CN113659897A - Sliding mode control method of permanent magnet linear synchronous motor - Google Patents

Sliding mode control method of permanent magnet linear synchronous motor Download PDF

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CN113659897A
CN113659897A CN202110918272.XA CN202110918272A CN113659897A CN 113659897 A CN113659897 A CN 113659897A CN 202110918272 A CN202110918272 A CN 202110918272A CN 113659897 A CN113659897 A CN 113659897A
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sliding mode
permanent magnet
formula
synchronous motor
error
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CN113659897B (en
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王秀平
王楠
曲春雨
姜胜龙
葛子荻
刘万明
姚舜禹
郭子琦
王雅慧
胡永壮
王长华
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Shenyang Institute of Engineering
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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
    • H02P21/0003—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
    • 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
    • H02P25/06—Linear motors
    • H02P25/064—Linear motors of the synchronous type
    • 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
    • H02P6/006—Controlling linear motors
    • 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

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Abstract

The invention discloses a sliding mode control method of a permanent magnet linear synchronous motor, which ensures that a system is converged in limited time by constructing an integral sliding mode surface, ensures that the system has good control precision and faster convergence speed by designing a second-order nonsingular terminal sliding mode controller, and simultaneously weakens the buffeting problem of the system; and finally, by a self-adaptive control method, a sliding mode tracking controller with parameter shaping is designed, the upper bound of uncertainty factors is estimated, and the system is ensured to have good convergence speed, robustness and anti-interference capability.

Description

Sliding mode control method of permanent magnet linear synchronous motor
Technical Field
The invention relates to the field of motor control, in particular to a sliding mode control method of a permanent magnet linear synchronous motor.
Background
In recent years, the permanent magnet linear motor has been widely applied in the fields of industrial automation, transportation and the like due to the advantages of simple structure, high thrust density, high feeding speed, easiness in maintenance, high positioning accuracy and the like. However, when the PMLSM moves linearly, it is easily affected by non-linear factors such as changes in system parameters, end effects, and external load disturbances. Therefore, an effective control method must be designed to improve the control accuracy and the anti-interference capability of the system.
Among the numerous nonlinear control methods, sliding mode variable structure control is of particular interest, and its robustness to uncertainty and parameter variations, fast dynamic response, and ease of implementation are attractive features. However, the phenomenon of buffeting, i.e. high frequency amplitude limited oscillations, is a major drawback of this method, which leads to degradation of tracking performance and robustness. Replacing the sign function with a softer approximation function such as a hyperbolic tangent and saturation function may mitigate the buffeting effect of some systems, but this may reduce the robustness of the system. Using smooth control is another method for counteracting buffeting, however, it is not feasible for all control inputs. Another approach is online disturbance estimation, but its accuracy depends on the sampling step size. High-order sliding mode control can obviously reduce buffeting effect, and one special high-order sliding mode control is second-order sliding mode control. The method is based on the application of a sign function on the time derivative of the control rate, and obtains the control input by utilizing integration, thereby reducing the problem of buffeting. Compared with the traditional sliding mode control, the second-order sliding mode control has the advantages of limited time stability, expansion of the degree of relative sliding variables in the sliding mode control, reduction of buffeting, improvement of precision of a closed-loop system and the like.
In the conventional sliding mode control, a linear sliding plane is usually selected, so that the tracking error gradually converges to zero after the system reaches the sliding mode. The speed of progressive convergence can be achieved by adjusting the sliding mode face parameters, but in any case the tracking error does not converge to zero in a finite time.
Therefore, it is necessary to develop an adaptive integral second-order nonsingular terminal sliding mode control method for a permanent magnet linear synchronous motor.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a sliding mode control method of a permanent magnet linear synchronous motor.
In order to achieve the purpose, the invention is implemented according to the following technical scheme:
a sliding mode control method of a permanent magnet linear synchronous motor comprises the following steps:
s1, converting three-phase winding current i of the permanent magnet linear synchronous motora、ib、icObtaining a current signal i under a two-phase static dq coordinate system through Clark coordinate transformationd、iqCombining the displacement and the speed of the motor to obtain a motion equation of the permanent magnet linear synchronous motor under the dq coordinate;
s2, constructing an integral sliding mode surface, and designing an integral second-order nonsingular rapid terminal sliding mode controller based on the integral sliding mode surface;
s3, designing an adaptive sliding mode tracker to control the permanent magnet linear synchronous motor on the basis of the integral second-order nonsingular fast terminal sliding mode controller in the step S2.
