CN109532509B - Magnetic-levitation train suspension control method based on sliding mode variable structure control - Google Patents

Magnetic-levitation train suspension control method based on sliding mode variable structure control Download PDF

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CN109532509B
CN109532509B CN201811622146.4A CN201811622146A CN109532509B CN 109532509 B CN109532509 B CN 109532509B CN 201811622146 A CN201811622146 A CN 201811622146A CN 109532509 B CN109532509 B CN 109532509B
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suspension
magnetic
control
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equation
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CN109532509A (en
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王成杰
陈涛
许平洋
王小明
伍星
杨颖�
刘志刚
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Southwest Jiaotong University
CRRC Zhuzhou Locomotive Co Ltd
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CRRC Zhuzhou Locomotive Co Ltd
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B60—VEHICLES IN GENERAL
    • B60L—PROPULSION OF ELECTRICALLY-PROPELLED VEHICLES; SUPPLYING ELECTRIC POWER FOR AUXILIARY EQUIPMENT OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRODYNAMIC BRAKE SYSTEMS FOR VEHICLES IN GENERAL; MAGNETIC SUSPENSION OR LEVITATION FOR VEHICLES; MONITORING OPERATING VARIABLES OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRIC SAFETY DEVICES FOR ELECTRICALLY-PROPELLED VEHICLES
    • B60L13/00—Electric propulsion for monorail vehicles, suspension vehicles or rack railways; Magnetic suspension or levitation for vehicles
    • B60L13/04—Magnetic suspension or levitation for vehicles
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B61—RAILWAYS
    • B61B—RAILWAY SYSTEMS; EQUIPMENT THEREFOR NOT OTHERWISE PROVIDED FOR
    • B61B13/00—Other railway systems
    • B61B13/08—Sliding or levitation systems
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

The invention discloses a maglev train suspension control method based on sliding mode variable structure control, which comprises the steps of establishing a system dynamic model equation according to a dynamic equation of a maglev train suspension control system, an electromagnet voltage equation and the change of a track action surface; analyzing and designing a magnetic suspension control system by utilizing a linearization theory, and approximating the obtained nonlinear equation set to a linearization model; judging the stability of the system according to the approximation model, selecting a proper state variable, then selecting a directly measurable state variable as a feedback quantity, and carrying out state feedback control; analyzing and utilizing the basic principle of a sliding mode control algorithm, establishing a magnetic suspension control system model based on sliding mode control, and adopting a proper control law to provide the design structure of the suspension controller. The invention can improve the performance of the controller in the suspension control system of the maglev train and can well meet the requirements of quick response, stability and anti-interference performance in the suspension process of the maglev train.

Description

Magnetic-levitation train suspension control method based on sliding mode variable structure control
Technical Field
The invention relates to the technical field of levitation control, in particular to a levitation control method of a magnetic-levitation train based on sliding mode variable structure control.
Background
The suspension control technology is one of the core and key technologies of the magnetic-levitation train, and is the foundation and the premise for realizing the normal operation of the magnetic-levitation train. The suspension control of the electromagnet of the domestic magnetic-levitation train basically adopts the lead-lag compensation technology in the classical control theory, so that the electromagnet is dynamically stabilized on a certain set air gap. This technique is capable of producing a stable magnetic levitation in most cases, but is not sufficient for the performance aspects of damping, stability margin, anti-disturbances, etc. required by passenger vehicles. Particularly, with the continuous improvement of the speed of the magnetic-levitation train, the increase of the complexity of the control system and the improvement of various performance requirements of the train, the PID control adopted by the traditional levitation controller can not meet the requirements of the technical development of the magnetic-levitation train.
The traditional PID controller of the maglev train has the working principle that a nonlinear system of the traditional PID controller of the maglev train is subjected to linearization processing to obtain a linearization model for state feedback control. The defect is that the working range is limited, the control performance of the controller is good only near the working point, and when the system is far away from the working point, the system can not achieve the expected control effect and even is unstable; meanwhile, the robustness is poor, the controller is designed based on an ideal state, and when the system encounters the problems of random disturbance, track irregularity, coupled vibration of a vehicle rail, passing through a curve, a vertical curve of the track and the like, a system model and parameters change, so that the controller cannot overcome the perturbation of system parameters on the control performance. This requires a more advanced and efficient control scheme, improving and perfecting the levitation control technology of the maglev train.
