CN104360596B - Limited time friction parameter identification and adaptive sliding mode control method for electromechanical servo system - Google Patents

Abstract

A limited time friction parameter identification and adaptive sliding mode control method for an electromechanical servo system includes: establishing an electromechanical servo system model, and initializing a system status and related control parameters; establishing a LuGre model of a nonlinear friction link; designing a nonlinear sliding mode and a sliding mode control gain adaptive law; designing a limited time parameter recognition method and designing parameter adaptive laws; designing a status observer, and designing an adaptive nonlinear sliding mode controller according to a system equation and the parameter adaptive laws. The limited time friction parameter identification and adaptive sliding mode control method for the electromechanical servo system has the advantages that friction parameter identification and convergence speed is increased, quick tracking is achieved, and tracking precision is effectively improved.

Description

Limited time friction parameter identification and self-adaptive sliding mode control method for electromechanical servo system

Technical Field

The invention relates to a friction parameter identification and self-adaptive sliding mode control method of an electromechanical servo system, in particular to a limited time rapid parameter identification and self-adaptive nonlinear sliding mode control method of a LuGre friction model.

Background

The friction is a complex and unavoidable nonlinear behavior in an electromechanical servo system, and often causes the problems of steady-state error, low-speed crawling, discontinuous limit cycle and bidirectional motion and the like of the system. Non-linear friction, which is present in almost all electromechanical servo systems, is one of the main obstacles to improving the performance of the servo system, and therefore, in order to improve the operating performance of the servo system, an appropriate compensation control method should be adopted to reduce or eliminate the influence of friction on the performance of the servo system.

Various advanced motion control methods are applied to the compensation control aspect of the friction nonlinear link at present, and mainly comprise adaptive control, robust control, sliding mode control, intelligent control methods and the like. The method can effectively solve the problem of uncertainty of friction parameters based on a model self-adaptive control technology, realizes real-time update through online parameter identification, can realize higher control precision without depending on high gain, can accurately describe static characteristics and dynamic characteristics of friction, and particularly has good effect on dynamic characteristic compensation of a friction link.

Because the LuGre friction model needs more identification parameters and is difficult to identify, and the convergence time of the identified parameters is longer, the method adopts a Finite-time identification method (Finite-time identification method), and the method enables parameter estimation errors to be converged in an exponential form through self-adaptive gain and closed-loop system excitation, thereby greatly accelerating the parameter convergence speed and improving the parameter convergence precision to a certain extent. Because the tracking speed and the overshoot are difficult to reach a balance in the traditional self-adaptive control, a Non-linear sliding mode surface (Non-linear sliding surface) is adopted to replace the traditional linear sliding mode surface, the nonlinear sliding mode can be ensured to have higher convergence speed under the condition of no overshoot, and meanwhile, the self-adaptive law of control gain is designed, so that the gain can be updated in real time, and the convergence speed and the tracking error can be ensured to achieve better effects.

Disclosure of Invention

The invention provides a limited time parameter identification and self-adaptive nonlinear sliding mode control method based on a LuGre friction model, aiming at overcoming the problems of too long parameter identification convergence time and low control precision of an electromechanical servo system during friction compensation, and better realizing quick identification and compensation of friction. The invention adopts the finite time parameter identification technology, combines the nonlinear sliding mode and the sliding mode gain self-adaptive technology, designs the self-adaptive nonlinear sliding mode control method, accelerates the parameter identification convergence speed in the friction model, effectively improves the control precision, and enables the tracking error to converge to the vicinity of the zero point in a shorter time.

The method comprises the following concrete implementation steps:

the finite time friction parameter identification and self-adaptive sliding mode control method of the electromechanical servo system comprises the following steps:

step 1, establishing an electromechanical servo system model with friction as shown in formula (1), and initializing a system state and related control parameters;

in the formula: j is the moment of inertia equivalent to the motor shaft; theta and omega respectively represent the actual position and the rotating speed of the output shaft of the motor; t is the input torque of the motor; t is_fIs the total friction torque; t is_dIs the set of various disturbances of the system;

step 2, establishing a LuGre friction model, wherein the expression of the LuGre friction model is as follows;

in the formula: z represents the average deformation of the bristles between the contact surfaces; sigma₀Representing the stiffness coefficient of the bristles; sigma₁Representing the sliding damping coefficient of the bristles; sigma₂Represents a viscous friction coefficient; g (omega)>0, a non-linear function, which can describe different friction effects; f_c，F_sRespectively representing Comlumbb friction torque and maximum static friction torque; omega_sRepresents stribeck velocity;

step 3, designing a nonlinear sliding mode surface;

