CN114400943B - Position-sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion - Google Patents

Position-sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion Download PDF

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CN114400943B
CN114400943B CN202210055757.5A CN202210055757A CN114400943B CN 114400943 B CN114400943 B CN 114400943B CN 202210055757 A CN202210055757 A CN 202210055757A CN 114400943 B CN114400943 B CN 114400943B
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CN114400943A (en
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钱林方
孙乐
佟明昊
邹权
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Nanjing University of Science and Technology
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/18—Estimation of position or speed
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/13—Observer control, e.g. using Luenberger observers or Kalman filters
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/22—Current control, e.g. using a current control loop
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P25/00—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details
    • H02P25/02—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details characterised by the kind of motor
    • H02P25/022—Synchronous motors

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Abstract

The invention discloses a position sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion. On the one hand, a position and velocity estimate of the rocker arm is obtained by a current loop-convex optimization-based position sensorless algorithm. On the other hand, a kalman filter is used to obtain an estimate of the position and velocity of the other rocker arm in the velocity loop based on the rocker arm equations of motion. Generally, the current loop position estimation is more accurate, however, the rotating speed estimation of the surface-mounted permanent magnet motor under the condition of low-speed overload is very noisy, and at the moment, the rotating speed estimation obtained by the speed loop Kalman filter is used as control feedback to obtain a better servo control effect. The method provided by the invention realizes data fusion by using a Kalman filter, takes the position estimation of a current loop as position feedback, takes the speed estimation of a speed loop as speed feedback, and further realizes three-loop drive control without a position sensor for a rocker servo system.

Description

Position-sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion
Technical Field
The invention belongs to a position sensor-free driving control technology of a rocker servo mechanism, and particularly relates to a position sensor-free rocker servo control method based on disturbance-resistant Kalman data fusion.
Background
The rocker mechanism is a typical application of a servo motor, and is widely applied to various robot joints at present. Many joint type rocker arm mechanisms are compact in structure and do not have position sensors, so that the drive control technology without the position sensors is particularly key for the high-precision rocker arm mechanism. The rocker arm is influenced by load torque, and the motion condition is complex, so that the improvement of the position identification precision has important significance.
In recent years, a position estimation method based on convex optimization is used to estimate the position of the motor. This method estimates the motor position without switching between low and high speed conditions. In contrast, for conventional position estimation, different position estimation methods need to be employed in low and high speed situations. The convex optimization based position estimation method is a new sensorless control strategy. According to the theory of convex optimization, the position and velocity can be found by finding the minimum of the loss function. The method can be applied to both low and high speed situations. It should be noted that for convex optimization, the position is observed at low speed, requiring the injection of high frequency signals. However, this method does not require digital demodulation and filtering.
However, the servo drive control without the position sensor has a key problem that in a low-speed overload situation, because the d-q axis inductances of the servo motors are very close, that is, the salient pole ratio is low, the noise of the rotating speed estimation is large, it is difficult to support high-quality speed loop control, and further, it is difficult to realize position servo drive control.
Disclosure of Invention
The invention aims to provide a position sensor-free rocker arm servo control method based on disturbance-resistant Kalman data fusion, which is used for realizing position sensor-free drive control of a rocker arm servo mechanism.
The technical scheme for realizing the purpose of the invention is as follows: a position sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion comprises the following steps:
step 1, identifying the position of a rocker arm based on a current loop position sensorless control algorithm;
step 2, observing the position and the rotating speed of the rocker arm by adopting a Kalman state observer based on a rocker arm motion equation;
and 3, controlling the motion control system by adopting three modes of position, speed and current, wherein the position feedback of the position loop adopts current loop position estimation, and the speed feedback of the speed loop adopts speed estimation of the speed loop.
Preferably, the specific method for identifying the position of the rocker arm based on the current loop position sensorless control algorithm is as follows:
inputting the current under the alpha-beta shafting, the voltage under the alpha-beta shafting and the estimated rotating speed at the last moment into a current loop position estimation module, calculating the current rotor position by the current loop position estimation module according to a loss function through a Newton iteration method, filtering the estimated fluctuation caused by noise by the rotor position through a phase-locked loop to obtain the motor rotor position and the rotating speed estimated value of the current loop, and calculating the angle position of the rocker arm according to the current loop position and the speed estimated value.
