CN107450352A - The simulation control method of non-linear Backstepping Controller based on Matlab - Google Patents
The simulation control method of non-linear Backstepping Controller based on Matlab Download PDFInfo
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- CN107450352A CN107450352A CN201710839078.6A CN201710839078A CN107450352A CN 107450352 A CN107450352 A CN 107450352A CN 201710839078 A CN201710839078 A CN 201710839078A CN 107450352 A CN107450352 A CN 107450352A
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- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B17/00—Systems involving the use of models or simulators of said systems
- G05B17/02—Systems involving the use of models or simulators of said systems electric
Abstract
The invention discloses a kind of simulation control method of the non-linear Backstepping Controller based on Matlab, comprise the following steps：(1) mathematical modeling of maglev ball system is established；Ignore other perturbed forces that bead is subject to；(2) the non-linear Backstepping Controller model of magnetic suspension system；It is theoretical according to LaSalle consistency, it was therefore concluded that all tracking errors are all asymptotic to level off to zero, and the stability of a system is proven, and the control rate of Backstepping Controller is given by formula (8,9,10).Maglev ball system of the invention with nonlinearity characteristic is few with good stability, high performance position tracking ability, effective, the modeling accuracy height for the method taken, external interference.
Description
Technical field
The present invention relates to control technology field, and in particular to a kind of emulation of the non-linear Backstepping Controller based on Matlab
Control method.
Background technology
With the development of automatic control technology, magnetic levitation technology is progressively set up, it by mechanics, control learn, electronics
A variety of subject technologies such as, electromagnetism, dynamics and computer science organically combine, and are typical electromechanical integration
Technology.Its cardinal principle is to overcome gravity using electromagnetic force, holder is suspended in the air, and " suspension " this characteristic makes its tool
Standby many advantages of being different from conventional art：
(1) Mechanical Contact is not present, entirely without abrasion, corresponding speed is fast, and loss is reduced, the service life of extension device；
(2) it is salable, without lubrication, (high speed, vacuum, ultra-clean etc.) can work in particular circumstances；(3) it is with small vibration, positioning, control essence
Degree is high, securely and reliably；(4) low in energy consumption, noise is small, cleanliness without any pollution.
Above advantage makes magnetic levitation technology fundamentally change traditional support modes, in traffic, metallurgy, aviation boat
My god, medical treatment, machinery, exhibition etc. various aspects contain wide application prospect, carry out the research of magnetic levitation technology and assault fortified position by
The development in numerous new and high technology forward positions can be driven, promotes the birth of NPD projects, the progress and expanding economy to science have emphatically
The realistic meaning wanted.
At present, the external research to magnetic levitation technology has been set up certain theory and method and is in fast development
In, effective application is obtained in a variety of high-precision end industrial circles and aerospace field, such as magnetic suspension lathe, magnetic suspension shaft
Hold, magnetic suspension train, magnetic levitation elevator etc..Although domestic magnetic levitation technology research is started late, horizontal to fall behind relatively, pass through
The developmental research of itself and the technological cooperation with foreign countries, significant progress has been achieved in fields such as magnetic suspension trains.
The core of magnetic levitation technology is magnetic levitation control technology, and good controller determines the stability and entirety of system
Performance.Because control structure is simple, controller parameter is conveniently adjusted for traditional control, and development is the most ripe, using also the widest
It is general.But due to the high-precision requirement of magnetic suspension system, working environment is complicated and changeable and magnetic field non-linear in itself, uses tradition
Method magnetic suspension system be controlled will cause that system response time is slow, and stability is poor, can not meet that control requires.With
The development of modern control technology, the research of magnetic suspension system have been gradually introduced robust control, intelligent control, optimum control and a variety of
Method is combined the advanced control methods such as control.
