CN105182745A - Mechanical-arm servo-system neural-network full-order sliding mode control method with dead-zone compensation - Google Patents

Abstract

A mechanical-arm servo-system neural-network full-order sliding mode control method with dead-zone compensation is disclosed. Aiming at a mechanical arm servo system which contains a dynamic execution mechanism and is with unknown dead-zone input, a full-order sliding mode control method is used and a neural network is combined so as to design the mechanical-arm servo-system neural-network full-order sliding mode control method with the dead-zone compensation. A dead zone is converted into a linear time-varying system, and then the neural network is used to approach an unknown function so as to compensate an additional influence of a traditional unknown dead zone and an unknown parameter of the system. In addition, a full-order sliding mode surface is designed so as to guarantee rapid and stable convergence of the system; generation of a differential term is avoided in an actual control system so that buffeting is improved and a singular problem is solved. The invention provides the control method which can improve a buffeting problem of the sliding mode surface, solve the singular problem and can effectively compensate a system unknown dynamic parameter and unknown dead zone input so that rapid and stable control of the system is realized.

Description

A kind of mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation

Technical field

The present invention relates to a kind of mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation, particularly with the mechanical arm servo system control method of the input of unknown dead band and the unknown dynamic parameter of system.

Background technology

Mechanical arm servo-drive system, as the increasingly automated equipment of one, 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.But the input of unknown dead band 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, sliding formwork control etc.

Sliding formwork controls to be considered to an effective robust control method in and external disturbance uncertain at resolution system.The advantages such as sliding-mode control has that algorithm is simple, fast response time, to external world noise and Parameter Perturbation strong robustness.Therefore, sliding-mode control is widely used in every field.Contrast conventional linear sliding formwork controls, and the superiority of TSM control is that his finite time is received.But TSM control discontinuous switching characteristic in itself will cause the buffeting of system, becomes the obstacle that TSM control is applied in systems in practice.In order to address this problem, the method for many improvement is suggested in succession, such as high_order sliding mode control method, observer control method.In these methods, choosing of sliding-mode surface all obtains according to idealized system parameter depression of order.Recently, a kind of full-order sliding mode control method is suggested, and this method well avoids buffeting problem and makes system input signal more level and smooth in the response of system.

But in the method for above-mentioned proposition, the dynamic model parameters of mechanical system all must be known in advance.Therefore, the uncertain factor of system and the input of unknown dead band can affect the direct application of mechanical arm servo-drive system.As everyone knows, because neural network approaches the ability of any smooth function in an arbitrary accuracy compacted, therefore it has been widely used in the non-intellectual of disposal system and nonlinear problem.For these reasons, many adaptive neural network control methods are used to control nonlinearity and the mechanical arm system containing the input of unknown dead band.

Summary of the invention

Dead band cannot be avoided to input systematic influence and there is the deficiency that sliding formwork controls buffeting problem to overcome existing mechanical arm servo-drive system, the invention provides a kind of mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation, achieve the effective compensation to unknown dead band, improve buffeting problem and the singular problem of system, ensure the convergence of system fast and stable.

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

With a mechanical arm servo-drive system neural network full-order sliding mode control method for dead area compensation, comprise 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 formula of 1.1 mechanical arm servo-drive systems is:

$M_H\left(q\right)\stackrel{\·\·}{q}+C_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+D_H\stackrel{\·}{q}+G_H\left(q\right)=T\left(\mathrm{\τ}\right)---\left(1\right)$

Wherein, q, with be respectively the position of joint of mechanical arm, speed and acceleration; M _h, C _hand D _hrepresent the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G _hrepresent gravity item; τ is control signal; T (τ) is dead band, is expressed as:

$T\left(\mathrm{\&tau;}\right)=\left\{\begin{array}{ccc}{g}_r(\mathrm{\&tau;}-b_r),& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ 0,& if& b_l<\mathrm{\&tau;}<b_r\\ g_1(\mathrm{\&tau;}-b_l),& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.---\left(2\right)$

Wherein, g _rand g _lrepresent the left slope in dead band and right slope respectively; b _rand b _lrepresent the unknown width parameter in dead band, meet g _r> 0, g _l> 0, b _r> 0, b _l< 0;

Formula (2) is expressed as by 1.2:

T(τ)＝g(t)τ+b(t)(3)

