CN107589666A - A kind of maglev train system control method of the sliding formwork control based on power Reaching Law - Google Patents
A kind of maglev train system control method of the sliding formwork control based on power Reaching Law Download PDFInfo
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Abstract
The present invention relates to a kind of maglev train system control method of the sliding formwork control based on power Reaching Law.The present invention based on single magnet arrangement magnetic suspension system model, input/output relation is linearized on the basis of, the control planning that the sliding formwork control of power Reaching Law is applied in single magnet arrangement magnetic suspension system is analyzed, and the stability of this method is analyzed using Li Ya spectrum promise husband's Theory of Stability.Matlab simulation results show that the sliding formwork control of power Reaching Law has more preferable control performance than traditional sliding formwork control, buffet obvious weakening, position signalling error is smaller, and dynamic characteristic is more preferable, and robustness is stronger.
Description
Technical field
The present invention relates to a kind of maglev train system control method, more particularly, to a kind of cunning based on power Reaching Law
The maglev train system control method of mould control.
Background technology
Maglev train system is the nonlinear system of a single-degree-of-freedom open-loop unstable, easily by system parameter disturbance and
The influence of external disturbance, the control to system bring difficulty.Although using the PID controller of routine system can be made stable,
System is poor to the adaptability of Parameters variation；Therefore Sakalli et al. is controlled using fuzzy, and feature is to be not required to accurately
System model, and parameter can on-line tuning, the dynamic performance and stability of satisfaction, but fuzzy control can be obtained
Fuzzy rule can only obtain by rule of thumb and algorithm is more complicated；Then Ulbig et al. uses forecast Control Algorithm, due to its robustness
It is strong and to model accuracy it is less demanding the advantages of, can effectively handle that controlled device is non-linear, probabilistic advantage, but lack
Point is that amount of calculation is larger；Uswarman et al. recognizes that sliding formwork control is insensitive, online without system to Parameters variation and disturbance
Identification, the advantages of physics realization is simple, it is controlled using sliding formwork control, still, obvious buffeting occurs in sliding formwork control
Problem, suspension object is set to occur positional fluctuation phenomenon when reaching stable state.
Buffeted to weaken, improve the dynamic property of system, reaching law method can be used, the present invention uses power Reaching Law
Sliding formwork control (PAR-SMC) strategy magnetic suspension system is controlled, simulation result shows the sliding formwork using power Reaching Law
Control not only enables a system to stable suspension, buffets obvious weakening, and have good tracking performance.
Sliding formwork control (SMC) is a kind of special nonlinear Control, and it shows as the discontinuity of control, can be according to system
Current state, purposefully constantly change, makes system be moved according to predetermined " sliding mode " state trajectory, and with other controls
Method difference processed is that system " structure " is not fixed.Although sliding mode can carry out parameter designing and with the parameter of object and
Disturb it is unrelated, this cause sliding formwork control have quick response, to Parameters variation and disturbance it is insensitive, without system on-line identification,
The features such as physics realization is simple.
The content of the invention
The above-mentioned technical problem of the present invention is mainly what is be addressed by following technical proposals：
A kind of maglev train system control method of the sliding formwork control based on power Reaching Law, comprises the following steps：
Step 1, to establish single magnet magnetic suspension system dynamic model equation as follows
Wherein, m is the quality of suspended substance, and g is acceleration of gravity, and ε (t) is suspension spacing, and N is coil turn, and i is coil
Electric current, F (i, ε) are electromagnetic attraction, μ0For space permeability, A is the area of single magnetic pole, and R is electromagnet winding resistance, fdTo be outer
Disturb on boundary；System control targe is by control voltage u (t), so as to control coil electric current i (t), finally realizes that object exports ε
(t) preferable track is tracked, t is time variation amount；
OrderDefinition status variable x1=ε,x3=i represents electromagnet suspension spacing, speed respectively
And electric current, wherein,For floating velocity, k is constant；It is as follows then to obtain magnetic suspension system Nonlinear state space model
Y=x1 (2d)
