A rigid aircraft composite adaptive pre-determined time attitude fault-tolerant control method

Abstract

The application discloses a rigid aircraft composite adaptive predetermined time attitude fault-tolerant control method, constructs a rigid aircraft kinematics and dynamics model based on modified Rodrigues parameters, and considers the actuator fault problem in the model; constructs a virtual controller in the form of hyperbolic tangent, avoids the controller singularity problem caused by derivation of the virtual controller; adopts a fuzzy logic system to approximate nonlinear uncertain items of the system, and designs an adaptive law to estimate the upper limit of composite disturbance; on this basis, a composite adaptive predetermined time attitude fault-tolerant controller is designed. The application can realize that the rigid aircraft attitude converges to the neighborhood of the original point within a predetermined time under the conditions of uncertain rotational inertia, external disturbance and actuator fault, wherein the upper limit of the minimum convergence time can be predetermined by adjusting a simple parameter.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of rigid spacecraft attitude control, and aims at the problems of uncertain moment of inertia, external disturbance and actuator failure of rigid spacecraft system in complex space environment, and proposes a rigid spacecraft composite adaptive predetermined time attitude fault-tolerant control method. BACKGROUND

[0002] Aerospace technology is one of the most rapidly developing and most influential fields of science and technology in the 21st century, and is also an important embodiment of a country's scientific and technological level and comprehensive national power. As a core component of aerospace systems, rigid spacecraft is an important carrier of aerospace activities. Whether it is the docking of "Shenzhou 11" spaceship and "Tiangong 2" space station, or the construction of global GPS positioning and navigation system, the carrier cannot be separated from rigid spacecraft. Attitude control of spacecraft is the premise for normal work of the payload of spacecraft, and is also the fundamental guarantee for accurate completion of the task of spacecraft.

[0003] Attitude control of spacecraft refers to the process of keeping the spacecraft in a given direction with a certain accuracy or maneuvering from the original direction to the target direction relative to a certain reference coordinate system. However, when rigid spacecraft operates in complex space environment for a long time, it is inevitably affected by uncertain moment of inertia, external disturbance and actuator failure, which brings severe challenges to the realization of high-speed and high-precision attitude control. In order to improve the control performance of spacecraft system, finite time attitude control and fixed time attitude control are proposed and rapidly developed due to their excellent performance in rapid convergence and high control precision. These two control strategies significantly improve the dynamic response performance by realizing system stability in a limited time, and have attracted widespread attention in high-speed and high-precision application fields such as aerospace. However, these two control strategies have the following two limitations: (i) the convergence time is highly dependent on the initial conditions of the system, the larger the initial value, the longer the convergence time, which leads to poor performance of the control system in dealing with large range of initial errors; (ii) the convergence time is a complex function of multiple control parameters, and it is difficult to establish a clear relationship between the convergence time and the control parameters, which increases the complexity of system parameter tuning. For time-critical tasks such as spacecraft and space station rendezvous and docking, these limitations may directly affect the efficiency and success of the task.

[0004] To overcome the limitations of finite / fixed-time control, the concept of pre-defined time stabilization was further proposed, which has the key advantage that the minimum upper bound of the system convergence time can be accurately determined by adjusting only one control parameter. With this advantage, scholars have successfully applied pre-defined time control to practical systems such as robotic arms, aircraft, unmanned vehicles, servo motors, etc. However, the existing pre-defined time control research mainly has the following two problems: (i) overestimating the upper bound of the system convergence time, i.e. or 2Ts, Ts is the pre-defined time constant; (ii) introducing some piecewise continuous functions to solve the controller singularity problem, thereby increasing the complexity of stability analysis. Recently, scholars have proposed several pre-defined time standards, with the upper limit of the convergence time being Ts, instead of or 2Ts, which will be beneficial to practical applications.

[0005] On the other hand, rigid aircrafts are inevitably subject to actuator aging and failure problems during long-term operation in harsh environments, especially the failure of the attitude control system, which is often fatal. These problems may cause the pre-designed control law to fail, seriously affecting the subsequent attitude adjustment and control of the aircraft. Therefore, in the design of the control scheme, it is crucial to fully consider the fault risk and long-term reliability of the control system to ensure the safety and mission success of the aircraft. However, most of the existing aircraft attitude fault-tolerant control methods can only guarantee the asymptotic or finite-time convergence of the attitude, and it is difficult to achieve high-speed and high-precision convergence of the attitude under actuator failure.

[0006] Therefore, in view of the problems of uncertain moment of inertia, external disturbance and actuator failure faced by rigid aircraft systems during long-term operation in complex space environments, a composite adaptive pre-defined time attitude fault-tolerant control method for rigid aircrafts is designed, which can ensure the safety and reliability of the aircraft system while improving the steady-state control accuracy and transient response speed of the aircraft, and has important academic value and good application prospect. SUMMARY

[0007] In order to overcome the problems of uncertain moment of inertia, external disturbance and actuator failure existing in the existing rigid aircraft attitude control system, the application provides a compound adaptive predetermined time attitude fault-tolerant control method, first, the kinematics and dynamics models of the rigid aircraft are constructed based on the modified Rodrigues parameters, and the actuator failure problem is considered in the model; secondly, the virtual controller in the hyperbolic tangent form is constructed, which avoids the singularity problem of the controller caused by the derivation of the virtual controller; then, the fuzzy logic system is used to approximate the nonlinear uncertain items of the system, and the adaptive law is designed to estimate the upper bound of the compound disturbance. On this basis, a compound adaptive predetermined time attitude fault-tolerant controller is designed to ensure that the attitude of the aircraft can converge to the neighborhood of the origin within a predetermined time, wherein the convergence time can be predetermined by adjusting a simple parameter.

[0008] In order to solve the above technical problems, the technical scheme is as follows:

[0009] A rigid aircraft compound adaptive predetermined time attitude fault-tolerant control method, comprising the following steps:

[0010] Step 1, the kinematics and dynamics models of the rigid aircraft are established, the system state and the control parameters are initialized, and the process is as follows:

[0011] 1.1 The kinematics equation of the rigid aircraft system is:

[0012]

[0013] Wherein, σ=[σ1,σ2,σ3] T Indicates the attitude of the rigid aircraft; σ1, σ2, σ3 are respectively mapped on the x, y, z axes of the space rectangular coordinate system; Is the derivative of σ; ω∈R 3 Is the angular velocity of the rigid aircraft; G(σ)∈R 3×3 Is the Jacobian matrix, which is defined as I3 is R 3×3 Unit matrix; σ T Is the transpose of σ; σ × Indicates:

[0014]

[0015] 1.2 The dynamics equation of the rigid aircraft system is:

[0016]

[0017] Wherein, J∈R 3×3 Is the moment of inertia matrix of the aircraft; Is the angular acceleration of the aircraft; d∈R 3represents external disturbance, satisfying ||d||≤d m , d m is an unknown constant; ||d|| represents the two-norm of d; ω × represents

[0018]

[0019] E(t) = diag{e1(t), e2(t), e3(t)} is a 3x3 diagonal matrix, representing the actuator efficiency matrix, e i (t) ∈ (0, 1], i = 1, 2, 3 represents the health condition of the i-th actuator: when e i (t) = 1, it represents that the i-th actuator is normal; when 0 < e i (t) < 1, it represents that the i-th actuator is partially failed; when e i (t) = 0, it represents that the i-th actuator has completely lost control effect; u = [u1, u2, u3] T is the control torque; is the actuator bias fault, satisfying u m is an unknown constant; represents the two-norm of

[0020] 1.3 The moment of inertia matrix J satisfies J = J0 + ΔJ, where J0 ∈ R 3×3 and ΔJ ∈ R 3×3 respectively represent the nominal part and uncertain part of J, then formula (3) is rewritten as:

[0021]

[0022] Further, we get:

[0023]

[0024] Define The rigid aircraft kinematics and dynamics equations are re-described as:

[0025]

[0026] where, is the nonlinear uncertain part of the system, is regarded as the total disturbance part of the system;

[0027]

[0028] Step 2, for the rigid aircraft system with inertia uncertainty, external disturbance and actuator fault, based on the idea of backstepping recursion, the required virtual controller is designed;

[0029] Step 3, design a predetermined time adaptive fuzzy controller. Further, the procedure of step 2 is as follows:

[0030] Design a virtual controller α = [α1, α2, α3] T is:

[0031]

[0032]

[0033] where, T s > 0 is a predetermined time constant, 0 < γ < 1, κ > 0, b > 1, β1> 0, β2> 0 are normal numbers; sig γ (·) = |·| γ sgn(·), sgn(·) represents a sign function.

[0034] Further, the procedure of step 3 is as follows:

[0035] 3.1 Define a fuzzy logic system as:

[0036] Λ i = W i T S i (Z n ) + ε i , i = 1, 2, 3 (10)

[0037] where, is a system lumped uncertainty, is an input vector, S i (X n ) ∈ R 3 is a basis function, W i T is the transpose of W i , W i ∈ R 3 is a weight vector, ε i is an approximation error, satisfying |ε i | ≤ ε N , i = 1, 2, 3, ε N is a very small normal number;

[0038] 3.2 Define a compound disturbance D' = [D1', D2', D3'] T is:

[0039] D' = ε + D (11)

[0040] where, ε = [ε1, ε2, ε3] TThe composite interference D′ satisfies ||D′||≤d a d a It is an unknown positive constant;

[0041] 3.3 Considering the composite pre-set time controller is designed as follows:

[0042]

[0043] Where z = [z1, z2, z3] T =x2-α is the virtual error; Let W1 be a 3×3 diagonal matrix, ||W1||, ||W2||, ||W3|| be the L2 norms of W1, W2, and W3 respectively, and θ = max{||W1|| 2 ,||W2|| 2 ,||W3|| 2} is ||W1|| 2 ,||W2|| 2 ,||W3|| 2 The maximum value of d; l It is a positive integer that satisfies This is an estimate of θ. For d l The estimated value; All are positive constants, and their values ​​satisfy: τ > 0, h1 = k1.

[0044] 3.4 Design and The adaptive update laws are as follows:

[0045]

[0046] Where λ, m1, m2, c1, and c2 are all positive constants, and their values ​​satisfy: λ > 0. They are respectively The derivative of z; ||z|| represents the second norm of z.

[0047] Furthermore, the method also includes the following steps;

[0048] Step 4, Proof of stability at the predetermined time, the process is as follows:

[0049] 4.1 Based on the idea of ​​inversion recursion, the following Lyapunov function is designed:

[0050]

[0051] Differentiating equation (15), we get:

[0052]

[0053] 4.2 Based on the idea of backstepping recursion, the following Lyapunov function is designed:

[0054]

[0055] The derivative of formula (17) is obtained as follows:

[0056]

[0057] Wherein, And is the parameter estimation error, is a normal number;

[0058] 4.3 Based on the idea of backstepping recursion, the following Lyapunov function is designed:

[0059]

[0060] The derivative of formula (19) is obtained, and formula (13) and formula (14) are substituted into formula (19) to obtain:

[0061]

[0062] Wherein, is a normal number, and Θ (γ) = (γ / 2) · ((2-γ) / 2) (2- γ ) / γ is a normal number;

[0063] Based on the above analysis, the rigid aircraft attitude can converge to a small enough neighborhood within a given time T s .

[0064] The application designs a composite adaptive predetermined time attitude fault-tolerant control method for a rigid aircraft, designs a hyperbolic tangent form virtual controller based on the idea of backstepping recursion, avoids the singularity problem of the controller caused by the derivative of the virtual controller, estimates and compensates the system nonlinear uncertainty items including the rotational inertia uncertainty, external disturbance and actuator failure by introducing a fuzzy logic system, designs an adaptive law to estimate the system composite disturbance upper bound including the external disturbance and the approximation error of the fuzzy logic system, and designs a composite adaptive predetermined time attitude fault-tolerant controller on this basis, so that the rigid aircraft attitude can converge to a neighborhood near the origin within a given time, and the minimum convergence time upper bound can be determined in advance by adjusting a simple parameter.

[0065] The present technical concept: for the rigid aircraft system with uncertain moment of inertia, external disturbance and actuator failure, firstly, the kinematics and dynamics models of the rigid aircraft are constructed based on the modified Rodrigues parameters, and the actuator failure problem is considered in the model; secondly, the virtual controller in the form of hyperbolic tangent is constructed, which avoids the singularity problem of the controller caused by the derivation of the virtual controller; then, the fuzzy logic system is used to approximate the nonlinear uncertain terms of the system, and the adaptive law is designed to estimate the upper bound of the composite disturbance. On this basis, a composite adaptive pre-determined time attitude fault-tolerant controller is designed to ensure that the aircraft attitude can converge to the neighborhood of the origin within a predetermined time, and the convergence time can be predetermined by adjusting a simple parameter.

[0066] The beneficial effects of the present application are: in the case of rigid aircraft with uncertain moment of inertia, external disturbance and actuator failure, the aircraft attitude converges within a predetermined time, and the minimum convergence time upper bound is only related to one control parameter. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 The control flowchart of the present application is shown in the figure;

[0068] Figure 2 The rigid aircraft attitude graph under Ts=10, 15, 20 seconds is shown in the figure;

[0069] Figure 3 The rigid aircraft angular velocity graph under Ts=10, 15, 20 seconds is shown in the figure;

[0070] Figure 4 The rigid aircraft virtual error graph under Ts=10, 15, 20 seconds is shown in the figure;

[0071] Figure 5 The rigid aircraft control torque graph under Ts=10, 15, 20 seconds is shown in the figure;

[0072] Figure 6 The rigid aircraft attitude graph under Ts=15 seconds with different κ values is shown in the figure;

[0073] Figure 7 The rigid aircraft control torque graph under Ts=15 seconds with different κ values is shown in the figure. DETAILED DESCRIPTION

[0074] The present application will be further described below with reference to the accompanying drawings.

[0075] Reference Figures 1-7 A rigid aircraft composite adaptive pre-determined time attitude fault-tolerant control method, comprising the following steps:

[0076] Step 1, establish the kinematics and dynamics model of the rigid aircraft, initialize the system state and control parameters, the process is as follows:

[0077] 1.1 The kinematics equation of the rigid aircraft system is:

[0078]

[0079] where σ = [σ1, σ2, σ3] T represents the attitude of the rigid aircraft; σ1, σ2, σ3 are the values mapped on the space rectangular coordinate system x, y, z axes respectively; is the derivative of σ; ω ∈ R 3 is the angular velocity of the rigid aircraft; G(σ) ∈ R 3×3 is the Jacobian matrix, defined as I3 is the R 3×3 unit matrix; σ T is the transpose of σ; σ × represents:

[0080]

[0081] 1.2 The dynamics equation of the rigid aircraft system is:

[0082]

[0083] where J ∈ R 3×3 is the rotational inertia matrix of the aircraft; is the angular acceleration of the aircraft; d ∈ R 3 represents the external disturbance, satisfying ||d||≤d m , d m is an unknown normal number; ||d|| represents the two norm of d; ω × represents:

[0084]

[0085] E(t) = diag{e1(t), e2(t), e3(t)} is a 3 × 3 diagonal matrix, representing the actuator efficiency matrix, e i (t) ∈ (0, 1], i = 1, 2, 3 represents the health status of the i-th actuator: when e i (t) = 1, the i-th actuator is normal, when 0 < e i (t) < 1, the i-th actuator is partially failed, when e i (t) = 0, the i-th actuator completely loses control effect; u = [u1, u2, u3] T is the control torque; is the actuator bias fault, satisfying um is an unknown constant; denotes the two-norm of

[0086] 1.3 The inertia matrix J satisfies J = J0 + ΔJ, where J0 ∈ R 3×3 and ΔJ ∈ R 3×3 denote the nominal and uncertain parts of J, respectively, then equation (3) can be rewritten as:

[0087]

[0088] Further, we have

[0089]

[0090] Define The rigid aircraft kinematics and dynamics equations are rewritten as:

[0091]

[0092] where, is the system nonlinear uncertain part, is considered as the total disturbance part of the system;

[0093]

[0094] Step 2, for the rigid aircraft system with inertia uncertainty, external disturbance and actuator faults, based on the idea of backstepping recursion, the required virtual controller is designed as follows:

[0095] The virtual controller α = [α1, α2, α3] T is designed as:

[0096]

[0097]

[0098] where, T s > 0 is a predetermined time constant, 0 < γ < 1, κ > 0, b > 1, β1 > 0, β2 > 0 are normal numbers; sig γ (·) = |·| γ sgn(·), sgn(·) denotes the sign function;

[0099] Step 3, design a predetermined time adaptive fuzzy controller, the process is as follows:

[0100] 3.1 Define the fuzzy logic system as:

[0101] Λi = W i T S i (Z n )+ ε i , i = 1, 2, 3 (10)

[0102] where, is the system lumped uncertainty, is the input vector, S i (X n ) e R 3 is the basis function, W i T is the transpose of W i , W i e R 3 is the weight vector, ε i is the approximation error, satisfying | ε i | < ε N , i = 1, 2, 3, ε N is a small positive number;

[0103] 3.2 Define the composite disturbance D' = [D1', D2', D3'] T as:

[0104] D' = ε + D (11)

[0105] where, ε = [ε1, ε2, ε3] T , the composite disturbance D' satisfies ||D' || < d a , d a is an unknown positive number;

[0106] 3.3 Consider the composite scheduled time controller is designed as:

[0107]

[0108] where, z = [z1, z2, z3] T = x2- α is the virtual error; is a 3 x 3 diagonal matrix, ||W1||, ||W2||, ||W3|| are the 2-norms of W1, W2, W3, respectively, θ = max{||W1||, 2 ||W2|| 2 ||W3|| 2} is the maximum of ||W1||, 2 ||W2|| 2 ||W3|| 2 ; d l is a positive number satisfying is the estimate of θ, for d l the estimated value of d are all positive constants, and their values satisfy: τ > 0, h1 = k1,

[0109] 3.4 Design and The adaptive update laws of and are respectively:

[0110]

[0111] where λ, m1, m2, c1, c2 are all positive constants, and their values satisfy: λ > 0, are the derivatives of and respectively; ||z|| represents the two-norm of z;

[0112] Step 4, the predetermined time stability proof, the process is as follows:

[0113] 4.1 Based on the idea of inversion recursion, the following Lyapunov function is designed:

[0114]

[0115] Taking the derivative of equation (15), we get:

[0116]

[0117] 4.2 Based on the idea of inversion recursion, the following Lyapunov function is designed:

[0118]

[0119] Taking the derivative of equation (17), we get:

[0120]

[0121] where and are the parameter estimation errors, is a positive constant;

[0122] 4.3 Based on the idea of inversion recursion, the following Lyapunov function is designed:

[0123]

[0124] Taking the derivative of equation (19) and substituting equations (13) and (14), we get:

[0125]

[0126] where ​is a positive constant, Θ(γ) = (γ / 2) · ((2-γ) / 2) (2- γ ) / γ is a positive constant;

[0127] Based on the above analysis, the rigid spacecraft attitude can converge to a small enough neighborhood within a given time T s .

[0128] To verify the effectiveness of the proposed method, the method is simulated for rigid spacecraft system. System initialization conditions and control parameters are set as follows: the sampling step is 0.001 seconds; the system initial value is selected as: σ(0) = [0.2, 0.1, -0.1] T , ω(0) = [-1, 0, 1] T radian / second; the nominal part of the moment of inertia matrix J0 = [20, 1.2, 0.9; 1.2, 17, 1.4; 0.9, 1.4, 15] kg*m2, the uncertain part of the moment of inertia matrix ΔJ = diag[sin(0.1t), 2sin(0.2t), 3sin(0.3t)] kg*m2; the external disturbance d(t) = [0.2sin(0.1t), 0.3sin(0.2t), 0.5sin(0.2t)] T N*m, actuator bias fault N*m, actuator efficiency matrix E(t) = diag[0.85, 0.85, 0.85]; the virtual controller, the actual controller and the adaptive law parameters are set as follows: To verify the predetermined time convergence performance of the proposed control method, three different Ts values are selected, i.e. Ts = 10 seconds, 15 seconds and 20 seconds.

[0129] Figures 2-7 are the simulation results of the proposed rigid spacecraft composite adaptive predetermined time attitude fault-tolerant control method on system (7). Among them Figure 2 and Figure 3 are the rigid spacecraft attitude effect diagram and angular velocity effect diagram under Ts = 10, 15, 20 seconds, respectively, from which Figure 2 and Figure 3 it can be seen that the spacecraft attitude and angular velocity can quickly converge within a given time Ts. Figure 4 and Figure 5 are the virtual error effect diagram and control moment effect diagram of the rigid spacecraft under Ts = 10, 15, 20 seconds, respectively, from which Figure 4 and Figure 5 similar conclusions can be obtained. From Figures 2-5It can be seen that the smaller the Ts value, the faster the convergence speed of the aircraft system, but the larger the control torque. In order to further verify the influence of the control parameter κ on the system performance, we fix Ts = 15 seconds, and the corresponding simulation results are shown in Figs. 6 and 7. It can be seen from Figs. 6 and 7 that the smaller κ value can accelerate the convergence speed of the aircraft system, but also leads to a larger control torque. Therefore, the selection of Ts and κ should be balanced between the convergence speed and the control torque according to the actual task requirements. In summary, even in the case of considering the uncertainty of the moment of inertia of the rigid aircraft system, external disturbance and actuator failure, the proposed control method can still show good transient and steady-state performance. Figure 6 and 7 Figure 6 It can be seen that the smaller the Ts value, the faster the convergence speed of the aircraft system, but the larger the control torque. In order to further verify the influence of the control parameter κ on the system performance, we fix Ts = 15 seconds, and the corresponding simulation results are shown in Figs. 6 and 7. It can be seen from Figs. 6 and 7 that the smaller κ value can accelerate the convergence speed of the aircraft system, but also leads to a larger control torque. Therefore, the selection of Ts and κ should be balanced between the convergence speed and the control torque according to the actual task requirements. In summary, even in the case of considering the uncertainty of the moment of inertia of the rigid aircraft system, external disturbance and actuator failure, the proposed control method can still show good transient and steady-state performance.

[0130] The above is the excellent optimization effect of the embodiment of the present application. Obviously, the present application is not limited to the above embodiment, and various modifications can be made without departing from the spirit and scope of the present application. The control method designed by the present application has good control effect on the rigid aircraft system with uncertain moment of inertia, external disturbance and actuator failure, and can effectively improve the attitude control accuracy and rapidity of the system.​

Claims

1. A rigid aircraft composite adaptive pre-determined time attitude fault-tolerant control method, the method comprising the steps of: Step 1, establishing the kinematics and dynamics model of the rigid aircraft, initializing the system state and control parameters, the process is as follows: 1.1 The kinematic equation of the rigid aircraft system is: (1)； wherein represents the attitude of the rigid aircraft; are respectively the values mapped on the axes of a spatial orthogonal coordinate system ; is the derivative of ; is the angular velocity of the rigid aircraft; is the Jacobian matrix defined as ; is the identity matrix ; is the transpose of ; is represented as: (2)； 1.2 The dynamic equation of the rigid aircraft system is: (3)； wherein is the moment of inertia matrix of the aircraft; is the angular acceleration of the aircraft; denotes external disturbances, satisfying , is an unknown normal number; denotes the two-norm of ; denotes is given by: (4)； Let be a diagonal matrix, representing the actuator efficiency matrix, Let be the health status of the th actuator: when , the th actuator is normal, when , the th actuator is partially failed, and when , the th actuator has completely lost control effect; be the control torque; be the actuator bias fault, satisfying , be an unknown constant; be the two-norm of 1.3 Moment of inertia matrix satisfies where and denote the nominal and uncertain parts of respectively, then equation (3) is rewritten as: (5)； Further, we have: (6)； Definitions The rigid aircraft kinematics and dynamics equations are re-described as: (7)； wherein, is the system nonlinear uncertainty part, is considered as the total disturbance part of the system; ; Step 2, for the rigid aircraft system with inertia uncertainty, external disturbance and actuator fault, based on the backstepping recursive idea, the required virtual controller is designed; the process is as follows: Designing virtual controllers To: (8)； (9)； wherein , , , , , is a predetermined time constant, , is a positive number; , denotes the sign function; Step 3, design a predetermined time adaptive fuzzy controller.

2. A rigid aircraft composite adaptive pre-determined time attitude fault tolerant control method as in claim 1, characterized by, The process of step 3 is as follows: 3.1 Define the fuzzy logic system as: (10)； wherein, is the system lumped uncertainty, is the input vector, is the basis function, is the transpose of is the weight vector, is the approximation error, satisfying , is a very small positive number; 3.2 Defining composite interference is: (11)； wherein , complex interference satisfies , is an unknown constant; 3.3 Consider the composite predetermined time controller designed as: (12)； wherein is a virtual error; is a diagonal matrix, are respectively the two-norm of is the maximum value of is a positive constant satisfying ; is an estimate of is an estimate of are all positive constants, and their values satisfy: , , , , ; 3.4 Design and The adaptive update laws for and are respectively: (13)； (14)； wherein, are normal numbers, and their values satisfy: , , , , are the derivatives of respectively; denotes the two-norm of .

3. The rigid aircraft composite adaptive predetermined time attitude fault-tolerant control method according to claim 1, further comprising the following steps: Step 4, predetermined time stability proof, the process is as follows: 4.1 Based on the backstepping recursive idea, the first step is to design the following Lyapunov function: (15)； Take the derivative of formula (15) to get: (16)； 4.2 Based on the backstepping recursive idea, the second step is to design the following Lyapunov function: (17)； Take the derivative of formula (17) to get: (18)； wherein and is a parameter estimation error, is a constant; 4.3 Based on the backstepping recursive idea, the third step is to design the following Lyapunov function: (19)； Take the derivative of formula (19), and put formula (13) and formula (14) into it, to get: (20)； wherein is a normal number, is a normal number; Based on the above analysis, the rigid aircraft attitude is able to converge to a sufficiently small neighborhood within a pre-specified time .

Images