CN105186957A

CN105186957A - Asynchronous motor speed sensorless rotor flux linkage estimation method

Info

Publication number: CN105186957A
Application number: CN201510563804.7A
Authority: CN
Inventors: 漆星; 王群京; 李国丽; 方圆; 武琼
Original assignee: Anhui University
Current assignee: Anhui University
Priority date: 2015-09-07
Filing date: 2015-09-07
Publication date: 2015-12-23

Abstract

The invention discloses a method for estimating the flux linkage of the rotor without a speed sensor of an asynchronous motor. The parameter optimization method is adopted, so that there is only one stable point on the phase plane of the control scheme, and finally the motor will be stable in the state where the flux linkage is established, and the motor is in the rotating state. It can also be started when the motor is running, the torque of the motor will not be lost, and the torque can always be output according to people's requirements.

Description

A sensorless rotor flux linkage estimation method for asynchronous motors

技术领域technical field

本发明涉及异步电机磁链估算方法领域，具体是一种异步电机无速度传感器转子磁链估算方法。The invention relates to the field of estimation methods for asynchronous motor flux linkage, in particular to a method for estimating rotor flux linkage without a speed sensor of an asynchronous motor.

背景技术Background technique

异步电机的无速度传感器矢量控制技术在工业生产中应用广泛。在中高速段，无速度传感器控制性能可以和有速度传感器媲美，而在低速范围内，无速度传感器却存在着带载能力弱，速度辨识精度低等问题，关于其稳定性问题也是实际工程中急需解决的内容。现有的稳定性分析主要集中在电机静止起动的工况下。而在实际场合，同样存在着电机旋转状态起动的工况，例如多电机转速-转矩匹配，牵引列车的带速重投功能等。此类问题一直未被人们很好的解决。当今市场上的变频器多不具有这方面的功能，即使少数品牌的变频器声称具有其功能，但其实际效果也不理想。以日本安川公司生产的变频器A1000为例，其在无速度传感器模式(开环矢量控制模式)下，并不支持转矩模式功能，其原因就在于无法很好的解决异步电机的旋转状态起动问题。The speed sensorless vector control technology of asynchronous motor is widely used in industrial production. In the middle and high speed section, the control performance of the sensorless sensor can be compared with that of the sensor with the speed sensor, but in the low speed range, the sensorless sensor has the problems of weak loading capacity and low speed identification accuracy. Urgently need to solve the content. Existing stability analysis mainly focuses on the condition of static starting of the motor. In actual situations, there are also working conditions where the motors are started in rotation, such as multi-motor speed-torque matching, and the speed re-start function of traction trains. Such problems have not been well resolved by people. Most inverters on the market today do not have this function, even if a few brands of inverters claim to have this function, their actual effect is not ideal. Take the inverter A1000 produced by Yaskawa Corporation of Japan as an example, it does not support the torque mode function in the speed sensorless mode (open-loop vector control mode), the reason is that it cannot solve the problem of starting the asynchronous motor in the rotating state. question.

现有技术的异步电机无速度传感器控制转子磁链角估算方案，大多都可以在其相平面上找到两个或者两个以上的稳定点，其中只有一个稳定点是电机可以输出转矩的状态，称之为磁链建立的状态。其他的稳定点，往往是电机无法输出转矩的状态，称之为磁链崩溃状态。人们总是希望电机无论初始状态如何，最终都能稳定在磁链建立状态，以保证电机总是能够输出转矩。而不要稳定在磁链崩溃的状态，因为这个状态不能让电机输出转矩。然而，在电机旋转起动的时候，若采用传统的转子磁链角估算方案，电机总是会稳定磁链崩溃的状态，无法输出转矩。Most of the prior art asynchronous motor speed sensorless control rotor flux angle estimation schemes can find two or more stable points on its phase plane, of which only one stable point is the state where the motor can output torque. It is called the state where the magnetic link is established. Other stable points are often the state where the motor cannot output torque, which is called the state of flux linkage collapse. People always hope that no matter what the initial state of the motor is, it will eventually be stable in the state of flux linkage establishment, so as to ensure that the motor can always output torque. Instead of stabilizing in a state where the flux linkage collapses, because this state cannot allow the motor to output torque. However, when the motor rotates and starts, if the traditional rotor flux angle estimation scheme is used, the motor will always be in a state of stable flux collapse and cannot output torque.

发明内容Contents of the invention

本发明的目的是提供一种异步电机无速度传感器转子磁链估算方法，以实现电机在旋转状态时起动转矩不丢失。The purpose of the present invention is to provide a speed sensorless rotor flux linkage estimation method of an asynchronous motor, so as to realize that the starting torque is not lost when the motor is in a rotating state.

