CN107490966B - Aircraft finite time self-adaptive attitude control method based on improved power approach law - Google Patents

Abstract

An aircraft finite time self-adaptive attitude control method based on an improved power approximation law is designed by utilizing a sliding mode control method based on the improved power approximation law and combining self-adaptive control aiming at the problem of aircraft attitude stability with centralized uncertainty. The design of the terminal sliding mode surface is to ensure the finite time convergence of the system and reduce the buffeting problem in the practical control system through an improved power approximation law. In addition, adaptive control is a feedback control system for intelligently adjusting its own characteristics according to environmental changes so that the system can operate in an optimum state according to some set criteria. The invention provides a control method which can reduce the buffeting problem of a sliding mode surface and control moment, and can realize the consistency of limited time of a system and final bounding under the condition that the system has uncertainty and interference.

Description

Aircraft finite time self-adaptive attitude control method based on improved power approach law

Technical Field

The invention relates to an aircraft finite time self-adaptive attitude control method based on an improved power approach law, in particular to an aircraft attitude control method with external interference and rotation inertia matrix uncertainty.

Background

The flight control system is the core of the unmanned aerial vehicle, and the unmanned aerial vehicle needs to complete autonomous flight and has good control characteristics on an inner loop (attitude loop) and an outer loop (horizontal position and height loop). The flight control law design of the drone determines its flight performance. These properties include various flight properties, such as: takeoff and landing performance, operation flight performance, flight safety reliability, flight monitoring performance, system automation, maintainability and the like. The performance requirements of the unmanned aerial vehicle flight control system are more and more complex, and a classical control method is difficult to process and coordinate the multivariable input and output characteristics of the system. With the development of modern control theory, the sliding mode variable structure control as a typical nonlinear control method can effectively improve the stability and maneuverability of the aircraft, thereby improving the task execution capacity. Therefore, the sliding mode variable structure control method for researching the unmanned aerial vehicle attitude system 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. Therefore, the sliding mode control method is widely applied to various fields. Compared with the traditional linear sliding mode control, the terminal sliding mode control has the advantage of limited time convergence. However, the discontinuous switching characteristic of the terminal sliding mode control in nature will cause the buffeting of the system, and the terminal sliding mode control becomes an obstacle to the application of the terminal sliding mode control in the practical system. To solve this problem, many improved methods are proposed in succession, such as a high-order sliding mode control method, an observer control method. Recently, an improved power approximation law has been proposed that provides a good reduction in the jitter problem and a smoother system input signal in the system response.

However, in most of the proposed methods described above, both the kinematic and the kinetic model parameters of the aircraft attitude system must be known in advance. Therefore, the proposed method cannot be directly applied to attitude control of an aircraft when there is an uncertainty factor in the system. It is known that adaptive control has been widely used for the problem of uncertain system control because it can intelligently adjust its own characteristics of a feedback control system according to environmental changes so that the system can operate in an optimal state according to some set criteria. For the reasons described above, a number of adaptive control methods are used to control spacecraft systems.

Disclosure of Invention

In order to overcome the defects of unknown nonlinearity and sliding mode control buffeting in the existing aircraft attitude control system, the invention provides an aircraft finite time self-adaptive attitude control method based on an improved power approximation law, and a control method for realizing the consistency and final bounding of the finite time of the system under the condition that the system has uncertainty and interference.

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

an aircraft finite time self-adaptive attitude control method based on an improved power approach law comprises the following steps:

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

1.1 the dynamic model expression form of the aircraft attitude control system is as follows:

wherein,

angular velocity and angular acceleration of the aircraft, respectively, × is an operator applying the operator × to a ═ a₁,a₂,a₃]^TCan obtain a^×＝[0,-a₃,a₂；a₃,0,-a₁；-a₂,a₁,0]；J∈R^3×3Is the rotational inertia matrix of the aircraft u ∈ R³And d (t) ∈ R³Control moment and external disturbance;

1.2 the representation form of the kinematics model of the aircraft attitude control system is as follows:

wherein the unit quaternion

Describing the attitude of an aircraft and satisfying

Are each q₀And q is_vDerivative of I ∈ R^3×3Is a 3 × 3 identity matrix;

