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

Full-order sliding mode control method for neural network of mechanical arm servo system for guaranteeing transient performance

Technical Field

The invention relates to a full-order sliding mode control method of a neural network of a mechanical arm servo system for ensuring transient performance, which is suitable for position tracking control of the mechanical arm servo system with a system model uncertainty item.

Background

As a highly automated device, a mechanical arm servo system is widely applied to high-performance systems such as robots, aviation aircrafts and the like, and how to realize the quick and accurate control of the mechanical arm servo system becomes a hotspot problem. However, system uncertainties are widely present in robot servo systems and often result in reduced or even failure of the control system. In order to solve the control problem of the mechanical arm servo system, many control methods exist, such as PID control, adaptive control, sliding mode control, and the like.

Sliding mode control is considered to be an effective robust control method in solving system uncertainty and external disturbances. The sliding mode control method has the advantages of simple algorithm, high response speed, strong robustness to external noise interference and parameter perturbation and the like. Therefore, the sliding mode control method is widely applied to various fields. Compared with the traditional linear sliding mode control, the terminal sliding mode control has the advantage of limited time convergence. However, the discontinuous switching characteristic of the terminal sliding mode control in nature will cause the buffeting of the system, and the terminal sliding mode control becomes an obstacle to the application of the terminal sliding mode control in the practical system. To solve this problem, many improved methods are successively proposed, such as a high-order sliding mode control method, an observer control method, and the like. In these methods, the selection of the slip-form surfaces is made by reducing the order of the ideal system parameters. Recently, a full-order sliding mode control method has been proposed, which well avoids the problem of chattering in the response of the system and makes the system input signal smoother. However, in the proposed method, the transient performance of the system is affected to some extent in order to eliminate the buffeting, for example, the rise time becomes long.

There are many methods for securing transient performance, such as blf (barrier lyapunov) control, ppc (describedpowerformancecontrol) method, and fc (funnelcontrol) method. The BLF control method can restrict the state variable of the system to indirectly limit the tracking error of the system, but the expression form of the Lyapunov function in the method is complex and the function is required to be guaranteed to be microminiature. The PPC uses new error variables to guarantee the steady-state error specified by the system, but has a singular value problem. The FC puts forward a virtual control variable related to the tracking error and applies the variable to the nonsingular terminal sliding mode control.

Disclosure of Invention

In order to solve the problems of low response speed, input buffeting control and the like in the conventional sliding mode control method of the mechanical arm servo system, the invention provides a full-order sliding mode control method of a neural network of the mechanical arm servo system, which ensures transient performance, accelerates the response speed of the system, improves the buffeting problem of control input and ensures the rapid and stable convergence of the system based on the condition that the parameters of a system model are uncertain.

The technical scheme proposed for solving the technical problems is as follows:

a full-order sliding mode control method for a neural network of a mechanical arm servo system for guaranteeing transient performance comprises the following steps:

step 1, establishing a dynamic model of a mechanical arm servo system, initializing a system state, sampling time and control parameters, and carrying out the following processes:

1.1 the dynamic model expression of the mechanical arm servo system is as follows:

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

wherein the ratio of q,andthe position, the speed and the acceleration of the mechanical arm joint are respectively; m_H，C_HAnd D_HRespectively representing a symmetrical positive definite inertia matrix, a centrifugal Coriolis matrix and a diagonal positive definite matrix of a damping friction coefficient of each joint; g_HRepresents a gravity term; u is a control signal;

1.2 because the system parameters in the formula (1) can not be accurately obtained due to the influence of measurement noise, load variation and external interference, the actual system parameters are 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 the estimated valueAndrepresents a known moiety; Δ M_H(q)，ΔD_HAnd Δ G_H(q) represents a system agnostic term;

step 2, designing a required neural network based on a mechanical arm servo system containing system model uncertainty items, wherein the process is as follows:

definition of theta^*For an ideal weight coefficient matrix, the nonlinear uncertainty function f is approximated as:

f＝θ^*Tphi (x) + (3) wherein, (. alpha)^TRepresents transposition;represents an input vector; phi (x) is [ phi ]₁(x),φ₂(x),…φ_m(x)]^TIs a basis function of the neural network; represents the approximation error of the neural network and satisfies | | | | ≦_N，_NIt is a positive constant; phi is a_i(x) Is taken as the 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_iKernel parameters representing a gaussian function; sigma_iRepresents the width of a gaussian function; exp (·) represents an exponential function based on a natural constant e;

step 3, calculating a system tracking error and an FC error, and designing a full-order sliding mode surface, wherein the process is as follows:

3.1 define the system tracking error as

e＝q_d-q(5)

Wherein q is_dThe expected trajectory is derived for the second order; the first and second order differentials of equation (5) are expressed as:

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

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

3.2 define FC error as

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

Wherein,

F_Φ(t)＝₀exp(-a₀t)+_∞(9)

wherein, a₀Is a constant that is positive in value,₀≥_∞＞0，|e(0)|＜F_Φ(0) (ii) a The first order differential and the second order differential of equation (8) are expressed as:

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

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

wherein, ${\mathrm{\Φ}}_F=\frac{1}{{(F_{\mathrm{\Φ}}-|\left|e\right|\left|\right)}^2},{\stackrel{\·}{F}}_{\mathrm{\Φ}}=-a_0{\mathrm{\δ}}_0\exp (-a_{0}t),{\stackrel{\·\·}{F}}_{\mathrm{\Φ}}=-{a_0}^2{\mathrm{\δ}}_0\exp (-a_{0}t),$

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

3.3 definition of the full step slip form surface as

$s={\stackrel{\·\·}{s}}_1+c_2\mathrm{sgn}\left({\stackrel{\·}{s}}_1\right){\left|{\stackrel{\·}{s}}_1\right|}^{{\mathrm{\α}}_2}+c_1\mathrm{sgn}\left(s_1\right){\left|s_1\right|}^{{\mathrm{\α}}_1}---\left(12\right)$

Wherein, c₁And c₂Are two positive constants chosen to ensure a polynomial p²+c₂p+c₁All the characteristics of (A) are rooted in the left half part of the complex plane to ensure the stability of the system, and 0 < α₁＜1，0＜α₂< 1, they are selected by the following polynomial:

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

wherein, α_n+1＝1，α_nα (1-,1), and ∈ (0, 1);

and 4, designing a neural network full-order sliding mode controller based on the mechanical arm system containing the system model uncertainty according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (1), the neural network full-order sliding-mode controller is designed as:

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

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

v＝-(k_d+k_T+η)sgn(s)(17)

wherein, c_iAnd α_iIs a constant, i ═ 1,2, as defined in formula (12);representing an estimated weight coefficient matrix; k is a radical of_d，k_TAnd η are both constants, and will be explained later;

4.2 design regulation rule of weight coefficient matrix of neural network:

wherein, is a positive definite diagonal matrix;

4.3 substituting equation (14) into (1) yields the following equation:

wherein,representing weight estimation errors of the neural network;representing a system disturbance term and is bounded, then assume d (q, t) ≦ l_dAnd isWherein l_dIs a bounded constant; k is a radical of_TIs selected such that k is satisfied when T > 0_T≥Tl_d；

By the equations (1), (12), (14) - (17) and (19), the full-order sliding mode surface is expressed as the following equation:

s＝d(q,t)+u_n(20)

bringing formula (17) into formula (16) gives:

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

at u_n(0) In the case of 0, the following equation is obtained:

k_T≥Tl_d≥T|u_n(t)|_max≥T|u_n(t)|(22)

4.4 design Lyapunov function:

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

the derivation is performed on equation (12):

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

bringing formula (16) into formula (24) gives:

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

differentiating equation (23) yields:

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

bringing formula (22) into formula (26) ifThe system is determined to be stable.

Based on the condition that the parameters of the system model are uncertain, the full-order sliding mode control method of the neural network of the mechanical arm servo system for ensuring transient performance is designed by utilizing the full-order sliding mode and the neural network, the response speed of the system is accelerated, buffeting of sliding mode control input is improved, and the system is ensured to be fast and stable to converge.

