CN107807527A - The adaptive super-twisting sliding mode control method of gyroscope adjustable gain - Google Patents

The adaptive super-twisting sliding mode control method of gyroscope adjustable gain Download PDF

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CN107807527A
CN107807527A CN201711085826.2A CN201711085826A CN107807527A CN 107807527 A CN107807527 A CN 107807527A CN 201711085826 A CN201711085826 A CN 201711085826A CN 107807527 A CN107807527 A CN 107807527A
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CN107807527B (en
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冯治琳
费峻涛
王欢
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Changzhou Campus of Hohai University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

The invention discloses a kind of adaptive super-twisting sliding mode control method of gyroscope adjustable gain, adaptive super-twisting sliding mode controller is designed with the method that supertwist control is combined using Equivalent Sliding Mode control, and design the adaptive law of super-twisting sliding mode controller parameter and gyroscope system uncertain parameter, stability analysis is finally carried out using Lyapunov function pair gyroscopes system, it is ensured that system asymptotic stability.Condition known to the first derivative border of distracter is not limited the present invention in by traditional Second Order Sliding Mode Control, ensure convergence of the sliding-mode surface under noisy condition, and combine the advantages that control of high-order super-twisting sliding mode can effectively suppress to buffet, control improves systematic function, microthrust test system is improved to uncertain and external interference robustness, ensures the stability of system.

Description

The adaptive super-twisting sliding mode control method of gyroscope adjustable gain
Technical field
The present invention relates to a kind of adaptive super-twisting sliding mode control method of gyroscope adjustable gain, belong to gyroscope Control technology 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, microthrust test controls main Problem is compensation foozle and measurement angular speed.By the research and development of decades, although microthrust test is in structure design and essence Degree etc. achieves significant progress, but due to its design principle limitation in itself and the technique machining accuracy limit of itself System so that the development of microthrust test is difficult to obtain qualitative leap.In order to improve microthrust test systematic function, its robustness is improved, it is domestic Advanced control method is applied in the control research of microthrust test by outer many scholars, it is proposed that different control methods.
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 adjustable gain Adaptive super-twisting sliding mode control method, by Self Adaptive Control with super-twisting sliding mode control be combined, on-line identification microthrust test The unknown parameter and angular speed of system, the complexity of gain selection in the presence of external interference is simplified, improve system external circle and do Disturb and probabilistic robustness.
In order to solve the above technical problems, the present invention provides a kind of adaptive super-twisting sliding mode control of gyroscope adjustable gain Method processed, comprises the 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) adaptive super-twisting sliding mode controller is designed with the method that supertwist control is combined using Equivalent Sliding Mode control, Design control law is as follows：
U=ueq+usw (9)
Wherein, u is control law, ueqFor Equivalent control law, uswFor switching law；
5) adaptive law of super-twisting sliding mode controller parameter and gyroscope system uncertain parameter is designed, and is used Lyapunov function pair gyroscopes system carries out 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) define the lumped parameter uncertainty and external interference of systemFor：
Formula (5) is expressed as：
Wherein：
Derivative meetδ is the uncertain upper dividing value with external interference derivative of lumped parameter.
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 law ueqSolution procedure it is as follows：
Sliding-mode surface derivation can be obtained：
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 law u are obtainedeqFor：
The switching law uswDesign is as follows：
Wherein, k1, k2For super-twisting sliding mode controller parameter, and k1＞ 0, k2＞ 0, and
Then control law is：
The adaptive law of foregoing super-twisting sliding mode controller parameter is：
Wherein,For k1Initial value, γ1,β1,γ2,β2It is normal number with χ；
The adaptive law of the gyroscope system uncertain parameter is：
Wherein,Meet：
For parameter estimating error；
The Lyapunov functions are chosen for：
Wherein, V is Lyapunov functions, and M, N, P is adaptive fixed gain, and meets M=MT＞ 0, N=NT＞ 0, P= PT＞ 0, is positive definite symmetric matrices, and tr { } representing matrix asks mark computing, V0(η)=ηTP η,WithFor optimized parameter,
P meets：
The beneficial effects of the present invention are：High-order super-twisting sliding mode is controlled and is combined with Self Adaptive Control, and is utilized Lyapunov Theory of Stability and Second Order Sliding Mode thought design Adaptive Second-Order Super-Twisting sliding mode controllers, Yi Jiwei The adaptive law of gyro unknown parameter and angular speed, be not only able to ensure system can in finite time Fast Convergent, reach Stable state, and it is capable of according to Adaptive Identification method the unknown parameter of real-time update estimating system online, solve system Unknown parameter problem, the movement locus for having reached system being capable of the accurate purpose of quick track reference track.The present invention carries High system external circle interference and probabilistic robustness, and effectively suppressed using Super-Twisting high_order sliding mode controls Control input is buffeted, and the advantages that strong robustness, the quick tracking to reference locus is realized, so as to improve systematic function.
