CN103616818B - The neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope - Google Patents

Abstract

The invention discloses a kind of neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope, the invention reside in and overall fast terminal sliding formwork is controlled to combine with adaptive control, according to Liapunov stability method design overall situation fast terminal sliding formwork control law, enable system state at very short Finite-time convergence to equilibrium point, utilize adaptive control to pick out angular velocity and other systematic parameter of gyroscope simultaneously, further consider the situation of the upper bound the unknown when model error and external interference, adaptive learning is carried out in the upper bound of the function utilizing fuzzy neural network to learn to the indeterminate of gyroscope system and external interference, achieve to modeling error and uncertain noises from motion tracking.The present invention is while guarantee speed of convergence and tracking performance, and interference has stronger robustness and adaptive ability to external world.

Description

The neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope

Technical field

The present invention relates to the control system of gyroscope, specifically the neural overall fast terminal sliding-mode control of a kind of adaptive fuzzy of gyroscope.

Background technology

Gyroscope is the fundamental measurement element of inertial navigation and inertial guidance system, and because of its huge advantage in volume and cost, gyroscope is widely used in Aeronautics and Astronautics, automobile, biomedicine, military affairs and consumer electronics field.But, because the error in design and manufaction exists and thermal perturbation, the difference between original paper characteristic and design can be caused, reduce the performance of gyroscope system.In addition, gyroscope itself belongs to multi-input multi-output system and systematic parameter exists uncertain and is subject to the impact of external environment.Compensation foozle and measured angular speed become the subject matter that gyroscope controls, and are necessary to carry out dynamic compensation and adjustment to gyroscope system.And in the stability contorting that traditional control method concentrates on driving shaft oscillation amplitude and frequency and diaxon frequency matching, the defect of gyroscope dynamic equation can not be solved well.

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.Adaptive control be when the model knowledge of controlled device or environmental knowledge know entirely even do not know little about it, enable system automatically work in optimum or the running status close to optimum, provide the control performance of high-quality.But the robustness of adaptive control disturbance is to external world very low, system is easily made to become unstable.Sliding mode variable structure control be the special nonlinear Control of a class in essence, its non-linear behavior is the uncontinuity controlled, this control strategy and other difference controlled are that the structure of system is not fixed, but on purpose constantly can change according to the current state of system according to system in dynamic process, force system to be moved according to the state trajectory of predetermined sliding mode.The shortcoming of the method is, after state trajectory arrives sliding-mode surface, to be difficult to strictly slide towards equilibrium point along sliding-mode surface, but to pass through back and forth in sliding-mode surface both sides, thus produces vibration.

Summary of the invention

The micro-gyrotron trajectory track that the present invention is directed to containing modeling error and uncertain noises controls, propose a kind of neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope, based on the estimated value of Liapunov stability method design gyroscope parameter matrix and the adaptive algorithm of fuzzy neural network weights, guarantee the global stability of whole control system, improve the reliability of system and the robustness to Parameters variation.

The technical solution used in the present invention is:

The neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope, comprises the following steps;

1) mathematical model building gyroscope system is:

$\stackrel{\·\·}{q}+(D+2\mathrm{\Ω})\stackrel{\·}{q}+\mathrm{Kq}=u+f---\left(3\right)$

Wherein, q is the position vector of mass at driving shaft and sensitive axis diaxon of gyroscope, is the output of gyroscope system; U is the control inputs of gyroscope; D is damping matrix; K contains the natural frequency of diaxon and the stiffness coefficient of coupling; Ω is angular speed matrix; F is parameter uncertainty and the external disturbance of system;

2) building overall fast terminal sliding-mode surface s is:

$s=\stackrel{\·}{e}+\mathrm{\αe}+{\mathrm{\βe}}^{p_2/p_1}---\left(6\right)$

Wherein, α=diag (α ₁, α ₂), β=diag (β ₁, β ₂) be sliding-mode surface constant; E=q-q _rfor tracking error; q _rfor mass is along the ideal position output vector of diaxon; Q is two shaft position output vectors of gyroscope; p ₁, p ₂(p ₁> p ₂) be positive odd number;

3) the neural overall fast terminal sliding mode controller of adaptive fuzzy is built:

3-1) for described gyroscope system, adopt the sliding-mode surface of formula (6), overall fast terminal sliding formwork control law U is made up of three control laws:

U＝u ₀+u ₁+u ₂(7)

