CN115459667A - Permanent magnet synchronous motor sensorless sliding mode control method based on improved approach law - Google Patents

Permanent magnet synchronous motor sensorless sliding mode control method based on improved approach law Download PDF

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CN115459667A
CN115459667A CN202211394827.6A CN202211394827A CN115459667A CN 115459667 A CN115459667 A CN 115459667A CN 202211394827 A CN202211394827 A CN 202211394827A CN 115459667 A CN115459667 A CN 115459667A
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sliding
sliding mode
law
permanent magnet
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CN115459667B (en
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郑剑锋
王思宁
牛辉
章明
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Liyang Chang Technology Transfer Center Co ltd
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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
    • H02P21/24—Vector control not involving the use of rotor position or rotor speed sensors
    • H02P21/26—Rotor flux based control
    • 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
    • H02P21/0003—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H02P21/0007—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control using sliding mode control
    • 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
    • H02P21/05—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation specially adapted for damping motor oscillations, e.g. for reducing hunting
    • 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
    • H02P21/14—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
    • 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
    • H02P25/024—Synchronous motors controlled by supply frequency
    • 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
    • H02P27/00—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage
    • H02P27/04—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage
    • H02P27/06—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage using dc to ac converters or inverters
    • H02P27/08—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage using dc to ac converters or inverters with pulse width modulation
    • 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
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/14—Electronic commutators
    • H02P6/16—Circuit arrangements for detecting position
    • H02P6/18—Circuit arrangements for detecting position without separate position detecting elements
    • H02P6/182—Circuit arrangements for detecting position without separate position detecting elements using back-emf in windings
    • 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
    • H02P2203/00—Indexing scheme relating to controlling arrangements characterised by the means for detecting the position of the rotor
    • H02P2203/03—Determination of the rotor position, e.g. initial rotor position, during standstill or low speed operation
    • 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
    • H02P2203/00—Indexing scheme relating to controlling arrangements characterised by the means for detecting the position of the rotor
    • H02P2203/09—Motor speed determination based on the current and/or voltage without using a tachogenerator or a physical encoder
    • 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
    • H02P2207/00—Indexing scheme relating to controlling arrangements characterised by the type of motor
    • H02P2207/05—Synchronous machines, e.g. with permanent magnets or DC excitation

Abstract

The invention relates to the technical field of motors, in particular to a sensorless sliding mode control method of a permanent magnet synchronous motor based on an improved approach law, which comprises the steps of constructing a speed regulating system of the permanent magnet synchronous motor; improving a constant speed term and an index term of an index approximation law of the sliding mode controller, and smoothing a sign function; judging whether the sliding mode controller of the improved index approach law meets the accessibility condition according to the stability of the Lyapunov; improving a switching function of the sliding-mode observer, calculating a stator current error equation, designing a sliding-mode control law, and analyzing stability. The invention adds the function which takes the absolute value of the speed error as a variable to the constant speed term and the exponential term respectively; the sign function in the exponential approximation law is improved and optimized, the problems of buffeting and large overshoot can be weakened after smoothing treatment, and the response speed and the dynamic quality of a control system are improved; and then, a sign function in the traditional sliding-mode observer is replaced by a saturation function, and the stability of the system is analyzed by utilizing a Lyapunov stability criterion.

Description

Permanent magnet synchronous motor sensorless sliding mode control method based on improved approach law
Technical Field
The invention relates to the technical field of motors, in particular to a sensorless sliding mode control method of a permanent magnet synchronous motor based on an improved approach law.
Background
At present, a Permanent Magnet Synchronous Motor (PMSM) is widely applied to the fields of aerospace, robots, new energy automobiles and the like due to the advantages of small size, simple structure, high efficiency, flexibility, diversity and the like. However, since the three-phase PMSM is a strongly coupled and nonlinear multivariable system, in order to improve the dynamic performance of the three-phase PMSM speed regulation system and weaken buffeting, the sliding mode control based on the approach law is widely applied due to the advantages of insensitivity to disturbance and parameters, high response speed, strong robustness and the like.
In the prior art, the sliding mode control of the permanent magnet synchronous motor of an improved index approaching law introduces the absolute value of a speed error into a constant speed term of a traditional index approaching law
Figure 919072DEST_PATH_IMAGE001
The system response time is shortened, buffeting of the state variables before the state variables reach the origin is weakened, and the overshoot of the initial moment of the motor rotating speed is large.
In order to realize a high-performance control system, it is very important to obtain accurate rotor position and rotation speed information, in the conventional control system, the rotor position and speed information can be obtained by using mechanical sensors such as an optical encoder, but the installation of the mechanical sensors increases the use cost, and the limitation of conditions such as temperature, humidity and vibration can impose strict requirements on the use environment, so that a great number of students pay extensive attention to and research the application of sensorless control in the permanent magnet synchronous motor; a novel improved sliding mode observer in the prior art introduces an amplification coefficient K before a traditional low-pass filter, amplifies a back electromotive force signal, replaces a sign function with an arc tangent function, introduces a recursive least square adaptive filter to replace the traditional low-pass filter, and finally introduces 1/K in an arc tangent operation module to restore the practicality of an estimated value, so that the problem that the system generates high-frequency buffeting due to sudden change of a switching function of the traditional sliding mode observer at a zero point is solved, but the design process of the system is complex.
Disclosure of Invention
Aiming at the defects of the existing algorithm, the invention solves the problems of buffeting and large overshoot; a sliding mode controller based on an improved exponential approximation law is designed on a speed ring to replace a traditional exponential approximation law sliding mode controller, and a speed error absolute value is newly added to each of a constant speed term and an exponential term
Figure 791213DEST_PATH_IMAGE001
As a function of a variable; meanwhile, a sign function in the traditional exponential approaching law is improved and optimized, and after smoothing, the buffeting phenomenon can be weakened, the response speed is increased, and the dynamic quality of a system is controlled; and then, a sign function in the traditional sliding-mode observer is replaced by a saturation function, and the stability of the system is analyzed by utilizing a Lyapunov stability criterion.
The technical scheme adopted by the invention is as follows: a permanent magnet synchronous motor sensorless sliding mode control method based on an improved approach law comprises the following steps:
step one, through rotor structure of three-phase permanent magnet synchronous motor, permanent magnet synchronous motord-qConstructing a permanent magnet synchronous motor speed regulating system by a mathematical model, clark and Park conversion, vector control and SVPWM control under an axis coordinate system;
step two, improving a constant speed term and an index term of an index approximation rule of a sliding mode controller in a permanent magnet synchronous motor speed regulating system, and smoothing a sign function to obtain an improved index approximation rule; and judging whether the sliding mode controller of the improved index approaching law meets the accessibility condition according to the Lyapunov stability;
further, the formula for improving the constant speed term and the index term of the index approach law of the sliding mode controller is as follows:
Figure 518997DEST_PATH_IMAGE002
wherein the content of the first and second substances,sis a function of the surface of the sliding mode,
Figure 211010DEST_PATH_IMAGE003
in order to be a rotational speed error,sign(s)in order to be a function of the sign,
Figure 682442DEST_PATH_IMAGE004
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential approximation term.
Further, the formula for smoothing the sign function is as follows:
Figure 409090DEST_PATH_IMAGE005
wherein the content of the first and second substances,
Figure 42197DEST_PATH_IMAGE006
is a normal number, and is,sis a sliding mode surface function.
