CN102508434A - Adaptive fuzzy sliding mode controller for micro gyroscope - Google Patents

Abstract

The invention discloses an adaptive fuzzy sliding mode controller for a micro gyroscope, which is characterized by comprising an adaptive fuzzy control system based on a reference module, and a switching control system. In the adaptive fuzzy control system, a sliding mode surface is used as the input of the fuzzy controller, and the weight is automatically adjusted by a dynamic adaptive law so as to realize fuzzy approximation to an equivalent control law. The adaptive fuzzy sliding mode controller for the micro gyroscope of the invention has the following benefits that: when the system is in a steady state, the dynamic performance of the micro gyroscope is an ideal mode, and the manufacturing error and the environment interference are compensated; an adaptive algorithm designed based on a Lyapunov method can ensure the global asymptotic stability of the whole closed-loop system; and with adaptive adjustment on the upper bound of the approximation error, buffeting is reduced obviously.

Description

The adaptive fuzzy sliding mode controller that is used for gyroscope

Technical field

The present invention relates to the design of Controller of gyroscope, the application of particularly a kind of method of adaptive fuzzy sliding mode control on the gyroscope controller.

Background technology

Gyroscope is the inertial sensor of measured angular speed, compares with traditional gyroscope, and gyroscope has huge advantage on volume and cost, therefore wide application market is arranged, such as on navigational guidance, consumer electronics, navigation and national defence.But, because the error in the manufacturing process exists and the influence of environment temperature, cause the difference between original paper characteristic and the design, cause existing the stiffness coefficient and the ratio of damping of coupling, reduced the sensitivity and the precision of gyroscope.In addition, itself belongs to multi-input multi-output system gyroscope, has the fluctuation that causes to systematic parameter of uncertainty and the external interference of parameter, and the compensation foozle becomes the subject matter that gyroscope is controlled with external interference.

Traditional design of Fuzzy Controller does not rely on the model of controlled device, but it relies on control expert or operator's experimental knowledge, and it is not easy to the self-teaching and the adjustment of controlled variable, thereby is difficult to guarantee the stability of control system.

Summary of the invention

The present invention provides a kind of adaptive fuzzy sliding mode controller that is used for gyroscope for addressing the above problem, and it can effectively improve stability, reliability and the dynamic quality of system, and has effectively reduced chattering phenomenon.

Adaptive fuzzy control method is the fuzzy logic system with adaptive learning algorithm, and its learning algorithm is to rely on data message to adjust the parameter of fuzzy logic system, and it can guarantee the stability of control system.At present, controlling schemes wherein all be with the sum of errors error rate as fuzzy input variable, so in reality, need more fuzzy rule, improved the complexity of system.

For addressing this problem; Mentality of designing of the present invention is: with sliding-mode surface as fuzzy input variable; Adaptive Fuzzy Control and fuzzy sliding mode tracking control can be combined like this; Draw both advantages; Can adjust weights automatically; Restrain through dynamic self-adapting and to produce gratifying system responses, and can as fuzzy sliding mode tracking control, reduce the quantity of fuzzy rule significantly.In a word, method of adaptive fuzzy sliding mode control of the present invention can not rely on the model of system, has the effect of simplifying the Fuzzy control system structural complexity.Wherein, also through the adaptive algorithm of Lyapunov method CONTROLLER DESIGN parameter, real-time online is adjusted controller parameter for it, thereby further guarantees the stability of system's overall situation.

In brief, the present invention is:

Design a desirable gyroscope dynamic model, as the system reference track, controlled target is to guarantee reference locus on the actual gyroscope trajectory track, reaches a kind of desirable dynamic perfromance, compensation foozle and environmental interference.Method of adaptive fuzzy sliding mode control of the present invention comprises Adaptive Fuzzy Control and switching controls.In the design of Adaptive Fuzzy Control, at first design the Integral Sliding Mode face, with of the input of this sliding-mode surface, adjust weights automatically with adaptive fuzzy controller as fuzzy controller, come the fuzzy equivalent control rule of approaching through the dynamic self-adapting rule; In the switching controls design, compensate the error between equivalent control rule and the fuzzy controller with switch controller, and the self-adaptation evaluated error upper bound, because there is online adjusting in it, therefore weakened buffeting significantly.Adaptive algorithm adopts based on the design of Lyapunov stability approach, thereby guarantees the overall progressive stability of gyroscope trajectory track coideal model and system, has improved the reliability of system.

Up to now, method of adaptive fuzzy sliding mode control of the present invention is not applied to as yet in the control of gyroscope.

Technical scheme of the present invention provides a kind of adaptive fuzzy sliding mode controller that is used for gyroscope, it is characterized in that: it comprises based on the self-adaptive fuzzy control system of reference model and handover control system; In self-adaptive fuzzy control system, adopt the input of sliding-mode surface, and adopt dynamic self-adapting rule adjustment weights automatically as fuzzy controller, thus the fuzzy equivalent control rule of approaching.

