Improved PMSM Speed Control Using Backstepping: Stability and Performance Analysis

Permanent Magnet Synchronous Motors (PMSMs) have become indispensable in various electromechanical applications in recent decades due to their high efficiency, compact size, and superior power density. These motors are widely used in electric vehicles, industrial robotics, aerospace propulsion systems, and renewable energy applications such as solar and wind power. Their ability to deliver high performance with precise control makes them ideal for energy-efficient and high-precision applications [1, 2].

Despite their advantages, PMSMs exhibit nonlinear behavior that introduces significant control challenges. These include abrupt changes in electrical load, thermal variations, and magnetic field saturation, which can hinder performance stability under varying operating conditions [3, 4].

The proportional-integral (PI) controller remains the most commonly used approach due to its simplicity and ease of implementation. Nevertheless, PI control struggles in nonlinear systems, failing to adequately compensate for dynamic parameter variations and sudden disturbances, leading to degraded tracking performance and stability issues. MPC offers improved adaptability to system changes but is computationally intensive, making it less suitable for real-time applications requiring fast response. Similarly, SMC is highly robust against disturbances but suffers from chattering effects, which can negatively impact motor efficiency and stability in high-precision applications [5-9].

Backstepping control has emerged as a promising alternative for handling nonlinear systems. By structuring the control process into sequential stages, each stabilized before proceeding to the next. Backstepping ensures a systematic and robust approach to nonlinear control. Grounded in Lyapunov stability theory, this method effectively adapts to system variations, making it particularly well-suited for PMSM applications [10, 11].

This paper proposes an improved backstepping control strategy for PMSMs, incorporating advanced mathematical analysis and MATLAB/SIMULINK simulations. A comparative performance evaluation is conducted against conventional PI control to assess response time, tracking accuracy, and robustness under varying load conditions and external disturbances. The results demonstrate that backstepping reduces settling time by 45% and improves tracking accuracy by 30% relative to PI control. Additionally, the method effectively mitigates torque disturbances up to ±5 Nm, reinforcing its reliability for industrial applications demanding high stability and precision.

This section details the application of backstepping control in regulating PMSM. Figure 1 illustrates the overall structure of the PMSM speed control system using the backstepping approach. The proposed control scheme ensures speed regulation while maintaining stability through sequential control steps. The system architecture comprises the PMSM dynamic model, current controllers, and a backstepping-based regulator, which dynamically adjusts the control voltages $V_d$ and $V_q$ based on Lyapunov stability conditions [17].

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Figure 1. Structure of PMSM speed adjustment by backstepping

The PMSM model can be rewritten in Eq. (8):

$\left\{\begin{aligned} \frac{d i_d}{d t} & =-\frac{R_s}{L_d} i_d+\frac{P \Omega L_q}{L_d} i_q+\frac{1}{L_d} V_d \\ \frac{d i_q}{d t} & =-\frac{R_s}{L_q} i_q-\frac{P \Omega L_d}{L_q} i_d-\frac{P \Phi_f \Omega}{L_q}+\frac{1}{L_q} V_q \\ \frac{d \Omega}{d t} & =\frac{P\left(L_d-L_d\right)}{J} i_d i_q+\frac{P \Phi_f}{J} i_q-\frac{f}{J} \Omega-\frac{1}{J} C_r\end{aligned}\right.$          (8)

The fundamental backstepping control method is cascading first-order subsystem stabilization utilizing the Lyapunov stability criteria. This strategy improves system resilience and asymptotic stability [18].

While backstepping control is highly effective in handling nonlinearities and disturbances, it can be computationally demanding due to the multiple steps involved in error computation and Lyapunov function evaluation. This increases the overall processing time, which can be a limiting factor for real-time applications, such as embedded systems or motor control with high-speed requirements. In practice, to mitigate these computational challenges, optimization techniques and hardware acceleration (such as parallel processing or the use of GPUs) may be employed to reduce the computational burden. Moreover, reducing-order models or approximations of the Lyapunov function can help achieve real-time performance while maintaining system stability [19].