Further, the formula of the improved exponential approximation law is as follows:
Figure 752664DEST_PATH_IMAGE007
(14)
wherein the content of the first and second substances,sis a function of the surface of the sliding mode,
Figure 762208DEST_PATH_IMAGE008
in order to be the error of the rotation speed,sign(s)in order to be a function of the sign,
Figure 343362DEST_PATH_IMAGE009
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential-approximation term,
Figure 412949DEST_PATH_IMAGE010
is a normal number.
Further, the specifically determining that the improved index approach law is satisfied according to the lyapunov stability by the sliding mode controller includes:
defining the Lyapunov function:
Figure 345133DEST_PATH_IMAGE011
(15)
wherein, the first and the second end of the pipe are connected with each other,Vis a function of the Lyapunov function,sis a sliding mode surface function;
the method is obtained according to a Lyapunov function and an improvement exponent approximation law formula:
Figure 158368DEST_PATH_IMAGE012
wherein, the first and the second end of the pipe are connected with each other,sis a sliding mode surface function;
Figure 125187DEST_PATH_IMAGE013
is the error of the rotating speed;
Figure 849166DEST_PATH_IMAGE014
in order to approach the coefficients of the coefficients,kis a coefficient of the exponential approximation term,
Figure 268646DEST_PATH_IMAGE015
is a normal number.
And step three, improving a switching function of the sliding mode observer, calculating a stator current error equation, designing a sliding mode control law, and analyzing stability.
Further, the switching function of the sliding mode observer is improved by adopting a saturation functionsat(s) Replacing switching functionssign(s)，sat(s) The formula of (1) is:
Figure 619993DEST_PATH_IMAGE016
wherein the content of the first and second substances,
Figure 175739DEST_PATH_IMAGE017
is a boundary layer of the magnetic recording medium,sis a function of the surface of the sliding mode,kis the coefficient of the exponential approximation term.
Further, the formula for calculating the stator current error equation is:
Figure 55970DEST_PATH_IMAGE018
wherein the content of the first and second substances,
Figure 962746DEST_PATH_IMAGE019
current observation error;
Figure 117784DEST_PATH_IMAGE020
is a stator resistor;
Figure 793616DEST_PATH_IMAGE021
is a stator inductance;
Figure 375907DEST_PATH_IMAGE022
to extend the back emf;sat() Is a saturation function;Kis the gain factor of the sliding mode observer.
Further, a sliding mode control law is designed, and equivalent control quantity is carried out, specifically comprising:
designing a sliding mode control law, wherein the formula is as follows:
Figure 769980DEST_PATH_IMAGE023
wherein, the first and the second end of the pipe are connected with each other,
Figure 463129DEST_PATH_IMAGE024
；
Figure 259047DEST_PATH_IMAGE025
is a sliding mode control law function;sat() Is a saturation function;Kis a sliding mode observer gain coefficient;
when the state variable of the sliding-mode observer reaches the sliding modeFlour
Figure 12239DEST_PATH_IMAGE026
And the state of the sliding-mode observer is always kept on the sliding-mode surface, and according to the equivalent control principle of sliding-mode control, the control quantity at the moment is regarded as equivalent control quantity to obtain:
Figure 628028DEST_PATH_IMAGE027
wherein the content of the first and second substances,
Figure 121939DEST_PATH_IMAGE028
to extend the back emf;sat() Is a saturation function;Kis a sliding mode observer gain coefficient;
because the actual control quantity is a discontinuous high-frequency switching signal, in order to extract a continuous extended back electromotive force estimation value, a low-pass filter is added to obtain the extended back electromotive force estimation value, and the formula is as follows:
Figure 772363DEST_PATH_IMAGE029
wherein the content of the first and second substances,
Figure 696457DEST_PATH_IMAGE030
is the time constant of the low-pass filter,
Figure 799542DEST_PATH_IMAGE031
to extend the back emf;
Figure 834494DEST_PATH_IMAGE032
expanding the back electromotive force estimation value;sat() Is a saturation function;
obtaining rotor position information by an arc tangent function method, wherein the formula is as follows:
Figure 339425DEST_PATH_IMAGE033
wherein, the first and the second end of the pipe are connected with each other,
Figure 434420DEST_PATH_IMAGE034
the estimated value of the rotor position before adding the compensation angle is obtained;
Figure 24801DEST_PATH_IMAGE035
expanding the back electromotive force estimated value;
Figure 863444DEST_PATH_IMAGE036
is an arctangent function;
the angle compensation is added to the rotor position information to make up for the position angle estimation error caused by the low-pass filter delay, and the formula of the rotor position information after the angle compensation is added is as follows:
Figure 488460DEST_PATH_IMAGE037
wherein the content of the first and second substances,
Figure 488778DEST_PATH_IMAGE038
is the cut-off frequency of the low-pass filter;
Figure 566455DEST_PATH_IMAGE039
the estimated value of the rotating speed is obtained;
Figure 208789DEST_PATH_IMAGE040
the estimated value of the rotor position before adding the compensation angle is obtained;
Figure 688312DEST_PATH_IMAGE041
the rotor position estimate after the addition of the compensation angle.
Further, the formula of the rotation speed estimation value is as follows:
Figure 125109DEST_PATH_IMAGE042
wherein, the first and the second end of the pipe are connected with each other,
Figure 427433DEST_PATH_IMAGE043
the estimated value of the rotating speed is obtained;
Figure 607879DEST_PATH_IMAGE044
expanding the back electromotive force estimated value;
Figure 207488DEST_PATH_IMAGE045
is a permanent magnet flux linkage.
The invention has the beneficial effects that:
1. the index approaching law of the sliding mode controller is improved, the convergence time is shorter in the process that a motion point reaches a balance point from an initial value, buffeting is obviously improved, overshoot is not generated at the initial moment of the rotating speed of the motor, overshoot at the moment of sudden loading is minimum, the dynamic quality of approaching motion is improved, and the dynamic response speed and the problem precision are improved.
2. The sliding mode controller designed by improving the approximation rule can improve the dynamic quality of a controlled system, and compared with the sliding mode control of the permanent magnet synchronous motor of the traditional index approximation rule and the improved index approximation rule, the sliding mode controller has the advantages of higher response speed, smaller overshoot, improved robustness and rapidity of the system, weakened buffeting of the improved sliding mode observer and improved disturbance resistance of the control system.
3. The sliding mode observer is insensitive to parameter change and strong in robustness, the problem that the mechanical speed sensor cannot guarantee system stability due to the fact that the working environment inside the motor is complex when a vector control system of the permanent magnet synchronous motor adopts the mechanical speed sensor to detect the position and the rotating speed information of a rotor is solved through the combination of the sliding mode improved index approach law speed controller and the sliding mode observer, when the motor is started from zero speed to a reference speed, the time for reaching the stable state is shortened, overshoot does not exist, the fluctuation range of errors of the actual rotating speed and the estimated rotating speed is reduced, the waveform jitter of a position estimated value is smaller, the estimated value is close to the true value, and the waveform is smooth and does not have the phenomenon of jitter in the whole motor operation process.
