CN103472725A - Control method of neural network full adjustment based on nominal controller - Google Patents

Abstract

The invention discloses a control method of a neural network full adjustment based on a nominal controller. The control method mainly comprises the two steps that a trace tracking controller and a neural network full adjustment compensating controller are designed based on a nominal value model, and the control output ends of the two controllers are combined to be used as the control input end of a micro-gyroscope. According to the control method of the neural network full adjustment based on the nominal controller, the advantage of a model control method is used, meanwhile, the powerful approximation capability of a neural network is used, model errors and an outside disturbance effect are estimated and compensated on line in real time, the tracking performance and the robustness of a system can be greatly improved, the adaptive algorithms of a neural network weight and the center and the sound stage width of a gaussian function are designed based on a Lyapunov stability theory, and the global stability of a closed-loop system and the boundedness of control input can be guaranteed.

Description

The full control method of regulating of a kind of neural network based on nominal controller

Technical field

The present invention relates to the control method of gyroscope, particularly relate to the full control method of regulating of a kind of neural network based on nominal controller.

Background technology

Micro-mechanical gyroscope (MEMS Gyroscope) is the inertial sensor of the sense angular speed that is used for of utilizing microelectric technique and micro-processing technology to process.It detects angular velocity by the micromechanical component of a vibration of being made by silicon, so the very easy miniaturization of micro-mechanical gyroscope and batch production, has cost and hangs down and the characteristics such as volume is little.In recent years, micro-mechanical gyroscope is being paid close attention to nearly in a lot of application, for example, gyroscope coordinate micro-machine acceleration transducer for inertial navigation, at digital camera for stabilized image, for wireless inertial mouse of computer etc.But the impact due to inevitable mismachining tolerance and environment temperature in manufacturing process, can cause the difference between original paper characteristic and design, causes gyroscope to have parameter uncertainty, is difficult to set up accurate mathematical model.The external disturbance effect of adding in working environment be can not ignore, and make the trajectory track control of gyroscope be difficult to realize, and robustness is lower.Traditional control method is the nominal value parameter designing based on gyroscope fully, and ignore the effect of quadrature error and external disturbance, although system is still stable in most cases, but it is far undesirable to follow the trail of effect, this controller for the single environment design has very large use limitation.

The domestic research for gyroscope mainly concentrates on structural design and manufacturing technology aspect at present, and above-mentioned mechanical compensation technology and driving circuit research, oscillation trajectory with advanced control method compensation foozle and control mass seldom appears, to reach the control fully of gyroscope and the measurement of angular velocity.The typical mechanism of domestic research gyroscope is Southeast China University's instrumental science and engineering college and Southeast China University's micro inertial instrument and advanced navigation techniques key lab.

International article has various advanced control methods is applied in the middle of the control of gyroscope, and adaptive control and sliding-mode control are typically arranged.These advanced methods have compensated on the one hand and have made the quadrature error that error causes, have realized on the other hand the TRAJECTORY CONTROL to gyroscope.But the adaptive control robustness of disturbance to external world is very low, easily makes system become unstable.

As can be seen here, above-mentioned existing gyroscope in the use, obviously still has inconvenience and defect, and remains to be further improved.The problem existed in the use in order to solve existing gyroscope, relevant manufacturer there's no one who doesn't or isn't seeks solution painstakingly, but have no for a long time applicable design always, by development, is completed.

Summary of the invention

The object of the invention is to, overcome the defect that existing gyroscope control method exists, particularly improve the gyroscope system having that model is uncertain, under the various disturbed conditions such as Parameter Perturbation and external disturbance power, to the tracking performance of ideal trajectory and the robustness of whole system, and provide a kind of neural network based on nominal controller full control method of regulating.

The object of the invention to solve the technical problems realizes by the following technical solutions, and the full control method of regulating of a kind of neural network based on nominal controller proposed according to the present invention comprises the following steps:

(1) ideal power of setting up gyroscope is learned equation;

(2) set up the non-dimension kinetics equation of gyroscope;

(3) design nominal controller, described nominal controller is based on the contrail tracker of the nominal value modelling of gyroscope, be specially, and in the kinetics equation due to gyroscope, D, K, Ω is unknown parameter, and the nominal value of establishing them is D ₀, K ₀, Ω ₀, by the control of controller output u ₁be designed to,

$u_1={\stackrel{..}{q}}_d-k_v\stackrel{.}{e}-k_{p}e+D_0\stackrel{.}{q}+K_{0}q+{2\mathrm{\Ω}}_0\stackrel{.}{q};---\left(7\right)$

Wherein, q _dfor the ideal trajectory of gyroscope, the movement locus that q is gyroscope, e=q-q _dfor tracking error, k _v, k _pfor the constant matrices of positive definite symmetry, $k_p=\left[\begin{array}{cc}{\mathrm{\α}}^2& 0\\ 0& {\mathrm{\α}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\α}& 0\\ 0& 2\mathrm{\α}\end{array}\right],$ α is constant;

(4) the design neural network is regulated compensating controller entirely, is specially:

4-1) Definition Model error f is:

$f=\tilde{D}\stackrel{\·}{q}+\tilde{K}q+2\widetilde{\mathrm{\Ω}}\stackrel{\·}{q}+d---\left(10\right)$

4-2) the design neural network is regulated compensating controller entirely, by neural network, approaches unknown model error f (x), the output of neural network

be the estimated value of model error f (x), the present invention adopts the RBF neural network, the output of neural network

for,

$\hat{f}\left(x\right)={\widehat{\mathrm{\θ}}}^T\widehat{\mathrm{\φ}}\left(x\right)$

Neural network is regulated entirely to the output u of compensating controller ₂be designed to

$u_2=-\hat{f}\left(x\right)$

Wherein, $\tilde{D}=D_0-D,\tilde{K}=K_0-K,\widetilde{\mathrm{\Ω}}={\mathrm{\Ω}}_0-\mathrm{\Ω};$ X is the neural network input vector, $x=\left[\begin{array}{c}e\\ \stackrel{\·}{e}\end{array}\right],$ θ is the neural network weight vector, the estimated value of neural network weight vector, the mix vector that φ (x) is each neural network hidden node output, it is the estimated value of φ (x);

(5) nominal controller and neural network are regulated to compensating controller entirely and combine, as the control inputs of gyroscope, i.e. u=u ₁+ u ₂(19)

Wherein, $u_1={\stackrel{\·\·}{q}}_d-k_v\stackrel{\·}{e}-k_{p}e+D_0\stackrel{\·}{q}+K_{0}q+{2\mathrm{\Ω}}_0\stackrel{\·}{q},u_2=-{\widehat{\mathrm{\θ}}}^T\widehat{\mathrm{\φ}}\left(x\right);$

