Fayssal E. Benmohamed*
| Abdelmounaim Marouf | Ismail K. Bousserhane | Abdelfatah Nasri
© 2026 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
In this research, we unveil a novel speed estimation schema for a two-wheel drive propulsion system. This paper presents a study on the Model Reference Adaptive System (MRAS) as applied to electric vehicles (EVs) with two independent wheels. Our proposed propulsion system is powered by two induction motor (IM) that drive both the front. We incorporate an electronic differential to calculate speed references for both wheels, which ensures robust control of the vehicle's behavior on the road. We implemented Field Oriented Control (FOC) to achieve excellent dynamic response along with precise and reliable control of both speed and torque. We simulated the MRAS estimator of our electric vehicle using the MATLAB Simulink environment, and the results were encouraging, demonstrating the effectiveness of our proposed control strategy.
electric vehicle, induction motor, Model Reference Adaptive System, implemented Field Oriented Control
Modern control systems for electric motors play a crucial role in advancing industrial and transportation applications, especially as we move towards fully electrified propulsion systems. With the increasing demand for technologies that are high-performing, energy-efficient, and dependable, it's essential to create innovative solutions that blend energy efficiency, dynamic performance, and operational strength [1].
In the electric vehicle world, these innovations have led to better energy efficiency and a longer driving range, all while maintaining reliable performance in different conditions [2]. They've also shown to be quite effective in tackling the precision loss or instability problems that traditional methods often face at lower speeds [3].
The world of electric vehicles (EVs) has seen remarkable changes over the last few decades. We've moved from basic prototypes to highly advanced propulsion systems that feature cutting-edge digital controls and top-notch power electronics. The arrival of robust control strategies, like Field Oriented Control (FOC), has been a game-changer, allowing for impressive dynamic performance and precise management of speed and torque [3, 4]. FOC is well-known for its significant impact on high-performance induction motor (IM) drives. It allows for precise control of torque and flux, delivering dynamic performance that rivals that of separately excited DC motors [5].
Historically, early control methods leaned on simplified models and traditional techniques, but they ran into some challenges, especially when it came to low-speed or regenerative operations [3]. These challenges have sparked a lot of research into sensorless estimation, which helps cut costs and boost system reliability in tough industrial settings [4, 6].
This research primarily aims to accurately estimate state variables like speed and magnetic flux, all without relying on mechanical sensors [6]. This move towards “sensorless” systems is all about reducing possible failure points and enhancing the overall durability of drives [4]. In this context, Adaptive Observers and Model Reference Adaptive Systems (MRAS) have been shown to be quite effective at providing reliable estimations, even when faced with complex operating conditions [3]. Recent contributions have shown that MRAS schemes, especially those tailored for estimating rotor speed, can really enhance stability at low speeds and during regenerative operation [7].
In a broader sense, incorporating these estimation and adaptation algorithms into the electromechanical control environment has also had a positive impact on various other fields, including railway systems, wind turbines, and Ev’s. In these areas, achieving optimal energy conversion and management is crucial for enhancing overall system performance [1].
Various techniques are available for the estimation of Ev’s speed [5], rotor flux-based [8], MRAS based on reactive power, extended MRAS [4, 6, 9, 10], observer-based [11], rotor slot harmonics-based [12], and methods. Out of these. MRAS is popular due to its simplicity, requirement of less computation time, and good stability.
In this piece, we’re looking at the cutting-edge approaches to designing an estimation system for electric propulsion applications, with a special emphasis on sensorless speed control of an electric vehicle based on an IM using MRAS. active power has been proposed as a reference model; the rotor speed will be estimated from an adaptive model.
The precise illustrates the electrical propulsion system of a two-wheel electric vehicle (EV) is shown in Figure 1. The system is powered by a battery that supplies energy to a central control system. This control unit generates control signals for two independent voltage-source inverters, each driving a three-phase electric motor connected to a wheel (left wheel and right wheel). The independent control of each wheel enables differential torque distribution, improving vehicle maneuverability and stability. The dashed lines and angle α represent the steering or wheel orientation with respect to the vehicle frame [4, 5, 10, 13, 14].
