CN107808025B - Method and system for inhibiting thermally induced deformation of spacecraft structure - Google Patents

Method and system for inhibiting thermally induced deformation of spacecraft structure Download PDF

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CN107808025B
CN107808025B CN201710800936.6A CN201710800936A CN107808025B CN 107808025 B CN107808025 B CN 107808025B CN 201710800936 A CN201710800936 A CN 201710800936A CN 107808025 B CN107808025 B CN 107808025B
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左博
范立佳
杨松
郭高峰
罗继强
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Beijing Institute of Spacecraft System Engineering
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Abstract

The invention discloses a method and a system for inhibiting thermally induced deformation of a spacecraft structure, wherein the method comprises the following steps: carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow; according to the given temperature field and the thermal deformation inhibition requirement, the control heat flow is obtained through the functional relation between the structural displacement and the space heat flow load and the control heat flow; and applying the control heat flow on the surface of the thin-wall rod piece to change the temperature distribution of the thin-wall rod piece and inhibit the structure of the spacecraft from thermally induced deformation. Therefore, the invention can utilize the existing temperature control equipment on the aircraft to counteract the unfavorable thermal deformation by changing the temperature distribution of the structure, and has the advantages of simple control, high reliability and convenient engineering.

Description

Method and system for inhibiting thermally induced deformation of spacecraft structure
Technical Field
The invention belongs to the technical field of spacecrafts, and particularly relates to a method and a system for inhibiting thermally induced deformation of a spacecraft structure.
Background
Aiming at the problem of thermally induced deformation of the structure of a spacecraft, at present, the inhibition mode generally adopted at home and abroad is as follows: and a piezoelectric actuator is arranged on the surface or in the structure, and the deformation of the structure is actively controlled by the piezoelectric actuator.
However, although the active control method of the piezoelectric actuator has the advantages of fast response and high control precision, the active control method also has the disadvantages of high energy consumption, easy failure of the interface, low reliability and the like, and the disadvantages prevent the application of the piezoelectric actuator to the spacecraft structure with extremely high reliability requirements to a certain extent.
Disclosure of Invention
The technical problem of the invention is solved: the method and the system for inhibiting the thermally induced deformation of the spacecraft structure overcome the defects of the prior art, counteract the unfavorable thermal deformation by changing the temperature distribution of the structure, and have the advantages of simple structure, high reliability and convenience in engineering.
In order to solve the technical problem, the invention discloses a method for inhibiting thermally induced deformation of a spacecraft structure, which comprises the following steps:
carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow;
according to the given temperature field and the thermal deformation inhibition requirement, the control heat flow is obtained through the functional relation between the structural displacement and the space heat flow load and the control heat flow;
and applying the control heat flow on the surface of the thin-wall rod piece to change the temperature distribution of the thin-wall rod piece and inhibit the structure of the spacecraft from thermally induced deformation.
In the method for suppressing the thermally induced deformation of the spacecraft structure, the applying the control heat flow on the surface of the thin-wall rod piece to change the temperature distribution of the thin-wall rod piece and suppress the thermally induced deformation of the spacecraft structure includes:
arranging a controlled heating sheet on the surface of the thin-wall rod piece;
and controlling the controlled heating plate to generate local heat flow consistent with the control heat flow, changing the temperature distribution of the thin-wall rod piece, and inhibiting the structure of the spacecraft from thermally induced deformation.
In the method for suppressing the thermally induced deformation of the spacecraft structure, the step of obtaining the control heat flow through a functional relationship between the structure displacement, the space heat flow load and the control heat flow according to the given temperature field and the thermal deformation suppression requirement comprises the following steps:
analyzing the sensitivity of the displacement field according to the given temperature field and the thermal deformation inhibition requirement to obtain the deviation and the sensitivity of the current displacement and the target displacement;
and according to the deviation and the sensitivity of the current displacement and the target displacement, performing optimal solution on the functional relationship between the structural displacement and the space heat flow load and the control heat flow by adopting a Gaussian-Newton algorithm to obtain the control heat flow.
In the above method for suppressing thermally induced deformations of a spacecraft structure,
determining the structural displacement corresponding to the control heat flow, and judging whether the control deviation meets the control precision required by the thermal deformation inhibition requirement;
and if the control deviation does not meet the control accuracy required by the thermal deformation inhibition requirement, returning to recalculate the control heat flow until the control deviation meets the control accuracy required by the thermal deformation inhibition requirement.
