CN108873927A - A kind of nonsingular set time Adaptive Attitude Tracking control method of rigid aircraft - Google Patents

Abstract

A kind of nonsingular set time Adaptive Attitude Tracking control method of rigid aircraft, probabilistic rigid aircraft attitude stabilization problem is concentrated for having, nonsingular set time adaptive controller is devised in conjunction with adaptive technique using sliding-mode control；The design of nonsingular set time sliding-mode surface not only guarantees the set time convergence of system mode, but also solves singular value problem；It is not always known in addition, adaptive updates rule is used to estimating system, including external interference and the uncertain upper bound of rotary inertia, therefore always uncertain upper bound information is not necessarily to be known in advance.The present invention realizes the control of the set time uniform ultimate bounded of Attitude Tracking error and angular speed error under external interference and the uncertain factor of rotary inertia.

Description

Nonsingular fixed time self-adaptive attitude tracking control method for rigid aircraft

Technical Field

The invention relates to a nonsingular fixed time self-adaptive attitude tracking control method for a rigid aircraft, in particular to an attitude tracking control method for a rigid aircraft with external interference and uncertain rotational inertia matrix.

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 affected by various external disturbances and uncertainty in the moment of inertia matrix. In order to effectively maintain the performance of the system, it needs to be robust to external interference and uncertainty of the rotational inertia matrix. The sliding mode variable structure control is a typical nonlinear control method, can effectively improve the stability and the maneuverability of a rigid aircraft, and has stronger robustness, thereby improving the task execution capacity. Therefore, the sliding mode variable structure control method for researching the attitude control system of the rigid aircraft has very important significance.

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 self-adaptive control means that the controller can modify the self control parameters to adapt to the dynamic characteristics of the system and external disturbance so as to obtain satisfactory dynamic performance and enable the system to achieve optimal control. The method is suitable for both linear systems and nonlinear systems, and mainly aims at controlling the uncertainty of the systems. The research object of the adaptive control is a system which has a certain degree of uncertainty and is easily interfered by the external environment. For the reasons described above, a number of adaptive control methods are used to control a spacecraft rigid aircraft system.

Therefore, the fixed time sliding mode control technology and the self-adaptive control method are effectively combined, the influence of external interference and uncertainty of system parameters on the control performance of the system is reduced, and the fixed time control of the attitude of the rigid aircraft is realized.

Disclosure of Invention

In order to overcome the problem of unknown nonlinearity of the existing attitude control system of the rigid aircraft, the invention provides a nonsingular fixed time self-adaptive attitude tracking control method of the rigid aircraft, which realizes the control of consistent and bounded fixed time of the attitude tracking error and the angular velocity error of the rigid aircraft system under the condition that the external interference and the uncertainty of the rotational inertia exist in the system.

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

a nonsingular fixed time self-adaptive attitude tracking 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 satisfyq₁,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 is an element of R³And d ∈ R³Control moment and external disturbance; 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 satisfyΩ_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(11)

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) - (11), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

whereinAndare each e_vAnd e₄A derivative of (a);is e_vTransposing;andare respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;andrespectively 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 (14) is rewritten as:

further obtaining:

1.6 differentiating equation (12) yields:

whereinIs a total indeterminate set, satisfiesAnd c is₁,c₂,c₃Is a normal number;is omega_eTransposing;is e_vThe second derivative of (a);

step 2, designing a required slip form surface aiming at a rigid aircraft system with uncertain external disturbance and moment inertia, wherein the process is as follows:

the nonsingular fixed time sliding mode surface is selected as follows:

wherein, and sgn (e)_i) Are all sign functions, λ₁＞0,λ₂＞0,a₂＞1， Is e_iI ═ 1,2, 3;

step 3, designing a nonsingular fixed time self-adaptive controller, wherein the process is as follows:

3.1 design fixed time controller:

wherein L＝[L₁,L₂,L₃]^T，S＝[S₁,S₂,S₃]^T，Γ＝diag(Γ₁,Γ₂,Γ₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry;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 diagonal matrix with 3 multiplied by 3 symmetry; r is more than 0₁＜1，r₂＞1，Are respectively c₁,c₂,c₃(ii) an estimate of (d);

3.2 design update law of adaptive parameters:

η therein₁,η₂,η₃,ε₁,ε₂,ε₃Is a normal number;are respectively asA derivative of (a);is composed ofThe two-norm of (a) is,is composed ofIs two norms, | | Ω_e| is omega_eA second norm of (d);

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:

whereinS^TIs the transpose of S;

derivation of equation (26) yields:

wherein k_3min＝min{k₃₁,k₃₂,k₃₃Min {. cndot } represents the minimum value;is the derivative of S; delta₁,δ₂,δ₃Is a normal number;

determining that 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:

the derivation of equation (28) yields:

wherein γ₂An 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 control of the system by applying the nonsingular fixed time self-adaptive attitude tracking control method of the rigid aircraft under the factors of external interference and uncertain rotational inertia, and ensures that the fixed time of the attitude tracking error and the angular velocity error of the rigid aircraft system is consistent and finally bounded. The technical conception of the invention is as follows: aiming at a rigid aircraft system containing external interference and uncertain rotary inertia, a nonsingular self-adaptive fixed time controller is designed by utilizing a sliding mode control method and combining self-adaptive control. The design of the nonsingular fixed time sliding mode surface not only ensures the fixed time convergence of the system state, but also solves the problem of singular value. In addition, based on the designed adaptive update law, the total uncertain upper bound information is not required to be known in advance. According to the invention, under the conditions that external interference and uncertain rotational inertia exist in the system, the attitude tracking error and the angular velocity error of the system are consistent in fixed time and finally are controlled in a bounded mode.

