signal. This effectively reduced the impact caused by system parameter variation and load disturbances, as well as improved the overall control performance further.

3.1. Quantitative Design of 2DOF Speed Controller

Figure 3 displayed the block diagram for the 2DOF speed controlling architecture of the induction motor drive system [17]. A feedforward controller was adopted to achieve the effect of 2DOF control, and the transfer function of feedforward controller $G_f(s)$ is shown in Equation (14).

$$G_f(s)={\displaystyle \frac{n_{1}s+n_0}{m_{1}s+m_0}}$$

(14)

where n₀, n₁, m₀, and m₁ are the parameters of the forward controller of the two-degrees-of-freedom (2DOF) controller.

Furthermore, the transfer function of the P-I controller can be expressed with Equation (15).

$$G_s(s)={\displaystyle \frac{K_{p}s+K_i}{s}}$$

(15)

The transfer function of the mechanical equation for motor can be expressed with Equation (16).

$$G_m(s)={\displaystyle \frac{1}{J_{m}s+B_m}}={\displaystyle \frac{b}{s+a}}$$

(16)

Among them, $a=B_m/J_m$ and $b=1/J_m$. The torque constant $k_t$ in Equation (12) was integrated with the speed controller $G_s(s)$ in Equation (15) and became Equation (17).

$$G_s^{\prime }(s)=k_t\cdot G_s(s)={\displaystyle \frac{K_p^{\prime }s+K_i^{\prime }}{s}}$$

(17)

Among them, $K_p^{\prime }=k_t{K}_p$ and $K_i^{\prime }=k_t{K}_i$.

Since there was a lower correlation between the feedforward controller $G_f(s)$ and the load variation, this controller was suitable for completing speed command tracking, while the speed controller $G_s^{\prime }(s)$ was used for load regulation. The design for the feedforward controller $G_f(s)$ kept the numerator of the original transfer function and offset the numerator of the transfer function in $\Delta {\omega }_d$ against $\Delta {\omega }_{rm}$ [17], which enabled characteristic of well speed command following.

The performance of the motor drive system was crucial in the controller design, which was not only relevant to the performance of the system’s stability and accuracy of speed command tracking, but also directly impacted the characteristics of dynamic response for the system. In view of this, to ensure the rationality and validity of the controller design, it was necessary to specify the performance requirements of the motor drive system precisely. The four critical performance metrics of the motor drive system are listed below, which serve as the foundation of the quantitative design controller [3,17] to improve the overall control performance of the system.

(1)

The steady-state error of step response for speed command and load disturbance was zero [3,17].

(2)

There was no overshoot in step response for speed command [18,19].

(3)

The response time for the step command following was defined as the $90\%$ time required $t_r$= 0.15 s for the response to rise from zero to its final value.

(4)

The maximum speed drop caused by variation in the unit step load was $\Delta {\omega }_{dm}$ = 30 rpm.

Summing up the four performance metrics above, four nonlinear equations can be established. Then, nonlinear equations can be solved with Matlab software (R2023b version). The parameters and variables of the feedforward controller $G_f(s)$ and the speed controller $G_s(s)$ can then be derived, realizing the compliance of performance required for controller design [3,17].

3.2. Robust 2DOF Controller with Fixed Weighting Factor

Based on the 2DOF speed controller designed in Section 3.1, although it provided good capability in speed command tracking and load regulation, the original control performance would not be maintained if there were variations in system parameters. To improve such a problem, the robust controller architecture for disturbance elimination as shown in Figure 4 [17] was adopted in this paper. In case of zero variation in system parameters, the compensation signal of the robust controller was zero. Therefore, there was no impact on the system. Once the system parameters or load varied, the robust controller generated a corresponding compensation signal to reduce the impact from disturbance against the system. This maintained the control performance and stability of the system. The magnitude of compensation signal was determined by the weighting factor. Ideally, the disturbance would be offset completely when w = 1. However, ideal results may not be achieved in actual situations due to factors such as excessive controlling force. The control performance of such a robust controller is verified through simulation results below.

