CN103345155B - The self-adaptation back stepping control system and method for gyroscope - Google Patents

Abstract

The invention discloses a kind of self-adaptation back stepping control system and method for gyroscope, described control system comprises reference locus module, M signal generation module, self-adaptation back stepping control device, parameter adaptive mechanism module, Attitude rate estimator module, and the gyroscope system with unknown parameter.The diaxon oscillation trajectory that control system of the present invention controls gyroscope follows the tracks of upper given reference locus, estimate the unknown parameter of gyroscope simultaneously, comprise extraneous input angle speed, the adaptive algorithm of control law and parameter designs based on Lyapunov stability theory, ensure that the global stability of control system and the Asymptotic Behavior For Some of tracking error, jamproof robust item is also added in control law, improve the antijamming capability of system, the present invention is that the expansion of gyroscope range of application provides good basis.

Description

The adaptive back stepping control system and method for gyroscope

Technical field

The present invention relates to the adaptive back stepping control system and method for the control system of gyroscope and method, more particularly to gyroscope.

Background technology

Gyroscope（MEMSGyroscope）It is the inertial sensor for being used for sensing angular speed processed using microelectric technique and micro-processing technology.It detects angular speed by the micromechanical component of a vibration being made up of silicon, therefore micro-mechanical gyroscope is very easy to miniaturization and produced in batches, with cost is low and the features such as small volume.In recent years, micro-mechanical gyroscope is nearly paid close attention in many applications, for example, gyroscope coordinates micro-machine acceleration transducer to be used to stablize image, for wireless inertial mouse of computer etc. for inertial navigation, in digital camera.But, due to the influence of inevitable mismachining tolerance and environment temperature during the manufacturing, the difference between original paper characteristic and design can be caused, causes gyroscope to there is parameter uncertainty, it is difficult to set up accurate mathematical modeling.Along with the external disturbance effect in working environment be can not ignore so that the trajectory track control of gyroscope is difficult to, and robustness is relatively low.Traditional control method is based entirely 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, tracking effect is far undesirable, this to have very big use limitation for the controller that single environment is designed.

In terms of the domestic research for gyroscope is concentrated mainly on structure design and manufacturing technology at present, and above-mentioned mechanical compensation technology and drive circuit research, seldom appearance compensates the oscillation trajectory of foozle and control mass with advanced control method, to reach the complete control and the measurement of angular speed 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, which has, is applied to various advanced control methods among the control of gyroscope, typically there is Self Adaptive Control and sliding-mode control.On the one hand these advanced methods compensate for quadrature error caused by fabrication error, on the other hand realize the TRAJECTORY CONTROL to gyroscope.But traditional Self Adaptive Control design process is complex, computationally intensive, and the robustness disturbed to external world is very low, system is become unstable.

As can be seen here, above-mentioned existing gyroscope is using upper, it is clear that has still suffered from inconvenience and defect, and has urgently been further improved.In order to solve existing gyroscope in problem present on use, relevant manufactures there's no one who doesn't or isn't painstakingly seek solution, but for a long time have no that applicable design is developed completion always.

The content of the invention

It is an object of the invention to, the defect for overcoming existing gyroscope control method to exist, particularly in the case where there are the various disturbed conditions such as uncertain model, Parameter Perturbation and outside noise, to improve gyroscope system to the tracking performance of reference locus and estimating the design process of extraneous angular speed and simplified control system, and provide a kind of adaptive back stepping control system and method for gyroscope.

The object of the invention to solve the technical problems realized using following technical scheme, the adaptive back stepping control system of gyroscope, including：

Reference locus module（101）, the reference locus for exporting the shaft vibration of gyroscope two, including position, speed and acceleration signal；

M signal generation module（102）, for receiving the output of reference locus and gyroscope system, and produce the output of the M signal in adaptive back stepping control device design process；

Adaptive back stepping control device（103）, for receiving M signal, and produce adaptive back stepping control device output；

Parameter adaptive mechanism module（104）, for receiving M signal, and produce adaptive back stepping control device parameter more new algorithm；

Attitude rate estimator module（105）, for receiving the output signal of parameter adaptive mechanism module, and produce extraneous Attitude rate estimator value output；

Gyroscope system（106）, the mathematical modeling of controlled device, it is contemplated that the influence of mechanical noise receives adaptive back stepping control device output signal, exports position and the velocity information of gyroscope vibrating mass.