Wherein, $g\left(t\right)=\left\{\begin{array}{ccc}{g}_r\left(t\right),& if& \mathrm{\&tau;}>0\\ g_l\left(t\right),& if& \mathrm{\&tau;}\&le;0\end{array}\right.$ And $b\left(t\right)=\left\{\begin{array}{ccc}-g_r{b}_r,& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ -g\left(t\right)\mathrm{\&tau;},& if& b_1<\mathrm{\&tau;}<b_r\\ -g_l{b}_l,& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.;$

1.3 through types (2) and (3), definition:

B(t)＝(g(t)-g _l(t))τ+b(t)(4)

Then formula (3) is represented as:

T(τ)＝g _l(t)τ+B(t)(5)

Therefore, relational expression is obtained || B (t) ||≤B _m=max ((2*g _l+ g _r) b _r, (2*g _l+ g _r) b _l);

1.4 owing to existing measurement noises, the impact of load variations and external interference, and the systematic parameter in formula (1) can not obtain accurately, therefore the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\&Delta;M}}_H\left(q\right)$

$C_H(q,\stackrel{\&CenterDot;}{q})={\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})+{\mathrm{\&Delta;C}}_H(q,\stackrel{\&CenterDot;}{q})$

$D_H={\hat{D}}_H+{\mathrm{\&Delta;D}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\&Delta;G}}_H\left(q\right)---\left(6\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system containing the input of unknown dead band and unknown parameter, the neural network of design, process is as follows:

Definition θ ^*for ideal weight matrix of coefficients, then nonlinear uncertain function f approached for:

F=θ ^{* T}φ (x)+ε (7) wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function:

${\mathrm{\&phi;}}_i\left(x\right)=\exp \&lsqb;-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\&sigma;}}_i^2}\&rsqb;,i=1,2,\mathrm{...},n---\left(8\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, computing system tracking error, design full-order sliding mode face, process is as follows:

3.1 define system tracking errors are

e＝q _d-q(9)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (9) and second-order differential are represented as:

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;}{q}---\left(10\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{e}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(11\right)$

3.2 therefore, and in order to avoid singular problem, full-order sliding mode face will be defined as:

$s=\stackrel{\&CenterDot;\&CenterDot;}{e}+c_{2}sgn\left(\stackrel{\&CenterDot;}{e}\right)|\stackrel{\&CenterDot;}{e}{|}^{{\mathrm{\&alpha;}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure the stability of system; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1=\mathrm{\&alpha;},n=1\\ {\mathrm{\&alpha;}}_{i-1}=\frac{{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{i+1}}{2{\mathrm{\&alpha;}}_{i+1}-{\mathrm{\&alpha;}}_i},i=2,...,n,\&ForAll;n\&GreaterEqual;2\end{array}\right.---\left(13\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm system containing the input of unknown dead band, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (1), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}\mathrm{\&tau;}=\frac{1}{g_l}\left({\hat{M}}_H\left(q\right)\right({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}\left(\stackrel{\&CenterDot;}{e}\right){\left|\stackrel{\&CenterDot;}{e}\right|}^{\mathrm{\&alpha;}2}+c_1\mathrm{sgn}\left(e\right){\left|e\right|}^{\mathrm{\&alpha;}1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H(q\left)\right)\end{array}---\left(14\right)$

$u_0={\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(15\right)$

${\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=v---\left(16\right)$

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); k _d, k _tbe all constant with η, and will be described afterwards;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}=\mathrm{\&Gamma;}\mathrm{\&phi;}\left(x\right)s^T---\left(18\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (14) to be brought in (1) and is obtained following equation by 4.3:

$s={\mathrm{\&theta;}}^{{*}^T}\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}-{\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+u_n={\widetilde{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}+u_n=d(q,t)+u_n---\left(19\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

Through type (1), formula (12), formula (14)-Shi (17) and formula (19), full-order sliding mode face is expressed as following equation:

s＝d(q,t)+u _n(20)

Formula (17) is brought in formula (16) and obtains:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\left(t_0\right)+\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\right)e^{t-t_0}\\ -\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\end{array}---\left(21\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)|(22)

4.4 design Liapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(23\right)$

Carry out differentiate to formula (12) to obtain:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\&CenterDot;}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(24\right)$

Formula (16) is brought in formula (24) and obtains:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)-(k_d+k_T+\mathrm{\&eta;})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(25\right)$

Carry out differential to formula (23) to obtain:

$\stackrel{\&CenterDot;}{V}=s^T\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)s^T-(k_d+k_T+\mathrm{\&eta;})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(26\right)$

Formula (22) is brought in formula (26), if then decision-making system is stable.

