CN108267961A - Quadrotor total state constrained control method based on symmetrical time-varying tangential type constraint liapunov function - Google Patents
Quadrotor total state constrained control method based on symmetrical time-varying tangential type constraint liapunov function Download PDFInfo
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Abstract
A kind of quadrotor total state constrained control method based on symmetrical time-varying tangential type constraint liapunov function, for the dynamic system of quadrotor, select a kind of symmetrical time-varying tangential type constraint liapunov function, a kind of quadrotor total state constrained control method based on symmetrical time-varying tangential type constraint liapunov function of design.The design of symmetrical time-varying tangential type constraint liapunov function is to ensure that the state of system and output can be limited in certain range, avoid excessive overshoot, 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 total state constrained control method based on symmetrical time-varying tangential type constraint liapunov function, and system is made to have preferable dynamic response process.
Description
Technical field
The present invention relates to a kind of full shapes of quadrotor based on symmetrical time-varying tangential type constraint liapunov function
State constrained control method makes quadrotor system have preferable dynamic response process.
Background technology
The one kind of quadrotor as rotary aircraft, with its is small, mobility is good, design is simple, system
The advantages that of low cost is made, has attracted the extensive concern of domestic and international university, research institution, company.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
System does not know robustness with external disturbance etc..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 causes 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
In order to overcome the shortcomings of that the mapping of existing quadrotor system is poor, the present invention provides one kind to be based on
The quadrotor total state constrained control method of symmetrical time-varying tangential type liapunov function reduces overshoot and surpasses
Between timing, 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 total state constrained control based on symmetrical time-varying tangential type constraint liapunov function
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, represent 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 represent three positions of the quadrotor under inertial coodinate system, U respectivelyfRepresent that quadrotor flies
The input torque of row device, m are the quality of quadrotor, and g represents acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain:
Dynamic model in 1.3 quadrotor rotation processes is:
Wherein, τx,τy,τzThe moment components of each axis on body coordinate system, I are represented respectivelyxx,Iyy,IzzMachine is represented respectively
The component of the rotary inertia of each axis under body coordinate system, × expression multiplication cross, ωpRepresent rolling angular speed, ωqRepresent pitch angle
Speed, ωrRepresent yaw rate,Represent rolling angular acceleration,Represent pitching angular acceleration,Represent that yaw angle adds
Speed;
Low-speed operations or floating state are in view of aircraft, attitude angle variation is smaller, 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, 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, calculation 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, zdRepresent the desired signal of z;
2.2 design constraint liapunov functionsAnd solve its first derivative:
Wherein, Kb1For time-varying parameter, meet Kb1>|e1|max, | e1|maxFor | e1| maximum value, α1For void
Intend controlled quentity controlled variable, expression formula is:
Wherein, k11For normal number;
Formula (10) is substituted into formula (9), is obtained:
Wherein,
2.3 design liapunov function V12For:
Wherein, Ks1For time-varying parameter, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value,
The first derivative of solution formula (12), can;:
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 represented respectively;
2.6 design constraint liapunov functionsOne is solved respectively
Order derivative obtains:
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
For:
Its, middle k21,k31For normal number;
Formula (19) is substituted into formula (18), is obtained:
Wherein,
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
By formula (23), (6) substitute into formula (22), respectively:
2.8 by formula (24), (25) separately design ux,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, x4dRepresent the desired value of φ, x5dRepresent the desired value of θ, x6d
Represent the desired value of ψ, e4Represent the tracking error of φ, e5Represent the tracking error of θ, e6Represent 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:
Wherein,
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
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).
Quadrotor total state the present invention is based on symmetrical time-varying tangential type constraint liapunov function is limited control
Method processed 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
Tangential type constrains the quadrotor total state constrained control method of liapunov function.Symmetrical time-varying tangential type constraint Lee
The design of Ya Punuofu functions is to ensure that the state of system and output can be limited in certain range, avoid excessive
Overshoot, while arrival time can also be reduced.So as to improve the dynamic response performance of quadrotor system.
Beneficial effects of the present invention are:Total state is limited, and 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.
The positioner that Fig. 5 is the present invention inputs schematic diagram.
The posture angle controller that Fig. 6 is the present invention inputs schematic diagram.
Fig. 7 is the control flow schematic diagram of the present invention.
Specific embodiment
The present invention will be further described below in conjunction with the accompanying drawings.
