An Adaptive Bandpass Full-Order Observer with a Compensated PLL for Sensorless IPMSMs

Abstract

Model-based sensorless control of interior permanent-magnet synchronous motors (IPMSMs) typically employs an estimation observer with embedded position information, followed by a position extraction process. Although a type-2 phase-locked loop (PLL) is widely adopted for position and speed extraction, it suffers from steady-state tracking errors under variable-speed operation, leading to torque bias in IPMSM torque control. To mitigate this issue, this paper first proposes an adaptive bandpass full-order observer in the stationary reference frame. Subsequently, a Kalman filter (KF)-based compensation strategy is introduced for the PLL to eliminate tracking errors while maintaining system stability. Experimental validation on a 300 kW platform confirms the effectiveness of the proposed sensorless torque control algorithm, demonstrating significant reductions in position error and torque fluctuations during acceleration and deceleration.

1. Introduction

Permanent magnet synchronous motors (PMSMs) are increasingly utilized in electric vehicles, home appliances, and rail transportation due to their high efficiency and power density [1,2,3]. However, reliability concerns, particularly position sensor failures in voltage source inverter (VSI)-driven interior PMSM (IPMSM) systems, hinder lifecycle cost reduction. As a result, sensorless control, which eliminates the need for additional hardware, has emerged as a promising solution to enhance system robustness [4]. This approach is especially critical for high-power applications such as mining trucks, cranes, and military vehicles, where reliable torque output under dynamic operating conditions is essential [5]. Model-based methods [6,7,8,9,10,11] are usually adapted for medium- and high-speed sensorless control, which estimates rotor position using motor model-derived back electromotive force (EMF) or flux observers.

Among model-based techniques, the model reference adaptive system (MRAS) offers simplicity but suffers from parameter sensitivity [12]. The sliding mode observer (SMO) provides robustness; recent research focuses on mitigating its low-pass filter (LPF) delay and chattering effects, such as adjusting the gain according to the switching delay [13], the fast terminal sliding mode observer [14] or the fuzzy logic controller-based SMO [15]. The full-order observer incorporates the extended EMF (EEMF) as a state variable and introduces feedback gain into the observer, ensuring system stability and dynamic performance across the full speed range [16,17]. Additionally, learning observer-based strategies has also been explored for EMF estimation [18,19], but their high-frequency switching components introduce chattering. Data-driven methods, while promising [20], demand extensive training datasets and high-performance controllers (e.g., DSP C6000 and dSPACE) due to computational complexity [21].

A full-order observer’s advantage lies in its ability to incorporate system dynamics through feedback gain tuning, enabling precise pole placement for desired frequency-domain responses. The EEMF is viewed as a state instead of a disturbance, reducing the high bandwidth requirement in comparison with the linear disturbance observer [16]. High-performance sensorless control can thus be achieved by tailoring the observer gains to the specific characteristics of the system. A high-bandwidth adaptive full-order observer is designed in [17] with a cascaded low-frequency feedback loop for speed estimation. The author in [22] proposes a pole-placement-based full-order observer feedback gain design method to address low-speed instability in induction motor sensorless control. An adaptive full-order observer gain is designed to suppress the resistance mismatch effect for sensorless variable flux reluctance motors [23]. To further improve EEMF estimation accuracy, a two-degree-of-freedom structure-based backstepping full-order observer is developed in [24] to suppress the DC error for IPMSM. The observer ensures 0 dB gain and zero phase delay at the operating frequency. However, due to its asymmetric frequency-domain response between the positive and negative frequency components, its estimation accuracy may be compromised. To overcome this limitation, this paper proposes an adaptive bandpass full-order observer with a symmetrical frequency-domain response while maintaining 0 dB gain and zero phase delay at the operating frequency, thereby improving estimation accuracy.

For position extraction, conventional approaches like the gradient descent algorithm [17] or arctan(−e_α/e_β) suffer from noise sensitivity. The phase-locked loop (PLL), particularly the type-2 quadrature PLL, is a preferred alternative [10,25,26,27,28,29]. However, it suffers from steady-state errors during frequency ramps. While the type-3 structure feedforward PLLs [10] and mechanical model-based compensation [27] have been proposed, they introduce stability risks or depend on inertia (J) and friction (B) parameters.

