CN104932271B - A kind of neutral net full-order sliding mode control method of mechanical arm servo-drive system - Google Patents

Abstract

A kind of neutral net full-order sliding mode control method of mechanical arm servo-drive system, for the mechanical arm servo-drive system containing Dynamic Execution mechanism, using full-order sliding mode control method, in conjunction with neutral net, design a mechanism arm servo-drive system neutral net full-order sliding mode control method.The design in full-order sliding mode face is, in order to ensure the finite time convergence control of system, and to eliminate buffeting and singular problem by avoiding the occurrence of differential term in actual control system.In addition, neutral net is the uncertainty for the unknown nonlinear of approximation system and inside and outside disturbance.The present invention provides a kind of fast and stable control that can be eliminated the buffeting problem and singular problem of sliding-mode surface, and the control method that energy effective compensation system unknown nonlinear and inside and outside are disturbed, realize system.

Description

Neural network full-order sliding mode control method of mechanical arm servo system

Technical Field

The invention relates to a neural network full-order sliding mode control method of a mechanical arm servo system, in particular to a control method of a mechanical arm servo system with a dynamic execution mechanism and unknown dynamic parameters of the system.

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, unknown dynamic parameters and external disturbances are widely present in the robot servo system, which often results in reduced efficiency and 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. 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 proposed in succession, such as a high-order sliding mode control method, an observer control method. 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 avoids the problem of chattering well in the response of the system and makes the input signal of the system smoother.

However, in most of the proposed methods, the dynamic model parameters of the mechanical system must be known in advance. Therefore, the proposed method cannot be directly applied to the control of the robot arm when there is an uncertainty factor in the system. As is well known, neural networks have been widely used to deal with system unknowns and non-linearity problems due to their ability to approximate any smooth function to a tight set of arbitrary accuracies. For the above reasons, many adaptive neural network control methods are used to control highly nonlinear robotic arm systems.

Disclosure of Invention

In order to overcome the defects of unknown nonlinearity and sliding mode control buffeting of the existing mechanical arm servo system, the invention provides a neural network full-order sliding mode control method of the mechanical arm servo system with a dynamic execution mechanism, which eliminates the buffeting problem and singularity problem of the system and ensures the rapid and stable convergence of the system.

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 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 form of the mechanical arm servo system is as follows:

wherein the ratio of q,andthe position, velocity and acceleration of the joints of the robot arm, M (q),d respectively represents 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 (q) represents a gravity term; τ represents a torque input vector for the joint;

1.2 when considering a dynamic actuator, equation (1) is re-expressed as:

wherein, is a vector of armature voltage inputs;represents an electromagnetic torque vector, wherein J_mAnd D_mRespectively representing an inertia diagonal matrix and a torsional damping coefficient; k_τ＝diag(K_τ1,K_τ2,…,K_τn) Then the torque constant of the diagonal matrix; q. q.s_mRepresents the motor angular position vector; tau is_LRepresents a vector of motor load torques;a diagonal matrix representing n-joint transmission gears;

1.3 due to the influence of measurement noise, load change and external interference, the system parameters in the formula (2) cannot be accurately obtained; then, the actual system parameters are rewritten as:

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 with unknown parameters, wherein the process is as follows:

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

f＝θ^*Tφ(x)+ (4)

wherein,represents an input vector; phi (x))＝[φ₁(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:

wherein, c_iKernel parameters representing a gaussian function; sigma_iThe width of the gaussian function is represented;

step 3, calculating the tracking error of the control system, 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 (6)

wherein q is_dThe expected trajectory is derived for the second order; then the first and second order differentials of equation (6) are expressed as follows:

3.2 then the full-order slip-form surface will be defined as:

wherein, c₁And c₂Is a positive constant, selected to ensure a polynomial p²+c₂All features of p + c rooted in the left half of the complex plane to ensure system stability α₁And α₂Is selected by the following polynomial:

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 a mechanical arm system containing a dynamic execution mechanism according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (2), the neural network full-order sliding-mode controller is designed as:

v＝-(k_d+k_T+η)sgn(s) (14)

wherein, c_iAnd α_iIs a constant, i ═ 1,2, as defined in formula (9); k is a radical of_d，k_TAnd η are both constants;

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

wherein, is a positive definite diagonal matrix;

4.3 bringing equation (11) into equation (2) 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_TIs selected to satisfy K when K > 0_T≥Tl_d；

4.4 by equation (2), equation (9), equations (11) - (14), and equation (16), the full-order sliding-mode surface is written as follows:

s＝d(q,t)+u_n(17)

4.5 bringing formula (14) into formula (13):

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)| (19)

4.6 design Lyapunov function:

the derivation of equation (9) yields:

bringing formula (13) into formula (21) gives:

differentiating equation (20) yields:

bringing formula (19) into formula (23) ifThe system is determined to be stable.

