Sensorless Stator Field-Oriented Controlled IM Drive at Low Speed with 𝑅_𝑟 Estimator

Abstract

This paper pertains to a technique of a sensorless indirect stator field-oriented induction motor control, which prevents the accumulative errors incurred by the integrator and the problem relating to the stability of the control system caused by the stator resistance susceptible to temperature variations while conducting the flux estimation directly and computing the synchronous rotary speed. The research adds an adaptive flux observer to estimate the speed of the rotor and uses the fixed trace algorithm (FTA) to execute an online estimation of the slip difference, thereby improving the system of stability under the low rotary speed at regenerating mode and the influence of the rotor resistance on the slip angle. Finally, the paper conducts simulations by Simulink of MATLAB and practices to verify the correctness of the result the paper presents.

1. Introduction

Induction motors are broadly applied to industrial machines due to their simple configuration and easy maintenance. However, the induction motor is the nonlinear and time-varying system, so the variable-transmission drive system of the early-stage induction motor has not been easily carried out until the field-oriented control theory was developed by Blaschke in the year 1972 and the rapid development of the algorithm technique for semiconductor came to overcome these problems [1–3]. To save the space and maintain and decrease the influence of promiscuous signals of the speed sensor on the reliability and the antinoise capability of the control system of the induction motor, researches on sensorless speed controller have been claimed incessantly [4–7]. It can be classified into the flux estimation mode and the model reference adaptive mode [4]. The former conducts the estimation by using an open-loop method, so it is susceptible to variations of motor parameters. The latter adopts the structure of the model adaptive mode in order to obtain a preferable system response and accuracy of estimation [1, 8]. However, when this estimation system is operated in the low rotary speed at regenerating mode, unstable areas still exist, which would affect the sturdiness of the entire sensorless speed control system [5]. In order to operate the entire estimation system in the low rotary speed at regenerating mode, this research adds the observer and uses the Routh Stability Criterion to obtain the observer feedback gain under the stabilized system [1, 6]. Further, the stator field-oriented control configuration is relatively simple, so it is often applied to the vehicle traction control system. The system can be classified into a direct control and an indirect control. The former adopts the integration induction voltage to estimate the flux directly and compute the synchronous speed. However, the integrator accumulates errors easily, and the generation of incorrect parameters of the stator resistance affects the estimation result as well. Therefore, this research adopts the latter. The shortcoming of the latter is that the accuracy of estimating the slip difference is affected as the slip estimation is susceptible to the rotor resistance which is subjected to temperature changes, and thence the conversion of coordinates and the stability of the system are also affected. Accordingly, the subject research adopts the rotor observer by the fixed trace algorithm (FTA) to estimate the rotor resistance [2, 9].

Therefore, the subject research applies an indirect stator field-oriented structure to control the induction motor, adopts the adaptive observer to estimate the rotor speed of the motor for the demand for low rotary speed regenerating mode, and uses the fixed trace algorithm (FTA) to estimate the rotor resistance, thereby increasing the control stability and accuracy of the induction motor.

2. Methodology

2.1. Dynamic of the Induction Motor

Under the random 𝑑-𝑞 coordinate system, the electrical equations for the stator side and the rotor side of the induction motor can be, respectively, expressed by 𝑅_𝑠+𝐿_𝑠𝑝𝐈+𝜔𝐿_𝑠𝐉^⇀𝑖^𝑎_𝑠+𝐿_𝑚𝑝𝐈+𝜔𝐿_𝑚𝐉^⇀𝑖^𝑎_𝑟=𝑣^⇀𝑎_𝑠𝐿_𝑚𝑝𝐈+𝐿_𝑚𝜔−𝜔_𝑟𝐉^⇀𝑖^𝑎_𝑠+𝑅_𝑟+𝐿_𝑟𝑝𝐈+𝐿_𝑟𝜔−𝜔_𝑟𝐉^⇀𝑖^𝑎_𝑟=0. (1)(2)

