CN109932902A - A kind of output limited control method of quadrotor aircraft - Google Patents

Abstract

An output-limited control method for a quadrotor aircraft. Aiming at the dynamic system of the quadrotor aircraft, a symmetric time-invariant tangent constrained Lyapunov function is selected, and a symmetric time-invariant tangent constrained Lyapunov function is designed. The output limited control method of the quadrotor aircraft. The design of the symmetric time-invariant tangent-constrained Lyapunov function is to ensure that the output of the system can be limited to a certain range, avoid excessive overshoot, and at the same time reduce the arrival time. Thereby, the dynamic response performance of the quadrotor aircraft system is improved. The invention provides an output-limited control method for a quadrotor aircraft based on a symmetrical time-invariant tangent constraint Lyapunov function, so that the system has a better dynamic response process.

Description

A kind of output limited control method of quadrotor aircraft

technical field

The invention relates to an output-limited control method for a quadrotor aircraft, which enables the quadrotor aircraft system to have a better dynamic response process.

Background technique

As a kind of rotary-wing aircraft, quadrotor aircraft has attracted extensive attention from universities, research institutions and companies at home and abroad due to its advantages of small size, good maneuverability, simple design and low manufacturing cost. However, due to the small size and light weight of the quadrotor aircraft, it is vulnerable to external interference during flight, and how to achieve high-performance motion control for the quadrotor aircraft has become a hot issue. For the control problem of quadrotor aircraft, there are many control methods, such as PID control, active disturbance rejection control, sliding mode control, backstepping control and so on.

Among them, backstepping control has been widely used in nonlinear systems, and its advantages include fast response speed, convenient implementation, robustness to system uncertainties and external disturbances, etc. The traditional backstepping control only considers the steady-state performance of the quadrotor, and does not pay too much attention to its transient response performance. Therefore, the traditional backstepping control method makes the application of the quadrotor aircraft system in practical situations greatly hindered. To solve this problem, a backstepping control method based on constrained Lyapunov function is proposed, which can effectively improve the transient performance of quadrotor aircraft systems in practical situations.

SUMMARY OF THE INVENTION

In order to overcome the shortcoming of the poor transient performance of the existing quadrotor aircraft system, the present invention provides an output limited control method of the quadrotor aircraft based on the symmetric time-invariant tangent constraint Lyapunov function, which reduces the overshoot. and overshoot time, so that the quadrotor aircraft system has a good dynamic response performance.

The technical solutions proposed to solve the above technical problems are as follows:

A method for limiting the output of a quadrotor aircraft, comprising the following steps:

Step 1, establish the dynamic model of the quadrotor aircraft system, set the initial value, sampling time and control parameters of the system, the process is as follows:

1.1 Determine the transition matrix T from the quadrotor system-based body coordinate system to the Earth-based inertial coordinate system:

Among them, φ, θ, ψ are the roll angle, pitch angle, and yaw angle of the quadrotor aircraft, respectively, indicating the rotation angle of the aircraft around each coordinate axis of the inertial coordinate system in turn;

1.2 The dynamic model of the quadrotor during translation is as follows:

Among them, x, y, z respectively represent the three positions of the quadrotor in the inertial coordinate system, U _f represents the input torque of the quadrotor, m is the mass of the quadrotor, g represents the acceleration of gravity,

Substitute equation (1) into equation (2) to get:

1.3 The dynamic model during the rotation of the quadrotor is:

Among them, τ _x , τ _y , τ _z respectively represent the moment components of each axis on the body coordinate system, I _xx , I _yy , I _zz respectively represent the components of the moment of inertia of each axis in the body coordinate system, × represents the cross product, ω _p represents the roll angular velocity, ω _q represents the pitch angular velocity, ω _r represents the yaw angular velocity, is the roll angular acceleration, represents the pitch angle acceleration, represents the yaw angular acceleration;