With reference to Fig. 1-Fig. 7, a kind of quadrotor based on symmetrical time-varying tangential type constraint liapunov function is complete
State constraint 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, represent 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 represent three positions of the quadrotor under inertial coodinate system, U respectivelyfRepresent that quadrotor flies
The input torque of row device, m are the quality of quadrotor, and g represents acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain:
Dynamic model in 1.3 quadrotor rotation processes is:
Wherein, τx,τy,τzThe moment components of each axis on body coordinate system, I are represented respectivelyxx,Iyy,IzzMachine is represented respectively
The component of the rotary inertia of each axis under body coordinate system, × expression multiplication cross, ωpRepresent rolling angular speed, ωqRepresent pitch angle
Speed, ωrRepresent yaw rate,Represent rolling angular acceleration,Represent pitching angular acceleration,Represent that yaw angle adds
Speed;
Low-speed operations or floating state are in view of aircraft, attitude angle variation is smaller, 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, 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, calculation 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, zdRepresent the desired signal of z;
2.2 design constraint liapunov functionsAnd solve its first derivative:
Wherein, Kb1For time-varying parameter, meet Kb1>|e1|max, | e1|maxFor | e1| maximum value, α1For void
Intend controlled quentity controlled variable, expression formula is:
Wherein, k11For normal number;
Formula (10) is substituted into formula (9), is obtained:
Wherein,
2.3 design liapunov function V12For:
Wherein, Ks1For time-varying parameter, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value, solve formula (12) single order
Derivative 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 represented respectively;
2.6 design constraint liapunov functionsOne is solved respectively
Order derivative obtains:
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
For:
Wherein, k21,k31For normal number;
Formula (19) is substituted into formula (18), is obtained:
Wherein,
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
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, x4dRepresent the desired value of φ, x5dRepresent the desired value of θ, x6d
Represent the desired value of ψ, e4Represent the tracking error of φ, e5Represent the tracking error of θ, e6Represent 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:
Wherein,
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
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 present invention gives emulation knot of the control method 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+0.1sint,kb4=kb5=kb6=2+0.1sint;Formula (12), formula (21) and formula (31) ks1=ks2=ks3=3.5+
0.1sint,ks4=ks5=ks6=4+0.1sint.
From Fig. 1 and Fig. 2 it is found that system output has good transient response, arrival time is 5.06 seconds, and overshoot is
0.0015。
From Fig. 3 and Fig. 4 it is found that system mode has good transient response, arrival time is 5.48 seconds, overshoot 0.
In conclusion the quadrotor total state based on symmetrical time-varying tangential type constraint liapunov function is limited
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
Above-described embodiment is limited to, without departing from essence spirit of the present invention and the premise 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 total state constrained control side based on symmetrical time-varying tangential type constraint liapunov function
Method, which 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, represent 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 represent three positions of the quadrotor under inertial coodinate system, U respectivelyfRepresent quadrotor
Input torque, m be quadrotor quality, g represent acceleration of gravity,
Formula (1) is substituted into formula (2) to obtain:
Dynamic model in 1.3 quadrotor rotation processes is:
Wherein, τx,τy,τzThe moment components of each axis on body coordinate system, I are represented respectivelyxx,Iyy,IzzRepresent that body is sat respectively
The component of the rotary inertia of each axis under mark system, × expression multiplication cross, ωpRepresent rolling angular speed, ωqRepresent rate of pitch,
ωrRepresent yaw rate,Represent rolling angular acceleration,Represent pitching angular acceleration,Represent yaw angular acceleration;
Low-speed operations or floating state are in view of aircraft, attitude angle variation is smaller, 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, 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, calculation 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, zdRepresent the desired signal of z;
2.2 design constraint liapunov functionsAnd solve its first derivative:
Wherein, Kb1For time-varying parameter, 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 time-varying parameter, meet Ks1>|s1|max, | s1|maxFor | s1| maximum value,
The first derivative of solution formula (12), can;:
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 represented respectively;
2.6 design constraint liapunov functionsIts single order is solved respectively to lead
Number, obtains:
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:
Its, middle k21,k31For normal number;
Formula (19) is substituted into formula (18), is obtained:
Wherein,
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
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, x4dRepresent the desired value of φ, x5dRepresent the desired value of θ, x6dRepresent ψ
Desired value, e4Represent the tracking error of φ, e5Represent the tracking error of θ, e6Represent 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:
Wherein,
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
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 (4)
Publication number | Priority date | Publication date | Assignee | Title |
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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 |
CN110687787A (en) * | 2019-10-11 | 2020-01-14 | 浙江工业大学 | Mechanical arm system self-adaptive control method based on time-varying asymmetric obstacle Lyapunov function |
CN110750050A (en) * | 2019-10-11 | 2020-02-04 | 浙江工业大学 | Neural network-based mechanical arm system preset performance control method |
WO2021237500A1 (en) * | 2020-05-27 | 2021-12-02 | 深圳市大疆创新科技有限公司 | Flight control method and device |
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2018
- 2018-02-11 CN CN201810142264.9A patent/CN108267961A/en not_active Withdrawn
Cited By (5)
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 |
CN110687787A (en) * | 2019-10-11 | 2020-01-14 | 浙江工业大学 | Mechanical arm system self-adaptive control method based on time-varying asymmetric obstacle Lyapunov function |
CN110750050A (en) * | 2019-10-11 | 2020-02-04 | 浙江工业大学 | Neural network-based mechanical arm system preset performance control method |
WO2021237500A1 (en) * | 2020-05-27 | 2021-12-02 | 深圳市大疆创新科技有限公司 | Flight control method and device |
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