CN107942672A - It is a kind of based on it is symmetrical when constant obstacle liapunov function the limited backstepping control method of quadrotor output - Google Patents
It is a kind of based on it is symmetrical when constant obstacle liapunov function the limited backstepping control method of quadrotor output Download PDFInfo
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Abstract
It is a kind of based on it is symmetrical when constant obstacle liapunov function the limited backstepping control method of quadrotor output, for the dynamic system of quadrotor, constant obstacle liapunov function when selecting a kind of symmetrical, design it is a kind of based on it is symmetrical when constant obstacle liapunov function the limited backstepping control method of quadrotor output.The design of constant obstacle liapunov function is to ensure that the output of system can be limited in certain scope, avoid excessive overshoot, while can also reduce arrival time when symmetrical.So as to improve the dynamic response performance of quadrotor system.The present invention provide it is a kind of based on it is symmetrical when constant obstacle liapunov function the limited backstepping control method of quadrotor output, make system that there is preferable dynamic response process.
Description
Technical Field
The invention relates to a four-rotor aircraft output limited backstepping control method based on a symmetric time invariant barrier Lyapunov function, which enables a four-rotor aircraft system to have a better dynamic response process.
Background
The four-rotor aircraft is one of the rotor aircraft, and attracts wide attention of universities, research institutions and companies at home and abroad due to the advantages of small volume, good maneuverability, simple design, low manufacturing cost and the like. However, since quad-rotor aircraft are small and lightweight and are susceptible to external disturbances during flight, how to achieve high performance motion control of quad-rotor aircraft has become a hot issue. For the control problem of the four-rotor aircraft, many control methods exist, such as PID control, active disturbance rejection control, sliding mode control, backstepping control, and the like.
Among them, the backstepping control has been widely applied to nonlinear systems, and its advantages include fast response speed, convenient implementation, robustness to system uncertainty and external interference, etc. The traditional backstepping control only considers the steady-state performance of the four-rotor aircraft and does not pay more attention to the transient response performance of the four-rotor aircraft. Therefore, the conventional backstepping control method greatly hinders the practical application of the four-rotor aircraft system. To solve this problem, a backstepping control method based on the barrier lyapunov function is proposed, which in practical situations can effectively improve the transient performance of a four-rotor aircraft system.
Disclosure of Invention
In order to improve the transient performance of a four-rotor aircraft system, the invention provides a four-rotor aircraft output limited step control method based on a symmetric time invariant barrier Lyapunov function, so that the overshoot and the overshoot time are reduced, and the four-rotor aircraft system has good dynamic response performance.
The technical scheme proposed for solving the technical problems is as follows:
a four-rotor aircraft output limited backstepping control method based on a symmetric time invariant barrier Lyapunov function comprises the following steps:
step 1, establishing a dynamic model of a four-rotor aircraft system, and setting initial values, sampling time and related control parameters of the system, wherein the process is as follows:
1.1 determining a transfer matrix T from a body coordinate system based on a quad-rotor aircraft system to an inertial coordinate based on the earth:
wherein phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle of the four-rotor aircraft and represent angles of the unmanned aerial vehicle rotating around each coordinate axis of an inertial coordinate system in sequence;
1.2 dynamic model in the translational process of the four-rotor aircraft is as follows:
wherein x, y and z respectively represent three positions of the four-rotor aircraft under an inertial coordinate system, U f Representing the input torque of the quad-rotor aircraft, m being the mass of the quad-rotor aircraft, g representing the gravitational acceleration,
substituting the formula (1) into the formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein τ is x ,τ y ,τ z Respectively representing the moment components, I, of the axes in the coordinate system of the machine body xx ,I yy ,I zz Respectively representing the components of the moment of inertia of each axis in the body coordinate system, x representing the cross product, ω p Representing roll angular velocity, ω q Representing pitch angle velocity, ω r Which is indicative of the yaw rate of the vehicle,which is indicative of the roll angular acceleration,a pitch angle acceleration is represented and,representing yaw angular acceleration;
