CN108536018B - Four-rotor aircraft self-adaptive control method based on inverse proportion function enhanced double-power approach law and fast terminal sliding mode surface - Google Patents

Four-rotor aircraft self-adaptive control method based on inverse proportion function enhanced double-power approach law and fast terminal sliding mode surface Download PDF

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CN108536018B
CN108536018B CN201810519781.3A CN201810519781A CN108536018B CN 108536018 B CN108536018 B CN 108536018B CN 201810519781 A CN201810519781 A CN 201810519781A CN 108536018 B CN108536018 B CN 108536018B
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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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Abstract

A self-adaptive control method of a four-rotor aircraft based on an inverse proportion function enhanced double power 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. The method combines inverse proportion function enhanced double power approach law sliding mode control and rapid terminal sliding mode control, can increase approach speed when the sliding mode surface is far away, reduces buffeting, improves the rapidness and robustness of the system, realizes rapid and stable control, realizes limited time control of tracking error, and solves the problem that the tracking error tends to 0 only when the time tends to be infinite in the traditional sliding mode surface. The interference boundary is estimated through self-adaptation, and the stability of the system is improved.

Description

Four-rotor aircraft self-adaptive control method based on inverse proportion function enhanced double-power approach law and fast terminal sliding mode surface
Technical Field
The invention relates to a self-adaptive control method of a four-rotor aircraft based on an inverse proportion function enhanced double-power approach law and a fast 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. Because the four-rotor aircraft can encounter external environment interference in flight, interference and compensation are carried out on the interference boundary through self-adaptation, and the stability of the system is improved.
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 method adopts the rapid terminal sliding mode control and the inverse proportion function-based enhanced double-power 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 the limited time control. Meanwhile, interference and compensation are carried out on the interference boundary through self-adaption, and the stability of the system is improved.
The technical scheme proposed for solving the technical problems is as follows:
a self-adaptive control method of a four-rotor aircraft based on an inverse proportion function enhanced double power 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 BDA0001674512610000021
where ψ, θ, φ are the yaw and pitch angles, respectively, of the aircraftElevation angle, roll angle, 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 BDA0001674512610000022
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 BDA0001674512610000023
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 BDA0001674512610000024
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 BDA0001674512610000031
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674512610000032
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 BDA0001674512610000033
Wherein
Figure BDA0001674512610000034
Figure BDA0001674512610000035
Figure BDA0001674512610000036
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 BDA0001674512610000037
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 BDA0001674512610000038
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674512610000041
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure BDA00016745126100000411
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 BDA0001674512610000042
Figure BDA0001674512610000043
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdThe conductive desired signals are x, y, z, phi, theta, psi, respectively,
Figure BDA0001674512610000044
i＝1，2,3,4,5,6，Di，c0i，c1i，c2i，ei，
Figure BDA0001674512610000045
respectively corresponding ith element;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674512610000046
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674512610000047
order to
Figure BDA0001674512610000048
Formula (12) is simplified to formula (13)
Figure BDA0001674512610000049
But due to the presence of alpha (e)
Figure BDA00016745126100000410
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 BDA0001674512610000051
wherein q isi(e),αi(e),βi(e) Q (e), alpha (e), beta (e) respectively,
Figure BDA0001674512610000052
combining formula (13) and formula (14) to obtain:
Figure BDA0001674512610000053
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674512610000054
3.3 design enhanced approach law
Figure BDA0001674512610000055
Wherein
Figure BDA0001674512610000056
N-1(X) is the inverse of N (X), k1＞0，k2＞0，β1＞1，0＜β2Less than 1, 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 BDA0001674512610000057
Wherein B is-1(X) is the inverse of B (X),
Figure BDA0001674512610000058
Figure BDA0001674512610000059
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure BDA00016745126100000510
Figure BDA00016745126100000511
Figure BDA00016745126100000512
step 4, property specification, the process is as follows:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure BDA0001674512610000061
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA0001674512610000062
the buffeting of the system is reduced.
The technical conception of the invention is as follows: aiming at a four-rotor aircraft system, a four-rotor aircraft self-adaptive control method based on inverse proportion function enhanced double power approximation law and a fast terminal sliding mode surface is designed by combining double power approximation law sliding mode control and fast 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 the inverse proportion function enhanced approach law, the approach speed can be increased when the sliding mode face is far away, buffeting can be reduced, the rapidness and robustness of the system are improved, and rapid and stable control is achieved. Meanwhile, interference and compensation are carried out on the interference boundary through self-adaption, and the stability of the system is improved.
The invention has the beneficial effects that: compared with the traditional double-power approach law sliding mode control, the method can increase the approach speed when the system is far away from the sliding mode, 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. Meanwhile, interference and compensation are carried out on the interference boundary through self-adaption, and the stability of the system is improved.
Drawings
Fig. 1 is a schematic diagram of a position tracking effect of a four-rotor aircraft, wherein a dotted line represents "1" type enhanced double-power-law adaptive control under a linear sliding mode surface, and a dotted line represents inverse proportion function "mu" type enhanced double-power-law adaptive control under a fast terminal sliding mode surface.
