CN107807527B - Self-adaptive super-distortion sliding mode control method with adjustable micro gyroscope gain - Google Patents

Self-adaptive super-distortion sliding mode control method with adjustable micro gyroscope gain Download PDF

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CN107807527B
CN107807527B CN201711085826.2A CN201711085826A CN107807527B CN 107807527 B CN107807527 B CN 107807527B CN 201711085826 A CN201711085826 A CN 201711085826A CN 107807527 B CN107807527 B CN 107807527B
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micro
gyroscope
sliding mode
adaptive
parameters
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CN107807527A (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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    • 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
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Abstract

The invention discloses a self-adaptive super-distortion sliding mode control method with adjustable gain of a micro gyroscope, which is characterized in that a self-adaptive super-distortion sliding mode controller is designed by adopting a method of combining equivalent sliding mode control and super-distortion control, the self-adaptive law of parameters of the super-distortion sliding mode controller and uncertain parameters of a micro gyroscope system is designed, and finally, an L yapunov function is adopted to carry out stability analysis on the micro gyroscope system to ensure the asymptotic stability of the system.

Description

Self-adaptive super-distortion sliding mode control method with adjustable micro gyroscope gain
Technical Field
The invention relates to a self-adaptive super-distortion sliding mode control method with adjustable gain of a micro gyroscope, and belongs to the technical field of control of micro gyroscopes.
Background
The gyroscope is the basic measurement element of the inertial navigation and inertial guidance system. The micro gyroscope has great advantages in the aspects of cost, volume, structure and the like, so that the micro gyroscope is widely applied to civil and military fields of navigation, spaceflight, aviation and oil field survey and development, navigation and positioning of land vehicles and the like. The main problems of micro-gyroscope control are compensation of manufacturing errors and measurement of angular velocity, which can lead to differences between the original characteristics and the design due to errors in design and manufacturing and the influence of temperature, thus leading to a reduction in sensitivity and accuracy of the gyroscope system. Through research and development of decades, the micro gyroscope makes remarkable progress in structural design, precision and the like, but due to the limitation of the design principle and the limitation of the process machining precision, the development of the micro gyroscope is difficult to make a qualitative leap. In order to improve the performance of a micro gyroscope system and improve the robustness of the micro gyroscope system, many scholars at home and abroad apply an advanced control method to the control research of the micro gyroscope and propose different control methods.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a self-adaptive super-distortion sliding mode control method with adjustable gain of a micro gyroscope.
In order to solve the technical problem, the invention provides a self-adaptive super-distortion sliding mode control method with adjustable gain of a micro gyroscope, which comprises the following steps:
1) simplifying the micro-gyroscope system into a damped oscillation system consisting of a mass block and a spring, and establishing a dimensionless mathematical model of the micro-gyroscope system;
2) designing a reference model;
3) designing a slip form surface;
4) the adaptive super-distortion sliding mode controller is designed by adopting a method of combining equivalent sliding mode control and super-distortion control, and the design control law is as follows:
u＝ueq+usw(9)
wherein u is the control laweqTo an equivalent control law, uswIs a switching control law;
5) designing self-adaptation laws of parameters of the supertorsion sliding mode controller and uncertain parameters of the micro-gyroscope system, and performing stability analysis on the micro-gyroscope system by adopting L yapunov functions to ensure asymptotic stability of the system.
