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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- 238000000034 method Methods 0.000 title claims abstract description 31
- 238000009415 formwork Methods 0.000 title abstract 5
- 230000005291 magnetic effect Effects 0.000 claims abstract description 34
- 239000000725 suspension Substances 0.000 claims abstract description 30
- 238000005339 levitation Methods 0.000 claims description 18
- 238000013459 approach Methods 0.000 claims description 16
- 230000001133 acceleration Effects 0.000 claims description 15
- 230000036461 convulsion Effects 0.000 claims description 6
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- 230000000087 stabilizing effect Effects 0.000 claims description 4
- 230000009897 systematic effect Effects 0.000 claims description 4
- 230000035699 permeability Effects 0.000 claims description 3
- 230000005294 ferromagnetic effect Effects 0.000 claims description 2
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- 238000004088 simulation Methods 0.000 abstract description 9
- 235000021170 buffet Nutrition 0.000 abstract 1
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- 230000003313 weakening effect Effects 0.000 abstract 1
- 230000004907 flux Effects 0.000 description 4
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- 241000287196 Asthenes Species 0.000 description 2
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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 invention relates to a maglev train system control method, in particular to a maglev train system control method based on power approach law sliding mode control.
Background
The magnetic suspension train system is a single-degree-of-freedom open-loop unstable nonlinear system, is easily influenced by system parameter disturbance and external interference, and brings difficulty to the control of the system. Although the system can be stabilized by adopting the conventional PID controller, the adaptability of the system to parameter change is poor; therefore, the sakali et al uses fuzzy PID to control, and is characterized in that an accurate system model is not needed, parameters can be adjusted on line, satisfactory system dynamic performance and stable performance can be obtained, but fuzzy rules of fuzzy control can only be obtained by experience and the algorithm is complex; therefore, the prediction control method applied by Ulbig and the like can effectively process the advantages of nonlinearity and uncertainty of the controlled object due to the advantages of strong robustness and low requirement on model precision, but has the defect of large calculated amount; uswarman et al realize that the sliding mode control has the advantages of insensitivity to parameter change and disturbance, no need of on-line identification of a system and simple physical realization, and adopt the sliding mode control for control, but the sliding mode control has obvious buffeting problem, so that a position fluctuation phenomenon occurs when a suspended object reaches a steady state.
In order to weaken buffeting and improve the dynamic performance of the system, an approach law method can be adopted, the power approach law sliding mode control (PAR-SMC) strategy is adopted to control the magnetic suspension system, and simulation results show that the system can stably suspend and obviously weaken buffeting by using the power approach law sliding mode control and has good tracking performance.
Sliding Mode Control (SMC) is a special nonlinear control that exhibits control discontinuities that can be purposefully varied to move the system according to a predetermined "sliding mode" state trajectory, depending on the current state of the system, unlike other control methods in which the system "architecture" is not fixed. Although the sliding mode can be subjected to parameter design and is independent of parameters and disturbance of an object, the sliding mode control has the characteristics of quick response, insensitivity to parameter change and disturbance, no need of system online identification, simple physical implementation and the like.
Disclosure of Invention
The technical problem of the invention is mainly solved by the following technical scheme:
a power-order approach law-based sliding mode control maglev train system control method comprises the following steps:
step 1, establishing a dynamic model equation of a single-magnetic ferromagnetic suspension system as follows
Wherein m is the mass of the suspension body, g is the gravity acceleration, epsilon (t) is the suspension distance, N is the number of turns of the coil, i is the current of the coil, F (i, epsilon) is the electromagnetic attraction force, mu0For vacuum permeability, A is the area of the single magnetic pole, R is the resistance of the electromagnet winding, fdExternal interference; the system control target is to control the coil current i (t) by controlling the voltage u (t) so as to finally realize that the object output epsilon (t) tracks an ideal track, wherein t is a time variation;
order toDefining a state variable x1＝ε,x3I represents the levitation pitch, velocity and current of the electromagnet, respectively, wherein,is the suspension velocity, k is a constant; then a space model of the nonlinear state of the magnetic suspension system is obtained as follows
