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CN109557933B - A state-constrained control method for rigid aircraft based on Lomborg observer - Google Patents

A state-constrained control method for rigid aircraft based on Lomborg observer Download PDF

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CN109557933B
CN109557933B CN201811423199.3A CN201811423199A CN109557933B CN 109557933 B CN109557933 B CN 109557933B CN 201811423199 A CN201811423199 A CN 201811423199A CN 109557933 B CN109557933 B CN 109557933B
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CN109557933A (en
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陈强
陈中天
何熊熊
孙明轩
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Zhejiang University of Technology ZJUT
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    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
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Abstract

一种基于龙伯格观测器的刚性飞行器状态约束控制方法，针对存在外部干扰和无角速度测量的刚性飞行器，提出龙伯格观测器估计未知状态量，因此无需知道飞行器的角速度。使用了适用于约束和非约束情况的改进型障碍李雅普诺夫函数实现了状态约束，再结合反步控制设计了刚性飞行器状态约束控制方法。本发明在外界干扰和和无角速度测量的情况下，保证了飞行器姿态观测误差和跟踪误差能够达到一致最终有界，并且状态变量受到约束。

Figure 201811423199

A state-constrained control method for a rigid aircraft based on the Lomborg observer. For a rigid aircraft with external disturbances and no angular velocity measurement, the Lomborg observer is proposed to estimate the unknown state quantity, so it is not necessary to know the angular velocity of the aircraft. The state constraint is realized by using the improved obstacle Lyapunov function suitable for constrained and unconstrained situations, and the state constraint control method of rigid aircraft is designed combined with backstepping control. The present invention ensures that the attitude observation error and the tracking error of the aircraft can reach a consistent and final bound, and the state variables are constrained under the condition of external interference and no angular velocity measurement.

Figure 201811423199

Description

Rigid aircraft state constraint control method based on Longberger observer
Technical Field
The invention relates to a rigid aircraft state constraint control method based on a Luenberger observer, which is an all-state constraint output feedback attitude tracking control method designed for a rigid aircraft with external interference and no angular velocity measurement.
Background
A rigid aircraft is a nonlinear, strong-coupling, multi-input and multi-output complex system, and a plurality of external disturbance moments affect the aircraft at any moment in flight, such as radiation moment, gravity gradient moment, geomagnetic moment and the like. And in many cases, the angular velocity signal of the aircraft may contain a lot of noise, and even sensor damage may result in the angular velocity signal not being accurately obtained. Therefore, the attitude control method independent of the angular velocity information has strong practical significance.
As the level of task refinement performed increases, it is not sufficient to focus solely on the steady-state accuracy of the aircraft. To ensure transient performance and stability of the system, the system state and the amplitude of the output are usually constrained. During the operation of the system, if the constraint condition is violated, the performance of the system may be reduced and even a safety problem may occur. The barrier lyapunov function method is a constraint control method, and the basic principle is that when a variable approaches a boundary of a region, the value of the lyapunov function tends to be infinite, so that the constraint of the variable is ensured. The conventional logarithmic barrier lyapunov function is not suitable for the unconstrained case, whereas the modified barrier lyapunov function may be suitable for both the constrained and unconstrained cases. The improved barrier Lyapunov function is used for not only restraining variables, but also effectively improving transient and steady-state performance of the system.
The backstepping control method is a recursion design control method based on the Lyapunov theorem, and a feedback control law and a Lyapunov function can be designed together in the process of gradual recursion. The backstepping method can reduce the difficulty of designing the controller by gradually recursion when designing the high-order controller. One of the main advantages of the backstepping control is that it avoids eliminating some of the useful non-linearities and achieves high accuracy control performance. The lunberg observer is a state observation proposed by lunberg, kalman and buchsi, etc., and can estimate the angular velocity information of the aircraft that cannot be obtained by using the observer, thereby realizing the design of a feedback controller without angular velocity information.