Further, the equation of motion of the permanent magnet linear synchronous motor in dq coordinate at step S1 is as follows:
Figure BDA0003206437110000021
wherein ,ud、uqD-axis voltage and q-axis voltage, respectively; rsResistance for each phase winding; i.e. id、iqD-axis current and q-axis current, respectively; l isd、LqD-axis inductance and q-axis inductance respectively; v is the mover speed; tau is a polar distance; psifIs an excitation flux linkage; feIs electromagnetic thrust; n ispIs the number of pole pairs; kfIs the thrust coefficient; the state equation of the permanent magnet linear synchronous motor obtained according to the motion equation of the permanent magnet linear synchronous motor is as follows:
Figure BDA0003206437110000031
wherein ,x1Is displacement; x is the number of2Is the speed;
Figure BDA0003206437110000032
is the acceleration; u is the control rate of the controller, and u is iq(ii) a ξ (t) is a function of the time of the unknown uncertain disturbance.
Further, the specific steps of constructing the integral sliding mode surface in step S2 are as follows:
the position tracking error is defined as:
Figure BDA0003206437110000033
wherein ,
Figure BDA0003206437110000034
for the reference displacement, the lyapunov function is constructed:
Figure BDA0003206437110000035
to V1The derivation can be:
Figure BDA0003206437110000036
the reference speed is set as:
Figure BDA0003206437110000037
substituting formula (6) into formula (5):
Figure BDA0003206437110000038
the speed error is defined as:
Figure BDA0003206437110000039
in order to ensure that the controller has good tracking accuracy and convergence speed, the integral sliding mode surface is defined as follows:
Figure BDA00032064371100000310
wherein ,k2，k3，k4Both are positive integers, q and p are positive odd numbers, and q < p.
Further, the step S2 of designing the integral second-order nonsingular fast terminal sliding mode controller based on the integral sliding mode surface includes the specific steps of:
if the initial error is equal to zero, then the tracking problem can be converted to the problem of error s (t) 0 on the sliding-mode surface for all t ≧ 0, and if the system trajectory can reach s (t) 0 on the sliding-mode surface, then on e2(t)＝0 and
Figure BDA0003206437110000041
in time, the system can still be kept on the sliding surface, once the tracking error reaches the end s, which is 0 end sliding surface, the following results are obtained:
Figure BDA0003206437110000042
at the same time, when
Figure BDA0003206437110000043
Then, obtaining:
Figure BDA0003206437110000044
Figure BDA0003206437110000045
the positive definite lyapunov function is constructed as:
Figure BDA0003206437110000046
derivation of equation (13) and substitution of equation (12) yields:
Figure BDA0003206437110000047
equation (14) shows that once the error trajectory reaches the sliding-mode surface (9), the error signal converges progressively to the origin; in fact, the error state is bounded due to the Lyapunov function V2Is positive, the time derivative of which is semi-negative, and is therefore positive for V2(∞)∈R+Existence of
Figure BDA0003206437110000048
Depending on the degree of bounding of the error signal,
Figure BDA0003206437110000049
is continuous, so according to the barbalt theorem,
Figure BDA00032064371100000410
obtained from formula (9)
Figure BDA00032064371100000411
Eventually, the error signal converges progressively to zero;
in second order sliding mode control, the purpose of the control is to make s and
Figure BDA00032064371100000412
converge to the origin, i.e.
Figure BDA00032064371100000413
Therefore, the integral second-order nonsingular terminal sliding mode surface is designed as follows:
Figure BDA0003206437110000051
where b is a positive coefficient controlling the rate of decrease of s, derived from equation (15):
Figure BDA0003206437110000052
one necessary condition for the error state to remain on the switch surface is
Figure BDA0003206437110000053
Thus obtaining a characteristic polynomial (11);
if the first and second derivatives of the error derivative are defined as:
Figure BDA0003206437110000054
Figure BDA0003206437110000055
by substituting formulae (17) and (18) into formula (16):
Figure BDA0003206437110000056
when in use
Figure BDA0003206437110000057
And then obtaining an equivalent control signal, wherein the equivalent controller and the switching controller are designed as follows:
Figure BDA0003206437110000058
Figure BDA0003206437110000059
wherein ,k5，k6＞0，0＜β＜1，
Figure BDA00032064371100000510
At the rate of handover control
Figure BDA00032064371100000511
In a discontinuous sign function
Figure BDA00032064371100000512
Acting on the first derivative of the control rate, and obtaining a continuous controller without buffeting after integration;
considering equations (2) and (9), in combination with equations (20) and (21), the total control rate is defined as:
Figure BDA00032064371100000513
formula (20), formula (21), and formula (22) are substituted in formula (19) to obtain:
Figure BDA0003206437110000061
constructing a positive definite lyapunov function:
Figure BDA0003206437110000062
taking the derivative of equation (24) and substituting (23) therein yields:
Figure BDA0003206437110000063
by making
Figure BDA0003206437110000064
And negative determination is carried out, so that the stability of the integral second-order nonsingular fast terminal sliding mode controller is ensured.
