CN104216284A - Limit time cooperative control method of mechanical arm servo system - Google Patents

Abstract

A limit time cooperative control method of a mechanical arm servo system includes: building the dynamic model of the mechanical arm servo system, and initiating the system state, sampling time and related control parameters; approximating the nonlinear input dead zone linearity in the system into a simple time-varying system according to the differential mean value theorem, and deriving a mechanical arm servo system model with an unknown dead zone; calculating the tracking error, the non-singular terminal sliding mode surface and the first derivative of a control system; selecting a neural network approximation unknown function on the basis of the mechanical arm servo system model with the unknown dead zone, designing a neural network limit time cooperative controller according to the system tracking error and the non-singular terminal sliding mode surface, and updating a neural network weight matrix. By the method, additional inverted compensation of a nonlinear dead zone can be avoided, sliding model high-frequency buffeting is reduced, and limit time fast tracking of the mechanical arm servo system is achieved.

Description

The finite time cooperative control method of mechanical arm servo-drive system

Technical field

The finite time cooperative control method of the mechanical arm servo-drive system that finite time cooperative control method, particularly and nonlinear dead-zone uncertain with system model that the present invention designs a mechanism arm servo-drive system input.

Background technology

Mechanical arm servo-drive system is widely used in robot, the contour performance system of aviation aircraft.The accurate fast control how realizing mechanical arm servo-drive system has become a hot issue.But dead-time voltage link is extensively present in mechanical arm servo-drive system, the efficiency of control system is often caused to reduce or even lost efficacy.For the control problem of mechanical arm servo-drive system, there is a lot of control method, such as PID controls, adaptive control, and sliding formwork controls, non-singular terminal sliding formwork control etc.

The advantages such as sliding-mode control has that algorithm is simple, fast response time, to external world noise and Parameter Perturbation strong robustness, are therefore widely used in the fields such as robot, motor, aircraft.But sliding formwork controls to there is inevitable shortcoming, such as there is high frequency and buffet.On the basis that sliding formwork controls, cooperative control method is suggested.The method can make system works under constant frequency, thus avoids high frequency buffeting.In addition, in order to improve the speed of convergence of system tracking error, generally on the basis of linear sliding mode, increasing nonlinear terms, being realized the rapid track and control of system by design kinematic nonlinearity sliding-mode surface, being called non-singular terminal sliding-mode control.

Summary of the invention

The present invention will overcome the above-mentioned shortcoming of prior art, a kind of finite time cooperative control method of mechanical arm servo-drive system is provided, theoretical and the non-singular terminal sliding formwork thought in conjunction with Collaborative Control, design finite time controller, ensures that system exports in finite time the accurate tracking of desired trajectory.

The finite time cooperative control method of mechanical arm servo-drive system of the present invention, comprises following steps:

Step 1, sets up the dynamic model such as formula mechanical arm servo-drive system (1) Suo Shi, initialization system state, sampling time and associated control parameters;

$M\left(x\right)\stackrel{..}{x}+V_m(x,\stackrel{.}{x})\stackrel{.}{x}+G\left(x\right)+F\left(\stackrel{.}{x}\right)=\mathrm{\τ}-{\mathrm{\τ}}_d---\left(1\right)$

Wherein be respectively position, speed and acceleration; M (x) ∈ R ^{n × n}for symmetric positive definite matrix; for Ke Shili; G (x) ∈ R ⁿfor gravity; represent friction force; τ _d∈ R ^{n × 1}for external disturbance; τ ∈ R ^{n × 1}for dead band output valve, be expressed as:

$\mathrm{\&tau;}=D\left(u\left(t\right)\right)=\left\{\begin{array}{ccc}{g}_r\left(u\right)& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ 0& \mathrm{if}& b_l<u\left(t\right)<b_r\\ g_l\left(u\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right.---\left(2\right)$

Wherein u (t) ∈ R is working control signal; g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, and meet b _l<0, b _r>0.

