CN108233781B - Direct current motor self-adaptive inversion sliding mode control method based on disturbance observer - Google Patents

Direct current motor self-adaptive inversion sliding mode control method based on disturbance observer Download PDF

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CN108233781B
CN108233781B CN201810048153.1A CN201810048153A CN108233781B CN 108233781 B CN108233781 B CN 108233781B CN 201810048153 A CN201810048153 A CN 201810048153A CN 108233781 B CN108233781 B CN 108233781B
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sliding mode
interference
brushless motor
direct current
motor
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CN108233781A (en
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庄浩
张登峰
王聪
李军
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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
    • 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
    • H02P23/00—Arrangements or methods for the control of AC motors characterised by a control method other than vector control
    • H02P23/0004—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust 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
    • H02P23/00—Arrangements or methods for the control of AC motors characterised by a control method other than vector control
    • H02P23/0004—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H02P23/0009—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
    • H02P23/00—Arrangements or methods for the control of AC motors characterised by a control method other than vector control
    • H02P23/12—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
    • 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
    • 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 sliding mode control method for self-adaptive inversion of a direct current brushless motor based on a nonlinear disturbance observer, which adopts the nonlinear disturbance observer to observe and compensate unmodeled dynamic and external load disturbance of the direct current brushless motor. And for the motor system after interference compensation, a controller is designed by adopting a self-adaptive inversion sliding mode method, so that the stability of the whole direct-current brushless motor system is ensured. The invention solves the defects that inversion control needs accurate modeling information of a controlled object and can not overcome disturbance by using inversion sliding mode control, and improves the robustness of the system. A nonlinear disturbance observer is used for observing and compensating unmodeled dynamic and external load disturbance of the direct current brushless motor, buffeting level of inversion sliding mode control is reduced, and control precision is improved. And designing a self-adaptive law for the upper bound of the interference observation error, estimating the upper bound of the interference observation error, and taking the estimated value of the upper bound of the interference observation error as a sliding mode switching gain to ensure the stability of the whole direct current brushless motor system.

Description

Direct current motor self-adaptive inversion sliding mode control method based on disturbance observer
Technical Field
The invention belongs to the technical field of motor servo control, and particularly relates to a sliding mode control method for self-adaptive inversion of a brushless direct current motor based on a nonlinear disturbance observer.
Background
The brushless direct current motor has the advantages of a traditional direct current motor, such as good mechanical property and speed regulation property, large starting torque, strong overload capacity, convenient adjustment, good dynamic property and the like, and has a series of characteristics of simple structure, reliable operation, convenient maintenance and the like of an alternating current motor, so that the brushless direct current motor is widely applied to a plurality of high-tech fields, such as laser processing, robots, numerical control machines and the like. The current classical PID three-loop (position loop, speed loop and current loop) control method is the main control method of the dc brushless motor. However, brushless dc motors suffer from unmodeled dynamic and external load disturbances, which can degrade control performance and even destabilize the control system. Therefore, the conventional three-loop control based on the PID cannot meet the requirement of high-performance control, and a more advanced control method needs to be researched.
At present, control methods for a brushless direct current motor include control methods such as PID control, inversion control, fuzzy control, sliding mode control and the like. The direct current brushless motor is used as a multivariable and nonlinear control system, a motor model cannot be accurately measured, and the traditional PID control method cannot achieve an ideal control effect in a wide speed regulation range and when load disturbance frequently changes suddenly. The inversion design decomposes a complex nonlinear system into subsystems with the order not exceeding the system order, and then designs a Lyapunov function and an intermediate virtual control quantity of each subsystem, however, the traditional inversion control needs accurate modeling information of a motor and cannot overcome disturbance. The implementation of fuzzy control depends on the experience of the operator, and the application range is limited. The sliding mode variable structure control has the advantages of rapidity, strong robustness, simple realization and the like. In conventional sliding mode control, a large switching gain is often needed to eliminate external interference and uncertainty, and the large switching gain causes a serious buffeting problem and deteriorates a control effect.
In summary, the disadvantages of the existing control technology of the dc brushless motor mainly include the following points:
1. influence of the unmodeled dynamic state of the direct current brushless motor and interference of an external load on control of the direct current brushless motor is ignored.
2. The traditional inversion control method needs accurate modeling information of the direct current brushless motor and can not overcome disturbance.
3. The traditional sliding mode control usually needs a large switching gain to eliminate external interference and uncertainty, and the large switching gain causes a serious buffeting problem and deteriorates the control effect.
4. The PID control algorithm has poor robustness and cannot meet the control requirement of high-precision dynamic performance application occasions.
