CN112713828A - Active sliding mode control method and device for fractional order chaotic system of permanent magnet synchronous motor - Google Patents

Active sliding mode control method and device for fractional order chaotic system of permanent magnet synchronous motor Download PDF

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CN112713828A
CN112713828A CN202011463473.7A CN202011463473A CN112713828A CN 112713828 A CN112713828 A CN 112713828A CN 202011463473 A CN202011463473 A CN 202011463473A CN 112713828 A CN112713828 A CN 112713828A
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fractional order
permanent magnet
synchronous motor
magnet synchronous
sliding mode
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刘欣蒙
胡茗茗
李明
李晓婷
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Shandong Xiaoya Group Co ltd
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/0003—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

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Abstract

The invention discloses a fractional order chaotic system active sliding mode control method of a permanent magnet synchronous motor, which is characterized in that on the basis of a mathematical model of the permanent magnet synchronous motor, a fractional order chaotic system of the permanent magnet synchronous motor containing an uncertain item is modeled to obtain a fractional order chaotic system model of a driving system, a novel fractional order integral sliding mode surface is selected by combining the methods of active control and sliding mode control, and an active sliding mode controller is designed based on a fractional order Lyapunov stability theory. Under the control of the controller, the synchronization between the fractional order permanent magnet synchronous motor chaotic system and the response system is realized, and experiments show that the fractional order chaotic phenomenon in the permanent magnet synchronous motor system can be effectively inhibited through the control of the controller, the control capability of an error system is improved, and the control of the fractional order chaotic synchronization of the permanent magnet synchronous motor has high robustness.

Description

Active sliding mode control method and device for fractional order chaotic system of permanent magnet synchronous motor
Technical Field
The invention relates to the field of nonlinear control of motors, in particular to a method and a device for actively sliding mode control of a fractional order chaotic system of a permanent magnet synchronous motor.
Background
Permanent magnet synchronous motors are typically multivariable, strongly coupled nonlinear systems as important loads for power systems.
In the prior art, a permanent magnet synchronous motor can exhibit chaotic behaviors under specific parameters and working conditions, usually represented by intermittent oscillation of torque and rotating speed, irregular electromagnetic noise of a system and the like, the irregular motions can seriously affect the stable operation of the system, and the chaotic behaviors of the permanent magnet synchronous motor can influence the precision control and the robustness of the control. After the PMSM model is proposed, how to control the chaotic behavior in the permanent magnet synchronous motor system has become a focus of attention. At present, most of the PMSM integer order chaotic systems are researched, but in practical application, fractional orders are more common than integer orders, so that the research on the fractional order chaotic synchronous control of the permanent magnet synchronous motor can realize the accurate control of the permanent magnet synchronous motor.
Disclosure of Invention
In order to solve the above problems, the present invention provides an active sliding mode control method for a fractional order chaotic system of a permanent magnet synchronous motor, comprising:
constructing a fractional order chaotic system model of a driving system:
Figure BDA0002829528590000011
obtaining a response system model corresponding to the driving system as follows:
Figure BDA0002829528590000012
wherein y (t) ═ y1，y2，y3)TΔ a is an uncertain parameter matrix, g (y, t) is a known nonlinear function, Δ g (y, t) is an uncertain part of the nonlinear function, and u (t) is (u)1，u2，u3)TIs a controller;
and obtaining a corresponding error system by calculating the difference between the response system and the driving system, wherein the error system comprises:
Figure BDA0002829528590000013
constructing fractional order slip form surfaces
Figure BDA0002829528590000021
And solving equivalent control according to the condition of the active sliding mode and the assumption that uncertain items delta f (x, t) and delta g (y, t) are bounded:
Figure BDA0002829528590000022
switching control:
Figure BDA0002829528590000023
the control input
Figure BDA0002829528590000024
The controller is designed as follows:
u(t)＝G(t)-ΔAy(t)-g(y，t)-Δg(y，t)+f(x，t)+Δf(x，t) (4)
defining according to an active sliding mode control law:
Figure BDA0002829528590000025
wherein K is a constant gain vector,
Figure BDA0002829528590000026
satisfies the conditions
Figure BDA0002829528590000027
s is a fractional order slip form surface;
according to control input
Figure BDA0002829528590000028
And solving the controller.