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, is a simple time-varying system by the non-linear input dead band linear-apporximation in system, derives the mechanical arm servo system models with unknown dead band;

2.1 according to Order Derivatives in Differential Mid-Value Theorem, there is ξ _l∈ (-∞, b _l) and ξ _r∈ (b _r,+∞) make

g _l(u)＝g′ _l(ξ′ _l)(u-b _l) (3)

Wherein ξ ' _l∈ (-∞, b _l], g ' _l(ξ _l) be function g _lu () is at ξ ' _lthe inverse at place;

g _r(u)＝g′ _r(ξ′ _r)(u-b _r) (4)

Wherein ξ ' _r∈ [b _l,+∞), g ' _r(ξ _r) be function g _ru () is at ξ ' _rthe inverse at place;

According to formula (3) and formula (4), formula (2) can be rewritten as

Wherein

$\mathrm{\&rho;}\left(u\right)=\left\{\begin{array}{ccc}-g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)b_r& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ -[g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)+g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)]u\left(t\right)& \mathrm{if}& b_l<u\left(t\right)<b_r\\ -g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right..---\left(6\right)$

Wherein ξ _l∈ (-∞, b _l], ξ _r∈ [b _l,+∞),

And

2.2 by formula (1) and formula (5) can be with the mechanical arm servo system models in unknown dead band:

Wherein | ρ (u) |≤ρ _n, ρ _nbe unknown normal number, meet ρ _n=(g _r1+ g _l1) max{b _r, b _l.

Step 3, the tracking error of calculating control system, non-singular terminal sliding-mode surface and first order derivative thereof;

The tracking error of 3.1 definition control system is

e(t)＝x _d-x (11)

Wherein x _dfor second order can lead desired trajectory.

3.2 in the process of design collaboration controller, and the collaborative flow pattern of definition is:

$s=\stackrel{.}{e}\left(t\right)+\mathrm{\&lambda;e}\left(t\right)---\left(12\right)$

Wherein λ is the normal number of influential system convergence.

According to the non-singular terminal sliding-mode surface shown in such as formula (13), design finite time collaborative controller, makes systematic error trend towards the collaborative flow pattern shown in formula (12) fast;

$q{\stackrel{.}{s}}^{p/r}+s=0---\left(13\right)$

Wherein q is positive definite constant, p>0, r>0,1<p/r<2.

The differentiate of 3.3 pairs of formulas (12), convolution (6) and formula (11) can be derived

Wherein non-linear unknown function h is

$h=M({\stackrel{..}{x}}_d+\mathrm{\&lambda;}\stackrel{.}{e})+V_m({\stackrel{.}{x}}_d+\mathrm{\&lambda;e})+G+F-\mathrm{\&rho;}\left(u\right).---\left(15\right)$

Step 4, based on the mechanical arm servo system models with unknown dead band, selects neural network to approach unknown function, and according to system tracking error, non-singular terminal sliding-mode surface, design neural network finite time collaborative controller, upgrades neural network weight matrix;

4.1 according to formula (13) and formula (14), and the expression formula of controller is

$u={\hat{W}}_a^T\mathrm{\&phi;}\left(x\right)+k{s}^{r/p}+\mathrm{\&delta;sgn}\left(s\right)---\left(16\right)$

Wherein for ideal weight estimated value, for Base Function, x is neural network input vector, k=q ^-1be controling parameters, δ is a positive constant, meets ε _nbe a normal number, represent the upper limit of neural network approximate error, τ _nfor the upper limit constant of external disturbance, for neural network weight evaluated error.

The right value update rule of 4.2 design neural networks is:

${\stackrel{\stackrel{.}{^}}{W}}_a=K_C\mathrm{\&phi;}\left(x\right)s^T---\left(17\right)$

Wherein K _cfor positive definite diagonal matrix, φ (x) is taken as following Gaussian function usually:

$\mathrm{\&phi;}\left(x\right)=\exp (-\frac{{|x-c|}^2}{2b^2})---\left(18\right)$

Wherein c=[c ₁, c ₂..., c _n] ^tit is the center of Gaussian function; B is the width of Gaussian function; 0< φ (x)≤1 can be released by formula (18).

Step 5, design Lyapunov function with then can prove that all signals in closed-loop control system are all uniformly bounded.Meanwhile, systematic error e can at Finite-time convergence to equilibrium point e=0.

