CN108964544B - Double-time scale sliding mode control system and method for permanent magnet linear synchronous motor - Google Patents
Double-time scale sliding mode control system and method for permanent magnet linear synchronous motor Download PDFInfo
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Abstract
The invention discloses a double-time scale sliding mode control system and method for a permanent magnet linear synchronous motor, and belongs to the field of linear motor control. Firstly, establishing a mathematical model of a permanent magnet linear synchronous motor in a two-phase rotation orthogonal coordinate system; secondly, establishing the model as a permanent magnet linear synchronous motor double-time scale model; then, in order to improve the robustness of the system to external disturbance, sliding mode control laws corresponding to the fast subsystem and the slow subsystem are respectively designed by adopting a quasi-sliding mode method and an approach law method, and then the time scales of the two subsystems are unified to synthesize the combined control law of the permanent magnet linear synchronous motor. And finally, analyzing the stability of the system by applying the Lyapunov stability theory. The most important characteristic of the invention is that the designed double-time scale sliding mode controller enables the controlled system of the permanent magnet linear synchronous motor to have better static performance and good and rapid dynamic performance, and enables the system to have strong robustness to external disturbance.
Description
Technical Field
The invention relates to a double-time scale sliding mode control system and method for a permanent magnet linear synchronous motor, and belongs to the technical field of linear synchronous motor control.
Background
The permanent magnet linear synchronous motor, as a linear induction motor, has the advantages of low rotational inertia, small volume and weight, high efficiency, easy maintenance, high reliability and the like besides the excellent advantages of the linear motor, and is generally applied to high-precision alternating current servo systems. Although the linear motor has special advantages compared with the conventional rotating motor, the linear motor is similar to the rotating motor, is a complex control object with high coupling, multivariable, nonlinearity and time-varying property, is influenced by nonlinear factors such as thrust fluctuation and friction in practical application, has weak resistance to external interference of different degrees, and brings great difficulty to the research of a control strategy. In order to greatly improve the control performance and the control precision of the permanent magnet linear synchronous motor, the traditional control strategy is difficult to meet the performance requirement of a permanent magnet linear synchronous motor control system, so that the research of a new control method is very meaningful, at present, the most common control strategy is PID closed-loop control in the classical control field, and the traditional control system is low in dynamic response speed and poor in control precision.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a double-time scale sliding mode control system and method for a permanent magnet linear synchronous motor, which can effectively improve the high dynamic response speed of the control system, have strong robustness on parameter perturbation and external disturbance, and are easy to design and implement.
In order to achieve the purpose, the invention adopts the technical scheme that: a permanent magnet linear synchronous motor double-time scale sliding mode control system comprises a slow subsystem sliding mode surface module and a fast subsystem sliding mode surface module, wherein the slow subsystem sliding mode surface module is connected with a combined control law module through a slow subsystem sliding mode control law module; the fast subsystem sliding mode surface module is connected with a combined control law module through the fast subsystem sliding mode control law module; the combined control law module is connected with a permanent magnet linear synchronous motor, the permanent magnet linear synchronous motor is also connected with an external interference signal, the permanent magnet linear synchronous motor is connected to the slow subsystem sliding mode surface module, and an error signal of the actual speed and the given speed of the permanent magnet linear synchronous motor is transmitted to the slow subsystem sliding mode surface module; the permanent magnet linear synchronous motor is connected to the fast subsystem sliding mode surface module through the fast variable estimation module, the currents id and iq of the permanent magnet linear synchronous motor are transmitted to the fast variable estimation module, and the fast variable current components idf and iqf obtained by subtracting the slow variable current components ids and iqs through the fast variable estimation module are transmitted to the fast subsystem sliding mode surface module.
