CN108388116A - The quadrotor that liapunov function is constrained based on symmetrical time-varying tangential type exports constrained control method - Google Patents
The quadrotor that liapunov function is constrained based on symmetrical time-varying tangential type exports constrained control method Download PDFInfo
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Abstract
A kind of quadrotor output constrained control method constraining liapunov function based on symmetrical time-varying tangential type, for the dynamic system of quadrotor, a kind of symmetrical time-varying tangential type constraint liapunov function, a kind of quadrotor constraining liapunov function based on symmetrical time-varying tangential type of design is selected to export constrained control method.The design of symmetrical time-varying tangential type constraint liapunov function is to avoid excessive overshoot in a certain range to ensure that the output of system can limit, while can also reduce arrival time.So as to improve the dynamic response performance of quadrotor system.The present invention provides a kind of quadrotor output constrained control method constraining liapunov function based on symmetrical time-varying tangential type, and system is made to have preferable dynamic response process.
Description
Technical field
The present invention relates to a kind of quadrotor outputs constraining liapunov function based on symmetrical time-varying tangential type
Constrained control method makes quadrotor system have preferable dynamic response process.
Background technology
The one kind of quadrotor as rotary aircraft, it is small with its, mobility is good, design is simple, system
The advantages that of low cost is made, the extensive concern of domestic and international university, research institution, company has been attracted.However, since quadrotor is flown
Device is small and light-weight, is in-flight vulnerable to external disturbance, how to realize the High Performance Motion Control to quadrotor
Have become a hot issue.For the control problem of quadrotor, there are many control methods, such as PID control,
Active Disturbance Rejection Control, sliding formwork control, Reverse Step Control etc..
Wherein Reverse Step Control has been widely used for nonlinear system, and advantage includes fast response time, easy to implement, right
The uncertain robustness etc. with external disturbance of system.Traditional Reverse Step Control only considers the stability of quadrotor
Can, there is no pay close attention to its transient response performance too much.Therefore, traditional backstepping control method makes quadrotor system
Application in a practical situation has very big obstruction.To solve this problem, the Reverse Step Control based on constraint liapunov function
Method is suggested, and this method can effectively improve the mapping of quadrotor system in a practical situation.
Invention content
Mapping in order to overcome the shortcomings of existing quadrotor system is poor, and the present invention provides one kind to be based on
The limited step control method of quadrotor output of symmetrical time-varying tangential type constraint liapunov function, reduces overshoot
With the overshoot time, making quadrotor system tool, there are one good dynamic response performances.
In order to solve the above-mentioned technical problem the technical solution proposed is as follows：
A kind of quadrotor output constrained control side constraining liapunov function based on symmetrical time-varying tangential type
Method includes the following steps：
Step 1, the dynamic model of quadrotor system, initial value, sampling time and the control of initialization system are established
Parameter processed, process are as follows：
1.1 determine from the body coordinate system based on quadrotor system to the transfer square of the inertial coordinate based on the earth
Battle array T：
Wherein, φ, θ, ψ are roll angle, pitch angle, the yaw angle of quadrotor respectively, indicate aircraft successively around
The angle of each reference axis rotation of inertial coodinate system；
Dynamic model during the translation of 1.2 quadrotors is as follows：
Wherein, x, y, z indicate three positions of the quadrotor under inertial coodinate system, U respectivelyfIndicate that quadrotor flies
The input torque of row device, m are the quality of quadrotor, and g indicates acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain：
Dynamic model in 1.3 quadrotor rotation processes is：
Wherein, τx,τy,τzRespectively represent the moment components of each axis on body coordinate system, Ixx,Iyy,IzzMachine is indicated respectively
The component of the rotary inertia of each axis under body coordinate system, × indicate multiplication cross, ωpIndicate rolling angular speed, ωqIndicate pitch angle
Speed, ωrIndicate yaw rate,Indicate rolling angular acceleration,Indicate pitching angular acceleration,Indicate that yaw angle adds
Speed；
In view of aircraft is in low-speed operations or floating state, attitude angle variation is smaller, it is believed thatTherefore formula (4) is rewritten as：
Simultaneous formula (3) and formula (5), the kinetic model for obtaining quadrotor are：
Wherein, ux=cos φ sin θ cos ψ+sin φ sin ψs, uy=cos φ sin θ sin ψ-sin φ cos ψ；
1.4, according to formula (6), define φ, and the desired value of θ is：
Wherein, φdFor the expected signal value of φ, θdFor θ expected signal values, arcsin is arcsin function；
Step 2, in each sampling instant, calculating position tracking error and its first derivative；Posture angle tracking is calculated to miss
Difference and its first derivative；Design position and posture angle controller, process are as follows：
2.1 define z tracking errors and its first derivative：
Wherein, zdIndicate the desired signal of z；
2.2 design constraint liapunov functionsAnd solve its first derivative：
Wherein, Kb1For time-varying parameter, meet Kb1>|e1|,α1For virtual controlling amount, expression formula is：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
2.3 design liapunov function V12For：
The first derivative of solution formula (12), obtains：
Wherein
Formula (14) and formula (6) are substituted into formula (13), obtained：
2.4 design Uf：
Wherein, k12For normal number；
2.5 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.6 design constraint liapunov functionsSolve it respectively
First derivative obtains：
Wherein, Kb2For time-varying parameter, meet Kb2>|e2|；Kb3For time-varying parameter, meet Kb3>|e3|；α2,α3For virtual controlling amount, expression formula is:
Wherein, k21,k31For normal number；
Formula (19) is substituted into formula (18), is obtained:
2.7 design liapunov function V22,V32
The first derivative of solution formula (21), obtains：
Wherein
By formula (23), (6) substitute into formula (22), respectively：
2.8 separately design u by formula (24), (25)x,uy：
Wherein, k22,k32For normal number；
2.9 define posture angle tracking error and its first derivative：
Wherein, j=4,5,6, x4=φ, x5=θ, x6=ψ, x4dIndicate the desired value of φ, x5dIndicate the desired value of θ, x6d
Indicate the desired value of ψ, e4Indicate the tracking error of φ, e5Indicate the tracking error of θ, e6Indicate the tracking error of ψ；
2.10 design constraint liapunov functionAnd solve its first derivative：
Wherein, kbjFor time-varying parameter, meet Kbj>|ej|；αjFor the virtual controlling amount of attitude angle, table
It is up to formula：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function：
The first derivative of solution formula (31), obtains：
Wherein
Formula (33) and formula (6) are substituted into formula (32), respectively：
2.12 separately design τ by formula (34), (35), (36)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (16) is substituted into formula (15) by 3.1, is obtained：
Formula (26) is substituted into formula (24), (25) by 3.2, is obtained：
3.3 wushu (37) substitute into formula (34), (35), (36), obtain：
3.4 know that quadrotor system is stable by (38), (39), (40).
The quadrotor that liapunov function is constrained the present invention is based on symmetrical time-varying tangential type exports constrained control
Method improves the mapping of system, reduces overshoot and arrival time.
The present invention technical concept be：For the dynamic system of quadrotor, design is a kind of to be based on symmetrical time-varying
The quadrotor that tangential type constrains liapunov function exports constrained control method.Symmetrical time-varying tangential type constrains Li Ya
The design of Pu Nuofu functions is to avoid excessive overshoot in a certain range to ensure that the output of system can limit, together
When can also reduce arrival time.So as to improve the dynamic response performance of quadrotor system.
Beneficial effects of the present invention are：Overshoot is reduced, arrival time is reduced, improves mapping.
Description of the drawings
Fig. 1 is the position tracking effect diagram of the present invention.
Fig. 2 is the attitude angle tracking effect schematic diagram of the present invention.
Fig. 3 is that the positioner of the present invention inputs schematic diagram.
Fig. 4 is that the posture angle controller of the present invention inputs schematic diagram.
Fig. 5 is the control flow schematic diagram of the present invention.
Specific implementation mode
The present invention will be further described below in conjunction with the accompanying drawings.
- Fig. 5 referring to Fig.1, a kind of quadrotor constraining liapunov function based on symmetrical time-varying tangential type are defeated
Go out constrained control method, includes the following steps：
Step 1, the dynamic model of quadrotor system, initial value, sampling time and the control of initialization system are established
Parameter processed, process are as follows：
1.1 determine from the body coordinate system based on quadrotor system to the transfer square of the inertial coordinate based on the earth
Battle array T：
Wherein, φ, θ, ψ are roll angle, pitch angle, the yaw angle of quadrotor respectively, indicate aircraft successively around
The angle of each reference axis rotation of inertial coodinate system；
Dynamic model during the translation of 1.2 quadrotors is as follows：
Wherein, x, y, z indicate three positions of the quadrotor under inertial coodinate system, U respectivelyfIndicate that quadrotor flies
The input torque of row device, m are the quality of quadrotor, and g indicates acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain：
Dynamic model in 1.3 quadrotor rotation processes is：
Wherein, τx,τy,τzRespectively represent the moment components of each axis on body coordinate system, Ixx,Iyy,IzzMachine is indicated respectively
The component of the rotary inertia of each axis under body coordinate system, × indicate multiplication cross, ωpIndicate rolling angular speed, ωqIndicate pitch angle
Speed, ωrIndicate yaw rate,Indicate rolling angular acceleration,Indicate pitching angular acceleration,Indicate that yaw angle adds
Speed；
In view of aircraft is in low-speed operations or floating state, attitude angle variation is smaller, it is believed thatTherefore formula (4) is rewritten as：
Simultaneous formula (3) and formula (5), the kinetic model for obtaining quadrotor are：
Wherein, ux=cos φ sin θ cos ψ+sin φ sin ψs, uy=cos φ sin θ sin ψ-sin φ cos ψ；
1.4, according to formula (6), define φ, and the desired value of θ is：
Wherein, φdFor the expected signal value of φ, θdFor θ expected signal values, arcsin is arcsin function；
Step 2, in each sampling instant, calculating position tracking error and its first derivative；Posture angle tracking is calculated to miss
Difference and its first derivative；Design position and posture angle controller, process are as follows：
2.1 define z tracking errors and its first derivative：
Wherein, zdIndicate the desired signal of z；
2.2 design constraint liapunov functionsAnd solve its first derivative：
Wherein, Kb1For time-varying parameter, meet Kb1>|e1|,α1For virtual controlling amount, expression formula is：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
2.3 design liapunov function V12For：
The first derivative of solution formula (12), obtains：
Wherein
Formula (14) and formula (6) are substituted into formula (13), obtained：
2.4 design Uf：
Wherein, k12For normal number；
2.5 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.6 design constraint liapunov functionsSolve it respectively
First derivative obtains：
Wherein, Kb2For time-varying parameter, meet Kb2>|e2|；Kb3For time-varying parameter, meet Kb3>|e3|；α2,α3For virtual controlling amount, expression formula is:
Wherein, k21,k31For normal number；
Formula (19) is substituted into formula (18), is obtained:
2.7 design liapunov function V22,V32
The first derivative of solution formula (21), obtains：
Wherein
By formula (23), (6) substitute into formula (22), respectively：
2.8 separately design u by formula (24), (25)x,uy：
Wherein, k22,k32For normal number；
2.9 define posture angle tracking error and its first derivative：
Wherein, j=4,5,6, x4=φ, x5=θ, x6=ψ, x4dIndicate the desired value of φ, x5dIndicate the desired value of θ, x6d
Indicate the desired value of ψ, e4Indicate the tracking error of φ, e5Indicate the tracking error of θ, e6Indicate the tracking error of ψ；
2.10 design constraint liapunov functionAnd solve its first derivative：
Wherein, kbjFor time-varying parameter, meet Kbj>|ej|；αjFor the virtual controlling amount of attitude angle, table
It is up to formula：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function：
The first derivative of solution formula (31), obtains：
Wherein
Formula (33) and formula (6) are substituted into formula (32), respectively：
2.12 separately design τ by formula (34), (35), (36)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (16) is substituted into formula (15) by 3.1, is obtained：
Formula (26) is substituted into formula (24), (25) by 3.2, is obtained：
3.3 wushu (37) substitute into formula (34), (35), (36), obtain：
3.4 know that quadrotor system is stable by (38), (39), (40).
The feasibility of extracting method in order to verify, the emulation knot that The present invention gives the control methods on MATLAB platforms
Fruit：
Parameter is given below：M=1.1kg, g=9.81N/kg in formula (2)；In formula (4), Ixx=1.22kgm2, Iyy=
1.22kg·m2, Izz=2.2kgm2；Z in formula (8), formula (17) and formula (27)d=1, xd=1, yd=1, ψd=0.5；Formula
(10), k in formula (19) and formula (29)11=2, k21=2, k31=2, k41=2, k51=2, k61=2；Formula (16), formula (26) and formula
(37) k in12=2, k22=2, k32=2, k42=2, k52=2, k62=2；Formula (9), formula (18) and formula (28) kb1=1.5+
0.1sint,kb2=1.5+0.1sint, kb3=1.5+0.1sint, kb4=2+0.1sint, kb5=2+0.1sint, kb6=2+
0.1sint。
From Fig. 1 and Fig. 2 it is found that it is 5.04 seconds that system, which has good transient response, arrival time, overshoot is
0.0005。
In conclusion constraining the limited control of quadrotor output of liapunov function based on symmetrical time-varying tangential type
Method processed can effectively improve the mapping of quadrotor system.
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, in the premise without departing from essence spirit of the present invention and without departing from range involved by substantive content of the present invention
Under it can be made it is various deformation be implemented.
Claims (1)
1. a kind of quadrotor constraining liapunov function based on symmetrical time-varying tangential type exports constrained control method,
It is characterized in that, the control method includes the following steps：
Step 1, the dynamic model of quadrotor system, initial value, sampling time and the control ginseng of initialization system are established
Number, process are as follows：
1.1 determine from the body coordinate system based on quadrotor system to the transfer matrix T of the inertial coordinate based on the earth：
Wherein, φ, θ, ψ are roll angle, pitch angle, the yaw angle of quadrotor respectively, indicate aircraft successively around inertia
The angle of each reference axis rotation of coordinate system；
Dynamic model during the translation of 1.2 quadrotors is as follows：
Wherein, x, y, z indicate three positions of the quadrotor under inertial coodinate system, U respectivelyfIndicate quadrotor
Input torque, m be quadrotor quality, g indicate acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain：
Dynamic model in 1.3 quadrotor rotation processes is：
Wherein, τx,τy,τzRespectively represent the moment components of each axis on body coordinate system, Ixx,Iyy,IzzIndicate that body is sat respectively
The component of the rotary inertia of each axis under mark system, × indicate multiplication cross, ωpIndicate rolling angular speed, ωqIndicate rate of pitch,
ωrIndicate yaw rate,Indicate rolling angular acceleration,Indicate pitching angular acceleration,Indicate yaw angular acceleration；
In view of aircraft is in low-speed operations or floating state, attitude angle variation is smaller, it is believed thatTherefore formula (4) is rewritten as：
Simultaneous formula (3) and formula (5), the kinetic model for obtaining quadrotor are：
Wherein, ux=cos φ sin θ cos ψ+sin φ sin ψs, uy=cos φ sin θ sin ψ-sin φ cos ψ；
1.4, according to formula (6), define φ, and the desired value of θ is：
Wherein φdFor the expected signal value of φ, θdFor θ expected signal values, arcsin is arcsin function；
Step 2, in each sampling instant, calculating position tracking error and its first derivative；Calculate posture angle tracking error and
Its first derivative；Design position and posture angle controller, process are as follows：
2.1 define z tracking errors and its first derivative：
Wherein, zdIndicate the desired signal of z；
2.2 design constraint liapunov functionsAnd solve its first derivative：
Wherein, Kb1For time-varying parameter, meet Kb1>|e1|,α1For virtual controlling amount, expression formula is：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
2.3 design liapunov function V12For：
The first derivative of solution formula (12), obtains：
Wherein
Formula (14) and formula (6) are substituted into formula (13), obtained：
2.4 design Uf：
Wherein, K12For normal number；
2.5 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.6 design constraint liapunov functionsIts single order is solved respectively to lead
Number, obtains：
Wherein, Kb2For time-varying parameter, meet Kb2>|e2|；Kb3For time-varying parameter, meet Kb3>|e3|；
α2,α3For virtual controlling amount, expression formula is:
Wherein, k21,k31For normal number；
Formula (19) is substituted into formula (18), is obtained:
2.7 design liapunov function V22,V32
The first derivative of solution formula (21), obtains：
Wherein
By formula (23), (6) substitute into formula (22), respectively：
2.8 separately design u by formula (24), (25)x,uy：
Wherein, k22,k32For normal number；
2.9 define posture angle tracking error and its first derivative：
Wherein, j=4,5,6, x4=φ, x5=θ, x6=ψ, x4dIndicate the desired value of φ, x5dIndicate the desired value of θ, x6dIndicate ψ
Desired value, e4Indicate the tracking error of φ, e5Indicate the tracking error of θ, e6Indicate the tracking error of ψ；
2.10 design constraint liapunov functionAnd solve its first derivative：
Wherein, kbjFor time-varying parameter, meet Kbj>|ej|；αjFor the virtual controlling amount of attitude angle, expression formula
For：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function：
The first derivative of solution formula (31), obtains：
Wherein
Formula (33) and formula (6) are substituted into formula (32), respectively：
2.12 separately design τ by formula (34), (35), (36)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (16) is substituted into formula (15) by 3.1, is obtained：
Formula (26) is substituted into formula (24), (25) by 3.2, is obtained：
3.3 wushu (37) substitute into formula (34), (35), (36), obtain：
3.4 know that quadrotor system is stable by (38), (39), (40).
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