Disclosure of Invention
Meanwhile, in order to reduce the switching gain of the traditional inversion sliding mode control, reduce the buffeting level of the traditional inversion sliding mode control, improve the control precision of the direct-current brushless motor, estimate the unmodeled dynamic state and the external load interference of the motor by using a nonlinear interference observer, compensate the interference, and replace the upper bound β of the conservative switching gain by using smaller switching gain, namely the interference observation error
Figure GDA0002452480430000021
Aiming at the problem that the upper bound β of the interference observation error is difficult to determine, designing an adaptive law of the upper bound of the interference observation error, estimating the upper bound β of the interference observation error, and taking the estimated value of the upper bound of the interference observation error as the switching gain of the inversion sliding mode control of the direct current brushless motor, thereby ensuring the stability of the whole direct current brushless motor system and further reducing the buffeting level.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a sliding mode control method for self-adaptive inversion of a brushless direct current motor based on a nonlinear disturbance observer is characterized by comprising the following steps:
step1, establishing a mathematical model of the direct current brushless motor.
And 2, under the condition that the unmodeled dynamic state and the external load interference of the direct-current brushless motor are considered, designing the controller by adopting an inversion-based sliding mode control method.
And 3, configuring a nonlinear disturbance observer to observe unmodeled dynamic and external load disturbance of the direct current brushless motor and compensate.
And 4, designing a self-adaptive law of an interference observation error upper bound for the direct current brushless motor system introduced with the nonlinear interference observer, estimating the interference observation error upper bound, and taking an estimated value of the interference observation error upper bound as a switching gain of inversion sliding mode control to ensure the stability of the whole direct current brushless motor system.
Step1, establishing a mathematical model of the DC brushless motor
The mechanical equation of motion of the dc brushless motor is:
Figure GDA0002452480430000022
in the formula: t iseFor electromagnetic torque, Te＝ktI(t)，ktIs a torque coefficient, TLThe load torque is J, the rotational inertia of the motor is J, the damping coefficient of the motor is B, w (t) is the angular velocity of the motor, and I (t) is the bus current.
Voltage balance equation of dc brushless motor:
Figure GDA0002452480430000031
in the formula: r 'is armature winding resistance, R' is 2R, R is phase resistance, L 'is armature winding inductance, L' is 2(L-M), L is self-inductance of each phase winding, M is mutual inductance between each two phase windings, k iseIs the motor back electromotive force coefficient.
The dynamic system equation of the brushless direct current motor obtained by the formula (1) and the formula (2) is as follows:
Figure GDA0002452480430000032
selecting a state variable x1(t)＝θ(t)，x2(t)＝w(t)，
Figure GDA0002452480430000033
The equation of state of the dc brushless motor can be expressed as:
Figure GDA0002452480430000034
in the formula:
Figure GDA0002452480430000035
and 2, under the condition that the unmodeled dynamic state and the external load interference of the direct-current brushless motor are considered, designing the controller by adopting an inversion sliding mode control method.
1) Under the condition that the direct current brushless motor does not model dynamics and external load interference, the state equation of the motor is as follows:
Figure GDA0002452480430000036
let total interference be F (t) ═ Δ a1*x3(t)+Δa2*x2(t) + Δ b u + d (t), the equation of state is:
Figure GDA0002452480430000037
wherein
Figure GDA0002452480430000038
Figure GDA0002452480430000039
As an upper bound of total interference, Δ a1*x3(t)+Δa2*x2(t) + Δ b × u is unmodeled dynamics, d (t) is external load disturbance.
2) Design of inverse sliding mode controller
Error of the system is defined:
Figure GDA0002452480430000041
wherein: delta1And delta2For designed virtual control quantity, xdIs given a position signal.
Step1 first error subsystem: e.g. of the type1＝x1(t)-xd
Figure GDA0002452480430000042
Defining a first Lyapunov function:
Figure GDA0002452480430000043
Figure GDA0002452480430000044
designing a first virtual control quantity delta1：
Figure GDA0002452480430000045
From formulas (11) and (10):
Figure GDA0002452480430000046
if e2When the value is equal to 0, then
Figure GDA0002452480430000047
v1Asymptotically stable, so that the design needs to be carried out next step, and the virtual control quantity delta is introduced2Let e2Tending to zero.
Step2 second error subsystem: e.g. of the type2＝x2(t)-δ1
Figure GDA0002452480430000048
Defining a second Lyapunov function:
Figure GDA0002452480430000049
Figure GDA00024524804300000410
design a second virtual controlSystem quantity delta2：
Figure GDA00024524804300000411
From formulae (13) and (16):
Figure GDA0002452480430000051
from formulae (15) and (17):
Figure GDA0002452480430000052
if e3When the value is equal to 0, then
Figure GDA0002452480430000053
v2Asymptotically stable, so that the design of the next step is required to make e3Tending to zero.
Step3 third error subsystem: e.g. of the type3＝x3(t)-δ2
Figure GDA0002452480430000054
Defining a sliding mode switching surface:
s＝ce2+e3，(c＞0) (20)
then
Figure GDA0002452480430000055
A third Lyapunov function is defined:
Figure GDA0002452480430000056
Figure GDA0002452480430000057
the design control law is as follows:
Figure GDA0002452480430000058
and (3) stability analysis of an inversion sliding mode controller:
substitution of control law (23)
Figure GDA0002452480430000059
In expression (22):
Figure GDA00024524804300000510
the entire motor system is thus stable.
