CN108563125B - Self-adaptive control method of four-rotor aircraft based on exponential enhancement type power approach law and fast terminal sliding mode surface - Google Patents

Self-adaptive control method of four-rotor aircraft based on exponential enhancement type power approach law and fast terminal sliding mode surface Download PDF

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CN108563125B
CN108563125B CN201810519670.2A CN201810519670A CN108563125B CN 108563125 B CN108563125 B CN 108563125B CN 201810519670 A CN201810519670 A CN 201810519670A CN 108563125 B CN108563125 B CN 108563125B
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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 exponential enhancement type 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 exponential enhancement type power approximation law sliding mode control and rapid terminal sliding mode control, can increase the approximation speed when the sliding mode surface is far away, can reduce buffeting, improves the rapidity and robustness of the system, realizes rapid and stable control, can realize limited time control of tracking errors, and solves the problem that the tracking errors tend to 0 only when the time tends to be infinite in the traditional sliding mode surface. Meanwhile, the interference boundary is estimated through self-adaptation, and the stability of the system is improved.

Description

Self-adaptive control method of four-rotor aircraft based on exponential enhancement type 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 exponential enhancement type 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 the traveling device enables many dangerous high-altitude operations to be easy and safe, so that the traveling device can deter 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 exponential enhancement type 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 exponential enhancement type 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 BDA0001674495390000021
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 BDA0001674495390000031
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 BDA0001674495390000032
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 BDA0001674495390000033
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 BDA0001674495390000034
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674495390000035
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 BDA0001674495390000036
Wherein
Figure BDA0001674495390000037
Figure BDA0001674495390000041
Figure BDA0001674495390000042
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 BDA0001674495390000043
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 BDA0001674495390000044
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674495390000045
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure BDA00016744953900000411
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:
Figure BDA0001674495390000046
Figure BDA0001674495390000047
Figure BDA0001674495390000048
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 BDA0001674495390000049
i＝1，2,3,4,5,6，Di，c0i，c1i，c2i，ei，
Figure BDA00016744953900000410
are respectively the corresponding ith element；
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674495390000051
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674495390000052
let α (e) be e + λ2·sigα1(e),
Figure BDA0001674495390000053
Formula (12) is simplified to formula (13)
Figure BDA0001674495390000054
But due to the presence of alpha (e)
Figure BDA0001674495390000055
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 BDA0001674495390000056
wherein q isi(e),αi(e),βi(e) Q (e), alpha (e), beta (e) respectively,
Figure BDA0001674495390000057
combining formula (13) and formula (14) to obtain:
Figure BDA0001674495390000058
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674495390000059
3.3 design enhanced approach law
Figure BDA00016744953900000510
Wherein
Figure BDA0001674495390000061
N-1(X) is the inverse of N (X), k1＞0，0＜β1Less 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 BDA0001674495390000062
Wherein B is-1(X) is the inverse of B (X),
Figure BDA0001674495390000063
Figure BDA0001674495390000064
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure BDA0001674495390000065
Figure BDA0001674495390000066
Figure BDA0001674495390000067
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 BDA0001674495390000068
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA0001674495390000069
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 an exponential enhancement type power approach law and a fast terminal sliding mode surface is designed by combining power approach 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 exponential enhancement type approach law, the approach speed can be increased when the sliding mode face is far away, buffeting can be reduced, the rapidness and the 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 power 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. 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 power approximation rule adaptive control under a linear sliding mode surface, and a dotted line represents exponential 'mu' type enhanced power approximation rule adaptive control under a fast terminal sliding mode surface.
Fig. 2 is a schematic diagram of attitude tracking effect of a four-rotor aircraft, wherein a dotted line represents '1' type enhanced power approximation rule adaptive control of a linear sliding mode surface, and a dotted line represents index 'mu' type enhanced power approximation rule adaptive control of a fast terminal sliding mode surface.
Fig. 3 is a schematic diagram of the input of a position controller for enhanced 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 fast terminal sliding mode surface adaptive control of a four-rotor aircraft based on an exponential 'mu' type enhanced power approximation law.
Fig. 5 is an input schematic diagram of an attitude controller for '1' -type enhanced 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 of a four-rotor aircraft fast terminal sliding mode surface based on exponential 'mu' type enhanced power approximation law adaptive control.
Fig. 7 is a schematic diagram of local amplification of input of an attitude controller for '1' -type enhanced power-order approximation 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 exponential 'mu' type enhanced power approximation law adaptive control.
Fig. 9 is an estimation of the bounds of the position disturbance of the four-rotor aircraft fast terminal sliding mode surface based on exponential "mu" type enhanced power approximation 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 exponential 'mu' type enhanced power approximation 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 self-adaptive control method of a quadrotor aircraft based on an exponential enhanced 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 BDA0001674495390000091
