CN108828937B - Finite time control method of four-rotor aircraft based on exponential enhancement type exponential approaching law and fast terminal sliding mode surface - Google Patents

Finite time control method of four-rotor aircraft based on exponential enhancement type exponential approaching law and fast terminal sliding mode surface Download PDF

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CN108828937B
CN108828937B CN201810519744.2A CN201810519744A CN108828937B CN 108828937 B CN108828937 B CN 108828937B CN 201810519744 A CN201810519744 A CN 201810519744A CN 108828937 B CN108828937 B CN 108828937B
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sliding mode
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CN108828937A (en
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陈强
陈凯杰
陶玫玲
胡轶
吴春
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Zhejiang University of Technology ZJUT
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

A four-rotor aircraft finite time control method based on an exponential enhancement type exponential approaching 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; and 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, the method combines index enhancement type index approach law-based sliding mode control and rapid terminal sliding mode control, can increase approach speed when the system is far away from a sliding mode surface, can reduce buffeting, improves 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 time tends to infinity in the traditional sliding mode surface.

Description

Finite time control method of four-rotor aircraft based on exponential enhancement type exponential approaching law and fast terminal sliding mode surface
Technical Field
The invention relates to a four-rotor aircraft finite time control method based on an exponential enhancement type exponential approaching 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.
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 exponential enhancement type exponential approaching law, avoids the singularity problem by switching the control idea, accelerates the approaching speed of a system to the sliding mode surface, reduces buffeting and realizes the limited time control.
The technical scheme proposed for solving the technical problems is as follows:
a four-rotor aircraft finite time control method based on an exponential enhancement type exponential approaching 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 BDA0001674500690000021
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 BDA0001674500690000022
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 BDA0001674500690000023
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 BDA0001674500690000024
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 BDA0001674500690000031
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674500690000032
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 BDA0001674500690000033
Wherein
Figure BDA0001674500690000034
Figure BDA0001674500690000035
Figure BDA0001674500690000036
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 BDA0001674500690000037
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 BDA0001674500690000038
wherein
Figure BDA0001674500690000039
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 BDA0001674500690000041
Figure BDA0001674500690000042
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 BDA0001674500690000043
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674500690000044
order to
Figure BDA0001674500690000049
Formula (12) is simplified to formula (13)
Figure BDA0001674500690000046
But due to the presence of alpha (e)
Figure BDA0001674500690000047
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 BDA0001674500690000048
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 BDA0001674500690000051
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674500690000052
3.3 design enhanced approach law
Figure BDA0001674500690000053
Wherein
Figure BDA0001674500690000054
N-1(X) is the inverse of N (X), k1＞0，k2More than 0, more than 0 and less than 1, more than 0, more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure BDA0001674500690000055
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 BDA0001674500690000056
The derivation is performed on both sides of the function to obtain:
Figure BDA0001674500690000057
because of the fact that
Figure BDA0001674500690000058
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 BDA0001674500690000059
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA00016745006900000510
the buffeting of the system is reduced.
The technical conception of the invention is as follows: aiming at a four-rotor aircraft system, by combining index approximation law sliding mode control and rapid terminal sliding mode control, a four-rotor aircraft finite time control method based on an index enhancement type index approximation law and a rapid terminal sliding mode surface is designed. 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.
The invention has the beneficial effects that: compared with the traditional index 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 four-rotor aircraft, in which a dotted line represents the control of the conventional exponential approach law, and a dotted line represents the finite-time control of the four-rotor aircraft based on the exponential enhancement type exponential approach law and the fast terminal sliding mode surface.
Fig. 2 is a schematic diagram of position tracking error of a quadrotor, wherein a dotted line represents conventional exponential approximation law control, and a dotted line represents finite time control of the quadrotor based on exponential enhancement type exponential approximation law and a fast terminal sliding mode surface.
Fig. 3 is a schematic diagram of the attitude angle tracking effect of a quad-rotor aircraft, in which a dotted line represents the conventional exponential approach law control, and a dotted line represents the finite time control of the quad-rotor aircraft based on the exponential enhancement type exponential approach law and the fast terminal sliding mode surface.
Fig. 4 is a schematic diagram of attitude angle tracking error of a quadrotor, wherein a dotted line represents traditional exponential approach law control, and a dotted line represents finite time control of the quadrotor based on exponential enhancement type exponential approach law and a fast terminal sliding mode surface.
FIG. 5 is a schematic diagram of position controller input under finite time control of a quadrotor aircraft based on an exponential enhancement type exponential approaching law and a fast terminal sliding mode surface.
Fig. 6 is a schematic diagram of the position controller inputs under conventional exponential approach law control for a four-rotor aircraft.
FIG. 7 is an input schematic diagram of an attitude angle controller under finite-time control of a four-rotor aircraft based on an exponential enhancement type exponential approaching law and a fast terminal sliding mode surface.