Fig. 2 is a schematic diagram of an attitude tracking effect of a four-rotor aircraft, wherein a dotted line represents "1" type enhanced double-power-law adaptive control of a linear sliding mode surface, and a dotted line represents inverse proportion function "mu" type enhanced double-power-law adaptive control of a fast terminal sliding mode surface.
Fig. 3 is a schematic input diagram of a position controller for enhanced double power law adaptive control of a '1' type under a linear sliding mode surface of a four-rotor aircraft.
Fig. 4 is an input schematic diagram of a position controller for a four-rotor aircraft fast terminal sliding mode surface based on inverse proportion function 'mu' type enhanced double power approach law adaptive control.
Fig. 5 is an input schematic diagram of an attitude controller for "1" -type enhanced double-power-law adaptive control under a linear sliding mode surface of a four-rotor aircraft.
Fig. 6 is an input schematic diagram of an attitude controller for four-rotor aircraft fast terminal sliding mode surface based on inverse proportion function 'mu' type enhanced double power approach law adaptive control.
Fig. 7 is a schematic diagram of local amplification of input of an attitude controller for "1" -type enhanced double-power approach law adaptive control under a linear sliding mode surface of a four-rotor aircraft.
Fig. 8 is a schematic diagram of local amplification of input of an attitude controller of a four-rotor aircraft fast terminal sliding mode surface based on inverse proportion function 'mu' type enhanced double power approach law adaptive control.
Fig. 9 is an estimation of the bounds of the position disturbance of the fast terminal sliding mode surface of the quadrotor based on the inverse proportional function 'mu' type enhanced double power approach law adaptive control.
Fig. 10 is an estimation of the boundary of attitude disturbance of a four-rotor aircraft fast terminal sliding mode surface based on inverse proportional function 'mu' type enhanced double power approach law adaptive control.
FIG. 11 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. 1-11, a four-rotor aircraft adaptive control method based on an inverse proportion function enhanced double power approach law and a 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 BDA0001674512610000071
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 BDA0001674512610000081
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 BDA0001674512610000082
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 BDA0001674512610000083
respectively represent machine coordinate systemThe attitude angular acceleration component of each axis;
considering that the aircraft is in a low-speed flight or hovering state, consider
Figure BDA0001674512610000084
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674512610000085
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 BDA0001674512610000091
Wherein
Figure BDA0001674512610000092
Figure BDA0001674512610000093
Figure BDA0001674512610000094
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 BDA0001674512610000095
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 BDA0001674512610000096
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674512610000097
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure BDA0001674512610000099
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 BDA0001674512610000098
Figure BDA0001674512610000101
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdThe conductive desired signals are x, y, z, phi, theta, psi, respectively,
Figure BDA00016745126100001012
i＝1，2,3,4,5,6，Di，c0i，c1i，c2i，ei，
Figure BDA0001674512610000102
respectively corresponding ith element;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674512610000103
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674512610000104
order to
Figure BDA0001674512610000105
Formula (12) is simplified to formula (13)
Figure BDA0001674512610000106
But due to the presence of alpha (e)
Figure BDA0001674512610000107
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 BDA0001674512610000108
wherein qi (e), αi(e),βi(e) Q (e), alpha (e), beta (e) respectively,
Figure BDA0001674512610000109
combining formula (13) and formula (14) to obtain:
Figure BDA00016745126100001010
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA00016745126100001011
3.3 design enhanced approach law
Figure BDA0001674512610000111
Wherein
Figure BDA0001674512610000112
N-1(X) is the inverse of N (X), k1＞0，k2＞0，β1＞1，0＜β2Less than 1, 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 BDA0001674512610000113
Wherein B is-1(X) is the inverse of B (X),
Figure BDA0001674512610000114
Figure BDA0001674512610000115
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure BDA0001674512610000116
Figure BDA0001674512610000117
Figure BDA0001674512610000118
step 4, property specification, the process is as follows:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure BDA0001674512610000119
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA00016745126100001110
the buffeting of the system is reduced.
In order to verify the effectiveness of the method, the invention provides a contrast of a sliding mode control method of a rapid terminal sliding mode surface based on a inverse proportion function mu-type enhanced double power approximation law and a sliding mode control method of a linear sliding mode surface 1-type enhanced double power approximation law:
wherein the 1-type enhanced double power approximation law is
Figure BDA00016745126100001111
Figure BDA0001674512610000121
For more efficient comparison, all parameters of the system are consistent, i.e. xd＝yd＝zd＝2、ψd0.5, fast terminal sliding mode surface parameters: lambda [ alpha ]1＝0.5、λ2＝2、α1＝2、α21.1, epsilon 0.3, linear slip-form face: lambda [ alpha ]10.5, "μ" type enhanced proximity law parameter: k is a radical of1＝1、k2＝10、δ＝0.1、p＝1、γ＝1、μ＝10，β1＝1.3、β20.7, enhanced approximation rule parameter of "1": k is a radical of1＝1、k2＝10、δ＝0.1、p＝1、γ＝1、β1＝1.3、β20.7, adaptive initial value setting
Figure BDA0001674512610000122
Figure BDA0001674512610000123
p0i＝p1i＝p2i＝0.1，ε0i＝ε1i＝ε2i0.001, 1,2,3,4,5,6, interference parameter: dx＝dy＝dz＝0.2sin(0.2t)、
Figure BDA0001674512610000124
Parameters of the four-rotor aircraft: 1.1 and Ixx＝1.22、Iyy＝1.22、Izz2.2, g 9.81, sampling parameters: t is ts＝0.007，N＝5000。
As can be seen from fig. 1 and 2, the adaptive control of the quadrotor aircraft based on the inverse proportion function enhanced double power approach law and the fast terminal sliding mode surface can reach the expected position more quickly; with reference to fig. 3-8, the self-adaptive control of the quadrotor aircraft based on the inverse proportional function enhanced double power approach law and the fast terminal sliding mode surface has smaller buffeting. Fig. 9 and 10 can see the effectiveness of the estimation of the adaptive epipolar.
In conclusion, the four-rotor aircraft self-adaptive control based on the inverse proportion function enhanced double-power approach 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 (1)

