Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter

Abstract

This article proposes a double-pre-warped Tustin bi-quad filter (DPT-BQF) to suppress mechanical resonance for non-contact integrated permanent magnet vernier motors (NIPMVMs). First, this article analyzes how conventional notch filters suffer from distortion in notch frequency, bandwidth, and depth when discretized with low-precision methods, which degrades filtering performance and weakens resonance suppression. To address this issue, the inherent limitations of the pre-warped Tustin discretization for BQFs are discussed. On this basis, a double pre-warped Tustin method is proposed by compensating for the bandwidth distortion, and its discretization performance is comprehensively evaluated. Furthermore, the principle, parameter design, and robustness range of the filter are deeply analyzed in the discrete domain. Finally, experimental validation on an NIPMVM test platform demonstrates the effectiveness and feasibility of the proposed method.

1. Introduction

The non-contact integrated permanent magnet vernier motor combines the advantages of magnetic gears and permanent magnet vernier motors (PMVM). With two-stage speed reduction, it can meet the requirements of industrial applications such as AGVs and robotic joints [1,2]. However, since the torque transmission of the magnetic gear is sinusoidal in nature, it eventually induces speed resonance, which is an issue that urgently needs to be suppressed [3,4].

Resonance suppression methods can be classified into mechanical methods [5], active methods [6,7,8], and passive methods [9,10]. In [5], an electroactive composite material is attached to the surface of a flexible single-link, generating a force opposite to the resonance. However, this approach requires additional structural design and viscoelastic materials. In [6], explicit model predictive control is applied for resonance suppression, where an observer provides feedback of the states. This approach essentially increases system damping to suppress resonance. However, it reduces system robustness and complicates the controller design process.

The most common passive suppression method is to use filters for resonance attenuation. However, the suppression performance is degraded by digital signal processing and relies on the accuracy of the identified resonance frequency [11]. To address this issue, adaptive notch filters capable of rapidly identifying the resonance center frequency can provide an effective solution [12]. Similarly, ref. [13] adopts an online adaptive notch filter for resonance mitigation, wherein the resonance frequency is identified with respect to the rotational speed via a fast FFT algorithm. Consequently, the notch filter parameters are updated in real time to enhance the suppression performance. However, when the notch filter is enabled, digital delay may cause a shift in the resonance frequency. In this regard, ref. [14] analyzes the resonance mechanism of a two-inertia system. To accurately identify the resonance frequency, the SNR for online FFT is improved by increasing the sampling period of the speed loop and reducing the output saturation of the speed loop. However, this method requires intentionally inducing a high-SNR resonance segment prior to each identification process, imposing significant limitations on its practical applicability.

To enable more flexible filter configuration and address the notch distortion caused by discretization, the impact of discretization methods on the suppression performance of bi-quad filters (BQFs) has been analyzed. The results indicate that as the degrees of freedom increase, the design of the BQF becomes more complex [9]. To address this, the resonance frequencies of the two-inertia system can be identified offline in advance, and a complete BQF parameter-tuning procedure can be designed based on these results [15]. However, this tuning method primarily considers the open-loop characteristics of the system. Once the speed controller is engaged, the tuned parameters may not achieve the intended effect or may even become ineffective. Furthermore, after the speed controller is introduced, a shift in the resonance frequency relative to the open-loop system may occur, which interferes with the notch frequency tuning [16].

To address this issue, a DPT-BQF is proposed, and the operating principle, parameter design, and robustness range of the filter are thoroughly analyzed in the discrete domain within a closed-loop system, achieving effective resonance suppression for NIPMVMs and other two-inertia systems.

The specific contributions of this paper are as follows: (1) Compared with the conventional BQF, the proposed method mathematically establishes the mapping relationship between the actual bandwidth and the theoretical bandwidth and provides a clear quantitative stability range for the notch filter. (2) From the implementation perspective, the proposed DPT-BQF has a simple structure. It only requires modification of the discretization coefficients during digital implementation and is, therefore, easy to implement. (3) The proposed DPT-BQF is general and can be applied to NIPMVMs as well as other resonance-prone applications, such as inflatable Savonius wind turbines [17]. The robust notch-filter design process in this paper is closely based on the NIPMVM model and is therefore system-specific; however, the underlying design concept can also be extended to other similar applications.

