CN107703757A - The super-twisting sliding mode control method of gyroscope system - Google Patents
The super-twisting sliding mode control method of gyroscope system Download PDFInfo
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Abstract
The invention discloses a kind of super-twisting sliding mode control method of gyroscope system, the super-twisting sliding mode controller for the method design gyroscope system being combined using Equivalent Sliding Mode control with supertwist control, then using the Lyapunov function pair gyroscope systems of D-quadratic form when being disturbed by constant value disturbance and variate, carry out stability analysis, it is ensured that system asymptotic stability.The present invention can effectively suppress to buffet caused by common sliding Mode Algorithm, and sliding variable and its first derivative can be made to converge to zero in finite time, so as to ensure that the track of gyroscope can accurately and effectively track its reference locus, ensure system Globally asymptotic, the robustness of improvement system, improve sensitivity and the accuracy of system.
Description
Technical field
The present invention relates to a kind of super-twisting sliding mode control method of gyroscope system, belong to the control technology of gyroscope
Field.
Background technology
Gyro is inertial navigation and the fundamental measurement element of inertial guidance system.Microthrust test is because it is in cost, volume, structure
Etc. big advantage be present, so as to be widely used in navigation, space flight, aviation and oil field exploration exploitation and land vehicle
Navigation and positioning etc. in civilian, military field.Because it has the influence of error and temperature in design and manufacture, original can be caused
Difference between part characteristic and design, so as to cause the reduction of gyroscope system sensitivity and precision, the master of gyroscope control
It is compensation foozle and measurement angular speed to want problem.By the research and development of decades, although gyroscope is in structure design
Significant progress is achieved with precision etc., but limitation and technique machining accuracy itself due to its design principle in itself
Limitation so that the development of gyroscope be difficult to obtain qualitative leap.
The measurement of angular speed and the compensation of foozle are the subject matter of microthrust test control, and traditional control method is main
Solve drive shaft oscillation amplitude and the stable control of frequency and the matching problem of two axle frequencies, it is impossible to efficiently solve micro- top
Deficiency and defect existing for spiral shell.
The content of the invention
The technical problems to be solved by the invention are the defects of overcoming prior art, there is provided a kind of gyroscope system surpasses
Sliding-mode control is distorted, can effectively suppress control input buffeting, make control input more smooth, and sliding formwork can be made
Variable and its first derivative make system reach Asymptotic Stability state, improve the items of system in Finite-time convergence to zero
Energy.
In order to solve the above technical problems, the present invention provides a kind of super-twisting sliding mode control method of gyroscope system, bag
Include following steps：
1) gyroscope system is reduced into one has damped oscillation system by what mass and spring were formed, establishes micro- top
The dimensionless mathematical modeling of spiral shell instrument system；
2) design reference model；
3) sliding-mode surface is designed；
4) slided using Equivalent Sliding Mode control with the supertwist for the method design gyroscope system that supertwist control is combined
Mould controller, design control law are as follows：
U=ucon+udis (9)
Wherein, u is control law, uconFor continuous control portion, the equivalent control term u of system is taken aseq, udisFor discontinuous control
Part, it is taken as system switching control item usw；
5) entered using the Lyapunov function pair gyroscope systems of D-quadratic form when being disturbed by constant value disturbance and variate
Row stability analysis, it is ensured that system asymptotic stability.
