The evolution of the battery EV car is fundamentally tied to advancements in its core propulsion technology. At the heart of this system lies the electric traction motor, with the Permanent Magnet Synchronous Motor (PMSM) being a predominant choice due to its high power density, efficiency, and torque capability. However, achieving precise, robust, and dynamic control of the PMSM is a critical challenge, directly impacting the driving performance, range, and safety of the battery EV car. Traditional control paradigms, while effective in ideal conditions, often struggle with inherent motor nonlinearities, parameter variations due to temperature and aging, and unpredictable load disturbances encountered in real-world driving.
This article presents a novel cascaded control strategy designed to address these challenges for PMSM drives in a battery EV car. We propose the integration of a Model-Free Predictive Control (MFPC) framework with a high-performance Exponential Extended State Observer (EESO), applied in a cascaded structure for both speed and current regulation. This approach aims to eliminate dependency on precise motor parameters, enhance disturbance rejection, and improve both transient and steady-state performance, contributing to a more responsive and reliable powertrain for the modern battery EV car.
Mathematical Model of PMSM and Traditional Control Limitations
The dynamic behavior of a PMSM in a rotating (d-q) reference frame is described by the following equations. The voltage equations are given by:
$$ u_d = R_s i_d + L_d \frac{di_d}{dt} – \omega_e L_q i_q $$
$$ u_q = R_s i_q + L_q \frac{di_q}{dt} + \omega_e L_d i_d + \omega_e \psi_f $$
The electromagnetic torque is produced according to:
$$ T_e = \frac{3}{2} n_p [ \psi_f i_q + (L_d – L_q) i_d i_q ] $$
For a surface-mounted PMSM (SPMSM) commonly used in many battery EV car applications, \( L_d = L_q = L_s \), simplifying the torque equation to \( T_e = \frac{3}{2} n_p \psi_f i_q \). The mechanical motion is governed by:
$$ J \frac{d\omega_m}{dt} = T_e – T_L – B\omega_m $$
where the key parameters and variables are summarized in the table below.
| Symbol | Description | Typical Unit |
|---|---|---|
| \( u_d, u_q \) | d- and q-axis stator voltages | V |
| \( i_d, i_q \) | d- and q-axis stator currents | A |
| \( R_s \) | Stator resistance | Ω |
| \( L_d, L_q, L_s \) | d- and q-axis, synchronous inductance | H |
| \( \psi_f \) | Permanent magnet flux linkage | Wb |
| \( \omega_e, \omega_m \) | Electrical and mechanical rotor speed | rad/s |
| \( n_p \) | Number of pole pairs | – |
| \( T_e, T_L \) | Electromagnetic and load torque | Nm |
| \( J \) | Combined rotor and load inertia | kg·m² |
| \( B \) | Viscous friction coefficient | Nm·s/rad |
Classical control methods like Field-Oriented Control (FOC) rely heavily on the accuracy of these parameters (\( R_s, L_s, \psi_f, J \)). In a battery EV car, these parameters are subject to significant variation: resistance changes with temperature, inductance saturates with current, and inertia effectively changes with passenger load. This mismatch degrades performance. While Finite Control Set Model Predictive Control (FCS-MPC) offers dynamic performance, its computational burden and sensitivity to parameter changes remain concerns for the cost-sensitive and reliability-demanding battery EV car market.
Foundation of Model-Free Predictive Control (MFPC) and Extended State Observer (ESO)
The proposed strategy is built upon the ultralocal model concept, which forgoes a detailed physical model. For a first-order single-input-single-output (SISO) system, the dynamics can be represented in a model-free form:
$$ \dot{y} = F + \alpha u $$
Here, \( y \) is the system output, \( u \) is the control input, \( \alpha \) is a non-physical tuning gain, and \( F \) is a lumped term representing all internal dynamics and external disturbances. For the current control of a PMSM in a battery EV car, we can define the outputs as the d- and q-axis currents. The key advantage is that the controller does not need explicit values for \( R_s \) or \( L_s \); instead, it actively estimates and compensates for the lumped disturbance \( F \).
A traditional Linear Extended State Observer (ESO) is used to estimate \( F \). It treats the disturbance as an extended state. For the current loop, with state \( x_1 = i_{dq} \) and extended state \( x_2 = F_{dq} \), a standard ESO is constructed as:
$$ \begin{aligned}
e &= i_{dq} – \hat{i}_{dq} \\
\dot{\hat{i}}_{dq} &= \hat{F}_{dq} + \alpha u_{dq} + \beta_1 e \\
\dot{\hat{F}}_{dq} &= \beta_2 e
\end{aligned} $$
Where \( \hat{i}_{dq} \) and \( \hat{F}_{dq} \) are the estimated current and disturbance, and \( \beta_1, \beta_2 \) are observer gains. The estimated disturbance \( \hat{F}_{dq} \) is then fed back for compensation in the MFPC law: \( u_{dq} = (\dot{y}^* – \hat{F}_{dq}) / \alpha \), where \( \dot{y}^* \) is the desired derivative of the output, often derived from a tracking error. This traditional MFPC-ESO method improves robustness but can suffer from slow convergence and limited estimation accuracy during aggressive transients, which are common in a dynamically driven battery EV car.
