CN105929694A - Adaptive neural network nonsingular terminal sliding mode control method for micro gyroscope - Google Patents

Abstract

The invention discloses an adaptive neural network nonsingular terminal sliding mode control method for a micro gyroscope. The method includes the steps of the establishing a mathematical model of the micro gyroscope, approximating the sum of the dynamic characteristics and external disturbance of the micro gyroscope by using a neural network control method, designing an adaptive neural network nonsingular terminal sliding mode device based on a dynamic surface; and controlling the micro gyroscope by using the adaptive neural network nonsingular terminal sliding mode device based on the dynamic surface. Through the method, a micro gyroscope system can rapidly reach a stable state, and manufacturing error and environment interference can be compensated. The algorithm designed based on dynamic surface method reduces parameters introduced, simplifies calculation and minimizes buffeting. Meanwhile, a nonsingular terminal sliding mode is introduced in the method to ensure that the system state converges in the sliding phase for a finite time and the control rules have no negative exponential terms, so that the effectiveness of the system can be improved.

Description

Micro-gyroscope adaptive neural network nonsingular terminal sliding mode control method

Technical Field

The invention relates to the technical field of micro gyroscope dynamic control, in particular to a micro gyroscope adaptive neural network nonsingular terminal sliding mode control method based on a dynamic surface.

Background

The micro gyroscope is a sensor for measuring the angular velocity of an inertial navigation and inertial guidance system, which is widely applied to the military and civil fields of navigation and positioning of aviation, aerospace, navigation and land vehicles, exploration and development of oil fields and the like. Micro gyroscopes have a great advantage in terms of volume and cost compared to traditional gyroscopes. However, due to the existence of errors in the manufacturing process and the influence of the external environment temperature, differences between the characteristics of the original piece and the design are caused, so that the coupled stiffness coefficient and damping coefficient exist, and the sensitivity and precision of the micro gyroscope are reduced. In addition, since the gyroscope itself belongs to a mimo system, there is uncertainty in parameters and system parameters are liable to fluctuate under external disturbances, and therefore, reduction of system chattering becomes one of the main problems in the control of the micro gyroscope.

Disclosure of Invention

The invention aims to provide a micro-gyroscope adaptive neural network nonsingular terminal sliding mode control method which has the characteristics of low buffeting, high reliability and high robustness to parameter change.

The technical scheme adopted by the invention is as follows: a micro-gyroscope adaptive neural network nonsingular terminal sliding mode control method comprises the following steps:

step one, establishing a mathematical model of the micro gyroscope:

secondly, approximating the sum of the dynamic characteristic of the micro gyroscope and external interference by using a neural network control method;

designing a non-singular terminal sliding mode controller of the adaptive neural network based on the dynamic surface;

and step four, controlling the micro gyroscope by using the adaptive neural network nonsingular terminal sliding mode controller designed in the step three.

The principle of the invention is as follows: the non-singular terminal sliding mode control method of the self-adaptive neural network based on the dynamic surface is applied to the micro gyroscope, a noise-carrying approximately ideal micro gyroscope dynamic model is designed to be used as a system reference track, and the whole non-singular terminal sliding mode control of the self-adaptive neural network based on the dynamic surface guarantees that the actual micro gyroscope track tracks the upper reference track, so that an ideal dynamic characteristic is achieved, manufacturing errors and environmental interference are compensated, and buffeting of the system is reduced. According to the parameters of the micro gyroscope and the input angular rate, a parameter-adjustable dynamic surface controller and a self-adaptive neural network controller are designed, a tracking error signal of a system is used as an input signal of the controller, the initial value of the parameters of the controller is set at will, the tracking error is guaranteed to be converged to zero, and meanwhile, all parameter estimation values are converged to the true value.

In the first step, the mathematical model of the micro gyroscope is as follows:

$\left\{\begin{array}{c}m\stackrel{\·\·}{x}+d_{xx}\stackrel{\·}{x}+d_{xy}\stackrel{\·}{y}+k_{xx}x+k_{xy}y=u_x+2{\mathrm{m\Ω}}_z\stackrel{\·}{y}\\ m\stackrel{\·\·}{y}+d_{xy}\stackrel{\·}{x}+d_{yy}\stackrel{\·}{y}+k_{xy}x+k_{yy}y=u_y-2{\mathrm{m\Ω}}_z\stackrel{\·}{x}\end{array}\right.$

wherein x and y represent the displacement of the micro gyroscope in the direction of X, Y axes, respectively, and d_xx、d_yyRespectively, the elastic coefficient, k, of the X, Y axial spring of the micro-gyroscope_xx、k_yyDamping coefficient of the micro-gyroscope in the direction of the X, Y axis, d_xy、k_xyFor coupling parameters caused by processing errors, etc., m is the mass of the mass block of the micro-gyroscope, Ω_zAngular velocity of self-rotation of mass u_x、u_yInput control forces of X, Y axes, respectively, in the form ofIs in the form of a first derivative of a parametric representation ofThe second derivative of the parametric representation of (a);

preferably, the invention performs non-dimensionalization on the model to obtain a non-dimensionalized model:

both sides of the equation are divided by m at the same time, and the dimensionless model is then:

model is rewritten to vector form:

$\stackrel{\·\·}{q}+D\stackrel{\·}{q}+Kq=u-2\mathrm{\Ω}\stackrel{\·}{q}$

wherein u is a dynamic surface control law,

considering system parameter uncertainty and external interference, the model is written as:

$\stackrel{\·\·}{q}+(D+\mathrm{\Δ}D)\stackrel{\·}{q}+(K+\mathrm{\Δ}K)q=u-2\mathrm{\Ω}\stackrel{\·}{q}+d$

wherein Δ D, Δ K are parameter perturbations, D is external interference;

writing the model as a state equation:

$\left\{\begin{array}{c}{\stackrel{\·}{q}}_1=q_2\\ {\stackrel{\·}{q}}_2=-(D+\mathrm{\Δ}D+2\mathrm{\Ω})\stackrel{\·}{q}-(K+\mathrm{\Δ}K)q+u+d\end{array}\right.$

wherein q is₁＝q，

For ease of calculation, q ═ x will be defined₁，x₁、x₂Is an input variable;

the state equation of the micro-gyroscope model becomes the following equation:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=f+u\end{array}\right.$

wherein f is the sum of the dynamic characteristic of the gyroscope and external interference, and:

f＝-(D+ΔD+2Ω)x₂-(K+ΔK)x₁+d。

a Radial Basis Function (RBF) neural network has a forward three-tier network topology. Wherein, the input layer is only the signal receiving layer and does not perform any signal processing. The dimension of the input layer is related to the dimension of the specific signal, e.g. x, which is a four-dimensional vector, for the input signal of the neural network in this example, so the input layer of the RBF network has four input nodes. The intermediate layer is a hidden layer, implements the nonlinear mapping function of the signal, and maps the signal from the input space to a hidden layer space with higher dimension and linearly separable signal characteristics. And the output layer performs weighted summation operation to generate RBF network output.

Preferably, in the second step of the present invention, the RBF neural network principle is usedTo approximate f, comprising the steps of:

using x (t) as input vector of RBF neural network, wherein x (t) includes x₁、x₂The input variable of (1); let radial basis vector h ═ h of RBF neural network₁,h₂,h₃，…h_m]^TWherein h is_iIs a gaussian basis function, i.e.:

$h_i=\exp (-\frac{\left|\right|x\left(t\right)-c_i|{|}^2}{2b_i^2}),i=1,2,...,m$

wherein c is ═ c₁,c₂,c₃,…c_m]^TThe central vector of the network hidden layer node is the same as the dimension of the input vector;

b＝[b₁,b₂,b₃,…b_m]^Tis the base width vector of the network hidden layer node, m is the number of hidden layer neurons, the weight from the input layer to the hidden layer of the RBF network is 1, and the weight vector from the network hidden layer to the output layer is W ═ W₁,w₂,w₃,…w_m]^T；

The RBF network output is:

y＝h^T*W

h^Tis a transpose of the radial basis function;

c of RBF neural network_iAnd b_iKeeping the RBF neural network fixed, and only adjusting the network weight W, so that the output of the RBF neural network is in a linear relation with the hidden layer output;

the output of the RBF neural network is:

$\hat{f}=h^T\hat{W}$

using outputs of neural networksTo approximate the dynamic characteristic f of the gyroscope;

an optimal approximation constant is defined and,W^*

$W^{*}=\arg \underset{W\∈\mathrm{\Ω}}{min}\[sup|\hat{f}-f|\]$

Ω is the set of W.

Order to

$\tilde{W}=\hat{W}-W^{*}$

Then:

f＝h^TW^*+

$f-\hat{f}=h^T{W}^{*}+\mathrm{\ϵ}-h^T\hat{W}=-h^T\tilde{W}+\mathrm{\ϵ}$

where is the approximation error of the neural network, for a given arbitrary small constant (> 0), the following inequality holds: l f-h^TW^*|≤。

Preferably, step three of the present invention comprises the steps of:

defining position errors

z₁＝x₁-x_1d

Wherein x_1dIs a command reference signal, then

${\stackrel{\·}{z}}_1={\stackrel{\·}{x}}_1-{\stackrel{\·}{x}}_{1d}$

Defining the Lyapunov function asWhereinIs z₁Is transposed, then

${\stackrel{\·}{V}}_1={z_1}^T{\stackrel{\·}{z}}_1={z_1}^T({\stackrel{\·}{x}}_1-{\stackrel{\·}{x}}_{1d})={z_1}^T(x_2-{\stackrel{\·}{x}}_{1d})$

To ensureIntroduction ofIs x₂Virtual control quantity of (2), defining

${\stackrel{\‾}{x}}_2=-c_1{z}_1+{\stackrel{\·}{x}}_{1d}$

c₁Is a constant greater than 0;

to overcome microDividing explosion phenomenon, introducing low-pass filter (α)₁Is a low-pass filterAbout input ofAnd (2) output of time, and satisfies:

$\left\{\begin{array}{c}\mathrm{\τ}{\stackrel{\·}{\mathrm{\α}}}_1+{\mathrm{\α}}_1={\stackrel{\‾}{x}}_2\\ {\mathrm{\α}}_1\left(0\right)={\stackrel{\‾}{x}}_2\left(0\right)\end{array}\right.$

where τ is the time constant of the filter, a constant greater than 0, α₁Is the output of a low pass filter, α₁(0)、Are respectively α₁Andinitial value of (a):

${\stackrel{\·}{\mathrm{\α}}}_1=\frac{{\stackrel{\‾}{x}}_2-{\mathrm{\α}}_1}{\mathrm{\τ}}$

the resulting filtering error is:

$y_2={\mathrm{\α}}_1-{\stackrel{\‾}{x}}_2$

defining a virtual control error: z is a radical of₂＝x₂-α₁Then, then

The second Lyapunov function is defined as:

$V_2=\frac{1}{2}{z_2}^T{z_2}^2$

to ensure

The non-singular terminal sliding mode surface of the dynamic surface designed for the micro gyroscope is defined as follows:

$s={\[s_1,s_2\]}^T=z_1+\frac{1}{\mathrm{\β}}{z_2}^{p_1/p_2}$

wherein β ═ diag (β)₁,β₂) Is the slip form surface constant, β₁,β₂Are all normal numbers, and are all positive numbers,p₁,p₂is positive odd number and 1 < p₁/p₂＜2；

To make it possible toFor a micro-gyroscope system, a non-singular terminal sliding mode surface of a dynamic surface is adopted to design a non-singular terminal of the dynamic surfaceThe end sliding mode control law is as follows:

u＝u₀+u₁+u₂+u₃

wherein,

$u_0=-f+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1$

$u_1=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right){\stackrel{\&CenterDot;}{z}}_1$

$u_2=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right)\frac{s}{\left|\right|s|{|}^2}({z_1}^T{z}_2+\frac{1}{2})$

$u_3=-\mathrm{\&rho;}\frac{{\&lsqb;s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\&rsqb;}^T}{\left|\right|s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)|{|}^2}\left|\right|s\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\left|\right|$

at this time, the neural network is used for outputtingAnd (3) approaching the dynamic characteristic f of the gyroscope, wherein the updated control law is as follows:

u＝u₀'+u₁+u₂+u₃

wherein:

the invention has the beneficial effects that:

the self-adaptive neural network controller based on the dynamic surface is utilized to dynamically control the micro-gyroscope, so that the micro-gyroscope system can reach a steady state at a high speed, the dynamic characteristic of the micro-gyroscope is a mode which tends to be ideal, and the manufacturing error and the environmental interference can be compensated. The algorithm designed based on the dynamic surface method reduces the introduced parameters, simplifies the calculation degree, reduces buffeting and improves the effectiveness of the system.

Drawings

FIG. 1 is a schematic diagram of a simplified model of a micro-gyroscope according to the invention;

FIG. 2 is a schematic diagram of the principles of the present invention;

FIG. 3 is a time domain response plot of error in an embodiment of the present invention;

FIG. 4 is a graphical representation of the time domain response of the x-axis control force and the y-axis control force in an embodiment of the present invention.

Detailed Description

The present invention will be better understood and implemented by those skilled in the art by the following detailed description of the embodiments of the present invention with reference to the accompanying drawings, which are not intended to limit the present invention.

Referring to fig. 2, the method for controlling the micro gyroscope adaptive fuzzy sliding mode based on the dynamic surface design includes the following steps:

step one, establishing a mathematical model of the micro gyroscope:

secondly, approximating the sum of the dynamic characteristic of the micro gyroscope and external interference by using a neural network control method;

designing a non-singular terminal sliding mode controller of the adaptive neural network based on the dynamic surface;

and step four, controlling the micro gyroscope by using the adaptive neural network nonsingular terminal sliding mode controller designed in the step three.

Examples

The first step is carried out:

referring to fig. 1, a typical micro-gyroscope is composed of the following parts: the mass block comprises a mass block, supporting springs along the X-axis direction and the Y-axis direction, an electrostatic driving device and a sensing device, wherein the electrostatic driving device drives the mass block to vibrate along the driving shaft direction, and the sensing device can detect the displacement and the speed of the mass block in the detection shaft direction.

Then, the mathematical model of the micro gyroscope established in step one is:

$\left\{\begin{array}{c}m\stackrel{\&CenterDot;\&CenterDot;}{x}+d_{xx}\stackrel{\&CenterDot;}{x}+d_{xy}\stackrel{\&CenterDot;}{y}+k_{xx}x+k_{xy}y=u_x+2{\mathrm{m\&Omega;}}_z\stackrel{\&CenterDot;}{y}\\ m\stackrel{\&CenterDot;\&CenterDot;}{y}+d_{xy}\stackrel{\&CenterDot;}{x}+d_{yy}\stackrel{\&CenterDot;}{y}+k_{xy}x+k_{yy}y=u_y-2{\mathrm{m\&Omega;}}_z\stackrel{\&CenterDot;}{x}\end{array}\right.---\left(1\right)$

wherein x and y represent the displacement of the micro gyroscope in the direction of X, Y axes, respectively, and d_xx、d_yyRespectively, the elastic coefficients of X, Y axial direction springs, k_xx、k_yyX, Y damping coefficients in the axial direction, d_xy、k_xyIs a coupling parameter caused by machining error, m is the mass of the gyroscope mass block, omega_zAngular velocity of self-rotation of mass u_x、u_yInput control forces of X, Y axes, respectively, in the form ofIs in the form of a first derivative of a parametric representation ofIs the second derivative of the parametric representation of (a).

Since the equations have unit quantities in addition to numerical quantities, the complexity of the design of the controller is increased. The vibration frequency of the mass block in the gyroscope model reaches KHz magnitude, while the autorotation angular velocity of the mass block is only several degrees and one hour magnitude, and the great difference of magnitude can bring inconvenience to simulation. In order to solve the problem of large difference between different unit quantities and different magnitude orders, dimensionless processing can be performed on the equation.

Both sides of the equation are divided by m at the same time, and the dimensionless model is then:

model is rewritten to vector form:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+D\stackrel{\&CenterDot;}{q}+Kq=u-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}---\left(3\right)$

wherein u is a dynamic surface control law,

considering system parameter uncertainty and external interference, the model can be written as:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+(D+\mathrm{\&Delta;}D)\stackrel{\&CenterDot;}{q}+(K+\mathrm{\&Delta;}K)q=u-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}+d---\left(4\right)$

wherein Δ D, Δ K are parameter perturbations, D is external interference;

it is written in the form of a state equation:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{q}}_1=q_2\\ {\stackrel{\&CenterDot;}{q}}_2=-(D+\mathrm{\&Delta;}D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}-(K+\mathrm{\&Delta;}K)q+u+d\end{array}\right.---\left(5\right)$

wherein q is₁＝q，

For ease of calculation, q ═ x will be defined₁，x₁、x₂Is an input variable;

the state equation becomes the following equation:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=f+u\end{array}\right.---\left(6\right)$

wherein f is the sum of the dynamic characteristic of the gyroscope and external interference, and:

f＝-(D+ΔD+2Ω)x₂-(K+ΔK)x₁+d (7)

preferably, the neural network principle is introduced in the step two, andapproaching f, specifically comprising the following steps:

a Radial Basis Function (RBF) neural network has a forward three-tier network topology. Wherein, the input layer is only the signal receiving layer and does not perform any signal processing. The dimension of the input layer is related to the dimension of the specific signal, e.g. x, which is a four-dimensional vector, for the input signal of the neural network in this example, so the input layer of the RBF network has four input nodes. The intermediate layer is a hidden layer, implements the nonlinear mapping function of the signal, and maps the signal from the input space to a hidden layer space with higher dimension and linearly separable signal characteristics. And the output layer performs weighted summation operation to generate RBF network output.

In the RBF network structure, x (t) is the input vector of the network. Let the radial basis vector h of RBF network be [ h ═ h₁,h₂,h₃,…h_m]^TWherein h is_iIs a Gaussian base function, i.e.

$h_i=\exp (-\frac{\left|\right|x\left(t\right)-c_i|{|}^2}{2b_i^2}),i=1,2,...,m---\left(8\right)$

Wherein c is ═ c₁,c₂,c₃,…c_m]^TIs the central vector of the network hidden layer node, and is the same as the input dimension.

b＝[b₁,b₂,b₃,…b_m]^TIs the base width vector of the network hidden layer node, which determines the size of the area, m is the number of hidden layer neurons, the weight from the input layer to the hidden layer of the RBF network is 1, and the weight vector from the network hidden layer to the output layer is W ═ W₁,w₂,w₃,…w_m]^T。

The RBF network outputs are

y＝h^T*W (9)

h^TIs the transpose of the radial basis function.

In practical application, the RBF network has more than one node c_iAnd b_iHoldingAnd if the network weight W is fixed and only the network weight W is adjusted, the output of the RBF network and the hidden layer output are in a linear relation.

Using the strong approximation characteristic of the neural network and the output of the neural networkTo approximate the dynamic characteristic f of the gyroscope:

the output of the RBF neural network is:

$\hat{f}=h^T\hat{W}---\left(10\right)$

defining an optimal approximation constant, W^*

$W^{*}=\arg \underset{W\&Element;\mathrm{\&Omega;}}{min}\&lsqb;sup|\hat{f}-f|\&rsqb;---\left(11\right)$

Ω is the set of W.

Order to

$\tilde{W}=\hat{W}-W^{*}---\left(12\right)$

Then:

f＝h^TW^*+ (13)

$f-\hat{f}=h^T{W}^{*}+\mathrm{\&epsiv;}-h^T\hat{W}=-h^T\tilde{W}+\mathrm{\&epsiv;}---\left(14\right)$

among these are the approximation errors of the neural network. For any given small constant (> 0), the following inequality holds: l f-h^TW^*|≤。

Preferably, the third step specifically comprises the following steps:

defining position errors

z₁＝x₁-x_1d(15)

Wherein x_1dIs a command signal, then

${\stackrel{\&CenterDot;}{z}}_1={\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_{1d}---\left(16\right)$

Defining the Lyapunov function asWhereinIs z₁Is transposed, then

${\stackrel{\&CenterDot;}{V}}_1={z_1}^T{\stackrel{\&CenterDot;}{z}}_1={z_1}^T({\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_{1d})={z_1}^T(x_2-{\stackrel{\&CenterDot;}{x}}_{1d})---\left(17\right)$

To ensureIntroduction ofIs x₂Virtual control quantity of (2), defining

${\stackrel{\&OverBar;}{x}}_2=-c_1{z}_1+{\stackrel{\&CenterDot;}{x}}_{1d}---\left(18\right)$

c₁Is a constant greater than 0;

in order to overcome the phenomenon of differential explosion, a low-pass filter is introduced:

get α₁Is a low-pass filterAbout input ofThe output of the time-of-day,

and satisfies the following conditions:

where τ is the time constant of the filter, a constant greater than 0, α₁Is the output of a low pass filter, α₁(0)、Are respectively α₁Andinitial value of (a):

from (16) can be obtained:

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=\frac{{\stackrel{\&OverBar;}{x}}_2-{\mathrm{\&alpha;}}_1}{\mathrm{\&tau;}}---\left(20\right)$

the generated filtering error is

$y_2={\mathrm{\&alpha;}}_1-{\stackrel{\&OverBar;}{x}}_2---\left(21\right)$

Virtual control error:

z₂＝x₂-α₁(22)

then:

${\stackrel{\&CenterDot;}{z}}_2=f+u-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(23\right)$

the true control input appears in the above equation.

The second Lyapunov function is defined as:

$V_2=\frac{1}{2}{z_2}^T{z_2}^2---\left(24\right)$

to ensure

The non-singular terminal sliding mode surface of the dynamic surface designed for the micro gyroscope is defined as follows:

$s={\&lsqb;s_1,s_2\&rsqb;}^T=z_1+\frac{1}{\mathrm{\&beta;}}{z_2}^{p_1/p_2}---\left(25\right)$

wherein β ═ diag (β)₁,β₂) Is the slip form surface constant, β₁,β₂Are all normal numbers, and are all positive numbers,p₁,p₂is positive odd number and 1 < p₁/p₂＜2

To make it possible toFor a micro-gyroscope system, a dynamic surface nonsingular terminal sliding mode surface is adopted, and the dynamic surface nonsingular terminal sliding mode control law is designed as follows:

u＝u₀+u₁+u₂+u₃(26)

wherein,

$u_0=-f+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(27\right)$

$u_1=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right){\stackrel{\&CenterDot;}{z}}_1---\left(28\right)$

$u_2=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right)\frac{s}{\left|\right|s|{|}^2}({z_1}^T{z}_2+\frac{1}{2})---\left(29\right)$

$u_3=-\mathrm{\&rho;}\frac{{\&lsqb;s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\&rsqb;}^T}{\left|\right|s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)|{|}^2}\left|\right|s\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\left|\right|---\left(30\right)$

at this time, the neural network is used for outputtingAnd (3) approaching the dynamic characteristic f of the gyroscope, wherein the updated overall control law is as follows:

u＝u₀'+u₁+u₂+u₃(31)

wherein:

${u^{\&prime;}}_0=-\hat{f}+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(32\right)$

the specific principle is shown in fig. 2.

The stability of the system proved as follows:

and considering the position tracking error, the virtual control error, the wave filtering error and the approximation error of the RBF neural network. Defining the Lyapunov function as

$V=\frac{1}{2}{z_1}^T{z}_1+\frac{1}{2}{s}^{T}s+\frac{1}{2}{y_2}^T{y}_2+\frac{1}{2\mathrm{\&gamma;}}{\tilde{W}}^T\tilde{W}---\left(33\right)$

In the formula z₁For tracking errors and their correlation functions, s is the sliding mode surface, y₂Is the error of the filtering and is,is the RBF neural network parameter error. γ is a number greater than 0.

Theorem: and if V (0) is less than or equal to p and p is greater than 0, all signals of the closed-loop system are converged and bounded.

The derivative of the Lyapunov function is:

$\stackrel{\&CenterDot;}{V}={z_1}^T{\stackrel{\&CenterDot;}{z}}_1+s^T\stackrel{\&CenterDot;}{s}+{y_2}^T{\stackrel{\&CenterDot;}{y}}_2+\frac{1}{\mathrm{\&gamma;}}{\tilde{W}}^T\stackrel{\&CenterDot;}{\hat{W}}---\left(34\right)$

wherein

${\stackrel{\&CenterDot;}{z}}_1={\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_{1d}=x_2-{\stackrel{\&CenterDot;}{x}}_{1d}=z_2+{\mathrm{\&alpha;}}_1-{\stackrel{\&CenterDot;}{x}}_{1d}=z_2+y_2+{\stackrel{\&OverBar;}{x}}_2-{\stackrel{\&CenterDot;}{x}}_{1d}---\left(35\right)$

$\stackrel{\&CenterDot;}{s}={\stackrel{\&CenterDot;}{z}}_1+\frac{p_1}{p_2}\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right){\stackrel{\&CenterDot;}{z}}_2---\left(36\right)$

${\stackrel{\&CenterDot;}{y}}_2={\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1-{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}}_2=\frac{{\stackrel{\&OverBar;}{x}}_2-{\mathrm{\&alpha;}}_1}{\mathrm{\&tau;}}-{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}}_2=\frac{-y_2}{\mathrm{\&tau;}}-{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}}_2=\frac{-y_2}{\mathrm{\&tau;}}+c_1{\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}---\left(37\right)$

The adaptive law of the neural network available from the lyapunov stability is:

$\stackrel{\&CenterDot;}{\hat{W}}=\mathrm{\&gamma;}\frac{p_1}{p_2}\left|\right|s\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\left|\right|h\left(x\right)---\left(38\right)$

from the above, it can be obtained:

$\stackrel{\&CenterDot;}{V}=-{{\mathrm{rz}}_1}^T{z}_1\&le;0---\left(39\right)$

the tracking track reaches the sliding mode surface in a limited time and stays on the sliding mode surface.Semi-negative nature of (c) ensures that both V, s are bounded. From equation (36), it can be seen thatIs also bounded. Since V (0) is bounded, 0 ≦ V (t ≦ V (0), it may be inferredIs bounded. According to the Barbalt's theorem and its deduction, s (t) will tend to zero, i.e.And then also have

Matlab simulation experiments were performed as follows.

A main program is designed through Matlab/Simulink software by combining a dynamic model of a micro-gyroscope sensor and a design method of non-singular terminal sliding mode control of an adaptive neural network based on a dynamic surface, and as shown in FIG. 2, a self-adaptive dynamic sliding mode controller, a controlled object micro-mechanical gyroscope and parameter dimensionalization calculation are written into subprograms by utilizing the characteristics of an S Function and are respectively placed in a plurality of S-functions.

From the prior literature, a set of parameters of the micro-gyroscope is chosen as follows:

the parameters of a set of micro-gyroscopes were chosen as follows:

m＝1.8×10^-7kg,k_xx＝63.955N/m,k_yy＝95.92N/m,k_xy＝12.779N/m

d_xx＝1.8×10^-6Ns/m,d_yy＝1.8×10^-6Ns/m,d_xy＝3.6×10^-7Ns/m

suppose the input angular velocity is Ω_z100rad/s, reference frequency ω₀1000 Hz. The non-dimensionalized parameters of the gyroscope are obtained as follows:

ω_x ²＝355.3,ω_y ²＝532.9,ω_xy＝70.99,d_xx＝0.01,d_yy＝0.01,d_xy＝0.02,Ω_Z＝0.01。

the reference model is selected as follows: r is₁＝sin(4.17t)，r₂＝1.2sin(5.11t)。

The initial conditions were set as: x is the number of₁₁(0)＝0.01,x₁₂(0)＝0,x₁₂(0)＝0.01,x₂₂(0)＝0.

Selecting parameters according to a control law as follows:

c₁₁＝2500,c₁₂＝2500；r₁＝1,r₂＝1；

γ₁＝500,γ₂＝500；tol₁＝0.01,tol₂＝0.01.

taking interference items: [ sin (5 t); sin (2t) ].

c₁＝[-1,-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8,1]

c＝[c₁；c₁；c₁；c₁]

b＝0.03

c_jIs the coordinate vector of the central point of the Gaussian base function of the jth neuron of the hidden layer.

b_jThe width of the gaussian basis function for the jth neuron in the hidden layer.

In the sliding mode control law, p is taken as the sliding mode surface parameter₁＝99，p₂When the sliding mode is 51, β is diag (1,1), the nonsingular terminal sliding mode surface is s-e₁+e₂ ^99/51。

The results of the experiment are shown in fig. 3 and 4:

the variation of the error between the actual output and the expected output is shown in fig. 3, and the result shows that the actual output can perfectly track the expected output in a short time, and the error is close to zero and is stable.

The control force input value curve is shown in fig. 4, and the result shows that the dynamic surface sliding mode controller successfully reduces the introduction of parameters, so that the buffeting of the system is obviously reduced.

The method is applied to the non-singular terminal sliding mode control of the adaptive neural network based on the dynamic surface of the micro gyroscope, and the non-singular terminal sliding mode method of the adaptive neural network based on the dynamic surface design is adopted to control the micro gyroscope, so that the buffeting is effectively reduced, and the tracking speed is improved. Under the condition that system parameters are unknown, various parameters of the system can be effectively estimated, and the stability of the system is ensured. The dynamic surface technology is introduced into the traditional self-adaptive backward-pushing technology, so that the advantages of the original backward-pushing technology are kept, the number of parameters is reduced, the problem of parameter expansion is avoided, and the complexity of calculation is obviously reduced. Meanwhile, a neural network self-adaptive method is introduced into the controller to well approach the dynamic performance of the gyroscope.

In addition, a nonsingular terminal sliding mode is introduced, so that the finite time convergence of the system state in the sliding stage is ensured, and the control law is free of negative exponential terms. And the stability of the whole system is proved on the basis of the Lyapunov stability theory. By using the system, buffeting of the system can be effectively reduced, manufacturing errors and environmental interference are compensated, and sensitivity and robustness of the system are improved.

The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes, which are made by using the contents of the present specification and the accompanying drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (4)

1. A micro-gyroscope adaptive neural network nonsingular terminal sliding mode control method comprises the following steps:

step one, establishing a mathematical model of the micro gyroscope:

secondly, approximating the sum of the dynamic characteristic of the micro gyroscope and external interference by using a neural network control method;

designing a non-singular terminal sliding mode controller of the adaptive neural network based on the dynamic surface;

and step four, controlling the micro gyroscope by using the adaptive neural network nonsingular terminal sliding mode controller designed in the step three.

2. The method of claim 1, wherein in step one, the mathematical model of the micro-gyroscope is:

$\left\{\begin{array}{c}m\stackrel{\&CenterDot;\&CenterDot;}{x}+d_{xx}\stackrel{\&CenterDot;}{x}+d_{xy}\stackrel{\&CenterDot;}{y}+k_{xx}x+k_{xy}y=u_x+2{\mathrm{m\&Omega;}}_z\stackrel{\&CenterDot;}{y}\\ m\stackrel{\&CenterDot;\&CenterDot;}{y}+d_{xy}\stackrel{\&CenterDot;}{x}+d_{yy}\stackrel{\&CenterDot;}{y}+k_{xy}x+k_{yy}y=u_y-2{\mathrm{m\&Omega;}}_z\stackrel{\&CenterDot;}{x}\end{array}\right.$

wherein x and y represent the displacement of the micro gyroscope in the direction of X, Y axes, respectively, and d_xx、d_yyRespectively, the elastic coefficient, k, of the X, Y axial spring of the micro-gyroscope_xx、k_yyDamping coefficient of the micro-gyroscope in the direction of the X, Y axis, d_xy、k_xyFor coupling parameters, m is the mass of the micro-gyroscope, Ω_zAngular velocity of self-rotation of mass u_x、u_yInput control forces of X, Y axes, respectively, in the form ofIs in the form of a first derivative of a parametric representation ofThe second derivative of the parametric representation of (a);

carrying out non-dimensionalization on the model to obtain a non-dimensionalized model:

both sides of the equation are divided by m at the same time, and the dimensionless model is then:

model is rewritten to vector form:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+D\stackrel{\&CenterDot;}{q}+Kq=u-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}$

wherein u is a dynamic surface control law,

considering system parameter uncertainty and external interference, the model is written as:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+(D+\mathrm{\&Delta;}D)\stackrel{\&CenterDot;}{q}+(K+\mathrm{\&Delta;}K)q=u-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}+d$

wherein Δ D, Δ K are parameter perturbations, D is external interference;

writing the model as a state equation:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{q}}_1=q_2\\ {\stackrel{\&CenterDot;}{q}}_2=-(D+\mathrm{\&Delta;}D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}-(K+\mathrm{\&Delta;}K)q+u+d\end{array}\right.$

wherein q is₁＝q，

For ease of calculation, q ═ x will be defined₁，x₁、x₂Is an input variable;

the state equation of the micro-gyroscope model becomes the following equation:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=f+u\end{array}\right.$

wherein f is the sum of the dynamic characteristic of the gyroscope and external interference, and:

f＝-(D+ΔD+2Ω)x₂-(K+ΔK)x₁+d。

3. the method as claimed in claim 2, wherein in step two, the RBF neural network principle is usedTo approximate f, comprising the steps of:

using x (t) as input vector of RBF neural network, and setting radial basis vector h of RBF neural network as [ h ]₁,h₂,h₃,…h_m]^TWherein h is_iIs a gaussian basis function, i.e.:

$h_i=\exp (-\frac{\left|\right|x\left(t\right)-c_i|{|}^2}{2b_i^2}),i=1,2,...,m$

wherein c is ═ c₁,c₂,c₃,…c_m]^TThe central vector of the network hidden layer node is the same as the dimension of the input vector; b ═ b₁,b₂,b₃,…b_m]^TIs the base width vector of the network hidden layer node, which determines the size of the area, m is the number of hidden layer neurons, the weight from the input layer to the hidden layer of the RBF network is 1, and the weight vector from the network hidden layer to the output layer is W ═ W₁，w₂，w₃，…w_m]^T；

