CN105549395A - Dead zone compensating control method for mechanical arm servo system with guaranteed transient performance - Google Patents

Abstract

A dead zone compensating control method for a mechanical arm servo system with guaranteed transient performance comprises following steps: a dynamic model of a mechanism arm servo system is established, and the system state, sampling time and control parameters are initialized; according to the differential mean value theorem, a non-linear input dead zone in the system is linearly approximated as a simple time-varying system, and a mechanism arm servo system model with unknown dead zones is deduced; a bounded function limiting the tracking error transient performance is introduced; a conversion error variable is defined by use of a error conversion method; a virtual control quantity of the system is designed by use of the Lyapunov method; an unknown virtue control quantity is estimated by use of a neural network; a first-stage wave filter is added to avoid problems such as inversion complexity blast. The invention provides a method for effectively compensating the influence of unknown dead zone input on the system and avoiding the problem of complexity blast brought by the inversion method, and for increasing the system transient tracking performance and guarantying stable tracking control of position signals.

Description

Ensure the mechanical arm servo-drive system dead time compensation control method of mapping

Technical field

The present invention relates to a kind of mechanical arm servo-drive system dead time compensation control method ensureing mapping, particularly with the flexible mechanical arm servo system self-adaptive control method in non-linear input dead band.

Background technology

Mechanical arm servo-drive system is widely used in robot, the contour performance system of aviation aircraft, and the accurate fast control how realizing mechanical arm servo-drive system has become a hot issue.Wherein, flexible mechanical arm is few, lightweight due to materials, and the advantage such as to consume energy low is subject to increasing attention.But unknown 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.Therefore, be improve control performance, for the compensation of nonlinear dead-zone and control method essential.The method of traditional solution dead-time voltage is generally set up inversion model or the approximate inverse model in dead band, and by estimating the bound parameter designing adaptive controller in dead band, with the nonlinear impact in deadband eliminating.But in the nonlinear system such as mechanical arm servo-drive system, the inversion model in dead band often not easily accurately obtains.For the unknown dead-time voltage input existed in system, based on Order Derivatives in Differential Mid-Value Theorem through line linearity, become a simple linear time varying system, avoid ancillary relief, thus unknown function and unknown parameter can be approached by a simple neural network.

For the control problem of mechanical arm servo-drive system, there is a lot of control method, such as PID controls, adaptive control, sliding formwork control etc.Sliding formwork controls to be considered to an effective robust control method in and external disturbance uncertain at resolution system.But the sliding formwork discontinuous switching characteristic controlled in itself will cause the buffeting of system, becomes the obstacle that sliding formwork controls to apply in systems in practice.The method of inversion and sliding formwork control to combine by someone, but the stable state that the method also can only realize system controls, and cannot carry out fast, following the tracks of completely to system.Therefore, propose a kind of mechanical arm servo-drive system dead time compensation control method ensureing mapping, introduce the bound function limiting tracking error transient response, by error conversion method, define a transformed error variable, the guarantee transient response problem of tracking error is converted into the boundedness problem of this error variance.Adopt the Lyapunov method, the virtual controlling amount of design system, and for avoiding the problems such as the complicated degree f explosion of inverting, add firstorder filter, thus ensureing boundedness and the uniform convergence of transformed error variable, the system that draws exports the tracking performance completely fast in whole interval.

Summary of the invention

In order to overcome the dead-time voltage of existing mechanical arm servo-drive system, model parameter uncertainty, and the deficiency of the complexity blast that brings of the method for inversion etc., the present invention proposes a kind of project organization that the mechanical arm servo-drive system dead time compensation control method of mapping simplifies controller that ensures, achieve the mechanical arm system Position Tracking Control of band unknown dead band input, ensure that system stability fast transient is followed the tracks of.

In order to the technical scheme solving the problems of the technologies described above proposition is as follows:

Ensure a mechanical arm servo-drive system dead time compensation control method for mapping, described control method comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is

$\left\{\begin{array}{c}I\stackrel{\·\·}{q}+K(q-\mathrm{\θ})+MgLsin\left(q\right)=0\\ J\stackrel{\·\·}{\mathrm{\θ}}-K(q-\mathrm{\θ})=v\left(u\right)\end{array}\right.---\left(1\right)$

Wherein, q and θ is respectively the angle of robot linkage and motor; I is the inertia of connecting rod; J is the inertia of motor; K is spring rate; M and L is quality and the length of connecting rod respectively; U is control signal; V (u) is dead band, is expressed as:

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

Wherein g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, meet b _l< 0, b _r> 0;

Definition x ₁=q, x ₃=θ, formula (1) is rewritten as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=-\frac{MgL}{I}sin\left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ {\stackrel{\&CenterDot;}{x}}_3=x_4\\ {\stackrel{\&CenterDot;}{x}}_4=\frac{1}{J}v\left(u\right)+\frac{K}{J}(x_1-x_3)\\ y=x_1\&CenterDot;\end{array}\right.---\left(3\right)$

Wherein, y is system output trajectory;

1.2 defining variable z ₁=x ₁, z ₂=x ₂, $z_4=-x_2\frac{MgL}{I}cos\left(x_1\right)-\frac{K}{I}(x_2-x_4),$ Then formula (3) is rewritten into

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=f_1\left(\stackrel{\&OverBar;}{z}\right)+b_{1}v\left(u\right)\\ y=z_1\end{array}\right.---\left(4\right)$