2. NIPMVM and Notch Structure

The NIPMVM studied is structurally composed of a PMVM rigidly connected to a magnetic gear. The magnetic gear can be simplified and modeled as a two-inertia system, and its equation can be expressed as follows:

$$\begin{array}{ccc}(J_{i}s+B_i)·{\omega }_{mi}& =& T_e-g_m^{-1}{T}_c-d_s·(p_i{\omega }_{mi}-p_o{\omega }_{mo})\\ (J_{o}s+B_o)·{\omega }_{mo}& =& T_c-T_L+d_s{g}_m·(p_i{\omega }_{mi}-p_o{\omega }_{mo})\end{array},$$

(1)

where T_e, T_c, and T_L are the electromagnetic torque, magnetic gear transmission torque, and load torque. p_i,o are the pole pair numbers of the inner and outer rotors of the NIPMVM, which are 4 and 24, respectively. J_i,o, ω_mi,o, and B_i,o ≈ 0 represent the rotational inertia, mechanical angular velocity, and friction coefficient of the inner and outer rotors, respectively. d_s ≈ 0 is the damping constant, and s denotes the differential operator.

Furthermore, a simplification is performed, and the simplified variables are shown as T_ε = g_mT_e, J_ε = $g_m^2$J_i, ω_mε = $g_m^{-1}$ω_mi, θ_eε = s⁻¹ (ω_mε − ω_mo), and k_eff = p_oT_maxcos(θ_err). In addition, the motion and simplified model of the NIPMVM are illustrated in Figure 1. It is noteworthy that the equivalent anti-resonant frequency and natural resonant frequency of the NIPMVM can be derived from the simplified two-inertia model, as ${\omega }_{a\epsilon }^2$ = k_eff/J_o, ${\omega }_{n\epsilon }^2$ = ${\omega }_{a\epsilon }^2$ (γ_ε + 1), and γ_ε = J_o/J_ε, respectively. In this regard, a structure where the BQF is connected in series between the speed loop and current loop controllers is adopted in this article to suppress the resonance.

3. Resonance Suppression Based on DPT-BQF

The discretization methods traditionally used for BQF mainly include the back Euler (BE-BQF), the Tustin (T-BQF), and the pre-warped Tustin (PT-BQF) methods. However, the notch filters still suffer from low center-frequency accuracy. To solve this, this article proposes a double-pre-warped discretization that significantly reduces the distortion of the filter.

3.1. Pre-Warped Tustin Bi-Quad Filter

When a BQF is discretized using the back Euler or Tustin method, significant deviations often occur. To mitigate such discretization distortion, a commonly used approach, the pre-warped Tustin method, is first introduced. After applying this method, the resulting discretized model GPT-BQF(z) is given by

$$\begin{array}{l}{G}_{\mathrm{PT}\text{-}\mathrm{BQF}}\left(z\right)=\\ {\displaystyle \frac{\left(k^2+{\omega }_T^2+k_{T2}{\omega }_{T}k\right)z^2-2\left(k^2-{\omega }_T^2\right)z+\left(k^2+{\omega }_T^2-k_{T2}{\omega }_{T}k\right)}{\left(k^2+{\omega }_T^2+k_{T1}{\omega }_{T}k\right)z^2-2\left(k^2-{\omega }_T^2\right)z+\left(k^2+{\omega }_T^2-k_{T1}{\omega }_{T}k\right)}}\end{array},$$