A control method of a permanent magnet linear synchronous motor double-time scale sliding mode control system comprises the following steps:
A. establishing a mathematical model;
B. establishing a double-time scale model;
C. designing a sliding mode control law;
D. synthesizing a combined controller;
E. and (5) analyzing the stability.
The invention has the beneficial effects that: the control system has strong robustness to interference and can realize accurate tracking of a given speed signal; the permanent magnet linear synchronous motor model is decomposed into a fast-slow subsystem through singular perturbation, a sliding mode controller respectively adopts a quasi-sliding mode method and an approach law method, the inherent buffeting phenomenon of sliding mode control is greatly improved, and the control quantity is close to zero after the control system enters a stable state.
Drawings
FIG. 1 is a basic operation diagram of a permanent magnet linear synchronous motor according to the present invention;
FIG. 2 is a block diagram of a dual time scale sliding mode control system of a permanent magnet linear synchronous motor;
FIG. 3 is a comparison of the speed response curves of the dual time scale sliding mode control and PID control of the present invention;
FIG. 4 is a comparison of the speed response curves of the dual time scale control of the present invention and a general sliding mode control;
FIG. 5 is a schematic diagram of an electromagnetic wave thrust curve of the dual time scale sliding mode control of the present invention;
FIG. 6 is an electromagnetic thrust of a general sliding mode control;
fig. 7 is a schematic diagram of three-phase currents of the motor under the control of the double-time scale sliding mode according to the invention.
In the figure: 1. the system comprises a slow subsystem sliding mode surface module, a fast subsystem sliding mode surface module, a 3 slow subsystem sliding mode control law module, a 4 fast subsystem sliding mode control law module, a 5 combined control law module, a 6 permanent magnet linear synchronous motor, 7 interference and 8 fast variable estimation module.
Detailed Description
The invention will be further explained with reference to the drawings.
As shown in fig. 2, the dual-time scale sliding mode control system of the permanent magnet linear synchronous motor comprises a slow subsystem sliding mode surface module 1 and a fast subsystem sliding mode surface module 2, wherein the slow subsystem sliding mode surface module 1 is connected with a combined control law module 5 through a slow subsystem sliding mode control law module 3; the fast subsystem sliding mode surface module 2 is connected with a combined control law module 5 through a fast subsystem sliding mode control law module 4; the combined control law module 5 is connected with a permanent magnet linear synchronous motor 6, the permanent magnet linear synchronous motor 6 is also connected with external interference 7, the permanent magnet linear synchronous motor 6 is connected to the slow subsystem sliding mode surface module 1, and error signals of the actual speed and the given speed of the permanent magnet linear synchronous motor 6 are transmitted to the slow subsystem sliding mode surface module 1; the permanent magnet linear synchronous motor 6 is connected to the fast subsystem sliding mode surface module 2 through the fast variable estimation module 8, and the current i of the permanent magnet linear synchronous motor 6 is converted into the current id、iqTransmitted to a fast variable estimation module 8, passes through the fast variable estimation module 8 and a slow variable current component ids、iqsFast varying current component i obtained by subtractiondf、iqfAnd transmitting to the sliding mode surface module 2 of the quick subsystem.