Wherein, $\stackrel{\&OverBar;}{z}={\&lsqb;z_1,z_2,z_3\&rsqb;}^T,f_1\left(\stackrel{\&OverBar;}{z}\right)=\frac{MgL}{I}sin\left(z_1\right)(z_2^2-\frac{K}{J})-\left(\frac{MgL}{I}cos\right(z_1)+\frac{K}{J}+\frac{K}{I})z_3,b_1=\frac{K}{IJ};$

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, comprise following process;

Linear process is carried out in 2.1 pairs of non-linear unknown dead bands

Wherein | ω (u) |≤ω _n, ω _nthat unknown positive number meets ω _n=(g ' _r+ g ' _l) max{b _r, b _land

2.2 according to Order Derivatives in Differential Mid-Value Theorem, then

Wherein $g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_r},{\mathrm{\&xi;}}_r\&Element;\&lsqb;b_r+\mathrm{\&infin;});$

Wherein $g_1^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_l},{\mathrm{\&xi;}}_l\&Element;(-\mathrm{\&infin;},b_l\&rsqb;;$

Then

$\mathrm{\&omega;}\left(u\right)=\left\{\begin{array}{cc}-g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)b_r& u\left(u\right)\&GreaterEqual;b_r\\ -\&lsqb;{g^{\&prime;}}_l\left({\mathrm{\&xi;}}_l\right)+g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)\&rsqb;u\left(t\right)& b_l<u\left(t\right)<b_r\\ -g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)b_l& u\left(t\right)\&le;b_l\end{array}\right.---\left(9\right)$

Formula (4), by formula (6) and formula (9), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu\\ y=z_1\end{array}\right.---\left(10\right)$

Wherein, $\stackrel{\&OverBar;}{z}=\&lsqb;z_1,z_2,z_3,z_4\&rsqb;,m\left(\stackrel{\&OverBar;}{z}\right)=f_1\left(\stackrel{\&OverBar;}{z}\right)+b_1*\mathrm{\&omega;}\left(u\right),$

Step 3, approaches uncertainty by neural network, and process is as follows:

Definition continuous function is:

h(X)＝W ^*Tφ(X)+ε(11)

Wherein W ^{* T}∈ R ^{n1 × n2}desirable weight matrix, φ (X) ∈ R ^{n1 × n2}be the function of desirable neural network, ε is the evaluated error of neural network, meets | ε | and≤ε _n, φ (X) functional form is:

$\mathrm{\&phi;}\left(X\right)=\frac{a}{b+\exp (-X/c)}+d---\left(12\right)$

Wherein, a, b, c, d are suitable constant;

Step 4, computing system transient control performance function and error conversion, process is as follows:

During 4.1 system transients control, controller input signal is:

u(t)＝ρ(F _φ(t),ψ(t),||e(t)||)×e(t)(13)

Wherein, e (t)=y-y _d, y _dbe desirable pursuit path, e (t) is tracking error, and ψ (t) is zoom factor, F _φt () is the border of error variance, || e (t) || be euclideam norm, in order to ensure that e (t) develops in border, time-varying gain ρ (.) is:

$\mathrm{\&rho;}\left(t\right)=\frac{1}{F_{\mathrm{\&phi;}}\left(t\right)-\left|\right|e\left(t\right)\left|\right|}---\left(14\right)$

The border of 4.2 design error variablees is:

Wherein, a continuous print positive function, to t>=0, have then

F _φ(t)＝δ ₀exp(-a ₀t)+δ _∞(16)

Wherein δ ₀>=δ _∞> 0, and | e (0) | < F _φ(0);

4.3 definition transient control error variances are:

$s_1=\frac{e\left(t\right)}{F_{\mathrm{\&phi;}}\left(t\right)-\left|\right|e\left(t\right)\left|\right|}---\left(17\right)$

Step 5, calculate system transients Properties Control dummy variable in the method for inversion, Dynamic sliding mode face and differential, process is as follows:

5.1 definition transient control dummy variable and differential thereof:

Definition error variance:

e＝y-y _d(18)

Wherein, y _dbe the ideal movements track of this system, y is that real system exports;

Then, formula (15) differentiate is obtained:

${\stackrel{\&CenterDot;}{s}}_1\left(t\right)=F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F(z_2-{\stackrel{\&CenterDot;}{y}}_d)-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_{F}e---\left(19\right)$

Wherein, φ _f=1/ (F _φ-|| e||) ²;

5.2 definition Liapunov functions:

$V_1=\frac{1}{2}{s_1}^2---\left(20\right)$

To V ₁differentiate obtains:

${\stackrel{\&CenterDot;}{V}}_1=s_1\&lsqb;F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F(z_2-{\stackrel{\&CenterDot;}{y}}_d)-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_{F}e\&rsqb;---\left(21\right)$

5.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_2={\stackrel{\&CenterDot;}{y}}_d-\frac{k_1{s}_1}{F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F}+\frac{{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}e}{F_{\mathrm{\&phi;}}}---\left(22\right)$

Wherein, k ₁for constant, and k ₁> 0;

The variable α that 5.4 definition one are new ₁, allow virtual controlling amount be τ by time constant ₁firstorder filter:

${\mathrm{\&tau;}}_1{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1+{\mathrm{\&alpha;}}_1={\stackrel{\&OverBar;}{z}}_2,{\mathrm{\&alpha;}}_1\left(0\right)={\stackrel{\&OverBar;}{z}}_2\left(0\right)---\left(23\right)$

5.5 definition filtering errors then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=\frac{{\stackrel{\&OverBar;}{z}}_2-{\mathrm{\&alpha;}}_1}{{\mathrm{\&tau;}}_1}=-\frac{y_2}{{\mathrm{\&tau;}}_1}---\left(24\right)$

