CN105278331A - Robust-adaptive neural network H-infinity control method of MEMS gyroscope - Google Patents

Abstract

The present invention discloses a robust-adaptive neural network H-infinity control method of a MEMS gyroscope. A controller is designed based on a Riccati equation, and includes two parts of a basic neural network controller constructed by utilizing a strong online approximating capability of the neural network and a robust control item used for overcoming influences of external disturbance and parameter uncertainty on MEMS gyroscope system output tracking errors and ensuring system closed-loop stabilization. Parameters in a adaptive adjustment neural network system based on a Lyapunov stability theory are adopted, thus to ensure stability of the system. The controller is based on the Riccati equation, such that non-linear phenomena in the system are compensated, the precise tracking aim is achieved, stability of the system and robustness to external disturbance are raised, and industrial utility values are achieved.

Description

A kind of infinite control method of robust adaptive neural network H of microthrust test

Technical field

The present invention relates to a kind of control method of gyroscope system, particularly relate to the infinite control method of robust adaptive neural network H of a kind of MEMS micro-top instrument.

Background technology

Micro-mechanical gyroscope (MEMSGyroscope) is the inertial sensor of the sense angular speed that is used for utilizing microelectric technique and micro-processing technology to process.It carrys out detection angle speed by the micromechanical component of a vibration of being made up of silicon, and therefore micro-mechanical gyroscope is very easy to miniaturization and batch production, has the features such as the low and volume of cost is little.In recent years, micro-mechanical gyroscope is paid close attention in many applications nearly, such as, gyroscope coordinates micro-machine acceleration transducer to be used for inertial navigation, in digital camera for stabilized image, wireless inertial mouse etc. for computer.But, due to the impact of mismachining tolerance inevitable in manufacturing process and environment temperature, the difference between original paper characteristic and design can be caused, cause gyroscope to there is parameter uncertainty, be 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 of gyroscope control to be difficult to realize, and robustness is lower.Traditional control method is completely based on the nominal value parameter designing of gyroscope, 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 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, little appearance advanced control method compensates the oscillation trajectory of foozle and Mass Control block, to reach the measurement of control and angular velocity completely to gyroscope.The typical mechanism of studies in China 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 and is applied in the middle of the control of gyroscope by various advanced control method, typically has adaptive control and sliding-mode control.These advanced methods compensate for the quadrature error that fabrication error causes on the one hand, achieve the TRAJECTORY CONTROL to gyroscope on the other hand.But the robustness of adaptive control disturbance is to external world very low, system is easily made to become unstable.

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

Summary of the invention

In order to solve the defect that in prior art, gyroscope control method exists, particularly overcome the impact on micro-gyroscope control system of parameter uncertainty and external interference, the infinite control method of robust adaptive neural network H of microthrust test of the present invention does not need the mathematical models knowing system, can the uncertain and external interference of compensating parameter, greatly improve the dynamic property of system, reduce the uncertain impact on tracking performance, raising system is to the robustness of Parameters variation and external interference, thus make ideal trajectory given on gyroscope two track shaft energy high precision tracking.In order to solve the problem, the technical solution used in the present invention is:

The infinite control method of robust adaptive neural network H of microthrust test, is characterized in that: adopt the controller based on Riccati equation design to comprise two parts: Part I is basic nerve network controller; Part II is robust control item, specifically comprises the following steps:

(1), desirable kinetic model is set up,

Design reference model is the sine wave of two different frequencies: x _m=A ₁sin (w ₁t), y _m=A ₂sin (w ₂t), wherein w ₁≠ w ₂and all non-vanishing, being rewritten into vector form is: wherein, A ₁, A ₂be the amplitude of gyroscope on x, y two change in coordinate axis direction respectively, t is the time, w ₁and w ₂be respectively the vibration frequency that gyroscope is given on x, y two change in coordinate axis direction; Vector form is: ${\stackrel{\·\·}{q}}_m+k_m{q}_m=0;$ Wherein $q=\left[\begin{array}{c}{x}_m\\ y_m\end{array}\right],k_m=\left[\begin{array}{cc}{w}_1^2& 0\\ 0& w_2^2\end{array}\right];$

(2), gyroscope system dynamics model is set up,

Set up the non-dimension vector model of gyroscope system, (1)

Wherein $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the position vector of gyroscope mass on x, y-axis direction; $u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$ For the control inputs of gyroscope on x, y-axis direction; $\mathrm{\Ω}=\left[\begin{array}{cc}0& -{\mathrm{\Ω}}_z\\ {\mathrm{\Ω}}_z& 0\end{array}\right]$ For angular velocity matries; $D=\left[\begin{array}{cc}{d}_{\mathrm{yy}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right]$ For damping matrix, $K=\left[\begin{array}{cc}{w}_x^2& w_{\mathrm{xy}}\\ w_{\mathrm{xy}}& w_y^2\end{array}\right]$ It is the coefficient entry containing gyroscope fixed frequency, stiffness coefficient and coupling stiffness coefficient; $d=\left[\begin{array}{c}{d}_1\\ d_2\end{array}\right]$ For indeterminate and the external interference of system, if do not consider external interference, and parameter D, Ω, K _bknown, then controller can be designed to: wherein, k=(k _v, k _p) ^tfor controller parameter matrix, for the tracking error matrix of system;

