CN108388118A - The quadrotor total state constrained control method of liapunov function is constrained based on asymmetric time-varying tangential type - Google Patents

The quadrotor total state constrained control method of liapunov function is constrained based on asymmetric time-varying tangential type Download PDF

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CN108388118A
CN108388118A CN201810142245.6A CN201810142245A CN108388118A CN 108388118 A CN108388118 A CN 108388118A CN 201810142245 A CN201810142245 A CN 201810142245A CN 108388118 A CN108388118 A CN 108388118A
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quadrotor
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陈强
胡忠君
胡轶
吴春
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Zhejiang University of Technology ZJUT
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

A kind of quadrotor total state constrained control method constraining liapunov function based on asymmetric time-varying tangential type, for the dynamic system of quadrotor, select a kind of asymmetric time-varying tangential type constraint liapunov function, a kind of quadrotor total state constrained control method constraining liapunov function based on asymmetric time-varying tangential type of design.The design of asymmetric time-varying tangential type constraint liapunov function, which is state in order to ensure system and output, can limit and avoid excessive overshoot in a certain range, 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 constraining liapunov function based on asymmetric time-varying tangential type, and system is made to have preferable dynamic response process.

Description

The quadrotor of liapunov function is constrained based on asymmetric time-varying tangential type Total state constrained control method
Technical field
The present invention relates to a kind of quadrotor constraining liapunov function based on asymmetric time-varying tangential type is complete State constraint 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 a kind of bases In the quadrotor total state constrained control method of asymmetric time-varying tangential type liapunov function, reduce overshoot And arrival 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 limited control of the quadrotor total state constraining liapunov function based on asymmetric time-varying tangential type Method processed, 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 respectively：
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 define q11：
2.3 design constraint liapunov function V11：
Wherein, Ka1,Kb1For time-varying parameter：
Wherein, | e1|maxFor | e1| maximum value；
2.4 solve formula (10) first derivative, obtain：
Wherein,α1For virtual controlling amount, Expression formula is：
Wherein, β1,k11For normal number；
Formula (13) is substituted into formula (12), is obtained：
2.5 define q12：
2.6 design constraint liapunov function V12：
Wherein, Kd1,Kc1For time-varying parameter：
Wherein, | s1|maxFor | s1| maximum value；
Solution formula (16) first derivative, obtains：
Wherein,Expression formula is
Formula (19) and formula (6) are substituted into formula (18), obtained：
2.7 design Uf：
Wherein, k12For normal number；
2.8 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.9 define q21,q31：
2.10 design constraint liapunov function V21,V31：
Wherein, Ka2,Kb2,Ka3,Kb3For time-varying parameter：
Wherein, | e2|maxFor | e2| maximum value, | e3|maxFor | e3| maximum value；
2.11 solve formula (25) first derivative, obtain：
Wherein, α2,α3For virtual controlling amount, expression formula is：
Wherein, k21,k31,β2,β3For normal number；
Formula (28) is substituted into formula (27), is obtained：
2.12 defining q22,q32：
2.13 design liapunov function V22,V32：
Wherein, Kc2,Kd2,Kc3,Kd3For time-varying parameter：
Wherein, | s2|maxFor | s2| maximum value, | s3|maxFor | s3| maximum value；
Solution formula (32) first derivative, obtains：
Wherein, Expression formula is
Formula (35) and formula (6) are substituted into formula (34), obtained：
2.14 designing ux,uy：
Wherein, k22,k32For time-varying parameter；
2.15 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；
2.16 defining qj1：
2.17 design constraint liapunov function Vj1：
Wherein, Kaj,KbjFor time-varying parameter：
Wherein, | ej|maxFor | ej| maximum value；
2.18 solve formula (40) first derivative, obtain：
Wherein,αjFor virtual controlling amount, expression formula is：
Wherein, kj1,βjFor normal number；
Formula (43) is substituted into formula (42), is obtained：
2.19 defining qj2：
2.20 design liapunov function Vj2：
Wherein, Kdj,KcjFor time-varying parameter, satisfaction-Kcj<sj<Kdj；
Solution formula (46) first derivative, obtains：
Wherein,
Formula (48) and formula (6) are substituted into formula (47), obtained：
2.21 design τ by formula (49)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (21) is substituted into formula (20) by 3.1, is obtained：
Formula (37) is substituted into formula (36) by 3.2, is obtained：
Formula (50) is substituted into formula (49) by 3.3, is obtained
3.4 know that quadrotor system is stable by (51), (52), (53).
The quadrotor total state that liapunov function is constrained the present invention is based on asymmetric time-varying tangential type is limited 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 asymmetric Become the quadrotor total state constrained control method of tangential type constraint liapunov function.Asymmetric time-varying tangential type is about The design of beam liapunov function, which is state in order to ensure system and output, can limit in a certain range, avoid Big 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, 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, a kind of quadrotor constraining liapunov function based on asymmetric time-varying tangential type Total state 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 respectively：
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 define q11：
2.3 design constraint liapunov function V11：
Wherein, Ka1,Kb1For time-varying parameter：
Wherein, | e1|maxFor | e1| maximum value；
2.4 solve formula (10) first derivative, obtain：
Wherein,α1For virtual controlling amount, Expression formula is：
Wherein, β1,k11For normal number；
Formula (13) is substituted into formula (12), is obtained：
2.5 define q12：
2.6 design constraint liapunov function V12：
Wherein, Kd1,Kc1For time-varying parameter：
Wherein, | s1|maxFor | s1| maximum value；
Solution formula (16) first derivative, obtains：
Wherein,Expression formula is
Formula (19) and formula (6) are substituted into formula (18), obtained：
2.7 design Uf：
Wherein, k12For normal number；
2.8 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.9 define q21,q31：
2.10 design constraint liapunov function V21,V31：
Wherein, Ka2,Kb2,Ka3,Kb3For time-varying parameter：
Wherein, | e2|maxFor | e2| maximum value, | e3|maxFor | e3| maximum value；
2.11 solve formula (25) first derivative, obtain：
Wherein, α2,α3For virtual controlling amount, expression formula is：
Wherein, k21,k31,β2,β3For normal number；
Formula (28) is substituted into formula (27), is obtained：
2.12 defining q22,q32：
2.13 design liapunov function V22,V32：
Wherein, Kc2,Kd2,Kc3,Kd3For time-varying parameter：
Wherein, | s2|maxFor | s2| maximum value, | s3|maxFor | s3| maximum value；
Solution formula (32) first derivative, obtains：
Wherein, Expression formula is
Formula (35) and formula (6) are substituted into formula (34), obtained：
2.14 designing ux,uy：
Wherein, k22,k32For time-varying parameter；
2.15 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；
2.16 defining qj1：
2.17 design constraint liapunov function Vj1：
Wherein, Kaj,KbjFor time-varying parameter：
Wherein, | ej|maxFor | ej| maximum value；
2.18 solve formula (40) first derivative, obtain：
Wherein,αjFor virtual controlling amount, expression formula is：
Wherein, kj1,βjFor normal number；
Formula (43) is substituted into formula (42), is obtained：
2.19 defining qj2：
2.20 design liapunov function Vj2：
Wherein, Kdj,KcjFor time-varying parameter, satisfaction-Kcj<sj<Kdj；
Solution formula (46) first derivative, obtains：
Wherein
Formula (48) and formula (6) are substituted into formula (47), obtained：
2.21 design τ by formula (49)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (21) is substituted into formula (20) by 3.1, is obtained：
Formula (37) is substituted into formula (36) by 3.2, is obtained：
Formula (50) is substituted into formula (49) by 3.3, is obtained
3.4 know that quadrotor system is stable by (51), (52), (53).
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 (22) and formula (38)d=1, xd=1, yd=1, ψd=0.5；Formula (13), k in formula (29) and formula (43)11=2, k21=2, k31=2, k41=2, k51=2, k61=2；Formula (21), formula (37) and formula (50) k in12=2, k22=2, k32=2, k42=2, k52=2, k62=2；Formula (10), formula (26) and formula (41) kb1=kb2=kb3 =kb4=kb5=kb6=2+0.1sint, ka1=ka2=ka3=ka4=ka5=ka6=1.5+0.1sint；Formula (17), formula (33) With formula (45) kd1=kd2=kd3=kd4=kd5=kd6=3.5+0.1sint, kc1=kc2=kc3=kc4=kc5=kc6=4+ 0.1sint；
From Fig. 1 and Fig. 2 it is found that it is 4.56 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 4.96 seconds that system mode, which has good transient response, arrival time, overshoot 0.
In conclusion based on asymmetric time-varying tangential type constrain liapunov function quadrotor total state by Limit 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. a kind of quadrotor total state constrained control constraining liapunov function based on asymmetric time-varying tangential type 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, indicate unmanned plane 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 unmanned plane 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 respectively：
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 define q11：
2.3 design constraint liapunov function V11：
Wherein, Ka1,Kb1For time-varying parameter：
Wherein, | e1|maxFor | e1| maximum value；
2.4 solve formula (10) first derivative, obtain：
Wherein,α1For virtual controlling amount,
Its expression formula is：
Wherein, β1,k11For normal number；
Formula (13) is substituted into formula (12), is obtained：
2.5 define q12：
2.6 design constraint liapunov function V12：
Wherein, Kd1,Kc1For time-varying parameter：
Wherein, | s1|maxFor | s1| maximum value；
Solution formula (16) first derivative, obtains：
Wherein, Expression formula is
Formula (19) and formula (6) are substituted into formula (18), obtained：
2.7 design Uf：
Wherein, k12For normal number；
2.8 define x, and y tracking errors are respectively e2,e3, then have：
Wherein, xd,ydX, the desired signal of y are indicated respectively；
2.9 define q21,q31：
2.10 design constraint liapunov function V21,V31：
Wherein, Ka2,Kb2,Ka3,Kb3For time-varying parameter：
Wherein, | e2|maxFor | e2| maximum value, | e3|maxFor | e3| maximum value；
2.11 solve formula (25) first derivative, obtain：
Wherein, α2,α3For virtual controlling amount, expression formula is：
Wherein, k21,k31,β2,β3For normal number；
Formula (28) is substituted into formula (27), is obtained：
2.12 defining q22,q32：
2.13 design liapunov function V22,V32：
Wherein, Kc2,Kd2,Kc3,Kd3For time-varying parameter：
Wherein, | s2|maxFor | s2| maximum value, | s3|maxFor | s3| maximum value；
Solution formula (32) first derivative, obtains：
Wherein, Expression formula is
Formula (35) and formula (6) are substituted into formula (34), obtained：
2.14 designing ux,uy：
Wherein, k22,k32For time-varying parameter；
2.15 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；
2.16 defining qj1：
2.17 design constraint liapunov function Vj1：
Wherein, Kaj,KbjFor time-varying parameter：
Wherein, | ej|maxFor | ej| maximum value；
2.18 solve formula (40) first derivative, obtain：
Wherein,αjFor virtual controlling amount, expression formula is：
Wherein, kj1,βjFor normal number；
Formula (43) is substituted into formula (42), is obtained：
2.19 defining qj2：
2.20 design liapunov function Vj2：
Wherein, Kdj,KcjFor time-varying parameter, satisfaction-Kcj<sj<Kdj；
Solution formula (46) first derivative, obtains：
Wherein,
Formula (48) and formula (6) are substituted into formula (47), obtained：
2.21 design τ by formula (49)x,τy,τz：
Wherein, k42,k52,k62For normal number；
Step 3, the stability of quadrotor system is verified, process is as follows：
Formula (21) is substituted into formula (20) by 3.1, is obtained：
Formula (37) is substituted into formula (36) by 3.2, is obtained：
Formula (50) is substituted into formula (49) by 3.3, is obtained
3.4 know that quadrotor system is stable by (51), (52), (53).
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CN109613829A (en) * 2018-02-11 2019-04-12 浙江工业大学 A kind of quadrotor total state constrained control method

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CN107368089B (en) * 2017-07-11 2019-12-03 浙江工业大学 A kind of quadrotor nonlinear sliding mode posture control method based on double exponential type functions
CN107450324B (en) * 2017-09-05 2019-10-22 西北工业大学 Consider the hypersonic aircraft adaptive fusion method of angle of attack constraint

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CN109613829A (en) * 2018-02-11 2019-04-12 浙江工业大学 A kind of quadrotor total state constrained control method
CN109375639A (en) * 2018-11-27 2019-02-22 浙江工业大学 A kind of rigid aircraft posture restraint tracking and controlling method based on asymmetric modified obstacle liapunov function

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