The foregoing dimensionless mathematical modeling for establishing gyroscope system comprises the following steps：
Newton's law in 1-1) being according to rotation, considers influence of the various foozles to micro- spiral shell top instrument, obtains
The mathematical modeling of gyroscope is：
Wherein, m is the quality of mass, and x, y are mass in drive shaft and the position vector of the axle of sensitive axis two, dxx,dyy
Represent x, the damped coefficient of the axles of y two, kxx,kyyIt is x respectively, the spring constant of the axles of y two, ux,uyIt is to represent x, the control of the axles of y two is defeated
Enter, kxy, dxyIt is coupling spring coefficient and damped coefficient, Ω caused by foozlezRepresent the angle in gyroscope working environment
Speed,It is Coriolis force；
1-2) by the both sides of the mathematical modulo pattern (1) of gyroscope simultaneously divided by gyroscope mass quality m, reference
Length q0, square ω of the resonant frequency of two axles0 2, the mathematical modeling for obtaining nondimensionalization is as follows：
The expression formula of each characteristic is：
Symbol " → " represents that the amount on the symbol left side is replaced with the amount on the right of symbol；
The mathematical modulo pattern (2) of nondimensionalization 1-3) is rewritten as vector form：
1-4) consider the parameter uncertainty and external interference of gyroscope system, the mathematical modeling of gyroscope system is repaiied
It is changed to：
Wherein, Δ D be inertial matrix D+2 Ω unknown parameter uncertainty, Δ K be matrix K unknown parameter not
Certainty, d are external interferences；
1-5) definition is uncertain and external interference is：
Then formula (5) is expressed as：
Wherein：
Foregoing reference model is：
Reference model chooses stable pure oscillation, order：
qr1=A1sin(ω1T), qr2=A2sin(ω2T),
Wherein, A1, A2For the amplitude of vibration, ω1, ω2For the frequency of vibration.
Foregoing sliding-mode surface s is designed as：
Wherein, c is sliding-mode surface constant, s1,s2For s two components, e is tracking error,
Wherein,For the output trajectory of gyroscope system,For the desired trajectory of gyroscope system.
Foregoing equivalent control term ueqSolution procedure it is as follows：
To sliding-mode surface derivation, obtain：
In the case where not considering external interference, obtained by formula (4)：
Formula (13) is updated into formula (12) to obtain：
OrderThus equivalent controller, equivalent control term u are obtainedeqFor：
The switching control item uswDesign is as follows：
Wherein, k1＞ 0, k2＞ 0；
Then control law is：
Under foregoing constant value disturbance, k1＞ 0, k2＞ 0；
Under the variate disturbance,
Wherein, δ is the boundary value of uncertain and external interference derivative.
The beneficial effects of the present invention are：
(1) present invention employs the control of equivalent control and super-twisting control algolithms system designed in conjunction
Rule, equivalent control ensure that system mode is moved in sliding-mode surface, super-twisting sliding formwork controls realize it is uncertain and
The robustness of external interference, it restrained effectively control input buffeting；
(2) present invention takes the influence of interference into account, and can make sliding variable and its first derivative limited
For Fast Convergent to zero, its reference locus can accurately and effectively be tracked by ensuring that the movement locus of control system in time,
Control system Global asymptotic stability is ensure that, improves robustness, sensitivity and the accuracy of control system.
(3) present invention employs D-quadratic form Lyapunov function pairs to system when being disturbed by constant value disturbance and variate
System has carried out stability analysis, ensures system Asymptotic Stability, having reached the movement locus of system can accurately and quickly track
The purpose of reference locus.
Brief description of the drawings
Fig. 1 is the simplified model figure of gyroscope system of the present invention；
Fig. 2 is the super-twisting System with Sliding Mode Controller structured flowcharts of gyroscope system of the present invention；
Fig. 3 is the tracking curves of microthrust test X-axis and Y-axis in present example；
Fig. 4 is the speed tracing curve of gyroscope system in present example；
Fig. 5 be present example in gyroscope system X-axis, Y-axis control input curve；
Fig. 6 is microthrust test system X-axis and Y-axis track following error curve in present example；
Fig. 7 is the speed tracing error curve of gyroscope system in present example；
Fig. 8 is the sliding-mode surface convergence curve in microthrust test system both direction in present example.
Embodiment
The invention will be further described below.Following examples are only used for the technical side for clearly illustrating the present invention
Case, and can not be limited the scope of the invention with this.
The mathematical modeling of one, gyroscopes：
Micro- spiral shell top instrument is generally by the mass hung by resilient material, electrostatic drive and sensing device further three parts
Composition.As shown in Figure 1 one can be reduced to has damped oscillation system by what mass and spring were formed, which show
The z-axis micromachined vibratory gyroscope model of simplification under cartesian coordinate system.