To address this issue, this article proposes an adaptive bandpass full-order observer with a compensated PLL. The primary contributions of this article are as follows:

• Development of an adaptive bandpass full-order observer for stationary-frame EEMF estimation;
• Quantitative analysis of PLL-induced position errors and their impact on torque output during acceleration and deceleration;
• Proposal of a Kalman filter (KF)-based PLL compensation strategy that eliminates steady-state errors without increasing system order or requiring mechanical parameters;
• Experimental validation on a 300 kW IPMSM platform, which demonstrates significant position error compensation and torque fluctuation reduction under dynamic conditions

2. Sensorless Control of IPMSM

2.1. Model of Full-Order Observer

The current model of IPMSM is

$$\left[\begin{array}{l}{u}_d\hfill \\ u_q\hfill \end{array}\right]=\left[\begin{array}{cc}{R}_s+\rho L_d& -{\omega }_e{L}_q\\ {\omega }_e{L}_d& R_s+\rho L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}0\\ {\omega }_e\psi \end{array}\right]$$

(1)

where i_dq and u_dq are the stator currents and voltages, R_s is the stator resistance, L_d and L_q are the d and q axis inductances, Ψ is the permanent magnet flux, ρ is the differential operator, and ω_e is the electric speed.

To obtain an asymmetrical stationary frame model, Equation (1) is symmetrized as follows [17]:

$$\left[\begin{array}{l}{u}_d\hfill \\ u_q\hfill \end{array}\right]=\left[\begin{array}{cc}{R}_s+\rho L_d& -{\omega }_e{L}_q\\ {\omega }_e{L}_q& R_s+\rho L_d\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}0\\ (L_d-L_q)({\omega }_e{i}_d-{\dot{i}}_q)+{\omega }_e\psi \end{array}\right].$$

(2)

Transferring Equation (2) with the inverse Park transform, the IPMSM model in the stationary reference frame can be represented as follows:

$${\mathit{u}}_{\alpha \beta }=\left[\left(R_s+\rho L_d\right)I+\Delta L{\omega }_{e}J\right]{\mathit{i}}_{\alpha \beta }+{\mathit{e}}_{\alpha \beta }$$

(3)

with the extended EMF (EEMF)

$${\mathit{e}}_{\alpha \beta }=\left\{\left(L_d-L_q\right)\left({\omega }_e{i}_d-{\dot{i}}_q\right)+{\omega }_e\psi \right\}\left[\begin{array}{l}-\sin {\theta }_e\hfill \\ \cos {\theta }_e\hfill \end{array}\right]$$

(4)

where

$$\mathbf{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathbf{J}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],\Delta L=L_d-L_q.$$

u_αβ = [u_α u_β]^T is the stator voltage, i_αβ = [i_α i_β]^T is the stator current, and θ_e represents the rotor position.

Extending the e_αβ to be a state variable, the full-order model of the IPMSM can be constructed as follows:

$$\begin{array}{l}\dot{x}=A_{c}x+B_{c}u\\ y=C_{c}x\end{array}$$

(5)

where

$$\begin{array}{l}x={\left[{\mathit{i}}_{\alpha \beta }\hspace{1em}{\mathit{e}}_{\alpha \beta }\right]}^{\mathrm{T}},u={\mathit{u}}_{\alpha \beta },y={\mathit{i}}_{\alpha \beta },A_c=\left[\begin{array}{cc}-I{\displaystyle \frac{R_s}{L_d}}+J{\omega }_e{\displaystyle \frac{\Delta L}{L_d}}& \hfill -{\displaystyle \frac{I}{L_d}}\\ 0& \hfill J{\omega }_e\end{array}\right],\\ B_c={\left[{\displaystyle \frac{I}{L_d}}\hspace{1em}0\right]}^{\mathrm{T}},C_c=\left[1\hspace{1em}0\right].\end{array}$$

(6)

Then, the full-order observer can be obtained as follows:

$$\dot{\widehat{x}}=A_c\widehat{x}+B_{c}u+G_c\left[y-C_c\widehat{x}\right]$$

(7)

where the ‘ˆ’ represents the estimated values, and G_c is the feedback gain matrix.

For convenience in subsequent analysis, the model is represented using a complex vector form, and the full-order observer is written as follows:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{\widehat{i}}_{\alpha \beta }\\ {\widehat{e}}_{\alpha \beta }\end{array}\right]=\left[\begin{array}{cc}{A}_c& -B_c\\ 0& j{\omega }_e\end{array}\right]\left[\begin{array}{c}{\widehat{i}}_{\alpha \beta }\\ {\widehat{e}}_{\alpha \beta }\end{array}\right]+B_c{u}_{\alpha \beta }+\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]{\tilde{i}}_{\alpha \beta }$$

(8)

where

$$A_c=-{\displaystyle \frac{R_s}{L_d}}+j{\displaystyle \frac{{\widehat{\omega }}_e\Delta L}{L_d}},B_c={\displaystyle \frac{1}{L_d}}.$$

(9)

2.2. Feedback Gain Matrix Design of Full-Order Observer

The feedback gain matrix is designed based on the expected bandpass characteristic of the observer. According to Equation (8), the estimated current can be written as follows:

$${\widehat{i}}_{\alpha \beta }={\displaystyle \frac{A_c-G_1}{s-A_c-G_1}}{i}_{\alpha \beta }+{\displaystyle \frac{B_c}{s-A_c-G_1}}(u_{\alpha \beta }-{\widehat{e}}_{\alpha \beta }).$$

(10)

According to Equation (3), the relationship between the motor current and the EEMF is

$$i_{\alpha \beta }={\displaystyle \frac{B_c}{s-A_c}}(u_{\alpha \beta }-e_{\alpha \beta }).$$

(11)

Assuming the estimated speed is equal to the actual speed, the transfer function of the observer can be derived as follows:

$$H(s)={\displaystyle \frac{{\widehat{e}}_{\alpha \beta }}{e_{\alpha \beta }}}={\displaystyle \frac{G_2\cdot B_c}{\left(s-j{\widehat{\omega }}_e\right)\left(s-A_c-G_1\right)+G_2\cdot B_c}}.$$

(12)

For the second-order system, a zero point at s = 0 provides suppression of DC bias and low-frequency disturbances. Assume the denominator of Equation (12) is

$$G_2\cdot B_c=K({\widehat{\omega }}_e)s$$

(13)

where K(${\widehat{\omega }}_e$) represents a quantity related to speed. The second-order system with two conjugate roots can be expressed as follows:

$$s^2+2\xi {\omega }_{n}s+{\omega }_n^2$$

(14)

where ξ is the damping ratio, and ω_n is the natural frequency. Considering the numerator of Equation (12), assuming ${\omega }_n={\widehat{\omega }}_e$, then it can be obtained that

$$A_c+G_1=-j{\omega }_e,G_2\cdot B_c=2k{\omega }_e\sqrt{{\displaystyle \frac{2L_d-L_q}{L_d}}}s$$

(15)

The feedback gain matrix is designed as follows:

$$G_1={\displaystyle \frac{R_s}{L_d}}-j{\widehat{\omega }}_e{\displaystyle \frac{2L_d-L_q}{L_d}},G_2=2k{\widehat{\omega }}_e\sqrt{L_d(2L_d-L_q)}s.$$

(16)

where k is the adjustable parameter of the bandpass observer. Then the transfer function is simplified as follows:

$$H(s)={\displaystyle \frac{2k{\widehat{\omega }}_e\sqrt{\frac{2L_d-L_q}{L_d}}s}{s^2+2k{\widehat{\omega }}_e\sqrt{\frac{2L_d-L_q}{L_d}}s+{\widehat{\omega }}_e^2}}.$$

(17)

Given the estimated speed equals to the actual speed at the operating frequency s = jω_e, the frequency-domain response is

$$H(j{\omega }_e)={\displaystyle \frac{2jk{\omega }_e^2\sqrt{\frac{2L_d-L_q}{L_d}}}{-{\omega }_e^2+2jk{\omega }_e^2\sqrt{\frac{2L_d-L_q}{L_d}}+{\omega }_e^2}}=1\angle 0^{°}.$$

(18)

H(jω_e) indicates the proposed bandpass observer extracts the EEMF without magnitude attenuation or phase delay. Figure 1 presents the Bode diagram of the observer at different speeds with k = 0.1. The magnitude response remains 0 dB and the phase response is 0° at different speeds, which is consistent with the numerical analysis. Figure 2 shows the Bode diagram of the observer with different k at ω_e = 200 rad/s. The results demonstrate that this single adjustable parameter exclusively governs the observer’s attenuation characteristics for both low- and high-frequency components. Notably, the bandwidth of the bandpass filter exhibits a proportional increase with higher k values.