In formula, y is the output suspension spacing of system,For the acceleration of system；
The object output y and control input u of wherein system is not contacted directly, can not directly design sliding mode controller；For
The relation between y and u is obtained, I/O linearization is carried out to system first；
OrderAnd y is differentiated
Wherein,For the output floating velocity of system,WithIt is the output suspension acceleration of system,WithFor system
Output suspension acceleration；
OrderAndThen system (1) is converted into
In definition (4), d >=0, | d |≤D, D are the upper bound；
Step 2, definition sliding-mode surface and Reaching Law are as follows：
It is x to take preferable position signallingd, then the velocity error of error and system is respectively e=xd-x1,System
The acceleration error of system and the acceleration error difference of system
Wherein,The speed of respectively preferable spacing, acceleration and acceleration,Respectively system is defeated
Go out the speed of spacing, acceleration and acceleration；
Defining sliding-mode surface function is
Wherein, c1,c2It is sliding formwork coefficient, is constant, and c1＞ 0, c2＞ 0；
(5) formula is differentiated
(4) formula is brought into (6) formula and produced
Power Reaching Law is
Wherein, k is the speed of system motion point convergence diverter surface s=0 in power Reaching Law, and α is power convergence constant, and
The ＞ α ＞ 0 of k ＞ 0,1, sign function
Step 3, based on step 1 and step 2, define the control law of sliding formwork control of the power Reaching Law of magnetic suspension system such as
Under：
It is equivalent by (7), (8) formula, and in perfect condition s=0, i.e.,Under the conditions of, it can obtain
Wherein, c1,c2, k, α are constant, and c1＞ 0, c2The ＞ α ＞ 0 of ＞ 0, k ＞ 0,1；
And system stable condition：
Utilize the stablizing effect of Lyapunov stability criterias checking PAR-SMC methods；
Lyapunov functions are chosen first
Wherein, V (s) is the scalar function of positive definite, and s is the sliding-mode surface function with continuous single order local derviation；
(9) are brought into (7), can be obtained
Seeking V first derivative and bringing (11) into obtain
And the condition for making magnetic suspension system stable is exactlyFrom (12) formula, the control method meets
The stable condition of system.
Therefore, the invention has the advantages that：The present invention has preferable stability, compared with sliding formwork control, in certain journey
The buffeting of system is weakened on degree, there is more preferable tracking performance, and the overshoot of system is smaller, and it is steady can faster to reach system
Fixed, speed tracing curve is more steady, has more preferable control performance than traditional sliding formwork control, buffets obvious weakening, position letter
Number error is smaller, and dynamic characteristic is more preferable, and robustness is stronger.
Brief description of the drawings
Accompanying drawing 1 is a kind of schematic diagram of the present invention.
Accompanying drawing 2 is position trace plot in the embodiment of the present invention.
Accompanying drawing 3 is site error trace plot in the embodiment of the present invention.
Accompanying drawing 4 is medium velocity trace plot of the embodiment of the present invention.
Accompanying drawing 5 is input voltage curve map in the embodiment of the present invention.
Embodiment
Below by embodiment, and with reference to accompanying drawing, technical scheme is described in further detail.
Embodiment：
First, the model of lower maglev train system is introduced first.
Typical magnetic suspension system structure is broadly divided into normally-on suction-type and superconduction repulsion type at present.Germany takes the lead in for 1969
The TR series often magnetic-type aerotrain of conductivity type is worked out, Japan then develops low speed magnetic suspension train in HSST normally-on suction-types
And released superconduction type MLX01, the Shanghai Maglev train that China Shanghai went out in equipment Imported From German EMH Company Technology design in 2003 in 1972
And 2015 annual proper motion research and development Changsha magnetic suspension train be normal conductivity type magnetic suspension train.The magnetcisuspension of this kind of normally-on suction-type
Floating train system is made up of multiple single magnet arrangements, is studied herein with magnetic suspension train list magnet arrangement, its schematic diagram is such as
Fig. 1, it is mainly made up of the suspension magnet on aerotrain track, train body bottom.
In Fig. 1, m is the quality of suspended substance, and g is acceleration of gravity, and ε (t) is suspension spacing, and N is coil turn, and i is line
Loop current, F (i, ε) are electromagnetic attraction, and A is the area of single magnetic pole, fdFor external interference.System control targe is by controlling electricity
U (t) is pressed, so as to control coil electric current i, finally realizes that object exports ε (t) and tracks preferable track.