Preferably, the method for acquiring the rotor position comprises the following steps:
constructing an alpha-beta axis voltage equation under a static coordinate system of the permanent magnet synchronous motor:
Figure BDA0003476143230000021
wherein v is α Is the alpha-axis voltage, v β Is the beta axis voltage, R is the winding phase resistance, p is the differential operator, i α And i β Is an alpha-beta axis current, omega re Is the rotor speed, L α (θ re )、L β (θ re )、L αβ (θ re ) Is an intermediate variable of the inductance in the alpha-beta coordinate system, theta re Is the rotor position, L α ＝L ∑ +L Δ cos2θ re ，L β ＝L ∑ -L Δ cos2θ re ，L αβ ＝L Δ sin2θ re Wherein
Figure BDA0003476143230000022
L d Is d-axis inductance, L q Is a q-axis inductor;
order to
Figure BDA0003476143230000023
Establishing a loss function based on a voltage equation:
Figure BDA0003476143230000024
wherein, T s Is the sampling time, i α (k) And i β (k) The current of the k-th alpha axis and beta axis, i α (k-1) and i β (k-1) are the k-1 st alpha-axis and beta-axis currents, ω, respectively re (k-1) is the electrical angular velocity of the rotor at the k-1 st time, L a (θ re (k))、L β (θ re (k) Is L α (θ re )、L β (θ re ) In discrete form, T pk (Δθ re ) Is a rotation operation in an alpha-beta axis, delta theta re The angle of rotation of the rotor in one adopted period is adopted;
adding a penalty term to the loss function, and constructing:
Figure BDA0003476143230000025
wherein the content of the first and second substances,
Figure BDA0003476143230000026
estimating the position of a motor rotor, wherein K is a penalty coefficient;
according to the convex optimization theory, the estimation quantity corresponding to the minimum point of the loss function is solved by using a Newton iteration method
Figure BDA0003476143230000027
As a rotor position estimator, the iterative method is:
Figure BDA0003476143230000028
preferably, the kalman state observer comprises two parts, a prediction equation and an update equation, of the form:
the prediction equation:
Figure BDA0003476143230000031
Figure BDA0003476143230000032
updating an equation:
Figure BDA0003476143230000033
Figure BDA0003476143230000034
Figure BDA0003476143230000035
wherein, y k In order to be a vector of measured values,
Figure BDA0003476143230000036
as model predictor vectors, K k Is the Kalman gain matrix, u k-1 Is the motor drive control quantity, delta k-1 Is the model uncertainty compensation quantity, A k Is a state matrix of an equation of motion formed from the equations of motion
Figure BDA0003476143230000037
Evaluating ^ on predictor vector>
Figure BDA0003476143230000038
Obtaining a partial derivative, P k Is a covariance matrix, C is an observation matrix, Q and R are respectively a process noise covariance matrix and a measurement noise covariance matrix, I is a unit matrix, and Kalman data fusion pass type ^ is based on>
Figure BDA0003476143230000039
Measured value y k And the model predictor->
Figure BDA00034761432300000310
Fused with each other to obtain a predicted value vector estimate->
Figure BDA00034761432300000311
Preferably, the current loop position and speed estimated values obtained in step 1 are used as the measured value vector y in the updating equation k 。
Preferably, the predictor vector of the velocity-loop kalman filter is estimatedMeter
Figure BDA00034761432300000312
The obtained comparison error is used for correcting the model uncertain compensation quantity delta in the step 2 k-1 。
Preferably, the error between the current loop position estimation value and the speed loop position estimation value is subjected to proportional integral processing and then is used as the model uncertain compensation quantity delta k-1 。
Compared with the prior art, the invention has the following remarkable advantages: the invention makes up the defect of inaccurate position estimation under the condition of low-speed overload of the permanent magnet servo motor by utilizing the position estimation of the speed ring at a specific position.
Drawings
FIG. 1 is a step diagram of position sensorless rocker servo control based on Kalman data fusion in an embodiment of the application.
FIG. 2 is a schematic block diagram of a position sensorless servo rocker control of an embodiment of the present application.
Detailed Description
As shown in fig. 1 and 2, a position sensorless rocker arm servo control method based on disturbance-rejection kalman data fusion specifically comprises the following steps:
step 1, identifying the position of a rocker arm based on a current loop position sensorless control algorithm, wherein the specific method comprises the following steps:
the current under the alpha-beta shaft system, the voltage instruction under the alpha-beta shaft system and the estimated rotating speed omega at the last moment are used re Inputting a current loop position estimation module, and calculating the current rotor position by the current loop position estimation module according to the loss function through a Newton iteration method; and the rotor position filters the estimation fluctuation caused by the noise through a phase-locked loop to obtain the motor rotor position and the rotating speed estimation value of the current loop, and then the angle position of the rocker arm is calculated according to the current loop position and the speed estimation value.