The Sliding Mode Variable Structure Control (Sliding Mode Variable Structure Control) has the advantages of fast dynamic response, no overshoot and strong robustness, and the Control Structure is simple and is easy to realize in engineering. Therefore, further research on the suspension controller of the magnetic-levitation train is necessary, and the existing control algorithm is improved and perfected.
Disclosure of Invention
In view of the above problems, an object of the present invention is to provide a method for controlling levitation of a magnetic levitation train based on sliding mode variable structure control, which can improve the performance of a controller in a levitation control system of the magnetic levitation train and can well meet the requirements of fast response, stability and anti-interference performance during the levitation process of the magnetic levitation train. The technical scheme is as follows:
a magnetic-levitation train suspension control method based on sliding mode variable structure control comprises the following steps:
step 1: constructing a dynamic model equation of a suspension system of the magnetic-levitation train:
Figure BDA0001927126060000021
in the formula: m is the equivalent mass of the suspension electromagnet, and z (t) is the distance from the magnetic pole surface to the reference plane; g is the acceleration of gravity; f. ofdDisturbance is an external dry disturbance; f (i, c) is the electromagnetic attraction force at the moment t; u (t) is the electromagnet winding voltage; r is equivalent winding resistance of the electromagnet; h (t) is the distance from the suspension rail to the reference plane; mu.s0Air permeability; c (t) is a suspended air gap; i (t) is the coil current; a is the area of the electromagnetic iron pole; n electromagnet winding turns;
step 2: carrying out linearization treatment on a dynamic model of the magnetic suspension system:
the above formula is placed at the equilibrium point (z)0,c0,h0,i0) And performing Taylor series expansion near the position, and neglecting a high-order term of the Taylor expansion formula to obtain a linearization form of the electromagnetic attraction in the neighborhood of the balance point as follows:
F(i,c)＝F(i0,c0)+△F＝F(i0,c0)+ki△i(t)-kc△c(t)
in the formula: z is a radical of0，c0，h0And i0The distance from the magnetic pole surface at the balance point to the reference plane, the suspension air gap, the distance from the suspension rail to the reference plane and the coil current are respectively; f (i)0,c0) The electromagnetic force generated by the magnetic poles at the balance point, △ F the variation of the electromagnetic force near the balance point, △ i (t) the variation of the current near the balance point, △ c (t) the variation of the gap near the balance point, and kiAnd kcRespectively a system current stiffness coefficient and a displacement stiffness coefficient, and
Figure BDA0001927126060000022
Figure BDA0001927126060000023
Fi(i0,c0) And Fc(i0,c0) The partial derivatives of the electromagnetic force at the balance point to the coil current and the suspension air gap respectively;
the electromagnet winding voltage equation is developed:
Figure BDA0001927126060000024
in the formula: l is0In order to balance the point inductance,
Figure BDA0001927126060000025
u0＝Ri0△ i, △ c are coil current and suspension air gap variation respectively;
the linear equation of the dynamic model equation set of the magnetic suspension system is as follows:
Figure BDA0001927126060000031
in the formula:
Figure BDA0001927126060000032
is the double differential of the distance of the magnetic pole surface to the reference plane with respect to time;
Figure BDA0001927126060000033
△ h (t) is the distance variation of the suspension rail to the reference plane;
and step 3: acquiring a relative reference model equation of the magnetic suspension system:
irrespective of the change of the active surface of the track, let h0And ≡ 0, △ z (t) △ c (t), and performing a lagrange transform on the linear equation of the dynamic model equation system to obtain a relative reference model equation of the system, wherein:
Figure BDA0001927126060000034
in the formula: c(s), Fd(S), I (S), U (S) are respectively the floating air gap value, the interference force, the,Coil current and coil voltage; l is inductance;
and 4, step 4: establishing a magnetic suspension control system model based on sliding mode control:
without considering external interference, by the output current I of the electromagnet0Is the system input quantity and the output voltage U after passing through the position sensor0For output quantity, an input-output transfer function of the magnetic suspension system is established:
Figure BDA0001927126060000035
in the formula: k is a radical ofSSelecting system state variables for position sensor gain
Figure BDA0001927126060000036
x represents the output voltage U obtained after the electromagnet is detected by the position sensor0，I0Is an input variable;
the system transfer function is transformed into a state space form:
Figure BDA0001927126060000037
in the formula:
Figure BDA0001927126060000038
is the first derivative of the state variable X; y is an output state quantity; a is11、a12、a21、a22、b1、b2、c1And c2 is the matrix gain factor;
designing a system switching function, and according to a magnetic suspension system state space equation, knowing that the state variable of the system is as follows:
Figure BDA0001927126060000039
systematic error e and error rate of change
Figure BDA00019271260600000310
Respectively as follows:
Figure BDA00019271260600000311
in the formula: r is a reference input quantity;
the switching function of the system is taken as:
Figure BDA0001927126060000041
in the formula: c ═ C1]Representing a gain matrix, c being a undetermined constant and c>0；
Figure BDA0001927126060000045
Is a systematic error matrix;
adopting an exponential approximation law method for weakening buffeting, and setting a system approximation law as slaw, the method comprises the following steps:
Figure BDA0001927126060000042
ξ, k are undetermined constants;
then according to the switching function, the following results are obtained:
Figure BDA0001927126060000043
obtaining the control law of the system according to the state space form of the system transfer function and the above formula:
Figure BDA0001927126060000044
in the formula: u is the output control quantity of the system controller;
and (4) adjusting the input voltage according to the switching function and the output control quantity of the sliding mode control law, so that the electromagnet is adjusted to realize the stable suspension of the train through current.
The invention has the beneficial effects that: the invention combines the specific situation of the suspension controller, and on the basis of analyzing the system modeling and control effect of the traditional suspension controller and the control algorithm of the existing suspension controller, firstly proposes that a sliding mode control algorithm which has a relatively simple structure, higher response speed and good control performance on disturbance perturbation is adopted in the magnetic suspension controller; the method can well meet the requirements of quick response, stability and anti-interference performance in the suspension process of the magnetic-levitation train.
Drawings
Fig. 1 is a structural diagram of a single electromagnet suspension system of a magnetic-levitation train.
Fig. 2 is a diagram of a single electromagnet levitation system relative to a reference model.
Figure 3 is a block diagram of electromagnet position-velocity-acceleration feedback control.
FIG. 4 is a diagram of a magnetic levitation system with a sliding mode variable structure controller.
FIG. 5 is a waveform diagram of a simulation result of a levitation gap of the PID controller.
FIG. 6 is a diagram of PID control levitation gap position response after disturbance addition.
FIG. 7 is a graph of levitation system position response based on a sliding mode variable controller.
FIG. 8 is a dynamic real-time position response diagram of a suspension gap under sliding mode control.
Detailed Description
The invention is described in further detail below with reference to the figures and specific embodiments. According to a dynamic equation of a suspension control system of a magnetic-levitation train, an electromagnet voltage equation and changes of a track action surface (mainly the distance of an air gap), a system dynamic model equation is established; analyzing and designing a magnetic suspension control system by utilizing a linearization theory, and approximating the obtained nonlinear equation set to a linearization model; judging the stability of the system according to the approximation model, selecting a proper state variable, then selecting the state variable which can be directly measured as a feedback quantity, and carrying out state feedback control; analyzing and utilizing the basic principle of a sliding mode control algorithm, establishing a magnetic suspension control system model based on sliding mode control, and adopting a proper control law to provide the design structure of the suspension controller.