The main objective is to control velocity by choosing expressions for subsystems $\left(d_i / d_t\right)$ and $\left(d i_q / d t\right)$, utilizing stator currents $\left(i_d\right.$ and $\left.i_q\right)$ as intermediate variables The goal is to manage PMSM speed while preserving system stability by determining voltage commands $\left(V_d\right.$ and $\left.V_q\right)$ [20].

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Figure 2. Internal structure of backstepping regulator block

Figure 2 presents the internal configuration of the backstepping regulator block. The system employs a multi-stage feedback mechanism to generate the required control voltages based on stator currents $\left(i_d, i_q\right)$ and rotor speed measurements. This hierarchical control strategy enables real-time stability adjustments by progressively minimizing system errors through Lyapunov-based adaptation.

Step 1-Calculation of the control law $V_d^{r e f}$

The first step in designing the backstepping control law involves regulating the direct axis current $i_d$. The regulation error is defined by Eq. (9):

$i_d^{r e f}=0, e_1=i_d^{r e f}-i_d$           (9)

The dynamic equation of error derived from Eq. (9) is shown in Eq. (10):

$\dot{e}_1=\dot{i}_d^{\text {ref }}-\dot{i}_d=\dot{i}_d^{\text {ref }}+\frac{R_s}{L_d} i_d-\frac{P \Omega L_q}{L_d} i_q-\frac{1}{L_d} V_d$           (10)

We employ the Lyapunov function $V_1$ as a form of energy to ensure the error converges to zero and the current is controlled, as shown in Eq. (11):

$V_1=\frac{1}{2} e_1^2$           (11)

The derivative of the function is given by Eq. (12):

$\dot{V}_1=e_1 \dot{e}_1=e_1\left(\dot{i}_d^{r e f}+\frac{R_s}{L_d} i_d-\frac{P \Omega L_q}{L_d} i_q-\frac{1}{L_d} V_d\right)$           (12)

For the derivative of $V_1$ to be negative, the following form is used (as introduced by the Backstepping method), as shown in Eq. (13):

$\dot{V}_1=e_1\left(k_1 e_1+\frac{R_s}{L_d} i_d-\frac{P \Omega L_q}{L_d} i_q-\frac{1}{L_d} V_d\right)=-k_1 e_1^2$           (13)

This equation forces the current $i_d$ to follow its reference, and the voltage $V_d$ controls the subsystem to guarantee Lyapunov stability. From this, we derive the reference voltage $i_d^{\text {ref }}$, as given in Eq. (14):

$V_d^{r e f}=L_d\left(k_1 e_1+\frac{R_s}{L_d} i_d-\frac{P \Omega L_q}{L_d} i_q\right)$           (14)

Step 2-Calculation of the Virtual Control Law $i_q^{\text {ref }}$ :

The second step in the backstepping control design focuses on regulating the rotor velocity $\Omega$. The velocity error is defined by Eq. (15):

$e_1=\Omega_{r e f}-\Omega n$           (15)

The derivative of this error is expressed as follows in Eq. (16):

$\dot{e}_1=\dot{\Omega}_{r e f}-\dot{\Omega}$          (16)

Eq. (17) shows how the Lyapunov function approach creates the virtual control law.

$\dot{e}_2=\dot{\Omega}_{r f}-\frac{3}{2}\left(\frac{p\left(L_d-L_q\right)}{J} i_d+\frac{p \Phi_f}{J}\right) i_q+\frac{f}{J} \Omega+\frac{1}{J} C_r$          (17)

The goal is for the error to converge to zero by choosing $i_q$ as the virtual control. The extended Lyapunov function is defined as shown in Eq. (18):

$V_2=V_1+\frac{1}{2} e_2^2=\frac{1}{2}\left[e_1^2+e_2^2\right]$          (18)

The time derivative is given by Eq. (19):

$\begin{aligned} & \dot{V}_2=\dot{V}_1+e_2 \dot{e}_2 \\ & =-k_1 e_1^2+e_2\left[f_2\left(x_1, x_2\right)+g_1\left(x_1, x_2\right) x_3-\dot{a}_1\right]\end{aligned}$          (19)