Proposed Cascaded MFPC with Exponential Extended State Observer (EESO)
To overcome the limitations of the linear ESO, we introduce an Exponential Extended State Observer (EESO) and deploy it within a cascaded MFPC structure for both the inner current loop and the outer speed loop of the battery EV car’s PMSM drive.
Exponential Extended State Observer (EESO) Design
The core innovation is the introduction of a nonlinear gain \( k \) that varies with the observer estimation error. This gain amplifies the correction terms when the error is large, leading to faster convergence, and reduces to unity as the error approaches zero to prevent overshoot. The EESO dynamics are defined as:
$$ \begin{aligned}
e &= x – \hat{x}_1 \\
\dot{\hat{x}}_1 &= \hat{x}_2 + \alpha u + k_1 l_1 e \\
\dot{\hat{x}}_2 &= k_2 l_2 e
\end{aligned} $$
The nonlinear gains \( k_1 \) and \( k_2 \) are designed as functions of the magnitude of the observation error \( |e| \). A practical and effective design uses a tangent function for smooth nonlinearity:
$$ k_1 = 1 + \tan\left(\Theta \frac{|x – \hat{x}_1|}{lim}\right) $$
$$ k_2 = k_1^2 $$
Here, \( \Theta \) (typically \( \pi/4 \)) scales the nonlinear effect, and \( lim \) is a boundary layer parameter related to the expected maximum error. This formulation ensures that during large transients—such as sudden acceleration or load change in a battery EV car—the observer reacts much more aggressively to estimate the rapidly changing disturbance, thereby improving the controller’s ability to compensate in real-time.
Cascaded Control Structure for Battery EV Car Drives
We apply the MFPC-EESO principle to both control loops in a cascaded architecture, which is the standard for high-performance motor drives in a battery EV car.
1. Inner Loop: MFPC-EESO for Current Control
The d- and q-axis currents are controlled independently using the ultralocal model \( \dot{i}_{dq} = F_{dq} + \alpha_c u_{dq} \). A dedicated EESO, as per the equations above, estimates the current \( \hat{i}_{dq} \) and the lumped disturbance \( \hat{F}_{dq} \) which includes cross-coupling, back-EMF, and resistance drop. The control law is then:
$$ u_{dq}^* = \frac{\dot{i}_{dq}^* – \hat{F}_{dq}}{\alpha_c} $$
Where \( \dot{i}_{dq}^* \) is generated from a simple proportional controller on the current tracking error: \( \dot{i}_{dq}^* = K_{p,i}(i_{dq}^* – \hat{i}_{dq}) \). The estimated current \( \hat{i}_{dq} \) is used for feedback, further filtering out measurement noise.
2. Outer Loop: MFPC-EESO for Speed Control
Similarly, the speed loop is constructed using an ultralocal model: \( \dot{\omega}_m = F_{\omega} + \alpha_{\omega} T_e^* \). Here, \( T_e^* \) is the reference torque (proportional to \( i_q^* \)), and \( F_{\omega} \) lumps together load torque disturbance, friction, and inertia effects. A separate EESO is designed for the speed loop:
$$ \begin{aligned}
e_{\omega} &= \omega_m – \hat{\omega}_m \\
\dot{\hat{\omega}}_m &= \hat{F}_{\omega} + \alpha_{\omega} T_e^* + k_{\omega 1} l_{\omega 1} e_{\omega} \\
\dot{\hat{F}}_{\omega} &= k_{\omega 2} l_{\omega 2} e_{\omega}
\end{aligned} $$
The speed control law becomes:
$$ i_q^* = \frac{1}{K_T} \left( \frac{\dot{\omega}_m^* – \hat{F}_{\omega}}{\alpha_{\omega}} \right) $$
where \( K_T = \frac{3}{2} n_p \psi_f \) and \( \dot{\omega}_m^* = K_{p,\omega}(\omega_m^* – \hat{\omega}_m) \). The d-axis current reference \( i_d^* \) is typically set to zero for an SPMSM. This cascaded MFPC-EESO structure effectively creates a “model-free” alternative to the traditional PI-based cascade control, with inherently better disturbance rejection capabilities crucial for a battery EV car experiencing varying road loads.
Parameter Tuning and Stability
The tuning process is systematic. First, for the EESO, the linear gains \( l_1 \) and \( l_2 \) (for both current and speed observers) are designed by placing the poles of the linearized observer (with \( k=1 \)) at a desired location \( -\lambda_c \) in the continuous domain. This yields:
$$ l_1 = -2\lambda_c, \quad l_2 = \lambda_c^2 $$
For discrete-time implementation with sampling time \( T_s \), the equivalent pole in the z-domain is \( z = 1 – \lambda_c T_s \). Stability requires \( |z| < 1 \), which guides the selection of the observer bandwidth \( \lambda_c \). The nonlinear parameters \( \Theta \) and \( lim \) are then tuned to shape the transient convergence without inducing instability. The control gain \( \alpha \) is a scalar that can be tuned experimentally for optimal response, significantly simplifying the process compared to tuning multiple PI parameters that are coupled to motor parameters. This simpler tuning enhances the adaptability of the control system across different PMSM models in a battery EV car platform.
