CN104682805B - Full-order sliding mode variable structure position servo control method for permanent magnet synchronous motor based on extended state observer - Google Patents

Abstract

A full-order sliding mode variable structure position servo control method for permanent magnet synchronous motors based on extended state observers, comprising: establishing a permanent magnet synchronous motor system, initializing system states and related control parameters; designing extended state observers; A full-order sliding mode controller is designed to eliminate the chattering problem in sliding mode control and ensure that the system state can quickly and stably converge to zero. The present invention proposes a full-order sliding mode variable structure position servo control method, which can improve the chattering problem of sliding mode control and improve system control accuracy, and ensure the rapid and accurate tracking of the output position of the motor to the expected trajectory.

Description

Full-order sliding mode variable structure position servo for permanent magnet synchronous motor based on extended state observer service control method

technical field

The invention relates to a full-order sliding mode variable structure position servo control method for a permanent magnet synchronous motor, in particular to a full-order sliding mode control method for a permanent magnet synchronous motor system whose partial states and upper bounds of nonlinear uncertain items are unknown.

Background technique

In the permanent magnet synchronous motor, due to the high control gain and the existence of the sign function in the traditional control method, there is a certain chattering problem. For the high-performance permanent magnet synchronous motor position servo system, how to weaken the chattering phenomenon in sliding mode control is a key technical problem to be solved urgently, which affects the precise positioning and position tracking performance of the motor system, and even affects the motor in severe cases. The system itself causes damage. In order to solve the problem of chattering in sliding mode control, reduce the adverse effects of chattering in permanent magnet synchronous motors, and improve the performance of the system, it is necessary to adopt an appropriate control method to achieve fast and accurate tracking of the output position of the motor to the desired trajectory .

At present, in the research of eliminating chattering, various improved sliding mode control methods have been proposed, such as using saturated functions instead of sign functions to design controllers, integral time-varying sliding mode controllers and adaptive sliding mode controllers. In addition, in recent years, the combination of disturbance observer and extended state observer with sliding mode control has been proposed for speed control and chattering-free sliding mode control of permanent magnet synchronous motors. The controller is a full-order sliding mode controller. Compared with the traditional reduced-order sliding mode controller, the advantage is that the control signal is continuous, which can effectively avoid the chattering phenomenon of sliding mode control. Aiming at the permanent magnet synchronous motor position servo system with unknown friction torque and model uncertain items, the present invention designs a permanent magnet synchronous motor full-order sliding mode variable structure position servo control method based on the extended state observer, and realizes that the output position of the motor is expected. Fast and precise tracking of trajectories.

Contents of the invention

In order to overcome the deficiencies of the chattering phenomenon in the sliding mode control of the permanent magnet synchronous motor position servo system with unknown friction torque and model uncertain items, the present invention provides a full-order sliding mode of permanent magnet synchronous motor based on the extended state observer The variable structure position servo control method can better avoid the chattering phenomenon of sliding mode control. The extended state observer is used to estimate the system state and uncertain items, and a full-order sliding mode control method is designed based on the estimated value to suppress the chattering problem in sliding mode control, and realize the fast and accurate tracking of the motor output position to the desired trajectory.

The technical scheme proposed in order to solve the above technical problems is as follows:

A full-order sliding mode variable structure position servo control method for a permanent magnet synchronous motor based on an extended state observer, comprising the following steps:

Step 1, establish a permanent magnet synchronous motor system, initialize the system state and control parameters;

1.1, in the d/q rotating coordinate system, the voltage equation, torque equation and motion equation of the permanent magnet synchronous motor are respectively:

T _e ＝1.5p _n ψ _f i _q (2)

Among them, u _d and u _q are the components of the stator voltage on the _d and q axes respectively; id and i _q are the components of the stator current on the d and q axes respectively; R is the stator resistance; L _d and L _q are respectively The components of the stator inductance on the d and q axes; p _n is the number of pole pairs; ω is the angular _velocity of the rotor; J is the moment of inertia; B is the friction coefficient; T _e is the electromagnetic torque; T _L is the load torque; Permanent magnet fundamental wave excitation flux linkage;

1.2, from equations (1)-(3), the second-order dynamic equation of the permanent magnet synchronous motor position loop is obtained as

Among them, b=1.5p _n ψ _f /J, d is the disturbance composed of unknown friction torque and load torque, d=-(T _L +Bω)/J;

1.3, according to the design idea of the extended state observer, the state variable x _i , i=1,2,3, let x ₁ =θ,

x ₂ ＝ω, and define the extended state x ₃ ＝a(t), then formula (4) can be written as the following equivalent form

y = x ₁ = θ (6)

in, For a given q-axis current reference input, b ₀ is the estimated value of b, u is the control input, y is the actual output position of the permanent magnet synchronous motor;

