Design of a Robust Controller for Speed Sensorless Brushless DC Motor Drive

Abstract

This paper proposes a robust controller combined with extension theory (ET) and applies it to the speed control of a brushless DC motor (BLDCM) drive system. The controller uses the motor’s speed error and its rate of change as characteristic values to establish ET classical domain and neighborhood domain models. Appropriate weights are set, and then the real-time speed error and its rate of change are used as input to calculate their correlation function with the ET model. The system can automatically determine the optimal proportional–integral (P-I) parameters, enabling the controller to perform real-time adaptive adjustments in response to the system’s nonlinear and time-varying characteristics. This overcomes the shortcomings of traditional fixed-gain P-I controllers with insufficient response during speed tracking and load changes. Furthermore, the system is equipped with a synchronous reference frame-based sliding mode observer (SRF-SMO) to achieve speed sensorless control. A power factor correction (PFC) circuit and over-voltage and over-current protection circuits are added to the drive system to improve the power quality at the power source and the operational safety of the drive system. To verify the effectiveness of the proposed controller, this paper implements the control algorithm on a 32-bit floating-point digital signal processor (DSP) TMS320F28335 using PSIM SimCoder’s automatic code generation technology. The experimental results show that compared to three different sliding mode controllers (SMCs) designed with constant speed reaching law (CSRL), exponential reaching law (ERL), and extension theory combined with exponential reaching law (ETERL), the proposed extension theory robust speed controller exhibits superior control performance in speed command tracking and load regulation response. This demonstrates that the proposed robust controller not only possesses stronger anti-disturbance capability, smaller speed drop, and shorter recovery time but additionally, the power factor of the drive system under rated load can reach above 0.98, and the protection mechanism can be activated under both over-voltage and over-current conditions, allowing the drive system to operate safely.

1. Introduction

With the booming development of high-precision industrial and electric vehicle technologies, the performance indicators of drive systems have become the key to determining equipment superiority. Due to their high torque-to-inertia ratio, excellent power density, and low maintenance costs, brushless DC motors (BLDCMs) [1] have gradually replaced traditional DC motors and induction motors, becoming the core of automation equipment, robotic arms, and electric vehicles. To fully exploit the performance of BLDCMs, the field-oriented control (FOC) [2] technique is widely adopted. By employing coordinate transformations to decouple the torque and flux of AC motors, it achieves control characteristics similar to those of DC motors.

Although the FOC architecture is quite mature, in practical engineering applications, the speed controller predominantly utilizes the traditional proportional–integral (P-I) controller [3]. While this type of controller is simple in structure and easy to implement, its parameter design is typically optimized only for a specific operating point of the system. However, the brushless DC motor (BLDCM) is inherently a nonlinear system, frequently facing challenges such as load inertia variations and external disturbances during practical operation. Therefore, when facing nonlinear factors, a fixed-gain P-I controller often struggles to balance the rapidity and stability of the transient response. It is prone to exhibiting sluggishness or overshoot, which restricts the control clarity of the overall system.

To break through the performance bottlenecks of controllers, both academia and industry have proposed various robust control methods. Among them, the sliding mode controller (SMC) [4] has received widespread attention due to its excellent robustness against internal parameter variations and external disturbances. However, the switching action of the traditional SMC near the sliding surface easily induces a high-frequency chattering phenomenon, which is liable to cause wear and tear on the mechanical structure. Although adopting the exponential reaching law (ERL) or the constant speed reaching law (CSRL) can improve some characteristics, it often requires a trade-off between convergence speed and overshoot, making it difficult to achieve optimization. Active disturbance rejection control (ADRC) has also been applied to BLDC motor drives, demonstrating strong disturbance rejection capability without requiring an accurate system model [5]. However, ADRC requires careful bandwidth tuning, and its estimation performance may degrade under rapid disturbance variations, limiting control accuracy in highly dynamic operating conditions. On the other hand, while the fuzzy logic controller (FLC) [6] can handle the uncertainties of nonlinear systems, its control performance heavily relies on a massive rule base established by expert experience. Furthermore, the complex computations place a significant burden on the digital signal processor during implementation, easily causing delays in the control loop. In recent years, AI-assisted control methods, including neural networks, machine learning, and reinforcement learning-based controllers, have demonstrated promising performance in nonlinear motor control [7]. However, such approaches typically require substantial computational resources and training data, posing challenges for real-time implementation on embedded platforms such as DSPs. In contrast, the ET-based adaptive method proposed in this study offers a computationally efficient and interpretable alternative that is well-suited for resource-constrained industrial motor drive systems. Furthermore, as international regulations governing green energy and grid power quality, notably the IEC 61000-3-2 [8] standard, become increasingly stringent, the design of high-performance motor drive systems can no longer be confined solely to back-end speed regulation. The power quality at the front-end input is of equal paramount importance. Conventional diode rectifier circuits generate significant harmonic currents and yield a low power factor (PF), which not only induces reactive power losses but also compromises grid stability. Consequently, the development of a drive system that simultaneously delivers high-dynamic-response speed control and excellent input power quality has emerged as a critical research imperative in modern power electronics.

Building upon the aforementioned analysis and to satisfy the rigorous demands of high-performance drive systems, this study proposes a robust speed controller (RSC) [9]. This approach integrates extension theory (ET) [10] with a P-I controller to overcome the limited adaptability of traditional P-I schemes. The controller defines the motor’s speed error and its rate of change as key state variables. By employing the concepts of the classical domain and neighborhood domain within ET to formulate a correlation function, it evaluates the real-time dynamic state of the system and calculates the correlation degree. Consequently, the P-I parameters are dynamically adjusted. This architecture retains the structural simplicity of the P-I controller while equipping it with adaptive capabilities, achieving rapid and overshoot-free speed tracking under load fluctuations. Notably, the controller architecture and classical domain settings are adapted from a prior simulation-based study [11], which utilized the same Mode 1 classical domain structure. However, because the practical implementation incorporates sensorless control, a PFC circuit, and hardware protection functions not present in the simulation, the classical domain ranges and parameter values in this paper have been empirically calibrated based on experimental results.

