CN112434370A - Characteristic modeling method for error-free compression of flexible aircraft - Google Patents

Characteristic modeling method for error-free compression of flexible aircraft Download PDF

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CN112434370A
CN112434370A CN202011264265.4A CN202011264265A CN112434370A CN 112434370 A CN112434370 A CN 112434370A CN 202011264265 A CN202011264265 A CN 202011264265A CN 112434370 A CN112434370 A CN 112434370A
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flexible aircraft
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孟斌
唐青原
王晓磊
解永春
吴宏鑫
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Beijing Institute of Control Engineering
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Abstract

The invention relates to a flexible aircraft error-free compression characteristic modeling method, which comprises the following steps: 1) establishing a dynamic equation of a controlled object of the flexible aircraft; 2) converting flexible aircraft dynamics into an accurate feedback linearization standard form; 3) solving the time scale of the flexible aircraft, and 4) selecting a sampling period T; 5) establishing a rigid body modal equation; 6) establishing a third-order characteristic model; 7) giving a boundary of the characteristic model coefficient; beginning with step 8), cycling through each control period; 8) identifying coefficients of the characteristic model by adopting a projection gradient method or a projection least square method; 9) designing a three-order adaptive control law; 10) returning to the step 8), entering the next control period.

Description

Characteristic modeling method for error-free compression of flexible aircraft
Technical Field
The invention relates to the field of aerospace, in particular to a flexible aircraft error-free compression feature modeling method.
Background
The characteristic modeling theory is proposed by Wu hong Xin academy in the 80 th century, and after more than 40 years of research, important progress is made in theory and application, a set of complete adaptive control theory and method with strong practicability is formed, and the method has important application prospect. Feature modeling is one of the key problems of the feature model theory. The characteristic modeling means that a high-order continuous mathematical model of a controlled object is converted into a low-order discrete characteristic model, and a coefficient boundary is given, so that a basis is provided for low-order control design. For aircraft with flexible attachments, passive control methods are currently used in engineering for control. The passive control method is a low-order feedback control method for controlling only the rigid body mode. The characteristic model theory can provide a mechanism for designing a low-order control law by a high-order controlled object. Specifically aiming at the high-order dynamics of the flexible aircraft, the mechanism of the passive control design can be given by utilizing a characteristic model theory. The method is used for solving the problem of establishing a low-order characteristic model of the high-order flexible aircraft. The problem of modeling the characteristics of flexible aircraft has been studied to some extent. The problems that exist at present are: when a nonlinear function in aircraft dynamics is compressed into coefficients of a feature model, errors exist, namely, the established feature model has unmodeled errors, which can cause the reduction of control precision; the bounds of the characteristic model coefficients only give theoretical research results, and no specific bounds of the coefficients are given. The invention solves the problems, provides a characteristic modeling method for error-free compression of the flexible aircraft, and provides a boundary of characteristic model coefficients, thereby providing a foundation for low-order control of the flexible aircraft. The method provided by the invention can be used for flexible hypersonic aircrafts, such as spacecraft with large flexible accessories and other flexible aircrafts, and has better universality.
Disclosure of Invention
The technical problem solved by the invention is as follows: the method overcomes the defects of the prior art, provides an error-free compression characteristic modeling method for the flexible aircraft, and solves the problem of unmodeled dynamics of a characteristic model of the flexible aircraft by providing an error-free compression method for high-order dynamics of the flexible aircraft.
The technical scheme of the invention is as follows: a flexible aircraft error-free compression feature modeling method comprises the following steps:
1) establishing a dynamic equation of a controlled object of the flexible aircraft;
2) converting flexible aircraft dynamics into an accurate feedback linearization standard form;
3) the time scale of the flexible aircraft is obtained,
4) selecting a sampling period T;
5) establishing a rigid body modal equation;
6) establishing a third-order characteristic model;
7) giving a boundary of the characteristic model coefficient;
beginning with step 8), cycling through each control period;
8) identifying coefficients of the characteristic model by adopting a projection gradient method or a projection least square method;
9) designing a three-order adaptive control law;
10) returning to the step 8), entering the next control period.
The specific form of establishing the flexible aircraft controlled object kinetic equation is as follows:
Figure BDA0002775615560000021
wherein the content of the first and second substances,
x1＝[φθψ]T
ws＝[wx wy wz]T
phi, theta, psi denote roll, pitch and yaw attitude angles, wx,wy,wzRespectively representing roll, pitch and yaw attitude angular velocities,
Figure BDA0002775615560000031
representing a transformation matrix;
Figure BDA0002775615560000032
representing the angular velocity antisymmetric array, eta, of the spacecraft centerbodyL∈RlR represents the real number domain, l represents the order of the flexural mode, ηR∈RlThe modal coordinate arrays, xi, of the left and right solar wings respectivelyL，ξRModal damping coefficient, w, of the left and right solar wings, respectivelyL，wRModal frequencies, F, of the left and right solar wings, respectivelysL∈R3×l,FsR∈R3×lRepresenting flexible and rigid coupling matrices, TsRepresenting an array of external moments acting on the spacecraft, IsRepresenting a spacecraft inertia array; consider 0 ≦ ψ<In the case of 90 deg., C (x) when cos ψ ≠ 01) Nonsingular; memory body inertia matrix
Rs＝Is-FFT
Wherein the content of the first and second substances,
F＝[FsL FsR]∈R3×2l。
the method for converting the flexible aircraft dynamics into the accurate feedback linearization standard form specifically comprises the following steps:
Figure BDA0002775615560000033
wherein the content of the first and second substances,
z1＝x1，
z2＝C(x1)ws，
Figure BDA0002775615560000034
Figure BDA0002775615560000035
Figure BDA0002775615560000036
Figure BDA0002775615560000037
Figure BDA0002775615560000038
Figure BDA0002775615560000041
Bη(z1)＝[-FsL 2ξLwLFsL-FsR 2ξRwRFsR]TC(z1)-1。
the specific process for solving the time scale of the flexible aircraft comprises the following steps:
Figure BDA0002775615560000042
wherein the content of the first and second substances,
f＝a2(z1,z2)z2+aη(z1)η
g＝b(z1)
q＝Aηη+Bη(z1)z2
Figure BDA0002775615560000043
the selected sampling period T
Figure BDA0002775615560000044
。
The concrete form of establishing the rigid body modal equation is as follows:
Figure BDA0002775615560000045
wherein the content of the first and second substances,
Figure BDA0002775615560000046
the specific form of the third-order characteristic model is as follows:
x1(k)＝f1(k)x1(k-1)+f2(k)x1(k-2)+f3(k)x1(k-3)+g0(k)Ts(k-1)+g1(k)Ts(k-2)
where k is 1,2, …, coefficient f1(k),f2(k),f3(k),g0(k),g1(k) Obtained by identification in step 8). Bounds of the feature model coefficients:
f1ii(k)∈[-4,2],i＝1,2,3
f2ii(k)∈[1,3],i＝1,2,3
f3ii(k)∈[-2,0],i＝1,2,3
fsij(k)∈[-1,1],s,i,j＝1,2,3,i≠j
i.e., f1(k) Is of the diagonal element of [ -4,2]，f2(k) Is a diagonal element of [1,3 ]]，f3(k) Is of the diagonal element of [ -2,0]Their off-diagonal elements all belong to [ -1,1](ii) a The coefficient bound is solved by the characteristic model theory.
The specific process of identifying the coefficient of the characteristic model by adopting the projection gradient algorithm in the step 8) is as follows:
note the book
Figure BDA0002775615560000051
Wherein the content of the first and second substances,
Figure BDA0002775615560000052
respectively correspond to f1,f2,f3,g0,g1In the identification of (a) a (b),
Figure BDA0002775615560000053
the projection gradient algorithm is adopted as follows;
firstly, calculating parameters to be identified by adopting a gradient method:
Figure BDA0002775615560000054
wherein λ is1>0,λ2>0, is the parameter to be adjusted, the size of which will affect the parameter convergence speed;
the initial values are:
Figure BDA0002775615560000055
Φ(0)＝015×1
the first 9 rows of recognition results are then projected into the set below,
Figure BDA0002775615560000056
finally, to
Figure BDA0002775615560000057
First order filtering is performed:
Figure BDA0002775615560000058
wherein F is more than or equal to 0θ≤1，FθIs the parameter to be adjusted, the size of which will affect the size of the parameter change.
The specific process of identifying the coefficient of the characteristic model by adopting the projection least square method in the step 8) is as follows: note the book
Figure BDA0002775615560000061
Wherein the content of the first and second substances,
Figure BDA0002775615560000062
respectively correspond to f1,f2,f3,g0,g1In the identification of (a) a (b),
Figure BDA0002775615560000063
the projection least squares algorithm is adopted as follows:
firstly, calculating parameters to be identified by adopting a least square method:
Figure BDA0002775615560000064
Figure BDA0002775615560000065
P(k)＝(I15×15-K(k)ΦT(k-1))P(k-1)
wherein x is1(k) As shown in formula (2), measured by the navigation system; the initial values are:
Figure BDA0002775615560000066
Φ(0)＝015×1
P(0)＝I15×15
then, the identification result is projected and filtered with a formula (1) and a formula (2) in a projection gradient method.
The third-order adaptive control law is designed as follows:
Figure BDA0002775615560000067
wherein the content of the first and second substances,
Figure BDA0002775615560000068
identified in step 8).
Compared with the prior art, the invention has the advantages that:
(1) the characteristic modeling method without the compression error of the flexible aircraft provided by the invention is characterized in that firstly, the flexible aircraft dynamics is converted into an input and output accurate feedback linearization form, and then the flexible mode is solved to establish differential equation description of the rigid body mode of the flexible aircraft, wherein the description has an affine linear form for the rigid body mode. The method solves the problem that in the existing flexible aircraft feature modeling, a feature model is directly established by using high-order dynamics of the flexible aircraft, so that a complex nonlinear function needs to be compressed into a coefficient of the feature model to generate a compression error.
Aiming at the characteristics of high-order nonlinear dynamics of the flexible spacecraft, an input and output accurate feedback linearization equation of the flexible spacecraft is established by establishing a differential homoembryo transformation matrix; by solving the flexible mode, the differential equation description of the rigid body mode of the flexible aircraft is established; and finally, establishing a third-order characteristic model of the flexible aircraft according to the characteristic model theory. The established differential equation description is an affine linear form of a rigid body mode and does not contain a flexible mode; and in the process of converting from the high-order nonlinear dynamics of the flexible aircraft, nonlinear compression errors are not generated, so that the established characteristic model does not contain unmodeled dynamics. Therefore, according to the characteristic model provided by the invention, a self-adaptive control law is designed, and the control precision can be improved.
(2) The characteristic modeling method for the flexible aircraft error-free compression provides a specific boundary of the characteristic model coefficient. The problem that the existing feature modeling only gives a theoretical result is solved. The characteristic modeling method without error compression converts the flexible aircraft dynamics into an input and output accurate feedback linearization form, and gives a specific boundary of a coefficient by using the relation between the accurate feedback linearization form and a time scale. Thereby providing a foundation for low-level control of flexible aircraft in engineering.
(3) The method provided by the invention is suitable for flexible hypersonic aircrafts, such as spacecraft with large flexible accessories and other flexible aircrafts, and has better universality.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The invention provides a characteristic modeling method for error-free compression of a flexible aircraft, aiming at the defects of the prior art, and the method is realized by steps (1) to (10) as shown in FIG. 1.
Step (1) establishing a dynamic equation of a controlled object of the flexible aircraft,
Figure BDA0002775615560000071
wherein the content of the first and second substances,
x1＝[φ θ ψ]T (2)
ws＝[wx wy wz]T
phi, theta, psi denote roll, pitch and yaw attitude angles, wx,wy,wzRespectively representing roll, pitch and yaw attitude angular velocities,
Figure BDA0002775615560000081
is a transformation matrix of the image data to be transformed,
Figure BDA0002775615560000082
representing the angular velocity antisymmetric array, eta, of the spacecraft centerbodyL∈Rl(R represents the real number domain, l represents the order of the flexural mode), ηR∈RlThe modal coordinate arrays, xi, of the left and right solar wings respectivelyL，ξRModal damping coefficient, w, of the left and right solar wings, respectivelyL，wRModal frequencies, F, of the left and right solar wings, respectivelysL∈R3×l,FsR∈R3×lRepresenting flexible and rigid coupling matrices, TsRepresenting an array of external moments acting on the spacecraft, IsRepresenting an array of spacecraft inertias. Here, 0 ≦ ψ<The 90 ° case (the quaternion equation is used for the study when ψ is 90 °). As can be seen from a simple calculation, C (x) is calculated when cos ψ ≠ 01) Is not unusual. Memory body inertia matrix
Rs＝Is-FFT
Wherein the content of the first and second substances,
F＝[FsL FsR]∈R3×2l
step (2) the flexible aircraft dynamics is converted into an accurate feedback linearization standard form,
Figure BDA0002775615560000091
wherein the content of the first and second substances,
z1＝x1 (5)
z2＝C(x1)ws (6)
Figure BDA0002775615560000092
Figure BDA0002775615560000093
Figure BDA0002775615560000094
Figure BDA0002775615560000095
Figure BDA0002775615560000096
Figure BDA0002775615560000097
Bη(z1)＝[-FsL 2ξLwLFsL -FsR 2ξRwRFsR]TC(z1)-1
the flexible aircraft accurate feedback linearization standard formal equation (4) is derived through the following process.
Firstly, establishing a differential homomorphic transformation matrix of the flexible aircraft,
Figure BDA0002775615560000098
wherein, C (x)1) Given in equation (3). Can prove the transformation TfAre micro-isoembryonal. Therefore we can use the transformation TfAnd (3) carrying out differential homomorphic transformation on the flexible aircraft kinetic equation formula (1). Then, T is carried out on the aircraft dynamics equation formula (1)fConversion, i.e. left-multiplying T by the state of equation (1)fTo obtain a new state
Figure BDA0002775615560000101
As shown in equation (5) -equation (7). Finally, for the state
Figure BDA0002775615560000102
And (4) derivation is carried out, and then the formula (4) can be obtained through derivation.
Step (3) obtaining the time scale of the flexible aircraft,
Figure BDA0002775615560000103
wherein the content of the first and second substances,
Figure BDA0002775615560000104
and (4) selecting a sampling period.
The sampling period T is selected according to the following equation (9),
Figure BDA0002775615560000105
wherein, TscaleObtained from equation (8).
And (5) establishing a rigid body modal equation.
Establishing a rigid body modal equation,
Figure BDA0002775615560000106
wherein the content of the first and second substances,
Figure BDA0002775615560000107
Figure BDA0002775615560000108
Figure BDA0002775615560000109
Figure BDA00027756155600001010
the derivation process of equation (11) is as follows:
at aη(z1) Under the condition of full rank of the line, the flexible mode can be solved by the 2 nd equation of the formula (4) and the matrix equation solution theory,
Figure BDA00027756155600001011
by taking the derivative of the 2 nd equation of equation (4),
Figure BDA0002775615560000111
by substituting equation 3 of equation (4) into equation (13),
Figure BDA0002775615560000112
further, by substituting the formula (12) into the formula (14), the formula (11) can be derived.
Step (6) establishing a third-order characteristic model,
x1(k)＝f1(k)x1(k-1)+f2(k)x1(k-2)+f3(k)x1(k-3)+g0(k)Ts(k-1)+g1(k)Ts(k-2) (15)
wherein x is1Given in equation (2), k is 1,2, …, coefficient f1(k),f2(k),f3(k),g0(k),g1(k) Identified in step (8).
The formula (15) is derived by the following method. By substituting equation (1) of equation (4) into equation (11) and by the feature model theory, a third-order feature model (15) can be derived.
And (7) giving the boundary of the characteristic model coefficient.
Figure BDA0002775615560000113
I.e., f1(k) Is of the diagonal element of [ -4,2]，f2(k) Is a diagonal element of [1,3 ]]，f3(k) Is of the diagonal element of [ -2,0]Their off-diagonal elements all belong to [ -1,1]。
The coefficient bound is solved by the characteristic model theory.
Next, starting from step (8), a loop is performed for each control period with k equal to 1,2, ….
And (8) identifying the coefficient of the characteristic model by adopting a projection gradient method or a projection least square method.
Note the book
Figure BDA0002775615560000114
Wherein the content of the first and second substances,
Figure BDA0002775615560000115
respectively correspond to f1,f2,f3,g0,g1In the identification of (a) a (b),
Figure BDA0002775615560000116
the projection gradient algorithm is as follows.
Firstly, calculating parameters to be identified by adopting a gradient method:
Figure BDA0002775615560000121
wherein λ is1>0,λ2>0, is the parameter to be adjusted, the size of which will affect the parameter convergence speed; x is the number of1(k) As shown in formula (2), measured by the navigation system; the initial values are:
Figure BDA0002775615560000122
Φ(0)＝015×1
the first 9 rows of recognition results are then projected into the set below,
Figure BDA0002775615560000123
finally, to
Figure BDA0002775615560000124
First order filtering is performed:
Figure BDA0002775615560000125
wherein F is more than or equal to 0θ≤1，FθIs the parameter to be adjusted, the size of which will affect the size of the parameter change.
The projection least squares algorithm is as follows:
firstly, calculating parameters to be identified by adopting a least square method:
Figure BDA0002775615560000126
Figure BDA0002775615560000127
P(k)＝(I15×15-K(k)ΦT(k-1))P(k-1)
wherein x is1(k) As shown in formula (2), measured by the navigation system; the initial values are:
Figure BDA0002775615560000128
Φ(0)＝015×1
P(0)＝I15×15
then, the identification result is projected and filtered with the formula (18) and the formula (19) in the projection gradient method.
And (9) designing a third-order adaptive control law.
In the present invention, the following adaptive control laws can be designed,
Figure BDA0002775615560000131
wherein the content of the first and second substances,
Figure BDA0002775615560000132
identified in step (8).
And (10) circularly entering the step (8) and entering the next control period.
The invention is not described in detail and is within the knowledge of a person skilled in the art.

Claims (11)

1. A flexible aircraft error-free compression feature modeling method is characterized by comprising the following steps:
1) establishing a dynamic equation of a controlled object of the flexible aircraft;
2) converting flexible aircraft dynamics into an accurate feedback linearization standard form;
3) the time scale of the flexible aircraft is obtained,
4) selecting a sampling period T;
5) establishing a rigid body modal equation;
6) establishing a third-order characteristic model;
7) giving a boundary of the characteristic model coefficient;
beginning with step 8), cycling through each control period;
8) identifying coefficients of the characteristic model by adopting a projection gradient method or a projection least square method;
9) designing a three-order adaptive control law;
10) returning to the step 8), entering the next control period.
2. The method of modeling features for error-free compression of a flexible aircraft according to claim 1, wherein: the specific form of establishing the flexible aircraft controlled object kinetic equation is as follows:
Figure FDA0002775615550000011
wherein the content of the first and second substances,
x1＝[φ θ ψ]T
ws＝[wx wy wz]T
phi, theta, psi denote roll, pitch and yaw, respectivelyAttitude angle, wx,wy,wzRespectively representing roll, pitch and yaw attitude angular velocities,
Figure FDA0002775615550000021
representing a transformation matrix;
Figure FDA0002775615550000022
representing the angular velocity antisymmetric array, eta, of the spacecraft centerbodyL∈RlR represents the real number domain, l represents the order of the flexural mode, ηR∈RlThe modal coordinate arrays, xi, of the left and right solar wings respectivelyL，ξRModal damping coefficient, w, of the left and right solar wings, respectivelyL，wRModal frequencies, F, of the left and right solar wings, respectivelysL∈R3×l,FsR∈R3×lRepresenting flexible and rigid coupling matrices, TsRepresenting an array of external moments acting on the spacecraft, IsRepresenting a spacecraft inertia array; consider 0 ≦ ψ<In the case of 90 deg., C (x) when cos ψ ≠ 01) Nonsingular; memory body inertia matrix
Rs＝Is-FFT
Wherein the content of the first and second substances,
F＝[FsL FsR]∈R3×2l。
3. the method of modeling features for error-free compression of a flexible aircraft according to claim 2, wherein: the method for converting the flexible aircraft dynamics into the accurate feedback linearization standard form specifically comprises the following steps:
Figure FDA0002775615550000023
wherein the content of the first and second substances,
z1＝x1，
z2＝C(x1)ws，
Figure FDA0002775615550000024
Figure FDA0002775615550000025
Figure FDA0002775615550000026
Figure FDA0002775615550000027
Figure FDA0002775615550000028
Figure FDA0002775615550000031
Bη(z1)＝[-FsL 2ξLwLFsL -FsR 2ξRwRFsR]TC(z1)-1。
4. the method of modeling features of a flexible aircraft without error compression of claim 3, wherein: the specific process for solving the time scale of the flexible aircraft comprises the following steps:
Figure FDA0002775615550000032
wherein the content of the first and second substances,
f＝a2(z1,z2)z2+aη(z1)η
g＝b(z1)
q＝Aηη+Bη(z1)z2
Figure FDA0002775615550000033
5. the method of modeling features of a flexible aircraft without error compression of claim 4, wherein: the selected sampling period T
Figure FDA0002775615550000034
。
6. The method of modeling features of a flexible aircraft without error compression of claim 5, wherein: the concrete form of establishing the rigid body modal equation is as follows:
Figure FDA0002775615550000035
wherein the content of the first and second substances,
Figure FDA0002775615550000036
Figure FDA0002775615550000037
Figure FDA0002775615550000038
Figure FDA0002775615550000039
。
7. the method of modeling features for error-free compression of a flexible aircraft according to claim 6, wherein: the specific form of the third-order characteristic model is as follows:
x1(k)＝f1(k)x1(k-1)+f2(k)x1(k-2)+f3(k)x1(k-3)+g0(k)Ts(k-1)+g1(k)Ts(k-2)
where k is 1,2, …, coefficient f1(k),f2(k),f3(k),g0(k),g1(k) Obtained by identification in step 8).
8. The method of modeling features for error-free compression of a flexible aircraft according to claim 7, wherein: bounds of the feature model coefficients:
f1ii(k)∈[-4,2],i＝1,2,3
f2ii(k)∈[1,3],i＝1,2,3
f3ii(k)∈[-2,0],i＝1,2,3
fsij(k)∈[-1,1],s,i,j＝1,2,3,i≠j
i.e., f1(k) Is of the diagonal element of [ -4,2]，f2(k) Is a diagonal element of [1,3 ]]，f3(k) Is of the diagonal element of [ -2,0]Their off-diagonal elements all belong to [ -1,1](ii) a The coefficient bound is solved by the characteristic model theory.
9. The method of modeling features for error-free compression of a flexible aircraft according to claim 8, wherein: the specific process of identifying the coefficient of the characteristic model by adopting the projection gradient algorithm in the step 8) is as follows:
note the book
Figure FDA0002775615550000041
Wherein the content of the first and second substances,
Figure FDA0002775615550000042
respectively correspond to f1,f2,f3,g0,g1In the identification of (a) a (b),
Figure FDA0002775615550000043
the projection gradient algorithm is adopted as follows;
firstly, calculating parameters to be identified by adopting a gradient method:
Figure FDA0002775615550000044
wherein λ is1>0,λ2>0, is the parameter to be adjusted, the size of which will affect the parameter convergence speed;
the initial values are:
Figure FDA0002775615550000045
Φ(0)＝015×1
the first 9 rows of recognition results are then projected into the set below,
Figure FDA0002775615550000051
finally, to
Figure FDA0002775615550000052
First order filtering is performed:
Figure FDA0002775615550000053
wherein F is more than or equal to 0θ≤1，FθIs the parameter to be adjusted, the size of which will affect the size of the parameter change.
10. The method of modeling features for error-free compression of a flexible aircraft according to claim 9, wherein: the specific process of identifying the coefficient of the characteristic model by adopting the projection least square method in the step 8) is as follows: note the book
Figure FDA0002775615550000054
Wherein the content of the first and second substances,
Figure FDA0002775615550000055
respectively correspond to f1,f2,f3,g0,g1In the identification of (a) a (b),
Figure FDA0002775615550000056
the projection least squares algorithm is adopted as follows:
firstly, calculating parameters to be identified by adopting a least square method:
Figure FDA0002775615550000057
Figure FDA0002775615550000058
P(k)＝(I15×15-K(k)ΦT(k-1))P(k-1)
wherein x is1(k) As shown in formula (2), measured by the navigation system; the initial values are:
Figure FDA0002775615550000059
Φ(0)＝015×1
P(0)＝I15×15
then, the identification result is projected and filtered with a formula (1) and a formula (2) in a projection gradient method.
11. The method of modeling features for error-free compression of a flexible aircraft according to claim 10, wherein: the third-order adaptive control law is designed as follows:
Figure FDA0002775615550000061
wherein the content of the first and second substances,
Figure FDA0002775615550000062
identified in step 8).
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