The invention designs a novel magnetic suspension controller based on a sliding mode control algorithm by planning the modeling process, closed-loop control, state feedback control, simulation platform function design and the like of the suspension control system of the medium-low speed magnetic suspension train, so that the robustness, dynamic response speed, convergence control speed, accuracy, external interference suppression and other aspects of the controller better meet the requirements of the development of the magnetic suspension train. The method comprises the following steps:
①, establishing a simulation model and a PID suspension controller model of the magnetic suspension train, analyzing the control result of the magnetic suspension train system, and obtaining the system damping, vibration frequency and stable working range mainly by analyzing the voltage/current and position-speed-acceleration feedback of the control system;
② A sliding mode controller of the suspension control system of the magnetic suspension controller is designed according to a specific vehicle type, firstly, an input and output transfer function is established according to the magnetic suspension control system added with a position sensor, a switching function and a sliding mode control law of the system are selected and designed, secondly, the passing current of an electromagnet is adjusted by outputting the input voltage of a control quantity adjusting system, so that the stable suspension of the train is realized, and finally, uncertain factors such as climbing resistance, external pressure and the like are simulated by adding interference variables, so that the anti-interference capability of the suspension control system is improved.
The method comprises the following specific steps:
step 1: constructing a dynamic model equation of a suspension system of the magnetic-levitation train:
assuming that the magnetic permeability in the magnetic pole is infinite and the magnetic potential is uniformly distributed on the air gap, neglecting the leakage flux of the winding, the electromagnetic attractive force F (i, c) at the time t is:
Figure BDA0001927126060000051
in the formula: mu.s0Has a value of 4 pi 10 for air permeability-7H·m-1(ii) a t is time; c (t) is a suspended air gap; i (t) is the coil current magnitude; a is the area of the electromagnetic iron pole; n electromagnet winding turns; and B is magnetic induction intensity. As shown in the formula (1), the functional relationship between the electromagnetic attraction and the air gap is a nonlinear inverse square relationship, i.e. the instability of the suspension system of the maglev train is illustrated, which needs to be carried outAnd (5) controlling.
And obtaining a voltage equation according to an electromagnet winding loop in the single electromagnet suspension system as follows:
Figure BDA0001927126060000052
in the formula: u (t) is the electromagnet coil voltage; r is equivalent winding resistance of the electromagnet; h (t) is the distance from the suspension rail to the reference plane;
Figure BDA0001927126060000061
is the differential of the coil current with respect to time;
Figure BDA0001927126060000062
the derivative of the floating air gap with respect to time.
The mechanical balance equation of the electromagnet in the vertical direction is as follows:
Figure BDA0001927126060000063
in the formula: m is the equivalent mass of the suspension electromagnet, and the reference size is 330 kg;
Figure BDA0001927126060000064
is the double differential of the distance of the magnetic pole surface to the reference plane with respect to time; g is the acceleration of gravity; f. ofdThe external interference is 1500N. In combination with the model of the single-electromagnet suspension system in fig. 1, the changes of the action surface of the guide rail are considered as follows:
z(t)＝h(t)+c(t) (4)
in the formula: z (t) is the distance of the pole surface from the reference plane.
In conclusion, a dynamic model of the maglev train suspension system is obtained according to the mechanics, circuit and electromagnetic equations of the single-electromagnet suspension system:
Figure BDA0001927126060000065
step 2: carrying out linearization treatment on a dynamic model of the magnetic suspension system:
the model is a nonlinear system, which usually requires a linearization process of the dynamic model, as shown in equation (5). Placing formula (5) at equilibrium point (z)0,c0,h0,i0) And performing Taylor series expansion on the vicinity of the position to obtain:
Figure BDA0001927126060000066
in the formula: z is a radical of0，c0，h0And i0The distance from the magnetic pole surface at the balance point to the reference plane, the suspension air gap, the distance from the suspension rail to the reference plane and the coil current are respectively; f (i)0,c0) The electromagnetic force generated by the magnetic pole is large and small; fi(i0,c0)，Fc(i0,c0) △ F is the variation of the electromagnetic force near the balance point, △ i (t) is the variation of the current near the balance point, △ c (t) is the variation of the gap near the balance point, and α (△ i, △ c) is the Taylor expansion high-order term about the current and the gap at the balance point.