3.1, for the proposed electromechanical servo system, the equation can also be expressed as follows:

wherein x is₁，x₂Indicating the output position and the rotating speed of the motor;

3.2 defining servo system position tracking error as

e＝θ-θ_ref(6) Wherein, theta_refIs a reference position signal;

3.3, setting the non-linear sliding mode surface as

Wherein F ∈ R is the linear portion in the sliding mode surface, P ∈ R is the normal number for adjusting the damping rate, ψ (s, θ) is a non-positive exponential function, as shown by equation (5),is the element in the first row and the second column of the system matrix in the state equation, equation (7) can be simplified to

Wherein F- ψ (s, θ) P;

step 4, designing a self-adaptive law of sliding mode control gain;

due to the influence of a system model and an external environment, a sliding mode gain k cannot be obtained accurately, a parameter self-adaption law is designed, and the expression is as follows:

wherein k is_s>0 is a constant;

step 4, designing a finite time parameter identification method;

4.1, the original system equation can be rewritten as follows:

wherein x is₁，x₂Respectively representing the actual position and the rotating speed of the motor; u represents an input torque;

4.2, order

Equation (10) can be written as follows:

wherein

4.3, for equation (11), let its state estimate be:

wherein,is an estimate of x; theta⁰Is an initial estimate of θ; k is a radical of_ω>0, a constant matrix;

4.4, defining auxiliary variables:

taking the filtered outputω(t₀) Is ω at t₀The value of the time is

η(t₀) To estimate the errorAt t₀The value of the time of day.

4.5, define variables Q and C, and the dynamic equation is as follows:

let t_cAt time Q (t)_c)>0, then there is the following adaptation law:

whereinIs an estimated value of theta^T>0 is used to guarantee at t₀≤t≤t_cTime of flightIs not increasing and is from t_cThe time instant converges to zero rapidly in an exponential manner, and the convergence rate has a lower bound (t) ═ lambda_min(Q (t)); therefore, for some parameters to be identified in the equations (1) to (4) which are parameters to be estimated in the equation (10), the identification speed can be increased by the equation (16);

4.6, settingIs an estimate of, andcombined formulas (1) - (4), (1)0) And (16) designing the following parameter adaptive law:

wherein,k_ζ>0，k_J>0，γ_ζ>0，γ_J>0 is a constant;is an estimate of state z; see formula (22);

step 5, designing a state observer and a self-adaptive control law;

5.1, represented by formula (6) and formula (8)

The nonlinear sliding mode surface can be rewritten as

s＝ω- (23)

5.2, since the deformation z of the bristles in the model is not measurable, a double closed-loop observer is designed, the expression of which is as follows:

whereinIs an estimate of state z; k is a radical of₀>0，k₁>0 is a constant;

5.3, the self-adaptive control law can be selected according to the following formulas:

5.4, designing Lyapunov function

The self-adaptive law, the state observer and the self-adaptive control law of each parameter are brought into the known state, the closed-loop system is asymptotically converged, and the tracking error is converged to zero in a short time.

The invention designs the finite time parameter identification and self-adaptive nonlinear sliding mode control method based on the LuGre friction model by combining the finite time parameter identification technology, the nonlinear sliding mode and the sliding mode gain self-adaptive technology, and realizes the quick convergence and the precision control of the friction parameter identification in the electromechanical servo system.