为了达到上述目的，本发明所采用的技术方案为：In order to achieve the above object, the technical scheme adopted in the present invention is:

一种异步电机无速度传感器转子磁链估算方法，其特征在于：包括以下步骤：A method for estimating the rotor flux linkage without a speed sensor of an asynchronous motor, characterized in that it comprises the following steps:

(1)、首先计算出电机d-q轴的反电动势：(1), first calculate the counter electromotive force of the d-q axis of the motor:

${e e}_{d d} = = {R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q},,$

${e e}_{q q} = = {R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d},,$

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

i_sd：电机d轴定子电流i _sd : motor d-axis stator current

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

R_r：电机转子电阻R _r : motor rotor resistance

L_m：电机互感L _m : Motor mutual inductance

ψ_rd：电机d轴转子磁链ψ _rd : motor d-axis rotor flux linkage

ψ_rq：电机q轴转子磁链ψ _rq : Motor q-axis rotor flux linkage

ω：电机转速，由步骤(4)算得，并代回步骤(1)ω: motor speed, calculated from step (4) and replaced by step (1)

(2)、根据步骤(1)的结果计算磁链的幅值：(2), calculate the magnitude of flux linkage according to the result of step (1):

$\frac{{dψ dψ}_{r r}}{d d t t} = = {e e}_{d d},,$

ψ_r＝∫e_d，ψ _r =∫e _d ,

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

(3)、根据步骤(2)的结果计算转子同步转速：(3), calculate rotor synchronous speed according to the result of step (2):

$\frac{{dθ dθ}_{s the s}}{d d t t} = = {ω ω}_{s the s} = = \frac{{e e}_{q q} - - {δe δe}_{d d}}{{ψ ψ}_{r r}}$

θ_s：转子磁链角度θ _s : rotor flux angle

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

δ：计算转子同步角速度所需的可调参数δ: An adjustable parameter required to calculate the synchronous angular velocity of the rotor

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

(4)、根据步骤(3)计算出电机实际转速(4) Calculate the actual speed of the motor according to step (3)

$ω ω = = {ω ω}_{s the s} - - \frac{{i i}_{s the s q q} {R R}_{r r}}{{ψ ψ}_{r r}}$

ω：电机实际转速ω: the actual speed of the motor

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

R_r：电机转子电阻R _r : motor rotor resistance

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

(5)、将步骤(3)算出的转子同步转速代入电机的状态方程：(5), the rotor synchronous speed calculated by step (3) is substituted into the state equation of the motor:

${f f}_{11} ((x x)) = = \frac{{L L}_{m m}}{{T T}_{r r}} {i i}_{s the s d d} - - \frac{11}{{T T}_{r r}} {ψ ψ}_{r r d d} + + ((\frac{{R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d} - - δ δ (({R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q}}{{ψ ψ}_{r r}} - - ω ω)) {ψ ψ}_{r r q q},,$

$\begin{matrix} {f f}_{22} ((x x)) = = \frac{{L L}_{m m}}{{T T}_{r r}} {i i}_{s the s q q} - - \frac{11}{{T T}_{r r}} {ψ ψ}_{r r q q} - - ((\frac{{R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d} - - δ δ (({R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q}))}{{ψ ψ}_{r r}} - - ω ω)) {ψ ψ}_{r r d d},, \\ {f f}_{33} ((x x)) = = {R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q},, \end{matrix}$

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

R_r：电机转子电阻R _r : motor rotor resistance

i_sd：电机d轴定子电流i _sd : motor d-axis stator current

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

L_m：电机互感L _m : Motor mutual inductance

ψ_rd：电机d轴转子磁链ψ _rd : motor d-axis rotor flux linkage

ψ_rq：电机q轴转子磁链ψ _rq : Motor q-axis rotor flux linkage

ω：电机转速ω: motor speed

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

(6)、计算状态方程雅克布矩阵的特征多项式：(6), Calculate the characteristic polynomial of the Jacobian matrix of the state equation:

det(sI-J)＝s³+a₂s²+a₁s+a₀，det(sI-J)=s ³ +a ₂ s ² +a ₁ s+a ₀ ,

a₀、a₁、a₂：特征多项式系数a ₀ , a ₁ , a ₂ : characteristic polynomial coefficients

其中in

$\{\begin{matrix} {a a}_{22} = = \frac{{R R}_{r r}}{{L L}_{m m}} + + δ δ ω ω \\ {a a}_{11} = = {ω ω}_{s the s} (({ω ω}_{s the s} + + δ δ \frac{{R R}_{r r}}{{L L}_{m m}})) \\ {a a}_{00} = = \frac{{R R}_{r r}}{{L L}_{m m}} {ω ω}_{11}^{22} \end{matrix},,$

