CN109188910B - Adaptive neural network fault-tolerant tracking control method of rigid aircraft - Google Patents

Abstract

A self-adaptive neural network fault-tolerant tracking control method of a rigid aircraft is used for designing a fixed time sliding mode surface aiming at the attitude tracking problem of the rigid aircraft with centralized uncertainty and ensuring the fixed time convergence of the state; a neural network is introduced to approximate a total uncertain function, and an adaptive fixed time controller is designed. The method realizes the final bounded control of consistent fixed time of the attitude tracking error and the angular velocity error of the aircraft system under the factors of external interference, uncertain rotational inertia, actuator saturation and faults.

Description

Adaptive neural network fault-tolerant tracking control method of rigid aircraft

Technical Field

The invention relates to a self-adaptive neural network fault-tolerant tracking control method for a rigid aircraft, in particular to a rigid aircraft attitude tracking method with external interference, uncertain rotational inertia matrix, saturated actuator and faults.

Background

Rigid aircraft attitude control systems play an important role in the healthy, reliable movement of rigid aircraft. In a complex aerospace environment, a rigid aircraft attitude control system can be influenced by various external interferences and faults such as aging and failure of the rigid aircraft during long-term continuous tasks. In order to maintain the performance of the system effectively, it is necessary to make it robust against external interference and actuator failure; in addition, the rigid aircraft has uncertain rotational inertia matrix, so that the control saturation is also a problem which often occurs to the aircraft. In summary, when the rigid aircraft performs a task, a fault-tolerant control method with high precision and stable convergence of the system in a short time is needed.

Sliding mode control is considered to be an effective robust control method in solving system uncertainty and external disturbances. The sliding mode control method has the advantages of simple algorithm, high response speed, strong robustness to external noise interference and parameter perturbation and the like. Terminal sliding mode control is an improvement over conventional sliding mode control, which can achieve limited time stability. However, existing limited time techniques to estimate convergence time require knowledge of the initial information of the system, which is difficult for the designer to know. In recent years, a fixed time technique has been widely used, and a fixed time control method has an advantage of conservatively estimating the convergence time of a system without knowing initial information of the system, as compared with an existing limited time control method.

The neural network is one of linear parameterized approximation methods and can be replaced by any other approximation method, such as an RBF neural network, a fuzzy logic system, and the like. By utilizing the property that a neural network approaches uncertainty and effectively combining a fixed time sliding mode control technology, the influence of external interference and system parameter uncertainty on the system control performance is reduced, and the fixed time control of the attitude of the rigid aircraft is realized.

Disclosure of Invention

In order to solve the problem of unknown nonlinearity of the existing attitude control system of the rigid aircraft, the invention provides a fault-tolerant tracking control method of an adaptive neural network of the rigid aircraft, and the control method realizes the fixed time consistency and the final bounded control method of the system state under the conditions of external interference, uncertain rotational inertia, saturated actuator and fault of the system.

The technical scheme proposed for solving the technical problems is as follows:

a self-adaptive neural network fault-tolerant tracking control method for a rigid aircraft comprises the following steps:

step 1, establishing a kinematics and dynamics model of a rigid aircraft, initializing system states and control parameters, and carrying out the following processes:

1.1 the kinematic equation for a rigid aircraft system is:

wherein q is_v＝[q₁,q₂,q₃]^TAnd q is₄Vector part and scalar part of unit quaternion respectively and satisfy

q₁,q₂,q₃Respectively mapping values on x, y and z axes of a space rectangular coordinate system;

are each q_vAnd q is₄A derivative of (a); omega belongs to R³Is the angular velocity of the rigid aircraft; i is₃Is R^3×3An identity matrix;

expressed as:

1.2 the kinetic equation for a rigid aircraft system is:

wherein J ∈ R^3×3Is the rotational inertia matrix of the rigid aircraft;

is the angular acceleration of the rigid aircraft; u ═ u₁,u₂,u₃]^T∈R³And d ∈ R³Control moment and external disturbance; d ═ diag (D)₁,D₂,D₃)∈R^3×3Is an actuator efficiency matrix with 3 multiplied by 3 symmetrical opposite angles, and satisfies that D is more than 0_i(t)≤1,i＝1,2,3；sat(u)＝[sat(u₁),sat(u₂),sat(u₃)]^TActual control moment, sat (u), generated for the actuator_i) Is an actuator with saturation characteristics, denoted sat (u)_i)＝sgn(u_i)min{u_mi,|u_i|}， u_miFor maximum available control torque, sgn (u)_i) Is a sign function, min { u }_mi,|u_i| is the minimum of the two; to represent control constraints, sat (u) is denoted as sat (u) ═ g (u) + d_s(u)， g(u)＝[g₁(u₁),g₂(u₂),g₃(u₃)]^T，g_i(u_i) As a function of hyperbolic tangent

d_s(u)＝[d_s1(u₁),d_s2(u₂),d_s3(u₃)]^TIs an approximate error vector; according to the median theorem, g_i(u_i)＝m_iu_i， 0＜m_iLess than or equal to 1; definition of H ═ DM ═ diag (δ)₁m₁,δ₂m₂,δ₃m₃)∈R^3×3Is a 3X 3 symmetric diagonal matrix, M ═ diag (M)₁,m₂,m₃)∈R^3×3Is a 3 multiplied by 3 symmetric diagonal matrix; dsat (u) is re-expressed as: dsat (u) ═ Hu + Dd_s(u) satisfy0＜h₀≤D_im_i≤1,i＝1,2,3，h₀Is an unknown normal number; omega^×Expressed as:

1.3 the desired kinematic equation for a rigid aircraft system is:

wherein q is_dv＝[q_d1,q_d2,q_d3]^TAnd q is_d4A vector part and a scalar part which are respectively a desired unit quaternion and satisfy

Ω_d∈R³A desired angular velocity;

are each q_dv,q_d4The derivative of (a) of (b),

is q_dvTransposing;

expressed as:

1.4 relative attitude motion of rigid aircraft described by quaternion:

Ω_e＝Ω-CΩ_d (12)

wherein e_v＝[e₁,e₂,e₃]^TAnd e₄A vector part and a scalar part of the attitude tracking error respectively; omega_e＝[Ω_e1,Ω_e2,Ω_e3]^T∈R³Is the angular velocity error;

is a corresponding directional cosine matrix and satisfies | | | C | | | | | | | ═ 1 and

is the derivative of C;

according to equations (1) - (12), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

wherein

And

are each e_vAnd e₄A derivative of (a);

is e_vTransposing;

and

are respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;

and

respectively expressed as:

1.5 rotational inertia matrix J satisfies J ═ J₀+ Δ J, wherein J₀And Δ J represents the nominal and indeterminate portions of J, respectively, equation (15) is rewritten as:

further obtaining:

1.6 differentiating the formula (13) gives:

wherein

Is e_vThe second derivative of (a);

step 2, aiming at a rigid aircraft system with external disturbance, uncertain rotational inertia, saturated actuator and fault, designing a required sliding mode surface, and comprising the following steps:

selecting a sliding mode surface S ═ S at fixed time₁,S₂,S₃]^T∈R³Comprises the following steps:

wherein the content of the first and second substances,

λ₁and λ₂Is a normal number; r is₁＝a₁/b₁，a₁,b₁Is a normal number, satisfies a₁＞b₁，i＝1,2,3；sgn(e₁),sgn(e₂),sgn(e₃) Are all sign functions; s_au＝[S_au1,S_au2,S_au3]^TExpressed as:

wherein

r₂＝a₂/b₂，a₂,b₂Is positive odd number, satisfies a₂＜b₂；

0＜r₂Less than 1, epsilon is a very small normal number;

step 3, designing a neural network fixed time controller, and the process is as follows:

3.1 define the neural network as:

G_i(X_i)＝W_i ^*TΦ(X_i)+ε_i (23)

wherein G ═ G₁,G₂,G₃]^TIs an uncertain set;

for an input vector, [ phi ]_i(X_i)∈R⁴Being basis functions of neural networks, W_i ^*∈R⁴The ideal weight vector is defined as:

wherein W_i∈R⁴Is a weight vector, ε_iTo approximate the error, | ε_i|≤ε_N,i＝1,2,3，ε_NIs a very small normal number;

is W_i ^*Taking the set of all the minimum values;

3.2 consider that the fixed time controller is designed to:

wherein

Is a diagonal matrix of 3 x 3 symmetry,

is theta_iIs equal to [ phi (X) ]₁),Φ(X₂),Φ(X₃)]^T；K₁＝diag(k₁₁,k₁₂,k₁₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₂＝diag(k₂₁,k₂₂,k₂₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₃＝diag(k₃₁,k₃₂,k₃₃)∈R^3×3Is a symmetric diagonal matrix; k is a radical of₁₁,k₁₂,k₁₃,k₂₁,k₂₂,k₂₃,k₃₁,k₃₂,k₃₃Is a normal number; r is more than 0₃＜1,r₄＞1；

sgn(S₁),sgn(S₂),sgn(S₃) Is a sign function;

||W_i ^*i is W_i ^*A second norm of (d);

3.3 design update law is:

wherein gamma is_i＞0,p_i＞0,i＝1,2,3，

Is composed of

Derivative of phi(X_i) Sigmoid function chosen as follows:

wherein l₁,l₂,l₃And l₄To approximate the parameter, [ phi ] (X)_i) Satisfies the relation 0 < phi (X)_i)＜Φ₀And is and

is the maximum of the two;

step 4, the stability of the fixed time is proved, and the process is as follows:

4.1 demonstrates that all signals of the rigid aircraft system are consistent and finally bounded, and the Lyapunov function is designed to be of the form:

wherein

S^TIs the transpose of S;

is that

Transposing;

differentiating equation (28) yields:

wherein

Is the minimum of the two;

is composed of

A second norm of (d);

thus, all signals of the rigid aircraft system are consistent and ultimately bounded;

4.2 demonstrate fixed time convergence, designing the Lyapunov function to be of the form:

differentiating equation (30) yields:

wherein

min{k₁₁,k₁₂,k₁₃}， min{k₂₁,k₂₂,k₂₃All the values are the minimum value of the three; upsilon is₂An upper bound value greater than zero;

based on the above analysis, the attitude tracking error and the angular velocity error of the rigid aircraft system are consistent at a fixed time and are finally bounded.

The invention realizes the stable tracking of the system by applying the self-adaptive neural network tracking control method under the factors of external interference, uncertain rotational inertia, actuator saturation and fault, and ensures that the system state realizes the consistent fixed time and is bounded finally. The technical conception of the invention is as follows: aiming at a rigid aircraft system with external interference, uncertain rotational inertia, saturated actuator and faults, a sliding mode control method is utilized, and a neural network is combined to design a self-adaptive neural network controller. The design of the fixed-time sliding mode surface ensures the fixed-time convergence of the system state. The invention realizes the control method that the fixed time of the attitude tracking error and the angular speed error of the system is consistent and finally bounded under the conditions that the system has external interference, uncertain rotational inertia, saturated actuator and faults.

The invention has the beneficial effects that: under the conditions that external interference exists in the system, the rotational inertia is uncertain, the actuator is saturated and has faults, the fixed time consistency of the attitude tracking error and the angular speed error of the system is finally bounded, and the convergence time is irrelevant to the initial state of the system.

Drawings

FIG. 1 is a schematic representation of the attitude tracking error of a rigid aircraft of the present invention;

FIG. 2 is a schematic diagram of the angular velocity error of the rigid vehicle of the present invention;

FIG. 3 is a schematic view of a slip-form surface of the rigid aircraft of the present invention;

FIG. 4 is a schematic illustration of the rigid aircraft control moments of the present invention;

FIG. 5 is a schematic illustration of a rigid aircraft parameter estimation of the present invention;

FIG. 6 is a control flow diagram of the present invention.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

Referring to fig. 1-6, a method for adaptive neural network fault-tolerant tracking control of a rigid aircraft, the method comprising the steps of:

step 1, establishing a kinematics and dynamics model of a rigid aircraft, initializing system states and control parameters, and carrying out the following processes:

1.7 the kinematic equation for a rigid aircraft system is:

wherein q is_v＝[q₁,q₂,q₃]^TAnd q is₄Vector part and scalar part of unit quaternion respectively and satisfy

q₁,q₂,q₃Respectively mapping values on x, y and z axes of a space rectangular coordinate system;

are each q_vAnd q is₄A derivative of (a); omega belongs to R³Is the angular velocity of the rigid aircraft; i is₃Is R^3×3An identity matrix;

expressed as:

1.8 the kinetic equation for a rigid aircraft system is:

wherein J ∈ R^3×3Is the rotational inertia matrix of the rigid aircraft;

is the angular acceleration of the rigid aircraft; u ═ u₁,u₂,u₃]^T∈R³And d ∈ R³Control moment and external disturbance; d ═ diag (D)₁,D₂,D₃)∈R^3×3Is an actuator efficiency matrix with 3 multiplied by 3 symmetrical opposite angles, and satisfies that D is more than 0_i(t)≤1,i＝1,2,3；sat(u)＝[sat(u₁),sat(u₂),sat(u₃)]^TActual control moment, sat (u), generated for the actuator_i) Is an actuator with saturation characteristics, denoted sat (u)_i)＝sgn(u_i)min{u_mi,|u_i|}， u_miFor maximum available control torque, sgn (u)_i) Is a sign function, min { u }_mi,|u_i| is the minimum of the two; to represent control constraints, sat (u) is denoted as sat (u) ═ g (u) + d_s(u)， g(u)＝[g₁(u₁),g₂(u₂),g₃(u₃)]^T，g_i(u_i) As a function of hyperbolic tangent

d_s(u)＝[d_s1(u₁),d_s2(u₂),d_s3(u₃)]^TIs an approximate error vector; according to the median theorem, g_i(u_i)＝m_iu_i， 0＜m_iLess than or equal to 1; definition of H ═ DM ═ diag (δ)₁m₁,δ₂m₂,δ₃m₃)∈R^3×3Is a 3X 3 symmetric diagonal matrix, M ═ diag (M)₁,m₂,m₃)∈R^3×3Is a 3 multiplied by 3 symmetric diagonal matrix; dsat (u) is re-expressed as: dsat (u) ═ Hu + Dd_s(u) satisfies 0 < h₀≤D_im_i≤1,i＝1,2,3，h₀Is an unknown normal number; omega^×Expressed as:

1.9 rigid aircraft systems the desired kinematic equation is:

wherein q is_dv＝[q_d1,q_d2,q_d3]^TAnd q is_d4A vector part and a scalar part which are respectively a desired unit quaternion and satisfy

Ω_d∈R³A desired angular velocity;

are each q_dv,q_d4The derivative of (a) of (b),

is q_dvTransposing;

expressed as:

1.10 relative attitude motion of rigid aircraft described by quaternion:

Ω_e＝Ω-CΩ_d (12)

wherein e_v＝[e₁,e₂,e₃]^TAnd e₄A vector part and a scalar part of the attitude tracking error respectively; omega_e＝[Ω_e1,Ω_e2,Ω_e3]^T∈R³Is the angular velocity error;

is a corresponding directional cosine matrix and satisfies | | | C | | | | | | | ═ 1 and

is the derivative of C;

according to equations (1) - (12), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

wherein

And

are each e_vAnd e₄A derivative of (a);

is e_vTransposing;

and

are respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;

and

respectively expressed as:

1.11 rotational inertia matrix J satisfies J ═ J₀+ Δ J, wherein J₀And Δ J represents the nominal and indeterminate portions of J, respectively, equation (15) is rewritten as:

further obtaining:

1.12 differentiating equation (13) yields:

wherein

Is e_vThe second derivative of (a);

step 2, aiming at a rigid aircraft system with external disturbance, uncertain rotational inertia, saturated actuator and fault, designing a required sliding mode surface, and comprising the following steps:

selecting a sliding mode surface S ═ S at fixed time₁,S₂,S₃]^T∈R³Comprises the following steps:

wherein the content of the first and second substances,

λ₁and λ₂Is a normal number; r is₁＝a₁/b₁，a₁,b₁Is a normal number, satisfies a₁＞b₁，i＝1,2,3；sgn(e₁),sgn(e₂),sgn(e₃) Are all sign functions; s_au＝[S_au1,S_au2,S_au3]^TExpressed as:

wherein

r₂＝a₂/b₂，a₂,b₂Is positive odd number, satisfies a₂＜b₂；

ε is a very small normal number;

step 3, designing a neural network fixed time controller, and the process is as follows:

3.1 define the neural network as:

G_i(X_i)＝W_i ^*TΦ(X_i)+ε_i (23)

wherein G ═ G₁,G₂,G₃]^TIs an uncertain set;

for an input vector, [ phi ]_i(X_i)∈R⁴Being basis functions of neural networks, W_i ^*∈R⁴The ideal weight vector is defined as:

wherein W_i∈R⁴Is a weight vector, ε_iTo approximate the error, | ε_i|≤ε_N,i＝1,2,3，ε_NIs a very small normal number;

is W_i ^*Taking the set of all the minimum values;

3.2 consider that the fixed time controller is designed to:

wherein

Is a diagonal matrix of 3 x 3 symmetry,

is theta_iIs equal to [ phi (X) ]₁),Φ(X₂),Φ(X₃)]^T；K₁＝diag(k₁₁,k₁₂,k₁₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₂＝diag(k₂₁,k₂₂,k₂₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₃＝diag(k₃₁,k₃₂,k₃₃)∈R^3×3Is a symmetric diagonal matrix; k is a radical of₁₁,k₁₂,k₁₃,k₂₁,k₂₂,k₂₃,k₃₁,k₃₂,k₃₃Is a normal number; r is more than 0₃＜1,r₄＞1；

sgn(S₁),sgn(S₂),sgn(S₃) Is a sign function;

||W_i ^*i is W_i ^*A second norm of (d);

3.3 design update law is:

wherein gamma is_i＞0,p_i＞0,i＝1,2,3，

Is composed of

Derivative of (2), phi (X)_i) Sigmoid function chosen as follows:

wherein l₁,l₂,l₃And l₄To approximate the parameter, [ phi ] (X)_i) Satisfies the relation 0 < phi (X)_i)＜Φ₀And is and

is the maximum of the two;

step 4, the stability of the fixed time is proved, and the process is as follows:

4.1 demonstrates that all signals of the rigid aircraft system are consistent and finally bounded, and the Lyapunov function is designed to be of the form:

wherein

S^TIs the transpose of S;

is that

Transposing;

differentiating equation (28) yields:

wherein

Is the minimum of the two;

is composed of

A second norm of (d);

thus, all signals of the rigid aircraft system are consistent and ultimately bounded;

4.2 demonstrate fixed time convergence, designing the Lyapunov function to be of the form:

differentiating equation (30) yields:

wherein

min{k₁₁,k₁₂,k₁₃}， min{k₂₁,k₂₂,k₂₃All the values are the minimum value of the three; upsilon is₂An upper bound value greater than zero;

based on the above analysis, the attitude tracking error and the angular velocity error of the rigid aircraft system are consistent at a fixed time and are finally bounded.

In order to verify the effectiveness of the method, the method carries out simulation verification on the aircraft system. The system initialization parameters are set as follows:

initial values of the system: q (0) ([ 0.3, -0.2, -0.3, 0.8832)]^T,Ω(0)＝[1,0,-1]^TRadian/second; q. q.s_d(0)＝[0,0,0,1]^T(ii) a Desired angular velocity

Radian/second; nominal part J of the rotational inertia matrix₀＝[40,1.2,0.9；1.2,17,1.4；0.9,1.4,15]Kilogram square meter, uncertainty Δ J of inertia matrix, diag [ sin (0.1t),2sin (0.2t),3sin (0.3t)](ii) a External perturbation d (t) ═ 0.2sin (0.1t),0.3sin (0.2t),0.5sin (0.2t)]^T(ii) newton-meters; the parameters of the slip form face are as follows: lambda [ alpha ]₁＝0.5,λ₂＝0.5,

The parameters of the controller are as follows:

K₁＝K₂＝K₃＝ 0.5I₃(ii) a The update law parameters are as follows: gamma ray_i＝1,p_i＝0.1,i＝1,2,3，

The parameters of the sigmoid function are selected as follows: l₁＝4,l₂＝8,l₃＝10,l₄-0.5. Maximum control moment u_miAt 10 n m, the actuator efficiency value was selected as:

the response schematic diagrams of the attitude tracking error and the angular velocity error of the rigid aircraft are respectively shown in fig. 1 and fig. 2, and it can be seen that both the tracking attitude error and the angular velocity error can be converged to a zero region of a balance point in about 3.8 seconds; the response diagram of the sliding mode surface of the rigid aircraft is shown in fig. 3, and it can be seen that the sliding mode surface can be converged into a zero region of a balance point in about 3.2 seconds; the control moment of the rigid aircraft is shown in fig. 4, and it can be seen that the control moment is limited to within 10 n m; the parameter estimation response diagrams are respectively shown in fig. 5.

Therefore, the sliding mode surface with fixed time designed by the invention effectively solves the problem of singular value; under the conditions that external interference exists in the system, the rotational inertia is uncertain, the actuator is saturated and has faults, the fixed time consistency of the attitude tracking error and the angular speed error of the system is finally bounded, and the convergence time is irrelevant to the initial state of the system.

While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.

Claims (1)

1. A self-adaptive neural network fault-tolerant tracking control method of a rigid aircraft is characterized by comprising the following steps: the method comprises the following steps:

step 1, establishing a kinematics and dynamics model of a rigid aircraft, initializing system states and control parameters, and carrying out the following processes:

1.1 the kinematic equation for a rigid aircraft system is:

wherein q is_v＝[q₁,q₂,q₃]^TAnd q is₄Vector part and scalar part of unit quaternion respectively and satisfy

q₁,q₂,q₃Respectively mapping values on x, y and z axes of a space rectangular coordinate system;

are each q_vAnd q is₄A derivative of (a); omega belongs to R³Is the angular velocity of the rigid aircraft; i is₃Is R^3×3An identity matrix;

expressed as:

1.2 the kinetic equation for a rigid aircraft system is:

wherein J ∈ R^3×3Is the rotational inertia matrix of the rigid aircraft;

is the angular acceleration of the rigid aircraft; u ═ u₁,u₂,u₃]^T∈R³And d ∈ R³Control moment and external disturbance; d ═ diag (D)₁,D₂,D₃)∈R^3×3Is an actuator efficiency matrix with 3 multiplied by 3 symmetrical opposite angles, and satisfies that D is more than 0_i(t)≤1,i＝1,2,3；sat(u)＝[sat(u₁),sat(u₂),sat(u₃)]^TActual control moment, sat (u), generated for the actuator_i) Is an actuator with saturation characteristics, denoted sat (u)_i)＝sgn(u_i)min{u_mi,|u_i|}，u_miFor maximum available control torque, sgn (u)_i) Is a sign function, min { u }_mi,|u_i| is the minimum of the two; to represent control constraints, sat (u) is denoted as sat (u) ═ g (u) + d_s(u)，g(u)＝[g₁(u₁),g₂(u₂),g₃(u₃)]^T，g_i(u_i) As a function of hyperbolic tangent

d_s(u)＝[d_s1(u₁),d_s2(u₂),d_s3(u₃)]^TIs an approximate error vector; according to median valueTheorem of g_i(u_i)＝m_iu_i，0＜m_iLess than or equal to 1; definition H DM diag (D)₁m₁,D₂m₂,D₃m₃)∈R^3×3Is a 3X 3 symmetric diagonal matrix, M ═ diag (M)₁,m₂,m₃)∈R^{3 ×3}Is a 3 multiplied by 3 symmetric diagonal matrix; dsat (u) is re-expressed as: dsat (u) ═ Hu + Dd_s(u) satisfies 0 < h₀≤D_im_i≤1,i＝1,2,3，h₀Is an unknown normal number; omega^×Expressed as:

1.3 the desired kinematic equation for a rigid aircraft system is:

wherein q is_dv＝[q_d1,q_d2,q_d3]^TAnd q is_d4A vector part and a scalar part which are respectively a desired unit quaternion and satisfy

Ω_d∈R³A desired angular velocity;

are each q_dv,q_d4The derivative of (a) of (b),

is q_dvTransposing;

expressed as:

1.4 relative attitude motion of rigid aircraft described by quaternion:

Ω_e＝Ω-CΩ_d (12)

wherein e_v＝[e₁,e₂,e₃]^TAnd e₄A vector part and a scalar part of the attitude tracking error respectively; omega_e＝[Ω_e1,Ω_e2,Ω_e3]^T∈R³Is the angular velocity error;

is a corresponding directional cosine matrix and satisfies | | | C | | | | | | | ═ 1 and

is the derivative of C;

according to equations (1) - (12), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

wherein

And

are each e_vAnd e₄A derivative of (a);

is e_vTransposing;

and

are respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;

and

respectively expressed as:

1.5 rotational inertia matrix J satisfies J ═ J₀+ Δ J, wherein J₀And Δ J represents the nominal and indeterminate portions of J, respectively, equation (15) is rewritten as:

further obtaining:

1.6 differentiating the formula (13) gives:

wherein

Is e_vThe second derivative of (a);

step 2, aiming at a rigid aircraft system with external disturbance, uncertain rotational inertia, saturated actuator and fault, designing a required sliding mode surface, and comprising the following steps:

selecting a sliding mode surface S ═ S at fixed time₁,S₂,S₃]^T∈R³Comprises the following steps:

wherein the content of the first and second substances,

λ₁and λ₂Is a normal number; r is₁＝a₁/b₁，a₁,b₁Is a normal number, satisfies a₁＞b₁，i＝1,2,3；sgn(e₁),sgn(e₂),sgn(e₃) Are all sign functions; s_au＝[S_au1,S_au2,S_au3]^TExpressed as:

wherein

r₂＝a₂/b₂，a₂,b₂Is positive odd number, satisfies a₂＜b₂；

0＜r₂Less than 1, epsilon is a very small normal number;

step 3, designing a neural network fixed time controller, and the process is as follows:

3.1 define the neural network as:

G_i(X_i)＝W_i ^*TΦ(X_i)+ε_i (23)

wherein G ═ G₁,G₂,G₃]^TIs an uncertain set;

as an input vector, phi (X)_i)∈R⁴Being basis functions of neural networks, W_i ^*∈R⁴The ideal weight vector is defined as:

wherein W_i∈R⁴Is a weight vector, ε_iTo approximate the error, | ε_i|≤ε_N,i＝1,2,3，ε_NIs a very small normal number;

is W_i ^*Taking the set of all the minimum values;

3.2 consider that the fixed time controller is designed to:

wherein

Is a diagonal matrix of 3 x 3 symmetry,

is theta_iIs equal to [ phi (X) ]₁),Φ(X₂),Φ(X₃)]^T；K₁＝diag(k₁₁,k₁₂,k₁₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₂＝diag(k₂₁,k₂₂,k₂₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry; k₃＝diag(k₃₁,k₃₂,k₃₃)∈R^3×3Is a symmetric diagonal matrix; k is a radical of₁₁,k₁₂,k₁₃,k₂₁,k₂₂,k₂₃,k₃₁,k₃₂,k₃₃Is a normal number; r is more than 0₃＜1,r₄＞1；

sgn(S₁),sgn(S₂),sgn(S₃) Is a sign function;

||W_i ^*i is W_i ^*A second norm of (d);

3.3 design update law is:

wherein gamma is_i＞0,p_i＞0,i＝1,2,3，

Is composed of

Derivative of (2), phi (X)_i) Sigmoid function chosen as follows:

wherein l₁,l₂,l₃And l₄To approximate the parameter, [ phi ] (X)_i) Satisfies the relation 0 < phi (X)_i)＜Φ₀And is and

is the maximum of the two;

step 4, the stability of the fixed time is proved, and the process is as follows:

4.1 demonstrates that all signals of the rigid aircraft system are consistent and finally bounded, and the Lyapunov function is designed to be of the form:

wherein

S^TIs the transpose of S;

is that

Transposing;

differentiating equation (28) yields:

wherein

Is the minimum of the two;

is composed of

A second norm of (d);

thus, all signals of the rigid aircraft system are consistent and ultimately bounded;

4.2 demonstrate fixed time convergence, designing the Lyapunov function to be of the form:

differentiating equation (30) yields:

wherein

min{k₁₁,k₁₂,k₁₃}，min{k₂₁,k₂₂,k₂₃All the values are the minimum value of the three; upsilon is₂An upper bound value greater than zero;

based on the above analysis, the attitude tracking error and the angular velocity error of the rigid aircraft system are consistent at a fixed time and are finally bounded.

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