Newton's law in being according to rotation, considers the influence to micro- spiral shell top such as various foozles, then by micro-
The nondimensionalization processing of gyroscope, the mathematical modeling for finally giving gyroscope are：
Wherein, m is the quality of mass, and x, y are mass in drive shaft and the position vector of the axle of sensitive axis two, dxx,dyy
Represent x, the damped coefficient of the axles of y two, kxx,kyyIt is x respectively, the spring constant of the axles of y two, ux,uyIt is to represent x, the control of the axles of y two is defeated
Enter, kxy, dxyIt is coupling spring coefficient and damped coefficient, Ω caused by foozlezRepresent the angle in gyroscope working environment
Speed,It is Coriolis force.
The mathematical modulo pattern (1) of gyroscope is a kind of form for having dimension, that is to say, that the physical quantity in equation is not only
Consider numerical values recited, and including the uniformity of its physical unit is also contemplated that, therefore add the complexity of controller design
Degree.The intrinsic frequency scope of the axle of gyroscope two is typically in kHz scopes, but input angular velocity may be in the several years per hour
To in the range of the several years per second, there is very big magnitude and distinguish in both, it is not easy to realize numerical simulation in time.In order to solve
Two above problem, it is necessary to which nondimensionalization processing is carried out to model.Nondimensionalization is very valuable in numerical simulation, and it can make
When two big time frame differences be present, numerical simulation is easily realized, while it can be various microthrust test systems
Design provides a unified mathematical formulae.
By the both sides of formula (1) simultaneously divided by microthrust test mass quality m, reference length q0, the resonant frequency of two axles
Square ω0 2, it is as follows to obtain nondimensionalization model：
The expression formula of each characteristic is：
Symbol " → " represents that the amount on the symbol left side is replaced with the amount on the right of symbol.
Nondimensionalization modular form (2) contains two equations, adds the difficulty and complexity of controller design.Therefore have
It is necessary that model is subjected to equivalent transformation, the equivalent transformation of model be beneficial to the design of controller and the stability analysis of system and
The application of various advanced control methods.Then formula (2) is rewritten as following vector form：
The equivalent mould of the parameter uncertainty and external interference of consideration system, then the microthrust test system according to described by (4) formula
Type, gyroscope system model may be modified such that：
In formula, Δ D be inertial matrix D+2 Ω unknown parameter uncertainty, Δ K be matrix K unknown parameter not
Certainty, d are external interferences.
Further formula (5) is represented by：
Have in formula：
Supertwist (super-twisting) System with Sliding Mode Controller of two, gyroscopes
The super-twisting System with Sliding Mode Controller structured flowcharts of gyroscope are as shown in Figure 2.
The control problem of gyroscope system is to control the track following problem of gyroscope system, and the target of control is
Design a suitable control law u so that the output q of system can in finite time quickly and accurately track reference track
qr.The present invention will control to be combined with supertwist control algolithm using Equivalent Sliding Mode come design control law u, selects following control
Rule：
U=ucon+udis (9)
In formula, uconFor continuous control part, the equivalent control term u of system can be taken aseq, wherein ueqFor ensureing system
State is on sliding-mode surface, udisFor discontinuous control section, realize to external interference and probabilistic robust control and weaken system
The buffeting of system, it can be taken as system switching control item usw, switching control herein realized with supertwist control algolithm.
Defining sliding-mode surface is：
In formula, c is sliding-mode surface constant, s1,s2For s two components, e is tracking error, wherein：
In formula, q is the output trajectory of gyroscope system,For the desired trajectory of gyroscope system, it is expected
Stable pure oscillation is chosen in track, wherein：qr1=A1sin(ω1T), qr2=A2sin(ω2T), A1, A2For the amplitude of vibration,
ω1, ω2For the frequency of vibration.
Equivalent controller is designed first：
Sliding-mode surface derivation can be obtained：
In the case where not considering external interference, the mathematical modeling of microthrust test system system can be described as (4) formula, according to formula
(4), it is represented by following form：
Formula (13) is updated into formula (12) to obtain：
OrderIt can thus be concluded that equivalent controller：
Based on supertwist control algolithm, design switching law is：
Wherein, k1＞ 0, k2＞ 0.