1.3 suppose the rotational inertia matrix J ═ J₀+ Δ J, wherein J₀And Δ J represent the nominal and indeterminate portions of J, respectively, then equation (1) is rewritten as:

1.4 to more easily describe the attitude dynamics controller design of an aircraft, let

Substituting formula (2) to obtain:

wherein,

differentiating equation (5) yields:

wherein,

are respectively P and q_vFirst and second derivatives of;

after the formula (5) and the formula (6) are substituted into the formula (4), P is simultaneously multiplied on two sides of the formula^TObtaining:

wherein, J^*＝P^TJ₀P and inertia matrix J due to rotation^*Is a skew symmetric positive definite matrix, then the matrix

Satisfying the following oblique symmetry relationship:

at the same time J^*The following inequalities are satisfied:

J_minand J_maxIs a normal number and represents J^*Lower and upper bounds of (1);

is a set of interference and uncertainty, satisfying | | T_d||≤γ₀Φ，Φ＝1+||ω||+||ω||²And gamma is₀Is a normal number;

step 2, under the condition that the moment of inertia is uncertain and external disturbance exists, designing a required sliding mode surface based on an attitude control system of the aircraft, wherein the process is as follows:

2.1 selection of slip form surface s ∈ R³Comprises the following steps:

wherein α and β are normal numbers;

r₁and r₂Is a positive odd number and 0<r₁<r₂(ii) a Function sig (q)_v)^rIs defined as sig (q)_v)^r＝[|q_v1|^rsign(q_v1),|q_v2|^rsign(q_v2),|q_v3|^rsign(q_v3)]^T；

Derivation of equation (10) yields:

wherein,

is the derivative of s; | q_vL is q_vAbsolute value of (d);

if q is_vj0, j is 1,2,3 and

wherein q is_vjJ is 1,2,3 is q_vThe jth element in the vector; to avoid the occurrence of singularities, which arise due to the presence of the negative fractional power r-1, the first derivative of s is changed to:

wherein q is_vr∈R³Is defined as:

wherein ∈ is a small constant, | ∈ | is the absolute value of ∈;

is q_vjA derivative of (a);

then, it is obtained from formula (7), formula (10) and formula (12):

wherein,

step 3, designing an improved power approximation law, wherein the process is as follows:

3.1 define the improved power approximation law as:

wherein theta is more than 0 and less than 1; k is more than 0; mu is more than 0 and less than 1;

sign(s) is a sign function; s_jJ is 1,2,3 is the j-th element in the s-vector; | s_jL is s_jJ is the absolute value of 1,2, 3; s is the norm of s;

step 4, designing a finite time self-adaptive sliding mode controller, and the process is as follows:

4.1 consider that the finite time adaptive sliding mode controller is designed to:

wherein, P is the range of PCounting; f is the norm of F; the | | | Ps | | | is the norm of Ps;

is gamma₀(ii) an estimate of (d);

4.2 design update law of adaptive parameters:

wherein, c₀And₀is a normal number;

4.3 design Lyapunov function:

wherein,

s^Tis the transpose of s;

the derivation is performed on equation (20) and is obtained according to equation (8):

for any normal number

The following inequalities exist:

thus, formula (21) is expressed as:

wherein, according to formula (9), the following is obtained:

according to

And

obtaining:

due to the presence of the following inequality:

therefore, from equations (25) and (26), it follows:

wherein,

the sliding mode surface is limited in time and finally bounded as obtained by the formula (27); thus, the convergence field Δ s is expressed as:

the slip form (10) is expressed as:

wherein, η_jIs a normal number, satisfies | η_j|≤Δs；

Then, equation (29) is written in two forms:

or

From formula (30) or formula (31), if

Or

The sliding mode surface of the formula (30) or the formula (31) has a similar structure to that of the formula (10), and therefore, the posture quaternion q is obtained_vjCan converge to the following region within a limited time:

from the equations (32) and (33), the attitude quaternion q is obtained_vjThe finite time convergence domain of (c) is:

is obtained from formula (29)

Can converge to:

according to

Obtained by the formula (2)

Wherein | ω | purple_∞And

are respectively omega and

infinite norm of (d); at the same time, the user can select the desired position,

the time required for the operation of the device to be carried out is limited,

therefore, consider equation (5) and the assumption

Where det (T) is the determinant of T, the following are obtained:

wherein, ω is_jJ is 1,2,3 is the jth element of the ω vector;

based on the analysis, the sliding mode surface s and the attitude quaternion q of the aircraft_vjAnd angular velocity ω_jIs locally finite time consistent and finally bounded.