2.3. Stability Analysis

The stability of the proposed observer is analyzed by the EEMF convergence. The EEMF error dynamics can be derived from Equation (17) as follows:

$${\displaystyle \frac{{\tilde{e}}_{\alpha \beta }}{e_{\alpha \beta }}}={\displaystyle \frac{-s^2-{\widehat{\omega }}_e^2}{s^2+2k{\widehat{\omega }}_e\sqrt{\frac{2L_d-L_q}{L_d}}s+{\widehat{\omega }}_e^2}}.$$

(19)

The characteristic equation of Equation (19) can be represented as follows:

$$s^2+2k{\widehat{\omega }}_e\sqrt{{\displaystyle \frac{2L_d-L_q}{L_d}}}s+{\widehat{\omega }}_e^2=0.$$

(20)

The two eigenvalues of Equation (20) is

$$s_{1,2}={\omega }_e\left(-k\sqrt{{\displaystyle \frac{2L_d-L_q}{L_d}}}\pm \sqrt{{\displaystyle \frac{2L_d-L_q}{L_d}}{k}^2-1}\right).$$

(21)

For the experimental IPMSM parameters presented in

Table 1. Parameters of IPMSM.

Parameter	Value	Parameter	Value
Rated Power	300 kW	Resistance	0.004375 Ω
Rated Speed	2000 r/min	d-axis inductance	0.4570 mH
Rated Voltage	800 V	q-axis inductance	0.5256 mH
Pole Pairs	6	Flux linkage	0.18247 Wb

, the real part of Equation (21) remains negative when 0 < k <1. It can be proved that the proposed observer is stable at different speeds.

2.4. Parameter Sensitivity Analysis

The IPMSM model in Equation (3) can be expressed as follows:

$${\displaystyle \frac{d{i}_{\alpha \beta }}{dt}}=(-{\displaystyle \frac{R_s}{L_d}}+j{\displaystyle \frac{{\omega }_e\Delta L}{L_d}})i_{\alpha \beta }+{\displaystyle \frac{1}{L_d}}(u_{\alpha \beta }-e_{\alpha \beta }).$$

(22)

Considering the parameter mismatches, the model is denoted as follows:

$${\displaystyle \frac{d{i}_{\alpha \beta }^{\prime }}{dt}}=A_c^{\prime }{i}_{\alpha \beta }^{\prime }+B_c^{\prime }(u_{\alpha \beta }^{\prime }-e_{\alpha \beta }^{\prime })$$

(23)

where the superscript ‘′’ represents the coefficient and variable with parameter mismatches. As the parameters change, both the real current and reference voltage are affected accordingly.

Substituting Equation (23) into Equation (8), the transfer function between the estimated EEMF and the real EEMF can be expressed as follows:

$${\widehat{e}}_{\alpha \beta }(s)={\displaystyle \frac{s(B_c-B_c^{\prime })+A_c{B}_c^{\prime }-B_c{A}_c^{\prime }}{B_c^{\prime }}}{\displaystyle \frac{H\left(s\right)}{B_c}}{i}_{\alpha \beta }^{\prime }(s)+H\left(s\right)e_{\alpha \beta }(s).$$

(24)

When the parameters are accurate, Equation (24) is equivalent to Equation (12). Substituting s = jω_e, the EEMF error with parameter mismatches is

$$\Delta {\widehat{e}}_{\alpha \beta }(s)={\displaystyle \frac{s(B_c-B_c^{\prime })+A_c{B}_c^{\prime }-B_c{A}_c^{\prime }}{B_c^{\prime }}}{\displaystyle \frac{H\left(s\right)}{B_c}}{i}_{\alpha \beta }^{\prime }(s).$$

(25)

When ΔR_s ≠ 0, the coefficient relationship is

$$B_c^{\prime }=B_c,A_c^{\prime }-A_c=-{\displaystyle \frac{\Delta R_s}{L_d}}.$$

(26)

Then, the estimated EEMF error is

$$\Delta {\widehat{e}}_{\alpha \beta }(s)=-\Delta R_s{i}_{\alpha \beta }^{\prime }(s).$$

(27)

When ΔL_d ≠ 0, the relationship between the coefficients is

$$A_c-A_c^{\prime }{\displaystyle \frac{B_c}{B_c^{\prime }}}=j{\widehat{\omega }}_e(1-{\displaystyle \frac{B_c}{B_c^{\prime }}}).$$

(28)