Step 2, expand state observer design;

Let z _i , i=1, 2, 3 be the observed values of the state variable x _i in formula (5), and define the observation error as ε _i = z _{i -xi} , then the expression of the nonlinear extended state observer is :

Among them, β ₁ , β ₂ , β ₃ are observer gains, β ₁ , β ₂ , β ₃ >0.fal(·) is a continuous power function with a linear segment near the origin, the expression is:

Among them, δ represents the interval length of the linear segment, δ>0, 0<α _i <1, i=1,2,3, sign(ε ₁ ) is a sign function, and the expression is:

Step 3, design of full-order sliding mode controller based on extended state observer;

3.1, define the tracking error e as

e = yy _d = x ₁ -y _d (9)

where y _d is the desired trajectory;

Then the first-order and second-order derivatives of the tracking error e are respectively

with

3.2, according to formula (9)-(11), design the following full-order sliding mode surface s:

Among them, λ ₁ and λ ₂ are control parameters, λ ₁ >0, λ ₂ >0;

Substitute formulas (9)-(11) into formula (12) to get

According to formula (13), the full-order sliding mode controller based on the extended state observer is designed as

u ₂ =-k sgn(s) (17)

Among them, T≥0, k=k _d +k _T +η, η, k _d , k _T are controller parameters, η>0, k _d >0, k _T >0;

3.3, Substituting formula (14)-(17) into formula (13), we have

s ₂ =u ₁ +(x ₃ -z ₃ )+λ ₂ (x ₂ -z ₂ )+λ ₁ (x ₁ -z ₁ ) =u1+d(x,z) (18)

Among them, d(x,z)=(x ₃ -z ₃ )+λ ₂ (x ₂ -z ₂ )+λ ₁ (x ₁ -z ₁ ), and satisfy d(x,z)≤l _d ,

l _d =l ₃ +λ ₂ l ₂ +λ ₁ l ₁ ;

Deriving formula (18) to get

3.4, Design Lyapunov function:

V＝0.5s ² (20)

Substitute formula (5), (12), (14)-(17) into formula (20), if The system is judged to be stable.

The present invention combines the extended state observer technology and the full-order sliding mode control technology to design a permanent magnet synchronous motor full-order sliding mode variable structure position servo controller based on the extended state observer to suppress the chattering problem in the sliding mode control, And realize the fast and accurate tracking of the output position of the motor to the desired trajectory.

The technical idea of the present invention is that the chattering problem inevitably occurs in the traditional sliding mode control. Aiming at the permanent magnet synchronous motor position servo system with unknown friction torque and model uncertain items, the extended state observer is used to estimate the system state and uncertain items, and a full-order sliding mode control method is designed based on the estimated value, and a The full-order sliding mode variable structure position servo control method of permanent magnet synchronous motor based on the extended state observer can suppress the chattering problem in the sliding mode control, and the designed full-order sliding mode controller can ensure that the tracking error e will converge to zero stably. The invention provides a full-order sliding mode variable structure position servo control method capable of improving the chattering problem of sliding mode control and improving system control precision, so as to ensure the rapid and accurate tracking of the output position of the motor to the desired trajectory.

The invention has the advantages of realizing fast and accurate tracking of the position of the permanent magnet synchronous motor and effectively eliminating chattering problems in sliding mode control.

Description of drawings

1 is a flowchart of a full-order sliding mode variable structure position servo control method for permanent magnet synchronous motors based on an extended state observer;

Fig. 2 is the block diagram of the permanent magnet synchronous motor position servo control system of the present invention;

Fig. 3 is the schematic diagram of the position tracking effect of the permanent magnet synchronous motor of the present invention;

Fig. 4 is a schematic diagram of the tracking error of the system of the present invention;

FIG. 5 is a schematic diagram of control signals of the present invention.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings.

Referring to Figures 1-5, a full-order sliding mode variable structure position servo control method for a permanent magnet synchronous motor based on an extended state observer includes the following steps:

Step 1, establish a permanent magnet synchronous motor system, initialize the system state and related control parameters;

1.1, in the d/q rotating coordinate system, the voltage equation, torque equation and motion equation of the permanent magnet synchronous motor are respectively:

T _e ＝1.5p _n ψ _f i _q (2)

Among them, u _d and u _q are the components of the stator voltage on the _d and q axes respectively; id and i _q are the components of the stator current on the d and q axes respectively; R is the stator resistance; L _d and L _q are respectively The components of the stator inductance on the d and q axes; p _n is the number of pole pairs; ω is the angular _velocity of the rotor; J is the moment of inertia; B is the friction coefficient; T _e is the electromagnetic torque; T _L is the load torque; Permanent magnet fundamental wave excitation flux linkage;