So the control law for obtaining gyroscope system is：
Three, stability analyses
It will consider that the parameter uncertainty of system and gyroscope system mathematic model (6) formula of external interference substitute into (12)
Formula obtains：
(17) formula substitution (18) formula is obtained：
Consider two kinds of situations：
(1), chooseDisturbed for constant value, (19) formula is converted accordingly, can transform to：
Wherein, k1＞ 0, k2＞ 0,Order：
Because k1＞ 0, k2＞ 0, its proper polynomial areΔ is variable, therefore A is
Hurwitz matrixes, so to any positive definite symmetric matrices Q, there must be a positive definite symmetric matrices P, meet Lyapunov equations：
ATP+PA=-Q (22)
Consider that D-quadratic form function V (x, y) is alternative Lyapunov functions：
In above formula：Solved using chain rule (Chain Rule)UsingHave：
Have to track derivations of the V along system (20)：
Because Q is positive definite symmetric matrices, have：
Therefore system is stable.
(2) if, takeDisturbed for variate, the stability of a system proves as follows：
Formula (19) is converted accordingly, can be changed to：
Wherein,(δ is the border of uncertain and external interference derivative
Value), k1,k2Value meet following formula：
Take positive definite symmetric matricesThe Lyapunov functions of D-quadratic form are：
WhereinOrder：
Utilize inequalityIt is and rightDerivation has：
Have to V along formula (27) system path derivation：
Make Q=- (ATP+PA+δ2CTC+PBBTP), then
Now have：
If Q ＞ 0,K when Q is positive definite is obtained by the Shur properties mended1And k2Span be formula (28), can
Know that system can be in Finite-time convergence to origin.
Four, experiment simulations are analyzed
In order to verify the feasibility of the gyroscope system super-twisting sliding mode scheme designed by the present invention, MATLAB is now utilized
Simulation software carries out numerical simulation experiment.
The parameter selection of gyroscope experiment simulation is as follows：
M=1.8 × 10-7Kg, kxx=63.955N/m, kyy=95.92N/m, kxy=12.779N/m
dxx=1.8 × 10-6Ns/m, dyy=1.8 × 10-6Ns/m,dxy=3.6 × 10-7Ns/m
It is assumed that the angular speed of input is Ωz=100rad/s, nondimensionalization processing is carried out to gyroscope system, chooses ginseng
It is q to examine length0=1 μm, reference frequency ω0=1000Hz, the dimensionless group for obtaining gyroscope system are as follows：
ωx 2=355.3, ωy 2=532.9, ωxy=70.99, dxx=0.01
dyy=0.01, dxy=0.002, Ωz=0.1
If the primary condition of system is：X (0)=1.0,Y (0)=0.5,The two axle phases of microthrust test
Hope that running orbit is：qr1=sin (π t), qr2=cos (0.5 π t), in sliding formwork control, take the parameter c=10 of sliding-mode surface.When micro-
Gyro system Parameter Perturbation 10%, external interference take white noise signal d=[0.5*randn (1,1)；0.5*randn (1,1)],
Take δ=10, in Super-Twisting switching laws, take k1=20, k2=30, setting simulation time is 60s.
Simulation result is as shown in Figures 3 to 8.
Fig. 3 is the tracking curves of the X-axis that gyroscope system obtains under super-twisting sliding mode control and Y-axis, wherein,
Solid line is reference locus, and dotted line is actual path.As seen from Figure 3, under super-twisting sliding mode control, system can be quickly accurate
Ground track reference track, control effect are good.
Fig. 4 is the speed tracing curve of gyroscope system, wherein, solid line is reference velocity, and dotted line is actual speed.From
It can be seen from the figure that, the speed tracing of system equally can quickly realize the accurate tracking to reference velocity in finite time.
Fig. 5 is gyroscope system X-axis, Y-axis control input curve, as seen from Figure 5, is controlled using super-twisting sliding mode
It can effectively suppress to buffet, improve systematic function.
Fig. 6 is X-axis and Y-axis track following error, as seen from Figure 6, the rail of system is made using super-twisting sliding mode control
Mark tracking error can rapidly go to zero in finite time, and convergence rate is very fast.
Fig. 7 is the speed tracing error in X-axis and Y-axis, as seen from Figure 7, makes system using super-twisting sliding mode control
Speed tracing error can rapidly be gone to zero in finite time, convergence rate is very fast.
