CN105573119A - Mechanical arm servo system neural network full-order sliding-mode control method for guaranteeing transient performance - Google Patents

Abstract

The invention provides a mechanical arm servo system neural network full-order sliding-mode control method for guaranteeing transient performance. Aiming at a mechanical arm servo system containing a dynamic executing mechanism and uncertain items for a system model, utilizing a full-order sliding-mode control method, and combined with a neural network, the invention designs a mechanical arm servo system neural network full-order sliding-mode control method for guaranteeing transient performance. The mechanical arm servo system neural network full-order sliding-mode control method compensates the uncertainty caused by system model parameters by approaching an unknown function through the neural network. Besides, the design of a full-order sliding-mode surface is used for guaranteeing quick and stable convergence of the system, and improving the buffeting problem by increasing a filter in the practical control system. The control method provided by the invention can improve and control the input buffeting problem, and improves the response speed for guaranteeing the transient performance so as to realize quick and stable control of the system.

Description

A kind of mechanical arm servo-drive system neural network full-order sliding mode control method ensureing mapping

Technical field

The present invention relates to a kind of mechanical arm servo-drive system neural network full-order sliding mode control method ensureing mapping, be applicable to the Position Tracking Control of the mechanical arm servo-drive system with system model indeterminate.

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 systematic uncertainty 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 its finite time convergence control.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 etc.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.Such as, but in the method for above-mentioned proposition, have impact on system transients performance to a certain extent to eliminate buffeting, the rise time is elongated.

Ensure that the method for mapping has a lot, such as BLF (barrierLyapunovfunction) controls, PPC (prescribedperformancecontrol) method and FC (funnelcontrol) method.BLF control method can the indirect restriction system tracking error of constrained system state variable, but Lyapunov function expression-form more complicated in method, and need to ensure that function can be micro-.PPC uses new error variance to ensure the steady-state error that system is specified, but there is singular value problem.FC proposes a virtual controlling variable relevant to tracking error, and is applied to by variable in the control of non-singular terminal sliding formwork.

Summary of the invention

In order to overcome the problems such as the slow and control inputs buffeting of the response speed that exists in existing mechanical arm servo system sliding-mode control method, the invention provides a kind of mechanical arm servo-drive system neural network full-order sliding mode control method ensureing mapping, based on the situation of system model Parameter uncertainties, accelerate system response time, improve the buffeting problem of control inputs, 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:

Ensure a mechanical arm servo-drive system neural network full-order sliding mode control method for mapping, 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:

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

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; U is control signal;

1.2 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:

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

$C_H(q,\stackrel{\·}{q})={\hat{C}}_H(q,\stackrel{\·}{q})+{\mathrm{\ΔC}}_H(q,\stackrel{\·}{q})$

$D_H={\hat{D}}_H+{\mathrm{\ΔD}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\ΔG}}_H\left(q\right)---\left(2\right)$

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 system model indeterminate, 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 (3) wherein, () ^trepresent transposition; 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{\φ}}_i\left(x\right)=\exp \[-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\σ}}_i^2}\],i=1,2,...,n---\left(4\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _irepresent the width of Gaussian function; The exponential function that exp () representative is the end with natural constant e;

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

3.1 define system tracking errors are

e＝q _d-q(5)

Wherein, q _dfor second order can lead desired trajectory; Then the first differential of formula (5) and second-order differential are represented as:

$\stackrel{\·}{e}={\stackrel{\·}{q}}_d-\stackrel{\·}{q}---\left(6\right)$

$\stackrel{\·\·}{e}={\stackrel{\·\·}{q}}_d-\stackrel{\·\·}{q}---\left(7\right)$

3.2 definition FC errors are

$s_1=\frac{e}{F_{\mathrm{\Φ}}\left(t\right)-\left|\right|e\left|\right|}---\left(8\right)$

Wherein,

F _Φ(t)＝δ ₀exp(-a ₀t)+δ _∞(9)

Wherein, a ₀a positive constant, δ ₀>=δ _∞> 0, | e (0) | < F _Φ(0); Then the first differential of formula (8) and second-order differential are denoted respectively as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_1=\frac{F_{\mathrm{\&Phi;}}\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}e}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e\end{array}---\left(10\right)$

$\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{s}}_1=F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+H_1\end{array}---\left(11\right)$

Wherein, ${\mathrm{\&Phi;}}_F=\frac{1}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2},{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-a_0{\mathrm{\&delta;}}_0\exp (-a_{0}t),{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-{a_0}^2{\mathrm{\&delta;}}_0\exp (-a_{0}t),$

${\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F=\frac{-2({\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}-\stackrel{\&CenterDot;}{e}\&CenterDot;sign(e\left)\right)}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^3},H_1=F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e};$