Under the random coordinate, the stator flux 𝜙^⇀𝑎_𝑠 and the rotor flux 𝜙^⇀𝑎_𝑟 can be, respectively, expressed by 𝜙^⇀𝑎_𝑠=𝐿_𝑠^⇀𝑖^𝑎_𝑠+𝐿_𝑚^⇀𝑖^𝑎_𝑟,_⇀𝜙^𝑎_𝑟=𝐿_𝑚^⇀𝑖^𝑎_𝑠+𝐿_𝑟^⇀𝑖^𝑎_𝑟, (3)(4) wherein the stator voltage is 𝑣^⇀𝑎_𝑠=𝑣^𝑎_𝑑𝑠𝑣^𝑎_𝑞𝑠^𝑇, the stator current is ^⇀𝑖^𝑎_𝑠=𝑖^𝑎_𝑑𝑠𝑖^𝑎_𝑞𝑠^𝑇, the rotor current is ^⇀𝑖^𝑎_𝑟=𝑖^𝑎_𝑑𝑟𝑖^𝑎_𝑞𝑟^𝑇, the stator flux is 𝜙^⇀𝑎_𝑠=𝜙^𝑎_𝑑𝑠𝜙^𝑎_𝑞𝑠^𝑇, the rotor flux is 𝜙^⇀𝑎_𝑟=𝜙^𝑎_𝑑𝑟𝜙^𝑎_𝑞𝑟^𝑇, the motor parameters, such as 𝑅_𝑆, 𝑅_𝑟, 𝐿_𝑆, 𝐿_𝑟, and 𝐿_𝑚 are, respectively, are denoted by the stator resistance, the rotor resistance, the stator inductance, the rotor inductance, and the mutual inductance, 𝜔 and 𝜔_𝑟 are denoted by the speed of the random angle and the speed of the rotor angle, 𝑝=𝑑/𝑑𝑡, 𝐈=¹⁰₀₁, and 𝐉=⁰⁻¹₁₀.

Formula (3) can be rewritten by ^⇀𝑖^𝑎_𝑟=(1/𝐿_𝑚)(𝜙^⇀𝑎_𝑠−𝐿_𝑠^⇀𝑖^𝑎_𝑠), then substitute it into equations (1) and (2); then convert the formula into the synchronous coordinate system (𝜔=𝜔_𝑒), and thence get 𝑅_𝑠𝐈^⇀𝑖^𝑒_𝑠+𝑝𝐈+𝜔_𝑒𝐉𝜙^⇀𝑒_𝑠=𝑣^⇀𝑒_𝑠,−𝐿_𝑠𝐿_𝑚𝑅_𝑟+𝜎𝐿_𝑟𝑝𝐈+𝜎𝐿_𝑟𝜔_𝑠𝑙𝐉^⇀𝑖^𝑒_𝑠+𝐿_𝑟𝐿_𝑚1𝜏_𝑟+𝑝𝐈+𝜔_𝑠𝑙𝐉𝜙^⇀𝑒_𝑠=0. (5)(6)

Finally, formulas (5) and (6) are arranged into the dynamic as follows: 𝑝𝐗=𝐴𝐗+𝐵𝑣^⇀𝑒_𝑠,_⇀𝑖^𝑒_𝑠=𝐶𝐗, (7)(8) wherein 𝐗=^⇀𝑖𝜙^𝑒_𝑠^⇀𝑒_𝑠^𝑇,𝐴𝐴=¹¹𝐴₁₂𝐴₂₁𝐴₂₂,1𝐵=𝐿_𝜎1^𝑇,,𝐴𝐶=10₁₁𝑅=−^𝑠𝐿_𝑠+1𝜏_𝑟𝜎𝐈−𝜔_𝑒−𝜔_𝑟𝐴𝐉,₁₂=1𝐿_𝜎1𝜏_𝑟𝐈−𝜔_𝑟𝐉,𝐴₂₁=−𝑅_𝑠𝐴𝐈,₂₂=−𝜔_𝑒𝜏𝐉,_𝑟=𝐿_𝑟𝑅_𝑟,𝐿_𝜎=𝐿_𝑠−𝐿_𝑚²𝐿_𝑟, (9) and 𝜔_𝑒 is the speed of the synchronous angle.

2.2. Full-Order Adaptive Speed Estimator

Formulas (7) and (8) are added into the full-order flux observer and arranged as 𝑝𝐗=𝐴𝐗+𝐵𝑣^⇀𝑒_𝑠+𝐻^⇀𝑖^𝑒_𝑠−^⇀𝑖^𝑒_𝑠⇀𝑖^𝑒_𝑠=𝐶𝐗, (10)(11) wherein, the symbol “” represents an estimated value, 𝐴=^𝐴11𝐴12_𝐴21𝐴22, 𝐴₁₁=−(𝑅_𝑠/𝐿_𝑠+1/𝜏_𝑟𝜎)𝐈−(𝜔_𝑒−𝜔_𝑟)𝐉, 𝐴₁₂=(1/𝐿_𝜎)((1/𝜏_𝑟)𝐈−𝜔_𝑟𝐉), 𝐻₁=ℎ₁𝐈+ℎ₂𝐉, 𝐻₂=ℎ₃𝐈+ℎ₄𝐉, and 𝐻=^𝐻1_𝐻2 represents the observer feedback gain matrix.