Fig. 8 is sliding-mode surface convergence curve of the microthrust test system under Super-Twisting sliding formwork controls in both direction,
It can be seen that sliding-mode surface function can quickly go to zero in finite time, sliding stability region is reached.
Described above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art
For member, without departing from the technical principles of the invention, some improvement and deformation can also be made, these are improved and deformation
Also it should be regarded as protection scope of the present invention.
Claims (6)
1. the super-twisting sliding mode control method of gyroscope system, it is characterised in that comprise the following steps：
1) gyroscope system is reduced into one has damped oscillation system by what mass and spring were formed, establishes gyroscope
The dimensionless mathematical modeling of system；
2) design reference model；
3) sliding-mode surface is designed；
4) the super-twisting sliding mode control for the method design gyroscope system being combined using Equivalent Sliding Mode control with supertwist control
Device processed, design control law are as follows：
U=ucon+udis (9)
Wherein, u is control law, uconFor continuous control portion, the equivalent control term u of system is taken aseq, udisFor discontinuous control unit
Point, it is taken as system switching control item usw；
5) carried out steady using the Lyapunov function pair gyroscope systems of D-quadratic form when being disturbed by constant value disturbance and variate
Qualitative analysis, it is ensured that system asymptotic stability.
2. the super-twisting sliding mode control method of gyroscope system according to claim 1, it is characterised in that the foundation
The dimensionless mathematical modeling of gyroscope system comprises the following steps：
Newton's law in 1-1) being according to rotation, considers influence of the various foozles to micro- spiral shell top instrument, obtains micro- top
The mathematical modeling of spiral shell instrument is：
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Wherein, m is the quality of mass, and x, y are mass in drive shaft and the position vector of the axle of sensitive axis two, dxx,dyyRepresent
The damped coefficient of the axle of x, y two, kxx,kyyIt is x respectively, the spring constant of the axles of y two, ux,uyIt is to represent x, the control input of the axles of y two,
kxy, dxyIt is coupling spring coefficient and damped coefficient, Ω caused by foozlezRepresent the angle speed in gyroscope working environment
Degree,It is Coriolis force；
1-2) by the both sides of the mathematical modulo pattern (1) of gyroscope simultaneously divided by gyroscope mass quality m, reference length
q0, square ω of the resonant frequency of two axles0 2, the mathematical modeling for obtaining nondimensionalization is as follows：
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<mo>,</mo>
<mfrac>
<msub>
<mi>d</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msub>
<mi>d</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>,</mo>
<mfrac>
<msub>
<mi>d</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msub>
<mi>d</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<msub>
<mi>k</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mrow>
<msup>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msubsup>
<mi>&omega;</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<mfrac>
<msub>
<mi>k</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mrow>
<msup>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msub>
<mi>&omega;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>,</mo>
<mfrac>
<msub>
<mi>k</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mrow>
<msup>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msubsup>
<mi>&omega;</mi>
<mi>y</mi>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<mfrac>
<msub>
<mi>&Omega;</mi>
<mi>z</mi>
</msub>
<mrow>
<msub>
<mi>m&omega;</mi>
<mn>0</mn>
</msub>
</mrow>
</mfrac>
<mo>&RightArrow;</mo>
<msub>
<mi>&Omega;</mi>
<mi>z</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
Symbol " → " represents that the amount on the symbol left side is replaced with the amount on the right of symbol；
The mathematical modulo pattern (2) of nondimensionalization 1-3) is rewritten as vector form：
<mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>D</mi>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>K</mi>
<mi>q</mi>
<mo>=</mo>
<mi>u</mi>
<mo>-</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
1-4) consider the parameter uncertainty and external interference of gyroscope system, the mathematical modeling modification of gyroscope system
For：
<mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>D</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>K</mi>
<mo>)</mo>
</mrow>
<mi>q</mi>
<mo>=</mo>
<mi>u</mi>
<mo>+</mo>
<mi>d</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, Δ D is the uncertainty of inertial matrix D+2 Ω unknown parameter, and Δ K is the uncertain of the unknown parameter of matrix K
Property, d is external interference；
1-5) definition is uncertain and external interference is：
Then formula (5) is expressed as：
Wherein：
<mrow>
<mi>q</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>D</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>d</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>d</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>d</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>d</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>K</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msubsup>
<mi>&omega;</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
</mtd>
<mtd>
<msub>
<mi>&omega;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&omega;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
<mtd>
<msubsup>
<mi>&omega;</mi>
<mi>y</mi>
<mn>2</mn>
</msubsup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>u</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mi>y</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>&Omega;</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<msub>
<mi>&Omega;</mi>
<mi>z</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&Omega;</mi>
<mi>z</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
3. the super-twisting sliding mode control method of gyroscope system according to claim 2, it is characterised in that the reference
Model is：
<mrow>
<msub>
<mi>q</mi>
<mi>r</mi>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow>
<mi>r</mi>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow>
<mi>r</mi>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Reference model chooses stable pure oscillation, order：
qr1=A1sin(ω1T), qr2=A2sin(ω2T),
Wherein, A1, A2For the amplitude of vibration, ω1, ω2For the frequency of vibration.