Firstly, the moment of inertia $J_m$ and viscosity coefficient $B_m$ are defined, respectively, in Equations (18) and (19). Among them, ${\overline{J}}_m$ and ${\overline{B}}_m$ represent the moment of inertia and viscosity coefficient under normal conditions, while $\Delta J_m$ and $\Delta B_m$ are the corresponding changes, respectively. Moreover, it was assumed that the torque constant was not affected by parameter variation; thus, $k_t={\overline{k}}_t$.

$$J_m={\overline{J}}_m+\Delta J_m$$

(18)

$$B_m={\overline{B}}_m+\Delta B_m$$

(19)

According to the system architecture shown in Figure 4, Mason’s gain formula can be utilized to derive transfer functions $G_{dr}^{*}(s)$ and $G_{dd}^{*}(s)$ [17] of the speed command tracking and load regulation, which are shown in Equations (20) and (21), respectively.

$$\begin{gathered} G_{dr}^{*}(s)={\displaystyle \frac{\Delta {\omega }_{rm}}{\Delta {\omega }_d^{*}}}\left|\begin{array}{l}\\ \Delta T_L=0\end{array}\right. \\ ={\displaystyle \frac{G_f(s)G_s(s)k_t{G}_m(s)}{1-\left(w-k_t{G}_s(s)G_p(s)-w{k}_t{G}_p(s)G_c(s)\right)}} \end{gathered}$$

(20)

$$\begin{gathered} G_{dd}^{*}(s)={\displaystyle \frac{\Delta {\omega }_{rm}}{\Delta T_L}}\left|\begin{array}{l}\\ \Delta {\omega }_{rm}^{*}=0\end{array}\right. \\ ={\displaystyle \frac{G_m(s)(1-w)}{1-\left(w-k_t{G}_s(s)G_p(s)-w{k}_t{G}_p(s)G_c(s)\right)}} \end{gathered}$$

(21)

By substituting Equations (18) and (19) into Equations (20) and (21), the simplified and equivalent block diagram of transfer function can be derived as shown in Figure 5 [17].

The setting range of weighting factors was between 0 and 1. In theory, when the weighting factor is set at 1, the amount of disturbance $\Delta J_m$, $\Delta B_m$ and the load variation $\Delta T_L$ in the block diagram as per Figure 5 are compensated fully. The control performance of the induction motor drive system would be the same as the 2DOF speed control system of the original design, but not affected by variations in the parameters and load of the motor. However, if complete suppression of disturbance is intended and the weighting factor is set at 1, once nonlinear factors such as delay or the control force limiter exist in the system, the control force required would be extremely large, which may easily cause system instability. In view of this, it was advised not to set the weighting factor too high. Instead, the trade-off between control performance and control force should be considered comprehensively to reduce the impact on the system. In the content below, the impact of different weighting factors on the control performance of the system when the moment of inertia $J_m$ varies is explored first. The simulation results serve as a basis of reference for adjusting weighting factors with EFT. Figure 6 displays the waveform obtained from the simulation for the response of motor speed command tracking and current with different weighting factors when $J_m=10{\overline{J}}_m$, where control force saturation and system delay conditions are considered under variations in the speed step command $\Delta {\omega }_{rm}^{*}$ = 100 rpm. Figure 7 displays the waveform obtained from simulation for the response of the motor drive system with different weighting factors when $J_m=10{\overline{J}}_m$ and the load variation is $\Delta T_L=2\mathrm{N}\cdot \mathrm{m}$, where control force saturation and system delay conditions are considered.

Additionally, the nominal case labeled “e” in the response waveform in the simulation results in Figure 6a and Figure 7a demonstrates that the system’s steady-state error has been completely and effectively removed. Furthermore, the step response of speed command tracking shows no overshoot. As a result, the controller’s design is said to meet the requirements under both conditions.

As reflected in simulation results in Figure 6 and Figure 7, under variations in moment of inertia, while control force saturation and delay exist in the system, instability and excessive current are present in the system when the weighting factor w = 0.93. Thus, the increase in the weighting factor w could improve the response of speed command tracking and the performance of load regulation. However, there is a possibility of system instability. As a result, to ensure system stability, there should be a balance between the control performance and control force; hence, the adequate adjustment of weighting factor w to optimize the overall control performance. To solve the problems mentioned above, an adjustment of weighting factors with EFT for a robust 2DOF controller was proposed in this paper. The block diagram of the system architecture is shown in Figure 8 [20]. Among them, the reference model is the transfer function between speed command input $\Delta {\omega }_{rm}^{*}$ and speed output $\Delta {\omega }_{rm}$.

