CN108828938A - Finite time control method of four-rotor aircraft based on inverse proportion function enhanced index approach law and fast terminal sliding mode surface - Google Patents
Finite time control method of four-rotor aircraft based on inverse proportion function enhanced index approach 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 an inverse proportion function enhanced 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 inverse proportion function enhanced index 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 be infinite in the traditional sliding mode surface.
Description
Technical field
The quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface that the present invention relates to a kind of
Aircraft finite-time control method.
Background technique
Quadrotor is since structure is simple, the feature of mobility strong, flying method uniqueness causes domestic and foreign scholars
And the extensive concern of scientific research institution, and rapidly become one of the hot spot studied in the world at present.Compared to Fixed Wing AirVehicle, rotation
Rotor aircraft can be vertically moved up or down, low to environmental requirement, not need runway, be reduced costs, and have huge commercial value.Fly
The development of row device makes the high altitude operation of many danger become light safety, causes to deter to other countries in military aspect, in the people
Working efficiency is greatly increased with aspect.Quadrotor has stronger flexibility, can realize movement and hovering at any time
Fast transition, and can be with the aerial mission of the competent more challenge of lesser damage risk.In field of scientific study, due to four
Rotor craft has the dynamic characteristic of non-linear, drive lacking, close coupling, and researcher is often as theoretical research, method
The experimental vehicle of verifying.Small-sized quadrotor is relied on, vehicle flight control system is built, carries out aircraft high-performance fortune
Dynamic control research, is the hot research field of current academia.
The characteristics of Reaching Law sliding formwork control, can be achieved on discontinuous control, and sliding mode is programmable, and be
System parameter and disturbance are not associated with.Reaching Law sliding formwork can not only rationally design the speed for reaching sliding-mode surface, reduce the approach stage
Time, improve the robustness of system, and can effectively weaken the buffeting problem in sliding formwork control.Currently, in quadrotor control
It is fewer using Reaching Law sliding formwork control in field processed.Enhanced Reaching Law is further speeded up on the basis of traditional Reaching Law
The velocity of approach that system reaches sliding-mode surface makes to buffet simultaneously smaller.
Summary of the invention
In order to overcome traditional sliding-mode surface to cannot achieve finite-time control and further speed up the velocity of approach of Reaching Law
The problem of with buffeting is reduced, is approached present invention employs fast terminal sliding formwork control and based on inverse proportion function enhanced index
Rule, avoids singularity problem by the thought of switching control, accelerates the velocity of approach that system reaches sliding-mode surface, reduces and tremble
Vibration, realizes finite-time control.
In order to solve the above-mentioned technical problem the technical solution proposed is as follows：
A kind of quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Duration control method includes the following steps：
Step 1, it determines from the body coordinate system based on quadrotor to the transfer of the inertial coodinate system based on the earth
Matrix；
Wherein ψ, θ, φ are yaw angle, pitch angle, the roll angle of aircraft respectively, indicate aircraft around successively inertial coordinate
It is the angle of each axis rotation, TψIndicate the transfer matrix of ψ, TθIndicate the transfer matrix of θ, TφIndicate the transfer matrix of φ；
Step 2, quadrotor kinetic model is analyzed according to newton Euler's formula, process is as follows：
2.1, have during translation：
Wherein x, y, z respectively indicates position of the quadrotor under inertial coodinate system, and m indicates the quality of aircraft, and g indicates weight
Power acceleration, mg indicate gravity suffered by quadrotor, the resultant force U that four rotors generater；
2.2, have in rotation process：
Wherein τx、τy、τzRespectively represent each axis moment components on body coordinate system, Ixx、Iyy、IzzRespectively represent body seat
Each axis rotary inertia component fastened is marked, × indicate multiplication cross, wp、wq、wrRespectively represent each axis attitude angle speed on body coordinate system
Component is spent,Respectively represent each axis posture component of angular acceleration on body coordinate system；
In view of aircraft is under low-speed operations or floating state, attitude angle variation is smaller, it is believed thatThen rotation process Chinese style (3) is expressed as formula (4)
2.3, joint type (1), (2), (4), shown in the kinetic model such as formula (5) for obtaining aircraft
Wherein Ux、Uy、UzThe input quantity of respectively three positioners；
According to formula (5), decoupling computation is carried out to position and attitude relationship, it is as a result as follows：
Wherein φdFor the expected signal value of φ, θdFor the expected signal value of θ, ψdFor the expected signal value of ψ, arcsin function
It is arcsin function, arctan function is arctan function；
Formula (5) can also be write as 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, tracking error is calculated, controller is designed according to fast terminal sliding-mode surface and its first derivative, process is such as
Under：
3.1, define tracking error and its first differential and second-order differential：
E=X1-Xd (8)
Wherein, Xd=[xd,yd,zd,φd,θd,ψd]T, xd,yd,zd,φd,θd,ψdRespectively x, y, z, φ, θ, ψ's leads
Desired signal；
3.2, design fast terminal sliding-mode surface：
Wherein, sigα(x)=| x |αSign (x), α1> α2> 1, λ1> 0, λ2> 0；
Derivation is carried out to formula (11), is obtained：
It enablesFormula (12) is reduced to formula (13)
But due to existing in α (e)Negative power time item, when α (e)=0 and β (e) ≠ 0 will lead to singularity problem；
Consider the method for switching control：
Wherein qi(e),αi(e),βiIt (e) is respectively q (e), i-th of element of α (e), β (e), i=1,2,3,4,5,6；
Joint type (13) and formula (14), obtain：
Joint type (7), formula (10) and formula (15), obtain：
3.3, design enhanced Reaching Law
WhereinN-1It (X) is the inverse matrix of N (X),
k1> 0, k20,0 < δ < 1 of >, γ > 0, μ > 1, p are positive integer；
3.4, joint type (16) and formula (17) obtain controller
Wherein B-1It (X) is the inverse matrix of B (X).