Step 3-Calculation of the final control law $V_q^{\text {ref }}$

In this step, the final control law is derived to determine the reference voltage for the entire system. The objective is to stabilize the system by extending the control strategy from the virtual control stage to the actual control input. The regulation error is given by Eq. (20):

$e_3=i_q^{r e f}-i_q$          (20)

The dynamic equations of error are given by Eq. (21):

$\dot{e}_3=\dot{i}_q^{r e f}-i_q=\dot{i}_q^{r e f}+\frac{R_s}{L_q} i_q+\frac{p \Omega L_d}{L_q} i_d-\frac{1}{L_q} V_q$          (21)

The Lyapunov function is extended as shown in Eq. (22):

$V_3=V_1+V_2+\frac{1}{2} e_3^2=\frac{1}{2}\left[e_1^2+e_2^2+e_3^2\right]$          (22)

The derivative is given by Eq. (23):

$\dot{V}_3=\dot{V}_1+\dot{V}_2+e_3 \dot{e}_3$          (23)

By choosing $\dot{V}_3$ to be semi-definite negative, we obtain Eq. (24):

$\dot{V}_3=-k_1 e_1^2-k_2 e_2^2-k_3 e_3^2 \leq 0$          (24)

From this, we obtain the final control law for $V_q^{\text {ref }}$, as given in Eq. (25):

$V_q^{r e f}=L_q\left[k_3 e_3+\frac{J}{p \Phi_f}\left(\left(k_2-\frac{f}{J}\right)\left(-\frac{p \Phi_f}{J} i_q+\frac{f}{J} \Omega+\frac{1}{J} C_r\right)+k_2 \dot{\Omega}_{r e f}+\ddot{\Omega}_{r e f}\right)\right]$          (25)

This final control law ensures that the quadrature-axis current $i_q$ tracks its reference value accurately, leading to precise speed regulation. Consequently, the reference voltages are determined to guarantee the asymptotic stability of the entire PMSM control system. The backstepping control approach achieves a robust and optimized overall system performance by systematically stabilizing each subsystem.

The simulation results confirm the superiority of Backstepping control in regulating PMSM speed under various operating conditions.

Compared to MPC and SMC, the backstepping control method offers several advantages. While MPC is more adaptive and can optimize control inputs over a predictive horizon, it tends to be computationally expensive, which limits its application in real-time systems. On the other hand, SMC provides robust disturbance rejection but suffers from chattering effects, which can impact system stability and efficiency, especially in high-precision applications. The backstepping control, however, balances robustness and precision, ensuring system stability without the computational complexity of MPC or the chattering effects of SMC. Moreover, the ability of backstepping to handle nonlinearities directly, without the need for linear approximations, makes it a superior choice for systems with significant nonlinear dynamics, such as PMSM.

The key observations include:

• Fast dynamic response: The backstepping controller achieves a rapid response time of t=0.01 s, significantly outperforming conventional PI controllers.

• No overshoot or instability: The controller ensures smooth tracking of the reference speed without exceeding set points.

• High robustness to load variations: Under a load torque of 5 N.m, the PMSM speed remains stable, demonstrating strong disturbance rejection capabilities.

• Efficient speed reversal: The system successfully transitions between positive and negative speed directions without oscillatory behavior.

• Decoupled control of torque and flux: Similar to a DC motor, the Backstepping approach effectively separates torque and flux components, improving efficiency and stability.

Overall, these results validate the effectiveness of the Backstepping method in handling nonlinear PMSM dynamics and improving system reliability across different operating scenarios. A detailed performance comparison between the Backstepping control method and the conventional PI controller reveals significant advantages in dynamic response, stability, and robustness.

The results indicate that Backstepping reduces settling time by 50%, current regulation by 40%, and flux variations during speed transitions significantly decrease.