The technical conception of the invention is as follows: aiming at a mechanical arm servo system which comprises a dynamic execution mechanism and system model uncertainty, a full-order sliding mode control method of the neural network of the mechanical arm servo system is designed by utilizing a full-order sliding mode control method and combining a neural network. And the uncertainty caused by the system model parameters is compensated by approximating an unknown function through a neural network. In addition, the design of the full-order sliding mode surface is to ensure the fast and stable convergence of the system, and improve the buffeting problem by adding a filter in the actual control system. The invention provides a control method for improving the problem of control input buffeting, accelerating the response speed and ensuring the transient performance, and realizes the quick and stable control of a system.

The invention has the advantages that: and uncertainty is allowed to exist in system model parameters, transient performance is guaranteed, buffeting is improved, and rapid and stable convergence is realized.

Drawings

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

Fig. 1(b) is a schematic diagram of the position tracking error of the first joint of the present invention.

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

FIG. 2(b) is a schematic diagram of the position tracking error of the second joint according to the present invention.

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

Fig. 3(b) is a schematic diagram of the controller input of the second joint of the present invention.

FIG. 4 is a control flow diagram of the present invention.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

Referring to fig. 1(a) -4, a full-order sliding mode control method for a neural network of a mechanical arm servo system for ensuring transient performance comprises the following steps:

step 1, establishing a dynamic model of a mechanical arm servo system, initializing a system state, sampling time and control parameters, and carrying out the following processes:

1.1 the dynamic model expression of the mechanical arm servo system is as follows:

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

wherein the ratio of q,andthe position, the speed and the acceleration of the mechanical arm joint are respectively; m_H，C_HAnd D_HRespectively representing a symmetrical positive definite inertia matrix, a centrifugal Coriolis matrix and a diagonal positive definite matrix of a damping friction coefficient of each joint; g_HRepresents a gravity term; u is a control signal;

1.2 because the system parameters in the formula (1) can not be accurately obtained due to the influence of measurement noise, load variation and external interference, the actual system parameters are 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 the estimated valueAndrepresents a known moiety; Δ M_H(q)，ΔD_HAnd Δ G_H(q) represents a system agnostic term;

step 2, designing a required neural network based on a mechanical arm servo system containing system model uncertainty items, wherein the process is as follows:

definition of theta^*For an ideal weight coefficient matrix, the nonlinear uncertainty function f is approximated as:

f＝θ^*Tφ(x)+(3)

wherein, (.)^TRepresents transposition;represents an input vector; phi (x) is [ phi ]₁(x),φ₂(x),…φ_m(x)]^TIs a basis function of the neural network; represents the approximation error of the neural network and satisfies | | | | ≦_N，_NIt is a positive constant; phi is a_i(x) Is taken as the 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_iKernel parameters representing a gaussian function; sigma_iRepresents the width of a gaussian function; exp (·) represents an exponential function based on a natural constant e;

step 3, calculating a system tracking error and an FC error, and designing a full-order sliding mode surface, wherein the process is as follows:

3.1 define the system tracking error as

e＝q_d-q(5)

Wherein q is_dThe expected trajectory is derived for the second order; the first and second order differentials of equation (5) are expressed 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 define FC error as

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

Wherein,

F_Φ(t)＝₀exp(-a₀t)+_∞(9)

wherein, a₀Is a constant that is positive in value,₀≥_∞＞0，|e(0)|＜F_Φ(0) (ii) a The first order differential and the second order differential of equation (8) are expressed 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 of the full step slip form surface as

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

Wherein, c₁And c₂Are two positive constants chosen to ensure a polynomial p²+c₂p+c₁All the characteristics of (A) are rooted in the left half part of the complex plane to ensure the stability of the system, and 0 < α₁＜1，0＜α₂< 1, they are selected by the following polynomial:

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

wherein, α_n+1＝1，α_nα (1-,1), and ∈ (0, 1);

and 4, designing a neural network full-order sliding mode controller based on the mechanical arm system containing the system model uncertainty according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (1), the neural network full-order sliding-mode controller is designed as:

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

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

v＝-(k_d+k_T+η)sgn(s)(17)

wherein, c_iAnd α_iIs a constant, i ═ 1,2, as defined in formula (12);representing an estimated weight coefficient matrix; k is a radical of_d，k_TAnd η are both constants, and will be explained later;

4.2 design regulation rule of weight coefficient matrix of neural network:

wherein, is a positive definite diagonal matrix;

4.3 substituting equation (14) into (1) yields the following equation:

wherein,representing weight estimation errors of the neural network;representing a system disturbance term and is bounded, then assume d (q, t) ≦ l_dAnd isWherein l_dIs a bounded constant; k is a radical of_TIs selected as requiredK is satisfied when T > 0_T≥Tl_d；

By the equations (1), (12), (14) - (17) and (19), the full-order sliding mode surface is expressed as the following equation:

s＝d(q,t)+u_n(20)

bringing formula (17) into formula (16) gives:

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

at u_n(0) In the case of 0, the following equation is obtained:

k_T≥Tl_d≥T|u_n(t)|_max≥T|u_n(t)|(22)

4.4 design Lyapunov function:

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

the derivation is performed on equation (12):

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

bringing formula (16) into formula (24) gives:

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

differentiating equation (23) yields:

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

bringing formula (22) into formula (26) ifThe system is determined to be stable.

To obtain the corresponding system parameter values in equation (2), we give the following robot servo system expressions for 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, $a_{11}=(m_1+m_2){r_1}^2+m_2{r_2}^2+2m_2{r}_1{r}_{2}cos\left(q_2\right)+J,$ a₁₂＝m₂r₂+m₂r₁r₂cos(q₂)，b₁₂＝m₂r₁r₂sin(q₂)，c₁＝(m₁+m₂)r₁cos(q₂)+m₂r₂cos(q₁+q₂)，c₂＝m₂r₂cos(q₁+q₂)， ${\mathrm{\&tau;}}_d=\left[\begin{array}{c}{\mathrm{\&tau;}}_{d_1}\\ {\mathrm{\&tau;}}_{d_2}\end{array}\right].$

in order to verify the effectiveness of the proposed method, the invention provides a comparison of two methods:

s1: a common neural network sliding mode control method;

s2: a neural network full-order sliding mode control method for ensuring transient performance;

for more efficient comparison, all parameters of the system are consistent, i.e.: q. q.s₁(0)＝0.1，q₂(0)＝0.1，J_m＝diag(0.67×10^-4,0.42×10^-4)，D_m1, and given a system perturbation of:system control messageThe number parameters are: k_τ＝diag(19/40,19/80)，＝diag(50,50)，α₁＝7/20，α₂＝2/5，c₁＝5000，c₂1900, T1; the actual parameters of the mechanical arm are as follows: r is₁＝0.2，r₂＝0.18，m₁＝2.3，m₂＝0.6，J₁＝0.02，J₂0.003, 9.8, k is_d+k_T+ η equals 10, the neural network contains 15 nodes, i.e. n equals 15, the neural network has a width of σ_i4, i-1, 2, …, 15; and the tracking signal is: y is_d1＝y_d2＝sin(2πt)。

As can be seen from the first graph, for the first joint, in the period from 1.3 seconds to 1.5 seconds, the S2 method tracks the descending track more quickly than the S1 method, so that the transient performance is effectively ensured, and the tracking error is controlled within the interval of +/-0.05 radian without jumping; as can be seen from the second graph, for the second joint, in the period from 1.2 seconds to 1.4 seconds, the S2 method tracks the descending track more quickly than the S1 method, the transient performance is effectively guaranteed, and the tracking error is controlled within the interval of +/-0.07 radian without jumping; as can be seen from fig. three, the control input of method S1 has a severe buffeting phenomenon, and the control input of method S2 is smoother than that of S1, which effectively improves the buffeting problem.

While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.

