CN108427278A - Based on it is symmetrical when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method - Google Patents

Based on it is symmetrical when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method Download PDF

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CN108427278A
CN108427278A CN201810212222.8A CN201810212222A CN108427278A CN 108427278 A CN108427278 A CN 108427278A CN 201810212222 A CN201810212222 A CN 201810212222A CN 108427278 A CN108427278 A CN 108427278A
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陈强
胡忠君
许昌源
吴春
胡轶
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Zhejiang University of Technology ZJUT
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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Abstract

It is a kind of based on it is symmetrical 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 symmetrical, design it is a kind of based on it is symmetrical 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 symmetrical 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 symmetrical when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method, make system have preferable dynamic response process.

Description

Four-rotor aircraft all-state limited control method based on symmetric time-varying exponential tangent composite constraint Lyapunov function
Technical Field
The invention relates to a four-rotor aircraft all-state limited control method based on a symmetric time-varying exponential tangent composite constraint 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 traditional backstepping control method greatly hinders the practical application of the four-rotor aircraft system. To solve this problem, a backstepping control method based on a constrained lyapunov function is proposed, which is effective in practical situations to improve the transient performance of a four-rotor aircraft system.
Disclosure of Invention
In order to overcome the defect of poor transient performance of the conventional four-rotor aircraft system, the invention provides a four-rotor aircraft all-state limited control method based on a symmetric time-varying exponential tangent composite constraint 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 all-state limited control method based on a symmetric time-varying exponential tangent composite constraint 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 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 aircraft 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, UfThe input torque of the four-rotor aircraft is represented, m is the mass of the four-rotor aircraft, and g represents the gravity acceleration;
substituting formula (1) into formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein, taux,τy,τzRespectively representing the moment components, I, of the axes in the coordinate system of the machine bodyxx,Iyy,IzzRespectively representing the components of the moment of inertia of each axis in the coordinate system of the body, x represents the cross product, omegapRepresenting roll angular velocity, ωqRepresenting pitch angle velocity, ωrWhich 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 aircraft is in a low-speed flight or hovering state, consider
Thus, equation (4) is rewritten as:
combining the vertical type (3) and the formula (5), and obtaining a dynamic model of the four-rotor aircraft as follows:
wherein u isx＝cosφsinθcosψ+sinφsinψ,uy＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, the desired value of θ is:
wherein phi isdIs the desired signal value of phi, thetadFor 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 isdA desired signal representing z;
2.2 design constraints Lyapunov functionAnd solving for its first derivative:
wherein, Kb1Is a time-varying parameter, satisfies Kb1>|e1|max，|e1|maxIs | e1The maximum value of l is the sum of,
α1the expression is the virtual control quantity: :
wherein k is11Is a normal number;
substituting formula (10) for formula (9) to obtain:
2.3 design Lyapunov function V12Comprises the following steps:
wherein, Ks1Is a time-varying parameter, satisfies Ks1>|s1|max，|s1|maxIs | s1The maximum value of |;
solving the first derivative of equation (12) to obtain:
wherein, the expression is as follows:
substituting formula (14) and formula (6) for formula (13) yields:
2.4 design Uf：
Wherein k is12Is a normal number;
2.5 define x, y tracking errors as e, respectively2,e3Then, there are:
wherein x isd,ydRespectively representing expected signals of x and y;