3.3. Extendable Fuzzy Theory (EFT)

In the Cantor set, an element either belongs to or does not belong to a set. Therefore, the range of the Cantor set {0, 1} can be used to solve a two-valued problem. In contrast to the standard set, the fuzzy set allows for describing concepts in which the boundary is not explicit. It concerns not only whether an element belongs to the set but also to what degree it belongs. The range of a fuzzy set is [0, 1]. However, the extendable fuzzy set extends the fuzzy set from [0, 1] to ($-\infty ,\infty$). As a result, it allows us to define a set including any data in the domain [19]. The purpose of EFT is to explore the extension capability in matters, where analysis and discussion can be conducted from qualitative and quantitative perspectives to solve the problem of contradictions in matters. Such a theory was based on two foundations, namely the matter–element theory and extension mathematics. The former focused on the possibility of variation in matters, together with properties of matter–element conversion and matter–element transformation; the latter applied the extension set and correlation function as the computational core [21]. Specifically, EFT represented the message from matters via the matter–element model, and stipulated the matter relationships between quality and quantity via matter–element conversion. Through identification of correlation functions, the impact of quality and quantity on matter was analyzed, where the level of correlation between matter characteristics was clarified. Comparisons of the standard sets, fuzzy sets, and extendable fuzzy sets are shown in

Table 1. Three different sorts of mathematical sets.

Compared Item	Cantor Set	Fuzzy Set	Extendable Fuzzy Set
Research objects	Data variables	Linguistic variables	Contradictory problems
Model	Mathematics model	Fuzzy mathematics model	Matter–element model
Descriptive function	Transfer function	Membership function	Correlation function
Descriptive property	Precision	Ambiguity	Extension
Range of set	${\mathrm{C}}_{\mathrm{S}}(\mathrm{x})$ $\in$ {0, 1}	${\mu }_{\mathrm{S}}(\mathrm{x})$ $\in$ [0, 1]	${\mathrm{K}}_{\mathrm{S}}(\mathrm{x})$ $\in$ (−∞, ∞)

.

3.3.1. Extendable Fuzzy Matter–Element Model

EFT processed problems based on the matter–element model. Should the structure of the matter–element model be represented with a mathematical function, it can be expressed per Equation (22) [22].

$$R=(N,C,V)$$

(22)

Among them, R is the basic element to describe matter (i.e., matter–element), while N, C, and V are the three elements constituting the matter–element model. N represents the name of the matter, C represents the characteristic of the matter, and V represents the characteristic value of the matter.

When matter N provides numerous characteristics points, it can be expressed through n characteristics $c_1$, $c_2$, …, $c_n$ and the corresponding characteristic value $v_1$, $v_1$, …, $v_n$; thus, the matter–element model can be represented per Equation (23).

$$R=\left[\begin{array}{l}{R}_1\\ R_2\\ ⋮\\ R_n\end{array}\right]=\left[\begin{array}{lll}N,& c_1,& v_1\\ & c_2,& v_2\\ & ⋮& ⋮\\ & c_n,& v_n\end{array}\right]$$

(23)

3.3.2. Definition of Classical Domain and Neighborhood Domain of EFT

In case the characteristic value is a range, such a range can be referred to as the classical domain $F_0=<a,b>$ and contained in the neighborhood domain $F=<d,e>$, i.e., $F_0\in F$, where point f is any point on interval F. The corresponding matter–element $R_0$ for $F_0=<a,b>$ can be expressed as in Equation (24).

$$R_0=(F_0,c_i,v_i)=\left[\begin{array}{lll}\begin{array}{l}{F}_0,\\ \\ \\ \end{array}& \begin{array}{c}{c}_1,\\ c_2,\\ ⋮\\ c_n\end{array}& \begin{array}{c}<a_1,b_1>\\ <a_2,b_2>\\ ⋮\\ <a_n,b_n>\end{array}\end{array}\right]$$

(24)

In Equation (24), c_i is the characteristic of $F_0$, and $v_i$ is the characteristic value of $c_i$, i.e., the classical domain. However, for $F=<d,e>$, the corresponding matter–element $R_F$ can be expressed as in Equation (25). Among them, $c_j$ is the characteristic of F, and $v_j$ is the characteristic value of $c_j$, i.e., the neighborhood domain.