(3), the design of the infinite controller of robust adaptive neural network H

Due to the parameter D of gyroscope system, Ω, K _bthe unknown, then the controller based on Riccati equation design comprises two parts: Part I is nerve network controller in formula, the input of neural network, for system can measuring-signal; represent the weight vector of neural network; M represents the number of RBF network hidden layer node; ξ (X) is hidden layer radial basis function output vector;

Another part is H infinite robust control item u _r=v ₁+ v ₂, wherein, v ₂=κ sgn (B ^tpX), λ > 0, κ > 0, matrix P is positive definite, and is meet following Riccati equation $\mathrm{PA}+A^{T}P+Q-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P+\frac{1}{{\mathrm{\ρ}}^2}{\mathrm{PBB}}^{T}P=0$ Solution;

(4), based on lyapnov stability theorem design neural network weight adaptive law Control performance standard be: $\underset{0}{\overset{T}{\∫}}{X}^T\mathrm{QX}\≤X^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}\left|\right|d\left|\right|t,$ Design lyapunov function is designed to: wherein P is tried to achieve by Riccati equation, and η is normal number, the derivative of Lyapunov function against time is: $\stackrel{\·}{V}=\frac{1}{2}{\stackrel{\·}{X}}^T\mathrm{PX}+\frac{1}{2}{X}^{T}P\stackrel{\·}{X}+\frac{1}{\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}\right),$ For realizing control objectives $\underset{0}{\overset{T}{\∫}}{X}^T\mathrm{QX}\≤X^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}\left|\right|d\left|\right|t,$ Design neural network parameter adaptive law is: $\stackrel{\stackrel{\·}{^}}{\mathrm{\θ}}=\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}=\mathrm{\η\ξ}{X}^T\mathrm{PB}.$

The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, is characterized in that: in step (1), for making system can accurate tracking reference locus, realize following H _∞tracking performance index:

$\underset{0}{\overset{T}{\∫}}{X}^T\mathrm{QX}\≤X^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}\left|\right|d\left|\right|t---\left(2\right)$

In formula, e=q _m-q is the tracking error of system, and T ∈ [0, ∞), d ∈ L ₂[0, T] is neural network approximate error, Q and P is two positive definite matrixes, be the error vector of parameter, η > 0, ρ > 0 is two given parameters.

The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, it is characterized in that: step (2) is about setting up gyroscope system dynamics model, concrete steps are as follows: the desirable dynamic perfromance of gyroscope is a kind of noenergy loss, two between centers, without the stable sine-wave oscillation of Dynamic Coupling, can be described below:

x _m＝A ₁sin(w ₁t)

(3)

y _m＝A ₂sin(w ₂t)

Desirable dynamic perfromance track is not only the reference model of system, being also the input signal of adaptive law, converging to the necessary condition of true value in order to meet parameter, the continuation of excitation, must ensure that reference locus comprises two different frequencies, therefore have w ₁≠ w ₂,

Reference model is write as vector form:

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

In formula, $q=\left[\begin{array}{c}{x}_m\\ y_m\end{array}\right],k_m=\left[\begin{array}{cc}{w}_1^2& 0\\ 0& w_2^2\end{array}\right],$

If do not consider external interference, and parameter D, Ω, K _bit is known, then controller can be designed to:

$u^{*}=(D+2\mathrm{\Ω})\stackrel{\·}{q}+K_{b}q+{\stackrel{\·\·}{q}}_m-k^{T}X---\left(5\right)$

In formula, k=(k _v, k _p) ^tcontroller parameter matrix, for the tracking error matrix of system,

Formula (5) being brought into the closed loop equation that formula (1) can obtain system is:

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

In formula, k _v, k _pfor $k_p=\left[\begin{array}{cc}{\mathrm{\α}}^2& 0\\ 0& {\mathrm{\α}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\α}& 0\\ 0& 2\mathrm{\α}\end{array}\right],$ E=q _m-q is system tracking error,

As can be seen from formula (6), the dynamic and static characteristic of closed-loop system, namely the tracking performance of gyroscope is by k _v, k _pdetermine, as long as select suitable parameter vector k _v, k _pvalue, can make polynomial expression h (s)=s ²+ k _vs+k _proot be positioned at left half open plane, then i.e. system stability, control task completes smoothly.

The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, is characterized in that:

With the output u of adaptive neural network system _c, approach the desirable control inputs u of gyroscope system ^*, neural network control device is designed to: u=u _c(X| θ)+u _r(8),

In formula, u _rfor H _∞robust control item,

u _r＝v ₁+v ₂(9)

$v_1=\frac{1}{\mathrm{\λ}}{B}^T\mathrm{PX}---\left(10\right)$

v ₂＝κsgn(B ^TPX)(11)

Wherein, λ > 0, κ > 0, matrix P is positive definite, and is the solution meeting following Riccati equation, $\mathrm{PA}+A^{T}P+Q-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P+\frac{1}{{\mathrm{\ρ}}^2}{\mathrm{PBB}}^{T}P=0---\left(12\right).$ The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, is characterized in that:

Based on Riccati equation $\mathrm{PA}+A^{T}P+Q-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P+\frac{1}{{\mathrm{\ρ}}^2}{\mathrm{PBB}}^{T}P=0$ The controller of design, its control objectives is: $\underset{0}{\overset{T}{\∫}}{X}^T\mathrm{QX}\≤X^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}\left|\right|d\left|\right|t$ (2)。