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 BDA0001674495390000092
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 BDA0001674495390000093
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively represent the coordinate system of the machine bodyRotational inertia component of each axis, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure BDA0001674495390000094
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 BDA0001674495390000095
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674495390000101
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 BDA0001674495390000102
Wherein
Figure BDA0001674495390000103
Figure BDA0001674495390000104
Figure BDA0001674495390000105
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 BDA0001674495390000106
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 BDA0001674495390000107
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674495390000108
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure BDA0001674495390000109
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 BDA0001674495390000111
Figure BDA0001674495390000112
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 BDA0001674495390000113
i＝1，2,3,4,5,6，Di，c0i，c1i，c2i，ei，
Figure BDA0001674495390000114
respectively corresponding ith element;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674495390000115
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674495390000116
order to
Figure BDA0001674495390000117
Formula (12) is simplified to formula (13)
Figure BDA0001674495390000118
But due to the presence of alpha (e)
Figure BDA0001674495390000119
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 BDA0001674495390000121
wherein q isi(e),αi(e),βi(e) Q (e), alpha (e), beta (e) respectively,
Figure BDA0001674495390000122
combining formula (13) and formula (14) to obtain:
Figure BDA0001674495390000123
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674495390000124
3.3 design enhanced approach law
Figure BDA0001674495390000125
Wherein
Figure BDA0001674495390000126
N-1(X) is the inverse of N (X), k1＞0，0＜β1Less 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 BDA0001674495390000127
Wherein B is-1(X) is the inverse of B (X),
Figure BDA0001674495390000128
Figure BDA0001674495390000129
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure BDA00016744953900001210
Figure BDA00016744953900001211
Figure BDA00016744953900001212
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 BDA0001674495390000131
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA0001674495390000132
the buffeting of the system is reduced.
In order to verify the effectiveness of the method, the invention provides a comparison between a sliding mode control method of a rapid terminal sliding mode surface based on an exponential mu-type enhanced power approximation law and a sliding mode control method of a linear sliding mode surface based on a 1-type enhanced power approximation law:
wherein, in the enhanced power approximation law of '1' type
Figure BDA0001674495390000133
Figure BDA0001674495390000134
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、α2Linear sliding form (1.1) and epsilon (0.3)Dough making: lambda [ alpha ]10.5, "μ" type enhanced proximity law parameter: k is a radical of1＝10、δ＝0.1、p＝1、γ＝1、μ＝10、β10.7, enhanced approximation rule parameter of "1": k is a radical of1＝10、δ＝0.1、p＝1、γ＝1、β10.7, adaptive initial value setting
Figure BDA0001674495390000135
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 BDA0001674495390000136
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 exponential enhanced power approximation law and the fast terminal sliding mode surface can reach the expected position more quickly; with reference to fig. 3-8, the adaptive control of the quadrotor aircraft based on the exponentially enhanced power-order 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 adaptive control of the four-rotor aircraft based on the exponential enhancement type 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 exponential enhancement type 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 FDA0003073260290000011
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 FDA0003073260290000012
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 FDA0003073260290000013
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 FDA0003073260290000014
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 FDA0003073260290000021
Then the formula (3) is represented as the formula (4) in the rotation process
Figure FDA0003073260290000022
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 FDA0003073260290000023
Wherein
Figure FDA0003073260290000024
Figure FDA0003073260290000025
Figure FDA0003073260290000026
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 FDA0003073260290000027
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;
considering further the case where interference exists, equation (5) can be written in a matrix form as follows:
Figure FDA0003073260290000028
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure FDA0003073260290000031
B(X)＝diag(1,1,1,b1,b2,b3),U＝[Ux,Uy,Uz,τx,τy,τz]T，
Figure FDA0003073260290000032
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 FDA0003073260290000033
Figure FDA0003073260290000034
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 FDA0003073260290000035
Di，c0i，c1i，c2i，ei，
Figure FDA0003073260290000036
respectively corresponding ith element;
3.2, designing a quick terminal sliding mode surface:
Figure FDA0003073260290000037
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure FDA0003073260290000038
order to
Figure FDA0003073260290000039
Formula (12) is simplified to formula (13)
Figure FDA00030732602900000310
But because of
Figure FDA00030732602900000311
In existence of
Figure FDA00030732602900000312
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 FDA0003073260290000041
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 FDA0003073260290000042
conjunctive formula (7), formula (10) and formula (15) yields:
Figure FDA0003073260290000043
3.3 design enhanced approach law
Figure FDA0003073260290000044
Wherein
Figure FDA0003073260290000045
N-1(X) is the inverse of N (X), k1＞0，0＜β1Less 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 FDA0003073260290000046
Wherein B is-1(X) is the inverse of B (X),
Figure FDA0003073260290000047
Figure FDA0003073260290000048
respectively corresponding ith element;
the adaptive law is designed as follows:
Figure FDA0003073260290000049
Figure FDA00030732602900000410
Figure FDA00030732602900000411
step 4, enhanced property specification, the process is as follows:
when the system moves away from the sliding mode, | s | is large, N(s) approaches δ,
Figure FDA0003073260290000051
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure FDA0003073260290000052
the buffeting of the system is reduced.
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