Disclosure of Invention
In order to solve the problem of attitude constraint control of a rigid aircraft without angular velocity information, the invention provides a rigid aircraft state constraint control method based on a Robert observer, which can realize that attitude observation errors and tracking errors of a rigid aircraft system can be consistent and finally bounded under the condition that the system has external interference and no angular velocity information.
The technical scheme proposed for solving the technical problems is as follows:
a rigid aircraft state constraint control method based on a Longberger observer comprises the following steps:
step 1, establishing a kinematics and dynamics model of the rigid aircraft based on the modified rodgers parameter, wherein the process is as follows:
1.1 the kinematic equation for a rigid aircraft system is:
Figure BDA0001880991520000021
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;
Figure BDA0001880991520000022
is the derivative of σ, σTIs the transpose of σ; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
Figure BDA0001880991520000023
form G is
Figure BDA0001880991520000024
It has the property of
Figure BDA0001880991520000025
G | | | is the two-norm of G;
1.2 the kinetic equation for a rigid aircraft system is:
Figure BDA0001880991520000026
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;
Figure BDA0001880991520000027
is the derivative of ω, representing the angular acceleration of the rigid vehicle; u is an element of R3And d ∈ R3Respectively control moment and external disturbance; omega×The form is as follows:
Figure BDA0001880991520000028
1.3 pairs
Figure BDA0001880991520000029
Derivation and substitution into formula (3) to obtain
Figure BDA0001880991520000031
Wherein L ═ G-1，
Figure BDA0001880991520000032
Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;
Figure BDA0001880991520000033
A×in the form of:
Figure BDA0001880991520000034
Figure BDA0001880991520000035
d′＝GJ-1d is less than or equal to d and satisfies | | | d' | | |mWherein d ismIs a normal number;
step 2, aiming at a rigid aircraft system with external interference and no angular velocity measurement, designing a controller, and carrying out the following process:
2.1 design the Lorberg State observer, let x1＝[x11,x12,x13]T＝σ，
Figure BDA0001880991520000036
Figure BDA0001880991520000037
The output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Figure BDA0001880991520000038
order to
Figure BDA0001880991520000039
Then changing the formula (9) into a state space form
Figure BDA00018809915200000310
Wherein
Figure BDA00018809915200000311
k1,k2Are two normal numbers; according to the lyapunov theorem, as long as the matrix a is a helvets matrix, for any symmetric matrix Q there must be a positive definite matrix P such that the following holds:
ATP+PA＝-2Q (12)
the form of the designed lunberger observer is as follows:
Figure BDA0001880991520000041
wherein
Figure BDA0001880991520000042
Figure BDA0001880991520000043
Are respectively x1And x2Is determined by the estimated value of (c),
Figure BDA0001880991520000044
is that
Figure BDA0001880991520000045
A derivative of (a);
Figure BDA0001880991520000046
replacing x with the variable in E (x)
Figure BDA0001880991520000047
The value of time; h is the gain matrix of the observer, of the form:
Figure BDA0001880991520000048
wherein h is1,h2δ is a normal number;
Figure BDA0001880991520000049
order to
Figure BDA00018809915200000410
Definition of
Figure BDA00018809915200000411
Subtracting the formula (12) from the formula (10) to obtain the observation error of the observer
Figure BDA00018809915200000412
Wherein
Figure BDA00018809915200000413
Is xeThe derivative of (a) of (b),
Figure BDA00018809915200000414
satisfy the requirement of
Figure BDA00018809915200000415
Figure BDA00018809915200000416
Is that
Figure BDA00018809915200000417
Two norm of (M ═ M)1,m2,m3]T，miI ═ 1,2,3 is the constant of normal numbers, | | | M | | | is the two-norm of M, | | | x |eIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
Figure BDA00018809915200000418
wherein sigmadIs the desired pose; α is a virtual control law of the form
Figure BDA00018809915200000419
Wherein
Figure BDA0001880991520000051
kb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,
Figure BDA0001880991520000052
is z1Transposing; c. C1Is a normal number;
Figure BDA0001880991520000053
is σdA derivative of (a);
the controller is designed as follows:
Figure BDA0001880991520000054
wherein c is2Is a normal number;
Figure BDA0001880991520000055
kb2is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,
Figure BDA0001880991520000056