The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, is characterized in that:

In step (4), Riccati equation $\mathrm{PA}+A^{T}P+Q-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P+\frac{1}{{\mathrm{\ρ}}^2}{\mathrm{PBB}}^{T}P=0$ (12)

Wherein, Riccati equation (12) exists the condition of separating is 2 ρ ²>=λ,

Bring formula (8) into formula (1), can obtain

$\stackrel{\·\·}{q}=-(D+2\mathrm{\Ω})\stackrel{\·}{q}-K_{b}q+u_c\left(X\right|\mathrm{\θ})+v_1+v_2+d---\left(13\right)$

Formula (5) is out of shape:

${\stackrel{\·\·}{q}}_m=u^{*}-(D+2\mathrm{\Ω})\stackrel{\·}{q}-K_{b}q-k^{T}X---\left(14\right)$

Formula (14) is deducted formula (13) to obtain:

$\stackrel{\·\·}{e}=u^{*}-k^{T}X-u_c\left(X\right|\mathrm{\θ})-v_1-v_2-d---\left(15\right)$

Write formula (15) as vector form to obtain:

$\stackrel{\·}{X}=\mathrm{AX}+B[u^{*}-u_c\left(X\right|\mathrm{\θ})-v_1-v_2-d]---\left(16\right)$

In formula, $A=\left[\begin{array}{cc}0& I\\ -k_v& -k_p\end{array}\right],B=\left[\begin{array}{c}0\\ I\end{array}\right]$

The optimized parameter of defined parameters vector θ is θ ^*, be defined as

${\mathrm{\θ}}^{*}=\arg {\min }_{\mathrm{\θ}\∈\mathrm{\Ω}}\left[{\sup }_{x\∈U_c}\right|u_c\left(X\right|\mathrm{\θ})-u^{*}\left(q)\right)\left|\right]---\left(17\right)$

In formula, Ω is the suitable bounded aggregate comprising θ,

Definition neural network minimum approximation error is

w＝u ^*(q)-u _c(X|θ ^*)(18)

Arrange I: there is a constant κ > 0, make || (w (X)) _i||≤κ, 1≤i≤2 in formula,

Formula (7), (18) are brought into formula (16) and are obtained: $\stackrel{\·}{X}=\mathrm{AX}-B{\widetilde{\mathrm{\θ}}}^T\mathrm{\ξ}\left(X\right)+\mathrm{Bw}-{\mathrm{Bv}}_1-{\mathrm{Bv}}_2-\mathrm{Bd}$ (19)

In formula, for parameter estimating error,

The adaptive law of Selecting All Parameters vector θ is:

$\stackrel{\stackrel{\·}{^}}{\mathrm{\θ}}=\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}=\mathrm{\η\ξ}{X}^T\mathrm{PB}---\left(20\right)$

In formula, η > 0 is the law of learning of parameter.

The infinite control method of robust adaptive neural network H of aforesaid a kind of microthrust test, it is characterized in that: step (5), based on lyapunov function, the adaptive law of design variable element, the track of reference model on the track following making gyroscope system, the global asymptotic stability of guarantee system

Definition lyapunov function: (21)

In formula, P is tried to achieve by Riccati equation, and η is normal number,

The derivative of V to the time is asked to obtain: $\stackrel{\·}{V}=\frac{1}{2}{\stackrel{\·}{X}}^T\mathrm{PX}+\frac{1}{2}{X}^{T}P\stackrel{\·}{X}+\frac{1}{\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}\right)---\left(22\right)$ Formula (19) is brought into formula (22) to obtain:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}[X^T{A}^T-{\mathrm{\ξ}}^T\widetilde{\mathrm{\θ}}{B}^T+{\mathrm{\ω}}^T{B}^T-{v^T}_1{B}^T-{v^T}_2{B}^T-d^T{B}^T]\mathrm{PX}\\ +\frac{1}{2}{X}^{T}P[\mathrm{AX}-B{\widetilde{\mathrm{\θ}}}^T\mathrm{\ξ}+\mathrm{Bw}-{\mathrm{Bv}}_1-{\mathrm{Bv}}_2-\mathrm{Bd}]+\frac{1}{\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}\right)\end{array}---\left(23\right)$

Formula (10), (11) are brought into (23) and are obtained:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}{X}^T(A^{T}P+\mathrm{PA}-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P)X+X^T\mathrm{PBw}-X^T\mathrm{PBd}\\ -\mathrm{\κ}{X}^T\mathrm{PBsgn}\left(B^T\mathrm{PX}\right)+\mathrm{tr}[{\widetilde{\mathrm{\θ}}}^T(\frac{1}{\mathrm{\η}}\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}-\mathrm{\ξ}{X}^T\mathrm{PB})\end{array}---\left(24\right)$

Formula (20) is brought into formula (24) to obtain:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}{X}^T(A^{T}P+\mathrm{PA}-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P)X+X^T\mathrm{PBw}-X^T\mathrm{PBd}\\ -\mathrm{\κ}{X}^T\mathrm{PBsgn}\left(B^T\mathrm{PX}\right)\end{array}---\left(25\right)$

According to Riccati equation (12) with arrange I:

$\begin{array}{c}\stackrel{\·}{V}=-\frac{1}{2}{X}^{T}QX-\frac{1}{2{\mathrm{\ρ}}^2}{X}^T{\mathrm{PBB}}^{T}PX+X^{T}PBw\\ -{\mathrm{\κX}}^{T}PB\mathrm{sgn}\left(B^{T}PX\right)-X^{T}PBd\\ =-\frac{1}{2}{X}^{T}QX-\frac{1}{2}{(\frac{1}{\mathrm{\ρ}}{B}^{T}PX+\mathrm{\ρ}d)}^T(\frac{1}{\mathrm{\ρ}}{B}^{T}PX+\mathrm{\ρ}d)+\frac{1}{2}{\mathrm{\ρ}}^2{d}^{T}d\\ +X^{T}PBw-{\mathrm{\κX}}^{T}PB\mathrm{sgn}\left(B^{T}PX\right)\\ \≤-\frac{1}{2}{X}^{T}QX+\frac{1}{2}{\mathrm{\ρ}}^2{d}^{T}d-\mathrm{\κ}\underset{i=1}{\overset{2}{\Σ}}\left|{\left(B^{T}PX\right)}_i\right|+\underset{i=1}{\overset{2}{\Σ}}\left|w_i\right|\·\left|{\left(B^{T}PX\right)}_i\right|\\ \≤-\frac{1}{2}{X}^{T}QX+\frac{1}{2}{\mathrm{\ρ}}^2\left|\right|d|{|}^2\end{array}---\left(26\right)$

(26) formula is obtained from 0 to T integration

$V\left(T\right)-V\left(0\right)\≤-\frac{1}{2}{\∫}_0^T{X}^T\mathrm{QXdt}+\frac{1}{2}{\mathrm{\ρ}}^2{\∫}_0^T{\left|\right|d\left|\right|}^2\mathrm{dt}---\left(27\right)$

Due to V (T) >=0, so (27) formula turns to

$\begin{array}{c}\frac{1}{2}{\∫}_0^T{X}^T\mathrm{QXdt}\≤V\left(0\right)+\frac{1}{2}{\mathrm{\ρ}}^2{\∫}_0^T{\left|\right|d\left|\right|}^{2}t\\ =\frac{1}{2}{X}^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{2\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+\frac{1}{2}{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}{\left|\right|d\left|\right|}^2\mathrm{dt}\end{array}---\left(28\right)$

Namely tracking error obtains H _∞control performance standard, control task completes.

The beneficial effect that the present invention reaches: the present invention works out the infinite control method of a kind of robust adaptive neural network H, utilize the approximation capability that neural network is powerful, the desirable controller of online approximating gyroscope system, do not need the mathematical models of controlled device, the dynamic perfromance of system can be improved well.Meanwhile, add robust item in the controller, substantially increase the robustness of the interference of system external circle and Parameters variation, improve the tracking performance of system.Controller based on the design of Riccati equation also further compensates the non-linear phenomena of system.The right value update algorithm of this method, based on the design of Lyapunov stability theory, can ensure the global stability of closed-loop system.

The present invention has the following advantages:

1, adopt the unknown term in robust adaptive neural network H infinite control method online approximating gyroscope system, do not need the mathematical models knowing system.

2, the infinite control method of robust adaptive neural network H is adopted to control gyroscope system, combine the advantage of the infinite control technology of ANN (Artificial Neural Network) Control, adaptive control, robust control and H, minimizing Parameters variation and external interference, on the impact of systematic error, improve dynamic perfromance and the robustness of system greatly.

3, controling parameters of the present invention can adaptive learning and adjustment, by its constantly oneself's adjustment, after system reaches stable state, the good tracking performance of whole system can be realized, obtain satisfied dynamic perfromance can and the robustness of interference and parameter uncertainty to external world.

4, based on the infinite controller of robust adaptive neural network H of Riccati equation design, reduce the impact of gyroscope non-linear phenomena on tracking error, adjusted by parameter adaptive, obtain given tracking error performance index.

5, the neural network weight adaptive law algorithm designed based on Lyapnov can ensure the asymptotic stability of whole closed-loop system.

Accompanying drawing explanation

Fig. 1 is principle assumption diagram of the present invention.

Fig. 2 adopts the trace plot on the gyroscope x of robust adaptive neural network H infinite control method, y-axis direction.

Fig. 3 adopts the error curve diagram on the gyroscope x of robust adaptive neural network H infinite control method, y-axis direction.

Fig. 4 adopts the control inputs figure on the gyroscope x of robust adaptive neural network H infinite control method, y-axis direction.

Embodiment

Below in conjunction with accompanying drawing, the invention will be further described.Following examples only for technical scheme of the present invention is clearly described, and can not limit the scope of the invention with this.

As shown in Figure 1, the infinite control method of robust adaptive neural network H of microthrust test of the present invention, comprises the following steps:

(1) dynamic model of gyroscope is set up

Controlled device is two axle gyroscope systems, and arranging gyroscope can rotate with angular velocity at the uniform velocity respectively in x, y-axis both direction, and centrifugal force can be ignored, and after non-dimension and equivalent transformation, the dynamic equation obtaining gyroscope is as follows:

Differential equation form after the non-dimension of gyroscope is:

$\stackrel{\·\·}{q}=-(D+2\mathrm{\Ω})\stackrel{\·}{q}-\mathrm{Kq}+u+d---\left(1\right)$

Wherein $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the position vector of gyroscope mass on x, y-axis direction; $u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$ For the control inputs of gyroscope on x, y-axis direction; $\mathrm{\Ω}=\left[\begin{array}{cc}0& -{\mathrm{\Ω}}_z\\ {\mathrm{\Ω}}_z& 0\end{array}\right]$ For angular velocity matries; $D=\left[\begin{array}{cc}{d}_{\mathrm{yy}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right]$ For damping matrix, $K=\left[\begin{array}{cc}{w}_x^2& w_{\mathrm{xy}}\\ w_{\mathrm{xy}}& w_y^2\end{array}\right]$ It is the coefficient entry containing gyroscope fixed frequency, stiffness coefficient and coupling stiffness coefficient; $d=\left[\begin{array}{c}{d}_1\\ d_2\end{array}\right]$ For indeterminate and the external interference of system.

For making system can accurate tracking reference locus, following H be realized _∞tracking performance index:

$\underset{0}{\overset{T}{\∫}}{X}^T\mathrm{QX}\≤X^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}\left|\right|d\left|\right|t---\left(2\right)$

In formula, e=q _m-q is the tracking error of system, and T ∈ [0, ∞), d ∈ L ₂[0, T] is neural network approximate error, Q and P is two positive definite matrixes, be the error vector of parameter, η > 0, ρ > 0 is two given parameters.

(2) reference model of gyroscope system is determined

The desirable dynamic perfromance of gyroscope is a kind of noenergy loss, and two between centers are without the stable sine of Dynamic Coupling

Vibration, can be described below:

x _m＝A ₁sin(w ₁t)

(3)

y _m＝A ₂sin(w ₂t)

Desirable dynamic perfromance track is not only the reference model of system, is also the input signal of adaptive law, in order to

Meet the necessary condition that parameter converges to true value, the continuation of excitation, must ensure that reference locus comprises two not

Same frequency, therefore have w ₁≠ w ₂.

Reference model is write as vector form:

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

In formula, $q=\left[\begin{array}{c}{x}_m\\ y_m\end{array}\right],k_m=\left[\begin{array}{cc}{w}_1^2& 0\\ 0& w_2^2\end{array}\right].$

If do not consider external interference, and parameter D, Ω, K _bit is known, then controller can be designed to:

$u^{*}=(D+2\mathrm{\Ω})\stackrel{\·}{q}+K_{b}q+{\stackrel{\·\·}{q}}_m-k^{T}X---\left(5\right)$

In formula, k=(k _v, k _p) ^tcontroller parameter matrix, for the tracking error matrix of system.

Formula (5) being brought into the closed loop equation that formula (1) can obtain system is:

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

In formula, k _v, k _pfor $k_p=\left[\begin{array}{cc}{\mathrm{\α}}^2& 0\\ 0& {\mathrm{\α}}^2\end{array}\right],k_v=\left[\begin{array}{cc}2\mathrm{\α}& 0\\ 0& 2\mathrm{\α}\end{array}\right],$ E=q _m-q is system tracking error.As can be seen from formula (6), the dynamic and static characteristic of closed-loop system, namely the tracking performance of gyroscope is by k _v, k _pdetermine.As long as select suitable parameter vector k _v, k _pvalue, can make polynomial expression h (s)=s ²+ k _vs+k _proot be positioned at left half open plane, then i.e. system stability, control task completes smoothly.

But, the parameter D of gyroscope system, Ω, k _bthe unknown, therefore in formula (5), the desirable controller of definition cannot be implemented.But formula (5) and the closed-loop system equation obtained thereof enlighten to us: utilize RBF neural to approach the desirable controller of gyroscope online in real time, thus do not require that we know the accurate dynamic model of system.Therefore, RBF neural controls and H by first _∞robust control technique combines, and proposes a kind of based on H _∞the neural network control device of robust technique.

(3) the infinite controller of robust adaptive neural network H is designed

For approaching desirable controller u ^*rBF neural export can be expressed as:

$u_c\left(X\right|\mathrm{\θ})={\widehat{\mathrm{\θ}}}^T\mathrm{\ξ}\left(X\right)---\left(7\right)$

In formula, X is the input of neural network, for system can measuring-signal; represent the weight vector of neural network; M represents the number of RBF network hidden layer node; ξ (X) is hidden layer radial basis function output vector.

Neural network control device can be designed to:

u＝u _c(X|θ)+u _r(8)

In formula, u _rfor H _∞robust control item.

u _r＝v ₁+v ₂(9)

$v_1=\frac{1}{\mathrm{\λ}}{B}^T\mathrm{PX}---\left(10\right)$

v ₂＝κsgn(B ^TPX)(11)

Wherein, λ > 0, κ > 0, matrix P is positive definite, and is the solution meeting following Riccati equation.

$\mathrm{PA}+A^{T}P+Q-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P+\frac{1}{{\mathrm{\ρ}}^2}{\mathrm{PBB}}^{T}P=0---\left(12\right).$

Wherein, Riccati equation (12) exists the condition of separating is 2 ρ ²>=λ.