The control method of adaptive back stepping control system based on gyroscope, comprises the following steps,

1）Based on Based Inverse Design Method, the dimensionless kinetic model of gyroscope is set up；

2）Design adaptive back stepping control device；

3）Design parameter adaptive algorithm；

4）Attitude rate estimator module（105）Based on the step 3）Adaptive algorithm calculate input angle speed real-time estimate.

Foregoing step 1）, the dimensionless kinetic model of gyroscope is set up, is specially：

1-1）Consider the presence of mechanical noise, the dimensionless vector form of the kinetics equation of two axle gyroscopes is：

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

In formula, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the oscillation trajectory of gyroscope, $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ The control input of gyroscope, $d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right],$ $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],$ $K=\left[\begin{array}{cc}{k}_{\mathrm{xx}}& k_{\mathrm{xy}}\\ k_{\mathrm{xy}}& k_{\mathrm{yy}}\end{array}\right],$ $\mathrm{\Ω}=\left[\begin{array}{cc}0& -{\mathrm{\Ω}}_z\\ {\mathrm{\Ω}}_z& 0\end{array}\right]$

X, y represent that the vibrating mass of gyroscope is vibrating the position of two axles respectively；d_xx,d_xy,d_yyFor the internal damping coefficient of gyroscope；k_xx,k_xy,k_yyFor the internal resilience coefficient of gyroscope；Ω_zIt is the input angle speed of gyroscope；d_x,d_yRepresent the internal mechanical noise of gyroscope system；

1-2）According to back-stepping design technology, by the kinetic simulation pattern of gyroscope（2）Following form is transformed to,

$\left\{\begin{array}{c}{\stackrel{\·}{X}}_1=X_2\\ {\stackrel{\·}{X}}_2=-(D+2\mathrm{\Ω})X_2-K{X}_1+u+d\end{array}\right.---\left(4\right)$

In formula, X₁=q,

Foregoing step 2）, adaptive back stepping control device is designed, is specially：

2-1）Centre generation signaling module（102）, receive reference locus module（101）The position signalling Y of the reference locus of output_d, rate signalAcceleration signalWith gyroscope system（107）Position q and speedOutput signal, generates the M signal in adaptive back stepping control device design process

The M signal is designed as：Tracking error e₁, e₁=q-Y_d

Virtual controlling amount α₁,Wherein c₁Represent any symmetric positive definite matrix deviation e₂, e₂=X₂-α₁

2-2）Adaptive back stepping control device（103）Receive M signalThe control signal u of gyroscope is produced, control signal is designed as,

$u=-c_2{e}_2-e_1+\hat{D}(e_2+{\mathrm{\α}}_1)+\hat{K}(e_1+Y_d)+\widehat{\mathrm{\Ω}}({2e}_2+2{\mathrm{\α}}_1)+{\stackrel{\·}{\mathrm{\α}}}_1-\mathrm{\ρsgn}\left(e_2\right)---\left(17\right)$

In formula, c₂For any symmetric positive definite matrix；Respectively gyroscope kinetic simulation pattern（2）In three parameters D, K, Ω estimate, ρ for gyroscope system internal mechanical noise upper bound ρ >=| | d | |, sgn () represent sign function；

Foregoing step 3）, design parameter adaptive algorithm, specially：Based on Lyapunov Theory of Stability design parameter adaptation mechanism modules（104）, make the more new algorithm of its output adaptive back stepping control device parameter,