5.6 estimate by neural network

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=-{W_1}^T{\mathrm{\&phi;}}_1\left(X_1\right)+{\mathrm{\&epsiv;}}_1---\left(25\right)$

Wherein $X_1={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_1}^T,{s_2}^T\&rsqb;}^T\&Element;R^5;$

Step 6, for formula (4), design virtual controlling amount;

6.1 definition error variances

s _i＝z _i-α _i-1,i＝2,3(26)

The first differential of formula (15) is:

${\stackrel{\&CenterDot;}{s}}_i=z_{i+1}-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_{i-1},i=2,3---\left(27\right)$

6.2 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_3=-k_2{s}_2-{\hat{W}}_1^T{\mathrm{\&phi;}}_1\left(X_1\right)-{\widehat{\mathrm{\&mu;}}}_1-F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F{s}_1---\left(28\right)$

Wherein, k ₂for constant and k ₂> 0, the estimated value of ε, w ₁estimated value;

6.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_4=-k_3{s}_3-{\hat{W}}_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\widehat{\mathrm{\&mu;}}}_2-s_2---\left(29\right)$

Wherein, k ₃for constant and k ₃> 0, the estimated value of ε, w ₂estimated value;

The variable α that 6.4 definition one are new _i, allow virtual controlling amount be τ by time constant _ifirstorder filter:

${\mathrm{\&tau;}}_i{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i+{\mathrm{\&alpha;}}_i={\stackrel{\&OverBar;}{z_i}}_{+1},{\mathrm{\&alpha;}}_i\left(0\right)={\stackrel{\&OverBar;}{z_i}}_{+1}\left(0\right)---\left(30\right)$

6.5 definition $y_{i+1}={\mathrm{\&alpha;}}_i-{\stackrel{\&OverBar;}{z_i}}_{+1},$ Then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i=\frac{{\stackrel{\&OverBar;}{z_i}}_{+1}-{\mathrm{\&alpha;}}_i}{{\mathrm{\&tau;}}_i}=-\frac{y_{i+1}}{{\mathrm{\&tau;}}_i}---\left(31\right)$

6.6 estimate by neural network

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2=-W_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\mathrm{\&epsiv;}}_2---\left(32\right)$

$-m+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3=-W_3^T{\mathrm{\&phi;}}_3\left(X_3\right)-{\mathrm{\&epsiv;}}_3---\left(33\right)$

Wherein, for ideal weight W _iestimated value, $X_i={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_i}^T,{s_{i+1}}^T\&rsqb;}^T\&Element;R^5$ In;

Step 7, CONTROLLER DESIGN inputs, and process is as follows:

7.1 definition error variances

s ₄＝z ₄-α ₃(34)

The first differential of calculating formula (20) is:

${\stackrel{\&CenterDot;}{s}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3---\left(35\right)$

7.2 CONTROLLER DESIGN input u:

$u=-k_5\left(\mathrm{\&eta;}sgn\right(s_4)+k_4{s}_4+W_3^T{\mathrm{\&phi;}}_3\left(X_3\right)+{\widehat{\mathrm{\&mu;}}}_3+s_3)---\left(36\right)$

Wherein, for ideal weight W ₃estimated value, k ₅>=1/n, ε ₃estimated value;

7.3 design adaptive rates:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{W}}}_j=K_j{\mathrm{\&phi;}}_j\left(X_{j+1}\right)s_{j+1}\\ \widehat{\mathrm{\&mu;}}=v_{\mathrm{\&mu;}}{s}_{j+1}\end{array}\right.---\left(37\right)$

Wherein, K _jadaptive matrix, v _{μ > 0};

Step 8, design Lyapunov function, process is as follows:

$V=\frac{1}{2}{S_1}^2+\frac{1}{2}\underset{i=2}{\overset{4}{\&Sigma;}}(s_i^2+y_i^2+{\tilde{W}}_{i-1}^T{K}_{i-1}^T{\tilde{W}}_{i-1}+\frac{1}{v_{\mathrm{\&mu;}}}{{\mathrm{\&mu;}}_j}^2)---\left(38\right)$

Wherein, w ^*it is ideal value;

Carry out differentiate to formula (26) to obtain:

$\stackrel{\&CenterDot;}{V}=\underset{i=1}{\overset{4}{\&Sigma;}}{s}_i{\stackrel{\&CenterDot;}{s}}_i-\underset{i=2}{\overset{4}{\&Sigma;}}\left({\tilde{W}}_{i-1}^T{K}_{i-1}^T{\hat{W}}_{i-1}^T\right)+\underset{j=2}{\overset{3}{\&Sigma;}}{v}_{\mathrm{\&mu;}}^{-1}{\widetilde{\mathrm{\&mu;}}}_j{\stackrel{\&CenterDot;}{\widehat{\mathrm{\&mu;}}}}_j+\underset{i=2}{\overset{4}{\&Sigma;}}{y}_i{\stackrel{\&CenterDot;}{y}}_i---\left(39\right)$

If then decision-making system is stable.

The present invention, in the situation of consideration unknown input dead band, designs a kind of flexible mechanical arm servo-drive system dead time compensation control method ensureing mapping, realize the stable of system and follow the tracks of fast, ensure that tracking error is at finite time convergence control.

