CN109557933B - Rigid aircraft state constraint control method based on Longberger observer - Google Patents

Rigid aircraft state constraint control method based on Longberger 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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    • G05—CONTROLLING; REGULATING
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    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • 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

A rigid aircraft state constraint control method based on a Roberter observer is provided, aiming at a rigid aircraft with external interference and no angular velocity measurement, the Roberter observer estimates unknown state quantity, and therefore the angular velocity of the aircraft does not need to be known. An improved barrier Lyapunov function suitable for constrained and unconstrained conditions is used for realizing state constraint, and a rigid aircraft state constraint control method is designed by combining with backstepping control. The method ensures that the attitude observation error and the tracking error of the aircraft can be consistent and finally bounded under the conditions of external interference and no angular velocity measurement, and the state variable is constrained.

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. A rigid aircraft state constraint control method based on a Longberger observer is characterized by comprising 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 FDA0003098625600000011
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;
Figure FDA0003098625600000012
is the derivative of σ, σTIs the transpose of σ; g represents a directional cosine matrix; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
Figure FDA0003098625600000013
form G is
Figure FDA0003098625600000014
It has the property of
Figure FDA0003098625600000015
G | | | is the two-norm of G;
1.2 the kinetic equation for a rigid aircraft system is:
Figure FDA0003098625600000016
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;
Figure FDA0003098625600000017
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 FDA0003098625600000018
ω1,ω2,ω3three normal numbers are indicated;
1.3 pairs
Figure FDA0003098625600000019
Derivation and substitution into formula (3) to obtain
Figure FDA0003098625600000021
Wherein E represents a new state quantity, L ═ G-1，
Figure FDA0003098625600000022
Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;
Figure FDA0003098625600000023
Λ×in the form of:
Figure FDA0003098625600000024
Figure FDA0003098625600000025
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 FDA0003098625600000026
x1Representing an observer first variable; x is the number of2Representing an observer second variable; the output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Figure FDA0003098625600000027
e (x) represents a new state quantity;
Figure FDA0003098625600000028
replacing x with the variable in E (x)
Figure FDA0003098625600000029
The value of time;
order to
Figure FDA00030986256000000210
Then changing the formula (9) into a state space form
Figure FDA00030986256000000211
Wherein
Figure FDA00030986256000000212
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 FDA0003098625600000031
wherein
Figure FDA0003098625600000032
Figure FDA0003098625600000033
Are respectively x1And x2Is determined by the estimated value of (c),
Figure FDA0003098625600000034
is that
Figure FDA0003098625600000035
A derivative of (a);
Figure FDA0003098625600000036
replacing x with the variable in E (x)
Figure FDA0003098625600000037
The value of time; h is the gain matrix of the observer, of the form:
Figure FDA0003098625600000038
wherein h is1,h2δ is a normal number;
Figure FDA0003098625600000039
H1representation is a gain sub-matrix 1, H2Denoted gain sub-matrix 2;
order to
Figure FDA00030986256000000310
Definition of
Figure FDA00030986256000000311
For observer error, the first equation of equation (10) is used to subtract equation (13) to obtain
Figure FDA00030986256000000312
Wherein
Figure FDA00030986256000000313
In order to observe the error by the observer,
Figure FDA00030986256000000314
is xeThe derivative of (a) of (b),
Figure FDA00030986256000000315
Figure FDA00030986256000000316
represents an estimation error, satisfies
Figure FDA00030986256000000317
Figure FDA00030986256000000318
Is that
Figure FDA00030986256000000319
M is a two norm ofm1,m2,m3]TM denotes a positive vector, MiIs a normal number, i ═ 1,2,3, | | | M | | | is the two-norm of M, | | | xeIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
Figure FDA00030986256000000320
wherein z is1Representing virtual variables 1, z2Representing virtual variables 2, σdIs the desired pose; α is a virtual control law of the form
Figure FDA0003098625600000041
Wherein
Figure FDA0003098625600000042
Φ1Representing controller parameters 1, kb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,
Figure FDA0003098625600000043
is z1Transposing; c. C1Is a normal number;
Figure FDA00030986256000000410
is σdA derivative of (a);
the controller is designed as follows:
Figure FDA0003098625600000044
wherein c is2Is a normal number;
Figure FDA0003098625600000045
Φ2representing controller parameters 2, kb2Is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,
Figure FDA0003098625600000046
is z2Transposing;
Figure FDA0003098625600000047
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 FDA0003098625600000048
wherein ln is a natural logarithm; e, natural constant; p represents a positive definite matrix;
derivation of equation (19) and substitution of equations (12), (14), (17), and (18) yields:
Figure FDA0003098625600000049
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 FDA0003098625600000051
wherein
Figure FDA0003098625600000052
C represents a normal quantity, λmax(P) is the maximum eigenvalue of matrix P;
Figure FDA0003098625600000053
μ represents a normal amount;
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 FDA0003098625600000054
μ0Representing a normal quantity, solving equation (21) for 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 FDA0003098625600000055
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
Figure FDA0003098625600000056
by the same derivation, z is obtained2Eventually converging to the following neighborhood:
Figure FDA0003098625600000057
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.
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