Rigid aircraft predetermined time adaptive neural network attitude stabilization method

Abstract

A rigid aircraft predetermined time adaptive neural network attitude stabilization method comprises the following steps: establishing kinematics and dynamics models of a rigid aircraft system, introducing a hysteresis quantizer to reduce buffeting in a quantized signal, and estimating and compensating lumped uncertain items of the system by using a neural network; a virtual controller is designed based on an inversion recursion idea, and the problem of controller singularity caused by derivation of the virtual controller is avoided; on the basis, a preset time adaptive neural network attitude controller is designed. According to the method, under the condition that inertia uncertainty, external interference and quantitative input exist, the rigid aircraft attitude is converged to the neighborhood near the original point within the preset time, and the upper bound of the minimum convergence time can be predetermined by adjusting a simple parameter.

Description

Technical Field

[0001] The present invention belongs to the technical field of rigid aircraft attitude control. Aiming at the inertial uncertainty, external interference and quantized input problems existing in the rigid aircraft system when operating in a complex space environment, a rigid aircraft predetermined time adaptive neural network attitude stabilization method with quantized input is proposed. Background Art

[0002] In recent years, the attitude control of rigid spacecraft has attracted extensive attention in the control community due to its wide application in space missions such as deep space exploration, rendezvous and docking, and earth observation. However, the highly nonlinear dynamics of the spacecraft itself, the complex space environment, and the increasing requirements for orbital activities have increased the difficulty of designing the spacecraft attitude controller. To this end, scholars have proposed many attitude control methods, such as PID control, sliding mode control, and adaptive control. However, the above attitude control methods can only guarantee the asymptotic convergence of the spacecraft attitude, which means that the time for the spacecraft attitude to converge to the equilibrium point may be infinite, which is difficult to meet the high-quality mission requirements of the spacecraft.

[0003] Compared with asymptotic control law, finite / fixed time control can provide faster convergence speed, higher control accuracy and better robustness to disturbances and uncertainties. To this end, finite / fixed time attitude control methods can improve the execution efficiency of rigid aircraft. However, the system convergence time of these two control methods cannot be preset, and the convergence time is a complex expression of controller parameters, which may be conservative and inaccurate for practical applications. In order to solve the above problems, scholars further proposed the concept of predetermined time stability. The biggest advantage of predetermined time control compared to finite / fixed time control is that it establishes a clear relationship between control parameters and convergence time, and realizes arbitrary adjustment of system convergence time. This method not only improves the convergence performance of the closed-loop system, but also can pre-set the convergence time of the system according to actual engineering needs. For example, in the spacecraft rendezvous and docking mission, there are usually clear requirements for the system convergence time. If the convergence time is too long, it may cause the mission to fail. However, the traditional predetermined time control method mainly avoids the controller singularity problem by introducing piecewise continuous functions, which increases the complexity of system stability analysis.

[0004] On the other hand, input quantization is an important consideration in vehicle control, especially for plug-and-play rigid micro-aircraft, which are composed of independent functional modules and transmit data through wireless networks. Due to the limited data transmission communication rate of low-cost wireless networks, communication delays and data packet loss may occur, resulting in system performance degradation. Quantized control has been widely used in networked control systems as an effective strategy to reduce the communication burden. However, most of the existing quantized control methods for rigid aircraft can only achieve asymptotic / finite time convergence of the aircraft system.

[0005] Therefore, in view of the inertial uncertainty, external interference and quantized input problems existing in the rigid aircraft system when operating in a complex space environment, a predetermined time adaptive neural network attitude stabilization method is designed to achieve the convergence of the aircraft attitude within a predetermined time and the minimum convergence time upper bound is only related to one control parameter. It has important academic value and good application prospects. Summary of the invention

[0006] In order to overcome the inertial uncertainty, external interference and quantized input problems existing in the existing rigid aircraft attitude control system, the present invention proposes a rigid aircraft predetermined time adaptive neural network attitude stabilization method. In the controller design process, a neural network is used to approximate the lumped uncertainty of the system, and a hysteresis quantizer is introduced to reduce the jitter in the quantized signal. On this basis, a predetermined time adaptive neural network attitude controller is designed to ensure that the aircraft attitude can converge to a neighborhood near the origin within a predetermined time, where the convergence time can be predetermined by adjusting a simple parameter.