Further, the control method is further comprising the steps of：
Step 4, property illustrates, process is as follows：
4.1, it was demonstrated that the accessibility of sliding formwork：
Design liapunov functionDerivation is carried out to this function both sides, is obtained：
BecausePerseverance is greater than 0, so formula (18) perseverance meets the accessibility of sliding formwork, system energy less than 0
Reach sliding-mode surface；
4.2, enhanced effect explanation：
When system is far from sliding-mode surface | s | very big, N (s) approaches δ,System
Velocity of approach is accelerated；When system is close to sliding-mode surface | s | approach 0, N (s) approaches μ,
The buffeting of system reduces.
Technical concept of the invention is：For quadrotor system, association index Reaching Law sliding formwork control and fast
Fast TSM control devises a kind of four rotations based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Rotor aircraft finite-time control method.Fast terminal sliding-mode surface is able to achieve the finite-time control of tracking error, solves biography
The problem of time tends to be infinite in system sliding-mode surface, and error just tends to 0.It not only can be remote based on the enhanced Reaching Law of inverse proportion function
It can increase velocity of approach when from sliding-mode surface, and buffeting can be reduced, improve the rapidity and robustness of system, realize fast and stable
Control.
Beneficial effects of the present invention are：The robustness of system is enhanced, compared with traditional exponentially approaching rule sliding formwork control,
Not only velocity of approach can be can increase when far from sliding-mode surface, and buffeting can be reduced, shorten the arrival time of sliding mode, from
And system is made quickly to realize stable convergence.In addition to this, the present invention utilizes fast terminal sliding formwork, solves in traditional sliding-mode surface
The problem of time tends to be infinite, and error just tends to 0, realizes finite-time control.
Detailed description of the invention
Fig. 1 is the position tracking effect diagram of quadrotor, and dotted line represents conventional indexes reaching law control,
Dotted line represents the quadrotor finite time based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Control.
Fig. 2 is the position tracking error schematic diagram of quadrotor, and dotted line represents conventional indexes reaching law control,
Dotted line represents the quadrotor finite time based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Control.
Fig. 3 is the attitude angle tracking effect schematic diagram of quadrotor, and dotted line represents conventional indexes Reaching Law control
System, when quadrotor of the dotted line representative based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Between control.
Fig. 4 is the posture angle tracking error schematic diagram of quadrotor, and dotted line represents conventional indexes Reaching Law control
System, when quadrotor of the dotted line representative based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Between control.
Fig. 5 is that the quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Positioner under time control inputs schematic diagram.
Fig. 6 is that the positioner under the conventional indexes reaching law control of quadrotor inputs schematic diagram.
Fig. 7 is that the quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Posture angle controller under time control inputs schematic diagram.
Fig. 8 is that the posture angle controller under the conventional indexes reaching law control of quadrotor inputs schematic diagram.
Fig. 9 is control flow schematic diagram of the invention.
Specific embodiment
The present invention will be further described with reference to the accompanying drawing.
- Fig. 9 referring to Fig.1, a kind of four rotations based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Rotor aircraft finite-time control method, includes the following steps：
Step 1, it determines from the body coordinate system based on quadrotor to the transfer of the inertial coodinate system based on the earth
Matrix；
Wherein ψ, θ, φ are yaw angle, pitch angle, the roll angle of aircraft respectively, indicate aircraft around successively inertial coordinate
It is the angle of each axis rotation, TψIndicate the transfer matrix of ψ, TθIndicate the transfer matrix of θ, TφIndicate the transfer matrix of φ；
Step 2, quadrotor kinetic model is analyzed according to newton Euler's formula, process is as follows：
2.1, have during translation：
Wherein x, y, z respectively indicates position of the quadrotor under inertial coodinate system, and m indicates the quality of aircraft, and g indicates weight
Power acceleration, mg indicate gravity suffered by quadrotor, the resultant force U that four rotors generater；
2.2, have in rotation process：
Wherein τx、τy、τzRespectively represent each axis moment components on body coordinate system, Ixx、Iyy、IzzRespectively represent body seat
Each axis rotary inertia component fastened is marked, × indicate multiplication cross, wp、wq、wrRespectively represent each axis attitude angle speed on body coordinate system
Component is spent,Respectively represent each axis posture component of angular acceleration on body coordinate system；
In view of aircraft is under low-speed operations or floating state, attitude angle variation is smaller, it is believed thatThen (3 are represented by formula (4) to rotation process Chinese style
2.3, joint type (1), (2), (4), shown in the kinetic model such as formula (5) for obtaining aircraft
Wherein Ux、Uy、UzThe input quantity of respectively three positioners；
According to formula (5), decoupling computation is carried out to position and attitude relationship, it is as a result as follows：
Wherein φdFor the expected signal value of φ, θdFor the expected signal value of θ, ψdFor the expected signal value of ψ, arcsin function
It is arcsin function, arctan function is arctan function；
Formula (5) can also be write as 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, tracking error is calculated, controller is designed according to fast terminal sliding-mode surface and its first derivative, process is such as
Under：
3.1, define tracking error and its first differential and second-order differential：
E=X1-Xd (8)
Wherein, Xd=[xd,yd,zd,φd,θd,ψd]T, xd,yd,zd,φd,θd,ψdRespectively x, y, z, φ, θ, ψ's leads
Desired signal；
3.2, design fast terminal sliding-mode surface：
Wherein, sigα(x)=| x |αSign (x), α1> α2> 1, λ1> 0, λ2> 0；
Derivation is carried out to formula (11), is obtained：
It enablesFormula (12) is reduced to formula (13)
But due to existing in α (e)Negative power time item, when α (e)=0 and β (e) ≠ 0 will lead to singularity problem；
Consider the method for switching control：
Wherein qi(e),αi(e),βiIt (e) is respectively q (e), i-th of element of α (e), β (e), i=1,2,3,4,5,6；
Joint type (13) and formula (14), obtain：
Joint type (7), formula (10) and formula (15), obtain：
3.3, design enhanced Reaching Law
WhereinN-1It (X) is the inverse matrix of N (X), k1> 0, k2
0,0 < δ < 1 of >, γ > 0, μ > 1, p are positive integer；
3.4, joint type (16) and formula (17) obtain controller
Wherein B-1It (X) is the inverse matrix of B (X)；
Step 4, property illustrates, process is as follows：
4.1, it was demonstrated that the accessibility of sliding formwork：
Design liapunov functionDerivation is carried out to this function both sides, is obtained：
BecausePerseverance is greater than 0, so formula (18) perseverance meets the accessibility of sliding formwork, system energy less than 0
Reach sliding-mode surface；
4.2, enhanced effect explanation：
When system is far from sliding-mode surface | s | very big, N (s) approaches δ,System
Velocity of approach is accelerated；When system is close to sliding-mode surface | s | approach 0, N (s) approaches μ,
The buffeting of system reduces.
For the validity for verifying proposed method, The present invention gives be based on inverse proportion function enhanced index Reaching Law sliding formwork
The comparison of control method and conventional indexes Reaching Law sliding-mode control：
In order to more effectively compare, all parameters of system are all consistent, i.e. xd=yd=zd=20, ψd=0.5,
Sliding-mode surface parameter：λ1=0.2, λ2=0.7, α1=2, α2=1.1, ε=0.1, Reaching Law parameter：k1=0.6, k2=0.8, δ=
0.5, p=1, γ=1, μ=2, quadrotor parameter：M=0.625, L=0.1275, Ixx=2.3 × 10-3、Iyy=2.4
×10-3、Izz=2.6 × 10-3, g=10, sampling parameter：ts=0.007, N=5000.
From Fig. 1-Fig. 4 as can be seen that based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Quadrotor finite-time control can faster reach desired location；In conjunction with figure 5-8, it is based on the enhanced finger of inverse proportion function
The quadrotor finite-time control of number Reaching Law and fast terminal sliding-mode surface has smaller buffeting.
In conclusion the quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface
Finite-time control can reduce the tracking time while reducing and buffeting, and tracking performance be improved, so that system quickly enters surely
Fixed convergence.