considering that the unmanned aerial vehicle is in a low-speed flight or hovering state, the attitude angle change is small, and the unmanned aerial vehicle is considered to beThus, equation (4) is rewritten as:
the dynamic model of the four-rotor aircraft is obtained by combining the vertical type (3) and the formula (5):
wherein u x ＝cosφsinθcosψ+sinφsinψ,u y ＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, the desired value of θ is:
wherein phi d Is the desired signal value of phi, theta d For the theta desired signal value, arcsin is an arcsine function;
step 2, calculating a position tracking error and a first derivative thereof at each sampling moment; calculating an attitude angle tracking error and a first derivative thereof; designing a position and attitude angle controller, and the process is as follows:
2.1 defines the z tracking error and its first derivative:
wherein z is d A desired signal representing z;
2.2 design obstacle Lyapunov functionAnd solving for its first derivative:
wherein K b1 Is e 1 Satisfies K b1 >|e 1 | max ，|e 1 | max Is | e 1 The maximum value of the value of l,α 1 the expression is the virtual control quantity:
wherein k is 11 Is a normal number;
substituting formula (10) for formula (9) yields:
2.3 design of Lyapunov function V 12 Comprises the following steps:
solving the first derivative of equation (12) to obtain:
wherein
Formula (13) is substituted with formula (14) and formula (6), yielding:
2.4 design U f ：
Wherein k is 12 Is a normal number;
2.5 define x, y tracking errors as e, respectively 2 ,e 3 Then, there are:
wherein x d ,y d Respectively representing expected signals of x and y;
2.6 design obstacle Lyapunov functionRespectively solving the first derivative to obtain:
wherein K b2 Is e 2 Boundary of (1), satisfies K b2 >|e 2 | max ，|e 2 | max Is | e 2 The maximum value of |; k b3 Is e 3 Boundary of (1), satisfies K b3 >|e 3 | max ，|e 3 | max Is | e 3 The maximum value of |;α 2 ,α 3 the expression is the virtual control quantity:
wherein k is 21 ,k 31 Is a normal number;
substituting equation (19) into equation (18) yields:
2.7 design Lyapunov function V 22 ,V 32
Solving the first derivative of equation (21) yields:
wherein
Substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectively x ,u y ：
Wherein k is 22 ,k 32 Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j =4,5,6,x 4 ＝φ,x 5 ＝θ,x 6 ＝ψ，x 4d Denotes the expected value, x, of phi 5d Denotes the desired value, x, of theta 6d Indicating the expected value, e, of 4 Denotes the tracking error of phi, e 5 Denotes the tracking error of theta, e 6 A tracking error representing ψ;
2.10 design obstacle Lyapunov functionAnd solving its first derivative:
wherein k is j Being a normal number, K bj Is e j Boundary of (2) satisfyingIs | e j The maximum value of |;α j the expression of the virtual control quantity of the attitude angle is as follows:
wherein k is j1 Is a normal number;
substituting equation (29) into equation (28) yields:
2.11 design obstacle Lyapunov function:
solving the first derivative of equation (31) yields:
wherein
Substituting the formula (33) and the formula (6) into the formula (32) respectively yields:
2.12 design τ by equations (34), (35), and (36), respectively x ,τ y ,τ z ：
Wherein k is 42 ,k 52 ,k 62 Is a normal number.
Further, the method comprises the following steps:
step 3, verifying the stability of the four-rotor aircraft system;
3.1 substituting formula (16) for formula (15) to obtain:
3.2 substituting formula (26) for formula (24), (25) to obtain:
3.3 substituting formula (37) for formulae (34), (35) and (36) to obtain
3.4 the four-rotor aircraft system is stable as can be seen from (38), (39), (40).
The four-rotor aircraft output limited backstepping control method based on the symmetric time invariant barrier Lyapunov function improves the transient performance of the system and reduces overshoot and arrival time.
The technical conception of the invention is as follows: aiming at a dynamics system of a four-rotor aircraft, a four-rotor aircraft output limited backstepping control method based on a symmetric time invariant barrier Lyapunov function is designed. The design of the symmetric invariant barrier Lyapunov function is to ensure that the output of the system can be limited within a certain range, avoid excessive overshoot, and reduce the arrival time. Thereby improving the dynamic response performance of the four-rotor aircraft system.
The invention has the advantages that: the overshoot is reduced, the arrival time is reduced, and the transient performance is improved.
Drawings
FIG. 1 is a schematic diagram of the position tracking effect of the present invention.
Fig. 2 is a schematic view of the tracking effect of the attitude angle of the present invention.
FIG. 3 is a schematic diagram of the position controller input of the present invention.
FIG. 4 is a schematic diagram of the input of the attitude angle controller according to the present invention.