Claims (1)

1. A full-order sliding mode control method for a neural network of a mechanical arm servo system for ensuring transient performance is characterized by comprising the following steps: the control method comprises the following steps:

step 1, establishing a dynamic model of a mechanical arm servo system, initializing a system state, sampling time and control parameters, and carrying out the following processes:

1.1 the dynamic model expression of the mechanical arm servo system is as follows:

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

wherein the ratio of q,andthe position, the speed and the acceleration of the mechanical arm joint are respectively; m_H，C_HAnd D_HRespectively representing a symmetrical positive definite inertia matrix, a centrifugal Coriolis matrix and a diagonal positive definite matrix of a damping friction coefficient of each joint; g_HRepresents a gravity term; u is a control signal;

1.2 because the system parameters in the formula (1) can not be accurately obtained due to the influence of measurement noise, load variation and external interference, the actual system parameters are 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 the estimated valueAndrepresents a known moiety; Δ M_H(q)，ΔD_HAnd Δ G_H(q) represents a system agnostic term;

step 2, designing a required neural network based on a mechanical arm servo system containing system model uncertainty items, wherein the process is as follows:

definition of theta^*For an ideal weight coefficient matrix, the nonlinear uncertainty function f is approximated as:

f＝θ^*Tφ(x)+(3)

wherein, (.)^TRepresents transposition;represents an input vector; phi (x) is [ phi ]₁(x),φ₂(x),…φ_m(x)]^TIs a basis function of the neural network; represents the approximation error of the neural network and satisfies | | | | ≦_N，_NIt is a positive constant; phi is a_i(x) Is taken as the 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_iKernel parameters representing a gaussian function; sigma_iRepresents the width of a gaussian function; exp (·) represents an exponential function based on a natural constant e;

step 3, calculating a system tracking error and an FC error, and designing a full-order sliding mode surface, wherein the process is as follows:

3.1 define the system tracking error as

e＝q_d-q(5)

Wherein q is_dThe expected trajectory is derived for the second order; the first and second order differentials of equation (5) are expressed 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 define FC error as

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

Wherein,

F_Φ(t)＝₀exp(-a₀t)+_∞(9)

wherein, a₀Is a constant that is positive in value,₀≥_∞＞0，|e(0)|＜F_Φ(0) (ii) a The first order differential and the second order differential of equation (8) are expressed 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 of the full step slip form surface as

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

Wherein, c₁And c₂Are two positive constants chosen to ensure a polynomial p²+c₂p+c₁All the characteristics of (A) are rooted in the left half part of the complex plane to ensure the stability of the system, and 0 < α₁＜1，0＜α₂< 1, they are selected by the following polynomial:

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

wherein, α_n+1＝1，α_nα (1-,1), and ∈ (0, 1);

and 4, designing a neural network full-order sliding mode controller based on the mechanical arm system containing the system model uncertainty according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (1), the neural network full-order sliding-mode controller is designed as:

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

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

v＝-(k_d+k_T+η)sgn(s)(17)

wherein, c_iAnd α_iIs a constant, i-1, 2,has been defined in formula (12);representing an estimated weight coefficient matrix; k is a radical of_d，k_TAnd η are both constants, and will be explained later;

4.2 design regulation rule of weight coefficient matrix of neural network:

wherein, is a positive definite diagonal matrix;

4.3 substituting equation (14) into (1) yields the following equation:

wherein,representing weight estimation errors of the neural network;representing a system disturbance term and is bounded, then assume d (q, t) ≦ l_dAnd isWherein l_dIs a bounded constant; k is a radical of_TIs selected such that k is satisfied when T > 0_T≥Tl_d；

By the equations (1), (12), (14) - (17) and (19), the full-order sliding mode surface is expressed as the following equation:

s＝d(q,t)+u_n(20)

bringing formula (17) into formula (16) gives:

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

at u_n(0) In the case of 0, the following equation is obtained:

k_T≥Tl_d≥T|u_n(t)|_max≥T|u_n(t)|(22)

4.4 design Lyapunov function:

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

the derivation is performed on equation (12):

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

bringing formula (16) into formula (24) gives:

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

differentiating equation (23) yields:

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

bringing formula (22) into formula (26) ifThe system is determined to be stable.