And 3, a nonlinear disturbance observer is configured to estimate and compensate the total disturbance F (t) of the direct current brushless motor.
According to the inverse sliding mode control law (23) in the step2, gain is switched
Figure GDA00024524804300000511
Depending on the upper bound of the total interference f (t), and if a conservative approach is used, a sufficiently large switching gain is selected
Figure GDA00024524804300000512
To ensure the stability of the system but to introduce severe jitter. Therefore, it is necessary to estimate the total interference f (t) and compensate the total interference f (t), so as to reduce the influence of the interference.
The non-linear disturbance observer is configured according to equation of state (6) of the dc brushless motor taking into account unmodeled dynamics of the motor and external load disturbances.
It is assumed that the change of the total disturbance with respect to the dynamics of the non-linear disturbance observer is slow, i.e.
Figure GDA0002452480430000061
Let the interference observation error be:
Figure GDA0002452480430000062
wherein
Figure GDA0002452480430000063
For the purpose of the observation error of the disturbance,
Figure GDA0002452480430000064
is an observed value of the disturbance.
Define the non-linear disturbance observer as:
Figure GDA0002452480430000065
in the formula: p (x) ═ L1x3(t)，L1＞0。
The dynamic equation for the interference observation error is:
Figure GDA0002452480430000066
because L is1If the error rate is more than 0, the interference observation error converges exponentially.
Let the compensation control law be
Figure GDA0002452480430000067
The output control law of the inversion sliding mode controller is u1The total control law is that u is u1+u2Then, after introducing the disturbance observer compensation, the state equation (6) of the electric machine becomes:
Figure GDA0002452480430000068
after the introduction of the non-linear disturbance observer,
Figure GDA0002452480430000069
the expression varies as:
Figure GDA0002452480430000071
control according to inverse sliding mode controllerLaw (23) for designing a new control law u for an electric machine system incorporating a non-linear disturbance observer1：
Figure GDA0002452480430000072
β therein is the interference observation error
Figure GDA0002452480430000073
The upper bound of (c).
After a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode controller is inverted:
substituting the control law (29) into
Figure GDA0002452480430000074
In expression (28)
Figure GDA0002452480430000075
Therefore, after a nonlinear disturbance observer is introduced, the whole motor system is stable.
And 4, designing a self-adaptive law of an interference observation error upper bound for the direct current brushless motor system introduced with the nonlinear interference observer, estimating the interference observation error upper bound, and taking an estimated value of the interference observation error upper bound as a switching gain of inversion sliding mode control to ensure the stability of the whole direct current brushless motor system.
Definition of
Figure GDA0002452480430000076
As an estimate of the interference observation error upper bound β:
estimation error
Figure GDA0002452480430000077
Comprises the following steps:
Figure GDA0002452480430000078
definition of
Figure GDA0002452480430000079
The parameter adaptation law of (1) is as follows:
Figure GDA00024524804300000710
the upper bound estimation error dynamic equation is
Figure GDA00024524804300000711
Order to
Figure GDA00024524804300000712
After a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode control system of the self-adaptive inversion of the direct current brushless motor is as follows:
defining the Lyapunov function:
Figure GDA0002452480430000081
Figure GDA0002452480430000082
therefore, the direct current brushless motor self-adaptive inversion sliding mode control system based on the nonlinear disturbance observer is stable.
To sum up: the final output control law u of the adaptive inversion sliding mode controller1Comprises the following steps:
Figure GDA0002452480430000083
the compensation control law is as follows:
Figure GDA0002452480430000084
therefore, the final total control law of the invention is u-u1+u2，
Figure GDA0002452480430000085
The invention relates to an observer based on nonlinear interferenceFor the direct current brushless motor, the inversion control method is combined with the sliding mode variable structure control method, so that the use range of the inversion control method can be enlarged, the direct current brushless motor has certain robustness to unmodeled dynamic and external load interference, meanwhile, a nonlinear interference observer is adopted to estimate total interference F (t), the total interference F (t) is compensated by the total interference estimation value, the influence of the interference is reduced, and the upper bound β of smaller switching gain, namely interference observation error replaces conservative switching gain
Figure GDA0002452480430000086
Finally, aiming at the interference observation error upper bound β which is difficult to determine, designing an adaptive law of the interference observation error upper bound β, and taking the estimation value of the interference observation error upper bound as the switching gain of the DC brushless motor inverse sliding mode control after interference compensation, the whole DC brushless motor system is ensured to be stable, and the switching gain, namely the buffeting level, is further reduced.
Compared with the prior art, the invention has the advantages that:
(1) the invention uses an inversion method to decompose the DC brushless motor into three subsystems, thereby simplifying the design of the controller.
(2) The inversion control method and the sliding mode control are combined, the defects that the traditional inversion control needs accurate modeling information and disturbance cannot be overcome are overcome, and the robustness of the system is improved.
(3) The invention adopts the nonlinear disturbance observer to estimate the total disturbance F (t), compensates the total disturbance F (t) by using the disturbance estimation value, reduces the influence of the disturbance, and replaces the conservative switching gain by the smaller switching gain, namely the upper bound β of the disturbance observation error
Figure GDA0002452480430000091
Therefore, the switching gain of the traditional inversion sliding mode control is reduced, the buffeting level of the traditional inversion sliding mode control is reduced, and the control precision is improved.
(4) Aiming at the problem that the upper bound β of the interference observation error is difficult to determine, the invention designs an adaptive law of the upper bound of the interference observation error, and takes the estimation value of the upper bound of the interference observation error as the switching gain of the inversion sliding mode control of the DC brushless motor after interference compensation, thereby ensuring the stability of the whole DC brushless motor system and further reducing the switching gain, namely the buffeting level.
(5) Therefore, the sliding mode control method for the self-adaptive inversion of the brushless direct current motor based on the nonlinear disturbance observer can be applied to the field of motors.
Is characterized in that: firstly, the position control system of the direct-current brushless motor is decomposed into three subsystems by using an inversion method, so that the design of a controller is simplified; then, an inversion control method and sliding mode control are combined, the defects that the traditional inversion control needs accurate modeling information of a motor and disturbance cannot be overcome are overcome, and the robustness of the system is improved; then, a nonlinear disturbance observer is adopted to estimate the total disturbance of the DC brushless motor position control system, the disturbance estimation value is used to compensate the total disturbance, the influence of the disturbance is reduced, and the conservative switching gain is replaced by smaller switching gain, namely the upper bound of disturbance observation error, so that the switching gain of the traditional sliding mode inversion control is reduced, the buffeting level of the traditional sliding mode inversion control is reduced, and the control precision is improved; and finally, designing a self-adaptive law of the upper bound of the interference observation error aiming at the difficulty in determining the upper bound of the interference observation error, and taking an estimated value of the upper bound of the interference observation error as a switching gain of the inverse sliding mode control of the DC brushless motor after interference compensation to ensure the stability of the whole DC brushless motor system and further reduce the switching gain, namely the buffeting level.
Drawings
Fig. 1 is a block diagram of a dc brushless motor control system.
Fig. 2 is a tracking error curve diagram based on non-linear disturbance observer dc brushless motor adaptive inversion sliding mode control.
Fig. 3 is a graph of tracking error of a conventional dc brushless motor controlled by an inverse sliding mode.
Fig. 4 is a tracking error graph based on a nonlinear disturbance observer dc brushless motor inverse sliding mode control.
Fig. 5 is a control voltage graph based on non-linear disturbance observer dc brushless motor adaptive inversion sliding mode control.
Fig. 6 is a control voltage graph of a conventional dc brushless motor with inverse sliding mode control.
Fig. 7 is a control voltage graph based on a nonlinear disturbance observer dc brushless motor inverse sliding mode control.
Detailed Description
The present invention will be further described with reference to the accompanying drawings.
A sliding mode control system of adaptive inversion of a brushless DC motor based on a nonlinear disturbance observer is disclosed, as shown in FIG. 1, the nonlinear disturbance observer estimates the total disturbance of the brushless DC motor according to the state of the motor and the control voltage and compensates the total disturbance. And finally, combining a self-adaptive inversion sliding mode controller to obtain the control voltage of the motor and control the direct current brushless motor to track the position given signal xd。
The technical scheme specifically realizes the following steps:
step1, establishing a mathematical model of the DC brushless motor
The mechanical equation of motion of the dc brushless motor is:
Figure GDA0002452480430000101
in the formula: t iseFor electromagnetic torque, Te＝ktI(t)，ktIs a torque coefficient, TLThe load torque is J, the rotational inertia of the motor is J, the damping coefficient of the motor is B, w (t) is the angular velocity of the motor, and I (t) is the bus current.
Voltage balance equation of dc brushless motor:
Figure GDA0002452480430000102
in the formula: r 'is armature winding resistance, R' is 2R, R is phase resistance, L 'is armature winding inductance, L' is 2(L-M), L is self-inductance of each phase winding, M is mutual inductance between each two phase windings, k iseIs the motor back electromotive force coefficient.
The dynamic system equation of the brushless direct current motor obtained by the formula (1) and the formula (2) is as follows:
Figure GDA0002452480430000103
selecting a state variable x1(t)＝θ(t)，x2(t)＝w(t)，
Figure GDA0002452480430000111
The equation of state of the dc brushless motor can be expressed as:
Figure GDA0002452480430000112
in the formula:
Figure GDA0002452480430000113
and 2, under the condition that the unmodeled dynamic state and the external load interference of the direct-current brushless motor are considered, designing the controller by adopting an inversion sliding mode control method.
