CN108845586B - Finite time control method of four-rotor aircraft based on hyperbolic sine enhanced constant-speed approach law and fast terminal sliding mode surface - Google Patents

Finite time control method of four-rotor aircraft based on hyperbolic sine enhanced constant-speed approach law and fast terminal sliding mode surface Download PDF

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CN108845586B
CN108845586B CN201810519604.5A CN201810519604A CN108845586B CN 108845586 B CN108845586 B CN 108845586B CN 201810519604 A CN201810519604 A CN 201810519604A CN 108845586 B CN108845586 B CN 108845586B
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sliding mode
formula
mode surface
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aircraft
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CN108845586A (en
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陈强
陈凯杰
胡轶
吴春
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Zhejiang University of Technology ZJUT
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
    • G05D1/0825—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability using mathematical models
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft

Abstract

A finite time control method of a four-rotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface comprises the following steps: step 1, determining a transfer matrix from a body coordinate system based on a four-rotor aircraft to an inertial coordinate system based on the earth; step 2, analyzing a four-rotor aircraft dynamic model according to a Newton Euler formula; and 3, calculating a tracking error, and designing a controller according to the fast terminal sliding mode surface and the first derivative thereof. Aiming at a four-rotor aircraft system, enhanced constant-speed approach law sliding mode control and rapid terminal sliding mode control based on hyperbolic sine are combined, approach speed can be increased when the system is far away from a sliding mode surface, buffeting can be reduced, rapidness and robustness of the system are improved, rapid and stable control is achieved, limited time control of tracking errors can be achieved, and the problem that the tracking errors tend to 0 only when time tends to be infinite in a traditional sliding mode surface is solved.