Further, the adaptive sliding mode tracker in step S3 includes:
considering equations (2) and (9), let us assume the uncertainty terms ξ (t) and ξ (t)
Figure BDA0003206437110000065
Bounded, but upper bound, positions, then the adaptive sliding mode tracker is designed:
Figure BDA0003206437110000066
wherein ,
Figure BDA0003206437110000067
is a lambda estimation variable, and the adaptive law design is as follows:
Figure BDA0003206437110000068
when delta is more than 0, the accessibility criterion of the sliding mode surface, s and
Figure BDA0003206437110000069
converge to zero, taking into account the estimation error:
Figure BDA00032064371100000610
the estimation error is derived and equation (27) is substituted into it to yield:
Figure BDA00032064371100000611
substituting the adaptive terminal sliding mode tracker control rate substitution formula (22) into the formula (19) again to obtain:
Figure BDA0003206437110000071
defining the Lyapunov function:
Figure BDA0003206437110000072
the following is derived from equation (31):
Figure BDA0003206437110000073
substituting formula (30) into formula (32) yields:
Figure BDA0003206437110000074
because s is less than or equal to s,
Figure BDA0003206437110000075
therefore, equation (32) is rewritten as:
Figure BDA0003206437110000076
adding one term and subtracting one term in the formula (34)
Figure BDA0003206437110000077
Obtaining:
Figure BDA0003206437110000078
by making
Figure BDA0003206437110000079
And negative determination is carried out, so that the stability of the sliding mode tracking controller is ensured.
Compared with the prior art, the method ensures the convergence of the system in limited time by constructing the integral sliding mode surface, ensures the system to have good control precision and faster convergence speed by designing the second-order nonsingular terminal sliding mode controller, and simultaneously weakens the buffeting problem of the system; and finally, by a self-adaptive control method, a sliding mode tracking controller with parameter shaping is designed, the upper bound of uncertainty factors is estimated, and the system is ensured to have good convergence speed, robustness and anti-interference capability.
Drawings
Fig. 1 is a schematic diagram of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. The specific embodiments described herein are merely illustrative of the invention and do not limit the invention.
As shown in fig. 1, the present embodiment provides a sliding mode control method for a permanent magnet linear synchronous motor, which includes the following specific steps:
s1, converting three-phase winding current i of the permanent magnet linear synchronous motora、ib、icObtaining a current signal i under a two-phase static dq coordinate system through Clark coordinate transformationd、iqCombining the displacement and the speed of the motor to obtain a motion equation of the permanent magnet linear synchronous motor under the dq coordinate;
by using idThe simplified PMLSM model is, with a control strategy of 0:
Figure BDA0003206437110000081
wherein ,ud、uqD-axis voltage and q-axis voltage, respectively; rsResistance for each phase winding; i.e. id、iqD-axis current and q-axis current, respectively; l isd、LqD-axis inductance and q-axis inductance respectively; v is the mover speed; tau is a polar distance; psifIs an excitation flux linkage; feIs electromagnetic thrust; n ispIs the number of pole pairs; kfIs the thrust coefficient; according to the motion equation of PMLSM, the state equation of PMLSM is obtained as follows:
Figure BDA0003206437110000082
wherein ,x1Is displacement; x is the number of2Is the speed;
Figure BDA0003206437110000083
is the acceleration; u is the control rate of the controller, and u is iq(ii) a ξ (t) is a function of the time of the unknown uncertain disturbance.