1.2, from equations (1)-(3), the second-order dynamic equation of the permanent magnet synchronous motor position loop can be obtained as

Among them, b=1.5p _n ψ _f /J, d is the disturbance composed of unknown friction torque and load torque, d=-(T _L +Bω)/J;

1.3, according to the design idea of the extended state observer, let x ₁ = θ, x ₂ = ω, and define the extended state x ₃ = a(t), then the formula (4) can be written as the following equivalent form

y = x ₁ = θ(6)

in, For a given q-axis current reference input, b ₀ is the estimated value of b, u is the control input, y is the actual output position of the permanent magnet synchronous motor;

Step 2, expand state observer design;

Let z _i , i=1, 2, 3 be the observed values of the state variable x _i in formula (5) respectively, define the observation error as ε _i = z _{i -xi} , then the nonlinear expansion state designed in the present invention The observer expression is:

Among them, β ₁ , β ₂ , β ₃ are observer gains, β ₁ , β ₂ , β ₃ >0.fal(·) is a continuous power function with a linear segment near the origin, the expression is:

Among them, δ represents the interval length of the linear segment, δ>0, 0<α _i <1, i=1,2,3, sign(ε ₁ ) is a sign function, and the expression is:

Step 3, design of full-order sliding mode controller based on extended state observer;

3.1, define the tracking error as

e = yy _d = x ₁ -y _d (9)

Then the first and second derivatives of e are

with

3.2, according to formula (9)-(11), design the following full-order sliding mode surface

Among them, λ ₁ and λ ₂ are control parameters, λ ₁ >0, λ ₂ >0;

Substitute formulas (9)-(11) into formula (12) to get

According to formula (13), the full-order sliding mode controller based on the extended state observer is designed as

u ₂ =-ksgn(s) (17)

Among them, T≥0, k=k _d +k _T +η, η, k _d , k _T are controller parameters, η>0, k _d >0, k _T >0;

3.3, Substituting formula (14)-(17) into formula (13), we have

s ₂ =u ₁ +(x ₃ -z ₃ )+λ(x ₂ -z ₂ )+λ ₁ (x ₁ -z ₁ ) =u1+d(x,z) (18)

Among them, d(x,z)=(x ₃ -z ₃ )+λ ₂ (x ₂ -z ₂ )+λ ₁ (x ₁ -z ₁ ), and satisfy d(x,z)≤l _d ,

l _d =l ₃ +λ ₂ l ₂ +λ ₁ l ₁ ;

Deriving formula (18) to get

3.4, Design Lyapunov function:

V＝0.5s ² (20)

Substitute formula (5), (12), (14)-(17) into formula (20), if The system is judged to be stable.

In order to verify the effectiveness of the proposed method, the present invention is based on the full-order sliding mode control based on extended state observer (FSMC+ESO) represented by formula (14)-(17) ) control effect, and compared with the reduced-order sliding mode control based on extended state observer (RSMC+ESO) effect. Some parameters of the permanent magnet synchronous motor system, extended state observer and sliding mode controller used in the simulation are designed as follows: The parameters of the permanent magnet synchronous motor are set as: rated power P = 0.2kW, rated speed ω = 3000r/min, permanent magnet Flux linkage ψ _f =0.371Wb, number of pole pairs p _n =4, dq-axis inductance L _d =L _q =30mH, moment of inertia J=0.17kg·cm ² , viscous damping coefficient B=0.001N·m/(r/ min); the extended state observer parameters are set as: β ₁ = β ₂ = β ₃ = 100, δ = 0.01, b ₀ = 10; the controller parameters are respectively set as: k = 20, λ ₂ = 2, λ ₁ = 5, T=0.01.

Figure 3 shows the comparison of the sinusoidal tracking effects of the two control methods RSMC+ESO and FSMC+ESO when the load T _L =2Nm. It can be seen from Fig. 3 that the FSMC+ESO control method designed in the present invention can realize the fast and effective tracking of the actual system output to the sinusoidal signal of the expected trajectory. The FSMC+ESO method has a faster tracking speed for the sinusoidal signal than the RSMC+ESO method. It can be seen from Figure 4 that the tracking error of the FSMC+ESO method tends to a stable range of [-0.01,0.01] after 2s, while the tracking error of the RSMC+ESO method tends to a stable range of [-0.005,0.005] after 3s, FSMC+ The steady-state error of the sinusoidal tracking of the ESO method is slightly larger than that of the RSMC+ESO method. It can be seen from Figure 5 that the chattering of the control signal of the FSMC+ESO control method is significantly smaller than that of the RSMC+ESO method. On the whole, under the action of the full-order sliding mode controller based on the extended state observer, not only the chattering problem in the sliding mode control can be effectively eliminated, but also the tracking error of the system can be stably converged to zero.