Description

Finite time control method of four-rotor aircraft based on hyperbolic sine enhanced constant-speed approach law and fast terminal sliding mode surface
Technical Field
The invention relates to a finite time control method of a four-rotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a rapid terminal sliding mode surface.
Background
The four-rotor aircraft has attracted wide attention of domestic and foreign scholars and scientific research institutions due to the characteristics of simple structure, strong maneuverability and unique flight mode, and is rapidly one of the hotspots of international research at present. Compared with a fixed-wing aircraft, the rotary-wing aircraft can vertically lift, has low requirement on the environment, does not need a runway, reduces the cost and has great commercial value. The development of aircrafts makes many dangerous high-altitude operations easy and safe, so as to cause deterrence to other countries in the military aspect and greatly increase the working efficiency in the civil aspect. The four-rotor aircraft has strong flexibility, can realize rapid transition of motion and hovering at any time, and can be competent for more challenging flight tasks with less damage risk. In the field of scientific research, because a four-rotor aircraft has the dynamic characteristics of nonlinearity, under-actuation and strong coupling, researchers often use the four-rotor aircraft as an experimental carrier for theoretical research and method verification. An aircraft flight control system is built by relying on a small four-rotor aircraft to carry out high-performance motion control research on the aircraft, and the method is a hot research field of the current academic world.
The approach law sliding mode control has the characteristics that discontinuous control can be realized, the sliding mode is programmable and is not related to system parameters and disturbance. The approach law sliding mode not only can reasonably design the speed of reaching the sliding mode surface, reduce the time of the approach stage, improve the robustness of the system, but also can effectively weaken the buffeting problem in the sliding mode control. Currently, in the field of four-rotor control, approach law sliding mode control is less used. The enhanced approach law further accelerates the approach speed of the system to the sliding mode surface and simultaneously enables the buffeting to be smaller on the basis of the traditional approach law.
Disclosure of Invention
In order to solve the problems that the traditional sliding mode surface can not realize limited time control, further accelerate the approaching speed of an approaching law and reduce buffeting, the invention adopts the rapid terminal sliding mode control and the hyperbolic sine enhanced constant speed approaching law, avoids the singularity problem through the switching control idea, accelerates the approaching speed of a system to the sliding mode surface, reduces buffeting and realizes limited time control.
The technical scheme proposed for solving the technical problems is as follows:
a finite time control method of a four-rotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface comprises the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a four-rotor aircraft to an inertial coordinate system based on the earth;
Figure BDA0001674482780000021
where psi, theta, phi are the yaw, pitch, roll angles of the aircraft, respectively, representing the angle of rotation of the aircraft about each axis of the inertial frame in sequence, TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing a four-rotor aircraft dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure BDA0001674482780000022
wherein x, y and z respectively represent the position of the four rotors under an inertial coordinate system, m represents the mass of the aircraft, g represents the gravity acceleration, mg represents the gravity borne by the four rotors, and the resultant force U generated by the four rotorsr；
2.2, the rotation process comprises the following steps:
Figure BDA0001674482780000023
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively representing the rotational inertia component of each axis on the coordinate system of the machine body, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure BDA0001674482780000024
respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the change of the attitude angle is small when the aircraft is in a low-speed flight or hovering state, the change is considered to be
Figure BDA0001674482780000031
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674482780000032
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Figure BDA0001674482780000033
Wherein
Figure BDA0001674482780000034
Figure BDA0001674482780000035
Figure BDA0001674482780000036
Ux、Uy、UzThe input quantities of the three position controllers are respectively;
according to the formula (5), decoupling calculation is carried out on the position and posture relation, and the result is as follows:
Figure BDA0001674482780000037
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor desired signal values of ψ, the arcsin function is an arcsine function and the arctan function is an arctangent function;
equation (5) can also be written in matrix form as follows:
Figure BDA0001674482780000038
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674482780000039
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T
Step 3, calculating a tracking error, and designing a controller according to the fast terminal sliding mode surface and the first derivative thereof, wherein the process is as follows:
3.1, defining the tracking error and its first and second differentials:
e＝X1-Xd (8)
Figure BDA0001674482780000041
Figure BDA0001674482780000048
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdConductive desired signals of x, y, z, phi, theta, psi, respectively;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674482780000042
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674482780000043
order to
Figure BDA0001674482780000044
Formula (12) is simplified to formula (13)
Figure BDA0001674482780000045
But due to the presence of alpha (e)
Figure BDA0001674482780000046
When α (e) is 0 and β (e) is not equal to 0, the negative power term of (a) causes a singularity problem;
consider the method of handover control:
Figure BDA0001674482780000047
wherein q isi(e),αi(e),βi(e) The i-th element, i ═ 1,2,3,4,5,6, q (e), α (e), β (e), respectively;
combining formula (13) and formula (14) to obtain:
Figure BDA0001674482780000051
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674482780000052
3.3 design enhanced approach law
Figure BDA0001674482780000053
Wherein
Figure BDA0001674482780000054
N-1(X) is an inverse matrix of N (X), k is more than 0, delta is more than 0 and less than 1, gamma is more than 0, mu is more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure BDA0001674482780000055
Wherein B is-1(X) is the inverse of B (X).
Further, the control method further includes the steps of:
step 4, property specification, the process is as follows:
4.1, proving accessibility of sliding forms:
designing Lyapunov functions
Figure BDA0001674482780000056
The derivation is performed on both sides of the function to obtain:
Figure BDA0001674482780000057
because of the fact that
Figure BDA0001674482780000058
The constant is larger than 0, so the formula (18) is constantly smaller than 0, the accessibility of the sliding mode is met, and the system can reach the sliding mode surface;
4.2, enhanced effect description:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure BDA0001674482780000059
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA00016744827800000510
the buffeting of the system is reduced.
The technical conception of the invention is as follows: aiming at a four-rotor aircraft system, a hyperbolic sine enhanced constant-speed approach law and rapid terminal sliding mode surface-based four-rotor aircraft finite time control method is designed by combining constant-speed approach law sliding mode control and rapid terminal sliding mode control. The quick terminal sliding mode surface can realize the limited time control of the tracking error, and solves the problems that the time tends to be infinite and the error tends to be 0 in the traditional sliding mode surface. Based on a hyperbolic sine enhanced approach law, the approach speed can be increased when the system is far away from a sliding mode surface, buffeting can be reduced, the rapidness and robustness of the system are improved, and rapid and stable control is realized.