ZJ Yang, Y Fukushima establish a kind of robust nonlinear output feedback controller using Kalman Filtering, lead to
Cross disturbance observer and effectively compensated for Errors and external disturbance, before the good position tracking ability of system is ensured
Put, be effectively improved the stability of system.The bright grade of Lie group of Central South University is based on Mixed Sensitivity control theory, to weighting
Function battle array carries out reasonable selection, devises a kind of controller of maglev ball system, the modeling accuracy for solving magnetic suspension system
The problems such as low and external interference.HM Gutierrez, PI Ro are directed to the non-linear and high bandwidth requirements of magnetic suspension servo-drive system,
The robustness of sliding mode controller is improved by changing sliding formwork parameter, realizes and long-range quick tracking control is carried out to magnetic suspension system
System.In view of the limitation of single control method, several control methods are combined and are applied to magnetic suspension system by many scholars.
Rong-Jong Wai are inspired by sliding mode control strategy, design a kind of Adaptive Fuzzy Neural-network Control model, without aiding in controlling
Device processed, the buffeting problem of sliding formwork control is effectively overcomed by adaptive real-time learning algorithm, system is provided simultaneously with well
Stability and rapidity.It is pre- to introduce radial base neural net on the basis of Sliding mode control theory by Faa-Jeng Lin
Survey device to estimate the uncertain factor of system, reach preferable transient control and the ability of tracking to periodic.
And between past more than ten years, research of the domestic and international researcher to non-linear Reverse Step Control is also gradually risen.It is non-thread
Property Reverse Step Control be a kind of recursiveness design method, ensure system robustness under the premise of can solve the not true of system very well
It is qualitative and non-linear.
At present, a kind of Simulation Control side of the non-linear Backstepping Controller based on Matlab of good stability is lacked
Method.
The content of the invention
It is an object of the invention to provide a kind of non-linear Backstepping Controller based on Matlab of good stability
Simulation control method.
To reach above-mentioned purpose, present invention employs following technical proposal：The present invention's is a kind of based on the non-thread of Matlab
The simulation control method of property Backstepping Controller, comprises the following steps：
(1) mathematical modeling of maglev ball system is established
Ignore other perturbed forces that bead is subject to, the kinetics equation of bead vertical direction can be obtained by Newton's second law：
X is magnetic pole to the distance of bead barycenter, unit m in formula；M be bead quality, unit kg；F (i, x) is electricity
Magnetic force, unit N；G is acceleration of gravity, unit m/s2；
Had by the Kirchhoff's law of magnetic circuit, Biot-Savart law and law of conservation of energy：
μ in formula0For air permeability, μ0=4 π × 10-7H/m；A is that the pole-face of iron core accumulates, unit m2；N is electromagnetism iron wire
Enclose the number of turn；X is bead barycenter to the instantaneous distance of electromagnet, unit m；I is the transient current in electromagnet coil, and unit is
A；
OrderThen the formula of electromagnetic force is：
Suffered to make a concerted effort to be zero when bead is in reference position, the gravity of bead is equal to the upward electromagnetism that bead is subject to
Ferreous, the acceleration of bead is 0, then has：
Consider whole maglev ball system, can be obtained according to Kirchhoff's second law：
L in formula1For magnet spool inductance, unit H；L0For the inductance of reference position, unit H；X is bead
To the distance of electromagnet, unit is；I is the transient current in electromagnet winding, and unit A, R are the equivalent resistance of electromagnet,
Unit is Ω；
In summary formula, maglev ball system can be represented using following Nonlinear System of Equations：
If the state variable of system is x1=x, x2=x ', x3=i, then maglev ball system can be rewritten as following non-
System of linear equations：Equation Chapter 2Section 8
(2) the non-linear Backstepping Controller model of magnetic suspension system
If quantity of state xiDesired value be xid, then tracking error ei=xi-xid, i=1,2,3.ρ hereinafteriAnd kiAll it is
Positive number, the weight coefficient used as controller；
Step1：Define Lyapunov functionsThen
Take
Then：
Step2：
Define Lyapunov functionsThen
Take
Then
Step3：
Define Lyapunov functionsThen
Take
Then
It is theoretical according to LaSalle consistency, it was therefore concluded that all tracking error ei, i=1,2,3 is asymptotic to level off to
Zero, the stability of a system is proven, and the control rate of Backstepping Controller is given by formula (8,9,10).