Wherein,

u ₀＝a+(D+2Ω)v+Kq，

D, K, Ω are three parameter matrixs of gyroscope,

$u_1=-W\frac{s}{\left|\right|s\left|\right|},W=\mathrm{diag}(w_1,w_2);$

W is sliding mode controller parameter;

for the parameter uncertainty of system and the upper bound of external disturbance f;

3-2) due to three parameter matrix D of gyroscope, K, Ω are unknown, according to Adaptive Control Theory, use estimated value alternate parameter matrix D, K, Ω, and the adaptive algorithm designing three estimated values, online real-time update estimated value, then control law u ₀be adjusted to u' ₀:

$u_0^{\′}=a+(\hat{D}+2\widehat{\mathrm{\Ω}})v+\hat{K}q;$

3-3) according to Fuzzy Neural Network Theory, fuzzy neural network is adopted to approach the parameter uncertainty of system and upper bound ρ (t) of external disturbance f, and design the adaptive algorithm of fuzzy neural network weights, the output of online real-time update fuzzy neural network, the output of fuzzy neural network for:

Wherein, the weights of fuzzy neural network, the normalization confidence level that φ (X) is fuzzy neural network, $X={\left[\begin{array}{cc}q& \stackrel{\·}{q}\end{array}\right]}^T$ For the input of fuzzy neural network,

Then control law u ₂be adjusted to u' ₂:

$u_2^{\′}=-\widehat{\mathrm{\ρ}}\left(t\right)\frac{s}{\left|\right|s\left|\right|};$

3-4) with described step 3-2) and step 3-3) adjustment after control law u' ₀and u' ₂, replace step 3-1) in control law u ₀and u ₂, obtain the control law U'' of the neural overall fast terminal sliding mode controller of adaptive fuzzy

U''＝u' ₀+u ₁+u' ₂；

3-5) using the control law U'' of neural for adaptive fuzzy overall fast terminal sliding mode controller as the control inputs of gyroscope system, bring in the mathematical model of gyroscope system, realize the tracing control to gyroscope system.

Aforesaid gyroscope parameter matrix D, the adaptive algorithm of the estimated value of K, Ω and the adaptive algorithm of fuzzy neural network weights design based on Lyapunov stability theory:

Lyapunov function is:

$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\widetilde{\mathrm{\Ω}}{P}^{-1}{\widetilde{\mathrm{\Ω}}}^T\right\}+\frac{1}{2}{\mathrm{\η}}^{-1}{\tilde{w}}^T\tilde{w}$

Wherein, the mark of tr () representing matrix; M, N, P are adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix, and η is fuzzy neural network learning rate, the evaluated error of neural network weight, be respectively parameter matrix D, the parameter estimating error of K, Ω,

In order to ensure the derivative of Lyapunov function choose gyroscope parameter matrix D, the adaptive algorithm of the estimated value of K, Ω is:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{D}}}^T=-\frac{1}{2}M(\stackrel{\·}{q}{s}^T+s{\stackrel{\·}{q}}^T)\\ {\stackrel{\·}{\hat{K}}}^T=-\frac{1}{2}N({\mathrm{qs}}^T+{\mathrm{sq}}^T)\\ {\stackrel{\·}{\widehat{\mathrm{\Ω}}}}^T=-\frac{1}{2}P(2\stackrel{\·}{q}{s}^T-2s{\stackrel{\·}{q}}^T)\end{array}\right.$

The adaptive algorithm of fuzzy neural network weights is:

Compared with prior art, beneficial effect of the present invention is embodied in: first, the proposition that overall situation fast terminal sliding formwork controls solves the optimal problem of convergence time, it combines traditional sliding formwork and controls the advantage with TSM control in the process of sliding mode design, also use the concept arrived fast simultaneously in the arrival stage, guarantee tracking error at shorter Finite-time convergence to zero; Secondly, when all parameters of gyroscope and angular speed all regard unknown variable as, for control and the parameter measurement problem of gyroscope, devise a kind of novel Adaptive Identification method, the online angular velocity of real-time update gyroscope and the estimated value of other systematic parameter; Finally, online adaptive study can be carried out to the upper bound of uncertain system and external interference by fuzzy neural network, and can buffeting be reduced, achieve to modeling error and uncertain noises from motion tracking.