$$R_F=(F,c_j,v_j)=\left[\begin{array}{lll}\begin{array}{l}F,\\ \\ \\ \end{array}& \begin{array}{c}{c}_1,\\ c_2,\\ ⋮\\ c_n\end{array}& \begin{array}{c}<a_1,b_1>\\ <a_2,b_2>\\ ⋮\\ <a_n,b_n>\end{array}\end{array}\right]$$

(25)

3.3.3. Distance and Rank Value

In classical mathematics, the calculation of distance is based on the absolute value of the difference between points. In EFT, however, the relationship between certain point f and the interval $F_0=<a,b>$ in the real domain is presented in function format, which is expressed as Equation (26).

$$\rho (f,F_0)=\left|f-{\displaystyle \frac{a+b}{2}}\right|-{\displaystyle \frac{b-a}{2}}$$

(26)

Other than the consideration of correlations between points and intervals, the point–interval or interval–interval relationships also require consideration. Therefore, $F_0=<a,b>$ and $F=<d,e>$ are set as two intervals in the real domain, respectively, and interval $F_0$ is within interval $F$. The location values of point f, interval $F_0$, and interval $F$ can be expressed as Equation (27).

$$D(f,F_0,F)=\left\{\begin{array}{c}\rho (f,F)-\rho (f,F_0)\\ -1\end{array}\right.\begin{array}{c},f\notin F_0\\ ,f\in F_0\end{array}$$

(27)

3.3.4. Correlation Function

The correlation function $K(f)$ was composed by dividing the distance from Equation (26) by the location value from Equation (27), which can be expressed as Equation (28).

$$K(f)={\displaystyle \frac{\rho (f,F_0)}{D(f,F_0,F)}}$$

(28)

Among them, the correlation function contains its maximum value when $f=(a+b)/2$, which is referred to as the elementary correlation function and is illustrated in Figure 9. In addition, $K(f)<-1$ represents that point f is out of the interval F, $K(f)<1$ represents that point f is within the interval $F_0$, while $-1<K(f)<0$ represents that the point falls in the extension domain.

3.4. Selection of Variable Weighting Factor Characteristics for Induction Motor Drive System

In Section 3.2, the simulation results indicated that upon the existence of nonlinear factors and parameter variation in the system, the fixed weighting factors could cause system instability and oscillation easily. To allow the weight factor of the 2DOF controller for the induction motor drive system to be adjusted along with speed variation, where the balance between control performance and control force could be obtained for better effect in control, the EFT-based smart control strategy was adopted in this paper for dynamic adjustment of the weighting factors. Such a control strategy divided the speed difference between the induction motor speed within the actual speed range (0 rpm to 1680 rpm) and its speed command $e(n)\triangleq {\omega }_{rm}^{*}(n)-{\widehat{\omega }}_{rm}(n)$, and the variation rate for such speed difference $\dot{e}\triangleq e(n+1)-e(n)$ into 20 categories. From Figure 10, it can be observed that the command error in speed difference was greater in categories Z1~Z4; hence, the greater oscillation in amplitude. The command error in speed difference was less in categories Z17~Z20; hence, there was less oscillation in amplitude. Taking the Z1 category for example, although the speed error continued to increase, the rate of speed change reduced progressively till such a rate became zero at point M (i.e., $\dot{e}\cdot =\cdot 0$), yet the speed error $e$ reached the maximum value. Therefore, the weighting factor should increase for enhancing the compensation amount, which reduces the impact of parameter variation. The analysis for other categories was implemented in a similar manner. Summing the above with results from the dynamic analysis in Figure 10, the matter–element models of classical domains in 20 categories were established according to extendable matter–element theory. The corresponding variation amount in weighting factor $\Delta w$ is shown in

Table 2. Extendable fuzzy matter–element models in 20 categories and variation amount in corresponding weight factor $\Delta w$.