3.3 definition full-order sliding mode faces are

$s={\stackrel{\&CenterDot;\&CenterDot;}{s}}_1+c_2\mathrm{sgn}\left({\stackrel{\&CenterDot;}{s}}_1\right){\left|{\stackrel{\&CenterDot;}{s}}_1\right|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}\left(s_1\right){\left|s_1\right|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be two positive constants, choosing of they ensures polynomial expression p ²+ c ₂p+c ₁whole characteristic roots in the left-half of complex plane to ensure the stability of system; 0 < α ₁< 1,0 < α ₂< 1, choosing of they is realized 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 system model indeterminate, 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:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}({\stackrel{\&CenterDot;}{s}}_1){\left|{\stackrel{\&CenterDot;}{s}}_1\right|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}(s_1){\left|s_1\right|}^{{\mathrm{\&alpha;}}_1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(14\right)$

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

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); weight coefficient matrix is estimated in representative; 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:

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:

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, then suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; k _tchoose be require meet k when T > 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:

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

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 situation of system model Parameter uncertainties, utilize full-order sliding mode and neural network, design ensures the mechanical arm servo-drive system neural network full-order sliding mode control method of mapping, accelerates system response time, improve the buffeting of sliding formwork control inputs, ensure the convergence of system fast and stable.

Technical conceive of the present invention is: for containing Dynamic Execution mechanism, and the mechanical arm servo-drive system containing system model indeterminate, utilize full-order sliding mode control method, again in conjunction with neural network, design a kind of mechanical arm servo-drive system neural network full-order sliding mode control method ensureing mapping.Approach unknown function by neural network, compensate for the uncertainty that system model parameter causes.In addition, the design in full-order sliding mode face is the fast and stable convergence in order to ensure system, and improves buffeting problem by increasing wave filter in the control system of reality.The invention provides one and improve control inputs buffeting problem, and accelerate the control method that response speed ensures mapping, the fast and stable realizing system controls.

Advantage of the present invention is: allow system model parameter to there is indeterminate, ensure mapping, improve 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 (a)-Fig. 4, a kind of mechanical arm servo-drive system neural network full-order sliding mode 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 formula of 1.1 mechanical arm servo-drive systems is:

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

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; U is control signal;

1.2 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:

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

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

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

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\&Delta;G}}_H\left(q\right)---\left(2\right)$

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 system model indeterminate, 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(3)

Wherein, () ^trepresent transposition; 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:

${\mathrm{\&phi;}}_i\left(x\right)=\exp \&lsqb;-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\&sigma;}}_i^2}\&rsqb;,i=1,2,...,n---\left(4\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _irepresent the width of Gaussian function; The exponential function that exp () representative is the end with natural constant e;

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

3.1 define system tracking errors are

e＝q _d-q(5)

Wherein, q _dfor second order can lead desired trajectory; Then the first differential of formula (5) and second-order differential are represented as:

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;}{q}---\left(6\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{e}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(7\right)$

3.2 definition FC errors are

$s_1=\frac{e}{F_{\mathrm{\&Phi;}}\left(t\right)-\left|\right|e\left|\right|}---\left(8\right)$

Wherein,

F _Φ(t)＝δ ₀exp(-a ₀t)+δ _∞(9)

Wherein, a ₀a positive constant, δ ₀>=δ _∞> 0, | e (0) | < F _Φ(0); Then the first differential of formula (8) and second-order differential are denoted respectively as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_1=\frac{F_{\mathrm{\&Phi;}}\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}e}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e\end{array}---\left(10\right)$

$\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{s}}_1=F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+H_1\end{array}---\left(11\right)$

Wherein, ${\mathrm{\&Phi;}}_F=\frac{1}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2},{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-a_0{\mathrm{\&delta;}}_0\exp (-a_{0}t),{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-{a_0}^2{\mathrm{\&delta;}}_0\exp (-a_{0}t),$

${\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F=\frac{-2({\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}-\stackrel{\&CenterDot;}{e}\&CenterDot;sign(e\left)\right)}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^3},H_1=F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e};$

3.3 definition full-order sliding mode faces are

$s={\stackrel{\&CenterDot;\&CenterDot;}{s}}_1+c_2\mathrm{sgn}\left({\stackrel{\&CenterDot;}{s}}_1\right){\left|{\stackrel{\&CenterDot;}{s}}_1\right|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}\left(s_1\right){\left|s_1\right|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be two positive constants, choosing of they ensures polynomial expression p ²+ c ₂p+c ₁whole characteristic roots in the left-half of complex plane to ensure the stability of system; 0 < α ₁< 1,0 < α ₂< 1, choosing of they is realized 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 system model indeterminate, 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:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}({\stackrel{\&CenterDot;}{s}}_1){\left|{\stackrel{\&CenterDot;}{s}}_1\right|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}(s_1){\left|s_1\right|}^{{\mathrm{\&alpha;}}_1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(14\right)$

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

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); weight coefficient matrix is estimated in representative; 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:

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:

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, then suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; k _tchoose be require meet k when T > 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:

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

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, $a_{11}=(m_1+m_2){r_1}^2+m_2{r_2}^2+2m_2{r}_1{r}_{2}cos\left(q_2\right)+J,$ a ₁₂＝m ₂r ₂+m ₂r ₁r ₂cos(q 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 two kinds of methods and contrasts:

S1: general neural Sliding-Mode Control Based on Network method;

S2: the neural network full-order sliding mode control method ensureing mapping;

Contrast in order to more effective, all parameters of system are all consistent, that is: q ₁(0)=0.1, q ₂(0)=0.1, 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=7/20, α ₂=2/5, c ₁=5000, c ₂=1900, 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; And tracking signal is: y _d1=y _d2=sin (2 π t).