(2)

where k = ω_n/tan(ω_nT_s/2). It is worth noting that the structure of the PT-BQF is identical to that of the T-BQF except for the difference in the coefficient k. Then, substituting z = e^jωTs to analyze its frequency characteristics yields

$$\begin{array}{l}{G}_{\mathrm{PT}\text{-}\mathrm{BQF}}\left(e^{\mathrm{j}\omega T_s}\right)=\\ {\displaystyle \frac{\left(k^2+{\omega }_T^2\right)\cos \left(\omega T_s\right)-\left(k^2-{\omega }_T^2\right)+\mathrm{j}{k}_{T2}{\omega }_{T}k\sin \left(\omega T_s\right)}{\left(k^2+{\omega }_T^2\right)\cos \left(\omega T_s\right)-\left(k^2-{\omega }_T^2\right)+\mathrm{j}{k}_{T1}{\omega }_{T}k\sin \left(\omega T_s\right)}}\end{array},$$

(3)

According to (3), the Bode plots for ω_n = 2π × 350 rad/s, b_t = 350 rad/s, and x_t = –30 dB are obtained when the sampling period is stepped from 0.0002 s to 0.001 s, as shown in Figure 2.

As shown, the pre-warped Tustin, due to its high accuracy at the notch frequency, effectively prevents the notch frequency from shifting after discretization. As the sampling period increases, the PT-BQF maintains both its notch frequency and depth without deviation. However, the accuracy of the PT-BQF deteriorates away from the notch frequency, leading to a narrowing of the bandwidth as the sampling period increases, which, likewise, affects the performance of the filter.

3.2. Double-Pre-Warped Tustin Bi-Quad Filter

This article introduces improvements to mitigate the distortion of the notch bandwidth. The DPT-BQF is proposed.

First, the correction method for the bandwidth of the DPT-BQF is introduced. Let the actual bandwidth of the BQF be Ω, and the magnitude gain at frequencies ω_n ± Ω/2 be x_Ω/2 dB. According to (3), its amplitude frequency characteristic can be shown as

$$\begin{array}{ccc}\hfill \gamma & =& \left(k^2+{\omega }_T^2\right)\cos \left(\omega T_s\right)-\left(k^2-{\omega }_T^2\right)\hfill \\ \hfill \eta & =& b_{T}k\sin \left(\omega T_s\right)\hfill \\ \hfill x_{\Omega /2}& =& 20\lg \sqrt{{\gamma }^2+{\left({10}^{x_T/20}\eta \right)}^2/{\gamma }^2+{\eta }^2}\hfill \end{array},$$

(4)

It can be further simplified as

$${\left[\left(k^2+{\omega }_T^2\right)\cos \left(\omega T_s\right)-\left(k^2-{\omega }_T^2\right)\right]}^2{\displaystyle \frac{1-{10}^{x_{\Omega /2}/10}}{{10}^{x_{\Omega /2}/10}-{10}^{x_T/10}}}={\left[b_{T}k\sin \left(\omega T_s\right)\right]}^2,$$

(5)

Then, by substituting ω = ω_n ± Ω/2 into the above equation, it can be further simplified as

$$\begin{array}{ccc}\hfill \cos \left(\Omega T_s\right)& =& {\lambda }_{\Omega /2}^2{\left(k^2+{\omega }_T^2\right)}^2-b_T^2{k}^2/{\lambda }_{\Omega /2}^2{\left(k^2+{\omega }_T^2\right)}^2+b_T^2{k}^2\hfill \\ \hfill {\lambda }_{\Omega /2}^2& =& 1-{10}^{x_{\Omega /2}/10}/{10}^{x_{\Omega /2}/10}-{10}^{x_T/10}\hfill \end{array},$$

(6)

Finally, using trigonometric identities, the mapping relationship between the ideal notch bandwidth b_T and the actual notch bandwidth Ω is provided in Appendix A, and the final result can be expressed as

$$b_T={\displaystyle \frac{{\lambda }_{\Omega /2}\left(k^2+{\omega }_t^2\right)}{k}}\tan \left({\displaystyle \frac{\Omega T_s}{2}}\right),$$

(7)