y＝x1(2d)
Wherein y is the output levitation distance of the system,is the acceleration of the system;
the object output y of the system is not directly connected with the control input u, and a sliding mode controller cannot be directly designed; firstly, carrying out input and output linearization on the system in order to obtain the relation between y and u;
order toAnd differentiates y
Wherein,is the output levitation speed of the system,andare the output levitation acceleration of the system,andoutputting the suspension acceleration for the system;
order toAnd isThe system (1) is converted into
In the definition formula (4), D is more than or equal to 0, | D | < D, and D is an upper bound;
step 2, defining a sliding mode surface and an approach law as follows:
taking the ideal position signal as xdThe error and the speed error of the system are e ═ x respectivelyd-x1，Systematic acceleration error and systematic jerk error respectively
Wherein,velocity, acceleration and jerk at the desired intervals,respectively outputting the speed, the acceleration and the jerk of the space by the system;
defining a sliding mode surface function as
Wherein, c1,c2Are sliding mode coefficients, all being constant, and c1＞0,c2＞0；
Differentiating the formula (5) to obtain
Bringing formula (4) into formula (6) to obtain
The power approximation law is
Wherein k is the speed of the system moving point approaching switching surface s being 0 in the power approaching law, α is a power approaching constant, k is more than 0,1 is more than α is more than 0, and the sign function
And 3, defining the control law of the sliding mode control of the power approach law of the magnetic suspension system based on the steps 1 and 2 as follows:
is equivalent to the formulas (7) and (8), and in an ideal state s is 0, namelyUnder the condition of obtaining
Wherein, c1,c2K, α are both constants, and c1＞0,c2＞0，k＞0,1＞α＞0；
And the system stable condition is as follows:
verifying the stabilizing effect of the PAR-SMC method by utilizing a Lyapunov stability criterion;
firstly, selecting a Lyapunov function
Wherein V(s) is a positive definite scalar function, and s is a sliding mode surface function with continuous first-order partial derivatives;
bringing (9) into (7) to obtain
The first derivative of V is taken and taken into (11)
The condition for stabilizing the magnetic suspension system isAs can be seen from the equation (12), this control method satisfies the condition for system stability.
Therefore, the invention has the following advantages: the method has better stability, weakens buffeting of the system to a certain extent compared with sliding mode control, has better tracking performance, has smaller overshoot of the system, can more quickly achieve system stability, has more stable speed tracking curve, has better control performance compared with the traditional sliding mode control, obviously weakens buffeting, has smaller position signal error, has better dynamic characteristic and has stronger robustness.
Drawings
Fig. 1 is a schematic diagram of the present invention.
Fig. 2 is a graph of position tracking in an embodiment of the present invention.
Fig. 3 is a graph of position error tracking in an embodiment of the present invention.
Fig. 4 is a graph of velocity tracking in an embodiment of the invention.
Fig. 5 is a graph of input voltage in an embodiment of the invention.
Detailed Description
The technical scheme of the invention is further specifically described by the following embodiments and the accompanying drawings.
Example (b):
firstly, a model of a magnetic levitation train system is introduced.
The current typical magnetic levitation system structure is mainly divided into a normal attraction force type and a superconductive repulsion force type. The TR series permanent-conducting magnetic levitation train is firstly researched in 1969 in Germany, the HSST permanent-conducting suction type medium-low speed magnetic levitation train is subsequently researched in Japan, and the superconducting MLX01 is introduced in 1972, and the Shanghai magnetic levitation train designed by introducing the Germany technology in 2003 in Shanghai of China and the Gangsha magnetic levitation train self-developed in 2015 are both permanent-conducting magnetic levitation trains. This kind of magnetic suspension train system with normal conductive attraction is composed of a plurality of single magnet structures, which are studied herein, and its schematic diagram is shown in fig. 1, which is mainly composed of suspension train tracks and suspension magnets on the bottom of train body.
In FIG. 1, m is the mass of the suspension, g is the gravitational acceleration, ε (t) is the suspension spacing, N is the number of coil turns, i is the coil current, F (i, ε) is the electromagnetic attraction, A is the area of the single magnetic pole, F is the magnetic field, gdIs an external disturbance. The control target of the system is to control the coil current i by controlling the voltage u (t), and finally realize that the object output epsilon (t) tracks an ideal track.