The technical conception of the invention is as follows: the electromechanical servo system inevitably generates a friction phenomenon during operation. Aiming at an electromechanical servo system with nonlinear friction, a limited time parameter identification and nonlinear sliding mode technology is combined, a limited time parameter identification and adaptive nonlinear sliding mode control method based on a LuGre friction model is designed, and the problems of too low convergence speed and low tracking precision of friction parameter identification in the traditional adaptive control are solved. The convergence speed of the parameters in the identification process is greatly accelerated by adopting a finite time parameter identification technology, the control effect is enabled to act on a servo system faster and more effectively, compared with the traditional linear sliding mode, the tracking effect can be realized faster under the condition of the same control precision by adopting a nonlinear sliding mode technology, the tracking error is enabled to converge to a zero point in shorter time, and meanwhile, the self-adaptive sliding mode gain technology is adopted, so that the method can work stably under different working conditions and disturbances, the sliding mode gain parameters are automatically adjusted, and the tracking precision is effectively ensured. The invention provides a self-adaptive nonlinear sliding mode control method capable of accelerating the identification convergence speed of friction parameters and improving the control precision, which ensures that when nonlinear friction exists in a system, the quick compensation of friction in an electromechanical servo system and the quick tracking of system output are realized.

The invention has the advantages that: the method can greatly accelerate the convergence speed of parameter identification when nonlinear friction exists in the electromechanical servo system, realize quick tracking and effectively improve the tracking precision.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a graph illustrating the position tracking effect of the electromechanical servo system of the present invention;

FIG. 3 is a plot of a comparison of the tracking error of the system of the present invention;

FIG. 4 is a comparison graph of the parameter identification speed of the present invention;

fig. 5 is a graph of adaptive control gain curves according to the present invention.

Detailed Description

Referring to fig. 1-5, a finite time friction parameter identification and adaptive sliding mode control method for an electromechanical servo system comprises the following steps:

step 1, establishing an electromechanical servo system model with friction as shown in formula (1), and initializing a system state and related control parameters;

in the formula: j is the moment of inertia equivalent to the motor shaft; theta and omega respectively represent the actual position and the rotating speed of the output shaft of the motor; t is the input torque of the motor; t is_fIs the total friction torque; t is_dIs the set of various disturbances of the system;

step 2, establishing a LuGre friction model, wherein the expression of the LuGre friction model is as follows;

in the formula: z represents the average deformation of the bristles between the contact surfaces; sigma₀Indicating bristleStiffness coefficient of wool; sigma₁Representing the sliding damping coefficient of the bristles; sigma₂Represents a viscous friction coefficient; g (omega)>0, a non-linear function, which can describe different friction effects; f_c，F_sRespectively representing Comlumbb friction torque and maximum static friction torque; omega_sRepresents stribeck velocity;

step 3, designing a nonlinear sliding mode surface;

3.1, for the proposed electromechanical servo system, the equation can also be expressed as follows:

wherein x is₁，x₂Indicating the output position and the rotating speed of the motor;

3.2 defining servo system position tracking error as

e＝θ-θ_ref(6) Wherein, theta_refIs a reference position signal;

3.3, setting the non-linear sliding mode surface as

Wherein F ∈ R is the linear portion in the sliding mode surface, P ∈ R is the normal number for adjusting the damping rate, ψ (s, θ) is a non-positive exponential function, as shown by equation (5),is the element in the first row and the second column of the system matrix in the state equation, equation (7) can be simplified to

Wherein F- ψ (s, θ) P;

step 4, designing a self-adaptive law of sliding mode control gain;

due to the influence of a system model and an external environment, a sliding mode gain k cannot be obtained accurately, a parameter self-adaption law is designed, and the expression is as follows:

wherein k is_s>0 is a constant;

step 4, designing a finite time parameter identification method;

4.1, the original system equation can be rewritten as follows:

wherein x is₁，x₂Respectively representing the actual position and the rotating speed of the motor; u represents an input torque;

4.2, order

Equation (10) can be written as follows:

wherein

4.3, for equation (11), let its state estimate be:

wherein,is an estimate of x; theta⁰Is an initial estimate of θ; k is a radical of_ω>0, a constant matrix;

4.4, defining auxiliary variables:

taking the filtered outputω (t0) is the value of ω at time t0, which is

η(t₀) To estimate the errorAt t₀The value of the time of day.