(6) design Lyapunov function, choose adaptive law,

Described Lyapunov function V is,

$V=\frac{1}{2}{x}^T\mathrm{Px}+\frac{1}{2}\mathrm{tr}\left(\widetilde{\mathrm{\θ}}\mathrm{\γ}{\widetilde{\mathrm{\θ}}}^T\right)+\frac{1}{2}{\widetilde{\mathrm{\φ}}}^T\mathrm{\Λ}\widetilde{\mathrm{\φ}}---\left(21\right)$

Described adaptive law is designed to,

$\stackrel{\·}{\widetilde{\mathrm{\θ}}}=\stackrel{\·}{\widehat{\mathrm{\θ}}}={\mathrm{\γ}}^{-1}\widehat{\mathrm{\φ}}{x}^T\mathrm{PB}---\left(24\right)$

$\stackrel{\·}{\widetilde{\mathrm{\φ}}}=\stackrel{\·}{\widehat{\mathrm{\φ}}}={\mathrm{\Λ}}^{-1}\widehat{\mathrm{\θ}}{B}^T\mathrm{Px}---\left(25\right)$

Wherein, γ, Λ, P, Q is symmetric positive definite matrix, γ, Λ>0, P and Q meet PA+A ^tp=-Q, $A=\left[\begin{array}{cc}0& I\\ -K_p& {-K}_v\end{array}\right];$ for the evaluated error of neural network weight vector,

for the ideal value of neural network weight vector,

for the evaluated error of neural network hidden node output vector,

ideal value for the output of neural network hidden node;

(7) adaptive law of described step (6) carried out to discretize, obtains the center of neural network weight, Gaussian function and the adaptive algorithm of sound stage width, guarantee the global stability of closed-loop system, mean with following:

The adaptive algorithm of neural network weight is, ${\widehat{\mathrm{\θ}}}_i(n+1)={\widehat{\mathrm{\θ}}}_i\left(n\right)-{\mathrm{\η}}_1\frac{\∂\hat{f}}{\∂{\mathrm{\θ}}_i^T}{x}^T\left(n\right)P\left(n\right)B\left(n\right)$

The adaptive algorithm at the center of Gaussian function is, ${\hat{c}}_i(n+1)={\hat{c}}_i\left(n\right)+{\mathrm{\η}}_3\frac{\∂\widehat{\mathrm{\φ}}\left(n\right)}{\∂{{\hat{c}}_i}^T}{\widehat{\mathrm{\θ}}B}^{T}P\left(n\right)x\left(n\right)$

The adaptive algorithm of the sound stage width of Gaussian function is, ${\widehat{\mathrm{\σ}}}_i(n+1)={\widehat{\mathrm{\σ}}}_i\left(n\right)+{\mathrm{\η}}_4\frac{\∂\widehat{\mathrm{\φ}}\left(n\right)}{\∂{{\widehat{\mathrm{\σ}}}_i}^T}{\widehat{\mathrm{\θ}}B}^{T}P\left(n\right)x\left(n\right),$

Wherein, the sampling pointer that n is discrete variable, the i row that i is i sound stage width and center vector,

for the output of RBF neural network, η ₁, η ₃, η ₄, be the scalar that can suitably regulate.

Aforesaid ideal power is learned equation and is described below:

x _d＝A ₁sin(ω ₁t),y _d＝A ₂sin(ω ₂t) （3）

ω wherein ₁, ω ₂respectively the vibration frequency of gyroscope on x axle and y direction of principal axis, ω ₁≠ ω ₂, and all non-vanishing, A ₁, A ₂be respectively the amplitude of gyroscope on x axle and y direction of principal axis, t is the time;

Ideal power is learned to equation to be write as vector form and is:

${\stackrel{\·\·}{q}}_d+k_d{q}_d=0---\left(4\right)$

Wherein, $q=\left[\begin{array}{c}{x}_d\\ y_d\end{array}\right]$ For the ideal movements track, $k_d=\left[\begin{array}{cc}{\mathrm{\ω}}_1^2& 0\\ 0& {\mathrm{\ω}}_2^2\end{array}\right].$

The aforesaid non-dimension kinetics equation of setting up gyroscope is:

2-1) consider foozle and external interference effect, the kinetics equation of diaxon gyroscope is:

$\begin{array}{c}m\stackrel{\·\·}{x}+d_{\mathrm{xx}}\stackrel{\·}{x}+d_{\mathrm{xy}}\stackrel{\·}{y}+k_{\mathrm{xx}}x+k_{\mathrm{xy}}y=u_x+d_x+2m{\mathrm{\Ω}}_z\stackrel{\·}{y}\\ m\stackrel{\·\·}{y}+d_{\mathrm{xy}}\stackrel{\·}{x}+d_{\mathrm{yy}}\stackrel{\·}{y}+k_{\mathrm{xy}}x+k_{\mathrm{yy}}y=u_y+d_y-2m{\mathrm{\Ω}}_z\stackrel{\·}{x}\end{array}---\left(1\right)$

Wherein, the quality that m is mass; X, y is respectively the position of mass along driving shaft and sensitive axis; d _xx, d _xy, d _yyfor the ratio of damping of gyroscope, k _xx, k _xy, k _yyelasticity coefficient for gyroscope; Ω _zthe angular velocity in the gyroscope working environment; u _x, u _yit is control inputs; d _x, d _yit is external interference;

2-2) model is carried out to non-dimension processing, obtains the vector form of the kinetics equation of non-dimension,

${\stackrel{\·\·}{q}}^{*}+D^{*}{\stackrel{\·}{q}}^{*}+K^{*}{q}^{*}=u^{*}-2{\mathrm{\Ω}}^{*}{\stackrel{\·}{q}}^{*}+d^{*}$

Wherein, $q^{*}=\frac{q}{q_0},q=\left[\begin{array}{c}x\\ y\end{array}\right],D^{*}=\frac{D}{{\mathrm{<figure-callout\; id="0"\; label="mw"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">mw</figure-callout>}}_0},D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],K=\left[\begin{array}{cc}{w}_x^2& w_{\mathrm{xy}}\\ w_{\mathrm{xy}}& w_y^2\end{array}\right],$