Accompanying drawing explanation

Fig. 1 is the simplified model schematic diagram of gyroscope system in the present invention;

Fig. 2 is the theory diagram of the neural overall fast terminal sliding-mode control of adaptive fuzzy of gyroscope in the present invention;

Fig. 3 is structure of fuzzy neural network figure in the present invention;

Fig. 4 is X in specific embodiments of the invention, Y-axis location tracking curve;

Fig. 5 is X in specific embodiments of the invention, Y-axis location tracking graph of errors;

Fig. 6 is overall fast terminal sliding-mode surface convergence curve in specific embodiments of the invention;

Fig. 7 is X in specific embodiments of the invention, Y-axis control inputs response curve;

Fig. 8 is gyroscope systematic parameter d in specific embodiments of the invention _xx, d _xy, d _yyand ω _x ², ω _xy, ω _y ²adaptive Identification curve;

Fig. 9 is gyroscope angular velocity Ω in specific embodiments of the invention _zadaptive Identification curve;

Figure 10 is X in specific embodiments of the invention, Y-axis upper bound change curve.

Embodiment

Above-mentioned explanation is only general introduction of the present invention, in order to technological means of the present invention can be better understood, and can be implemented according to the content of instructions, below in conjunction with accompanying drawing and preferred embodiment, the neural overall fast terminal sliding-mode control of the adaptive fuzzy of the gyroscope proposed according to the present invention is elaborated.

The present invention is achieved in the following ways:

One, the mathematical model of gyroscope system is built

As shown in Figure 1, according to the Newton's law rotated in system, take into account manufacturing defect and mismachining tolerance, then pass through the nondimensionalization process of model, the lumped-parameter structure mathematical model obtaining actual gyroscope is:

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

Wherein, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the mass of gyroscope is in the position vector of driving shaft and sensitive axis diaxon, be the output of gyroscope system; $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ For the control inputs of gyroscope diaxon; $d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right]$ For external disturbance effect; $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right]$ For damping matrix, wherein, d _xx, d _yyfor the ratio of damping of diaxon, d _xyfor Coupling Damping coefficient; $K=\left[\begin{array}{cc}{\mathrm{\ω}}_x^2& {\mathrm{\ω}}_{\mathrm{xy}}\\ {\mathrm{\ω}}_{\mathrm{xy}}& {\mathrm{\ω}}_y^2\end{array}\right],$ Wherein, $\sqrt{\frac{k_{\mathrm{xx}}}{{{\mathrm{m\ω}}_0}^2}}\→{\mathrm{\ω}}_x,\sqrt{\frac{k_{\mathrm{yy}}}{{{\mathrm{m\ω}}_0}^2}}\→{\mathrm{\ω}}_y,\frac{k_{\mathrm{xy}}}{{{\mathrm{m\ω}}_0}^2}\→{\mathrm{\ω}}_{\mathrm{xy}},$ ω ₀for the natural frequency of diaxon, k _xx, k _yyfor the stiffness coefficient of diaxon, k _xyfor the stiffness coefficient of coupling; $\mathrm{\Ω}=\left[\begin{array}{cc}0& -{\mathrm{\Ω}}_z\\ {\mathrm{\Ω}}_z& 0\end{array}\right]$ For angular speed matrix, Ω _zfor the angular speed in gyroscope working environment, it is a unknown quantity.

The parameter uncertainty of consideration system and external disturbance, then can be shown as following form by gyroscope system table according to the mathematical model of gyroscope:

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

In formula, Δ D is the uncertainty of the unknown parameter of inertial matrix D+2 Ω, and Δ K is the uncertainty of the unknown parameter of inertial matrix K, and d represents external interference.

Further, formula (2) can be write as:

$\stackrel{\·\·}{q}+(D+2\mathrm{\Ω})\stackrel{\·}{q}+\mathrm{Kq}=u+f---\left(3\right)$

In formula, f represents parameter uncertainty and the external disturbance of system, meets:

$f=d-\mathrm{\ΔD}\stackrel{\·}{q}-\mathrm{\ΔKq}---\left(4\right)$

Generally, following hypothesis can be done to the uncertainty of system unknown parameter and external disturbance:

||f(t)||＜ρ(t)(5)

The upper bound that ρ (t) is parameter uncertainty and external disturbance f.

Two, overall fast terminal sliding-mode surface is built

The control problem that the present invention considers is the tracking problem of gyroscope, and the target of control is exactly that the suitable control law of design one makes system output q reach ideal trajectory q in finite time _rperfect tracking.

As shown in Figure 2, for the track following of gyroscope system, overall fast terminal sliding-mode surface s is designed to:

$s={[s_1,s_2]}^T=\stackrel{\·}{e}+\mathrm{\αe}+{\mathrm{\βe}}^{p_2/p_1}---\left(6\right)$

In formula, α=diag (α ₁, α ₂), β=diag (β ₁, β ₂) be sliding-mode surface constant, e=q-q _r=[x-x _r, y-y _r] ^ttracking error, the derivative of tracking error, q _rfor mass is along the ideal position output vector of diaxon, q is the position output vector of gyroscope; p ₁, p ₂(p ₁> p ₂) be positive odd number.