Neglecting the taylor expansion high order terms, the linearized form of the equilibrium point neighborhood can be obtained as:
Figure BDA0001927126060000067
in the formula: c. C0,i0Air gap and current at the balance point, respectively; f (i)0,c0) The electromagnetic force generated by the magnetic pole is large and small; k is a radical ofi,kcRespectively is a system current rigidity coefficient and a displacement rigidity coefficient, and the magnitude is as follows:
Figure BDA0001927126060000071
Figure BDA0001927126060000072
similarly, the winding voltage equation in equation (2) can be expanded:
Figure BDA0001927126060000073
in the formula: l is0In order to balance the point inductance,
Figure BDA0001927126060000074
u0＝Ri0△ i, △ c are current and gap variation, respectively.
The linear equation of the dynamic model equation set of the magnetic suspension system shown in the formula (5) is:
Figure BDA0001927126060000075
in the formula:
Figure BDA0001927126060000076
the double differential of the distance of the pole surface to the reference plane with respect to time;
Figure BDA0001927126060000077
is the differential of the coil current with respect to time.
And step 3: acquiring a relative reference model equation of the magnetic suspension system:
let h0The equation of the relative reference model of the system obtained by carrying out the Laplace transform on the equation (9) is that the equation is equal to 0, △ z (t) ═ △ c (t), namely the mathematical model under the condition that the change of the orbit action surface is not considered, is called as the relative reference model:
Figure BDA0001927126060000078
in the formula: c(s), Fd(s), I(s), U(s) are respectively the gap value, the interference force, the coil current and the coil voltage in the complex frequency domain. The construction of the relative reference model is shown in fig. 2.
The characteristic equation of the system is (ms) obtained by analyzing the equation transfer function in the formula (10)2-kc) (Ls + R) ═ 0, according to the Laus criterion the system characteristic root distribution condition can be obtainedIn this case, since it can be determined that the system is an unstable system of three stages, feedback control is required.
At present, a single-iron suspension control system is mainly designed by adopting a state feedback method. The current is used as a control quantity, a proper state variable is selected, and then the state quantity which can be directly measured is selected as a feedback quantity. Taking position-velocity-acceleration feedback as an example, choose
Figure BDA0001927126060000079
The state variables, △ c,
Figure BDA00019271260600000710
as a feedback variable, the control current can then be expressed as:
Figure BDA00019271260600000711
in the formula: k is a radical ofp,kv,kaIs a reference coefficient of the variable.
The system state equation obtained according to equations (9) and (11) is:
Figure BDA0001927126060000081
the system control block diagram is obtained according to the system state equation (12), as shown in fig. 3. The performance of the state feedback closed loop system is kp,kv,kaThree parameters are jointly determined.
The sliding mode variable structure control is introduced, and the basic principle of the sliding mode variable structure controller of the suspension system of the magnetic-levitation train is to output control quantity and adjust input voltage according to a designed switching function and a sliding mode control law, so that the stable suspension of the train is realized by adjusting the electromagnet through current.
Firstly, the external interference is not considered, and the output current I of the electromagnet is used0Is the system input quantity and the output voltage U after passing through the position sensor0For output quantity, an input-output transfer function of the magnetic suspension system is established:
Figure BDA0001927126060000082
in the formula: k is a radical ofSSelecting system state variables for position sensor gain
Figure BDA0001927126060000083
x represents the output voltage U obtained after the electromagnet is detected by the position sensor0The input variable is I0Then the system transfer function is transformed into a state space form:
Figure BDA0001927126060000084
in the formula:
Figure BDA0001927126060000085
is the first derivative of the state variable X; y is an output state quantity; a, b and c are matrix gain coefficients.
Constructing a block diagram of a magnetic levitation control system with a sliding mode controller, as shown in fig. 4, wherein r is a reference input, u is an output control quantity of the system controller, y is a position output of the system, x is an output voltage after passing through a position sensor,
Figure BDA0001927126060000086
the first derivative of x.
Designing a system switching function, and according to a magnetic suspension system state space equation, the state variable of the system is as follows:
Figure BDA0001927126060000087
the system error and the error change rate are respectively:
Figure BDA0001927126060000088
the switching function of the system is taken as:
Figure BDA0001927126060000089
in the formula: c ═ C1],
Figure BDA00019271260600000810
Adopting an exponential approximation law method for weakening buffeting, and setting a system approximation law as slaw, the method comprises the following steps:
Figure BDA0001927126060000091
further according to formula (17):
Figure BDA0001927126060000092
in order to ensure the stability of the sliding mode of the system, the polynomial p + c is Hurwitz stable, and the stable c range is c >0 according to the Laus criterion.