Preferably, in said handover control system, adopt switch controller to compensate the error between said equivalent control rule and the said fuzzy controller.

Preferably, the adaptive algorithm of the parameter of said self-adaptive fuzzy control system adopts the design of Lyapunov method, with the overall progressive stability of reference model and The whole control system on the assurance gyroscope trajectory track.

Preferably, in said adaptive fuzzy control system output when anti-fuzzy adaptive weights defined as <img file = "BDA0000106089750000031.GIF" he = "45" img-content = "drawing " img-format =" tif " inline =" yes " orientation =" portrait " wi =" 183 "/> adaptive algorithm: <img file =" BDA0000106089750000032.GIF " he =" 65 " img-content = "drawing" img-format = "tif" inline = "yes" orientation = "portrait" wi = "349" /> designing its Lyapunov function: <maths num="0001"> <![CDATA[<math> <mrow> <msub> <mi> V </ mi> <mn> 1 </ mn> </ msub> <mrow> <mo> (</ mo> <mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, </ mo> <mover> <mi> α < / mi> <mo> ~ </ mo> </ mover> <mo>) </ mo> </ mrow> <mo> = </ mo> <mfrac> <mn> 1 </ mn> <mn> 2 </ mn> </ mfrac> <msup> <mi> s </ mi> <mn> 2 </ mn> </ msup> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 1 </ mn> </ msub> </ mrow> </ mfrac> <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover > <mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>, </ mo> </ mrow> < / math>]]> </maths> where η <sub > 1 </sub> is a positive real number; said switching control system, adaptive switching gain is defined as <img file = "BDA0000106089750000034.GIF" he = "49" img-content = "drawing" img-format = "tif" inline = "yes" orientation = "portrait" wi = "169" /> adaptive algorithm is: < maths num = "0002"> <! [CDATA [<math> <mrow> <mover> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> < mo> &CenterDot; </ mo> </ mover> <mo> = </ mo> <mover> <mover> <mi> E </ mi> <mo> ^ </ mo> </ mover> <mo> & CenterDot ; </ mo> </ mover> <mo> = </ mo> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub> <mo> | </ mo> < mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo> | </ mo> <mo>, </ mo> </ mrow> </ math>]]> </maths> designing its Lyapunov function: <maths num="0003"> <! [CDATA [<math> <mrow> <mi > V </ mi> <mrow> <mo> (</ mo> <mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, </ mo> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>, </ mo> <mover> < mi> E </ mi> <mo> ~ </ mo> </ mover> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>) </ mo> </ mrow> <mo> = </ mo> <msub> <mi> V </ mi> <mn> 1 </ mn> </ msub> <mrow> <mo> ( </ mo> <mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, </ mo > <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn > 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub> </ mrow> </ mfrac> < msup> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mn> 2 </ mn> </ msup> <mo> = </ mo> <mfrac> <mn > 1 </ mn> <mn> 2 </ mn> </ mfrac> <msup> <mi> s </ mi> <mn> 2 </ mn> </ msup> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 1 </ mn> </ msub> </ mrow> </ mfrac> <msup> <mover> <mi> α </ mi> < mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub > </ mrow> </ mfrac> <msup> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mn> 2 </ mn> </ msup> <mo> , </ mo> </ mrow> </ math>]]> </maths> where η <sub > 2 </sub> is a positive real number.

Preferably, said reference model adopts two between centers not have the stable sine-wave oscillation of Dynamic Coupling: x _m=A ₁Sin (w ₁T), y _m=A ₂Sin (w ₂T), its differential equation is:

Preferably, said sliding-mode surface is the Integral Sliding Mode face: $s=\stackrel{\&CenterDot;}{q}\left(t\right)-{\&Integral;}_0^t[{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m\left(\mathrm{\&tau;}\right)-k_1\stackrel{\&CenterDot;}{e}\left(\mathrm{\&tau;}\right)-k_{2}e\left(\mathrm{\&tau;}\right)]\mathrm{d\&tau;},$ Tracking error e (t)=q (t)-q wherein _m(t), k ₁, k ₂Be the positive constant of non-zero.

The beneficial effect that is used for the adaptive fuzzy sliding mode controller of gyroscope of the present invention is: after system reached stable state, the dynamic perfromance of gyroscope was a kind of idealized model, had compensated foozle and environmental interference; Can guarantee the overall progressive stability of whole closed-loop system based on the adaptive algorithm of Lyapunov method design; Owing to add the self-adaptation in the approximate error upper bound is regulated, significantly reduced buffeting.

Description of drawings

Fig. 1 is the simplified model synoptic diagram of gyroscope in the specific embodiment of the present invention;

Fig. 2 is a gyroscope Adaptive Fuzzy Sliding Mode Control system architecture diagram in the specific embodiment of the present invention;

Fig. 3 is X, the Y-axis tracking effect curve map of gyroscope in the specific embodiment of the present invention;

Fig. 4 is a sliding-mode surface curve map in the specific embodiment of the present invention;

Fig. 5 is handoff gain gyrostatic control input u fixedly time the in the specific embodiment of the present invention _x, u _yChange curve;

Fig. 6 is gyrostatic control input u when the handoff gain self-adaptation is regulated in the specific embodiment of the present invention _x, u _yChange curve;

Fig. 7 is handoff gain E self-adaptation estimation curve figure in the specific embodiment of the present invention.