Then, the estimated EEMF error is

$$\Delta {\widehat{e}}_{\alpha \beta }(s)={\displaystyle \frac{j{\widehat{\omega }}_e(\frac{B_c}{B_c^{\prime }}-1)+A_c-A_c^{\prime }\frac{B_c}{B_c^{\prime }}}{B_c}}{i}_{\alpha \beta }^{\prime }=0.$$

(29)

When ΔL_d ≠ 0, the relationship between the coefficients is

$$B_c^{\prime }=B_c,A_c+j{\widehat{\omega }}_e{\displaystyle \frac{L_q}{L_d}}=A_c^{\prime }+j{\widehat{\omega }}_e{\displaystyle \frac{L_q^{\prime }}{L_d}}.$$

(30)

Then, the estimated EEMF error is

$$\Delta {\widehat{e}}_{\alpha \beta }(s)={\displaystyle \frac{A_c-A_c^{\prime }}{B_c}}{i}_{\alpha \beta }^{\prime }(s)={\displaystyle \frac{j{\widehat{\omega }}_e}{L_d}}\Delta L_q{i^{\prime }}_{\alpha \beta }(s).$$

(31)

Transforming the stationary frame EEMF to the synchronous frame, Equations (27) and (31) are transformed into

$$\Delta {\widehat{e}}_{dq}(s)=\Delta R_s{i}_{dq}^{\prime }(s)$$

(32)

$$\Delta {\widehat{e}}_{dq}(s)=j{\omega }_e\Delta L_q{i}_{dq}^{\prime }(s).$$

(33)

In the synchronous frame, the position error of PLL is

$$\Delta E\approx -{\displaystyle \frac{{\widehat{e}}_d}{{\widehat{e}}_q}}.$$

(34)

With the stator resistance mismatch and q-axis inductance mismatch, the position errors are

$$\Delta E\approx -{\displaystyle \frac{\Delta R_s{i}_d^{\prime }}{{\widehat{e}}_q+\Delta R_s{i}_q^{\prime }}}$$

(35)

$$\Delta E\approx {\displaystyle \frac{\Delta L_q{\omega }_e{i}_q^{\prime }}{{\widehat{e}}_q+\Delta L_q{\omega }_e{i}_d^{\prime }}}.$$

(36)

Figure 3 presents the position error at 200 rad/s under R_s and L_q mismatches. As shown in Figure 3a,b, the impact of ΔR_s on position error is negligible. In contrast, Figure 3c,d indicate that a mismatch in L_q results in a significant position error, which is proportional to both i_q and ΔL_q. Notably, i_d has no influence on the position error.

3. Analysis and Compensation of PLL

3.1. Conventional PLL Performance with Ramp Input Speed

The block diagram of sensorless control is shown in Figure 4. The IPMSM is controlled using field-oriented control (FOC) and a PI current regulator. To mitigate measurement noise, a PLL is generally employed to extract position and speed in sensorless control systems. By normalizing the EEMF, the PLL retains consistent tracking performance with a fixed PI controller, even as the EEMF amplitude varies with speed. Figure 5 illustrates the diagram of the conventional normalized PLL in the stationary frame. The time-domain expression of the EEMF error is

$$\Delta E={\displaystyle \frac{-{\widehat{e}}_{\alpha }\cos {\widehat{\theta }}_e-{\widehat{e}}_{\beta }\sin {\widehat{\theta }}_e}{\sqrt{{\widehat{e}}_{\alpha }^2+{\widehat{e}}_{\beta }^2}}}=-\sin (\Delta {\theta }_e)\approx -\Delta {\theta }_e.$$

(37)

Then the equivalent small signal diagram of Figure 5 is illustrated in Figure 6, and the open loop transfer functions of the position and speed are

$$G_O(s)={\displaystyle \frac{{\widehat{\theta }}_e(s)}{{\theta }_e(s)}}={\displaystyle \frac{k_{p}s+k_i}{s^2}}$$

(38)

$${\widehat{\omega }}_e(s)=s{\widehat{\theta }}_e(s).$$

(39)

The error transfer function of Figure 6 is

$${\Phi }_{\mathrm{er}}(s)={\displaystyle \frac{\Delta E(s)}{{\theta }_e(s)}}={\displaystyle \frac{s^2}{s^2+k_{p}s+k_i}}.$$

(40)

For a typical step input speed, where ω_e = c, and a ramp input speed, where ω_e = at, the corresponding position inputs are c/s² and a/s³, respectively. Using the final value theorem, the steady-state position error for the step input speed is calculated as θ_ess1, and the error for the ramp input speed is calculated as θ_ess2:

$${\theta }_{\mathrm{ess}1}=\underset{s\to 0}{\lim } s{\Phi }_{\mathrm{er}}(s){\displaystyle \frac{c}{s^2}}=0$$