Drawings
FIG. 1 is a flow chart of a sensorless sliding mode control method of a permanent magnet synchronous motor based on an improved approach law according to the invention;
FIG. 2 is a schematic control block diagram of a prior art PMSM speed control system;
FIG. 3 is a graph of the coordinate relationship of the Clark transformation and Park transformation of the prior art;
FIG. 4 is a graph comparing the switching function traces of the present invention with comparative examples 1 and 2;
FIG. 5 is a partially enlarged comparative plot of the switching function of the present invention versus comparative examples 1 and 2;
FIG. 6 is a graph comparing the controller output u motion trajectory of the present invention with that of comparative examples 1 and 2;
FIG. 7 is a diagram of a surface-mounted three-phase PMSM vector control simulation model based on a sliding mode speed controller;
FIG. 8 is a diagram of a simulation model of the controller of comparative example 1;
FIG. 9 is a diagram of a simulation model of the controller of comparative example 2;
FIG. 10 is a diagram of a controller simulation model of the present invention;
FIG. 11 is a waveform diagram of three-phase current of the controller of comparative example 1;
FIG. 12 is a waveform diagram of three-phase current of the controller of comparative example 2;
FIG. 13 is a three-phase current waveform of the controller of the present invention;
FIG. 14 is a graph comparing the machine speeds of the present invention with comparative examples 1 and 2;
FIG. 15 is a graph comparing motor torques of the present invention with comparative examples 1 and 2;
FIG. 16 is a schematic block diagram of PMSM sliding mode vector control of the governor system of the present invention;
FIG. 17 is a functional block diagram of a sliding-mode observer algorithm;
FIG. 18 is a PMSM dual closed loop system simulation model;
fig. 19 is a waveform diagram of the sliding mode observed rotational speed estimation of comparative example 1;
FIG. 20 is a waveform diagram of the sliding mode observed speed estimation of the present invention;
fig. 21 is a waveform diagram of a rotation speed estimation error of comparative example 1;
FIG. 22 is a plot of a speed estimation error waveform of the present invention;
FIG. 23 is a graph of the rotor position actual value and the estimated waveform of comparative example 1;
FIG. 24 is a graph of the rotor position actual versus estimated waveforms of the present invention;
FIG. 25 is a rotor position estimation error map of comparative example 1;
FIG. 26 is a rotor position estimation error map of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings and examples, which are simplified schematic drawings and which illustrate only the basic structure of the invention and, therefore, only show the structures associated with the invention.
As shown in fig. 1, a sensorless sliding mode control method for a permanent magnet synchronous motor based on an improved approach law includes the following steps:
constructing a permanent magnet synchronous motor speed regulating system through a rotor structure of a three-phase permanent magnet synchronous motor, a mathematical model under a d-q axis coordinate system of the permanent magnet synchronous motor, clark and Park conversion, vector control and SVPWM control;
fig. 2 is a schematic control block diagram of a speed control system of a permanent magnet synchronous motor, wherein the speed control system of the permanent magnet synchronous motor comprises a permanent magnet synchronous motor structure, a mathematical model of the permanent magnet synchronous motor, coordinate transformation, a vector control principle and an SVPWM principle;
according to different positions on a permanent magnet rotor of the permanent magnet synchronous motor, the rotor structure of the three-phase permanent magnet synchronous motor can be divided into a surface-mounted structure and a built-in structure.
Permanent magnet synchronous machine mathematical model:
for the sake of simple analysis, the three-phase permanent magnet synchronous motor is assumed to be an ideal motor, and the following assumptions are satisfied:
1. ignoring saturation of the motor core;
2. eddy current and hysteresis loss in the motor are not counted;
3. the current in the motor is symmetrical three-phase sine wave current.
Selective surface-mounted permanent magnet synchronous motord-qMathematical model under the axis coordinate system, PMSM stator voltage equation:
Figure 815186DEST_PATH_IMAGE046
wherein, the first and the second end of the pipe are connected with each other,
Figure 601877DEST_PATH_IMAGE047
respectively stator current and stator voltage
Figure 320434DEST_PATH_IMAGE048
An on-axis component;
Figure 774549DEST_PATH_IMAGE049
respectively stator inductance in
Figure 818729DEST_PATH_IMAGE050
An on-axis component;Ris a stator resistor;
Figure 827136DEST_PATH_IMAGE051
the number of pole pairs of the motor is;
Figure 349384DEST_PATH_IMAGE052
mechanical angular velocity;
Figure 126847DEST_PATH_IMAGE053
is a permanent magnet flux linkage.
Electromagnetic torque expression:
Figure 545190DEST_PATH_IMAGE054
（2）
wherein, the first and the second end of the pipe are connected with each other,T e in order to be an electromagnetic torque,
Figure 572052DEST_PATH_IMAGE055
the number of the pole pairs of the motor is,
Figure 363903DEST_PATH_IMAGE056
is a permanent magnet flux linkage, and is provided with a permanent magnet,
Figure 792610DEST_PATH_IMAGE057
for stator current inqThe component on the axis.
PMSM equation of motion expression:
Figure 913013DEST_PATH_IMAGE058
（3）
wherein, the first and the second end of the pipe are connected with each other,Jis the moment of inertia;
Figure 161592DEST_PATH_IMAGE059
in order to be a load torque,
Figure 25643DEST_PATH_IMAGE060
the number of the pole pairs of the motor is,
Figure 43277DEST_PATH_IMAGE061
is a permanent magnet flux linkage, and is characterized in that,
Figure 334581DEST_PATH_IMAGE062
for stator current inqThe component on the axis of the light beam,
Figure 70456DEST_PATH_IMAGE063
is the mechanical angular velocity.
In order to simplify the mathematical model of the three-phase permanent magnet synchronous motor in the natural coordinate system, the coordinate transformation adopted generally includes stationary coordinate transformation (Clark transformation) and synchronous rotating coordinate transformation (Park transformation), and the coordinate relationship between them is shown in fig. 3 below, in whichABCIs a natural coordinate system and is characterized in that,d-qin order to rotate the coordinate system synchronously,
Figure 472619DEST_PATH_IMAGE064
is a stationary coordinate system.
Clark transformation:
will be a natural coordinate systemABCTransformation to a stationary coordinate system
Figure 344760DEST_PATH_IMAGE065
Is called Clark transformation, from fig. 2a Clark transformation formula can be derived:
Figure 806965DEST_PATH_IMAGE066
(4)
wherein, the first and the second end of the pipe are connected with each other,
Figure 30136DEST_PATH_IMAGE067
representing variables such as motor voltage, current or flux linkage;
Figure 235989DEST_PATH_IMAGE068
is a coordinate transformation matrix, which can be expressed as:
Figure 228216DEST_PATH_IMAGE069
（5）
the coefficient before transforming matrix is 2/3, which is obtained by using the unchanged amplitude as the constraint condition.