Embodiment

Following specific embodiments of the invention is described in further detail.

One, the kinetics equation of gyroscope

General little gyrotron comprises three ingredients: by the mass that resilient material supported, and electrostatic drive and sensing apparatus.Static driven circuit major function is the constant of amplitude when driving and keeping little gyrotron vibration, i.e. constant amplitude oscillation; Sensing circuit is used for the position and the speed of perceived quality piece.Gyroscope can be reduced to one has a vibration-damping system by what mass and spring constituted.Fig. 1 has shown little gyrotron model of under cartesian coordinate system, simplifying.As far as Z axle gyroscope, can think that mass is limited in the x-y plane, to move, and can not move along the Z axle.In fact, because the existence of manufacturing defect and mismachining tolerance can cause the additional dynamic coupling of x axle and y axle, like the stiffness coefficient and the ratio of damping of coupling.Take into account foozle, the lumped parameter mathematical model of actual gyroscope is:

$m\stackrel{\&CenterDot;\&CenterDot;}{x}+d_{\mathrm{xx}}\stackrel{\&CenterDot;}{x}+d_{\mathrm{xy}}\stackrel{\&CenterDot;}{y}+k_{\mathrm{xx}}x+k_{\mathrm{xy}}y=u_x+2m{\mathrm{\&Omega;}}_z\stackrel{\&CenterDot;}{y}$ (1)

$m\stackrel{\&CenterDot;\&CenterDot;}{y}+d_{\mathrm{xy}}\stackrel{\&CenterDot;}{x}+d_{\mathrm{yy}}\stackrel{\&CenterDot;}{y}+k_{\mathrm{xy}}x+k_{\mathrm{yy}}y=u_y+2m{\mathrm{\&Omega;}}_z\stackrel{\&CenterDot;}{x}$

M is the quality of mass, and x, y are the coordinate of mass in rotation system, d _Xx, d _YyBe respectively the ratio of damping of x axle and y axle, k _Xx, k _YyBe respectively the spring constant of x axle and y axle, d _Xy, k _XyBe respectively ratio of damping and the spring constant of coupling of coupling, close and be called quadrature error, u _x, u _yBe the control input of diaxon,

It is Coriolis force.

The mathematical model of little gyrotron of formula (1) expression is a kind of dimension form that has, and promptly each physical quantity in the formula not only will be considered numerical values recited, also will take the consistance of each physical quantity unit into account, has therefore increased the complexity of design of Controller.The natural frequency scope of little gyrotron diaxon is generally in the KHz scope, and input angular velocity maybe be only the several years per hour in the scope of several years per second, the time frame difference that both existence are very big is difficult for realizing numerical simulation.In order to solve above two problems, be necessary model is carried out non-dimension processing.The both sides of formula (1) are together divided by reference mass m, reference length q ₀, and the resonant frequency of diaxon square

The non-dimension model that obtains gyroscope is:

$\stackrel{\&CenterDot;\&CenterDot;}{x}+d_{\mathrm{xx}}\stackrel{\&CenterDot;}{x}+d_{\mathrm{xy}}\stackrel{\&CenterDot;}{y}+{w_x}^{2}x+w_{\mathrm{xy}}y=u_x+2{\mathrm{\&Omega;}}_z\stackrel{\&CenterDot;}{y}$ (2)

$\stackrel{\&CenterDot;\&CenterDot;}{y}+d_{\mathrm{xy}}\stackrel{\&CenterDot;}{x}+d_{\mathrm{yy}}\stackrel{\&CenterDot;}{y}+w_{\mathrm{xy}}x+{w_y}^{2}y=u_y-2{\mathrm{\&Omega;}}_z\stackrel{\&CenterDot;}{x}$

Wherein:

$\frac{d_{\mathrm{xx}}}{{\mathrm{<figure-callout\; id="0"\; label="mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">mw</figure-callout>}}_0}\&RightArrow;d_{\mathrm{xx}},\frac{{\mathrm{<figure-callout\; id="0"\; label="d\; xy\; mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{xy}}}{{\mathrm{mw}}_{}}\&RightArrow;d_{\mathrm{xy}},\frac{{\mathrm{<figure-callout\; id="0"\; label="d\; yy\; mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{yy}}}{{\mathrm{mw}}_{}}\&RightArrow;d_{\mathrm{yy}},\frac{k_{\mathrm{xx}}}{{{\mathrm{<figure-callout\; id="0"\; label="mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">mw</figure-callout>}}_0}^2}\&RightArrow;{w_x}^2,\frac{k_{\mathrm{xy}}}{{{\mathrm{<figure-callout\; id="0"\; label="mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">mw</figure-callout>}}_0}^2}\&RightArrow;w_{\mathrm{xy}},\frac{k_{\mathrm{yy}}}{{{\mathrm{<figure-callout\; id="0"\; label="mw"\; filenames="111107144817.png"\; state="\{\{state\}\}">mw</figure-callout>}}_0}^2}\&RightArrow;{{\mathrm{<figure-callout\; id="2"\; label="w\; y"\; filenames="111107144817.png"\; state="\{\{state\}\}">w</figure-callout>}}_y}^2,\frac{{\mathrm{\&Omega;}}_z}{w_0}\&RightArrow;{\mathrm{\&Omega;}}_z$