The above set forth is the excellent optimization effect shown by an embodiment of the present invention. Obviously, the present invention is not limited to the above-mentioned embodiment. It can be implemented in various modifications. The proposed control scheme is effective for the position servo system of permanent magnet synchronous motor with unknown friction torque and model uncertainty. Under the action of the proposed controller, the chattering problem in sliding mode control can be suppressed. The actual motor The output position can quickly and accurately track the desired trajectory.

Claims (1)

1. A full-order sliding mode variable structure position servo control method for permanent magnet synchronous motors based on extended state observer, is characterized in that: comprise the following steps: Step 1, establish a permanent magnet synchronous motor system, initialize the system state and control parameters; 1.1, in the d/q rotating coordinate system, the voltage equation, torque equation and motion equation of the permanent magnet synchronous motor are respectively: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; Ri</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\frac{{\mathrm{<span\; class="notranslate">\; di</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}}\\ {\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; Ri</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&omega;p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&omega;p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}\frac{{\mathrm{<span\; class="notranslate">\; di</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}}{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ T _e ＝1.5p _n ψ _f i _q (2) ${\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; J</span>}\frac{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; B</span>}\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ Among them, u _d and u _q are the components of the stator voltage on the _d and q axes respectively; id and i _q are the components of the stator current on the d and q axes respectively; R is the stator resistance; L _d and L _q are respectively The components of the stator inductance on the d and q axes; p _n is the number of pole pairs; ω is the angular _velocity of the rotor; J is the moment of inertia; B is the friction coefficient; T _e is the electromagnetic torque; T _L is the load torque; Permanent magnet fundamental wave excitation flux linkage; 1.2, from equations (1)-(3), the second-order dynamic equation of the permanent magnet synchronous motor position loop is obtained as $\left\{\begin{array}{c}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}\\ \stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; bi</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; d</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ Among them, b=1.5p _n ψ _f /J, d is the disturbance composed of unknown friction torque and load torque, d=-(T _L +Bω)/J; 1.3, according to the design idea of the extended state observer, the state variable x _i , i=1,2,3, let x ₁ =θ, x ₂ ＝ω, and define the extended state x ₃ ＝a(t), then formula (4) can be written as the following equivalent form $\left\{\begin{array}{c}{\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; u</span>}\\ {\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ y=x ₁ =θ (6) where, For a given q-axis current reference input, b ₀ is the estimated value of b, u is the control input, y is the actual output position of the permanent magnet synchronous motor; Step 2, the design of the extended state observer; Let z _i , i=1, 2, 3 be the observed values of the state variable x _i in formula (5), and define the observation error as ε _i = z _{i -xi} , then the expression of the nonlinear extended state observer is : $\left\{\begin{array}{c}{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; z</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; z</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; u</span>}\\ {\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; z</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ Among them, β ₁ , β ₂ , β ₃ are observer gains, β ₁ , β ₂ , β ₃ >0, fal(·) is a continuous power function with a linear segment near the origin, the expression is: $\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}{{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; ,</span>& <span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}\\ <span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; g</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& <span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ Among them, δ represents the interval length of the linear segment, δ>0, 0<α _i <1, i=1,2,3, sign(ε ₁ ) is a sign function, and the expression is: $\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; g</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right.<span\; class="notranslate">\; ;</span>$ Step 3, design of full-order sliding mode controller based on extended state observer; 3.1, define the tracking error e as e = yy _d = x ₁ -y _d (9) where y _d is the desired trajectory; Then the first-order and second-order derivatives of the tracking error e are respectively $\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$ with $\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$ 3.2, according to formula (9)-(11), design the following full-order sliding mode surface s: $\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$ Among them, λ ₁ and λ ₂ are control parameters, λ ₁ >0, λ ₂ >0; Substitute formulas (9)-(11) into formula (12) to get $\begin{array}{c}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; e</span>}\\ <span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$ According to formula (13), the full-order sliding mode controller based on the extended state observer is designed as $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$ ${\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Tu</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$ u ₂ =-ksign(s) (17) Among them, T≥0, k=k _d +k _T +η, η, k _d , k _T are controller parameters, η>0, k _d >0, k _T >0; 3.3, Substituting formula (14)-(17) into formula (13), we have $\begin{array}{c}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$ Among them, d(x,z)=(x ₃ -z ₃ )+λ ₂ (x ₂ -z ₂ )+λ ₁ (x ₁ -z ₁ ), and satisfy d(x,z)≤l _d , l _d =l ₃ +λ ₂ l ₂ +λ ₁ l ₁ ; Deriving formula (18) to get $\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Tu</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$ 3.4, Design Lyapunov function: V＝0.5s ² (20) Substitute formula (5), (12), (14)-(17) into formula (20), if The system is judged to be stable.