Bring formula (8) into formula (1), can obtain

$\stackrel{\·\·}{q}=-(D+2\mathrm{\Ω})\stackrel{\·}{q}-K_{b}q+u_c\left(X\right|\mathrm{\θ})+v_1+v_2+d---\left(13\right)$

Formula (5) is out of shape:

${\stackrel{\·\·}{q}}_m=u^{*}-(D+2\mathrm{\Ω})\stackrel{\·}{q}-K_{b}q-k^{T}X---\left(14\right)$

Formula (14) is deducted formula (13) to obtain:

$\stackrel{\·\·}{e}=u^{*}-k^{T}X-u_c\left(X\right|\mathrm{\θ})-v_1-v_2-d---\left(15\right)$

Write formula (15) as vector form to obtain:

$\stackrel{\·}{X}=\mathrm{AX}+B[u^{*}-u_c\left(X\right|\mathrm{\θ})-v_1-v_2-d]---\left(16\right)$

In formula, $A=\left[\begin{array}{cc}0& I\\ -k_v& -k_p\end{array}\right],B=\left[\begin{array}{c}0\\ I\end{array}\right]$

The optimized parameter of defined parameters vector θ is θ ^*, be defined as

${\mathrm{\θ}}^{*}=\arg {\min }_{\mathrm{\θ}\∈\mathrm{\Ω}}\left[{\sup }_{x\∈U_c}\right|u_c\left(X\right|\mathrm{\θ})-u^{*}\left(q)\right)\left|\right]---\left(17\right)$ In formula, Ω is the suitable bounded aggregate comprising θ.

Definition neural network minimum approximation error is

w＝u ^*(q)-u _c(X|θ ^*)(18)

Arrange I: there is a constant κ > 0, make || (w (X)) _i||≤κ, 1≤i≤2 in formula.

Formula (7), (18) are brought into formula (16) and are obtained:

$\stackrel{\·}{X}=\mathrm{AX}-B{\widetilde{\mathrm{\θ}}}^T\mathrm{\ξ}\left(X\right)+\mathrm{Bw}-{\mathrm{Bv}}_1-{\mathrm{Bv}}_2-\mathrm{Bd}---\left(19\right)$

In formula, for parameter estimating error.

The adaptive law of Selecting All Parameters vector θ is:

$\stackrel{\stackrel{\·}{^}}{\mathrm{\θ}}=\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}=\mathrm{\η\ξ}{X}^T\mathrm{PB}---\left(20\right)$

In formula, η > 0 is the law of learning of parameter.

Theorem 1 control object is (1) formula, if get control law u for (8) formula, the adaptive law of parameter θ gets (20) formula, then control program can obtain drawing a conclusion:

(1)θ∈Ω,q,e,u∈L _∞

(2) for the horizontal ρ of given suppression, tracking error reaches H _∞tracking performance index (2).Based on lyapunov function, the adaptive law of design variable element, the track of reference model on the track following making gyroscope system, ensures the global asymptotic stability of system.

Definition lyapunov function:

$V=\frac{1}{2}{X}^T\mathrm{PX}+\frac{1}{2\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\widetilde{\mathrm{\θ}}\right)---\left(21\right)$

In formula, P is tried to achieve by Riccati equation, and η is normal number.

The derivative of V to the time is asked to obtain:

$\stackrel{\·}{V}=\frac{1}{2}{\stackrel{\·}{X}}^T\mathrm{PX}+\frac{1}{2}{X}^{T}P\stackrel{\·}{X}+\frac{1}{\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}\right)---\left(22\right)$

Formula (19) is brought into formula (22) to obtain:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}[X^T{A}^T-{\mathrm{\ξ}}^T\widetilde{\mathrm{\θ}}{B}^T+{\mathrm{\ω}}^T{B}^T-{v^T}_1{B}^T-{v^T}_2{B}^T-d^T{B}^T]\mathrm{PX}\\ +\frac{1}{2}{X}^{T}P[\mathrm{AX}-B{\widetilde{\mathrm{\θ}}}^T\mathrm{\ξ}+\mathrm{Bw}-{\mathrm{Bv}}_1-{\mathrm{Bv}}_2-\mathrm{Bd}]+\frac{1}{\mathrm{\η}}\mathrm{tr}\left({\widetilde{\mathrm{\θ}}}^T\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}\right)\end{array}---\left(23\right)$

Formula (10), (11) are brought into (23) and are obtained:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}{X}^T(A^{T}P+\mathrm{PA}-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P)X+X^T\mathrm{PBw}-X^T\mathrm{PBd}\\ -\mathrm{\κ}{X}^T\mathrm{PBsgn}\left(B^T\mathrm{PX}\right)+\mathrm{tr}[{\widetilde{\mathrm{\θ}}}^T(\frac{1}{\mathrm{\η}}\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}-\mathrm{\ξ}{X}^T\mathrm{PB})\end{array}---\left(24\right)$

Formula (20) is brought into formula (24) to obtain:

$\begin{array}{c}\stackrel{\·}{V}=\frac{1}{2}{X}^T(A^{T}P+\mathrm{PA}-\frac{2}{\mathrm{\λ}}{\mathrm{PBB}}^{T}P)X+X^T\mathrm{PBw}-X^T\mathrm{PBd}\\ -\mathrm{\κ}{X}^T\mathrm{PBsgn}\left(B^T\mathrm{PX}\right)\end{array}---\left(25\right)$

According to Riccati equation (12) with arrange I:

$\begin{array}{c}\stackrel{\·}{V}=-\frac{1}{2}{X}^{T}QX-\frac{1}{2{\mathrm{\ρ}}^2}{X}^T{\mathrm{PBB}}^{T}PX+X^{T}PBw\\ -{\mathrm{\κX}}^{T}PB\mathrm{sgn}\left(B^{T}PX\right)-X^{T}PBd\\ =-\frac{1}{2}{X}^{T}QX-\frac{1}{2}{(\frac{1}{\mathrm{\ρ}}{B}^{T}PX+\mathrm{\ρ}d)}^T(\frac{1}{\mathrm{\ρ}}{B}^{T}PX+\mathrm{\ρ}d)+\frac{1}{2}{\mathrm{\ρ}}^2{d}^{T}d\\ +X^{T}PBw-{\mathrm{\κX}}^{T}PB\mathrm{sgn}\left(B^{T}PX\right)\\ \≤-\frac{1}{2}{X}^{T}QX+\frac{1}{2}{\mathrm{\ρ}}^2{d}^{T}d-\mathrm{\κ}\underset{i=1}{\overset{2}{\Σ}}\left|{\left(B^{T}PX\right)}_i\right|+\underset{i=1}{\overset{2}{\Σ}}\left|w_i\right|\·\left|{\left(B^{T}PX\right)}_i\right|\\ \≤-\frac{1}{2}{X}^{T}QX+\frac{1}{2}{\mathrm{\ρ}}^2\left|\right|d|{|}^2\end{array}---\left(26\right)$

(26) formula is obtained from 0 to T integration

$V\left(T\right)-V\left(0\right)\≤-\frac{1}{2}{\∫}_0^T{X}^T\mathrm{QXdt}+\frac{1}{2}{\mathrm{\ρ}}^2{\∫}_0^T{\left|\right|d\left|\right|}^2\mathrm{dt}---\left(27\right)$

Due to V (T) >=0, so (27) formula turns to

$\begin{array}{c}\frac{1}{2}{\∫}_0^T{X}^T\mathrm{QXdt}\≤V\left(0\right)+\frac{1}{2}{\mathrm{\ρ}}^2{\∫}_0^T{\left|\right|d\left|\right|}^{2}t\\ =\frac{1}{2}{X}^T\left(0\right)\mathrm{PX}\left(0\right)+\frac{1}{2\mathrm{\η}}{\widetilde{\mathrm{\θ}}}^T\left(0\right)\widetilde{\mathrm{\θ}}\left(0\right)+\frac{1}{2}{\mathrm{\ρ}}^2\underset{0}{\overset{T}{\∫}}{\left|\right|d\left|\right|}^2\mathrm{dt}\end{array}---\left(28\right)$

Namely tracking error obtains H _∞control performance standard, control task completes.

(6) simulation analysis

According to above robust adaptive neural network H _∞the algorithm controlled, carries out numerical simulation to control system in MATLAB/Simulink.The gyroscope parameter of emulation experiment is as follows:

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

In l-G simulation test, the original state of controlled device is got [0000], and reference locus is x _m=sin (6.17t), y _m=sin (5.11t), uncertain and interference total amount d=1* [randn (1,1), randn (1,1)] ^tμ N, the gain of robust item is taken as λ=0.1, κ=500 respectively.Choosing node in hidden layer for RBF neural is 11, and learning rate is η=30.The optimum configurations of formula (12) Riccati equation is: Q=10*I, ρ=2.Dynamic simulation original program, obtains following simulation result:

Fig. 2 is the pursuit path figure of X under the infinite control of robust adaptive neural network H, Y-axis, and solid line is desired trajectory, and dotted line is actual path, and as seen from the figure, tracking effect is better, and through after a while, system can follow the tracks of desired movement locus.Fig. 3 is the tracking error figure of X under the infinite control of robust adaptive neural network H, Y-axis, as can be seen from the figure, substantially converges to zero, and keep this motion through very short time error curve.Fig. 4 is the control inputs figure of X under the infinite control of robust adaptive neural network H, Y-axis.

As can be seen from above analogous diagram, the infinite control method of robust adaptive neural network H of gyroscope that the present invention proposes can make tracking error very rapid convergence to zero, there is good tracking performance, and interference and Parameters variation have good robustness to external world, substantially increase the tracking performance of system.

More than show and describe ultimate principle of the present invention, principal character and advantage.The technician of the industry should understand; the present invention is not restricted to the described embodiments; what describe in above-described embodiment and instructions just illustrates principle of the present invention; without departing from the spirit and scope of the present invention; the present invention also has various changes and modifications, and these changes and improvements all fall in the claimed scope of the invention.Application claims protection domain is defined by appending claims and equivalent thereof.