When the electromagnetic force of the magnetic suspension system is analyzed and calculated, the magnetic field is difficult to accurately calculate, so that the magnetic flux completely penetrates through the iron core except for the air gap, and no magnetic leakage phenomenon is caused, such as an electromagnetic force equation, an electromagnet coil voltage equation and a motion equation of a suspension body are required to be calculated in the assumption that the magnetic flux completely penetrates through the iron core, as shown in fig. 1. According to the ampere loop theorem, the method can obtain
Wherein mu0Is a vacuum magnetic permeability. The magnetic flux density at the levitation pitch obtained from the formula (1) is
Where Φ is the magnetic flux. The electromagnet winding inductance is
Electromagnetic attraction force at time t is
The voltage equation of the electromagnet winding loop is
Wherein R is the electromagnet winding resistance. If the direction of the downward force is positive, the kinematic equation of the electromagnet in the vertical direction can be obtained as
Dynamic model equation of single electromagnet magnetic suspension system can be obtained in conclusion
Order toDefining a state variable x1＝ε,x3I represents the suspension distance, speed and current of the electromagnet respectively, and k is a constant. Then obtaining the nonlinear state space model of the magnetic suspension system
y＝x1(11)
Can be simplified into
Wherein
As can be seen from equations (12) and (13), the object output y is not directly connected to the control input u, and the sliding mode controller cannot be directly designed. To obtain the relationship between y and u, the system is first linearized in input and output. Order toAnd differentiates y
Order toAnd isThen
If D is more than or equal to 0, and | D | is less than or equal to D. Taking the ideal position signal as xdIf the error is e ═ xd-x1， Defining a sliding mode function as
Wherein c is1,c2Are all constants. Differentiating the above equation to obtain
Bringing formula (15) into formula (17)
Sliding mode control law u (t) is based on ideal stateObtained by the equivalent method, the control law of the traditional sliding mode control can be obtained
Wherein η is the gain of the sign function, and η ≧ 0.
In order to verify the stability of the traditional sliding mode control method, a Lyapunov function is taken asThen
When in useWhereas s ≡ 0, according to the lasale invariance principle, the closed-loop system becomes progressively stable, when t → ∞ s ≡ 0, and the speed of convergence of s depends on η.
According to the expression of the control law, when the interference D is large, in order to ensure the robust performance of the system, a sufficiently large interference upper bound needs to be ensured, and the large upper bound D can cause buffeting.
The discontinuous switching characteristic of the conventional sliding mode control itself also causes a chattering phenomenon of the system, and this phenomenon must be practically present. However, the elimination of buffeting corresponds to the elimination of the disturbance and perturbation resistance of the sliding mode variable structure, which is impossible, so that buffeting can be weakened only to a certain extent.
Second, the following describes the control principle of PAR-SMC.
The sliding mode movement comprises two processes of approach movement and sliding mode movement. The system approaches the switching surface from any initial state until the motion reaching the switching surface is an approach motion, i.e., a process in which the approach motion is s → 0. According to the SMC principle, the accessibility condition of the sliding mode only guarantees the requirement that a moving point at any position of a state space reaches a switching surface within a limited time, no limitation is made on the specific track of the approach movement, and the dynamic quality of a system can be improved by adopting an approach law method. In order to reduce buffeting in the SMC control effect and improve the dynamic performance of a system, a power-approximation-law sliding mode control (PAR-SMC) method is adopted for control.
The power approximation law is
To reduce chattering, the value of α may be adjusted to ensure that the system state moves away from the sliding mode, while moving toward the sliding mode at a greater speed, and that the system state moves toward the sliding mode, while ensuring a smaller control gain.
The control law of PAR-SMC can be obtained from equations (19) and (21)
The stability effect of the PAR-SMC method is verified by applying the Lyapunov stability criterion, and the expression (22) is brought into the expression (18) to obtain
Similarly, taking the Lyapunov function asThen there is
And the condition of system stability is
It can be seen that the system satisfies the stability condition.
Third, the SMC and PAR-SMC simulation experiments using the above method are described below.
And (3) performing stable control on the maglev train model by using a sliding mode control strategy of a power approximation law, realizing simulation in Matlab, and using the physical parameters in the table 1 in a system.
TABLE 1 magnetic levitation System parameters
Table 1 Magnetic levitation system parameters
Considering that the initial state of the maglev train system is x (0) ([ 0.016,0,0.1 ]), and the input reference signal of the system is generally r ═ 0.01m, under the condition, comparing the traditional Sliding Mode Control (SMC) and the sliding mode control (PAR-SMC) method of the power approximation law, applying the method to a maglev train system model (12), and then performing parameter design and simulation.
The parameter of the traditional SMC controller in simulation is designed to be c1＝144,c224, η, 100, and the parameter of PAR-SMC controller is c1＝144,c2Building a simulation model in Matlab, programming a controller and an object program by using an S function form, wherein the simulation model is 24, η is 100, k is 40, α is 0.6, and in a simulation result, a position tracking curve of a conventional SMC and a PAR-SMC is shown in fig. 2, a position error tracking curve is shown in fig. 3, a speed tracking curve is shown in fig. 4, and an input voltage signal is shown in fig. 5.