Figure 1. Electrical propulsion system of two wheels electric vehicle (EV) chain
2.1 Induction motor model
In the dq-reference frame that rotates synchronously with the rotor flux, the dynamic voltage equations for the three-phase squirrel-cage IM are represented as follows [1, 11].
$\begin{gathered}\frac{d}{d t}\left[\begin{array}{l}i_{d s} \\ i_{q s} \\ \emptyset_{d r} \\ \emptyset_{q r}\end{array}\right]=\left[\begin{array}{cccc}-A_1 & \omega_e & A_2 & A_3 \omega_r \\ \omega_e & -A_1 & -A_3 \omega_r & A_2 \\ A_4 & 0 & -A_5 & -\omega_{s l} \\ 0 & A_4 & \omega_{s l} & -A_5\end{array}\right]\left[\begin{array}{l}i_{d s} \\ i_{q s} \\ \emptyset_{d r} \\ \emptyset_{q r}\end{array}\right] \\ +\frac{1}{\sigma L_s}\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{l}v_{d s} \\ v_{q s}\end{array}\right]\end{gathered}$ (1)
and
$\left[\begin{array}{l}i_{d s} \\ i_{q s}\end{array}\right]=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]\left[\begin{array}{l}i_{d s} \\ i_{q s} \\ \emptyset_{d r} \\ \emptyset_{q r}\end{array}\right]$ (2)
where,
$\begin{gathered}A_1=\frac{1}{\sigma L_s}\left(R_s+\frac{L_m^2}{L_r \tau_r}\right), A_2=\frac{1}{\sigma L_s}\left(\frac{L_m}{L_r \tau_r}\right), A_3 \\ =\frac{1}{\sigma L_s}\left(\frac{L_m}{L_r}\right), A_4=\frac{L_m}{\tau_r}, A_5=\frac{1}{\tau_r}\end{gathered}$
The parameters of the induction motor are provided in Appendix.
2.2 Mechanical loads of the studied two wheels electric vehicle
The vehicle's mechanical load is made up of three resistive torques, which consist of:
$\mathrm{T}_{ {aero }}=\frac{1}{2} \rho \mathrm{SC}_{\mathrm{x}} \mathrm{R}_{\mathrm{r}}^2 . \omega_{\mathrm{r}}^2$ (3)
$\mathrm{T}_{ {slope }}=\mathrm{Mg}. \sin \alpha \mathrm{R}_{\mathrm{r}}$ (4)
$\mathrm{T}_{\max }=\mathrm{Mgf}_{\mathrm{r}} \mathrm{R}_{\mathrm{r}}$ (5)
We obtain the total resistive torque:
$T_{{ev }}=T_{{slope }}+T_{ {tyre }}+T_{ {aero }}$ (6)
The way the traction system is modeled allows for the use of controls like vector control and speed control, which are essential for keeping the system stable overall [13, 14]. The EV parameters are included in the Appendix
The diagram in Figure 2 displays vector Control, which is often referred to as FOC, is crucial for maximizing the performance of drives. Its ease of use and swift dynamic response really set it apart [1]. This approach allows for decoupled control of torque and flux, similar to a DC machine, which is why it’s been widely adopted in high-performance drives. The beauty of vector control lies in its excellent dynamic performance and its capability to independently control torque and flux through the d-axis (for flux) and q-axis (for torque) currents when things are steady [1].
Figure 2. Schematic diagram of indirect flux vector control of induction motor (IM)
To effectively implement IFOC, we start by transforming the three-phase stator quantities (abc) into a synchronous reference frame (dq) that is aligned with the rotor flux [4]. This method provides independent control over the flux using the direct axis current (ids) and the torque through the quadrature axis current (iqs). The correct orientation of the reference frame requires the slip frequency, calculated as [5].