The remainder of this paper is organized as follows: Section 2 derives the mathematical model of the BLDCM and introduces the FOC architecture. Section 3 elaborates on the proposed control system, encompassing the PFC circuit, the sliding mode speed estimator [12], and the design of the extension robust speed controller. Section 4 presents the experimental results, benchmarking the proposed method against SMCs employing various reaching laws to validate the overall system performance [13] and protection mechanisms. Finally, Section 5 concludes this study. The main contributions of this paper are summarized as follows: (1) An ET-based P-I controller (ETPIC) is proposed as the core speed controller, achieving real-time adaptive parameter tuning based on correlation function calculations; (2) an SRF-SMO is employed for sensorless speed estimation, eliminating the need for mechanical sensors; and (3) a three-phase six-switch active PFC circuit is integrated at the front end to improve input power quality. The PFC circuit and hardware protection schemes serve as auxiliary modules to enhance system stability and industrial applicability.

2. BLDCM and Traditional FOC System

This section explores in detail the mathematical model and control architecture of the BLDCM drive system. First, through coordinate transformations [14], the dynamic equations [15] of the motor in the synchronous rotating coordinate system (SRCS) are derived to establish the theoretical foundation for the subsequent controller design. Subsequently, the traditional drive system architecture based on FOC is introduced, and the performance limitations of the traditional P-I controller in the face of system parameter variations are analyzed. This serves as a comparative basis for the robust control strategy proposed in this study.

2.1. Dynamic Model of BLDCM

The dynamic mathematical model of the BLDCM in the d-, q-axis SRCS can be described by applying a coordinate transformation to convert the three-phase stationary coordinate system into the d-, q-axis SRCS. Assuming that the rotor flux direction aligns with the d-axis, and neglecting magnetic circuit saturation and eddy current losses, its stator voltage dynamic equations can be expressed as Equations (1) and (2).

$$v_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q$$

(1)

$$v_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{L}_d{i}_d+{\omega }_e{\lambda }_m$$

(2)

where $v_d$ and $v_q$ are the d- and q-axis voltages; $i_d$ and $i_q$ are the d- and q-axis currents; $R_s$ is the stator resistance; $L_d$ and $L_q$ are the d- and q-axis inductances, respectively; ${\omega }_e$ is the electrical angular velocity; and ${\lambda }_m$ is the permanent magnet flux linkage.

In terms of electromagnetic torque, although the interior permanent magnet motor exhibits saliency characteristics $L_d\ne L_q$, its theoretical output torque consists of two parts: permanent magnet torque and reluctance torque, which can be expressed as Equation (3).

$$T_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{P}{2}}[{\lambda }_m{i}_q+(L_d-L_q)i_d{i}_q]$$

(3)

However, to simplify the control complexity, this study adopts FOC and its associated control strategy $i_d=0$. Under this control $i_d=0$, the reluctance torque term in Equation (3) will become zero. At this time, if the torque constant is defined, it can be expressed as Equation (4).

$$K_t={\displaystyle \frac{3}{2}}{\displaystyle \frac{P}{2}}{\lambda }_m$$

(4)

Then, the electromagnetic torque equation can be simplified to Equation (5).

$$T_e=K_t{i}_q$$

(5)

As can be seen from Equation (5), under the $i_d=0$ control architecture, the electromagnetic torque T_e is proportional to the q-axis current, making the torque control of the AC motor as linear and intuitive as that of a traditional DC motor. Furthermore, the mechanical dynamic equation describes the relationship between the motor speed and the load, which can be expressed as Equation (6).

$$T_e=J_m{\displaystyle \frac{d{\omega }_r}{dt}}+B_m{\omega }_r+T_L$$

(6)

where $J_m$ is the moment of inertia; $B_m$ is the viscous friction coefficient; ${\omega }_r$ is the mechanical speed; and $T_L$ is the load torque.

2.2. Traditional FOC System

The motor drive system investigated in this study adopts the FOC architecture, and its system block diagram is shown in Figure 1. The main components include: P-I controllers for the speed loop and current loop, coordinate transformation, decoupling control circuit, space vector pulse width modulation (SVPWM), and a three-phase inverter. In this system, the acquisition methods of key feedback signals are mainly divided into two parts: current sensing and speed estimation. First, regarding current sensing, the three-phase stator currents $i_a$, $i_b$ and $i_c$ are measured through a Hall effect current sensor (HX-25P, LEM USA Inc., Milwaukee, WI, USA) with a conversion ratio of 1 V/6.25 A. After the analog voltage signal output by the sensor is sampled by the analog-to-digital converter, it is converted into a digital signal and fed back into the digital signal processor. Second, to reduce hardware costs and enhance the system’s reliability in harsh environments, this system does not use a traditional encoder; instead, it employs a speed estimator to obtain speed information. This estimator utilizes four state variables after coordinate transformation: the direct-axis voltage command $v_d$, the quadrature-axis voltage command $v_q$, and the direct-axis current $i_d$ and quadrature-axis current $i_q$. Through mathematical model calculations of the BLDCM, it estimates the rotor magnetic field position $\widehat{\theta }$ and speed ${\widehat{\omega }}_r$ to achieve sensorless speed control.