Three, the neural overall fast terminal sliding mode controller of adaptive fuzzy is built

For gyroscope system, adopt the sliding-mode surface that formula (6) describes, overall fast terminal sliding formwork control law U is designed to be made up of three control laws:

U＝u ₀+u ₁+u ₂(7)

Wherein,

u ₀＝a+(D+2Ω)v+Kq(8)

$u_1=-W\frac{s}{\left|\right|s\left|\right|}---\left(9\right)$

$u_2=-\mathrm{\ρ}\left(t\right)\frac{s}{\left|\right|s\left|\right|}---\left(10\right)$

In formula, W is sliding mode controller parameter; W=diag (w ₁, w ₂), w _i> 0, i=1,2, and have:

$v={\stackrel{\·}{q}}_r-\mathrm{\αe}-{\mathrm{\βe}}^{p_2/p_1}---\left(11\right)$

$a=\stackrel{\·}{v}={\stackrel{\·\·}{q}}_r-\mathrm{\α}\stackrel{\·}{e}-\frac{p_2}{p_1}\mathrm{\βdiag}({e_1}^{p_2/p_1-1},{e_2}^{p_2/p_1-1})\stackrel{\·}{e}---\left(12\right)$

Due to three parameter matrix D of gyroscope, K, Ω are unknown, so the control law shown in formula (7) cannot be implemented.According to Adaptive Control Theory, by three gyroscope parameter matrixs in formula (7) respectively by their estimated value substitute, and design the adaptive algorithm of three estimated values, online real-time update estimated value, so control law u shown in formula (8) ₀u' can be adjusted to ₀:

$u_0^{\′}=a+(\hat{D}+2\widehat{\mathrm{\Ω}})v+\hat{K}q---\left(13\right)$

Control law U becomes U', U'=u' ₀+ u ₁+ u ₂.

The parameter estimating error of definition D, K, Ω is respectively:

$\tilde{D}=\hat{D}-D$

$\tilde{K}=\hat{K}-K---\left(14\right)$

$\widehat{\mathrm{\Ω}}=\widehat{\mathrm{\Ω}}-\mathrm{\Ω}$

Using after adjustment control law U' as the control inputs of gyroscope system, to be updated in the mathematical model of the gyroscope system that formula (3) represents:

$\stackrel{\·\·}{q}+(D+2\mathrm{\Ω}+\mathrm{\ΔD})\stackrel{\·}{q}+(K+\mathrm{\ΔK})q=a+(\hat{D}+2\widehat{\mathrm{\Ω}})v+\hat{K}q+u_1+u_2+d---\left(15\right)$

$\⇒(\stackrel{\·\·}{q}-a)+(D+2\mathrm{\Ω})\stackrel{\·}{q}-(\hat{D}+2\widehat{\mathrm{\Ω}})v+\mathrm{Kq}-\hat{K}q=-\mathrm{\ΔD}\stackrel{\·}{q}-\mathrm{\ΔKq}+u_1+u_2+d$

$\⇒\stackrel{\·}{s}=(\hat{D}+2\widehat{\mathrm{\Ω}})v-(D+2\mathrm{\Ω})\stackrel{\·}{q}+\hat{K}q+u_1+u_2+f$

$\⇒\stackrel{\·}{s}=(\hat{D}+2\widehat{\mathrm{\Ω}})v-[(\hat{D}+2\widehat{\mathrm{\Ω}})-(\tilde{D}+2\widetilde{\mathrm{\Ω}})]\stackrel{\·}{q}+\tilde{K}q+u_1\text{+}{u}_2+f$

$\⇒\stackrel{\·}{s}=(\hat{D}+2\widehat{\mathrm{\Ω}})v-(\hat{D}+2\widehat{\mathrm{\Ω}})\stackrel{.}{q}+(\tilde{D}+2\widetilde{\mathrm{\Ω}})\stackrel{\·}{q}+\tilde{K}q+u_1+u_2+f$

$\⇒\stackrel{\·}{s}=-(\hat{D}+2\widehat{\mathrm{\Ω}})s+\tilde{D}\stackrel{\·}{q}+2\widetilde{\mathrm{\Ω}}\stackrel{\·}{q}+\tilde{K}q+u_1+u_2+f---\left(16\right)$

