CN109167538B - Control method for inhibiting torque ripple of permanent magnet brushless direct current motor based on double-layer structure - Google Patents

Control method for inhibiting torque ripple of permanent magnet brushless direct current motor based on double-layer structure Download PDF

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CN109167538B
CN109167538B CN201810976515.3A CN201810976515A CN109167538B CN 109167538 B CN109167538 B CN 109167538B CN 201810976515 A CN201810976515 A CN 201810976515A CN 109167538 B CN109167538 B CN 109167538B
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permanent magnet
phase
magnet brushless
phase winding
direct current
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CN109167538A (en
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李胜
刘辽雪
张磊
赵飞
刘源
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Nanjing University of Science and Technology
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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
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/10—Arrangements for controlling torque ripple, e.g. providing reduced torque ripple
    • 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
    • H02P6/34—Modelling or simulation for control purposes

Abstract

The invention discloses a control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure. The method comprises the following steps: establishing a mathematical model of the permanent magnet brushless direct current motor: establishing a mathematical model of a three-phase permanent magnet brushless direct current motor connected in a two-conduction star manner in a three-phase static coordinate system; establishing a biological intelligence-based double-layer controller: firstly, establishing a primary controller, namely an iterative learning controller; and then establishing a secondary controller, namely a robust sliding mode controller. The invention can rapidly keep the electromagnetic torque stable, effectively overcome the influence of external disturbance and parameter perturbation, ensure the control dynamic performance and robustness of the permanent magnet brushless direct current motor and improve the control precision of the permanent magnet brushless direct current motor.

Description

Control method for inhibiting torque ripple of permanent magnet brushless direct current motor based on double-layer structure
Technical Field
The invention relates to the technical field of motor control, in particular to a control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure.
Background
The permanent magnet brushless direct current motor is rapidly and widely applied along with the development of permanent magnet materials, power electronics and modern control technology in recent years, and is gradually and widely applied in important fields of robots, aerospace, new energy automobiles, national defense and the like. The DC motor has the characteristics of high operating efficiency, good speed regulation performance and the like. However, due to the structure of the motor body and the control strategy thereof, the problem of torque pulsation is caused to a certain extent, and the torque performance requirement is often not met when the motor runs at a low speed, so that the application of the motor in high-precision occasions is limited.
The torque ripple externally presented by the permanent magnet brushless direct current motor is mainly cogging torque ripple and commutation torque ripple. In the last decade or so, torque ripple suppression methods are roughly classified into two categories: firstly, through reasonable design of a motor body, a stator and rotor structure and a winding form are mainly included, so that the cogging torque is reduced as much as possible, and the counter electromotive force is close to an ideal waveform; secondly, the voltage or the current passing through the winding is adjusted through a control strategy to make up the deviation between the motor body and the inverter and the ideal characteristics, the method is more widely applied on the premise of not changing the structure of the motor body, but the influence of external disturbance and parameter perturbation exists, the control dynamic performance and robustness of the permanent magnet brushless direct current motor cannot be ensured, and the control precision of the permanent magnet brushless direct current motor is reduced.
Disclosure of Invention
The invention aims to provide a control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure, which has high control precision, good robustness and good dynamic performance.
The technical solution for realizing the purpose of the invention is as follows: a control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure comprises the following steps:
step 1: establishing a mathematical model of the permanent magnet brushless direct current motor: establishing a mathematical model of a three-phase permanent magnet brushless direct current motor connected in a two-conduction star manner in a three-phase static coordinate system;
step 2: establishing a biological intelligence-based double-layer controller: firstly, establishing a primary controller, namely an iterative learning controller; and then establishing a secondary controller, namely a robust sliding mode controller.