S2, constructing an integral sliding mode surface, and designing an integral second-order nonsingular rapid terminal sliding mode controller based on the integral sliding mode surface;
the position tracking error is defined as:
Figure BDA0003206437110000091
wherein ,
Figure BDA0003206437110000092
for the reference displacement, the lyapunov function is constructed:
Figure BDA0003206437110000093
to V1The derivation can be:
Figure BDA0003206437110000094
the reference speed is set as:
Figure BDA0003206437110000095
substituting (6) into (5):
Figure BDA0003206437110000096
the position loop is thus progressively stabilized. The speed error is defined as:
Figure BDA0003206437110000097
in order to ensure that the controller has good tracking accuracy and convergence speed, the integral sliding mode surface is defined as follows:
Figure BDA0003206437110000098
wherein ,k2，k3，k4Both are positive integers, q and p are positive odd numbers, and q < p, if the initial error is equal to zero, then the tracking problem can be converted to the problem of error on the sliding-mode face, s (t) ≧ 0. If the system trajectory can reach the slip form plane where s (t) is 0, then at e2(t)＝0 and
Figure BDA0003206437110000099
the system can still remain on the slip-form face.
Once the tracking error reaches the end s-0 end sliding mode surface, we can get:
Figure BDA00032064371100000910
at the same time, when
Figure BDA00032064371100000911
Then, it is possible to obtain:
Figure BDA00032064371100000912
Figure BDA0003206437110000101
the positive definite lyapunov function is constructed as:
Figure BDA0003206437110000102
by taking the derivative of equation (13) and substituting (12), one can obtain:
Figure BDA0003206437110000103
equation (14) shows that once the error trajectory reaches the sliding-mode surface (9), the error signal converges asymptotically to the origin. In practice, the error state is bounded. Due to the Lyapunov function V2Is positive, the time derivative of which is semi-negative, and is therefore positive for V2(∞)∈R+Existence of
Figure BDA0003206437110000104
Depending on the degree of bounding of the error signal,
Figure BDA0003206437110000105
is continuous, so according to the Barbalt theorem, it can be found that
Figure BDA0003206437110000106
At the same time, we can obtain from formula (9)
Figure BDA0003206437110000107
Eventually, the error signal converges asymptotically to zero.
In second order sliding mode control, the purpose of the control is to make s and
Figure BDA0003206437110000108
converge to the origin, i.e.
Figure BDA0003206437110000109
Therefore, the integral second-order nonsingular terminal sliding mode surface is designed as follows:
Figure BDA00032064371100001010
where b is a positive coefficient controlling the rate of decrease of s, derived from equation (15):
Figure BDA00032064371100001011
one necessary condition for the error state to remain on the switch surface is
Figure BDA00032064371100001012
Thus, a characteristic polynomial (11) is obtained.
If the first and second derivatives of the error derivative are defined as:
Figure BDA0003206437110000111
Figure BDA0003206437110000112
substituting equations (17) and (18) into (16) can yield:
Figure BDA0003206437110000113
when in use
Figure BDA0003206437110000114
And obtaining an equivalent control signal, which is a necessary criterion for ensuring that the error state reaches the switching surface. The equivalent controller and the switching controller are designed as follows:
Figure BDA0003206437110000115
Figure BDA0003206437110000116
wherein ,k5，k6＞0，0＜β＜1，
Figure BDA0003206437110000117
At the rate of handover control
Figure BDA0003206437110000118
In a discontinuous sign function
Figure BDA0003206437110000119
Acting on the first derivative of the control rate. The controller obtained after integration is continuous and buffeting-free.
Considering a permanent magnet linear motor system (2) and an integral sliding mode surface (9), combining equations (20) and (21), the total control rate is defined as:
Figure BDA00032064371100001110
by substituting formulae (20), (21) and (22) into formula (19):
Figure BDA00032064371100001111
constructing a positive definite lyapunov function:
Figure BDA00032064371100001112
taking the derivative of equation (24) and substituting (23) therein yields:
Figure BDA0003206437110000121
due to the time derivative of the Lyapunov function
Figure BDA0003206437110000122
Is negative and therefore equation (24) is gradually reduced, the switching plane s (t) and its time derivative converge to the origin, and the controller is stable.