According to the model reference adaptive system, formulas (7) and (8) are the model reference and formulas (10) and (11) are the adjustable model. To make a subtraction between formulas (7) and (10) as well as a subtraction between formulas (8) and (11), respectively, get 𝑝𝑒=𝐴𝑒+Δ𝐴𝐗−𝐻^⇀𝑖^𝑒_𝑠−^⇀𝑖^𝑒_𝑠𝑒_𝑖=𝐶𝑒, (12)(13) wherein, in formulas (12) and (13), 𝑒=𝑒_𝑖𝑒_𝜙^𝑇, 𝑒_𝑖=^⇀𝑖^𝑒_𝑠−^⇀𝑖^𝑒_𝑠, 𝑒_𝜙=𝜙^⇀𝑒_𝑠−𝜙^⇀𝑒_𝑠, Δ𝐴=^{Δ𝐴11Δ𝐴12}₀₀, Δ𝐴₁₁=𝐉Δ𝜔_𝑟, Δ𝐴₁₂=−𝐉(1/𝐿_𝜎)Δ𝜔_𝑟, and Δ𝜔_𝑟=𝜔_𝑟−𝜔_𝑟.

Incorporate the current difference calculated by formula (13) into formula (12) and get the following formula after the process of linearization: 𝑝𝑒=(𝐴−𝐻𝐶)𝑒+𝐖, (14) wherein 𝑊 is denoted by 𝐉𝐖=^⇀𝑖^𝑒_𝑠−1𝐿_𝜎^⇀𝜙^𝑒_𝑠Δ𝜔_𝑟0. (15)

The structure of the rotor speed 𝜔_𝑟 estimator as shown in Figure 1 includes a linear feedforward block and a nonlinear feedback block, wherein the nonlinear function 𝐹, set as the subject speed estimator, is defined by 𝜔_𝑟=𝐾_𝑝𝜔+𝐾_𝑖𝜔𝑑𝑡𝑒^𝑇_𝑖𝐉^⇀𝑖^𝑒_𝑠−1𝐿_𝜎^⇀𝜙^𝑒_𝑠. (16)

Figure 1 

Model reference adaptive system block.

In formula (16), 𝐾_𝑝𝜔 is the proportional gain and 𝐾_𝑖𝜔 is the integration gain.

With respect to the selection of the feedback gain of the observer, formulas (7) and (10) are converted into the synchronous coordinate system (𝜔=𝜔_𝑒) while deducing and rewriting them as 𝑝^⇀𝑖^𝑒_𝑠=𝐴^⇀₁₁𝑖^𝑒_𝑠+𝐴^⇀₁₂𝜙^𝑒_𝑠+𝑣^⇀𝑒_𝑠𝑝𝜙^⇀𝑒_𝑠=𝐴^⇀₂₁𝑖^𝑒_𝑠+𝐴^⇀₂₂𝜙^𝑒_𝑠𝑝^⇀𝑖^𝑒_𝑠=𝐴₁₁^⇀𝑖^𝑒_𝑠+𝐴𝜙₁₂^⇀𝑒_𝑠+𝑣^⇀𝑒_𝑠+𝐻₁^⇀𝑖^𝑒_𝑠−^⇀𝑖^𝑒_𝑠𝑝𝜙^⇀𝑒_𝑠=𝐴₂₁^⇀𝑖^𝑒_𝑠+𝐴𝜙₂₂^⇀𝑒_𝑠+𝐻₂^⇀𝑖^𝑒_𝑠−^⇀𝑖^𝑒_𝑠. (17)(18) Subtract formula (17) from formula (18) and then get an error equation as follows: 𝑠𝐈−𝐴₁₁−𝐻₁𝑒_𝑖=𝐴₁₂𝑒_𝜙+Δ𝐴₁₁^⇀𝑖^𝑒_𝑠+Δ𝐴𝜙₁₂^⇀𝑒_𝑠𝑠𝐈−𝐴₂₂𝑒_𝜙=𝐴₂₁−𝐻₂𝑒_𝑖. (19) The following formula is obtained from formula (19): 𝑒_𝑖=𝐺(𝑠)𝐉^⇀𝑖^𝑒_𝑠−1𝐿_𝜎^⇀𝜙^𝑒_𝑠, (20) wherein 𝐺(𝑠)=𝐺₁₁𝐺₁₂𝐺₂₁𝐺₂₂,𝑒_𝑖=𝑒_𝑑𝑒_𝑞^𝑇. (21) Under a steady state, 𝑒_𝑞=𝐺₂₂𝑖̂^𝑒_𝑑𝑠−1𝐿_𝜎𝜙^𝑒_𝑑𝑠Δ𝜔_𝑟. (22)