4. the super-twisting sliding mode control method of gyroscope system according to claim 3, it is characterised in that the sliding formwork
Face s is designed as：
<mrow>
<mi>s</mi>
<mo>=</mo>
<mi>c</mi>
<mi>e</mi>
<mo>+</mo>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mo>&lsqb;</mo>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, c is sliding-mode surface constant, s1,s2For s two components, e is tracking error,
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>e</mi>
<mo>=</mo>
<mi>q</mi>
<mo>-</mo>
<msub>
<mi>q</mi>
<mi>r</mi>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mi>x</mi>
<mo>-</mo>
<msub>
<mi>q</mi>
<mrow>
<mi>r</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>y</mi>
<mo>-</mo>
<msub>
<mi>q</mi>
<mrow>
<mi>r</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>r</mi>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mi>r</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mover>
<mi>y</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mi>r</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,For the output trajectory of gyroscope system,For the desired trajectory of gyroscope system.
5. the super-twisting sliding mode control method of gyroscope system according to claim 4, it is characterised in that described equivalent
Control item ueqSolution procedure it is as follows：
To sliding-mode surface derivation, obtain：
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mi>c</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mover>
<mi>e</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>=</mo>
<mi>c</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>-</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>r</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
In the case where not considering external interference, obtained by formula (4)：
<mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mo>=</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<mi>K</mi>
<mi>q</mi>
<mo>+</mo>
<mi>u</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
Formula (13) is updated into formula (12) to obtain：
<mrow>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mi>c</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<mi>K</mi>
<mi>q</mi>
<mo>+</mo>
<mi>u</mi>
<mo>-</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>r</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
OrderThus equivalent controller, equivalent control term u are obtainedeqFor：
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>e</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>-</mo>
<mi>c</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>K</mi>
<mi>q</mi>
<mo>+</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>r</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
The switching control item uswDesign is as follows：
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>s</mi>
<mi>w</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msqrt>
<mrow>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
</mrow>
</msqrt>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mo>&Integral;</mo>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>&tau;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, k1＞ 0, k2＞ 0；
Then control law is：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>u</mi>
<mo>=</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>d</mi>
<mi>i</mi>
<mi>s</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>e</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>s</mi>
<mi>w</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>-</mo>
<mi>c</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>K</mi>
<mi>q</mi>
<mo>+</mo>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>r</mi>
</msub>
<mo>-</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msqrt>
<mrow>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
</mrow>
</msqrt>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mo>&Integral;</mo>
<mi>sgn</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>&tau;</mi>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
6. the super-twisting sliding mode control method of gyroscope system according to claim 5, it is characterised in that the constant value
Under disturbance, k1＞ 0, k2＞ 0；
Under the variate disturbance,
Wherein, δ is the boundary value of uncertain and external interference derivative.
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Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109240083A (en) * | 2018-09-28 | 2019-01-18 | 河海大学常州校区 | The adaptive fuzzy super-twisting sliding mode control method of gyroscope system |
| CN109742941A (en) * | 2019-01-16 | 2019-05-10 | 武汉工程大学 | DC-DC converter chaotic control method, system and medium based on supertwist control |
| CN109917645A (en) * | 2019-02-21 | 2019-06-21 | 河海大学常州校区 | The double feedback fuzzy neural network super-twisting sliding mode Control System Design methods of microthrust test |
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