Brief description of the drawings
Fig. 1 is the simplified model figure of microthrust test system of the present invention；
Fig. 2 is the adjustable adaptive super-twisting sliding mode Control system architecture block diagram of microthrust test system gain of the present invention.
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, microthrust tests：
Micro-vibration gyroscope is generally by the mass hung by resilient material, electrostatic drive and sensing device further three Part forms.As shown in Figure 1 one can be reduced to has damped oscillation system by what mass and spring were formed, and it is aobvious The z-axis micromachined vibratory gyroscope model of the simplification under cartesian coordinate system is shown.
Newton's law in being according to rotation, considers the influence to microthrust test such as various foozles, then by micro- The nondimensionalization processing of gyro, the mathematical modeling for finally giving microthrust test 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 speed in microthrust test working environment Degree,It is Coriolis force.
Mathematical modeling (1) formula of micro-mechanical gyroscope is a kind of form for having dimension, adds the complexity of controller design Degree, it is not easy to realize numerical simulation.In order to solve the problems, such as two above, it is necessary to carry out nondimensionalization processing to model.
By the both sides of formula (1) simultaneously divided by microthrust test mass of foundation block quality m, reference length q0, the resonance frequency of two axles Square ω of rate0 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.
Non-dimensional model (2) progress equivalent transformation 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, microthrust test 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：
Wherein,Lumped parameter for system is uncertain and external interference, its derivative meet(δ is lump The upper dividing value of parameter uncertainty and external interference derivative, δ are positive constant).
Adaptive supertwist (super-twisting) System with Sliding Mode Controller of two, microthrust test adjustable gains
The adaptive super-twisting System with Sliding Mode Controller structured flowchart of microthrust test adjustable gain is as shown in Figure 2.
The present invention will be controlled using Equivalent Sliding Mode and combined with Super-Twisting control algolithms come design control law u, choosing Select following control law.
U=ueq+usw (9)
Wherein, ueqFor Equivalent control law, uswFor switching law, switching law herein uses Super- Twisting sliding formwork controls design.
Designing sliding-mode surface is：
Wherein, c is sliding-mode surface constant, s1,s2For s two components, e,Respectively tracking error and tracking error is led Number, and：
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.
Sliding-mode surface derivation can be obtained：
Equivalent controller is designed first：
In the case where not considering external interference, the mathematical modeling of gyroscope system can be described as (4) formula, according to formula (4), it is represented by following form：
(13) formula substitution (12) formula is obtained：
OrderIt can thus be concluded that equivalent controller：
Using Super-Twisting sliding formwork controls, by switching law uswIt is designed as：
So the control law for obtaining microthrust test system is：
In formula, k1, k2For super-twisting sliding mode controller parameter, and k1＞ 0, k2＞ 0, andIt can design certainly Adapt to rule k1,k2, so that s andIn Finite-time convergence to zero.
K in design formula (16)1,k2Adaptive law be：
Wherein,For k1Initial value, γ1,β1,γ2,β2Be normal number with χ, then s andIt is null solution Uniformly asymptotic stadbility.
Three, adaptive laws design and stability analysis
Due to the D in microthrust test non-dimensional model, the yield value of tri- parameters of K, Ω and controller is unknown or can not Accurately obtain, so the control law of formula (17) can not directly be implemented.Therefore, according to the general thoughts of Self Adaptive Control, design The adaptive algorithm of microthrust test unknown parameter and the adaptive law of controller gain, online real-time update estimate, ensure system Stability.
(6) formula substitution (12) formula is obtained：
Formula (17) is substituted into formula (19) and obtained：
Formula (20) is converted accordingly, can be changed to：
Amount of orientation
Order：
η derivations can be obtained：
For the system of reality, the D in microthrust test non-dimensional model, K, tri- parameters of Ω are unknown or can not be accurate Really obtain, so the control law of formula (15) can not directly be implemented.Therefore, according to the general thoughts of Self Adaptive Control, using D, K, Ω estimateTo replace unknown true value D, K, Ω, and the adaptive algorithm of three parameters is designed, it is online real-time Estimate is updated, ensures the stability of system.