Because the displacement range of mass is in the submillimeter scope, so the desirable 1 μ m of reasonable reference length; The natural frequency of the diaxon of gyroscope is generally in kilohertz range, so the desirable 1KHz of reference frequency.

Get with vector form rewrite model (2):

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

Wherein $q=\left[\begin{array}{c}x\\ y\end{array}\right],$ $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],$ $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],$ $K=\left[\begin{array}{cc}{{\mathrm{\&omega;}}_x}^2& {\mathrm{\&omega;}}_{\mathrm{xy}}\\ {\mathrm{\&omega;}}_{\mathrm{xy}}& {{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="111107144817.png"\; state="\{\{state\}\}">y</figure-callout>}}}^2\end{array}\right],$ $\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -{\mathrm{\&Omega;}}_z\\ {\mathrm{\&Omega;}}_{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="111107144817.png"\; state="\{\{state\}\}">z</figure-callout>}}& 0\end{array}\right]$

Because the desirable dynamic perfromance of gyroscope is a kind of noenergy loss, two between centers do not have the stable sine-wave oscillation of Dynamic Coupling, can be described below:

$q_m=\left[\begin{array}{c}{x}_m\\ y_m\end{array}\right]=\left[\begin{array}{c}{A}_1\sin \left({\mathrm{\&omega;}}_{1}t\right)\\ A_2\sin \left({\mathrm{\&omega;}}_{2}t\right)\end{array}\right]---\left(4\right)$

So we select desirable dynamic perfromance q _mTrack as a reference.

Controlled target is to seek the appropriate control rule to make reference locus on the gyroscope trajectory track, improves the dynamic perfromance of actual gyroscope.

Two, the adaptive control system of gyroscope

Gyroscope Adaptive Fuzzy Sliding Mode Control system architecture diagram as shown in Figure 2.

The definition tracking error does

e(t)＝q(t)-q _m(t) (5)

The defining integration sliding-mode surface

$s=\stackrel{\&CenterDot;}{q}\left(t\right)-{\&Integral;}_0^t[{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m\left(\mathrm{\&tau;}\right)-k_1\stackrel{\&CenterDot;}{e}\left(\mathrm{\&tau;}\right)-k_{2}e\left(\mathrm{\&tau;}\right)]\mathrm{d\&tau;}---\left(6\right)$

K wherein ₁, k ₂Be the positive constant of non-zero.

If the sliding mode control in the ideal state, then

is

$\stackrel{\&CenterDot;\&CenterDot;}{e}\left(t\right)+k_1\stackrel{\&CenterDot;}{e}\left(t\right)+k_{2}e\left(t\right)=0---\left(7\right)$

Can find out, through suitable definite k ₁, k ₂, tracking error will trend towards zero with exponential form.

Suppose that D, K, Ω are known, then can controlled rate do according to formula (3) (6)

$u^{*}\left(t\right)=(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}\left(t\right)+\mathrm{Kq}\left(t\right)+{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m\left(t\right)-k_1\stackrel{\&CenterDot;}{e}\left(t\right)-k_{2}e\left(t\right)---\left(8\right)$

When D, K, Ω the unknown, u ^*(t) be difficult to be used for practical application, therefore, we can adopt fuzzy system to approach u ^*(t).

Get α _iBe adjustable parameter, the reverse gelatinization is carried out in fuzzy control, then fuzzy controller is output as:

u _fz(s，α)＝α ^Tξ (9)

α=[α wherein ₁, α ₂..., α _m] ^T, ξ=[ξ ₁, ξ ₂..., ξ _m] ^T

ξ _iBe defined as

${\mathrm{\&xi;}}_i=\frac{w_i}{{\mathrm{\&Sigma;}}_{i=1}^m{w}_i}---\left(10\right)$

According to omnipotent approximation theorem, exist the Fuzzy control system of an optimum to approach u ^*(t).

u ^*(t)＝u _fz(s，α ^*)+ε＝α ^*Tξ+ε (11)

Wherein ε is an approximate error, satisfies | ε | and＜E.

Adopt Fuzzy control system to approach u ^*(t)

$u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})={\widehat{\mathrm{\&alpha;}}}^T\mathrm{\&xi;}---\left(12\right)$

Wherein

Be α ^*Estimated value.