Category	Extendable Fuzzy Matter–Element Model	$\mathbf{Variation}\mathbf{Amount}\mathbf{in}\mathbf{Weighting}\mathbf{Factor}\mathbf{\Delta }\mathit{w}$
Z1	$F_1=\left[\begin{array}{l}\begin{array}{lll}{F}_1& e& <0,1600>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,96385>\end{array}\end{array}\right]$	−0.3
Z2	$F_2=\left[\begin{array}{l}\begin{array}{lll}{F}_2& e& <0,1600>\end{array}\\ \begin{array}{lll}& \dot{e}& <-96385,0>\end{array}\end{array}\right]$	−0.3
Z3	$F_3=\left[\begin{array}{l}\begin{array}{lll}{F}_3& e& <\text{−}1600,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <\text{−}96385,0>\end{array}\end{array}\right]$	0.3
Z4	$F_4=\left[\begin{array}{l}\begin{array}{lll}{F}_4& e& <-1600,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,96385>\end{array}\end{array}\right]$	0.3
Z5	$F_5=\left[\begin{array}{l}\begin{array}{lll}{F}_5& e& <0,1300>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,78313>\end{array}\end{array}\right]$	−0.25
Z6	$F_6=\left[\begin{array}{l}\begin{array}{lll}{F}_6& e& <0,1300>\end{array}\\ \begin{array}{lll}& \dot{e}& <-78313,0>\end{array}\end{array}\right]$	−0.25
Z7	$F_7=\left[\begin{array}{l}\begin{array}{lll}{F}_7& e& <-1300,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <-78313,0>\end{array}\end{array}\right]$	0.25
Z8	$F_8=\left[\begin{array}{l}\begin{array}{lll}{F}_8& e& <-1300,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,78313>\end{array}\end{array}\right]$	0.25
Z9	$F_9=\left[\begin{array}{l}\begin{array}{lll}{F}_9& e& <0,1000>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,60240>\end{array}\end{array}\right]$	−0.15
Z10	$F_{10}=\left[\begin{array}{l}\begin{array}{lll}{F}_{10}& e& <0,1000>\end{array}\\ \begin{array}{lll}& \dot{e}& <-60240,0>\end{array}\end{array}\right]$	−0.15
Z11	$F_{11}=\left[\begin{array}{l}\begin{array}{lll}{F}_{11}& e& <-1000,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <-60240,0>\end{array}\end{array}\right]$	0.15
Z12	$F_{12}=\left[\begin{array}{l}\begin{array}{lll}{F}_{12}& e& <-1000,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,60240>\end{array}\end{array}\right]$	0.15
Z13	$F_{13}=\left[\begin{array}{l}\begin{array}{lll}{F}_{13}& e& <0,700>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,42168>\end{array}\end{array}\right]$	−0.1
Z14	$F_{14}=\left[\begin{array}{l}\begin{array}{lll}{F}_{14}& e& <0,700>\end{array}\\ \begin{array}{lll}& \dot{e}& <\text{−}42168,0>\end{array}\end{array}\right]$	−0.1
Z15	$F_{15}=\left[\begin{array}{l}\begin{array}{lll}{F}_{15}& e& <-700,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <-42618,0>\end{array}\end{array}\right]$	0.1
Z16	$F_{16}=\left[\begin{array}{l}\begin{array}{lll}{F}_{16}& e& <-700,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,42168>\end{array}\end{array}\right]$	0.1
Z17	$F_{17}=\left[\begin{array}{l}\begin{array}{lll}{F}_{17}& e& <0,400>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,24029>\end{array}\end{array}\right]$	−0.01
Z18	$F_{18}=\left[\begin{array}{l}\begin{array}{lll}{F}_{18}& e& <0,400>\end{array}\\ \begin{array}{lll}& \dot{e}& <-24096,0>\end{array}\end{array}\right]$	−0.01
Z19	$F_{19}=\left[\begin{array}{l}\begin{array}{lll}{F}_{19}& e& <-400,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <-24096,0>\end{array}\end{array}\right]$	0.01
Z20	$F_{20}=\left[\begin{array}{l}\begin{array}{lll}{F}_{20}& e& <-400,0>\end{array}\\ \begin{array}{lll}& \dot{e}& <0,24096>\end{array}\end{array}\right]$	0.01

. The neighborhood domain established with the maximum and minimum value of the classical domain for each characteristic is shown in Equation (29).

$$R_F=(F,c_j,v_j)\left[\begin{array}{l}\begin{array}{lll}F& e& <-1600,1600>\end{array}\\ \begin{array}{lll}& \dot{e}& <-96385,96385>\end{array}\end{array}\right]$$

(29)

3.5. Procedures of EFT-Based Dynamic Adjustment of Weighting Factors

To achieve the purpose of dynamically adjusting the weighting factor of the robust speed controller for the induction motor drive system, the speed difference between the motor speed command and the actual speed, as well as the variation rate for such a speed difference, served as the two characteristics in this paper. Through the distance and rank value in EFT, the correlation degree with each category was calculated. The greatest correlation was selected and classified as the corresponding characteristic category for determining a suitable weighting factor. The procedures of determining weighting factors with EFT are described in detail below.