Further, the step 1 of establishing a mathematical model of the permanent magnet brushless dc motor specifically includes:
the back electromotive force of the permanent magnet brushless direct current motor is trapezoidal wave, and a mathematical model of the three-phase permanent magnet brushless direct current motor connected in a two-two conduction star manner is established in a three-phase static coordinate system;
first, the following settings are made:
(1) the three-phase windings are symmetrical, and the air gaps are uniform;
(2) eddy current and magnetic hysteresis loss are not counted, and a magnetic circuit is not saturated;
(3) the three-phase counter electromotive force is ideal trapezoidal wave and has equal amplitude;
(4) the inductance does not change with the change of the spatial position;
based on the setting, the voltage balance equation of the three-phase winding of the permanent magnet brushless direct current motor stator is as follows:
Figure BDA0001777567800000021
in the formula: v is a differential operator, and v is d/dt; u. ofa,ub,ucPhase voltages for the three-phase windings; i.e. ia,ib,icPhase current for a three-phase winding; e.g. of the typea,eb,ecIs the counter electromotive force of the three-phase winding; ra,Rb,RcResistance of three-phase winding; setting the three-phase winding to be symmetrical, Ra＝Rb＝Rc＝R；La，Lb，LcSelf-inductance of the three-phase windings respectively; mab、Mac、MbcMutual inductance between the A-phase winding and the B-phase winding, between the A-phase winding and the C-phase winding, and between the B-phase winding and the C-phase winding, Mba、Mca、McbMutual inductance between the B-phase winding and the A-phase winding, between the C-phase winding and the A-phase winding, and between the C-phase winding and the B-phase winding, respectively, and Mab＝Mba，Mac＝Mca，Mbc＝Mcb；
The back electromotive force adopts magnet steel surface-mounted rotor structure for trapezoidal wave's permanent magnet brushless DC motor, and the equivalent air gap length of motor is the constant, and consequently stator three-phase winding's self-inductance is the constant, and two liang of mutual inductances of three-phase winding are also the constant, and both all are irrelevant with the position, promptly: l isa＝La＝La＝Ls；Mab＝Mba＝Mac＝Mca＝Mba＝Mcb＝M；
The formula (1) transforms into:
Figure BDA0001777567800000022
the permanent magnet brushless direct current motor stator three-phase winding connection mode is star connection, and the relation between phase currents is as follows:
ia+ib+ic＝0 (3)
obtained by the formula (3):
Mib+Mic＝-Mia
Mia+Mic＝-Mib (4)
Mia+Mib＝-Mic
substituting the formulas (3) and (4) into the formula (2) to obtain:
Figure BDA0001777567800000031
wherein the effective inductance L ═ Ls-M；
Electromagnetic torque T generated by stator winding of permanent magnet brushless direct current motoreThe expression is as follows:
Te＝(eaia+ebib+ecic)/ω (6)
wherein ω is the motor angular velocity;
when the stator current is square wave and the counter electromotive force is trapezoidal wave, the passing time of the square wave current is 120 electrical angles in each half period, the flat top part of the trapezoidal wave counter electromotive force is also 120 electrical angles, and the square wave current and the trapezoidal wave counter electromotive force are synchronous; and at any moment, the stator current only passes through two phases, so the electromagnetic power P of the permanent magnet brushless direct current motor is as follows:
P＝(eaia+ebib+ecic)＝2EsIs (7)
wherein E issTo conduct the counter-electromotive force of one phase, IsTo conduct stator current of one phase;
according to formulae (6) and (7):
Te＝P/ω＝2EsIs/ω (8)
the relationship between the electromagnetic torque and the motor rotating speed is as follows:
Figure BDA0001777567800000032
Figure BDA0001777567800000033
wherein T iseIs the electromagnetic torque, in Nm; t isLLoad torque in Nm; b is a damping coefficient, and omega is the angular speed of the motor rotor; j is the moment of inertia of the motor, and theta is the angle rotated by the motor per unit time.