Adopt switching controls rule u _Vs(s) compensate u ^*(t) and Between error, then overhead control rule does

$u\left(t\right)=u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})+u_{\mathrm{vs}}\left(s\right)---\left(13\right)$

With formula (13) substitution formula (3),

$\stackrel{\&CenterDot;\&CenterDot;}{q}\left(t\right)=-(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}\left(t\right)-\mathrm{Kq}\left(t\right)+u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})+u_{\mathrm{vs}}\left(s\right)---\left(14\right)$

Get by formula (6), (8) and (14)

$\stackrel{\&CenterDot;}{s}=-(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}\left(t\right)-\mathrm{Kq}+u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})$

$+u_{\mathrm{vs}}\left(s\right)-u^{*}\left(t\right)+(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}\left(t\right)+\mathrm{Kq}\left(t\right)---\left(15\right)$

$=u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})+u_{\mathrm{vs}}\left(s\right)-u^{*}\left(t\right)$

According to formula (11),

${\tilde{u}}_{\mathrm{fz}}={\hat{u}}_{\mathrm{fz}}-u^{*}={\hat{u}}_{\mathrm{fz}}-{u_{\mathrm{fz}}}^{*}-\mathrm{\&epsiv;}---\left(16\right)$

Definition

the formula (16) becomes

${\tilde{u}}_{\mathrm{fz}}={\widetilde{\mathrm{\&alpha;}}}^T\mathrm{\&xi;}-\mathrm{\&epsiv;}---\left(17\right)$

Definition Lyapunov function

$V_1(s\left(t\right),\widetilde{\mathrm{\&alpha;}})=\frac{1}{2}{s}^2\left(t\right)+\frac{1}{2{\mathrm{\&eta;}}_1}{\widetilde{\mathrm{\&alpha;}}}^T\widetilde{\mathrm{\&alpha;}}---\left(18\right)$

η wherein ₁Be positive real number.

The both sides differentiate is got

${\stackrel{\&CenterDot;}{V}}_1(s\left(t\right),\widetilde{\mathrm{\&alpha;}})=s\left(t\right)\stackrel{\&CenterDot;}{s}\left(t\right)+\frac{1}{2{\mathrm{\&eta;}}_1}({\widetilde{\mathrm{\&alpha;}}}^T\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}}+{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}}}^T\widetilde{\mathrm{\&alpha;}})=s\left(t\right)(u_{\mathrm{fz}}(s,\widehat{\mathrm{\&alpha;}})+u_{\mathrm{vs}}\left(s\right)-u^{*}\left(t\right))+\frac{1}{{\mathrm{\&eta;}}_1}{\widetilde{\mathrm{\&alpha;}}}^T\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}}$

$=s\left(t\right)({\widetilde{\mathrm{\&alpha;}}}^T\mathrm{\&xi;}+u_{\mathrm{vs}}\left(s\right)-\mathrm{\&epsiv;})+\frac{1}{{\mathrm{\&eta;}}_1}{\widetilde{\mathrm{\&alpha;}}}^T\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}}={\widetilde{\mathrm{\&alpha;}}}^T(s\left(t\right)\mathrm{\&xi;}+\frac{1}{{\mathrm{\&eta;}}_1}\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}})+s\left(t\right)(u_{\mathrm{vs}}\left(s\right)-\mathrm{\&epsiv;})---\left(19\right)$

Which because <img file = "BDA0000106089750000093.GIF" he = "52" img-content = "drawing" img-format = "tif" inline = "yes" orientation = "portrait" wi = "80" /> is a scalar, so <maths num="0032"> <! [CDATA [<math> <mrow> <msup> <mover> <mi> α </ mi> <mo> ~ < / mo> </ mover> <mi> T </ mi> </ msup> <mover> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> & CenterDot ; </ mo> </ mover> <mo> = </ mo> <msup> <mrow> <mo> (</ mo> <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> <mo>) </ mo> </ mrow> <mi> T </ mi> </ msup> <mo> = </ mo> <msup> <mover> <mover > <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>, </ mo> </ mrow> </ math>]]> </maths> is <maths num =" 0033 "> <! [CDATA [<math> <mrow> <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> <mo > + </ mo> <msup> <mover> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> < mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> = </ mo> <mn> 2 </ mn > <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mover> <mi> & alpha ; </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> </ mrow> </ math>]]> </maths> In order to achieve <img file = "BDA0000106089750000096.GIF" he = "59" img-content = "drawing" img-format = "tif" inline = "yes" orientation = "portrait" wi = "294" /> using the following self- adaptation and switching control law

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}}=-{\mathrm{\&eta;}}_{1}s\left(t\right)\mathrm{\&xi;}---\left(20\right)$

u _vs(s)＝-E(t)sgn(s(t)) (21)

Then

${\stackrel{\&CenterDot;}{V}}_1(s\left(t\right),\widetilde{\mathrm{\&alpha;}})=-E\left|s\left(t\right)\right|-\mathrm{\&epsiv;s}\left(t\right)\&le;-E\left|s\left(t\right)\right|+\left|\mathrm{\&epsiv;}\right|\left|s\left(t\right)\right|$ (22)

$=-(E-|\mathrm{\&epsiv;}\left|\right)\left|s\left(t\right)\right|\&le;0$

In switching controls,,, then can produce big buffeting if the choosing of E (t) value is excessive because handoff gain E (t) is difficult to confirm that it is definite in working control, often to pass through experience; If the choosing of E (t) value is too small, then control system is unstable.