The performance of backstepping control was analyzed at both low and high speeds. The backstepping controller demonstrated superior responsiveness at low speeds, achieving faster convergence to the reference speed with minimal overshoot. The system maintained smooth operation despite rapid load changes, showcasing its effectiveness in precise low-speed regulation. At high speeds, backstepping control showed enhanced stability and tracking accuracy compared to traditional control methods. It handled rapid speed transitions effectively without losing accuracy, ensuring smooth and efficient motor operation across various speeds.

These results confirm that backstepping control provides robust performance at low and high speeds, making it ideal for applications requiring precise speed regulation across the entire speed range. Future work will conduct hardware-in-the-loop (HIL) testing further to validate the performance of the backstepping control strategy.

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Figure 18. Performance comparison of control methods

Figure 18 compares backstepping, MPC, and SMC control methods based on three key performance metrics: settling time (s), steady state error, and overshoot (%). The data highlights the superior performance of Backstepping control across all metrics. Specifically:

Settling time: Backstepping achieves the shortest settling time, demonstrating faster convergence than MPC and SMC.

Steady state error: Backstepping also shows the lowest steady-state error, ensuring greater accuracy in maintaining the reference speed.

Overshoot: Backstepping outperforms MPC and SMC by maintaining the lowest overshoot percentage, indicating better stability under varying conditions.

These improvements illustrate why Backstepping control is more efficient in systems requiring fast response, minimal error, and robust performance.

HIL simulation offers a bridge between the simulation environment and real-world systems, allowing real-time validation of the control strategy. The PMSM model will be implemented on a real-time simulator, and control signals will be sent to a physical PMSM system to evaluate the performance of the backstepping controller under actual operational conditions. This testing will help assess the robustness and effectiveness of the control strategy, ensuring that the backstepping control method is suitable for deployment in practical applications, such as electric vehicle propulsion systems and industrial automation.

One crucial aspect to consider when applying backstepping control to PMSM is the variation of motor parameters, particularly resistance and inductance. These parameters can change due to environmental factors such as temperature fluctuations or mechanical wear.

The findings indicate that backstepping control outperforms MPC and SMC in key performance metrics. Specifically, backstepping control achieves a 44.44% improvement in settling time and a 33.33% improvement in disturbance rejection compared to MPC and SMC, respectively. This makes backstepping an ideal choice for applications requiring fast, accurate, and stable performance, such as PMSM motor control in industrial automation and electric vehicles.

In this study, we analyzed the effect of these variations on system performance. The results showed that backstepping control remained stable despite significant changes in resistance and inductance. However, more considerable variations in motor parameters slightly affected the system's response time and steady-state accuracy. To mitigate these effects, adaptive techniques or real-time recalibration of the control gains (k₁, k₂, and k₃) could be employed to ensure optimal performance even when motor parameters change.

Additionally, backstepping control can be extended to ensure coordinated operation between multiple motors in multi-motor systems, such as those used in electric vehicles or industrial automation. The challenge lies in synchronizing the motor speeds and torques to ensure efficient system performance. Backstepping control can be applied to each motor individually, with the control laws designed to guarantee global system stability. This coordination is achieved by creating a distributed control strategy that ensures all motors operate within their stability limits while sharing the load. By integrating backstepping with a cooperative control approach, multi-motor systems can achieve improved performance, load distribution, and fault tolerance, even under dynamic operational conditions.

This superior performance is attributed to the adaptive nature of the Backstepping approach, which effectively manages nonlinearities and external disturbances.

Furthermore, backstepping's ability to maintain system stability under various operating conditions highlights its suitability for high-performance applications requiring precise speed and torque control.

Table 2 clearly illustrates that Backstepping significantly improves transient and steady-state performance compared to the PI controller. While PI control is computationally more uncomplicated, it struggles with load variations and exhibits higher torque overshoot, making it less suitable for high-precision applications. In contrast, Backstepping control provides smoother responses, minimal steady-state errors, and robust adaptation to dynamic changes, making it ideal for advanced motor control applications such as electric vehicle propulsion and industrial automation.

Table 2. Performance comparison between backstepping and PI control strategies