In the method for inhibiting the thermally induced deformation of the spacecraft structure, the functional relation between the structure displacement and the space heat flow load and the control heat flow is determined by the following steps:
adopting Fourier unit to obtain space heat flow load Q of spacecraft frame structures(t) and controlling the Heat flow Qc(t) temperature response under influence:
Figure BDA0001401533450000021
Figure BDA0001401533450000022
wherein, the formulas (1) and (2) are transient heat conduction finite element equations formed by the average temperature and the perturbation temperature of the Fourier unit respectively;
Figure BDA0001401533450000023
T0represents the average temperature; t ismRepresenting the perturbation temperature; c represents a heat capacity matrix; k0And KmRespectively representing heat conduction matrixes corresponding to an average temperature equation and a perturbation temperature equation; r (T)0) Representing a radiation matrix, proportional to the third power of the average temperature;
Figure BDA0001401533450000024
and
Figure BDA0001401533450000025
respectively representing space heat flow load vectors of an average temperature equation and a perturbation temperature equation;
Figure BDA0001401533450000039
and
Figure BDA00014015334500000310
respectively representing control heat flow load vectors of an average temperature equation and a perturbation temperature equation;
according to a static finite element equation of the structure, obtaining the structure displacement u (t):
Ku(t)＝F(T(t))···(3)
wherein K is a stiffness matrix of the structure; f (T) represents a structural equivalent temperature load; t (T) [ [ T ]0(t)]T [Tm(t)]T]；
Determining the functional relationship between the structural displacement and the space heat flow load and the control heat flow:
u(t)＝F[Qs(t),Qc(t),t]。
in the method for suppressing the thermally induced deformation of the spacecraft structure, the optimal solution of the functional relationship between the structure displacement and the space heat flow load and the control heat flow is performed by using a gaussian-newton algorithm according to the deviation and the sensitivity between the current displacement and the target displacement to obtain the control heat flow, and the method includes the following steps:
converting the functional relation between the structural displacement and the space heat flow load and the control heat flow into nonlinear optimal control of a discrete time system to obtain a target function;
and (3) performing optimization solution on the objective function by adopting a Gaussian-Newton algorithm:
order: the iterative relationship of the k step of the gauss-newton algorithm is:
Figure BDA0001401533450000031
then, the objective function is
Figure BDA0001401533450000032
The Taylor expansion of the points is:
Figure BDA0001401533450000033
wherein,
Figure BDA0001401533450000034
is to QcGradient operator;
Figure BDA0001401533450000035
Figure BDA0001401533450000036
wherein, the Jacobian matrix
Figure BDA0001401533450000037
Order:
Figure BDA0001401533450000038
then the process of the first step is carried out,
Figure BDA0001401533450000041
substituting equations (5) and (6) into equation (7) and ignoring the second derivative term of V yields:
Figure BDA0001401533450000042
from equations (4) and (8), the iterative format of the gauss-newton algorithm can be obtained:
Figure BDA0001401533450000043
Figure BDA0001401533450000044
when in use
Figure BDA0001401533450000045
And
Figure BDA0001401533450000046
when the conditions are met, determining iterative convergence, and solving the control heat flow:
Figure BDA0001401533450000047
Figure BDA0001401533450000048
wherein c is the number of control variables.
In the method for inhibiting the thermally induced deformation of the spacecraft structure, the jacobian matrix is determined by the following steps:
to control the heat flow QcIs the control variable dkFor formulas (1), (2) and (3) with respect to dkThe partial derivatives are calculated to obtain:
Figure BDA0001401533450000049
Figure BDA00014015334500000410
Figure BDA00014015334500000411
wherein,
Figure BDA00014015334500000412
represents a pair control variable dkPartial derivatives of (d);
Figure BDA00014015334500000413
and
Figure BDA00014015334500000414
respectively represent the average temperature vector T0And perturbation of the temperature vector TmPartial derivatives of (d);
the following equations (9), (10) and (11) can be collated:
Figure BDA00014015334500000415
Figure BDA00014015334500000416
Figure BDA00014015334500000417
wherein Q is0And QmRespectively as follows:
Figure BDA0001401533450000051
Figure BDA0001401533450000052
using the Wilson-theta method, using the average temperature sensitivity at time t
Figure BDA0001401533450000053
And perturbation of temperature sensitivity
Figure BDA0001401533450000054
Average temperature sensitivity at t + Δ t
Figure BDA0001401533450000055
And perturbation of temperature sensitivity
Figure BDA0001401533450000056
Obtaining:
Figure BDA0001401533450000057
Figure BDA0001401533450000058
substituting the solving results of the formulas (15) and (16) into the formula (14), and solving to obtain a Jacobian matrix
Figure BDA0001401533450000059
Correspondingly, the invention also discloses a system for inhibiting the thermally induced deformation of the spacecraft structure, which comprises the following components:
the first calculation module is used for carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow;
the second resolving module is used for solving the control heat flow through the functional relation between the structural displacement and the space heat flow load and the control heat flow according to the given temperature field and the thermal deformation inhibition requirement;
and the control module is used for applying the control heat flow on the surface of the thin-wall rod piece, changing the temperature distribution of the thin-wall rod piece and inhibiting the structure of the spacecraft from thermally induced deformation.