Generally, the upper dividing value of uncertain factor and external interference is difficult to or cannot predicts at all, according to the feature of fuzzy neural network, fuzzy neural network can be adopted to come upper bound ρ (t) of approximating parameter uncertainty and external disturbance f, ensure system stability and tracking characteristics.As shown in Figure 3, fuzzy neural network is the network that fuzzy system and neural network are combined and formed, it is give Indistinct Input signal and fuzzy weighting value by the neural network of routine in itself, and its learning algorithm is Learning Algorithm or its popularization normally.Fuzzy neural network is made up of input layer, obfuscation layer, fuzzy reasoning layer and output layer, utilizes the upper bound of fuzzy neural network approximating parameter uncertainty and external disturbance, is described as:

$\widehat{\mathrm{\ρ}}\left(t\right)={\hat{w}}^T\mathrm{\φ}\left(X\right)---\left(17\right)$

Wherein, $X={\left[\begin{array}{cc}q& \stackrel{\·}{q}\end{array}\right]}^T$ For the input of fuzzy neural network, it is measurable signal in system; the weights of fuzzy neural network, online real-time update; φ (X) is called the normalization confidence level of fuzzy neural network; being the output of fuzzy neural network, is the estimation to the f upper bound.

Assuming that the weight w of the optimum fuzzy neural network of existence one group, make to set up with lower inequality:

|ε(X)|＝|w ^Tφ(X)-ρ(t)|ε ^*(18)

Wherein, ε (X) the best approximation error that is parameter uncertainty and external disturbance upper bound ρ (t).

Parameter uncertainty and external disturbance f are also the function of time t,

Assuming that upper bound ρ (t) of parameter uncertainty and external disturbance f (t) meets following condition:

ρ(t)-||f(t)||＞ε ₀＞ε ^*(19)

Wherein for very little positive number.

Based on this, the control law u shown in formula (10) ₂u' can be adjusted to ₂:

$u_2^{\′}=-\widehat{\mathrm{\ρ}}\left(t\right)\frac{s}{\left|\right|s\left|\right|}---\left(20\right)$

Like this, the control law U' after adjustment is adjusted again, becomes U'', U''=u' ₀+ u ₁+ u' ₂, U'' is the control law of the neural overall fast terminal sliding mode controller of adaptive fuzzy.

Getting Lyapunov function is:

$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}\frac{1}{2}\mathrm{tr}\left\{\widetilde{\mathrm{\Ω}}{P}^{-1}{\widetilde{\mathrm{\Ω}}}^T\right\}+\frac{1}{2}{\mathrm{\η}}^{-1}{\tilde{w}}^T\tilde{w}---\left(21\right)$

In formula, the mark of tr (A) representing matrix A; be the evaluated error of neural network weight, because w is definite value, have m, N, P are adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix, and η is fuzzy neural network learning rate.

In order to ensure the derivative of Lyapunov function design adaptive algorithm be respectively:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{D}}}^T=-\frac{1}{2}M(\stackrel{\·}{q}{s}^T+s{\stackrel{\·}{q}}^T)\\ {\stackrel{\·}{\hat{K}}}^T=-\frac{1}{2}N({\mathrm{qs}}^T+{\mathrm{sq}}^T)\\ {\stackrel{\·}{\widehat{\mathrm{\Ω}}}}^T=-\frac{1}{2}P(2\stackrel{\·}{q}{s}^T-2s{\stackrel{\·}{q}}^T)\end{array}\right.---\left(22\right)$

The Weight number adaptively algorithm of design fuzzy neural network is:

$\stackrel{\·}{\hat{w}}=\mathrm{\η}\left|\right|s\left|\right|\mathrm{\φ}\left(X\right)=---\left(23\right)$

Using the control law U'' of neural for adaptive fuzzy overall fast terminal sliding mode controller as the control inputs of gyroscope system, bring in the mathematical model of gyroscope system,