For magnetic suspension system electromagnetic force when analyzing calculating, it is extremely difficult that magnetic field, which is carried out, accurately to calculate, therefore
Assuming that in addition to air gap, magnetic flux completely extends across iron core, magnetic leakage free phenomenon, such as Fig. 1, it is necessary to electromagnetic force equation, electromagnet coil electricity
The equation of motion of pressure equation and suspended substance is calculated.It can be obtained according to Ampère circuital theorem
Wherein μ0For space permeability.It is by formula (1) the spacing magnetic flux density that must can suspend
Wherein Φ is magnetic flux.Then electromagnet winding inductance is
It is in the electromagnetic attraction of t
The voltage equation of electromagnet winding loop is
Wherein R is electromagnet winding resistance.The direction for providing downward force is that just, then can obtain electromagnet in vertical direction
Kinematical equation be
The dynamic model equation of Single electromagnet magnetic suspension system can to sum up be obtained
OrderDefinition status variable x1=ε,x3=i represents electromagnet suspension spacing, speed respectively
And electric current, k are constant.Then obtain magnetic suspension system Nonlinear state space model
Y=x1 (11)
It can be reduced to
Wherein
From formula (12) and (13), object output y and control input u is not contacted directly, can not directly design sliding formwork
Controller.To obtain the relation between y and u, I/O linearization is carried out to system first.OrderAnd y is differentiated
OrderAndThen
Assuming that d >=0, | d |≤D.It is x to take preferable position signallingd, then error is e=xd-x1, Defining sliding formwork function is
Wherein c1,c2It is constant.Above formula is differentiated
Formula (15) is brought into formula (17) to obtain
Sliding formwork control ratio u (t) is by perfect conditionEquivalent method obtain, the control law of traditional sliding formwork control can be obtained
For
Wherein η is the gain of sign function, and η >=0.
To verify the stability of traditional sliding-mode control, the Lyapunov functions are taken to beThen
WhenWhen, s ≡ 0, according to LaSalle principles of invariance, closed-loop system asymptotically stability, as t → ∞, s ≡ 0,
And s convergence rates depend on η.
Known by control law expression formula, when disturbing d larger, to ensure the robust performance of system, must ensured sufficiently large
The upper bound is disturbed, and larger upper bound D can cause to buffet.
The discontinuous switching characteristic of traditional sliding formwork control in itself also results in system and chattering phenomenon, and this phenomenon occurs
Certainly exist in practice.But the disturbance rejection and the ability of anti-perturbation buffeted equivalent to sliding moding structure is eliminated are eliminated, this is can not
Can, so buffeting can only be weakened to a certain extent.
2nd, lower PAR-SMC control principles are described below.
Sliding formwork motion includes convergence motion and sliding formwork moves two processes.System from arbitrary initial state trend diverter surface,
Moved until reaching the motion of diverter surface for convergence, i.e. convergence motion is the process of s → 0.According to SMC principles, sliding formwork accessibility
Condition only ensures to reach the requirement of diverter surface in finite time by state space optional position motor point, and is moved for convergence
Specific track do not impose any restrictions, the dynamic quality of system can be improved using the method for Reaching Law.In order to weaken SMC control effects
Buffeting in fruit, improve the dynamic property of system, now controlled using sliding formwork control (PAR-SMC) method of power Reaching Law
System.
Power Reaching Law is
Buffeted to reduce, can adjust α values, it is ensured that, can be with larger speed when system mode is away from sliding-modes
Convergence sliding-modes, and during system mode convergence sliding-modes, it is ensured that less control gain.
The control law that PAR-SMC can be obtained by formula (19) and (21) is
With the stablizing effect of Lyapunov stability criterias checking PAR-SMC methods, bringing formula (22) into formula (18) can
Similarly, the Lyapunov functions are taken to beThen have
And the condition of the stability of a system is
System meets stability condition as can be seen here.
3rd, it is below SMC and PAR-SMC emulation experiments using the above method.
Stability contorting is carried out to magnetic suspension train model with the sliding mode control strategy of power Reaching Law, and it is real in Matlab
Now emulate, physical parameter in table 1 is used in system.
The magnetic suspension system parameter of table 1
Table 1 Magnetic levitation system parameters
Consider that maglev train system original state is x (0)=[0.016,0,0.1], the input reference signal one of system
As be r=0.01m, on this condition more traditional sliding formwork control (SMC) and power Reaching Law sliding formwork control (PAR-SMC)
Method, and be used in maglev train system model (12), then carry out parameter designing and emulation.