A control method of a permanent magnet linear synchronous motor double-time scale sliding mode control system comprises the following steps:
A. establishing a mathematical model;
basic working principle of the permanent magnet linear synchronous motor: with reference to the rotating synchronous motor, the rotating synchronous motor is cut along the radius direction of the rotating synchronous motor, and then the circumference of the rotating synchronous motor is horizontally laid along the linear direction, so that a mechanical structure similar to that of the linear motor can be obtained, and therefore, the permanent magnet linear synchronous motor can be regarded as developed by the rotating synchronous motor. The permanent magnet linear synchronous motor is mainly divided into a primary part and a secondary part, wherein the former corresponds to a stator part of the rotary synchronous motor, and the latter corresponds to a rotor part of the rotary synchronous motor. In order to generate an excitation magnetic field, N, S permanent magnets which are longitudinally magnetized are sequentially and evenly distributed at intervals along the secondary stage. In order to generate an air gap magnetic field, tooth grooves provided with three-phase armature windings are distributed on the primary iron core, and when the linear motor is connected with a power supply, a traveling wave magnetic field is generated to drive the linear motor to horizontally move along the guide rail; the basic working principle is shown in figure 1;
a.1, establishing a dynamic model:
the dynamic equation of the permanent magnet linear synchronous motor is composed of a voltage equation, a flux linkage equation, an electromagnetic thrust equation and a motion equation; the idea of applying vector control to a mathematical model of a permanent magnet linear synchronous motor on a two-phase synchronous rotation orthogonal coordinate system is obtained through coordinate transformation, so that the electromagnetic thrust is in direct proportion to a quadrature axis current component iqIs given by a given value i of the direct-axis current componentdSetting the value to zero, and obtaining a dynamic model of the simplified permanent magnet linear synchronous motor on a dq coordinate system as shown in the following formula (1):
wherein id、iq、ud、uqThe current and voltage values of d and q axes are respectively, L is inductance, R is resistance value of rotor winding, ω ═ π v/τ is angular velocity of rotor, v is velocity, τ is polar distance of magnetic pole,
is a permanent magnet flux linkage, Fe is electromagnetic thrust, M is carrier mass, B is viscous friction coefficient, FLFor load torque, KFIs an electromagnetic thrust coefficient, and is expressed by the following formula (2)) Shown in the figure:
wherein p is the magnetic pole pair number of the motor;
a.2, establishing a state equation:
rewriting the dynamic model of the permanent magnet linear synchronous motor obtained in the step A.1 into a state equation form, as shown in the following formula (3):
wherein the state variables are v and i ═ id iq]TThe controlled variable is u ═ ud uq]T；
A.3, standard form of singular perturbation:
the mathematical model under the two-phase rotation orthogonal coordinate system shows that the nonlinear coupling degree between the current and between the current and the speed is very large, so that a corresponding method is required to realize linear decoupling; considering that the value of the electrical time constant L/R is far less than that of the mechanical time constant M/B, the standard form of the motor singular perturbation can be obtained by taking σ ═ LB/MR as a perturbation parameter with a small value in the singular perturbation system as shown in formula (4):
B. establishing a double-time scale model;
b.1, establishing a slow subsystem model:
since the value of σ is very small, assuming perturbation parameter σ → 0, the full-order system model of the original permanent magnet linear synchronous motor is shown as the following formula (5):
wherein v iss,ids,iqs,uds,uqsRespectively representing slow-varying components corresponding to v, i and u; substituting the current value obtained by solving the second fraction in the formula (5) into the first fraction to obtain an expression of the slow subsystem model as shown in the formula (6):
wherein the content of the first and second substances,
udsand uqsFor the slow subsystem control signal, KFIs the electromagnetic thrust coefficient, M is the carrier mass,
is a permanent magnet flux linkage, FLThe rotor is a load torque, tau is the polar distance of a magnetic pole, B is a viscous friction coefficient, L is inductance, and R is the resistance value of a rotor winding;
b.2, establishing a fast subsystem model:
compared with the slow subsystem, let v be constant,
the available fast-changing current component is as shown in equation (7):
and (3) taking the fast time scale gamma as t/sigma, and finally obtaining a mathematical model of the fast subsystem as shown in formula (8):
wherein if＝[idf iqf]T,udf、uqfIs a control signal of the fast subsystem, KFIs the electromagnetic thrust coefficient, M is the carrier mass,