Then, the discretized DPT-BQF model G_PT-BQF(z) is deduced as

$$\begin{array}{l}{G}_{\mathrm{DPT}\text{-}\mathrm{BQF}}\left(z\right)=\\ {\displaystyle \frac{\left(k^2+{\omega }_T^2+k_{T2}{\omega }_{T}k\right)z^2-2\left(k^2-{\omega }_T^2\right)z+\left(k^2+{\omega }_T^2-k_{T2}{\omega }_{T}k\right)}{\left(k^2+{\omega }_T^2+k_{T1}{\omega }_{T}k\right)z^2-2\left(k^2-{\omega }_T^2\right)z+\left(k^2+{\omega }_T^2-k_{T1}{\omega }_{T}k\right)}}\end{array},$$

(8)

where the parameter b_T in the k_T1 is tuned according to (7).

The tuning procedure of the DPT-BQF is as follows: (a) Obtain the resonance frequency ω_T using the online resonance identification method [18]. (b) Select the Ω, and map it to the tuning parameter b_T according to (7). (c) Choose an appropriate notch depth parameter x_T and, together with b_T, determine the parameters k_T1 and k_T2. (d) Finally, using k = ω_T/tan(ω_TT_s/2) and according to (8), complete the notch filter design, where the filter output y can be expressed as

$$\begin{array}{ccc}\hfill y\left(k\right)& =& {\displaystyle \frac{\begin{array}{r}\left(a_1+a_3\right)u\left(k\right)-2a_{2}u\left(k-1\right)+\left(a_1-a_3\right)u\left(k-2\right)\\ +2a_{2}y\left(k-1\right)-\left(a_1-a_4\right)y\left(k-2\right)\end{array}}{a_1+a_4}}\hfill \\ \hfill a_1& =& k^2+{\omega }_T^2,a_2=k^2-{\omega }_T^2,a_3=k_{T2}{\omega }_{T}k,a_4=k_{T1}{\omega }_{T}k\hfill \end{array},$$

(9)

At this point, the design and tuning of the DPT-BQF has been completed, and the digital implementation of the DPT-BQF can be achieved according to (9).

3.3. Performance Analysis of the DPT-BQF

(1) Notch frequency: By differentiating the amplitude–frequency characteristic of the DPT-BQF to determine its notch frequency according to (8), the following is obtained:

$$\begin{array}{l}{\displaystyle \frac{d\left|G_{\mathrm{DPT}\text{-}\mathrm{BQF}}\right|}{d\omega }}={\displaystyle \frac{\phi }{{\gamma }^2+{\eta }^2}}\\ \gamma =\left(k^2+{\omega }_T^2\right)\cos \left(\omega T_s\right)-\left(k^2-{\omega }_T^2\right)\\ \eta =k_{T1}{\omega }_{T}k{T}_s\sin \left(\omega T_s\right)\\ \phi =2\gamma \eta \left({\displaystyle \frac{k_{T2}^2}{k_{T1}^2}}-1\right)\left[\left(k^2+{\omega }_T^2\right)T_s\sin \left(\omega T_s\right)\eta +\cos \left(\omega T_s\right)\gamma \right]\end{array},$$

(10)

It can further be solved to obtain:

$$\cos \left(\omega T_s\right)={\displaystyle \frac{k^2-{\omega }_T^2}{k^2+{\omega }_T^2}}\Rightarrow {\displaystyle \frac{{\omega }_T}{k}}=\tan \left({\displaystyle \frac{\omega T_s}{2}}\right)\Rightarrow \omega ={\omega }_T,$$

(11)

It can be seen that the DPT-BQF, benefiting from the pre-warped tuning of k, achieves an accurate mapping between the actual notch frequency ω and the tuned notch frequency ω_T, effectively preventing notch frequency deviation.

(2) Notch depth: As analyzed above, the DPT-BQF achieves accurate mapping between the actual notch frequency ω and the tuned notch frequency ω_T by pre-correcting the discretization coefficients. Furthermore, analyzing the notch depth at the frequency ω_T yields:

$$\begin{array}{cc}\hfill {\left|G_{\mathrm{DPT}\text{-}\mathrm{BQF}}\right|}_{\left|\omega ={\omega }_T\right.}& ={\displaystyle \frac{{\gamma }^2+k_{T2}{\eta }^2}{{\gamma }^2+k_{T1}{\eta }^2}}={\displaystyle \frac{k_{T2}}{k_{T1}}}={10}^{{\displaystyle \raisebox{1ex}{$x_T$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}\hfill \\ \hfill \gamma & =\left(k^2+{\omega }_T^2\right)\cos \left({\omega }_T{T}_s\right)-\left(k^2-{\omega }_T^2\right)\hfill \\ \hfill \eta & ={\omega }_{T}k\sin \left(\omega T_s\right)\hfill \end{array},$$

(12)

As shown, the notch depth of the DPT-BQF at ω_T is x_T = 20lg|G_PT-BQF|_{|ω = ωT}, i.e., the magnitude gain at the notch frequency. Therefore, when using the pre-warped Tustin method for discretization, the actual notch depth is consistent with the tuned value, and no distortion occurs.

Hence, the DPT-BQF, benefiting from the pre-warping of the coefficient k and the proposed bandwidth pre-warping, improves the discretization accuracy of the notch frequency, notch depth, and notch bandwidth in a comprehensive manner. This enables the controller to achieve precise suppression of resonance. Finally, to illustrate the discretization accuracy of the DPT-BQF more intuitively, the Bode plots are also presented in Figure 3. As shown, the notch frequency, bandwidth, and depth of the DPT-BQF exhibit no significant deviation.

4. Parameter Design and Analysis of the DPT-BQF

4.1. Parameter Design in a Closed-Loop System

The system performance after inserting a BQF is analyzed from the perspective of a closed-loop system. First, the closed-loop transfer function of the NIPMVM after inserting the DPT-BQF can be derived as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\omega }_{mi}}{{\omega }_{ref}}}=& \frac{\left(d_3{z}^3+d_2{z}^2+d_{1}z\right)\left(b_2{z}^2+b_{1}z+b_0\right)\left[\left(k_p+k_i{T}_s\right)z-k_p\right]}{k_t\left(z-1\right)\left(c_3{z}^3+c_2{z}^2+c_{1}z+c_0\right)\left(a_2{z}^2+a_{1}z+a_0\right)}\hfill \\ & +\left[\left(k_p+k_i{T}_s\right)z-k_p\right]\left(d_3{z}^3+d_2{z}^2+d_{1}z\right)\left(b_2{z}^2+b_{1}z+b_0\right)\hfill \\ \hfill {\displaystyle \frac{{\omega }_o}{{\omega }_{ref}}}=& \frac{g_m{k}_{eff}{T}_s^3{z}^3\left(b_2{z}^2+b_{1}z+b_0\right)\left[\left(k_p+k_i{T}_s\right)z-k_p\right]}{k_t\left(z-1\right)\left(c_3{z}^3+c_2{z}^2+c_{1}z+c_0\right)\left(a_2{z}^2+a_{1}z+a_0\right)}\hfill \\ & +\left[\left(k_p+k_i{T}_s\right)z-k_p\right]\left(d_3{z}^3+d_2{z}^2+d_{1}z\right)\left(b_2{z}^2+b_{1}z+b_0\right)\hfill \\ & \begin{array}{llll}{a}_2=k^2+{\omega }_T^2+k_{T1}{\omega }_{T}k& b_2=k^2+{\omega }_T^2+k_{T2}{\omega }_{T}k& c_3=J_{\epsilon }{J}_o+(J_{\epsilon }+J_o)k_{eff}{T}_s^2& d_3=g_m^2\left(J_o{T}_s+k_{eff}{T}_s^3\right)\\ a_1=-2\left(k^2-{\omega }_T^2\right)& b_1=-2\left(k^2-{\omega }_T^2\right)& c_2=-3J_{\epsilon }{J}_o-(J_{\epsilon }+J_o)k_{eff}{T}_s^2& d_2=-2g_m^2{J}_o{T}_s\\ a_0=k^2+{\omega }_T^2-k_{T1}{\omega }_{T}k& b_0=k^2+{\omega }_T^2-k_{T2}{\omega }_{T}k& c_1=3J_{\epsilon }{J}_o,c_0=-J_{\epsilon }{J}_o& d_1=g_m^2{J}_o{T}_s\end{array}\hfill \end{array},$$