Traditional SMC controller parameters are designed as c in emulation1=144, c2=24, η=100；The parameter of PAR-SMC controllers
It is taken as c1=144, c2=24, η=100, k=40, α=0.6, simulation model is built in Matlab, compiled using S function form
Writing controller and object program, in simulation result, traditional SMC and PAR-SMC position tracking curve such as Fig. 2, site error with
Track curve such as Fig. 3, speed tracing curve such as Fig. 4, input voltage signal are as shown in Figure 5.
From the graph as can be seen that under the identical time in 2 and Fig. 3, the overshoot of traditional sliding formwork control is larger, regulating time
It is longer, and shake larger, and use the sliding formwork control of power Reaching Law substantially to shake that weakening is a lot, and overshoot is smaller, when stable
Between it is shorter, and the position tracking error of the sliding formwork control of power Reaching Law does not have a larger fluctuation, and traditional sliding formwork control
Middle tracking error is larger.As can be seen here, the sliding-mode control of power Reaching Law more can the effective given position letter of tracking system
Number, dynamic property is more preferable, and robustness is stronger.
4 it can be seen that from the graph, after being improved with PAR-SMC control methods, obtained rate curve is substantially than tradition
SMC controlling curve it is smooth, this largely illustrate PAR-SMC control methods can guarantee that train operation stationarity and
Security.
From fig. 5, it can be seen that PAR-SMC input control voltage is smaller than traditional SMC control voltage, come from Energy Angle
See, the shortcomings that energy resource consumption caused by too high voltages when PAR-SMC avoids input is big.To sum up Fig. 2,3,4,5 are understood, PAR-
SMC can reach more satisfactory state, make the stable suspension of system, weaken system chatter, improve dynamic performance.
Specific embodiment described herein is only to spirit explanation for example of the invention.Technology belonging to the present invention is led
The technical staff in domain can be made various modifications or supplement to described specific embodiment or be replaced using similar mode
Generation, but without departing from the spiritual of the present invention or surmount scope defined in appended claims.
Claims (1)
1. a kind of maglev train system control method of the sliding formwork control based on power Reaching Law, comprises the following steps：
Step 1, to establish single magnet magnetic suspension system dynamic model equation as follows
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Wherein, m is the quality of suspended substance, and g is acceleration of gravity, and ε (t) is suspension spacing, and N is coil turn, and i is coil electricity
Stream, F (i, ε) are electromagnetic attraction, μ0For space permeability, A is the area of single magnetic pole, and R is electromagnet winding resistance, fdFor the external world
Interference；System control targe is by control voltage u (t), so as to control coil electric current i (t), finally realizes that object exports ε (t)
Preferable track is tracked, t is time variation amount；
OrderDefinition status variable x1=ε,x3=i represents electromagnet suspension spacing, speed and electricity respectively
Stream, wherein,For floating velocity, k is constant；It is as follows then to obtain magnetic suspension system Nonlinear state space model
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OrderAnd y is differentiated
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Wherein,For the output floating velocity of system,WithIt is the output suspension acceleration of system,WithFor the defeated of system
Go out the acceleration that suspends；
OrderAndThen system (1) is converted into
<mrow>
<mover>
<mi>y</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mo>=</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mi>u</mi>
<mo>+</mo>
<mi>d</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
In definition (4), d >=0, | d |≤D, D are the upper bound；
Step 2, definition sliding-mode surface and Reaching Law are as follows：
It is x to take preferable position signallingd, then the velocity error of error and system is respectively e=xd-x1,System
The acceleration error of acceleration error and system is distinguished
Wherein,The speed of respectively preferable spacing, acceleration and acceleration,Respectively system outlet chamber
Away from speed, acceleration and acceleration；