The arranged close-loop system of the full-order adaptive speed estimator is shown in Figure 2.

Figure 2 

Torque-speed plane.

3. Indirect Stator Field-Oriented Control

For conducting an indirect stator field-oriented control, it needs to compute 𝜔_𝑠𝑙 during the control because the synchronous speed (𝜔_𝑒) needs to estimate rotor speed (𝜔_𝑟) and the slip (𝜔_𝑠𝑙) together. In the synchronous 𝑑-𝑞 coordinate system, when the equation derived from the stator current and the stator flux which are set as the state variables makes the 𝑞-axis component of the stator flux reach zero to acquire the slip difference speed, and the stator flux is in accordance with 𝑑-axis of the synchronous reference coordinates frame. Hence, the slip difference speed and the 𝑑-axis stator flux can be expressed respectively, as follows: 𝜔_𝑠𝑙=𝐿_𝑆1+𝜎𝜏_𝑟𝑠𝑖^𝑒_𝑞𝑠𝜏_𝑟𝜙^𝑒_𝑑𝑠−𝐿_𝜎𝑖^𝑒_𝑑𝑠,1+𝜏_𝑟𝑠1+𝜎𝜏_𝑟𝑠𝜙^𝑒_𝑑𝑠=𝑖^𝑒_𝑑𝑠−𝜎𝜏_𝑟𝜔_𝑠𝑙1+𝜎𝜏_𝑟𝑠𝑖^𝑒_𝑞𝑠. (23)(24)

In the paper, use formula (23) to estimate 𝜔_𝑠𝑙. Furthermore, because formula (24) includes the 𝑞-axis component, the feedforward compensation quantity is added and defined by formula (25) to execute the decoupling control. Consider 𝑖^𝑒_𝑑𝑞=𝜎𝜏_𝑟𝜔_𝑠𝑙1+𝜎𝜏_𝑟𝑠𝑖^𝑒_𝑞𝑠. (25)

After adding the feedforward compensation quantity, formula (24) can be renewed as 𝜙^𝑒_𝑑𝑠=𝐿_𝑠1+𝜎𝜏_𝑟𝑠1+𝜏_𝑟𝑠𝑖^𝑒_𝑞𝑠. (26)

Herein, the electromagnetic torque 𝑇_𝑒 under the stator field-oriented control can be expressed by 𝑇_𝑒=3𝑃4𝑖^𝑒_𝑞𝑠𝜙^𝑒_𝑑𝑠. (27)

And formula (5) is renewed into the following equations: 𝑣^𝑒_𝑑𝑠=𝑅_𝑠𝑖^𝑒_𝑑𝑠+𝑝𝜙^𝑒_𝑑𝑠𝑣^𝑒_𝑞𝑠=𝑅_𝑠𝑖^𝑒_𝑞𝑠+𝜔_𝑒𝜙^𝑒_𝑑𝑠. (28)(29)

As shown in formula (29), it includes the 𝑑-axis component, so the feedforward compensation quantity 𝜔_𝑒𝜙^𝑒_𝑑𝑠 is added for subjecting the axis to the decoupling control.