The aforementioned establishment of the dimensionless mathematical model of the micro-gyroscope system comprises the following steps:
1-1) according to Newton's law in the rotation system, comprehensively considering the influence of various manufacturing errors on the micro gyroscope, and obtaining a mathematical model of the micro gyroscope as follows:
Figure BDA0001460057000000021
where m is the mass of the mass, x, y are the position vectors of the mass at both the drive and sense axes, dxx,dyyExpressing the damping coefficients, k, of the x, y axesxx,kyySpring constants, u, of the x, y axes, respectivelyx,uyIs a control input representing two axes x, y, kxy，dxyIs coupling due to manufacturing errorsSpring and damping coefficients, ΩzRepresenting the angular velocity in the operating environment of the micro-gyroscope,
Figure BDA0001460057000000022
is the coriolis force;
1-2) dividing both sides of the mathematical model formula (1) of the micro-gyroscope by the mass m of the mass block of the micro-gyroscope simultaneously, the reference length q0Square of resonance frequency of two axes omega0 2The dimensionless mathematical model is obtained as follows:
Figure BDA0001460057000000023
the expression for each dimensionless quantity is:
Figure BDA0001460057000000024
the symbol "→" indicates that the amount to the left of the symbol is replaced with the amount to the right of the symbol;
1-3) rewriting the dimensionless mathematical model (2) into vector form:
Figure BDA0001460057000000025
1-4) considering the parameter uncertainty and the external interference of the micro-gyroscope system, modifying the mathematical model of the micro-gyroscope system into:
Figure BDA0001460057000000026
wherein, Δ D is the uncertainty of the unknown parameters of the inertial matrix D +2 Ω, Δ K is the uncertainty of the unknown parameters of the matrix K, and D is the external interference;
1-5) lumped parameter uncertainty and external interference defining system
Figure BDA0001460057000000027
Comprises the following steps:
Figure BDA0001460057000000031
formula (5) is represented as:
Figure BDA0001460057000000032
wherein:
Figure BDA0001460057000000033
Figure BDA0001460057000000034
is satisfied with
Figure BDA0001460057000000035
The upper bound of the lumped parameter uncertainty and the external disturbance derivative.
The aforementioned reference model is:
Figure BDA0001460057000000036
selecting stable sinusoidal oscillation by the reference model, and enabling:
qr1＝A1sin(ω1t)，qr2＝A2sin(ω2t)，
wherein A is1，A2Being amplitude of oscillation, ω1，ω2Is the frequency of the oscillation.
The slip form surface s is designed as follows:
Figure BDA0001460057000000037
wherein c is a sliding mode surface constant, s1,s2Is the two components of s, e is the tracking error,
Figure BDA0001460057000000038
wherein the content of the first and second substances,
Figure BDA0001460057000000039
is the output track of the micro-gyroscope system,
Figure BDA00014600570000000310
is the desired trajectory of the micro-gyroscope system.
The equivalent control law ueqThe solution process of (2) is as follows:
derivation of the slip form surface can be obtained:
Figure BDA00014600570000000311
without considering external interference, the method is obtained by the formula (4):
Figure BDA00014600570000000312
substituting equation (13) into equation (12) yields:
Figure BDA00014600570000000313
order to
Figure BDA0001460057000000041
From this, an equivalent controller, an equivalent control law u, is obtainedeqComprises the following steps:
Figure BDA0001460057000000042
the switching control law uswThe design is as follows:
Figure BDA0001460057000000043
wherein k is1，k2Is a supertwist sliding mode controller parameter, and k1＞0,k2Is greater than 0, and
Figure BDA0001460057000000044
the control law is:
Figure BDA0001460057000000045
the adaptive law of the parameters of the supertwist sliding mode controller is as follows:
Figure BDA0001460057000000046
wherein the content of the first and second substances,
Figure BDA0001460057000000047
is k1Initial value of (a), gamma1,β1,γ2,β2And χ is a normal number;
the self-adaptive law of uncertain parameters of the micro-gyroscope system is as follows:
Figure BDA0001460057000000048
wherein the content of the first and second substances,
Figure BDA0001460057000000049
satisfies the following conditions:
Figure BDA0001460057000000051
Figure BDA0001460057000000052
estimating an error for the parameter;
the L yapunov function was chosen as:
Figure BDA0001460057000000053
wherein V is L yapunov function, M, N, P is adaptive fixed gain, and M is M ═ MT＞0,N＝NT＞0,P＝PTGreater than 0, is a positive definite symmetric matrix, tr {. cndot.) represents the trace-solving operation of the matrix, V0(η)＝ηTPη，
Figure BDA0001460057000000054
And
Figure BDA0001460057000000055
in order to optimize the parameters of the process,
p satisfies:
Figure BDA0001460057000000056
Figure BDA0001460057000000057
Figure BDA0001460057000000058
the method has the advantages that high-order Super-twist sliding mode control is combined with self-adaptive control, a self-adaptive second-order Super-twist sliding mode controller and a self-adaptive law of unknown parameters and angular velocities of the micro-gyroscope are designed by utilizing L yapunov stability theory and a second-order sliding mode idea, the system can be ensured to be rapidly converged in a limited time to reach a stable state, the unknown parameters of the system can be updated and estimated on line in real time according to a self-adaptive identification method, the problem of the unknown parameters of the system is solved, and the purpose that the motion trajectory of the system can accurately and rapidly track a reference trajectory is achieved.