R_r：电机转子电阻R _r : motor rotor resistance

L_m：电机互感L _m : Motor mutual inductance

ω：电机转速ω: motor speed

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

(7)、根据劳斯稳定判据，选择合适的可调参数，使得状态方程具有全局渐近稳定性，相平面只有一个稳定点，最终选择可调参数为：(7) According to the Routh stability criterion, select the appropriate adjustable parameters, so that the state equation has a global asymptotic stability, and the phase plane has only one stable point, and finally select the adjustable parameters as:

$δ δ = = \sqrt{22} * * s the s i i g g n no (({ω ω}_{r r})) . .$

ω：电机转速，由步骤(4)算得ω: motor speed, calculated from step (4)

sign(ω)：转速的符号，正转为1，反转为-1sign(ω): the sign of the speed, forward rotation is 1, reverse rotation is -1

本发明采用参数优化的方法，使得控制方案的相平面上只有一个稳定点，而这个稳定点就是电机磁链建立的状态。即无论电机初始处于何种状态，因为没有其他的稳定点。最终电机都会稳定在磁链建立的状态，电机在旋转状态时也可以起动，电机的转矩不会丢失，总能按照人们的要求输出转矩。The present invention adopts the method of parameter optimization, so that there is only one stable point on the phase plane of the control scheme, and this stable point is the state where the flux linkage of the motor is established. That is, no matter what state the motor is in initially, because there is no other stable point. In the end, the motor will be stable in the state where the flux linkage is established, and the motor can also be started when it is rotating. The torque of the motor will not be lost, and the torque can always be output according to people's requirements.

附图说明Description of drawings

图1为本发明方法得到的相平面图。Fig. 1 is the phase plane diagram that the method of the present invention obtains.

图2为现有技术异步电机无速度传感器矢量控制方案图。Fig. 2 is a schematic diagram of a speed sensorless vector control scheme of an asynchronous motor in the prior art.

图3为本发明磁链估算方案图。Fig. 3 is a scheme diagram of flux linkage estimation in the present invention.

图4为具体实施方式中正反转切换逻辑流程框图。Fig. 4 is a logic block diagram of forward and reverse switching in a specific embodiment.

具体实施方式Detailed ways

一种异步电机无速度传感器转子磁链估算方法，包括以下步骤：A method for estimating the rotor flux linkage of an asynchronous motor without a speed sensor, comprising the following steps:

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

R_r：电机转子电阻R _r : motor rotor resistance

i_sd：电机d轴定子电流i _sd : motor d-axis stator current

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

L_m：电机互感L _m : Motor mutual inductance

ψ_rd：电机d轴转子磁链ψ _rd : motor d-axis rotor flux linkage

ψ_rq：电机q轴转子磁链ψ _rq : Motor q-axis rotor flux linkage

$\frac{{dψ dψ}_{r r}}{d d t t} = = {e e}_{d d},,$

ψ_r＝∫e_d，ψ _r =∫e _d ,

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

$\frac{{dθ dθ}_{s the s}}{d d t t} = = {ω ω}_{s the s} = = \frac{{e e}_{q q} - - {δe δ e}_{d d}}{{ψ ψ}_{r r}}$

θ_s：转子磁链角度θ _s : rotor flux angle

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

ω：电机实际转速ω: the actual speed of the motor

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

R_r：电机转子电阻R _r : motor rotor resistance

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

e_d：电机d轴反电动势e _d : motor d-axis back electromotive force

e_q：电机q轴反电动势e _q : Motor q-axis back electromotive force

R_r：电机转子电阻R _r : motor rotor resistance

i_sd：电机d轴定子电流i _sd : motor d-axis stator current

i_sq：电机q轴定子电流i _sq : Motor q-axis stator current

L_m：电机互感L _m : Motor mutual inductance

ψ_rd：电机d轴转子磁链ψ _rd : motor d-axis rotor flux linkage

ψ_rq：电机q轴转子磁链ψ _rq : Motor q-axis rotor flux linkage

ω：电机转速ω: motor speed

ψ_r：电机转子磁链幅值ψ _r : Motor rotor flux amplitude

其中in

R_r：电机转子电阻R _r : motor rotor resistance

L_m：电机互感L _m : Motor mutual inductance

ω：电机转速ω: motor speed

ω_s：转子同步角速度ω _s : rotor synchronous angular velocity

$δ δ = = \sqrt{22} * * s the s i i g g n no (({ω ω}_{r r})) . .$

ω：电机转速，由步骤(4)算得ω: motor speed, calculated from step (4)

sign(ω)：转速的符号，正转表示1，反转表示-1sign(ω): the sign of the speed, forward rotation means 1, reverse rotation means -1

选择合适的可调参数可使相平面图如图1所示。Selecting the appropriate adjustable parameters can make the phase plane diagram shown in Figure 1.