Lyapunov functions V is elected as：

$V=\frac{1}{2}{e_1}^T{e}_1+\frac{1}{2}{e_2}^T{e}_2+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_D}^{-1}\tilde{D}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_K}^{-1}\tilde{K}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_{\mathrm{\Ω}}}^{-1}\widetilde{\mathrm{\Ω}}{\widetilde{\mathrm{\Ω}}}^T\right\}$

In formula, γ_D,γ_K,γ_ΩFor adaptive gain, the mark of tr () representing matrix, Represent parameter estimating error；

Adaptive algorithmRespectively：

${\stackrel{\stackrel{\·}{^}}{D}}^T=-\frac{1}{2}{\mathrm{\γ}}_D[(e_2+{\mathrm{\α}}_1){e_2}^T+e_2{(e_2+{\mathrm{\α}}_1)}^T]$

${\stackrel{\stackrel{\·}{^}}{K}}^T=-\frac{1}{2}{\mathrm{\γ}}_K[(e_1+Y_d){e_2}^T+e_2{(e_1+Y_d)}^T]$

${\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}}^T={\mathrm{\γ}}_{\mathrm{\Ω}}[e_2{(e_2+{\mathrm{\α}}_1)}^T-(e_2+{\mathrm{\α}}_1){e_2}^T]$

According to parameter adaptive mechanism module（104）Output, online real-time update.

Foregoing step 4）, input angle speed Ω_zReal-time estimateFor

${\widehat{\mathrm{\Ω}}}_z\left(t\right)={\widehat{\mathrm{\Ω}}}_z\left(0\right)+{\∫}_0^t\frac{1}{2}[\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}\left(\mathrm{2,1}\right)-\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}\left(\mathrm{1,2}\right)]$

In formula,Represent Ω_zEstimation initial value,Difference representing matrixThe first row two row and the second row one arrange element.

Compared with prior art, advantage is the present invention：

（1）The control method that Based Inverse Design Method is combined with adaptive technique is employed, the design process of micro-gyroscope control system is both simplified, the new approaches of micro-gyroscope control system design is opened, the performance and robustness of control system is improved again；

（2）The parameter of controller uses adaptive algorithm, can on-line control parameter control system, adaptive algorithm based on Lyapunov Theory of Stability design, it is ensured that the global stability of closed-loop system；

（3）Control to gyroscope need not be set up on the basis of object Accurate Model, save the expense of modeling.

Brief description of the drawings

Fig. 1 is principle assumption diagram of the invention；

Fig. 2 is the tracking curves of the gyroscope drive shaft based on the present invention；

Fig. 3 is the tracking curves of the gyroscope sensitive axis based on the present invention；

Fig. 4 for the present invention in gyroscope drive shaft control input curve；

Fig. 5 for the present invention in gyroscope sensitive axis control input curve；

Fig. 6 is the control system in the present invention to the estimation curve of gyroscope parameter and extraneous input angular velocity estimation curve.

Embodiment

Further to illustrate that the present invention, to reach the technological means and effect that predetermined goal of the invention is taken, below in conjunction with accompanying drawing and preferred embodiment, the adaptive back stepping control system and method according to gyroscope proposed by the present invention is described in detail as rear.

As shown in figure 1, the adaptive back stepping control system of gyroscope, including：

Reference locus module 101, the reference locus for exporting the shaft vibration of gyroscope two, including position, speed and acceleration signal；

M signal generation module 102, for receiving the output of reference locus and gyroscope system, and produces the output of the M signal in adaptive back stepping control device design process；

Adaptive back stepping control device 103, for receiving M signal, and produces adaptive back stepping control device output；

Parameter adaptive mechanism module 104, for receiving M signal, and produces adaptive back stepping control device parameter more new algorithm；

Attitude rate estimator module 105, for receiving the output signal of parameter adaptive mechanism module, and produces extraneous Attitude rate estimator value output；

Gyroscope system 106, controlled device mathematical modeling, it is contemplated that the influence of mechanical noise, receive adaptive back stepping control device output signal, and produce position and the rate signal output of oscillation trajectory.