${w_x}^2=\frac{k_{\mathrm{xx}}}{\mathrm{<figure-callout\; id="0"\; label="m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">m</figure-callout>}{w_{}}^{}{{}_0}^2},{w_y}^2=\frac{k_{\mathrm{<figure-callout\; id="0"\; label="yy\; m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">yy</figure-callout>}}}{m{w_{}}^{}},w_{\mathrm{xy}}=\frac{k_{\mathrm{<figure-callout\; id="0"\; label="xy\; m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">xy</figure-callout>}}}{m{w_{}}^{}},u^{*}=\frac{\mathrm{<figure-callout\; id="0"\; label="u\; m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">u</figure-callout>}}{m{w_{}}^{}},u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],{\mathrm{\&Omega;}}^{*}=\frac{\mathrm{\&Omega;}}{w_0},$

$\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -{\mathrm{\&Omega;}}_z\\ {\mathrm{\&Omega;}}_{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">z</figure-callout>}}& 0\end{array}\right],d^{*}=\frac{\mathrm{<figure-callout\; id="0"\; label="d\; m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">d</figure-callout>}}{m{w_{}}^{}},d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right];$

2-3) for convenience of calculation, again with q, replace q ^*, with D, replace D ^*, with K, replace K ^*, with u, replace u ^*, with Ω, replace Ω ^*, with d, replace d ^*,

Above-mentioned non-dimension vector equation is rewritten as:

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

Wherein, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the movement locus of gyroscope, $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ Control inputs for gyroscope.Aforesaid RBF neural network is selected three-decker, input layer, hidden layer and output layer; But described input layer receives measuring-signal input x; Described hidden layer adopts the output after the gaussian basis function calculates Nonlinear Mapping; Described output layer obtains the output of whole RBF network by the output of each hidden node of weighting, as follows with mathematical description RBF neural network model:

$\begin{array}{c}{y}_i=\underset{j=1}{\overset{n_2}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{ij}}{\mathrm{\&phi;}}_j,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;n_2\\ {\mathrm{\&phi;}}_j\left(x\right)=\exp \left(\right||x-c_j||/{\mathrm{\&sigma;}}_j),j=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;n_3\end{array}---\left(12\right)$

Wherein, n ₂, n ₃mean respectively hidden node number and output layer node number; ω _ijmean the network weight of i concealed nodes to j output; y _imean the output of RBF neural network; φ _j(x) be hidden node output; c _j, σ _jthe center vector and the sound stage width that mean respectively each hidden node;

The RBF neural network model is write as vector form:

y＝θ ^Tφ(x) （13）

θ ^t=[ω _ij] vector that forms for network weight, φ (x)=[φ _j(x) vector] formed for each hidden node output.

Compared with prior art, advantage is in the present invention:

(1) dynamic perfromance of gyroscope is a kind of idealized model, has compensated foozle and environmental interference;

(2) adaptive algorithm of the center based on Lyapunov method design neural network weight, Gaussian function and sound stage width can guarantee the boundedness of Global asymptotic stability and the control inputs of whole closed-loop system;

(3) advantage of the control method based on the nominal value modelling and the powerful approximation capability of neural network are combined, estimate in real time online and compensating Modeling error and external disturbance effect, greatly improve the dynamic perfromance of gyroscope, improved the robustness that system is disturbed to external world;

(4) the present invention does not need to be based upon on the basis of object Accurate Model to the control of gyroscope, has saved the expense of modeling, more is suitable for practicality, and has the value on industry.。

The accompanying drawing explanation

Fig. 1 is micro-gyrotron model that the present invention simplifies;

The tracking curves that Fig. 2 is gyroscope x axle in specific embodiments of the invention;

The tracking curves that Fig. 3 is gyroscope y axle in specific embodiments of the invention;

Fig. 4 is gyroscope tracking error response curve in specific embodiments of the invention;

Fig. 5 is that in specific embodiments of the invention, x axle neural network is regulated the curve map of approximate model error entirely;

Fig. 6 is that in specific embodiments of the invention, y axle neural network is regulated the curve map of approximate model error entirely.

Embodiment

Reach for further setting forth the present invention technological means and the effect that predetermined goal of the invention is taked, below in conjunction with accompanying drawing and preferred embodiment, its embodiment of the full adjustment control method of a kind of neural network based on nominal controller, structure, feature and effect thereof that foundation the present invention is proposed, be described in detail as follows.

The full control method of regulating of neural network based on nominal controller of the present invention, comprise the steps:

(1) set up the non-dimension kinetic model of micro-gyro

Take into account foozle and external interference effect, the kinetics equation of micro-mechanical gyroscope is:

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

In formula, the quality that m is mass; X, y is respectively the position of mass along driving shaft and sensitive axis; d _xx, d _xy, d _yyfor the ratio of damping of gyroscope, k _xx, k _xy, k _yyfor the elasticity coefficient of gyroscope, when unknown and slow, become; Ω _zbeing the angular velocity in the gyroscope working environment, is also unknown quantity; u _x, u _yit is control inputs; d _x, d _yit is the external interference effect.

Model is carried out to non-dimension processing, define non-dimension time t ^*=w ₀t, equation (1) both sides are with square w divided by the diaxon natural frequency ₀ ², reference length q ₀with mass quality m, obtain the vector form of the kinetics equation of non-dimension

${\stackrel{\&CenterDot;\&CenterDot;}{q}}^{*}+D^{*}{\stackrel{\&CenterDot;}{q}}^{*}+K^{*}{q}^{*}=u^{*}-2{\mathrm{\&Omega;}}^{*}{\stackrel{\&CenterDot;}{q}}^{*}+d^{*}$

Wherein, $q^{*}=\frac{\mathrm{<figure-callout\; id="0"\; label="q\; q"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">q</figure-callout>}}{q_{}},q=\left[\begin{array}{c}x\\ y\end{array}\right],D^{*}=\frac{\mathrm{<figure-callout\; id="0"\; label="D\; mw"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">D</figure-callout>}}{{\mathrm{mw}}_{}},D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],K=\left[\begin{array}{cc}{w}_x^2& w_{\mathrm{xy}}\\ w_{\mathrm{xy}}& w_y^2\end{array}\right],$

${w_x}^2=\frac{k_{\mathrm{xx}}}{m{{\mathrm{<figure-callout\; id="0"\; label="w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">w</figure-callout>}}_0}^2},{w_y}^2=\frac{k_{\mathrm{yy}}}{m{{\mathrm{<figure-callout\; id="0"\; label="w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">w</figure-callout>}}_0}^2},w_{\mathrm{xy}}=\frac{k_{\mathrm{xy}}}{m{{\mathrm{<figure-callout\; id="0"\; label="w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">w</figure-callout>}}_0}^2},u^{*}=\frac{u}{m{{\mathrm{<figure-callout\; id="0"\; label="w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">w</figure-callout>}}_0}^2{\mathrm{<figure-callout\; id="0"\; label="q"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">q</figure-callout>}}_0},u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],{\mathrm{\&Omega;}}^{*}=\frac{\mathrm{\&Omega;}}{{\mathrm{<figure-callout\; id="0"\; label="w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">w</figure-callout>}}_0},$