(41)

$${\theta }_{\mathrm{ess}2}=\underset{s\to 0}{\lim } s{\Phi }_{\mathrm{er}}(s){\displaystyle \frac{a}{s^3}}={\displaystyle \frac{a}{k_i}}.$$

(42)

Therefore, the conventional PLL tracks the step input speed with zero steady-state error [10]. However, for the ramp input speed, there is a constant position error related to the ramp and the k_i of the PLL. According to Equation (39), in the acceleration and deceleration process.

$${\omega }_{\mathrm{ess}}={\displaystyle \frac{d({\widehat{\theta }}_e+{\theta }_{\mathrm{ess}})}{dt}}-{\widehat{\omega }}_e={\displaystyle \frac{d({\theta }_{\mathrm{ess}})}{dt}}.$$

(43)

From Equations (41)–(43), the steady-state speed error for step input and ramp input speeds are both zero. While the conventional PLL performs well for step input speeds, it requires a position compensation when applied to the frequency-ramped IPMSM.

3.2. Impact of Position Error on Torque Control

With the estimated position error ${\tilde{\theta }}_e$, the feedback ${\widehat{i}}_{dq}$ in Figure 7 is

$$\begin{array}{c}{\widehat{i}}_d=i_d\cos ({\tilde{\theta }}_e)-i_q\sin ({\tilde{\theta }}_e)\\ {\widehat{i}}_q=i_q\cos ({\tilde{\theta }}_e)+i_d\sin ({\tilde{\theta }}_e).\end{array}$$

(44)

The torque of the IPMSM is

$$T_e={\displaystyle \frac{3}{2}}{n}_p\left[\psi i_q+(L_d-L_q)i_d{i}_q\right]$$

(45)

where n_p is the number of pole pairs. For the maximum torque per ampere (MTPA) control of IPMSMs, the trajectory of reference i_dq is determined by a LUT. The current trajectory caused by different position errors is shown in Figure 8. Curve PQ represents the current limit, curve OR denotes the reference MTPA, curve ST illustrates the maximum torque per voltage (MTPV) trajectory, while T₁ and T₂ represent the constant torque curves. A set of dashed lines ORn represents the MTPA offset trajectory with different position errors. The deviation angle of the current increases as the position error increases. The operating point will move from the expected point A to point B, and the torque will decrease from T₁ to T₂.

3.3. Feedforward PLL

The feedforward PLL of SPMSM proposed in [10] and the QPLL proposed in [27] are shown in Figure 9a, b, respectively. The closed-loop transfer function of Figure 9a is

$${\Phi }_1(s)={\displaystyle \frac{{\widehat{\theta }}_e(s)}{{\theta }_e(s)}}={\displaystyle \frac{(k_p+{\omega }_c)s^2+(k_i+k_p{\omega }_c)s+k_i{\omega }_c}{s^3+(k_p+{\omega }_c)s^2+(k_i+k_p{\omega }_c)s+k_i{\omega }_c}}$$

(46)

where ω_c is the cutoff frequency of the low-pass filter in the feed-forward path. The addition of the feedforward loop increases the order of the PLL by introducing an extra parameter, which may raise the risk of instability. In contrast, the QPLL shown in Figure 9b relies on the differentiation of current signals and mechanical parameters, which compromises its accuracy.

3.4. KF-Based PLL Compensation Strategy

Based on the analysis above, the conventional PLL requires a position compensation to maintain the command torque for frequency-ramped IPMSMs. According to Equation (43), the compensation position θ_CP is derived as follows:

$${\theta }_{CP}={\displaystyle \frac{{\omega }_e(k)-{\omega }_e(k-N)}{N\cdot T_s\cdot k_i}}.$$

(47)

Typically, a buffer size N >> 1 is assigned to reduce instantaneous errors. However, during program execution, storing N data points takes a significant amount of time. If N is too large, it may reduce the accuracy of the position estimation and negatively affect motor performance. In addition, the estimated speed contains a considerable ripple, which needs to be filtered before calculating θ_CP.