(13)

Furthermore, keeping the controller parameters unchanged, the notch bandwidth is tuned to b_t = 12 × ω_t rad/s, and the notch depth to x_t = −20 dB. To compare the performance differences when tuning the notch frequency based on the previously mentioned vibration frequency and natural resonance frequency, the performance comparison results for notch frequencies ω_t of 31.9 Hz and 34.9 Hz are shown in Figure 4.

As shown in Figure 4a, the performance at vibration and natural resonance frequency is evaluated. Without the notch filter, the dominant poles yield a damping ratio of only 0.11. With the filter applied, the damping ratio increases to 0.503 at the vibration frequency and 0.405 at the natural resonance frequency. Moreover, the Bode plot results indicate comparable suppression performance under both conditions, with slightly improved attenuation at the vibration frequency.

4.2. Robustness Analysis of the System with the DPT-BQF

For convenient tuning of DPT-BQF parameters, it is meaningful to analyze the tuning range of notch parameters under the premise of system stability. Then, the robust parameter range of the closed-loop system in (13) is shown in Figure 5.

It should be added that, when plotting Figure 5, the controller parameter tuning method from [19] is adopted. In this case, z₁ = 0.9, p₁ = p₂ = 0.5, and the sampling period is 0.0001 s.

First, Figure 5a shows the poles for ω_t = 5 to 42 Hz. When the frequency is in the low range, the system remains stable; however, when the frequency increases to 42 Hz, a pair of poles moves outside the unit circle. Figure 5b shows the poles for x_t = 0 to −32 dB. Notably, from (6), x_t should not be set at −3 dB. When x_t < −3 dB, one pair of poles lies on the boundary and moves inside the unit circle as the notch depth increases. However, another pole moves outside the stability circle when the notch depth reaches −32 dB. Finally, Figure 5c shows the poles for b_t = 0 to 15.7 times the notch frequency. When the bandwidth reaches 15.7 times, the system becomes unstable.

In summary, the tuning of filter parameters significantly affects system stability. An excessively small or large notch frequency, depth, or bandwidth may cause instability. Therefore, the robust parameter range should be analyzed, and parameters should be configured within this range.

For convenient parameter selection, the detailed parameter configuration procedure is given as follows:

• First, after the load and controller parameters are determined, the resonance frequency is obtained using the parameter identification method in [18], and this frequency is used as the notch frequency of the filter.
• According to (13), and following the form shown in Figure 5, the system pole-zero trajectories under different notch frequencies, notch depths, and bandwidths are plotted.
• The parameters are then selected according to the actual trajectories under different system parameters. Specifically, the notch frequency is selected based on the identification result, and it only needs to be ensured that it lies within the stable range. The notch bandwidth and notch depth can be repeatedly tuned within the stable range to obtain better performance. It is worth noting that, since the stable range is generally limited, the tuning process is convenient.

5. Experimental Verification

To evaluate the effectiveness of the proposed DPT-BQF for the NIPMVM, experiments were conducted on the test platform shown in Figure 6. The NIPMVM is driven by the inverter based on IPM IM513-L6A. Then, the TMS320F28335 is used to implement the proposed method. The notch frequency is tuned according to the identified actual frequency [18], set to ω_t = 34.9 Hz, while the other notch parameters are set as follows: notch bandwidth b_t = 15 × ω_t rad/s and notch depth x_t = −22 dB. It is worth noting that, to verify the notch distortion issue, the initial parameters of all notch filters are identical.

To verify that the DPT method achieves the highest accuracy for the BQF, a frequency-sweep signal with an amplitude of 1 A and a frequency range of 0–500 Hz is used. The notch frequency is set to ω_t = 200 Hz, the bandwidth to b_t = 200 Hz, and the depth to x_t = −30 dB. The results are shown in Figure 7.