Drawings
FIG. 1 is a simplified block diagram of a micro gyroscope system of the present invention;
FIG. 2 is a structural block diagram of a gain-adjustable adaptive super-twisted sliding mode control system of the micro-gyroscope system of the present invention.
Detailed Description
The invention is further described below. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The mathematical model of the micro gyroscope is as follows:
the micro-vibration gyroscope is generally composed of a mass suspended by an elastic material support, an electrostatic driving device and a sensing device. It can be simplified to a damped oscillatory system of masses and springs as shown in figure 1, which shows a simplified z-axis micromechanical vibrating gyroscope model in a cartesian coordinate system.
According to Newton's law in a rotation system, the influence of various manufacturing errors and the like on the micro gyroscope is comprehensively considered, and then through the non-dimensionalization processing of the micro gyroscope, the mathematical model of the micro gyroscope is finally obtained as follows:
Figure BDA0001460057000000061
where m is the mass of the mass, x, y are the position vectors of the mass at both the drive and sense axes, dxx,dyyExpressing the damping coefficients, k, of the x, y axesxx,kyySpring constants, u, of the x, y axes, respectivelyx,uyIs a control input representing two axes x, y, kxy，dxyIs the coupling spring coefficient and damping coefficient, omega, caused by manufacturing errorszRepresenting the angular velocity in the micro-gyroscope operating environment,
Figure BDA0001460057000000062
is the coriolis force.
The mathematical model (1) of the micromechanical gyroscope is in a dimensional form, so that the design complexity of the controller is increased, and numerical simulation is not easy to realize. In order to solve the above two problems, it is necessary to perform a dimensionless process on the model.
Dividing the two sides of the formula (1) by the mass m of the micro gyroscope basic mass block at the same time, and obtaining the reference length q0Square of resonance frequency of two axes omega0 2The dimensionless model is obtained as follows:
Figure BDA0001460057000000063
the expression for each dimensionless quantity is:
Figure BDA0001460057000000064
the symbol "→" indicates that the quantities to the left of the symbol are replaced with the quantities to the right of the symbol.
And (3) performing equivalent transformation on the dimensionless model (2) and rewriting the dimensionless model into the following vector form:
Figure BDA0001460057000000071
considering the parameter uncertainty and the external interference of the system, according to the equivalent model of the micro-gyroscope system described by the formula (4), the micro-gyroscope system model can be modified as follows:
Figure BDA0001460057000000072
in the formula, Δ D is the uncertainty of the unknown parameter of the inertia matrix D +2 Ω, Δ K is the uncertainty of the unknown parameter of the matrix K, and D is the external interference.
Further equation (5) can be expressed as:
Figure BDA0001460057000000073
in the formula:
Figure BDA0001460057000000074
Figure BDA0001460057000000075
wherein the content of the first and second substances,
Figure BDA0001460057000000076
the derivative of the system is satisfied with the lumped parameter uncertainty and the external interference
Figure BDA0001460057000000077
(positive constants for upper bound values of lumped parameter uncertainty and external disturbance derivative).
Adaptive super-twisting (super-twisting) sliding mode control system with adjustable micro-gyroscope gain
The structural block diagram of the micro-gyroscope gain adjustable adaptive super-twisting sliding mode control system is shown in fig. 2.
The invention designs a control law u by combining equivalent sliding mode control and a Super-Twisting control algorithm, and selects the following control law.
u＝ueq+usw(9)
Wherein u iseqTo an equivalent control law, uswFor the switching control law, the switching control law is designed by adopting Super-Twisting sliding mode control.
The design slip form surface is:
Figure BDA0001460057000000078
wherein c is a sliding mode surface constant, s1,s2The sum of the two components of s, e,
Figure BDA0001460057000000079
the derivative of the tracking error and tracking error, respectively, and:
Figure BDA0001460057000000081
in the formula, q is the output track of the micro gyroscope system,
Figure BDA0001460057000000082
selecting a stable sinusoidal oscillation for the desired trajectory of the micro-gyroscope system, wherein: q. q.sr1＝A1sin(ω1t)，qr2＝A2sin(ω2t)，A1，A2Being amplitude of oscillation, ω1，ω2Is the frequency of the oscillation.