The invention has the beneficial effects that: the designed sliding mode surface with fixed time effectively solves the problem of singular value; under the conditions that external interference and rotational inertia uncertainty exist in the system, the consistent and final bounded attitude tracking error and angular velocity error of the system in fixed time are realized.

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 non-singular fixed-time adaptive attitude tracking control method for a rigid aircraft, the control 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:

1.1 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 satisfyq₁,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 is an element of R³And d ∈ R³Control moment and external disturbance; 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 satisfyΩ_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(11)

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) - (11), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

whereinAndare each e_vAnd e₄A derivative of (a);is e_vTransposing;andare respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;andrespectively 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 (14) is rewritten as:

further obtaining:

1.6 differentiating equation (12) yields:

whereinIs a total indeterminate set, satisfiesAnd c is₁,c₂,c₃Is a normal number;is omega_eTransposing;is e_vThe second derivative of (a);

step 2, designing a required slip form surface aiming at a rigid aircraft system with uncertain external disturbance and moment inertia, wherein the process is as follows:

the nonsingular fixed time sliding mode surface is selected as follows:

wherein, and sgn (e)_i) Are all sign functions, λ₁＞0,λ₂＞0,a₂＞1， Is e_iI ═ 1,2, 3;

step 3, designing a nonsingular fixed time self-adaptive controller, wherein the process is as follows:

3.1 design fixed time controller:

wherein L＝[L₁,L₂,L₃]^T，S＝[S₁,S₂,S₃]^T，Γ＝diag(Γ₁,Γ₂,Γ₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry;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 diagonal matrix with 3 multiplied by 3 symmetry; r is more than 0₁＜1，r₂＞1，Are respectively c₁,c₂,c₃(ii) an estimate of (d);

3.2 design update law of adaptive parameters:

η therein₁,η₂,η₃,ε₁,ε₂,ε₃Is a normal number;are respectively asA derivative of (a);is composed ofThe two-norm of (a) is,is composed ofIs two norms, | | Ω_e| is omega_eA second norm of (d);

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:

whereinS^TIs the transpose of S;

derivation of equation (26) yields:

wherein k_3min＝min{k₃₁,k₃₂,k₃₃Min {. cndot } represents the minimum value;is the derivative of S; delta₁,δ₂,δ₃Is a normal number;

determining that 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:

the derivation of equation (28) yields:

wherein γ₂An 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 rigid 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_d(0)＝[0,0,0,1]^T(ii) a Desired angular velocityRadian/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＝1,λ₂＝1,a₁＝1.5,a₂1.5; the parameters of the controller are as follows:the update law parameters are as follows:

the response diagrams of the attitude quaternion and the angular velocity of the rigid aircraft are respectively shown in fig. 1 and fig. 2, and it can be seen that both the attitude quaternion and the angular velocity can be converged into a zero region of a balance point within about 5 seconds; the sliding mode surface response diagram 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 seconds; the control moment and parameter estimation response diagrams of the rigid aircraft are shown in fig. 4 and 5, respectively.

Therefore, the sliding mode surface with fixed time designed by the invention effectively solves the problem of singular value; under the condition that external interference and rotational inertia uncertainty exist in the system, the attitude tracking error and the angular speed error of the system are consistent in fixed time and are 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 nonsingular fixed time self-adaptive attitude 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 satisfyq₁,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 is an element of R³And d ∈ R³Control moment and external disturbance; 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 satisfyΩ_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(11)

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) - (11), the rigid aircraft attitude tracking error dynamics and kinematics equations are:

whereinAndare each e_vAnd e₄A derivative of (a);is e_vTransposing;andare respectively omega_dAnd Ω_eA derivative of (a); (omega)_e+CΩ_d)^×And omega^×Equivalence;andrespectively 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 (14) is rewritten as:

further obtaining:

1.6 differentiating equation (12) yields:

wherein

Is a total indeterminate set, satisfiesAnd c is₁,c₂,c₃Is a normal number;is omega_eTransposing;is e_vThe second derivative of (a);

step 2, designing a required slip form surface aiming at a rigid aircraft system with uncertain external disturbance and moment inertia, wherein the process is as follows:

the nonsingular fixed time sliding mode surface is selected as follows:

wherein, and sgn (e)_i) Are all sign functions, λ₁＞0,λ₂＞0,a₂＞1， Is e_iI ═ 1,2, 3;

step 3, designing a nonsingular fixed time self-adaptive controller, wherein the process is as follows:

3.1 design fixed time controller:

wherein L＝[L₁,L₂,L₃]^T，S＝[S₁,S₂,S₃]^T，Γ＝diag(Γ₁,Γ₂,Γ₃)∈R^3×3Is a diagonal matrix with 3 multiplied by 3 symmetry;i＝1,2,3；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 diagonal matrix with 3 multiplied by 3 symmetry; r is more than 0₁＜1，r₂＞1，Are respectively c₁,c₂,c₃(ii) an estimate of (d);

3.2 design update law of adaptive parameters:

η therein₁,η₂,η₃,ε₁,ε₂,ε₃Is a normal number;are respectively asA derivative of (a);is composed ofThe two-norm of (a) is,is composed ofIs two norms, | | Ω_e| is omega_eA second norm of (d);

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:

whereinS^TIs the transpose of S;

derivation of equation (26) yields:

wherein k_3min＝min{k₃₁,k₃₂,k₃₃Min {. cndot } represents the minimum value;is the derivative of S; delta₁,δ₂,δ₃Is a normal number;

determining that 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:

the derivation of equation (28) yields:

wherein γ₂An 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.