Park transformation:
will be a stationary coordinate system
Figure 586954DEST_PATH_IMAGE070
Transformation to a synchronous rotating coordinate systemd-qThe coordinate transformation of (2) is called Park transformation, and a Park coordinate transformation formula can be obtained according to fig. 2:
Figure 297422DEST_PATH_IMAGE071
(6)
wherein the content of the first and second substances,
Figure 41387DEST_PATH_IMAGE072
is a coordinate transformation matrix, which can be expressed as:
Figure 589917DEST_PATH_IMAGE073
（7）
wherein the content of the first and second substances,
Figure 393925DEST_PATH_IMAGE074
transforming the angle for the coordinate;
will synchronously rotate the coordinate systemd-qTransformation to a stationary coordinate system
Figure 60530DEST_PATH_IMAGE065
The coordinate transformation of (2) is called inverse Park transformation, and can be expressed as:
Figure 608186DEST_PATH_IMAGE075
(8)
wherein, the first and the second end of the pipe are connected with each other,
Figure 43847DEST_PATH_IMAGE076
is a coordinate transformation matrix, which can be expressed as:
Figure 18756DEST_PATH_IMAGE077
（9）
wherein, the first and the second end of the pipe are connected with each other,
Figure 438236DEST_PATH_IMAGE078
transforming the angle for the coordinate;
the vector control is based on the coordinate transformation theory, the basic principle of the vector control is that the PMSM is equivalent to a direct current motor according to the magnetic field equivalent principle, and the stator current vector is decomposed into exciting current in a rotor rotating coordinate systemi d And torque currenti q Then the torque control of the PMSM is realized by respectively controlling the two; PMSM vector control adopts two closed-loop control links of speed and current, firstly, a position signal acquired by a sensor is calculated to obtain the rotating speed of a motor, then the rotating speed is compared with a reference speed to obtain an adjusting deviation, then the adjusting deviation is used as an input value of a sliding mode observer (SMC), and a control strategy is used for obtaining an input valuei q A reference value of (d); secondly, the collected motor stator current is obtained through coordinate transformationi d 、i q (ii) a Finally willi d =0 and obtained by PI controlleri q Obtained by transforming reference values with coordinatesi d 、i q Respectively comparing to obtain regulating deviation, and then using PI controller to make regulation deviationi d 、i q The control of the motor is realized.
The SVPWM control strategy is to control the converter according to the switching of the space voltage (current) vector of the converter, and the idea is to adopt the switching of the space voltage vector of the inverter to obtain a quasi-circular rotating magnetic field, so that the alternating current motor obtains better control performance than the SPWM algorithm under the condition of low switching frequency; the SVPWM algorithm is actually a particular combination of switching firing sequences and pulse width magnitudes corresponding to the three-phase voltage source inverter power devices in the ac machine that will produce three-phase less distorted sinusoidal current waveforms in the stator coils that differ from each other by 120 electrical degrees.
Step two, improving a constant speed term and an index term of an index approximation rule of a sliding mode controller in a permanent magnet synchronous motor speed regulating system, and smoothing a sign function to obtain an improved index approximation rule; and judging whether the sliding mode controller of the improved index approach law meets the accessibility condition according to the stability of the Lyapunov;
sliding mode control philosophy:
the sliding mode control is a control strategy of a variable structure control system, the control is discontinuous, and is a switching characteristic which enables the structure of the system to change along with time, the characteristic enables the system to move up and down along a specified state track with small amplitude and high frequency under a certain condition, and the sliding mode control needs to meet the following conditions:
1. a sliding mode exists;
2. satisfies the accessibility condition, and is arranged on the sliding form surface
Figure 789583DEST_PATH_IMAGE079
The other points of movement will reach the slip-form surface in a limited time, i.e.
Figure 613838DEST_PATH_IMAGE080
；
3. The stability of the sliding mode movement is ensured.
The motion of a sliding mode variable structure control system is generally composed of two parts, as shown in fig. 3: the first partABThe method is a normal movement outside a sliding mode surface, and is an approaching movement stage from approaching the sliding mode surface to reaching; the second partBCIs arranged near and along the slip-form surface
Figure 25228DEST_PATH_IMAGE081
The movement of (2); the normal movement phase must be satisfied
Figure 666425DEST_PATH_IMAGE082
The system state space variable can reach the sliding mode surface from any unknown initial state within a limited time under the accessibility condition; therefore, the approximation law function can be designed to ensure the quality of the normal motion phase.
Traditional exponential approximation law:
the index approach law is proposed for the first time in 1996 by the high-rise academy of propylene in China (referred to as comparative example 1), and the expression is as follows:
Figure 555883DEST_PATH_IMAGE083
(10)
wherein the content of the first and second substances,sis a sliding mode surface function;sign(s)is a sign function;
Figure 700557DEST_PATH_IMAGE084
is an approximation coefficient;
in the prior art, an expression of an improved index approach law (referred to as a comparative example 2 for short) provided in sliding mode control of a permanent magnet synchronous motor of the improved index approach law is as follows:
Figure 282848DEST_PATH_IMAGE085
（11）
wherein the content of the first and second substances,sis a sliding mode surface function;
Figure 676920DEST_PATH_IMAGE086
in order to be the error of the rotation speed,sign(s)is a sign function;
Figure 370070DEST_PATH_IMAGE087
is an approximation coefficient.
From equation (10), it can be seen that the conventional index approximation law includes two parts, namely an isovelocity term and an index term
Figure 165987DEST_PATH_IMAGE088
The system state can be ensured to approach to the sliding mode at a larger speed when the sliding mode surface is far away from the sliding mode surface, but the approach of the motion point to the switching surface is an asymptotic process, and the simple exponential approach cannot ensure the arrival within a limited time, so that the constant velocity term is added
Figure 388021DEST_PATH_IMAGE089
Can ensure that when s approaches the sliding mode surface, the approaching speed is
Figure 3810DEST_PATH_IMAGE090
Rather than zero, ensuring arrival within a limited time; in the exponential approximation law, in order to satisfy the requirement of fast approximation and simultaneously weaken buffeting, reasonable increase should be madeqWhile reducing
Figure 235071DEST_PATH_IMAGE090
。
When the state variable of the traditional index approach law is far away from the sliding mode surface, the cut-in is not stable, and the convergence speed is low; when the moving point reaches the balance point from the initial value, the convergence time is longer, the buffeting is larger, the dynamic quality of the approaching movement is poor, the dynamic response speed is slow, the overshoot is large when the load is started and suddenly loaded, and the disturbance resistance capability is poor.
In order to solve the defects of the traditional exponential approximation law, an exponential approximation law formula is improved, and the formula is as follows:
Figure 882566DEST_PATH_IMAGE091
(12)
wherein the content of the first and second substances,sis a function of the surface of the sliding mode,
Figure 806660DEST_PATH_IMAGE092
in order to be the error of the rotation speed,sign(s)in the form of a function of the sign,
Figure 909745DEST_PATH_IMAGE093
in order to approach the coefficients of the coefficients,kcoefficients that are exponential approximations;
absolute value of speed error
Figure 944697DEST_PATH_IMAGE094
Respectively introducing into an isovelocity term and an index term, and analyzing to obtain that: when the motion trail of the system state variable is far away from the sliding mode surface,
Figure 715207DEST_PATH_IMAGE094
relatively large, the speed change term approaches zero and is mainly acted by the variable index term
Figure 810202DEST_PATH_IMAGE095
The speed tends to the sliding mode surface, so that the approaching speed is increased; when the sliding mode surface is approached, the approaching rate of the exponential term approaches to 0, and the approaching coefficient of the speed change term is
Figure 135004DEST_PATH_IMAGE096
And the sliding mode control law acts to enable the state variable to enter the sliding mode surface and move towards the original point, the process enables the speed change term to be continuously reduced and finally to be stabilized at the original point, the buffeting phenomenon is restrained, and the defects of the existing index approaching law are overcome.