Technical conceive of the present invention is: can not survey for state, and with the mechanical arm servo-drive system that unknown dead band inputs, utilizes Order Derivatives in Differential Mid-Value Theorem to optimize dead space arrangements, become a simple linear time varying system.Again in conjunction with neural network, inverting dynamic surface sliding formwork control and transformed error variable mapping control, the dead time compensation control method of the arm servo-drive system that designs a mechanism.Pass through Order Derivatives in Differential Mid-Value Theorem, make dead band continuously differentiable, utilize error transform, obtain new error variance, combined by the method for inversion and sliding formwork and design virtual controlling variable, the complexity explosion issues brought for avoiding the method for inversion, adds firstorder filter, and utilize neural network to estimate the derivative of virtual controlling amount, the position transient tracking realizing system controls.The invention provides a kind of can effectively avoid the method for inversion to bring complexity explosion issues and the dysgenic compensating control method of dead band input to system, realize the tenacious tracking of system and improve mapping.

Beneficial effect of the present invention is: avoid the complexity explosion issues that dead band inversion calculation operates and the method for inversion is intrinsic, simplifies controller architecture, improves system transients tracking performance and also ensures that the tenacious tracking of position signalling controls.

Accompanying drawing explanation

Fig. 1 is the schematic diagram of nonlinear dead-zone of the present invention;

Fig. 2 is the schematic diagram of tracking effect of the present invention;

Fig. 3 is the schematic diagram of tracking error of the present invention;

Fig. 4 is the schematic diagram of controller of the present invention input;

Fig. 5 is control flow chart of the present invention.

Embodiment

Below in conjunction with accompanying drawing, the present invention will be further described.

With reference to Fig. 1-Fig. 5, a kind of mechanical arm servo-drive system dead time compensation control method ensureing mapping, comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is

$\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q}+K(q-\mathrm{\&theta;})+MgLsin\left(q\right)=0\\ J\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}-K(q-\mathrm{\&theta;})=v\left(u\right)\end{array}\right.---\left(1\right)$

Wherein, q and θ is respectively the angle of robot linkage and motor; I is the inertia of connecting rod; J is the inertia of motor; K is spring rate; M and L is quality and the length of connecting rod respectively; U is control signal; V (u) is dead band, is expressed as:

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

Wherein g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, meet b _l< 0, b _r> 0;

Definition x ₁=q, x ₃=θ, formula (1) is rewritten as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=-\frac{MgL}{I}sin\left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ {\stackrel{\&CenterDot;}{x}}_3=x_4\\ {\stackrel{\&CenterDot;}{x}}_4=\frac{1}{J}v\left(u\right)+\frac{K}{J}(x_1-x_3)\\ y=x_1\&CenterDot;\end{array}\right.---\left(3\right)$

Wherein, y is system output trajectory;

1.2 defining variable z ₁=x ₁, z ₂=x ₂, $z_4=-x_2\frac{MgL}{I}cos\left(x_1\right)-\frac{K}{I}(x_2-x_4),$ Then formula (3) is rewritten into

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=f_1\left(\stackrel{\&OverBar;}{z}\right)+b_{1}v\left(u\right)\\ y=z_1\end{array}\right.---\left(4\right)$

Wherein, $\stackrel{\&OverBar;}{z}={\&lsqb;z_1,z_2,z_3\&rsqb;}^T,f_1\left(\stackrel{\&OverBar;}{z}\right)=\frac{MgL}{I}sin\left(z_1\right)(z_2^2-\frac{K}{J})-\left(\frac{MgL}{I}cos\right(z_1)+\frac{K}{J}+\frac{K}{I})z_3,b_1=\frac{K}{IJ};$

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, comprise following process;

Linear process is carried out in 2.1 pairs of non-linear unknown dead bands

Wherein | ω (u) |≤ω _n, ω _nthat unknown positive number meets ω _n=(g ' _r+ g ' _l) max{b _r, b _land

2.2 according to Order Derivatives in Differential Mid-Value Theorem, then

Wherein $g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_r},{\mathrm{\&xi;}}_r\&Element;\&lsqb;b_r+\mathrm{\&infin;});$

Wherein $g_1^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_l},{\mathrm{\&xi;}}_l\&Element;(-\mathrm{\&infin;},b_l\&rsqb;;$

Then

$\mathrm{\&omega;}\left(u\right)=\left\{\begin{array}{cc}-g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)b_r& u\left(u\right)\&GreaterEqual;b_r\\ -\&lsqb;{g^{\&prime;}}_l\left({\mathrm{\&xi;}}_l\right)+g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)\&rsqb;u\left(t\right)& b_l<u\left(t\right)<b_r\\ -g_l^{\&prime;}\left({\mathrm{\&xi;}}_l\right)b_l& u\left(t\right)\&le;b_l\end{array}\right.---\left(9\right)$

Formula (4), by formula (6) and formula (9), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu\\ y=z_1\end{array}\right.---\left(10\right)$

Wherein, $\stackrel{\&OverBar;}{z}=\&lsqb;z_1,z_2,z_3,z_4\&rsqb;,m\left(\stackrel{\&OverBar;}{z}\right)=f_1\left(\stackrel{\&OverBar;}{z}\right)+b_1*\mathrm{\&omega;}\left(u\right),$

Step 3, approaches uncertainty by neural network, and process is as follows:

Definition continuous function is:

h(X)＝W ^*Tφ(X)+ε(11)

Wherein W ^{* T}∈ R ^{n1 × n2}desirable weight matrix, φ (X) ∈ R ^{n1 × n2}be the function of desirable neural network, ε is the evaluated error of neural network, meets | ε | and≤ε _n, φ (X) functional form is:

$\mathrm{\&phi;}\left(X\right)=\frac{a}{b+\exp (-X/c)}+d---\left(12\right)$

Wherein, a, b, c, d are suitable constant;