The invention designs a neural network full-order sliding mode control method of a mechanical arm servo system based on unknown nonlinear factors, a full-order sliding mode and a neural network, realizes stable control of the system, eliminates buffeting of sliding mode control, and ensures rapid and stable convergence of the system.

The technical conception of the invention is as follows: aiming at a mechanical arm servo system containing a dynamic execution mechanism, a full-order sliding mode control method of a neural network of the mechanical arm servo system is designed by utilizing a full-order sliding mode control method and combining a neural network. The design of the full-order sliding mode surface is to ensure the rapid and stable convergence of the system, and eliminate the buffeting and singularity problems by avoiding the occurrence of differential terms in an actual control system. In addition, neural networks are used to approximate the unknown non-linearities of the system and the uncertainty of internal and external disturbances. The invention provides a control method which can eliminate buffeting and singularity of a sliding mode surface, can effectively compensate unknown dynamic parameters of a system and internal and external disturbance, and realizes quick and stable control of the system.

The invention has the advantages that: and buffeting is eliminated, the unmonamic parameters of the system and uncertain disturbance items inside and outside are compensated, and rapid and stable convergence is realized.

Drawings

Fig. 1 is a schematic diagram of the position tracking effect when k is 10 according to the present invention, where (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 2 is a schematic diagram of the position tracking error when k is 10 according to the present invention, in which (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 3 is a schematic diagram of velocity tracking when k is 10 according to the present invention, in which (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 4 is a schematic diagram of controller inputs when k is 10 according to the present invention, where (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 5 is a schematic diagram showing the position tracking effect when k is 40 according to the present invention, where (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 6 is a schematic diagram of the position tracking error when k is 40 according to the present invention, in which (a) denotes the joint 1 and (b) denotes the joint 2.

Fig. 7 is a schematic diagram of velocity tracking when k is 40 according to the present invention, in which (a) denotes a joint 1 and (b) denotes a joint 2.

Fig. 8 is a schematic diagram of controller inputs when the present invention is set to k-40, where (a) denotes the joint 1 and (b) denotes the joint 2.

FIG. 9 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 to 9, a neural network full-order sliding mode control method for a mechanical arm servo system includes 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 form of the mechanical arm servo system is as follows:

wherein the ratio of q,andthe position, velocity and acceleration of the joints of the robot arm, M (q),d respectively represents 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 (q) represents a gravity term; τ represents a torque input vector for the joint;

1.2 when considering a dynamic actuator, equation (1) is re-expressed as:

wherein, is aA vector of armature voltage inputs;represents an electromagnetic torque vector, wherein J_mAnd D_mRespectively representing an inertia diagonal matrix and a torsional damping coefficient; k_τ＝diag(K_τ1,K_τ2,…,K_τn) Then the torque constant of the diagonal matrix; q. q.s_mRepresents the motor angular position vector; tau is_LRepresents a vector of motor load torques;a diagonal matrix representing n-joint transmission gears;

1.3 due to the influence of measurement noise, load change and external interference, the system parameters in the formula (2) cannot be accurately obtained; then, the actual system parameters are rewritten as:

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 with unknown parameters, wherein the process is as follows:

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

f＝θ^*Tφ(x)+ (4)

wherein,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 typically taken as the following gaussian function:

wherein, c_iKernel parameters representing a gaussian function; sigma_iThe width of the gaussian function is represented;

step 3, calculating a tracking error of the control system, and designing a full-order sliding mode surface;

3.1 define the system tracking error as:

e＝q_d-q (6)

wherein q is_dThe expected trajectory is derived for the second order; then the first and second order differentials of equation (6) are expressed as follows:

3.2 then the full-order slip-form surface will be defined as:

wherein, c₁And c₂Is a positive constant, selected to ensure a polynomial p²+c₂All features of p + c rooted in the left half of the complex plane to ensure system stability α₁And α₂Is selected by the following polynomial:

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 a mechanical arm servo system containing a dynamic execution mechanism according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (2), the neural network full-order sliding-mode controller is designed as:

v＝-(k_d+k_T+η)sgn(s) (14)

wherein, c_iAnd α_i(i ═ 1,2) is a constant, which has been defined in formula (9); 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 bringing equation (11) into equation (2) yields the following equation:

wherein,representing weight estimation errors of the neural network;representing a system disturbance term and being bounded, then we assume d (q, t) ≦ l_dAnd isWherein l_dIs a bounded constant; k_TIs selected to satisfy K when K > 0_T≥Tl_d；

4.4 by equation (2), equation (9), equations (11) - (14), and equation (16), the full-order sliding-mode surface is written as follows:

s＝d(q,t)+u_n(17)

4.5 bringing formula (14) into formula (13):

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)| (19)

4.6 design Lyapunov function:

the derivation of equation (9) yields:

bringing formula (13) into formula (21) gives:

differentiating equation (20) yields:

bringing formula (19) into formula (23) 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:

wherein,a₁₂＝m₂r₂+m₂r₁r₂cos(q₂)，b₁₂＝m₂r₁r₂sin(q₂)，c₁＝(m₁+m₂)r₁cos(q₂)+m₂r₂cos(q₁+q₂)，c₂＝m₂r₂cos(q₁+q₂)，

in order to verify the effectiveness of the proposed method, the invention provides a comparison between a neural network-based finite time control (NNFTC) method and a neural network full-order sliding mode control (CFNSMC) method:

for more efficient comparison, all parameters of the system are consistent, i.e.: q. q.s₁(0)＝0.5，q₂(0)＝0.5，J_m＝diag(0.67×10^-4,0.42×10^-4)，D_m1, and given a system perturbation of:the system control signal parameters are: k_τ＝diag(19/40,19/80)，＝diag(50,50)，α₁＝13/27，α₂＝13/27，c₁＝100，c₂40, 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, and 9.8. Further, let k be k_d+k_T+ η, and comparing the two control methods at two different k values, respectively.

The first condition is as follows: k 10

Since the robotic arm system we are tracking is two joints, we are tracking q_d1Sin (2 π t) and q_d2Sin (2 pi t). As can be seen from fig. 1 and 2, the CFNSMC method and the NNFTC method have an approximate tracking effect when tracking a joint; however, the CFNSMC method has better tracking effect than the NNFTC method when tracking joint 2. As is evident from the trace rate diagram of fig. 3, the trace rate curve of the CFNSMC method is smoother than that obtained by the NNFTC method. In fig. 4, the NNFTC method has a significant buffeting effect, whereas the CFNSMC method eliminates the buffeting effect.

Case two: k is 40

From fig. 5 and 6 we see that the NNFTC method has even better tracking effect than the CFNSMC method when tracking joint 1; however, when joint 2 is tracked, it is inferior to the CFNSMC method. In addition, we have seen from fig. 6 and 7 that the CFNSMC method is much smoother than the NNFTC method in both the control input signal and the tracking speed curves. Also, comparing fig. 1 and 5, it is clear that the CFNSMC method is more robust than the NNFTC method in the face of different gains k.

In summary, compared with the NNFTC method, the CFNSMC method is less sensitive to different control gains k, i.e. has stronger robustness; and has better capability of eliminating buffeting on the control signal and the speed tracking signal.

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 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 form of the mechanical arm servo system is as follows:

$M\left(q\right)\stackrel{\·\·}{q}+C(q,\stackrel{\·}{q})\stackrel{\·}{q}+D\stackrel{\·}{q}+G\left(q\right)=\mathrm{\τ}---\left(1\right)$

wherein the ratio of q,andthe position, velocity and acceleration of the joints of the robot arm, m (q),d respectively represents 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 (q) represents a gravity term; τ represents a torque input vector for the joint;

1.2 when considering a dynamic actuator, equation (1) is re-expressed as:

$M_H\left(q\right)\stackrel{\·\·}{q}+C_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+D_{H}q+G_H\left(q\right)q=u---\left(2\right)$

wherein, is a vector of armature voltage inputs;represents an electromagnetic torque vector, wherein J_mAnd D_mRespectively representing an inertia diagonal matrix and a torsional damping coefficient; k_τ＝diag(K_τ1,K_τ2,...,K_τn) Then the torque constant of the diagonal matrix; q. q.s_mRepresents the motor angular position vector; tau is_LRepresents a vector of motor load torques;a diagonal matrix representing n-joint transmission gears;