Because the sign function in the exponential approximation law is a discontinuous switch function, buffeting is increased, and the sign function is smoothed:
Figure 973647DEST_PATH_IMAGE097
(13)
wherein, the first and the second end of the pipe are connected with each other,
Figure 598663DEST_PATH_IMAGE098
is a normal number and takes value
Figure 864560DEST_PATH_IMAGE099
；
The improvement index approach law at this time is:
Figure 207816DEST_PATH_IMAGE100
(14)
wherein, the first and the second end of the pipe are connected with each other,sis a function of the surface of the sliding mode,
Figure 584571DEST_PATH_IMAGE101
in order to be the error of the rotation speed,
Figure 64094DEST_PATH_IMAGE102
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential-approximation term,
Figure 500891DEST_PATH_IMAGE103
is a normal number;
and (3) stability analysis:
according to the Lyapunov stability criterion, the method comprises the following steps: arrival conditions of slip form
Figure 65865DEST_PATH_IMAGE104
Defining the Lyapunov function:
Figure 246310DEST_PATH_IMAGE105
(15)
the following equation (14) is derived:
Figure 329409DEST_PATH_IMAGE106
(16)
the system is asymptotically stable, and the sliding mode controller for improving the exponential approximation law meets the accessibility condition.
Comparing the performance analysis of the approximation law in the comparative example 1 and the comparative example 2 with the performance analysis of the improved index approximation law of the invention;
for the following equation of state:
Figure 937108DEST_PATH_IMAGE107
(17)
wherein, the first and the second end of the pipe are connected with each other,A、Bis a system matrix parameter;
respectively analyzing the performance of the traditional index approach law and the improved index approach law, and designing a sliding mode surface function as follows:
Figure 989378DEST_PATH_IMAGE108
(18)
wherein, the first and the second end of the pipe are connected with each other,Cthe parameters of the sliding mode surface are obtained;
the following is derived from equation (18):
Figure 973514DEST_PATH_IMAGE109
(19)
the control output of the systemuThe expression is as follows:
Figure 162050DEST_PATH_IMAGE110
(20)
wherein the content of the first and second substances,
Figure 940650DEST_PATH_IMAGE111
，x1、x2 are respectively two state variables of the system,A、Bis a system matrix parameter; the embodiment sets up:
Figure 480216DEST_PATH_IMAGE112
，
Figure 2464DEST_PATH_IMAGE113
，Cis the parameter of the sliding mode surface,
Figure 45506DEST_PATH_IMAGE114
，slawfor the approach law, the initial value of the state variable is set to
Figure 995008DEST_PATH_IMAGE115
。
Respectively carrying out simulation in MATLAB, and improving parameter setting in an index approaching law:
Figure 21870DEST_PATH_IMAGE116
the approach law selects the same parameters:
Figure 82230DEST_PATH_IMAGE117
。
FIGS. 4-6 show the switching function trajectory, switching function local amplification and control output of the present invention and comparative examples 1 and 2, respectivelyuComparing the tracks; the comparison of the approaching law performance shows that: according to the improved exponential approximation rule, in the process that the moving point reaches the equilibrium point from the initial value, the response speed is fastest, the convergence time is shortest, the buffeting is obviously improved, the dynamic quality of the approximation movement is improved, and the dynamic response speed and the steady-state precision are improved.
Designing a sliding mode speed controller, and the process is as follows:
defining PMSM system state variables:
Figure 245358DEST_PATH_IMAGE118
（21）
wherein, the first and the second end of the pipe are connected with each other,
Figure 365760DEST_PATH_IMAGE119
the reference rotating speed of the motor is a constant;
Figure 614339DEST_PATH_IMAGE120
is the actual rotational speed.
According to the formulae (3) and (21):
Figure 475460DEST_PATH_IMAGE121
（22）
definition of
Figure 899620DEST_PATH_IMAGE122
Then equation (22) can be rewritten as:
Figure 925344DEST_PATH_IMAGE123
（23）
defining a first order linear sliding mode surface function as:
Figure 661219DEST_PATH_IMAGE124
(24)
wherein c >0 is a parameter to be designed.
The formula (23) is derived:
Figure 328961DEST_PATH_IMAGE125
(25)
the available controller output expression:
Figure 201102DEST_PATH_IMAGE126
(26)
thus, the q-axis reference current is found to be:
Figure 397728DEST_PATH_IMAGE127
(27)
wherein the content of the first and second substances,
Figure 620899DEST_PATH_IMAGE128
is composed ofqA shaft reference current;cin order to design the parameters to be designed,c>0；
Figure 826752DEST_PATH_IMAGE129
is the constant velocity approach term coefficient;
Figure 818979DEST_PATH_IMAGE130
is an exponential approach term coefficient;
Figure 717665DEST_PATH_IMAGE131
the absolute value of the rotation speed error is obtained.
And (3) simulation comparison of a sliding mode speed controller:
a surface-mounted three-phase PMSM vector control simulation model based on a sliding mode speed controller is built in MATLAB/Simulink, the built system simulation model is shown in FIG. 7, wherein the speed ring uses the sliding mode controller based on an improved index approach law to replace a traditional sliding mode speed controller, a PI regulator is still adopted as a current ring, the approach law sliding mode speed controllers of a comparative example 1 and a comparative example 2 and the improved sliding mode speed controller simulation model of the invention are shown in the following FIGS. 8-10, and motor parameters and simulation conditions used in simulation are set as shown in the following Table 1:
TABLE 1 Motor parameter table
Figure 431062DEST_PATH_IMAGE132
In order to verify the superiority of the sliding mode speed controller designed by the invention, simulation conditions are set as follows: reference rotational speed is
Figure 909448DEST_PATH_IMAGE133
Initial moment load torque
Figure 21760DEST_PATH_IMAGE134
At t =0.2s, the load torque suddenly increases to
Figure 825768DEST_PATH_IMAGE135
Simulation time is set to 0.4s, sliding mode controller parameters are set to c =35,
Figure 23531DEST_PATH_IMAGE136
；
comparing the invention with the index approach law sliding mode speed controllers provided in comparative example 1 and comparative example 2, the simulation result is shown in the following fig. 11-13; it can be seen that in the first three-phase current stabilization phase, the control response speed of the comparative example 1 is the slowest, the current variation amplitude tends to be 0 and the maximum, the current is stabilized at 0.07s, the stabilization process tends to be the longest, the three-phase current waveform in the whole process is zigzag, the number of burrs is large, and the waveform is uneven; the exponential approximation law of the comparative example 2 is that the phase current a value is about 15.53A at the initial moment, the phase current b peak value reaches 27.48A, the current change amplitude is large, the response speed is high, the current curve is more stable than the traditional curve and tends to be stable at 0.45 s; the phase a current value in the improved index approach law current waveform of the invention is about 8.36A, the phase b current peak value reaches 16.52A, the current variation amplitude is minimum and is stabilized at 0.023s, and the three-phase current waveform curve is smooth and presents a standard sine waveform.
In the process that the three-phase current tends to be stable for the second time after the load is suddenly applied to the system for 0.2s, the control of the comparative example 1 cannot reach the specific torque in time, the current change amplitude is large in the stable process, about 2.3A, and the control is not stable enough and is stabilized at the position of 0.3 s; the current variation amplitude of comparative example 2 stabilized at 0.28s with an exponential approach law stabilization of about 1.2A; the improvement index approach law of the invention is stabilized at 0.26s, and the stabilization time is shortest.