Further, the establishing of the primary controller, that is, the iterative learning controller in step 2 is specifically as follows:
the dynamic process of setting the controlled object is as follows:
Figure BDA0001777567800000041
x, y and u in the formula (11) are respectively a state, an output variable and an input variable of the system at the moment t; f and g are vector functions with the same dimension as the state and the output respectively, and the structure and the parameters are unknown;
setting ud(t) is the expected control quantity, the control targets of the iterative learning are as follows: given a desired output yd(t) and initial State x for each runk(0) At a given time T e 0, T]Can repeatedly run according to a set learning control algorithm, so that the control input quantity uk(t) can trend toward a desired control quantity, and the output quantity of the system can trend toward a desired output quantityyd(t)；
First order controller tracking error e at kth runk(t) is:
ek(t)＝yd(t)-yk(t) (12)
wherein, yk(t) is the actual output quantity at the time t during the kth operation;
an iterative learning rule with a forgetting factor is adopted, and the control rate is designed as follows:
uk+1(t)＝(1-α)uk(t)+Φek(t)+Γek+1(t) (13)
wherein k is 1, 2, 3 …;
u (t): the output quantity of the first-order controller is also the input quantity of the second-order controller;
α: a forgetting factor;
ek(t): speed error, ek(t) n _ ref-n _ actual, where n _ ref is the desired speed and n _ actual is the actual speed;
phi: open loop gain;
f: closed loop gain;
setting control variable u ═ IsThe state variable x ═ ω and y ═ TeThe equation of state from the combination of equations (8) and (10) is:
Figure BDA0001777567800000042
to ensure the convergence of the iterative learning of the system, the following conditions must be satisfied:
Figure BDA0001777567800000051
wherein rho is the radius of the motor rotor;
therefore, according to the above convergence adjustment, it is only necessary that the open-loop gain and the closed-loop gain are greater than zero, and
Figure BDA0001777567800000052
further, the establishing of the secondary controller, i.e. the robust sliding mode controller, in step 2 is specifically as follows:
the following settings are made:
(1) the initial value of the state is on the sliding mode surface; i.e., f (0) ═ Cx (0);
(2) f (t) → 0 when t → infinity, ensuring that the designed slip form surface can finally reach the expected slip form surface;
in order to meet the requirements of system control performance and the setting, the design time-varying sliding mode surface s is as follows:
s＝Cx-Cx(0)e-βt (17)
wherein x is a state quantity, C is a sliding mode surface constant, and beta is a time constant;
the preset system state equation of the sliding mode controller is designed as follows:
Figure BDA0001777567800000053
where the uncertainty terms Δ A and Δ B are represented by J, B and TLParameter induction; load torque TLChanges occur so f is equal to TLA system disturbance quantity as a controlled object;
selecting the Lyapunov function as V ═ s22, deriving the Lyapunov function
Figure BDA0001777567800000054
To make the system meet the Lyapunov stability criterion, i.e.
Figure BDA0001777567800000055
Therefore, the robust sliding mode controller is designed as follows:
U＝-g(t)(CB)-1sgn(s) (19)
wherein g (t) is a function continuously derivable first order over time t ∈ [0, ∞ ], sgn(s) is a sign function;
to reduce buffeting, the function sgn(s) is replaced by a saturation function sat(s),
Figure BDA0001777567800000056
wherein delta → 0+。
Compared with the prior art, the invention has the following remarkable advantages: (1) the first-level controller adopts iterative learning control, and can quickly track a given expected speed; (2) the secondary controller adopts robust sliding mode control, so that the dynamic performance and robustness of the system can be improved; (3) based on a biological intelligent double-layer structure controller, the torque pulsation of the permanent magnet brushless direct current motor can be effectively inhibited under the condition of ensuring the stability and robustness of the system.
Drawings
Fig. 1 is a block diagram of a control system of a permanent magnet brushless dc motor in the control method for suppressing torque ripple of the permanent magnet brushless dc motor based on a double-layer structure according to the present invention.
Fig. 2 is an equivalent circuit diagram of the permanent magnet brushless dc motor of the present invention.
Fig. 3 is a schematic diagram of back electromotive force and winding current of the permanent magnet brushless dc motor according to the present invention.
Fig. 4 is a schematic diagram of a two-layer controller according to the present invention.
Detailed Description
The following describes a control method for suppressing torque ripple of a permanent magnet brushless dc motor based on a two-layer structure in detail with reference to the accompanying drawings and embodiments.
The invention discloses a control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure, which is characterized by comprising the following steps of:
step 1: establishing a mathematical model of the permanent magnet brushless direct current motor: establishing a mathematical model of a three-phase permanent magnet brushless direct current motor connected in a two-conduction star manner in a three-phase static coordinate system;
step 2: establishing a biological intelligence-based double-layer controller: firstly, establishing a primary controller, namely an iterative learning controller; and then establishing a secondary controller, namely a robust sliding mode controller.
Further, as shown in fig. 1, which is a block diagram of a control system of a permanent magnet brushless dc motor according to the present invention, the step 1 of establishing a mathematical model of the permanent magnet brushless dc motor specifically includes the following steps:
the back electromotive force of the permanent magnet brushless direct current motor is trapezoidal wave, and a mathematical model of the three-phase permanent magnet brushless direct current motor connected in a two-two conduction star manner is established in a three-phase static coordinate system;
first, the following settings are made:
(1) the three-phase windings are symmetrical, and the air gaps are uniform;
(2) eddy current and magnetic hysteresis loss are not counted, and a magnetic circuit is not saturated;
(3) the three-phase counter electromotive force is ideal trapezoidal wave and has equal amplitude;
(4) the inductance does not change with the change of the spatial position.