With

instead of E (t), then

$u_{\mathrm{vs}}\left(s\right)=-\hat{E}\left(t\right)\mathrm{sgn}\left(s\left(t\right)\right)---\left(23\right)$

Where

to estimate the switching term gain.

The definition evaluated error

$\tilde{E}\left(t\right)=\hat{E}\left(t\right)-E\left(t\right)---\left(24\right)$

Definition Lyapunov function

$V(s\left(t\right),\widetilde{\mathrm{\&alpha;}},\widetilde{E\left(t\right)})=V_1(s\left(t\right),\widetilde{\mathrm{\&alpha;}})+\frac{1}{2{\mathrm{\&eta;}}_2}{\tilde{E}}^2=\frac{1}{2}{s}^2\left(t\right)+\frac{1}{2{\mathrm{\&eta;}}_1}{\widetilde{\mathrm{\&alpha;}}}^T\widetilde{\mathrm{\&alpha;}}+\frac{1}{2{\mathrm{\&eta;}}_2}{\tilde{E}}^2---\left(25\right)$

Wherein

To the both sides differentiate

$\stackrel{\&CenterDot;}{V}(s\left(t\right),\widetilde{\mathrm{\&alpha;}},\tilde{E}\left(t\right))={\stackrel{\&CenterDot;}{V}}_1(s\left(t\right),\widetilde{\mathrm{\&alpha;}})+\frac{1}{{\mathrm{\&eta;}}_2}\tilde{E}\stackrel{\&CenterDot;}{\tilde{E}}={\widetilde{\mathrm{\&alpha;}}}^T(s\left(t\right)\mathrm{\&xi;}+\frac{1}{{\mathrm{\&eta;}}_1}\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&alpha;}}})+s\left(t\right)(u_{\mathrm{vs}}\left(s\right)-\mathrm{\&epsiv;})+\frac{1}{{\mathrm{\&eta;}}_2}\tilde{E}\stackrel{\&CenterDot;}{\tilde{E}}$

$=-\hat{E}\left(t\right)\left|s\left(t\right)\right|-\mathrm{\&epsiv;s}\left(t\right)+\frac{1}{{\mathrm{\&eta;}}_2}(\hat{E}-E)\stackrel{\&CenterDot;}{\tilde{E}}$

$=-\hat{E}\left(t\right)\left(\right|s\left(t\right)|-\frac{1}{{\mathrm{\&eta;}}_2}\stackrel{\&CenterDot;}{\tilde{E}})-\mathrm{\&epsiv;s}\left(t\right)+\frac{1}{{\mathrm{\&eta;}}_2}E\stackrel{\&CenterDot;}{\tilde{E}}---\left(26\right)$

In order to make $\stackrel{\&CenterDot;}{V}(s\left(t\right),\widetilde{\mathrm{\&alpha;}},\tilde{E}\left(t\right))\&le;0,$ The definition adaptive law

$\stackrel{\&CenterDot;}{\tilde{E}}=\stackrel{\&CenterDot;}{\hat{E}}={\mathrm{\&eta;}}_2\left|s\left(t\right)\right|---\left(27\right)$

Then

$\stackrel{\&CenterDot;}{V}(s\left(t\right),\widetilde{\mathrm{\&alpha;}},\tilde{E}\left(t\right))=-\mathrm{\&epsiv;s}\left(t\right)-E\left|s\left(t\right)\right|\&le;\left|\mathrm{\&epsiv;}\right|\left|s\left(t\right)\right|-E\left|s\left(t\right)\right|$ (28)

$=-(E-|\mathrm{\&epsiv;}\left|\right)\left|s\left(t\right)\right|\&le;0$

Following formula is carried out integration

${\&Integral;}_0^t\stackrel{\&CenterDot;}{V}\left(\mathrm{\&tau;}\right)\mathrm{d\&tau;}=V\left(t\right)-V\left(0\right)\&le;-{\&Integral;}_0^t(E-|\mathrm{\&epsiv;}\left|\right)\left|s\left(\mathrm{\&tau;}\right)\right|\mathrm{d\&tau;}$

Get

$V\left(t\right)+{\&Integral;}_0^t(E-|\mathrm{\&epsiv;}\left|\right)\left|s\left(\mathrm{\&tau;}\right)\right|\mathrm{d\&tau;}\&le;V\left(0\right)$

Because V (0) bounded and V (t) are the non-limited function that increases, therefore have

<math> <mrow> <msub> <mi>lim</mi> <mrow> <mi>t</mi> <mo>&amp;RightArrow;</mo> <mo>&amp;infin;</mo> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </msubsup> <mrow> <mo>(</mo> <mi>E</mi> <mo>-</mo> <mo>|</mo> <mi>&amp;epsiv;</mi> <mo>|</mo> <mo>)</mo> </mrow> <mo>|</mo> <mi>s</mi> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>|</mo> <mi>d&amp;tau;</mi> <mo>&lt;;</mo> <mo>&amp;infin;</mo> </mrow></math>

According to the Barbalat theorem, we can obtain lim _{T → ∞}(E-| ε |) | s (t) |=0, mean lim _{T → ∞}S (t)=0.