is z2Transposing;
Figure BDA0001880991520000057
is the derivative of α;
step 3, proving the stability of the attitude system of the rigid aircraft, wherein the process is as follows:
3.1 proving that the attitude observation error and the tracking error of the rigid aircraft system are consistent and finally bounded, and designing an improved obstacle Lyapunov function into the following form:
Figure BDA0001880991520000058
wherein ln is a natural logarithm; e, natural constant;
the derivation of equation (218) and the substitution of equations (12), (14), (17) and (18) yields:
Figure BDA0001880991520000059
wherein η is a normal number; i H2Is H2A second norm of (d); p is the two-norm of P;
equation (20) is simplified to:
Figure BDA00018809915200000510
wherein
Figure BDA00018809915200000511
λmax(P) is the maximum eigenvalue of matrix P;
Figure BDA00018809915200000512
therefore, according to the Lyapunov theorem, the attitude observation error and the tracking error of the rigid aircraft system can be consistent and finally bounded;
3.2 proving that the rigid aircraft state quantities are constrained:
order to
Figure BDA0001880991520000061
Solving equation (28) yields the following inequality:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
wherein V (0) is the output value of V;
combining formulae (19) and (22) to obtain
Figure BDA0001880991520000062
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
Figure BDA0001880991520000063
by the same derivation, z is obtained2Eventually converging to the following neighborhood:
Figure BDA0001880991520000064
as seen from formulae (24) and (25), z1And z2Are respectively subjected to kb1And kb2And (4) combining the system description to make all state quantities of the rigid aircraft be restrained.
Under the conditions that external interference exists in the rigid aircraft and no angular velocity measurement exists, the state constraint control method of the rigid aircraft based on the Longberg observer is designed by combining the Longberg observer, the backstepping control method and the improved barrier Lyapunov function, and high-precision control and state constraint of the system are achieved.
The technical conception of the invention is as follows: aiming at a rigid aircraft with external interference and no angular velocity measurement, a Longberg observer is provided for estimating unknown state quantity, and a state constraint control method is designed by combining backstepping control and an improved barrier Lyapunov function, so that the attitude observation error and the tracking error of the rigid aircraft can be consistent and bounded finally.
The invention has the advantages that: under the conditions that external interference exists in the system and no angular velocity measurement exists, the observation error and the tracking error of the system can be consistent and finally bounded, and the state quantity of the aircraft can be restrained.
Drawings
FIG. 1 is a diagram of the effect of attitude tracking of a rigid aircraft of the present invention;
FIG. 2 is a schematic representation of the rigid aircraft attitude tracking error of the present invention;
FIG. 3 is a schematic illustration of the rigid aircraft control input torque of the present invention;
FIG. 4 is a schematic view of the rigid aerial vehicle observation error of the present invention;
FIG. 5 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 to 5, a rigid aircraft state constraint control method based on a lunberger observer includes the following steps:
step 1, establishing a kinematics and dynamics model of the rigid aircraft based on the modified rodgers parameter, wherein the process is as follows:
1.1 the kinematic equation for a rigid aircraft system is:
Figure BDA0001880991520000071
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;
Figure BDA0001880991520000072
is the derivative of σ, σTIs the transpose of σ; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
Figure BDA0001880991520000073
form G is
Figure BDA0001880991520000074
It has the property of
Figure BDA0001880991520000075
G | | | is the two-norm of G;
1.2 the kinetic equation for a rigid aircraft system is:
Figure BDA0001880991520000081
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;
Figure BDA0001880991520000082
is the derivative of ω, representing the angular acceleration of the rigid vehicle; u is an element of R3And d ∈ R3Respectively control moment and external disturbance; omega×The form is as follows:
Figure BDA0001880991520000083
1.3 pairs
Figure BDA0001880991520000084
Derivation and substitution into formula (3) to obtain
Figure BDA0001880991520000085
Wherein L ═ G-1，
Figure BDA0001880991520000086
Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;
Figure BDA0001880991520000087
A×in the form of:
Figure BDA0001880991520000088
Figure BDA0001880991520000089
d′＝GJ-1d is less than or equal to d and satisfies | | | d' | | |mWherein d ismIs a normal number;
step 2, aiming at a rigid aircraft system with external interference and no angular velocity measurement, designing a controller, and carrying out the following process:
2.1 design the Lorberg State observer, let x1＝[x11,x12,x13]T＝σ，
Figure BDA00018809915200000810
Figure BDA00018809915200000811
The output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Figure BDA00018809915200000812
order to
Figure BDA00018809915200000813
Then changing the formula (9) into a state space form
Figure BDA00018809915200000814
Wherein
Figure BDA0001880991520000091
k1,k2Are two normal numbers; according to the lyapunov theorem, as long as the matrix a is a helvets matrix, for any symmetric matrix Q there must be a positive definite matrix P such that the following holds:
ATP+PA＝-2Q (12)
the form of the designed lunberger observer is as follows:
Figure BDA0001880991520000092
wherein
Figure BDA0001880991520000093
Figure BDA0001880991520000094
Are respectively x1And x2Is determined by the estimated value of (c),
Figure BDA0001880991520000095
is that
Figure BDA0001880991520000096
A derivative of (a);
Figure BDA0001880991520000097
replacing x with the variable in E (x)
Figure BDA0001880991520000098
The value of time; h is the gain matrix of the observer, of the form:
Figure BDA0001880991520000099
wherein h is1,h2δ is a normal number;
Figure BDA00018809915200000910
order to
Figure BDA00018809915200000911
Definition of
Figure BDA00018809915200000912
Subtracting the formula (12) from the formula (10) to obtain the observation error of the observerTo
Figure BDA00018809915200000913
Wherein
Figure BDA00018809915200000914
Is xeThe derivative of (a) of (b),
Figure BDA00018809915200000915
satisfy the requirement of
Figure BDA00018809915200000916
Figure BDA00018809915200000917
Is that
Figure BDA00018809915200000918
Two norm of (M ═ M)1,m2,m3]T，miI ═ 1,2,3 is the constant of normal numbers, | | | M | | | is the two-norm of M, | | | x |eIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
Figure BDA0001880991520000101
wherein sigmadIs the desired pose; α is a virtual control law of the form
Figure BDA0001880991520000102
Wherein
Figure BDA0001880991520000103
kb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,
Figure BDA0001880991520000104
is z1Transposing; c. C1Is a normal number;
Figure BDA0001880991520000105
is σdA derivative of (a);
the controller is designed as follows:
Figure BDA0001880991520000106
wherein c is2Is a normal number;
Figure BDA0001880991520000107
kb2is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,
Figure BDA0001880991520000108
is z2Transposing;
Figure BDA0001880991520000109
is the derivative of α;
step 3, proving the stability of the attitude system of the rigid aircraft, wherein the process is as follows:
3.1 proving that the attitude observation error and the tracking error of the rigid aircraft system are consistent and finally bounded, and designing an improved obstacle Lyapunov function into the following form:
Figure BDA00018809915200001010
wherein ln is a natural logarithm; e, natural constant;
the derivation of equation (218) and the substitution of equations (12), (14), (17) and (18) yields:
Figure BDA00018809915200001011
wherein η is a normal number; i H2Is H2A second norm of (d); p is the two-norm of P;
equation (20) is simplified to:
Figure BDA0001880991520000111
wherein
Figure BDA0001880991520000112
λmax(P) is the maximum eigenvalue of matrix P;
Figure BDA0001880991520000113
therefore, according to the Lyapunov theorem, the attitude observation error and the tracking error of the rigid aircraft system can be consistent and finally bounded;
3.2 proving that the rigid aircraft state quantities are constrained:
order to
Figure BDA0001880991520000114
Solving equation (28) yields the following inequality:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
wherein V (0) is the output value of V;
combining formulae (19) and (22) to obtain
Figure BDA0001880991520000115
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
Figure BDA0001880991520000116
by the same derivation, z is obtained2Finally converge toThe following neighborhoods:
Figure BDA0001880991520000117
as seen from formulae (24) and (25), z1And z2Are respectively subjected to kb1And kb2And (4) combining the system description to make all state quantities of the rigid aircraft be restrained.
