A method for constrained unmanned aerial vehicle system sliding mode fault-tolerant control under actuator failure

Abstract

The application discloses a novel fault-tolerant control algorithm of a sliding mode for an actuator fault of a multi-rotor aircraft with state constraints. In the case that an actuator failure fault occurs in a quad-rotor unmanned aerial vehicle system with state constraints, a fault-tolerant control method is designed by combining barrier Lyapunov functions and sliding mode control. A non-singular fast terminal sliding mode is constructed to replace a traditional sliding mode surface for faults and external disturbances, so that the system has strong robustness and can realize convergence in a short finite time. In the case that state constraints exist in actual flight of the quad-rotor aircraft, barrier Lyapunov functions are introduced in the design of a control law, the sliding mode surface is constrained in a certain range in the verification of stability of the algorithm, so that the output state of the aircraft is guaranteed not to violate the constraint condition. The application is used for fault-tolerant flight control of the quad-rotor unmanned aerial vehicle with the actuator failure.

Description

TECHNICAL FIELD

[0001] The application relates to a kind of multi-rotor unmanned aerial vehicle systems with state constraints and actuator failures, and designs a sliding mode fault-tolerant control algorithm based on barrier Lyapunov function, belonging to the fault-tolerant control technical field of uncertain nonlinear system. BACKGROUND

[0002] In recent decades, with the rapid development of energy power technology, composite material technology, microelectronic technology and other industries, multi-rotor unmanned aerial vehicle technology has been greatly improved. Because of its strong adaptability, simple operation, low cost, vertical take-off and other advantages, multi-rotor unmanned aerial vehicle has been widely used in military, industry, agriculture, search and rescue and other fields, and has brought a lot of convenience to human life. However, some special working environment or long time use may cause the unmanned aerial vehicle to fail, and any small failure may cause damage to the unmanned aerial vehicle, resulting in huge property loss and even casualties. Therefore, fault-tolerant control algorithm becomes more and more important and gradually becomes the research focus.

[0003] Fault-tolerant control can be generally divided into active fault-tolerant control and passive fault-tolerant control. Passive fault-tolerant control mainly improves the robustness of the controller to reduce the sensitivity to the possible failure of the early prediction, and does not need to obtain fault information online to reconstruct the control law. When unknown faults occur, the effect of passive fault-tolerant control is not ideal. The control idea of active fault-tolerant control is to obtain fault information through a fault diagnosis module, and then adjust the structure and parameters of the control law according to the obtained fault information. Compared with passive fault-tolerant control, active fault-tolerant control does not need to sacrifice the control performance to achieve fault-tolerant control effect, and can better complete fault detection and isolation. Therefore, active fault-tolerant control has more obvious advantages and is currently receiving more attention.

[0004] Actuator failure is a common fault of multi-rotor unmanned aerial vehicle. In recent years, some scholars have made a lot of achievements in fault-tolerant control of unmanned aerial vehicle actuator failure, and have proposed many practical control algorithms, such as sliding mode control, adaptive control, predictive control and the like. Sliding mode control algorithm has good robustness in dealing with parameter perturbation and external disturbance.

[0005] However, in actual flight, there are often some constraints on the aircraft, such as attitude angle and flight speed, which often have a constraint range. When the range is exceeded, there may be great risk. Sliding mode control cannot well handle the problem of state constraints. To solve the problem of state constraints, some researchers have proposed the method of barrier Lyapunov function. By constructing a barrier Lyapunov function, the boundary of the independent variable can be set. When the independent variable approaches the boundary value, the function will tend to infinity, thereby ensuring that the independent variable will not exceed the constraint range. The combination of barrier Lyapunov function and sliding mode control can well solve the problem of fault-tolerant control of unmanned aerial vehicle with state constraints. At present, many scholars are doing research in this regard. SUMMARY

[0006] In view of the above research background, a sliding mode fault-tolerant control algorithm based on barrier Lyapunov function is proposed for a multi-rotor unmanned aerial vehicle system with state constraints and actuator faults. An adaptive observer is designed to obtain accurate fault information. In order to ensure better dynamic performance, a non-singular fast terminal sliding mode surface is designed to replace the traditional sliding mode surface, so that the system has strong robustness and can converge in finite time. Considering the constraint conditions of quadrotor aircraft in actual flight, barrier Lyapunov function is introduced when designing the control law, which not only verifies the stability of the system, but also limits the sliding mode surface within a specified range, so as to avoid the problem of excessive overshoot and violation of constraint conditions.

[0007] Technical scheme: A new sliding mode fault-tolerant control method for quadrotor unmanned aerial vehicle with state constraints and actuator partial failure. Its characteristics are as follows: first, accurate fault information is obtained through an adaptive observer, and then a non-singular fast terminal sliding mode surface is designed to replace the traditional sliding mode surface according to the fault information and system error, so that the system has strong robustness and the error can converge in finite time, improving the control performance of the system; then, a sliding mode control law is designed according to the state constraints and barrier Lyapunov function, which verifies the stability of the system while ensuring that the overshoot will not be too large and violating the constraint conditions, and finally forms a fault-tolerant controller, including the following specific steps:

[0008] Step 1) Determine the system model and constraint conditions, including the following steps:

[0009] Step 1.1) Determine the system model, as shown in equation (1):

[0010]

[0011] Where x i1 =(x, y, z, φ, θ, ψ) T And x i2= (u, v, w, p, q, r) T respectively represent the position and velocity state of the quadrotor at time t; u i (t) is the control input; d i is the external disturbance received by the quadrotor, satisfying |d i |≤ζ, ζ is a known constant; f i (x) and g i (x) are known continuous vector-valued functions, representing the inherent nonlinear dynamics of the quadrotor and the effect of the control input on the quadrotor state, respectively; i = 1, 2, 3, 4, 5, 6;

[0012] Step 1.2) Determine the constraint condition. Considering that in actual flight tasks, the state of the quadrotor will usually have a limited range, exceeding which may cause danger, the quadrotor state constraint is shown in equation (2):

[0013]

[0014] wherein, and K ci are constants, representing the upper and lower bounds of x i1 (t), respectively; for any x i1 (t) satisfying equation (2), there always exists a normal g0 number satisfying 0 < g0≤|g (x)| for any i To satisfy the constraint condition, the expected value y i of y id and its derivatives have upper and lower bounds, i.e., there exist constants Y i0 , Y i1 ,..., Y in such that hold for any t≥t0; wherein y id is the expected value of the output y i ;

[0015] Step 2) Determine the fault model of the quadrotor system, including the following steps:

[0016] Step 2.1) Determine the fault model

[0017] Given that u i (t) is the control input of the i-th channel, when an actuator failure fault occurs, equation (1) can be rewritten as:

[0018]

[0019] wherein u iF (t) represents the control input after the actuator failure of the ith channel, as shown in equation (4):

[0020]

[0021] where σ i represents the failure rate of the ith channel, and satisfies 0≤σ i <1; when σ i = 0, the ith channel is working normally, and when 0 < σ i <1, the ith channel has a failure but is still working;

[0022] Step 2.2) Determine the fault information

[0023] Write equation (1) in the form of state space:

[0024]

[0025] where F(x) is the nonlinear part of the system; U is the control vector; d and D are the bounded disturbance and coefficient matrix respectively; E = diag{σ1, σ2, …, σ6} represents the failure rate of the actuator; I is the unit matrix.

[0026] Construct the following output observer for equation (5):

[0027]

[0028] where and are the observed values of X(t), Y(t) and E respectively; define as the observation error of the output; K is the output feedback matrix, which satisfies that all the eigenvalues of the new coefficient matrix configured are negative; η is the switching function, whose expression is as follows:

[0029]

[0030] Take Q = diag{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, and configure the corresponding symmetric positive definite matrix P, which satisfies R satisfies (RC) T = PD; define

[0031]

[0032] where Γ e is symmetric positive definite, which is the observer gain matrix; L satisfies C T L T= PG (I-E), thus converging to obtain accurate fault information, thereby completing the reconstruction;

[0033] Step 3) Designing the sliding mode surface, including the following steps:

[0034] Step 3.1) According to the information of the quadrotor unmanned aerial vehicle, the error of the system is defined as shown in equation (9):

[0035]

[0036] Wherein, e i1 (t) and e i2 (t) are the position error variable and the speed error variable at time t, respectively;

[0037] Step 3.2) Design the sliding mode surface function according to the demand, in order to make the state of the unmanned aerial vehicle not violate the constraint condition, it is necessary to ensure that when the sliding mode function is bounded, the error variable is also bounded, so the designed sliding mode surface is shown in equation (10):

[0038]

[0039] Wherein, k il and k i2 are normal numbers, in order to avoid singularity, α i and β i satisfy 1 < p i < 2 and β i < α i ; sign(·) is a sign function, that is:

[0040]

[0041] Step 4) Designing the fault-tolerant control law, including the following steps:

[0042] Step 4.1) Constructing barrier Lyapunov function, according to the constraint condition of the system, the designed barrier function is shown in equation (12):

[0043]

[0044] Wherein κ ai = K ci -Y i 0, κ ai , κ bi is the boundary of the sliding mode surface, when the value of s i approaches the boundary, the value of the function will tend to infinity; in order to simplify, λ is used instead of λ(s i ) in the following;

[0045] Step 4.2) Design of fault-tolerant control law, respectively, the derivation of formula (10), (12) is:

[0046]

[0047]

[0048] In order to make the system slide on the sliding surface, let Substitute formula (3), (9), (10), (13) into formula (14) to obtain the equivalent control law:

[0049]

[0050] Considering the uncertainty and disturbance existing in practical application, it is also necessary to change the characteristics of In order to ensure that the system converges in finite time, that is, to meet formula (16):

[0051] t r ≤t0+h ln V (16)

[0052] Where, t r Convergence time, t0 is the initial time, h>0; by transforming formula (16) can be obtained:

[0053]

[0054] According to formula (17), the switching control law is designed as:

[0055]

[0056] Finally, the complete fault-tolerant control law is:

[0057]

[0058] Step 5) According to the running state of the multi-rotor unmanned aerial vehicle system, select appropriate parameters to complete the fault-tolerant control.

[0059] Beneficial effects: the present application proposes a sliding mode fault-tolerant control method for the actuator partial failure fault of the multi-rotor unmanned aerial vehicle system with state limit, when the quad-rotor unmanned aerial vehicle exists actuator partial failure fault, a sliding mode fault-tolerant control method is proposed by combining barrier lyapunov function, so that the unmanned aerial vehicle system can run normally after the actuator failure; the fault information is obtained by using the adaptive observer; the non-singular fast terminal sliding surface is used instead of the traditional sliding surface to improve the convergence speed and improve the robustness; when designing the control law, the barrier lyapunov function is used to construct the boundary, which solves the problem of state limit while ensuring stability, and has the following specific advantages:

[0060] (1) The designed adaptive observer can accurately obtain fault information, and the fault-tolerant control law can adjust parameters according to the fault information, and the sliding mode control itself has strong robustness, which makes the system have stronger ability to deal with faults and disturbances;

[0061] (2) According to the system output error, a non-singular fast terminal sliding mode surface is designed to replace the traditional sliding mode surface, which improves the response speed and robustness of the system, and makes the controller have better practicability;

[0062] (3) The method of introducing barrier Lyapunov function is used to set the boundary value of the sliding mode surface, so as to limit the range of output state, and ensure that the limit condition will not be violated due to excessive overshoot.

[0063] The method provided by the application is a sliding mode fault-tolerant control method for the actuator partial failure fault of the multi-rotor unmanned aerial vehicle system with state limitation, has certain application significance, is easy to implement, has good real-time performance, high accuracy, can effectively improve the safety of the control system, has strong operability, saves time, is more efficient, and can be widely applied to the actuator fault-tolerant control of the multi-rotor unmanned aerial vehicle system. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 is a flowchart of the method of the application;

[0065] Figure 2 is a schematic diagram of a quad-rotor model and its coordinate system;

[0066] Figure 3 is a roll angle curve diagram of the Qball-X4 quad-rotor unmanned aerial vehicle when the actuator fails;

[0067] Figure 4 is an X-axis position curve diagram of the Qball-X4 quad-rotor unmanned aerial vehicle when the actuator fails;

[0068] Figure 5 is a Y-axis position curve diagram of the Qball-X4 quad-rotor unmanned aerial vehicle when the actuator fails;

[0069] Figure 6 is a Z-axis position curve diagram of the Qball-X4 quad-rotor unmanned aerial vehicle when the actuator fails; DETAILED DESCRIPTION

[0070] The application will be further explained in combination with the drawings.

[0071] As Figure 1As shown, a sliding mode fault-tolerant control method for actuator partial failure in a state-constrained quadrotor unmanned aerial vehicle (UAV) is proposed. The method is characterized by: when an actuator partial failure occurs in the UAV system, a non-singular fast terminal sliding mode fault-tolerant control method is proposed, enabling the UAV system to operate normally after an actuator failure and allowing the error to converge within a finite time; then, a sliding mode control law is designed based on state constraints and a barrier Lyapunov function, ultimately forming a fault-tolerant controller. The specific steps include:

[0072] Step 1) Determine the system model and constraints, including the following steps:

[0073] Step 1.1) Determine the system model, as shown in equation (1):

[0074]

[0075] Where, x il = (x, y, z, φ, θ, ψ) T and x i2 = (u, v, w, p, q, r) T Represent the position and velocity state of the quadcopter UAV at time t, respectively; u i (t) represents the control input; d i For the external disturbances experienced by the drone, satisfying |d i |≤ζ, where ζ is a known constant; f i (x) and g i (x) is a known continuous vector-valued function, representing the inherent nonlinear dynamic behavior of the UAV and the influence of the control input on the UAV's state, respectively; i = 1, 2, 3, 4, 5, 6;

[0076] Step 1.2) Determine the constraints. Considering that in actual flight missions, the state of the UAV will usually have a limited range, and exceeding this range may cause danger, the UAV state constraints are as shown in equation (2):

[0077]

[0078] in, and K ci Let x be a constant, representing x respectively. i1 The upper and lower bounds of (t); for any x i1 When (t) satisfies equation (2), there always exists a normal g0 number that satisfies for any Both have 0 < g and 0 ≤ |g i (x)|;To satisfy the constraints, y i Expected value y id Its derivatives have upper and lower bounds, meaning there exist constants. Yi0 , Y i1 ,..., Y in such that holds for any t≥ t0; where, y id is the expected value of the output y i .

[0079] Step 2) determining the fault model of the UAV system, comprising the following steps:

[0080] Step 2.1) determining the fault model

[0081] Given that u i (t) is the control input of the i-th channel, when an actuator failure fault occurs, formula (1) can be rewritten as:

[0082]

[0083] wherein u i F (t) represents the control input after the i-th channel occurs an actuator failure fault, as shown in formula (4):

[0084]

[0085] wherein σ i represents the failure rate of the i-th channel, and satisfies 0≤ σ i <1; when σ i = 0, the i-th channel works normally, when 0 < σ i <1, the i-th channel occurs a failure fault but still works;

[0086] Step 2.2) determining the fault information

[0087] Write formula (1) in the form of state space:

[0088]

[0089] wherein F(x) is the nonlinear part of the system; U is the control vector; d and D are the bounded disturbance and coefficient matrix respectively; E = diag{σ1, σ 2,.. , σ6} represents the failure rate of the actuator; I is the unit matrix.

[0090] Construct the following output observer for formula (5):

[0091]

[0092] wherein, and are the observed values of X(t), Y(t) and E respectively; define is the output observation error; K is the output feedback matrix, which satisfies the new coefficient matrix configured is negative; η is a switching function, whose expression is as follows:

[0093]

[0094] Take Q=diaf{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, and configure the corresponding symmetric positive definite matrix P, which satisfies R satisfies (RC) T = pD; define the constructed adaptive law after the system is configured with the poles as follows:

[0095]

[0096] Wherein, Γ e is symmetric positive definite, and is the observer gain matrix; L satisfies C T L T = PG(I-E), so that the accurate fault information can be obtained by convergence, so as to complete reconstruction;

[0097] Step 3) design the sliding mode surface, including the following steps:

[0098] Step 3.1) according to the information of the quadrotor unmanned aerial vehicle, the error of the system is defined, as shown in equation (9):

[0099]

[0100] Wherein, e i1 (t) and e i2 (t) are the position error variable and the speed error variable at time t respectively;

[0101] Step 3.2) design the sliding mode surface function according to the demand, in order to make the state of the unmanned aerial vehicle not violate the constraint condition, it is necessary to ensure that when the sliding mode function is bounded, the error variable is also bounded, so the designed sliding mode surface is as shown in equation (10):

[0102]

[0103] Wherein, k i1 and k i2 are normal numbers, in order to avoid singularity, α i and β i satisfy 1<β i <2 and β i <α i ; sign(·) is a sign function, that is:

[0104]

[0105] Step 4) Designing fault-tolerant control law, including the following steps:

[0106] Step 4.1) Constructing barrier Lyapunov function, according to the system constraints, the designed barrier function is shown in equation (12):

[0107]

[0108] Where κ au = K ci - Y i0 , κ ai ,κ bi is the boundary of the sliding surface, when the value of s i approaches the boundary, the value of the function tends to infinity; in order to simplify, λ(s i ) is replaced by λ later;

[0109] Step 4.2) Designing fault-tolerant control law, respectively, the derivative of equation (10), (12) is:

[0110]

[0111]

[0112] In order to make the system slide on the sliding surface, let Substituting equation (3), (9), (10), (13) into equation (14) can get the equivalent control law:

[0113]

[0114] Considering the uncertainty and disturbance existing in practical application, it is also necessary to change the characteristics of by designing switching control law: in order to ensure the system converges in finite time, that is to meet equation (16):

[0115] t r ≤t0+h ln V (16)

[0116] Where, t r is the convergence time, t0 is the initial time, h > 0; by transforming equation (16) can get:

[0117]

[0118] According to equation (17), the switching control law is designed as:

[0119]

[0120] Finally, the complete fault-tolerant control law is obtained by combining equations (15) and (18):

[0121]

[0122] Step 5) According to the operating state of the multi-rotor unmanned aerial vehicle system, appropriate parameters are selected to complete the fault-tolerant control thereof.

[0123] The above only describes the preferred embodiments of the present application, and it should be noted that those skilled in the art can make several improvements and refinements without departing from the principles of the present application, and these improvements and refinements should also be considered within the scope of protection of the present application.

[0124] Table 1: Qball-X4 body parameter value table

[0125]

[0126]

[0127] The effectiveness of the implementation scheme is illustrated by actual case simulation.

[0128] To verify the effectiveness of the method, the experimental device Qball-X4 quad-rotor unmanned aerial vehicle developed by the Canadian Quanser company for quad-rotor unmanned aerial vehicle control is used as the application object, and the body parameters of Qball-X4 are shown in Table 1.

[0129] In order to establish the mathematical model of the quad-rotor unmanned aerial vehicle Qball-X4, the structure of the quad-rotor unmanned aerial vehicle Qball-X4 is simplified, and the simplified structure is shown in Figure 2 The X-shaped frame is used and the body coordinate system O b -X b Y b Z b , the ground coordinate system of the quad-rotor is O g -X g Y g Z g .

[0130] Generally, the quad-rotor aircraft system has six-dimensional variables, i.e. (X, Y, Z, ψ, θ, φ), wherein X, Y, Z are position variables, ψ is the yaw angle, θ is the pitch angle, and φ is the roll angle. Ω = [p, q, r] T and V = [u, v, w] T represent the Euler angle speed and linear speed, respectively.

[0131] According to the Newton-Euler formula, the dynamic equation of the system can be introduced as follows:

[0132]

[0133] where k φ , k θ , k ψ are the drag coefficients, g is the gravity acceleration, and the control variables are defined as follows:

[0134]

[0135]

[0136] where F i (i = 1, 2, 3, 4) and τ i (i = 1, 2, 3, 4) are the drag and torque of the propeller, respectively.

[0137] The initial Euler angles and position coordinates are set as Θ(0) = [-0.2, -0.2, 0.5] T (rad), P(0) = [0, 1, 0.3] T (m), the initial linear velocity and angular velocity are set as V(0) = [0, 0, 0] T (m / s), The desired position target is set as [y 1d , y 2d , y 3d ] T = [0.5, 0.5, 0.8] T (m), and the desired attitude target is set as [y 4d , y 5d , y 6d ] T = [0.4sin(t), 0.3cos(t), 0.3] T (rad), and the constraints of position and attitude are set as follows:

[0138]

[0139] To verify the effectiveness of the control law, white noise with an upper limit of ζ = 0.5 is set as the disturbance, and a 30% failure fault is injected in the roll channel from the 5thsecond, i.e., E = diag{0, 0, 0, 0.3, 0, 0}. According to the actual situation, the parameters in the control law are set as: α i = 2, β i = 1.67, k i1 = 1, k i2 = 1, q = 5.

[0140] The proposed control law is compared with the traditional sliding mode control and backstepping method, respectively, Figure 3 is the dynamic diagram of the roll angle under fault conditions for the three methods, Figures 4-6The displacement curves of the three methods in the x, y and z directions in the fault state are respectively shown in the following figures.

[0141] The simulation results show that the unmanned aerial vehicle system sliding mode fault-tolerant control algorithm designed by the application can have strong fault-tolerant capability to external disturbance and actuator failure, the convergence of the control law has good rapidity and accuracy, and the state limit problem can be well handled. As can be seen from the figure, when the fault occurs, the backstepping method does not exceed the constraint range, but the fault-tolerant capability is poor, and the tracking performance is poor; the traditional sliding mode method can realize convergence, but the overshoot is large, and cannot be controlled within the constraint range; the method proposed in the case has stronger robustness, lower sensitivity to faults, faster convergence speed, and meets the constraint condition, and the effect is more excellent. In summary, for the four-rotor unmanned aerial vehicle system with state limit and actuator failure, the fault-tolerant control method simulated in the case is effective.

Claims

1.A method for a sliding mode fault-tolerant control for partial actuator failure of quadrotor unmanned aerial vehicle with state constraints, characterized in that: In the presence of actuator partial failure faults of unmanned aerial vehicle system, a nonsingular fast terminal sliding mode fault-tolerant control method is proposed, which makes the unmanned aerial vehicle system run normally after the actuator failure and makes the error converge in finite time. Then, according to the state restriction combined with barrier Lyapunov function, a sliding mode control law is designed, and finally a fault-tolerant controller is formed, including the following specific steps: Step 1) Determine the system model and constraint conditions, including the following steps: Step 1.1) Determine the system model, as shown in equation (1): where x i1 = (x, y, z, φ, θ, ψ) T and x i2 = (u, v, w, p, q, r) T denote the position and velocity states of the quadrotor UAV at time t, respectively; u i (t) is the control input; d i is the external disturbance experienced by the UAV, satisfying |d i |≤ ζ, ζ being a known constant; f i (x) and g i (x) are known continuous vector-valued functions representing the intrinsic nonlinear dynamics of the UAV and the effect of the control input on the UAV states, respectively; i = 1, 2, 3, 4, 5, 6; Step 1.2) Determine the constraint condition. Considering that in actual flight tasks, the state of the unmanned aerial vehicle will usually have a limited range, and beyond this range, danger may occur. The state constraint of the unmanned aerial vehicle is shown in equation (2): wherein, and κ ci are constants, respectively, representing the upper and lower bounds of x i1 (t); for any x i1 (t) satisfying equation (2), there always exists a normal g0number satisfying 0 < g0≤ |g i (x)|; to satisfy the constraint condition, the expected value y i of y id and its derivatives of each order have upper and lower bounds, i.e., there exist constants such that hold for any t ≥ t0; wherein y id is the expected value of the output y i ;​ Step 2) Determine the fault model of the unmanned aerial vehicle system, including the following steps: Step 2.1) Determine the fault model Known u i (t) is the control input for the i-th channel, when the actuator failure fault occurs, formula (1) can be rewritten as: wherein, represents the control input after the actuator failure of the ith channel, as shown in equation (4): wherein σ i represents the failure rate of the i-th channel, and satisfies 0≤σ i <1; when σ i =0, the i-th channel is working normally, and when 0<σ i <1, the i-th channel has a failure but is still working; Step 2.2) Determine the fault information Write equation (1) in the form of state space: Where F(x) is the nonlinear part of the system; U is the control vector; d and D are the bounded disturbance and coefficient matrix respectively; E = diag{σ1, σ2,..., σ6} represents the failure rate of the actuator; I is the unit matrix; For equation (5), the following output observer is constructed: wherein, and are the observed values of X(t), Y(t) and E, respectively; define as the output observation error; K is the output feedback matrix, which satisfies that the eigenvalues of the configured new coefficient matrix are all negative; η is a switching function, and its expression is as follows: Take Q = diag{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, and configure the corresponding symmetric positive definite matrix P, which satisfies R satisfies (RC) T = PD; define the configuration adaptive law after the system configuration pole as follows: where Γ e symmetric positive definite, is the observer gain matrix; L satisfies C T L T = PG(I-E), thus the accurate fault information can be obtained by convergence, and the reconstruction is completed. Step 3) Design the sliding surface, including the following steps: Step 3.1) According to the information of the quadrotor unmanned aerial vehicle, the error of the system is defined as shown in equation (9): where e i1 (t) and e i2 (t) are the position error variable and the velocity error variable at time t, respectively. Step 3.2) Design the sliding surface function according to the requirements. In order to ensure that the state of the unmanned aerial vehicle does not violate the constraint condition, it is necessary to ensure that the error variable is bounded when the sliding mode function is bounded. Therefore, the designed sliding surface is shown in equation (10): where k i1 and k i2 are positive constants, and a i and b i are positive constants satisfying 1 < b i < 2 and b i < a i ; sign(·) is the sign function, i.e., Step 4) Design the fault-tolerant control law, including the following steps: Step 4.1) Construct the barrier Lyapunov function. According to the system constraint condition, the designed barrier function is shown in equation (12): wherein κ ai = κ ci - Y i0 , κ ai , κ bi is the boundary of the sliding surface, when s i is close to the boundary, the value of the function will tend to infinity; in order to simplify, λ(s i ) is uniformly replaced by λ later Step 4.2) Design of fault-tolerant control law. Derive equation (10), (12) respectively: To make the system slide on the sliding surface, let Substituting equations (3), (9), (10), (13) into equation (14), the equivalent control law is obtained as Considering the uncertainties and disturbances existing in practical applications, it is also necessary to change the switching control law by design The characteristics of the system are as follows: in order to ensure the convergence of the system in a finite time, that is, to meet formula (16): t r ≤ t0+ h ln V (16) where t r is the convergence time, t0is the initial time, h > 0; by deforming equation (16) we obtain: According to equation (17), the switching control law is designed as: Finally, the complete fault-tolerant control law is: Step 5) According to the running state of the multi-rotor unmanned aerial vehicle system, select appropriate parameters to complete the fault-tolerant control of it.

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