Based on the setting, the voltage balance equation of the three-phase winding of the permanent magnet brushless direct current motor stator is as follows:
Figure BDA0001777567800000071
in the formula: v is a differential operator, and v is d/dt; u. ofa,ub,ucPhase voltages for the three-phase windings; i.e. ia,ib,icPhase current for a three-phase winding; e.g. of the typea,eb,ecIs the counter electromotive force of the three-phase winding; ra,Rb,RcResistance of three-phase winding; setting the three-phase winding to be symmetrical, Ra＝Rb＝Rc＝R；La，Lb，LcSelf-inductance of the three-phase windings respectively; mab、Mac、MbcMutual inductance between the A-phase winding and the B-phase winding, between the A-phase winding and the C-phase winding, and between the B-phase winding and the C-phase winding, Mba、Mca、McbMutual inductance between the B-phase winding and the A-phase winding, between the C-phase winding and the A-phase winding, and between the C-phase winding and the B-phase winding, respectively, and Mab＝Mba，Mac＝Mca，Mbc＝Mcb；
The permanent magnet brushless direct current motor with the counter electromotive force of trapezoidal waves adopts a magnetic steel surface-mounted rotor structure, and because the magnetic conductivity of a permanent magnet is close to air, the equivalent air gap length of the motor is constant, the self-inductance of a stator three-phase winding is constant, the two-two mutual inductance of the three-phase winding is also constant, and the two mutual inductances are independent of the position, namely: l isa＝La＝La＝Ls；Mab＝Mba＝Mac＝Mca＝Mba＝Mcb＝M；
Equation (1) can be transformed into:
Figure BDA0001777567800000072
the permanent magnet brushless direct current motor stator three-phase winding connection mode is star connection, and the relation between phase currents is as follows:
ia+ib+ic＝0 (3)
from formula (3):
Mib+Mic＝-Mia
Mia+Mic＝-Mib (4)
Mia+Mib＝-Mic
the formula (3) and the formula (4) are substituted into the formula (2) to obtain the final product:
Figure BDA0001777567800000073
wherein the effective inductance L ═ Ls-M；
According to the above equation (5), an equivalent circuit diagram of the permanent magnet brushless dc motor shown in fig. 2 can be obtained, wherein the electromagnetic torque of the permanent magnet brushless dc motor is generated by the electromagnetic induction action of the current passed through the three-phase stator winding and the magnetic field between the rotor magnetic steels;
the equivalent circuit of the permanent magnet brushless direct current motor is as follows:
Ux＝Rix+Lpix+ex (6)
wherein x is a, b, c, and:
Figure BDA0001777567800000081
electromagnetic torque T generated by stator winding of permanent magnet brushless direct current motoreThe expression is as follows:
Te＝(eaia+ebib+ecic)/ω (8)
wherein ω is the motor angular velocity;
when the stator current is square wave and the counter electromotive force is trapezoidal wave, the passing time of the square wave current is 120 electrical angles in each half period, the flat top part of the trapezoidal wave counter electromotive force is also 120 electrical angles, and the square wave current and the trapezoidal wave counter electromotive force are synchronous; and at any moment, the stator current only passes through two phases, so the electromagnetic power P of the permanent magnet brushless direct current motor is as follows:
P＝(eaia+ebib+ecic)＝2EsIs (9)
wherein E issTo conduct the counter-electromotive force of one phase, IsTo conduct stator current for one phase.
According to the formulae (8) and (9):
Te＝P/ω＝2EsIs/ω (10)
according to Newton's second law of motion, the relationship between the electromagnetic torque and the motor speed is as follows:
Figure BDA0001777567800000082
Figure BDA0001777567800000083
wherein T iseIs the electromagnetic torque, in Nm; t isLLoad torque in Nm; b is a damping coefficient, and omega is the angular speed of the motor rotor; j is the moment of inertia of the motor, and theta is the angle rotated by the motor per unit time.
