CN108388127A - Based on it is asymmetric when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method - Google Patents

Abstract

It is a kind of based on it is asymmetric when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method, for the dynamic system of quadrotor, varying index tangent compound constraint liapunov function when selecting a kind of asymmetric, design it is a kind of based on it is asymmetric when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method.The design of the compound constraint liapunov function of varying index tangent, which is state in order to ensure system and output, when asymmetric 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 provide it is a kind of based on it is asymmetric when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method, make system have preferable dynamic response process.

Description

Based on it is asymmetric when the compound constraint liapunov function of varying index tangent four rotations Rotor aircraft 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 The quadrotor total state constrained control method of the compound liapunov function of varying index tangent when asymmetric, is reduced 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：

It is a kind of based on it is asymmetric when the compound constraint liapunov function of varying index tangent the full shape of quadrotor 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 respectively_fIndicate 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, τ_x,τ_y,τ_zRespectively represent the moment components of each axis on body coordinate system, I_xx,I_yy,I_zzMachine is indicated respectively The component of the rotary inertia of each axis under body coordinate system, × indicate multiplication cross, ω_pIndicate rolling angular speed, ω_qIndicate pitch angle Speed, ω_rIndicate 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, u_x=cos φ sin θ cos ψ+sin φ sin ψs, u_y=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, φ_dFor the expected signal value of φ, θ_dFor 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, z_dIndicate the desired signal of z；

2.2 define q₁₁：

2.3 design constraint liapunov function V₁₁：

Wherein, K_a1,K_b1For 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, K_d1,K_c1For 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 U_f：

Wherein, k₁₂For normal number；

2.8 define x, and y tracking errors are respectively e₂,e₃, then have：

Wherein, x_d,y_dX, the desired signal of y are indicated respectively；

2.9 define q₂₁,q₃₁：

2.10 design constraint liapunov function V₂₁,V₃₁：

Wherein, K_a2,K_b2,K_a3,K_b3For 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, Its 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, K_c2,K_d2,K_c3,K_d3For 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 u_x,u_y：

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, x_4dIndicate the desired value of φ, x_5dIndicate the desired value of θ, x_6d 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 q_j1：

2.17 design constraint liapunov function V_j1：

Wherein, K_aj,K_bjFor time-varying parameter：

Wherein, | e_j|_maxFor | e_j| maximum value；

2.18 solve formula (40) first derivative, obtain：

Wherein,α_jFor virtual controlling amount,

Its expression formula is：

Wherein, k_j1,β_jFor normal number；

Formula (43) is substituted into formula (42), is obtained：

2.19 defining q_j2：

2.20 design liapunov function V_j2：

Wherein K_dj,K_cjFor time-varying parameter, satisfaction-K_cj<s_j<K_dj；

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,τ_y,τ_z：

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, it is a kind of based on it is asymmetric when the compound constraint liapunov function of varying index tangent four rotations Rotor aircraft 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 respectively_fIndicate 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, τ_x,τ_y,τ_zRespectively represent the moment components of each axis on body coordinate system, I_xx,I_yy,I_zzMachine is indicated respectively The component of the rotary inertia of each axis under body coordinate system, × indicate multiplication cross, ω_pIndicate rolling angular speed, ω_qIndicate pitch angle Speed, ω_rIndicate 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, u_x=cos φ sin θ cos ψ+sin φ sin ψs, u_y=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, φ_dFor the expected signal value of φ, θ_dFor 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, z_dIndicate the desired signal of z；

2.2 define q₁₁：

2.3 design constraint liapunov function V₁₁：

Wherein, K_a1,K_b1For 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, K_d1,K_c1For 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 U_f：

Wherein, k₁₂For normal number；

2.8 define x, and y tracking errors are respectively e₂,e₃, then have：

Wherein, x_d,y_dX, the desired signal of y are indicated respectively；

2.9 define q₂₁,q₃₁：

2.10 design constraint liapunov function V₂₁,V₃₁：

Wherein, K_a2,K_b2,K_a3,K_b3For 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, Its 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, K_c2,K_d2, K_c3,K_d3For 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 u_x,u_y：

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, x_4dIndicate the desired value of φ, x_5dIndicate the desired value of θ, x_6d 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 q_j1：

2.17 design constraint liapunov function V_j1：

Wherein, K_aj,K_bjFor time-varying parameter：

Wherein, | e_j|_maxFor | e_j| maximum value；

2.18 solve formula (40) first derivative, obtain：

Wherein,α_jFor virtual controlling amount, Its expression formula is：

Wherein, k_j1, β_jFor normal number；

Formula (43) is substituted into formula (42), is obtained：

2.19 defining q_j2：

2.20 design liapunov function V_j2：

Wherein K_dj,K_cjFor time-varying parameter, satisfaction-K_cj<s_j<K_dj；

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, τ_y,τ_z：

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), I_xx=1.22kgm², I_yy= 1.22kg·m², I_zz=2.2kgm²；Z in formula (8), formula (22) and formula (38)_d=1, x_d=1, y_d=1, ψ_d=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) k_b1=k_b2= k_b3=k_b4=k_b5=k_b6=2+0.1sint, k_a1=k_a2=k_a3=k_a4=k_a5=k_a6=1.5+0.1sint；Formula (17), formula (33) and formula (45) k_d1=k_d2=k_d3=k_d4=k_d5=k_d6=3.5+0.1sint, k_c1=k_c2=k_c3=k_c4=k_c5=k_c6= 4+ 0.1sint；

From Fig. 1 and Fig. 2 it is found that it is 4.2 seconds that system output, which has good transient response, arrival time, overshoot 0.

From Fig. 3 and Fig. 4 it is found that it is 4.56 seconds that system mode, which has good transient response, arrival time, overshoot 0.

In conclusion based on it is asymmetric when the 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 asymmetric when the compound constraint liapunov function of varying index tangent quadrotor total state Constrained control 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 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 respectively_fIndicate 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, τ_x,τ_y,τ_zRespectively represent the moment components of each axis on body coordinate system, I_xx,I_yy,I_zzIndicate that body is sat respectively The component of the rotary inertia of each axis under mark system, × indicate multiplication cross, ω_pIndicate rolling angular speed, ω_qIndicate rate of pitch, ω_rIndicate 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, u_x=cos φ sin θ cos ψ+sin φ sin ψs, u_y=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, φ_dFor the expected signal value of φ, θ_dFor 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, z_dIndicate the desired signal of z；

2.2 define q₁₁：

2.3 design constraint liapunov function V₁₁：

Wherein, K_a1,K_b1For time-varying parameter：

Wherein, | e₁|_maxFor | e₁| maximum value；

2.4 solve formula (10) first derivative, obtain：

Wherein,α₁For virtual controlling amount, table It is up to formula：

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, K_d1,K_c1For time-varying parameter：

Wherein, | s₁|_maxFor | s₁| maximum value；

Solution formula (16) first derivative, obtains：

Wherein,

Table It is up to formula

Formula (19) and formula (6) are substituted into formula (18), obtained：

2.7 design U_f：

Wherein, k₁₂For normal number；

2.8 define x, and y tracking errors are respectively e₂,e₃, then have：

Wherein, x_d,y_dX, the desired signal of y are indicated respectively；

2.9 define q₂₁,q₃₁：

2.10 design constraint liapunov function V₂₁,V₃₁：

Wherein, K_a2,K_b2,K_a3,K_b3For 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 For：

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, K_c2,K_d2,K_c3,K_d3For 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 u_x,u_y：

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, x_4dIndicate the desired value of φ, x_5dIndicate the desired value of θ, x_6dIndicate ψ Desired value, e₄Indicate the tracking error of φ, e₅Indicate the tracking error of θ, e₆Indicate the tracking error of ψ；

2.16 defining q_j1：

2.17 design constraint liapunov function V_j1：

Wherein, K_aj,K_bjFor time-varying parameter：

Wherein, | e_j|_maxFor | e_j| maximum value；

2.18 solve formula (40) first derivative, obtain：

Wherein,α_jFor virtual controlling amount, table It is up to formula：

Wherein, k_j1,β_jFor normal number；

Formula (43) is substituted into formula (42), is obtained：

2.19 defining q_j2：

2.20 design liapunov function V_j2：

Wherein K_dj,K_cjFor time-varying parameter, satisfaction-K_cj<s_j<K_dj；

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,τ_y,τ_z：

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).