Therefore formula (15), which can arrange, is：
So control law (17) is changed into：
Designed according to Lypunov Theory of Stabilityk1,k2Adaptive algorithm, define D, K, Ω parameter estimates Count errorRespectively：
Choosing Lyapunov functions is：
In formula, M, N, P are adaptive fixed gain, and meet M=MT＞ 0, N=NT＞ 0, P=PT＞ 0, it is that positive definite is symmetrical Matrix, tr { } representing matrix ask mark computing, V0(η)=ηTP η,WithFor optimized parameter,
Take
By the control law of formula (25) substitute into that consideration system is uncertain and kinetics equation (6) formula of external interference in and abbreviation ：
Formula (12) is substituted into (29) to obtain：
According to definition of the formula (26) to parameter estimating error, formula (30) can abbreviation be further：
V seeks first derivative to the time, has：
Wherein,Formula (31) substitution (32) is had：
Due to D=DT, K=KT, Ω=- ΩT, and(scalar), therefore：
It can similarly obtain：
Therefore, formula (33), which can arrange, is：
To ensureDesign firstParameter update law be：
So have：
OrderTherefore have：
AndSo：
And because：
Make Q=- (ATP+PAT+PBBTP+δ2CTC), above formula, which can arrange, is：
Parameter is substituted into obtain：
Then obtaining the condition that Q is positive definite according to matrix theory is：
To sum up, formula (38), which can arrange, is：
By positive definite quadratic form function V0(η)=ηTP η can be obtained：
Wherein, λmin(P), λmax(P) representing matrix P minimal eigenvalue and eigenvalue of maximum,
Then：
It can be obtained by formula (42)：
Wherein, λmin(Q) minimal eigenvalue for being matrix Q, η1, η2For η two components,
By：
It can obtain：||η||2≥|η1|
Then：
Wherein：
So formula (44) can arrange and be again：
Wherein,
Assuming that using adaptive law formula (18), k1,k2Equal bounded, sufficiently large constant then always be presentSo thatThen：
Wherein：
By formula (18)Adaptive law expression formula substitute into above formula (51) ξ=0, this up-to-date style (52) is changed into：
From Lyapunov Theory of Stability, as long as meetingThen η, k1,k2It is consistent asymptotic stability in equalization point , and the first derivative of sliding-mode surface s and sliding-mode surfaceCan be in Finite-time convergence to zero, for k1,k2Value condition, In systemBefore, k1,k2Value will the constant speed increase in the presence of adaptive law formula (18), therefore by limited Time, k1,k2Its value condition can be met.
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 adaptive super-twisting sliding mode control method of gyroscope adjustable gain, 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) adaptive super-twisting sliding mode controller, design are designed with the method that supertwist control is combined using Equivalent Sliding Mode control Control law is as follows：
U=ueq+usw (9)
Wherein, u is control law, ueqFor Equivalent control law, uswFor switching law；
5) adaptive law of super-twisting sliding mode controller parameter and gyroscope system uncertain parameter is designed, and is used Lyapunov function pair gyroscopes system carries out stability analysis, it is ensured that system asymptotic stability.
2. the adaptive super-twisting sliding mode control method of gyroscope adjustable gain according to claim 1, its feature exist In the 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 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：
<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msup> <msub> <mi>&amp;omega;</mi> <mi>x</mi> </msub> <mn>2</mn> </msup> <mi>x</mi> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;Omega;</mi> <mi>z</mi> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>d</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msup> <msub> <mi>&amp;omega;</mi> <mi>y</mi> </msub> <mn>2</mn> </msup> <mi>y</mi> <mo>=</mo> <msub> <mi>u</mi> <mi>y</mi> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>&amp;Omega;</mi> <mi>z</mi> </msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
The expression formula of each characteristic is：
<mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mrow> <msub> <mi>m&amp;omega;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mfrac> <msub> <mi>d</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mrow> <msub> <mi>m&amp;omega;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>&amp;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&amp;omega;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>&amp;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&amp;omega;</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msubsup> <mi>&amp;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&amp;omega;</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msub> <mi>&amp;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&amp;omega;</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msubsup> <mi>&amp;omega;</mi> <mi>y</mi> <mn>2</mn> </msubsup> <mo>,</mo> <mfrac> <msub> <mi>&amp;Omega;</mi> <mi>z</mi> </msub> <mrow> <msub> <mi>m&amp;omega;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msub> <mi>&amp;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>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>D</mi> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>K</mi> <mi>q</mi> <mo>=</mo> <mi>u</mi> <mo>-</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mover> <mi>q</mi> <mo>&amp;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>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>D</mi> <mo>)</mo> </mrow> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mi>K</mi> <mo>+</mo> <mi>&amp;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) define the lumped parameter uncertainty and external interference of systemFor：
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>&amp;omega;</mi> <mi>x</mi> <mn>2</mn> </msubsup> </mtd> <mtd> <msub> <mi>&amp;omega;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;omega;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>&amp;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>&amp;Omega;</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;Omega;</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;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> </mrow>
Derivative meetδ is the uncertain upper dividing value with external interference derivative of lumped parameter.