$\omega_s=\omega_m+\frac{L_m}{T_r} \cdot \frac{i_{q s}}{\emptyset_r}$ (7)
The MRAS has gained traction in IM drives for sensorless control applications, primarily because it can accurately estimate speed and flux without relying on additional mechanical sensors [4]. In the active power based MRAS approach, the reference model calculates active power using stator currents and voltages, while the adjustable model derives the same variable based on rotor speed. The difference between these two models is processed by an adaptation mechanism [15], often a PI controller, which adjusts the estimated rotor speed until everything converges smoothly [5, 10, 16].
This method stands out for its robustness and ease of use, as it avoids relying on flux models that can be quite sensitive to changes in parameters [9]. Furthermore, active power MRAS has been successfully implemented in electric vehicle propulsion, where it enhances the effectiveness of field-oriented control schemes by providing dependable speed estimation across various operating conditions [5]. You can find the overall structure of MRAS illustrated in Figure 3.
Figure 3. Basic configuration of Model Reference Adaptive System (MRAS)
The MRAS includes three essential blocks: the reference model, the adjustable model, and the adaptation mechanism as present in Figure 3 [15-18]. The reference model and adjustable model generate identical outputs, but they do it in distinct ways. To illustrate, the reference model follows a specific equation yref; whereas the adjustable model uses yest. The error from the two models is directed to the adaptation mechanism, which produces the estimated quantity (x). To accurately estimate x from the MRAS, the reference model must be independent of x, while the adjustable model needs to be dependent on it [3, 17-20].
Figure 4 shows the complete IM drive along with the proposed MRAS speed estimator. The speed estimation algorithm, which is outlined by a dotted line, consists of three primary blocks: the "Reference Model," the "Adjustable Model," and the "Adaptation Mechanism." We'll go into detail about the design of these blocks in the next subsections [3].
Figure 4. Proposed Model Reference Adaptive System (MRAS) based on active power for speed estimation
In the d-q synchronously rotating reference frame, the voltages equations of the IM can be articulated as [3]:
$\begin{gathered}V_{q r}=R_s i_{q s}+\omega_e \sigma L_s i_{d s}+p \sigma L_s\dot{\iota_{qs}}+\frac{L_m}{L_r}\left(\dot{\omega_e {\emptyset}_{d r}}+p \emptyset_{q r}\right)\end{gathered}$ (8)
$\begin{gathered}V_{d s}=R_s i_{d s}-\omega_e \sigma L_s i_{q s}+p \sigma L_s\dot{\iota_{ds}}-\frac{L_m}{L_r}\left(\dot{\omega_e {\emptyset}_{q r}}-p \emptyset_{d r}\right)\end{gathered}$ (9)
From a practical perspective, the instantaneous active power can provide really useful real-time data on the steady-state and dynamic behaviors of IM. This makes it a strong candidate for estimating IM parameters using a model reference adaptive control scheme. The active power for the IM can be calculated in a stationary reference frame using the following equation [5, 16]:
$P_1=\left(v_{d s} i_{d s}+v_{q s} i_{q s}\right)$ (10)
Use of Eq. (10) also has the advantage of dispensing with the voltage sensors. Moreover, due to the filtering aspects, it is always easy to deal with the reference quantities.