Further, in step (1), the simulating, verifying of the controller of magnetic levitation ball system performance is tested in Matlab environment
Middle progress, model parameter are chosen for：
M=0.028kg, R=13 Ω, L1=118mH, x0=0.05m, i0=1.2A, K=4.774 × 10-4Nm2/A2；
In step (2), controller parameter selection is：
ρ1=100, ρ2=0.1, ρ3=1, k1=100, k2=1, k3=100.
Further, in step (1), emulation uses sine curve around initial position as desired trajectory, just
The amplitude of string pursuit path is 0.02m, cycle 1s.
Further, in step (1), the maglev ball system includes coil, iron core, generating laser, sensing
Device, controller, performs device and small ball, are provided with coil on the iron core, the controller respectively with generating laser, pass
Sensor is connected with performs device, and small ball is provided with below the iron core, is set between the sensor and generating laser
It is equipped with small ball.
Beneficial effect：The present invention has good stability, high performance position tracking ability, and the method taken has
Effect, modeling accuracy is high, and external interference is few.
Compared with prior art, the invention has the advantages that：
(1) present invention carries out model emulation in Matlab, bead is tracked sine curve desired trajectory, finally
Realize the stable suspersion of bead and quick and precisely track.Simulation result shows the validity of taken method, can ensure that and is
The stability of system simultaneously has good performance of dynamic tracking.
(2) the features such as present invention is directed to mission nonlinear and unstability, devises a kind of non-linear Backstepping Controller, with
Maglev ball system chooses sine curve of the bead around initial position as desired trajectory as research object,
Control and simulation analysis are tracked in Matlab to it.Simulation result shows that used non-linear Backstepping Controller ensures
Maglev ball system with nonlinearity characteristic is with good stability and high performance position tracking ability.
Brief description of the drawings
Fig. 1 is the structural representation of the maglev ball system of the present invention；
The magnetic suspension system that Fig. 2 is the present invention emulates trace plot；
The magnetic suspension system that Fig. 3 is the present invention emulates tracking error figure.
Embodiment
The present invention is further described by following examples, but should be noted that the scope of the present invention is not implemented by these
Any restrictions of example.
Embodiment 1
A kind of simulation control method of non-linear Backstepping Controller based on Matlab of the present invention, comprises the following steps：
(1) mathematical modeling of maglev ball system is established
Ignore other perturbed forces that bead is subject to, the kinetics equation of bead vertical direction can be obtained by Newton's second law：
X is magnetic pole to the distance of bead barycenter, unit m in formula；M be bead quality, unit kg；F (i, x) is electricity
Magnetic force, unit N；G is acceleration of gravity, unit m/s2；
Had by the Kirchhoff's law of magnetic circuit, Biot-Savart law and law of conservation of energy：
μ in formula0For air permeability, μ0=4 π × 10-7H/m；A is that the pole-face of iron core accumulates, unit m2；N is electromagnetism iron wire
Enclose the number of turn；X is bead barycenter to the instantaneous distance of electromagnet, unit m；I is the transient current in electromagnet coil, and unit is
A；
OrderThen the formula of electromagnetic force is：
Suffered to make a concerted effort to be zero when bead is in reference position, the gravity of bead is equal to the upward electromagnetism that bead is subject to
Ferreous, the acceleration of bead is 0, then has：
Consider whole maglev ball system, can be obtained according to Kirchhoff's second law：
L in formula1For magnet spool inductance, unit H；L0For the inductance of reference position, unit H；X is bead
To the distance of electromagnet, unit is；I is the transient current in electromagnet winding, and unit A, R are the equivalent resistance of electromagnet,
Unit is Ω；
In summary formula, maglev ball system can be represented using following Nonlinear System of Equations：
If the state variable of system is x1=x, x2=x ', x3=i, then maglev ball system can be rewritten as following non-
System of linear equations：Equation Chapter 2Section 8
Emulation uses the sine curve around initial position to be as desired trajectory, the amplitude of sinusoidal tracking track
0.02m, cycle 1s.
