CN107561931A - Nonlinear sliding mode pose control method of quadrotor aircraft based on single exponential function - Google Patents
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Abstract
A nonlinear sliding mode pose control method of a quadrotor unmanned aerial vehicle system based on a single exponential function is designed by utilizing the nonlinear sliding mode control method aiming at the quadrotor unmanned aerial vehicle system containing a dynamic execution mechanism. The design of the sliding mode surface is to ensure the rapid and stable convergence of the system. In addition, the sliding mode surface is designed by adopting a nonlinear function, so that the robustness and the tracking precision of the system can be improved. The invention provides a nonlinear sliding mode pose control method of a four-rotor aircraft based on a single exponential function, and the method can be used for realizing the rapid and stable control of a system.
Description
Technical Field
The invention relates to a nonlinear sliding mode pose control method of a four-rotor aircraft based on a single exponential function, which is used for meeting the performance requirement of the four-rotor unmanned aerial vehicle on the rapid and accurate tracking of reference input.
Background
An Unmanned Aerial Vehicle (UAV) is an aircraft that achieves autonomous flight by remote control or based on its own sensors. With the continuous maturation of the technology, the method is applied to many fields of civil use and military use. Unmanned aerial vehicle can divide into fixed wing and rotor two kinds, and fixed wing unmanned aerial vehicle's advantage is energy efficiency height, and consequently the flight distance is all longer with the time, and small-size glider also controls more easily when the wind speed is not big, but when they were designed into unmanned form, the operability was relatively poor. Compared with the prior art, the rotor unmanned aerial vehicle has strong operability and mobility, can conveniently complete take-off, landing and other actions, and has lower requirements on the working environment. The four-rotor unmanned aerial vehicle is taken as one of the rotor unmanned aerial vehicles, and attracts wide attention of domestic and foreign universities, research institutions and companies due to the advantages of small size, good maneuverability, simple design, low manufacturing cost and the like. The rotor unmanned aerial vehicle is very suitable for monitoring, reconnaissance and other civil and military fields. In the civil field, the rotor unmanned aerial vehicle is mainly applied to disaster relief, ground monitoring, high-altitude aerial photography and the like; because its concealment is high, the good reliability also is used for military fields such as battlefield control, military reconnaissance. Unmanned aerial vehicle research is also a major hotspot in foreign countries, wherein the united states aeronautics and astronautics authority (NASA) has developed ten-rotor unmanned planes for strategic detection and scientific research, and also developed a technology that enables large unmanned planes and manned aircraft to fly safely in national airspace together; the research of unmanned aerial vehicles in China is also highly regarded, and the fifth part of high-end equipment planned by thirteen five is mentioned in item 18: the industrialization of a stem-branch airplane, a helicopter, a general airplane and an unmanned aerial vehicle is promoted. Therefore, the unmanned aerial vehicle has extremely high strategic, scientific research and commercial values for the research of the unmanned aerial vehicle.
Sliding mode control is considered to be an effective robust control method in solving system uncertainty and external disturbances. The sliding mode control method has the advantages of simple algorithm, high response speed, strong robustness to external noise interference and parameter perturbation and the like. Therefore, the sliding mode control method is widely applied to various fields. Compared with the traditional linear sliding mode control, the nonlinear sliding mode control has the advantages that the performance requirement of rapid tracking is met, the synchronous control precision is higher, and the robustness of the system is higher. Therefore, the nonlinear sliding mode is utilized to control the quad-rotor unmanned aerial vehicle system, and the method has important theoretical and practical significance.
Disclosure of Invention
In order to meet the performance requirement of the quad-rotor unmanned aerial vehicle on the rapid reference input tracking, and simultaneously have higher synchronous control precision, so that the system has higher robustness, the invention provides the nonlinear sliding mode pose control method of the quad-rotor unmanned aerial vehicle based on the single exponential function, so that the robustness and the synchronous control precision of the system are enhanced, and the rapid and stable convergence of the system is ensured.