Defining sliding-mode surface function is
<mrow>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mi>e</mi>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, c1,c2It is sliding formwork coefficient, is constant, and c1＞ 0, c2＞ 0；
(5) formula is differentiated
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>+</mo>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mo>=</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
(4) formula is brought into (6) formula and produced
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mi>u</mi>
<mo>-</mo>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mi>u</mi>
<mo>-</mo>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Power Reaching Law is
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<mi>k</mi>
<mo>|</mo>
<mi>s</mi>
<msup>
<mo>|</mo>
<mi>&alpha;</mi>
</msup>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, k is the speed of system motion point convergence diverter surface s=0 in power Reaching Law, and α is power convergence constant, and k ＞
0,1 ＞ α ＞ 0, sign function
Step 3, based on step 1 and step 2, the control law for defining the sliding formwork control of the power Reaching Law of magnetic suspension system is as follows：
It is equivalent by (7), (8) formula, and in perfect condition s=0, i.e.,Under the conditions of, it can obtain
<mrow>
<mi>u</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>&lsqb;</mo>
<mo>-</mo>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>k</mi>
<mo>|</mo>
<mi>s</mi>
<msup>
<mo>|</mo>
<mi>&alpha;</mi>
</msup>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>D</mi>
<mi> </mi>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, c1,c2, k, α are constant, and c1＞ 0, c2The ＞ α ＞ 0 of ＞ 0, k ＞ 0,1；
And system stable condition：
Utilize the stablizing effect of Lyapunov stability criterias checking PAR-SMC methods；
Lyapunov functions are chosen first
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, V (s) is the scalar function of positive definite, and s is the sliding-mode surface function with continuous single order local derviation；
(9) are brought into (7), can be obtained
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<mi>k</mi>
<mo>|</mo>
<mi>s</mi>
<msup>
<mo>|</mo>
<mi>&alpha;</mi>
</msup>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>D</mi>
<mi> </mi>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>d</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Seeking V first derivative and bringing (11) into obtain
<mrow>
<mover>
<mi>V</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mi>s</mi>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mo>-</mo>
<mi>k</mi>
<mo>|</mo>
<mi>s</mi>
<msup>
<mo>|</mo>
<mrow>
<mi>&alpha;</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>-</mo>
<mi>D</mi>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
<mo>-</mo>
<mi>d</mi>
<mi>s</mi>
<mo>&le;</mo>
<mo>-</mo>
<mi>k</mi>
<mo>|</mo>
<mi>s</mi>
<msup>
<mo>|</mo>
<mrow>
<mi>&alpha;</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&le;</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
And the condition for making magnetic suspension system stable is exactlyFrom (12) formula, the control method meets system
Stable condition.
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| CN108549241A (en) * | 2018-05-28 | 2018-09-18 | 浙江工业大学 | Quadrotor self-adaptation control method based on arc tangent enhanced double power Reaching Laws and fast terminal sliding-mode surface |
| CN108563126A (en) * | 2018-05-28 | 2018-09-21 | 浙江工业大学 | Quadrotor self-adaptation control method based on hyperbolic sine enhanced power Reaching Law and fast terminal sliding-mode surface |
| CN108563128A (en) * | 2018-05-28 | 2018-09-21 | 浙江工业大学 | Quadrotor self-adaptation control method based on index enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN108776485A (en) * | 2018-05-28 | 2018-11-09 | 浙江工业大学 | Quadrotor self-adaptation control method based on arc tangent enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN108804823A (en) * | 2018-06-11 | 2018-11-13 | 中国民用航空飞行学院 | Magnetic-levitation train vertical dynamics control system, information data processing terminal |
| CN108803638A (en) * | 2018-05-28 | 2018-11-13 | 浙江工业大学 | Quadrotor self-adaptation control method based on tanh enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN109532509A (en) * | 2018-12-28 | 2019-03-29 | 西南交通大学 | A kind of magnetic floating train suspending control method based on Sliding mode variable structure control |