The present invention is based on the input of unknown dead band and unknown nonlinear factor, full-order sliding mode and neural network, design the mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation, realizes system stability and control, improve the buffeting that sliding formwork controls, ensure the convergence of system fast and stable.

Technical conceive of the present invention is: for containing Dynamic Execution mechanism, and with the mechanical arm servo-drive system that unknown dead band inputs, utilize full-order sliding mode control method, then in conjunction with neural network, design a kind of mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation.Dead band is converted into a linear time varying system, then approaches unknown function by neural network, compensate for traditional added influence in unknown dead band and the unknown parameter of system.In addition, the design in full-order sliding mode face is the fast and stable convergence in order to ensure system, and by avoiding occurring differential term to improve buffeting and singular problem in the control system of reality.The invention provides a kind of buffeting problem and the solution singular problem that can improve sliding-mode surface, and the control method of the unknown dynamic parameter of energy effective compensation system and the input of unknown dead band, the fast and stable realizing system controls.

Advantage of the present invention is: the non-dynamic parameter of bucking-out system and the input of unknown dead band, improves buffeting problem, realizes fast and stable convergence.

Accompanying drawing explanation

The position tracking effect schematic diagram that Fig. 1 (a) is joint one of the present invention.

The position tracking error schematic diagram that Fig. 1 (b) is joint one of the present invention.

The position tracking effect schematic diagram that Fig. 2 (a) is joint two of the present invention.

The position tracking error schematic diagram that Fig. 2 (b) is joint two of the present invention.

The controller input schematic diagram that Fig. 3 (a) is joint one of the present invention.

The controller input schematic diagram that Fig. 3 (b) is joint two of the present invention.

Fig. 4 is control flow schematic diagram of the present invention.

Embodiment

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

With reference to Fig. 1-Fig. 4, a kind of mechanical arm servo-drive system neural network full-order sliding mode control method with dead area compensation, 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 formula of 1.1 mechanical arm servo-drive systems is:

$M_H\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+D_H\stackrel{\&CenterDot;}{q}+G_H\left(q\right)=T\left(\mathrm{\&tau;}\right)---\left(1\right)$

Wherein, q, with be respectively the position of joint of mechanical arm, speed and acceleration; M _h, C _hand D _hrepresent the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G _hrepresent gravity item; τ is control signal; T (τ) is dead band, is expressed as:

$T\left(\mathrm{\&tau;}\right)=\left\{\begin{array}{ccc}{g}_r(\mathrm{\&tau;}-b_r),& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ 0,& if& b_l<\mathrm{\&tau;}<b_r\\ g_1(\mathrm{\&tau;}-b_l),& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.---\left(2\right)$

Wherein, g _rand g _lrepresent the left slope in dead band and right slope respectively; b _rand b _lrepresent the unknown width parameter in dead band, meet g _r> 0, g _l> 0, b _r> 0, b _l< 0;

Formula (2) is expressed as by 1.2:

T(τ)＝g(t)τ+b(t)(3)

Wherein, $g\left(t\right)=\left\{\begin{array}{ccc}{g}_r\left(t\right),& if& \mathrm{\&tau;}>0\\ g_l\left(t\right),& if& \mathrm{\&tau;}\&le;0\end{array}\right.$ And $b\left(t\right)=\left\{\begin{array}{ccc}-g_r{b}_r,& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ -g\left(t\right)\mathrm{\&tau;},& if& b_1<\mathrm{\&tau;}<b_r\\ -g_l{b}_l,& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.;$

1.3 through types (2) and (3), definition:

B(t)＝(g(t)-g _l(t))τ+b(t)(4)

Then formula (3) is represented as:

T(τ)＝g _l(t)τ+B(t)(5)

Therefore, relational expression is obtained || B (t) ||≤B _m=max ((2*g _l+ g _r) b _r, (2*g _l+ g _r) b _l);

1.4 owing to existing measurement noises, the impact of load variations and external interference, and the systematic parameter in formula (1) can not obtain accurately, therefore the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\&Delta;M}}_H\left(q\right)$

$C_H(q,\stackrel{\&CenterDot;}{q})={\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})+{\mathrm{\&Delta;C}}_H(q,\stackrel{\&CenterDot;}{q})$

$D_H={\hat{D}}_H+{\mathrm{\&Delta;D}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\&Delta;G}}_H\left(q\right)---\left(6\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system containing the input of unknown dead band and unknown parameter, the neural network of design, process is as follows:

Definition θ ^*for ideal weight matrix of coefficients, then nonlinear uncertain function f approached for:

f＝θ ^*Tφ(x)+ε(7)

Wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function:

${\mathrm{\&phi;}}_i\left(x\right)=\exp \&lsqb;-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\&sigma;}}_i^2}\&rsqb;,i=1,2,\mathrm{...},n---\left(8\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, computing system tracking error, design full-order sliding mode face, process is as follows:

3.1 define system tracking errors are

e＝q _d-q(9)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (9) and second-order differential are represented as:

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;}{q}---\left(10\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{e}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(11\right)$

3.2 therefore, and in order to avoid singular problem, full-order sliding mode face will be defined as:

$s=\stackrel{\&CenterDot;\&CenterDot;}{e}+c_{2}sgn\left(\stackrel{\&CenterDot;}{e}\right)|\stackrel{\&CenterDot;}{e}{|}^{{\mathrm{\&alpha;}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure the stability of system; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1=\mathrm{\&alpha;},n=1\\ {\mathrm{\&alpha;}}_{i-1}=\frac{{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{i+1}}{2{\mathrm{\&alpha;}}_{i+1}-{\mathrm{\&alpha;}}_i},i=2,...,n,\&ForAll;n\&GreaterEqual;2\end{array}\right.---\left(13\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm system containing the input of unknown dead band, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (1), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}\mathrm{\&tau;}=\frac{1}{g_l}\left({\hat{M}}_H\left(q\right)\right({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}\left(\stackrel{\&CenterDot;}{e}\right){\left|\stackrel{\&CenterDot;}{e}\right|}^{\mathrm{\&alpha;}2}+c_1\mathrm{sgn}\left(e\right){\left|e\right|}^{\mathrm{\&alpha;}1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H(q\left)\right)\end{array}---\left(14\right)$

$u_0={\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(15\right)$

${\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=v---\left(16\right)$

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); k _d, k _tbe all constant with η, and will be described afterwards;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}=\mathrm{\&Gamma;}\mathrm{\&phi;}\left(x\right)s^T---\left(18\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (14) to be brought in (1) and is obtained following equation by 4.3:

$s={\mathrm{\&theta;}}^{{*}^T}\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}-{\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+u_n={\widetilde{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}+u_n=d(q,t)+u_n---\left(19\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

Through type (1), formula (12), formula (14)-Shi (17) and formula (19), full-order sliding mode face is expressed as following equation:

s＝d(q,t)+u _n(20)

Formula (17) is brought in formula (16) and obtains:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\left(t_0\right)+\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\right)e^{t-t_0}\\ -\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\end{array}---\left(21\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)|(22)

4.4 design Liapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(23\right)$

Carry out differentiate to formula (12) to obtain:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\&CenterDot;}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(24\right)$

Formula (16) is brought in formula (24) and obtains:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)-(k_d+k_T+\mathrm{\&eta;})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(25\right)$

Carry out differential to formula (23) to obtain:

$\stackrel{\&CenterDot;}{V}=s^T\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)s^T-(k_d+k_T+\mathrm{\&eta;})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(26\right)$

Formula (22) is brought in formula (26), if then decision-making system is stable.

In order to obtain the corresponding system parameter value in formula (2), we provide the mechanical arm servo-drive system expression formula in following two joints:

$\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{12}& a_{22}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_1\\ {\stackrel{\&CenterDot;\&CenterDot;}{q}}_2\end{array}\right]+\left[\begin{array}{cc}-b_{12}{\stackrel{\&CenterDot;}{q}}_1& -2b_{12}{\stackrel{\&CenterDot;}{q}}_2\\ 0& b_{12}{\stackrel{\&CenterDot;}{q}}_2\end{array}\right]\left[\begin{array}{c}{\stackrel{\&CenterDot;}{q}}_1\\ {\stackrel{\&CenterDot;}{q}}_2\end{array}\right]+\left[\begin{array}{c}{c}_{1}g\\ c_{2}g\end{array}\right]=\left[\begin{array}{c}{\mathrm{\&tau;}}_1\\ {\mathrm{\&tau;}}_2\end{array}\right]+\left[\begin{array}{c}{\mathrm{\&tau;}}_{d_1}\\ {\mathrm{\&tau;}}_{d_2}\end{array}\right]---\left(27\right)$