现有技术中的异步电机无速度传感器矢量控制方案如图2所示，图2中，1为转子磁链角估计方案，2为电机转速的PI调节器，3、4为电流的调节器，5为park变换装置，对应的6为park逆变换装置，7为SVPWM调制器，8为CLARK变换装置，9为逆变器，10是所控制的异步电机。The speed sensorless vector control scheme of asynchronous motors in the prior art is shown in Figure 2. In Figure 2, 1 is the rotor flux angle estimation scheme, 2 is the PI regulator of the motor speed, 3 and 4 are the current regulators, 5 is a park conversion device, the corresponding 6 is a park inverse conversion device, 7 is a SVPWM modulator, 8 is a CLARK conversion device, 9 is an inverter, and 10 is a controlled asynchronous motor.

应用本发明方法的转子磁链角估计方案如图3所示：The rotor flux angle estimation scheme applying the method of the present invention is as shown in Figure 3:

输入为异步电机在α-β坐标系下的电压v_α、v_β，以及电流i_α、i_β，输出为估算转子磁链角θ。i_α、i_β乘以定子电阻R_s(1和2)，与v_α、v_β相减，得到反电动势e_α、e_β，e_α、e_β经过矢量回转器3，得到d-q坐标系下的反电动势e_d、e_q，e_d经过积分器5得到电机转子磁链幅值ψ_R，同时，e_d乘以参数δ与e_q相减，再除以转子磁链幅值ψ_R，得到转子同步转速ω_s，并根据ω_s计算出电机实际转速ω，注意到ω和δ的计算互为影响，构成环路。将ω_s经过积分器6得到转子磁链角度θ_s，而ω则用来计算下一时刻的参数δ。The input is the voltage v _α , v _β , and the current i _α , i _β of the asynchronous motor in the α-β coordinate system, and the output is the estimated rotor flux angle θ. i _α and i _β are multiplied by the stator resistance R _s (1 and 2), and subtracted from v _α and v _β to obtain counter electromotive forces e _α and e _β , and e _α and e _β pass through the vector gyrator 3 to obtain the dq coordinate system The counter electromotive force ed and _e _q under the current condition, _ed passes through the integrator 5 to obtain the motor rotor flux amplitude ψ _R , and at the same time, _ed is multiplied by the parameter δ Subtract it from e _q , and then divide it by the rotor flux amplitude ψ _R to get the rotor synchronous speed ω _s , and calculate the actual motor speed ω according to ω _s , and notice that the calculation of ω and δ influence each other to form a loop. Pass ω _s through the integrator 6 to obtain the rotor flux angle θ _s , and ω is used to calculate the parameter δ at the next moment.

采用本发明方法时，有一个需要注意的地方：在电机从正转到反转切换的过程中，有可能会导致参数δ的误动作，因此，采取了一个正反转切换逻辑进行改进。如图4所示，其步骤为：When adopting the method of the present invention, there is a point that needs attention: in the process of switching the motor from forward to reverse, it may cause a malfunction of the parameter δ. Therefore, a forward and reverse switching logic is adopted for improvement. As shown in Figure 4, the steps are:

1、若电机当前估算转速为正转，且数值较高(大于6rad/s)，则δ为正。1. If the current estimated speed of the motor is forward rotation and the value is relatively high (greater than 6rad/s), then δ is positive.

2、若电机当前估算转速为反转，且数值较高(小于-6rad/s)，则δ为负。3、若当前电机转速较低(大于-6rad/s且小于6rad/s)，应根据i_q来判断，若i_q大于0，则δ为正，若i_q小于0，则δ为负。2. If the current estimated speed of the motor is reverse and the value is relatively high (less than -6rad/s), then δ is negative. 3. If the current motor speed is low (greater than -6rad/s and less than 6rad/s), it should be judged according to i _q , if i _q is greater than 0, then δ is positive, if i _q is less than 0, then δ is negative.