The method is based on the aircraft finite time self-adaptive attitude control method of the improved power approximation law under the factors of the uncertainty of the rotation inertia matrix and the external interference, realizes the stable control of the system, reduces buffeting of sliding mode control, and ensures that the system realizes the consistency of finite time and is bounded finally.

The technical conception of the invention is as follows: aiming at an aircraft control system containing the uncertainty of a rotation inertia matrix and external interference, a finite-time self-adaptive attitude control method of an aircraft based on an improved power approach law is designed by combining a sliding mode control method of the improved power approach law and self-adaptive control. The sliding mode surface design based on the improved power approximation law is to ensure that a system can stably converge to the neighborhood of an original point in limited time, and the buffeting is reduced by improving the power approximation law. In addition, the self-adaptive control can intelligently adjust the feedback control system of the self characteristics according to the environmental change so that the system can work in the optimal state according to some set standards. The invention provides a method which can reduce the buffeting problem of a sliding mode surface and a control moment, and realize the consistency of limited time of a system and final bounding under the condition that the system has uncertainty and interference.

The invention has the beneficial effects that: and buffeting is reduced, and the finite time consistency of the system is realized and finally bounded under the condition that uncertainty and interference exist in the system.

Drawings

Fig. 1 is a schematic diagram of a sliding mode surface based on different approaches of the present invention, wherein (a) represents method one, (b) represents method two, and (c) represents method three.

Fig. 2 is a schematic diagram of control torque based on different approach laws, wherein (a) represents method one, (b) represents method two, and (c) represents method three.

Fig. 3 is a diagram showing quaternion of aircraft attitude based on different approaches according to the present invention, wherein (a) shows method one, (b) shows method two, and (c) shows method three.

Fig. 4 is a schematic diagram of angular velocities based on different approach laws according to the present invention, wherein (a) represents method one, (b) represents method two, and (c) represents method three.

Fig. 5 is a schematic diagram of parameter estimation based on different approaches according to the present invention, in which (a) represents method one, (b) represents method two, and (c) represents method three.

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, an aircraft finite time adaptive attitude control method based on an improved power approach law includes the following steps:

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

1.1 the dynamic model expression form of the aircraft attitude control system is as follows:

wherein the sum of the values of ω,

angular velocity and angular acceleration of the aircraft, respectively, × is an operator applying the operator × to a ═ a₁,a₂,a₃]^TCan obtain a^×＝[0,-a₃,a₂；a₃,0,-a₁；-a₂,a₁,0]；J∈R^3×3Is the rotational inertia matrix of the aircraft u ∈ R³And d (t) ∈ R³Control moment and external disturbance;

1.2 the representation form of the kinematics model of the aircraft attitude control system is as follows:

wherein the unit quaternion

Describing the attitude of an aircraft and satisfying

Are each q₀And q is_vDerivative of I ∈ R^3×3Is a 3 × 3 identity matrix;

1.3 suppose the rotational inertia matrix J ═ J₀+ Δ J, wherein J₀And Δ J represent the nominal and indeterminate portions of J, respectively, then equation (1) is rewritten as:

1.4 toThe design of the attitude dynamics controller of the aircraft is described more conveniently

Substituting formula (2) to obtain:

wherein,

differentiating equation (5) yields:

wherein,

are respectively P and q_vFirst and second derivatives of;

after the formula (5) and the formula (6) are substituted into the formula (4), P is simultaneously multiplied on two sides of the formula^TObtaining:

wherein, J^*＝P^TJ₀P and inertia matrix J due to rotation^*Is a skew symmetric positive definite matrix, then the matrix

Satisfying the following oblique symmetry relationship:

at the same time J^*The following inequalities are satisfied:

J_minand J_maxIs a normal number and represents J^*Lower and upper bounds of (1);

is a set of interference and uncertainty, satisfying | | T_d||≤γ₀Φ，Φ＝1+||ω||+||ω||²And gamma is₀Is a normal number;

step 2, under the condition that the moment of inertia is uncertain and external disturbance exists, designing a required sliding mode surface based on an attitude control system of the aircraft, wherein the process is as follows:

2.1 selection of slip form surface s ∈ R³Comprises the following steps:

wherein α and β are normal numbers;

r₁and r₂Is a positive odd number and 0<r₁<r₂(ii) a Function sig (q)_v)^rIs defined as sig (q)_v)^r＝[|q_v1|^rsign(q_v1),|q_v2|^rsign(q_v2),|q_v3|^rsign(q_v3)]^T；

Derivation of equation (10) yields:

wherein,

is the derivative of s; | q_vL is q_vAbsolute value of (d);

if q is_vj0, j is 1,2,3 and

wherein q is_vjJ is 1,2,3 is q_vThe jth element in the vector; to avoid the occurrence of singularities, which arise due to the presence of the negative fractional power r-1, the first derivative of s is changed to:

wherein q is_vr∈R³Is defined as:

wherein ∈ is a small constant, | ∈ | is the absolute value of ∈;

is q_vjA derivative of (a);

then, it is obtained from formula (7), formula (10) and formula (12):

wherein,

step 3, designing an improved power approximation law, wherein the process is as follows:

3.1 define the improved power approximation law as:

wherein theta is more than 0 and less than 1; k is more than 0; mu is more than 0 and less than 1;

sign(s) is a sign function; s_jJ is 1,2,3 is the j-th element in the s-vector; | s_jL is s_jJ is the absolute value of 1,2, 3; s is the norm of s;

step 4, designing a finite time self-adaptive sliding mode controller, and the process is as follows:

4.1 consider that the finite time adaptive sliding mode controller is designed to:

wherein, P is the norm of P; f is the norm of F; the | | | Ps | | | is the norm of Ps;

is gamma₀(ii) an estimate of (d);

4.2 design update law of adaptive parameters:

wherein, c₀And₀is a normal number;

4.3 design Lyapunov function:

wherein,

s^Tis the transpose of s;

the derivation is performed on equation (20) and is obtained according to equation (8):

for any normal number

The following inequalities exist:

thus, formula (21) is expressed as:

wherein, according to formula (9), the following is obtained:

according to

And

obtaining:

due to the presence of the following inequality:

therefore, from equations (25) and (26), it follows:

wherein,

the sliding mode surface is limited in time and finally bounded as obtained by the formula (27); thus, the convergence field Δ s is expressed as:

the slip form (10) is expressed as:

wherein, η_jIs a normal number, satisfies | η_j|≤Δs；

Then, equation (29) is written in two forms:

or

From formula (30) or formula (31), if

Or

The sliding mode surface of the formula (30) or the formula (31) has a similar structure to that of the formula (10), and therefore, the posture quaternion q is obtained_vjCan converge to the following region within a limited time:

is represented by formula (32) and formula(33) Obtaining the attitude quaternion q_vjThe finite time convergence domain of (c) is:

is obtained from formula (29)

Can converge to:

according to

Obtained by the formula (2)

Wherein | ω | purple_∞And

are respectively omega and

infinite norm of (d); at the same time, the user can select the desired position,

the time required for the operation of the device to be carried out is limited,

therefore, consider equation (5) and the assumption

Where det (T) is the determinant of T, the following are obtained:

wherein, ω is_jJ is 1,2,3 is the jth element of the ω vector;

based on the analysis, the sliding mode surface s and the attitude quaternion q of the aircraft_vjAnd angular velocity ω_jIs locally finite time consistent and finally bounded.