The RBF network output is:

y＝h^T*W

h^Tis a transpose of the radial basis function;

c of RBF neural network_iAnd b_iKeeping the RBF neural network fixed, and only adjusting the network weight W, so that the output of the RBF neural network is in a linear relation with the hidden layer output;

the output of the RBF neural network is:

$\hat{f}=h^T\hat{W}$

using outputs of neural networksTo approximate the dynamic characteristic f of the gyroscope;

defining an optimal approximation constant, W^*

$W^{*}=\arg \underset{W\&Element;\mathrm{\&Omega;}}{min}\&lsqb;sup|\hat{f}-f|\&rsqb;$

Ω is a set of W;

order to

$\tilde{W}=\hat{W}-W^{*}$

Then:

f＝h^TW^*+

$f-\hat{f}=h^T{W}^{*}+\mathrm{\&epsiv;}-h^T\hat{W}=-h^T\tilde{W}+\mathrm{\&epsiv;}$

where is the approximation error of the neural network, for a given arbitrary small constant (> 0), the following inequality holds: l f-h^TW^*|≤。

4. The method of claim 3, wherein step three comprises the steps of:

defining position errors

z₁＝x₁-x_1d

Wherein x_1dIs a command reference signal, then

${\stackrel{\&CenterDot;}{z}}_1={\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_{1d}$

Defining the Lyapunov function asWhereinIs z₁Is transposed, then

${\stackrel{\&CenterDot;}{V}}_1={z_1}^T{\stackrel{\&CenterDot;}{z}}_1={z_1}^T({\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_{1d})={z_1}^T(x_2-{\stackrel{\&CenterDot;}{x}}_{1d})$

To ensureIntroduction ofIs x₂Virtual control quantity of (2), defining

${\stackrel{\&OverBar;}{x}}_2=-c_1{z}_1+{\stackrel{\&CenterDot;}{x}}_{1d}$

c₁Is a constant greater than 0;

in order to overcome the phenomenon of differential explosion, a low-pass filter is introduced, namely α₁Is a low-pass filterAbout input ofAnd (2) output of time, and satisfies:

$\left\{\begin{array}{c}\mathrm{\&tau;}{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1+{\mathrm{\&alpha;}}_1={\stackrel{\&OverBar;}{x}}_2\\ {\mathrm{\&alpha;}}_1\left(0\right)={\stackrel{\&OverBar;}{x}}_2\left(0\right)\end{array}\right.$

where τ is the time constant of the filter, a constant greater than 0, α₁Is the output of a low pass filter, α₁(0)、Are respectively α₁Andinitial value of (a):

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=\frac{{\stackrel{\&OverBar;}{x}}_2-{\mathrm{\&alpha;}}_1}{\mathrm{\&tau;}}$

the resulting filtering error is:

$y_2={\mathrm{\&alpha;}}_1-{\stackrel{\&OverBar;}{x}}_2$

defining a virtual control error: z is a radical of₂＝x₂-α₁Then, then

The second Lyapunov function is defined as:

$V_2=\frac{1}{2}{z_2}^T{z_2}^2$

to ensure

The non-singular terminal sliding mode surface of the dynamic surface designed for the micro gyroscope is defined as follows:

$s={\&lsqb;s_1,s_2\&rsqb;}^T=z_1+\frac{1}{\mathrm{\&beta;}}{z_2}^{p_1/p_2}$

wherein β ═ diag (β)₁,β₂) Is the slip form surface constant, β₁,β₂Are all normal numbers, and are all positive numbers,p₁,p₂is positive odd number, and 1<p₁/p₂<2；

To make it possible toFor a micro-gyroscope system, a dynamic surface nonsingular terminal sliding mode surface is adopted, and the dynamic surface nonsingular terminal sliding mode control law is designed as follows:

u＝u₀+u₁+u₂+u₃

wherein,

$u_0=-f+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1$

$u_1=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right){\stackrel{\&CenterDot;}{z}}_1$

$u_2=-\frac{p_2}{p_1}\mathrm{\&beta;}diag\left({z_2}^{1-p_1/p_2}\right)\frac{s}{\left|\right|s|{|}^2}({z_1}^T{z}_2+\frac{1}{2})$

$u_3=-\mathrm{\&rho;}\frac{{\&lsqb;s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\&rsqb;}^T}{\left|\right|s^T\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)|{|}^2}\left|\right|s\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}diag\left({z_2}^{p_1/p_2-1}\right)\left|\right|$

at this time, the neural network is used for outputtingAnd (3) approaching the dynamic characteristic f of the gyroscope, wherein the updated overall control law is as follows:

u＝u₀'+u₁+u₂+u₃

wherein: