A quadrotor unmanned aerial vehicle sliding mode fault-tolerant control method with delay state constraint

Abstract

This invention discloses an active fault-tolerant control algorithm for unmanned aerial vehicles (UAVs) based on non-singular terminal fast sliding mode. Considering the actuator failure of a quadrotor UAV with state constraints, an active fault-tolerant control algorithm is proposed by combining a predetermined performance function with sliding mode control. An observer is designed for actuator failure, which can acquire accurate fault information of the system within a finite time. A predetermined performance function and a transfer function are designed according to the constraints, and an error transformation system is completed based on the fault value. A sliding mode surface is designed for the transformed system, thus ultimately forming a complete sliding mode fault-tolerant controller. This invention improves the convergence speed of the system by designing a non-singular fast terminal sliding mode surface; it ensures that the aircraft's state will not exceed the constraint boundary after a predetermined time using the predetermined performance function; and it allows the aircraft to enter the specified constraint range on time and complete path tracking in any initial state due to the introduction of the transfer function. This invention is used for fault-tolerant flight control of a class of quadrotor UAVs with state constraints and actuator failures.

Description

Technical Field

[0001] This invention relates to a sliding mode fault-tolerant control algorithm based on a predetermined performance function for a quadrotor unmanned aerial vehicle system with dynamic state constraints and actuator failures, belonging to the field of active fault-tolerant control technology for uncertain nonlinear systems. Background Technology

[0002] In recent decades, with the rapid development of computer technology, communication technology, materials technology, and other industries, quadcopter drone technology has also made great strides. Due to their small size, low cost, and flexible use, quadcopter drones are widely used in aerial photography, cargo transportation, military reconnaissance, grain dispersal, and many other fields, bringing great convenience to people's lives. However, the working environment of drones is complex, and they are prone to malfunctions after prolonged operation and wear. Due to their high coupling, even small faults or parameter perturbations can lead to crashes or even injuries. Therefore, research on fault-tolerant control algorithms for quadcopter drones has become a hot topic.

[0003] Fault-tolerant control can be broadly categorized into active fault tolerance and passive fault tolerance. Passive fault tolerance focuses on improving the robustness of the controller, ensuring system stability even when pre-estimated faults occur. However, passive fault tolerance lacks a fault diagnosis module and has many limitations. Active fault tolerance, on the other hand, requires online observation of fault information before reconstructing the control law based on the actual fault. Because active fault tolerance can adjust to the actual situation, it is more effective than passive fault tolerance and can handle a wider range of fault types. Therefore, active fault tolerance algorithms have greater advantages and are currently used in many applications.

[0004] In recent years, many researchers have made significant progress in the field of fault-tolerant control. They have designed numerous effective control algorithms, such as sliding mode control, neural network algorithms, and adaptive control, to address actuator failures in quadcopter UAVs, and have combined and improved these algorithms. Sliding mode control algorithms offer advantages such as strong robustness and fast response speed, demonstrating significant strength in handling external disturbances and system faults. Currently, they are often combined with fault diagnosis methods for fault-tolerant control of UAVs.

[0005] However, due to the complex working environment of UAVs or certain special operational requirements, it is necessary to constrain their states to ensure safe and efficient flight mission completion. For example, attitude angles and positions often have a constrained range that must not be exceeded during flight. Sliding mode control cannot effectively handle state constraints. To address this issue, some researchers have proposed the method of predefined performance functions. By constructing a predefined performance function, boundaries and trends are set for the independent variables, and new variables are introduced to complete the error transformation, converting the constrained system into an unconstrained system, thus ensuring that the independent variables do not exceed the constraint range. Combining predefined performance functions with sliding mode control can effectively solve the fault-tolerant control problem of UAVs with state constraints. Currently, many researchers are conducting research in this area. Summary of the Invention

[0006] Objective: To address the aforementioned research background, this invention proposes a novel sliding mode fault-tolerant control algorithm for quadrotor unmanned aerial vehicle (UAV) systems with state constraints and actuator faults. A finite-time observer is designed to quickly determine system parameter perturbations caused by faults. Considering the constraints present in actual flight missions, a predetermined performance function is designed to prevent excessive overshoot during the response process. New variables are introduced and error transformation is used to ensure compliance with constraints. A transfer function is also designed to relax the requirements for the UAV's initial state. To ensure faster convergence and better control performance, a non-singular fast terminal sliding mode surface is designed to replace the traditional sliding mode surface.

[0007] Technical Solution: A novel sliding mode fault-tolerant control method for quadrotor unmanned aerial vehicle (UAV) systems with actuator failure and state constraints. Its key features include: firstly, monitoring accurate fault values ​​using a finite-time observer; designing a predetermined performance function based on state constraints, and considering the initial state to design a transition function, allowing the system an adjustment period, and then completing the system transition accordingly; based on the fault information obtained from the observer, designing a non-singular fast terminal sliding mode to replace the traditional sliding surface for the transitioned system, and designing a control law, ultimately forming a fault-tolerant controller, including the following specific steps:

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

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

[0010]

[0011] Among them, X1=(x, y, z, φ, θ, ψ) T And X2 = (u, v, w, p, q, r) T Let U(t) represent the position and velocity state of the quadcopter UAV at time t; U(t) = [u1, u2, u3, u4]T To control the input, u i The value represents the square of the rotational speed of each propeller; D is the external disturbance experienced by the UAV, which has an unknown upper bound; F(X2) is a known continuous vector-valued function representing the influence of the UAV's inherent nonlinear dynamics on its state; G = diag{g1, g2, ..., g6} is the control channel, determined by the specific parameters of the UAV; L c Represents the input transformation matrix;

[0012]

[0013] Where l is the distance from the center of the propeller to the center of the fuselage, and c t c is the thrust coefficient of the propeller. d This is the torque coefficient of the propeller;

[0014] Step 1.2) Determine the fault model. In this invention, actuator faults are modeled as failure faults. When a fault occurs, the control input is U. f (t), the system model is shown in equation (3):

[0015]

[0016] Where I is a fourth-order identity matrix; E(t) = diag{σ1, σ2, σ3, σ4} is the failure matrix, σ i Let represent the failure rate of the i-th controller, and satisfy 0 ≤ σ. i <1; when σ i When σ = 0, it indicates that the i-th actuator is working normally; when 0 < σ i When <1, the i-th channel experiences partial failure but continues to function.

[0017] Step 1.3) Determine the fault information, separate the unknown terms in formula (3), and the part concerning X2(t) can be rewritten in the following form:

[0018]

[0019] Where, Γ1={l 11 , l 12 , ..., l 16 ] T Let Γ1 be the gain vector, where each element is a positive real number; all uncertainties are contained in δ, where δ = [δ1, δ2, ..., δ6]. T The observer is established as shown in equation (5):

[0020]

[0021] in, Let X2 be the observed value, and the observation error be defined as... A simple calculation can yield a value about X. e Auxiliary systems:

[0022]

[0023] D * As input, the corresponding observer is set as shown in equation (7):

[0024]

[0025] in, D represents * The observed values, Γ2, Γ3, and Γ4 are all gain vectors, where all elements are positive real numbers, and k1 and k2 are two positive odd numbers satisfying k1 < k2, sig p (x)=sgn(x)*|x| p sig p (X)=[sig p (x1), sig p (x2), ..., sig p (x n )] T Where sgn(·) represents the sign function, i.e.:

[0026]

[0027] Therefore, X in the auxiliary system e D * and can The internal convergence, among which, l3 and l4 represent the largest elements in Γ3 and Γ4 respectively. Then, the accurate fault information can be calculated according to formula (4) to complete the reconstruction.

[0028] Step 2) Determine the constraints and complete the error transformation, including the following steps:

[0029] Step 2.1) Design the predetermined performance function. Considering the complex working environment of the quadcopter UAV, which is prone to collisions or rollovers, it is necessary to constrain its attitude angles and position states. The constraint conditions are shown in Equation (9):

[0030]

[0031] in, Let these be the lower and upper bound vectors, respectively. Formula (9) shows that for any i = 1, 2, ..., 6, we have... They are respectively K (t), X1(t), The elements in the equation; the tracking error is defined as:

[0032] E e =X1-X d (10)

[0033] Among them, X d =[x 1d x 2d , ..., x 6d ] T For the set desired trajectory vector, E e = [e1, e2, ..., e6] T Combined with formulas (9) and (10), the corresponding predetermined performance functions are set as shown in formula (11):

[0034]

[0035] in, μ i0 μ i∞ , κ i and All are positive numbers;

[0036] Step 2.2) Design the transfer function. Since the initial state of the UAV may not be within the specified boundaries, a settling time is required. Let the settling time for each state be T. i The transfer function is established as shown in equation (12):

[0037]

[0038] in For an nth-order differentiable function, considering the adjustment stage, the constraints are modified as shown in equation (13):

[0039]

[0040] To ensure the system meets the constraints, design a monotonically increasing value with a range of [value missing]. The invertible function S between i (ε), and the tracking error is transformed as follows:

[0041]

[0042] For S i Taking the inverse of (ε) yields:

[0043]

[0044] Step 3) Design the sliding surface: To ensure that the UAV is not affected by the initial state and can be adjusted to the constraint range within a specified time, the transformation error ε must be ensured.i Since the sliding surface converges, the design is as shown in equation (16):

[0045]

[0046] Where, k i1 and k i2 For positive constants, p is used to avoid singularity. i and q i Satisfying 1 < q i <2 and q i <p i The definition of sig(·) is the same as in formula (7);

[0047] Step 4) Design the fault-tolerant control law. First, design the Lyapunov function as shown in equation (17):

[0048]

[0049] The sliding mode control law consists of two parts: the equivalent control law and the switching control law, namely:

[0050] u i =u ieq +u isw (18)

[0051] make The equivalent control law is obtained as follows:

[0052]

[0053] To enhance the robustness of the system and make it converge, let The switching control law is set as shown in equation (20):

[0054]

[0055] Among them, z i =v i e i , L e + =L e T (L e L e T ) -1 L represents e The generalized inverse, λ i and γ i All are positive numbers, representing gain;

[0056] Step 5) Select appropriate parameters based on the operating status of the multi-rotor UAV system to complete its fault-tolerant control.

[0057] Beneficial Effects: This invention proposes a sliding mode fault-tolerant control method for multi-rotor unmanned aerial vehicle (UAV) systems with state constraints and actuator failures. When a quadrotor UAV experiences partial actuator failure, a sliding mode fault-tolerant control method is proposed, combining a delayed predetermined performance function to ensure the UAV system can continue operating normally after an actuator failure. A finite-time observer is used to quickly acquire fault information. A predetermined performance function and transfer function are designed to define the constraint boundaries of the system state, preventing collisions or rollovers due to excessive overshoot. A non-singular fast terminal sliding mode surface is used instead of a traditional sliding mode surface to ensure high convergence speed and strong robustness. Specific advantages include:

[0058] (1) The designed finite-time observer can quickly and accurately diagnose fault values ​​and provide accurate information for controller reconfiguration, which makes the system less sensitive to faults and disturbances;

[0059] (2) Based on the system output error, a non-singular fast terminal sliding surface was designed to replace the traditional sliding surface, which makes the system response faster and more robust, and improves the dynamic performance of the control algorithm.

[0060] (3) Design a predetermined performance function and introduce a transfer function to introduce new variables to complete the transformation. Define the boundaries of each item to avoid the danger of rollover or collision. It is also more lenient in terms of initial state conditions than a general performance function.

[0061] The method proposed in this invention is a sliding mode fault-tolerant control method for actuator failure in a state-constrained quadrotor UAV system. It has certain application significance, is easy to implement, has good real-time performance and high accuracy, can effectively improve the safety of the control system, is highly operable, saves time, and is more efficient. It can be widely used in actuator fault-tolerant control of quadrotor UAV systems. Attached Figure Description

[0062] Figure 1 This is a flowchart of the method of the present invention;

[0063] Figure 2 This is a schematic diagram of a quadcopter model and its coordinate system;

[0064] Figure 3 This is a graph showing the X-axis tracking error when the Qball quadcopter drone's actuator malfunctions.

[0065] Figure 4 This is a graph showing the Y-axis tracking error when the Qball quadcopter drone's actuator malfunctions.

[0066] Figure 5 This is a graph showing the Z-axis tracking error when the Qball quadcopter drone's actuator malfunctions.

[0067] Figure 6 This is a curve showing the roll angle tracking error when the Qball quadcopter drone's actuator malfunctions.

[0068] Figure 7 This is a pitch tracking error curve of a Qball quadcopter UAV when the actuator fails;

[0069] Figure 8 This is a curve showing the yaw angle tracking error when the Qball quadcopter drone's actuator malfunctions. Detailed Implementation

[0070] The invention will now be further explained with reference to the accompanying drawings.

[0071] like Figure 1 As shown, a sliding mode fault-tolerant control method for a quadcopter UAV with partial actuator failure is characterized by: When a state-constrained UAV system experiences partial actuator failure, an active fault-tolerant control method combining a predetermined performance function with non-singular fast terminal sliding mode is proposed. This ensures the UAV system can operate normally when actuator failure occurs and that the state error remains within the specified constraint range under any initial state. A finite-time observer is constructed to quickly acquire fault and disturbance information. Based on this information, the control algorithm is reconstructed, ultimately forming a fault-tolerant controller. The specific steps include:

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

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

[0074]

[0075] Among them, X1=(x, y, z, φ, θ, ψ) T And X2 = (u, v, w, p, q, r) T Let U(t) represent the position and velocity state of the quadcopter UAV at time t; U(t) = [u1, u2, u3, u4] T To control the input, u i The value represents the square of the rotational speed of each propeller; D is the external disturbance experienced by the UAV, which has an unknown upper bound; F(X2) is a known continuous vector-valued function representing the influence of the UAV's inherent nonlinear dynamics on its state; G = diag{g1, g2, ..., g6} is the control channel, determined by the specific parameters of the UAV; L c Represents the input transformation matrix;

[0076]

[0077] Where l is the distance from the center of the propeller to the center of the fuselage, and c t c is the thrust coefficient of the propeller. d This is the torque coefficient of the propeller;

[0078] Step 1.2) Determine the fault model. In this invention, actuator faults are modeled as failure faults. When a fault occurs, the control input is U. f (t), the system model is shown in equation (3):

[0079]

[0080] Where I is a fourth-order identity matrix; E(t) = diag{σ1, σ2, σ3, σ4} is the failure matrix, σ i Let represent the failure rate of the i-th controller, and satisfy 0 ≤ σ. i <1; when σ i When σ = 0, it indicates that the i-th actuator is working normally; when 0 < σ i When <1, the i-th channel experiences partial failure but continues to function.

[0081] Step 1.3) Determine the fault information, separate the unknown terms in formula (3), and the part concerning X2(t) can be rewritten in the following form:

[0082]

[0083] Where, Γ1=[l 11 , l 12 , ..., l 16 ] T Let Γ1 be the gain vector, where each element is a positive real number; all uncertainties are contained in δ, where δ = [δ1, δ2, ..., δ6]. T The observer is established as shown in equation (5):

[0084]

[0085] in, Let X2 be the observed value, and the observation error be defined as... A simple calculation can yield a value about X. e Auxiliary systems:

[0086]

[0087] D * As input, the corresponding observer is set as shown in equation (7):

[0088]

[0089] in, D represents * The observed values, Γ2, Γ3, and Γ4 are all gain vectors, where all elements are positive real numbers, and k1 and k2 are two positive odd numbers satisfying k1 < k2, sig p (x)=sgn(x)*|x| p sig p (X)=[sig p (x1), sig p (x2), ..., sig p (x n )] T Where sgn(·) represents the sign function, i.e.:

[0090]

[0091] Therefore, X in the auxiliary system e D * and can The internal convergence, among which, l3 and l4 represent the largest elements in Γ3 and Γ4 respectively. Then, the accurate fault information can be calculated according to formula (4) to complete the reconstruction.

[0092] Step 2) Determine the constraints and complete the error transformation, including the following steps:

[0093] Step 2.1) Design the predetermined performance function. Considering the complex working environment of the quadcopter UAV, which is prone to collisions or rollovers, it is necessary to constrain its attitude angles and position states. The constraint conditions are shown in Equation (9):

[0094]

[0095] in, Let these be the lower and upper bound vectors, respectively. Formula (9) shows that for any i = 1, 2, ..., 6, we have... They are respectively The elements in the equation; the tracking error is defined as:

[0096] E e =X1-X d (10)

[0097] Among them, X d =[x 1d x 2d , ..., x 6d ] T For the set desired trajectory vector, E e = [e1, e2, ..., e6] TCombined with formulas (9) and (10), the corresponding predetermined performance functions are set as shown in formula (11):

[0098]

[0099] in, μ i0 μ i∞ , κ i and All are positive numbers;

[0100] Step 2.2) Design the transfer function. Since the initial state of the UAV may not be within the specified boundaries, a settling time is required. Let the settling time for each state be T. i The transfer function is established as shown in equation (12):

[0101]

[0102] in For an nth-order differentiable function, considering the adjustment stage, the constraints are modified as shown in equation (13):

[0103]

[0104] To ensure the system meets the constraints, design a monotonically increasing value with a range of [value missing]. The invertible function S between i (ε), and the tracking error is transformed as follows:

[0105]

[0106] For S i Taking the inverse of (ε) yields:

[0107]

[0108] Step 3) Design the sliding surface: To ensure that the UAV is not affected by the initial state and can be adjusted to the constraint range within a specified time, the transformation error ε must be ensured. i Since the sliding surface converges, the design is as shown in equation (16):

[0109]

[0110] Where, k i1 and k i2 For positive constants, p is used to avoid singularity. i and q i Satisfying 1 < q i <2 and q i <p iThe definition of sig(·) is the same as in formula (7);

[0111] Step 4) Design the fault-tolerant control law. First, design the Lyapunov function as shown in equation (17):

[0112]

[0113] The sliding mode control law consists of two parts: the equivalent control law and the switching control law, namely:

[0114] u i =u ieq +u isw (18)

[0115] make The equivalent control law is obtained as follows:

[0116]

[0117] To enhance the robustness of the system and make it converge, let The switching control law is set as shown in equation (20):

[0118]

[0119] Among them, z i =v i e i , L e + =L e T (L e L e T ) -1 L represents e The generalized inverse, λ i and γ i All are positive numbers, representing gain;

[0120] Step 5) Select appropriate parameters based on the operating status of the multi-rotor UAV system to complete its fault-tolerant control.

[0121] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

[0122] The effectiveness of the implementation plan is illustrated below using a real-world case simulation.

[0123] To verify the effectiveness of the method, the Qball quadcopter drone, a quadcopter drone controlled by Quanser Inc. of Canada, was used as the simulation verification object. Table 1 shows the relevant parameters of Qball.

[0124] To establish a dynamic model of the Qball aircraft, the structure of the quadcopter UAV Qball is simplified, and its simplified structure is as follows: Figure 2 As shown. The quadcopter UAV Qball uses a cross-shaped frame. To facilitate research, a body coordinate system O is established. b -X b Y b Z b and ground coordinate system O g -X g Y g Z g .

[0125] Table 1: Qball Body Parameter Values

[0126] parameter name value m Body mass 1.5kg ω Actuator bandwidth 15 rad / s <![CDATA[J z ]]> Yaw moment of inertia <![CDATA[0.08kg·m 2 ]]> <![CDATA[J x ]]> Rolling moment of inertia <![CDATA[0.04kg·m 2 ]]> <![CDATA[J y ]]> Pitch moment of inertia <![CDATA[0.04kg·m 2 ]]> <![CDATA[c t ]]> propeller thrust coefficient <![CDATA[2.35*10 -5 N·m 2 ]]> <![CDATA[c d ]]> propeller torque coefficient <![CDATA[5.7*10 -7 N·m·s 2 ]]> l Fuselage radius (half the wheelbase) 0.2m Tm Motor response time constant 0.011s

[0127] Typically, a quadcopter drone has six-dimensional variables: (X, Y, Z, ψ, θ, φ), where X, Y, and Z represent displacement 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 Let represent the Euler angular velocity and the linear velocity of displacement, respectively. According to the Newton-Euler formula, the dynamic equations of the system can be obtained:

[0128]

[0129] Where k φ k θ k ψ Let g be the tension coefficient and g be the acceleration due to gravity. The definition of its control quantity is as follows:

[0130]

[0131] Among them, u i (i = 1, 2, 3, 4) represents the square of the propeller speed.

[0132] To verify the effectiveness of the control law, based on the actual situation, we set the parameter in the control law as: α i =2,β i =1.67, k i1 =1,k i2=1, q=5. Considering that in practical applications, usually only the initial x and y coordinates of the aircraft may be outside the constraint range, requiring an adjustment time, this experiment only introduces transfer functions into the performance functions of the first two control channels, setting the initial adjustment time T. a The time interval is 1 second, and the transfer function is as follows:

[0133]

[0134] Where i = 1, 2.

[0135] Let the initial Euler angles and position coordinates be Θ(0) = [-0.2, 0.2, 0]. T (rad), P(0)=[-0.9, 1.9, 0] T (m), with initial linear velocity and angular velocity set as V(0) = [0, 0, 0]. T Ω(0) = [0, 0, 0] T The desired target position and orientation are set as follows: [x d y d , z d , ψ d ] T =[sin(0.2πt), cos(0.2πt), -0.3t, π / 3] T Considering disturbances during actual flight, the upper limit is set to... White noise is used as interference. Without loss of generality, taking motor 3 as an example, a 30% failure fault is injected starting from the 11th second, i.e., E = diag{0, 0, 0.3, 0}. Considering the constraints that the Qball UAV may encounter in actual flight conditions, the relevant parameters for each state performance function and transfer function are shown in Table 2:

[0136] Table 2: Relevant parameters of the performance function

[0137]

[0138] Where i = 1, 2, 3, j = 4, 5, and the constraints of the yaw angle are listed separately because they are different from those of other state variables.

[0139] To verify the superiority of the proposed algorithm, this method is compared with traditional sliding mode control methods. Figure 3-5 The curves show the position tracking error comparison between the two control methods under fault conditions. Figure 6-8 The curves show the attitude tracking error comparison between the two control methods under fault conditions.

[0140] Both displacement tracking and attitude tracking curves show that both methods can track the target curve and eventually converge. In comparison, the Qball UAV, under the control of the algorithm designed in this invention, can better suppress the impact of actuator faults. With the help of the fault observer, the control parameters can be reasonably adjusted according to the specific fault value, making it more targeted. Therefore, the tracking curve of the algorithm designed in this invention is smoother, and the oscillation amplitude after fault injection is smaller. Furthermore, this invention uses a fast terminal sliding surface and introduces a predetermined performance control method in the algorithm design. Therefore, the control algorithm in this chapter can effectively improve the dynamic performance of the system, with faster response speed and smaller overshoot. Moreover, from... Figure 3 and Figure 4 As can be seen, by introducing a transfer function to improve the predetermined performance control method, an initial settling time can be set for the UAV, effectively solving the problem that its initial state is not within the constraints. In summary, the fault-tolerant control method simulated in this case is effective for actuator failures in quadcopters with state constraints.

Claims

1. A sliding mode fault-tolerant control method for quadrotor unmanned aerial vehicles with actuator partial failure faults, characterized in that: When a state-constrained unmanned aerial vehicle (UAV) system experiences actuator failure, an active fault-tolerant control method combining a delayed predetermined performance function and non-singular fast terminal sliding mode is proposed. This method ensures the UAV system can operate normally even when actuator failure occurs and keeps the state error within the specified constraints under any initial state. A finite-time observer is constructed to quickly acquire fault and disturbance information. Based on this information, the control algorithm is reconstructed, ultimately forming a fault-tolerant controller. The specific steps include: Step 1) Determine the system model and fault information, including the following steps: Step 1.1) Determine the system model, as shown in equation (1): Among them, X1=(x, y, z, φ, θ, ψ) T And X2 = (u, v, w, p, q, r) T Let U(t) represent the position and velocity state of the quadcopter UAV at time t; U(t) = [u1, u2, u3, u4] T To control the input, u i The value represents the square of the rotational speed of each propeller; D is the external disturbance experienced by the UAV, which has an unknown upper bound; F(X2) is a known continuous vector-valued function representing the influence of the UAV's inherent nonlinear dynamics on its state; G = diag{g1, g2, ..., g6} is the control channel, determined by the specific parameters of the UAV; L c Represents the input transformation matrix; Where l is the distance from the center of the propeller to the center of the fuselage, and c t c is the thrust coefficient of the propeller. d This is the torque coefficient of the propeller; Step 1.2) Determine the fault model. In this invention, actuator faults are modeled as failure faults. When a fault occurs, the control input is U. f (t), the system model is shown in equation (3): Where I is a fourth-order identity matrix; E(t) = diag{σ1, σ2, σ3, σ4} is the failure matrix, σ i Let represent the failure rate of the i-th controller, and satisfy 0 ≤ σ. i <1; when σ i When σ = 0, it indicates that the i-th actuator is working normally; when 0 < σ i When <1, the i-th channel experiences partial failure but continues to function. Step 1.3) Determine the fault information, separate the unknown terms in formula (3), and the part concerning X2(t) can be rewritten in the following form: Where, Γ1=[l 11 , l 12 , ..., l 16 ] T Let Γ1 be the gain vector, where each element is a positive real number; all uncertainties are contained in δ, where δ = [δ1, δ2, ..., δ6]. T The observer is established as shown in equation (5): where represents the observed value of X2, the observation error is defined as A simple calculation can be obtained about X e auxiliary system: D * As input, set the corresponding observer, as shown in equation (7): in, D represents * The observed values, Γ2, Γ3, and Γ4 are all gain vectors, where all elements are positive real numbers, and k1 and k2 are two positive odd numbers that satisfy k l <k2, sig p (x)=sgn(x)*|x| p sig p (X)=[sig p (x1), sig p (x2), ..., sig p (x n )] T Where sgn(·) represents the sign function, i.e.: Therefore, X in the auxiliary system e D * and can The internal convergence, among which, l3 and l4 represent the largest elements in Γ3 and Γ4 respectively. Then, the accurate fault information can be calculated according to formula (4) to complete the reconstruction. Step 2) Determine the constraints and complete the error transformation, including the following steps: Step 2.1) Design the predetermined performance function. Considering the complex working environment of the quadcopter UAV, which is prone to collisions or rollovers, it is necessary to constrain its attitude angles and position states. The constraint conditions are shown in Equation (9): where denote the lower and upper bound vectors, respectively, and formula (9) indicates that for any i = 1, 2,..., 6 are given by K (t), X1(t), the elements in X(t); the tracking error is defined as: E e = X1- X d (10) Among them, X d =[x 1d x 2d , ..., x 6d ] T For the set desired trajectory vector, E e = [e1, e2, ..., e6] T Combined with formulas (9) and (10), the corresponding predetermined performance functions are set as shown in formula (11): wherein μ i0 , μ i∞ , κ i and are normal numbers; Step 2.2) Design the transfer function. Since the initial state of the UAV may not be within the specified boundaries, a settling time is required. Let the settling time for each state be T. i And establish the drift transfer function as shown in equation (12): where is an n-th order differentiable function, considering the adjustment phase, the constraint condition is modified as shown in equation (13): To ensure the system meets the constraints, design a monotonically increasing value with a range of [value missing]. The invertible function S between i (ε), and the tracking error is transformed as follows: S i (ε) Taking the inverse gives: Step 3) Design the sliding surface: To ensure that the UAV is not affected by the initial state and can be adjusted to the constraint range within a specified time, the transformation error ε must be ensured. i Since the sliding surface converges, the design is as shown in equation (16): Where, k i1 and k i2 For positive constants, p is used to avoid singularity. i and q i Satisfying 1 < q i <2 and q i <p i The definition of sig(·) is the same as in formula (7); Step 4) Design the fault-tolerant control law. First, design the Lyapunov function as shown in equation (17): The sliding mode control law consists of two parts: the equivalent control law and the switching control law, namely: u i = u ieq + u isw (18) Let The equivalent control law is obtained as follows: To enhance the robustness of the system and make it converge, let The switching control law is set as shown in equation (20): in, L e + =L e T (L e L e T ) -1 L represents e The generalized inverse, λ i and γ i All are positive numbers, representing gain; Step 5) Select appropriate parameters based on the operating status of the multi-rotor UAV system to complete its fault-tolerant control.

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