The control law of the system obtained from equations (14) and (19) is:
Figure BDA0001927126060000093
wherein c, ξ and k are undetermined constants.
And finally, respectively establishing a magnetic levitation PID control system simulation diagram and a sliding mode variable structure control system simulation diagram in an MATLAB/Simulink environment. The initial position of the suspension electromagnet of the maglev train is set to deviate from the balance point by about 8mm, and the suspension gap stability value is 9 mm.
Firstly, a magnetic levitation PID control system simulation is set up, and the real-time position response of a suspension body gap is shown in FIG. 5.
The simulation result graph shows that the clearance of the suspension system is gradually stabilized at about 9mm under the control, and the vertical fluctuation is within the range of +/-2 mm, so that the clearance standard of the suspension control of the train is met. The system is a static difference-free system. When a 1500N dynamic disturbance force is applied, the levitation gap fluctuation diagram is shown in FIG. 6. It can be seen that the dynamic performance is poor, and the control system is greatly influenced by interference. The suspension controller can be simply and conveniently designed by the control method, but the robustness is poor, and the stability margin is small.
Selecting a value of c according to experience, wherein the value of c is not suitable for being too large, ensuring the control speed and avoiding large static errors, ξ is slightly large, the static errors can be reduced, k is small, otherwise overshoot is easily caused, selecting undetermined parameters c, ξ and k of a sliding mode variable structure controller of the magnetic suspension system, wherein k is respectively c 80, ξ 120 and k 20, the sliding mode controller approaches to-120 sgn(s) -20s, adding a dynamic interference force with the size of about 1500N to control and simulate the magnetic suspension system, and the simulation result is shown in figure 7.
According to simulation results, the system realizes stable suspension, the system output is stable and has no overshoot, the overall response speed is high, the adjusting time is 0.08s, but the steady-state error of the system is large, the value is 18.5%, but the buffeting phenomenon is almost avoided, and the system has good robustness to parameter disturbance and external interference.
In order to verify the simulation result, a sliding mode variable structure controller of the maglev train suspension system is established in an S-Function module, a sliding mode variable structure control algorithm of the system is compiled, the control algorithm is moved into a DSP, an actual magnetic suspension sliding mode variable structure control system based on the DSP is established, the system is controlled in real time, and the dynamic real-time position response of the system is shown in FIG. 8.
From the real-time control result of fig. 8, the sliding mode variable structure controller can realize stable suspension, the system has no overshoot, the overall response time is fast, and the adjustment time is 0.1 s. However, the steady-state error of the system is large and reaches 24.1%, which is a necessary result caused by that the parameters of the sliding mode variable structure controller are determined by the linearized mathematical model and the designed approximation rule, and the linearized model established by the system and the actual model have certain errors.
In conclusion, the magnetic suspension control system based on sliding mode variable structure control has the advantages of good response speed, high anti-interference capability, high robustness, almost no buffeting and the like, and is a control method for optimizing the suspension control of the magnetic suspension train.