In terms of the control process, the system inputs the speed command ${\omega }_r$ and the estimated actual speed error ${\widehat{\omega }}_r$ into the P-I speed controller to generate the q-axis current command $i_q$. Subsequently, the current errors of the d, q-axis respectively pass through two independent P-I current controllers to generate the d, q-axis voltage commands. To eliminate the cross-coupling effect existing in the d and q-axis voltage equations during motor operation, this system introduces feed-forward decoupling compensation. After superimposing compensation voltages onto the outputs of the P-I controllers, the final voltage commands ${V_d}^{*}$ and ${V_q}^{*}$ are generated. Finally, the three-phase inverter is driven through coordinate transformation and SVPWM.

3. Proposed Control System

This section details the integrated hardware and control algorithm design for the BLDCM drive system proposed in this study. To clearly highlight the core innovations and improve readability, this section is structured into four main subsections: the front-end active PFC circuit, SRFC, the proposed ETPIC, and the hardware protection mechanisms. As illustrated in Figure 1, the developed sensorless BLDCM drive system mainly consists of three core control loops: (1) Power conversion and drive loop: To ensure excellent power quality, its input stage adopts a three-phase six-switch active PFC based on space vector modulation technology. This circuit converts the utility AC power into a stable DC link voltage and achieves unity PFC, thereby providing a stable energy source for the back-end inverter and the motor; (2) speed estimation loop: It employs a sliding mode estimator based on the synchronous reference frame [16]. By utilizing the voltage commands and feedback currents, which are transformed into the SRCS via coordinate transformation, it estimates the rotor speed ${\widehat{\omega }}_r$ and rotor position ${\widehat{\theta }}_e$ in real time to replace traditional mechanical speed and position sensors; and (3) speed control loop: It utilizes a robust controller (RC) that combines ET and a P-I controller. This controller takes the error between the speed command ${\omega }_r$ and the estimated speed ${\widehat{\omega }}_r$, e along with its rate of change $\dot{e}$, as input characteristics. By establishing classical domain and neighborhood domain models to divide the control intervals, it introduces a characteristic weight to calculate the correlation degree. Ultimately, the maximum correlation degree is selected to determine the current optimal control interval, the parameters k_p and k_i of the P-I controller are adjusted in real time, and the q-axis current command $i_q$ is output.

3.1. Three-Phase Active PFC

To enhance the power quality of the AC input for the drive system and provide a stable DC link voltage for the back-end inverter, this study configures a three-phase six-switch active PFC at the front-end power supply side. The architecture of this circuit is shown in Figure 2. The power stage utilizes the PM100CG1A065 intelligent power module (Mitsubishi Electric Corporation, Tokyo, Japan) [17], while the control strategy is based on voltage-oriented control (VOC) combined with SVPWM [18] to achieve unity PFC.

3.2. Sliding Mode Speed Estimator

This study adopts a sliding mode estimator based on the SRCS combined with a phase-locked loop (PLL) [16,19] to achieve sensorless speed estimation. By operating in the SRCS, back-EMF signals appear as DC quantities, eliminating the phase delay and amplitude attenuation introduced by low-pass filters in stationary-frame estimators. A sliding mode switching function drives the current estimation error to zero in finite time, after which a filter extracts the equivalent control effort to obtain the quadrature-axis back-EMF. The direct-axis back-EMF component, which reflects the angular error between estimated and actual rotor positions, is then regulated to zero by a PLL-based PI controller, yielding accurate closed-loop speed and position feedback for FOC.

3.3. Principles of P-I Controllers and Their Applications in FOC

Although fixed-gain P-I controllers are widely adopted in FOC-based BLDCM drives—where the d-axis current command is set to zero to implement MTPA [20] control—their inability to simultaneously optimize dynamic response and stability under time-varying, nonlinear motor parameters limits control performance. Therefore, this study introduces ET [21] to design an adaptive P-I controller that dynamically adjusts k_p and k_i based on the system’s current error state, enhancing robustness and control precision across varying operating conditions.

3.4. Fundamentals of ET

ET is a mathematical method for handling incompatible and nonlinear problems by quantitatively evaluating the correlation between system characteristics and predefined domains. Its core components are defined as follows. The extension model considers the object, characteristics, and value as three fundamental elements for describing the objective world, encapsulating them into a logical cell called a matter element. If N represents the name of the object, C represents the characteristic, and V represents the measured value of that characteristic, then a one-dimensional matter element R can be defined as Equation (7).

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

(7)

The multidimensional matter element model encapsulates an object’s characteristics C₁, C₂…, C_n (such as speed error, rate of change in speed error, etc.), and the corresponding values are V₁, V₂…, V_n, respectively. Then, the multidimensional matter element matrix R can be expressed as Equation (8). In this study, the speed error and its rate of change serve as the two core characteristics.

$$R=\left[\begin{array}{c}{R}_1\\ R_2\\ ⋮\\ R_n\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{l}N,\\ \\ \\ \end{array}& \begin{array}{c}{C}_1,\\ C_2,\\ ⋮\\ C_n\end{array}& \begin{array}{c}{V}_1\\ V_2\\ ⋮\\ V_n\end{array}\end{array}\right]$$

(8)

The classical domain F₀ = ⟨a, b⟩ defines the most satisfactory range for a characteristic, while the neighborhood domain F = ⟨d, e⟩ defines the maximum allowable variation range, where F₀ ∈ F. Their respective matter element models are expressed as Equations (9) and (10).

$$R_0=(F_0,C_i,V_i)=\left[\begin{array}{ccc}\begin{array}{l}{F}_0,\\ \\ \\ \end{array}& \begin{array}{c}{C}_1,\\ C_2,\\ ⋮\\ C_n\end{array}& \begin{array}{c}<a_1, b_1>\\ <a_2, b_2>\\ ⋮\\ <a_n, b_n>\end{array}\end{array}\right]$$

(9)

$$R_F=(F,C_j,V_j)=\left[\begin{array}{ccc}\begin{array}{l}F,\\ \\ \\ \end{array}& \begin{array}{c}{C}_1,\\ C_2,\\ ⋮\\ C_n\end{array}& \begin{array}{c}<d_1,e_1>\\ <d_2,e_2>\\ ⋮\\ <d_n,e_n>\end{array}\end{array}\right]$$

(10)

Based on these domains, the distance from a point f to F₀ and the rank value are defined in Equations (11) and (12), respectively. These are used to construct the elementary correlation function K(f) in Equation (13), which maps the current system state to a continuous quantitative value, providing the basis for adaptive parameter selection.