Derivation of the slip form surface can be obtained:
Figure BDA0001460057000000083
firstly, designing an equivalent controller:
without considering external disturbances, the mathematical model of the micro-gyroscope system can be described as equation (4), which, according to equation (4), can be expressed in the form:
Figure BDA0001460057000000084
substituting equation (13) into equation (12) to obtain:
Figure BDA0001460057000000085
order to
Figure BDA0001460057000000086
This results in an equivalent controller:
Figure BDA0001460057000000087
adopting Super-Twisting sliding mode control to control switchingswThe design is as follows:
Figure BDA0001460057000000088
therefore, the control law of the micro-gyroscope system is obtained as follows:
Figure BDA0001460057000000089
in the formula, k1，k2For supertwist sliding mode controlSystem parameters, and k1＞0,k2Is greater than 0, and
Figure BDA00014600570000000810
programmable adaptive law k1,k2So that s and
Figure BDA00014600570000000811
converging to zero in a finite time.
K in design formula (16)1,k2The adaptive law of (1) is as follows:
Figure BDA0001460057000000091
wherein the content of the first and second substances,
Figure BDA0001460057000000092
is k1Initial value of (a), gamma1,β1,γ2,β2And χ is a normal number, s and
Figure BDA0001460057000000093
is a zero solution consistent asymptotically stable.
Adaptive rule design and stability analysis
Since the three parameters of D, K, and Ω in the micro-gyroscope dimensionless model and the gain value of the controller are unknown or cannot be accurately obtained, the control law of equation (17) cannot be directly implemented. Therefore, according to the general idea of adaptive control, an adaptive algorithm of unknown parameters of the micro gyroscope and an adaptive law of controller gain are designed, the estimated value is updated on line in real time, and the stability of the system is ensured.
Substituting the formula (6) into the formula (12) to obtain:
Figure BDA0001460057000000094
substituting formula (17) into formula (19) to obtain:
Figure BDA0001460057000000095
the corresponding transformation of equation (20) can be changed to:
Figure BDA0001460057000000096
vector taking
Figure BDA0001460057000000097
Order:
Figure BDA0001460057000000098
derivation of η yields:
Figure BDA0001460057000000101
for an actual system, three parameters of D, K, and Ω in the micro-gyroscope dimensionless model are unknown or cannot be accurately obtained, so the control law of equation (15) cannot be directly implemented. Thus, according to the general idea of adaptive control, the estimated values of D, K, omega are used
Figure BDA0001460057000000102
The method is used for replacing unknown true values D, K and omega, and designing a self-adaptive algorithm of three parameters to update an estimated value on line in real time, so that the stability of the system is ensured.
Formula (15) can therefore be arranged as:
Figure BDA0001460057000000103
therefore, the control law (17) is:
Figure BDA0001460057000000104
designed according to L ypunov stability theory
Figure BDA0001460057000000105
k1,k2Defining the parameter estimation error of D, K and omega
Figure BDA0001460057000000106
Respectively as follows:
Figure BDA0001460057000000107
the L yapunov function was chosen as:
Figure BDA0001460057000000108
wherein M, N, P are adaptive fixed gains, and M is equal to MT＞0,N＝NT＞0,P＝PTGreater than 0, is a positive definite symmetric matrix, tr {. cndot.) represents the trace-solving operation of the matrix, V0(η)＝ηTPη，
Figure BDA0001460057000000109
And
Figure BDA00014600570000001010
in order to optimize the parameters of the process,
get
Figure BDA00014600570000001011
Substituting the control law of the formula (25) into a dynamic equation (6) considering the uncertainty of the system and the external interference and simplifying the equation to obtain:
Figure BDA0001460057000000111
substituting equation (12) into (29) yields:
Figure BDA0001460057000000112
equation (30) can be further reduced to, based on the definition of the parameter estimation error by equation (26):
Figure BDA0001460057000000113
v first derivative over time, having:
Figure BDA0001460057000000114
wherein the content of the first and second substances,
Figure BDA0001460057000000115
substituting (32) into equation (31) has:
Figure BDA0001460057000000116
because D is equal to DT,K＝KT,Ω＝-ΩTAnd is and
Figure BDA0001460057000000117
(scalar), therefore:
Figure BDA0001460057000000118
the same can be obtained:
Figure BDA0001460057000000119
thus, formula (33) can be arranged as:
Figure BDA00014600570000001110
to ensure
Figure BDA0001460057000000121
First of all design
Figure BDA0001460057000000122
The parameter adaptation law of (1) is as follows:
Figure BDA0001460057000000123
therefore, the method comprises the following steps:
Figure BDA0001460057000000124
order to
Figure BDA0001460057000000125
Therefore, there are:
Figure BDA0001460057000000126
and is
Figure BDA0001460057000000127
Therefore:
Figure BDA0001460057000000128
and because:
Figure BDA0001460057000000129
let Q be ═ ATP+PAT+PBBTP+2CTC) The above formula can be organized as:
Figure BDA0001460057000000131
substituting the parameters to obtain:
Figure BDA0001460057000000132
then the condition that Q is positive according to the matrix theory is:
Figure BDA0001460057000000133
in summary, equation (38) can be arranged as:
Figure BDA0001460057000000134
from an orthodefinite quadratic function V0(η)＝ηTP η can be:
Figure BDA0001460057000000135
wherein λ ismin(P)，λmax(P) represents the minimum eigenvalue and the maximum eigenvalue of the matrix P,
then:
Figure BDA0001460057000000136
from formula (42):
Figure BDA0001460057000000137
wherein λ ismin(Q) is the minimum eigenvalue of the matrix Q, η1，η2Are the two components of η and,
consists of:
Figure BDA0001460057000000138
can obtain | | η | non-woven phosphor2≥|η1|
Then:
Figure BDA0001460057000000141
wherein:
Figure BDA0001460057000000142
therefore, the formula (44) can be further arranged as:
Figure BDA0001460057000000143
wherein the content of the first and second substances,
Figure BDA0001460057000000144
assume the adaptation law (18), k1,k2Are bounded so that there is always a sufficiently large constant
Figure BDA0001460057000000145
So that
Figure BDA0001460057000000146
Thus:
wherein:
Figure BDA0001460057000000148
general formula (18)
Figure BDA0001460057000000149
When the adaptive law expression (2) is substituted into the above expression (51) to obtain ξ being equal to 0:
Figure BDA0001460057000000151
from the L yapunov theory of stability, provided that it satisfies
Figure BDA0001460057000000152
η, k1,k2Is consistently and progressively stabilized at the equilibrium point, and the slip-form surface s and the first derivative of the slip-form surface
Figure BDA0001460057000000154
Can converge to zero in a finite time, for k1,k2At the system, in the system
Figure BDA0001460057000000153
Front, k1,k2Will increase equally rapidly under the action of the adaptive law (18), so that over a finite time, k1,k2The value condition can be satisfied.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (3)

1. The self-adaptive super-distortion sliding mode control method with adjustable gain of the micro gyroscope is characterized by comprising the following steps of:
1) simplifying the micro-gyroscope system into a damped oscillation system consisting of a mass block and a spring, and establishing a dimensionless mathematical model of the micro-gyroscope system;
2) designing a reference model;
3) designing a slip form surface; the slip form surface s is designed as follows:
Figure FDA0002478072340000011
wherein c is a sliding mode surface constant, s1,s2Is the two components of s, e is the tracking error,
Figure FDA0002478072340000012
wherein the content of the first and second substances,
Figure FDA0002478072340000013
is the output track of the micro-gyroscope system,
Figure FDA0002478072340000014
is the desired trajectory of the micro-gyroscope system;
4) the adaptive super-distortion sliding mode controller is designed by adopting a method of combining equivalent sliding mode control and super-distortion control, and the design control law is as follows:
u＝ueq+usw(9)
wherein u is the control laweqTo an equivalent control law, uswIs a switching control law;
the equivalent control law ueqThe solution process of (2) is as follows:
derivation of the slip form surface can be obtained:
Figure FDA0002478072340000015
without considering the external interference, there are:
Figure FDA0002478072340000016
substituting equation (13) into equation (12) yields:
Figure FDA0002478072340000017
order to
Figure FDA0002478072340000018
From this, an equivalent controller, an equivalent control law u, is obtainedeqComprises the following steps:
Figure FDA0002478072340000019
the switching control law uswThe design is as follows:
Figure FDA00024780723400000110
wherein k is1，k2Is a supertwist sliding mode controller parameter, and k1＞0,k2> 0, and k2A micro gyroscopeUpper bound values of system lumped parameter uncertainty and external interference derivative;