1) Under the condition that the direct current brushless motor does not model dynamics and external load interference, the state equation of the motor is as follows:
Figure GDA0002452480430000114
let total interference be F (t) ═ Δ a1*x3(t)+Δa2*x2(t) + Δ b u + d (t), the equation of state is:
Figure GDA0002452480430000115
wherein
Figure GDA0002452480430000116
Figure GDA0002452480430000117
As an upper bound of total interference, Δ a1*x3(t)+Δa2*x2(t) + Δ b × u is unmodeled dynamics, d (t) is external load disturbance.
2) Design of inverse sliding mode controller
Error of the system is defined:
Figure GDA0002452480430000118
wherein: delta1And delta2For designed virtual control quantity, xdIs given a position signal.
Step1 first error subsystem: e.g. of the type1＝x1(t)-xd
Figure GDA0002452480430000119
Defining a first Lyapunov function:
Figure GDA0002452480430000121
Figure GDA0002452480430000122
designing a first virtual control quantity delta1：
Figure GDA0002452480430000123
From formulas (11) and (10):
Figure GDA0002452480430000124
if e2When the value is equal to 0, then
Figure GDA0002452480430000125
v1Asymptotically stable, so that the design needs to be carried out next step, and the virtual control quantity delta is introduced2Let e2Tending to zero.
Step2 second error subsystem: e.g. of the type2＝x2(t)-δ1
Figure GDA0002452480430000126
Defining a second Lyapunov function:
Figure GDA0002452480430000127
Figure GDA0002452480430000128
designing a second virtual control quantity delta2：
Figure GDA0002452480430000129
From formulae (13) and (16):
Figure GDA00024524804300001210
from formulae (15) and (17):
Figure GDA00024524804300001211
if e3When the value is equal to 0, then
Figure GDA00024524804300001212
v2Asymptotically stable, so that the design of the next step is required to make e3Tending to zero.
Step3 third error subsystem: e.g. of the type3＝x3(t)-δ2
Figure GDA00024524804300001213
Defining a sliding mode switching surface:
s＝ce2+e3，(c＞0)(20)
then
Figure GDA0002452480430000131
A third Lyapunov function is defined:
Figure GDA0002452480430000132
Figure GDA0002452480430000133
the design control law is as follows:
Figure GDA0002452480430000134
and (3) stability analysis of an inversion sliding mode controller:
substitution of control law (23)
Figure GDA0002452480430000135
In expression (22):
Figure GDA0002452480430000136
the entire motor system is thus stable.
And 3, a nonlinear disturbance observer is configured to estimate and compensate the total disturbance F (t) of the direct current brushless motor.
According to the inverse sliding mode control law (23) in the step2, gain is switched
Figure GDA0002452480430000137
Depending on the upper bound of the total interference f (t), and if a conservative approach is used, a sufficiently large switching gain is selected
Figure GDA0002452480430000138
To ensure the stability of the system but to introduce severe jitter. Therefore, it is necessary to estimate the total interference f (t) and compensate the total interference f (t), so as to reduce the influence of the interference.
The non-linear disturbance observer is configured according to equation of state (6) of the dc brushless motor taking into account unmodeled dynamics of the motor and external load disturbances.
It is assumed that the change of the total disturbance with respect to the dynamics of the non-linear disturbance observer is slow, i.e.
Figure GDA0002452480430000139
Let the interference observation error be:
Figure GDA00024524804300001310
wherein
Figure GDA0002452480430000141
For the purpose of the observation error of the disturbance,
Figure GDA0002452480430000142
is an observed value of the disturbance.
Define the non-linear disturbance observer as:
Figure GDA0002452480430000143
in the formula: p (x) ═ L1x3(t)，L1＞0。
The dynamic equation for the interference observation error is:
Figure GDA0002452480430000144
because L is1If the error rate is more than 0, the interference observation error converges exponentially.
Let the compensation control law be
Figure GDA0002452480430000145
The output control law of the inversion sliding mode controller is u1The total control law is that u is u1+u2Then, after introducing the disturbance observer compensation, the state equation (6) of the electric machine becomes:
Figure GDA0002452480430000146
after the introduction of the non-linear disturbance observer,
Figure GDA0002452480430000147
the expression varies as:
Figure GDA0002452480430000148
designing a new control law u for a motor system introducing a nonlinear disturbance observer according to a control law (23) of an inverse sliding mode controller1：
Figure GDA0002452480430000149
β therein is the interference observation error
Figure GDA00024524804300001410
The upper bound of (c).
After a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode controller is inverted:
substituting the control law (29) into
Figure GDA00024524804300001411
In expression (28):
Figure GDA0002452480430000151
therefore, after a nonlinear disturbance observer is introduced, the whole motor system is stable.
And 4, designing a self-adaptive law of an interference observation error upper bound for the direct current brushless motor system introduced with the nonlinear interference observer, estimating the interference observation error upper bound, and taking an estimated value of the interference observation error upper bound as a switching gain of inversion sliding mode control to ensure the stability of the whole direct current brushless motor system.