$\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -{\mathrm{\&Omega;}}_z\\ {\mathrm{\&Omega;}}_{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">z</figure-callout>}}& 0\end{array}\right],d^{*}=\frac{\mathrm{<figure-callout\; id="0"\; label="d\; m\; w"\; filenames="HDA0000384165750000022.png"\; state="\{\{state\}\}">d</figure-callout>}}{m{w_{}}^{}},d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right];$

For convenience of calculation, again with q, replace q ^*, with D, replace D ^*, with K, replace K ^*, with u, replace u ^*, with Ω, replace Ω ^*, with d, replace d ^*,

Above-mentioned non-dimension vector equation is rewritten as:

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

Wherein, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the movement locus of gyroscope, $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ Control inputs for gyroscope.

(2) set up ideal power and learn model

The desirable dynamic perfromance of gyroscope is a kind of noenergy loss, and two between centers, without the stable sine-wave oscillation of Dynamic Coupling, can be described below:

x _d＝A ₁sin(ω ₁t),y _d＝A ₂sin(ω ₂t) （3）

ω wherein ₁, ω ₂respectively the vibration frequency of gyroscope on x axle and y direction of principal axis, ω ₁≠ ω ₂, and all non-vanishing, A ₁, A ₂be respectively the amplitude of gyroscope on x axle and y direction of principal axis, t is the time;

Reference model is write as vector form:

${\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+k_d{q}_d=0---\left(4\right)$

Wherein, $q=\left[\begin{array}{c}{x}_d\\ y_d\end{array}\right]$ For the ideal movements track, $k_d=\left[\begin{array}{cc}{\mathrm{\&omega;}}_1^2& 0\\ 0& {\mathrm{\&omega;}}_2^2\end{array}\right].$

(3) design nominal controller, be specially the contrail tracker of the nominal value model of design based on gyroscope,

If system is accurate, and without external interference, i.e. d=0, the output u of controller ₁can be designed as:

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000384165750000021.png,HDA0000384165750000022.png"\; state="\{\{state\}\}">u</figure-callout>}}_1={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-k_v\stackrel{\&CenterDot;}{e}-k_{p}e+D\stackrel{\&CenterDot;}{q}+Kq+2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}---\left(5\right)$

Output u by controller ₁as the control inputs of gyroscope, bring in equation (2), obtain the closed-loop system expression formula:

$\stackrel{\&CenterDot;\&CenterDot;}{e}+k_v\stackrel{\&CenterDot;}{e}+k_{p}e=0---\left(6\right)$

Wherein, q _dfor desirable gyroscope track, e=q-q _dfor tracking error, k _v, k _pfor the normal matrix of positive definite, symmetry, by the deviser, determined, generally get $k_p=\left[\begin{array}{cc}{\mathrm{\&alpha;}}^2& 0\\ 0& {\mathrm{\&alpha;}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\&alpha;}& 0\\ 0& 2\mathrm{\&alpha;}\end{array}\right],$ α is constant, can make system meet certain dynamic perfromance and stability.

But in fact, the D in the gyroscope kinetic model, K, Ω is unknown variable parameter, so establish their nominal value, is D ₀, K ₀, Ω ₀, the output u of controller so ₁become:

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000384165750000021.png,HDA0000384165750000022.png"\; state="\{\{state\}\}">u</figure-callout>}}_1={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-k_v\stackrel{\&CenterDot;}{e}-k_{p}e+D_0\stackrel{\&CenterDot;}{q}+K_{0}q+{2\mathrm{\&Omega;}}_0\stackrel{\&CenterDot;}{q}---\left(7\right)$

Output u by controller ₁as the control inputs of gyroscope, bring in equation (2), can obtain:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+D\stackrel{\&CenterDot;}{q}+\mathrm{Kq}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-k_v\stackrel{\&CenterDot;}{e}-k_{p}e+D_0\stackrel{\&CenterDot;}{q}+K_{0}q+{2\mathrm{\&Omega;}}_0\stackrel{\&CenterDot;}{q}-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}+d---\left(8\right)$

If $\tilde{D}=D_0-D,\tilde{K}=K_0-K,\widetilde{\mathrm{\&Omega;}}={\mathrm{\&Omega;}}_0-\mathrm{\&Omega;},$ Obtain closed-loop system:

$\stackrel{\&CenterDot;\&CenterDot;}{e}+k_v\stackrel{\&CenterDot;}{e}+k_{p}e=\tilde{D}\stackrel{\&CenterDot;}{q}+\tilde{K}q+2\widetilde{\mathrm{\&Omega;}}\stackrel{\&CenterDot;}{q}+d---\left(9\right)$

By formula (9), can be seen, existence due to uncertainty and the external interference of system, although still can guarantee systematically stability, but the dynamic property of gyroscope and robustness all can reduce greatly, therefore be necessary to design a compensating controller, improve systematically dynamic perfromance and tracking error.

(4) the design neural network is regulated compensating controller entirely, be specially,

F is as follows for the Definition Model error:

$f=\tilde{D}\stackrel{\&CenterDot;}{q}+\tilde{K}q+2\widetilde{\mathrm{\&Omega;}}\stackrel{\&CenterDot;}{q}+d---\left(10\right)$

(9) formula is rewritten into to state space form:

$\stackrel{\&CenterDot;}{x}=\mathrm{Ax}+\mathrm{Bf}---\left(11\right)$

Wherein, $x=\left[\begin{array}{c}e\\ \stackrel{\&CenterDot;}{e}\end{array}\right],A=\left[\begin{array}{cc}0& I\\ -k_p& -k_v\end{array}\right],B=\left[\begin{array}{c}0\\ I\end{array}\right],x\&Element;R^{4\&times;1},A\&Element;R^{4\&times;4},B\&Element;R^{4\&times;2},$ Hence one can see that, A, and B is constant vector, and x is only the function of f, and f () can be expressed as f (x) so, because neural network has powerful tracking power, therefore can utilize the approximate characteristic of neural network to identify unknown f (x).

The present invention selects the RBF neural network, selects three-decker: input layer, hidden layer and output layer.But input layer is accepted the measuring-signal input x in system; Hidden layer adopts the output after the gaussian basis function calculates Nonlinear Mapping; Output layer obtains the output of whole RBF neural network by the output of each hidden node of weighting, as follows with mathematical description RBF neural network model:

$\begin{array}{c}{y}_i=\underset{j=1}{\overset{n_2}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{ij}}{\mathrm{\&phi;}}_j,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;n_2\\ {\mathrm{\&phi;}}_j\left(x\right)=\exp \left(\right||x-c_j||/{\mathrm{\&sigma;}}_j),j=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;n_3\end{array}---\left(12\right)$

In formula, n ₂, n ₃mean respectively hidden node number and output layer node number, and the dimension of input signal x is designated as n ₁; ω _ijmean the network weight of i concealed nodes to j output; y _imean the output of RBF neural network; φ _j(x) be hidden node output; c _j, σ _jthe center vector and the sound stage width that mean respectively each hidden node.Existing document is verified, and the RBF network can approach with arbitrary accuracy the nonlinear function of arbitrary smooth, the online real-time update of center vector, sound stage width and weights of the RBF network in the present invention.

The vector form of RBF neural network model is:

y＝θ ^Tφ(x) （13）

Wherein, θ ^t=[ω _ij] vector that forms for network weight, φ (x)=[φ _j(x) vector] formed for each hidden node output, x is the neural network input vector.

Approach unknown model error f (x), the output of neural network by neural network

be the estimated value of model error f (x), neural network regulated entirely to the output u of compensating controller ₂be designed to

$u_2=-\hat{f}\left(x\right)$

According to RBF neural network model formula (13), can obtain the output of neural network

$\hat{f}\left(x\right)={\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)$

$u_2=-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)---\left(14\right)$

Wherein

for the estimated value of neural network weight vector,

estimated value for neural network concealed nodes output vector.

Can suppose a given constant matrices ε ₀, have a desirable weight vector θ ^*with a desirable the number of hidden nodes n ^*, make the ideal output of RBF neural network

meet:

$\max \left|\right|\hat{f}(x,{\mathrm{\&theta;}}^{*})-f\left(x\right)\left|\right|\&le;{\mathrm{\&epsiv;}}_0$

Definition η is the modeling error after the use neural network:

$\mathrm{\&eta;}=f\left(x\right)-\hat{f}(x,{\mathrm{\&theta;}}^{*})---\left(15\right)$

Wherein, η has upper bound η ₀:

${\mathrm{\&eta;}}_0=\sup \left|\right|f\left(x\right)-\hat{f}(x,{\mathrm{\&theta;}}^{*})\left|\right|---\left(16\right)$

Formula (11) can be write as:

$\begin{array}{c}\stackrel{\&CenterDot;}{x}=\mathrm{Ax}+B\{\hat{f}(x,{\mathrm{\&theta;}}^{*})-+[f\left(x\right)-\hat{f}(x,{\mathrm{\&theta;}}^{*})\left]\right\}\\ =\mathrm{Ax}+B[\hat{f}(x,{\mathrm{\&theta;}}^{*})+\mathrm{\&eta;}\end{array}---\left(17\right)$

In formula, θ ^*desirable neural network weight,

Can obtain desirable output according to RBF neural network model formula (13)

for:

$\hat{f}(x,{\mathrm{\&theta;}}^{*})={\mathrm{\&theta;}}^{*T}{\mathrm{\&phi;}}^{*}\left(x\right)---\left(18\right)$

θ wherein, φ (x) is known variables, needs to regulate θ, c _i, σ _iobtain tracking effect preferably.

(5) nominal controller and neural network are regulated to compensating controller entirely and combine, as the control inputs of gyroscope, so new control inputs can be expressed as:

u＝u ₁+u ₂ （19）

Wherein,

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000384165750000021.png,HDA0000384165750000022.png"\; state="\{\{state\}\}">u</figure-callout>}}_1={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-k_v\stackrel{\&CenterDot;}{e}-k_{p}e+D_0\stackrel{\&CenterDot;}{q}+K_{0}q+{2\mathrm{\&Omega;}}_0\stackrel{\&CenterDot;}{q}---\left(7\right)$

$u_2=-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)---\left(14\right)$

After bringing the gyroscope kinetics equation into, closed-loop system can be expressed as:

$\stackrel{\&CenterDot;\&CenterDot;}{e}+k_v\stackrel{\&CenterDot;}{e}+k_{p}e=\tilde{D}\stackrel{.}{q}+\tilde{K}q+2\widetilde{\mathrm{\&Omega;}}\stackrel{\&CenterDot;}{q}+d-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)$

State space form is: $\stackrel{\&CenterDot;}{x}=\mathrm{Ax}+\mathrm{Bf}-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)$

$\stackrel{\&CenterDot;}{x}\mathrm{Ax}+B[{\mathrm{\&theta;}}^{*T}{\mathrm{\&phi;}}^{*}\left(x\right)+\mathrm{\&eta;}-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)]---\left(20\right)$

In formula,

$A=\left[\begin{array}{cc}0& I\\ -k_p& -k_v\end{array}\right],B=\left[\begin{array}{c}0\\ I\end{array}\right],x\&Element;R^{4\&times;1},A\&Element;R^{4\&times;4},B\&Element;R^{4\&times;2},k_p=\left[\begin{array}{cc}{\mathrm{\&alpha;}}^2& 0\\ 0& {\mathrm{\&alpha;}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\&alpha;}& 0\\ 0& 2\mathrm{\&alpha;}\end{array}\right],$

Make α>0, A is stable matrix.

(6) design Lyapunov function, choose adaptive law,

The evaluated error of definition weight vector

for, the evaluated error of neural network hidden node output vector

for,

definition Lyapunov function V is:

$V=\frac{1}{2}{x}^T\mathrm{Px}+\frac{1}{2}\mathrm{tr}\left(\widetilde{\mathrm{\&theta;}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+\frac{1}{2}{\widetilde{\mathrm{\&phi;}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}---\left(21\right)$

In formula, γ, Λ, P, Q is symmetric positive definite matrix, γ, Λ>0, P and Q meet:

PA+A ^TP＝-Q （22）

To Lyapunov function V, differentiate obtains:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=\frac{1}{2}{x}^{T}P\stackrel{\&CenterDot;}{x}+\frac{1}{2}{\stackrel{\&CenterDot;}{x}}^T\mathrm{Px}+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}+x^T\mathrm{PB}{\mathrm{\&theta;}}^{*T}{\mathrm{\&phi;}}^{*T}\left(x\right)-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T{\widehat{\mathrm{\&phi;}}}^T\left(x\right)+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}+x^T\mathrm{PB}({\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}-{\widehat{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}-{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}+{\widetilde{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}})-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T{\widehat{\mathrm{\&phi;}}}^T\left(x\right)+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}+x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}-x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}+x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T{\widehat{\mathrm{\&phi;}}}^T\left(x\right)+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}-x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}+x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}\mathrm{\&gamma;}}{\widetilde{\mathrm{\&theta;}}}^T\right)+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T\widetilde{\mathrm{\&phi;}}-x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}+\mathrm{tr}\left(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}\\ =-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}+({\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T)\widetilde{\mathrm{\&phi;}}+[\mathrm{tr}(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}\mathrm{\&gamma;}}{\widetilde{\mathrm{\&theta;}}}^T-x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}})\end{array}+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}$

Push in process, due to

with

all very little, multiply each other less, therefore

item is 0.

Known $x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}=\mathrm{tr}\left(x^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\right)=\mathrm{tr}\left(\widehat{\mathrm{\&phi;}}{x}^T\mathrm{PB}{\widetilde{\mathrm{\&theta;}}}^T\right),$ Can obtain:

$\stackrel{\&CenterDot;}{V}=-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}+({\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T)\widetilde{\mathrm{\&phi;}}+\mathrm{tr}\left[(\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}-\widehat{\mathrm{\&phi;}}{x}^T\mathrm{PB}){\widetilde{\mathrm{\&theta;}}}^T\right]---\left(23\right)$

We set: ${\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}}^T\mathrm{\&Lambda;}-x^T\mathrm{PB}{\widehat{\mathrm{\&theta;}}}^T=0,\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}\mathrm{\&gamma;}-\widehat{\mathrm{\&phi;}}{x}^T\mathrm{PB}=0,$

Again due to $\widetilde{\mathrm{\&theta;}}=\widehat{\mathrm{\&theta;}}-{\mathrm{\&theta;}}^{*},\widetilde{\mathrm{\&phi;}}=\widehat{\mathrm{\&phi;}}-{\mathrm{\&phi;}}^{*},$ θ * and φ * are constant, therefore $\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}},\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&phi;}}}$

Choosing adaptive law has:

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{\&gamma;}}^{-1}\widehat{\mathrm{\&phi;}}{x}^T\mathrm{PB}---\left(24\right)$

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&phi;}}}={\mathrm{\&Lambda;}}^{-1}\widehat{\mathrm{\&theta;}}{B}^T\mathrm{Px}---\left(25\right)$

After choosing adaptive law:

$\stackrel{\&CenterDot;}{V}=-\frac{1}{2}{x}^T\mathrm{Qx}+{\mathrm{\&eta;}}^T{B}^T\mathrm{Px}---\left(26\right)$

By || η ^t||≤η ₀, || B ^tp||≤|| P|| has:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\&le;-\frac{1}{2}{\mathrm{\&lambda;}}_{\min }\left(Q\right){\left|\right|x\left|\right|}^2+{\mathrm{\&eta;}}_0{\mathrm{\&lambda;}}_{\max }\left(P\right)\left|\right|x\left|\right|\\ =-\frac{1}{2}\left|\right|x\left|\right|\left[{\mathrm{\&lambda;}}_{\min }\left(Q\right)\right|\left|x\right||-2{\mathrm{\&eta;}}_0{\mathrm{\&lambda;}}_{\max }\left(P\right)]\end{array}---\left(27\right)$

In order to make

to make so: ${\mathrm{\&lambda;}}_{\min }\left(x\right)\&GreaterEqual;\frac{{2\mathrm{\&lambda;}}_{\max }\left(P\right)}{\left|\right|x\left|\right|}{\mathrm{\&eta;}}_0,$

Wherein, λ _minand λ (P) _max(P) be respectively minimal eigenvalue and the eigenvalue of maximum of P,

According to mathematical discretization method, adaptive law is carried out to discretize, obtain the neural network weight adaptive algorithm of discrete form:

${\widehat{\mathrm{\&theta;}}}_i(n+1)={\widehat{\mathrm{\&theta;}}}_i\left(n\right)-{\mathrm{\&eta;}}_1\frac{\&PartialD;\hat{f}}{\&PartialD;{\mathrm{\&theta;}}_i^T}{x}^T\left(n\right)P\left(n\right)B\left(n\right)---\left(28\right)$

Obtain center and the sound stage width adaptive algorithm of the Gaussian function of discrete form:

${\hat{c}}_i(n+1)={\hat{c}}_i\left(n\right)+{\mathrm{\&eta;}}_3\frac{\&PartialD;\widehat{\mathrm{\&phi;}}\left(n\right)}{\&PartialD;{{\hat{c}}_i}^T}{\widehat{\mathrm{\&theta;}}B}^{T}P\left(n\right)x\left(n\right)---\left(29\right)$

${\widehat{\mathrm{\&sigma;}}}_i(n+1)={\widehat{\mathrm{\&sigma;}}}_i\left(n\right)+{\mathrm{\&eta;}}_4\frac{\&PartialD;\widehat{\mathrm{\&phi;}}\left(n\right)}{\&PartialD;{{\widehat{\mathrm{\&sigma;}}}_i}^T}{\widehat{\mathrm{\&theta;}}B}^{T}P\left(n\right)x\left(n\right)---\left(30\right)$

In formula, the sampling pointer that n is discrete variable, the i row that i is i sound stage width and center vector,

for the output of RBF neural network, η ₁, η ₃, η ₄, be the scalar that can suitably regulate.

Finally, carry out simulation analysis of computer

In order to show more intuitively the validity of the full control method of regulating of the neural network based on nominal controller that the present invention proposes, now utilize mathematical software MATLAB/SIMULINK to carry out computer simulation experiment to this control program.The parameter of choosing the gyroscope of non-dimension is:

$w_x^2=355.3,w_y^2=532.9,w_{\mathrm{xy}}=70.99,$

d _xx＝0.01,d _yy＝0.01,d _xy＝0.002,Ω _z＝0.1

The nominal value of supposing the gyroscope parameter is: D ₀=0.9*D, K ₀=0.9*K, Ω ₀=0;

Desirable x axle and y track shaft are: x _d=cos (6.17t), y _d=cos (5.11t);

The original state of gyroscope is: x (0)=[0; 0; 0; 0];

Get 11 concealed nodes, the initial value of each element of neural network weight vector is 0, and each initial value of sound stage width is designed to 2, and it is [1111] that the center initial value be take 0.2 step change from [1-1-1-1];

Adaptation law coefficient Q gets larger to guarantee to follow the trail of effect as far as possible, Q=10*I, and wherein I is the quadravalence matrix;

K _v, k _pin get α=2;

η in modus ponens (28), (29), (30) ₁=η ₃=η ₄=0.05;

External interference is got:

d＝[10*((sin(6.17*t)) ²+2*cos(6.17*t));10*((sin(5.11*t)) ²+2*cos(5.11*t))]；

Under the situation of above each controller parameter, the operation simulated program, obtain the simulation result curve of invention specific embodiment as shown in Figures 2 to 6.

Fig. 2 and Fig. 3 are respectively gyroscope diaxon tracking curves, and wherein, dotted line is ideal trajectory, the actual motion track that solid line is gyroscope, as seen from the figure, and the X of gyroscope, the Y-axis output trajectory can be followed the tracks of the track of coideal very soon;

Fig. 4 is diaxon tracking error response curve, can see that tracking error converges to rapidly zero;

In Fig. 5 and Fig. 6, solid line is realistic model error f, the output that dotted line is the RBF neural network

can prove that model error f can be regulated entirely by the RBF neural network

effectively approach, tracking error reduces greatly, and the robustness of effectively having improved the dynamic perfromance of gyroscope system and having disturbed to external world.Therefore, simulation results show the full validity of regulating of RBF neural network.

From above analogous diagram, can find out, the control method that the present invention proposes has good control effect to the track following of gyroscope, greatly improved tracking performance and the robustness of gyroscope system, the high precision of gyroscope diaxon oscillation trajectory is controlled theoretical foundation and From Math are provided.

The content be not described in detail in instructions of the present invention belongs to the known technical know-how of professional and technical personnel in the field.

The above, it is only preferred embodiment of the present invention, not the present invention is done to any large restriction in form, although the present invention discloses as above with preferred embodiment, yet not in order to limit the present invention, any those skilled in the art, within not breaking away from the technical solution of the present invention scope, when the technology contents that can utilize above-mentioned announcement is made a little change or is modified to the equivalent embodiment of equivalent variations, in every case be the content that does not break away from technical solution of the present invention, any simple modification of above embodiment being done according to technical spirit of the present invention, equivalent variations and modification, all still belong in the scope of our bright technical scheme.

Claims (4)

1. the full control method of regulating of the neural network based on nominal controller, is characterized in that, comprises the following steps:

(1) ideal power of setting up gyroscope is learned equation;

(2) set up the non-dimension kinetics equation of gyroscope;

(3) design nominal controller, described nominal controller is based on the contrail tracker of the nominal value modelling of gyroscope, be specially, and in the kinetics equation due to gyroscope, D, K, Ω is unknown parameter, and the nominal value of establishing them is D ₀, K ₀, Ω ₀, by the control of controller output u ₁be designed to,

$u_1={\stackrel{..}{q}}_d-k_v\stackrel{.}{e}-k_{p}e+D_0\stackrel{.}{q}+K_{0}q+{2\mathrm{\&Omega;}}_0\stackrel{.}{q};---\left(7\right)$

Wherein, q _dfor the ideal trajectory of gyroscope, the movement locus that q is gyroscope, e=q-q _dfor tracking error, k _v, k _pfor the constant matrices of positive definite symmetry, $k_p=\left[\begin{array}{cc}{\mathrm{\&alpha;}}^2& 0\\ 0& {\mathrm{\&alpha;}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\&alpha;}& 0\\ 0& 2\mathrm{\&alpha;}\end{array}\right],$ α is constant;

(4) the design neural network is regulated compensating controller entirely, is specially:

4-1) the Definition Model error is

$f=\tilde{D}\stackrel{\&CenterDot;}{q}+\tilde{K}q+2\widetilde{\mathrm{\&Omega;}}\stackrel{\&CenterDot;}{q}+d---\left(10\right)$

4-2) the design neural network is regulated compensating controller entirely, by neural network, approaches unknown model error f (x), the output of neural network

be the estimated value of model error f (x), the present invention adopts the RBF neural network, the output of neural network for,

$\hat{f}\left(x\right)={\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right)$

Neural network is regulated entirely to the output u of compensating controller ₂be designed to

$u_2=-\hat{f}\left(x\right)$

Wherein, $\tilde{D}=D_0-D,\tilde{K}=K_0-K,\widetilde{\mathrm{\&Omega;}}={\mathrm{\&Omega;}}_0-\mathrm{\&Omega;};$ X is the neural network input vector, $x=\left[\begin{array}{c}e\\ \stackrel{.}{e}\end{array}\right],$ θ is the neural network weight vector, the estimated value of neural network weight vector, the mix vector that φ (x) is each neural network hidden node output,

it is the estimated value of φ (x);

(5) nominal controller and neural network are regulated to compensating controller entirely and combine, as the control inputs of gyroscope, i.e. u=u ₁+ u ₂(19)

Wherein, $u_1={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-k_v\stackrel{\&CenterDot;}{e}-k_{p}e+D_0\stackrel{\&CenterDot;}{q}+K_{0}q+{2\mathrm{\&Omega;}}_0\stackrel{\&CenterDot;}{q},u_2=-{\widehat{\mathrm{\&theta;}}}^T\widehat{\mathrm{\&phi;}}\left(x\right);$

(6) design Lyapunov function, choose adaptive law,

Described Lyapunov function V is,

$V=\frac{1}{2}{x}^T\mathrm{Px}+\frac{1}{2}\mathrm{tr}\left(\widetilde{\mathrm{\&theta;}}\mathrm{\&gamma;}{\widetilde{\mathrm{\&theta;}}}^T\right)+\frac{1}{2}{\widetilde{\mathrm{\&phi;}}}^T\mathrm{\&Lambda;}\widetilde{\mathrm{\&phi;}}---\left(21\right)$

Described adaptive law is,

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{\&gamma;}}^{-1}\widehat{\mathrm{\&phi;}}{x}^T\mathrm{PB}---\left(24\right)$

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&phi;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&phi;}}}={\mathrm{\&Lambda;}}^{-1}\widehat{\mathrm{\&theta;}}{B}^T\mathrm{Px}---\left(25\right)$

Wherein, γ, Λ, P, Q is symmetric positive definite matrix, γ, Λ>0, P and Q meet PA+A ^tp=-Q, $A=\left[\begin{array}{cc}0& I\\ -K_p& {-K}_v\end{array}\right];$

for the evaluated error of neural network weight vector,

θ ^*for the ideal value of neural network weight vector,

for the evaluated error of neural network hidden node output vector,

ideal value for the output of neural network hidden node;

(7) adaptive law of described step (6) carried out to discretize, obtains the center of neural network weight, Gaussian function and the adaptive algorithm of sound stage width, guarantee the global stability of closed-loop system, mean with following:

The adaptive algorithm of neural network weight is, ${\widehat{\mathrm{\&theta;}}}_i(n+1)={\widehat{\mathrm{\&theta;}}}_i\left(n\right)-{\mathrm{\&eta;}}_1\frac{\&PartialD;\hat{f}}{\&PartialD;{\mathrm{\&theta;}}_i^T}{x}^T\left(n\right)P\left(n\right)B\left(n\right)$

The adaptive algorithm at the center of Gaussian function is, ${\hat{c}}_i(n+1)={\hat{c}}_i\left(n\right)+{\mathrm{\&eta;}}_3\frac{\&PartialD;\widehat{\mathrm{\&phi;}}\left(n\right)}{\&PartialD;{{\hat{c}}_i}^T}{\widehat{\mathrm{\&theta;}}B}^{T}P\left(n\right)x\left(n\right)$

The adaptive algorithm of the sound stage width of Gaussian function is, ${\widehat{\mathrm{\&sigma;}}}_i(n+1)={\widehat{\mathrm{\&sigma;}}}_i\left(n\right)+{\mathrm{\&eta;}}_4\frac{\&PartialD;\widehat{\mathrm{\&phi;}}\left(n\right)}{\&PartialD;{{\widehat{\mathrm{\&sigma;}}}_i}^T}{\widehat{\mathrm{\&theta;}}B}^{T}P\left(n\right)x\left(n\right),$

Wherein, the sampling pointer that n is discrete variable, the i row that i is i sound stage width and center vector,

for the output of RBF neural network, η ₁, η ₃, η ₄, be the scalar that can suitably regulate.

2. according to the full control method of regulating of a kind of neural network based on nominal controller claimed in claim 1, it is characterized in that, described ideal power is learned equation and is described below:

x _d＝A ₁sin(ω ₁t),y _d＝A ₂sin(ω ₂t) （3）

ω wherein ₁, ω ₂respectively the vibration frequency of gyroscope on x axle and y direction of principal axis, ω ₁≠ ω ₂, and all non-vanishing, A ₁, A ₂be respectively the amplitude of gyroscope on x axle and y direction of principal axis, t is the time;

Ideal power is learned to equation to be write as vector form and is:

${\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+k_d{q}_d=0---\left(4\right)$

Wherein, $q=\left[\begin{array}{c}{x}_d\\ y_d\end{array}\right]$ For the ideal movements track, $k_d=\left[\begin{array}{cc}{\mathrm{\&omega;}}_1^2& 0\\ 0& {\mathrm{\&omega;}}_2^2\end{array}\right].$

3. according to the full control method of regulating of a kind of neural network based on nominal controller claimed in claim 1, it is characterized in that, the described non-dimension kinetics equation of setting up gyroscope is:

2-1) consider foozle and external interference effect, the kinetics equation of diaxon gyroscope is:

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

Wherein, the quality that m is mass; X, y is respectively the position of mass along driving shaft and sensitive axis; d _xx, d _xy, d _yyfor the ratio of damping of gyroscope, k _xx, k _xy, k _yyelasticity coefficient for gyroscope; Ω _zthe angular velocity in the gyroscope working environment; u _x, u _yit is control inputs; d _x, d _yit is external interference;

2-2) model is carried out to non-dimension processing, obtains the vector form of the kinetics equation of non-dimension,

${\stackrel{\&CenterDot;\&CenterDot;}{q}}^{*}+D^{*}{\stackrel{\&CenterDot;}{q}}^{*}+K^{*}{q}^{*}=u^{*}-2{\mathrm{\&Omega;}}^{*}{\stackrel{\&CenterDot;}{q}}^{*}+d^{*}$

Wherein, $q^{*}=\frac{q}{q_0},q=\left[\begin{array}{c}x\\ y\end{array}\right],D^{*}=\frac{D}{{\mathrm{mw}}_0},D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],K=\left[\begin{array}{cc}{w}_x^2& w_{\mathrm{xy}}\\ w_{\mathrm{xy}}& w_y^2\end{array}\right],$

${w_x}^2=\frac{k_{\mathrm{xx}}}{m{w_0}^2},{w_y}^2=\frac{k_{\mathrm{yy}}}{m{w_0}^2},w_{\mathrm{xy}}=\frac{k_{\mathrm{xy}}}{m{w_0}^2},u^{*}=\frac{u}{m{w_0}^2{q}_0},u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],{\mathrm{\&Omega;}}^{*}=\frac{\mathrm{\&Omega;}}{w_0},$

$\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -{\mathrm{\&Omega;}}_z\\ {\mathrm{\&Omega;}}_z& 0\end{array}\right],d^{*}=\frac{d}{m{w_0}^2{q}_0},d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right]$

2-3) for convenience of calculation, again with q, replace q ^*, with D, replace D ^*, with K, replace K ^*, with u, replace u ^*, with Ω, replace Ω ^*, with d, replace d ^*,

Above-mentioned non-dimension vector equation is rewritten as:

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

Wherein, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the movement locus of gyroscope, $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ Control inputs for gyroscope.

4. according to the full control method of regulating of a kind of neural network based on nominal controller claimed in claim 1, it is characterized in that, described RBF neural network is selected three-decker, input layer, hidden layer and output layer; But described input layer receives measuring-signal input x; Described hidden layer adopts the output after the gaussian basis function calculates Nonlinear Mapping; Described output layer obtains the output of whole RBF network by the output of each hidden node of weighting, as follows with mathematical description RBF neural network model:

$\begin{array}{c}{y}_i=\underset{j=1}{\overset{n_2}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{ij}}{\mathrm{\&phi;}}_j,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;n_2\\ {\mathrm{\&phi;}}_j\left(x\right)=\exp \left(\right||x-c_j||/{\mathrm{\&sigma;}}_j),j=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;n_3\end{array}---\left(12\right)$

Wherein, n ₂, n ₃mean respectively hidden node number and output layer node number; ω _ijmean the network weight of i concealed nodes to j output; y _imean the output of RBF neural network; φ _j(x) be hidden node output; c _j, σ _jthe center vector and the sound stage width that mean respectively each hidden node;

The RBF neural network model is write as vector form:

y＝θ ^Tφ(x) （13）

θ ^t=[ω _ij] vector that forms for network weight, φ (x)=[φ _j(x) vector] formed for each hidden node output.

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