| CN109581877A (en) * | 2019-01-18 | 2019-04-05 | 浙江工业大学 | A kind of magnetic levitation ball sliding mode variable structure control method based on ESO |
| CN110007605A (en) * | 2019-05-20 | 2019-07-12 | 长沙学院 | A kind of Robust Predictive Control method of repulsion formula magnetic levitation system |
| CN110244558A (en) * | 2019-05-09 | 2019-09-17 | 重庆大学 | Method for Fuzzy Logic Controller Design based on EMS suspension system |
| CN110614924A (en) * | 2019-07-29 | 2019-12-27 | 中车株洲电力机车有限公司 | Online identification method and system for inductance parameters of suspension electromagnet of normally-conductive magnetic-levitation train |
| CN111142374A (en) * | 2020-01-03 | 2020-05-12 | 江西理工大学 | Speed control method of automatic driving system of suspension type permanent magnetic levitation train |
| CN113183769A (en) * | 2021-06-04 | 2021-07-30 | 西南交通大学 | Suspension control method for medium-low speed maglev train |
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| CN108803638B (en) * | 2018-05-28 | 2021-08-03 | 浙江工业大学 | Self-adaptive control method of four-rotor aircraft based on hyperbolic tangent enhanced rapid power approach law and rapid terminal sliding mode surface |
| CN108563126B (en) * | 2018-05-28 | 2021-08-03 | 浙江工业大学 | Self-adaptive control method of four-rotor aircraft based on hyperbolic sine enhanced power approximation law and fast terminal sliding mode surface |
| CN108776485A (en) * | 2018-05-28 | 2018-11-09 | 浙江工业大学 | Quadrotor self-adaptation control method based on arc tangent enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN108776485B (en) * | 2018-05-28 | 2021-08-03 | 浙江工业大学 | Self-adaptive control method of four-rotor aircraft based on arc tangent enhanced fast power approach law and fast terminal sliding mode surface |
| CN108803638A (en) * | 2018-05-28 | 2018-11-13 | 浙江工业大学 | Quadrotor self-adaptation control method based on tanh enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN108563126A (en) * | 2018-05-28 | 2018-09-21 | 浙江工业大学 | Quadrotor self-adaptation control method based on hyperbolic sine enhanced power Reaching Law and fast terminal sliding-mode surface |
| CN108549241A (en) * | 2018-05-28 | 2018-09-18 | 浙江工业大学 | Quadrotor self-adaptation control method based on arc tangent enhanced double power Reaching Laws and fast terminal sliding-mode surface |
| CN108563128A (en) * | 2018-05-28 | 2018-09-21 | 浙江工业大学 | Quadrotor self-adaptation control method based on index enhanced quick power Reaching Law and fast terminal sliding-mode surface |
| CN108804823A (en) * | 2018-06-11 | 2018-11-13 | 中国民用航空飞行学院 | Magnetic-levitation train vertical dynamics control system, information data processing terminal |
| CN109532509A (en) * | 2018-12-28 | 2019-03-29 | 西南交通大学 | A kind of magnetic floating train suspending control method based on Sliding mode variable structure control |
| CN109581877B (en) * | 2019-01-18 | 2021-12-17 | 浙江工业大学 | Magnetic levitation ball sliding mode variable structure control method based on ESO |
| CN109581877A (en) * | 2019-01-18 | 2019-04-05 | 浙江工业大学 | A kind of magnetic levitation ball sliding mode variable structure control method based on ESO |
| CN110244558B (en) * | 2019-05-09 | 2022-02-08 | 重庆大学 | Fuzzy controller design method based on EMS suspension system |
| CN110244558A (en) * | 2019-05-09 | 2019-09-17 | 重庆大学 | Method for Fuzzy Logic Controller Design based on EMS suspension system |
| CN110007605A (en) * | 2019-05-20 | 2019-07-12 | 长沙学院 | A kind of Robust Predictive Control method of repulsion formula magnetic levitation system |
| CN110007605B (en) * | 2019-05-20 | 2020-03-24 | 长沙学院 | Robust prediction control method of repelling magnetic suspension device |
| CN110614924B (en) * | 2019-07-29 | 2021-03-26 | 中车株洲电力机车有限公司 | Online identification method and system for inductance parameters of suspension electromagnet of normally-conductive magnetic-levitation train |
| CN110614924A (en) * | 2019-07-29 | 2019-12-27 | 中车株洲电力机车有限公司 | Online identification method and system for inductance parameters of suspension electromagnet of normally-conductive magnetic-levitation train |
| CN111142374A (en) * | 2020-01-03 | 2020-05-12 | 江西理工大学 | Speed control method of automatic driving system of suspension type permanent magnetic levitation train |
| CN111142374B (en) * | 2020-01-03 | 2023-04-28 | 江西理工大学 | Speed control method of automatic driving system of suspension type permanent magnet maglev train |
| CN113183769A (en) * | 2021-06-04 | 2021-07-30 | 西南交通大学 | Suspension control method for medium-low speed maglev train |
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