Definition of
Figure GDA0002452480430000152
As an estimate of the interference observation error upper bound β:
estimation error
Figure GDA0002452480430000153
Comprises the following steps:
Figure GDA0002452480430000154
definition of
Figure GDA0002452480430000155
The parameter adaptation law of (1) is as follows:
Figure GDA0002452480430000156
the upper bound estimation error dynamic equation is
Figure GDA0002452480430000157
Order to
Figure GDA0002452480430000158
After a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode control system of the self-adaptive inversion of the direct current brushless motor is as follows:
defining the Lyapunov function:
Figure GDA0002452480430000159
Figure GDA00024524804300001510
therefore, the direct current brushless motor self-adaptive inversion sliding mode control system based on the nonlinear disturbance observer is stable.
To sum up: the final output control law u of the adaptive inversion sliding mode controller1Comprises the following steps:
Figure GDA0002452480430000161
the compensation control law is as follows:
Figure GDA0002452480430000162
therefore, the final total control law of the invention is u-u1+u2，
Figure GDA0002452480430000163
Simulation verification is carried out on the direct current brushless motor self-adaptive inversion sliding mode control method based on the nonlinear disturbance observer.
The parameters of the direct current brushless motor are as follows: rated voltage 24v, rated current 11.4A, moment of inertia J1.98 e-4kg/m2The damping coefficient B of the motor is 0N.m.s, the self inductance L of each phase winding is 35uH, the mutual inductance M of each phase winding is 0, and the torque coefficient k ist0.058Nm/A, coefficient of counter electromotive force ke6.07v/krpm, winding resistance r of each phase is 0.31 omega, pole pair number p is 4, and load moment TL0.1N/m, load moment disturbance is DeltaTL0.05sin (t) N/m, the position command that the system expects to track is curve xd＝sin(3t)rad。
Parameters of the adaptive inversion sliding mode controller are set as follows: k is a radical of1＝300，k2300, c 300, h 300; the parameters of the non-linear disturbance observer are set as: l is1＝50；
The method aims at the traditional DC brushless motor inversion sliding mode control method to carry out simulation and set the total interference upper bound
Figure GDA0002452480430000164
The method aims at simulating a DC brushless motor inversion sliding mode control method based on a nonlinear disturbance observer and setting an upper bound of disturbance observation errors
Figure GDA0002452480430000165
FIG. 3 shows the position tracking error of the conventional inversion sliding mode control, and the maximum position tracking error in the steady state is 5.2 × e- 3rad。
FIG. 4 shows the position tracking error of inverse sliding mode control based on the nonlinear disturbance observer, and the maximum tracking error in the steady state is 1.2 × e-5And (7) rad. Compared with the graph 3, after the compensation effect of the nonlinear disturbance observer on the disturbance is introduced, the position tracking error is effectively reduced, and the control precision is improved.
FIG. 2 is a graph of the position tracking error of the present invention, with a maximum position tracking error of 2.8 × e at steady state-6rad, compared with the figure 4, the method disclosed by the invention estimates the interference observation error upper bound by using a self-adaptive method on the basis of the inverse sliding mode control method based on the nonlinear interference observer, and the estimated value of the interference observation error upper bound is used as the switching gain of the motor inverse sliding mode control, so that the tracking error is further reduced, and the control precision is further improved.
Fig. 6 is a control voltage curve diagram of the traditional inversion sliding mode control, the control voltage has a large buffeting level, the buffeting amplitude reaches 2.4v, even positive and negative jumps of the control voltage occur, the control voltage jumped greatly is easy to damage the direct current brushless motor, and the direct current brushless motor cannot be used practically.
Fig. 7 is a control voltage based on inverse sliding mode control of a nonlinear disturbance observer, the maximum buffeting amplitude of the control voltage is 0.1v, and compared with fig. 6, after the compensation effect of the nonlinear disturbance observer on disturbance is introduced, the buffeting amplitude of the control voltage is effectively reduced.
Fig. 5 is a control voltage curve diagram of the present invention, the maximum buffeting amplitude of the control voltage is 0.0045v, and compared with fig. 7, the present invention estimates the interference observation error upper bound by using a self-adaptive method on the basis of the inverse sliding mode control method based on the nonlinear interference observer, and the estimated value of the interference observation error upper bound is used as the switching gain of the motor inverse sliding mode control, so as to further reduce the buffeting level of the control voltage.
In an actual dc brushless motor control system, the average value U of the voltage across the armature winding of the motor is α U according to the PWM control principles，Usα is the rated voltage 24v of the power supply voltage, namely the motor, and is the duty ratio of PWM, 0 < α < 1, the maximum buffeting amplitude of the control voltage is 0.0045v, and the buffeting amplitude delta α of the corresponding PWM duty ratio is 0.01875%, so that the buffeting amplitude of the small PWM duty ratio can be ignored.