4.5, define variables Q and C, and the dynamic equation is as follows:

let t_cAt time Q (t)_c)>0, then there is the following adaptation law:

whereinIs an estimated value of theta^T>0 is used to guarantee at t₀≤t≤t_cTime of flightIs non-increasing and converges to zero exponentially and rapidly from time tc, with a convergence rate having a lower bound (t) ═ λ_min(Q (t)); therefore, for some parameters to be identified in the equations (1) to (4) which are parameters to be estimated in the equation (10), the identification speed can be increased by the equation (16);

4.6, settingIs an estimate of, andthe following parameter adaptation laws are designed by combining equations (1) - (4), (10) and (16):

wherein,kζ>0，kJ>0，γζ>0，γJ>0 is a constant;is an estimate of state z; see formula (22);

step 5, designing a state observer and a self-adaptive control law;

5.1, represented by formula (6) and formula (8)

The nonlinear sliding mode surface can be rewritten as

s＝ω- (23)

5.2, since the deformation z of the bristles in the model is not measurable, a double closed-loop observer is designed, the expression of which is as follows:

whereinIs an estimate of state z; k is a radical of₀>0，k₁>0 is a constant;

5.3, the self-adaptive control law can be selected according to the following formulas:

5.4, designing Lyapunov function

The self-adaptive law, the state observer and the self-adaptive control law of each parameter are brought into the known state, the closed-loop system is asymptotically converged, and the tracking error is converged to zero in a short time.

In order to verify the effectiveness of the proposed method, the present invention performs a simulation experiment on the control effect of the adaptive control law represented by equation (25) and the convergence effect of the parameter adaptive laws of equations (17) to (21). Initial conditions and some parameters in the experiment were set, namely: the moment of inertia J in the system equation is 0.9 kg.m²σ in Friction model₀＝30Nm，σ₁＝2.5Nm，σ₂＝0.2Nm(rad/s)^-1，ζ＝2.7，F_c＝0.28Nm，F_s＝0.34Nm，ω_s0.01 (rad)/s; for σ₀，σ₁，ζ，T_dThe initial values of J are set toζ⁰＝2.48，J⁰＝0.3kg·m². Furthermore, k in the state observer₀＝k₁0.01; adaptive sliding mode gain k_s50; in the nonlinear sliding mode surface, F is 20, P is 0.01,in law of parameter adaptationk_ζ＝5，k_J＝0.08，For reference position signals, the sinusoidal signal θ is taken for convenience_ref2sin (t), perturbation signal θ_refTdThe proposed control method was simulated at 0.1sin (4 π t).

As can be seen from FIG. 2, the finite time friction parameter identification and adaptive sliding mode control method designed by the invention can realize fast and effective tracking on the reference signal. It can be seen from the comparison of the position tracking errors in fig. 3 that the solution proposed by the present invention tends to a stable range of [ -0.00032,0.00032] after 0.5s, whereas the conventional adaptive control method tends to be stable in 13s, the stable range is [ -0.00065,0.00065], and during the stabilization process, there is severe buffeting, and the buffeting amplitude is gradually decreased from 0.08.

FIG. 4 shows the comparison of the respective adaptive parameter identification speeds, in FIG. 4a, for the bristle stiffness coefficient σ₀By contrast, it can be seen from the figure that the scheme provided by the present invention can converge to the true value at 0.5s with a convergence error of 0.00012, whereas the conventional adaptive control method converges to the true value at 12s with a convergence error of 0.005 and buffeting occurs before converging to the true value; from FIG. 4b, the sliding damping coefficient σ₁The comparison shows that the scheme provided by the invention converges to the true value at 0.5s, and is stabilized on the true value,the self-adaptive control method converges to the vicinity of a true value in 8s and the final convergence error is 0.006; from fig. 4c ζ ═ σ₁+σ₂The convergence comparison shows that the scheme of the invention converges to the true value in 0.3s, and the final convergence error is 0.005, while the traditional self-adaptive control method converges to the true value in 13s, and the final convergence error is 0.015; from the comparison of the moment of inertia J in fig. 4d, it can be seen that the scheme of the present invention converges to the true value in 0.8s and finally stabilizes on the true value, whereas the conventional adaptive control method converges to the true value in 14s, and the parameter adjustment can only stabilize to 0.73 anyway, and cannot converge to the true value of 0.9, and the stabilization error is 0.17; as can be seen from the disturbance tracking comparison in FIG. 4e, the method of the present invention can basically realize tracking compensation with tracking error of [ -0.025,0.025]However, the conventional adaptive control method cannot effectively track the disturbance. The comparison data shows that the limited time parameter identification provided by the invention can more effectively realize the quick identification of the friction parameter, greatly shorten the parameter identification time and improve the parameter convergence precision to a certain extent. Fig. 5 shows a case where the sliding mode adaptive gain curve k changes with changes in the respective parameters and the tracking error, and eventually converges to a final value with convergence of the parameters and stabilization of the tracking error. On the whole, under the limited time friction parameter identification and the self-adaptive sliding mode control method, the scheme provided by the invention can more quickly realize parameter identification and position tracking and effectively realize friction compensation control.