To Lyapunov function V along time t differentiate, and the Weight number adaptively algorithm of the parameters adaption algorithm of formula (22) and formula (23) is brought into, obtains:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=s^T\stackrel{\&CenterDot;}{s}+\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\stackrel{\&CenterDot;}{\tilde{D}}}^T\right\}+\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\stackrel{\&CenterDot;}{\tilde{K}}}^T\right\}+\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}-{\mathrm{\&eta;}}^{-1}{\tilde{w}}^T\stackrel{\&CenterDot;}{\hat{w}}\\ =s^T(u_1+u_2^{\&prime;}+f)+s^T[-(\hat{D}+2\widehat{\mathrm{\&Omega;}})]s+s^T\tilde{D}\stackrel{\&CenterDot;}{q}+\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\stackrel{\&CenterDot;}{\tilde{D}}}^T\right\}+s^T\tilde{K}q\\ +\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\stackrel{\&CenterDot;}{\tilde{K}}}^T\right\}+2s^T\widetilde{\mathrm{\&Omega;}}\stackrel{\&CenterDot;}{q}+\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}-{\mathrm{\&eta;}}^{-1}{\tilde{w}}^T\stackrel{\&CenterDot;}{\hat{w}}\\ =s^T(u_1+u_2^{\&prime;}+f)+s^T[-(\hat{D}+2\widehat{\mathrm{\&Omega;}})]s-{\tilde{w}}^T\left|\right|s\left|\right|\mathrm{\&phi;}\left(X\right)\\ =s^T{u}_1+s^T[-(\hat{D}+2\widehat{\mathrm{\&Omega;}})]s+s^{T}f-\left|\right|s\left|\right|[{\hat{w}}^T\mathrm{\&phi;}\left(X\right)-\mathrm{\&rho;}\left(t\right)+\mathrm{\&rho;}\left(t\right)]-{\tilde{w}}^T\left|\right|s\left|\right|\mathrm{\&phi;}\left(X\right)\\ \&le;-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|-\left|\right|s\left|\right|[\mathrm{\&rho;}\left(t\right)-|\left|f\right|\left|\right]-\left|\right|s\left|\right|[{\hat{w}}^T\mathrm{\&phi;}\left(X\right)-\mathrm{\&rho;}\left(t\right)]-{\tilde{w}}^T\left|\right|s\left|\right|\mathrm{\&phi;}\left(X\right)\\ =-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|-\left|\right|s\left|\right|[\mathrm{\&rho;}\left(t\right)-|\left|f\right||=-|\left|s\right|\left|\right[{\hat{w}}^T\mathrm{\&phi;}\left(X\right)-w^T\mathrm{\&phi;}\left(X\right)+\mathrm{\&epsiv;}\left(X\right)]-(w^T-{\hat{w}}^T)|\left|s\right||\mathrm{\&phi;}\left(X\right)\\ =-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|-\left|\right|s\left|\right|[\mathrm{\&rho;}\left(t\right)-|\left|f\right|\left|\right]-\left|\right|s\left|\right|\mathrm{\&epsiv;}\left(X\right)\\ \&le;-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|-\left|\right|s\left|\right|[\mathrm{\&rho;}\left(t\right)-|\left|f\right|\left|\right]+\left|\right|s\left|\right|\left|\mathrm{\&epsiv;}\left(X\right)\right|\\ =\left|\right|s\left|\right|\left\{\right|\mathrm{\&epsiv;}\left(X\right)|-[\mathrm{\&rho;}\left(t\right)-\left|\right|f\left|\right|\left]\right\}-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|\\ \&le;\left|\right|s\left|\right|({\mathrm{\&epsiv;}}^{*}-{\mathrm{\&epsiv;}}_0)-{\mathrm{\&lambda;}}_{\min }\left(W\right)\left|\right|s\left|\right|\&le;-\mathrm{\&xi;}\left|\right|s\left|\right|<0\end{array}---\left(24\right)$

In formula, ξ=λ _min(W)-(ε ^*-ε ₀) > 0 and || s|| ≠ 0, λ _min(W) be the smallest real eigenvalue of W.

Thus, can judge that designed controller ensure that the global stability of system based on Liapunov stability second method, and make the output tracking error of system at Finite-time convergence to zero.

Four, Computer Simulation

In order to show the validity of the neural overall fast terminal sliding-mode control of gyroscope adaptive fuzzy that the present invention proposes more intuitively, perceptive construction on mathematics/SIMULINK is now utilized to carry out computer simulation experiment to the present invention.

With reference to existing document, the parameter 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 ^-6N·s/m,d _yy＝1.8×10 ^-6N·s/m,d _xy＝3.6×10 ^-7N·s/m

Suppose that unknown input angular velocity is Ω _z=100rad/s, reference length is chosen for q ₀=1 μm, natural frequency ω ₀=1000Hz, after nondimensionalization, three parameter matrixs of gyroscope are:

$D=\left[\begin{array}{cc}0.01& 0.002\\ 0.002& 0.01\end{array}\right],K=\left[\begin{array}{cc}355.3& 70.99\\ 70.99& 532.9\end{array}\right],\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -0.1\\ 0.1& 0\end{array}\right]---\left(25\right)$