In order to verify the effectiveness of the proposed method, the invention provides three different methods for simulation comparison, as follows:

the method comprises the following steps: the finite time self-adaptive attitude control method based on the improved power approximation law comprises the following steps that (1) an approximation law expression is expressed as an expression (15) and an expression (16);

the second method comprises the following steps: the finite time self-adaptive attitude control method based on the exponential approximation law comprises the following expression:

the third method comprises the following steps: a finite time self-adaptive attitude control method based on a traditional approach law has the following expression:

for more efficient comparison, all parameters of the system are identical, i.e. the parameters of equations (14) and (15) are the same as equations (37) and (38), where K is 0.5, μ is 0.01, θ is 0.1,

and given system external disturbances d (t) 0.005 × sin (0.8t), cos (0.5t), cos (0.3t)]^TN.m, sliding mode parameters of α -0.1, β -0.1 and r₁＝3，r₂(ii) 5; the parameters of the adaptive update law are:₀＝0.01，

the actual parameters of the aircraft attitude system are as follows: j. the design is a square₀＝diag([140,120,130])kg·m²，ΔJ＝diag[sin(0.1t),2sin(0.2t),3sin(0.3t)]kg·m²，ω(0)＝[0,0,0]^Trad/s，q_v(0)＝[0.3,-0.3,0.2]^T，q₀(0) 0.8832; the parameters in equation (24) are: j. the design is a square_max＝560，₀1 is ═ 1; to avoid discontinuities of equation (18)

The resulting buffeting problem, applying a continuous term in the simulation

Alternatively, where ξ is a normal number, ξ ═ 0.0002.

Fig. 1 and 2 are schematic diagrams of the glide mode surface and the control moment response respectively based on different approach laws. If D(s) tends to 0.01 as | | s | | is larger, the expression (15) and the expression (37) are

Is 50, and is larger than K of formula (38) of 0.5. On the contrary, when s is less,

tending towards 0.5. This phenomenon causes the controller gain to vary in the range of 50 to 0.5. As shown in fig. 1 and 2, the convergence time of the sliding surfaces based on method one and method two is about 1.2 seconds, while the convergence time of the sliding surfaces based on method three is about 4.2 seconds. Obviously, the method I and the method II are superior to the method III, and the aircraft attitude system can have higher stability performance and shorter convergence time. In addition, | s in the formula (15)_j|^θThe existence of the first method, the second method and the third method can effectively reduce the buffeting problem.

The aircraft attitude quaternion and angular velocity response diagrams based on different approximation laws are shown in fig. 3 and fig. 4 respectively. The results show that all three methods can achieve finite time consistency and final bounding. The convergence time of the aircraft attitude quaternion based on method one and method two is about 10 seconds, which is 2 seconds faster than method three. Furthermore, the convergence time of the aircraft angular velocity based on method one and method two is about 11 seconds, which is approximately 3 seconds faster than method three. Through the above analysis, the convergence rate of the attitude quaternion and the angular velocity based on the first method and the second method is faster than the convergence rate of the attitude quaternion and the angular velocity based on the third method. A diagram of the response of parameter estimation based on different approach laws is shown in fig. 5. Referring to fig. 1-5, the method of the present invention achieves better control performance than the other two methods.

In summary, compared with the second method and the third method, the first method can achieve good control performance, and has better capability of reducing buffeting on the sliding mode surface and the control moment.

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. An aircraft finite time self-adaptive attitude control method based on an improved power approach law is characterized by comprising the following steps: the control method comprises the following steps:

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

1.1 the dynamic model expression form of the aircraft attitude control system is as follows:

wherein the sum of the values of ω,

angular velocity and angular acceleration of the aircraft, respectively, × is an operator applying the operator × to a ═ a₁,a₂,a₃]^TCan obtain a^×＝[0,-a₃,a₂；a₃,0,-a₁；-a₂,a₁,0]；J∈R^3×3Is the rotational inertia matrix of the aircraft; u. of∈R³And d (t) ∈ R³Control moment and external disturbance;

1.2 the kinematic model expression form of the aircraft attitude control system is as follows:

wherein the unit quaternion

Describing the attitude of an aircraft and satisfying

Are each q₀And q is_vA derivative of (a); i is₃∈R^3×3Is a 3 × 3 identity matrix;

1.3 suppose the rotational inertia matrix J ═ J₀+ Δ J, wherein J₀And Δ J represent the nominal and indeterminate portions of J, respectively, equation (1) is rewritten as:

1.4 to more easily describe the attitude dynamics controller design of an aircraft, let

Substituting formula (2) to obtain:

wherein,

differentiating equation (5) yields:

wherein,

are respectively P and q_vFirst and second derivatives of;

after the formula (5) and the formula (6) are substituted into the formula (4), P is simultaneously multiplied on two sides of the formula^TObtaining:

wherein, J^*＝P^TJ₀P and inertia matrix J due to rotation^*Is a skew symmetric positive definite matrix, then the matrix