The technical scheme proposed for solving the technical problems is as follows:
a nonlinear sliding mode pose control method of a quadrotor aircraft based on a single exponential function comprises the following steps:
step 1, establishing a dynamic model of a quad-rotor unmanned aerial vehicle system, initializing system state, sampling time and control parameters, and carrying out the following process:
1.1 the kinetic model of the quad-rotor unmanned aerial vehicle system is expressed in the form:
wherein, x, y, z represent unmanned aerial vehicle in the position of three coordinate axis under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and F represents the external force that closes that acts on unmanned aerial vehicle, and the resultant force U that produces including the gravity mg that unmanned aerial vehicle receives and four rotors F And T is a transfer matrix from a body coordinate system to an inertial coordinate system, and the expression form is as follows:
T＝[T 1 T 2 T 3 ] (2)
1.2 the moment balance equation in the rotation process of the unmanned aerial vehicle is as follows:
wherein, tau x 、τ y 、τ z Respectively represent each axial moment component I on the coordinate system of the machine body xx 、Ι yy 、Ι zz Respectively representing the rotational inertia components of each shaft on a body coordinate system, x represents cross product, l, m and n respectively represent the attitude angular velocity components of each shaft on the body coordinate system,respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the unmanned aerial vehicle is generally in a low-speed flight or hovering state, the attitude angle change is small and can be setBecause of the influence of measurement noise, power supply variation and external interference, and system parameters and states in the formulas (1) and (3) cannot be accurately obtained, the dynamic model of the unmanned aerial vehicle is expressed as:
wherein Respectively representing model uncertainty and external interference terms;
1.3, according to the formula (4), decoupling calculation is carried out on the position posture relation, and the result is as follows:
wherein arcsin is an arc sine function, and arctan is an arc tangent function;are each theta 1 ，θ 2 ，θ 3 The expected value of (d);
after decoupling calculation, the position and the attitude angle are independent respectively, and a position controller and an attitude angle controller are designed respectively by being divided into two subsystems, so that a clear idea is provided for a control strategy;
considering that the position and attitude angle equations are both of a second-order multiple-input multiple-output nonlinear system, and the attitude angle equation is more complex, equation (4) is expressed in the following form for the convenience of designing and describing the controller:
wherein X 1 ＝[x y z θ 1 θ 2 θ 3 ] T ,
B(X)＝[1 1 1 b 1 b 2 b 3 ] T ,U＝[U x U y U z τ x τ y τ z ] T A according to the model of the aircraft 11 ＝0 6*6 ，A 12 ＝I 6*6 ，
That is, formula (6) is equivalent to
Step 2, designing a required sliding mode surface based on a quadrotor unmanned aerial vehicle system with unknown parameters, wherein the process is as follows:
defining the system state tracking error as:
e＝X d -X (8)
whereinAs indicated in the form of a conductive desired signal,expressed as a conductive real signal, then the first and second differentials of equation (8) are expressed as:
defining the nonlinear sliding mode surface as:
wherein F is selected such that (A) 11 -A 12 T F) A constant with a stable eigenvalue and a pole with less damping; Ψ (y) is a nonlinear function that varies in dependence on the output to vary the damping of the system, and has a value in the range of [ - β,0 [ - β [ ]]Where β is a normal number, and therefore, where Ψ (y) takes the following single exponential form:
α is a normal number, P is a positive array and satisfies:
P(A 11 -A 12 T F) T +(A 11 -A 12 T F)P＝-W (13)
wherein W is a positive definite matrix;
when the slip form surface s =0, it is obtained according to equation (11):
the united-standing (7 a) and equation (14) in conjunction with the sliding surface model write the following system:
in order to prove the stability of the sliding mode surface, the stability of the formula (15) needs to be proved, and the lyapunov function is designed for the formula (15):
definition ofThen the
Since Ψ (y) <0, therefore
Because of W>, 0, then obtainThe system represented by formula (15) is stable;
and 3, designing a nonlinear sliding mode controller based on a quad-rotor unmanned aerial vehicle system according to a sliding mode control theory and a single exponential nonlinear function, wherein the process is as follows:
3.1 considering equation (7), the nonlinear sliding mode controller is designed to:
wherein K is a normal number, determines the convergence speed of the slip form surface, and
3.2 design Lyapunov function:
the derivation of equation (11) yields:
the derivation of equation (21) and the substitution of equation (22) yields:
according to the formula (8)
WhereinThe derivative is zero because the desired value is set to a constant; therefore, it is possible to
By substituting formula (20) for formula (25)
The system is determined to be stable.
The invention designs a nonlinear sliding mode pose control method of a quadrotor aircraft based on a single exponential type function based on nonlinear function and sliding mode control, realizes stable control of a system, enhances the precision of sliding mode control, and ensures rapid and stable convergence of the system.
The technical conception of the invention is as follows: aiming at a four-rotor unmanned aerial vehicle system with a dynamic execution mechanism, a nonlinear sliding mode pose control method of a four-rotor aircraft based on a single exponential function is designed by utilizing a nonlinear sliding mode control method. The design of the sliding mode surface is to ensure the rapid and stable convergence of the system. In addition, the sliding mode surface is designed by adopting a nonlinear function, so that the robustness and the tracking precision of the system can be improved. The invention provides a nonlinear sliding mode pose control method of a four-rotor aircraft based on a single exponential function, and the method can be used for realizing the rapid and stable control of a system.
The invention has the advantages that: the method meets the performance requirement of rapid tracking, has higher synchronous control precision, has higher robustness of the system, compensates the unmovable parameters of the system and uncertain disturbance items inside and outside, and realizes rapid and stable convergence.
Drawings
FIG. 1 is a position slip form surface k of the present invention 1 =1, attitude angle sliding mode face k 2 Schematic diagram of the position tracking effect when =10, x denotes the position of the x-axis, y denotes the position of the y-axis, and z denotes the position of the z-axis, where (a) denotes a linear sliding mode surface and (b) denotes a non-linear sliding mode surface;
FIG. 2 is a position slip form surface k of the present invention 1 =1, attitude angle sliding mode face k 2 A schematic diagram of tracking effects of a roll angle, a pitch angle and a yaw angle when the attitude angle is 10, wherein (a) represents a linear sliding mode surface, and (b) represents a nonlinear sliding mode surface;
FIG. 3 is a position slip form surface k of the present invention 1 =1, attitude angle sliding mode face k 2 Moment tracking diagram at time =10, [ tau ] x Representing the moment of the x-axis, τ y Representing moment of the y-axis, τ z Represents the moment of the z-axis, wherein (a) represents a linear sliding mode surface, and (b) represents a nonlinear sliding mode surface;
FIG. 4 is a position slip form surface k of the present invention 1 =1, attitude angle sliding mode surface k 2 A schematic of controller input at =10, x denotes controller input on the x-axis, y denotes controller input on the y-axis, and z denotes controller input on the z-axis, where (a) denotes a linear sliding mode surface and (b) denotes a non-linear sliding mode surface;
FIG. 5 is a control flow diagram of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1-5, a nonlinear sliding-mode pose control method for a quadrotor aircraft based on a single exponential function comprises the following steps:
step 1, establishing a dynamic model of a quad-rotor unmanned aerial vehicle system, initializing system state, sampling time and control parameters, and carrying out the following process:
1.1 the kinetic model of the quad-rotor unmanned aerial vehicle system is expressed in the form:
wherein, x, y, z represent unmanned aerial vehicle in the position of three coordinate axis under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and F represents the external force that closes that acts on unmanned aerial vehicle, and the resultant force U that produces including the gravity mg that unmanned aerial vehicle receives and four rotors F And T is a transfer matrix from a machine body coordinate system to an inertia coordinate system, and the expression form is as follows:
T＝[T 1 T 2 T 3 ] (2)
1.2 the moment balance equation in the rotation process of the unmanned aerial vehicle is as follows:
wherein, tau x 、τ y 、τ z Respectively represent each axial moment component I on the coordinate system of the machine body xx 、Ι yy 、Ι zz Respectively representing the rotational inertia components of each axis in the body coordinate system, x represents cross product, l, m, n respectively represent the attitude angular velocity components of each axis in the body coordinate system,respectively represent the organismAttitude angular acceleration components of each axis on the coordinate system;
considering that the unmanned aerial vehicle is generally in a low-speed flight or hovering state, the attitude angle change is small and can be setBecause of the influence of measurement noise, power supply variation and external interference, and system parameters and states in the formulas (1) and (3) cannot be accurately obtained, the dynamic model of the unmanned aerial vehicle is expressed as:
wherein Respectively representing model uncertainty and external interference terms;
1.3 according to the formula (4), decoupling calculation is carried out on the position and posture relation, and the result is as follows:
wherein arcsin is an arcsine function and arctan is an arctangent function;are each theta 1 ，θ 2 ，θ 3 The expected value of (a);
after decoupling calculation, the position and the attitude angle are independent respectively, and a position controller and an attitude angle controller are designed respectively by dividing the two subsystems, so that a clear idea is provided for a control strategy;
considering that the position and attitude angle equations are both of a second-order multiple-input multiple-output nonlinear system, and the attitude angle equation is more complex, equation (4) is expressed in the following form for the convenience of design and explanation of the controller:
wherein,
X 1 ＝[x y z θ 1 θ 2 θ 3 ] T ,
B(X)＝[1 1 1 b 1 b 2 b 3 ] T ,U＝[U x U y U z τ x τ y τ z ] T according to A corresponding to the model of the aircraft 11 ＝0 6*6 ，A 12 ＝I 6*6 ，
That is, formula (6) is equivalent to
Step 2, designing a required sliding mode surface based on a quad-rotor unmanned aerial vehicle system with unknown parameters, wherein the process is as follows:
defining the system state tracking error as:
e＝X d -X (8)
whereinAs indicated in the form of a conductive desired signal,expressed as a conductive real signal, then the first and second differentials of equation (8) can be expressed as:
defining the nonlinear sliding mode surface as:
wherein F is selected such that (A) 11 -A 12 T F) A constant with a stable eigenvalue and a pole with less damping; Ψ (y) is a nonlinear function that varies the damping of the system in dependence on the output variation, and has a range of [ - β,0 ]]Where β is a normal number, and therefore, where Ψ (y) takes the following single exponential form:
α is a normal number, P is a positive array and satisfies:
P(A 11 -A 12 T F) T +(A 11 -A 12 T F)P＝-W (13)
wherein W is a positive array;
when the slip form surface s =0, it is obtained from equation (11):
the united vertical (7 a) and formula (14) in combination with the sliding surface model write the following system:
in order to prove the stability of the sliding mode surface, the stability of the formula (15) needs to be proved, and the lyapunov function is designed for the formula (15):
definition ofThen
Since Ψ (y) <0, therefore
Because of W>, 0, then obtainThe system represented by formula (15) is stable;
step 3, designing a nonlinear sliding mode controller based on a quad-rotor unmanned aerial vehicle system according to a sliding mode control theory and a single-exponential nonlinear function, wherein the process is as follows:
3.1 considering equation (7), the nonlinear sliding-mode controller is designed to:
wherein K is a normal number, determines the convergence speed of the slip form surface, and
3.2 design Lyapunov function:
the derivation of equation (11) yields:
the derivation is performed on equation (21) and equation (22) is substituted to obtain:
according to the formula (8)
WhereinThe derivative is zero because the desired value is set to a constant; therefore, it is not only easy to use
By substituting formula (20) for formula (25)
The system is determined to be stable.
In order to verify the effectiveness of the method, the invention provides a comparison between a Linear Sliding Surface (LSS) control method and a nonlinear sliding surface (NLSS) control method, wherein the LSS control method comprises the following steps:
for more effective comparison, all parameters of the system are consistent, and table 1 gives the system model parameters (parameter settings in equations (3) - (5)) while the system initial state is set to 0, and the position reference value is given as x d ＝2m，y d ＝2m，z d =2m; the yaw angle reference value is given as θ 3d =0.5rad. The nonlinear function parameters α =2, β =0.1, p =1, f =2.5. In the position of the slip form surface k 1 =1, k in attitude sliding mode plane 2 ＝10。
TABLE 1
Comparing the two control methods under the condition that the same parameter control is equal, we can find that both the two methods can ensure the system convergence and have certain control accuracy, but compared with the LSS method, the NLSS method has better rapidity, can control the position and attitude angle to reach the expected value more quickly, and ensures the rapid and stable convergence of the system.
In summary, compared with the LSS method, the NLSS method has better position tracking accuracy and smaller controller amplitude.
While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.
Claims (1)
1. A nonlinear sliding mode pose control method of a quadrotor aircraft based on a single exponential function is characterized by comprising the following steps: the control method comprises the following steps:
step 1, establishing a dynamic model of a quad-rotor unmanned aerial vehicle system, initializing system state, sampling time and control parameters, and carrying out the following process:
1.1 the kinetic model of the quad-rotor unmanned aerial vehicle system is expressed in the form:
wherein, x, y, z represent unmanned aerial vehicle in the position of three coordinate axis under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and F represents the external force that closes that acts on unmanned aerial vehicle, and the resultant force U that produces including the gravity mg that unmanned aerial vehicle receives and four rotors F And T is a transfer matrix from a machine body coordinate system to an inertia coordinate system, and the expression form is as follows:
T＝[T 1 T 2 T 3 ] (2)
1.2 the moment balance equation in the rotation process of the unmanned aerial vehicle is as follows:
wherein, tau x 、τ y 、τ z Respectively represent each axial moment component I on the coordinate system of the machine body xx 、Ι yy 、Ι zz Each representsThe rotation inertia component of each axis on the body coordinate system, x represents cross product, l, m, n represent the attitude angular velocity component of each axis on the body coordinate system respectively,respectively representing the attitude angular acceleration components of all axes on the body coordinate system;
setting upUnited (1) - (3), the dynamic model expression of unmanned aerial vehicle is:
wherein Respectively representing model uncertainty and external interference terms;
1.3, according to the formula (4), decoupling calculation is carried out on the position posture relation, and the result is as follows:
wherein arcsin is an arc sine function, and arctan is an arc tangent function;are each theta 1 ，θ 2 ，θ 3 The expected value of (d);
after decoupling calculation, the position and the attitude angle are independent respectively and are divided into two subsystems for designing a position controller and an attitude angle controller respectively;
formula (4) is represented as follows:
wherein, X 1 ＝[x y z θ 1 θ 2 θ 3 ] T ,
B(X)＝[1 1 1 b 1 b 2 b 3 ] T ,U＝[U x U y U z τ x τ y τ z ] T According to A corresponding to the model of the aircraft 11 ＝0 6*6 ，A 12 ＝I 6*6 ，A 21 ＝0 6*6 ,
That is, formula (6) is equivalent to
Step 2, designing a required sliding mode surface based on a quad-rotor unmanned aerial vehicle system with unknown parameters, wherein the process is as follows:
defining the system state tracking error as:
e＝X d -X (8)
whereinAs indicated in the form of a conductive desired signal,expressed as a conductive real signal, then the first differential and two of equation (8)The order differential is expressed as:
defining the nonlinear sliding mode surface as:
wherein F is selected such that (A) 11 -A 12 T F) A constant with a stable eigenvalue and a pole with less damping; Ψ (y) is a nonlinear function that varies the damping of the system in dependence on the output variation, and has a range of [ - β,0 ]]Where β is a normal number, and therefore, where Ψ (y) takes the following single-exponential form:
α is a positive constant, P is a positive array and satisfies:
P(A 11 -A 12 T F) T +(A 11 -A 12 T F)P＝-W (13)
wherein W is a positive array;
when the slip form surface s =0, it is obtained according to equation (11):
the united-standing (7 a) and equation (14) in conjunction with the sliding surface model write the following system:
the lyapunov function is designed for equation (15):
definition ofThen the
Since Ψ (y) <0, therefore
Because of W>, 0, then obtainThe system represented by formula (15) is stable;
step 3, designing a nonlinear sliding mode controller based on a quad-rotor unmanned aerial vehicle system according to a sliding mode control theory and a single-exponential nonlinear function, wherein the process is as follows:
3.1 considering equation (7), the nonlinear sliding mode controller is designed to:
wherein K is a normal number, determines the convergence speed of the slip form surface, and
3.2 design Lyapunov function:
the derivation of equation (11) yields:
the derivation of equation (21) and the substitution of equation (22) yields:
according to the formula (8)
WhereinThe derivative is zero because the desired value is set to a constant; therefore, it is not only easy to use
By substituting formula (20) for formula (25)
The system is determined to be stable.
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