Using Eqs. (8) and (9) in Eq. (10) and simplifying the instantaneous value of P:
$\begin{aligned}P_2=\left[R_s i_{q s}+\omega_e \sigma L_s i_{d s}+p \sigma L_s\dot{\iota_{qs}}+\frac{L_m}{L_r}\left(\dot{\omega_e \emptyset_{d r}}+p \emptyset_{q r}\right) i_{d s}\right] \\+\left[R_s i_{d s}-\omega_e \sigma L_s i_{q s}+p \sigma L_s\dot{\iota_{ds}}+\frac{L_m}{L_r}\left(\dot{\omega_e \emptyset_{q r}}-p \emptyset_{d r}\right) i_{q s}\right]\end{aligned}$ (11)
It's important to point out that the expressions for P mentioned above don't include stator resistance, which is a Substituting the condition $\emptyset_{d r}=L_m i_{d s}$ and $\emptyset_{q r}=0$ for the (IFOC) IM drive, the more simplified expression P3 to:
$P_4=\frac{L_m^2}{L_r}\left(\omega_{s l}+\widehat{\omega_r}\right) i_{d s} i_{q s}+R_s\left(i_{d s}^2+i_{q s}^2\right)$ (12)
Figure 4 shows proposed MRAS based on active power for speed estimation of IM, where a PI controller adjusts the estimated rotor speed to match the outputs of a reference model P1 and an adjustable model P4 using electrical active power error.
Figure 5. Vector controlled IM drive with Model Reference Adaptive System (MRAS) based speed estimator
The complete control system is presented in Figure 5, which combines a vector-controlled IM drive with a speed estimator based on the MRAS to provide accurate and dependable sensorless speed regulation.
In order to characterize the behavior of the drive wheel system, simulations were performed on MATLAB software and the following results were obtained.
6.1 Testing a 10% slop in movement on right-hand turns at a speed of 100 km/h
Figure 6 demonstrates the simulation results for a vehicle navigating a right-hand bend at a steady speed of 100 km/h. Between t = 2.5 s and t = 4.5 s, the vehicle climbs a 10% incline while maintaining the same speed. Thereafter, from t = 5 to 9 seconds, it continues moving through right-hand bends. We assume that both motors operate without any disturbances during these maneuvers. In these situations, the drive wheels take different paths, leading to varying rotational speeds. Specifically, the right drive wheel turns more slowly than the left, as illustrated in subfigures:
(a)-Linear speed of right wheel
(b)-10℅slop at a steady speed of right wheel
(c)-Linear speed of Left drive wheel
(d)-10℅slop at a steady speed of left wheel
(e)-Linear speed of EV
(f)-The torques of EV
(g)-The torque of the right and left drive wheel
(h)-Quadrature and Direct flux
Figure 6. Simulation result for right hand-bands and 10 % slop at 100 km/h
6.2 Testing acceleration and deceleration with 10% slop
Figure 7 shows the speed profile displays repeated periods of acceleration, constant speed, and slowdown during the time span of 0 to 10 seconds. In the first second, the speed rises quickly from 0 km/h to roughly 40 km/h, after which it stays almost constant until about 2.5 s. It accelerates again to reach roughly 80 km/h by 3.5 s, followed by another steady-speed phase until about 4.5 s. After that, the speed rises to its maximum value of roughly 120 km/h at 5.5 s and stays virtually steady until 6.5 s. After a brief period of constant speed, there is a smooth deceleration phase during which the speed drops from 120 km/h to roughly 50 km/h between 6.5 and 8 seconds. Finally, the speed drops sharply from 50 km/h to 0 km/h between 9 s and 10 s, indicating a complete stop of the system and confirming controlled acceleration and deceleration behavior throughout the driving cycle.
(a)- Linear speed of the right wheel
(b)- Zoomed view of linear speed (estimation accuracy)
(c)- Torque components over time
(d)- Driver and ground flux
Figure 7. Testing acceleration and deceleration with 10% slop
Global Interpretation
A simplified qualitative comparison of Model Reference Adaptive System (MRAS)-based control strategies is presented in the Appendix.