Step 4, computing system transient control performance function and error conversion, process is as follows:

During 4.1 system transients control, controller input signal is:

u(t)＝ρ(F _φ(t),ψ(t),||e(t)||)×e(t)(13)

Wherein, e (t)=y-y _d, y _dbe desirable pursuit path, e (t) is tracking error, and ψ (t) is zoom factor, F _φt () is the border of error variance, || e (t) || be euclideam norm, in order to ensure that e (t) develops in border, time-varying gain ρ (.) is:

$\mathrm{\&rho;}\left(t\right)=\frac{1}{F_{\mathrm{\&phi;}}\left(t\right)-\left|\right|e\left(t\right)\left|\right|}---\left(14\right)$

The border of 4.2 design error variablees is:

Wherein, a continuous print positive function, to t>=0, have then

F _φ(t)＝δ ₀exp(-a ₀t)+δ _∞(16)

Wherein δ ₀>=δ _∞> 0, and | e (0) | < F _φ(0);

4.3 definition transient control error variances are:

$s_1=\frac{e\left(t\right)}{F_{\mathrm{\&phi;}}\left(t\right)-\left|\right|e\left(t\right)\left|\right|}---\left(17\right)$

Step 5, calculate system transients Properties Control dummy variable in the method for inversion, Dynamic sliding mode face and differential, process is as follows:

5.1 definition transient control dummy variable and differential thereof,

Definition error variance:

e＝y-y _d(18)

Wherein, y _dbe the ideal movements track of this system, y is that real system exports;

Then, formula (15) differentiate is obtained:

${\stackrel{\&CenterDot;}{s}}_1\left(t\right)=F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F(z_2-{\stackrel{\&CenterDot;}{y}}_d)-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_{F}e---\left(19\right)$

Wherein, φ _f=1/ (F _φ-|| e||) ²;

5.2 definition Liapunov functions:

$V_1=\frac{1}{2}{s_1}^2---\left(20\right)$

To V ₁differentiate obtains:

${\stackrel{\&CenterDot;}{V}}_1=s_1\&lsqb;F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F(z_2-{\stackrel{\&CenterDot;}{y}}_d)-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_{F}e\&rsqb;---\left(21\right)$

5.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_2={\stackrel{\&CenterDot;}{y}}_d-\frac{k_1{s}_1}{F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F}+\frac{{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}e}{F_{\mathrm{\&phi;}}}---\left(22\right)$

Wherein, k ₁for constant, and k ₁> 0;

The variable α that 5.4 definition one are new ₁, allow virtual controlling amount be τ by time constant ₁firstorder filter:

${\mathrm{\&tau;}}_1{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1+{\mathrm{\&alpha;}}_1={\stackrel{\&OverBar;}{z}}_2,{\mathrm{\&alpha;}}_1\left(0\right)={\stackrel{\&OverBar;}{z}}_2\left(0\right)---\left(23\right)$

5.5 definition filtering errors then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=\frac{{\stackrel{\&OverBar;}{z}}_2-{\mathrm{\&alpha;}}_1}{{\mathrm{\&tau;}}_1}=-\frac{y_2}{{\mathrm{\&tau;}}_1}---\left(24\right)$

5.6 estimate by neural network

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=-{W_1}^T{\mathrm{\&phi;}}_1\left(X_1\right)+{\mathrm{\&epsiv;}}_1---\left(25\right)$

Wherein $X_1={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_1}^T,{s_2}^T\&rsqb;}^T\&Element;R^5;$

Step 6, for formula (4), design virtual controlling amount, process is as follows:

6.1 definition error variances

s _i＝z _i-α _i-1,i＝2,3(26)

The first differential of formula (15) is:

${\stackrel{\&CenterDot;}{s}}_i=z_{i+1}-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_{i-1},i=2,3---\left(27\right)$

6.2 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_3=-k_2{s}_2-{\hat{W}}_1^T{\mathrm{\&phi;}}_1\left(X_1\right)-{\widehat{\mathrm{\&mu;}}}_1-F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F{s}_1---\left(28\right)$

Wherein, k ₂for constant and k ₂> 0, the estimated value of ε, w ₁estimated value;

6.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_4=-k_3{s}_3-{\hat{W}}_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\widehat{\mathrm{\&mu;}}}_2-s_2---\left(29\right)$

Wherein, k ₃for constant and k ₃> 0, the estimated value of ε, w ₂estimated value;

The variable α that 6.4 definition one are new _i, allow virtual controlling amount be τ by time constant _ifirstorder filter:

${\mathrm{\&tau;}}_i{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i+{\mathrm{\&alpha;}}_i={\stackrel{\&OverBar;}{z_i}}_{+1},{\mathrm{\&alpha;}}_i\left(0\right)={\stackrel{\&OverBar;}{z_i}}_{+1}\left(0\right)---\left(30\right)$

6.5 definition $y_{i+1}={\mathrm{\&alpha;}}_i-{\stackrel{\&OverBar;}{z_i}}_{+1},$ Then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i=\frac{{\stackrel{\&OverBar;}{z_i}}_{+1}-{\mathrm{\&alpha;}}_i}{{\mathrm{\&tau;}}_i}=-\frac{y_{i+1}}{{\mathrm{\&tau;}}_i}---\left(31\right)$

6.6 estimate by neural network

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2=-W_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\mathrm{\&epsiv;}}_2---\left(32\right)$

$-m+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3=-W_3^T{\mathrm{\&phi;}}_3\left(X_3\right)-{\mathrm{\&epsiv;}}_3---\left(33\right)$

Wherein, for ideal weight W _iestimated value, $X_i={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_i}^T,{s_{i+1}}^T\&rsqb;}^T\&Element;R^5$ In;

Step 7, CONTROLLER DESIGN inputs, and process is as follows:

7.1 definition error variances

s ₄＝z ₄-α ₃(34)

The first differential of calculating formula (20) is:

${\stackrel{\&CenterDot;}{s}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3---\left(35\right)$

7.2 CONTROLLER DESIGN input u:

$u=-k_5\left(\mathrm{\&eta;}sgn\right(s_4)+k_4{s}_4+W_3^T{\mathrm{\&phi;}}_3\left(X_3\right)+{\widehat{\mathrm{\&mu;}}}_3+s_3)---\left(36\right)$

Wherein, for ideal weight W ₃estimated value, k ₅>=1/n, ε ₃estimated value;

7.3 design adaptive rates:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{W}}}_j=K_j{\mathrm{\&phi;}}_j\left(X_{j+1}\right)s_{j+1}\\ \widehat{\mathrm{\&mu;}}=v_{\mathrm{\&mu;}}{s}_{j+1}\end{array}\right.---\left(37\right)$

Wherein, K _jadaptive matrix, v _{μ > 0};

Step 8, design Lyapunov function

$V=\frac{1}{2}{S_1}^2+\frac{1}{2}\underset{i=2}{\overset{4}{\&Sigma;}}(s_i^2+y_i^2+{\tilde{W}}_{i-1}^T{K}_{i-1}^T{\tilde{W}}_{i-1}+\frac{1}{v_{\mathrm{\&mu;}}}{{\mathrm{\&mu;}}_j}^2)---\left(38\right)$

Wherein, w ^*it is ideal value;

Carry out differentiate to formula (26) to obtain:

$\stackrel{\&CenterDot;}{V}=\underset{i=1}{\overset{4}{\&Sigma;}}{s}_i{\stackrel{\&CenterDot;}{s}}_i-\underset{i=2}{\overset{4}{\&Sigma;}}\left({\tilde{W}}_{i-1}^T{K}_{i-1}^T{\hat{W}}_{i-1}^T\right)+\underset{j=2}{\overset{3}{\&Sigma;}}{v}_{\mathrm{\&mu;}}^{-1}{\widetilde{\mathrm{\&mu;}}}_j{\stackrel{\&CenterDot;}{\widehat{\mathrm{\&mu;}}}}_j+\underset{i=2}{\overset{4}{\&Sigma;}}{y}_i{\stackrel{\&CenterDot;}{y}}_i---\left(39\right)$

If then decision-making system is stable.

For the validity of checking institute extracting method, The present invention gives the tracking performance of mechanical arm servo-drive system dead time compensation control method and the figure of tracking error that ensure mapping.The parameter of system initialization is: x ₁(0)=0, x ₂(0)=0, the parameter of neural network is: K=0.1, a=2, b=10, c=1, d=-1, is: δ to the boundary function parameter that mapping controls ₀=1, δ _∞=0.2, a ₀=0.3, the parameter of virtual controlling amount is: k ₁=1, k ₂=20, k ₃=20, k ₄=5, k ₅=1, the time constant parameter of firstorder filter is t ₂=t ₃=t ₄=0.02; System model parameter is Mgl=5, I=1, J=1, K=40, I=1; And dead band is:

$v\left(u\right(t\left)\right)=\left\{\begin{array}{cc}(1-0.3\sin (u\left)\right)(u-0.8),& u>0.8\\ 0,& -0.5<u<0.8\\ (0.8-0.2\cos (u\left)\right)(u+0.5),& u\&le;-0.5\end{array}\right..---\left(40\right)$

Follow the tracks of y _dthe signal of=0.5 (sin (t)+sin (0.5t)), as seen from Figure 2, ensures that the mechanical arm servo-drive system dead time compensation control method of mapping can well track movement locus; As can be seen from Figure 3, the tracking error of the method is very little, almost nil.As can be seen from Figure 4, with in the input control device situation of dead band, nonlinear dead-zone restriction is comparatively large, still can realize the tenacious tracking of system.Therefore, the invention provides one can the unknown dead band of effective compensation, the complexity explosion issues avoiding the method for inversion to bring demonstrate,proved system transients Properties Control method, realize the stable of system and follow 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. ensure a mechanical arm servo-drive system dead time compensation control method for mapping, it is characterized in that: described control method comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is

$\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q}+K(q-\mathrm{\&theta;})+MgLsin\left(q\right)=0\\ J\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}-K(q-\mathrm{\&theta;})=v\left(u\right)\end{array}\right.---\left(1\right)$

Wherein, q and θ is respectively the angle of robot linkage and motor; I is the inertia of connecting rod; J is the inertia of motor; K is spring rate; M and L is quality and the length of connecting rod respectively; U is control signal; V (u) is dead band, is expressed as:

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

Wherein g _l(u), g _ru () is unknown nonlinear function; b _land b _rfor the unknown width parameter in dead band, meet b _l< 0, b _r> 0; Definition x ₁=q, x ₃=θ, formula (1) is rewritten as

$\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=-\frac{MgL}{I}sin\left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ {\stackrel{\&CenterDot;}{x}}_3=x_4\\ {\stackrel{\&CenterDot;}{x}}_4=\frac{1}{J}v\left(u\right)+\frac{K}{J}(x_1-x_3)\\ y=x_1\end{array}.---\left(3\right)$

Wherein, y is system output trajectory;

1.2 defining variable z ₁=x ₁, z ₂=x ₂, $z_4=-x_2\frac{MgL}{I}cos\left(x_1\right)-\frac{K}{I}(x_2-x_4),$ Then formula (3) is rewritten into

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=f_1\left(\stackrel{\&OverBar;}{z}\right)+b_{1}v\left(u\right)\\ y=z_1\end{array}\right.---\left(4\right)$

Wherein, $\stackrel{\&OverBar;}{z}={\&lsqb;z_1,z_2,z_3\&rsqb;}^T,f_1\left(\stackrel{\&OverBar;}{z}\right)=\frac{MgL}{I}sin\left(z_1\right(z_2^2-\frac{K}{J})-(\frac{MgL}{I}cos\left(z_1\right)+\frac{K}{J}+\frac{K}{I})z_3,b_1=\frac{K}{IJ};$

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, comprise following process;

Linear process is carried out in 2.1 pairs of non-linear unknown dead bands

Wherein | ω (u) |≤ω _n, ω _nthat unknown positive number meets ω _n=(g ' _r+ g ' _l) max{b _r, b _land

2.2 according to Order Derivatives in Differential Mid-Value Theorem, then

Wherein $g_r^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_r},{\mathrm{\&xi;}}_r\&Element;\&lsqb;b_r,+\mathrm{\&infin;});$

Wherein $g_1^{\&prime;}\left({\mathrm{\&xi;}}_r\right)=\frac{\&part;v\left(\mathrm{\&xi;}\right)}{\&part;\mathrm{\&xi;}}{|}_{\mathrm{\&xi;}={\mathrm{\&xi;}}_l},{\mathrm{\&xi;}}_l\&Element;(b_r,+\mathrm{\&infin;}];$

Then

$\mathrm{\&omega;}\left(u\right)=\left\{\begin{array}{cc}-{g^{\&prime;}}_r\left({\mathrm{\&xi;}}_r\right)b_r& u\left(t\right)\&GreaterEqual;b_r\\ -\&lsqb;{g^{\&prime;}}_l\left({\mathrm{\&xi;}}_l\right)+{g^{\&prime;}}_r\left({\mathrm{\&xi;}}_r\right)\&rsqb;u\left(t\right)& b_l<u\left(t\right)<b_r\\ -{g^{\&prime;}}_l\left({\mathrm{\&xi;}}_l\right)b_l& u\left(t\right)\&le;b_l\end{array}\right.---\left(9\right)$

Formula (4), by formula (6) and formula (9), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2\\ {\stackrel{\&CenterDot;}{z}}_2=z_3\\ {\stackrel{\&CenterDot;}{z}}_3=z_4\\ {\stackrel{\&CenterDot;}{z}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu\\ y=z_1\end{array}\right.---\left(10\right)$

Wherein,

Step 3, approaches uncertainty by neural network, and process is as follows:

Definition continuous function is:

h(X)＝W ^*Tφ(X)+ε(11)

Wherein W ^{* T}∈ R ^{n1 × n2}desirable weight matrix, φ (X) ∈ R ^{n1 × n2}be the function of desirable neural network, ε is the evaluated error of neural network, meets | ε | and≤ε _n, φ (X) functional form is:

$\mathrm{\&phi;}\left(X\right)=\frac{a}{b+\exp (-X/c)}+d---\left(12\right)$

Wherein, a, b, c, d are suitable constant;

Step 4, computing system transient control performance function and error conversion, process is as follows:

During 4.1 system transients control, controller input signal is:

Wherein, e (t)=y-y ^d, y ^dbe desirable pursuit path, e (t) is tracking error, and ψ (t) is zoom factor, F _φt () is the border of error variance, || e (t) || be euclideam norm, in order to ensure that e (t) develops in border, time-varying gain ρ (.) is:

$\mathrm{\&rho;}\left(t\right)=\frac{1}{F_{\mathrm{\&phi;}}\left(t\right)-\left|\right|e\left(t\right)\left|\right|}---\left(14\right)$

The border of 4.2 design error variablees is:

Wherein, a continuous print positive function, to t>=0, have then

F _φ(t)＝δ ₀exp(-a ₀t)+δ _∞(16)

Wherein δ ₀>=δ _∞> 0, and | e (0) | < F _φ(0);

4.3 definition transient control error variances are:

Step 5, calculate system transients Properties Control dummy variable in the method for inversion, Dynamic sliding mode face and differential, process is as follows:

5.1 definition transient control dummy variable and differential thereof:

Definition error variance:

e(t)＝y-y _d(18)

Wherein, y _dbe the ideal movements track of this system, y is that real system exports;

Then, formula (15) differentiate is obtained:

Wherein, φ _f=1/ (F _φ-|| e||) ²;

5.2 definition Liapunov functions:

$V_1=\frac{1}{2}{s}_1^2---\left(20\right)$

To V ₁differentiate obtains:

5.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_2={\stackrel{\&CenterDot;}{y}}_d-\frac{k_1{s}_1}{F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_F}+\frac{{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&phi;}}e}{F_{\mathrm{\&phi;}}}---\left(22\right)$

Wherein, k ₁for constant, and k ₁> 0;

The variable α that 5.4 definition one are new ₁, allow virtual controlling amount be τ by time constant ₁firstorder filter:

${\mathrm{\&tau;}}_1{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1+{\mathrm{\&alpha;}}_1={\stackrel{\&OverBar;}{z}}_2,{\mathrm{\&alpha;}}_1\left(0\right)={\stackrel{\&OverBar;}{z}}_2\left(0\right)---\left(23\right)$

5.5 definition filtering errors then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=\frac{{\stackrel{\&OverBar;}{z}}_2-{\mathrm{\&alpha;}}_1}{{\mathrm{\&tau;}}_1}=-\frac{y_2}{{\mathrm{\&tau;}}_1}---\left(24\right)$

5.6 estimate by neural network

Wherein $X_1={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_1}^T,{s_2}^T\&rsqb;}^T\&Element;R^5;$

Step 6, for formula (4), design virtual controlling amount;

6.1 definition error variances

s _i＝z _i-α _i-1,i＝2,3(26)

The first differential of formula (15) is:

${\stackrel{\&CenterDot;}{s}}_i=z_{i+1}-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_{i-1},i=2,3---\left(27\right)$

6.2 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_3=-k_2{s}_2-{\hat{W}}_1^T{\mathrm{\&phi;}}_1\left(X_1\right)-{\widehat{\mathrm{\&mu;}}}_1-F_{\mathrm{\&phi;}}{\mathrm{\&phi;}}_{FS1}---\left(28\right)$

Wherein, k ₂for constant and k ₂> 0, the estimated value of ε, w ₁estimated value;

6.3 design virtual controlling amounts

${\stackrel{\&OverBar;}{z}}_4=-k_3{s}_3-{\hat{W}}_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\widehat{\mathrm{\&mu;}}}_2-s_2---\left(29\right)$

Wherein, k ₃for constant and k ₃> 0, the estimated value of ε, w ₂estimated value;

The variable α that 6.4 definition one are new _i, allow virtual controlling amount be τ by time constant _ifirstorder filter:

${\mathrm{\&tau;}}_i{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i+{\mathrm{\&alpha;}}_i={\stackrel{\&OverBar;}{z}}_{i+1},{\mathrm{\&alpha;}}_i\left(0\right)={\stackrel{\&OverBar;}{z}}_{i+1}\left(0\right)---\left(30\right)$

6.5 definition $y_{i+1}={\mathrm{\&alpha;}}_i-{\stackrel{\&OverBar;}{z}}_{i+1},$ Then

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_i=\frac{{\stackrel{\&OverBar;}{z}}_{i+1}-{\mathrm{\&alpha;}}_i}{{\mathrm{\&tau;}}_i}=-\frac{y_{i+1}}{{\mathrm{\&tau;}}_i}---\left(31\right)$

6.6 estimate by neural network

${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2=-W_2^T{\mathrm{\&phi;}}_2\left(X_2\right)-{\mathrm{\&epsiv;}}_2---\left(32\right)$

$-m+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3=-W_3^T{\mathrm{\&phi;}}_3\left(X_3\right)-{\mathrm{\&epsiv;}}_3---\left(33\right)$

Wherein, for ideal weight W _iestimated value, $X_i={\&lsqb;{y_d}^T,{{\stackrel{\&CenterDot;}{y}}_d}^T,{{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d}^T,{s_i}^T,{s_{i+1}}^T\&rsqb;}^T\&Element;R^5$ In;

Step 7, CONTROLLER DESIGN inputs, and process is as follows:

7.1 definition error variances

s ₄＝z ₄-α ₃(34)

The first differential of calculating formula (20) is:

${\stackrel{\&CenterDot;}{s}}_4=m\left(\stackrel{\&OverBar;}{z}\right)+nu-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_3---\left(35\right)$

7.2 CONTROLLER DESIGN input u:

$u=-k_5\left(\mathrm{\&eta;}\mathrm{sgn}\right(s_4)+k_4{s}_4+W_3^T{\mathrm{\&phi;}}_3(X_3)+{\widehat{\mathrm{\&mu;}}}_3+s_3)---\left(36\right)$

Wherein, for ideal weight W ₃estimated value, k ₅>=1/n, ε ₃estimated value;

7.3 design adaptive rates:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{W}}}_j=K_j{\mathrm{\&phi;}}_j\left(X_{j+1}\right)s_{j+1}\\ \stackrel{\&CenterDot;}{\widehat{\mathrm{\&mu;}}}=v_{\mathrm{\&mu;}}{s}_{j+1}\end{array}\right.---\left(37\right)$

Wherein, K _jadaptive matrix, v _{μ > 0};

Step 8, design Lyapunov function, process is as follows:

$V=\frac{1}{2}{s}_1^2+\frac{1}{2}\underset{i=2}{\overset{4}{\&Sigma;}}(s_i^2+y_i^2+{\tilde{W}}_{i-1}^T{K}_{i-1}^T{\tilde{W}}_{i-1}+\frac{1}{v_{\mathrm{\&mu;}}}{{\mathrm{\&mu;}}_j}^2)---\left(38\right)$

Wherein, w ^*it is ideal value;

Carry out differentiate to formula (26) to obtain:

$\stackrel{\&CenterDot;}{V}=\underset{i=1}{\overset{4}{\&Sigma;}}{s}_i{\stackrel{\&CenterDot;}{s}}_i-\underset{i=2}{\overset{4}{\&Sigma;}}\left({\tilde{W}}_{i-1}^T{K}_{i-1}^T{\tilde{W}}_{i-1}^T\right)+\underset{j=1}{\overset{3}{\&Sigma;}}{v}_{\mathrm{\&mu;}}^{-1}{\widetilde{\mathrm{\&mu;}}}_j{\stackrel{\&CenterDot;}{\widehat{\mathrm{\&mu;}}}}_j+\underset{i=2}{\overset{4}{\&Sigma;}}{y}_i{\stackrel{\&CenterDot;}{y}}_i)---\left(39\right)$

If then decision-making system is stable.