The invention has the beneficial effects that: compared with the traditional constant velocity approach law sliding mode control, the method can increase the approach speed when the system is far away from the sliding mode surface, reduce buffeting and shorten the arrival time of the sliding mode, thereby enabling the system to realize stable convergence more quickly. In addition, the invention utilizes the quick terminal sliding mode, solves the problems that the time tends to be infinite and the error tends to be 0 in the traditional sliding mode surface, and realizes the limited time control.
Drawings
Fig. 1 is a schematic diagram of the position tracking effect of a quadrotor aircraft, wherein a dotted line represents traditional constant velocity approach law control, and a dotted line represents limited time control of the quadrotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface.
Fig. 2 is a schematic diagram of position tracking error of a quadrotor aircraft, wherein a dotted line represents traditional constant velocity approach law control, and a dotted line represents limited time control of the quadrotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface.
FIG. 3 is a schematic diagram of position controller inputs under conventional constant velocity approach law control for a quad-rotor aircraft.
Fig. 4 is a schematic input diagram of a position controller under finite time control of a quadrotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface.
FIG. 5 is a schematic diagram of attitude angle controller inputs under conventional constant velocity approach law control for a quad-rotor aircraft.
Fig. 6 is an input schematic diagram of an attitude angle controller under finite time control of a quadrotor aircraft based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface.
FIG. 7 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. 1-7, a finite time control method of a four-rotor aircraft based on hyperbolic sine enhanced constant velocity approach law and fast terminal sliding mode surface includes the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a four-rotor aircraft to an inertial coordinate system based on the earth;
Figure BDA0001674482780000071
where psi, theta, phi are the yaw, pitch, roll angles of the aircraft, respectively, representing the angle of rotation of the aircraft about each axis of the inertial frame in sequence, TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing a four-rotor aircraft dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure BDA0001674482780000072
wherein x, y and z respectively represent the position of the four rotors under an inertial coordinate system, m represents the mass of the aircraft, g represents the gravity acceleration, mg represents the gravity borne by the four rotors, and the resultant force U generated by the four rotorsr；
2.2, the rotation process comprises the following steps:
Figure BDA0001674482780000073
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzAre respectively provided withRepresenting the rotational inertia component of each axis on the coordinate system of the machine body, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure BDA0001674482780000074
respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the change of the attitude angle is small when the aircraft is in a low-speed flight or hovering state, the change is considered to be
Figure BDA0001674482780000081
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674482780000082
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Figure BDA0001674482780000083
Wherein
Figure BDA0001674482780000084
Figure BDA0001674482780000085
Figure BDA0001674482780000086
Ux、Uy、UzThe input quantities of the three position controllers are respectively;
according to the formula (5), decoupling calculation is carried out on the position and posture relation, and the result is as follows:
Figure BDA0001674482780000087
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor desired signal values of ψ, the arcsin function is an arcsine function and the arctan function is an arctangent function;
equation (5) can also be written in matrix form as follows:
Figure BDA0001674482780000088
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674482780000091
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T
Step 3, calculating a tracking error, and designing a controller according to the fast terminal sliding mode surface and the first derivative thereof, wherein the process is as follows;
3.1, defining the tracking error and its first and second differentials:
e＝X1-Xd (8)
Figure BDA0001674482780000092
Figure BDA0001674482780000093
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdConductive desired signals of x, y, z, phi, theta, psi, respectively;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674482780000094
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674482780000095
order to
Figure BDA0001674482780000096
Formula (12) can be simplified to formula (13)
Figure BDA0001674482780000097
But due to the presence of alpha (e)
Figure BDA0001674482780000098
When α (e) is 0 and β (e) is not equal to 0, the negative power term of (a) causes a singularity problem;
consider the method of handover control:
Figure BDA0001674482780000101
wherein q isi(e),αi(e),βi(e) The i-th element, i ═ 1,2,3,4,5,6, q (e), α (e), β (e), respectively;
combining formula (13) and formula (14) to obtain:
Figure BDA0001674482780000102
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674482780000103
3.3 design enhanced approach law
Figure BDA0001674482780000109
Wherein
Figure BDA0001674482780000104
N-1(X) is an inverse matrix of N (X), k is more than 0, delta is more than 0 and less than 1, gamma is more than 0, mu is more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure BDA0001674482780000105
Wherein B is-1(X) is the inverse of B (X).
The control method further comprises the following steps:
step 4, property specification, the process is as follows:
4.1, proving accessibility of sliding forms:
designing Lyapunov functions
Figure BDA0001674482780000106
The derivation is performed on both sides of the function to obtain:
Figure BDA0001674482780000107
because of the fact that
Figure BDA0001674482780000108
The constant is larger than 0, so the formula (18) is constantly smaller than 0, the accessibility of the sliding mode is met, and the system can reach the sliding mode surface;
4.2, enhanced effect description:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure BDA0001674482780000111
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA0001674482780000112
the buffeting of the system is reduced.
In order to verify the effectiveness of the method, the invention provides a hyperbolic sine enhanced constant velocity approach law sliding mode control method compared with the traditional constant velocity approach law sliding mode control method:
for more efficient comparison, all parameters of the system are consistent, i.e. xd＝yd＝zd＝20、ψd0.5, slip form surface parameters: lambda [ alpha ]1＝0.2、λ2＝0.1、α1＝2、α21.1, epsilon 0.5, and the approach law parameter: k is a radical of11, δ is 0.1, p is 1, γ is 5, μ is 1.5, the four-rotor aircraft parameters: m 0.625, L0.1275, Ixx＝2.3×10-3、Iyy＝2.4×10-3、Izz＝2.6×10-3G ═ 10, sampling parameters: t is ts＝0.007，N＝5000。
As can be seen from fig. 1 and 2, the finite-time control of the quadrotor aircraft based on the hyperbolic sine enhanced constant-velocity approach law and the fast terminal sliding mode surface can reach the expected position more quickly; with reference to fig. 3-6, the finite time control of the quadrotor aircraft based on the hyperbolic sine enhanced constant velocity approach law and the fast terminal sliding mode surface has smaller buffeting.
In conclusion, the finite time control of the four-rotor aircraft based on the hyperbolic sine enhanced constant-speed approaching law and the fast terminal sliding mode surface can reduce the buffeting, reduce the tracking time and improve the tracking performance, so that the system can enter stable convergence more quickly.
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 (2)