Wherein, nondimensionalization process is, $\frac{d_{\mathrm{xx}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{xx}},\frac{d_{\mathrm{xy}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{xy}},\frac{d_{\mathrm{yy}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{yy}},\sqrt{\frac{k_{\mathrm{xx}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_x,$ $\frac{k_{\mathrm{xy}}}{{{\mathrm{m\&omega;}}_0}^2}\&RightArrow;{\mathrm{\&omega;}}_{\mathrm{xy}},\sqrt{\frac{k_{\mathrm{yy}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_y,\frac{{\mathrm{\&Omega;}}_z}{{\mathrm{\&omega;}}_0}\&RightArrow;{\mathrm{\&Omega;}}_z.$

In emulation experiment, the estimation initial value of gyroscope three parameter matrixs is taken as respectively:

adaptive gain M, N, P are taken as: M=N=P=diag (150,150); The ideal trajectory of diaxon is taken as respectively: x _r=sin (π t), y _r=cos (0.5 π t); The starting condition of system is taken as:

parameter uncertainty and the external disturbance of system are taken as:

f＝[0.5*randn(1,1);0.5*randn(1,1)]。

Sliding-mode surface parameter choose is: p ₁=5, p ₂=3, α ₁=α ₂=0.25, β ₁=β ₂=0.5; Sliding mode controller parameter W=diag (w ₁, w ₂) be taken as: W=diag (2,4) (w ₁=2, w ₂=4).

Structure of fuzzy neural network selects 2-10-25-1, and the random value between [-1,1] got by the initial value of neural network weight w, and the initial value of center vector and gaussian basis fat vector is got $C=\left(c_{\mathrm{ij}}\right)=\left[\begin{array}{ccc}0.5& 0.5& 0.5\\ 0.5& 0.5& 0.5\end{array}\right]$ With B=(b _ij)=[0.20.20.2] ^t, fuzzy neural network learning rate gets η=0.001.

Fig. 4 is X, the position curve of pursuit in Y direction that gyroscope adopts the neural overall fast terminal sliding-mode control of adaptive fuzzy and obtains, 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. 5 is the tracking error curve in X, Y direction, as can be seen from the figure, substantially converges to zero, and keep this motion through very short time error curve.

Fig. 6 is terminal sliding mode face convergence curve in gyroscope X, Y direction, can constantly level off to zero, show that system can arrive sliding mode stabilized zone, i.e. s=0 through terminal sliding mode face after a while.System is not subject to the impact of external disturbance and uncertain factor, and arrive sliding-mode surface by the very short time, control system will enter sliding formwork track, keeps this motion.Compared with controlling with traditional fast terminal sliding formwork, overall fast terminal sliding formwork controls the optimal problem solving convergence time, thus realizes system state and converge to equilibrium state quickly and accurately.

Fig. 7 is control inputs response curve in X, Y direction, as can be seen from the figure, adopts the control inputs of fuzzy neural network upper bound adaptive learning substantially not produce buffeting.

Fig. 8 is the Adaptive Identification curve of gyroscope systematic parameter, and result shows d _xx, d _xy, d _yyand ω _x ², ω _xy, ω _y ²these parameters not only can converge to respective true value soon, and overshoot is also less.Fig. 9 is gyroscope angular velocity Ω _zidentification curve, result shows that Attitude rate estimator finally converges to its true value.

Figure 10 is X, Y direction upper bound change curve, and upper bound change is the result learnt by fuzzy neural network.The external environment condition different according to system carries out adaptive learning to the upper bound, makes it can adapt to adaptive terminal System with Sliding Mode Controller well, reduces the generation that control system is buffeted simultaneously.

The above, it is only preferred embodiment of the present invention, not any in form large restriction is done to the present invention, although the present invention discloses as above with preferred embodiment, but and be not used to limit the present invention, any those skilled in the art, do not departing within the scope of technical solution of the present invention, make a little change when the technology contents of above-mentioned announcement can be utilized or be modified to the Equivalent embodiments of equivalent variations, in every case be the content not departing from technical solution of the present invention, according to any simple modification that technical spirit of the present invention is done above embodiment, equivalent variations and modification, all still belong in the scope of our bright technical scheme.