[0007] The technical solutions proposed to solve the above technical problems are as follows:

[0008] A method for pre-setting time adaptive neural network attitude stabilization of a rigid aircraft comprises the following steps:

[0009] Step 1: Establish the kinematic and dynamic models of the rigid aircraft and initialize the system state and control parameters. The process is as follows:

[0010] 1.1 The kinematic equation of the rigid aircraft system is:

[0011]

[0012] Where σ=[σ1,σ2,σ3] T Represents the attitude of the rigid aircraft; σ1, σ2, σ3 are the values ​​mapped on the x, y, and z axes of the spatial rectangular coordinate system respectively; is the derivative of σ; ω∈R 3 is the angular velocity of the rigid vehicle; G(σ)∈R 3×3is the Jacobian matrix, defined as I3 is the 3×3 identity matrix; σ T is the transpose of σ; × It is expressed as:

[0013]

[0014] 1.2 The dynamic equation of the rigid aircraft system is:

[0015]

[0016] Among them, J∈R 3×3 is the rotational inertia matrix of the aircraft; is the angular acceleration of the aircraft; d∈R 3 is the external disturbance; ω × It is expressed as:

[0017]

[0018] Q(u(t))∈R 3 is the quantized value of the control torque, u(t)∈R 3 is the control torque, Q(u(t)) is defined as:

[0019]

[0020] Among them, u1(t), u2(t) and u3(t) are the components of the control torque u(t);

[0021] The hysteresis quantizer is introduced to reduce the jitter in the quantized signal. i (t)), i=1,2,3 is defined as:

[0022]

[0023] in, u min >0 means Q(u i (t)) dead zone range, ρ represents the measure of quantization density, sgn(u i (t)) represents the sign function, and for u i The derivative of (t);

[0024] To facilitate controller design, the quantized input Q(u(t)) is further rewritten as:

[0025]

[0026] Wherein, M(u)=diag{M(u1),M(u2),M(u3)} is a 3×3 diagonal matrix; is the approximate error; M(u i )(i=1,2,3) and They satisfy the following inequalities:

[0027] 1-δ≤M(u i )≤1+δ (8)

[0028]

[0029] 1.3 The rotational inertia matrix J satisfies J=J0+ΔJ, where J0 and ΔJ represent the nominal part and the uncertain part of J respectively. Then equation (3) can be rewritten as:

[0030]

[0031] Further we get:

[0032]

[0033] Define ξ1=σ, The kinematic and dynamic equations of the rigid aircraft are reconstructed as:

[0034]

[0035] in, is the system uncertainty,

[0036]

[0037] Step 2: For the rigid aircraft system with inertial uncertainty, external disturbance and quantized input, the required virtual controller is designed based on the inverse recursive idea. The process is as follows:

[0038] Design virtual controller α=[α1,α2,α3] T for:

[0039] α=-k1ξ1-k2sig 1+γ (ξ1) (13)

[0040] Among them, 0<γ<1, k1=k 11 +k 12 , β1>0,β2>0 are positive constants;

[0041] Step 3, design a scheduled time adaptive neural network controller.

[0042] Further, the process of step 3 is as follows:

[0043] 3.1 Define neural network as:

[0044] Υi =W i T Φ i (Z n )+ε i (14)

[0045] in, is the total uncertainty of the system, is the input vector, Φ i (Z n )∈R 3 is the basis function, W i T W i The transpose of W i ∈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 very small positive constant;

[0046] 3.2 Considering the scheduled time the controller is designed to:

[0047]