The control method of the adaptive back stepping control system of gyroscope, comprises the following steps,

（1）Based on Based Inverse Design Method, the dimensionless kinetic model of gyroscope is set up

Acted in view of foozle and external interference, the kinetics equation of two axle micro-mechanical gyroscopes is：

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

In formula, m is the quality of vibrating machine part；X, y are respectively vibrating mass in the position of drive shaft and sensitive axis；d_xx,d_xy,d_yyFor the damped coefficient of gyroscope, k_xx,k_xy,k_yyFor the spring ratio of gyroscope；Ω_zIt is the angular speed in gyroscope working environment；u_x,u_yIt is control input；d_x,d_yIt is mechanical noise.Handled by nondimensionalization, the model of gyroscope is written as vector form,

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

In formula, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the oscillation trajectory of gyroscope, $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],$ $d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right],$ $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right],$ $K=\left[\begin{array}{cc}{k}_{\mathrm{xx}}& k_{\mathrm{xy}}\\ k_{\mathrm{xy}}& k_{\mathrm{yy}}\end{array}\right],$ $\mathrm{\Ω}=\left[\begin{array}{cc}0& -{\mathrm{\Ω}}_z\\ {\mathrm{\Ω}}_z& 0\end{array}\right],$ It can be reasonably assumed that mechanical noise d (t) boundeds, the upper bound is ρ, i.e., | | d (t) | |≤ρ.

Based on back-stepping design technology, equivalent transformation is carried out to the model of gyroscope first.Defined variable,

$X_1=q,X_2=\stackrel{\·}{q}---\left(3\right)$

Based on variable X₁,X₂, the kinetic simulation pattern of gyroscope（2）It is rewritten as

$\left\{\begin{array}{c}{\stackrel{\·}{X}}_1=X_2\\ {\stackrel{\·}{X}}_2=-(D+2\mathrm{\Ω})X_2-{\mathrm{KX}}_1+u+d\end{array}\right.---\left(4\right)$

（2）Design adaptive back stepping control device

The target of micro-gyroscope control system is so that the oscillation trajectory X of gyroscope system₁Reference locus is given in tracking, defining tracking error is：

e₁=X₁-Y_d（5）

Y_dFor reference locus.

Based on Based Inverse Design Method, start to design the adaptive back stepping control device of gyroscope below.

Step one:Generation signaling module 102 in the middle of design, makes it produce the M signal in adaptive back stepping control device design process

Tracking error e₁, e₁=X₁-Y_d

To tracking error e₁Derivation：

${\stackrel{\·}{e}}_1={\stackrel{\·}{X}}_1-{\stackrel{\·}{Y}}_d=X_2-{\stackrel{\·}{Y}}_d---\left(6\right)$

Based on formula（6）, virtual controlling amount α can be designed₁：

${\mathrm{\α}}_1=-c_1{e}_1+{\stackrel{\·}{Y}}_d---\left(7\right)$

In formula, c₁Represent any symmetric positive definite matrix, c₁=c₁ ^T＞ 0.

To tracking error systematic（6）Choose a Lyapunov functions V₁For：

$V_1=\frac{1}{2}{e_1}^T{e}_1---\left(8\right)$

V₁To time derivation, ${\stackrel{\·}{V}}_1={e_1}^T{\stackrel{\·}{e}}_1={e_1}^T(X_2-{\stackrel{\·}{Y}}_d)---\left(9\right)$

If X₂=α₁, then ${\stackrel{\·}{V}}_1=-{e_1}^T{c}_1{e}_1$

It is apparent fromMeet negative definiteness, therefore tracking error systematic（6）Asymptotically stable in the large, e₁Asymptotic convergence is to zero.

But, X₂It is not control input, is also a system variable, is not and α₁Moment is equal.