To illustrate the effectiveness of the proposed method, the invention provides a numerical simulation experiment of a rigid aircraft system, with a rotational inertia matrix of
Figure BDA0001880991520000121
External interference is
d＝1.5×10-3[3cos(0.8t)+1,1.5sin(0.8t)+3cos(0.8t),3sin(0.8t)+1]TCattle-rice (27)
The initial value of the attitude is
Figure BDA0001880991520000122
The desired attitude trajectory is σd＝[sin(2t),sin(2t+π),cos(2t)]T. The control parameters are selected as follows: k is a radical of1＝20,k2＝4,h1＝1,h2＝4.5,δ＝20,c1＝2,c2＝3,kb1＝0.4,kb2＝1.5。
Fig. 1 shows the attitude tracking effect of the rigid aircraft without angular velocity measurement, and fig. 2 shows the attitude tracking error of the rigid aircraft. As can be seen from fig. 1 and 2, the controller of the present invention can realize highly accurate attitude tracking control and does not require angular velocity information. Fig. 3 shows the control input torque u of the method according to the invention. Fig. 4 shows the observation error of the lunberger observer, and it can be seen from the figure that the observer of the method can realize accurate estimation of the state quantity.
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)

1.一种基于龙伯格观测器的刚性飞行器状态约束控制方法，其特征在于，所述控制方法包括以下步骤：1. a rigid aircraft state constraint control method based on Lumberg observer, is characterized in that, described control method comprises the following steps: 步骤1，建立基于修正罗德里格参数的刚性飞行器的运动学和动力学模型，过程如下：Step 1, establish the kinematics and dynamics model of the rigid aircraft based on the modified Rodrigue parameters. The process is as follows: 1.1刚性飞行器系统的运动学方程为：1.1 The kinematic equation of the rigid aircraft system is:
Figure FDA0003098625600000011
Figure FDA0003098625600000011
 其中σ＝[σ1,σ2,σ3]T为修正罗德里格参数，其描述了飞行器的姿态特征；
Figure FDA0003098625600000012
是σ的导数，σT是σ的转置；G表示方向余弦矩阵；ω∈R3是刚性飞行器的角速度；I3是R3×3单位矩阵；σ×的形式为：where σ=[σ 1 ,σ 2 ,σ 3 ] T is the modified Rodrigue parameter, which describes the attitude characteristics of the aircraft;
Figure FDA0003098625600000012
is the derivative of σ, σ T is the transpose of σ; G represents the direction cosine matrix; ω ∈ R 3 is the angular velocity of the rigid aircraft; I 3 is the R 3 × 3 identity matrix; σ × has the form:
Figure FDA0003098625600000013
Figure FDA0003098625600000013
 G形式为
Figure FDA0003098625600000014
其有性质
Figure FDA0003098625600000015
||G||是G的二范数；The form G is
Figure FDA0003098625600000014
its nature
Figure FDA0003098625600000015
||G|| is the second norm of G; 1.2刚性飞行器系统的动力学方程为：1.2 The dynamic equation of the rigid aircraft system is:
Figure FDA0003098625600000016
Figure FDA0003098625600000016
 其中J∈R3×3是刚性飞行器的转动惯量矩阵；
Figure FDA0003098625600000017
是ω的导数，表示刚性飞行器的角加速度；u∈R3和d∈R3分别为控制力矩和外部扰动；ω×形式为：where J∈R 3×3 is the rotational inertia matrix of the rigid aircraft;
Figure FDA0003098625600000017
is the derivative of ω, which represents the angular acceleration of the rigid aircraft; u ∈ R 3 and d ∈ R 3 are the control torque and external disturbance, respectively; ω × takes the form:
Figure FDA0003098625600000018
Figure FDA0003098625600000018
 ω1,ω2,ω3表示三个正常数；ω 1 , ω 2 , ω 3 represent three positive numbers; 1.3对
Figure FDA0003098625600000019
求导并代入式(3)，得1.3 pairs
Figure FDA0003098625600000019
Derivation and substituting into equation (3), we get
Figure FDA0003098625600000021
Figure FDA0003098625600000021
 其中E表示新的状态量，L＝G-1，
Figure FDA0003098625600000022
是L的导数；J-1是J的逆矩阵；
Figure FDA0003098625600000023
Λ×的形式为：where E represents the new state quantity, L=G -1 ,
Figure FDA0003098625600000022
is the derivative of L; J -1 is the inverse of J;
Figure FDA0003098625600000023
The form of Λ × is:
Figure FDA0003098625600000024
Figure FDA0003098625600000024
Figure FDA0003098625600000025
d′＝GJ-1d且满足||d′||≤dm，其中dm是一个正常数；
Figure FDA0003098625600000025
d'=GJ -1 d and satisfy ||d'||≤d m , where d m is a positive constant; 步骤2，针对带有外部干扰和无角速度测量的刚性飞行器系统，设计控制器，过程如下：Step 2, for the rigid aircraft system with external disturbance and no angular velocity measurement, design the controller, the process is as follows: 2.1设计龙伯格状态观测器，令x1＝[x11,x12,x13]T＝σ，
Figure FDA0003098625600000026
x1表示观测器第一变量；x2表示观测器第二变量；飞行器的输出为y＝σ，式(1)和(3)改写为：2.1 Design the Lomborg state observer, let x 1 =[x 11 ,x 12 ,x 13 ] T =σ,
Figure FDA0003098625600000026
x 1 represents the first variable of the observer; x 2 represents the second variable of the observer; the output of the aircraft is y=σ, and equations (1) and (3) are rewritten as:
Figure FDA0003098625600000027
Figure FDA0003098625600000027
 E(x)＝E表示新的状态量；
Figure FDA0003098625600000028
是将E(x)中的变量替换x为
Figure FDA0003098625600000029
时的值；E(x)=E represents the new state quantity;
Figure FDA0003098625600000028
is to replace the variable x in E(x) with
Figure FDA0003098625600000029
time value; 令
Figure FDA00030986256000000210
然后将式(9)改成状态空间形式make
Figure FDA00030986256000000210
Then change Equation (9) into state space form
Figure FDA00030986256000000211
Figure FDA00030986256000000211
 其中in
Figure FDA00030986256000000212
Figure FDA00030986256000000212
 k1,k2是两个正常数；根据李雅普诺夫定理，只要矩阵A是赫尔维茨矩阵，则对于任意对称矩阵Q，一定存在一个正定矩阵P使得下式成立：k 1 , k 2 are two constant numbers; according to Lyapunov's theorem, as long as the matrix A is a Hurwitz matrix, for any symmetric matrix Q, there must be a positive definite matrix P such that the following formula holds: ATP+PA＝-2Q (12)A T P+PA=-2Q (12) 设计的龙伯格观测器形式如下：The designed Lomborg observer has the following form:
Figure FDA0003098625600000031
Figure FDA0003098625600000031
 其中
Figure FDA0003098625600000032
Figure FDA0003098625600000033
分别为x1和x2的估计值，
Figure FDA0003098625600000034
是
Figure FDA0003098625600000035
的导数；
Figure FDA0003098625600000036
是将E(x)中的变量替换x为
Figure FDA0003098625600000037
时的值；H是观测器的增益矩阵，形式为：in
Figure FDA0003098625600000032
Figure FDA0003098625600000033
are the estimates of x 1 and x 2 , respectively,
Figure FDA0003098625600000034
Yes
Figure FDA0003098625600000035
the derivative of ;
Figure FDA0003098625600000036
is to replace the variable x in E(x) with
Figure FDA0003098625600000037
; H is the gain matrix of the observer in the form:
Figure FDA0003098625600000038
Figure FDA0003098625600000038
 其中h1,h2,δ都是正常数；
Figure FDA0003098625600000039
H1表示是增益子矩阵1，H2表示是增益子矩阵2；where h 1 , h 2 , δ are all positive numbers;
Figure FDA0003098625600000039
H 1 represents gain sub-matrix 1, and H 2 represents gain sub-matrix 2; 令
Figure FDA00030986256000000310
定义
Figure FDA00030986256000000311
为观测器观测误差，用式(10)第一个方程减去式(13)得到make
Figure FDA00030986256000000310
definition
Figure FDA00030986256000000311
For the observer observation error, the first equation of equation (10) is subtracted from equation (13) to get
Figure FDA00030986256000000312
Figure FDA00030986256000000312
 其中
Figure FDA00030986256000000313
为观测器观测误差，
Figure FDA00030986256000000314
是xe的导数，
Figure FDA00030986256000000315
Figure FDA00030986256000000316
表示估计误差，满足
Figure FDA00030986256000000317
Figure FDA00030986256000000318
是
Figure FDA00030986256000000319
的二范数，M＝[m1,m2,m3]T，M表示一个正向量，mi是正常数，i＝1,2,3，||M||是M的二范数，||xe||是xe的二范数；in
Figure FDA00030986256000000313
is the observer observation error,
Figure FDA00030986256000000314
is the derivative of x e ,
Figure FDA00030986256000000315
Figure FDA00030986256000000316
represents the estimation error, satisfying
Figure FDA00030986256000000317
Figure FDA00030986256000000318
Yes
Figure FDA00030986256000000319
The two-norm of , M=[m 1 , m 2 , m 3 ] T , M represents a positive vector, m i is a normal number, i=1, 2, 3, ||M|| is the two-norm of M , ||x e || is the second norm of x e ; 2.2设计控制器，首先定义虚拟变量：2.2 To design the controller, first define dummy variables:
Figure FDA00030986256000000320
Figure FDA00030986256000000320
 其中z1表示虚拟变量1，z2表示虚拟变量2，σd是期望姿态；α是虚拟控制律，其形式为where z 1 represents dummy variable 1, z 2 represents dummy variable 2, σ d is the desired attitude; α is the virtual control law, whose form is
Figure FDA0003098625600000041
Figure FDA0003098625600000041
 其中
Figure FDA0003098625600000042
Φ1表示控制器参数1，kb1是正常数，满足kb1≥||z1(0)||2，而||z1(0)||是z1初始值的二范数，
Figure FDA0003098625600000043
是z1的转置；c1是正常数；
Figure FDA00030986256000000410
是σd的导数；in
Figure FDA0003098625600000042
Φ 1 represents the controller parameter 1, k b1 is a positive number, satisfying k b1 ≥||z 1 (0)|| 2 , and ||z 1 (0)|| is the two-norm of the initial value of z 1 ,
Figure FDA0003098625600000043
is the transpose of z 1 ; c 1 is a positive constant;
Figure FDA00030986256000000410
is the derivative of σ d ; 控制器设计为：The controller is designed to:
Figure FDA0003098625600000044
Figure FDA0003098625600000044
 其中c2是正常数；
Figure FDA0003098625600000045
Φ2表示控制器参数2，kb2是正常数，满足kb2≥||z2(0)||2，而||z2(0)||是z2初始值的二范数，
Figure FDA0003098625600000046
是z2的转置；
Figure FDA0003098625600000047
是α的导数；where c 2 is a positive constant;
Figure FDA0003098625600000045
Φ 2 represents the controller parameter 2, k b2 is a positive number, satisfying k b2 ≥||z 2 (0)|| 2 , and ||z 2 (0)|| is the two-norm of the initial value of z 2 ,
Figure FDA0003098625600000046
is the transpose of z 2 ;
Figure FDA0003098625600000047
is the derivative of α; 步骤3，刚性飞行器姿态系统稳定性证明，其过程如下：Step 3, the stability proof of the rigid aircraft attitude system, the process is as follows: 3.1证明刚性飞行器系统的姿态观测误差和跟踪误差一致最终有界，设计改进型障碍李雅普诺夫函数为如下形式：3.1 Prove that the attitude observation error and tracking error of the rigid aircraft system are consistent and ultimately bounded, and the improved obstacle Lyapunov function is designed as follows:
Figure FDA0003098625600000048
Figure FDA0003098625600000048
 其中ln是自然对数；e自然常数；P表示一个正定矩阵；where ln is the natural logarithm; e is a natural constant; P represents a positive definite matrix; 对式(19 )求导并将式(12)、(14)、(17)和(18)代入得：Derivating equation (19) and substituting equations (12), (14), (17) and (18), we get:
Figure FDA0003098625600000049
Figure FDA0003098625600000049
 其中η是正常数；||H2||是H2的二范数；||P||是P的二范数；where η is a positive constant; ||H 2 || is the two-norm of H 2 ; ||P|| is the two-norm of P; 将式(20)化简得：Simplify equation (20) to get:
Figure FDA0003098625600000051
Figure FDA0003098625600000051
 其中
Figure FDA0003098625600000052
C表示一个正常量，λmax(P)是矩阵P的最大特征值；
Figure FDA0003098625600000053
μ表示一个正常量；in
Figure FDA0003098625600000052
C represents a normal quantity, λ max (P) is the largest eigenvalue of the matrix P;
Figure FDA0003098625600000053
μ represents a normal quantity; 因此，根据李雅普诺夫定理，刚性飞行器系统姿态观测误差和跟踪误差能够实现一致最终有界；Therefore, according to the Lyapunov theorem, the attitude observation error and tracking error of the rigid aircraft system can be uniform and ultimately bounded; 3.2证明刚性飞行器状态量受到约束：3.2 Prove that the rigid aircraft state quantities are constrained: 令
Figure FDA0003098625600000054
μ0表示一个正常量，解式(21 )得如下不等式：make
Figure FDA0003098625600000054
μ 0 represents a normal quantity, and the following inequality can be obtained by solving equation (21): 0≤V≤μ0+(V(0)-μ0)e-Ct (22)0≤V≤μ 0 +(V(0)-μ 0 )e -Ct (22) 其中V(0)是V的输出值；where V(0) is the output value of V; 结合式(19)和(22)，得Combining equations (19) and (22), we get
Figure FDA0003098625600000055
Figure FDA0003098625600000055
 通过解不等式(23)，得z1最终收敛到如下邻域：By solving inequality (23), z 1 finally converges to the following neighborhood:
Figure FDA0003098625600000056
Figure FDA0003098625600000056
 通过相同的推导，得z2最终收敛到如下邻域：Through the same derivation, z 2 finally converges to the following neighborhood:
Figure FDA0003098625600000057
Figure FDA0003098625600000057
 从式(24)和(25)看出，z1和z2分别受到kb1和kb2的约束，再结合系统描述，得刚性飞行器所有状态量都受到约束。It can be seen from equations (24) and (25) that z 1 and z 2 are constrained by k b1 and k b2 respectively. Combined with the system description, all state quantities of the rigid aircraft are constrained.
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