The invention has the following advantages:
the invention discloses a method for inhibiting the structure thermal deformation of a spacecraft, which comprises the steps of carrying out thermal deformation analysis on a structure finite element model of the spacecraft to obtain a functional relation between structure displacement and space heat flow load and control heat flow; according to the given temperature field and the thermal deformation inhibition requirement, the control heat flow is obtained through the functional relation between the structural displacement and the space heat flow load and the control heat flow; and applying the control heat flow on the surface of the thin-wall rod piece to change the temperature distribution of the thin-wall rod piece and inhibit the structure of the spacecraft from thermally induced deformation. Therefore, the invention can utilize the existing temperature control equipment on the aircraft to counteract the unfavorable thermal deformation by changing the temperature distribution of the structure, and has the advantages of simple control, high reliability and convenient engineering.
Drawings
FIG. 1 is a flow chart illustrating the steps of a method for suppressing thermally-induced deformations of a spacecraft structure according to an embodiment of the present invention;
fig. 2 is a schematic diagram of the numbering of the temperature control points of the bottom plate of a certain satellite service bay in an embodiment of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, common embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.
The invention discloses a method for inhibiting the structure of a spacecraft from thermally induced deformation, which counteracts the unfavorable thermal deformation by changing the temperature distribution of the structure.
Referring to fig. 1, a flowchart illustrating steps of a method for suppressing thermally-induced deformation of a spacecraft structure according to an embodiment of the present invention is shown. In this embodiment, the method for suppressing the thermally induced deformation of the spacecraft structure includes:
and 101, carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow.
In a preferred embodiment of the invention, the functional relationship between the displacement of the structure and the load of the space heat flow and the control heat flow can be determined by:
step S11, adopting Fourier unit to obtain space heat flow load Q of spacecraft frame structures(t) and controlling the Heat flow Qc(t) temperature response under influence:
Figure BDA0001401533450000061
Figure BDA0001401533450000062
wherein, the formulas (1) and (2) are transient heat conduction finite element equations composed of the average temperature and the perturbation temperature of the Fourier unit, namely an average temperature field equation and a perturbation temperature field equation;
Figure BDA0001401533450000063
T0represents the average temperature; t ismRepresenting the perturbation temperature; c represents a heat capacity matrix; k0And KmRespectively representing heat conduction matrixes corresponding to an average temperature equation and a perturbation temperature equation; r (T)0) Representing a radiation matrix, proportional to the third power of the average temperature;
Figure BDA0001401533450000071
and
Figure BDA0001401533450000072
respectively representing space heat flow load vectors of an average temperature equation and a perturbation temperature equation;
Figure BDA0001401533450000073
and
Figure BDA0001401533450000074
respectively representing control heat flow load vectors of an average temperature equation and a perturbation temperature equation; t represents the action time. In the present embodiment, equations (1) and (2) are ordinary differential equations that can be solved discretely in the time domain by the Wilson- θ method. Since equation (1) is non-linear, a Newton-Raphson iterative solution is also used at each time step. After the average temperature is found, equation (2) is linear and can be directly solved. Because the average temperature field equation and the perturbation temperature field equation are decoupled, the solution efficiency can be greatly improved by adopting the Fourier unit.