Design M signal, deviation e₂For：

e₂=X₂-α₁（10）

To e₂Carry out derivation：

${\stackrel{\·}{e}}_2={\stackrel{\·}{X}}_2-{\stackrel{\·}{\mathrm{\α}}}_1$

$=-(D+2\mathrm{\Ω})X_2-K{X}_1+u+d-{\stackrel{\·}{\mathrm{\α}}}_1$ （11）

$=-(D+2\mathrm{\Ω})(e_2+{\mathrm{\α}}_1)-K(e_1+Y_d)+u-{\stackrel{\·}{\mathrm{\α}}}_1+d$

$=-(D+2\mathrm{\Ω})e_2-{\mathrm{Ke}}_1-K{Y}_d-(D+2\mathrm{\Ω}){\mathrm{\α}}_1-{\stackrel{\·}{\mathrm{\α}}}_1+u+d$

Formula（11）In occur in that control input u.

Step 2:Adaptive back stepping control device 103 is designed, it is produced control signal, as the control input of gyroscope,

To formula（11）Choose Lyapunov functions V₂For：

$V_2=V_1+\frac{1}{2}{e_2}^T{e}_2---\left(12\right)$

It is rightThe derivation of progress time,

${\stackrel{\·}{V}}_2={e_1}^T(X_2-{\stackrel{\·}{Y}}_d)+{e_2}^T{\stackrel{\·}{e}}_2$

$={e_1}^T(-c_1{e}_1+e_2)+{e_2}^T[-(D+2\mathrm{\Ω})e_2-{\mathrm{Ke}}_1-{\mathrm{KY}}_d-(D+2\mathrm{\Ω}){\mathrm{\α}}_1-{\stackrel{\·}{\mathrm{\α}}}_1+u+d]---\left(13\right)$

$=-{e_1}^T{c}_1{e}_1+{e_1}^T{e}_2+{e_2}^T[-(D+2\mathrm{\Ω})e_2-{\mathrm{Ke}}_1-{\mathrm{KY}}_d-(D+2\mathrm{\Ω}){\mathrm{\α}}_1-{\stackrel{\·}{\mathrm{\α}}}_1+u+d]$

To ensureDesign control law u, as the control input of gyroscope,

$u=-c_2{e}_2-e_1+(D+2\mathrm{\Ω})e_2+K(e_1+Y_d)+(D+2\mathrm{\Ω}){\mathrm{\α}}_1+{\stackrel{\·}{\mathrm{\α}}}_1+v---\left(14\right)$

In formula, c₂=c₂ ^T＞ 0, v are robust, for compensating mechanical noise d influence,

V=- ρ sgn (e₂)（15）

By formula（14）、（15）Bring formula into（13）Obtain,

${\stackrel{\·}{V}}_2=-{e_1}^T{c}_1{e}_1-{e_2}^T{c}_2{e}_2+{e_2}^{T}d-\mathrm{\ρ}{e_2}^T\mathrm{sgn}\left(e_2\right)\≤0---\left(16\right)$

Formula（16）IndicateNegative definiteness, i.e. control system stability.

However, three parameters D, K, Ω of gyroscope are unknown, or can not accurately know, formula（14）Shown control law can not directly be implemented, it is necessary to improve.Thought based on self-adaptation control method, the respective estimates of present invention D, K, ΩThe adaptive algorithm of these three parameters is designed instead of unknown true value, and in invention further part.Then, improved adaptive control laws u is,

$u=-c_2{e}_2-e_1+\hat{D}(e_2+{\mathrm{\α}}_1)+\hat{K}(e_1+Y_d)+\widehat{\mathrm{\Ω}}({2e}_2+{2\mathrm{\α}}_1)+{\stackrel{\·}{\mathrm{\α}}}_1-\mathrm{\ρsgn}\left(e_2\right)---\left(17\right)$

Equation（17）The control input of the output signal, as gyroscope of as adaptive back stepping control device 103, obtained closed-loop system equation is：

$\left\{\begin{array}{c}{\stackrel{\·}{e}}_1=e_2+{\mathrm{\α}}_1-{\stackrel{\·}{Y}}_d\\ {\stackrel{\·}{e}}_2=[-c_2{e}_2-e_1+d-\mathrm{\ρsgn}\left(e_2\right)]+\tilde{D}(e_2+{\mathrm{\α}}_1)+\tilde{K}(e_1+Y_d)+\widetilde{\mathrm{\Ω}}({2e}_2+2{\mathrm{\α}}_1)\end{array}\right.---\left(18\right)$