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

(11)

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

(12)

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

(13)

3.5. Quantitative Design of the Speed Controller

To establish the mathematical foundation for parameter design, the closed-loop transfer function of the speed control loop is derived. It is worth noting that cogging torque and starting torque are not independently modeled in this study. Since the proposed method focuses on the dynamic response within the medium-to-high speed range, the dominant dynamics of the system are governed by electromagnetic torque and moment of inertia. High-frequency, small-amplitude nonlinear effects, such as cogging torque, have a negligible influence in this operating range. Furthermore, any unmodeled parasitic torques and nonlinear factors are collectively treated as equivalent external disturbances, which are effectively compensated for by the proposed ETPIC architecture. The P-I controller transfer function G_c(s) and plant transfer function are given in Equations (14) and (15), respectively. By approximating the inner current loop as unity gain—valid when its bandwidth exceeds five times that of the speed loop—the overall closed-loop transfer function G_close(s) is obtained as Equation (16) [22,23,24], as illustrated in Figure 3.

$$G_c(s)=k_p+{\displaystyle \frac{k_i}{s}}={\displaystyle \frac{k_{p}s+k_i}{s}}$$

(14)

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

(15)

$$G_{close}(s)={\displaystyle \frac{{\omega }_{rm}(s)}{{{\omega }^{*}}_{rm}(s)}}={\displaystyle \frac{G_c(s)G_m(s)}{1+G_c(s)G_m(s)}}={\displaystyle \frac{k_{p}s+k_i}{J_m{s}^2+(B_m+k_p)s+k_i}}$$

(16)

Based on G_close(s), four performance indicators are specified: zero steady-state error, minimal overshoot, fast rise time, and bounded speed drop under load disturbance. The corresponding nonlinear equations are solved numerically in MATLAB R2024b to obtain feasible P-I parameter ranges [22,24]. Since multiple parameter combinations may satisfy these time-domain specifications, the integral of absolute error (IAE) [25], defined in Equation (17), is adopted as the final optimization criterion. Unlike the integral of squared error (ISE) or the integral of time-weighted squared error (ITSE), which tend to assign excessive weight to larger transient errors, the IAE exhibits excellent sensitivity to small persistent errors throughout the control process. This characteristic provides a more balanced evaluation of the overall cumulative error for speed tracking in this application. The parameter set yielding the minimum IAE is selected as the baseline for subsequent ET-based adaptive tuning.

$$IAE={\displaystyle {\int }_0^{\infty } \left|e(t)\right|} dt$$

(17)

3.6. ET-Based P-I Controller Parameter Tuning

Using the IAE-optimized parameters from Section 3.5 as the tuning baseline, ET is applied to enable adaptive adjustment of k_p and k_i in real time. The speed error e and its rate of change ė are selected as the two core characteristics. As defined in

Table 1. Construction of the neighborhood domain for speed control based on ET.

Characteristic	Neighborhood Domain
Speed error (rpm)	−6~6
Rate of change in speed error (rpm/s)	−120~120

and

Table 2. Construction of classical domains for various characteristics under different operating modes based on ET.

	Characteristic	Speed Error (rpm)	Rate of Change in Speed Error (rpm/s)
Type
Mode 1		<−1.2,1.2>	<−24,24>
Mode 2		<−2.4,2.4>	<−48,48>
Mode 3		<−3.6,3.6>	<−72,72>
Mode 4		<−4.8,4.8>	<−96,96>
Mode 5		<−6,6>	<−120,120>

, a neighborhood domain encompassing all extreme operating conditions is partitioned into five classical domain modes, allowing the controller to select optimal parameters based on the maximum correlation degree principle.

3.6.1. Interval Subdivision and Parameter Design for Mode 1

Among all operating modes, Mode 1 represents a steady-state region where the error is extremely small and the system has converged close to the target value. Within this region, the system’s requirements for control precision are the most stringent, as even minor parameter variations could induce torque ripple. Driven by this consideration, to enhance the system’s control performance as it approaches a steady state, this study conducts a more meticulously detailed quantitative design specifically for Mode 1.

In terms of parameter design, this study utilizes the P-I controller parameters k_p = 0.04 and k_i = 0.365, which were obtained from the optimization of the IAE described in the previous section, as the baseline. Based on the minute variations in speed error and the rate of change in error, this region is further subdivided into 40 smaller classical domain intervals.

Table 3. Twenty representative classical domain intervals and parameter design results subdivided under Mode 1.

Extenics-Based Matter Element Model	Parameter
$R_1=\left[\begin{array}{l}\begin{array}{ccc}{F}_1& e& <0,0.3>\end{array}\\ \begin{array}{ccc} & \dot{e}& <0,6>\end{array}\end{array}\right]$	$k_p=0.04, k_i=0.365$
$R_2=\left[\begin{array}{l}\begin{array}{ccc}{F}_2& e& <0,0.4>\end{array}\\ \begin{array}{ccc} & \dot{e}& <0,8>\end{array}\end{array}\right]$	$k_p=0.0435, k_i=0.345$
$R_3=\left[\begin{array}{l}\begin{array}{ccc}{F}_3& e& <0.1,0.5>\end{array}\\ \begin{array}{ccc} & \dot{e}& <2,10>\end{array}\end{array}\right]$	$k_p=0.0470, k_i=0.325$
$R_4=\left[\begin{array}{l}\begin{array}{ccc}{F}_4& e& <0.2,0.6>\end{array}\\ \begin{array}{ccc} & \dot{e}& <4,12>\end{array}\end{array}\right]$	$k_p=0.0505, k_i=0.305$
$R_5=\left[\begin{array}{l}\begin{array}{ccc}{F}_5& e& <0.3,0.7>\end{array}\\ \begin{array}{ccc} & \dot{e}& <6,14>\end{array}\end{array}\right]$	$k_p=0.0540, k_i=0.285$
$R_6=\left[\begin{array}{l}\begin{array}{ccc}{F}_6& e& <0.4,0.8>\end{array}\\ \begin{array}{ccc} & \dot{e}& <8,16>\end{array}\end{array}\right]$	$k_p=0.0575, k_i=0.265$
$R_7=\left[\begin{array}{l}\begin{array}{ccc}{F}_7& e& <0.5,0.9>\end{array}\\ \begin{array}{ccc} & \dot{e}& <10,18>\end{array}\end{array}\right]$	$k_p=0.0610, k_i=0.245$
$R_8=\left[\begin{array}{l}\begin{array}{ccc}{F}_8& e& <0.6,1>\end{array}\\ \begin{array}{ccc} & \dot{e}& <12,20>\end{array}\end{array}\right]$	$k_p=0.0645, k_i=0.225$
$R_9=\left[\begin{array}{l}\begin{array}{ccc}{F}_9& e& <0.7,1.1>\end{array}\\ \begin{array}{ccc} & \dot{e}& <14,22>\end{array}\end{array}\right]$	$k_p=0.0680, k_i=0.205$
$R_{10}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{10}& e& <0.8,1.2>\end{array}\\ \begin{array}{ccc} & \dot{e}& <16,24>\end{array}\end{array}\right]$	$k_p=0.0715, k_i=0.185$
$R_{11}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{11}& e& <0,0.3>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-6,0>\end{array}\end{array}\right]$	$k_p=0.0490, k_i=0.300$
$R_{12}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{12}& e& <0,0.4>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-8,0>\end{array}\end{array}\right]$	$k_p=0.0530, k_i=0.285$
$R_{13}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{13}& e& <0.1,0.5>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-10,-2>\end{array}\end{array}\right]$	$k_p=0.0570, k_i=0.270$
$R_{14}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{14}& e& <0.2,0.6>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-12,-4>\end{array}\end{array}\right]$	$k_p=0.0610, k_i=0.255$
$R_{15}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{15}& e& <0.3,0.7>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-14,-6>\end{array}\end{array}\right]$	$k_p=0.0650, k_i=0.240$
$R_{16}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{16}& e& <0.4,0.8>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-16,-8>\end{array}\end{array}\right]$	$k_p=0.0690, k_i=0.225$
$R_{17}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{17}& e& <0.5,0.9>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-18,-10>\end{array}\end{array}\right]$	$k_p=0.0730, k_i=0.210$
$R_{18}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{18}& e& <0.6,1>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-20,-12>\end{array}\end{array}\right]$	$k_p=0.0780, k_i=0.195$
$R_{19}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{19}& e& <0.7,1.1>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-22,-14>\end{array}\end{array}\right]$	$k_p=0.0830, k_i=0.180$
$R_{20}=\left[\begin{array}{l}\begin{array}{ccc}{F}_{20}& e& <0.8,1.2>\end{array}\\ \begin{array}{ccc} & \dot{e}& <-24,-16>\end{array}\end{array}\right]$	$k_p=0.0880, k_i=0.165$

lists 20 representative sets of these classical domains along with their corresponding parameter design results. Table 3 presents only the 20 representative parameter sets corresponding to positive speed errors. The remaining 20 parameter sets correspond to negative speed errors with positive and negative speed error variations. Their controller gains are determined by adjusting the baseline controller parameters according to the corresponding classical domain ranges, following the same design procedure. Therefore, only the representative parameter sets are presented to avoid redundant tabulation. Through this subdivision mechanism, the controller parameters can be adjusted smoothly and continuously in response to minute changes in error, thereby enhancing overall control performance. The specific choice of dividing the Mode 1 interval into exactly 40 subintervals (20 positive and 20 negative) was determined through a trade-off analysis between this control smoothness and the computational limitations of the DSP. If a smaller total number of intervals is chosen (e.g., 10 or 20), the transition of control parameters becomes too discrete, leading to larger speed ripples during transient responses. Conversely, if an excessively large number is selected (e.g., 100), the increased number of condition-checking statements significantly raises the computational burden and memory footprint, potentially causing execution cycle overruns. Empirical testing demonstrated that 40 intervals provide sufficient resolution to ensure smooth parameter adaptation while maintaining the algorithm’s execution time well within the safe margin of the system’s sampling period.

3.6.2. Correlation Degree Calculation and Weight Settings

The purpose of setting weight values is to measure the relative importance of each system characteristic to the controller’s decision-making, allowing the extension inference to more accurately reflect the current dynamic requirements of the system. In this study, the weights for the two characteristics are both set to 0.5, meaning that the speed error and the rate of change in speed error possess equal influence over the selection of the final parameters.

In conjunction with the classical domain design described in the previous section, the system will utilize the correlation function to perform calculations across the 40 subdivided intervals during operation. Through this multi-characteristic weighted attribution mechanism, the controller can precisely determine the system status and derive the optimal parameter combination. The exceptional adaptive tuning capability of this controller distinguishes itself from traditional look-up table methods, which merely select parameters based on fixed conditions.