4. Design of the Observer Feedback Gain

As shown in Figure 2, the transfer function 𝐺₂₂ is arranged by 𝐺₂₂=𝑠³+𝑥𝑠²+𝑚+𝜔²_𝑒𝑠+𝜔_𝑒𝑛+𝜔_𝑒𝑥𝑠²+𝑥𝑠+𝑚−𝜔_𝑒𝑦−𝜔²_𝑒²+𝑦+2𝜔_𝑒𝑠+𝑛+𝜔_𝑒𝑥², (30) wherein 𝑥=ℎ₁+𝑅_𝑠𝐿_𝜎+1𝜎𝜏_𝑟𝑦=ℎ₂−𝜔_𝑟1𝑚=𝐿_𝜎1𝜏_𝑟𝑅_𝑠+ℎ₃+𝜔_𝑟ℎ₄1𝑛=𝐿_𝜎1𝜏_𝑟ℎ₄−𝜔_𝑟𝑅_𝑠+ℎ₃. (31)

To design the observer feedback gain, the molecular equation of 𝐺₂₂ is made in the Routh table as shown in (32) according to the Routh Stability Criterion. Consider 𝑠³1𝑚+𝜔²_𝑒𝑠²𝑥𝑛𝜔_𝑒+𝑥𝜔²_𝑒𝑠¹𝑚𝑥−𝑛𝜔_𝑒𝑥0𝑠𝑛𝜔_𝑒+𝑥𝜔²_𝑒0. (32) According to the Routh table, a steady-state condition can be made as follows: 𝑥>0𝑚𝑥>𝑛𝜔_𝑒𝑥𝜔_𝑒𝜔_𝑒−𝜔_𝑐>0 (33) wherein 𝜔_𝑐𝑛=−𝑥. (34)

To analyse the unstable-speed area clearly, the torque-speed chart is applied to present the relationship between the two elements. As shown in formula (23), under the steady state, the slip 𝜔_𝑠𝑙 can be arranged by 𝜔_𝑠𝑙=𝐿_𝜎𝑖^𝑒_𝑞𝑠𝜙^𝑒_𝑑𝑠−𝐿_𝜎𝑖^𝑒_𝑑𝑠. (35) From formula (35), 𝑖^𝑒_𝑞𝑠=𝜔_𝑠𝑙𝜙^𝑒_𝑑𝑠𝐿_𝜎−𝑖^𝑒_𝑑𝑠. (36) Incorporate formula (36) into formula (27) and get 𝑇_𝑒=3𝑃4𝜙^𝑒2_𝑑𝑠𝐿_𝜎−𝑖^𝑒_𝑑𝑠𝜙^𝑒_𝑑𝑠𝜔_𝑠𝑙=𝐾_𝑇𝜔_𝑠𝑙. (37) In formula (37), the slip 𝜔_𝑠𝑙=𝜔_𝑒−𝜔_𝑟 and 𝐾_𝑇 is a torque constant, which is presented by 𝐾_𝑇=3𝑃4𝜙^𝑒2_𝑑𝑠𝐿_𝜎−𝑖^𝑒_𝑑𝑠𝜙^𝑒_𝑑𝑠. (38) If the condition of 𝜔_𝑐=𝜔_𝑒 is satisfied, the incorporation of formula (33) can get 𝜔_𝑒=𝑅_𝑠+ℎ₃𝐿_𝜎𝑥𝜔_𝑟−ℎ₄𝜏_𝑟𝐿_𝜎𝑥. (39)

Finally, incorporate formula (39) into formula (37) and arrange it; the equation is got from the torque and the rotation speed as follows: 𝑇_𝑒=𝑎𝜔_𝑟+𝑏. (40) In formula (40), 𝑎 and 𝑏 are shown as follows: 𝑎=𝐾_𝑇𝑅_𝑠+ℎ₃𝐿_𝜎𝑥−1.𝑏=−𝐾_𝑇ℎ₄𝜏_𝑟𝐿_𝜎𝑥. (41)

Figure 2 shows the torque-speed chart, wherein the 𝑥-axis represents the rotor speed of the motor 𝜔_𝑟 and the 𝑦-axis represents the motor torque 𝑇_𝑒. In the figure, the gray part shows that, when the speed estimator is operated in the regenerating area, an unstable part will be generated during the speed estimation, and the steady-state condition of formula (33) is used to design the observer gain to reduce the unstable part.

