CN108829128B - Four-rotor aircraft finite time control method based on logarithm enhancement type exponential approaching law and fast terminal sliding mode surface - Google Patents
Four-rotor aircraft finite time control method based on logarithm enhancement type exponential approaching law and fast terminal sliding mode surface Download PDFInfo
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Abstract
A finite time control method of a four-rotor aircraft based on a logarithm enhancement type exponential 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, the method combines sliding mode control based on logarithm enhancement type exponential approach law 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
Technical Field
The invention relates to a finite time control method of a four-rotor aircraft based on a logarithm enhanced exponential 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.
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 logarithm-enhanced index-based 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 a logarithm enhancement type exponential approach law and a fast terminal sliding mode surface comprises the following steps:
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:
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:
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,
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
Then the formula (3) is represented as the formula (4) in the rotation process
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Wherein
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:
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:
wherein X1＝[x,y,z,φ,θ,ψ]T,
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)
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:
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
order to
Formula (12) is simplified to formula (13)
But due to the presence of alpha (e)
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:
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:
conjunctive formula (7), formula (10) and formula (15) yields:
3.3 design enhanced approach law
Wherein
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
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
The derivation is performed on both sides of the function to obtain:
because of the fact that
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,
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
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 finite time control method based on logarithmic enhancement type exponential approximation law and a fast terminal sliding mode surface is designed by combining exponential 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 logarithm 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 conventional exponential approach law control, and a dotted line represents finite time control of the four-rotor aircraft based on a logarithmic enhancement type exponential approach law and a 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 logarithmic enhancement type exponential approximation law and 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 conventional exponential approach law control, and a dotted line represents finite time control of the quad-rotor aircraft based on a logarithmically enhanced exponential approach law and a fast terminal sliding mode surface.
Fig. 4 is a schematic diagram of attitude angle tracking error of a quad-rotor aircraft, in which a dotted line represents conventional exponential approach law control, and a dotted line represents finite time control of the quad-rotor aircraft based on logarithmically enhanced 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 a logarithm enhanced exponential approach 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 a logarithmic enhancement type exponential approach 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 a logarithmically enhanced exponential approach law and a fast terminal sliding mode surface includes the following steps:
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:
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:
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,
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
Then the formula (3) is represented as the formula (4) in the rotation process
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Wherein
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:
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:
wherein X1＝[x,y,z,φ,θ,ψ]T,
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)
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:
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
order to
Formula (12) is simplified to formula (13)
But due to the presence of alpha (e)
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:
wherein q isi(e),αi(e),βi(e) I-th elements q (e), α (e), β (e), i-1, 2,3,4,5,6；
Combining formula (13) and formula (14) to obtain:
conjunctive formula (7), formula (10) and formula (15) yields:
3.3 design enhanced approach law
Wherein
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
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
The derivation is performed on both sides of the function to obtain:
because of the fact that
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,
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
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 based on a logarithm enhanced exponential approximation law and a traditional exponential approximation 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 logarithm enhanced exponential approximation law and the fast terminal sliding mode surface can reach the expected position more quickly; with reference to fig. 5-8, the limited-time control of the quadrotor based on the logarithm enhanced exponential approximation 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 logarithm enhanced exponential approximation 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 enters 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 finite time control method of a four-rotor aircraft based on a logarithm enhanced exponential 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;
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:
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, mg represents the gravity borne by the four rotors, and the sum of the four rotors is generatedForce Ur；
2.2, the rotation process comprises the following steps:
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,
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
2.3, connecting the vertical type (1), (2) and (4), and obtaining the dynamic model of the aircraft as shown in the formula (5)
Wherein
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:
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:
wherein X1＝[x,y,z,φ,θ,ψ]T,
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)
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:
wherein, sigα(x)＝|x|α·sign(x)，α1＞α2＞1，λ1＞0，λ2＞0；
Derivation of equation (11) yields:
order to
Formula (12) is simplified to formula (13)
But because of
In existence of
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:
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:
conjunctive formula (7), formula (10) and formula (15) yields:
3.3 design enhanced approach law
Wherein
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
Wherein B is-1(X) is the inverse of B (X).
2. The limited-time control method for the quadrotor aircraft based on the logarithmically enhanced exponential approach law and the fast terminal sliding mode surface according to claim 1, characterized by further comprising the following steps:
step 4, property specification, the process is as follows:
4.1, proving accessibility of sliding forms:
designing Lyapunov functions
The derivation is performed on both sides of the function to obtain:
because of the fact that
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,
the approach speed of the system is accelerated; when the system approaches the sliding mode, | s | approaches 0, N(s) approaches μ,
the buffeting of the system is reduced.
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