2.6 design constraints Lyapunov function Respectively solving the first derivative to obtain:
wherein, Kb2Is e2Boundary of (1), satisfies Kb2>|e2|max，|e2|maxIs | e2The maximum value of |; kb3Is e3Boundary of (1), satisfies Kb3>|e3|max，|e3|maxIs | e3The maximum value of |;
α2,α3for virtually controlling the quantity, tables thereofThe expression is as follows:
wherein k is21,k31Is a normal number;
substituting formula (19) for formula (18) to obtain:
2.7 design Lyapunov function V22,V32
Wherein, Ks2Is s is2Boundary of (1), satisfies Ks2>|s2|max，|s2|maxIs | s2The maximum value of |; wherein Ks3Is s is3Boundary of (1), satisfies Ks3>|s3|max，|s3|maxIs | s3The maximum value of |;
solving the first derivative of equation (21) to obtain:
wherein, the expression is as follows:
substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectivelyx,uy：
Wherein k is22,k32Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j is 4,5,6, x4＝φ,x5＝θ,x6＝ψ，x4dDenotes the expected value, x, of phi5dDenotes the desired value, x, of theta6dIndicating the desired value, e, of4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
2.10 design constraints Lyapunov functionAnd solving for its first derivative:
wherein k isjIs a normal number, KbjIs ejBoundary of (1), satisfies Kbj>|ej|max，|ej|maxIs | ejThe maximum value of |;
αjthe expression is a virtual control quantity of the attitude angle, and is as follows:
wherein k isj1Is a normal number;
substituting formula (29) for formula (28) to obtain:
2.11 design constraints Lyapunov function Vj2：
Wherein, KsjIs s isjBoundary of (1), satisfies Ksj>|sj|max，|sj|maxIs | sjThe maximum value of |;
solving the first derivative of equation (31) to obtain:
wherein, the expression is as follows:
substituting formula (32) with formula (33) and formula (6) respectively results in:
2.12 design of τ by equations (34), (35), (36), respectivelyx,τy,τz：
Wherein k is42,k52,k62Is a normal number;
step 3, verifying the stability of the four-rotor aircraft system, wherein the process is as follows:
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 formula (34), (35) and (36) to obtain:
3.4 by (38), (39), (40) the quad-rotor aircraft system is stable.
The four-rotor aircraft all-state limited control method based on the symmetric time-varying exponential tangent composite constraint 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 all-state limited control method based on a symmetric time-varying exponential tangent composite constraint Lyapunov function is designed. The design of the symmetric time-varying exponential tangent composite constraint Lyapunov function is to ensure that the state and 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 quad-rotor aircraft system.
The invention has the beneficial effects that: the full state is limited, 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 diagram illustrating the tracking effect of position and velocity according to the present invention.
Fig. 4 is a schematic diagram of the tracking effect of the attitude angular velocity according to the present invention.
FIG. 5 is a schematic diagram of the position controller input of the present invention.
FIG. 6 is a schematic diagram of the input of the attitude angle controller according to the present invention.
FIG. 7 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. 1-7, a four-rotor aircraft all-state limited control method based on a symmetric time-varying exponential tangent composite constraint 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 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 aircraft 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，UfThe input torque of the four-rotor aircraft is represented, m is the mass of the four-rotor aircraft, and g represents the gravity acceleration;
substituting formula (1) into formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein, taux,τy,τzRespectively representing the moment components, I, of the axes in the coordinate system of the machine bodyxx,Iyy,IzzRespectively representing the components of the moment of inertia of each axis in the coordinate system of the body, x represents the cross product, omegapRepresenting roll angular velocity, ωqRepresenting pitch angle velocity, ωrWhich 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 aircraft is in a low-speed flight or hovering state, the attitude angle change is small, and the attitude angle change is considered to be
Thus, equation (4) is rewritten as:
combining the vertical type (3) and the formula (5), and obtaining a dynamic model of the four-rotor aircraft as follows:
wherein u isx＝cosφsinθcosψ+sinφsinψ,uy＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, the desired value of θ is:
wherein phidIs the desired signal value of phi, thetadFor 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 isdA desired signal representing z;
2.2 design constraints Lyapunov functionAnd solving for its first derivative:
wherein, Kb1Is a time-varying parameter, satisfies Kb1>|e1|max，|e1|maxIs | e1The maximum value of l is the sum of,
α1the expression is the virtual control quantity:
wherein k is11Is a normal number;
substituting formula (10) for formula (9) to obtain:
2.3 design Lyapunov function V12Comprises the following steps:
wherein, Ks1Is a time-varying parameter, satisfies Ks1>|s1|max，|s1|maxIs | s1The maximum value of |;
solving the first derivative of equation (12) to obtain:
wherein, the expression is as follows:
substituting formula (14) and formula (6) for formula (13) yields:
2.4 design Uf：
Wherein k is12Is a normal number;
2.5 define x, y tracking errors as e, respectively2,e3Then, there are:
wherein x isd,ydRespectively representing expected signals of x and y;
2.6 design constraints Lyapunov function Respectively solving the first derivative to obtain:
wherein, Kb2Is e2Boundary of (1), satisfies Kb2>|e2|max，|e2|maxIs | e2The maximum value of |; kb3Is e3Boundary of (1), satisfies Kb3>|e3|max，|e3|maxIs | e3The maximum value of |;
α2,α3the expression is the virtual control quantity:
wherein k is21,k31Is a normal number;
substituting formula (19) for formula (18) to obtain:
2.7 design Lyapunov function V22,V32
Wherein, Ks2Is s is2Boundary of (1), satisfies Ks2>|s2|max，|s2|maxIs | s2The maximum value of |; wherein Ks3Is s is3Boundary of (1), satisfies Ks3>|s3|max，|s3|maxIs | s3The maximum value of |;
solving the first derivative of equation (21) to obtain:
wherein, the expression is as follows:
substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectivelyx,uy：
Wherein k is22,k32Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j is 4,5,6, x4＝φ,x5＝θ,x6＝ψ，x4dDenotes the expected value, x, of phi5dDenotes the desired value, x, of theta6dIndicating the desired value, e, of4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
2.10 design constraints Lyapunov functionAnd solving for its first derivative:
wherein k isjIs a normal number, KbjIs ejBoundary of (1), satisfies Kbj>|ej|max，|ej|maxIs | ejThe maximum value of |;
αjthe expression is a virtual control quantity of the attitude angle, and is as follows:
wherein k isj1Is a normal number;
substituting formula (29) for formula (28) to obtain:
2.11 design constraints Lyapunov function Vj2：
Wherein, KsjIs s isjBoundary of (1), satisfies Ksj>|sj|max，|sj|maxIs | sjThe maximum value of |;
solving the first derivative of equation (31) to obtain:
wherein, the expression is as follows:
substituting formula (32) with formula (33) and formula (6) respectively results in:
2.12 design of τ by equations (34), (35), (36), respectivelyx,τy,τz：
Wherein k is42,k52,k62Is a normal number;
step 3, verifying the stability of the four-rotor aircraft system, wherein the process is as follows:
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 formula (34), (35) and (36) to obtain:
3.4 by (38), (39), (40) the quad-rotor aircraft system is stable.
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: in the formula (2), m is 1.1kg, g is 9.81N/kg; in the formula (4), Ixx＝1.22kg·m2，Iyy＝1.22kg·m2，Izz＝2.2kg·m2(ii) a Z in the formulae (8), (17) and (27)d＝1，xd＝1，yd＝1，ψd0.5; k in the formulae (10), (19) and (29)11＝2,k21＝2,k31＝2,k41＝2,k51＝2,k612; k in the formulae (16), (26) and (37)12＝2,k22＝2,k32＝2,k42＝2,k52＝2,k622; formula (9), formula (18) and formula (28) kb1＝kb2＝kb3＝1.5+0.1sint,kb4＝kb5＝kb62+0.1 sint; formula (12), formula (21) and formula (31) ks1＝ks2＝ks3＝3.5+0.1sint,ks4＝ks5＝ks6＝4+0.1sint。
As can be seen from fig. 1 and 2, the system output has good transient characteristics, the arrival time is 4.46 seconds, and the overshoot is 0.
As can be seen from fig. 3 and 4, the system state has good transient characteristics, the arrival time is 4.82 seconds, and the overshoot amount is 0.
In conclusion, the four-rotor aircraft all-state limited control method based on the symmetric time-varying exponential tangent composite constraint Lyapunov function can effectively improve the transient performance of the four-rotor aircraft system in all states.
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 four-rotor aircraft all-state limited control method based on a symmetric time-varying exponential tangent composite constraint 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 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 aircraft 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, UfThe input torque of the four-rotor aircraft is represented, m is the mass of the four-rotor aircraft, and g represents the gravity acceleration;
substituting formula (1) into formula (2) to obtain:
1.3 the dynamic model of the four-rotor aircraft in the rotating process is as follows:
wherein, taux,τy,τzRespectively representing the moment components, I, of the axes in the coordinate system of the machine bodyxx,Iyy,IzzRespectively representing the components of the moment of inertia of each axis in the coordinate system of the body, x represents the cross product, omegapRepresenting roll angular velocity, ωqRepresenting pitch angle velocity, ωrWhich 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 aircraft is in a low-speed flight or hovering state, consider
Thus, equation (4) is rewritten as:
combining the vertical type (3) and the formula (5), and obtaining a dynamic model of the four-rotor aircraft as follows:
wherein u isx＝cosφsinθcosψ+sinφsinψ,uy＝cosφsinθsinψ-sinφcosψ；
1.4 according to equation (6), define φ, the desired value of θ is:
wherein phi isdIs the desired signal value of phi, thetadFor 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 isdA desired signal representing z;
2.2 design constraints Lyapunov functionAnd solving for its first derivative:
wherein, Kb1Is a time-varying parameter, satisfies Kb1>|e1|max，|e1|maxIs | e1The maximum value of l is the sum of,
α1the expression is the virtual control quantity: :
wherein k is11Is a normal number;
substituting formula (10) for formula (9) to obtain:
2.3 design Lyapunov function V12Comprises the following steps:
wherein, Ks1Is a time-varying parameter, satisfies Ks1>|s1|max，|s1|maxIs | s1The maximum value of |;
solving the first derivative of equation (12) to obtain:
wherein, the expression is as follows:
substituting formula (14) and formula (6) for formula (13) yields:
2.4 design Uf：
Wherein k is12Is a normal number;
2.5 define x, y tracking errors as e, respectively2,e3Then, there are:
wherein x isd,ydRespectively representing expected signals of x and y;
2.6 design constraints Lyapunov function Respectively solving the first derivative to obtain:
wherein, Kb2Is e2Boundary of (1), satisfies Kb2>|e2|max，|e2|maxIs | e2The maximum value of |; kb3Is e3Boundary of (1), satisfies Kb3>|e3|max，|e3|maxIs | e3The maximum value of |;
α2,α3the expression is the virtual control quantity:
wherein k is21,k31Is a normal number;
substituting formula (19) for formula (18) to obtain:
2.7 design Lyapunov function V22,V32
Wherein, Ks2Is s is2Boundary of (1), satisfies Ks2>|s2|max，|s2|maxIs | s2The maximum value of |; wherein Ks3Is s is3Boundary of (1), satisfies Ks3>|s3|max，|s3|maxIs | s3The maximum value of |;
solving the first derivative of equation (21) to obtain:
wherein, the expression is as follows:
substituting formulae (23) and (6) for formula (22) respectively results in:
2.8 design u by equations (24) and (25), respectivelyx,uy：
Wherein k is22,k32Is a normal number;
2.9 defines the attitude angle tracking error and its first derivative:
wherein j is 4,5,6, x4＝φ,x5＝θ,x6＝ψ，x4dDenotes the expected value, x, of phi5dDenotes the desired value, x, of theta6dIndicating the desired value, e, of4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
2.10 design constraints Lyapunov functionAnd solving for its first derivative:
wherein k isjIs a normal number, KbjIs ejBoundary of (1), satisfies Kbj>|ej|max，|ej|maxIs | ejThe maximum value of |;
αjthe expression is a virtual control quantity of the attitude angle, and is as follows:
wherein k isj1Is a normal number;
substituting formula (29) for formula (28) to obtain:
2.11 design constraints Lyapunov function Vj2：
Wherein, KsjIs s isjBoundary of (1), satisfies Ksj>|sj|max，|sj|maxIs | sjThe maximum value of |;
solving the first derivative of equation (31) to obtain:
wherein, the expression is as follows:
substituting formula (32) with formula (33) and formula (6) respectively results in:
2.12 design of τ by equations (34), (35), (36), respectivelyx,τy,τz：
Wherein k is42,k52,k62Is a normal number;
step 3, verifying the stability of the four-rotor aircraft system, wherein the process is as follows:
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 formula (34), (35) and (36) to obtain:
3.4 by (38), (39), (40) the quad-rotor aircraft system is stable.
CN201810212222.8A 2018-03-15 2018-03-15 Based on it is symmetrical when the compound constraint liapunov function of varying index tangent quadrotor total state constrained control method Withdrawn CN108427278A (en)

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