From the above analysis, when 𝜔_𝑐=0, the observer gain is ℎ₁𝑅=−^𝑠𝐿_𝜎+1𝜎𝜏_𝑟ℎ+𝑘,₂=𝑘𝜔_𝑟,ℎ₃ℎ=0,₄=𝜏_𝑟𝑅_𝑠𝜔_𝑟,𝑘>0, (42) wherein, to satisfy the steady-state condition, a positive integer 𝑘 larger than zero is selected and the observer feedback gain ℎ₁~ℎ₄ incorporates with 𝑥, 𝑦, 𝑚, and 𝑛, respectively, to get ()𝜔𝑥=𝑘,𝑦=𝑘−1_𝑟,1𝑚=𝐿_𝜎𝑅_𝑠𝜏_𝑟+𝜔²_𝑟𝜏_𝑟𝑅_𝑠,𝑛=0. (43)

5. Design of PI Gain Parameters of the Speed Estimator

In the single-input single-output close-loop system of Figure 3, 𝜂 represents the noise which affects the speed estimation, and the paper inferred from the influence of the noise that 𝜔𝜂=𝑖̂^𝑒_𝑑𝑠−1/𝐿_𝜎𝜙^𝑒_𝑑𝑠𝐾_𝑝𝜔+𝐾_𝑖𝜔/𝑠1+𝐺₂₂𝑖̂^𝑒_𝑑𝑠−(1/𝐿_𝜎)𝜙^𝑒_𝑑𝑠²𝐾_𝑝𝜔+𝐾_𝑖𝜔./𝑠 (44) To consider the influence of the high frequency noise on the speed estimation, incorporate 𝑠=𝑗∞ into formula (44) and infer that 𝜔𝜂_𝑠=𝑗∞=𝑖̂^𝑒_𝑑𝑠−1𝐿_𝜎𝜙^𝑒_𝑑𝑠𝐾_𝑝𝜔. (45) From formula (45), it can be found that the size of the proportional gain 𝐾_𝑝𝜔 can restrict the high frequency noise, so the proportional gain 𝐾_𝑝𝜔 selected smaller value to reduce the influence of the noise on the speed estimator. Considering the effect of the low frequency noise, it can incorporate 𝑠=𝑗0 into formula (44) and arrange it to get 𝜔𝜂_𝑠=𝑗0=1𝑖̂^𝑒_𝑑𝑠−(1/𝐿_𝜎)𝜙^𝑒_𝑑𝑠𝐺_22𝑠=𝑗0. (46) From formula (46), the adjustment to the proportional gain 𝐾_𝑝𝜔 or the integration gain 𝐾_𝐼𝜔 cannot improve the influence of the low frequency noise on the speed estimator. However, consider the influence of the proportional and integration gain of the motor acceleration/deceleration on the speed estimator (𝑅 can be set as an acceleration/deceleration velocity) and infer from formula (47) that a selection of the value of the integration gain 𝐾_𝐼𝜔 can affect the steady-state error of the speed estimation as the error 𝑒_𝑠 is set by acceleration/deceleration during a slope input speed. When the integration result is increased, the steady-state error of the speed is relatively decreased to increase the accuracy of the speed estimator. Consider 𝑒_𝑠=lim_𝑠→0𝑅𝑠×𝑠²×11+𝐺₂₂𝑖̂^𝑒_𝑑𝑠−(1/𝐿_𝜎)𝜙^𝑒_𝑑𝑠²𝐾_𝑝𝜔+𝐾_𝑖𝜔=𝑅/𝑠𝐾_𝑖𝜔𝑖̂^𝑒_𝑑𝑠−1/𝐿_𝜎𝜙^𝑒_𝑑𝑠²𝐺_22𝑠=0. (47)

Figure 3 

Close-loop system of the full-order adaptive speed estimator.