[0048] Where z = [z1, z2, z3] T =ξ2-α is the intermediate error; H=diag{1 / (1-δ),1 / (1-δ),1 / (1-δ)} is a 3×3 diagonal matrix, is a 3×3 diagonal matrix, ||W1||,||W2||,||W3|| are the bi-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 is the estimated value of θ; τ,λ1, λ2, λ3 are all positive numbers, and their values ​​satisfy: τ>0, sig 1-γ (z)=[|z1| 1-γ sgn(z1),|z2| 1-γ sgn(z2),|z3| 1-γ sgn(z3)] T , sig 1+γ (z)=[|z1| 1+γ sgn(z1),|z2|1+γ sgn(z2),|z3| 1+γ sgn(z3)] T ; sgn(z1), sgn(z2), sgn(z3) are all symbolic functions;

[0049] 3.3 Design the adaptive update law as:

[0050]

[0051] in, for The derivative of; κ, c, m are all positive constants, and their values ​​satisfy: κ>0,

[0052] Furthermore, the method further comprises the following steps:

[0053] Step 4, proof of stability over a predetermined period of time, the process is as follows:

[0054] 4.1 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0055]

[0056] By taking the derivative of formula (17), we can obtain:

[0057]

[0058] 4.2 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0059]

[0060] By taking the derivative of formula (19), we can obtain:

[0061]

[0062] in, is the parameter estimation error; is a positive constant;

[0063] 4.3 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0064]

[0065] By taking the derivative of formula (21), we can obtain:

[0066]

[0067] in, is a positive constant;

[0068] Based on the above analysis, the rigid aircraft attitude can be Converges to a small enough neighborhood.

[0069] The present invention designs a predetermined time adaptive neural network attitude stabilization method for a rigid aircraft with quantized input, constructs a hysteresis quantizer to reduce the jitter in the quantized signal, introduces a neural network to estimate and compensate for the system lumped uncertainty including inertial uncertainty, external interference and quantized input, designs a virtual controller based on the inverse recursive idea, and avoids the controller singularity problem caused by the derivation of the virtual controller. On this basis, a predetermined time adaptive neural network attitude controller is designed to achieve the rigid aircraft attitude being able to converge to a neighborhood near the origin within a predetermined time, wherein the upper limit of the minimum convergence time can be predetermined by adjusting a simple parameter.

[0070] The technical concept of the present invention is as follows: for a rigid aircraft system with inertial uncertainty, external interference and quantized input, the kinematic and dynamic models of the rigid aircraft system are first established, a hysteresis quantizer is introduced to reduce the jitter in the quantized signal, and a neural network approximation system is used to estimate and compensate for the system lumped uncertainty including inertial uncertainty, external interference and quantized input. On this basis, a predetermined time adaptive neural network attitude controller is designed to enable the rigid aircraft attitude to converge to a neighborhood near the origin within a predetermined time, where the upper limit of the minimum convergence time can be predetermined by adjusting a simple parameter.

[0071] The beneficial effects of the present invention are: in the presence of inertial uncertainty, external interference and quantized input of a rigid aircraft, the aircraft attitude can be converged within a predetermined time, and the upper limit of the minimum convergence time is only related to one control parameter. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 is a control flow chart of the present invention;

[0073] Figure 2 is the attitude diagram of the rigid aircraft at Tc = 6 seconds;

[0074] Figure 3 is the attitude diagram of the rigid aircraft at Tc = 10 seconds;

[0075] Figure 4 is the angular velocity diagram of the rigid aircraft at Tc = 6 seconds;

[0076] Figure 5 is the angular velocity diagram of the rigid aircraft at Tc = 10 seconds;

[0077] Figure 6 Quantified control torque diagram for a rigid aircraft at Tc = 6 seconds;

[0078] Figure 7 Quantified control torque diagram for a rigid vehicle at Tc = 10 seconds. DETAILED DESCRIPTION

[0079] The present invention will be further described below in conjunction with the accompanying drawings.

[0080] Reference Figure 1-Figure 7 , a method for pre-setting time adaptive neural network attitude stabilization of a rigid aircraft, comprising the following steps:

[0081] Step 1: Establish the kinematic and dynamic models of the rigid aircraft and initialize the system state and control parameters. The process is as follows:

[0082] 1.1 The kinematic equation of the rigid aircraft system is:

[0083]

[0084] Where σ=[σ1,σ2,σ3] T Represents the attitude of the rigid aircraft; σ1, σ2, σ3 are the values ​​mapped on the x, y, and z axes of the spatial rectangular coordinate system respectively; is the derivative of σ; ω∈R 3 is the angular velocity of the rigid vehicle; G(σ)∈R 3×3 is the Jacobian matrix, defined as I3 is the 3×3 identity matrix; σ T is the transpose of σ; × It is expressed as:

[0085]

[0086] 1.2 The dynamic equation of the rigid aircraft system is:

[0087]

[0088] Among them, J∈R 3×3 is the rotational inertia matrix of the aircraft; is the angular acceleration of the aircraft; d∈R 3 is the external disturbance; ω × It is expressed as:

[0089]

[0090] Q(u(t))∈R 3 is the quantized value of the control torque, u(t)∈R 3 is the control torque, Q(u(t)) is defined as:

[0091]

[0092] Among them, u1(t), u2(t) and u3(t) are the components of the control torque u(t);

[0093] The hysteresis quantizer is introduced to reduce the jitter in the quantized signal. i (t)), i=1,2,3 is defined as:

[0094]

[0095] in, u min >0 means Q(u i (t)) dead zone range, ρ represents the measure of quantization density, sgn(u i (t)) represents the sign function, and for u i The derivative of (t);

[0096] To facilitate controller design, the quantized input Q(u(t)) is further rewritten as:

[0097]

[0098] Wherein, M(u)=diag{M(u1),M(u2),M(u3)} is a 3×3 diagonal matrix; is the approximate error; M(u i )(i=1,2,3) and They satisfy the following inequalities:

[0099] 1-δ≤M(u i )≤1+δ (8)

[0100]

[0101] 1.3 The rotational inertia matrix J satisfies J=J0+ΔJ, where J0 and ΔJ represent the nominal part and the uncertain part of J respectively. Then equation (3) can be rewritten as:

[0102]

[0103] Further we get:

[0104]

[0105] Define ξ1=σ, The kinematic and dynamic equations of the rigid aircraft are reconstructed as:

[0106]

[0107] in, is the system uncertainty,

[0108] Step 2: For a rigid aircraft system with inertial uncertainty, external disturbances, and quantized inputs, the required virtual controller is designed based on the inverse recursive idea. The process is as follows:

[0109] Design virtual controller α=[α1,α2,α3] T for:

[0110] α=-k1ξ1-k2sig 1+γ (ξ1) (13)

[0111] Among them, 0<γ<1, k1=k 11 +k 12 , β1>0,β2>0 are positive constants;

[0112] Step 3, design a scheduled time adaptive neural network controller, the process is as follows:

[0113] 3.1 Define neural network as:

[0114] Υ i =W i T Φ i (Z n )+ε i (14)

[0115] in, is the total uncertainty of the system, is the input vector, Φ i (Z n )∈R 3 is the basis function, W i T W i The transpose of W i ∈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 very small positive constant;

[0116] 3.2 Considering the scheduled time the controller is designed to:

[0117]

[0118] Where z = [z1, z2, z3] T=ξ2-α is the intermediate error; H=diag{1 / (1-δ),1 / (1-δ),1 / (1-δ)} is a 3×3 diagonal matrix, is a 3×3 diagonal matrix, ||W1||,||W2||,||W3|| are the bi-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 is the estimated value of θ; τ,λ1, λ2, λ3 are all positive numbers, and their values ​​satisfy: τ>0, sig 1-γ (z)=[|z1| 1-γ sgn(z1),|z2| 1-γ sgn(z2),|z3| 1-γ sgn(z3)] T , sig 1+γ (z)=[|z1| 1+γ sgn(z1),|z2| 1+γ sgn(z2),|z3| 1+γ sgn(z3)] T ; sgn(z1), sgn(z2), sgn(z3) are all symbolic functions;

[0119] 3.3 Design the adaptive update law as:

[0120]

[0121] in, for The derivative of; κ, c, m are all positive constants, and their values ​​satisfy: κ>0,

[0122]

[0123] Step 4, proof of stability over a predetermined period of time, the process is as follows:

[0124] 4.1 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0125]

[0126] By taking the derivative of formula (17), we can obtain:

[0127]

[0128] 4.2 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0129]

[0130] By taking the derivative of formula (19), we can obtain:

[0131]

[0132] in, is the parameter estimation error; is a positive constant;

[0133] 4.3 Based on the inversion recursive idea, the following Lyapunov function is designed:

[0134]

[0135] By taking the derivative of formula (21), we can obtain:

[0136]

[0137] in, is a positive constant;

[0138] Based on the above analysis, the rigid aircraft attitude can be Converges to a small enough neighborhood.

[0139] To verify the effectiveness of the proposed method, this method is simulated for a rigid aircraft system. The 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) = [-0.037, 0.049, 0.061] T ,ω(0)=[0,0,0] T radians / second; the nominal part of the rotational 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 rotational inertia matrix ΔJ = diag[sin(0.1t), 2sin(0.2t), 3sin(0.3t)] kg*m2; the external disturbance d(t) = (||ω|| 2 +0.05)*[sin(0.8t),0.3cos(0.5t),sin(0.3t)] T Newton*m; virtual controller, actual controller and adaptive law parameters are set to β1 = 1, β2 = 1, κ = 1, τ = 1. In order to verify the scheduled time convergence performance of the proposed control method, two different sets of Tc values ​​are selected, namely Tc = 6 seconds and Tc = 10 seconds.

[0140] Figure 2-Figure 7 The figure is a simulation result of the proposed rigid aircraft predetermined time adaptive neural network attitude stabilization method with quantized input on system (12). Figure 2 and Figure 3 They are the attitude effect diagrams of the rigid aircraft under Tc = 6 seconds and Tc = 10 seconds. Figure 2 It can be seen that the actual stabilization time of the aircraft attitude is 5 seconds, which is less than the expected time. from Figure 3 It can be seen that when Tc is set to 10 seconds, the actual attitude stabilization time increases to 7 seconds; Figure 2 and Figure 3 It can be concluded that by adjusting a parameter Tc, the minimum upper bound of the attitude stabilization time can be determined in advance, and a smaller Tc value will increase the convergence speed of the attitude; Figure 4 and Figure 5 They are the angular velocity effect diagrams of the rigid aircraft under Tc = 6 seconds and Tc = 10 seconds. Figure 4 and Figure 5 A similar conclusion can be drawn again, that is, the actual convergence time of the angular velocity is always shorter than the predetermined time Figure 6 and Figure 7 They are the quantized control torque effect diagrams of the rigid aircraft under Tc = 6 seconds and Tc = 10 seconds. Figure 6 and Figure 7 It can be seen that the smaller the Tc value, the greater the control gain. Therefore, the convergence speed and control torque should be weighed when selecting Tc. In summary, even when the inertial uncertainty, external disturbance and quantized input of the rigid aircraft system are taken into account, the proposed control method can still show good control performance, including convergence within the predetermined time and high-precision control.

[0141] The above is an excellent optimization effect of an embodiment of the present invention. Obviously, the present invention is not limited to the above embodiment, and various modifications can be made to it without departing from the basic spirit of the present invention and not exceeding the scope of the essential content of the present invention. The control method designed by the present invention has a good control effect on the rigid aircraft system with inertial uncertainty, external interference and quantized input, and can effectively improve the attitude control accuracy and rapidity of the system.