The above description is a comparative example of the simulation of the present invention to show the good tracking control effect, but it is obvious that the present invention is not limited to the above example, and various modifications can be made without departing from the basic spirit of the present invention and the scope of the present invention is not exceeded. The control scheme provided by the invention has a good compensation control effect on the nonlinear friction in the electromechanical servo system, and under the action of the control scheme, the system can quickly realize tracking and has higher tracking precision.

Claims (1)

1. The finite time friction parameter identification and self-adaptive sliding mode control method of the electromechanical servo system comprises the following steps:

step 1, establishing an electromechanical servo system model with friction as shown in formula (1), and initializing a system state and related control parameters;

$\left\{\begin{array}{c}\stackrel{\·}{\mathrm{\θ}}=\mathrm{\ω}\\ J\stackrel{\·}{\mathrm{\ω}}=T-T_f-T_d\end{array}\right.---\left(1\right)$

in the formula: j is the moment of inertia equivalent to the motor shaft; theta and omega respectively represent the actual position and the rotating speed of the output shaft of the motor; t is the input torque of the motor; t is_fIs the total friction torque; t is_dIs the set of various disturbances of the system;

step 2, establishing a LuGre friction model, wherein the expression of the LuGre friction model is as follows;

$T_f={\mathrm{\σ}}_{0}z+{\mathrm{\σ}}_1\frac{dz}{dt}+{\mathrm{\σ}}_2\mathrm{\ω}---\left(2\right)$

$\frac{dz}{dt}=\mathrm{\ω}-\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}z---\left(3\right)$

${\mathrm{\σ}}_{0}g\left(\mathrm{\ω}\right)=F_c+(F_s-F_c)e^{-{(\mathrm{\ω}/{\mathrm{\ω}}_s)}^2}---\left(4\right)$

in the formula: z represents the average deformation of the bristles between the contact surfaces; sigma₀Representing the stiffness coefficient of the bristles; sigma₁Representing the sliding damping coefficient of the bristles; sigma₂Represents a viscous friction coefficient; g (omega) > 0, which is a nonlinear function and can describe different friction effects; f_c，F_sRespectively representing Comlumbb friction torque and maximum static friction torque；ω_sRepresents stribeck velocity;

step 3, designing a nonlinear sliding mode surface;

3.1, for the proposed electromechanical servo system, the equation can also be expressed as follows:

$\left[\begin{array}{c}{\stackrel{\·}{x}}_1\\ {\stackrel{\·}{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1/J\end{array}\right]T-\left[\begin{array}{c}0\\ 1/J\end{array}\right]T_f-\left[\begin{array}{c}0\\ 1/J\end{array}\right]T_d---\left(5\right)$

wherein x is₁，x₂Indicating the output position and the rotating speed of the motor;

3.2 defining servo system position tracking error as

e＝θ-θ_ref(6)

Wherein, theta_refIs a reference position signal;

3.3, setting the non-linear sliding mode surface as

$s=\left[\begin{array}{cc}F-\mathrm{\ψ}(s,\mathrm{\θ})a_{12}^{T}P& 1\end{array}\right]\left[\begin{array}{c}e\\ \stackrel{\·}{e}\end{array}\right]---\left(7\right)$

Wherein F ∈ R is the linear portion in the sliding mode surface, P ∈ R is the normal number for adjusting the damping rate, ψ (s, θ) is a non-positive exponential function, as shown by equation (5),is the element in the first row and the second column of the system matrix in the state equation, equation (7) can be simplified to

Wherein F- ψ (s, θ) P;

step 4, designing a self-adaptive law of sliding mode control gain;

due to the influence of a system model and an external environment, a sliding mode gain k cannot be obtained accurately, a parameter self-adaption law is designed, and the expression is as follows:

$\stackrel{\·}{k}=k_s\left|s\right|---\left(9\right)$

wherein k is_s0 is a constant;

step 4, designing a finite time parameter identification method;

4.1, the original system equation can be rewritten as follows:

$\left[\begin{array}{c}{\stackrel{\·}{x}}_1\\ {\stackrel{\·}{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{ccccc}-z& -\stackrel{\·}{z}& -x_2& -1& u\end{array}\right]\left[\begin{array}{c}\frac{{\mathrm{\σ}}_0}{J}\\ \frac{{\mathrm{\σ}}_1}{J}\\ \frac{{\mathrm{\σ}}_2}{J}\\ \frac{T_d}{J}\\ \frac{1}{J}\end{array}\right]---\left(10\right)$

wherein x is₁，x₂Respectively representing the actual position and the rotating speed of the motor; u represents an input torque;

4.2, order

Equation (10) can be written as follows:

$\stackrel{\·}{x}=f\left(x\right)+g(x,u)\mathrm{\θ}---\left(11\right)$

wherein

4.3, for equation (11), let its state estimate be:

$\stackrel{\·}{\hat{x}}=f\left(x\right)+g(x,u){\mathrm{\θ}}^0+k_{\mathrm{\ω}}(x-\hat{x})---\left(12\right)$

wherein,is an estimate of x; theta⁰Is an initial estimate of θ; k is a radical of_ωIs more than 0 and is a constant matrix;

4.4, defining auxiliary variables:

$\mathrm{\η}=x-\hat{x}-\mathrm{\ω}(\mathrm{\θ}-{\mathrm{\θ}}^0)---\left(13\right)$

taking the filtered outputω(t₀)＝0，ω(t₀) Is ω at t₀The value of the time is

$\stackrel{\·}{\mathrm{\η}}=-k_w\mathrm{\η},\mathrm{\η}\left(t_0\right)=e\left(t_0\right)---\left(14\right)$

η(t₀) To estimate the errorAt t₀A value of a time of day;

4.5, define variables Q and C, and the dynamic equation is as follows:

$\left\{\begin{array}{cc}\stackrel{\·}{Q}={\mathrm{\ω}}^T\mathrm{\ω},& Q\left(t_0\right)=0\\ \stackrel{\·}{C}={\mathrm{\ω}}^T({\mathrm{\ω\θ}}^0+x-\hat{x}-\mathrm{\η}),& C\left(t_0\right)=0\end{array}\right.---\left(15\right)$

let t_cAt time Q (t)_c) If > 0, the following adaptation law is applied:

$\stackrel{\·}{\widehat{\mathrm{\θ}}}=\mathrm{\Γ}(C-Q\widehat{\mathrm{\θ}}),\widehat{\mathrm{\θ}}\left(t_0\right)={\mathrm{\θ}}^0---\left(16\right)$

whereinIs an estimated value of theta^T> 0 for ensuring at t₀≤t≤t_cTime of flightIs not increasing and is from t_cThe time instant converges to zero rapidly in an exponential manner, and the convergence rate has a lower bound (t) ═ lambda_min(Q (t)); therefore, for some parameters to be identified in the equations (1) to (4) which are parameters to be estimated in the equation (10), the identification speed can be increased by the equation (16);

4.6, settingIs an estimate of, andthe following parameter adaptation laws are designed by combining equations (1) - (4), (10) and (16):

${\stackrel{\·}{\widehat{\mathrm{\σ}}}}_0=-k_{{\mathrm{\σ}}_0}{\hat{z}}_{0}s-(C-Q{\widehat{\mathrm{\σ}}}_0)=-k_{{\mathrm{\σ}}_0}{\hat{z}}_{0}s-{\mathrm{\γ}}_{{\mathrm{\σ}}_0}{\widetilde{\mathrm{\σ}}}_0---\left(17\right)$

