Second-Order Complex-Coefficient Flux Observer with Stator Resistance Estimation for Induction Motor Sensorless Drives

Abstract

The voltage model (VM) is widely used in sensorless induction motor drives owing to its structural simplicity and speed-independence. However, DC offsets in back electromotive force (BEMF) caused by measurement errors can degrade the accuracy of flux and speed estimation. To address this issue, this article proposes a second-order complex-coefficient flux observer (SCFO) that effectively eliminates DC offsets without introducing phase delay and amplitude attenuation, while maintaining excellent dynamic performance. Furthermore, to enhance the robustness of the flux observer against stator resistance variations, an improved stator resistance adaptive law based on the rotating reference frame is proposed. Ultimately, experimental validation on a 1.5 kW induction motor drive platform confirms the effectiveness of the proposed sensorless scheme.

1. Introduction

In recent decades, induction motor drives have been widely applied in industrial applications. In conventional vector control systems, rotor speed and position are obtained through mechanical sensors [1]. This scheme inevitably increases the complexity of mechanical structures and raises installation costs. To address these issues, many sensorless drive schemes have been proposed over the past two decades [2,3,4].

For high-performance induction motor sensorless drives, accurate flux estimation is crucial. The voltage model is widely used to estimate flux, and it is mainly applied in the following two sensorless schemes. The first is the model reference adaptive system (MRAS), which uses the voltage model as the reference model [5]. The second scheme employs methods such as inverse tangent calculation or phase-locked loop (PLL) to estimate rotor position and speed based on the flux estimated by the voltage model [6,7,8]. The performance of both sensorless schemes in estimating rotor position and speed depends heavily on the voltage model. According to the structure of the voltage model, flux estimation can be obtained by integrating BEMF. However, in practice, BEMF often contains DC offsets caused by measurement and sampling errors. The pure integration calculation, which is not recommended in practical applications, will make the estimated flux unbounded owing to the DC offset. To address these issues, a series of improvement schemes for voltage model flux estimators have been proposed. Existing solutions can be roughly categorized based on the introduction of reference flux information into two types: the closed-loop flux observer, which introduces reference flux information, while the open-loop flux observer is designed based on the desired frequency-domain characteristics. The most representative scheme of the former is the combined voltage and current model flux observer [9]. The observer introduces the flux estimated by the current model as the reference flux into the voltage model. In [10], a nonlinear flux observer is proposed for PMSM sensorless control, which also requires reference flux. However, unlike the observer mentioned above, the nonlinear flux observer has a more straightforward structure, as it does not require trigonometric function operations. In [11], the reference flux is introduced into the flux observer through proportional closed-loop control. To further eliminate the DC offset, a flux trajectory center correction method is designed. In [12], an offset estimator is proposed that uses a parallel structure of two closed-loop flux observers. It is worth mentioning that this observer requires a reference flux as the initial value for flux estimation. Although the closed-loop observer mentioned above addresses the issues of flux estimation saturation and offset through closed-loop feedback, this type of flux observer still requires precise reference flux information. When the reference flux is inaccurate, the performance of the closed-loop flux observer may deteriorate. The improved pure integrator scheme, which does not require the reference flux, avoids the issues mentioned earlier. A conventional improved pure integrator scheme uses a low-pass filter (LPF) instead of the pure integrator. While the improved scheme can slightly suppress DC offset, it also introduces significant phase delay and amplitude attenuation when the operating frequency is below the cutoff frequency. In [13], a flux observer based on the first-order LPF with the LPF pole proportional to the stator frequency is proposed. However, the proposed schemes suffer from the deficiency of requiring complicated flux vector rotation and amplification calculations. Another popular method for improving pure integrator schemes is to use generalized integrators instead of pure integrators. Compared to LPF, generalized integrators do not require magnitude and phase compensation. In [14], a flux observer based on the second-order generalized integrator (SOGI) is proposed. In [15,16,17,18], different high-order generalized integrator-based improved flux observers are proposed. It is noteworthy that only generalized integrators of third-order or higher can effectively eliminate DC offset, whereas SOGI can only suppress it. Although increasing the generalized integrator’s order improves the capability to mitigate DC offset and harmonics, it also increases the complexity of the observer structure. In [19,20], A complex-coefficient flux observer (CFO) that utilizes the orthogonality between flux and BEMF is proposed. Although the structure of the CFO is simpler than most of the above flux observers, its ability to suppress DC offset is limited.

In sensorless vector control systems of induction motors, stator resistance mismatch can lead to flux estimation errors and field orientation errors, resulting in a decline in sensorless control performance and even stability issues in the system. Accurate stator resistance parameters are crucial for achieving high-performance sensorless vector control [21]. Traditional methods use the stator resistance estimated offline in the flux observer, but the actual stator resistance often changes due to the temperature rise caused by long-term motor operation. To address this issue, numerous studies have proposed online stator resistance identification methods based on full-order adaptive observers [22,23] and full-order sliding mode observers [24,25,26] combined with model reference adaptive laws. The coupling between speed and stator resistance estimation makes the sensitivity of stator resistance dependent on the performance of stator frequency estimation.

In this article, a novel second-order complex-coefficient flux observer (SCFO) for DC offset elimination is proposed. The designed SCFO has a bandpass frequency characteristic, which eliminates the DC offset while avoiding amplitude attenuation and phase lag in the fundamental component. The introduction of a complex coefficient structure reduces the order of the flux observer, with the designed observer having an order of 2, making it simpler and easier to implement compared to flux observers based on high-order generalized integrators. In addition, detailed performance analysis and discretization implementation are also provided. In addition, an improved stator resistance adaptive law based on the rotating reference frame is proposed. The designed adaptive law does not require the participation of the stator frequency, effectively decoupling the stator frequency estimation and stator resistance estimation.

The outline of this brief is structured as follows. In Section 2 the voltage model of the induction motor is given. In Section 3, the SCFO is proposed, and the stability and discretization of the SCFO are also discussed in this Section. In Section 4, a stator resistance estimator is proposed. In Section 5, comparative experiments are conducted to verify the effectiveness of the proposed methods. Finally, Section 6 concludes this article.

2. Model of Induction Motor

The mathematical model of inverse-Γ equivalent circuit of induction motor in the αβ reference frame can be given as

$$\left\{\begin{array}{l}{\mathit{u}}_{\mathit{s}}={\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{s}}+{\mathit{L}}_{\sigma }\mathit{p}{\mathit{i}}_{\mathit{s}}+{\mathit{R}}_r{\mathit{i}}_{\mathit{s}}-{\mathit{R}}_r/{\mathit{L}}_{\mathrm{m}}\mathit{I}{\mathit{\psi }}_r+{\mathsf{\omega }}_r\mathit{J}{\mathit{\psi }}_r\\ \mathit{p}{\mathit{\psi }}_r={\mathit{R}}_r{\mathit{i}}_{\mathit{s}}-{\mathit{R}}_r/{\mathit{L}}_{\mathrm{m}}\mathit{I}{\mathit{\psi }}_r+{\mathsf{\omega }}_r\mathit{J}{\mathit{\psi }}_r\end{array}\right.$$

(1)

where ${\mathit{u}}_s={[{\mathit{u}}_{\alpha }{\mathit{u}}_{\beta }]}^T$, ${\mathit{i}}_s={[{\mathit{i}}_{\alpha }{\mathit{i}}_{\beta }]}^T$ and ${\mathit{\psi }}_{\mathit{r}}={[{\psi }_{\mathit{r}\alpha }{\psi }_{\mathit{r}\beta }]}^T$ are stator voltage vector, stator current vector, and rotor flux vector, respectively. R_s, R_r, L_m, L_σ are the stator resistance, rotor resistance, magnetizing inductance, and transient inductance, respectively. p is the differential operator symbol. ${\omega }_{\mathit{r}}$ is the electrical rotor angular speed, p_n is the number of pole pairs. According to (1), ${\mathit{\psi }}_{\mathit{r}}$ can be expressed as

$${\mathit{\psi }}_{\mathit{r}}={\displaystyle \int ({\mathit{u}}_s-{\mathit{R}}_s{\mathit{i}}_s-L_{\sigma }p{\mathit{i}}_s)dt}={\displaystyle \int {\mathit{e}}_{\mathit{r}}dt}$$

(2)

where ${\mathit{e}}_{\mathit{r}}$ is the BEMF vector. The above equation represents the traditional voltage-model flux observer.

3. Proposed Flux Observer

In this Section, the SCFO is proposed to overcome the shortcomings of conventional flux observers. The proposed observer can eliminate the DC offset and maintain excellent dynamic performance. The stability and discretization of SCFO are also discussed in this section.

3.1. Second-Order Complex-Coefficient Flux Observer

The traditional complex-coefficient filter [27], denoted as $G_c(s)$, functions as a low-pass filter and is ineffective at eliminating DC offset. Although the high-pass filter $G_h(s)$ has a simple structure and can remove DC offset, it introduces both magnitude and phase errors in the frequency response at the fundamental frequency.

$$\left\{\begin{array}{l}{G}_c(s)={\displaystyle \frac{k_c}{k_c+s-j{\widehat{\omega }}_f}}\\ G_h(s)={\displaystyle \frac{s}{s+{\omega }_c}}\end{array}\right.$$

(3)

where $k_c$ is gain of complex-coefficient filter, ${\omega }_c$ is the cut-off frequency of conventional high-pass filters. To effectively eliminate DC offset in flux estimation, a series of complex-coefficient high-pass filters is designed by combining two conventional filters. $G_{HPF}$ maintains the same magnitude-frequency response as a pure integrator at the fundamental frequency while effectively eliminating DC offset.

$$G_{HPF}={\displaystyle \frac{sL(s)}{sL(s)+N(s)(k\left|{\widehat{\omega }}_f\right|+jsks\mathit{i}gn({\widehat{\omega }}_f))}}$$

(4)

where k is the observer gain. HPF with distinct frequency characteristics can be acquired by setting diverse L(s) and N(s). By serially connecting the CFO with the designed HPF, the following flux observer can be obtained:

$${\displaystyle \frac{{\mathit{\psi }}_{\mathit{SCFO}}(\mathit{s})}{{\mathit{e}}_{\mathit{r}}(\mathit{s})}}={\displaystyle \frac{\mathit{sL}(\mathit{s})}{\mathit{sL}(\mathit{s})+\mathit{N}(\mathit{s})(\mathit{k}\left|{\widehat{\omega }}_f\right|+\mathit{jsksign}({\widehat{\omega }}_f))}}{\displaystyle \frac{1-\mathit{jksign}({\widehat{\omega }}_f)}{\mathit{s}+\mathit{k}\left|{\widehat{\omega }}_f\right|}}$$

(5)

To simplify the structure of the proposed flux observer, L(s) is set to unity. To extract the offset in the BEMF, N(s) is set to $1/(s+k\left|{\widehat{\omega }}_f\right|)$. Substituting $N(s)=1/(s+k\left|{\widehat{\omega }}_f\right|)$ and L(s) = 1 into (5), the transfer function of the proposed SCFO can be given as

$${\displaystyle \frac{{\mathit{\psi }}_{\mathit{SCFO}}(s)}{{\mathit{e}}_{\mathit{r}}(s)}}={\displaystyle \frac{s(1-jk\cdot s\mathit{i}gn({\widehat{\omega }}_f))}{s(s+k\left|{\widehat{\omega }}_f\right|)+jsks\mathit{i}gn({\widehat{\omega }}_f)+k\left|{\widehat{\omega }}_f\right|}}$$

(6)

The block diagrams of the proposed SCFO is shown in Figure 1.

3.2. Stability Analysis

The stability of the designed SCFO is analyzed through flux estimation error convergence, and ${\widehat{\omega }}_f>0$ is assumed to facilitate the stability proof. Define ${\mathit{\psi }}_{\mathit{r}}^{\ast }={\mathit{e}}_{\mathit{r}}/j{\omega }_f$, where ${\mathit{\psi }}_{\mathit{r}}^{\ast }$ is the flux directly calculated from ${\mathit{e}}_{\mathit{r}}$. Substituting ${\mathit{\psi }}_{\mathit{r}}^{\ast }$ into (6), one obtains

$${\displaystyle \frac{{\mathit{\psi }}_{SCFO}(s)}{{\mathit{\psi }}_{\mathit{r}}^{\ast }(s)}}={\displaystyle \frac{s(1-jk)j{\omega }_f}{s(s+k{\widehat{\omega }}_f)+jsk+k{\widehat{\omega }}_f}}$$

(7)

Considering ${\mathit{\psi }}_{SCFO}={\tilde{\mathit{\psi }}}_{\mathit{r}}+{\mathit{\psi }}_{\mathit{r}}^{\ast }$, where ${\tilde{\mathit{\psi }}}_{\mathit{r}}$ is the difference between the flux estimated by the SCFO and ${\mathit{\psi }}_{\mathit{R}}^{\ast }$, (7) can be expressed as

$${\displaystyle \frac{{\tilde{\mathit{\psi }}}_{\mathit{r}}(\mathit{s})+{\mathit{\psi }}_r^{\ast }(\mathit{s})}{{\mathit{\psi }}_r^{\ast }(\mathit{s})}}={\displaystyle \frac{\mathit{s}(1-\mathit{jk})\mathit{j}{\omega }_f}{\mathit{s}(\mathit{s}+\mathit{k}{\widehat{\omega }}_f)+\mathit{jsk}+\mathit{k}{\widehat{\omega }}_f}}$$

(8)

To derive the characteristic equation of the flux estimation error, one obtains

$${\tilde{\mathit{\psi }}}_{\mathit{r}}(\mathit{s})({\mathit{s}}^2+(\mathit{k}{\widehat{\omega }}_f+\mathit{jk})\mathit{s}+\mathit{k}{\widehat{\omega }}_f)={\mathit{\psi }}_r(\mathit{s})({\mathit{s}}^2+(\mathit{k}{\widehat{\omega }}_f+\mathit{jk})s+\mathit{k}{\widehat{\omega }}_f-\mathit{j}{\omega }_f\mathit{s}-\mathit{k}{\omega }_f\mathit{s})$$

(9)

When SCFO converges to the real motor system, substituting $s=j{\widehat{\omega }}_f$ and ${\omega }_f={\widehat{\omega }}_f$ into (9), the rotor flux estimation error can be derived as

$${\tilde{\psi }}_{\mathit{R}}(s)(s^2+(k{\widehat{\omega }}_f+jk)s+k{\widehat{\omega }}_f)=0$$

(10)

The eigenvalues of (10) can be expressed as

$$c_{1,2}={\displaystyle \frac{-jk-k{\widehat{\omega }}_f\pm \sqrt{{(jk+k{\widehat{\omega }}_f)}^2-4k{\widehat{\omega }}_f}}{2}}$$

(11)

Since the observer gains k > 0, it can be deduced that $\mathit{R}e(\sqrt{{(jk+k{\widehat{\omega }}_f)}^2-4k{\widehat{\omega }}_f})<k{\widehat{\omega }}_f$. Therefore, the real parts of both eigenvalues are negative, which guarantees the convergence of the flux estimation error.

3.3. Performance Analysis

Analysis of DC component: When the DC component is used as input for the SCFO, the following can be obtained

$${\mathit{\psi }}_{\mathit{SCFO},\mathit{dc}}(\mathit{s})=\mathit{s}{\displaystyle \frac{\mathit{s}(1-\mathit{jksign}({\widehat{\omega }}_f\left)\right)}{\mathit{s}(\mathit{s}+\mathit{k}\left|{\widehat{\omega }}_f\right|)+\mathit{jksign}({\widehat{\omega }}_f)+\mathit{k}\left|{\widehat{\omega }}_f\right|}}\cdot {\displaystyle \frac{{\mathit{e}}_{\mathit{offset}}}{\mathit{s}}}$$

(12)

Through the terminal value theorem, flux estimation error can be described as

$$\begin{array}{ll}\underset{\mathit{t}\to \infty }{\mathit{lim}}{\psi }_{SCFO,dc}(\mathit{t})& =\underset{\mathit{s}\to 0}{\mathit{lim}}\mathit{s}{\mathit{\psi }}_{\mathit{SCFO},\mathit{dc}}(\mathit{s})\\ & =\underset{\mathit{s}\to 0}{\mathit{lim}}\mathit{s}{\displaystyle \frac{\mathit{s}(1-\mathit{jksign}({\widehat{\omega }}_f\left)\right)}{\mathit{s}(\mathit{s}+\mathit{k}\left|{\widehat{\omega }}_f\right|)+\mathit{jksign}({\widehat{\omega }}_f)+\mathit{k}\left|{\widehat{\omega }}_f\right|}}\cdot {\displaystyle \frac{{\mathit{e}}_{\mathit{offset}}}{s}}=0\end{array}$$

(13)

From (13), it can be indicated that the proposed flux observer can theoretically eliminate the DC offset.

(2) Analysis of fundamental component: substituting $s=j{\widehat{\omega }}_f$ into (6), it can be obtained as

$${\mathit{G}}_{\mathit{SCFO}}(\mathit{j}{\widehat{\omega }}_f)={\displaystyle \frac{\mathit{j}{\widehat{\omega }}_f+\mathit{k}\left|{\widehat{\omega }}_f\right|}{(-{{\widehat{\omega }}_f}^2+\mathit{jk}{{\widehat{\omega }}_f}^2\times \mathit{sign}({\widehat{\omega }}_f\left)\right)-\mathit{k}\left|{\widehat{\omega }}_f\right|+\mathit{k}\left|{\widehat{\omega }}_f\right|}}={\displaystyle \frac{1}{\mathit{j}{\widehat{\omega }}_f}}$$

(14)

As shown in (14), the frequency domain response of the fundamental component of SCFO is consistent with that of the pure integrator.

(3) Analysis of higher-order harmonic component: substituting $s=jh{\widehat{\omega }}_f$ into (6), it can be obtained as

$${\mathit{G}}_{\mathit{SCFO}}(\mathit{jh}{\widehat{\omega }}_f)={\displaystyle \frac{\mathit{jh}{\widehat{\omega }}_f(1-\mathit{jksign}({\widehat{\omega }}_f\left)\right)}{-(\mathit{h}{\widehat{\omega }}_f{)}^2-\mathit{hk}\left|{\widehat{\omega }}_f\right|+\mathit{k}\left|{\widehat{\omega }}_f\right|+\mathit{jhk}{\widehat{\omega }}_f\left|{\widehat{\omega }}_f\right|}}$$

(15)

The magnitude response of flux estimation can be given as

$${\displaystyle \frac{\mathit{h}\sqrt{{\mathit{k}}^2+1}}{\sqrt{{(\mathit{k}-\mathit{hk}-{\mathit{h}}^2{\widehat{\omega }}_f)}^2+(\mathit{kh}{\widehat{\omega }}_f{)}^2}}}$$

(16)

As shown in (16), the ability of the SCFO to suppress harmonics mainly depends on the gain k. The performance of SCFO in suppressing harmonics is comparable to that of the CFO, and the conclusion can also be derived from Figure 2, which presents the Bode diagrams of the pure integrator, SCFO, and CFO with different gains. In Figure 2, the center frequency is set to 100 rad/s, and the gains k of the SCFO are set to 2. From Figure 2, it can be concluded that the flux estimation of the pure integrator will saturate due to its infinite gain at zero frequency. The DC gain of the CFO will decrease as the observer gain increases, while this trade-off inevitably degrades its filtering performance. Compared to the aforementioned flux observer, the proposed SCFO exhibits a band-pass characteristic, enabling it to effectively eliminate the DC component. As evidenced by the Bode plot and performance analysis, increasing the observer gain accelerates the convergence speed of the DC offset estimation but reduces its ability to suppress higher-order harmonics. The gain only affects the convergence speed of the DC offset estimation, while the DC component in the flux estimation will eventually converge to zero. Therefore, since the primary focus is on the observer’s capability to suppress higher-order harmonics, the gain should not be set too high and is generally selected within the range of 1 to 5.

3.4. Discrete Implementation

The SCFO designed in the continuous domain requires discretization. This process is accomplished using the backward Euler method, and the corresponding block diagram is illustrated in Figure 3. The discretization of SCFO is categorized into two parts. The first part involves the discretization of the offset estimation, and the second part involves the discretization of the rotor flux estimation. The discretization processes are shown as follows.

$$\left\{\begin{array}{lll}{\hat{\mathit{e}}}_{\mathit{offset}}(\mathit{i}+1)& =& {\hat{\mathit{e}}}_{\mathit{offset}}(\mathit{i})+{\mathit{T}}_{\mathit{s}}\mathit{k}[\mathit{jsign}({\widehat{\omega }}_f)({\mathit{e}}_r(\mathit{i}+1)-{\hat{\mathit{e}}}_{\mathit{offset}}(\mathit{i}))+{\mathit{\psi }}_{\mathit{SCFO}}(\mathit{i})\left|{\widehat{\omega }}_f\right|]\\ {\mathit{\psi }}_{\mathit{SCFO}}(\mathit{i}+1)& =& {\mathit{\psi }}_{\mathit{SCFO}}(\mathit{i})+{\mathit{T}}_{\mathit{s}}[-k\mathit{jsign}({\widehat{\omega }}_f)({\mathit{e}}_r(\mathit{i}+1)-{\hat{\mathit{e}}}_{\mathit{offset}}(\mathit{i}))-k{\mathit{\psi }}_{\mathit{SCFO}}(\mathit{i})\left|{\widehat{\omega }}_f\right|\\ & & +({\mathit{e}}_r(\mathit{i}+1)-{\hat{\mathit{e}}}_{\mathit{offset}}(\mathit{i}+1))]\end{array}\right.$$

(17)

where T_s is the discretization period. To further demonstrate the advantages of the proposed method, the computational complexity of the four observers after applying the backward difference method is shown in

Table 1. Comparison of Computational complexity.

Flux Observer	Order	Addition Operation	Multiplication Operation	Offset Suppression
CFO	1	6	8	medium
SOGI	2	8	12	medium
TOGI	3	12	16	Good
SCFO	2	10	12	Good

.

Both the SCFO and the Third-order generalized integrator exhibit good DC offset suppression capability, but the SCFO has lower computational complexity. Pseudo-code for the discrete implementation of SCFO is presented as Appendix A.

3.5. Analysis of Parameter Perturbation on the SCFO

The ideal BEMF and the fundamental component of the BEMF considering parameter variations are expressed as follows:

$$\left\{\begin{array}{l}{\mathit{e}}_{\mathit{r}} ={\mathit{u}}_s-{\mathit{R}}_s{\mathit{i}}_s-L_{\sigma }p{\mathit{i}}_s\\ {\mathit{e}}_{\mathit{r},\mathit{v}}={\mathit{u}}_s-({\mathit{R}}_s+\Delta {\mathit{R}}_s){\mathit{i}}_s-(L_{\sigma }+\Delta L_{\sigma })p{\mathit{i}}_s\end{array}\right.$$

(18)

As can be seen from (14), the proposed SCFO exhibits identical frequency characteristics with the pure integrator at the fundamental frequency. Additionally, variations in stator resistance and leakage inductance are reflected only in the fundamental component of the BEMF. Therefore, the impact of parameter variations on the flux estimation of the proposed method can be expressed as follows:

$${\mathit{\psi }}_{SCFO}-{\mathit{\psi }}_{\mathit{r}}=-\Delta {\mathit{R}}_s{\mathit{i}}_s/j{\omega }_f-\Delta L_{\sigma }{\mathit{i}}_s$$

(19)

In fact, for open-loop flux observers with frequency characteristics at the fundamental frequency identical to that of the pure integrator, such as CFO and SCFO, their sensitivity to stator resistance and leakage inductance is the same. The flux estimation errors caused by parameter deviations in these observers can be represented by (19). From the equation, it can be observed that at low speeds, the deviation of stator resistance has a significant impact on flux estimation, while at high speeds, the leakage inductance has a more pronounced effect.

4. Proposed Stator Resistance Estimator

The voltage equation of the induction motor in the rotating reference frame can be written as follows

$$\left\{\begin{array}{l}{\mathit{u}}_{\mathit{sd}}={\mathit{L}}_{\sigma }\mathit{p}{\mathit{i}}_{\mathit{sd}}+{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sd}}+\mathit{p}{\psi }_{\mathit{rd}}-{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sq}}-{\omega }_{\mathit{e}}{\psi }_{\mathit{rq}}\\ {\mathit{u}}_{\mathit{sq}}={\mathit{L}}_{\sigma }\mathit{p}{\mathit{i}}_{\mathit{sq}}+{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sq}}+\mathit{p}{\psi }_{\mathit{rq}}+{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sd}}+{\omega }_{\mathit{e}}{\psi }_{\mathit{rd}}\end{array}\right.$$

(20)

Through setting the derivative terms to zero, the steady-state voltage equations are derived as follows

$$\left\{\begin{array}{l}{\mathit{u}}_{\mathit{sd}}={\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sd}}-{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sq}}-{\omega }_{\mathit{e}}{\psi }_{\mathit{rq}}\\ {\mathit{u}}_{\mathit{sq}}={\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sq}}+{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sd}}+{\omega }_{\mathit{e}}{\psi }_{\mathit{rd}}\end{array}\right.$$

(21)

When the stator resistance is accurate, that is, when the field orientation is accurate, the following can be obtained

$$\left\{\begin{array}{l}{\psi }_{\mathit{rq}}=0\\ {\psi }_{\mathit{rd}}=L_m{\mathit{i}}_{sd}\end{array}\right.$$

(22)

Substituting (22) into (21) gives the following

$$\left\{\begin{array}{l}{\mathit{u}}_{\mathit{sd}}={\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sd}}-{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sq}}\\ {\mathit{u}}_{\mathit{sq}}={\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sq}}+{\mathit{L}}_{\sigma }{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sd}}+{\mathit{L}}_m{\omega }_{\mathit{e}}{\mathit{i}}_{\mathit{sd}}\end{array}\right.$$

(23)

The reference model of the traditional active power model is the dot product of the stator voltage vector and the stator current vector. However, when applying this reference model, it inevitably leads to the presence of stator frequency terms in the adjustable model. Therefore, an improved reference model can be designed as follows.

$$\mathit{Ref}=\left(L_{\sigma }{\mathit{u}}_{sq}{\mathit{i}}_{sq}\right)/L_s+{\mathit{u}}_{sd}{\mathit{i}}_{sd}$$

(24)

Substituting (23) into the above equation, the adjustable model in the model reference adaptive system can be obtained as follows:

$$\mathit{Adj}=\left(L_{\sigma }{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sq}}^2\right)/{\mathit{L}}_{\mathit{s}}+{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sd}}^2$$

(25)

From the above equation, it can be seen that the stator frequency does not appear in the adjustable model, thereby achieving the decoupling between stator frequency estimation and stator resistance estimation. When applying the PI adaptive law, the stator resistance adaptive rate can be written as follows:

$${\hat{\mathit{R}}}_{\mathit{s}}=\left({\mathit{k}}_p+{\displaystyle \frac{k_{\mathit{i}}}{p}}\right)\left(\left(L_{\sigma }{\mathit{u}}_{sq}{\mathit{i}}_{sq}\right)/L_s+{\mathit{u}}_{sd}{\mathit{i}}_{sd}-\left(L_{\sigma }{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sq}}^2\right)/{\mathit{L}}_{\mathit{s}}-{\mathit{R}}_{\mathit{s}}{\mathit{i}}_{\mathit{sd}}^2\right)$$

(26)

Block diagram of the stator resistance estimator is shown in Figure 4. It is noteworthy that, considering that the proportional gain may introduce noise into the estimation, the proportional gain is often set to 0, with only the integral gain being adjusted.

5. Experimental Results

In order to verify the effectiveness of the SCFO proposed in this article, experiments were conducted on the experimental platform shown in Figure 5. This platform consists of two identical induction motors, two AC servo drives, a host computer, etc. The two induction motors and the torque sensor are coupled together. The motor on the right side of the platform is used to test the proposed algorithm, while the other motor operates in torque mode to supply load torque. The two induction motors and the torque sensor are mechanically connected through couplings, with the motor on the right side of the platform designated for testing the proposed algorithm, while the other motor operates in torque mode to provide load torque. The nominal parameters of the test motor are listed in

Table 2. Parameters of the Tested Induction Motor.

Parameter	Value	Parameter	Value
rated speed	1500 rpm	magnetizing inductance	91 mH
rated voltage	220 V	stator leakage inductance	10 mH
rated current	6.3 A	stator resistance	1.21 Ω
pole pairs	2	rotor resistance	0.74 Ω

. An incremental encoder with 5000 pulses per revolution is installed on each induction motor. Two AC servo drives based on STM32F103 are employed to control the motors separately. The phase current measured by sampling resistance is used for current control. The stator voltage is not directly monitored; instead, the flux observer utilizes the reference voltage derived from the current controller. The method from [28] is applied to compensate for the inverter nonlinearity. Both the switching frequency and the sampling frequency are set to 10 kHz. A host computer is utilized for data observation and code programming. In the experiments, the PLL is used to estimate the position and speed. The block diagram of the sensorless-driven induction motor system using the proposed scheme is shown in Figure 6. In the experiment, the gain of the proposed method and the conventional method are both set to 2. The integral gain of the stator resistance estimator is set to 1. Five independent repeated experiments were conducted, and the results obtained are very similar. The position obtained by the encoder is used solely for comparison in experiments.

Figure 7 and Figure 8 show the experimental results of the conventional CFO and SCFO, respectively, under DC disturbances at 600 r/min and no load. As shown in Figure 7a,b, it is observed that there exists an apparent DC offset in the estimated rotor flux, and the estimated speed has fluctuations with a frequency of ω_f. When the DC disturbance is injected into the α-phase changes from 1 V to 2 V, the speed estimation error increases. The difference between the DC components of the α-phase rotor flux and the β-phase rotor flux increased from 10.2 mWb to 20.1 mWb. From Figure 8, it can be seen that the proposed SCFO maintains excellent speed estimation performance under DC disturbances. When a DC disturbance is injected, the speed estimation error undergoes a slight fluctuation and then rapidly converges. This transient process involves estimating the DC component in BEMF. Once the estimation of the DC offset is completed, there is minimal change in both the speed estimation error and the estimated flux compared to the state before the injection of the DC disturbance. The experiments indicate that the studied observer can identify and suppress DC disturbance rapidly.

In order to validate the suppression ability of the proposed method for DC offset within different speed domains, the experimental results of the two flux observers under speed variations are presented in Figure 9 and Figure 10, respectively. It can be observed from Figure 9 that at low speeds, the DC offset in the flux estimation and the fluctuation of the speed estimation error are relatively large in conventional CFO. With the increase in rotor speed, the DC offset in flux estimation and the fluctuations of the speed estimation error decrease, but there still exists a speed estimation error with fluctuations at a frequency of ω_f. In contrast, the speed estimation error of the proposed method is significantly smaller than that of the comparative method. As shown in Figure 10, the proposed method demonstrates excellent dynamic performance throughout the entire speed-up process.

The experimental results of the two flux observers under rated load at 400 rpm are shown in Figure 11 and Figure 12, respectively. From Figure 11a,b, it can be observed that when the load torque suddenly changes from 0 to 1 p.u., the speed estimation errors for the CFO methods increase. Conversely, the proposed method quickly estimates the DC component in the BEMF following an abrupt load change, resulting in smaller speed estimation errors compared to the conventional methods. The experiments demonstrated that the proposed method effectively achieves good DC offset elimination performance under rated load torque. Figure 13 presents the experimental results of the proposed method under rated load, with the speed ranging from 200 rpm to −200 rpm. From Figure 13, it can be observed that the speed estimation and flux estimation remain stable in the vicinity of 0 Hz frequency.

The experiment verified that the proposed SCFO has good dynamic performance at low frequencies. At 300 rpm and rated load, the experimental results for different initial stator resistance values are shown in Figure 14 and Figure 15, respectively. It can be observed from the figures that whether the initial stator resistance is 0.5 p.u. or 1.5 p.u., after enabling the stator resistance estimator, the stator resistance estimate quickly converges to the true stator resistance value. Once the stator resistance converges, the speed estimation error also converges simultaneously due to the convergence of the flux estimation.

6. Conclusions

In this article, A novel SCFO is proposed for sensorless induction motor drives. This flux observer effectively estimates the DC offset of the BEMF based on the orthogonality between the rotor flux and the BEMF, and it eliminates the DC offset in the flux estimation through feedforward compensation. The bandpass characteristic of the flux observer facilitates the elimination of the DC offset while maintaining consistency between the fundamental component of the flux estimation and that of the pure integrator. Furthermore, an improved stator resistance adaptive law based on the rotating reference frame is proposed. Finally, Comprehensive experimental results demonstrate that, compared to conventional methods, the proposed observer effectively eliminates the DC offset in flux estimation under both steady-state and dynamic conditions, thereby improving the accuracy of speed estimation.