Step S12, according to the static finite element equation of the structure, obtaining the structure displacement u (t):
Ku(t)＝F(T(t))···(3)
in this embodiment, the structure displacement u (t) can be obtained from the static finite element equation of the structure without considering the influence of the inertia force. Wherein K is a structureA stiffness matrix of (a); f (T) represents a structural equivalent temperature load; t (T) [ [ T ]0(t)]T [Tm(t)]T]。
Step S13, determining the functional relationship between the structure displacement and the space heat flow load and the control heat flow:
u(t)＝F[Qs(t),Qc(t),t]。
and 102, obtaining the control heat flow through the functional relation between the structural displacement, the space heat flow load and the control heat flow according to the given temperature field and the thermal deformation inhibition requirement.
In this embodiment, the sensitivity of the displacement field may be analyzed according to a given temperature field and thermal deformation suppression requirements, so as to obtain the deviation and sensitivity between the current displacement and the target displacement; and according to the deviation and the sensitivity of the current displacement and the target displacement, performing optimal solution on the functional relationship between the structural displacement and the space heat flow load and the control heat flow by adopting a Gaussian-Newton algorithm to obtain the control heat flow.
In this embodiment, the structural displacement corresponding to the control heat flow may be determined, and it is determined whether the control deviation satisfies the control accuracy required by the thermal deformation suppression requirement; and if the control deviation does not meet the control accuracy required by the thermal deformation inhibition requirement, returning to recalculate the control heat flow until the control deviation meets the control accuracy required by the thermal deformation inhibition requirement.
In a preferred embodiment of the present invention, the optimally solving the functional relationship between the structural displacement and the space heat flow load and the control heat flow by using a gaussian-newton algorithm according to the deviation and the sensitivity between the current displacement and the target displacement to obtain the control heat flow specifically may include:
and step S21, converting the functional relation between the structure displacement and the space heat flow load and the control heat flow into nonlinear optimal control of a discrete time system to obtain a target function.
In this embodiment, for u (t) ═ F [ Qs(t),Qc(t),t]By controlling the heat flow Q with minimum control energyc(t) to make the structure bitShifting u (t) over a period of time (t)0,tf) After tfTime of day and target displacement udThe deviation of (t) is minimal and can be expressed as an optimal control problem as follows:
equation of state
Figure BDA0001401533450000081
Figure BDA0001401533450000082
An objective function W:
Figure BDA0001401533450000083
for a given equation of state
Figure BDA0001401533450000084
Seeking an admission control for heat flow
Figure BDA0001401533450000085
The objective function W is made to take a minimum value,
Figure BDA0001401533450000086
i.e. the optimum control sought.
Preferably, the optimal control problem is a nonlinear control problem due to the presence of radiative heat transfer, the time interval (t) being0,tf) Equally divided into n intervals: (t)0,t1)，(t1,t2)，…(tr-1,tr)，(tn-1,tn＝tf) Wherein (r ═ 1,2, …, n). Thus, the nonlinear control system can be simplified as follows:
(1) u (t) only at a limited number of time points trSatisfies u (t)r)＝udr，trTarget displacement at time, tr∈(0,t)。
(2)Qc(t) at (t)r-1,tr) Linear change in time period, note Qcr＝Qc(tr)，trControl of heat flow at a point in time.
The nonlinear control problem of the continuous-time system can be transformed into a nonlinear optimal control problem of the discrete-time system:
the dynamic equation is as follows: u (t)r)＝F′[u(tr-1),Qs(tr-1),Qc(tr-1),tr-1],(r＝1,2,…,n)
An objective function:
Figure BDA0001401533450000087
setting:
u＝[u(t1)T u(t2)T … u(tn)T]T
Figure BDA0001401533450000088
Figure BDA0001401533450000089
then, the objective function can be expressed as:
Figure BDA00014015334500000912
wherein:
V＝ud-u
then, an allowance for controlling the heat flow is sought
Figure BDA00014015334500000913
Make the objective function W*Taking the minimum value of the number of the bits,
Figure BDA00014015334500000914
i.e. the optimum control of the discrete time system sought.
And step S22, performing optimization solution on the objective function by adopting a Gaussian-Newton algorithm.
In this embodiment, the gauss-newton algorithm is an iterative process, and the process of optimally solving the objective function by using the gauss-newton algorithm may be as follows:
order: the iterative relationship of the k step of the gauss-newton algorithm is:
Figure BDA0001401533450000091
then, the objective function W*In that
Figure BDA0001401533450000092
The Taylor expansion of the points is:
Figure BDA0001401533450000093
wherein, k is a variable,
Figure BDA0001401533450000094
is QcThe gradient operator of (2);
Figure BDA0001401533450000095
Figure BDA0001401533450000096
wherein, the Jacobian matrix
Figure BDA0001401533450000097
Order:
Figure BDA0001401533450000098
then the process of the first step is carried out,
Figure BDA0001401533450000099
substituting equations (5) and (6) into equation (7) and ignoring the second derivative term of V yields:
Figure BDA00014015334500000910
from equations (4) and (8), the iterative format of the gauss-newton algorithm can be obtained:
Figure BDA00014015334500000911
Figure BDA0001401533450000101
when in use
Figure BDA0001401533450000102
And
Figure BDA0001401533450000103
when the conditions are met, determining iterative convergence, and solving the control heat flow:
Figure BDA0001401533450000104
Figure BDA0001401533450000105
wherein c is the number of control variables.
In a preferred embodiment of the present invention, the jacobian matrix may be determined by:
to control the heat flow QcIs the control variable dkFor formulas (1), (2) and (3) with respect to dkThe partial derivatives are calculated to obtain:
Figure BDA0001401533450000106
Figure BDA0001401533450000107
Figure BDA0001401533450000108
wherein,
Figure BDA0001401533450000109
represents a pair control variable dkPartial derivatives of (d);
Figure BDA00014015334500001010
and
Figure BDA00014015334500001011
respectively represent the average temperature vector T0And perturbation of the temperature vector TmPartial derivatives of (d);
the following equations (9), (10) and (11) can be collated:
Figure BDA00014015334500001012
Figure BDA00014015334500001013
Figure BDA00014015334500001014
wherein Q is0And QmRespectively as follows:
Figure BDA00014015334500001015
Figure BDA00014015334500001016
in the present embodiment, to solve the coefficient matrix and the load directionThe partial derivative of the quantity can be obtained by firstly explicitly giving a corresponding sensitivity expression by a unit coefficient matrix and a load vector in a unit sense and then obtaining the partial derivative in the overall coordinate system through a group set. After solving equations (1) and (2), equations (12) and (13) are constant coefficient differential equations. Thus, the time domain dispersion can be realized by Wilson-theta method, and the average temperature sensitivity at t time can be used
Figure BDA0001401533450000111
And perturbation of temperature sensitivity
Figure BDA0001401533450000112
Average temperature sensitivity at t + Δ t
Figure BDA0001401533450000113
And perturbation of temperature sensitivity
Figure BDA0001401533450000114
Obtaining:
Figure BDA0001401533450000115
Figure BDA0001401533450000116
where θ is an intermediate variable. Substituting the solving results of the formulas (15) and (16) into the formula (14), and solving to obtain a Jacobian matrix
Figure BDA0001401533450000117
And 103, applying the control heat flow on the surface of the thin-wall rod piece, changing the temperature distribution of the thin-wall rod piece, and inhibiting the structure of the spacecraft from thermally induced deformation.
In this embodiment, controlled heating patches may be arranged on the thin-walled rod surface; and controlling the controlled heating plate to generate local heat flow consistent with the control heat flow, changing the temperature distribution of the thin-wall rod piece, and inhibiting the structure of the spacecraft from thermally induced deformation. The invention utilizes the existing temperature control equipment on the aircraft, and has the advantages of simple control, high reliability and convenient engineering.
Based on the above embodiments, the method for suppressing the thermally induced deformation of the spacecraft structure is described with reference to a specific example. Referring to fig. 2, a schematic numbering of temperature control points of the floor of a satellite service bay according to an embodiment of the present invention is shown. In this embodiment, when a normal vector of a bottom plate of a satellite service bay-Y + X is in a high-temperature working condition, the maximum included angle of the normal vector has 30 arc seconds, and it is necessary to suppress the change of the included angle of the normal vector of the bottom plate of the satellite service bay-Y + X. 29 temperature control points (as shown in fig. 2) selected near the service bay-Y + X bottom plate are used as a candidate temperature control point set, and the Z-direction movement of the service bay-Y + X bottom plate is restrained by adopting the method for restraining the thermally induced deformation of the spacecraft structure.