${\stackrel{\·}{\widehat{\mathrm{\σ}}}}_1=k_{{\mathrm{\σ}}_1}\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}{\hat{z}}_{1}s-(C-Q{\widehat{\mathrm{\σ}}}_1)=k_{{\mathrm{\σ}}_1}\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}{\hat{z}}_{1}s-{\mathrm{\γ}}_{{\mathrm{\σ}}_1}{\widetilde{\mathrm{\σ}}}_1---\left(18\right)$

$\stackrel{\·}{\widehat{\mathrm{\ζ}}}=-k_{\mathrm{\ζ}}\mathrm{\ω}s-(C-Q{\widehat{\mathrm{\σ}}}_1)=-k_{\mathrm{\ζ}}\mathrm{\ω}s-{\mathrm{\γ}}_{\mathrm{\ζ}}\widetilde{\mathrm{\ζ}}---\left(19\right)$

${\stackrel{\·}{\hat{T}}}_d=-k_{T_d}s-(C-Q{\widehat{\mathrm{\σ}}}_1)=-k_{T_d}s-{\mathrm{\γ}}_{T_d}{\tilde{T}}_d---\left(20\right)$

$\stackrel{\·}{\hat{J}}=-k_J\stackrel{\·}{\mathrm{\δ}}s-(C-Q{\widehat{\mathrm{\σ}}}_1)=-k_J\stackrel{\·}{\mathrm{\δ}}s-{\mathrm{\γ}}_J\tilde{J}---\left(21\right)$

wherein,k_ζ＞0，k_J＞0，γ_ζ＞0，γ_J> 0 is a constant; is an estimate of state z; see formula (22);

step 5, designing a state observer and a self-adaptive control law;

5.1, represented by formula (6) and formula (8)

$\mathrm{\δ}={\stackrel{\·}{\mathrm{\θ}}}_{ref}-(F-\mathrm{\ψ}(s,\mathrm{\θ}\left)P\right)e={\stackrel{\·}{\mathrm{\θ}}}_{ref}-\mathrm{\Γ}e---\left(22\right)$

The nonlinear sliding mode surface can be rewritten as

s＝ω- (23)

5.2, since the deformation z of the bristles in the model is not measurable, a double closed-loop observer is designed, the expression of which is as follows:

$\left\{\begin{array}{c}\frac{d{\hat{z}}_0}{dt}=\mathrm{\ω}-\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}{\hat{z}}_0-k_{0}s\\ \frac{d{\hat{z}}_1}{dt}=\mathrm{\ω}-\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}{\hat{z}}_1+k_1\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}s\end{array}\right.---\left(24\right)$

whereinIs an estimate of state z; k is a radical of₀＞0，k₁> 0 is a constant;

5.3, the self-adaptive control law can be selected according to the following formulas:

$T=-k\mathrm{sgn}\left(s\right)+{\widehat{\mathrm{\σ}}}_0{\hat{z}}_0-{\widehat{\mathrm{\σ}}}_1\frac{\left|\mathrm{\ω}\right|}{g\left(\mathrm{\ω}\right)}{\hat{z}}_1+\widehat{\mathrm{\ζ}}\mathrm{\ω}+{\hat{T}}_d+\hat{J}\stackrel{\·}{\mathrm{\δ}}---\left(25\right)$

5.4, designing Lyapunov function

$V=\frac{1}{2}{\mathrm{Js}}^2+\frac{1}{2k_0}{\mathrm{\σ}}_0{\tilde{z}}_0^2+\frac{1}{2k_1}{\mathrm{\σ}}_1{\tilde{z}}_1^2+\frac{1}{2k_{{\mathrm{\σ}}_0}}{\widetilde{\mathrm{\σ}}}_0^2+\frac{1}{2k_{{\mathrm{\σ}}_1}}{\widetilde{\mathrm{\σ}}}_1^2+\frac{1}{2k_{\mathrm{\ζ}}}{\widetilde{\mathrm{\ζ}}}^2+\frac{1}{2k_{T_d}}{\tilde{T}}_d^2+\frac{1}{2k_J}{\tilde{J}}^2---\left(26\right)$

The self-adaptive law, the state observer and the self-adaptive control law of each parameter are brought into the known state, the closed-loop system is asymptotically converged, and the tracking error is converged to zero in a short time.