Wherein, a ₁₁=(m ₁+ m ₂) r ₁ ²+ m ₂r ₂ ²+ 2m ₂r ₁r ₂cos (q ₂)+J, a ₁₂=m ₂r ₂+ m ₂r ₁r ₂cos (q ₂), a ₂₂=m ₂r ₂ ²+ J ₂, b ₁₂=m ₂r ₁r ₂sin (q ₂), c ₁=(m ₁+ m ₂) r ₁cos (q ₂)+m ₂r ₂cos (q ₁+ q ₂), c ₂=m ₂r ₂cos (q ₁+ q ₂), ${\mathrm{\&tau;}}_d=\left[\begin{array}{c}{\mathrm{\&tau;}}_{d_1}\\ {\mathrm{\&tau;}}_{d_2}\end{array}\right].$

The validity of extracting method in order to verify, The present invention gives following three kinds of methods and contrasts:

S1: the neural network full-order sliding mode control method not with dead area compensation;

S2: the neural network full-order sliding mode control method of band dead area compensation;

S3: the neural network finite-time control with dead area compensation: method is carried out.

Contrast in order to more effective, all parameter of system is all consistent, that is: q ₁(0)=0.5, q ₂(0)=0.5, J _m=diag (0.67 × 10 ^-4, 0.42 × 10 ^-4), D _m=diag (0.21,0.15), N=diag (9,1), and given system disturbance is: system control signal parameter is: K _τ=diag (19/40,19/80), Γ=diag (50,50), α ₁=13/27, α ₂=13/27, c ₁=100, c ₂=40, T=1; Mechanical arm actual parameter is: r ₁=0.2, r ₂=0.18, m ₁=2.3, m ₂=0.6, J ₁=0.02, J ₂=0.003, g=9.8, makes k=k _d+ k _t+ η=10; Neural network comprises 15 nodes, i.e. n=15; Neural network width is: σ _i=4, i=1,2 ..., 15; Deadzone parameter is: b _r=0.8, b _l=-0.5, g _r=1.2, g _l=0.5; And tracking signal is: y _d1=y _d2=0.5 (sin (2 π t)+cos (π t)).

From Fig. 1, we find out, owing to there is the impact of dead band input, S1 method is the poorest on the tracking effect in joint one, also there is maximum tracking error, and S3 method has better tracking effect than S1 method, and S2 method has best tracking effect.For joint 2, the existence due to dead band causes M in formula (1) _hthere is unusual appearance, therefore make S1 method lose efficacy; From Fig. 2, we obtain, and same S2 method has better tracking effect and less tracking error than S3 method.As can be seen from Figure 3, S3 method has obvious chattering phenomenon, but S2 method then effectively improves this chattering phenomenon.

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., with a mechanical arm servo-drive system neural network full-order sliding mode control method for dead area compensation, 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 formula of 1.1 mechanical arm servo-drive systems is:

$M_H\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+D_H\stackrel{\&CenterDot;}{q}+G_H\left(q\right)=T\left(\mathrm{\&tau;}\right)---\left(1\right)$

Wherein, q, with be respectively the position of joint of mechanical arm, speed and acceleration; M _h, C _hand D _hrepresent the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G _hrepresent gravity item; τ is control signal; T (τ) is dead band, is expressed as:

$T\left(\mathrm{\&tau;}\right)=\left\{\begin{array}{ccc}{g}_r(\mathrm{\&tau;}-b_r),& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ 0,& if& b_l<\mathrm{\&tau;}<b_r\\ g_1(\mathrm{\&tau;}-b_l),& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.---\left(2\right)$

Wherein, g _rand g _lrepresent the left slope in dead band and right slope respectively; b _rand b _lrepresent the unknown width parameter in dead band, meet g _r> 0, g _l> 0, b _r> 0, b _l< 0;

Formula (2) is expressed as by 1.2:

T(τ)＝g(t)τ+b(t)(3)

Wherein, $g\left(t\right)=\left\{\begin{array}{ccc}{g}_r\left(t\right),& if& \mathrm{\&tau;}>0\\ g_l\left(t\right),& if& \mathrm{\&tau;}\&le;0\end{array}\right.$ And $b\left(t\right)=\left\{\begin{array}{ccc}-g_r{b}_r,& if& \mathrm{\&tau;}\&GreaterEqual;b_r\\ -g\left(t\right)\mathrm{\&tau;},& if& b_1<\mathrm{\&tau;}<b_r\\ -g_l{b}_l,& if& \mathrm{\&tau;}\&le;b_1\end{array}\right.;$