As can be seen from figure mono-, for joint one, during 1.3 seconds to 1.5 seconds, S2 method is compared S1 method and is followed the tracks of more rapidly and gone up descending trajectory, effectively ensure that mapping, and tracking error controls, in the interval of ± 0.05 radian, saltus step not to occur; As can be seen from figure bis-, for joint two, during 1.2 seconds to 1.4 seconds, S2 method is compared S1 method and is followed the tracks of more rapidly and gone up descending trajectory, effectively ensure that mapping, and tracking error controls, in the interval of ± 0.07 radian, saltus step not to occur; There is serious chattering phenomenon as can be seen from the control inputs of figure tri-, method S1, and the control inputs of method S2 compares S1 more smoothly, effectively improves buffeting problem.

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 neural network full-order sliding mode 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 formula of 1.1 mechanical arm servo-drive systems is:

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

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; U is control signal;

1.2 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:

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

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

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

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\&Delta;G}}_H\left(q\right)---\left(2\right)$

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 system model indeterminate, 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(3)

Wherein, () ^trepresent transposition; 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:

${\mathrm{\&phi;}}_i\left(x\right)=\exp \&lsqb;-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\&sigma;}}_i^2}\&rsqb;,i=1,2,...,n---\left(4\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _irepresent the width of Gaussian function; The exponential function that exp () representative is the end with natural constant e;

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

3.1 define system tracking errors are

e＝q _d-q(5)

Wherein, q _dfor second order can lead desired trajectory; Then the first differential of formula (5) and second-order differential are represented as:

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;}{q}---\left(6\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{e}={\stackrel{\&CenterDot;\&CenterDot;}{q}}_d-\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(7\right)$

3.2 definition FC errors are

$s_1=\frac{e}{F_{\mathrm{\&Phi;}}\left(t\right)-\left|\right|e\left|\right|}---\left(8\right)$

Wherein,

F _Φ(t)＝δ ₀exp(-a ₀t)+δ _∞(9)

Wherein, a ₀a positive constant, δ ₀>=δ _∞> 0, | e (0) | < F _Φ(0); Then the first differential of formula (8) and second-order differential are denoted respectively as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_1=\frac{F_{\mathrm{\&Phi;}}\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}e}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e\end{array}---\left(10\right)$

$\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{s}}_1=F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}\\ =F_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;\&CenterDot;}{e}+H_1\end{array}---\left(11\right)$

Wherein, ${\mathrm{\&Phi;}}_F=\frac{1}{{(F_{\mathrm{\&Phi;}}-|\left|e\right|\left|\right)}^2},{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-a_0{\mathrm{\&delta;}}_0\exp (-a_{0}t),{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}=-{a_0}^2{\mathrm{\&delta;}}_0\exp (-a_{0}t),$ ${\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F=\frac{-2\left({\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}-\stackrel{\&CenterDot;}{e}\&CenterDot;sign\left(e\right)\right)}{{\left(F_{\mathrm{\&Phi;}}-\left|\right|e\left|\right|\right)}^3},H_1=F_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_F\stackrel{\&CenterDot;}{e}+{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e}-{\stackrel{\&CenterDot;\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{F}e-{\stackrel{\&CenterDot;}{F}}_{\mathrm{\&Phi;}}{\mathrm{\&Phi;}}_F\stackrel{\&CenterDot;}{e};$

3.3 definition full-order sliding mode faces are

$s={\stackrel{\&CenterDot;\&CenterDot;}{s}}_1+c_{2}sgn\left({\stackrel{\&CenterDot;}{s}}_1\right)|{\stackrel{\&CenterDot;}{s}}_1{|}^{{\mathrm{\&alpha;}}_2}+c_{1}sgn\left(s_1\right)|s_1{|}^{{\mathrm{\&alpha;}}_1}---\left(12\right)$

Wherein, c ₁and c ₂be two positive constants, choosing of they ensures polynomial expression p ²+ c ₂p+c ₁whole characteristic roots in the left-half of complex plane to ensure the stability of system; 0 < α ₁< 1,0 < α ₂< 1, choosing of they is realized 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 system model indeterminate, 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:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+c_2\mathrm{sgn}({\stackrel{\&CenterDot;}{s}}_1\left)\right|{\stackrel{\&CenterDot;}{s}}_1{|}^{{\mathrm{\&alpha;}}_2}+c_1\mathrm{sgn}\left(s_1\right)|s_1{|}^{{\mathrm{\&alpha;}}_1}\\ +u_0)+{\hat{C}}_H(q,\stackrel{\&CenterDot;}{q})\stackrel{\&CenterDot;}{q}+{\hat{D}}_H\stackrel{\&CenterDot;}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(14\right)$

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

v＝-(k _d+k _T+η)sgn(s)(17)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (12); weight coefficient matrix is estimated in representative; 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:

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:

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, then suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; k _tchoose be require meet k when T > 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:

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

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.