In practical situations, it is difficult to find the uncertainty terms ξ (t) and ξ (t) &
Figure BDA0003206437110000123
The upper bound of (c). Therefore, it is necessary to design the estimation parameters
Figure BDA0003206437110000124
S3, designing an adaptive sliding mode tracker to control the permanent magnet linear synchronous motor on the basis of the integral second-order nonsingular fast terminal sliding mode controller in the step S2;
consider a permanent magnet linear motor system (2) and an integral sliding form surface (9). Suppose the uncertainty term xi (t) and
Figure BDA0003206437110000125
bounded, but at an upper bound location. Then, designing a parameter setting terminal sliding mode tracking controller:
Figure BDA0003206437110000126
wherein ,
Figure BDA0003206437110000127
is a lambda estimation variable, and the adaptive law design is as follows:
Figure BDA0003206437110000128
when delta is more than 0, the accessibility criterion of the sliding mode surface, s and
Figure BDA0003206437110000129
converging to zero. Considering the estimation error:
Figure BDA00032064371100001210
derivation of the estimation error and substitution of equation (27) therein can yield:
Figure BDA00032064371100001211
substituting the adaptive terminal sliding mode tracker control rate substitution formula (22) into the formula (19) again to obtain:
Figure BDA0003206437110000131
defining the Lyapunov function:
Figure BDA0003206437110000132
the following is derived from equation (31):
Figure BDA0003206437110000133
substituting formula (30) into formula (32) yields:
Figure BDA0003206437110000134
because s is less than or equal to s,
Figure BDA0003206437110000135
therefore, equation (32) can be rewritten as:
Figure BDA0003206437110000136
adding one term and subtracting one term in the formula (34)
Figure BDA0003206437110000137
Obtaining:
Figure BDA0003206437110000138
therefore, due to parameter adjustment of the control rate of the terminal sliding mode tracker, the Lyapunov function (31) is gradually reduced, the sliding mode surface accessibility criterion is met, and the tracker is stable.
The technical solution of the present invention is not limited to the limitations of the above specific embodiments, and all technical modifications made according to the technical solution of the present invention fall within the protection scope of the present invention.

Claims (5)

1. A sliding mode control method of a permanent magnet linear synchronous motor is characterized by comprising the following steps:
s1, converting three-phase winding current i of the permanent magnet linear synchronous motora、ib、icObtaining a current signal i under a two-phase static dq coordinate system through Clark coordinate transformationd、iqCombining the displacement and the speed of the motor to obtain a motion equation of the permanent magnet linear synchronous motor under the dq coordinate;
s2, constructing an integral sliding mode surface, and designing an integral second-order nonsingular rapid terminal sliding mode controller based on the integral sliding mode surface;
s3, designing an adaptive sliding mode tracker to control the permanent magnet linear synchronous motor on the basis of the integral second-order nonsingular fast terminal sliding mode controller in the step S2.
2. The sliding-mode control method for the permanent magnet linear synchronous motor according to claim 1, wherein the motion equation of the permanent magnet linear synchronous motor in dq coordinate at step S1 is as follows:
Figure FDA0003206437100000011
wherein ,ud、uqD-axis voltage and q-axis voltage, respectively; rsResistance for each phase winding; i.e. id、iqD-axis current and q-axis current, respectively; l isd、LqD-axis inductance and q-axis inductance respectively; v is the mover speed; tau is a polar distance; psifIs an excitation flux linkage; feIs electromagnetic thrust; n ispIs the number of pole pairs; kfIs the thrust coefficient; the state equation of the permanent magnet linear synchronous motor obtained according to the motion equation of the permanent magnet linear synchronous motor is as follows:
Figure FDA0003206437100000012
wherein ,x1Is displacement; x is the number of2Is the speed;
Figure FDA0003206437100000013
is the acceleration; u is the control rate of the controller, and u is iq(ii) a ξ (t) is a function of the time of the unknown uncertain disturbance.
3. The sliding mode control method of the permanent magnet linear synchronous motor according to claim 1, wherein the concrete steps of constructing the integral sliding mode surface in the step S2 are as follows:
the position tracking error is defined as:
Figure FDA0003206437100000021
wherein ,
Figure FDA0003206437100000022
for the reference displacement, the lyapunov function is constructed:
Figure FDA0003206437100000023
to V1The derivation can be:
Figure FDA0003206437100000024
the reference speed is set as:
Figure FDA0003206437100000025
substituting formula (6) into formula (5):
Figure FDA0003206437100000026
the speed error is defined as:
Figure FDA0003206437100000027
in order to ensure that the controller has good tracking accuracy and convergence speed, the integral sliding mode surface is defined as follows:
Figure FDA0003206437100000028
wherein ,k2，k3，k4Both are positive integers, q and p are positive odd numbers, and q < p.