Satisfying the following oblique symmetry relationship:

at the same time J^*The following inequalities are satisfied:

wherein, J_minAnd J_maxIs a normal number and represents J^*Lower and upper bounds of (1);

is a set of interference and uncertainty, satisfying | | T_d||≤γ₀Φ，Φ＝1+||ω||+||ω||²And gamma is₀Is a normal number;

step 2, under the condition that the moment of inertia is uncertain and external disturbance exists, designing a required sliding mode surface based on an attitude control system of the aircraft, wherein the process is as follows:

2.1 selection of slip form surface s ∈ R³Comprises the following steps:

wherein α and β are normal numbers;

r₁and r₂Is a positive odd number and 0<r₁<r₂(ii) a Function sig (q)_v)^rIs defined as sig (q)_v)^r＝[|q_v1|^rsign(q_v1),|q_v2|^rsign(q_v2),|q_v3|^rsign(q_v3)]^T；

Derivation of equation (10) yields:

wherein,

is the derivative of s; | q_vL is q_vAbsolute value of (d); diag (| q)_v|^r-1)＝diag([|q_v1|^r-1,|q_v2|^r-1,|q_v3|^r-1])∈R^3×3；

If q is_vj0, j is 1,2,3 and

wherein q is_vjJ is 1,2,3 is q_vThe jth element in the vector; to avoid the occurrence of singularities, which arise due to the presence of the negative fractional power r-1, the first derivative of s is changed to:

wherein q is_vr∈R³Is defined as:

wherein ∈ is a small constant, | ∈ | is the absolute value of ∈;

is q_vjA derivative of (a);

then, it is obtained from formula (7), formula (10) and formula (12):

wherein,

step 3, designing an improved power approximation law, wherein the process is as follows:

3.1 define the improved power approximation law as:

wherein theta is more than 0 and less than 1; k is more than 0; mu is more than 0 and less than 1;

sign(s) is an s-sign function; s_jJ is 1,2,3 is the j-th element in the s-vector; | s_jL is s_jJ is the absolute value of 1,2, 3; s is the norm of s;

step 4, designing a finite time self-adaptive sliding mode controller, and the process is as follows:

4.1 consider that the finite time adaptive sliding mode controller is designed to:

wherein, P is the norm of P; f is the norm of F; the | | | Ps | | | is the norm of Ps;

is gamma₀(ii) an estimate of (d);

4.2 design update law of adaptive parameters:

wherein, c₀And₀is a normal number;

is composed of

A derivative of (a);

4.3 design Lyapunov function:

wherein,

s^Tis the transpose of s;

the derivation is performed on equation (20) and is obtained according to equation (8):

for any normal number

The following inequalities exist:

thus, formula (21) is expressed as:

wherein, according to formula (9), the following is obtained:

according to

And

obtaining:

due to the presence of the following inequality:

therefore, from equations (25) and (26), it follows:

wherein,

the sliding mode surface is limited in time and finally bounded as obtained by the formula (27); thus, the convergence field Δ s is expressed as:

the slip form (10) is expressed as:

wherein, η_jIs a normal number, satisfies | η_j|≤Δs；

Then, equation (29) is written in two forms:

or

From formula (30) or formula (31), if

Or

The sliding mode surface of the formula (30) or the formula (31) has a similar structure to that of the formula (10), and therefore, the posture quaternion q is obtained_vjCan converge to the following region within a limited time:

from the equations (32) and (33), the attitude quaternion q is obtained_vjThe finite time convergence domain of (c) is:

is obtained from formula (29)

Can converge to:

according to

Obtained by the formula (2)

Wherein | ω | purple_∞And

are respectively omega and

infinite norm of (d); at the same time, the user can select the desired position,

the time required for the operation of the device to be carried out is limited,

therefore, consider equation (5) and the assumption

Where det (T) is the determinant of T, the following are obtained:

wherein, ω is_jJ is 1,2,3 is the jth element of the ω vector;

based on the analysis, the sliding mode surface s and the attitude quaternion q of the aircraft_vjAnd angular velocity ω_jIs locally finite time consistent and finally bounded.

Images