Step 1: Construct the corresponding category of the matter–element model against the category of speed error $e$ and its variation rate $\dot{e}$.

$$R_n=(F,c,v)=\left[\begin{array}{lll}\begin{array}{l}{F}_0\\ \end{array}& \begin{array}{c}e\\ \dot{e}\end{array}& \begin{array}{c}<a_1,b_1>\\ <a_2,b_2>\end{array}\end{array}\right],n=1,2,\dots ,20$$

(30)

Step 2: Enter the speed difference $e$ and its variation rate $\dot{e}$ to be classified, where a matter–element model is established.

$$R_{new}=\left[\begin{array}{lll}\begin{array}{l}{F}_{new}\\ \end{array}& \begin{array}{c}e\\ \dot{e}\end{array}& \begin{array}{c}{v}_{new1}\\ v_{new2}\end{array}\end{array}\right]$$

(31)

Step 3: Calculate the speed difference $e$ and its variation rate $\dot{e}$ based on Equation (26), where the correlation function with each category is derived accordingly.

Step 4: Set the weights $w_1$ and $w_2$ of each characteristic to express the significance of each characteristic. With consideration that each characteristic contains equivalent significance in the system, set both the weights $w_1$ and $w_2$ to 0.5, and $w_1+w_2=1$.

Step 5: Calculate the correlation degree with each category.

$${\lambda }_n={\displaystyle \sum _{i=1}^2{W}_i{K}_{ni},n=1,2,\dots ,20}$$

(32)

Step 6: For convenience of classification, normalize the correlation degree of each category classified with Equation (33), which sets the correlation degree at <−1, 1>.

$$\left\{\begin{array}{l}{\lambda }_n^{’}={\displaystyle \frac{{\lambda }_n}{\left|{\lambda }_{\max }\right|}},if{\lambda }_n>0\\ {\lambda }_n^{’}={\displaystyle \frac{{\lambda }_n}{\left|-{\lambda }_{\max }\right|}},if{\lambda }_n<0\end{array}\right.$$

(33)

where ${\lambda }_{\max }$ and $-{\lambda }_{\max }$ are the maximum value and minimum value of the correlation degrees for each category classified, respectively.

Step 7: With the maximum correlation degree derived from calculation, identify the category to classify speed difference and its variation rate. Based on the category, determine the variation amount in the weighting factor, and calculate the new weighting factor again, i.e.,

$$w_{new}=w_{old}+\Delta w$$

(34)

Among them, $w_{old}$ is the weighting factor corresponding to the category with the maximum correlation degree derived from calculation in the previous period.