3.7. Parameter Tuning Process of the ET-Based P-I Controller

This section integrates the previously described matter element model design and characteristic weight settings to concretely explain the dynamic parameter tuning process of the proposed extension P-I controller. The presented control method takes the speed error and the rate of change in speed error of the BLDCM as the two core characteristics. Through ET, it calculates their distances and rank values to determine the correlation degree with each classical domain category. The category with the highest correlation degree is then selected to determine the optimal P-I controller parameters. The specific tuning steps are described below.

Step 1: Establish the matter element models to be evaluated: For each speed error e and speed error change rate $\dot{e}$ category, construct the corresponding matter element model.

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

(18)

Step 2: Establish the category matter element models: Input the speed error e and speed error change rate $\dot{e}$ to be categorized, and establish their matter element model.

$$R_{new}=\left[\begin{array}{ccc}\begin{array}{l}{F}_0\\ \end{array}& \begin{array}{c}e\\ \dot{e}\end{array}& \begin{array}{c}{V}_{new1}\\ V_{new2}\end{array}\end{array}\right], n=1,2,\dots ,20$$

(19)

Step 3: Calculate correlation function values: Input the newly sampled speed error and rate of change in speed error, and calculate their correlation functions with respect to each category according to Equation (13).

Step 4: Set characteristic weights: Set the weight values W₁ and W₂ for each characteristic to represent their relative importance. In this study, through the experimental test results, we know that both characteristics (speed error and the rate of change in speed error) are considered equally important. Therefore, the weight values W₁ and W₂ are both set to 0.5, satisfying W₁ + W₂ = 1. It should be noted that there is no rigid mathematical constraint for these values; rather, they serve as empirical design parameters. In practical applications, designers can flexibly adjust these weights based on specific performance requirements—for instance, assigning a higher value to W₁ to prioritize steady-state accuracy, or a higher value to W₂ to enhance the damping of transient responses.

Step 5: Calculate the correlation degree for each category: Take the correlation function values of each characteristic obtained from Step 3 and the weight values from Step 4 to perform a weighted summation according to Equation (20). This calculates the comprehensive correlation degree of the matter element to be evaluated with respect to each category model.

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

(20)

Step 6: Normalize the correlation degrees: To facilitate classification, the correlation degrees of each belonging category are normalized so that the correlation values fall within the range of <−1,1>. Here, ${\lambda }_{\max }$ and $-{\lambda }_{\max }$ represent the maximum and minimum values among the correlation degrees assigned to each category, respectively.

$$\left\{\begin{array}{ll}{\lambda }_n^{\prime }={\displaystyle \frac{{\lambda }_n}{\left|{\lambda }_{\max }\right|}}& ,\hspace{1em}if {\lambda }_n>0\\ {\lambda }_n^{\prime }={\displaystyle \frac{{\lambda }_n}{\left|-{\lambda }_{\max }\right|}}& ,\hspace{1em}if {\lambda }_n<0\end{array}\right.$$

(21)

Step 7: Decision-making and parameter output: Using the calculated normalized correlation degrees, determine the category to which the current speed error and rate of change in speed error belong based on the maximum correlation principle. Select the corresponding optimal P-I controller parameters to complete the adaptive parameter tuning.

4. Experimental Results and Analysis

To verify the feasibility and control performance of the proposed ET-based P-I controller applied to a BLDCM drive system, this study adopts a model-based design (MBD) workflow. Utilizing PSIM SimCoder automatic code generation technology, the algorithm is directly implemented on a 32-bit DSP, and a complete hardware experimental platform has been constructed.

First, the experimental hardware architecture and software development workflow are introduced, followed by a description of the test motor and measurement equipment. Subsequently, real-world experimental waveforms are analyzed and compared across several aspects: the PFC circuit, system protection mechanisms, speed tracking response, and load regulation response. This comprehensive evaluation demonstrates that the proposed motor drive system achieves outstanding dynamic control performance, input power quality, and operational reliability.

4.1. Digital Control Core and Development Work

The system employs a digital signal processor (TMS320F28335, Texas Instruments Inc., Dallas, TX, USA) as the core control chip [26]. This chip is a 32-bit floating-point DSP operating at a frequency of up to 150 MHz. It features built-in high-resolution pulse width modulation (PWM) and quadrature encoder pulse modules, making it highly suitable for high-end motor drive control applications. In terms of software implementation, an MBD workflow utilizing the PSIM SimCoder automatic code generation tool is adopted. First, the control block diagram is constructed and verified within the PSIM simulation environment to confirm that the control logic is correct. Subsequently, SimCoder automatically converts the graphical blocks into optimized C language code. This development workflow ensures that the hardware experimental results faithfully reflect the theoretical design, effectively eliminating human programming errors that may occur during manual coding. The control loop operates at a sampling frequency of 18 kHz. The proposed ET-based adaptive algorithm, including correlation function computation and parameter selection, completes execution within each control cycle on the 150 MHz DSP platform, confirming the real-time feasibility of the proposed method with low processor utilization.

Power Drive Circuit Design

The BLDCM drive system developed in this study utilizes the PM100CG1A065 intelligent power module (IPM) produced by Mitsubishi Electric for its inverter stage. This IPM features high integration and exceptional reliability, with a rated voltage capacity of up to 650 V and a maximum continuous current capability of 100 A, making it highly suitable for the medium-to-high power motor drive system configured in this research. Internally, the module integrates six sets of insulated gate bipolar transistor (IGBT) power switches capable of bidirectional power transmission, providing the drive stage with a high degree of flexibility. This IPM will be deployed separately in both the inverter and the PFC converter stages of the overall drive system.