Meanwhile, a nonlinear disturbance observer is adopted to estimate the total disturbance F (t), the total disturbance estimation value is used to compensate the total disturbance F (t), the influence of the disturbance is reduced, and the upper bound β of smaller switching gain, namely disturbance observation error, replaces the conservative switching gain
Figure GDA0002452480430000171
Thereby effectively reducing the control voltage buffeting level of the traditional inversion sliding mode control and effectively reducing the position tracking error. And finally, designing an adaptive law of an upper bound of the interference observation error aiming at the interference observation error, and taking an estimated value of the upper bound of the interference observation error as a switching gain of motor inversion sliding mode control, so that the stability of the whole direct current brushless motor system is ensured, the buffeting level of control voltage is further reduced, and the position tracking error is further reduced.

Claims (3)

1. A sliding mode control method for self-adaptive inversion of a brushless direct current motor based on a nonlinear disturbance observer is characterized by comprising the following steps:
step1, establishing a mathematical model of the direct current brushless motor, which specifically comprises the following steps:
establishing a mathematical model of the direct current brushless motor,
the mechanical equation of motion of the dc brushless motor is:
Figure FDA0002437977750000011
in the formula: t iseFor electromagnetic torque, Te＝ktI(t)，ktIs a torque coefficient, TLIs load torque, J is motor moment of inertia, B is motor damping coefficient, w (t) is motor angular velocity, I (t) is bus current,
voltage balance equation of dc brushless motor:
Figure FDA0002437977750000012
in the formula: r 'is armature winding resistance, R' is 2R, R is phase resistance, L 'is armature winding inductance, L' is 2(L-M), L is self-inductance of each phase winding, M is mutual inductance between each two phase windings, k iseIs the counter-electromotive force coefficient of the motor,
the dynamic system equation of the brushless direct current motor obtained by the formula (1) and the formula (2) is as follows:
Figure FDA0002437977750000013
selecting a state variable x1＝θ(t)，x2＝w(t)，
Figure FDA0002437977750000014
The equation of state of the dc brushless motor can be expressed as:
Figure FDA0002437977750000015
in the formula:
Figure FDA0002437977750000016
step2, under the condition of considering unmodeled dynamics and external load interference of the direct current brushless motor, designing the controller by adopting an inversion sliding mode control method, firstly, decomposing a direct current brushless motor position control system into three subsystems by adopting an inversion method, and simplifying the design of the controller; then, an inversion control method is combined with sliding mode control, and the defects that the traditional inversion control needs accurate modeling information of a motor and can not overcome disturbance are overcome by using the sliding mode control, and the method specifically comprises the following steps:
under the condition of considering the unmodeled dynamic state and the external load interference of the direct current brushless motor, the controller is designed by adopting an inversion sliding mode control method,
1) under the condition that the direct current brushless motor does not model dynamics and external load interference, the state equation of the motor is as follows:
Figure FDA0002437977750000021
let total interference be F (t) ═ Δ a1*x3(t)+Δa2*x2(t) + Δ b u + d (t), the equation of state is:
Figure FDA0002437977750000022
wherein
Figure FDA0002437977750000023
Figure FDA0002437977750000024
At the upper bound of the total interference, Δ a1*x3(t)+Δa2*x2(t) + Δ b × u is unmodeled dynamics, d (t) is external load disturbance,
2) design of inverse sliding mode controller
Error of the system is defined:
Figure FDA0002437977750000025
wherein: delta1And delta2For designed virtual control quantity, xdFor a given position signal, it is possible to,
step1 first error subsystem: e.g. of the type1＝x1(t)-xd
Figure FDA0002437977750000026
Defining a first Lyapunov function:
Figure FDA0002437977750000027
Figure FDA0002437977750000028
designing a first virtual control quantity delta1：
Figure FDA0002437977750000029
From formulas (11) and (10):
Figure FDA0002437977750000031
if e2When the value is equal to 0, then
Figure FDA0002437977750000032
v1Asymptotically stable, so that the design needs to be carried out next step, and the virtual control quantity delta is introduced2Let e2And the flow rate of the water tends to zero,
step2 second error subsystem: e.g. of the type2＝x2(t)-δ1
Figure FDA0002437977750000033
Defining a second Lyapunov function:
Figure FDA0002437977750000034
Figure FDA0002437977750000035
designing a second virtual control quantity delta2：
Figure FDA0002437977750000036
From formulae (13) and (16):
Figure FDA0002437977750000037
from formulae (15) and (17):
Figure FDA0002437977750000038
if e3When the value is equal to 0, then
Figure FDA0002437977750000039
v2Asymptotically stable, so that the design of the next step is required to make e3And the flow rate of the water tends to zero,
step3 third error subsystem: e.g. of the type3＝x3(t)-δ2
Figure FDA00024379777500000310
Defining a sliding mode switching surface:
s＝ce2+e3，(c＞0) (20)
then
Figure FDA00024379777500000311
A third Lyapunov function is defined:
Figure FDA00024379777500000312
Figure FDA00024379777500000313
the design control law is as follows:
Figure FDA0002437977750000041
and (3) stability analysis of an inversion sliding mode controller:
substitution of control law (23)
Figure FDA0002437977750000042
In expression (22):
Figure FDA0002437977750000043
therefore, the whole motor system is asymptotically converged and stable;
step3, a nonlinear disturbance observer is configured to observe unmodeled dynamic and external load disturbance of the direct current brushless motor, compensate the disturbance, reduce the influence of the disturbance, replace conservative switching gain by smaller switching gain, namely the upper bound of disturbance observation error, reduce the switching gain of the traditional inversion sliding mode control,
and 4, designing a self-adaptive law of an interference observation error upper bound for the direct current brushless motor system introduced with the nonlinear interference observer, estimating the interference observation error upper bound, and taking an estimated value of the interference observation error upper bound as a switching gain of inversion sliding mode control to ensure the stability of the whole direct current brushless motor system.