Preferably, two time points of 600s and 1200s are selected for control. The result is shown in table 1, the vector angle is controlled from 28.4 arc seconds (maximum angle point) to 5.9 arc seconds (maximum angle point), and the control effect is achieved:
Time uncontrolled angle (second angle) Controlling back angle (second angle)
600s 13.8 4.4
1200s 28.4 5.9
TABLE 1 service cabin-Y + X baseboard normal vector included angle control result table
On the basis of the embodiment of the method, the invention also discloses a system for inhibiting the thermally induced deformation of the spacecraft structure, which comprises the following steps: the first calculation module is used for carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow; the second resolving module is used for solving the control heat flow through the functional relation between the structural displacement and the space heat flow load and the control heat flow according to the given temperature field and the thermal deformation inhibition requirement; and the control module is used for applying the control heat flow on the surface of the thin-wall rod piece, changing the temperature distribution of the thin-wall rod piece and inhibiting the structure of the spacecraft from thermally induced deformation.
For the system embodiment, since it corresponds to the method embodiment, the description is relatively simple, and for the relevant points, refer to the description of the method embodiment section.
The embodiments in the present description are all described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other.
The above description is only for the best mode of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention.
Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (1)

1. A method for inhibiting thermally-induced deformation of a spacecraft structure, comprising:
carrying out thermal deformation analysis on a structural finite element model of the spacecraft to obtain a functional relation between structural displacement and space heat flow load and control heat flow;
according to the given temperature field and the thermal deformation inhibition requirement, the control heat flow is obtained through the functional relation between the structural displacement and the space heat flow load and the control heat flow; the method comprises the following steps: analyzing the sensitivity of the displacement field according to the given temperature field and the thermal deformation inhibition requirement to obtain the deviation and the sensitivity of the current displacement and the target displacement; according to the deviation and the sensitivity of the current displacement and the target displacement, a Gaussian-Newton algorithm is adopted to carry out optimization solution on the functional relation between the structural displacement and the space heat flow load and the control heat flow, and the control heat flow is obtained; determining the structural displacement corresponding to the control heat flow, and judging whether the control deviation meets the control precision required by the thermal deformation inhibition requirement; if the control deviation does not meet the control precision required by the thermal deformation inhibition requirement, the control heat flow is calculated again until the control deviation meets the control precision required by the thermal deformation inhibition requirement;
the control heat flow is applied to the surface of the thin-wall rod piece, the temperature distribution of the thin-wall rod piece is changed, and the thermal deformation of the spacecraft structure is inhibited; the method comprises the following steps: arranging a controlled heating sheet on the surface of the thin-wall rod piece; the controlled heating plate is controlled to generate local heat flow consistent with the control heat flow, the temperature distribution of the thin-wall rod piece is changed, and the thermal deformation of the spacecraft structure is inhibited;
wherein:
determining a functional relationship between structural displacement and space heat flow loading and control heat flow by:
adopting Fourier unit to obtain space heat flow load Q of spacecraft frame structures(t) and controlling the Heat flow Qc(t) temperature response under influence:
Figure FDA0003085752580000011
Figure FDA0003085752580000012
wherein, the formulas (1) and (2) are transient heat conduction finite element equations formed by the average temperature and the perturbation temperature of the Fourier unit respectively;
Figure FDA0003085752580000013
T0represents the average temperature; t ismRepresenting the perturbation temperature; c represents a heat capacity matrix; k0And KmRespectively representing heat conduction matrixes corresponding to an average temperature equation and a perturbation temperature equation; r (T)0) Representing a radiation matrix, proportional to the third power of the average temperature;
Figure FDA0003085752580000021
and
Figure FDA0003085752580000022
respectively representing space heat flow load vectors of an average temperature equation and a perturbation temperature equation;
Figure FDA0003085752580000023
and
Figure FDA0003085752580000024
respectively representing control heat flow load vectors of an average temperature equation and a perturbation temperature equation;
according to a static finite element equation of the structure, obtaining the structure displacement u (t):
Ku(t)＝F(T(t))…(3)
wherein K is a stiffness matrix of the structure; f (T) represents a structural equivalent temperature load; t (T) [ [ T ]0(t)]T [Tm(t)]T]；
Determining the functional relationship between the structural displacement and the space heat flow load and the control heat flow:
u(t)＝F[Qs(t),Qc(t),t]
for u (t) ═ F [ Qs(t),Qc(t),t]By controlling the heat flow Q with minimum control energyc(t) to displace the structure u (t) over a time period (t)0,tf) After tfTime of day and target displacement ud(t) is the smallest deviation, expressed as the following optimal control problem:
equation of state
Figure FDA0003085752580000025
Figure FDA0003085752580000026
An objective function W:
Figure FDA0003085752580000027
for a given equation of state
Figure FDA0003085752580000028
Seeking an admission control for heat flow
Figure FDA0003085752580000029
The objective function W is made to take a minimum value,
Figure FDA00030857525800000210
the optimal control is obtained;
time interval (t)0,tf) Equally divided into n intervals: (t)0,t1)，(t1,t2)，…(tr-1,tr)，(tn-1,tn＝tf)，r＝1,2,…,n；
The nonlinear control system is simplified as follows:
u (t) only at a limited number of time points trSatisfies u (t)r)＝udr，trTarget displacement at time point, Qc(t) at (t)r-1,tr) Linear change in time period, note Qcr＝Qc(tr)，trControl heat flow at time points, tr∈(0,t)；
Then, the nonlinear control problem of the continuous time system is converted into the nonlinear optimal control problem of the discrete time system:
the dynamic equation is as follows: u (t)r)＝F′[u(tr-1),Qs(tr-1),Qc(tr-1),tr-1],(r＝1,2,…,n)
An objective function:
Figure FDA0003085752580000031
setting:
u＝[u(t1)T u(t2)T … u(tn)T]T
Figure FDA0003085752580000032
Figure FDA0003085752580000033
then, the objective function can be expressed as:
Figure FDA0003085752580000034
wherein:
V＝ud-u
then, an allowance for controlling the heat flow is sought
Figure FDA0003085752580000035
Make the objective function W*Taking the minimum value of the number of the bits,
Figure FDA0003085752580000036
the optimal control of the discrete time system is obtained;
according to the deviation and the sensitivity of the current displacement and the target displacement, a Gaussian-Newton algorithm is adopted to carry out optimization solution on the functional relation between the structural displacement and the space heat flow load and the control heat flow to obtain the control heat flow, and the method comprises the following steps:
converting the functional relation between the structural displacement and the space heat flow load and the control heat flow into nonlinear optimal control of a discrete time system to obtain a target function;
and (3) performing optimization solution on the objective function by adopting a Gaussian-Newton algorithm:
order: the iterative relationship of the k step of the gauss-newton algorithm is:
Figure FDA0003085752580000037
then, the objective function is
Figure FDA0003085752580000038
The Taylor expansion of the points is:
Figure FDA0003085752580000039
wherein,
Figure FDA00030857525800000310
is to QcGradient operator;
Figure FDA00030857525800000311
Figure FDA0003085752580000041
wherein, the Jacobian matrix
Figure FDA0003085752580000042
Order:
Figure FDA0003085752580000043
then the process of the first step is carried out,
Figure FDA0003085752580000044
substituting equations (5) and (6) into equation (7) and ignoring the second derivative term of V yields:
Figure FDA0003085752580000045
from equations (4) and (8), the iterative format of the gauss-newton algorithm can be obtained:
Figure FDA0003085752580000046
Figure FDA0003085752580000047
when in use
Figure FDA0003085752580000048
And
Figure FDA0003085752580000049
when the conditions are met, determining iterative convergence, and solving the control heat flow:
Figure FDA00030857525800000410
Figure FDA00030857525800000411
wherein c is the number of control variables;
determining a jacobian matrix by:
to control the heat flow QcIs the control variable dkTo formula (1)) (2) and (3) with respect to dkThe partial derivatives are calculated to obtain:
Figure FDA00030857525800000412
Figure FDA00030857525800000413
Figure FDA00030857525800000414
wherein,
Figure FDA00030857525800000415
represents a pair control variable dkPartial derivatives of (d);
Figure FDA00030857525800000416
and
Figure FDA00030857525800000417
respectively represent the average temperature vector T0And perturbation of the temperature vector TmPartial derivatives of (d);
the following equations (9), (10) and (11) can be collated:
Figure FDA0003085752580000051
Figure FDA0003085752580000052
Figure FDA0003085752580000053
wherein Q is0And QmRespectively as follows:
Figure FDA0003085752580000054
Figure FDA0003085752580000055
using the Wilson-theta method, using the average temperature sensitivity at time t
Figure FDA0003085752580000056
And perturbation of temperature sensitivity
Figure FDA0003085752580000057
Average temperature sensitivity at t + Δ t
Figure FDA0003085752580000058
And perturbation of temperature sensitivity
Figure FDA0003085752580000059
Obtaining:
Figure FDA00030857525800000510
Figure FDA00030857525800000511
substituting the solving results of the formulas (15) and (16) into the formula (14), and solving to obtain a Jacobian matrix
Figure FDA00030857525800000512
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