In a further embodiment, the voltage equation of the permanent magnet synchronous motor in the static coordinate system (α - β) is as follows:
Figure BDA0003476143230000041
wherein v is α Is the alpha-axis voltage, v β Is the beta axis voltage, R is the winding phase resistance, p is the differential operator, L α (θ re )、L β (θ re )、L αβ (θ re ) Is an intermediate variable of inductance value in alpha-beta coordinate system, dependent on rotor position theta re Is changed i α And i β Is the alpha-beta axis current.
For simplicity, the voltage is written as
Figure BDA0003476143230000042
The loss function is established based on the voltage equation as:
Figure BDA0003476143230000043
wherein T is pk (Δθ re ) Is a rotation operation in the alpha-beta axis, T s Is the sampling time, i α (k) And i β (k) Is the k-th alpha-beta axis current, i α (k-1) and i β (k-1) is the k-1 st alpha-beta axis current, omega re (k-1) is the electrical angular velocity of the rotor at the k-1 st time, L a (θ re (k))、L β (θ re (k) Is L α (θ re )、L β (θ re ) In discrete form.
Considering that the position of the rotor at low speed can not change too fast, adding a penalty term to the loss function, and constructing:
Figure BDA0003476143230000044
according to the convex optimization theory, when the loss function takes the minimum value, the corresponding position estimator
Figure BDA0003476143230000045
Namely, the estimated value of the position closest to the true position, the minimum point pair of the loss function is solved by using a Newton iteration methodThe desired estimate->
Figure BDA0003476143230000046
The iteration method comprises the following steps:
Figure BDA0003476143230000047
step 2, observing the position of the rocker arm by adopting a Kalman state observer based on a rocker arm motion equation;
equation of motion of the rocker arm:
Figure BDA0003476143230000051
where J is the total moment of inertia translated to the motor shaft, θ rm Is the mechanical angular position of the motor shaft, having P theta rm ＝θ re Where P is the number of pole pairs of the motor, T e Is the electromagnetic torque, mgL arm Is the gravitational moment amplitude and B is the coefficient of friction. The above equation is written in the form of a state space:
Figure BDA0003476143230000052
Figure BDA0003476143230000053
written in discrete form are: x is the number of k ＝f(x k-1 ,u k-1 0), wherein
Figure BDA0003476143230000054
In the above formula, the compensation quantity δ is introduced in consideration of the variation and uncertainty of the model parameters k-1 And (3) constructing: x is the number of k ＝f(x k-1 ,u k-1 +δ k-1 ,0)。
The Kalman state observer comprises a prediction equation and an update equation, and is in the form of:
the prediction equation:
Figure BDA0003476143230000055
Figure BDA0003476143230000056
updating an equation:
Figure BDA0003476143230000057
Figure BDA0003476143230000058
Figure BDA0003476143230000059
wherein y is k In order to be a vector of measured values,
Figure BDA00034761432300000510
as model predictor vectors, K k Is the Kalman gain matrix, u k-1 Is the motor drive control quantity, delta k-1 Is the model uncertainty compensation quantity, A k Is a state matrix of an equation of motion formed from the equations of motion
Figure BDA00034761432300000511
Evaluating ^ on predictor vector>
Figure BDA00034761432300000512
Obtaining a partial derivative, P k Is a covariance matrix, C is an observation matrix, Q and R are respectively a process noise covariance matrix and a measurement noise covariance matrix, and Kalman data fusion pass formula
Figure BDA00034761432300000513
Measured value y k And model predictionValue->
Figure BDA00034761432300000514
Fused with each other to obtain a predicted value vector estimate->
Figure BDA00034761432300000515
The Kalman state observer takes the current loop position and speed estimated values obtained in the step 1 as measurement value vectors y in an updating equation k Introduced into the Kalman observer, i.e. as y k And velocity loop model predictor vector
Figure BDA00034761432300000516
And fusing to obtain the position and speed estimation of the speed ring.
