CN106774373B - A kind of quadrotor drone finite time Attitude tracking control method - Google Patents
A kind of quadrotor drone finite time Attitude tracking control method Download PDFInfo
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Abstract
A kind of quadrotor drone finite time Attitude tracking control method, the present invention relates to the modeling of quadrotor drone attitude control system and finite time Attitude tracking control methods.The disturbance torque that comprehensive analysis of the present invention quadrotor drone faces, unknown rotary inertia, the factors such as control output saturation and Actuators Failures failure devise passive fault tolerant control device based on parameter adaptive method, and make it have stability in finite time energy.Step of the present invention are as follows: step 1: the kinematics model of quadrotor drone Attitude Tracking is established；Step 2: the kinetic model of quadrotor drone Attitude Tracking is established；Step 3: the posture filtering error of quadrotor drone is defined；Step 4: design finite time Integral Sliding Mode face；Step 5: the finite time Attitude tracking control device of quadrotor drone is designed.The present invention is used for UAV Flight Control field.
Description
Technical Field
The invention relates to a finite time attitude tracking control method for a quad-rotor unmanned aerial vehicle.
Background
As a small unmanned aerial vehicle capable of taking off and landing vertically and hovering at a fixed point, the quad-rotor unmanned aerial vehicle has a wide application in the commercial and civil fields due to the advantages of simple mechanical structure, high safety, low use cost and the like, for example, oil and gas companies have been approved by the federal aviation administration to perform oil field exploration by using quad-rotors, and the unmanned aerial vehicle monitoring system is developed by Bladeworx in israel to protect jersey cold light rails from being damaged. In addition, four rotor unmanned aerial vehicle have also made very big development in the aspect of the application such as video aerial photography, agricultural plant protection, cargo handling.
As an under-actuated control system, the quad-rotor unmanned aerial vehicle has reliable attitude control which is an important condition and guarantee for completing various flight tasks. Factors influencing the stability of the attitude control system of the quad-rotor unmanned aerial vehicle are many, such as system inertia uncertainty, external wind moment interference, gyro moment caused by rotors and other interference moments; an actuating mechanism of the quad-rotor unmanned aerial vehicle is a brushless direct current motor, and partial failure faults can occur under the influence of a manufacturing process and a high-strength task; in addition, the brushless dc motor has an allowable maximum instantaneous current, and if the control signal is too large, the loading current of the motor is too large, which may burn the motor, so that the control design needs to consider the factor of saturation of the control output. These above-mentioned factors are constantly influencing four rotor unmanned aerial vehicle attitude control system's control performance, lead to the system unstable even.
At present, aiming at attitude control of a quad-rotor unmanned aerial vehicle, a plurality of control design methods such as PID (proportion integration differentiation) control, linear quadratic programming, adaptive robust control and the like exist, but some defects exist. On the one hand, the Control design methods only consider the above-mentioned partial influence factors, such as designing robust controllers (Journal: AIAAInfotech @ Aeroscope Conference; author: Steven L.Waslander and Carlos Wang; publication time: 2009; article title: Wind discovery and recommendation for quadrotor position Control; page code: 2009-; on the other hand, at present, research results aiming at attitude control of the unmanned aerial vehicle are all asymptotic time stability, and finite time stability is not involved, and the finite time control method has better practical application prospect due to the superiority of time optimization, rapid convergence and high-precision control performance.
Disclosure of Invention
The invention aims to solve the problem that the finite time attitude tracking of a quad-rotor unmanned aerial vehicle cannot be realized under the condition of facing various stability influence factors, and provides a finite time attitude tracking control method of the quad-rotor unmanned aerial vehicle.
A finite-time attitude tracking control method for a quad-rotor unmanned aerial vehicle comprises the following steps:
the method comprises the following steps: establishing a kinematics model for attitude tracking of the quad-rotor unmanned aerial vehicle;
step two: establishing a dynamic model for attitude tracking of the quad-rotor unmanned aerial vehicle;
step three: defining attitude filtering errors of the quad-rotor unmanned aerial vehicle according to the kinematic model established in the step one;
step four: designing a finite time integral sliding mode surface according to the attitude filtering error defined in the step three;
step five: and designing a finite time attitude tracking controller of the quad-rotor unmanned aerial vehicle according to the dynamics model established in the second step and the finite time integral sliding mode surface designed in the fourth step.
The invention has the beneficial effects that:
1. the method considers various factors influencing stability, such as interference torque, unknown rotary inertia, control output saturation, actuator failure fault and the like faced by the quad-rotor unmanned aerial vehicle in actual engineering, and carries out analysis and modeling;
2. the finite time attitude tracking control scheme provided by the invention has the advantages of simple structure, easy realization, passive fault tolerance performance and no need of detection, separation or even controller reconstruction process of fault information;
3. according to the finite time attitude tracking control scheme provided by the invention, the designed controller is independent of the boundaries of system rotational inertia information and interference moment by using a parameter self-adaptive method, so that the robustness of the unmanned aerial vehicle system is improved.
4. The finite time attitude tracking control scheme provided by the invention can realize the attitude tracking control of the quad-rotor unmanned aerial vehicle within finite time, and improve the transient performance and the steady-state performance of the unmanned aerial vehicle system.
Drawings
Fig. 1 is a four-rotor unmanned aerial vehicle attitude tracking control analysis flow chart.
Fig. 2 is a schematic diagram of a quad-rotor drone attitude dynamics modeling analysis.
Fig. 3 is a graph of attitude tracking error convergence.
Fig. 4 is a graph of angular velocity error convergence.
Fig. 5 is a graph showing changes in the parameter estimation values.
Fig. 6 is a graph showing a change in control torque.
Detailed Description
The first embodiment is as follows: the invention relates to a finite time attitude tracking control method of a quad-rotor unmanned aerial vehicle, which has the following conception:
firstly, establishing a kinematic model according to the relative motion of attitude tracking of the quad-rotor unmanned aerial vehicle; analyzing and modeling various factors influencing stability, such as interference torque, unknown rotary inertia, control output saturation, actuator failure fault and the like faced by the quadrotors in actual engineering, and establishing a dynamic model for attitude tracking of the quadrotor unmanned aerial vehicle;
secondly, defining an attitude filtering error based on a principle of stabilizing the attitude error in finite time; designing a finite time integral sliding mode surface based on a sliding mode control method insensitive to parameter change and disturbance, and reducing steady-state errors by introducing an integral term to inhibit constant interference;
thirdly, a passive fault-tolerant controller is designed based on a design principle with a simple structure, and the detection and separation of fault information and even the reconstruction process of the controller are not needed; and based on a design principle of improving the robustness of the unmanned aerial vehicle system, the designed controller is independent of the boundary of system rotational inertia information and interference moment by using a parameter self-adaptive method.
According to the above concepts, as shown in fig. 1, in combination with an embodiment of attitude tracking control of a quad-rotor unmanned aerial vehicle, a method for finite-time attitude tracking control of a quad-rotor unmanned aerial vehicle is specifically described, which includes the following steps:
the method comprises the following steps: establishing a kinematics model for attitude tracking of the quad-rotor unmanned aerial vehicle;
the specific process is as follows:
considering the attitude of a quad-rotor drone described by a quaternion, the relative motion tracked by the quad-rotor drone attitude can be expressed as:
wherein,respectively representing the attitude tracking error and the angular velocity error of the unmanned aerial vehicle body coordinate system relative to the expected coordinate system, and havingRespectively representing the body attitude and the angular speed of the unmanned aerial vehicle;respectively representing the desired attitude and the desired angular velocity of the drone, and ωd,Known and bounded;representing the attitude rotation matrix of the coordinate system of the unmanned aerial vehicle body relative to the expected coordinate system, and havingAnd C1;representing a quaternion multiplication; aTRepresenting a transpose of a vector or matrix;representing a real number domain; i represents a third order identity matrix; i | · | | represents a 2 norm of the vector or matrix; cross multiplication matrix
Based on the relative motion of unmanned aerial vehicle attitude tracking described by formula (1), the kinematics model for establishing the attitude tracking of the quad-rotor unmanned aerial vehicle is as follows:
wherein, E (E)v)＝(e0I+e×) And has | | | E (E)v)||＝1；e×A cross-multiplication matrix representing e; the top dot of the character represents the first derivative with respect to time;
step two: establishing a dynamic model for attitude tracking of the quad-rotor unmanned aerial vehicle;
the specific process is as follows:
the actuating mechanism of the quad-rotor unmanned aerial vehicle is a brushless direct current motor which has the maximum allowable instantaneous current, so that the constraint of control output saturation needs to be considered during control design in order to avoid motor burnout; partial failure faults can occur in the executing mechanism in the working process due to the influence of the manufacturing process and high-intensity tasks; in addition, quad-rotor unmanned aerial vehicles always suffer from the influence of disturbance moments such as external wind moment disturbance and gyro moment. The factors are considered in the four-rotor unmanned aerial vehicle attitude dynamics modeling analysis, and the specific modeling analysis is shown in fig. 2;
to sum up, consider the interference moment that four rotor unmanned aerial vehicle faced, unknown inertia, control output saturation and executor failure fault, establish four rotor unmanned aerial vehicle attitude tracking's kinetic model and do:
wherein the symmetric positive definite matrixRepresenting an unknown moment of inertia of the drone; delta is diag (delta)1,δ2,δ3) Failure matrix for actuator, 0 < deltai≤1,i＝1,2,3；Representing control instructions generated by a controller; sat (u) ═ sat (u)1),sat(u2),sat(u3)]TIndicating the controller output saturation characteristic, sat (u)i)＝sgn(ui)·min{|ui|,uimax}，uimaxRepresents the maximum output value of the ith control component, sgn (·) represents a sign function; defining theta as the portion of the controller output that exceeds the saturation amplitude, then sat (u) theta + u, then theta [ theta ]1,θ2,θ3]TWhereini＝1,2,3；(ωe+Cωd)×＝ω×All represent a cross-multiplication matrix of ω;represents omegaeA cross-product matrix of; diag (delta)1,δ2,δ3) Representing the main diagonal elements as delta respectively1,δ2,δ3A diagonal matrix of (a);
representing disturbance moment, including external wind moment and gyro moment caused by rotor, the disturbance moment is unknown but bounded, i.e. | | Γ | ≦ dΓ(1+||ω||)，dΓ> 0 is a constant;
the purpose of attitude control of the quad-rotor unmanned aerial vehicle is to design a finite time attitude tracking controller, so that the unmanned aerial vehicle can realize attitude tracking in finite time, namely an attitude error e and an angular velocity error omega for attitude tracking of the unmanned aerial vehicleeApproaching the origin in a finite time;
step three: defining attitude filtering errors of the quad-rotor unmanned aerial vehicle according to the kinematic model established in the step one;
the specific process is as follows:
defining the attitude filtering error of the quad-rotor unmanned aerial vehicle as follows:
wherein,to a virtual controller ·-1Representing the inverse of the matrix; r is more than 01＜1；K1＝diag(k11,k12,k13) Is a diagonal matrix, k1i> 0, i ═ 1,2, 3; power functionWherein
It can be shown that by stabilizing the attitude filtering error z, the attitude error e can be stabilized for a finite time TeInternally converging to the origin. From z equal to 0A kinematic model (2) of the quad-rotor unmanned aerial vehicle attitude tracking established according to step one, thus havingThe Lyapunov function is designed to be V for the error dynamic systeme＝eTe, obtaining the derivative
From the theory of finite time control, it can be known that the attitude error e will be in finite timeInternally converging to an origin, where e (0) represents an initial value of the attitude error; omega can be known from the formula (4)eWill also be in a limited time TeAnd (4) internally converging.
Therefore, the following steps can simultaneously realize the attitude error e and the angular velocity error omega only by considering the design of the finite time attitude tracking controller for stabilizing the attitude filtering error zeIs calmed for a limited time;
step four: designing a finite time integral sliding mode surface according to the attitude filtering error defined in the step three;
the specific process is as follows:
according to a sliding mode control method insensitive to parameter change and disturbance and based on the thought of stabilizing attitude filtering errors in finite time, a finite time integral sliding mode surface is designed, and steady-state errors are reduced by introducing an integral term to inhibit constant interference. Therefore, the finite time integral sliding mode is designed according to the attitude filtering error defined by equation (4) as follows:
wherein the power function sigp(z)＝[sigp(z1),sigp(z2),sigp(z3)]TWherein sigp(zi)＝|zi|psgn(zi) I is 1,2, 3; τ represents an integral variable; c ═ diag (c)1,c2,c3)，c1＞0,c2＞0,c3＞0；0＜p＜1。
The sliding mode surface designed according to formula (5) can prove that the system has a finite time convergence characteristic in the sliding mode phase, namely: when S is 0, the attitude filtering error z will converge to the origin within a finite time;
and (3) proving that: from S ═ 0Design the Lyapunov function asDerived by derivation
Wherein
According to the theory of finite time controlThe attitude filtering error z will be in finite timeInternally converges to the origin, where z (0) represents the initial value of the attitude filtering error. And further combining the results of the step three to obtain an attitude error e and an angular speed error omega of the attitude tracking of the quad-rotor unmanned aerial vehicleeAt a finite time Tz+TeInternally converging to the origin;
step five: designing a finite time attitude tracking controller of the quad-rotor unmanned aerial vehicle according to the dynamics model established in the step two and the finite time integral sliding mode surface designed in the step four;
the specific process is as follows:
and (3) considering the dynamic model (2) established in the step two and the finite time integral sliding mode surface (5) designed in the step four, obtaining the system sliding mode dynamic:
wherein,because the boundaries of the rotational inertia and the disturbance moment of the unmanned aerial vehicle are unknown, relevant parameters all need parameter adaptive estimation, and therefore, the following reasonable assumptions can be made by considering the forms of all the items in theta: | | | Θ | | is less than or equal to b Φ, Φ ═ 1+ | | ω | + | | | ω | | u2) When the parameter b is more than 0 unknown, self-adaptive estimation is needed;
in order to improve the transient performance and the steady-state performance of the system, according to the dynamics model (3) established in the step two and the finite time integral sliding mode surface (5) designed in the step four, the attitude tracking controller of the quad-rotor unmanned aerial vehicle is designed as follows:
wherein the gain matrix K is controlled2＝diag(k21,k22,k23)，K3＝diag(k31,k32,k33) And k is2i＞0，k3i＞0，i＝1,2,3；0＜r2Less than 1; power functionWhereini＝1,2,3；The estimated value of the parameter b is given by the following parameter adaptive updating law (8);
the design parameter adaptive updating law is as follows:
wherein λ > 0, η > 0 is a constant and λ η > 1 is satisfied.
It can be seen that the design of the fault-tolerant controller does not need any fault information detection, separation or even controller reconstruction process, and the saturated amplitude requirement of the actuator is considered in the design process; and the designed finite time attitude tracking controller (7) is independent of the system rotational inertia information and the interference moment boundary by using a parameter self-adaptive method, and the designed controller can be ensured to have certain robustness to interference and system uncertainty.
It can be shown that the quad-rotor unmanned aerial vehicle can realize attitude tracking within a limited time under the effects of the finite time attitude tracking controller designed by the formula (7) and the parameter adaptive updating law designed by the formula (8).
And (3) proving that: aiming at a dynamics model (3) of a quad-rotor unmanned aerial vehicle, a finite time integral sliding mode surface (5) and a system sliding mode dynamic state (6) are considered, and a Lyapunov function is designed as follows:
wherein, deltaminIs the minimum eigenvalue of the failure matrix delta.
The derivation of V along the system rail pair can be found:
wherein | · | purple sweet1Representing the 1 norm of the vector. For any vector, it holds | · | | non-woven phosphor1Is greater than or equal to | | · |, thus there is
Considering that for any phi > 0, the assembly is verticalWhere 0 < ζ < 1, thus scalingCan obtain the product
The above formula is substituted into the formula (9)
Wherein,Jmaxa maximum characteristic value representing the moment of inertia J;
according to the theory of limited time control, the attitude tracking control system of the quad-rotor unmanned aerial vehicle can be controlled in limited timeThe inner completion sliding mode approaches dynamic state, wherein V0The initial value of the Lyapunov function V is more than 0 and less than 1. Further combining the results of the third step and the fourth step, the quad-rotor unmanned aerial vehicle can be known to have limited time Tf+Tz+TeInternally realized attitude error e and angular velocity error omegaeAnd meanwhile, the posture tracking control within a limited time is realized.
The finite time attitude tracking control method of the quad-rotor unmanned aerial vehicle provides numerical simulation verification, and shows that when the quad-rotor unmanned aerial vehicle faces various stability influence factors, the provided control method can realize finite time attitude tracking and has better control performance, and the method specifically comprises the following steps:
the model parameters of the quad-rotor unmanned aerial vehicle are selected as follows:
disturbance torque
Uncertainty of moment of inertia
Initial value of angular velocity ω (0) [0.10-0.1 ]]Trad/s; initial attitude qv(0)＝[0.3-0.2 0.3 0.8832]T；
Controlling the output saturation amplitude uimax＝0.001Nm；；
The desired trajectory to track is:
initial value q of expected attitudedv(0)＝[0.7 0.5 0.4123 0.3]T，
Desired angular velocity ωd＝0.05×[sin(0.1t) 2sin(0.2t) 3sin(0.3t)]Trad/s；
The failure fault of the actuator is as follows:
the initial estimation values of the parameter self-adaptive updating law are all 0;
combining the requirements of the invention on controller design and parameter adaptive updating law in the formulas (4), (5), (7) and (8), respectively takingThe parameters are as follows: c. Ci＝0.15，p＝0.7；k1i＝0.2，r1＝0.6；k2i＝0.4，k3i＝0.1，r20.75; λ ═ 5, η ═ 1; in order to avoid buffeting of the symbolic function, simulation verification is carried outInstead of the sign function, where ρ is taken to be 0.01.
Fig. 3 and 4 are a graph showing the attitude error convergence curve and the angular velocity error convergence curve respectively when the quad-rotor unmanned aerial vehicle performs attitude tracking, and it can be seen that the unmanned aerial vehicle completes attitude tracking within 20 seconds, and the steady-state error can be controlled to be 10-6And 10-5And therefore has higher tracking accuracy; FIG. 5 is a graph of a change in the parameter estimate, indicating that the parameter estimate has finally converged to 0; fig. 6 is a graph of variation of the control torque, and it can be seen that the control torque is constrained within the range of 0.001Nm due to the limitation of the saturation amplitude of the control output, and the curve spiking phenomena occurring at 10 seconds, 15 seconds and 20 seconds correspond to the occurrence of failure faults, but the designed controller can overcome the problem, which indicates that the controller has good fault tolerance capability.
Other steps and parameters are the same as those in the first embodiment.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope of the present invention.
Claims (1)
1. A finite time attitude tracking control method for a quad-rotor unmanned aerial vehicle is characterized by comprising the following steps: the finite-time attitude tracking control method of the quad-rotor unmanned aerial vehicle comprises the following steps:
the method comprises the following steps: establishing a kinematics model for attitude tracking of the quad-rotor unmanned aerial vehicle;
the specific process is as follows:
describe four rotor unmanned aerial vehicle's gesture by the quaternion, the relative motion that represents four rotor unmanned aerial vehicle gesture tracking is:
wherein e isv，ωeRespectively representing the attitude tracking error and the angular velocity error of the unmanned aerial vehicle body coordinate system relative to the expected coordinate system, e0The attitude errors e are respectively evScalar and vector of (a); q. q.svAnd omega respectively represent the body attitude and angular velocity of the unmanned aerial vehicle, q0Q is each qvScalar and vector of (a); q. q.sdv，ωdRepresenting desired attitude and desired angular velocity, q, respectively, of the droned0、qdAre each qdvScalar and vector of (a); c represents the attitude rotation matrix of the drone body coordinate system relative to the desired coordinate system,representing a quaternion multiplication; q. q.s×A cross-product matrix of q;
based on the relative motion of unmanned aerial vehicle attitude tracking described by formula (1), the kinematics model for establishing the attitude tracking of the quad-rotor unmanned aerial vehicle is as follows:
wherein, E (E)v)＝(e0I+e×) And | | | E (E)v)||＝1，e×A cross-multiplication matrix representing e; i represents a third order identity matrix;
step two: establishing a dynamic model for attitude tracking of the quad-rotor unmanned aerial vehicle;
the specific process is as follows:
the method comprises the following steps of establishing a dynamics model for attitude tracking of the quad-rotor unmanned aerial vehicle:
whereinJ represents the unknown moment of inertia of the drone; delta is diag (delta)1,δ2,δ3) Failure matrix for actuator, 0 < deltai≤1,i＝1,2,3；u＝[u1,u2,u3]TRepresenting control instructions generated by a controller; sat (u) ═ sat (u)1),sat(u2),sat(u3)]TIndicating the controller output saturation characteristic, sat (u)i)＝sgn(ui)·min{|ui|,uimax}，uimaxRepresents the maximum output value of the ith control component, sgn (·) represents a sign function; defining theta as the part of controller output exceeding saturation amplitude, sat (u) is theta + u, and theta is theta1,θ2,θ3]TWherein(ωe+Cωd)×＝ω×All represent a cross-multiplication matrix of ω;is omegaeA cross-product matrix of;
gamma is disturbance moment and includes external wind moment and gyro moment caused by rotor wing, | gamma | ≦ dΓ(1+||ω||)，dΓ> 0 is a constant;
step three: defining attitude filtering errors of the quad-rotor unmanned aerial vehicle according to the kinematic model established in the step one;
the specific process is as follows:
defining the attitude filtering error of the quad-rotor unmanned aerial vehicle as follows:
wherein,is a virtual controller; a-1Representing the inverse of the matrix; r is more than 01＜1；K1＝diag(k11,k12,k13) Is a diagonal matrix, k1i> 0, i ═ 1,2, 3; power functionWherein
By stabilizing the attitude filtering error z, the attitude error e is stabilized for a finite time TeInternally converging to the origin; from z ═ 0According to the kinematics model (2) of the attitude tracking of the quad-rotor unmanned aerial vehicle established in the step one, thenThe Lyapunov function is designed to be V for the error dynamic systeme＝eTe, deriving to obtain:
attitude error e will be in finite timeInternally converging to an origin, where e (0) represents an initial value of the attitude error; omega can be known from the formula (4)eWill be in a limited time TeInner convergence;
step four: designing a finite time integral sliding mode surface according to the attitude filtering error defined in the step three;
the specific process is as follows:
designing a finite time integral sliding mode according to the attitude filtering error defined by equation (4) is as follows:
wherein the power function sigp(z)＝[sigp(z1),sigp(z2),sigp(z3)]TWherein sigp(zi)＝|zi|p sgn(zi) I is 1,2, 3; τ represents an integral variable; c ═ diag (c)1,c2,c3)，c1＞0,c2＞0,c3＞0；0＜p＜1；
The system has a finite time convergence characteristic in the sliding mode phase, namely: when S is equal to 0, the attitude filtering error z is in a finite time TzInternally converging to the origin;
from S ═ 0Design the Lyapunov function asThe derivation can be:
wherein
Attitude filtering error z in finite timeInternally converging to the origin, where z (0) represents the initial value of the attitude filtering error; and combining the results of the step three to obtain the attitude error e and the angular speed error omega of the attitude tracking of the quad-rotor unmanned aerial vehicleeAt Tz+TeConverge to the origin in time;
step five: designing a finite time attitude tracking controller of the quad-rotor unmanned aerial vehicle according to the dynamics model established in the step two and the finite time integral sliding mode surface designed in the step four;
the specific process is as follows:
system sliding mode dynamics:
wherein,the following is assumed:
||Θ||≤bΦ，Φ＝(1+||ω||+||ω||2)，b＞0；
according to the dynamics model (3) established in the step two and the finite time integral sliding mode surface (5) designed in the step four, designing a posture tracking controller of the quad-rotor unmanned aerial vehicle as follows:
wherein the gain matrix K is controlled2＝diag(k21,k22,k23)，K3＝diag(k31,k32,k33) And k is2i＞0，k3i＞0，i＝1,2,3；0＜r2Less than 1; power functionWherein Is an estimate of b, given by equation (8);
the design parameter adaptive updating law is as follows:
wherein lambda is more than 0, eta is more than 0 and is a constant, and lambda eta is more than 1;
under the action of a finite time attitude tracking controller designed by the formula (7) and a parameter self-adaptive updating law designed by the formula (8), the quad-rotor unmanned aerial vehicle realizes attitude tracking within finite time;
aiming at a dynamics model (3) of a quad-rotor unmanned aerial vehicle, a finite time integral sliding mode surface (5) and a system sliding mode dynamic state (6) are considered, and a Lyapunov function is designed as follows:
wherein, deltaminIs the minimum eigenvalue of the failure matrix delta;
the derivation of V along the system rail pair can be found:
wherein | · | purple sweet11 norm representing a vector; for any vector, | ·| non-conducting phosphor1If all of the conditions are satisfied, the following results are obtained:
for any phi > 0, the average value of the equation,where 0 < ζ < 1, thus scalingThe following can be obtained:
substituting the above formula into formula (9) can obtain:
wherein,Jmaxa maximum characteristic value representing the moment of inertia J;
four-rotor unmanned aerial vehicle attitude tracking control system in limited timeThe inner completion sliding mode approaches dynamic state, wherein V0Is the initial value of the Lyapunov function V, and theta is more than 0 and less than 1; combining the results of step three and step four, the quad-rotor drone is at Tf+Tz+TeRealizing attitude error e and angular velocity error omega in timeeAnd meanwhile, stabilization is realized, and attitude tracking control is realized.
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