1. A self-adaptive control method of a four-rotor aircraft based on an inverse proportion function enhanced double power 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 FDA0003073261100000011
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 FDA0003073261100000012
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 FDA0003073261100000013
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 FDA0003073261100000014
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 FDA0003073261100000021
Then the formula (3) is represented as the formula (4) in the rotation process
Figure FDA0003073261100000022
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 FDA0003073261100000023
Wherein
Figure FDA0003073261100000024
Figure FDA0003073261100000025
Figure FDA0003073261100000026
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 FDA0003073261100000027
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidDesired signal value of psi, arcsin functionThe number is an arcsine function, and the arctan function is an arctangent function;
considering further the case where interference exists, equation (5) can be written in a matrix form as follows:
Figure FDA0003073261100000028
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure FDA0003073261100000031
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure FDA0003073261100000032
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 FDA0003073261100000033
Figure FDA0003073261100000034
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θd,ψdThe conductive desired signals are x, y, z, phi, theta, psi, respectively,
Figure FDA0003073261100000035
i＝1，2,3,4,5,6，Di，c0i，c1i，c2i，ei，
Figure FDA0003073261100000036
respectively corresponding ith element;
3.2, designing a quick terminal sliding mode surface:
Figure FDA0003073261100000037
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure FDA0003073261100000038
order to
Figure FDA0003073261100000039
Formula (12) is simplified to formula (13)
Figure FDA00030732611000000310
But because of
Figure FDA00030732611000000311
In existence of
Figure FDA00030732611000000312
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 FDA0003073261100000041
wherein q isi(e),αi(e),βi(e) Q (e), α (e), β (e), i ═ 1,2,3,4,5, 6;
combining formula (13) and formula (14) to obtain:
Figure FDA0003073261100000042
conjunctive formula (7), formula (10) and formula (15) yields:
Figure FDA0003073261100000043
3.3 design enhanced approach law
Figure FDA0003073261100000044
Wherein
Figure FDA0003073261100000045
N-1(X) is the inverse of N (X), k1＞0，k2＞0，β1＞1，0＜β2Less than 1, 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, coupled (16) and (17), resulting in a controller:
Figure FDA0003073261100000046
wherein B is-1(X) is the inverse of B (X),
Figure FDA0003073261100000047
Figure FDA0003073261100000048
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure FDA0003073261100000049
Figure FDA00030732611000000410
Figure FDA00030732611000000411
step 4, property specification, the process is as follows:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure FDA0003073261100000051
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure FDA0003073261100000052
the buffeting of the system is reduced.
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