Preferably, the obtaining of the fractional order chaotic system model of the permanent magnet synchronous motor comprises:
mathematical model for permanent magnet synchronous motor
Figure BDA0002829528590000029
Performing radiation transformation and scale transformation to obtain a permanent magnet synchronous motor chaotic mathematical model considering uniform air gaps:
Figure BDA00028295285900000210
wherein idStator current, i, representing d-axisqStator current representing the q-axis, ω rotor angular velocity, udStator voltage, u, representing the d-axisqRepresenting the stator voltage of q axis, T is load torque, J is rotational inertia, beta is viscous damping coefficient, R is stator winding, LdStator inductance of d-axis, LqQ-axis stator inductance, psi excitation flux linkage, p pole pair number,
Figure BDA00028295285900000211
order to
Figure BDA0002829528590000031
Considering an uncertain item in a fractional order chaotic system of the permanent magnet synchronous motor and a controller u (t), obtaining a fractional order chaotic system model:
Figure BDA0002829528590000032
wherein x (t) ═ x1，x2，x3)TWhere A is a known constant matrix, f (x, t) is a known nonlinear function, and Δ f (x, t) is the uncertainty of the nonlinear function.
Preferably, setting parameters to enable the fractional order chaotic system model to be in a chaotic state; and taking alpha as 0.96, xi as 20 and sigma as 5.46, and setting parameters of a driving system as (a, b and c) as (30,2 and 15) to enable the fractional order chaotic system model to be in a chaotic state.
Preferably, the fractional order slip-form surface s (e (t)) is constructed:
Figure BDA0002829528590000033
where B is a constant gain matrix, and M is diag [ M ═ M [ ]1，m2，m3]，
Figure BDA0002829528590000034
Determining equivalent control for alpha-order Caputo operator on condition that the equivalent control keeps state track on fractional order sliding mode surface
Figure BDA0002829528590000035
To fractional order slip form surface
Figure BDA0002829528590000036
Both sides take the differential of order α with respect to time t:
Figure BDA0002829528590000037
when the system generates sliding mode motion, the equivalent control enables the state track to be kept on the fractional order sliding mode surface, and the requirement of meeting the requirement
Figure BDA0002829528590000038
Obtaining the equivalent control:
Figure BDA0002829528590000039
preferably, it is assumed that there are limits to the uncertainty terms Δ f (x, t) and Δ g (y, t) and that there are appropriate normal numbers ζ and μ that meet
|Δf(x，t)|≤ζ，|Δg(x，t)|≤μ；
In order to meet sliding mode conditions and ensure that sliding mode motion can be ensured to occur certainly, according to a fractional sliding mode control law, the switching control is obtained as follows:
Figure BDA00028295285900000310
wherein λ is>0，
Figure BDA00028295285900000311
ζ, μ, the evaluation value, and the unknown parameter, respectively.
Preferably, for unknown parameters
Figure BDA00028295285900000312
Using the fractional order adaptation rate, we obtain:
Figure BDA00028295285900000313
wherein eta is1>0，η2>0；
By the following Lyapunov function:
Figure BDA00028295285900000314
wherein the content of the first and second substances,
Figure BDA0002829528590000041
verifying whether the error system is asymptotically stable.
Preferably, given a positive V, the Lyapunov function is subjected to alpha-order differentiation, whether the result is negative is judged, and if the result is negative, the error system is gradually stable.
The invention provides an active sliding mode control device of a fractional order chaotic system of a permanent magnet synchronous motor, which comprises a control unit, a storage unit, a bus unit, an acquisition unit and a driving unit, wherein the control unit, the storage unit, the acquisition unit and the driving unit are connected through the bus, the acquisition unit acquires working parameters of the permanent magnet synchronous motor, the driving unit is electrically connected with the permanent magnet synchronous motor, the storage unit stores at least one instruction, and the control unit executes the instruction to realize the active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor to construct a controller.