According to the algorithm of above fuzzy self-adaption Sliding-Mode Control Based, in MATLAB/Simulink, control system is carried out numerical simulation.Little oscillation gyro instrument parameter of emulation experiment is following:

m＝1.8×10 ^-7kg，k _xx＝63.955N/m，k _yy＝95.92N/m，k _xy＝12.779N/m，

d _xx＝1.8×10 ^-6N·s/m，d _yy＝1.8×10 ^-6N·s/m，d _xy＝3.6×10 ^-7N·s/m

Unknown input angular velocity is assumed to Ω _z=100rad/s.Reference length is chosen for q ₀=1 μ m, reference frequency ω ₀=1000Hz, after the non-dimensionization, each gyroscope parameter is following:

ω _x ²＝355.3，ω _y ²＝532.9，ω _xy＝70.99，d _xx＝0.01，

d _yy＝0.01，d _xy＝0.002，Ω＝0.1

Come obfuscation to subordinate function in the sliding-mode surface employing following 5, promptly

μ _NM(s)＝exp[-((s+π/6)/(π/24)) ²]，μ _NS(s)＝exp[-((s+π/12)/(π/24)) ²]，

μ _ZO(s)＝exp[-(s/(π/24)) ²]，μ _PS(s)＝exp[-((s-π/12)/(π/24)) ²]，

μ _PM(s)=exp[-((s-π/6)/(pi/2 4)) ²], adopt 5 rules to approach μ ^*(t)

The original state of controlled device is got [0000], reference locus $q_m=\left[\begin{array}{c}{x}_m\\ y_m\end{array}\right]=\left[\begin{array}{c}0.2\sin \left(\mathrm{\&pi;t}\right)\\ 0.5\sin \left(\frac{\mathrm{\&pi;}}{2}t\right)\end{array}\right],$ $f\left(t\right)=\left[\begin{array}{c}20\cos \left(\mathrm{\&pi;t}\right)\\ 40\sin \left(\frac{\mathrm{\&pi;}}{4}t\right)\end{array}\right]$ Controller parameter is got η ₁=500, η ₂=0.5, E=100.

Simulation result such as Fig. 3, Fig. 4, Fig. 5, Fig. 6, shown in Figure 7.

The tracking effect curve as shown in Figure 3, the result shows that the X of actual gyroscope, Y-axis track can be good at following the trail of reference model, tracking error between the two changes in time can converge to zero soon, has improved the dynamic perfromance of actual gyroscope.

The sliding-mode surface curve as shown in Figure 4, the result show sliding-mode surface S1, S2 all be in very among a small circle the fluctuation, converge on 0 gradually.Then whole closed-loop system is progressive stable.

Handoff gain is gyrostatic control input u fixedly the time _x, u _yChange curve as shown in Figure 5, the result shows because handoff gain E be difficult to confirm, and is excessive if E value is selected in working control, then can produce big buffeting.In order to reduce buffeting, we have increased the self-adaptation estimation to E, and self-adaptation is regulated the size of E.

Gyrostatic control input u when the handoff gain self-adaptation is regulated _x, u _yChange curve as shown in Figure 6, the result shows owing to increased the self-adaptation of E is estimated, the size of having regulated E automatically, with gain E stationary phase relatively, chattering phenomenon obviously obtains very big improvement.

E switchable gain adaptive estimation curve shown in Figure 7, the results show that the adaptive law because E is

s in the small-scale fluctuations, E always showed a slow upward trend.Have only when s=0, E just can level off to a constant.

Above embodiment only is the present invention's a kind of embodiment wherein, and it describes comparatively concrete and detailed, but can not therefore be interpreted as the restriction to claim of the present invention.Should be pointed out that for the person of ordinary skill of the art under the prerequisite that does not break away from the present invention's design, can also make some distortion and improvement, these all belong to protection scope of the present invention.Therefore, the protection domain of patent of the present invention should be as the criterion with accompanying claims.

Claims (6)

1. be used for the adaptive fuzzy sliding mode controller of gyroscope, it is characterized in that: it comprises based on the self-adaptive fuzzy control system of reference model and handover control system; In self-adaptive fuzzy control system, adopt the input of sliding-mode surface, and adopt dynamic self-adapting rule adjustment weights automatically as fuzzy controller, thus the fuzzy equivalent control rule of approaching.