3. the adaptive super-twisting sliding mode control method of gyroscope adjustable gain according to claim 2, its feature exist In 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 adaptive super-twisting sliding mode control method of gyroscope adjustable gain according to claim 3, its feature exist In the sliding-mode surface s is designed as：
<mrow> <mi>s</mi> <mo>=</mo> <mi>c</mi> <mi>e</mi> <mo>+</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>&amp;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>&amp;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>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>r</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mn>2</mn> </mrow> </msub> <mo>&amp;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 adaptive super-twisting sliding mode control method of gyroscope adjustable gain according to claim 4, its feature exist In the Equivalent control law ueqSolution procedure it is as follows：
Sliding-mode surface derivation can be obtained：
<mrow> <mover> <mi>s</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>c</mi> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mover> <mi>e</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>c</mi> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;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>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>q</mi> <mo>&amp;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>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>c</mi> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mi>K</mi> <mi>q</mi> <mo>+</mo> <mi>u</mi> <mo>-</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;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 law 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>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>K</mi> <mi>q</mi> <mo>+</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;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 law 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>&amp;Integral;</mo> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>&amp;tau;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
Wherein, k1, k2For super-twisting sliding mode controller parameter, and k1＞ 0, k2＞ 0, and
Then control law is：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>u</mi> <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>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>K</mi> <mi>q</mi> <mo>+</mo> <msub> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;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>&amp;Integral;</mo> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>&amp;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 adaptive super-twisting sliding mode control method of gyroscope adjustable gain according to claim 5, its feature exist In the adaptive law of the super-twisting sliding mode controller parameter is：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>k</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> <msqrt> <mfrac> <msub> <mi>&amp;gamma;</mi> <mn>1</mn> </msub> <mn>2</mn> </mfrac> </msqrt> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>&amp;NotEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>k</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mn>2</mn> </msub> <msqrt> <mfrac> <msub> <mi>&amp;gamma;</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> </msqrt> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>&amp;NotEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mo>|</mo> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msub> <mover> <mi>k</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mi>&amp;chi;</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <msup> <mi>&amp;epsiv;</mi> <mn>2</mn> </msup> <mn>8</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;epsiv;k</mi> <mn>1</mn> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
Wherein,k1|T=0For k1Initial value, γ1,β1,γ2,β2It is normal number with χ；
The adaptive law of the gyroscope system uncertain parameter is：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mover> <mover> <mi>D</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>T</mi> </msup> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>M</mi> <mrow> <mo>(</mo> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <msup> <mi>s</mi> <mi>T</mi> </msup> <mo>+</mo> <mi>s</mi> <msup> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>T</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mover> <mover> <mi>K</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>T</mi> </msup> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>N</mi> <mrow> <mo>(</mo> <msup> <mi>qs</mi> <mi>T</mi> </msup> <mo>+</mo> <msup> <mi>sq</mi> <mi>T</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mover> <mover> <mi>&amp;Omega;</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>T</mi> </msup> <mo>=</mo> <mo>-</mo> <mi>P</mi> <mrow> <mo>(</mo> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <msup> <mi>s</mi> <mi>T</mi> </msup> <mo>-</mo> <mi>s</mi> <msup> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>T</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>
Wherein,Meet：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>D</mi> <mo>~</mo> </mover> <mo>=</mo> <mover> <mi>D</mi> <mo>^</mo> </mover> <mo>-</mo> <mi>D</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>K</mi> <mo>~</mo> </mover> <mo>=</mo> <mover> <mi>K</mi> <mo>^</mo> </mover> <mo>-</mo> <mi>K</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&amp;Omega;</mi> <mo>~</mo> </mover> <mo>=</mo> <mover> <mi>&amp;Omega;</mi> <mo>^</mo> </mover> <mo>-</mo> <mi>&amp;Omega;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
For parameter estimating error；
The Lyapunov functions are chosen for：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>V</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>s</mi> <mi>T</mi> </msup> <mi>s</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>t</mi> <mi>r</mi> <mo>{</mo> <mover> <mi>D</mi> <mo>~</mo> </mover> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>D</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>t</mi> <mi>r</mi> <mo>{</mo> <mover> <mi>K</mi> <mo>~</mo> </mover> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>K</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>t</mi> <mi>r</mi> <mo>{</mo> <mover> <mi>&amp;Omega;</mi> <mo>~</mo> </mover> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>&amp;Omega;</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>&amp;eta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>&amp;gamma;</mi> <mn>1</mn> </msub> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>*</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>&amp;gamma;</mi> <mn>2</mn> </msub> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>*</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
Wherein, V is Lyapunov functions, and M, N, P is adaptive fixed gain, and meets M=MT＞ 0, N=NT＞ 0, P=PT＞ 0, it is positive definite symmetric matrices, tr { } representing matrix asks mark computing, V0(η)=ηTP η,WithFor optimized parameter,
P meets：
<mrow> <mi>&amp;eta;</mi> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mo>|</mo> <mi>s</mi> <msup> <mo>|</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mi>sgn</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>l</mi> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow>
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