4.2. Test Motor and Experimental Equipment Configuration

Figure 4 shows a photograph of the physical BLDCM drive experimental platform established in this study. The main components and their functional descriptions are as follows: (1) Personal computer (PC): Used to run the PSIM software and the code composer studio (CCS) integrated development environment (IDE). It is responsible for algorithm development, program burning, and real-time monitoring of internal control variables within the DSP; (2) oscilloscope: utilized to capture the motor’s phase currents, DC link voltage, and rectifier input waveforms to verify both the dynamic and steady-state performance of the controller; (3) power resistor: the primary function is to help discharge the residual electrical charge in the DC link capacitors after experiments are completed, ensuring operational safety; (4) AM-2200H permanent magnet BLDCM [27]: Acts as the plant (controlled object). Its detailed electrical specifications and parameters are listed in

Table 4. Specifications of the adopted BLDCM.

Electrical Specifications	Value
Three-phase rated voltage	220 V
Three-phase rated current	9.8 A
Rated apparent power	2156 VA
Rated speed	6000 rpm
Operating frequency range	0~200 Hz
Number of poles	4
Stator resistance	0.15 Ω
d-axis inductance	2.39 mH
q-axis inductance	1.235 mH
Flux linkage	0.126 Wb
Moment of inertia	0.00145 kg·m2

; and (5) dynamometer: A magnetic powder brake coupled with an integrated torque measurement and display system is connected to the motor shaft end to serve as the mechanical load for the system. The applied load torque is determined directly from the torque readings displayed by the dynamometer during experiments. (6) DC power supply: Provides the necessary power to the dynamometer. By manually adjusting the output voltage of this power supply, the load torque exerted by the magnetic powder brake on the motor shaft can be varied; and (7) measurement instruments: include digital clamp-on voltmeters and current meters used for real-time measurement of the system’s voltage and current values.

4.3. AC-Side Power Factor Measurement

In this study, a three-phase six-switch PFC converter is employed to shape the input current, forcing it to track the input voltage and maintain a sinusoidal waveform in phase with the voltage. Figure 5a illustrates the experimental waveforms of the input voltage and current before PFC is applied. Measured by the instrument, the PF is only 0.815, with the corresponding measurement data displayed in Figure 5b. Conversely, Figure 6a shows the experimental input phase voltage and current waveforms after implementing the PFC controller. It can be observed from the figure that the current waveform has been significantly improved. Furthermore, instrumental measurements verify that the PF has been successfully elevated to 0.987, as shown by the measurement data in Figure 6b.

4.4. System Protection Mechanism Test Results

(1)

Over-voltage protection test

Because the drive system developed in this study integrates a PFC circuit, the DC link voltage is regulated at 370 V under normal operating conditions. To prevent over-voltage from damaging the DC link capacitors or the power switches, the over-voltage protection threshold of the system is set to 390 V. Figure 7a presents the experimental operation process of the over-voltage protection. The moment the detected DC link voltage exceeds 390 V, the protection mechanism is triggered, simultaneously disabling the switching signals for both the PFC circuit and the inverter. It can be observed from the waveforms that once the protection mechanism is activated, the DC link voltage immediately ceases to rise and undergoes natural decay, eventually stabilizing at approximately 311 V, which corresponds to the peak voltage of the utility grid after full-wave rectification. This validates the effectiveness of the over-voltage protection.

(2)

Over-current protection test

The over-current protection mechanism utilizes Hall effect current sensors to sample the three-phase input current signals in real time. After being processed by the hardware protection circuit, this signal is transmitted to the external interrupt protection pin of the DSP. The over-current protection threshold in this experiment is set to 5 A. The moment any phase current exceeds this predefined value, the program immediately executes the protection logic, cutting off the drive signals to the IPM. Figure 7b illustrates the experimental waveform of the over-current protection action. It demonstrates that at the exact instant the current hits 5 A, the switching control signals for both the PFC converter and the inverter are deactivated immediately, forcing the motor to stop running and thereby preventing components from burning out.

4.5. Experimental Testing and Comparison of Speed Control Performance

To verify the feasibility of the proposed ET-based P-I controller (ETPIC), speed command tracking and load regulation tests were conducted under various operating conditions. Furthermore, its control performance was compared against three distinct control strategies: the constant speed reaching law sliding mode controller (CSRLSMC), the exponential reaching law sliding mode controller (ERLSMC), and the ET-based exponential reaching law sliding mode controller (ETERLSMC) [28].

4.5.1. Speed Tracking Response

Figure 8 illustrates the speed command tracking responses under no-load conditions, where the speed commands undergo a small step change of 100 rpm at 1000 rpm, 2000 rpm, and 3000 rpm, respectively, while Figure 9 shows the speed command tracking responses under a fixed-load condition, where the speed commands undergo large step changes of 1000 rpm at 1000 rpm, 2000 rpm, and 3000 rpm, respectively. As observed from Figure 8 and Figure 9, the proposed ETPIC demonstrates superior dynamic tracking performance compared to the other three SMCs throughout the speed command tracking process. Furthermore, whether under minor speed command variations at no-load or large-scale speed command variations under load, the ETPIC converges to the target speed values more stably and rapidly than the ETERLSMC.

Table 5. Speed tracking comparison for load = 0 N-m.

Speed Command Change	ETPIC		CSRLSMC		ERLSMC		ETERLSMC
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
1000→1100 rpm	0 rpm	0.4 s	0 rpm	1.2 s	8 rpm	0.8 s	0 rpm	0.6 s
2000→2100 rpm	0 rpm	0.5 s	0 rpm	1.3 s	8 rpm	0.9 s	0 rpm	0.7 s
3000→3100 rpm	0 rpm	0.6 s	0 rpm	1.5 s	8 rpm	0.10 s	0 rpm	0.8 s

,

Table 6. Speed tracking comparison for load = 1 N-m.