As can be seen from fig. 2 and fig. 3, in the same time, overshoot of the conventional sliding mode control is large, the adjustment time is long, and jitter is large, while the sliding mode control using the power approximation law is significantly attenuated by much jitter, overshoot is smaller, the settling time is shorter, and the position tracking error of the sliding mode control using the power approximation law does not have large fluctuation, while the tracking error in the conventional sliding mode control is large. Therefore, the power approximation law sliding mode control method can effectively track the position signal given by the system, and is better in dynamic performance and stronger in robustness.
As can be seen from fig. 4, after the improvement is performed by using the PAR-SMC control method, the obtained speed curve is obviously smoother than the control curve of the conventional SMC, which greatly illustrates that the PAR-SMC control method can ensure the stability and safety of train operation.
As can be seen from fig. 5, the input control voltage of the PAR-SMC is smaller than that of the conventional SMC, and the PAR-SMC avoids the disadvantage of large energy consumption caused by the excessively high voltage at the input from the energy saving point of view. As can be seen from fig. 2, 3, 4, and 5, the PAR-SMC can achieve a more ideal state, so that the system can suspend stably, thereby reducing the buffeting of the system and improving the dynamic performance of the system.
The specific embodiments described herein are merely illustrative of the spirit of the invention. Various modifications or additions may be made to the described embodiments or alternatives may be employed by those skilled in the art without departing from the spirit or ambit of the invention as defined in the appended claims.
Claims (1)
1. A power-order approach law-based sliding mode control maglev train system control method comprises the following steps:
step 1, establishing a dynamic model equation of a single-magnetic ferromagnetic suspension system as follows
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Wherein m is the mass of the suspension body, g is the gravity acceleration, epsilon (t) is the suspension distance, N is the number of turns of the coil, i is the current of the coil, F (i, epsilon) is the electromagnetic attraction force, mu0For vacuum permeability, A is the area of the single magnetic pole, R is the resistance of the electromagnet winding, fdExternal interference; the system control target is to control the coil current i (t) by controlling the voltage u (t) so as to finally realize that the object output epsilon (t) tracks an ideal track, wherein t is a time variation;
order toDefining a state variable x1＝ε,x3I represents the levitation pitch, velocity and current of the electromagnet, respectively, wherein,is the suspension velocity, k is a constant; then a space model of the nonlinear state of the magnetic suspension system is obtained as follows
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y＝x1(2d)
Wherein y is the output levitation distance of the system,is the acceleration of the system;
the object output y of the system is not directly connected with the control input u, and a sliding mode controller cannot be directly designed; firstly, carrying out input and output linearization on the system in order to obtain the relation between y and u;
order toAnd differentiates y
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Wherein,is the output levitation speed of the system,andare the output levitation acceleration of the system,andoutputting the suspension acceleration for the system;
order toAnd isThe system (1) is converted into
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In the definition formula (4), D is more than or equal to 0, | D | < D, and D is an upper bound;
step 2, defining a sliding mode surface and an approach law as follows:
taking the ideal position signal as xdThe error and the speed error of the system are e ═ x respectivelyd-x1，Systematic acceleration error and systematic jerk error respectively
Wherein,velocity, acceleration and jerk at the desired intervals,respectively outputting the speed, the acceleration and the jerk of the space by the system;
defining a sliding mode surface function as
<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,c2Are sliding mode coefficients, all being constant, and c1＞0,c2＞0；
Differentiating the formula (5) to obtain
<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>
Bringing formula (4) into formula (6) to obtain
<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>
The power approximation 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 the system moving point approaching switching surface s being 0 in the power approaching law, α is a power approaching constant, k is more than 0,1 is more than α is more than 0, and the sign function
And 3, defining the control law of the sliding mode control of the power approach law of the magnetic suspension system based on the steps 1 and 2 as follows:
is equivalent to the formulas (7) and (8), and in an ideal state s is 0, namelyUnder the condition of obtaining
<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,c2K, α are both constants, and c1＞0,c2＞0，k＞0,1＞α＞0；
And the system stable condition is as follows:
verifying the stabilizing effect of the PAR-SMC method by utilizing a Lyapunov stability criterion;
firstly, selecting a Lyapunov function
<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 a positive definite scalar function, and s is a sliding mode surface function with continuous first-order partial derivatives;
bringing (9) into (7) to obtain
<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>
The first derivative of V is taken and taken into (11)
<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>
The condition for stabilizing the magnetic suspension system isAs can be seen from the equation (12), this control method satisfies the condition for system stability.
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