6. Estimation of Rotor Resistance by FTA

Rewrite formula (3) as ^⇀𝑖^𝑠_𝑠=(1/𝐿_𝑚)(𝜙^⇀𝑠_𝑟−𝐿_𝑟^⇀𝑖^𝑠_𝑟) and incorporate formula (2) to get 𝑝𝜙^𝑠_𝑑𝑟=−𝑅_𝑟𝑖^𝑠_𝑑𝑟−𝜔_𝑟𝜙^𝑠_𝑞𝑟𝑝𝜙^𝑠_𝑞𝑟=−𝑅_𝑟𝑖^𝑠_𝑞𝑟−𝜔_𝑟𝜙^𝑠_𝑑𝑟. (48)(49) Then rewrite formula (49) as 𝜔_𝑟=𝑝𝜙^𝑠_𝑞𝑟+𝑅_𝑟𝑖^𝑠_𝑞𝑟𝜙^𝑠_𝑑𝑟. (50) Incorporate formula (50) into formula (48) and arrange it to get the formula of the estimated rotor resistance 𝑅_𝑟 and thence rewrite it according to the fixed trace algorithm (FTA) as follows: 𝑅_𝑟=−𝑝𝜙^𝑠_𝑑𝑟𝜙^𝑠_𝑑𝑟+𝑝𝜙^𝑠_𝑞𝑟𝜙^𝑠_𝑞𝑟𝑖^𝑠_𝑑𝑟𝜙^𝑠_𝑑𝑟+𝑖^𝑠_𝑞𝑟𝜙^𝑠_𝑞𝑟 (51) or 𝑅_𝑟[𝑛]=[]−[𝑛]𝑅𝑟𝑛−1𝑟𝑎1+𝑟(𝑎[𝑛])²𝑅_𝑟[𝑛−1]𝑎[𝑛]−𝑏[𝑛], (52) wherein 𝑎[𝑛]=−(𝑖^𝑠_𝑑𝑟𝜙^𝑠_𝑑𝑟+𝑖^𝑠_𝑞𝑟𝜙^𝑠_𝑞𝑟), 𝑏[𝑛]=𝑝𝜙^𝑠_𝑑𝑟𝜙^𝑠_𝑑𝑟+𝑝𝜙^𝑠_𝑞𝑟𝜙^𝑠_𝑞𝑟, and 𝑟 represents the gain. This paper adopts the estimated rotor resistance 𝑅_𝑟 of this formula.

7. Results of Simulation and Experiment

To verify the feasibility and sturdiness of the subject system, an experimental platformas shown in Figure 5 is constituted. According to the structure of the system shown in Figure 4, it executed simulations and experiments by controlling different situations. First, the personal computer of Simulink program containing MATLAB is the speed sensorless vector controller, which functions as the simulation process. The result of the simulation drives the induction motor via PC-based control card which controls the inverter practically. The specification of the induction motor is shown in Table 1. During the experiment, there are three different rotor speeds, namely, ±1800 rpm, ±900 rpm, ±600 rpm, and ±300 rpm, and conduct the simulation of controlling the acceleration and deceleration under the nonloading and loading modes. Results are shown in Figures 6, 7, 8, and 9. Figures 10, 11, 12, and 13, respectively, show the simulation results while loading 1 N-m under the rotor speed of the induction motor commanded at ±300 rpm, ±900 rpm, ±600 rpm, and ±1800 rpm. Figures 6–9, respectively, show the nonloading practical data when the induction motor is at ±300 rpm, ±900 rpm, and ±1800 rpm. Figures 10–13, respectively, show the practical data of loading 1 N-m under the rotor speed of the induction motor commanded at ±300 rpm, ±900 rpm, and ±1800 rpm.

Table 1 

Figure 4 

Integral block of speed sensorless controller.

Figure 5 

The structure of the experimental platform.

(a)

(b)

(c)

(d)

Figure 6 

Simulation and Practical data of the induction motor at 300 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

(a)  

(b)

(c)

(d)

Figure 7 

Simulation and Practical data of the induction motor at 600 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

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(b)

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(d)

Figure 8 

Simulation and Practical data of the induction motor at 900 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

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(b)

(c)

(d)

Figure 9 

Simulation and Practical data of the induction motor at 1800 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

(a)

(b)

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(d)

Figure 10 

Loading simulation and Practical data of the induction motor at 300 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

(a)

(b)

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(d)

Figure 11 

Loading simulation and Practical data of the induction motor at 600 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

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(b)

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(d)

Figure 12 

Loading simulation and Practical data of the induction motor at 900 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

(a)

(b)

(c)

(d)

Figure 13 

Loading simulation and Practical data of the induction motor at 1800 rpm. (a) Actual motor speed. (b) Estimated motor speed. (c) Stator flux. (d) Estimated torque.

As shown from Figures 6–13, the upper part shows the simulation result and the lower part shows the experimental result, wherein (a) and (b) represent the respective comparisons between the simulations and experiments of the actual and estimated motor speed. The 𝑥-axis shows the time with the unit showing the seconds (sec) and the 𝑦-axis shows the rotor speed with the unit showing the speed per minute (rpm). The blue solid line represents the speed command, wherein 0-1 seconds can be the clockwise acceleration of the motor, 1-2 seconds can be the constant and clockwise rotation of the motor, 2-3 seconds can be the clockwise deceleration of the motor, 3-4 seconds can be the counterclockwise acceleration, 4-5 seconds can be the constant and counterclockwise rotation of the motor, and 5-6 seconds can be the counterclockwise deceleration of the motor. (c) is the motor estimated torque, wherein the 𝑥-axis shows the time with the unit showing the seconds (sec) and 𝑦-axis shows the motor torque with the unit showing the Newton meter (N-m). (d) is the stator flux circle, wherein the 𝑥-axis shows the stator flux of d-axis component and 𝑦-axis shows the stator flux of 𝑞-axis component.