4. Simulation Results

In this paper, Matlab/Simulink (R2023b version) was applied to simulate and verify the speed control of the induction motor drive system. Considering variation for the moment of inertia at $J_m=10{\overline{J}}_m$, system delay at 0.02 s (with said figure primarily determined by the electrical and mechanical parameters of the induction motor used in this paper), and limited control force, the speed control of the motor drive system was simulated using two methods proposed in this paper. These methods are the robust controller with a weighting factor adjusted using EFT and the robust controller adopting a fixed weighting factor. Figure 11 displays the responses of speed command tracking and current derived from simulation, where different weighting factors were adopted when the speed increased from 1000 rpm to 1100 rpm. From Figure 11a, it can be observed that for waveform a derived from simulation at 0.1 s to 0.13 s with an extendable fuzzy robust controller adopted, the rising curve was close to the simulated waveform d with a fixed weighting factor of zero (w = 0). Thus, it could be known that the initial adjustment of the weighting factor of the extendable fuzzy robust controller was zero, where the increase or decrease in the weighting factor was determined through the speed difference and its variation rate afterwards. Therefore, during tracking, the speed response of the extendable fuzzy robust controller presented a minor oscillation in the transient response. From this, it could be observed that the weighting factor was in balance between fast response and oscillation of suppression. Therefore, it was proved that the adoption of the extendable fuzzy robust controller can effectively select the suitable weighting factor, which prevents large oscillation caused when the fixed weighting factor of 0.93 is adopted for simulation waveform g. Moreover, the time of the steady-state response for the extendable fuzzy robust controller proposed to reach the final speed was even shorter than the speed response of simulation waveform f with a fixed weighting factor of 0.8 adopted. Figure 12 presents the speed response and current response loaded with $2\mathrm{N}\cdot \mathrm{m}$. From Figure 12a, it can be observed that, compared to the simulation waveforms d, e, f, and g with a fixed weighting factor adopted, the recovery time required for load regulation of the simulation waveform a with the weighting factor adjusted in an extension manner was distinctively shorter, and the speed drop was also smaller. Summing up the above, both the speed command tracking and control performance of load regulation derived from simulation for the induction motor drive system with the extendable fuzzy robust 2DOF speed controller proposed in this paper were better than the robust 2DOF controller adopting fixed weighting factors.

Figure 13a,b show the motor speed response and current response waveform as the speed command changes under $\Delta {\omega }_{rm}^{*}=100\mathrm{rpm}$ when an extendable fuzzy robust 2DOF controller and a traditional fuzzy logic robust 2DOF controller [23] are used for control. Figure 14a,b depict the load regulation response and current response waveforms of two method types under a load disturbance amount of $\Delta T_L=2\mathrm{N}\cdot \mathrm{m}$. Figure 13 and Figure 14 show that the extendable fuzzy robust 2DOF controller outperforms the traditional fuzzy logic robust 2DOF controller in terms of speed command tracking and load regulation response performance.

The control system block diagram in Figure 8 shows that the extendable fuzzy robust 2DOF controller proposed in this paper has two-degrees-of-freedom (2DOF). As a result, it can improve both the speed of command tracking and the response time of load regulation. Moreover, in terms of speed command tracking control, the reference model of predetermined speed command tracking has been established, and the extension theory has been used to automatically adjust the weighting factor (w) of the observer, preventing excessive overshoot or system instability caused by controller output saturation. Furthermore, the extension theory used does not necessitate a large amount of training data or complex calculations, is simpler to implement, and does not require large amounts of memory.

In addition, the response performance comparison in Figure 11 and Figure 12 compares the existing traditional robust controllers labeled d, e, f, and g in the figures. It does indeed show improved control performance, whether in terms of speed command tracking or load regulation response. In fact, as long as the motor parameters (electrical or mechanical) change significantly, using a traditional P-I or robust controller will result in the speed significantly overshooting or a longer recovery time. In this paper, the extendable fuzzy robust 2DOF controller was used to automatically adjust the observer’s weighting factor (w), reducing speed overshoot and speed recovery times. Figure 13 and Figure 14 compare the control performance of the extendable fuzzy robust 2DOF controller to the traditional fuzzy logic controller (FLC) mentioned in this paper. The figures show that the extendable fuzzy robust 2DOF controller proposed in this paper outperforms the existing FLC in terms of control. In Figure 13b, the response of curve c (light green line) is the same as that of f (dark green line), so it is indistinguishable from each other.

5. Conclusions

To improve the control performance of a robust speed controller with a fixed weighting factor for an induction motor drive system, the extendable fuzzy robust speed controller was proposed in this paper. The proposed controller could automatically adjust the weighting factors for the robust speed controller under variations in parameters, speed commands, and load disturbances of the drive system. Despite these adjustments, the controller performance of induction motor drive systems could still be maintained. The simulation results indicated that under circumstances of parameter variation in the drive system, delay characteristics, and saturated control force provided in the system, the robust controller adopting a fixed weighting factor could cause oscillation in speed response easily, and affect the system’s stability. Based on the characteristic of speed difference and its variation rate, the robust controller adopted in this paper could dynamically adjust the weighting factor to a better value. Therefore, such a controller could not only improve the response performance of speed command tracking and load regulation, but also suppress the system oscillation effectively, reducing the impact caused by parameter variation on the system, and enhancing the system’s stability, and hence its robustness.