FIG. 14 is a comparison graph of the rotating speed of the motor in three methods, after the motor of the exponential approximation law sliding mode speed controller in comparative example 1 is started at zero speed, the rotating speed reaches a reference value in about 0.1s, the rotating speed can reach 1178r/min, and obvious large overshoot exists, wherein the overshoot is 17.8%; when the load is suddenly changed from 0N.m to 10N.m in 0.2s, the rotating speed waveform is quickly deviated from a given value and is restored to be stable again in about 0.3s, the fluctuation peak value is 857.9r/min, and the overshoot is 14.2%; after the sliding mode speed controller motor of the comparative example 2 is started at zero speed, the rotating speed reaches a reference value in about 0.12s, the highest peak value of the rotating speed is 1128r/min, the overshoot is 12.8%, the overshoot is reduced by 28.1% compared with the traditional method, and after the load is suddenly changed in 0.2s, the fluctuation peak value of the rotating speed is 898.7r/min, and the overshoot is 10.1%; after the improved sliding mode speed controller motor is started at zero speed, the rotating speed reaches a reference value in about 0.07s, the rotating speed waveform has no overshoot phenomenon, when the load is suddenly changed in 0.2s, the rotating speed fluctuation peak value is 902.5r/min, the overshoot is 9.75%, the overshoot is reduced by 31.34% in comparison with the overshoot in the comparison ratio 1 and is reduced by 3.47% in comparison with the overshoot in the comparison ratio 2.
Fig. 15 is a comparison of motor torques for three methods, and the simulation shows that: at the initial moment, the electromagnetic torque response speed of the invention is fastest, the electromagnetic torque tends to be stable at 0.04s, no overshoot is generated in the process of tending to be stable, and the trend of the electromagnetic torque response speed is shortest; the response speed of the electromagnetic torque of comparative example 2 was the second, the settling time was 0.08s; the electromagnetic torque response speed of the comparative example 1 is the slowest and tends to be stable at 0.09s, and the waveform diagram shows that obvious burrs exist in the response waveform in the whole electromagnetic torque response process. In the sudden loading stage, the electromagnetic torque variation amplitude of the comparative example 1 is 3N.m, the overshoot is 30%, and the response waveform has obvious buffeting in the whole electromagnetic torque response process; the overshoot of comparative example 2 and the present invention is about 17%, tending to a shorter settling time than conventional, smooth waveform and no buffeting.
And step three, improving a switching function of the sliding-mode observer, calculating a stator current error equation, designing a sliding-mode control law, and performing stability analysis.
Fig. 16 is a schematic diagram of PMSM sliding mode vector control of a speed regulation system, a mechanical speed sensor is mostly used in a vector control system of a permanent magnet synchronous motor to detect position and rotation speed information of a rotor, and a non-speed sensor is gradually used for control because the internal working environment of the motor is complex and severe and the mechanical speed sensor cannot guarantee system stability.
For surface-mounted permanent magnet synchronous motor (
Figure 571187DEST_PATH_IMAGE137
) In a stationary coordinate system
Figure 272427DEST_PATH_IMAGE138
The current state equation of the mathematical model in (1) is:
Figure 512915DEST_PATH_IMAGE139
(28)
wherein, the first and the second end of the pipe are connected with each other,
Figure 932396DEST_PATH_IMAGE140
respectively stator current at
Figure 752584DEST_PATH_IMAGE141
The component of (a);
Figure 308330DEST_PATH_IMAGE142
respectively, stator voltages are at
Figure 719720DEST_PATH_IMAGE143
The component of (a);
Figure 892075DEST_PATH_IMAGE144
as stator inductance；
Figure 781534DEST_PATH_IMAGE145
To extend the back emf.
Figure 454436DEST_PATH_IMAGE146
（29）
Wherein:
Figure 36727DEST_PATH_IMAGE147
in order to be the electrical angular velocity,
Figure 430800DEST_PATH_IMAGE148
is a permanent magnet flux linkage, and is provided with a permanent magnet,
Figure 123949DEST_PATH_IMAGE149
is a position angle.
From the expressions (28) and (29), it can be seen that the expression of the extended back electromotive force is only a variable related to the rotation speed of the motor, and since the extended back electromotive force of the three-phase permanent magnet synchronous motor contains all information of the position and the rotation speed of the rotor of the motor, the rotation speed and the position information of the motor can be solved only by accurately acquiring the extended back electromotive force.
Based onsign(s) The traditional sliding mode observer control system of the function has larger buffeting caused by high-frequency signal switching, the invention improves the sliding mode observer and adopts a saturation functionsat(s) Replace the switching function in the conventional sliding-mode observersign(s)；
Function of saturationsat(s) Its expression is:
Figure 654288DEST_PATH_IMAGE150
(30)
wherein, the first and the second end of the pipe are connected with each other,
Figure 141901DEST_PATH_IMAGE151
the boundary layer is formed, and the essence of the boundary layer is that switching control is adopted outside the boundary layer; within the boundary layer, linear feedback control is adopted；kIs a saturation function coefficient;sis a sliding mode surface function.
In order to obtain an estimate of the extended back emf, the sliding-mode observer is designed as follows:
Figure 23269DEST_PATH_IMAGE152
（31）
wherein, the first and the second end of the pipe are connected with each other,
Figure 520109DEST_PATH_IMAGE153
is an observed value of the stator current;
Figure 170534DEST_PATH_IMAGE154
is a control input of the observer;sat() Is a saturation function;Kis the gain coefficient of the sliding-mode observer;
Figure 829048DEST_PATH_IMAGE155
is a stator resistor;
Figure 197712DEST_PATH_IMAGE156
is the stator inductance.
The stator current error equation is obtained by subtracting equations (31) and (28):
Figure 232665DEST_PATH_IMAGE157
Figure 3174DEST_PATH_IMAGE158
(32)
wherein the content of the first and second substances,
Figure 832590DEST_PATH_IMAGE159
current observation error;
Figure 688551DEST_PATH_IMAGE160
to extend the back emf;sat() Is a saturation function;
Figure 518405DEST_PATH_IMAGE161
is a stator resistor;
Figure 877842DEST_PATH_IMAGE162
is a stator inductance;Kis the gain factor of the sliding mode observer.
Fig. 17 is a schematic block diagram of the sliding-mode observer algorithm implementation, and the sliding-mode control law is designed as follows:
Figure 143738DEST_PATH_IMAGE163
（33）
wherein, the first and the second end of the pipe are connected with each other,
Figure 955836DEST_PATH_IMAGE164
；sat() Is a saturation function;Kis the gain coefficient of the sliding-mode observer;
Figure 598170DEST_PATH_IMAGE165
is a sliding mode control law function.
When the state variable of the observer reaches the sliding mode surface
Figure 77693DEST_PATH_IMAGE166
And then, the state of the observer is always kept on the sliding mode surface, and according to the equivalent control principle of sliding mode control, the control quantity at the moment can be regarded as equivalent control quantity, so that the following can be obtained:
Figure 514491DEST_PATH_IMAGE167
（34）
since the actual control quantity is discontinuous high-frequency switching signal, in order to extract continuous extended back electromotive force estimation value, a low-pass filter is usually added, namely
Figure 79464DEST_PATH_IMAGE168
（35）
Wherein the content of the first and second substances,
Figure 259910DEST_PATH_IMAGE169
is a low-pass filterAn inter constant;sat() Is a saturation function;Kis the gain coefficient of the sliding mode observer;
Figure 593939DEST_PATH_IMAGE170
the back emf estimate is extended.
Obtaining rotor position information by an arc tangent function method, wherein the formula is as follows:
Figure 936059DEST_PATH_IMAGE171
(36)
wherein the content of the first and second substances,
Figure 988329DEST_PATH_IMAGE172
expanding the back electromotive force estimated value;
Figure 706886DEST_PATH_IMAGE173
is a rotor position estimate;
Figure 161001DEST_PATH_IMAGE174
is an arctangent function.
However, since the equivalent control quantity is low-pass filtered, the amplitude and phase lag phenomenon will be generated by expanding the estimated value of the back electromotive force while filtering the high-frequency switching signal, so an angle compensation is needed to be added on the basis of calculating the rotor position by the formula (36) to compensate the position angle estimation error caused by the delay of the low-pass filter, that is, the position angle estimation error is caused by the delay of the low-pass filter
Figure 939601DEST_PATH_IMAGE175
（37）
Wherein the content of the first and second substances,
Figure 476237DEST_PATH_IMAGE176
is the cut-off frequency of the low-pass filter;
Figure 732906DEST_PATH_IMAGE177
is an arctangent function;
Figure 775948DEST_PATH_IMAGE178
is a rotation speed estimated value;
Figure 725450DEST_PATH_IMAGE179
the estimated value of the rotor position before adding the compensation angle is obtained;
Figure 486733DEST_PATH_IMAGE180
the rotor position estimate after the addition of the compensation angle.
The rotation speed information is obtained by differentiating equation (36), and the expression of the rotation speed estimation value is:
Figure 812672DEST_PATH_IMAGE181
（38）
wherein, the first and the second end of the pipe are connected with each other,
Figure 975800DEST_PATH_IMAGE182
the estimated value of the rotating speed is obtained;
Figure 96202DEST_PATH_IMAGE183
expanding the back electromotive force estimated value;
Figure 610360DEST_PATH_IMAGE184
is a permanent magnet flux linkage.
And (3) stability analysis:
the stability of the system is analyzed by applying the Lyapunov stability criterion, and the system can be known as the following formula (32):
Figure 474411DEST_PATH_IMAGE185
（39）
wherein the content of the first and second substances,Ras the resistance of the stator,L s in order to be the stator inductance, the inductance,sis a slip form surface and is provided with a plurality of slip forms,E s is the back electromotive force of the motor,Kin order to obtain the sliding mode gain of the sliding mode observer,sat(s) Is a saturation function.
Establishing a Lyapunov stability equation:
Figure 226467DEST_PATH_IMAGE186
(40)
wherein the content of the first and second substances,Ras the resistance of the stator,L s in order to be the stator inductance, the inductance,
Figure 783350DEST_PATH_IMAGE187
in order to be the function of Lyapunov,
Figure 519225DEST_PATH_IMAGE188
is the derivative of the Lyapunov function,sat(s) In order to be a function of the saturation,E s is the motor back electromotive force.
According to the stability condition, the following requirements are satisfied:
Figure 921387DEST_PATH_IMAGE189
(41)
from the formula (39):
Figure 796458DEST_PATH_IMAGE190
(42)
because of
Figure 524243DEST_PATH_IMAGE191
Then, then
Figure 747413DEST_PATH_IMAGE192
Then, then
Figure 953267DEST_PATH_IMAGE193
（43）
Simulation analysis:
as shown in fig. 18, for a PMSM dual closed-loop system simulation model built in MATLAB/Simulink, the motor parameters are consistent with the sliding mode speed controller simulation parameters, and the running process is designed as follows: the reference speed of the motor is 1000r/min, the simulation time is 0.4s, and a saturation function is adopted in a Sliding Mode Observer (SMO) algorithmsat() Symbolic function replacing traditional sliding-mode observersign()，sat() Is set to be [2-2 ]]In order to accelerate the simulation speed, a fixed-step-size ode3 (Bogacki-Shampine) algorithm is selected, and the simulation step size is set to be 2x10 -7 And s, simulating and comparing the traditional sliding-mode observer based on the traditional index approach law with the improved index approach law + improved sliding-mode observer of the invention for comparison.
From FIG. 19, which is a waveform diagram of the rotation speed of comparison 1, it can be seen that, when the motor is started from zero speed to reach the reference speed, the speed reaches the stable maximum speed of 1162r/min at about 0.1s, and the overshoot is 16.2%; fig. 20 is a waveform diagram of the modified rotation speed, and it can be seen from the diagram that the motor reaches a steady state in about 0.07s during the process from zero-speed starting to the reference speed, no overshoot occurs, and the waveform is smooth and has no buffeting phenomenon during the whole motor operation process.
Fig. 21 is a waveform diagram of the rotating speed estimation error of the conventional sliding-mode observer based on the conventional exponential approximation law, and it can be seen from the diagram that: when the traditional sliding-mode observer is adopted, under the steady state of the motor, the error fluctuation range of the actual rotating speed and the estimated rotating speed is-7.2-10.1 r/min, the rotating speed error is 17.3r/min, and the jitter amplitude is large; FIG. 22 shows that when the improved sliding-mode observer is adopted to make the motor in a steady state, the error between the actual rotating speed and the estimated rotating speed is 14.92r/min, and the jitter is obviously reduced. Compared with the traditional sliding mode observer, the improved sliding mode observer has the advantages that the motor rotating speed tracking curve has smaller buffeting, only a small amount of ripples exist, and the improved sliding mode observer has better dynamic performance.
FIGS. 23 and 24 are respectively a waveform of the rotor position angle estimation of the conventional sliding mode observer based on the conventional exponential approximation law and a waveform of the rotor position angle estimation of the present invention using the improved sliding mode observer;
FIGS. 25 and 26 are respectively a waveform diagram of an estimation error of a rotor position angle of a conventional sliding mode observer based on a conventional exponential approach law and a waveform diagram of an estimation error of a rotor position angle of the improved sliding mode observer according to the present invention;
compared with the traditional sliding-mode observer which can be based on the traditional exponential approximation law, the position estimation error of the traditional sliding-mode observer has larger burrs, and compared with the traditional sliding-mode observer, the position estimation value of the improved sliding-mode observer has small waveform jitter and is close to a real value.
In light of the foregoing description of the preferred embodiment of the present invention, many modifications and variations will be apparent to those skilled in the art without departing from the spirit and scope of the invention. The technical scope of the present invention is not limited to the content of the specification, and must be determined according to the scope of the claims.

Claims (9)

1. A permanent magnet synchronous motor sensorless sliding mode control method based on an improved approach law is characterized by comprising the following steps:
step one, through rotor structure of three-phase permanent magnet synchronous motor, permanent magnet synchronous motord-qConstructing a permanent magnet synchronous motor speed regulating system by a mathematical model, clark and Park conversion, vector control and SVPWM control under an axis coordinate system;
step two, improving a constant speed term and an index term of an index approximation rule of a sliding mode controller in a permanent magnet synchronous motor speed regulating system, and smoothing a sign function to obtain an improved index approximation rule; and judging whether the sliding mode controller of the improved index approach law meets the accessibility condition according to the stability of the Lyapunov;
and step three, improving a switching function of the sliding mode observer, calculating a stator current error equation, designing a sliding mode control law, and analyzing stability.