Considering that the aircraft is in a low-speed flight or hovering state, the attitude angle changes little, it is considered that So formula (4) is rewritten as:

Combining Equation (3) and Equation (5), the dynamic model of the quadrotor aircraft is obtained as:

where u _x =cosφsinθcosψ+sinφsinψ,u _y =cosφsinθsinψ-sinφcosψ;

1.4 According to formula (6), the expected values of φ and θ are defined as:

Among them, φ _d is the expected signal value of φ, θ _d is the expected signal value of θ, and arcsin is the inverse sine function;

Step 2, at each sampling moment, calculate the position tracking error and its first derivative; calculate the attitude angle tracking error and its first derivative; design the position and attitude angle controller, the process is as follows:

2.1 Define the z tracking error and its first derivative:

where z _d represents the desired signal of z;

2.2 Design Constrained Lyapunov Functions and solve for its first derivative:

Among them, K _b1 is the boundary of e ₁ , satisfying K _b1 >|e ₁ | _max , |e ₁ | _max is the maximum value of |e ₁ |, α ₁ is a virtual control quantity, and its expression is:

Among them, k ₁₁ is a positive number;

Substituting equation (10) into equation (9), we get:

2.3 Design the Lyapunov function V ₁₂ as:

Solving the first derivative of equation (12), we get:

in

Substituting equations (14) and (6) into equation (13), we get:

2.4 Design U _f :

Among them, k ₁₂ is a positive number;

2.5 Define the x and y tracking errors as e ₂ , e ₃ respectively, then there are:

e ₂ =xx _d , e ₃ =yy _d , Among them, x _d , y _d represent the desired signals of x and y, respectively;

2.6 Design Constrained Lyapunov Functions Solving its first derivative separately, we get:

Among them, K _b2 is the boundary of e ₂ , satisfying K _b2 >|e ₂ | _max , |e ₂ | _max is the maximum value of |e ₂ |; K _b3 is the boundary of e ₃ , satisfying K _b3 >|e ₃ | _max , |e ₃ | _max is the maximum value of |e ₃ |; α ₂ , α ₃ are virtual control variables, and their expressions are:

Among them, k ₂₁ , k ₃₁ are positive numbers;

Substituting equation (19) into equation (18), we get:

2.7 Designing Lyapunov functions V ₂₂ , V ₃₂

Solving the first derivative of equation (21), we get:

in

Substituting equations (23) and (6) into equation (22), we get:

2.8 Design u _x , u _y respectively by formulas (24) and (25):

Among them, k ₂₂ , k ₃₂ are positive numbers;

2.9 Define the attitude angle tracking error and its first derivative:

Among them, j=4, 5, 6, x ₄ = φ, x ₅ = θ, x ₆ = ψ, x _4d represents the expected value of φ, x _5d represents the expected value of θ, x _6d represents the expected value of ψ, and e ₄ represents φ The tracking error of , e ₅ represents the tracking error of θ, and e ₆ represents the tracking error of ψ;

2.10 Design Constrained Lyapunov Functions and solve for its first derivative:

Among them, k _j is a positive number, K _bj is the boundary of e _j , satisfy K _bj >|e _j | _max , and |e _j | _max is the maximum value of |e _j |; α _j is the virtual control amount of the attitude angle, and its expression is:

Among them, k _j1 is a positive number;

Substituting equation (29) into equation (28), we get:

2.11 Design constraint Lyapunov function:

Solving the first derivative of equation (31), we get:

in

Substituting equations (33) and (6) into equations (32), we get:

2.12 Design τ _x , τ _y , τ _z by formulas (34), (35), (36) respectively:

Among them, k ₄₂ , k ₅₂ , and k ₆₂ are positive numbers;

Step 3, verify the stability of the quadrotor aircraft system, the process is as follows:

3.1 Substitute equation (16) into equation (15), we get:

3.2 Substitute equation (26) into equations (24) and (25) to get:

3.3 Substituting equation (37) into equations (34), (35), (36), we get

3.4 The quadrotor system is known to be stable through (38), (39), (40).

The invention is based on the limited output control method of the quadrotor aircraft based on the symmetric time-invariant tangent constraint Lyapunov function, which improves the transient performance of the system and reduces the overshoot and the arrival time.

The technical idea of the present invention is: for the dynamic system of the quadrotor aircraft, a method for limiting the output of the quadrotor aircraft based on the symmetric time-invariant tangent constraint Lyapunov function is designed. The design of the symmetric time-invariant tangent-constrained Lyapunov function is to ensure that the output of the system can be limited to a certain range, avoid excessive overshoot, and at the same time reduce the arrival time. Thereby improving the dynamic response performance of the quadrotor aircraft system.

The beneficial effects of the invention are: reducing the overshoot, reducing the arrival time and improving the transient performance.

Description of drawings

FIG. 1 is a schematic diagram of the position tracking effect of the present invention.

FIG. 2 is a schematic diagram of an attitude angle tracking effect of the present invention.

FIG. 3 is a schematic diagram of the input of the position controller of the present invention.

FIG. 4 is a schematic diagram of the input of the attitude angle controller of the present invention.

FIG. 5 is a schematic diagram of a control flow of the present invention.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings.

1-5, a method for limiting the output of a quadrotor aircraft, comprising the following steps:

Step 1, establish the dynamic model of the quadrotor aircraft system, set the initial value, sampling time and control parameters of the system, the process is as follows:

1.1 Determine the transition matrix T from the quadrotor system-based body coordinate system to the Earth-based inertial coordinate system:

Among them, φ, θ, ψ are the roll angle, pitch angle, and yaw angle of the quadrotor aircraft, respectively, indicating the rotation angle of the aircraft around each coordinate axis of the inertial coordinate system in turn;

1.2 The dynamic model of the quadrotor during translation is as follows:

Among them, x, y, z respectively represent the three positions of the quadrotor in the inertial coordinate system, U _f represents the input torque of the quadrotor, m is the mass of the quadrotor, g represents the acceleration of gravity,

Substitute equation (1) into equation (2) to get:

1.3 The dynamic model during the rotation of the quadrotor is:

Among them, τ _x , τ _y , τ _z respectively represent the moment components of each axis on the body coordinate system, I _xx , I _yy , I _zz respectively represent the components of the moment of inertia of each axis in the body coordinate system, × represents the cross product, ω _p represents the roll angular velocity, ω _q represents the pitch angular velocity, ω _r represents the yaw angular velocity, is the roll angular acceleration, represents the pitch angle acceleration, represents the yaw angular acceleration;

Considering that the aircraft is in a low-speed flight or hovering state, the attitude angle changes little, it is considered that So formula (4) is rewritten as:

Combining Equation (3) and Equation (5), the dynamic model of the quadrotor aircraft is obtained as:

Among them, u _x =cosφsinθcosψ+sinφsinψ,u _y =cosφsinθsinψ-sinφcosψ;

1.4 According to formula (6), the expected values of φ and θ are defined as:

Among them, φ _d is the expected signal value of φ, θ _d is the expected signal value of θ, and arcsin is the inverse sine function;

Step 2, at each sampling moment, calculate the position tracking error and its first derivative; calculate the attitude angle tracking error and its first derivative; design the position and attitude angle controller, the process is as follows:

2.1 Define the z tracking error and its first derivative:

Among them, z _d represents the desired signal of z;

2.2 Design Constrained Lyapunov Functions and solve for its first derivative:

Among them, K _b1 is the boundary of e ₁ , satisfying K _b1 >|e ₁ | _max , |e ₁ | _max is the maximum value of |e ₁ |, α ₁ is a virtual control quantity, and its expression is:

Among them, k ₁₁ is a positive number;

Substituting equation (10) into equation (9), we get:

2.3 Design the Lyapunov function V ₁₂ as:

Solving the first derivative of equation (12), we get:

in

Substituting equations (14) and (6) into equation (13), we get:

2.4 Design U _f :

Among them, k ₁₂ is a positive number;

2.5 Define the x and y tracking errors as e ₂ , e ₃ respectively, then there are:

e ₂ =xx _d , e ₃ =yy _d ,

where x _d , y _d represent the desired signals of x and y, respectively;

2.6 Design Constrained Lyapunov Functions Solving its first derivative separately, we get:

Among them, K _b2 is the boundary of e ₂ , satisfying K _b2 >|e ₂ | _max , |e ₂ | _max is the maximum value of |e ₂ |; K _b3 is the boundary of e ₃ , satisfying K _b3 >|e ₃ | _max , |e ₃ | _max is the maximum value of |e ₃ |; α ₂ , α ₃ are virtual control variables, and their expressions are:

Among them, k ₂₁ , k ₃₁ are positive numbers;

Substituting equation (19) into equation (18), we get:

2.7 Designing Lyapunov functions V ₂₂ , V ₃₂

Solving the first derivative of equation (21), we get:

in

Substituting equations (23) and (6) into equation (22), we get:

2.8 Design u _x , u _y respectively by formulas (24) and (25):

Among them, k ₂₂ , k ₃₂ are positive numbers;

2.9 Define the attitude angle tracking error and its first derivative:

Among them, j=4, 5, 6, x ₄ = φ, x ₅ = θ, x ₆ = ψ, x _4d represents the expected value of φ, x _5d represents the expected value of θ, x _6d represents the expected value of ψ, and e ₄ represents φ The tracking error of , e ₅ represents the tracking error of θ, and e ₆ represents the tracking error of ψ;

2.10 Design Constrained Lyapunov Functions and solve for its first derivative:

where k _j is a positive number, and K _bj is the boundary of e _j , satisfying K _bj >|e _j | _max , and |e _j | _max is the maximum value of |e _j |; α _j is the virtual control amount of the attitude angle, and its expression is:

where k _j1 is a positive number;

Substituting equation (29) into equation (28), we get:

2.11 Design constraint Lyapunov function:

Solving the first derivative of equation (31), we get:

in

Substituting equations (33) and (6) into equations (32), we get:

2.12 Design τ _x , τ _y , τ _z by formulas (34), (35), (36) respectively:

where k ₄₂ , k ₅₂ , and k ₆₂ are positive numbers;

Step 3, verify the stability of the quadrotor aircraft system, the process is as follows:

3.1 Substitute equation (16) into equation (15), we get:

3.2 Substitute equation (26) into equations (24) and (25) to get:

3.3 Substituting equation (37) into equations (34), (35), (36), we get

3.4 The quadrotor system is known to be stable through (38), (39), (40).

In order to verify the feasibility of the proposed method, the present invention provides the simulation results of the control method on the MATLAB platform:

The parameters are given as follows: in formula (2), m=1.1kg, g=9.81N/kg; in formula (4), Ixx= _1.22kg · ^m2 , Iyy= _1.22kg · ^m2 , _Izz =2.2 kg·m ² ; in formula (8), formula (17) and formula (27), z _d =1, x _d =1, y _d =1, ψ _d =0.5; formula (10), formula (19) and In formula (29), k ₁₁ =2, k ₂₁ =2, k ₃₁ =2,k ₄₁ =2,k ₅₁ =2,k ₆₁ =2; in formula (16), formula (26) and formula (37) k ₁₂ =2,k ₂₂ =2,k ₃₂ =2,k ₄₂ =2,k ₅₂ =2,k ₆₂ =2; formula (9), formula (18) and formula (28) k _b1 =1.5,k _b2 = 1.5, k _b3 =1.5, k _b4 =2, k _b5 =2, k _b6 =2.

It can be seen from Figure 1 and Figure 2 that the system has good transient characteristics, the arrival time is 4.23 seconds, and the overshoot is 0.0245.

To sum up, the output-limited control method of quadrotor aircraft based on symmetric time-invariant tangent constrained Lyapunov function can effectively improve the transient performance of quadrotor aircraft system.

What has been described above is the excellent optimization effect exhibited by an embodiment of the present invention. Obviously, the present invention is not limited to the above-mentioned embodiment. It can be implemented in various variations.

Claims (1)

1. a kind of quadrotor exports constrained control method, which comprises the following steps:

Step 1, the dynamic model for establishing quadrotor system sets initial value, sampling time and the control ginseng of system 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 respectively indicate three positions of the quadrotor under inertial coodinate system, U_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 are as follows:

Wherein, τ_x, τ_y, τ_zRespectively represent the moment components of each axis on body coordinate system, I_xx, I_yy, I_zzRespectively indicate body seat 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, attitude angle variation is smaller, it is believed thatTherefore formula (4) is rewritten are as follows:

Joint type (3) and formula (5), obtain the kinetic model of quadrotor are as follows:

Wherein u_x=cos φ sin θ cos ψ+sin φ sin ψ, u_y=cos φ sin θ sin ψ-sin φ cos ψ；

1.4, according to formula (6), define φ, the desired value of θ are as follows:

Wherein, φ_dFor the expected signal value of φ, θ_dFor θ expected signal value, 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 error and its first derivative:

Wherein z_dIndicate the desired signal of z；

2.2 design constraint liapunov functionsAnd solve its first derivative:

Wherein, K_b1For e₁Boundary, meet K_b1> | e₁|_max, | e₁|_maxFor | e₁| maximum value,α₁Virtually to control Amount processed, expression formula are as follows:

Wherein, k₁₁For normal number；

Formula (10) are substituted into formula (9), are obtained:

2.3 design liapunov function V₁₂Are as follows:

The first derivative of solution formula (12), obtains:

Wherein

Formula (14) and formula (6) are substituted into formula (13), obtained:

2.4 design U_f:

Wherein, k₁₂For normal number；

2.5 define x, and y tracking error is respectively e₂, e₃, then have:

Wherein, x_d, y_dRespectively indicate x, the desired signal of y；

2.6 design constraint liapunov functionsOne is solved respectively Order derivative obtains:

Wherein, K_b2For e₂Boundary, meet K_b2> | e₂|_max, | e₂|_maxFor | e₂| maximum value；K_b3For e₃Boundary, meet K_b3 > | e₃|_max, | e₃|_maxFor | e₃| maximum value；α₂, α₃For virtual controlling amount, expression formula Are as follows:

Wherein, k₂₁, k₃₁For normal number；

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

2.7 design liapunov function V₂₂, V₃₂

The first derivative of solution formula (21), obtains:

Wherein

By formula (23), (6) substitute into formula (22), respectively:

2.8 by formula (24), and (25) separately design u_x, u_y:

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, 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；

2.10 design constraint liapunov functionAnd solve its first derivative:

Wherein, k_jFor normal number, K_bjFor e_jBoundary, meet K_bj> | e_j|_max, | e_j|_maxFor | e_j| maximum value；α_jFor the virtual controlling amount of attitude angle, expression formula are as follows:

Wherein, k_j1For normal number；

Formula (29) are substituted into formula (28), are obtained:

2.11 design constraint liapunov functions:

The first derivative of solution formula (31), obtains:

Wherein

Formula (33) and formula (6) are substituted into formula (32), respectively:

2.12 by formula (34), and (35), (36) separately design τ_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 (16) are substituted into formula (15) by 3.1, are obtained:

Formula (26) are substituted into formula (24), (25) by 3.2, are obtained:

3.3 wushu (37) substitute into formula (34), (35), (36), obtain

3.4 by (38), and (39), (40) know that quadrotor system is stable.