2. The sliding-mode control method for the adaptive inversion of the brushless direct-current motor based on the nonlinear disturbance observer according to claim 1, wherein the step3 specifically comprises: a nonlinear disturbance observer is configured to estimate and compensate the total disturbance F (t) of the DC brushless motor,
according to the inverse sliding mode control law (23) in the step2, gain is switched
Figure FDA0002437977750000044
Depending on the upper bound of the total interference f (t), and if a conservative approach is used, a sufficiently large switching gain is selected
Figure FDA0002437977750000045
To ensure the stability of the system, severe buffeting is caused, so the total interference f (t) needs to be estimated and compensated, the influence of interference is reduced,
configuring a non-linear disturbance observer according to equation of state (6) of the brushless DC motor under consideration of unmodeled dynamics of the motor and external load disturbances,
it is assumed that the change of the total disturbance with respect to the dynamics of the non-linear disturbance observer is slow, i.e.
Figure FDA0002437977750000046
Let the interference observe the error:
Figure FDA0002437977750000047
wherein
Figure FDA0002437977750000051
In order to interfere with the error of the observation,
Figure FDA0002437977750000052
as an observed value of the interference,
define the non-linear disturbance observer as:
Figure FDA0002437977750000053
in the formula: p (x) ═ L1x3(t)，L1＞0，
The dynamic equation for the interference observation error is therefore:
Figure FDA0002437977750000054
because L is1If the interference is more than 0, the interference observation error converges exponentially,
let the compensation control law be
Figure FDA0002437977750000055
The output control law of the inversion sliding mode controller is u1The total control law is that u is u1+u2Then, after introducing the disturbance observer compensation, the state equation (6) of the electric machine becomes:
Figure FDA0002437977750000056
after the introduction of the non-linear disturbance observer,
Figure FDA0002437977750000057
the expression varies as:
Figure FDA0002437977750000058
designing a new control law u for a motor system introducing a nonlinear disturbance observer according to a control law (23) of an inverse sliding mode controller1：
Figure FDA0002437977750000059
β therein is the interference observation error
Figure FDA00024379777500000510
The upper bound of (a) is,
after a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode controller is inverted:
substituting the control law (29) into
Figure FDA00024379777500000511
In expression (28):
Figure FDA0002437977750000061
therefore, after a nonlinear disturbance observer is introduced, the whole motor system is asymptotically stable.
3. The method for controlling the sliding-mode adaptive inversion of the brushless direct-current motor based on the nonlinear disturbance observer according to claim 2, wherein the step 4 specifically comprises: for a direct current brushless motor system introduced with a nonlinear disturbance observer, designing a self-adaptive law of an upper bound of disturbance observation errors, estimating the upper bound of the disturbance observation errors, taking the estimated value of the upper bound of the disturbance observation errors as a switching gain of inverse sliding mode control, ensuring the stability of the whole direct current brushless motor system,
definition of
Figure FDA0002437977750000062
To estimate the interference observation error upper bound β
Figure FDA0002437977750000063
Comprises the following steps:
Figure FDA0002437977750000064
definition of
Figure FDA0002437977750000065
The parameter adaptation law of (1) is as follows:
Figure FDA0002437977750000066
the upper bound estimation error dynamic equation is:
Figure FDA0002437977750000067
order to
Figure FDA0002437977750000068
After a nonlinear disturbance observer is introduced, the stability analysis of the sliding mode control system of the self-adaptive inversion of the direct current brushless motor is as follows:
defining the Lyapunov function:
Figure FDA0002437977750000069
Figure FDA00024379777500000610
therefore, the DC brushless motor self-adaptive inversion sliding mode control system based on the nonlinear disturbance observer is stable,
to sum up: obtaining the final output control law u of the adaptive inversion sliding mode controller1Comprises the following steps:
Figure FDA0002437977750000071
the compensation control law is as follows:
Figure FDA0002437977750000072
the final overall control law is therefore u-u1+u2，
Figure FDA0002437977750000073
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