In formula, $\tilde{D}=\hat{D}-D,\tilde{K}=\hat{K}-K,\widetilde{\mathrm{\Ω}}=\widehat{\mathrm{\Ω}}-\mathrm{\Ω},$ Represent parameter estimating error；

（3）Design the adaptive algorithm of controller parameter

Based on Lyapunov Theory of Stability design parameter adaptation mechanisms module 104, make the more new algorithm of its output adaptive back stepping control device parameter, first to formula（18）Closed-loop system, choose a Lyapunov functions V be：

$V=\frac{1}{2}{e_1}^T{e}_1+\frac{1}{2}{e_2}^T{e}_2+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_D}^{-1}\tilde{D}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_K}^{-1}\tilde{K}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{{{\mathrm{\γ}}_{\mathrm{\Ω}}}^{-1}\widetilde{\mathrm{\Ω}}{\widetilde{\mathrm{\Ω}}}^T\right\}---\left(19\right)$

In formula, γ_D,γ_K,γ_ΩFor adaptive gain, the mark of tr () representing matrix.

The derivation of time is carried out to V：

$\stackrel{\·}{V}=[-{e_1}^T{c}_1{e}_1-{e_2}^T{c}_2{e}_2+{e_2}^{T}d-\mathrm{\ρ}{e_2}^T\mathrm{sgn}\left(e_2\right)]$

$+{e_2}^T[\tilde{D}(e_2+{\mathrm{\α}}_1)+\tilde{K}(e_1+Y_d)+\widetilde{\mathrm{\Ω}}(2e_2+2{\mathrm{\α}}_1)]---\left(20\right)$

$+\mathrm{tr}\left\{{{\mathrm{\γ}}_D}^{-1}\tilde{D}{\stackrel{\stackrel{\·}{~}}{D}}^T\right\}+\mathrm{tr}\left\{{{\mathrm{\γ}}_K}^{-1}\tilde{K}{\stackrel{\stackrel{\·}{~}}{K}}^T\right\}+\mathrm{tr}\left\{{{\mathrm{\γ}}_{\mathrm{\Ω}}}^{-1}\widetilde{\mathrm{\Ω}}{\stackrel{\stackrel{\·}{~}}{\mathrm{\Ω}}}^T\right\}$

To ensure that the closed-loop system under adaptive controller can be stablized, design is neededAdaptive algorithm so that

Adaptive algorithm design is as follows：

${\stackrel{\stackrel{\·}{^}}{D}}^T=-\frac{1}{2}{\mathrm{\γ}}_D[(e_2+{\mathrm{\α}}_1){e_2}^T+e_2{(e_2+{\mathrm{\α}}_1)}^T]$

${\stackrel{\stackrel{\·}{^}}{K}}^T=-\frac{1}{2}{\mathrm{\γ}}_K[(e_1+Y_d){e_2}^T+e_2{(e_1+Y_d)}^T]---\left(21\right)$

${\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}}^T={\mathrm{\γ}}_{\mathrm{\Ω}}[e_2{(e_2+{\mathrm{\α}}_1)}^T-(e_2+{\mathrm{\α}}_1){e_2}^T]$

Obtained according to the adaptive algorithm of design,

$\stackrel{\·}{V}=[-{e_1}^T{c}_1{e}_1-{e_2}^T{c}_2{e}_2+{e_2}^{T}d-\mathrm{\ρ}{e_2}^T\mathrm{sgn}\left(e_2\right)]\≤0---\left(22\right)$

ParameterAccording to the adaptive algorithm of design, online real-time update.

In summary, the adaptive back stepping control device formula that the present invention is designed（17）With the adaptive algorithm formula of controller parameter（21）It ensure that the stability of gyroscope system.Because V meets radially unbounded, therefore closed-loop system is globally asymptotically stable.