2. The sensorless sliding-mode control method of the permanent magnet synchronous motor based on the improved approximation law according to claim 1, wherein the formula for improving the constant speed term and the index term of the index approximation law of the sliding-mode controller is as follows:
Figure 241124DEST_PATH_IMAGE001
wherein the content of the first and second substances,
Figure 396162DEST_PATH_IMAGE002
is a function of the surface of the sliding mode,
Figure 337573DEST_PATH_IMAGE003
in order to be the error of the rotation speed,sign(s)in order to be a function of the sign,
Figure 654285DEST_PATH_IMAGE004
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential approximation term.
3. The improved approximation law-based sensorless sliding mode control method for the permanent magnet synchronous motor according to claim 1, wherein a formula for smoothing a sign function is as follows:
Figure 251619DEST_PATH_IMAGE005
wherein the content of the first and second substances,
Figure 944769DEST_PATH_IMAGE006
is a normal number, and is,sis a sliding mode surface function.
4. The improved approximation law-based sensorless sliding-mode control method for the permanent magnet synchronous motor according to claim 1, wherein the formula of the improved exponential approximation law is as follows:
Figure 740687DEST_PATH_IMAGE007
wherein, the first and the second end of the pipe are connected with each other,sis a function of the surface of the sliding mode,
Figure 490949DEST_PATH_IMAGE008
in order to be the error of the rotation speed,sign(s)in order to be a function of the sign,
Figure 372318DEST_PATH_IMAGE009
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential-approximation term,
Figure 869158DEST_PATH_IMAGE010
is a normal number.
5. The sensorless sliding-mode control method of the permanent magnet synchronous motor based on the improved approximation law according to claim 1 is characterized in that a sliding-mode controller for judging the improved exponent approximation law according to the lyapunov stability meets accessibility conditions, and specifically comprises the following steps:
defining the Lyapunov function:
Figure 519582DEST_PATH_IMAGE011
(15)
wherein, the first and the second end of the pipe are connected with each other,Vis a function of the Lyapunov function,sis a sliding mode surface function;
the method is obtained according to the Lyapunov function and the improved exponential approximation law formula:
Figure 443676DEST_PATH_IMAGE012
wherein the content of the first and second substances,
Figure 484444DEST_PATH_IMAGE013
is a function of the surface of the sliding mode,
Figure 316134DEST_PATH_IMAGE014
in order to be a rotational speed error,sign(s)in order to be a function of the sign,
Figure 24327DEST_PATH_IMAGE015
in order to approach the coefficients of the coefficients,kis the coefficient of the exponential-approximation term,
Figure 119322DEST_PATH_IMAGE016
is a normal number.
6. The improved approximation law-based sensorless sliding mode control of permanent magnet synchronous motor according to claim 1The method is characterized in that the switching function of the sliding-mode observer is improved by adopting a saturation functionsat(s) Replacing switching functionssign(s)，sat(s) The formula of (1) is:
Figure 975282DEST_PATH_IMAGE017
wherein the content of the first and second substances,
Figure 548346DEST_PATH_IMAGE018
is a boundary layer of the magnetic recording medium,sis a function of the surface of the sliding mode,kis the coefficient of the exponential approximation term.
7. The improved approximation law-based sensorless sliding-mode control method for the permanent magnet synchronous motor according to claim 1, wherein a formula for calculating a stator current error equation is as follows:
Figure 173363DEST_PATH_IMAGE019
wherein the content of the first and second substances,
Figure 439259DEST_PATH_IMAGE020
current observation error;
Figure 516936DEST_PATH_IMAGE021
is a stator resistor;
Figure 439498DEST_PATH_IMAGE022
a stator inductor;
Figure 919021DEST_PATH_IMAGE023
to extend the back emf;sat() Is a saturation function;Kis the gain factor of the sliding-mode observer.
8. The improved approximation law-based sensorless sliding-mode control method for the permanent magnet synchronous motor according to claim 1 is characterized in that a sliding-mode control law is designed, and equivalent control quantities are carried out, and specifically the method comprises the following steps:
designing a sliding mode control law, wherein the formula is as follows:
Figure 355818DEST_PATH_IMAGE024
Figure 920792DEST_PATH_IMAGE025
when the state variable of the sliding-mode observer reaches the sliding-mode surface
Figure 835658DEST_PATH_IMAGE026
And the state of the sliding mode observer is always kept on the sliding mode surface, and according to the equivalent control principle of sliding mode control, the control quantity at the moment is regarded as equivalent control quantity to obtain:
Figure 435267DEST_PATH_IMAGE027
wherein, the first and the second end of the pipe are connected with each other,
Figure 42966DEST_PATH_IMAGE028
to extend the back emf;sat() Is a saturation function;Kis a sliding mode observer gain coefficient;
because the actual control quantity is a discontinuous high-frequency switching signal, in order to extract a continuous extended back electromotive force estimation value, a low-pass filter is added to obtain the extended back electromotive force estimation value, and the formula is as follows:
Figure 95235DEST_PATH_IMAGE029
wherein the content of the first and second substances,
Figure 813793DEST_PATH_IMAGE030
is the time constant of the low-pass filter,
Figure 267908DEST_PATH_IMAGE031
to extend the back emf;
Figure 249770DEST_PATH_IMAGE032
expanding the back electromotive force estimation value;sat() Is a saturation function;
obtaining the position information of the rotor by an arc tangent function method, wherein the formula is as follows:
Figure 789336DEST_PATH_IMAGE033
wherein the content of the first and second substances,
Figure 311584DEST_PATH_IMAGE034
the estimated value of the rotor position before adding the compensation angle is obtained;
Figure 620206DEST_PATH_IMAGE035
expanding the back electromotive force estimated value;
Figure 569707DEST_PATH_IMAGE036
is an arctangent function;
the angle compensation is added to the rotor position information to make up for the position angle estimation error caused by the low-pass filter delay, and the formula of the rotor position information after the angle compensation is added is as follows:
Figure 328060DEST_PATH_IMAGE037
wherein, the first and the second end of the pipe are connected with each other,
Figure 653999DEST_PATH_IMAGE038
is the cut-off frequency of the low-pass filter;
Figure 82707DEST_PATH_IMAGE039
the estimated value of the rotating speed is obtained;
Figure 203109DEST_PATH_IMAGE040
the estimated value of the rotor position before adding the compensation angle is obtained;
Figure 717267DEST_PATH_IMAGE041
the rotor position estimate after the addition of the compensation angle.
9. The improved approximation law-based sensorless sliding-mode control method for the permanent magnet synchronous motor according to claim 8, wherein the formula of the rotation speed estimation value is as follows:
Figure 784580DEST_PATH_IMAGE042
wherein the content of the first and second substances,
Figure 802215DEST_PATH_IMAGE043
the estimated value of the rotating speed is obtained;
Figure 359098DEST_PATH_IMAGE044
expanding the back electromotive force estimated value;
Figure 94973DEST_PATH_IMAGE045
is a permanent magnet flux linkage.
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