The maglev ball system includes coil, iron core, generating laser, sensor, controller, performs device and small steel
Ball, coil is provided with the iron core, the controller is connected with generating laser, sensor and performs device respectively, institute
State and small ball is provided with below iron core, small ball is provided between the sensor and generating laser.
The simulating, verifying experiment of the controller of magnetic levitation ball system performance is carried out in Matlab environment, and model parameter is chosen
For：
M=0.028kg, R=13 Ω, L1=118mH, x0=0.05m, i0=1.2A, K=4.774 × 10-4Nm2/A2；
In step (2), controller parameter selection is：
ρ1=100, ρ2=0.1, ρ3=1, k1=100, k2=1, k3=100.
(2) the non-linear Backstepping Controller model of magnetic suspension system
If quantity of state xiDesired value be xid, then tracking error ei=xi-xid, i=1,2,3.ρ hereinafteriAnd kiAll it is
Positive number, the weight coefficient used as controller；
Step1：Define Lyapunov functionsThen
Take
Then：
Step2：
Define Lyapunov functionsThen
Take
Then
Step3：
Define Lyapunov functionsThen
Take
Then
It is theoretical according to LaSalle consistency, it was therefore concluded that all tracking error ei, i=1,2,3 is asymptotic to level off to
Zero, the stability of a system is proven, and the control rate of Backstepping Controller is given by formula (8,9,10).
It is a complicated nonlinear system in itself in view of magnetic suspension, the present invention chooses maglev ball system as research
Object, for the free degree of its vertical direction, using non-linear backstepping control method, by small ball around initial position just
Chord curve is tracked control and simulation analysis to it as desired trajectory.Test result indicates that whole controller meets tool
There are the stability and rapidity requirement of the magnetic suspension system of nonlinearity, there is stronger robustness to be tracked with good dynamic
Ability.
As shown in figure 1, maglev ball system by coil, iron core, generating laser, sensor, controller, performs device,
The elements such as small ball form.Its system construction drawing is as shown in Figure 1.
Assuming that when electromagnet coil voltage is U0, electromagnetic force F and its own gravity that small ball is received balance each other, i.e.,
F=mg, steel ball then keep stable suspersion, and this levitation position is referred to as reference position.
X is the hoverheight of bead in Fig. 1, and bead be in reference position during original state, and controller is according to setting generation
Control signal, the control signal are converted into voltage signal input electromagnet through performs device, change the size of electromagnetic force, from
And change the hoverheight x of bead, bead is suspended in the given position of controller.Generating laser is irradiated to by bead simultaneously
On sensor, be converted to voltage output through processing and new control signal is produced according to the change of voltage to controller, controller, from
And form closed loop magnetic levitation ball closed-loop control system.
The Backstepping Controller that the present invention is integrated it can be seen from Fig. 2 and Fig. 3 simulation result can be to it is expected to track rail
Mark is accurately tracked, and error is very small, and actually by the Backstepping design method of nonlinear system, whole controller ensures
Magnetic suspension system with nonlinearity characteristic is with stable and high performance position tracking ability.
General principle, principal character and the advantages of the present invention of the present invention has been shown and described above.The technology of the industry
Personnel are it should be appreciated that the present invention is not limited to the above embodiments, and the simply explanation described in above-described embodiment and specification is originally
The principle of invention, without departing from the spirit and scope of the present invention, various changes and modifications of the present invention are possible, the present invention
Claimed scope is by appended claims, specification and its equivalent thereof.