The Kalman state observer estimates the predicted value vector of the speed ring Kalman filter
Figure BDA00034761432300000517
The obtained comparison error is used for correcting the model uncertain compensation quantity delta in the step 2 k-1 。/>
The speed and position estimated values obtained by the speed loop Kalman filter are greatly influenced by the uncertainty of the speed loop model parameters, while the current loop position estimated value is relatively less influenced by the parameter change, so that the error between the current loop position estimated value and the speed loop position estimated value is introduced and is used as a compensation quantity delta after proportional integral processing k-1 The compensation quantity corrects a speed ring estimation model in the Kalman filter, and the accuracy of the rotation speed estimation in the Kalman filter is ensured.
And 3, controlling the motion control system by adopting three modes of position, speed and current, estimating the position of a position loop by adopting current loop position feedback, and estimating the speed of a speed loop by adopting speed estimation.
Examples
The present invention will be further described with reference to the accompanying drawings. Servo for dragging rocker arm load by using permanent magnet synchronous motorFor example, as shown in fig. 2, the whole control system adopts a three-loop system consisting of a position loop, a speed loop and a current loop, and the current loop calculates a voltage reference value (u) * α ，u * β ) After PWM modulation, the permanent magnet motor is driven by the driver to drag the rocker arm. The rotation speed and the position information required by the three-loop control are both output of the Kalman observer. It should be noted that, in particular, at low speed operation, the current loop position estimation needs to be assisted by high frequency voltage signal injection to observe the rotor salient pole position.
Step 1, identifying the position of a rocker arm based on a current loop position sensorless control algorithm: the current under the alpha-beta shaft system, the voltage instruction under the alpha-beta shaft system and the estimated rotating speed omega at the last moment are used re Inputting a current loop position estimation module, and calculating the current rotor position and a Cost value by a Newton iteration method according to the loss function; and the rotor position filters the estimated fluctuation caused by the noise through a phase-locked loop to obtain the motor rotor position and the rotating speed estimated value of the current loop, and then the angle position of the rocker arm is calculated according to the motor rotor angle.
Step 2, observing the position and the rotating speed of the rocker arm based on the position ring: a Kalman state observer is adopted to observe the position of the rocker arm based on the motion equation of the rocker arm, and the Kalman state observer comprises a prediction equation and an update equation.
Using the current loop position and speed estimated value obtained in step 1 as the measured value vector y in step 2 k Introduced into the Kalman observer, i.e. as y k And velocity loop model predictor vector
Figure BDA0003476143230000061
And fusing to obtain the position and speed estimation of the speed ring.
Estimating predicted value vector of velocity ring Kalman filter
Figure BDA0003476143230000071
The obtained comparison error is used for correcting the model uncertain compensation quantity delta in the step 2 k-1 。
And 3, controlling the motion control system by adopting three modes of position, speed and current, wherein the position feedback of the position loop adopts current loop position estimation, and the speed feedback of the speed loop adopts speed estimation of the speed loop.
In particular, in the embodiments of the present application, the rocker arm stops at θ rm When the rotation speed is 0 and the gravity moment is maximum mgL at 90 DEG, the rotation speed is 0 arm The method belongs to a typical low-speed overload state, and because d-q axis inductance is close at the moment, the current loop estimation effect is poor, which is mainly reflected in that the noise of the rotation speed estimation is large, and high-quality rotation speed control is difficult to support, a speed loop Kalman filter is adopted to obtain stable rotation speed estimation so as to support the speed loop control, and the speed loop Kalman filter is easily influenced by the parameter change and uncertainty of the speed loop, so that errors obtained by comparing the position estimation of the speed loop Kalman filter with the current loop estimation are introduced to form a compensation item to correct a Kalman filter model, and further, accurate rotation speed estimation is obtained.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims (5)

1. A position sensor-free rocker arm servo control method based on disturbance rejection Kalman data fusion is characterized by comprising the following steps:
step 1, identifying the position of a rocker arm based on a current loop position sensorless control algorithm;
and 2, observing the position and the rotating speed of the rocker arm by adopting a Kalman state observer based on a rocker arm motion equation, wherein the Kalman state observer comprises a prediction equation and an update equation, and the form of the Kalman state observer is as follows:
the prediction equation:
Figure FDA0003944956890000011
Figure FDA0003944956890000012
updating an equation:
Figure FDA0003944956890000013
Figure FDA0003944956890000014
Figure FDA0003944956890000015
wherein, y k In order to be a vector of measured values,
Figure FDA0003944956890000016
as model predictor vectors, K k Is the Kalman gain matrix, u k-1 Is the motor drive control quantity, delta k-1 Is the model uncertainty compensation quantity, A k Is a state matrix of the equation of motion formed from the equation of motion
Figure FDA0003944956890000017
Vector estimation of predicted values
Figure FDA0003944956890000018
Obtaining a partial derivative, P k Is a covariance matrix, C is an observation matrix, Q and R are respectively a process noise covariance matrix and a measurement noise covariance matrix, I is an identity matrix, and Kalman data fusion pass formula
Figure FDA0003944956890000019
Measured value y k And model prediction values
Figure FDA00039449568900000110
Mutually fused to obtain predicted value vector estimation
Figure FDA00039449568900000111
Step 3, the motion control system adopts three-loop control of position, speed and current, wherein the position feedback of the position loop adopts current loop position estimation, and the speed feedback of the speed loop adopts speed estimation of the speed loop;
estimating predicted value vector of velocity ring Kalman filter
Figure FDA00039449568900000112
The obtained comparison error is used for correcting the model uncertain compensation quantity delta in the step 2 k-1 。
2. The disturbance-rejection Kalman data fusion-based position sensorless rocker arm servo control method according to claim 1, characterized in that the specific method for identifying the position of the rocker arm based on the current loop position sensorless control algorithm is as follows:
inputting the current under the alpha-beta shafting, the voltage under the alpha-beta shafting and the estimated rotating speed at the last moment into a current loop position estimation module, calculating the current rotor position by the current loop position estimation module through a Newton iteration method according to a loss function, filtering the estimated fluctuation caused by noise by the rotor position through a phase-locked loop to obtain the motor rotor position and the rotating speed estimated value of the current loop, and calculating the angle position of the rocker arm according to the current loop position and the speed estimated value.
3. The disturbance-rejection Kalman data fusion-based position sensor-free rocker arm servo control method according to claim 2, characterized in that the rotor position is obtained by:
constructing an alpha-beta axis voltage equation under a static coordinate system of the permanent magnet synchronous motor:
Figure FDA0003944956890000021
wherein v is α Is the alpha-axis voltage, v β Is the beta axis voltage, R is the winding phase resistance, p is the differential operator, i α And i β Is an alpha-beta axis current, omega re Is the rotor speed, L α (θ re )、L β (θ re )、L αβ (θ re ) Is an intermediate variable of the inductance in the alpha-beta coordinate system, theta re Is the rotor position, L α ＝L ∑ +L Δ cos2θ re ，L β ＝L ∑ -L Δ cos2θ re ，L αβ ＝L Δ sin2θ re Wherein
Figure FDA0003944956890000022
L d Is d-axis inductance, L q Is a q-axis inductor;
order to
Figure FDA0003944956890000023
Establishing a loss function based on a voltage equation:
Figure FDA0003944956890000024
wherein, T s Is the sampling time, i α (k) And i β (k) The current of the k-th alpha axis and beta axis, i α (k-1) and i β (k-1) are the k-1 st alpha-axis and beta-axis currents, ω, respectively re (k-1) is the electrical angular velocity of the rotor at the k-1 st time, L a (θ re (k))、L β (θ re (k) Is L α (θ re )、L β (θ re ) In discrete form, T pk (Δθ re ) Is a rotation operation in an alpha-beta axis, delta theta re The angle of rotation of the rotor in one adopted period is adopted;
adding a penalty term to the loss function, and constructing:
Figure FDA0003944956890000025
wherein the content of the first and second substances,
Figure FDA0003944956890000026
estimating the position of a motor rotor, wherein K is a penalty term coefficient;
according to the convex optimization theory, the estimation quantity corresponding to the minimum point of the loss function is solved by using a Newton iteration method
Figure FDA0003944956890000027
As a rotor position estimator, the iterative method is:
Figure FDA0003944956890000028
4. the position sensor-free rocker arm servo control method based on disturbance-resistant Kalman data fusion according to claim 1, characterized in that the current loop position and speed estimated values obtained in step 1 are used as measurement value vector y in the update equation k 。
5. The tamper-resistant Carl-based of claim 1The position-sensorless rocker arm servo control method for the Manchester data fusion is characterized in that errors of a current ring position estimated value and a speed ring position estimated value are subjected to proportional integral processing and then serve as model uncertain compensation quantity delta k-1 。
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