the control law is:
Figure FDA0002478072340000021
wherein D, omega and K are dimensionless mathematical model parameters of the micro gyroscope system;
5) designing self-adaptive laws of parameters of a supertorsion sliding mode controller and uncertain parameters of the micro-gyroscope system, and performing stability analysis on the micro-gyroscope system by adopting L yapunov functions to ensure asymptotic stability of the system;
the self-adaptive law of the parameters of the supertorsion sliding mode controller is as follows:
Figure FDA0002478072340000022
wherein the content of the first and second substances,
Figure FDA0002478072340000023
is k1Initial value of (a), gamma1,β1,γ2,β2And χ is a normal number;
the self-adaptive law of uncertain parameters of the micro-gyroscope system is as follows:
Figure FDA0002478072340000024
wherein the content of the first and second substances,
Figure FDA0002478072340000025
satisfies the following conditions:
Figure FDA0002478072340000026
Figure FDA0002478072340000031
estimating an error for the parameter;
the L yapunov function was chosen as:
Figure FDA0002478072340000032
wherein V is L yapunov function, M, N, P is adaptive fixed gain, and M is M ═ MT＞0,N＝NT＞0,P＝PTGreater than 0, is a positive definite symmetric matrix, tr {. cndot.) represents the trace-solving operation of the matrix, V0(η)＝ηTPη，
Figure FDA0002478072340000033
And
Figure FDA0002478072340000034
in order to optimize the parameters of the process,
p satisfies:
Figure FDA0002478072340000035
Figure FDA0002478072340000036
Figure FDA0002478072340000037
2. the micro-gyroscope gain-adjustable adaptive super-distortion sliding-mode control method according to claim 1, wherein the establishing of the dimensionless mathematical model of the micro-gyroscope system comprises the following steps:
1-1) according to Newton's law in the rotation system, comprehensively considering the influence of various manufacturing errors on the micro gyroscope, and obtaining a mathematical model of the micro gyroscope as follows:
Figure FDA0002478072340000038
wherein m is massMass of the mass, x, y being the position vectors of the mass at both the drive and sense axes, dxx,dyyExpressing the damping coefficients, k, of the x, y axesxx,kyySpring constants, u, of the x, y axes, respectivelyx,uyIs a control input representing two axes x, y, kxy，dxyIs the coupling spring coefficient and damping coefficient, omega, caused by manufacturing errorszRepresenting the angular velocity in the operating environment of the micro-gyroscope,
Figure FDA0002478072340000039
is the coriolis force;
1-2) dividing both sides of the mathematical model formula (1) of the micro-gyroscope by the mass m of the mass block of the micro-gyroscope simultaneously, the reference length q0Square of resonance frequency of two axes omega0 2The dimensionless mathematical model is obtained as follows:
Figure FDA0002478072340000041
the expression for each dimensionless quantity is:
Figure FDA0002478072340000042
the symbol "→" indicates that the amount to the left of the symbol is replaced with the amount to the right of the symbol;
1-3) rewriting the dimensionless mathematical model (2) into vector form:
Figure FDA0002478072340000043
1-4) considering the parameter uncertainty and the external interference of the micro-gyroscope system, modifying the mathematical model of the micro-gyroscope system into:
Figure FDA0002478072340000044
wherein, Δ D is the uncertainty of the unknown parameters of the inertial matrix D +2 Ω, Δ K is the uncertainty of the unknown parameters of the matrix K, and D is the external interference;
1-5) lumped parameter uncertainty and external interference defining system
Figure FDA0002478072340000045
Comprises the following steps:
Figure FDA0002478072340000046
formula (5) is represented as:
Figure FDA0002478072340000047
wherein:
Figure FDA0002478072340000048
Figure FDA0002478072340000049
is satisfied with
Figure FDA00024780723400000410
The upper bound of the lumped parameter uncertainty and the external disturbance derivative.
3. The micro-gyroscope gain-adjustable adaptive super-distortion sliding-mode control method according to claim 2, wherein the reference model is:
Figure FDA00024780723400000411
selecting stable sinusoidal oscillation by the reference model, and enabling:
qr1＝A1sin(ω1t)，qr2＝A2sin(ω2t)，
wherein A is1，A2Being amplitude of oscillation, ω1，ω2Is the frequency of the oscillation.
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