4. The sliding-mode control method for the permanent magnet linear synchronous motor according to claim 3, wherein the step S2 of designing the integral second-order nonsingular fast terminal sliding-mode controller based on the integral sliding-mode surface comprises the following specific steps:
if the initial error is equal to zero, then the tracking problem can be converted to the problem of error s (t) 0 on the sliding-mode surface for all t ≧ 0, and if the system trajectory can reach s (t) 0 on the sliding-mode surface, then on e2(t)＝0 and
Figure FDA0003206437100000029
in time, the system can still be kept on the sliding surface, once the tracking error reaches the end s, which is 0 end sliding surface, the following results are obtained:
Figure FDA0003206437100000031
at the same time, when
Figure FDA0003206437100000032
Then, obtaining:
Figure FDA0003206437100000033
Figure FDA0003206437100000034
the positive definite lyapunov function is constructed as:
Figure FDA0003206437100000035
derivation of equation (13) and substitution of equation (12) yields:
Figure FDA0003206437100000036
equation (14) shows that once the error trajectory reaches the sliding-mode surface (9), the error signal converges progressively to the origin; in fact, the error state is bounded due to the Lyapunov function V2Is positive, the time derivative of which is semi-negative, and is therefore positive for V2(∞)∈R+Existence of
Figure FDA0003206437100000037
Depending on the degree of bounding of the error signal,
Figure FDA0003206437100000038
is continuous, so according to the barbalt theorem,
Figure FDA0003206437100000039
obtained from formula (9)
Figure FDA00032064371000000310
Eventually, the error signal converges progressively to zero;
in second order sliding mode control, the purpose of the control is to make s and
Figure FDA00032064371000000314
converge to the origin, i.e.
Figure FDA00032064371000000311
Therefore, the integral second-order nonsingular terminal sliding mode surface is designed as follows:
Figure FDA00032064371000000312
where b is a positive coefficient controlling the rate of decrease of s, derived from equation (15):
Figure FDA00032064371000000313
one necessary condition for the error state to remain on the switch surface is
Figure FDA0003206437100000041
Thus obtaining a characteristic polynomial (11);
if the first and second derivatives of the error derivative are defined as:
Figure FDA0003206437100000042
Figure FDA0003206437100000043
by substituting formulae (17) and (18) into formula (16):
Figure FDA0003206437100000044
when in use
Figure FDA0003206437100000045
And then obtaining an equivalent control signal, wherein the equivalent controller and the switching controller are designed as follows:
Figure FDA0003206437100000046
Figure FDA0003206437100000047
wherein ,k5，k6＞0，0＜β＜1，
Figure FDA0003206437100000048
At the rate of handover control
Figure FDA0003206437100000049
In a discontinuous sign function
Figure FDA00032064371000000410
Acting on the first derivative of the control rate, and obtaining a continuous controller without buffeting after integration;
considering equations (2) and (9), in combination with equations (20) and (21), the total control rate is defined as:
Figure FDA00032064371000000411
formula (20), formula (21), and formula (22) are substituted in formula (19) to obtain:
Figure FDA00032064371000000412
constructing a positive definite lyapunov function:
Figure FDA00032064371000000413
taking the derivative of equation (24) and substituting (23) therein yields:
Figure FDA0003206437100000051
by making
Figure FDA0003206437100000052
And negative determination is carried out, so that the stability of the integral second-order nonsingular fast terminal sliding mode controller is ensured.
5. The sliding mode control method of a permanent magnet linear synchronous motor according to claim 4,
the step of designing the adaptive sliding mode tracker in the step S3 is as follows:
considering equations (2) and (9), let us assume the uncertainty terms ξ (t) and ξ (t)
Figure FDA0003206437100000053
Bounded, but upper bound, positions, then the adaptive sliding mode tracker is designed:
Figure FDA0003206437100000054
wherein ,
Figure FDA0003206437100000055
is a lambda estimation variable, and the adaptive law design is as follows:
Figure FDA0003206437100000056
when delta is more than 0, the accessibility criterion of the sliding mode surface, s and
Figure FDA0003206437100000057
converge to zero, taking into account the estimation error:
Figure FDA0003206437100000058
the estimation error is derived and equation (27) is substituted into it to yield:
Figure FDA0003206437100000059
substituting the adaptive terminal sliding mode tracker control rate substitution formula (22) into the formula (19) again to obtain:
Figure FDA00032064371000000510
defining the Lyapunov function:
Figure FDA0003206437100000061
the following is derived from equation (31):
Figure FDA0003206437100000062
substituting formula (30) into formula (32) yields:
Figure FDA0003206437100000063
because s is less than or equal to s,
Figure FDA0003206437100000064
therefore, equation (32) is rewritten as:
Figure FDA0003206437100000065
adding one term and subtracting one term in the formula (34)
Figure FDA0003206437100000066
Obtaining:
Figure FDA0003206437100000067
by making
Figure FDA0003206437100000068
And negative determination is carried out, so that the stability of the sliding mode tracking controller is ensured.
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