1.3 through types (2) and (3), definition:

B(t)＝(g(t)-g _l(t))τ+b(t)(4)

Then formula (3) is represented as:

T(τ)＝g _l(t)τ+B(t)(5)

Therefore, relational expression is obtained || B (t) ||≤B _m=max ((2*g _l+ g _r) b _r, (2*g _l+ g _r) b _l);

1.4 owing to existing measurement noises, the impact of load variations and external interference, and the systematic parameter in formula (1) can not obtain accurately, therefore the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\&Delta;M}}_H\left(q\right)$

$C_H(q,\stackrel{\&CenterDot;}{q})={\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})+{\mathrm{\&Delta;C}}_H(q,\stackrel{\&CenterDot;}{q})$

$D_H={\hat{D}}_H+{\mathrm{\&Delta;D}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\&Delta;G}}_H\left(q\right)---\left(6\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system containing the input of unknown dead band and unknown parameter, the neural network of design, process is as follows:

Definition θ ^*for ideal weight matrix of coefficients, then nonlinear uncertain function f approached for:

f＝θ ^*Tφ(x)+ε(7)

Wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function:

${\mathrm{\&phi;}}_i\left(x\right)=\exp \&lsqb;-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\&sigma;}}_i^2}\&rsqb;,i=1,2,\mathrm{...},n---\left(8\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, computing system tracking error, design full-order sliding mode face, process is as follows:

3.1 define system tracking errors are

e＝q _d-q(9)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (9) and second-order differential are represented as:

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;}{q}---\left(10\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{e}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(11\right)$

3.2 therefore, and in order to avoid singular problem, full-order sliding mode face will be defined as:

$s=\stackrel{\&CenterDot;\&CenterDot;}{e}+c_{2}sgn\left(\stackrel{\&CenterDot;}{e}\right)|\stackrel{\&CenterDot;}{e}{|}^{{\mathrm{\&alpha;}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure the stability of system; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1=\mathrm{\&alpha;},n=1\\ {\mathrm{\&alpha;}}_{i-1}=\frac{{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{i+1}}{2{\mathrm{\&alpha;}}_{i+1}-{\mathrm{\&alpha;}}_i},i=2,\mathrm{...},n,\&ForAll;n\&GreaterEqual;2\end{array}\right.---\left(13\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm system containing the input of unknown dead band, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (1), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}\mathrm{\&tau;}=\frac{1}{g_l}({\hat{M}}_H\left(q\right)({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}\left(\stackrel{\&CenterDot;}{e}\right){\left|\stackrel{\&CenterDot;}{e}\right|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}\left(e\right){\left|e\right|}^{{\mathrm{\&alpha;}}_1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H\left(q\right))\end{array}---\left(14\right)$

$u_0={\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(15\right)$

${\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=v---\left(16\right)$

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); k _d, k _tall constant with η;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}=\mathrm{\&Gamma;}\mathrm{\&phi;}\left(x\right)s^T---\left(18\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (14) to be brought in (1) and is obtained following equation by 4.3:

$s={\mathrm{\&theta;}}^{{*}^T}\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}-{\widehat{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+u_n={\widetilde{\mathrm{\&theta;}}}^T\mathrm{\&phi;}\left(x\right)+\mathrm{\&epsiv;}+u_n=d(q,t)+u_n---\left(19\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

Through type (1), formula (12), formula (14)-Shi (17) and formula (19), full-order sliding mode face is expressed as following equation:

s＝d(q,t)+u _n(20)

Formula (17) is brought in formula (16) and obtains:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\left(t_0\right)+\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\right)e^{t-t_0}\\ -\left(1/T\right)\left(k_d+k_T+\mathrm{\&eta;}\right)\mathrm{sgn}\left(s\right)\end{array}---\left(21\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)|(22)

4.4 design Liapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(23\right)$

Carry out differentiate to formula (12) to obtain:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n=\stackrel{\&CenterDot;}{d}(q,t)+{\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\&CenterDot;}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(24\right)$

Formula (16) is brought in formula (24) and obtains:

$\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)-(k_d+k_T+\mathrm{\&eta;})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(25\right)$

Carry out differential to formula (23) to obtain:

$\stackrel{\&CenterDot;}{V}=s^T\stackrel{\&CenterDot;}{s}=\stackrel{\&CenterDot;}{d}(q,t)s^T-(k_d+k_T+\mathrm{\&eta;})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(26\right)$

Formula (22) is brought in formula (26), if then decision-making system is stable.