From Figures 6–9 showing the results of simulations and experiments of the motor in the nonloading clockwise and counterclockwise rotations, Figure 17 can verify that, when the motor is operated at the acceleration/deceleration speed, the estimator can estimate the actual rotation speed of the motor accurately and catch the commanded speed of the motor rapidly. When the motor is operated at the constant speed, the estimator can estimate the actual rotation speed of the motor accurately and render the motor able to follow the commanded speed stably. In Figures 10–13, when the motor is loaded by 1 N-m, the estimator can also estimate the actual speed accurately and catch the commanded speed rapidly. The simulations and experiments shown in Figures 6–13 can verify that the speed estimator can possess a wide-range speed estimation regardless of the loading or nonloading condition.

Furthermore, for the operation under the low rotary speed regenerating test, Figure 14 and Figure 15 show respective data derived from simulating and practicing when the rotor speed to be set at 100 rpm and loading −1 N-m at the 3rd second, wherein Figure 14(a) is the simulation data without adding the observer feedback gain. Figure 14(b) is the simulation data with the addition of the observer feedback gain. Figure 15(a) is the practical data without adding the observer feedback gain. Figure 15(b) is the practical data with the addition of the observer feedback gain. From the data in Figures 14 and 15, when the verification controller is operated in the low rotary speed regenerating mode, it can select the adequate observer feedback gain and estimate the rotor speed correctly and steadily. As shown in Figure 16(a), if the rotor resistance is raised by 30% in 2-3 seconds due to the increased temperature, the estimation result of FTA to the rotor resistance is shown in the figure. The dotted line in the figure denotes the actual value of the rotor resistance of the motor, and the solid line in the figure denotes the estimated value of the rotor resistance giving the slip angle. When the rotor resistance estimator is not added, the generation of error on the synchronous angle can be read in Figure 16(b), wherein the dotted line denotes the reference value of the synchronous angle and the solid line denotes the estimated value of the synchronous angle. Figure 17 shows the data after adding the rotor resistance estimator, from which the synchronous angle can be accurately estimated. Figure 18 shows the estimation results of the simulation and experiment of the rotor resistance by the fixed trace algorithm (FTA), from which it can verify that FTA can estimate the rotor resistance accurately. Figure 19 shows that the result of simulation without FTA when actual rotor resistance is changed caused by temperature variations. It is unstable and it verifies the feasibility and sturdiness of the method set forth in the research.

(a)

(b)

Figure 14 

Simulation data of the speed estimation under the regenerating mode of the rotor speed at 100 rpm. (a) Without adding feedback gain. (b) Adding feedback gain.

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Figure 15 

Practical data of the speed estimation under the regenerating mode of the rotor speed at 100 rpm. (a) Without adding feedback gain. (b) Adding feedback gain.

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Figure 16 

Estimation result of synchronous angle 𝜃𝑒 without adding the rotor resistance estimator under variations of 𝑅𝑟. (a) Rotor resistance. (b) Synchronous angle.

(a)

(b)

Figure 17 

Estimation result of synchronous angle 𝜃𝑒 by adding the rotor resistance estimator under variations of 𝑅𝑟. (a) Rotor resistance. (b) Synchronous angle.

(a)

(b)

Figure 18 

Results of simulation and experimental of the rotor resistance estimator by FTA. (a) Simulation results, (b) Experimental results.

Figure 19 

Results of simulation without FTA when actual rotor resistance is changed disagree with controller parameter.

8. Conclusion

The subject sensorless indirect stator field-oriented control technique provides a set of standard designed steps to be operated under the low rotary speed at regenerating mode and the circumstance of the unstable system. This paper further considers that the property of changes of the rotor resistance caused by variations of temperature would affect the accuracy of estimating the synchronous angle, so this paper combines the FTA which conducts the online and instant estimation of the rotor resistance in order to solve the problem of incorrect estimation of the synchronous angle incurred by the changes of the rotor resistance. Finally, the simulation and practical results can verify the feasibility and sturdiness of the method set forth in the research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.