Although parameter adaptive mechanism module 104 has obtained the adaptive algorithm expression formula of each parameter, but it is the change rule of parameter, the instantaneous value for not illustrating parameter is how many, and it is exactly angular speed that gyroscope, which joins a parameter that is most important, needing most estimation, so design Attitude rate estimator module 105, it is received the output signal of parameter adaptive mechanism module 104, and produce the real-time estimate output of input angle speed

BecauseThe element that wherein Ω (1,2), Ω (2,1) difference representing matrix the Ω row of the first row two and the second row one are arranged, so Attitude rate estimator value module 105 is output as：

${\widehat{\mathrm{\Ω}}}_z\left(t\right)={\widehat{\mathrm{\Ω}}}_z\left(0\right)+{\∫}_0^t\frac{1}{2}[\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}\left(\mathrm{2,1}\right)-\stackrel{\stackrel{\·}{^}}{\mathrm{\Ω}}\left(\mathrm{1,2}\right)]---\left(23\right)$

In formula,Represent Ω_zEstimation initial value.

Finally, Computer Simulation is carried out

In the present embodiment, computer simulation experiment is carried out using mathematical software MatLab/Simulink, the parameter for choosing gyroscope is：

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

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

It is assumed that extraneous angular speed is Ω_zThree parameter matrixs of gyroscope after=100rad/s, nondimensionalization are：

$D=\left[\begin{array}{cc}0.01& 0.002\\ 0.002& 0.01\end{array}\right],$ $K=\left[\begin{array}{cc}355.3& 70.99\\ 70.99& 532.9\end{array}\right],$ $\mathrm{\Ω}=\left[\begin{array}{cc}0& -0.1\\ 0.1& 0\end{array}\right]$

The estimation initial value of three control parameter matrixes of adaptive back stepping control device is taken as respectively：

$\hat{D}\left(0\right)=0.9*D,\hat{K}\left(0\right)=0.9*K,\widehat{\mathrm{\Ω}}\left(0\right)=\mathrm{zeros}\left(\mathrm{2,2}\right)$

Reference locus is designed as：x_d=cos (ω₁t),y_d=cos (ω₂T), wherein ω₁=6.17, ω₂=5.11

Adaptive back stepping control device parameter c₁=c₂=20*I, I represent second order unit matrix.

The gain gamma of three adaptive algorithms_D=γ_K=γ_Ω=20,

Gyroscope is zero original state, and mechanical noise d is put aside, is set as zero, under above control parameter, runs simulated program, obtains the result figure of the specific embodiment of the invention.

Reference picture 2 and Fig. 3, wherein Fig. 2 are drive axle position aircraft pursuit course, and Fig. 3 is sensitive axis position tracking curve.In two figures, solid line is the output of gyroscope, and dotted line is to refer to oscillation trajectory, control system enables to two shaft positions of gyroscope to export, in the case where not knowing gyroscope parameter, given reference locus can be promptly tracked, the requirement of high-precision tracking has been reached.

Reference picture 4 and Fig. 5, two figures illustrate the curve of output of adaptive back stepping control device, and Fig. 4 is the control input curve of gyroscope drive shaft, and Fig. 5 is the control input curve of gyroscope sensitive axis.

Reference picture 6 is control system to the estimation curve of gyroscope parameter and extraneous input angular velocity estimation curve.Under upper, respectively K (1,1), K (1,2), K (2,2), Ω from left to right_zEstimation curve.From result figure as can be seen that the adaptive back stepping control system of the gyroscope of the present invention can correctly estimate unknown gyroscope parameter, and extraneous angular speed.

It can be seen that from above analogous diagram, control method proposed by the present invention has good control effect to the track following of gyroscope, the tracking performance and robustness of gyroscope system are substantially increased, the high-precision control to the shaft vibration track of gyroscope two provides theoretical foundation and Math.Adaptive control system can estimate unknown gyroscope parameter, including extraneous angular speed simultaneously, add the practicality of control system.