The present invention is in conjunction with Collaborative Control principle and non-singular terminal sliding formwork, and design finite time cooperative control method, ensures to export in finite time the accurate tracking of desired trajectory.

Technical conceive of the present invention is: for mechanical arm servo-drive system that is uncertain with model and nonlinear dead-zone input, Order Derivatives in Differential Mid-Value Theorem is utilized to optimize dead space arrangements, in conjunction with Collaborative Control, non-singular terminal sliding formwork and neural network, the finite time cooperative control method of the arm servo-drive system that designs a mechanism.By Order Derivatives in Differential Mid-Value Theorem, make dead band continuously differentiable, then approached the unknown function comprising dead band by neural network, eliminate the additional inverse of traditional dead band and compensate.Meanwhile, design finite time collaborative controller and ensure that the convergence of system tracking error fast and stable causes zero point.The invention provides a kind of high frequency that can improve in sliding formwork control and buffet problem, and effectively avoid the finite time cooperative control method of dead band input to systematic influence, the rapid track and control of mechanical arm servo-drive system can be realized.

Advantage of the present invention is: avoid the additional inverse compensation in dead band, realize system finite time convergence control, avoid high frequency to buffet.

Accompanying drawing explanation

Fig. 1 is nonlinear dead-zone of the present invention model

Fig. 2 is the tracking effect in mechanical arm servo-drive system joint 1,2 of the present invention

Fig. 3 is the tracking error of system of the present invention

Fig. 4 is controller input signal of the present invention

Fig. 5 is Collaborative Control algorithm flow chart of the present invention

Embodiment

With reference to accompanying drawing, the finite time cooperative control method of mechanical arm servo-drive system of the present invention, comprises the following steps:

Step 1, sets up the dynamic model such as formula mechanical arm servo-drive system (1) Suo Shi, initialization system state, sampling time and associated control parameters;

$M\left(x\right)\stackrel{..}{x}+V_m(x,\stackrel{.}{x})\stackrel{.}{x}+G\left(x\right)+F\left(\stackrel{.}{x}\right)=\mathrm{\&tau;}-{\mathrm{\&tau;}}_d---\left(1\right)$

Wherein be respectively position, speed and acceleration; M (x) ∈ R ^{n × n}for symmetric positive definite matrix; for Ke Shili; G (x) ∈ R ⁿfor gravity; represent friction force; τ _d∈ R ^{n × 1}for external disturbance; τ ∈ R ^{n × 1}for dead band output valve, be expressed as:

$\mathrm{\&tau;}=D\left(u\left(t\right)\right)=\left\{\begin{array}{ccc}{g}_r\left(u\right)& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ 0& \mathrm{if}& b_l<u\left(t\right)<b_r\\ g_l\left(u\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right.---\left(2\right)$

Wherein u (t) ∈ R is working control signal; g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, and meet b _l<0, b _r>0.

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, is a simple time-varying system by the non-linear input dead band linear-apporximation in system, derives the mechanical arm servo system models with unknown dead band;

2.1 according to Order Derivatives in Differential Mid-Value Theorem, there is ξ _l∈ (-∞, b _l) and ξ _r∈ (b _r,+∞) make

g _l(u)＝g′ _l(ξ′ _l)(u-b _l) (3)

Wherein ξ ' _l∈ (-∞, b _l], g ' _l(ξ _l) be function g _lu () is at ξ ' _lthe inverse at place;

g _r(u)＝g′ _r(ξ′ _r)(u-b _r) (4)

Wherein ξ ' _r∈ [b _l,+∞), g ' _r(ξ _r) be function g _ru () is at ξ ' _rthe inverse at place;

According to formula (3) and formula (4), formula (2) can be rewritten as

Wherein

$\mathrm{\&rho;}\left(u\right)=\left\{\begin{array}{ccc}-g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)b_r& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ -[g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)+g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)]u\left(t\right)& \mathrm{if}& b_l<u\left(t\right)<b_r\\ -g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right..---\left(6\right)$

Wherein ξ _l∈ (-∞, b _l], ξ _r∈ [b _l,+∞),

And

2.2 by formula (1) and formula (5) can be with the mechanical arm servo system models in unknown dead band:

Wherein | ρ (u) |≤ρ _n, ρ _nbe unknown normal number, meet ρ _n=(g _r1+ g _l1) max{b _r, b _l.