In this paper, the research highlights the potential for improving the stability of two-wheel vehicles by using two independent drive wheels for motion, all through a MRAS based on Active Power. The study examines a control method applied to an EV, utilizing MRAS estimator to ensure safe driving on slopes. The MATLAB simulation results demonstrate that this estimate structure enables a robust speed control loop, achieving excellent dynamic performance in electric vehicles. The proposed MRAS-based model ensures precise speed estimation and regulation of the drive wheels under both flat and curved road conditions. The estimated wheel speeds closely track the reference speeds, maintaining high accuracy. Furthermore, road inclinations do not adversely affect the stability of the drive motors, highlighting the reliability and robustness of the proposed estimator and control strategy.
|
EV |
Electric vehicle |
|
IM |
Induction motor |
|
MRAS |
Model Reference Adaptive System |
|
FOC |
Field Oriented Control |
|
IFOC |
Indirect Field Oriented Control |
|
a, b, c |
Three-phase stator reference frame |
|
d, q |
Direct–Quadrature reference frame |
|
$v_{d s}, v_{q s}$ |
Stator voltages in the d–q reference frame |
|
$i_{d s}, i_{q s}$ |
Stator currents in the d–q reference frame |
|
$\emptyset_{d s}, \emptyset_{q s}$ |
Stator flux components in the d–q reference frame |
|
$\emptyset_r$ |
Rotor flux |
|
$R_s$ |
Stator resistance |
|
$R_r$ |
Rotor resistance |
|
$L_s$ |
Stator inductance |
|
$L_r$ |
Rotor inductance |
|
$L_m$ |
Mutual inductance |
|
$\omega_s$ |
Synchronous angular speed |
|
$\omega_r$ |
Rotor angular speed |
|
$\widehat{\omega_r}$ |
Estimated rotor angular speed |
|
$\omega_{s l}$ |
Slip angular speed |
|
$T_{E V}$ |
Electric vehicle traction torque |
|
$T_{ {aero }}$ |
Aerodynamic torque |
|
$T_{{slop }}$ |
Slope-related torque |
|
$T_{{tyer }}$ |
Tire resisting torque |
|
J |
Equivalent moment of inertia |
|
$f$ |
Viscous friction coefficient |
|
$\alpha$ |
Road slope angle |
|
$P$ |
Instantaneous active power |
|
$PI$ |
Proportional–Integral controller |
|
$K_p$ |
Proportional gain |
|
$K_i$ |
Integral gain |
|
$\emptyset_d$ |
Direct-axis flux |
|
$\emptyset_q$ |
Quadrature-axis flux |
|
$\theta$ |
Electrical angle |
|
$\sigma$ |
Coefficient of dispersion |
|
$R_{ {roy }}$ |
Wheel radius |
|
g |
Gravitational acceleration |
|
M |
Vehicle mass |
|
$\rho$ |
Air density |
|
$S$ |
Vehicle frontal area |
|
$C_x$ |
Aerodynamic drag coefficient |
σ is the coefficient of dispersion is given by [13] :
$\sigma=1-\frac{L_m^2}{L_s L_r}$
Table 1. Simplified qualitative comparison of Model Reference Adaptive System (MRAS)-based control strategies
|
Criterion |
MRAS-Based Active Power |
MRAS-Based Reactive Power |
MRAS-Based Flux |
|
Tracking performance |
High |
Medium–High |
High |
|
Overshoot / oscillations |
Low |
Low–Medium |
Low |
|
Steady-state error |
Very low |
Low |
Low |
|
Implementation complexity |
Medium |
Medium |
High |
Table 2. Parameters of induction motor (IM) [13]
|
Parameter |
Value |
Unit |
|
$R_r$ $R_s$ $L_r$ $L_s$ $L_m$ $f$ $P$ $J$ |
0.003 0.0044 494.9e-6 496.1e-6 482e-6 50 2 1.1 |
Ω Ω H H H Hz - Kg.m2 |
Table 3. Parameters of EV [13]
|
Parameter |
Value |
Unit |
|
$R_{r o y}$ g $f$ M $\rho$ $S$ $C_x$ $\alpha$ |
0.25 – 0.30 9.806 0.015 1000 1.225 0.5 – 0.7 0.7 – 0.9 0 – 10° |
m m/s² N·m·s/rad Kg kg/m³ m² - Rad |
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