Described above is the excellent effect of optimization that one embodiment that the present invention provides is shown, it is clear that the present invention is not only
It is limited to above-described embodiment, without departing from essence spirit of the present invention and without departing from the premise of range involved by substantive content of the present invention
Under it can be made it is various deformation be implemented.
Claims (2)
1. a kind of quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface is limited
Between control method, which is characterized in that the control method includes the following steps：
Step 1, it determines from the body coordinate system based on quadrotor to the transfer matrix of the inertial coodinate system based on the earth；
Wherein ψ, θ, φ are yaw angle, pitch angle, the roll angle of aircraft respectively, indicate that aircraft is each around successively inertial coodinate system
The angle of axis rotation, TψIndicate the transfer matrix of ψ, TθIndicate the transfer matrix of θ, TφIndicate the transfer matrix of φ；
Step 2, quadrotor kinetic model is analyzed according to newton Euler's formula, process is as follows：
2.1, have during translation：
Wherein x, y, z respectively indicates position of the quadrotor under inertial coodinate system, and m indicates the quality of aircraft, and g indicates that gravity adds
Speed, mg indicate gravity suffered by quadrotor, the resultant force U that four rotors generater；
2.2, have in rotation process：
Wherein τx、τy、τzRespectively represent each axis moment components on body coordinate system, Ixx、Iyy、IzzRespectively represent body coordinate system
On each axis rotary inertia component, × indicate multiplication cross, wp、wq、wrRespectively represent each axis attitude angular velocity point on body coordinate system
Amount,Respectively represent each axis posture component of angular acceleration on body coordinate system；
In view of aircraft is under low-speed operations or floating state, it is believed that
Then rotation process Chinese style (3) is expressed as formula (4)
2.3, joint type (1), (2), (4), shown in the kinetic model such as formula (5) for obtaining aircraft
Wherein Ux、Uy、UzThe input quantity of respectively three positioners；
According to formula (5), decoupling computation is carried out to position and attitude relationship, it is as a result as follows：
Wherein φdFor the expected signal value of φ, θdFor the expected signal value of θ, ψdFor the expected signal value of ψ, arcsin function is anti-
SIN function, arctan function are arctan functions；
Formula (5) can also be write as 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, tracking error is calculated, controller is designed according to fast terminal sliding-mode surface and its first derivative, process is as follows：
3.1, define tracking error and its first differential and second-order differential：
E=X1-Xd (8)
Wherein, Xd=[xd,yd,zd,φd,θd,ψd]T, xd,yd,zd,φd,θd,ψdRespectively x, y, z, φ, θ, ψ's leads expectation
Signal；
3.2, design fast terminal sliding-mode surface：
Wherein, sigα(x)=| x |αSign (x), α1> α2> 1, λ1> 0, λ2> 0；
Derivation is carried out to formula (11), is obtained：
It enablesFormula (12) is reduced to formula (13)
But due to existing in α (e)Negative power time item, when α (e)=0 and β (e) ≠ 0 will lead to singularity problem；
Consider the method for switching control：
Wherein qi(e),αi(e),βiIt (e) is respectively q (e), i-th of element of α (e), β (e), i=1,2,3,4,5,6；
Joint type (13) and formula (14), obtain：
Joint type (7), formula (10) and formula (15), obtain：
3.3, design enhanced Reaching Law
Wherein c, N-1It (X) is the inverse matrix of N (X), k1> 0, k20,0 < δ < 1 of >, γ > 0, μ > 1, p are positive integer；
3.4, joint type (16) and formula (17) obtain controller
Wherein B-1It (X) is the inverse matrix of B (X).
2. the quadrotor based on inverse proportion function enhanced index Reaching Law and fast terminal sliding-mode surface as described in claim 1
Aircraft finite-time control method, which is characterized in that the control method is further comprising the steps of：
Step 4, property illustrates, process is as follows：
4.1, it was demonstrated that the accessibility of sliding formwork：
Design liapunov functionDerivation is carried out to this function both sides, is obtained：
BecausePerseverance is greater than 0, so formula (18) perseverance meets the accessibility of sliding formwork, system can reach less than 0
Sliding-mode surface；
4.2, enhanced effect explanation：
When system is far from sliding-mode surface | s | very big, N (s) approaches δ,System becomes
Nearly speed is accelerated；When system is close to sliding-mode surface | s | approach 0, N (s) approaches μ,System
The buffeting of system reduces.
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Application publication date: 20181116 Assignee: Foshan chopsticks Technology Co.,Ltd. Assignor: JIANG University OF TECHNOLOGY Contract record no.: X2024980000084 Denomination of invention: Finite time control method for quadcopter based on inverse proportional function enhanced exponential approaching law and fast terminal sliding mode surface Granted publication date: 20210618 License type: Common License Record date: 20240104 |