FIG. 8 is a schematic diagram of attitude angle controller inputs under conventional exponential approximation law control for a quad-rotor aircraft.
FIG. 9 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. 1-9, a finite-time control method of a four-rotor aircraft based on an exponential enhancement type exponential approaching 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 BDA0001674500690000071
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 BDA0001674500690000072
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 acceleration of gravity, and mg represents the position of the four rotorsResultant force U generated by four rotors under the action of gravityr；
2.2, the rotation process comprises the following steps:
Figure BDA0001674500690000081
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 BDA0001674500690000082
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 BDA0001674500690000083
Then the formula (3) is represented as the formula (4) in the rotation process
Figure BDA0001674500690000084
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 BDA0001674500690000085
Wherein
Figure BDA0001674500690000086
Figure BDA0001674500690000087
Figure BDA0001674500690000088
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 BDA0001674500690000091
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 BDA0001674500690000092
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure BDA0001674500690000093
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 BDA0001674500690000094
Figure BDA0001674500690000095
wherein, Xd＝[xd,yd,zd,φd,θd,ψd]T，xd,yd,zd,φd,θdψ x is a conductive desired signal of x, y, z, φ, θ, ψ, respectively;
3.2, designing a quick terminal sliding mode surface:
Figure BDA0001674500690000096
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure BDA0001674500690000097
order to
Figure BDA0001674500690000101
Formula (12) is simplified to formula (13)
Figure BDA0001674500690000102
But due to the presence of alpha (e)
Figure BDA0001674500690000103
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 BDA0001674500690000104
wherein q isi(e),αi(e),βi(e) Are each q(e) α (e), i (i) is 1,2,3,4,5, 6;
combining formula (13) and formula (14) to obtain:
Figure BDA0001674500690000105
conjunctive formula (7), formula (10) and formula (15) yields:
Figure BDA0001674500690000106
3.3 design enhanced approach law
Figure BDA0001674500690000107
Wherein
Figure BDA0001674500690000108
N-1(X) is the inverse of N (X), k1＞0，k2More than 0, more than 0 and less than 1, more than 0, more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure BDA0001674500690000109
Wherein B is-1(X) is the inverse of B (X);
step 4, property specification, the process is as follows:
4.1, proving accessibility of sliding forms:
designing Lyapunov functions
Figure BDA0001674500690000111
The derivation is performed on both sides of the function to obtain:
Figure BDA0001674500690000112
because of the fact that
Figure BDA0001674500690000113
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 BDA0001674500690000114
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure BDA0001674500690000115
the buffeting of the system is reduced.
In order to verify the effectiveness of the method, the invention provides a comparison between an index enhancement type index approach law sliding mode control method and a traditional index 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＝0.7、α1＝2、α21.1, 0.1, and the approach law parameter: k is a radical of1＝0.6、k20.8, δ -0.5, p-1, γ -1, μ -2, quad-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。
1-4, the finite-time control of the quadrotor aircraft based on the exponential enhancement type exponential approaching law and the fast terminal sliding mode surface can reach the expected position more quickly; with reference to fig. 5-8, the four-rotor aircraft finite time control based on the exponentially enhanced exponential approximation law and the fast terminal sliding mode surface has less buffeting.
In conclusion, the finite-time control of the four-rotor aircraft based on the exponential enhancement type exponential approaching law and the fast terminal sliding mode surface can reduce the buffeting and the tracking time at the same time, improve the tracking performance and enable the system to 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 an exponential enhancement type exponential approaching 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 FDA0002969897030000011
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 FDA0002969897030000012
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 FDA0002969897030000013
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 FDA0002969897030000014
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 FDA0002969897030000021
Then the formula (3) is represented as the formula (4) in the rotation process
Figure FDA0002969897030000022
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 FDA0002969897030000023
Wherein
Figure FDA0002969897030000024
Figure FDA0002969897030000025
Figure FDA0002969897030000026
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 FDA0002969897030000027
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 FDA0002969897030000028
wherein X1＝[x,y,z,φ,θ,ψ]T,
Figure FDA0002969897030000031
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 FDA0002969897030000032
Figure FDA0002969897030000033
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 FDA0002969897030000034
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
Figure FDA0002969897030000035
order to
Figure FDA0002969897030000036
Formula (12) is simplified to formula (13)
Figure FDA0002969897030000037
But because of
Figure FDA0002969897030000038
In existence of
Figure FDA0002969897030000039
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 FDA0002969897030000041
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 FDA0002969897030000042
conjunctive formula (7), formula (10) and formula (15) yields:
Figure FDA0002969897030000043
3.3 design enhanced approach law
Figure FDA0002969897030000044
Wherein
Figure FDA0002969897030000045
N-1(X) is the inverse of N (X), k1＞0，k2More than 0, more than 0 and less than 1, more than 0, more than 1, and p is a positive integer;
3.4, combined vertical (16) and formula (17), to obtain a controller
Figure FDA0002969897030000046
Wherein B is-1(X) is the inverse of B (X).
2. The finite-time control method for a quadrotor aircraft based on exponentially enhanced exponential approach law and fast terminal sliding-mode surfaces according to claim 1, characterized in that 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 FDA0002969897030000047
The derivation is performed on both sides of the function to obtain:
Figure FDA0002969897030000048
because of the fact that
Figure FDA0002969897030000049
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 FDA0002969897030000051
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
Figure FDA0002969897030000052
the buffeting of the system is reduced.
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