Claims

1. A method for attitude stabilization of a rigid aircraft using a predetermined time adaptive neural network, characterized in that the method comprises the following steps: Step 1: Establish the kinematic and dynamic models of the rigid aircraft and initialize the system state and control parameters. The process is as follows: 1.1 The kinematic equation of the rigid aircraft system is: Where σ=[σ1,σ2,σ3] T Represents the attitude of the rigid aircraft; σ1, σ2, σ3 are the values ​​mapped on the x, y, and z axes of the spatial rectangular coordinate system respectively; is the derivative of σ; ω∈R 3 is the angular velocity of the rigid vehicle; G(σ)∈R 3×3 is the Jacobian matrix, defined as I3 is the 3×3 identity matrix; σ T is the transpose of σ; × It is expressed as: 1.2 The dynamic equation of the rigid aircraft system is: Among them, J∈R 3×3 is the rotational inertia matrix of the aircraft; is the angular acceleration of the aircraft; d∈R 3 is the external disturbance; ω × It is expressed as: Q(u(t))∈R 3 is the quantized value of the control torque, u(t)∈R 3 is the control torque, Q(u(t)) is defined as: Among them, u1(t), u2(t) and u3(t) are the components of the control torque u(t); The hysteresis quantizer is introduced to reduce the jitter in the quantized signal. i (t)) is defined as, i = 1, 2, 3: in, u min represents Q(u i (t)) dead zone range, u min > 0, ρ represents the measure of quantized density, sgn(u i (t)) represents the sign function, and for ui The derivative of (t); To facilitate controller design, the quantized input Q(u(t)) is further rewritten as: Wherein, M(u)=diag{M(u1),M(u2),M(u3)} is a 3×3 diagonal matrix; is the approximate error; M(u i )and Satisfy the following inequalities respectively, i=1,2,3: 1-δ≤M(u i )≤1+δ (8) 1.3 The rotational inertia matrix J satisfies J=J0+ΔJ, where J0 and ΔJ represent the nominal part and the uncertain part of J respectively. Then equation (3) can be rewritten as: Further we get: definition The kinematic and dynamic equations of the rigid aircraft are reconstructed as: in, is the system uncertainty, Step 2: For a rigid aircraft system with inertial uncertainty, external disturbances, and quantized inputs, the required virtual controller is designed based on the inverse recursive idea. The process is as follows: Design virtual controller α=[α1,α2,α3] T for: α=-k1ξ1-k2sig 1+γ (ξ1) (13) Among them, 0<γ<1, k1=k 11 +k 12 , is a positive constant; Step 3, design a scheduled time adaptive neural network controller.

2. The method for attitude stabilization of a rigid aircraft using a predetermined time adaptive neural network as claimed in claim 1, wherein the process of step 3 is as follows: 3.1 Define neural network as: c i =W i T F i (Z n )+e i (14) in, is the total uncertainty of the system, i=1,2,3, is the input vector, Φ i (Z n )∈R 3 is the basis function, W i T W i The transpose of W i ∈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 very small positive constant; 3.2 Considering the scheduled time the controller is designed to: Where z = [z1, z2, z3] T =ξ2-α is the intermediate error; H=diag{1 / (1-δ),1 / (1-δ),1 / (1-δ)} is a 3×3 diagonal matrix, is a 3×3 diagonal matrix, ||W1||,||W2||,||W3|| are the bi-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 is the estimated value of θ; τ,λ1, λ2, λ3 are all positive numbers, and their values ​​satisfy: τ>0, sig 1-γ (z)=[|z1| 1-γ sgn(z1),|z2| 1-γ sgn(z2),|z3| 1-γ sgn(z3)] T , sig 1+γ (z)=[|z1| 1+γ sgn(z1),|z2| 1+γ sgn(z2),|z3| 1+γ sgn(z3)] T ; sgn(z1), sgn(z2), sgn(z3) are all symbolic functions; 3.3 Design the adaptive update law as: in, for The derivative of; κ, c, m are all positive constants, and their values ​​satisfy: κ>0, 3. A method for pre-setting time adaptive neural network attitude stabilization of a rigid aircraft as claimed in claim 1 or 2, the method further comprising the following steps: Step 4, proof of stability over a predetermined period of time, the process is as follows: 4.1 Based on the inversion recursive idea, the following Lyapunov function is designed: By taking the derivative of formula (17), we can obtain: 4.2 Based on the inversion recursive idea, the following Lyapunov function is designed: By taking the derivative of formula (19), we can obtain: in, is the parameter estimation error; is a positive constant; 4.3 Based on the inversion recursive idea, the following Lyapunov function is designed: By taking the derivative of formula (21), we can obtain: in, is a positive constant; Based on the above analysis, the rigid aircraft attitude can be Converges to a small enough neighborhood.

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