Claims

1. A method for estimating rotor flux linkage without a speed sensor of an asynchronous motor, is characterized in that: comprise the following steps:

(1), first calculate the counter electromotive force of the d-q axis of the motor:

{e e}_{d d} = = {R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q},,

{e e}_{q q} = = {R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d},,

e _d : motor d-axis back electromotive force

e _q : Motor q-axis back electromotive force

R _r : motor rotor resistance

i _sd : motor d-axis stator current

i _sq : Motor q-axis stator current

L _m : Motor mutual inductance

ψ _rd : motor d-axis rotor flux linkage

ψ _rq : Motor q-axis rotor flux linkage

ω: motor speed, calculated from step (4) and replaced by step (1)

(2), calculate the magnitude of flux linkage according to the result of step (1):

\frac{{dψ dψ}_{r r}}{d d t t} = = {e e}_{d d},,

ψ _r =∫e _d ,

e _d : motor d-axis back electromotive force

ψ _r : Motor rotor flux amplitude

(3), calculate rotor synchronous speed according to the result of step (2):

\frac{{dθ dθ}_{s the s}}{d d t t} = = {ω ω}_{s the s} = = \frac{{e e}_{q q} - - {δe δe}_{d d}}{{ψ ψ}_{r r}}

θ _s : rotor flux angle

ω _s : rotor synchronous angular velocity

δ: An adjustable parameter required to calculate the synchronous angular velocity of the rotor

e _d : motor d-axis back electromotive force

e _q : Motor q-axis back electromotive force

ψ _r : Motor rotor flux amplitude

(4) Calculate the actual speed of the motor according to step (3)

ω ω = = {ω ω}_{s the s} - - \frac{{i i}_{s the s q q} {R R}_{r r}}{{ψ ψ}_{r r}}

ω: the actual speed of the motor

ω _s : rotor synchronous angular velocity

R _r : motor rotor resistance

i _sq : Motor q-axis stator current

ψ _r : Motor rotor flux amplitude

(5), the rotor synchronous speed that step (3) calculates is substituted into the state equation of motor:

{f f}_{11} ((x x)) = = \frac{{L L}_{m m}}{{T T}_{r r}} {i i}_{s the s d d} - - \frac{11}{{T T}_{r r}} {ψ ψ}_{r r d d} + + ((\frac{{R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d} - - δ δ (({R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q}}{{ψ ψ}_{r r}} - - ω ω)) {ψ ψ}_{r r q q},,

{f f}_{22} ((x x)) = = \frac{{L L}_{m m}}{{T T}_{r r}} {i i}_{s the s q q} - - \frac{11}{{T T}_{r r}} {ψ ψ}_{r r q q} - - ((\frac{{R R}_{r r} {i i}_{s the s q q} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r q q} + + {ωψ ωψ}_{r r d d} - - δ δ (({R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q}))}{{ψ ψ}_{r r}} - - ω ω)) {ψ ψ}_{r r d d},,

{f f}_{33} ((x x)) = = {R R}_{r r} {i i}_{s the s d d} - - \frac{{R R}_{r r}}{{L L}_{m m}} {ψ ψ}_{r r d d} - - {ωψ ωψ}_{r r q q},,

e _d : motor d-axis back electromotive force

e _q : Motor q-axis back electromotive force

R _r : motor rotor resistance

i _sd : motor d-axis stator current

i _sq : Motor q-axis stator current

L _m : Motor mutual inductance

ψ _rd : motor d-axis rotor flux linkage

ψ _rq : Motor q-axis rotor flux linkage

ω: motor speed

ψ _r : Motor rotor flux amplitude

(6), Calculate the characteristic polynomial of the Jacobian matrix of the state equation:

det(sI-J)=s ³ +a ₂ s ² +a ₁ s+a ₀ ,

a ₀ , a ₁ , a ₂ : characteristic polynomial coefficients

in

\{\begin{matrix} {a a}_{22} = = \frac{{R R}_{r r}}{{L L}_{m m}} + + δ δ ω ω \\ {a a}_{11} = = {ω ω}_{s the s} (({ω ω}_{s the s} + + δ δ \frac{{R R}_{r r}}{{L L}_{m m}})) \\ {a a}_{00} = = \frac{{R R}_{r r}}{{L L}_{m m}} {ω ω}_{11}^{22} \end{matrix},,

a ₀ , a ₁ , a ₂ : characteristic polynomial coefficients

R _r : motor rotor resistance

L _m : Motor mutual inductance

ω: motor speed

ω _s : rotor synchronous angular velocity

(7) According to the Routh stability criterion, select the appropriate adjustable parameters, so that the state equation has a global asymptotic stability, and the phase plane has only one stable point, and finally select the adjustable parameters as:

δ δ = = \sqrt{22} * * s the s i i g g n no ((ω ω))

ω: motor speed, calculated from step (4)

sign(ω): The sign of the speed, forward rotation means 1, reverse rotation means -1.