is a permanent magnet flux linkage, FLThe rotor is a load torque, tau is the polar distance of a magnetic pole, B is a viscous friction coefficient, L is inductance, and R is the resistance value of a rotor winding;
and B.3, obtaining a double-time scale model:
by the formulas (6) and (8), a permanent magnet linear synchronous motor double-time scale model is obtained as shown in the formula (9):
C. designing a sliding mode control law;
c.1 slow subsystem sliding mode function:
will give a given velocity vsActual speed v of linear synchronous motor with permanent magnetsObtaining a speed error signal esAnd sending the data to a sliding mode surface module of the slow subsystem; the slow subsystem sliding mode surface module is used for generating a speed error signal e according to the speed error signalsObtaining a sliding mode function value S of the slow subsystem as shown in the formula (10)s(es)：
Wherein, CsIs a speed error coefficient, and Cs>0; and a speed error es＝v*-vsV is a given velocity signal;
c.2 slow subsystem equivalent control law:
the slow subsystem sliding mode control law module outputs a sliding mode function value S according to the sliding mode surfaces(es) Calculating to obtain a control signal us(ii) a In the formula (10), the external disturbance d is not consideredsAnd is and
at 0, the sliding mode function is derived to obtain the form shown in equation (11):
obtaining the equivalent control law of the slowness subsystem as shown in the formula (12):
after considering the applied disturbance ds, the switching robustness term is designed as shown in the following equation (13):
uss＝Kssign(Ss) (13)；
the joint vertical type (12) and the formula (13) can obtain the sliding mode control law of the slowness sub-system as shown in the formula (14):
in order to effectively reduce the chattering phenomenon, the sign function may be replaced by a saturation function, so that the slow subsystem control law equation (14) is rewritten as shown in the following equation (15):
wherein δ is the boundary layer thickness;
c.3, a sliding mode function of a fast subsystem:
the fast variable estimation module is used for converting the current id、iqRespectively with slowly varying current component ids、iqsSubtracting to obtain a fast-changing current component idf、iqfAnd sending the data to a fast subsystem sliding mode surface module; the fast subsystem sliding mode surface module is used for generating a fast variable current signal i according to the fast variable current signal idf、iqfObtaining a sliding mode function value S of the fast subsystem as shown in the formula (16)f；
Wherein, CfIs a slip form surface coefficient of the fast subsystem, and Cf>0；
C.4 equivalent control law of the fast subsystem:
the sliding mode control law module of the fast subsystem outputs a sliding mode function value S according to the sliding mode functionfCalculating to obtain a control signal uf(γ); in order to weaken buffeting phenomenon existing in the system, an approach law design is adopted
Wherein epsilonf＝diag(εdf,εqf)，Sf＝[S1f S2f]T，Kf＝diag(Kdf,Kqf)；
Solving the formula (17) to obtain the sliding mode control law of the fast subsystem as shown in the formula (18):
wherein the content of the first and second substances,
ufa control signal representing the fast subsystem, M is the carrier mass,
is a permanent magnet flux linkage, KFIs the electromagnetic thrust coefficient, tau is the polar distance of magnetic pole, B is the viscous friction coefficient, L is inductance, R is the resistance value of rotor winding, SfIs a sliding mode surface function of the fast subsystem;
D. synthesizing a combined controller;
will control signal usControl signal uf(gamma) sending to the combinationThe control law module is combined into a combined controller; the combined controller sends a control signal of the full-order system to the permanent magnet linear synchronous motor;
the joint type (14) and the formula (18) are combined, the system mainly plays a role in a slow subsystem, and the time scale can be unified into a slow-variable subsystem time scale t; and then adding the control laws shown in the formula (14) and the formula (18) to finally obtain a control signal of the full-order system shown in the formula (19):
u＝us+uf (19)；
E. and (3) stability analysis:
for the slowness sub-system (6), a lyapunov function is defined as (20):
to LsTaking the derivative of time, one can obtain:
wherein the content of the first and second substances,
satisfy the requirement of
The available slowness subsystem is stable;
for the promiscuous system (8), a lyapunov function is defined as formula (22):
to LfTaking the derivative of time to obtain
For inequality
Is solved as
It can be seen that Lf(t) exponential convergence to 0, the rate of convergence depending on KfThe fast subsystem is known to be numerically stable.
According to the method, stable control laws are respectively designed according to the speed system and the slow system according to the singular perturbation principle, and the obtained combined control law is stable.
In this embodiment, in order to verify the effectiveness and advantages of the designed system, the embodiment builds a model and performs simulation for a dual-time scale sliding mode control system and method for a permanent magnet linear synchronous motor: the parameters of the permanent magnet linear synchronous motor are set as follows, the viscous friction coefficient B is 0.22, the carrier mass M is 100kg, the pole pitch tau of the magnetic pole is 3.6cm, the pole pair p of the motor in the permanent magnet flux linkage is 3, and the parameters of the controller of the slow subsystem are set as follows Cs＝[0.4 0.4]T,KsBiag (0,81), δ 0.01; parameter settings for the fast subsystem controller are as follows Cf＝diag(1,1)，εf＝diag(30,30)，KfDiag (300); inputting a step signal with the speed of 1r/s, starting the permanent magnet linear synchronous motor in an idle load mode, and suddenly adding load disturbance of TL 100N when t is 0.5 s; the simulation results of the permanent magnet linear synchronous motor double-time scale sliding mode control system are shown in fig. 3 to 7; fig. 3 to 7 are speed response curves of double-time scale sliding mode control and PID control, speed response curves of double-time scale control and general sliding mode control, electromagnetic wave thrust curves of double-time scale sliding mode control, electromagnetic thrust curves of general sliding mode control, and three-phase current schematic curves of a motor under double-time scale sliding mode control, respectively; as can be seen from fig. 3, compared with PID control, the dual-time scale sliding mode control has faster dynamic response speed, better dynamic performance, and better robustness to external disturbance; as can be seen from fig. 4, compared with the general equivalent sliding mode control, the quasi-sliding mode method and the approach law method are adopted, the dynamic quality of the system is better, and the robustness to external interference is stronger; drawing(s)Compared with fig. 6, it can be known that compared with the general equivalent sliding mode control, the dual-time scale sliding mode control designed by the quasi-sliding mode method and the approach law method has stronger buffeting restraining capability, reduces the influence caused by thrust fluctuation, and simultaneously, when external load disturbance occurs, the electromagnetic thrust can overcome the interference of the electromagnetic thrust on the performance of the motor system; the schematic curve of the three-phase current of the motor under the double-time scale sliding mode control is shown in fig. 7, and the buffeting phenomenon of the system can be improved. It should be noted that the excellent performance exhibited by the present example is illustrative of the present invention and not limiting.
The design process and the idea of the permanent magnet linear synchronous motor double-time scale sliding mode control system and method are explained above. The method is established as a double-time scale model of the permanent magnet linear synchronous motor, sliding mode control laws corresponding to a fast subsystem and a slow subsystem are respectively designed by adopting a quasi-sliding mode method and an approach law method, then time scales of the two subsystems are unified, a combined controller of the permanent magnet linear synchronous motor is synthesized, and meanwhile, the stability of the system is analyzed by applying the Lyapuloff stability theory. Simulation results show that the control system not only has high dynamic response speed, but also has strong robustness to external disturbance, and can realize accurate tracking of given speed signals. In addition, the buffeting phenomenon of the sliding mode control is greatly improved.
Claims (1)
1. A permanent magnet linear synchronous motor double-time scale sliding mode control method is characterized by comprising a slow subsystem sliding mode surface module and a fast subsystem sliding mode surface module, wherein the slow subsystem sliding mode surface module is connected with a combined control law module through a slow subsystem sliding mode control law module; the fast subsystem sliding mode surface module is connected with a combined control law module through the fast subsystem sliding mode control law module; the combined control law module is connected with a permanent magnet linear synchronous motor, the permanent magnet linear synchronous motor is also connected with external interference, and the permanent magnet linear synchronous motor is connected to the slow subsystem sliding mode surface module; the permanent magnet linear synchronous motor is connected to the fast subsystem sliding mode surface module through the fast variable estimation module;
and comprises the steps of:
A. establishing a mathematical model;
a.1, establishing a dynamic model:
the dynamic equation of the permanent magnet linear synchronous motor is composed of a voltage equation, a flux linkage equation, an electromagnetic thrust equation and a motion equation; the idea of applying vector control to a mathematical model of a permanent magnet linear synchronous motor on a two-phase synchronous rotation orthogonal coordinate system is obtained through coordinate transformation, so that the electromagnetic thrust is in direct proportion to a quadrature axis current component iqSetting the given value of the direct-axis current component as zero, and obtaining a dynamic model of the simplified permanent magnet linear synchronous motor on a dq coordinate system as shown in the following formula (1):
wherein id、iq、ud、uqThe current and voltage values of d and q axes respectively, L is inductance, R is resistance value of rotor winding, ω ═ π v/τ is angular velocity of rotor, v is velocity, τ is polar distance of magnetic pole,
is a permanent magnet flux linkage, Fe is electromagnetic thrust, M is carrier mass, B is viscous friction coefficient, FLFor load torque, KFThe expression of the electromagnetic thrust coefficient is shown in the following formula (2):
wherein p is the magnetic pole pair number of the motor;
a.2, establishing a state equation:
rewriting the dynamic model of the permanent magnet linear synchronous motor obtained in the step A.1 into a state equation form, as shown in the following formula (3):
wherein the state variables are v and i ═ id iq]TThe controlled variable is u ═ ud uq]T；
A.3, standard form of singular perturbation:
taking sigma-LB/MR as perturbation parameters in a singular perturbation system, and obtaining a standard form of the singular perturbation of the motor as shown in a formula (4):
B. establishing a double-time scale model;
b.1, establishing a slow subsystem model:
since the value of σ is very small, assuming perturbation parameter σ → 0, the full-order system model of the original permanent magnet linear synchronous motor is shown as the following formula (5):
wherein v iss,ids,iqs,uds,uqsRespectively representing slow-varying components corresponding to v, i and u; substituting the current value obtained by solving the second fraction in the formula (5) into the first fraction to obtain an expression of the slow subsystem model as shown in the formula (6):
wherein the content of the first and second substances,
udsand uqsFor the slow subsystem control signal, KFIs the electromagnetic thrust coefficient, M is the carrier mass,
is a permanent magnet flux linkage, FLThe rotor is a load torque, tau is the polar distance of a magnetic pole, B is a viscous friction coefficient, L is inductance, and R is the resistance value of a rotor winding;
b.2, establishing a fast subsystem model:
compared with the slow subsystem, let v be constant,
the available fast-changing current component is as shown in equation (7):
and (3) taking the fast time scale gamma as t/sigma, and finally obtaining a mathematical model of the fast subsystem as shown in formula (8):
wherein if＝[idf iqf]T,udf、uqfIs a control signal of the fast subsystem, KFIs the electromagnetic thrust coefficient, M is the carrier mass,
is a permanent magnet flux linkage, FLThe rotor is a load torque, tau is the polar distance of a magnetic pole, B is a viscous friction coefficient, L is inductance, and R is the resistance value of a rotor winding;
and B.3, obtaining a double-time scale model:
by the formulas (6) and (8), a permanent magnet linear synchronous motor double-time scale model is obtained as shown in the formula (9):
C. designing a sliding mode control law;
c.1 slow subsystem sliding mode function:
will give a given velocity vsActual speed v of linear synchronous motor with permanent magnetsObtaining a speed error signal esAnd sending the data to a sliding mode surface module of the slow subsystem; the slow subsystem sliding mode surface module is used for generating a speed error signal e according to the speed error signalsObtaining a sliding mode function value S of the slow subsystem as shown in the formula (10)s(es)：
Wherein, CsIs a speed error coefficient, and Cs>0; and a speed error es＝v*-vsV is a given velocity signal;
S1s、S2sfor slow subsystem sliding mode function value SsTwo components written in the form of column vectors;
c.2 slow subsystem equivalent control law:
the slow subsystem sliding mode control law module outputs a sliding mode function value S according to the sliding mode surfaces(es) Calculating to obtain a control signal us(ii) a In the formula (10), the external disturbance d is not consideredsAnd is and
at 0, the sliding mode function is derived to obtain the form shown in equation (11):
wherein, C1s、C2sIs a speed error coefficient CsTwo components written in the form of column vectors; f (v)s) To relate to vsA function of (a);
obtaining the equivalent control law of the slowness subsystem as shown in the formula (12):
after considering the external disturbance ds, the designed switching robustness term is shown as the formula (13): u. ofss＝Kssign(Ss) (13)；
Wherein, KsRepresenting the sliding mode switching control gain of the slow subsystem;
the joint vertical type (12) and the formula (13) can obtain the sliding mode control law of the slowness sub-system as shown in the formula (14):
c.3, a sliding mode function of a fast subsystem:
the fast variable estimation module is used for converting the current id、iqRespectively with slowly varying current component ids、iqsSubtracting to obtain a fast-changing current component idf、iqfAnd sending the data to a fast subsystem sliding mode surface module; the fast subsystem sliding mode surface module is used for generating a fast variable current signal i according to the fast variable current signal idf、iqfObtaining a sliding mode function value S of the fast subsystem as shown in the formula (16)f；
Wherein, CfIs a slip form surface coefficient of the fast subsystem, and Cf>0；S1f、S2fFor the sliding mode function value S of the express subsystemfTwo components written in the form of column vectors;
c.4 equivalent control law of the fast subsystem:
the sliding mode control law module of the fast subsystem outputs a sliding mode function value S according to the sliding mode functionfCalculating to obtain a control signal uf(γ); in order to weaken buffeting phenomenon existing in the system, an approach law design is adopted
Wherein epsilonf＝diag(εdf,εqf)，Sf＝[S1f S2f]T，Kf＝diag(Kdf,Kqf)；εdf、εqfIs epsilonfTwo components on d-and q-axes, Kdf、KqfIs KfTwo components on the d-axis and q-axis; epsilonfAnd KfParameters of an approach law;
solving the formula (17) to obtain the sliding mode control law of the fast subsystem as shown in the formula (18):
wherein the content of the first and second substances,
ufa control signal representing the fast subsystem, M is the carrier mass,
is a permanent magnet flux linkage, KFIs the electromagnetic thrust coefficient, tau is the polar distance of magnetic pole, B is the viscous friction coefficient, L is inductance, R is the resistance value of rotor winding, SfIs a sliding mode surface function of the fast subsystem;
D. synthesizing a combined controller;
will control signal usControl signal uf(gamma) sending to a combination control law module to combine the combination controller; the combined controller sends a control signal of the full-order system to the permanent magnet linear synchronous motor;
the joint type (14) and the formula (18) are combined, the system mainly plays a role in a slow subsystem, and the time scales are unified into a slow-variable subsystem time scale t; then adding the control laws shown in the formula (14) and the formula (18) to obtain the formula(19) Control signals for the full-order system shown; u-us+uf (19)；
E. And (3) stability analysis:
for the slowness sub-system (6), a lyapunov function is defined as (20):
to LsTaking the derivative of time, one can obtain:
wherein the content of the first and second substances,
satisfy the requirement of
The slowness system is known to be stable;
Kds、Kqsis KsTwo components on the d-axis and q-axis;
for the promiscuous system (8), a lyapunov function is defined as formula (22):
to LfTaking the derivative of time, one can obtain:
for inequality
Is solved as
It can be seen that Lf(t) exponential convergence to 0, the rate of convergence depending on KfThe fast subsystem is known to be numerically stable; respectively designing stable control laws according to the speed system and the slow system according to the singular perturbation principle, wherein the obtained combined control law is stable, and Lf(t0) Represents the initial value, t, of the Lyapunov function of the fast subsystem0Indicating the initial time.
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| CN112468044B (en) * | 2020-11-10 | 2022-09-06 | 东南大学 | Model prediction control system and method for permanent magnet synchronous linear motor |
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