Claims (1)

1. A magnetic-levitation train suspension control method based on sliding mode variable structure control is characterized by comprising the following steps:
step 1: constructing a dynamic model equation set of a suspension system of the magnetic-levitation train:
Figure FDA0002305619640000011
in the formula: m is the equivalent mass of the suspension electromagnet, and z (t) is the distance from the magnetic pole surface to the reference plane; g is the acceleration of gravity; f. ofd(t) is external dry disturbance, and F (i, c) is electromagnetic attraction at the moment t; u (t) is the electromagnet winding voltage; r is equivalent winding resistance of the electromagnet; h (t) is the distance from the suspension rail to the reference plane; mu.s0Air permeability; c (t) is a suspended air gap; i (t) is the coil current; a is the area of the electromagnetic iron pole; n electromagnet winding turns;
step 2: carrying out linearization treatment on a dynamic model equation set of a suspension system of the magnetic-levitation train:
the above formula is placed at the equilibrium point (z)0,c0,h0,i0) And performing Taylor series expansion near the position, and neglecting a high-order term of the Taylor expansion formula to obtain a linearization form of the electromagnetic attraction in the neighborhood of the balance point as follows:
F(i,c)＝F(i0,c0)+△F＝F(i0,c0)+ki△i(t)-kc△c(t)
wherein z is0，c0，h0And i0The distance from the magnetic pole surface at the balance point to the reference plane, the suspension air gap, the distance from the suspension rail to the reference plane and the coil current are respectively; f (i)0,c0) The electromagnetic force generated by the magnetic poles at the balance point;
△ F is the variation of electromagnetic force near the balance point, △ i (t) is the variation of current near the balance point, △ c (t) is the variation of air gap near the balance point, k isiAnd kcRespectively a system current stiffness coefficient and a displacement stiffness coefficient, and
Figure FDA0002305619640000012
Fi(i0,c0) And Fc(i0,c0) The partial derivatives of the electromagnetic force at the balance point to the coil current and the suspension air gap respectively;
the electromagnet winding voltage equation is developed:
Figure FDA0002305619640000013
in the formula: l is0In order to balance the point inductance,
Figure FDA0002305619640000014
u0＝Ri0△ i, △ c are coil current and suspension air gap variation respectively;
the linear equation of the dynamic model equation set of the maglev train suspension system is as follows:
Figure FDA0002305619640000021
in the formula:
Figure FDA0002305619640000022
is the double differential of the distance of the magnetic pole surface to the reference plane with respect to time;
Figure FDA0002305619640000023
△ h (t) is the distance variation of the suspension rail to the reference plane;
and step 3: obtaining a relative reference model equation of a suspension system of the magnetic-levitation train:
irrespective of the change of the active surface of the track, let h0And ≡ 0, △ z (t) △ c (t), and performing a lagrange transform on the linear equation of the dynamic model equation system to obtain a relative reference model equation of the system, wherein:
Figure FDA0002305619640000024
in the formula: c(s), Fd(S), I (S), U (S) are respectively a complex frequency domain, namely a suspended air gap value, an interference force, a coil current and a coil voltage in the S domain; l is inductance;
and 4, step 4: establishing a magnetic suspension control system model based on sliding mode variable structure control:
without considering external interference, by the output current I of the electromagnet0Is the system input quantity and the output voltage U after passing through the position sensor0For the output quantity, an input-output transfer function of the suspension system of the magnetic-levitation train is established:
Figure FDA0002305619640000025
in the formula: k is a radical ofSIs the position sensor gain; selecting system state variables
Figure FDA0002305619640000026
x represents the output voltage U obtained after the electromagnet is detected by the position sensor0；I0Is an input variable;
the system transfer function is transformed into a state space form:
Figure FDA0002305619640000027
in the formula:
Figure FDA0002305619640000028
is the first derivative of the state variable X; y is an output state quantity; a is11、a12、a21、a22、b1、b2、c1And c2Is a matrix gain factor;
designing a system switching function, and knowing according to a state space equation of a suspension system of the magnetic-levitation train, the state variable of the system is as follows:
x1＝U0,
Figure FDA0002305619640000029
systematic error e and error rate of change
Figure FDA00023056196400000210
Respectively as follows:
e＝r-x1,
Figure FDA00023056196400000211
in the formula: r is a reference input quantity;
the switching function of the system is taken as:
Figure FDA0002305619640000031
in the formula: c ═ C1]Representing a gain matrix, c being a undetermined constant and c>0；
Figure FDA0002305619640000032
Is a systematic error matrix;
adopting an exponential approximation law method for weakening buffeting, and setting a system approximation law as slaw, the method comprises the following steps:
Figure FDA0002305619640000033
ξ, k are undetermined constants;
then according to the switching function, the following results are obtained:
Figure FDA0002305619640000034
the control law of the system is obtained according to the above equation and the state space form of the system transfer function, and can be expressed as:
Figure FDA0002305619640000035
in the formula: u is the output control quantity of the system controller;
and adjusting the input voltage according to the switching function and the sliding mode variable structure control law output control quantity, so that the electromagnet is adjusted to realize the stable suspension of the train through current.
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