Step 3, the tracking error of calculating control system, non-singular terminal sliding-mode surface and first order derivative thereof;

The tracking error of 3.1 definition control system is

e(t)＝x _d-x (11)

Wherein x _dfor second order can lead desired trajectory.

3.2 in the process of design collaboration controller, and the collaborative flow pattern of definition is:

$s=\stackrel{.}{e}\left(t\right)+\mathrm{\&lambda;e}\left(t\right)---\left(12\right)$

Wherein λ is the normal number of influential system convergence.

According to the non-singular terminal sliding-mode surface shown in such as formula (13), design finite time collaborative controller, makes systematic error trend towards the collaborative flow pattern shown in formula (12) fast;

$q{\stackrel{.}{s}}^{p/r}+s=0---\left(13\right)$

Wherein q is positive definite constant, p>0, r>0,1<p/r<2.

The differentiate of 3.3 pairs of formulas (12), convolution (6) and formula (11) can be derived

Wherein non-linear unknown function h is

$h=M({\stackrel{..}{x}}_d+\mathrm{\&lambda;}\stackrel{.}{e})+V_m({\stackrel{.}{x}}_d+\mathrm{\&lambda;e})+G+F-\mathrm{\&rho;}\left(u\right).---\left(15\right)$

Step 4, based on the mechanical arm servo system models with unknown dead band, selects neural network to approach unknown function, and according to system tracking error, non-singular terminal sliding-mode surface, design neural network finite time collaborative controller, upgrades neural network weight matrix;

4.1 according to formula (13) and formula (14), and the expression formula of controller is

$u={\hat{W}}_a^T\mathrm{\&phi;}\left(x\right)+k{s}^{r/p}+\mathrm{\&delta;sgn}\left(s\right)---\left(16\right)$

Wherein for ideal weight estimated value, for Base Function, x is neural network input vector, k=q ^-1be controling parameters, δ is a positive constant, meets ε _nbe a normal number, represent the upper limit of neural network approximate error, τ _nfor the upper limit constant of external disturbance, for neural network weight evaluated error.

The right value update rule of 4.2 design neural networks is:

${\stackrel{\stackrel{.}{^}}{W}}_a=K_C\mathrm{\&phi;}\left(x\right)s^T---\left(17\right)$

Wherein K _cfor positive definite diagonal matrix, φ (x) is taken as following Gaussian function usually:

$\mathrm{\&phi;}\left(x\right)=\exp (-\frac{{|x-c|}^2}{2b^2})---\left(18\right)$

Wherein c=[c ₁, c ₂..., c _n] ^tit is the center of Gaussian function; B is the width of Gaussian function; 0< φ (x)≤1 can be released by formula (18).

Step 5, design Lyapunov function with then can prove that all signals in closed-loop control system are all uniformly bounded.Meanwhile, systematic error e can at Finite-time convergence to equilibrium point e=0.

For the validity of checking institute extracting method, carry out following experiment:

Make x=[x ₁, x ₂] ^t, then formula (1) can be expressed as the mechanical arm servo-drive system with two joints, and expression is:

$\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}{\stackrel{..}{x}}_1\\ {\stackrel{..}{x}}_2\end{array}\right]+\left[\begin{array}{c}{E}_1\\ E_2\end{array}\right]+\left[\begin{array}{c}{F}_1\\ F_2\end{array}\right]+\left[\begin{array}{c}{d}_1\\ d_2\end{array}\right]=\left[\begin{array}{c}D\left(u_1\right)\\ D\left(u_2\right)\end{array}\right]---\left(19\right)$

Wherein P ₁₁=n ₁+ n ₂+ 2 γ cosx ₂, P ₁₂=n ₂+ γ cosx ₂, P ₂₁=n ₂+ 2 γ cosx ₂, P ₂₂=n ₂, $E_1=-\mathrm{\&gamma;}(2{\stackrel{.}{x}}_1{\stackrel{.}{x}}_2+{\stackrel{.}{x}}_2^2)\sin x_2,E_2=\mathrm{\&gamma;}{\stackrel{.}{x}}_1^2\sin x_2,$ F ₁=n ₁e ₁cosx ₁+ γ e ₁cos (x ₁+ x ₂), F ₂=γ e ₁cos (x ₁+ x ₂), γ=l ₂m ₁m ₂, e ₁=g/m ₁, g=9.8m/s ²acceleration of gravity; l ₁, l ₂for joint length; m ₁, m ₂represent quality; x ₁, x ₂for angular displacement; D=[d ₁d ₂] ^tfor its exterior disturbance, meet d ₁, d ₂∈ [-0.1,0.1]; As shown in Figure 1, D (u) is the output valve of controller u behind nonlinear dead-zone formula (28) Suo Shi;

$D\left(u\right)=\left\{\begin{array}{cc}(1-0.3\sin \left(u\right))(u-0.8),& u>0.8\\ 0,& -0.5<u<0.8\\ (0.8-0.2\cos \left(u\right))(u+0.5),& u\&le;-0.5\end{array}\right..---\left(20\right)$

Related parameter choosing in Controller gain variations is: in formula (12) $\mathrm{\&lambda;}=\left[\begin{array}{cc}1& 0\\ 0& 5\end{array}\right];$ K=[1 1], r=[3 3] in formula (16), p=[5 5], δ=1.5; In formula (17), K _cto be value be 1 positive definite diagonal matrix; Center c=[the c of Gaussian function numerical expression (18) ₁, c ₂..., c ₂₅], each element value is the random number in [-2,2], b=8.

Fig. 2 is mechanical arm servo-drive system joint angle displacement x ₁and x ₂follow the tracks of wanted signal x _d1=sin (0.02 π) and x _d2the design sketch of=cos (0.02 π), Fig. 3 represents the tracking error of system, and Fig. 4 represents controller input signal.As can be seen from Figures 2 and 3, the controller designed by formula (16) can follow the tracks of wanted signal fast in finite time, and tracking error can be tending towards 0 in about 3.5 seconds.As seen from Figure 4, controller signals is buffeted and is diminished after 2s.Therefore, finite time cooperative control method provided by the present invention, can not only avoid the additional inverse of nonlinear dead-zone to compensate, reduce sliding formwork high frequency and buffet problem, and the finite time that can realize mechanical arm servo-drive system be followed the tracks of fast.

What more than set forth is the excellent effect of optimization that an embodiment that the present invention provides shows, obvious the present invention is not just limited to above-described embodiment, do not depart from essence spirit of the present invention and do not exceed scope involved by flesh and blood of the present invention prerequisite under can do all distortion to it and implemented.

Claims (1)

1. the finite time cooperative control method of mechanical arm servo-drive system, comprises following steps:

Step 1, sets up the dynamic model such as formula mechanical arm servo-drive system (1) Suo Shi, initialization system state, sampling time and associated control parameters;

$M\left(x\right)\stackrel{..}{x}+V_m(x,\stackrel{.}{x})\stackrel{.}{x}+G\left(x\right)+F\left(\stackrel{.}{x}\right)=\mathrm{\&tau;}-{\mathrm{\&tau;}}_d---\left(1\right)$

Wherein be respectively position, speed and acceleration; M (x) ∈ R ^{n × n}for symmetric positive definite matrix; for Ke Shili; G (x) ∈ R ⁿfor gravity; represent friction force; τ _d∈ R ^{n × 1}for external disturbance; τ ∈ R ^{n × 1}for dead band output valve, be expressed as:

$\mathrm{\&tau;}=D\left(u\left(t\right)\right)=\left\{\begin{array}{ccc}{g}_r\left(u\right)& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ 0& \mathrm{if}& b_l<u\left(t\right)<b_r\\ g_l\left(u\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right.---\left(2\right)$

Wherein u (t) ∈ R is working control signal; g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, and meet b _l<0, b _r>0;

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, is a simple time-varying system by the non-linear input dead band linear-apporximation in system, derives the mechanical arm servo system models with unknown dead band;

2.1 according to Order Derivatives in Differential Mid-Value Theorem, there is ξ _l∈ (-∞, b _l) and ξ _r∈ (b _r,+∞) make

g _l(u)＝g′ _l(ξ′ _l)(u-b _l) (3)

Wherein ξ ' _l∈ (-∞, b _l], g ' _l(ξ _l) be function g _lu () is at ξ ' _lthe inverse at place;

g _r(u)＝g′ _r(ξ′ _r)(u-b _r) (4)

Wherein ξ ' _r∈ [b _l,+∞), g ' _r(ξ _r) be function g _ru () is at ξ ' _rthe inverse at place;

According to formula (3) and formula (4), formula (2) can be rewritten as

Wherein

$\mathrm{\&rho;}\left(u\right)=\left\{\begin{array}{ccc}-g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)b_r& \mathrm{if}& u\left(t\right)\&GreaterEqual;b_r\\ -[g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)+g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)]u\left(t\right)& \mathrm{if}& b_l<u\left(t\right)<b_r\\ -g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)& \mathrm{if}& u\left(t\right)\&le;b_l\end{array}\right..---\left(6\right)$

Wherein ξ _l∈ (-∞, b _l], ξ _r∈ [b _l,+∞),

And

2.2 by formula (1) and formula (5) can be with the mechanical arm servo system models in unknown dead band:

Wherein | ρ (u) |≤ρ _n, ρ _nbe unknown normal number, meet ρ _n=(g _r1+ g _l1) max{b _r, b _l;

Step 3, the tracking error of calculating control system, non-singular terminal sliding-mode surface and first order derivative thereof;

The tracking error of 3.1 definition control system is

e(t)＝x _d-x (11)

Wherein x _dfor second order can lead desired trajectory;

3.2 in the process of design collaboration controller, and the collaborative flow pattern of definition is:

$s=\stackrel{.}{e}\left(t\right)+\mathrm{\&lambda;e}\left(t\right)---\left(12\right)$

Wherein λ is the normal number of influential system convergence;

According to the non-singular terminal sliding-mode surface shown in such as formula (13), design finite time collaborative controller, makes systematic error trend towards the collaborative flow pattern shown in formula (12) fast;

$q{\stackrel{.}{s}}^{p/r}+s=0---\left(13\right)$

Wherein q is positive definite constant, p>0, r>0,1<p/r<2;

The differentiate of 3.3 pairs of formulas (12), convolution (6) and formula (11) can be derived

Wherein non-linear unknown function h is

$h=M({\stackrel{..}{x}}_d+\mathrm{\&lambda;}\stackrel{.}{e})+V_m({\stackrel{.}{x}}_d+\mathrm{\&lambda;e})+G+F-\mathrm{\&rho;}\left(u\right).---\left(15\right)$

Step 4, based on the mechanical arm servo system models with unknown dead band, selects neural network to approach unknown function, and according to system tracking error, non-singular terminal sliding-mode surface, design neural network finite time collaborative controller, upgrades neural network weight matrix;

4.1 according to formula (13) and formula (14), and the expression formula of controller is

$u={\hat{W}}_a^T\mathrm{\&phi;}\left(x\right)+k{s}^{r/p}+\mathrm{\&delta;sgn}\left(s\right)---\left(16\right)$

Wherein for ideal weight estimated value, for Base Function, x is neural network input vector, k=q ^-1be controling parameters, δ is a positive constant, meets ε _nbe a normal number, represent the upper limit of neural network approximate error, τ _nfor the upper limit constant of external disturbance, for neural network weight evaluated error;

The right value update rule of 4.2 design neural networks is:

${\stackrel{\stackrel{.}{^}}{W}}_a=K_C\mathrm{\&phi;}\left(x\right)s^T---\left(17\right)$

Wherein K _cfor positive definite diagonal matrix, φ (x) is taken as following Gaussian function usually:

$\mathrm{\&phi;}\left(x\right)=\exp (-\frac{{|x-c|}^2}{2b^2})---\left(18\right)$

Wherein c=[c ₁, c ₂..., c _n] ^tit is the center of Gaussian function; B is the width of Gaussian function; 0< φ (x)≤1 can be released by formula (18);

Step 5, design Lyapunov function with then can prove that all signals in closed-loop control system are all uniformly bounded; Meanwhile, systematic error e can at Finite-time convergence to equilibrium point e=0.