Performance Evaluation: Comparative Analysis
To validate the superiority of the proposed Cascaded MFPC-EESO, a comparative analysis with the traditional MFPC with Linear ESO (MFPC-LESO) is conducted via simulation and experimental prototype, with parameters representative of a light-duty battery EV car auxiliary drive or main traction system.
| Parameter | Value |
|---|---|
| DC Bus Voltage (\(U_{dc}\)) | 311 V |
| Stator Resistance (\(R_s\)) | 2.65 Ω |
| Stator Inductance (\(L_s\)) | 5.2 mH |
| PM Flux Linkage (\(\psi_f\)) | 0.048 Wb |
| Rotor Inertia (\(J\)) | 5.0×10⁻⁴ kg·m² |
| Pole Pairs (\(n_p\)) | 4 |
| Rated Speed | 1500 rpm |
| Rated Torque | 1.27 Nm |
The tests evaluate dynamic tracking, disturbance rejection, and steady-state performance under scenarios critical for a battery EV car.
Dynamic Speed Tracking
A speed profile commands the motor to track 500 rpm, 1500 rpm, and 900 rpm sequentially. The proposed C-MFPC-EESO demonstrates consistently faster settling time and reduced overshoot compared to MFPC-LESO. The q-axis current response with C-MFPC-EESO is significantly smoother, with lower peak current during transitions, indicating reduced stress on the power electronics of the battery EV car. The quantitative improvements are summarized below:
| Performance Metric | @ 500 rpm (MFPC-LESO) | @ 500 rpm (C-MFPC-EESO) | Improvement |
|---|---|---|---|
| Speed Settling Time | ~35 ms | ~18 ms | ≈ 49% faster |
| q-current Overshoot | ~42% | ~20% | ≈ 52% reduction |
| Speed Ripple (Steady-State) | ±15 rpm | ±7 rpm | ≈ 53% reduction |
Disturbance Estimation and Rejection
The performance of the EESO itself is a critical advantage. The following plot conceptually illustrates the estimation error \( e_F = F – \hat{F} \) for the q-axis disturbance during a speed step. The EESO (orange) converges to a precise estimate much faster and with smaller error magnitude than the linear ESO (blue), especially during the initial transient phase.
This superior estimation translates directly into better disturbance rejection. In a step-load torque disturbance test, where a sudden load is applied to the motor (simulating a change in gradient for the battery EV car), the C-MFPC-EESO strategy results in a smaller and shorter-lived speed dip. The recovery time is improved by over 30%, and the associated torque/current ripple during the stabilization period is markedly lower, enhancing passenger comfort and drivetrain smoothness in the battery EV car.
Robustness to Parameter Variation
A key test for any motor drive in a battery EV car is robustness. The stator resistance was intentionally increased by 100% in the simulation to model severe heating. While both model-free methods degrade less than a traditional PI controller, the C-MFPC-EESO maintained better speed regulation and lower steady-state error compared to MFPC-LESO. This is because the EESO’s faster and more accurate adaptation to the changed system dynamics (lumped into \( F \)) allows the MFPC law to compensate more effectively.
Conclusion and Impact on Battery EV Car Technology
This article has presented a comprehensive cascaded Model-Free Predictive Control strategy enhanced with an Exponential Extended State Observer for PMSM drives. The proposed C-MFPC-EESO method successfully addresses several key challenges in high-performance motor control for battery EV cars:
- Enhanced Dynamic Performance: The nonlinear EESO provides rapid and accurate estimation of lumped disturbances, enabling the predictive controller to achieve faster speed and current tracking with reduced overshoot and settling time.
- Superior Disturbance Rejection: The combined action of real-time disturbance estimation and feedforward compensation in the MFPC law offers strong rejection of load torque disturbances and internal parameter variations.
- Reduced Parameter Sensitivity: By utilizing an ultralocal model, the control design is effectively decoupled from the precise PMSM parameters (\(R_s, L_s, J\)), simplifying commissioning and improving robustness over the operating life of the battery EV car.
- Simplified Tuning: The reduction of tunable parameters to primarily observer bandwidth (\(\lambda_c\)) and control gain (\(\alpha\)) streamlines the calibration process compared to traditional multi-PI cascade tuning.
The simulation and experimental results confirm the tangible benefits of this approach, including smoother current waveforms, more agile speed response, and robust operation under varying conditions. Implementing such an advanced control strategy can contribute significantly to improving the driving experience, energy efficiency, and overall reliability of the next generation of battery EV cars. Future work may explore the integration of this control scheme with fault diagnosis algorithms and its optimization for very high-speed operation relevant to premium battery EV car platforms.