Speed Command Change	ETPIC		CSRLSMC		ERLSMC		ETERLSMC
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
1000→2000 rpm	0 rpm	1.5 s	0 rpm	Not	70 rpm	3.2 s	0 rpm	2.0 s

and

Table 7. Speed tracking comparison for load = 2 N-m.

Speed Command Change	ETPIC		CSRLSMC		ERLSMC		ETERLSMC
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
1000→2000 rpm	0 rpm	1.7 s	0 rpm	Not	50 rpm	4.0 s	0 rpm	2.1 s
2000→3000 rpm	0 rpm	1.6 s	0 rpm	Not	25 rpm	2.8 s	0 rpm	1.8 s

present the quantitative speed tracking performance of the four controllers—ETPIC, CSRLSMC, ERLSMC, and ETERLSMC—under loads of 0 N-m, 1 N-m, and 2 N-m, respectively. The key metrics compared are overshoot and settling time. Under no-load conditions (Table 5), for speed command changes of 1000→1100 rpm, 2000→2100 rpm, and 3000→3100 rpm, the proposed ETPIC achieves zero overshoot with settling times of 0.4 s, 0.5 s, and 0.6 s, respectively. CSRLSMC also achieves zero overshoot but requires longer settling times of 1.2 s, 1.3 s, and 1.5 s. ETERLSMC achieves zero overshoot with settling times of 0.6 s, 0.7 s, and 0.8 s, while ERLSMC exhibits a consistent overshoot of 8 rpm across all speed steps. These results confirm that the proposed ETPIC achieves the fastest settling response with zero overshoot under no-load conditions. Under a load of 1 N-m (Table 6), for the speed command change from 1000 to 2000 rpm, the proposed ETPIC maintains zero overshoot with a settling time of 1.5 s. ETERLSMC also achieves zero overshoot but with a longer settling time of 2.0 s. ERLSMC exhibits a significant overshoot of 70 rpm with a settling time of 3.2 s, while CSRLSMC fails to complete the speed tracking within the observation window (denoted as “Not”), demonstrating insufficient dynamic response under loaded conditions. Under a heavy load of 2 N-m (Table 7), for speed command changes from 1000 to 2000 rpm and 2000 to 3000 rpm, the proposed ETPIC consistently achieves zero overshoot with settling times of 1.7 s and 1.6 s, respectively. ETERLSMC achieves zero overshoot with settling times of 2.1 s and 1.8 s. ERLSMC produces overshoots of 50 rpm and 25 rpm with settling times of 4.0 s and 2.8 s, while CSRLSMC again fails to converge under both speed command changes. These results collectively confirm that the proposed ETPIC delivers the best overall speed tracking performance, achieving zero overshoot and the fastest settling response across all tested load conditions.

4.5.2. Experimental Analysis of Load Regulation Response

Figure 10, Figure 11, Figure 12 and Figure 13 display the experimental results of the load regulation response under conditions where the speed commands are set to 1000 rpm, 2000 rpm, and 3000 rpm respectively, while the load varies from 0 N-m to either 1 N-m or 2 N-m. From the experimental results, it can be observed that the CSRLSMC, due to its lack of integral action, fails to track the speed commands stably, causing the system to take up to 40 s to converge to the target speed. Furthermore, under load variations, its speed recovery capability is manifestly insufficient, exhibiting a larger speed drop and a delayed dynamic recovery. In comparison, although the ERLSMC possesses a faster speed command tracking capability, overshoots occur under certain operating conditions during its response process. This can lead to transient instability in the system, thereby compromising overall control stability. As for the ETERLSMC, while it effectively suppresses overshoots and enhances the tracking stability of speed commands, its dynamic response and speed recovery capability under load variations remain inadequate, making it difficult to simultaneously satisfy both rapid command tracking and disturbance rejection performance. In sharp contrast, the controller proposed in this study demonstrates the minimum speed drop and the fastest recovery speed across all the aforementioned testing conditions.

5. Conclusions

The three core innovations of this study are: (1) The ET-based P-I controller (ETPIC) for robust adaptive speed control; (2) the SRF-SMO for sensorless speed estimation; and (3) the integrated three-phase active PFC circuit for improved input power quality. This study successfully developed a high-performance BLDCM drive system by proposing an intelligent robust controller that combines ET with a P-I controller, integrated with a three-phase active PFC technique. This approach effectively resolves the issues of inadequate parameter adaptability associated with fixed-gain controllers in traditional FOC, as well as the poor input-side power quality inherent in conventional drive circuits. In terms of speed control performance, the experimental results confirm that the designed extension robust controller outperforms the CSRLSMC, ERLSMC, and ETERLSMC in both speed command tracking and load regulation responses. Particularly under abrupt and severe load torque variations, the proposed controller demonstrates a faster recovery speed and a smaller speed drop, validating its exceptional robustness in speed command tracking and disturbance rejection capabilities. Regarding controller implementation and power quality improvement, this study adopted an MBD workflow, utilizing PSIM SimCoder automatic code generation technology to efficiently implement the complex control algorithm on a DSP. Furthermore, by deploying a PFC converter at the AC input front end of the drive system, the experimental results show that the PF under rated load reaches above 0.98, and the input current exhibits a low-harmonic sinusoidal waveform, effectively improving energy efficiency and complying with power quality standards.

The proposed control strategy features a straightforward architecture, a low computational burden, and eliminates the need for extensive training datasets, making it easily realizable. Consequently, this study not only realizes an integrated drive system that simultaneously possesses superior dynamic speed response and high input power quality, but also demonstrates the feasibility of applying ET to the dynamic parameter allocation of nonlinear systems. It offers significant theoretical reference value and high potential for future applications in the design of intelligent self-adaptive controllers and the development of green energy drive technologies.