1.3 due to the influence of measurement noise, load change and external interference, the system parameters in the formula (2) cannot be accurately obtained; then, the actual system parameters are rewritten as:

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

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

$D_H={\hat{D}}_H+{\mathrm{\ΔD}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\ΔG}}_H\left(q\right)---\left(3\right)$

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 with unknown parameters, wherein the process is as follows:

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

f＝θ^*Tφ(x)+ (4)

wherein,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 \[-\frac{\left|\right|x-o_i\left|\right|}{2{\mathrm{\σ}}_i^2}\],i=1,2,...,m---\left(5\right)$

wherein o is_iKernel parameters representing a gaussian function; sigma_iThe width of the gaussian function is represented;

step 3, calculating the tracking error of the control system, 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 (6)

wherein q is_dThe expected trajectory is derived for the second order; then the first and second order differentials of equation (6) are expressed as follows:

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

$\stackrel{\·\·}{e}={\stackrel{\·\·}{q}}_d-\stackrel{\·\·}{q}---\left(8\right)$

3.2 then the full-order slip-form surface will be defined as:

$s=\stackrel{\·\·}{e}+c_{2}sgn\left(\stackrel{\·}{e}\right)|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\α}}_1}---\left(9\right)$

wherein, c₁And c₂Is a positive constant, selected to ensure a polynomial p²+c₂p+c₁All features of (a) are rooted in the left half of the complex plane to ensure system stability, α₁And α₂Is selected by the following polynomial:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\mathrm{\α},n=1\\ {\mathrm{\α}}_{i-1}=\frac{{\mathrm{\α}}_i{\mathrm{\α}}_{i+1}}{2{\mathrm{\α}}_{i+1}-{\mathrm{\α}}_i},i=2,...,n,\∀n\≥2\end{array}\right.---\left(10\right)$

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 a mechanical arm system containing a dynamic execution mechanism according to a full-order sliding mode and a neural network theory, wherein the process is as follows:

4.1 considering equation (2), the neural network full-order sliding-mode controller is designed as:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\·\·}{q}}_d+c_2\mathrm{sgn}(\stackrel{\·}{e}\left)\right|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_1\mathrm{sgn}\left(e\right)|e{|}^{{\mathrm{\α}}_1}+u_0)\\ +{\hat{C}}_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+{\hat{D}}_H\stackrel{\·}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(11\right)$

$u_0={\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(12\right)$

${\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n=v---\left(13\right)$

v＝-(k_d+k_T+η)sgn(s) (14)

wherein, c_iAnd α_iIs a constant, i ═ 1,2, as defined in formula (9); k is a radical of_d、k_TAnd η are both constants, T is a positive constant;

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

$\stackrel{\·}{\widehat{\mathrm{\θ}}}=\mathrm{\Γ}\mathrm{\φ}\left(x\right)s^T---\left(15\right)$

wherein, is a positive definite diagonal matrix;

4.3 substituting equation (11) into equation (2) yields the following equation:

$s={\mathrm{\θ}}^{*T}\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}-{\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+u_n={\widetilde{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}+u_n=d(q,t)+u_n---\left(16\right)$

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 at k_TK is satisfied when > 0_T≥Tl_d；

4.4 by equation (2), equation (9), equations (11) - (14), and equation (16), the full-order sliding-mode surface is written as follows:

s＝d(q,t)+u_n(17)

4.5 substitution of formula (14) into formula (13):

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\right(t_0)+(1/T\left)\right(k_d+k_T+\mathrm{\η}\left)\mathrm{sgn}\right(s\left)\right)e^{t-t_0}\\ -(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)\end{array}---\left(18\right)$

t₀is an initial time value, u_n(t) represents u at time t_nValue u_n(t₀) Represents t₀U of time_nA value;

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)| (19)

4.6 design Lyapunov function:

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

the derivation of equation (9) yields:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\·}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(21\right)$

obtained by substituting formula (13) into formula (21):

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)-(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(22\right)$

differentiating equation (20) yields:

$\stackrel{\·}{V}=s^T\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)s^T-(k_d+k_T+\mathrm{\η})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(23\right)$

by substituting formula (19) into formula (23) ifThe system is determined to be stable.