The content not being described in detail in description of the invention belongs to technological know-how known to professional and technical personnel in the field.

It is described above; only it is presently preferred embodiments of the present invention; but it is not limited to the present invention; any those skilled in the art; without departing from the scope of the present invention; when the technology contents using the disclosure above make a little change or are modified to the equivalent embodiment of equivalent variations, as long as being the content without departing from technical solution of the present invention, the protection domain of our bright technical scheme is still fallen within.

Claims (1)

1. the control method of the adaptive back stepping control system based on gyroscope, it is characterised in that：Comprise the following steps,

1) the adaptive back stepping control system of gyroscope is set up, including：

Reference locus module (101), the reference locus for exporting the shaft vibration of gyroscope two, including position, speed and acceleration signal；

M signal generation module (102), for receiving the output of reference locus and gyroscope system, and produces the output of the M signal in adaptive back stepping control device design process；

Adaptive back stepping control device (103), for receiving M signal, and produces adaptive back stepping control device output；

Parameter adaptive mechanism module (104), for receiving M signal, and produces adaptive back stepping control device parameter more new algorithm；

Attitude rate estimator module (105), for receiving the output signal of parameter adaptive mechanism module, and produces extraneous Attitude rate estimator value output；

Gyroscope system (106), the mathematical modeling of controlled device, it is contemplated that the influence of mechanical noise, receive adaptive back stepping control device output signal, export position and the velocity information of gyroscope vibrating mass；

2) Based Inverse Design Method is based on, the dimensionless kinetic model of gyroscope is set up；Specially：

2-1) consider the presence of mechanical noise, the dimensionless vector form of the kinetics equation of two axle gyroscopes is：

In formula,For the oscillation trajectory of gyroscope,The control input of gyroscope,

X, y represent that the mass of gyroscope is vibrating the position of two axles respectively；d_xx,d_xy,d_yyFor the internal damping coefficient of gyroscope；k_xx,k_xy,k_yyFor the internal resilience coefficient of gyroscope；Ω_zIt is the input angle speed of gyroscope；d_x,d_yRepresent the internal mechanical noise of gyroscope system；

2-2) according to back-stepping design technology, the dimensionless vector form formula (2) of the kinetics equation of gyroscope is transformed to following form,

In formula,

3) adaptive back stepping control device is designed；Specially：

Generation signaling module (102) in the middle of 3-1), receives the position signalling Y of the reference locus of reference locus module (101) output_d, rate signalAcceleration signalWith the position q and speed of gyroscope system (107)Output signal, generates the M signal e in adaptive back stepping control device design process₁,α₁,e₂,

The M signal is designed as：Tracking error e₁, e₁=q-Y_d

Virtual controlling amount α₁,Wherein c₁Represent any symmetric positive definite matrix

Deviation e₂, e₂=X₂-α₁

3-2) adaptive back stepping control device (103) receives M signal e₁,α₁,e₂,The control signal u of gyroscope is produced, control signal is designed as,

In formula, c₂For any symmetric positive definite matrix；Three parameters D, K, Ω estimate respectively in gyroscope kinetic simulation pattern (4), ρ are the upper bound of the internal mechanical noise of gyroscope system, sgn () expression sign functions

4) design parameter adaptive algorithm；Specially：Based on Lyapunov Theory of Stability design parameter adaptation mechanism modules (104), make the more new algorithm of its output adaptive back stepping control device parameter,

Lyapunov functions V is elected as：

In formula, γ_D,γ_K,γ_ΩFor adaptive gain, the mark of tr () representing matrix, Represent parameter estimating error；

Adaptive algorithmRespectively：

According to the output of parameter adaptive mechanism module (104), online real-time update；

5) Attitude rate estimator module (105) be based on the step 4) adaptive algorithm calculate input angle speed real-time estimate, input angle speed Ω_zReal-time estimateFor

In formula,Represent Ω_zEstimation initial value,Difference representing matrixThe first row two row and the second row one arrange element.