As shown in fig. 3, a diagram of back electromotive force and winding current of the permanent magnet brushless dc motor in the present invention is shown, and the relationship between back electromotive force and electromagnetic torque of the mathematical model of the permanent magnet brushless dc motor, and electrical angle and angular velocity of the motor is:
Figure BDA0001777567800000091
Figure BDA0001777567800000092
thus:
Figure BDA0001777567800000093
wherein k iseIs a back electromotive force constant, ktIs the electromagnetic torque constant;
electrical angle thetaeEqual to the mechanical angle theta of the motormMultiplication by the number of pole pairs:
θe＝0.5pθm (16)
function F (θ)e) Is a function of the current position of the rotor of the motor, which represents the trapezoidal waveform of the back electromotive force;
the motor rotor position function for one cycle is:
Figure BDA0001777567800000094
according to the mathematical relation between the rotating speed and the torque of the motor, at the same frequency point, the rotating speed pulsation reflects the torque pulsation condition, so that the rotating speed control precision of the motor and the steady-state performance of a system can be improved through closed-loop control of the rotating speed of the motor, and the performance of inhibiting the torque pulsation is improved.
Further, as shown in fig. 4, which is a schematic diagram of the two-layer structure controller in the present invention, the establishment of the two-layer controller based on biological intelligence in step 2 specifically includes the following steps:
step 2.1, establishing a primary controller, namely an iterative learning controller;
and 2.2, establishing a secondary controller, namely a robust sliding mode controller.
Further, the establishing of the primary controller, i.e., the iterative learning controller, in step 2.1 is specifically as follows:
the dynamic process of setting the controlled object is as follows:
Figure BDA0001777567800000101
x, y and u in the formula (18) are respectively a state, an output variable and an input variable of the system at the moment t; f and g are vector functions with the same dimension as the state and the output respectively, and the structure and the parameters are unknown;
setting ud(t) is the expected control quantity, the control targets of the iterative learning are as follows: given a desired output yd(t) and initial State x for each runk(0) At a given time T e 0, T]Can repeatedly run according to a certain learning control algorithm, so that the control input quantity uk(t) can trend toward a desired control quantity, and the output quantity of the system can trend toward a desired output quantity yd(t)；
At the kth run, the first order controller tracking error is:
ek(t)＝yd(t)-yk(t) (19)
wherein, ykAnd (t) is the actual output quantity at the time t in the k-th operation.
An iterative learning rule with a forgetting factor is adopted, and the control rate is designed as follows:
uk+1(t)＝(1-α)uk(t)+Φek(t)+Γek+1(t) (20)
wherein k is 1, 2, 3 …;
u (t): the output quantity of the first-order controller is also the input quantity of the second-order controller;
α: a forgetting factor;
ek(t): speed error, ek(t) n _ ref-n _ actual, n _ ref being the desired speed, n _ actual being the actual speed;
phi: open loop gain;
f: a closed loop gain.
Setting control variable u ═ IsThe state variable x ═ ω and y ═ TeThe equation of state available from the combination of equations (8) and (10) is:
Figure BDA0001777567800000111
to ensure the convergence of the iterative learning of the system, the following conditions must be satisfied:
Figure BDA0001777567800000112
wherein rho is the radius of the motor rotor;
therefore, according to the above convergence adjustment, it is only necessary that the open-loop gain and the closed-loop gain are greater than zero, and
Figure BDA0001777567800000113
further, the establishment of the secondary controller, i.e. the robust sliding-mode controller, described in step 2.2 is as follows:
the following settings are made:
(1) the initial value of the state is on the sliding mode surface; i.e., f (0) ═ Cx (0);
(2) f (t) → 0 when t → infinity, ensuring that the designed slip form surface can finally reach the expected slip form surface;
in order to meet the requirements of system control performance and the setting, the design time-varying sliding mode surface s is as follows:
s＝Cx-Cx(0)e-βt (24)
wherein x is a state quantity, C is a sliding mode surface constant, and beta is a time constant. The selection of C directly influences the response time of the system. C too small system response is too slow, C too big system is easy to overshoot;
considering that the system has disturbance, the preset system state equation of the designed sliding mode controller is as follows:
Figure BDA0001777567800000114
where the uncertainty terms Δ A and Δ B are represented by J, B and TLEqual parameter causes; load torque TLChanges occur so f is equal to TLA system disturbance quantity regarded as a controlled object;
selecting the Lyapunov function as V ═ s2(ii)/2, deriving the Lyapunov function
Figure BDA0001777567800000121
To make the system meet the Lyapunov stability criterion, i.e.
Figure BDA0001777567800000122
Therefore, the sliding mode controller is designed as follows:
U＝-g(t)(CB)-1sgn(s) (26)
where g (t) is a function of first order continuous derivatives for time t ∈ [0, ∞ ], and sgn(s) is a sign function.
To reduce buffeting, the function sgn(s) is replaced by a saturation function sat(s),
Figure BDA0001777567800000123
wherein delta → 0+A suitable delta is sufficiently good to dampen sliding mode buffeting.
In summary, the present invention provides a bi-layer controller based on biological intelligence for a permanent magnet brushless dc motor. The first-level controller of the double-layer structure controller adopts iterative learning control, can quickly track a given expected speed and well reduce the input error of the system; the two-stage controller adopts robust sliding mode control, so that the stability and robustness of the system can be ensured, and the torque ripple inhibition performance of the system is improved. The control method for inhibiting the torque pulsation of the permanent magnet brushless direct current motor based on the double-layer structure can quickly keep the electromagnetic torque stable, effectively overcome the influence of external disturbance and parameter perturbation, ensure the control dynamic performance and robustness of the permanent magnet brushless direct current motor and improve the control precision of the permanent magnet brushless direct current motor.

Claims (2)

1. A control method for inhibiting torque pulsation of a permanent magnet brushless direct current motor based on a double-layer structure is characterized by comprising the following steps:
step 1: establishing a mathematical model of the permanent magnet brushless direct current motor: establishing a mathematical model of a three-phase permanent magnet brushless direct current motor connected in a two-conduction star manner in a three-phase static coordinate system;
step 2: establishing a biological intelligence-based double-layer controller: firstly, establishing a primary controller, namely an iterative learning controller; then establishing a secondary controller, namely a robust sliding mode controller;
step 1, establishing a mathematical model of the permanent magnet brushless direct current motor, specifically as follows:
the back electromotive force of the permanent magnet brushless direct current motor is trapezoidal wave, and a mathematical model of the three-phase permanent magnet brushless direct current motor connected in a two-two conduction star manner is established in a three-phase static coordinate system;
first, the following settings are made:
(1) the three-phase windings are symmetrical, and the air gaps are uniform;
(2) eddy current and magnetic hysteresis loss are not counted, and a magnetic circuit is not saturated;
(3) the three-phase counter electromotive force is ideal trapezoidal wave and has equal amplitude;
(4) the inductance does not change with the change of the spatial position;
based on the setting, the voltage balance equation of the three-phase winding of the permanent magnet brushless direct current motor stator is as follows:
Figure FDA0003308492340000011
in the formula: v is a differential operator, and v is d/dt; u. ofa,ub,ucPhase voltages for the three-phase windings; i.e. ia,ib,icPhase current for a three-phase winding; e.g. of the typea,eb,ecIs the counter electromotive force of the three-phase winding; ra,Rb,RcResistance of three-phase winding; setting the three-phase winding to be symmetrical, Ra＝Rb＝Rc＝R；La，Lb，LcSelf-inductance of the three-phase windings respectively; mab、Mac、MbcMutual inductance between the A-phase winding and the B-phase winding, between the A-phase winding and the C-phase winding, and between the B-phase winding and the C-phase winding, Mba、Mca、McbMutual inductance between the B-phase winding and the A-phase winding, between the C-phase winding and the A-phase winding, and between the C-phase winding and the B-phase winding, respectively, and Mab＝Mba，Mac＝Mca，Mbc＝Mcb；
The back electromotive force adopts magnet steel surface-mounted rotor structure for trapezoidal wave's permanent magnet brushless DC motor, and the equivalent air gap length of motor is the constant, and consequently stator three-phase winding's self-inductance is the constant, and two liang of mutual inductances of three-phase winding are also the constant, and both all are irrelevant with the position, promptly: l isa＝Lb＝Lc＝Ls；Mab＝Mba＝Mac＝Mca＝Mbc＝Mcb＝M；
The formula (1) transforms into:
Figure FDA0003308492340000021
the permanent magnet brushless direct current motor stator three-phase winding connection mode is star connection, and the relation between phase currents is as follows:
ia+ib+ic＝0 (3)
obtained by the formula (3):
Figure FDA0003308492340000022
substituting the formulas (3) and (4) into the formula (2) to obtain:
Figure FDA0003308492340000023
wherein the effective inductance L ═ Ls-M；
Electromagnetic torque T generated by stator winding of permanent magnet brushless direct current motoreThe expression is as follows:
Te＝(eaia+ebib+ecic)/ω (6)
when the stator current is square wave and the counter electromotive force is trapezoidal wave, the passing time of the square wave current is 120 electrical angles in each half period, the flat top part of the trapezoidal wave counter electromotive force is also 120 electrical angles, and the square wave current and the trapezoidal wave counter electromotive force are synchronous; and at any moment, the stator current only passes through two phases, so the electromagnetic power P of the permanent magnet brushless direct current motor is as follows:
P＝(eaia+ebib+ecic)＝2EsIs (7)
wherein E issTo conduct the counter-electromotive force of one phase, IsTo conduct stator current of one phase;
according to formulae (6) and (7):
Te＝P/ω＝2EsIs/ω (8)
the relationship between the electromagnetic torque and the motor rotating speed is as follows:
Figure FDA0003308492340000024
Figure FDA0003308492340000025
wherein T iseIs the electromagnetic torque, in Nm; t isLLoad torque in Nm; b is a damping coefficient, and omega is the angular speed of the motor rotor; j is the rotational inertia of the motor, and theta is the angle rotated by the motor in unit time;
establishing a primary controller, namely an iterative learning controller, in the step 2 specifically comprises the following steps:
the dynamic process of setting the controlled object is as follows:
Figure FDA0003308492340000031
x, y and u in the formula (11) are respectively a state, an output variable and an input variable of the system at the moment t; f and g are vector functions with the same dimension as the state and the output respectively, and the structure and the parameters are unknown;
setting ud(t) is the expected control quantity, the control targets of the iterative learning are as follows: given a desired output yd(t) and initial State x for each runk(0) At a given time T e 0, T]Can repeatedly run according to a set learning control algorithm, so that the control input quantity uk(t) can trend toward a desired control quantity, and the output quantity of the system can trend toward a desired output quantity yd(t)；
Tracking error e of primary controller in k-th operationk(t) is:
ek(t)＝yd(t)-yk(t) (12)
wherein, yk(t) is the actual output quantity at the time t during the kth operation;
an iterative learning rule with a forgetting factor is adopted, and the control rate is designed as follows:
uk+1(t)＝(1-α)uk(t)+Φek(t)+Γek+1(t) (13)
wherein k is 1, 2, 3 …;
u (t): the output quantity of the first-stage controller is also the input quantity of the second-stage controller;
α: a forgetting factor;
ek(t): speed error, ek(t) n _ ref-n _ actual, where n _ ref is the desired speed and n _ actual is the actual speed;
phi: open loop gain;
f: closed loop gain;
setting control variable u ═ IsThe state variable x ═ ω and y ═ TeThe equation of state from the combination of equations (8) and (10) is:
Figure FDA0003308492340000041
to ensure the convergence of the iterative learning of the system, the following conditions must be satisfied:
Figure FDA0003308492340000042
wherein rho is the radius of the motor rotor;
therefore, depending on the convergence condition, only the open-loop gain and the closed-loop gain need to be greater than zero, and
Figure FDA0003308492340000043
2. the control method for suppressing torque ripple of a permanent magnet brushless dc motor based on a two-layer structure according to claim 1, wherein the step 2 of establishing a two-stage controller, i.e. a robust sliding mode controller, specifically comprises the following steps:
the following settings are made:
(1) the initial value of the state is on the sliding mode surface; i.e., f (0) ═ Cx (0);
(2) f (t) → 0 when t → infinity, ensuring that the designed slip form surface can finally reach the expected slip form surface;
in order to meet the requirements of system control performance and the setting, the design time-varying sliding mode surface s is as follows:
s＝Cx-Cx(0)e-βt (17)
wherein x is a state quantity, C is a sliding mode surface constant, and beta is a time constant;
the preset system state equation of the sliding mode controller is designed as follows:
Figure FDA0003308492340000044
where the uncertainty terms Δ A and Δ B are represented by J, B and TLParameter induction; load torque TLChanges occur so f is equal to TLA system disturbance quantity as a controlled object;
selecting the Lyapunov function as V ═ s22, deriving the Lyapunov function
Figure FDA0003308492340000045
To make the system meet the Lyapunov stability criterion, i.e.
Figure FDA0003308492340000046
Therefore, the robust sliding mode controller is designed as follows:
U＝-g(t)(CB)-1sgn(s) (19)
wherein g (t) is a function continuously derivable first order over time t ∈ [0, ∞ ], sgn(s) is a sign function;
to reduce buffeting, the function sgn(s) is replaced by a saturation function sat(s),
Figure FDA0003308492340000051
wherein delta → 0+。
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