As seen, only the DPT-BQF shows no distortion, with both frequency and bandwidth at the preset 200 Hz. In contrast, the BE-BQF shifts the notch frequency to 180 Hz with a depth reduction to −6 dB; the T-BQF shifts the frequency to 165 Hz and narrows the bandwidth to 120 Hz; and the PT-BQF maintains the frequency but also narrows the bandwidth to 120 Hz.

To verify the resonance suppression capability of the DPT-BQF at the rated speed, the comparison results of the systems with the BQF inserted, under an inner-rotor speed variation from 0 r/min−500 r/min−0 r/min, are shown in Figure 8.

The DPT-BQF, benefiting from higher accuracy, provides optimal resonance suppression across a wide range. Systems without a notch filter or with conventional notch filters exhibit significant resonance, which becomes more pronounced at high speeds. Simultaneously, severe resonance is also observed in the current. These speed variation experiments confirm the excellent mechanical resonance suppression performance of the DPT-BQF in the NIPMVM. Furthermore, to quantitatively analyze the performance of the DPT-BQF, a comparison table is provided, as shown in

Table 1. Performance comparison table of different BQFs.

	ωt Error (%)	bt Error (%)	xt Error (%)	iq Ripple (A)	nmo Ripple (r/min)	Computation Time (μs)
No filter	/	/	/	6	33	/
BE-BQF	10	/	80	2.9	17.3	3.21
T-BQF	17.5	40	0	3.2	24.2	3.31
PT-BQF	0	40	0	3.3	23.7	3.98
DPT-BQF	0	0	0	0.8	7.9	4.26

. The BE-BQF shows obvious notch frequency and notch depth errors, while the T-BQF suffers from both notch frequency and bandwidth errors. The PT-BQF eliminates the notch frequency error, but its bandwidth error remains 40%. In contrast, the proposed DPT-BQF achieves zero error in notch frequency, bandwidth, and depth. Moreover, it reduces the current ripple to 0.8 A and the speed ripple to 7.9 r/min, demonstrating better discretization accuracy and resonance suppression performance. Furthermore, to compare the computational cost of different BQFs, the computation time of each method is also provided in Table 1. It can be seen that even the DPT-BQF only requires 4.26 μs, indicating strong feasibility for real-time implementation.

Furthermore, experiments are conducted to validate the robustness of the filter parameters analyzed in Figure 5 and to provide a reference for parameter tuning. The results of sudden changes in parameters under steady-state conditions with the DPT-BQF at optimal tuning are shown in Figure 9.

The figure shows the results when the notch frequency, depth, and bandwidth are increased to ω_t = 38 Hz, x_t = −42 dB and b_t = 16 × ω_t rad/s. As seen when the notch frequency and bandwidth exceed the robust range, the system instantly becomes unstable. In contrast, increasing the notch depth still allows the system to operate stably, although the resonance suppression performance is significantly reduced. This is consistent with the analysis in Figure 5 above. As the notch frequency and bandwidth increase, a pair of poles moves along the real axis outside the stability circle, with larger values corresponding to greater distances from the stability boundary. Meanwhile, further decreasing the notch depth places the poles on the boundary; however, they essentially move along the boundary, minimally affecting stability but reducing resonance suppression performance. These results also validate the correctness of the discrete-domain analysis method proposed in this article and provide valuable guidance for BQF tuning.

6. Conclusions

This article presents a double-pre-warped Tustin bi-quad filter to suppress NIPMVM mechanical resonance. The notch distortion caused by traditional discretization is analyzed, and the mechanism of improved accuracy from double pre-warping of notch frequency and bandwidth is explained. In the discrete domain, the resonance suppression mechanism is studied from a closed-loop perspective, and a parameter tuning method ensuring system robustness is provided. Finally, experimental results confirm that the proposed DPT-BQF offers superior mechanical resonance suppression.