2. the adaptive fuzzy sliding mode controller that is used for gyroscope according to claim 1 is characterized in that: in said handover control system, adopt switch controller to compensate the error between said equivalent control rule and the said fuzzy controller.

3. the adaptive fuzzy sliding mode controller that is used for gyroscope according to claim 1 is characterized in that: the adaptive algorithm of the parameter of said self-adaptive fuzzy control system adopts the design of Lyapunov method.

4 according to claim 3, wherein the micro-gyroscope for adaptive fuzzy sliding mode controller, wherein: in said adaptive fuzzy control system output when anti-fuzzy adaptive weights defined as <img file = "FDA0000106089740000011.GIF" he = "55" id = "ifm0001" img-content = "drawing" img-format = "tif" inline = "yes" orientation = "portrait" wi = " 165 "/> adaptive algorithm: <img file =" FDA0000106089740000012.GIF " he =" 65 " id =" ifm0002 " img-content =" drawing " img-format =" tif " inline = "yes" orientation = "portrait" wi = "345" /> designing its Lyapunov function: <maths num="0001"> <! [CDATA [<math> <mrow> <msub> <mi> V </ mi> <mn> 1 </ mn> </ msub> <mrow> <mo> (</ mo> <mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, </ mo> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> < mo>) </ mo> </ mrow> <mo> = </ mo> <mfrac> <mn> 1 </ mn> <mn> 2 </ mn> </ mfrac> <msup> <mi> s < / mi> <mn> 2 </ mn> </ msup> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 1 </ mn> </ msub> </ mrow> </ mfrac> <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>, </ mo> </ mrow> </ math>]]> </maths> where η <sub > 1 </sub> is a positive real number; said switching control system, adaptive switching gain is defined as <img file = "FDA0000106089740000014.GIF" he = "60" id = "ifm0004 " img-content =" drawing " img-format =" tif " inline =" yes " orientation =" portrait " wi =" 169 "/> adaptive algorithm: <maths num =" 0002 " > <! [CDATA [<math> <mrow> <mover> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover > <mo> = </ mo> <mover> <mover> <mi> E </ mi> <mo> ^ </ mo> </ mover> <mo> &CenterDot; </ mo> </ mover> <mo > = </ mo> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub> <mo> | </ mo> <mi> s </ mi> <mrow> < mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo> | </ mo> <mo>, </ mo> </ mrow> </ math >]]> </maths> designing its Lyapunov function: <maths num="0003"> <! [CDATA [<math> <mrow> <mi> V </ mi> <mrow> <mo > (</ mo> <mi> s </ mi> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, < / mo> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>, </ mo> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>) </ mo> </ mrow > <mo> = </ mo> <msub> <mi> V </ mi> <mn> 1 </ mn> </ msub> <mrow> <mo> (</ mo> <mi> s </ mi > <mrow> <mo> (</ mo> <mi> t </ mi> <mo>) </ mo> </ mrow> <mo>, </ mo> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn > 2 </ mn> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub> </ mrow> </ mfrac> <msup> <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mn> 2 </ mn> </ msup> <mo> = </ mo> <mfrac> <mn> 1 </ mn> <mn> 2 < / mn> </ mfrac> <msup> <mi> s </ mi> <mn> 2 </ mn> </ msup> <mrow> <mo> (</ mo> <mi> t </ mi> < mo>) </ mo> </ mrow> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η < / mi> <mn> 1 </ mn> </ msub> </ mrow> </ mfrac> <msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mi> T </ mi> </ msup> <mover> <mi> α </ mi> <mo> ~ </ mo> </ mover> <mo> + </ mo> <mfrac> <mn> 1 </ mn> <mrow> <mn> 2 </ mn> <msub> <mi> η </ mi> <mn> 2 </ mn> </ msub> </ mrow> </ mfrac> <msup > <mover> <mi> E </ mi> <mo> ~ </ mo> </ mover> <mn> 2 </ mn> </ msup> <mo>, </ mo> </ mrow> </ math>]]> </maths> where η <sub > 2 </sub> is a positive real number.

5. the adaptive fuzzy sliding mode controller that is used for gyroscope according to claim 1 is characterized in that: said reference model adopts two between centers not have the stable sine-wave oscillation of Dynamic Coupling: x _m=A ₁Sin (w ₁T), y _m=A ₂Sin (w ₂T), its differential equation is:

6. the adaptive fuzzy sliding mode controller that is used for gyroscope according to claim 1 is characterized in that: said sliding-mode surface is the Integral Sliding Mode face: $s=\stackrel{\&CenterDot;}{q}\left(t\right)-{\&Integral;}_0^t[{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m\left(\mathrm{\&tau;}\right)-k_1\stackrel{\&CenterDot;}{e}\left(\mathrm{\&tau;}\right)-k_{2}e\left(\mathrm{\&tau;}\right)]\mathrm{d\&tau;},$ Tracking error e (t)=q (t)-q wherein _m(t), k ₁, k ₂Be the positive constant of non-zero.

Images