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 four-rotor aircraft output limited backstepping control method based on a symmetric time-invariant barrier lyapunov function includes the following steps:
step 1, establishing a dynamic model of a four-rotor aircraft system, and setting initial values, sampling time and related control parameters of the system, wherein the process is as follows:
1.1 determining a transfer matrix T from a body coordinate system based on a four-rotor aircraft system to an inertial coordinate based on the earth:
phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle of the four-rotor aircraft and represent angles of the unmanned aerial vehicle rotating around each coordinate axis of an inertial coordinate system in sequence;
1.2 dynamic model in the translational process of the four-rotor aircraft is as follows:
wherein x, y and z respectively represent three positions of the four-rotor aircraft in an inertial coordinate system, U f Representing the input torque of the quad-rotor aircraft, m being the mass of the quad-rotor aircraft, g representing the gravitational acceleration,
substituting the formula (1) into the formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein tau is x ,τ y ,τ z Respectively representing the moment components of each axis on the coordinate system of the machine body, I xx ,I yy ,I zz Respectively representing the components of the moment of inertia of each axis in the coordinate system of the body, x represents the cross product, omega p Representing the roll angular velocity, ω q Representing pitch angle velocity, ω r Which is indicative of the yaw rate,it is indicated that the roll angular acceleration is,a pitch angle acceleration is represented and,representing yaw angular acceleration;
in view ofThe unmanned plane is generally in a low-speed flight or hovering state, the attitude angle change is small, and the unmanned plane is considered to be in a low-speed flight or hovering stateThus, equation (4) is rewritten as:
the dynamic model of the four-rotor aircraft is obtained by combining the vertical type (3) and the formula (5):
wherein u is x ＝cosφsinθcosψ+sinφsinψ,u y ＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, the desired value of θ is:
wherein phi d Is the desired signal value of phi, theta d For the theta desired signal value, arcsin is an arcsine function;
step 2, calculating a position tracking error and a first derivative thereof at each sampling moment; calculating an attitude angle tracking error and a first derivative thereof; designing a position and attitude angle controller, and the process is as follows:
2.1 defines the z tracking error and its first derivative:
wherein z is d A desired signal representing z;
2.2 design obstacle Lyapunov functionAnd solving for its first derivative:
wherein K b1 Is e 1 Satisfies K b1 >|e 1 | max ，|e 1 | max Is | e 1 The maximum value of the value of l,α 1 the expression is the virtual control quantity:
wherein k is 11 Is a normal number;
substituting formula (10) for formula (9) to obtain:
2.3 design Lyapunov function V 12 Comprises the following steps:
solving the first derivative of equation (12) to obtain:
wherein
Formula (13) is substituted with formula (14) and formula (6), yielding:
2.4 design U f ：
Wherein k is 12 Is a normal number;
2.5 define x, y tracking errors as e, respectively 2 ,e 3 Then, there are:
wherein x d ,y d Respectively representing expected signals of x and y;
2.6 design obstacle Lyapunov functionRespectively solving the first derivative to obtain:
wherein K b2 Is e 2 Boundary of (1), satisfies K b2 >|e 2 | max ，|e 2 | max Is | e 2 The maximum value of |; k b3 Is e 3 Boundary of (1), satisfies K b3 >|e 3 | max ，|e 3 | max Is | e 3 The maximum value of |;α 2 ,α 3 the expression is the virtual control quantity:
wherein k is 21 ,k 31 Is a normal number;
substituting equation (19) into equation (18) yields:
2.7 design Lyapunov function V 22 ,V 32
Solving the first derivative of equation (21) yields:
wherein
Substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectively x ,u y ：
Wherein k is 22 ,k 32 Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j =4,5,6,x 4 ＝φ,x 5 ＝θ,x 6 ＝ψ，x 4d Denotes the expected value, x, of phi 5d Denotes the desired value, x, of theta 6d Indicating the desired value, e, of 4 Indicating a tracking error of phi, e 5 Denotes the tracking error of theta, e 6 A tracking error representing ψ;
2.10 design obstacle Lyapunov functionAnd solving for its first derivative:
wherein k is j Being a normal number, K bj Is e j Is satisfied withIs | e j The maximum value of |;α j the expression is a virtual control quantity of the attitude angle, and is as follows:
wherein k is j1 Is a normal number;
substituting equation (29) into equation (28) yields:
2.11 design obstacle Lyapunov function:
solving the first derivative of equation (31) yields:
whereinSubstituting formula (33) and formula (6) into formula (32) yields:
2.12 design of τ by equations (34), (35), (36), respectively x ,τ y ,τ z ：
Wherein k is 42 ,k 52 ,k 62 Is a normal number;
step 3, verifying the stability of the four-rotor aircraft system;
3.1 substitution of formula (16) for formula (15) gives:
3.2 substituting formula (26) for formula (24), (25) to obtain:
3.3 substituting formula (37) for formulae (34), (35) and (36) to obtain
3.4 the quad-rotor aircraft system is stable as shown by (38), (39), (40).
In order to verify the feasibility of the method, the invention provides the simulation result of the control method on an MATLAB platform:
the parameters are given as follows: m =1.1kg, g =9.81n/kg in formula (2); in the formula (4), I xx ＝1.22kg·m 2 ，I yy ＝1.22kg·m 2 ，I zz ＝2.2kg·m 2 (ii) a Z in the formulae (8), (17) and (27) d ＝0.5，x d ＝0.5，y d ＝0.5，ψ d =0.5; k in the formulae (10), (19) and (29) 11 ＝1,k 21 ＝1,k 31 ＝1,k 41 ＝1,k 51 ＝1,k 61 =1; k in the formulae (16), (26) and (37) 12 ＝1,k 22 ＝1,k 32 ＝1,k 42 ＝1,k 52 ＝1,k 62 =1; formula (9), formula (18) and formula (28) k b1 ＝1.87,k b2 ＝1.87,k b3 ＝1.87,k b4 ＝1.87,k b5 ＝1.87,k b6 ＝1.87。
As can be seen from fig. 1 and 2, the system has good transient characteristics, with an arrival time of 3.02 seconds and an overshoot of 0.0008.
In conclusion, the four-rotor aircraft output limited backstepping control method based on the symmetric time invariant barrier Lyapunov function can effectively improve the transient performance of a four-rotor aircraft system.
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 (2)
1. A four-rotor aircraft output limited backstepping control method based on a symmetric time invariant barrier Lyapunov function is characterized by comprising the following steps:
step 1, establishing a dynamic model of a four-rotor aircraft system, and setting initial values, sampling time and related control parameters of the system, wherein the process is as follows:
1.1 determining a transfer matrix T from a body coordinate system based on a quad-rotor aircraft system to an inertial coordinate based on the earth:
phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle of the four-rotor aircraft and represent angles of the unmanned aerial vehicle rotating around each coordinate axis of an inertial coordinate system in sequence;
1.2 dynamic model in the translational process of the four-rotor aircraft is as follows:
wherein x, y, z respectively represent a quad-rotor aircraftThree positions in the inertial frame, U f Representing the input torque of the quad-rotor aircraft, m being the mass of the quad-rotor aircraft, g representing the gravitational acceleration,
substituting the formula (1) into the formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein tau is x ,τ y ,τ z Respectively representing the moment components, I, of the axes in the coordinate system of the machine body xx ,I yy ,I zz Respectively representing the components of the moment of inertia of each axis in the body coordinate system, x representing the cross product, ω p Representing the roll angular velocity, ω q Representing pitch angle velocity, ω r Which is indicative of the yaw rate,which is indicative of the roll angular acceleration,the pitch angular acceleration is represented as,representing yaw angular acceleration;
considering that the unmanned aerial vehicle is in a low-speed flight or hovering state, the attitude angle change is small, and the unmanned aerial vehicle is considered to be in a low-speed flight or hovering stateThus, formula (4) is rewritten as:
the dynamic model of the four-rotor aircraft is obtained by combining the vertical type (3) and the formula (5):
wherein u x ＝cosφsinθcosψ+sinφsinψ,u y ＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, θ desired as:
wherein phi d Is the desired signal value of phi, theta d For the theta desired signal value, arcsin is an arcsine function;
step 2, calculating a position tracking error and a first derivative thereof at each sampling moment; calculating an attitude angle tracking error and a first derivative thereof; designing a position and attitude angle controller, and the process is as follows:
2.1 defines the z tracking error and its first derivative:
wherein z is d A desired signal representing z;
2.2 design obstacle Lyapunov functionAnd solving for its first derivative:
wherein K is b1 Is e 1 Boundary of (1), satisfies K b1 >|e 1 | max ，|e 1 | max Is | e 1 The maximum value of l is the sum of,α 1 the expression is the virtual control quantity:
wherein k is 11 Is a normal number;
substituting formula (10) for formula (9) to obtain:
2.3 design of Lyapunov function V 12 Comprises the following steps:
solving the first derivative of equation (12) yields:
wherein
Substituting formula (14) and formula (6) for formula (13) yields:
2.4 design U f ：
Wherein k is 12 Is a normal number;
2.5 define x, y tracking errors as e, respectively 2 ,e 3 Then, there are:
wherein x d ,y d Respectively representing expected signals of x and y;
2.6 design obstacle Lyapunov functionRespectively solving the first derivative to obtain:
wherein K b2 Is e 2 Boundary of (1), satisfies K b2 >|e 2 | max ，|e 2 | max Is | e 2 The maximum value of |; k b3 Is e 3 Boundary of (1), satisfies K b3 >|e 3 | max ，|e 3 | max Is | e 3 The maximum value of |;α 2 ,α 3 the expression is the virtual control quantity:
wherein k is 21 ,k 31 Is a normal number;
substituting equation (19) into equation (18) yields:
2.7 design Lyapunov function V 22 ,V 32
Solving the first derivative of equation (21) to obtain:
wherein
Substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectively x ,u y ：
Wherein k is 22 ,k 32 Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j =4,5,6,x 4 ＝φ,x 5 ＝θ,x 6 ＝ψ，x 4d Denotes the expected value, x, of phi 5d Denotes the desired value, x, of theta 6d Indicating the desired value, e, of 4 Denotes the tracking error of phi, e 5 Denotes the tracking error of θ, e 6 A tracking error representing ψ;
2.10 design obstacle Lyapunov functionAnd solving its first derivative:
wherein k is j Being a normal number, K bj Is e j Boundary of (1), satisfies K bj >|e j | max ，|e j | max Is | e j The maximum value of |;the expression is a virtual control quantity of the attitude angle, and is as follows:
wherein k is j1 Is a normal number;
substituting equation (29) into equation (28) yields:
2.11 design obstacle Lyapunov function:
solving the first derivative of equation (31) yields:
wherein
Substituting formula (33) and formula (6) into formula (32) yields:
2.12 design τ by equations (34), (35), and (36), respectively x ,τ y ,τ z ：
Wherein k is 42 ,k 52 ,k 62 Is a normal number.
2. A method of controlling limited backstepping of a four-rotor aircraft output based on the symmetric time-invariant barrier lyapunov function according to claim 1, further comprising the steps of:
step 3, verifying the stability of the four-rotor aircraft system;
3.1 substitution of formula (16) for formula (15) gives:
3.2 substituting formula (26) for formula (24), (25) to obtain:
3.3 substituting formula (37) for formula (34), (35) and (36) to obtain
3.4 the quad-rotor aircraft system is stable as shown by (38), (39), (40).
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| CN111679684A (en) * | 2020-06-16 | 2020-09-18 | 江苏师范大学 | Four-rotor unmanned aerial vehicle backstepping control method with input time lag |
| CN112327897A (en) * | 2020-11-04 | 2021-02-05 | 江苏师范大学 | Four-rotor unmanned aerial vehicle attitude control method with input dead zone |
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| CN108845493A (en) * | 2018-08-21 | 2018-11-20 | 曲阜师范大学 | The set time tracking and controlling method of mechanical arm system with output constraint |
| CN108845493B (en) * | 2018-08-21 | 2021-04-27 | 曲阜师范大学 | Fixed time tracking control method for mechanical arm system with output constraint |
| CN111679684A (en) * | 2020-06-16 | 2020-09-18 | 江苏师范大学 | Four-rotor unmanned aerial vehicle backstepping control method with input time lag |
| CN112327897A (en) * | 2020-11-04 | 2021-02-05 | 江苏师范大学 | Four-rotor unmanned aerial vehicle attitude control method with input dead zone |
| CN112327897B (en) * | 2020-11-04 | 2022-07-29 | 江苏师范大学 | Quad-rotor unmanned aerial vehicle attitude control method with input dead zone |
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Application publication date: 20180420 Assignee: Tongxiang Tujian Intelligent Technology Co.,Ltd. Assignor: JIANG University OF TECHNOLOGY Contract record no.: X2023980037543 Denomination of invention: A Output Constrained Backstepping Control Method for Quad Rotorcraft Based on Symmetric Time Invariant Barrier Lyapunov Function Granted publication date: 20200221 License type: Common License Record date: 20230705 |