The compensated PLL diagram is presented in Figure 10. The KF provides an optimal estimation of the current state by combining noisy sensor measurements with a predicted state estimate and its associated uncertainty [30]. The discrete-time state space model of the KF is

$$\begin{array}{l}{x}_k=F_k{x}_{k-1}+G_k{u}_k+w_k\hfill \\ y_k=H_k{x}_k+v_k\hfill \end{array}$$

(48)

where x_k is the state vector at time k, y_k is the measurement value at time k, w_k ~ N(0,Q) is the process noise, and v_k ~ N(0,R) is the measurement noise. w_k and v_k are assumed to be Gaussian with covariance matrices Q and R, respectively. u_k is the control input signal of the system, F_k is the state transition matrix, G_k is the control input matrix, and H_k is the measurement matrix. The KF structure involves two main steps and five key equations for computation. The prediction step obtains the a priori state estimate of the state x:

$${\widehat{x}}_k^{-}=F_{k-1}{\widehat{x}}_{k-1}+G_{k-1}{u}_{k-1}.$$

(49)

Then, the error covariance at the priori step is

$$P_k^{-}=F_{k-1}{P}_{k-1}{F}_{k-1}^T+Q$$

(50)

where P_k−1 is the error covariance at the posteriori step. The second step is to update the current estimate state and the error covariance by

$$K_k=P_k^{-}{H}_k^T{(H_k{P}_k^{-}{H}_k^T+R)}^{-1}$$

(51)

$${\widehat{x}}_k={\widehat{x}}_k^{-}+K_k(y_k-H_k{\widehat{x}}_k^{-})$$

(52)

$$P_k=(I-K_k{H}_k)P_k^{-}$$

(53)

where K_k is the Kalman gain. For the speed filter system, u_k is zero. F and H are equal to I. The model in Equation (48) is transferred to

$$\begin{array}{l}{x}_k=x_{k-1}+w_k\hfill \\ y_k=x_k+v_k.\hfill \end{array}$$

(54)

The effectiveness of noise and ripple filtering in the KF output depends on the selection of Q and R. A smaller Q results in reduced output ripple, and a larger R also decreases the ripple. However, the KF can introduce output delay for the ramp input speed. And the delay increases as the output smoothness improves. Therefore, a trade-off must be made between the values of Q and R. Figure 11 compares the response of the KF to step input under different parameters with white noise injected into the input signal for simulation. The parameter settings are KF1 (Q = 0.001, R = 0.005), KF2 (Q = 0.00001, R = 0.5), KF3 (Q = 0.0001, R = 5), KF4 (Q = 0.0001, R = 0.05), and KF5 (Q = 0.0001, R = 0.5). Figure 11a shows that KF5 exhibits better performance considering the overall filtering effect and delay time. Figure 11b,c compare the root mean square error (RMSE) and delay time, revealing that the errors of KF2 and KF3 are identical. This indicates that excessively reducing Q or significantly increasing R can decrease the error but at the cost of increased delay time. Additionally, the comparison between KF4 and KF5 suggests that a reduction in R leads to an increase in RMSE. Consequently, KF1 has the smallest delay but the largest RMSE.

4. Experimental Analysis

The experimental platform is shown in Figure 12. The IPMSM parameters are shown in Table 1, with a rated speed of 2000 r/min. The load machine is a 450 kW induction motor. The switching and the sample frequencies are 4 kHz and 8 kHz, respectively. Data is collected by the host computer from the controller via Ethernet at a frequency of 4 kHz. As shown in Figure 13, all sensorless control algorithms are implemented on the DSP TMS320F28379D. The observer and FOC control programs are executed on CPU1, while the KF and Ethernet transmission are programmed on CPU2. The motor starts using a variable reluctance (VR) resolver and switches to sensorless control at 300 r/min. In the following tests, the PLL parameters remain constant across different experiments (k_p = 200, k_i = 1000). The parameter k of the bandpass observer is 0.8. Figure 14 presents the experimental result of KF under KF1-5, which is consistent with the simulation results shown in Figure 11. Considering the trade-off between filtering performance and delay time, the KF parameters are set to KF5 (Q = 0.0001 and R = 0.5).

Performance of the adaptive bandpass full-order observer under low speed and rated speed is presented in Figure 15 and Figure 16, respectively. Figure 15 illustrates the estimated results at 300 r/min under 100 N·m and the rated load. The speed error contains small ripples resulting from the inverter nonlinearity. Even though the adaptive bandpass observer reduces the fifth and seventh harmonics of the estimated EEMF, they are not completely eliminated. Figure 16 presents the estimated results at 2000 r/min under 100 N·m and the rated load. As the speed increases, the fifth and seventh harmonics are further away from the fundamental frequency; therefore, the speed error fluctuations are smaller than that of the low speed. The estimated EEMFs exhibit consistent sinusoidal waveforms across both low- and high-speed ranges under all load conditions.

To verify the influences of the parameter mismatch on the observer, the corresponding experimental results are revealed in Figure 17. Since the parameters of the motor cannot be manually adjusted, the same effect is achieved by modifying the parameters used in the observer and FOC. Figure 17a,b illustrate the performance of the adaptive bandpass full-order observer under load with a 25% resistance mismatch and 25% d-axis inductance mismatch, respectively. There are no significant impacts on the estimation of position. With a 25% q-axis inductance error, Figure 17c shows that the position error is increased.

Figure 18 presents experimental results of the conventional PLL with speed variation and load variation. In regions A–C, acceleration and deceleration processes result in steady-state position error. In region D–I, the position error varies with the load variation without steady-state error, which results from the inverter nonlinearity. Meanwhile, the estimated speed remains accurate with both speed and load variations.

In order to validate the compensation effect of the proposed method, Figure 19, Figure 20, Figure 21 and Figure 22 present the comparison results of the conventional PLL and the compensated PLL in different conditions: (1) the speed reference begins with 500 r/min and changes to 1000 r/min with a ramp of 200 rad/s² under a load of 100 N·m; (2) the speed reference begins with 500 r/min and changes to 1000 r/min with a ramp of 200 rad/s² under the rated load; (3) the speed reference begins with 300 r/min and changes to 2000 r/min with a ramp of 200 rad/s² under a load of 100 N·m; (4) the speed reference begins with 300 r/min and changes to 2000 r/min with a ramp of 200 rad/s² under the rated load. The theoretical steady-state position error θ_ess, as given by (42), is 3.6°. The torque is calculated using (45), which adapts the rotor position converted by the AD2S1210. The estimated torque accuracy is affected by parameter mismatches, but since parameters remain consistent under identical operating conditions, the position error can be deduced from the estimated torque. As previously discussed, when the estimated position lags the actual rotor position, the estimated torque falls below the commanded value; conversely, when the estimated position leads the rotor position, the estimated torque exceeds the command value.

It Is observed that the position error and torque fluctuations In condition (1) with the conventional PLL are 12.9° and 17 N·m, respectively, whereas with the compensated PLL, they are reduced to 5.5° and 10 N·m, respectively. Similarly, in conditions (2), conditions (3) and conditions (4), the position error fluctuations are reduced from 14.2° to 7.6°, from 14.7° to 7.2° and from 19.75° to 12.27 °, respectively. While the torque fluctuation decreases from 17 N·m to 10 N·m in condition (1), from 96 N·m to 62 N·m in condition (2), from 23 N·m to 15 N·m in condition (3), and from 138 N·m to 68 N·m in condition (4), respectively.

Table 2. Compensation error in different conditions.

Working Condition	Compensation Position Error	Compensated Torque
Condition (1)	2.78%	7%
Condition (2)	8.33%	3.4%
Condition (3)	4.2%	8%
Condition (4)	3.89%	7%

presents the calculated position compensation errors and torque compensation values under different conditions. The position errors are within 10% under all conditions, which means the steady-state position error of the traditional PLL under ramp speed input aligns with theoretical calculations, and the compensated strategy significantly reduces the ramp position error. And the torque fluctuations caused by position error during dynamic processes are significantly reduced, and the estimated torque demonstrates the compensation’s impact on motor control. The compensated PLL effectively mitigates the position error under different conditions.

5. Conclusions

In this paper, an adaptive bandpass full-order observer was proposed for IPMSM sensorless control, which realizes EEMF estimation at the operating frequency with 0 dB gain and zero phase delay. And theoretical analysis revealed the conventional PLL tracking error of ramp input speed. The proposed KF-based PLL compensation strategy significantly enhanced the estimation accuracy during acceleration and deceleration processes. Experimental results demonstrated that the compensated position reduces torque fluctuations in dynamic processes. The outcomes under various load and speed conditions align closely with theoretical predictions. The compensated PLL structure can be extended to other stationary frame observers.

The proposed method is still sensitive to model parameter mismatches, particularly the q-axis inductance. Future work will focus on incorporating online parameter identification techniques to further improve observer robustness across the entire operating range.