Claims (7)

1. the infinite control method of robust adaptive neural network H of microthrust test, is characterized in that: adopt the controller based on Riccati equation design to comprise two parts: Part I is basic nerve network controller; Part II is robust control item, specifically comprises the following steps:

(1), desirable kinetic model is set up,

Design reference model is the sine wave of two different frequencies: x _m=A ₁sin (w ₁t), y _m=A ₂sin (w ₂t), wherein w ₁≠ w ₂and all non-vanishing, being rewritten into vector form is: wherein, A ₁, A ₂be the amplitude of gyroscope on x, y two change in coordinate axis direction respectively, t is the time, w ₁and w ₂be respectively the vibration frequency that gyroscope is given on x, y two change in coordinate axis direction; Vector form is: wherein

(2), gyroscope system dynamics model is set up,

Set up the non-dimension vector model of gyroscope system,

Wherein for the position vector of gyroscope mass on x, y-axis direction; for the control inputs of gyroscope on x, y-axis direction; for angular velocity matries; for damping matrix, it is the coefficient entry containing gyroscope fixed frequency, stiffness coefficient and coupling stiffness coefficient; for indeterminate and the external interference of system, if do not consider external interference, and parameter D, Ω, K _bknown, then controller can be designed to: wherein, k=(k _v, k _p) ^tfor controller parameter matrix, for the tracking error matrix of system;

(3), the design of the infinite controller of robust adaptive neural network H

Due to the parameter D of gyroscope system, Ω, K _bthe unknown, then the controller based on Riccati equation design comprises two parts: Part I is nerve network controller in formula, the input of neural network, for system can measuring-signal; represent the weight vector of neural network; M represents the number of RBF network hidden layer node; ξ (X) is hidden layer radial basis function output vector;

Another part is H infinite robust control item u _r=v ₁+ v ₂, wherein, v ₂=κ sgn (B ^tpX), λ > 0, κ > 0, matrix P is positive definite, and is meet following Riccati equation solution;

(4), based on lyapnov stability theorem design neural network weight adaptive law Control performance standard be: design lyapunov function is designed to: wherein P is tried to achieve by Riccati equation, and η is normal number, the derivative of Lyapunov function against time is: for realizing control objectives design neural network parameter adaptive law is:

2. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 1, is characterized in that: in step (1), for making system can accurate tracking reference locus, realize following H _∞tracking performance index:

In formula, for the tracking error of system, and T ∈ [0, ∞), d ∈ L ₂[0, T] is neural network approximate error, Q and P is two positive definite matrixes, be the error vector of parameter, η > 0, ρ > 0 is two given parameters.

3. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 2, is characterized in that: step (2) is about setting up gyroscope system dynamics model, and concrete steps are as follows:

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 _m＝A ₁sin(w ₁t)

y _m＝A ₂sin(w ₂t)(3)

Desirable dynamic perfromance track is not only the reference model of system, being also the input signal of adaptive law, converging to the necessary condition of true value in order to meet parameter, the continuation of excitation, must ensure that reference locus comprises two different frequencies, therefore have w ₁≠ w ₂,

Reference model is write as vector form:

In formula,

If do not consider external interference, and parameter D, Ω, K _bit is known, then controller can be designed to:

In formula, k=(k _v, k _p) ^tcontroller parameter matrix, for the tracking error matrix of system,

Formula (5) being brought into the closed loop equation that formula (1) can obtain system is:

In formula, k _v, k _pfor e=q _m-q is system tracking error,

As can be seen from formula (6), the dynamic and static characteristic of closed-loop system, namely the tracking performance of gyroscope is by k _v, k _pdetermine, as long as select suitable parameter vector k _v, k _pvalue, can make polynomial expression h (s)=s ²+ k _vs+k _proot be positioned at left half open plane, then i.e. system stability, control task completes smoothly.

4. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 3, is characterized in that:

With the output u of adaptive neural network system _c, approach the desirable control inputs u of gyroscope system ^*, neural network control device is designed to: u=u _c(X| θ)+u _r(8),

In formula, for H _∞robust control item,

u _r＝v ₁+v ₂(9)

v ₂＝κsgn(B ^TPX)(11)

Wherein, λ > 0, κ > 0, matrix P is positive definite, and is the solution meeting following Riccati equation,

5. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 4, is characterized in that:

Based on Riccati equation the controller of design, its control objectives is:

6. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 5, is characterized in that:

In step (4), Riccati equation

Wherein, Riccati equation (12) exists the condition of separating is 2 ρ ²>=λ,

Bring formula (8) into formula (1), can obtain

Formula (5) is out of shape:

Formula (14) is deducted formula (13) to obtain:

Write formula (15) as vector form to obtain:

In formula,

The optimized parameter of defined parameters vector θ is θ ^*, be defined as

In formula, Ω is the suitable bounded aggregate comprising θ,

Definition neural network minimum approximation error is

w＝u ^*(q)-u _c(X|θ ^*)(18)

Arrange I: there is a constant κ > 0, make || (w (X)) _i||≤κ, 1≤i≤2 in formula,

Formula (7), (18) are brought into formula (16) and are obtained:

In formula, for parameter estimating error,

The adaptive law of Selecting All Parameters vector θ is:

In formula, η > 0 is the law of learning of parameter.

7. the infinite control method of robust adaptive neural network H of a kind of microthrust test according to claim 6, it is characterized in that: step (5), based on lyapunov function, the adaptive law of design variable element, the track of reference model on the track following making gyroscope system, the global asymptotic stability of guarantee system

Definition lyapunov function:

In formula, P is tried to achieve by Riccati equation, and η is normal number,

The derivative of V to the time is asked to obtain:

Formula (19) is brought into formula (22) to obtain:

Formula (10), (11) are brought into (23) and are obtained:

Formula (20) is brought into formula (24) to obtain:

According to Riccati equation (12) with arrange I:

(26) formula is obtained from 0 to T integration

Due to V (T) >=0, so (27) formula turns to

Namely tracking error obtains H _∞control performance standard, control task completes.