Claims (2)

1. the neural overall fast terminal sliding-mode control of the adaptive fuzzy of gyroscope, is characterized in that, comprise the following steps;

1) mathematical model building gyroscope system is:

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

Wherein, q is the position vector of mass at driving shaft and sensitive axis diaxon of gyroscope, is the output of gyroscope system; U is the control inputs of gyroscope; D is damping matrix; K is the coefficient containing the natural frequency of diaxon and the stiffness coefficient of coupling; Ω is angular speed matrix; F is parameter uncertainty and the external disturbance of system;

2) building overall fast terminal sliding-mode surface s is:

$s=\stackrel{\&CenterDot;}{e}+\mathrm{\&alpha;}e+{\mathrm{\&beta;e}}^{p_2/p_1}---\left(6\right)$

Wherein, α=diag (α ₁, α ₂), β=diag (β ₁, β ₂) be sliding-mode surface constant; E=q-q _rfor tracking error; q _rfor mass is along the ideal position output vector of diaxon; p ₁, p ₂for positive odd number, wherein, p ₁> p ₂;

3) the neural overall fast terminal sliding mode controller of adaptive fuzzy is built:

3-1) for described gyroscope system, adopt the sliding-mode surface of formula (6), overall fast terminal sliding formwork control law U is made up of three control laws:

U＝u ₀+u ₁+u ₂(7)

Wherein,

u ₀＝a+(D+2Ω)v+Kq，

D, K, Ω are three parameter matrixs of gyroscope,

$u_1=-W\frac{s}{\left|\right|s\left|\right|},$ W＝diag(w ₁,w ₂)；

W is sliding mode controller parameter;

the parameter uncertainty that ρ (t) is system and the upper bound of external disturbance f;

3-2) due to three parameter matrix D of gyroscope, K, Ω are unknown, according to Adaptive Control Theory, use estimated value alternate parameter matrix D, K, Ω, and the adaptive algorithm designing three estimated values, online real-time update estimated value, then control law u ₀be adjusted to u' ₀:

$u_0^{\&prime;}=a+(\hat{D}+2\widehat{\mathrm{\&Omega;}})v+\hat{K}q;$

3-3) according to Fuzzy Neural Network Theory, fuzzy neural network is adopted to approach the parameter uncertainty of system and upper bound ρ (t) of external disturbance f, and design the adaptive algorithm of fuzzy neural network weights, the output of online real-time update fuzzy neural network, the output of fuzzy neural network for:

Wherein, the weights of fuzzy neural network, the normalization confidence level that φ (X) is fuzzy neural network, $X={\left[\begin{array}{cc}q& \stackrel{\&CenterDot;}{q}\end{array}\right]}^T$ For the input of fuzzy neural network,

Then control law u ₂be adjusted to u' ₂:

$u_2^{\&prime;}=-\widehat{\mathrm{\&rho;}}\left(t\right)\frac{s}{\left|\right|s\left|\right|};$

3-4) with described step 3-2) and step 3-3) adjustment after control law u' ₀and u' ₂, replace step 3-1) in control law u ₀and u ₂, obtain the control law U of the neural overall fast terminal sliding mode controller of adaptive fuzzy "

U”＝u' ₀+u ₁+u' ₂；

3-5) using the control law U of neural for adaptive fuzzy overall fast terminal sliding mode controller " as the control inputs of gyroscope system, bring in the mathematical model of gyroscope system, realize the tracing control to gyroscope system.

2. the neural overall fast terminal sliding-mode control of the adaptive fuzzy of gyroscope according to claim 1, it is characterized in that, described gyroscope parameter matrix D, the adaptive algorithm of the estimated value of K, Ω and the adaptive algorithm of fuzzy neural network weights design based on Lyapunov stability theory:

Lyapunov function is:

$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}tr\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}tr\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}+\frac{1}{2}tr\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\widetilde{\mathrm{\&Omega;}}}^T\right\}+\frac{1}{2}{\mathrm{\&eta;}}^{-1}{\tilde{w}}^T\tilde{w}$

Wherein, the mark of tr () representing matrix; M, N, P are adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix, and η is fuzzy neural network learning rate, the evaluated error of neural network weight, be respectively parameter matrix D, the parameter estimating error of K, Ω,

In order to ensure the derivative of Lyapunov function choose gyroscope parameter matrix D, the adaptive algorithm of the estimated value of K, Ω is:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{D}}}^T=-\frac{1}{2}M(\stackrel{\&CenterDot;}{q}{s}^T+s{\stackrel{\&CenterDot;}{q}}^T)\\ {\stackrel{\&CenterDot;}{\hat{K}}}^T=-\frac{1}{2}N(q{s}^T+s{q}^T)\\ {\stackrel{\&CenterDot;}{\widehat{\mathrm{\&Omega;}}}}^T=-\frac{1}{2}P(2\stackrel{\&CenterDot;}{q}{s}^T-2s{\stackrel{\&CenterDot;}{q}}^T)\end{array}\right.$

The adaptive algorithm of fuzzy neural network weights is: