CN108536162A - Based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state constrained control method - Google Patents
Based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state constrained control method Download PDFInfo
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
- G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
- G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
- G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
- G05D1/0825—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability using mathematical models
Abstract
It is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state constrained control method, for the dynamic system of quadrotor, not varying index tangent compound constraint liapunov function when selecting a kind of symmetrical, design it is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state constrained control method.To be state in order to ensure system and output can not limit and avoid excessive overshoot in a certain range for the design of the compound constraint liapunov function of varying index tangent when symmetrical, while can also reduce arrival time.So as to improve the dynamic response performance of quadrotor system.The present invention provide it is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state constrained control method, make system that there is preferable dynamic response process.
Description
Technical field
The present invention relates to it is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor
Aircraft total state 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 quadrotor total state constrained control method of constant tangential type liapunov function when constant symmetrical when symmetrical, is reduced
Overshoot and 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：
It is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent the full shape of quadrotor
State constrained control method, includes the following steps：
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, it is believed that Therefore 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 e1Boundary, meet Kb1>|e1|max, | e1|maxFor | e1| maximum value,α1For void
Quasi- controlled quentity controlled variable, expression formula are：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
Wherein,
2.3 design liapunov function V12For：
Wherein, Ks1For s1Boundary, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value；
The first derivative of solution formula (12), obtains：
Wherein, Expression formula is as follows
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 functions
Its first derivative is solved respectively, is obtained：
Wherein, Kb2For e2Boundary, meet Kb2>|e2|max, | e2|maxFor | e2| maximum value；Kb3For e3Boundary, meet Kb3>|
e3|max, | e3|maxFor | e3| maximum value；
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
Wherein, Ks2For s2Boundary, meet Ks2>|s2|max, | s2|maxFor | s2| maximum value；Wherein Ks3For s3Boundary,
Meet Ks3>|s3|max, | s3|maxFor | s3| maximum value；
The first derivative of solution formula (21), obtains：
Wherein, Expression formula is as follows
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：
Its, middle kjFor normal number, KbjFor ejBoundary, meet Kbj>|ej|max, | ej|maxFor | ej| maximum value；αjFor the virtual controlling amount of attitude angle, expression formula is：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function Vj2：
Wherein, KsjFor sjBoundary, meet Ksj>|sj|max, | sj|maxFor | sj| maximum value；
The first derivative of solution formula (31), obtains：
Wherein, Expression formula is
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：
Formula (37) is substituted into formula (34), (35), (36) by 3.3, is obtained
3.4 know that quadrotor system is stable by (38), (39), (40).
The present invention is based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor it is complete
State constraint control method improves the mapping of system, reduces overshoot and arrival time.
The present invention technical concept be：For the dynamic system of quadrotor, when design one kind being based on symmetrical not
The quadrotor total state constrained control method of the compound constraint liapunov function of varying index tangent.It is constant when symmetrical
The design of the compound constraint liapunov function of index tangent, which is state in order to ensure system and output, can be limited in one
In fixed range, excessive overshoot is avoided, while arrival time can also be reduced.So as to improve the dynamic of quadrotor system
Response performance.
Beneficial effects of the present invention are：Total state is limited, reduces overshoot, reduces arrival time, 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 the position and speed tracking effect schematic diagram of the present invention.
Fig. 4 is the attitude angular velocity tracking effect schematic diagram of the present invention.
Fig. 5 is that the positioner of the present invention inputs schematic diagram.
Fig. 6 is that the posture angle controller of the present invention inputs schematic diagram.
Fig. 7 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. 7 referring to Fig.1, it is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent four rotations
Rotor aircraft total state constrained control method, includes the following steps：
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 e1Boundary, meet Kb1>|e1|max, | e1|maxFor | e1| maximum value,α1For void
Quasi- controlled quentity controlled variable, expression formula are：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
Wherein,
2.3 design liapunov function V12For：
Wherein, Ks1For s1Boundary, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value；
The first derivative of solution formula (12), obtains：
Wherein Expression formula is as follows
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 functions
Its first derivative is solved respectively, is obtained：
Wherein, Kb2For e2Boundary, meet Kb2>|e2|max, | e2|maxFor | e2| maximum value；Kb3For e3Boundary, meet Kb3>|
e3|max, | e3|maxFor | e2| maximum value；
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
Wherein, Ks2For s2Boundary, meet Ks2>|s2|max, | s2|maxFor | s2| maximum value；Wherein Ks3For s3Boundary,
Meet Ks3>|s3|max, | s3|maxFor | s3| maximum value；
The first derivative of solution formula (21), obtains：
Wherein, Expression formula is as follows
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, kjFor normal number, KbjFor ejBoundary, meet Kbj>|ej|max, | ej|maxFor | ej| maximum value；αjFor the virtual controlling amount of attitude angle, expression formula is：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function Vj2：
Wherein, KsjFor sjBoundary, meet Ksj>|sj|max, | sj|maxFor | sj| maximum value；
The first derivative of solution formula (31), obtains：
Wherein, Expression formula is
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：
Formula (37) is substituted into formula (34), (35), (36) by 3.3, is obtained
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=kb2=kb3=
1.5,kb4=kb5=kb6=2；Formula (12), formula (21) and formula (31) ks1=ks2=ks3=3.5, ks4=ks5=ks6=4.
From Fig. 1 and Fig. 2 it is found that it is 5.02 seconds that system output, which has good transient response, arrival time, overshoot is
0.0004。
From Fig. 3 and Fig. 4 it is found that it is 5.44 seconds that system mode, which has good transient response, arrival time, overshoot 0.
In conclusion based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor
Total state constrained control method can effectively improve the mapping of quadrotor system total state.
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. it is a kind of based on it is symmetrical when the not compound constraint liapunov function of varying index tangent quadrotor total state
Constrained control method, which is characterized in that the described method comprises the following steps：
1.1 determine from the body coordinate system based on quadrotor system to the transfer matrix of the inertial coordinate based on the earth
T：
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, it is believed that Therefore 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 e1Boundary, meet Kb1>|e1|max, | e1|maxFor | e1| maximum value,α1Virtually to control
Amount processed, expression formula are：
Wherein, k11For normal number；
Formula (10) is substituted into formula (9), is obtained：
Wherein,
2.3 design liapunov function V12For：
Wherein, Ks1For s1Boundary, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value；
The first derivative of solution formula (12), obtains：
Wherein, Expression formula is as follows
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 functions
Its first derivative is solved respectively, is obtained：
Wherein, Kb2For e2Boundary, meet Kb2>|e2|max, | e2|maxFor | e2| maximum value；Kb3For e3Boundary, meet Kb3>|e3
|max, | e3|maxFor | e3| maximum value；
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
Wherein, Ks2For s2Boundary, meet Ks2>|s2|max, | s2|maxFor | s2| maximum value；Wherein Ks3For s3Boundary, meet
Ks3>|s3|max, | s3|maxFor | s3| maximum value；
The first derivative of solution formula (21), obtains：
Wherein, Expression formula is as follows
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：
Its, middle kjFor normal number, KbjFor ejBoundary, meet Kbj>|ej|max, | ej|maxFor | ej| maximum value；αjFor the virtual controlling amount of attitude angle, expression formula is：
Wherein, kj1For normal number；
Formula (29) is substituted into formula (28), is obtained：
2.11 design constraint liapunov function Vj2：
Wherein, KsjFor sjBoundary, meet Ksj>|sj|max, | sj|maxFor | sj| maximum value；
The first derivative of solution formula (31), obtains：
Wherein, Expression formula is
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：
Formula (37) is substituted into formula (34), (35), (36) by 3.3, is obtained
3.4 know that quadrotor system is stable by (38), (39), (40).
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109557933A (en) * | 2018-11-27 | 2019-04-02 | 浙江工业大学 | A kind of rigid aircraft state constraint control method based on imperial Burger observer |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109557933A (en) * | 2018-11-27 | 2019-04-02 | 浙江工业大学 | A kind of rigid aircraft state constraint control method based on imperial Burger observer |
| CN109557933B (en) * | 2018-11-27 | 2021-08-03 | 浙江工业大学 | Rigid aircraft state constraint control method based on Longberger observer |
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Application publication date: 20180914 |