1. A four-rotor aircraft finite time control method based on hyperbolic sine enhanced constant velocity approach law and a fast terminal sliding mode surface is characterized by comprising the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a four-rotor aircraft to an inertial coordinate system based on the earth;
Figure FDA0002965241810000011
wherein psi, theta and phi are respectively the yaw angle, pitch angle and roll angle of the aircraft, and represent the angle of the aircraft sequentially rotating around each axis of the inertial coordinate system, and TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing a four-rotor aircraft dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure FDA0002965241810000012
wherein x, y and z respectively represent the position of the four rotors under an inertial coordinate system, m represents the mass of the aircraft, g represents the gravity acceleration, mg represents the gravity borne by the four rotors, and the resultant force U generated by the four rotorsr；
2.2, the rotation process comprises the following steps:
Figure FDA0002965241810000013
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively representing each axis rotation on the coordinate system of the machine bodyComponent of moment of inertia, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure FDA0002965241810000014
respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the aircraft is in a low-speed flight or hovering state, consider
Figure FDA0002965241810000021
Then the formula (3) is represented as the formula (4) in the rotation process
Figure FDA0002965241810000022
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Figure FDA0002965241810000023
Wherein
Figure FDA0002965241810000024
Figure FDA0002965241810000025
Figure FDA0002965241810000026
Ux、Uy、UzThe input quantities of the three position controllers are respectively;
according to the formula (5), decoupling calculation is carried out on the position and posture relation, and the result is as follows:
Figure FDA0002965241810000027
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor desired signal values of ψ, the arcsin function is an arcsine function and the arctan function is an arctangent function;
equation (5) can also be written in matrix form as follows:
Figure FDA0002965241810000028
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure FDA0002965241810000031
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T；
Step 3, calculating a tracking error, and designing a controller according to the fast terminal sliding mode surface and the first derivative thereof, wherein the process is as follows:
3.1, defining the tracking error and its first and second differentials:
e＝X1-Xd (8)
Figure FDA0002965241810000032
Figure FDA0002965241810000033
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdConductive desired signals of x, y, z, phi, theta, psi, respectively;
3.2, designing a quick terminal sliding mode surface:
Figure FDA0002965241810000034
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure FDA0002965241810000035
order to
Figure FDA0002965241810000036
Formula (12) is simplified to formula (13)
Figure FDA0002965241810000037
But because of
Figure FDA0002965241810000038
In existence of
Figure FDA0002965241810000039
When α (e) is 0 and β (e) is not equal to 0, the negative power term of (a) causes a singularity problem; consider the method of handover control:
Figure FDA0002965241810000041
wherein q isi(e),αi(e),βi(e) The i-th element, i ═ 1,2,3,4,5,6, q (e), α (e), β (e), respectively;
combining formula (13) and formula (14) to obtain:
Figure FDA0002965241810000042
conjunctive formula (7), formula (10) and formula (15) yields:
Figure FDA0002965241810000043
3.3 design enhanced approach law
Figure FDA0002965241810000044
Wherein
Figure FDA0002965241810000045
N-1(X) is an inverse matrix of N (X), k is more than 0, delta is more than 0 and less than 1, gamma is more than 0, mu is more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure FDA0002965241810000046
Wherein B is-1(X) is the inverse of B (X).
2. The hyperbolic sine-enhanced constant velocity approach law and fast terminal sliding mode surface-based quad-rotor aircraft finite time control method according to claim 1, further comprising the steps of:
step 4, property specification, the process is as follows:
4.1, proving accessibility of sliding forms:
designing Lyapunov functions
Figure FDA0002965241810000047
The derivation is performed on both sides of the function to obtain:
Figure FDA0002965241810000048
because of the fact that
Figure FDA0002965241810000051
The constant is larger than 0, so the formula (18) is constantly smaller than 0, the accessibility of the sliding mode is met, and the system can reach the sliding mode surface;
4.2, enhanced effect description:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure FDA0002965241810000052
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure FDA0002965241810000053
the buffeting of the system is reduced.
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