Claims (4)
1. a kind of simulation control method of the non-linear Backstepping Controller based on Matlab, it is characterised in that comprise the following steps：
(1) mathematical modeling of maglev ball system is established
Ignore other perturbed forces that bead is subject to, the kinetics equation of bead vertical direction can be obtained by Newton's second law：
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Had by the Kirchhoff's law of magnetic circuit, Biot-Savart law and law of conservation of energy：
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Consider whole maglev ball system, can be obtained according to Kirchhoff's second law：
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If the state variable of system is x1=x, x2=x ', x3=i, then maglev ball system can be rewritten as following non-linear
Equation group：Equation Chapter 2 Section 8
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(2) the non-linear Backstepping Controller model of magnetic suspension system
If quantity of state xiDesired value be xid, then tracking error ei=xi-xid, i=1,2,3.ρ hereinafteriAnd kiAll for just
Number, the weight coefficient used as controller；
Step1：Define Lyapunov functionsThen
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Take
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<mi>k</mi>
<mn>1</mn>
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<mi>e</mi>
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</mrow>
Then：
Step2：
Define Lyapunov functionsThen
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</mfenced>
Take
Then
Step3：
Define Lyapunov functionsThen
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<mn>2</mn>
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</mtable>
</mfenced>
Take
Then
It is theoretical according to LaSalle consistency, it was therefore concluded that all tracking error ei, i=1,2,3 it is all asymptotic level off to zero, system
Stability is proven, and the control rate of Backstepping Controller is given by formula (8,9,10).
2. the simulation control method of the non-linear Backstepping Controller according to claim 1 based on Matlab, its feature exist
In：
In step (1), the simulating, verifying experiment of the controller of magnetic levitation ball system performance is carried out in Matlab environment, model ginseng
Number is chosen for：
M=0.028kg, R=13 Ω, L1=118mH, x0=0.05m, i0=1.2A, K=4.774 × 10-4Nm2/A2；
In step (2), controller parameter selection is：
ρ1=100, ρ2=0.1, ρ3=1, k1=100, k2=1, k3=100.
3. the simulation control method of the non-linear Backstepping Controller according to claim 1 based on Matlab, its feature exist
In：In step (1), emulation uses sine curve around initial position as desired trajectory, the width of sinusoidal tracking track
Spend for 0.02m, cycle 1s.
4. the simulation control method of the non-linear Backstepping Controller according to claim 1 based on Matlab, its feature exist
In：In step (1), the maglev ball system includes coil, iron core, generating laser, sensor, controller, performs dress
Put and small ball, be provided with coil on the iron core, the controller respectively with generating laser, sensor and performs device phase
Connect, be provided with small ball below the iron core, small ball is provided between the sensor and generating laser.
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| Application Number | Priority Date | Filing Date | Title |
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| CN201710839078.6A CN107450352A (en) | 2017-09-18 | 2017-09-18 | The simulation control method of non-linear Backstepping Controller based on Matlab |
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| Application Number | Priority Date | Filing Date | Title |
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| CN201710839078.6A CN107450352A (en) | 2017-09-18 | 2017-09-18 | The simulation control method of non-linear Backstepping Controller based on Matlab |
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| Publication Number | Publication Date |
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| 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 |
| CN116774588A (en) * | 2023-07-25 | 2023-09-19 | 石家庄铁道大学 | EMS type magnetic suspension train suspension frame double cross coupling self-adaptive backstepping control method |
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| 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 |
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| CN116774588A (en) * | 2023-07-25 | 2023-09-19 | 石家庄铁道大学 | EMS type magnetic suspension train suspension frame double cross coupling self-adaptive backstepping control method |
| CN116774588B (en) * | 2023-07-25 | 2024-01-23 | 石家庄铁道大学 | EMS type magnetic suspension train suspension frame double cross coupling self-adaptive backstepping control method |
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