The active sliding mode control method and device for the fractional order chaotic system of the permanent magnet synchronous motor have the following beneficial effects:
the invention provides a fractional order chaotic system active sliding mode control method of a permanent magnet synchronous motor, which constructs a fractional order chaotic system model of a driving system through a PMSM model, acquires a corresponding response system model (comprising a controller), obtains an error system by utilizing the difference between the response system and the driving system, constructs a fractional order sliding mode surface related to the error system, solves a control input according to the fractional order sliding mode surface, further solves the controller, controls the permanent magnet synchronous motor through the controller, can effectively solve the fractional order chaotic phenomenon generated in the running process of the permanent magnet synchronous motor, improves the control capability of the error system, improves the control precision, eliminates the chaotic influence, ensures that the control of the permanent magnet synchronous motor has stronger robustness, and ensures that the permanent magnet synchronous motor is more stable in the running process, the method can be widely applied to the field of precise control of the permanent magnet synchronous motor. The active sliding mode control device of the fractional order chaotic system of the permanent magnet synchronous motor provided by the invention realizes the control of the controller on the permanent magnet synchronous motor, has stronger control capability on an error system, improves the control precision, and eliminates the chaotic influence so as to have stronger robustness on the control of the permanent magnet synchronous motor.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the structures shown in the drawings without creative efforts.
Fig. 1 is a schematic diagram of a chaotic attractor of a fractional order chaotic system of a permanent magnet synchronous motor in an embodiment of the invention;
FIG. 2 shows a state variable x of a fractional order PMSM chaotic system and a response system in an embodiment of the present invention1 y1A track graph;
FIG. 3 shows a state variable x of a fractional order PMSM chaotic system and a response system in an embodiment of the present invention2 y2A track graph;
FIG. 4 shows a state variable x of a fractional order PMSM chaotic system and a response system in an embodiment of the present invention3 y3A track graph;
FIG. 5 shows an error variable e of the fractional order PMSM chaotic system and the response system in the embodiment of the present inventioniAn evolution schematic diagram;
FIG. 6 is a flowchart of an active sliding mode control method of a fractional order chaotic system of a permanent magnet synchronous motor according to an embodiment of the present invention;
fig. 7 is a schematic diagram of an active sliding mode control device of a fractional order chaotic system of a permanent magnet synchronous motor in the embodiment of the invention.
The implementation, functional features and advantages of the objects of the present invention will be further described with reference to the accompanying drawings.
Detailed Description
It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention is described below with reference to the accompanying drawings, wherein fig. 1 is a schematic diagram of a chaotic attractor of a fractional order chaotic system of a permanent magnet synchronous motor in an embodiment of the invention; FIG. 2 is a state variable x of a fractional order permanent magnet synchronous motor chaotic system and a response system in the embodiment of the invention1y1A track graph; FIG. 3 shows a state variable x of a fractional order PMSM chaotic system and a response system in an embodiment of the present invention2y2A track graph; FIG. 4 shows a fractional order PMSM chaotic system and response in an embodiment of the present inventionState variable x of the system3y3A track graph; FIG. 5 shows an error variable e of the fractional order PMSM chaotic system and the response system in the embodiment of the present inventioniAn evolution schematic diagram; fig. 6 is a flowchart of an active sliding-mode control method of a fractional chaotic system of a permanent magnet synchronous motor according to an embodiment of the present invention; fig. 7 is a schematic diagram of an active sliding mode control device of a fractional order chaotic system of a permanent magnet synchronous motor in the embodiment of the invention.
Referring to fig. 6, the invention provides an active sliding mode control method of a fractional order chaotic system of a permanent magnet synchronous motor, which comprises the following steps:
s100, constructing a fractional order chaotic system model of a driving system; the specific process is as follows: a mathematical model of the permanent magnet synchronous motor is as follows:
Figure BDA0002829528590000061
performing radiation transformation and scale transformation to obtain a permanent magnet synchronous motor chaotic mathematical model considering uniform air gaps:
Figure BDA0002829528590000062
wherein idStator current, i, representing d-axisqStator current representing the q-axis, ω rotor angular velocity, udStator voltage, u, representing the d-axisqRepresenting the stator voltage of q axis, T is load torque, J is rotational inertia, beta is viscous damping coefficient, R is stator winding, LdStator inductance of d-axis, LqQ-axis stator inductance, psi excitation flux linkage, p pole pair number,
Figure BDA0002829528590000063
order to
Figure BDA0002829528590000064
Considering uncertain items in fractional order chaotic system of permanent magnet synchronous motor and controlAnd (t) obtaining the following fractional order chaotic system model:
Figure BDA0002829528590000065
wherein x (t) ═ x1，x2，x3)TWhere A is a known constant matrix of order 3 × 3, f (x, t) is a known nonlinear function, and Δ f (x, t) is the uncertainty of the nonlinear function.
Setting parameters to enable the fractional order chaotic system model to be in a chaotic state; in the specific implementation process, alpha is 0.96, xi is 20, sigma is 5.46, parameters of a driving system are (a, b, c) are (30,2,15), the fractional order chaotic system model is in a chaotic state, and an initial value is randomly selected to be x (0), 1,5,7)T∈R3At this time, the fractional order chaotic system is in a chaotic state, and referring to fig. 1, a chaotic attractor of the permanent magnet synchronous motor can be obtained.
S200, acquiring a response system model corresponding to the driving system as follows:
Figure BDA0002829528590000066
wherein y (t) ═ y1，y2，y3)TIs a three-dimensional state vector, Δ a is an uncertain parameter matrix, g (y, t) is a known nonlinear function, Δ g (y, t) is an uncertain part of the nonlinear function, u (t) is (u ═ t)1，u2，u3)TIs a controller;
s300, calculating the difference between the response system and the driving system to obtain a corresponding error system as follows:
Figure BDA0002829528590000071
s400, in order to synchronize a driving system and a response system of the fractional order chaotic system model, constructing a fractional order sliding mode surface S (e (t)):
Figure BDA0002829528590000072
where B is a constant gain matrix, and M is diag [ M ═ M [ ]1，m2，m3]，
Figure BDA0002829528590000073
And calculating and acquiring control input for the alpha-order Caputo operator according to the fractional order sliding mode surface. And solving equivalent control and switching control according to the active sliding mode condition and the assumption that uncertain items delta f (x, t) and delta g (y, t) are bounded so as to obtain the control input.
To fractional order slip form surface
Figure BDA0002829528590000074
Both sides take the differential of order α with respect to time t:
Figure BDA0002829528590000075
when the system generates sliding mode motion, the equivalent control enables the state track to be kept on the switching surface, and the requirement of meeting the requirement
Figure BDA0002829528590000076
Thus, it is possible to provide
Figure BDA0002829528590000077
Obtaining the equivalent control:
Figure BDA0002829528590000078
since the values of the uncertainty factor tend to be small, given the bounds of uncertainty Δ f (x, t) and Δ g (y, t), there is a suitable normal zeta and μ fit to satisfy
|Δf(x，t)|≤ζ，|Δg(x，t)|≤μ；
According to fractional order, in order to meet the sliding mode condition and ensure that sliding mode motion can be ensured to occur certainlyA sliding mode control law, wherein the switching control is obtained as follows:
Figure BDA0002829528590000079
wherein λ is>0，
Figure BDA00028295285900000710
ζ, μ, the evaluation value, and the unknown parameter, respectively.
Whereby the control input
Figure BDA00028295285900000711
The results were:
Figure BDA00028295285900000712
s500, designing the controller as follows:
u(t)＝G(t)-ΔAy(t)-g(y，t)-Δg(y，t)+f(x，t)+Δf(x，t) (4)
defining according to an active sliding mode control law:
Figure BDA00028295285900000713
wherein K is a constant gain vector,
Figure BDA00028295285900000714
for control input and satisfying conditions
Figure BDA00028295285900000715
And s is a fractional sliding mode surface.
And the controller is solved by the control input. And substituting the result of the control input into the formula (4) to obtain the controller.
S600, verifying whether the error system comprising the controller is gradually stable. For unknown parameters here
Figure BDA0002829528590000081
Using the fractional order adaptation rate, we obtain:
Figure BDA0002829528590000082
wherein eta is1>0，η2>0；
Based on a fractional order Lyapunov stability theory: for general fractional order systems
Figure BDA0002829528590000083
It is assumed that a Lyapunov function V (p) exists and satisfies
Figure BDA0002829528590000084
When t is greater than or equal to t0It is progressively stable for all p (t). By the following Lyapunov function:
Figure BDA0002829528590000085
wherein the content of the first and second substances,
Figure BDA0002829528590000086
verifying whether the error system is asymptotically stable. In a specific implementation process, alpha-order differentiation is carried out on the Lyapunov function, whether a result is negative is judged, and if the result is negative, the error system is gradually stable. And performing alpha-order differentiation on the Lyapunov function to obtain:
Figure BDA0002829528590000087
substituting the control input to obtain:
Figure BDA0002829528590000088
thereby determining that the error system gradually converges and maintains on the fractional order sliding mode surface within a finite time.
Referring to fig. 7, the invention provides an active sliding mode control device of a fractional order chaotic system of a permanent magnet synchronous motor, which comprises a control unit, a storage unit, a bus unit, an acquisition unit and a driving unit, wherein the control unit, the storage unit, the acquisition unit and the driving unit are connected through the bus, the acquisition unit acquires working parameters of the permanent magnet synchronous motor, the driving unit is electrically connected with the permanent magnet synchronous motor, the storage unit stores at least one instruction, the control unit executes the instruction to realize a controller constructed by the active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor, the control unit controls the driving unit through the controller, and the driving unit drives the permanent magnet synchronous motor to work.
The active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor is used for experiments, and in the experiments, an uncertain matrix delta A is randomly and initially:
Figure BDA0002829528590000091
the initial conditions were set to:
(x1(0),x2(0),x3(0))T＝(3,2,10)T
(y1(0),y2(0),y3(0))T＝(-5,1,88)T
the controller parameters may be selected as:
Figure BDA0002829528590000092
Figure BDA0002829528590000093
M＝diag(2,3,2)；
Figure BDA0002829528590000094
the state trajectories of the fractional order chaotic system and the response system obtained through the experiment are shown in fig. 2, fig. 3 and fig. 4. From the numerical result, the state track of the fractional order chaotic system and the state track of the corresponding response system gradually reach the consistency with the time. The error state trajectories of the fractional order chaotic system and the response system are shown in fig. 5, and the numerical result can be used for obtaining that the error between the driving system and the response system gradually approaches zero along with the continuous increase of time, namely the two systems are synchronized.
The invention provides a fractional order chaotic system active sliding mode control method of a permanent magnet synchronous motor, which constructs a fractional order chaotic system model of a driving system through a PMSM model, and obtaining a corresponding response system model (comprising a controller), obtaining an error system by utilizing the difference between the response system and the driving system, constructing a fractional order sliding mode surface related to the error system, the control input is solved according to the fractional order sliding mode surface, the controller is solved, the permanent magnet synchronous motor is controlled by the controller, the fractional order chaos phenomenon generated in the running process of the permanent magnet synchronous motor can be effectively solved, the control capability of an error system is improved, the control precision is improved, and the chaos influence is eliminated, so that the control on the permanent magnet synchronous motor has stronger robustness, and the method can be widely applied to the field of precise control of the permanent magnet synchronous motor.
It should be noted that in the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. The word "comprising" does not exclude the presence of elements or steps not listed in a claim. The word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. The invention may be implemented by means of hardware comprising several distinct elements, and by means of a suitably programmed computer. In the unit claims enumerating several means, several of these means may be embodied by one and the same item of hardware. The usage of the words first, second and third, etcetera do not indicate any ordering. These words may be interpreted as names.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the appended claims be interpreted as including the preferred embodiment and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (8)

1. An active sliding mode control method of a fractional order chaotic system of a permanent magnet synchronous motor is characterized by comprising the following steps:
constructing a fractional order chaotic system model of a driving system:
Figure FDA0002829528580000011
obtaining a response system model corresponding to the driving system as follows:
Figure FDA0002829528580000012
wherein y (t) ═ y1，y2，y3)TΔ a is an uncertain parameter matrix, g (y, t) is a known nonlinear function, Δ g (y, t) is an uncertain part of the nonlinear function, and u (t) is (u)1，u2，u3)TIs a controller;
and obtaining a corresponding error system by calculating the difference between the response system and the driving system, wherein the error system comprises:
Figure FDA0002829528580000013
constructing fractional order slip form surfaces
Figure FDA0002829528580000014
And solving equivalent control according to the condition of the active sliding mode and the assumption that uncertain items delta f (x, t) and delta g (y, t) are bounded:
Figure FDA0002829528580000015
switching control:
Figure FDA0002829528580000016
the control input
Figure FDA0002829528580000017
The controller is designed as follows:
u(t)＝G(t)-ΔAy(t)-g(y，t)-Δg(y，t)+f(x，t)+Δf(x，t) (4)
defining according to an active sliding mode control law:
Figure FDA0002829528580000018
wherein K is a constant gain vector,
Figure FDA0002829528580000019
satisfies the conditions
Figure FDA00028295285800000110
s is a fractional order slip form surface;
according to control input
Figure FDA00028295285800000111
And solving the controller.
2. The active sliding-mode control method of the fractional order chaotic system of the permanent magnet synchronous motor according to claim 1, wherein the obtaining of the fractional order chaotic system model of the permanent magnet synchronous motor comprises:
mathematical model for permanent magnet synchronous motor
Figure FDA0002829528580000021
Performing radiation transformation and scale transformation to obtain a permanent magnet synchronous motor chaotic mathematical model considering uniform air gaps:
Figure FDA0002829528580000022
wherein idStator current, i, representing d-axisqStator current representing the q-axis, ω rotor angular velocity, udStator voltage, u, representing the d-axisqRepresenting the stator voltage of the q axis, T is the load torque, J is the moment of inertia, beta is the viscous damping coefficient, R is the stator winding, LdStator inductance of d-axis, LqQ-axis stator inductance, psi excitation flux linkage, p pole pair number,
Figure FDA0002829528580000023
order to
Figure FDA0002829528580000024
Considering an uncertain item in a fractional order chaotic system of the permanent magnet synchronous motor and a controller u (t), obtaining a fractional order chaotic system model:
Figure FDA0002829528580000025
wherein x (t) ═ x1，x2，x3)TWhere A is a known constant matrix, f (x, t) is a known nonlinear function, and Δ f (x, t) is the uncertainty of the nonlinear function.
3. The active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor according to claim 1, characterized in that parameters are set to enable a fractional order chaotic system model to be in a chaotic state; and taking alpha as 0.96, xi as 20 and sigma as 5.46, and setting parameters of a driving system as (a, b and c) as (30,2 and 15) to enable the fractional order chaotic system model to be in a chaotic state.
4. The active sliding-mode control method for the fractional order chaotic system of the permanent magnet synchronous motor according to claim 1, wherein the fractional order sliding-mode surface s (e (t)) is constructed:
Figure FDA0002829528580000026
where B is a constant gain matrix, and M is diag [ M ═ M [ ]1，m2，m3]，
Figure FDA0002829528580000027
The method comprises the steps that an alpha-order Caputo operator determines equivalent control under the condition that the state track is kept on a fractional order sliding mode surface by the equivalent control
Figure FDA0002829528580000028
To fractional order slip form surface
Figure FDA0002829528580000031
Both sides take the differential of order α with respect to time t:
Figure FDA0002829528580000032
when the system generates sliding mode motion, the equivalent control enables the state track to be kept on the fractional order sliding mode surface, and the requirement of meeting the requirement
Figure FDA0002829528580000033
Thus, it is possible to provide
Figure FDA0002829528580000034
Obtaining the equivalent control:
Figure FDA0002829528580000035
5. the active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor according to claim 4, wherein a proper normal number ζ and μ are satisfied assuming that there are limits on uncertainty terms Δ f (x, t) and Δ g (y, t)
|Δf(x，t)|≤ζ，|Δg(x，t)|≤μ；
In order to meet sliding mode conditions and ensure that sliding mode motion can be ensured to occur certainly, according to a fractional sliding mode control law, the switching control is obtained as follows:
Figure FDA0002829528580000036
wherein λ is>0，
Figure FDA0002829528580000037
ζ, μ, the evaluation value, and the unknown parameter, respectively.
6. The active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor according to claim 5, characterized in that unknown parameters are subjected to
Figure FDA0002829528580000038
Using the fractional order adaptation rate, we obtain:
Figure FDA0002829528580000039
wherein eta is1>0，η2>0；
By the following Lyapunov function:
Figure FDA00028295285800000310
wherein the content of the first and second substances,
Figure FDA00028295285800000311
verifying whether the error system is asymptotically stable.
7. The active sliding mode control method of the fractional order chaotic system of the permanent magnet synchronous motor according to claim 6, wherein a known positive V is determined, alpha order differentiation is performed on the Lyapunov function, whether the result is negative is judged, and if the result is negative, the error system is asymptotically stable.
8. The active sliding-mode control device of the fractional order chaotic system of the permanent magnet synchronous motor is characterized by comprising a control unit, a storage unit, a bus unit, an acquisition unit and a driving unit, wherein the control unit, the storage unit, the acquisition unit and the driving unit are connected through the bus, the acquisition unit acquires working parameters of the permanent magnet synchronous motor, the driving unit is electrically connected with the permanent magnet synchronous motor, the storage unit stores at least one instruction, and the control unit executes the instruction to realize the active sliding-mode control method of the fractional order chaotic system of the permanent magnet synchronous motor to construct a controller according to any one of claims 1 to 7.
CN202011463473.7A 2020-12-11 2020-12-11 Active sliding mode control method and device for fractional order chaotic system of permanent magnet synchronous motor Pending CN112713828A (en)

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