CN111552180A - Tether system satellite system tether deployment oscillation suppression control method - Google Patents

Tether system satellite system tether deployment oscillation suppression control method Download PDF

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CN111552180A
CN111552180A CN202010366720.5A CN202010366720A CN111552180A CN 111552180 A CN111552180 A CN 111552180A CN 202010366720 A CN202010366720 A CN 202010366720A CN 111552180 A CN111552180 A CN 111552180A
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tether
time
representing
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segment
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朱战霞
江华
王闯
赵素平
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Northwestern Polytechnical University
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Abstract

The invention discloses a tether deployment oscillation suppression control method of a tether satellite system, which comprises the steps of firstly, taking a two-body tether satellite system on a circular orbit as a research object, aiming at the characteristic that the tether satellite system cannot directly control the internal angle of a plane in the deployment process under-actuation, converting an optimal control problem into a parameter optimization problem with algebraic constraints by using a Gaussian pseudo-spectrum method, and calculating an optimal track by giving constraints of the deployment process and a terminal. On the basis, the optimal track is used as a reference track, a nonlinear model is adopted to predict tracking control, and the problem of track tracking control is researched under the condition of introducing observation errors. The invention adopts a nonlinear model prediction tracking control method to carry out tracking control on the optimal track, so that the tethered satellite system has anti-interference capability and realizes the optimal index in the unfolding process. In addition, the closed-loop control method is applied to the bead point model, and the method is verified to have strong applicability under the actual condition.

Description

Tether system satellite system tether deployment oscillation suppression control method
[ technical field ] A method for producing a semiconductor device
The invention belongs to the technical field of tethered satellite systems, and relates to a tether deployment oscillation suppression control method for a tethered satellite system.
[ background of the invention ]
Tethered satellite systems were originally proposed by russian's "father of space" ziolkovsky (Tsiolkovsky), who proposed the idea of attempting to achieve an "equatorial open-sky tower" (i.e., "space elevator") from earth directly to geosynchronous orbit by a tethered connection in 1895. In recent decades, rapid explosion in the aerospace field and continuous breakthrough in the material field have brought about the attention of researchers by virtue of the advantages of tethered satellite systems. The tethered satellite system has unique advantages in the fields of deep space power generation, space debris cleaning, space gravity simulation and the like. For example, by utilizing tethered satellite formation, deep space exploration, aurora observation, three-dimensional exploration, interferometry and the like can be realized; by utilizing the momentum exchange system, the generation of an artificial microgravity environment, the orbit change of a spacecraft, energy transmission, material transportation and the like can be realized; the tethered satellite system with the capturing mechanism can be used for capturing and removing space debris; by utilizing the electric power rope, the tethered satellite system can also be applied to large-scale space structure building, spacecraft maneuvering, orbit reduction of abandoned satellites, active removal of space debris and the like. In addition, the space tethered satellite system can be used for non-contact space close-range operation.
In the design and research of tethered satellite systems, safety considerations should be considered, and many challenging performance requirements are involved. Taking the case of a spatial tether interferometry system, where all satellites must be coordinated to maintain precise relative positions at all times, accurate relative attitude positioning is maintained while pointing accurately at the same target to ensure proper transmission of light. The engineering 'feasibility' is only the bottom line required by the design and control of the rope system, and the performance of the system is also crucial to the success or failure of the actual task. However, the problem of nonlinear optimal control of such complex systems still remains one of the difficulties in the control field. It is worth noting how the control force should be applied in practical applications. In the case of tethered satellite systems, the jet is not the preferred actuator, and it is simpler and more economical to control by adjusting tether tension, length, tether position (offset control), or a combination of these parameters. This control is not applicable to conventional satellite systems, but is unique to tethered satellite systems.
In the tethered satellite space experiments up to now in 1966, there were many items that ended up being defeated because of the tether deployment problem. In a TSS-1 experiment with a subsatellite thruster, when the system fails to deploy to 268m, the experimental data shows that the spacecraft in an active control mode is more suitable for a short tether system, and the performance is poor by a control method of the subsatellite thruster in the deployment of a longer tether. Therefore, the control method of the subsatellite with the thruster is not universal. In the TPE tether experiment, the tether is not kept in a tensioning state during the deployment process, large-amplitude oscillation is generated, and tether winding is generated during subsequent recovery. In the ATEx experiment with no thrusters for the subsatellite, the tether cannot be in a tensioning state due to oscillation, so that the tether is finally wound, the subsatellite deviates from an expected track due to too large oscillation of the tether, and the tether is only unfolded for 22m, so that a sensor feedback error signal is finally failed to cause task failure. In an SEDS-2 experiment, a tether satellite system is unfolded by adopting an open-loop prediction control method, and the relative speed of a tether reaches 7m/s at the moment of the completion of unfolding, so that a series of serious tether oscillations are caused. It can be seen from the above failed tether satellite experimental project that if the oscillation of the tether is suppressed and the thin tether is always in a tensioned state during the deployment of the tether, the problems of the tether winding, deviation from the orbit and transmission of error information will not occur. Therefore, the problem of oscillation suppression during the deployment of the tether becomes a research focus and difficulty that must be solved for the task.
Tethered satellite systems are characterized by a high degree of non-linearity, the control problem of which typically involves complex states, control constraints. The control method for the unfolding of the tether of the tethered satellite system can be divided into open-loop control and closed-loop control. When open-loop control is researched on the problem of unfolding control of the tethered satellite system, as the open-loop control needs to input a control law at the beginning moment of unfolding and cannot compensate disturbance in the unfolding process, most scholars select and use a relatively accurate tethered system dynamic model to improve the effectiveness of the designed control law. The closed-loop control can make up for the defect that the open-loop control cannot feed back and compensate disturbance in real time to a certain extent, but when the closed-loop control is carried out on the spreading process of the rope system satellite system, the problems of feedback, disturbance and the like need to be considered, so most scholars select a simpler rigid rod model as a research object. The model treats the tether as a non-stretchable rigid rod, which can greatly reduce the difficulty of controller design and simulation, but cannot accurately describe the dynamic characteristics of the tether in the process of deployment and recovery, particularly the oscillation characteristics under certain conditions, so that the requirements of tether oscillation suppression and control research in the process of deployment and recovery cannot be completely met. Therefore, it is necessary to design and verify the closed-loop controller on a more accurate model.
[ summary of the invention ]
The invention aims to solve the problems in the prior art and provides a tether deployment oscillation suppression control method for a tether satellite system.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
step 1, a dynamic model of a tethered satellite system. Firstly, neglecting the influence of attitude motion of a satellite and tether flexibility, establishing a dumbbell model of a two-body tether satellite system on a circular orbit through a second Lagrange equation, further researching a tether multi-body dynamics modeling method, and expanding a tether model to obtain a bead point model with higher precision. And deducing the unfolding mode of the bead point model based on the assumption that the number of the tied rope segments is not changed when the bead point model is unfolded.
And 2, planning an optimal path in the tether deployment process. Firstly, researching the optimal track development state of the target function by taking the internal angle of the surface as the target function; and then respectively taking the optimal time, the optimal time and the optimal tension as objective functions to carry out expansion trajectory planning, and comparing the system performance and stability under each objective function. And selecting a proper objective function according to the comparison conclusion, and performing optimal trajectory planning on the bead point model.
And 3, tracking and controlling the tether track. And (3) introducing nonlinear model predictive tracking control, firstly, carrying out tracking control by taking the optimal track of the dumbbell model in the step (2) as a reference track, and verifying the feasibility of the control method. And then, considering observation errors on the basis, applying disturbance in a white noise form to the internal angle of the surface in the expansion process, and further verifying the applicability of the nonlinear model prediction tracking control. And finally, tracking control is carried out by taking the optimal track of the bead point model as a reference track, and verification of closed-loop control aiming at the accurate model is completed.
Compared with the prior art, the invention has the following beneficial effects:
the invention adopts a nonlinear model prediction tracking control method to carry out tracking control on the optimal track, so that the tethered satellite system has anti-interference capability and realizes the optimal index in the unfolding process. Firstly, a two-body tethered satellite system on a circular orbit is taken as a research object, aiming at the characteristic that the under-actuated tethered satellite system cannot directly control the internal angle of the surface in the unfolding process, a Gaussian pseudo-spectrum method is used for converting the optimal control problem into a parameter optimization problem with algebraic constraints, and an optimal track is calculated by giving the constraints of the unfolding process and a terminal. On the basis, the optimal track is used as a reference track, a nonlinear model is adopted to predict tracking control, and the problem of track tracking control is researched under the condition of introducing observation errors; in addition, the closed-loop control method is applied to the bead point model, and the method is verified to be high in applicability under the actual condition.
[ description of the drawings ]
FIG. 1 is a schematic diagram of a two-body tethered satellite system of the present invention, wherein (a) is the relationship of three sets of coordinate systems and (b) is the relationship of the tethered satellite system to the orbital plane;
FIG. 2 is a schematic diagram of a tethered multibody kinetic model of the present invention wherein (a) is a plot of bead point model nodes and (b) is a schematic diagram of bead point model piecewise parameters;
FIG. 3 is a graph of the change in-plane angle with time of the present invention;
FIG. 4 is a graph of the in-plane angular velocity of the present invention as a function of time;
FIG. 5 is a graph of release cord length versus time for the present invention;
FIG. 6 is a graph of tether release rate over time in accordance with the present invention;
FIG. 7 is a graph of tether tension over time in accordance with the present invention;
FIG. 8 is a graph of release cord length versus time for the present invention;
FIG. 9 is a graph of the change in-plane angle with time of the present invention;
FIG. 10 is a graph of tether tension over time in accordance with the present invention;
FIG. 11 is a graph of tether release rate over time in accordance with the present invention;
FIG. 12 is a graph of release cord length versus time for the present invention;
FIG. 13 is a graph of the change in-plane angle with time of the present invention;
FIG. 14 is a graph of tether tension over time in accordance with the present invention;
FIG. 15 is a graph of tether release rate over time in accordance with the present invention;
FIG. 16 is a graph of release cord length versus time for the present invention;
FIG. 17 is a graph of tether tension over time in accordance with the present invention;
FIG. 18 is a graph of tether release rate over time in accordance with the present invention;
FIG. 19 shows the internal face angle θ of the present invention1A graph of time;
FIG. 20 shows the internal face angle θ of the present invention2Graph over time.
[ detailed description ] embodiments
The invention is described in further detail below with reference to the accompanying drawings:
the invention discloses a tether deployment oscillation suppression control method of a tethered satellite system, which comprises the following steps of:
step 1, establishing a dynamic model of a tethered satellite system;
firstly, neglecting the influence of attitude motion of a satellite and tether flexibility, establishing a dumbbell model of a two-body tether satellite system on a circular orbit through a second Lagrange equation, further researching a tether multi-body dynamics modeling method, and expanding the tether model to obtain a bead point model with higher precision. And deducing the unfolding mode of the bead point model based on the assumption that the number of the tied rope segments is not changed when the bead point model is unfolded.
The dumbbell model for the tethered satellite system comprises the following steps:
Figure BDA0002476946280000061
wherein θ represents an in-plane angle, θ 'represents an in-plane angular velocity, θ "represents an in-plane angular acceleration, ξ represents an original length of the dimensionless tether, ξ' represents a release velocity of the dimensionless tether, ξ" represents an acceleration of the dimensionless tether,
Figure BDA0002476946280000062
denotes the face external angle, T denotes tether tension, m denotes the total system mass, m1Denotes the mass of the mother and the star, m2The mass of the sub-star is represented,
Figure BDA0002476946280000063
representing the current release length of a tether, and omega representing the orbital angular velocity of the tethered satellite system;
as shown in FIG. 1, let the tethered satellite system operate undisturbed at radius R0On a circular track, the track inclination angle is
Figure BDA0002476946280000064
Orbital angular velocity of tethered satellite system
Figure BDA0002476946280000065
Wherein the gravitational constant is mue＝3.986005×1014m3/s2When modeling, the main star and the sub-star are regarded as particles, the tether is kept in a straight line state in the releasing process, the length is l, ξ represents the original length of the dimensionless tether,
Figure BDA0002476946280000066
lcindicating the maximum tether release length.
The tethered satellite bead point model is:
Figure BDA0002476946280000067
Figure BDA0002476946280000071
Figure BDA0002476946280000072
in the formula, the corner code N represents the number of nodes, N is 1,2, …, N-1, and N represents the total number of the current nodes; lnThe length of the nth segment of tether is shown,
Figure BDA0002476946280000073
the release speed of the nth segment of tether is shown,
Figure BDA0002476946280000074
representing the acceleration of the nth segment of the tether,
Figure BDA0002476946280000075
represents the n-th tether face outer angle,
Figure BDA0002476946280000076
represents the (n + 1) th tether face external angle,
Figure BDA0002476946280000077
indicates the out-of-plane angle of the tether segment n-1,
Figure BDA0002476946280000078
which is indicative of the out-of-plane angular velocity,
Figure BDA0002476946280000079
represents the out-of-plane angular velocity of the nth segment,
Figure BDA00024769462800000710
represents the n-th out-of-plane angular acceleration, θnDenotes the angle of the n-th face, θn+1Denotes the internal angle of the n +1 th segment, θn-1Indicates the internal angle of the n-1 th segment,
Figure BDA00024769462800000711
represents the angular velocity of the nth section of face,
Figure BDA00024769462800000712
denotes the angular acceleration of the nth section, TnRepresenting the nth tension, Tn+1Represents the n +1 th tension, Tn-1Showing the tension of the (n-1) th segment,
Figure BDA00024769462800000713
which represents the quality of the node n,
Figure BDA00024769462800000714
representing the quality of the node n +1,
Figure BDA00024769462800000715
representing forces other than gravitational tension, "-" represents the corresponding axis,
Figure BDA00024769462800000716
representing the other forces projected by the nth segment into the x-axis direction,
Figure BDA00024769462800000717
other forces projected to the x-axis direction by the (n + 1) th segment, x represents the x-axis of the orbital coordinate system,
Figure BDA00024769462800000718
representing the other forces projected by the nth segment into the y-axis direction,
Figure BDA00024769462800000719
other forces projected to the y-axis direction by the (n + 1) -th segment, y represents the y-axis of the orbit coordinate system,
Figure BDA00024769462800000720
representing the other forces projected by the nth segment in the z-axis direction,
Figure BDA00024769462800000721
indicating the other forces projected by the segment n +1 onto the z-axis direction, and z indicating the z-axis of the orbital coordinate system.
As shown in FIG. 2, the centroid of the view principal star is the system centroid O2With a track radius of R0. The total number of nodes is currently N,the serial numbers are increased from the subsatellite to the main star (B)1,B2,…,BN) Wherein the first node B1Position coincident with child star centroid, end node BNThe position coincides with the principal star centroid. The external force borne by the tether is intensively applied to the node during modeling, the node mass is determined by a centralized mass method according to the linear density of the tether, and the subsatellite mass is superposed to the first node B1The above. With tether density ρ, the discrete node mass is given by:
Figure BDA0002476946280000081
in the formula (I), the compound is shown in the specification,
Figure BDA0002476946280000082
represents the quality of node one, m2Denotes the subsatellite mass,/1Denotes the first strand length,/n-1The length of the n-1 th rope is shown.
In order to realize the selection of the subsequent control simulation, the number of discrete units is kept unchanged, and the unfolding mode of the tether retracting process is simulated by simultaneously changing the attributes (such as length and quality) of all the units. Firstly, considering that a bead point model is complex, if n is high, the later control method is difficult to verify, and therefore, taking n as 2 means that a tethered satellite system is regarded as a system in which two extensible rigid rods are connected with a mother satellite and a child satellite. In the subsequent control, only the suppression of the swinging of the internal angle in the unfolding process is considered, i.e. the order
Figure BDA0002476946280000083
Taking into account forces F other than gravity and tensionnOf (2), i.e. F n0. The system equation can be simplified as follows:
Figure BDA0002476946280000084
wherein the content of the first and second substances,
Figure BDA0002476946280000085
representing a first segment systemRope unwinding acceleration, θ1The internal angle of the first tether face is shown,
Figure BDA0002476946280000086
representing the in-plane angular velocity of the first segment of tether,
Figure BDA0002476946280000087
representing the angular acceleration, theta, in the plane of the first tether segment2A second segment of tether in-plane angle is shown,
Figure BDA0002476946280000091
representing the second segment tether in-plane angular velocity,
Figure BDA0002476946280000092
representing the second segment tether in-plane angular acceleration,
Figure BDA0002476946280000093
a quality of the node is represented by a quality,
Figure BDA0002476946280000094
representing node two quality, T1Indicating the tension on the first length of tether,
Figure BDA0002476946280000095
indicating the release rate of the first segment of tether.
Step 2, planning the optimal unfolding process of the tether of the tethered satellite system
Firstly, researching the optimal track development state of the target function by taking the internal angle of the surface as the target function; and then respectively taking the optimal time, the optimal time and the optimal tension as objective functions to carry out expansion trajectory planning, and comparing the system performance and stability under each objective function. And selecting a proper objective function according to the comparison conclusion, and performing optimal trajectory planning on the bead point model.
Tethered satellite systems are a typical under-actuated system, especially for the "dumbbell" model, whose control force-tether tension exists only in the second derivative equation of tether length. In the process of unfolding the tether, the tension of the tether mainly affects the unfolding speed of the tether and cannot directly affect the angle and the angular speed of the tether, so that it is difficult to directly optimally control the problems of tether oscillation and the like in the process of unfolding the tether. And converting the optimal control problem into a parameter optimization problem with algebraic constraints, and calculating an optimal track meeting the requirements by giving constraints of an expansion process and a terminal.
The state equation of the controlled object is expressed as:
Figure BDA0002476946280000096
wherein t represents a continuous time, x (t) represents a state quantity at time t, and u (t) represents a control quantity at time t;
the objective function for optimizing the performance index is expressed as follows:
Figure BDA0002476946280000097
where φ represents an optimization target, x (t)f) Represents tfState quantity of time, tfRepresenting the end time, and representing the end time node L by an angle code f to represent a state equation of the optimization object;
the path constraint c [ ] and the boundary constraint ψ [ ] are expressed as follows:
Figure BDA0002476946280000101
gaussian pseudospectral method with discrete time region tau ∈ < -1,1 [ ]]The inner Gaussian point is a matching point, and the time t ∈ [ t ] is continued0,tf]Conversion to τ is as follows:
Figure BDA0002476946280000102
in the formula, t0Denotes a start time, τ denotes a discrete time region;
differential equation of
Figure BDA0002476946280000103
And the objective function J can be expressed as:
Figure BDA0002476946280000104
Figure BDA0002476946280000105
wherein x (τ) represents a τ time state quantity, u (τ) represents a τ time control quantity, and x (1) represents a 1 time state quantity;
approximating the system state variables by N +1 Lagrange polynomials to obtain:
Figure BDA0002476946280000106
wherein X (τ) represents an N-order difference approximation function of X (τ), i represents a mark for distinguishing a difference node or a basis function, and X (τ)i) Representing the exact value of the state variable at the node of the time difference, τiRepresenting a time difference node, Li(τ) represents a Lagrangian difference basis function;
Figure BDA0002476946280000107
Figure BDA0002476946280000108
in the formula, τjRepresenting time difference value nodes, wherein the corner codes j represent labels for distinguishing different time difference value nodes;
approximating the control variable with N lagrangians to obtain:
Figure BDA0002476946280000111
wherein U (τ) represents an N-order difference approximation function of U (τ), U (τ)i) Indicating control variable at time node τiThe value of the accuracy of the measurement is determined,
Figure BDA0002476946280000112
representing a Lagrangian difference basis function;
Figure BDA0002476946280000113
Figure BDA0002476946280000114
derivation of equation (11) yields:
Figure BDA0002476946280000115
in the formula (I), the compound is shown in the specification,
Figure BDA0002476946280000116
denotes the time derivative of X (τ), X (τ)i) Indicating state variables at time node τiThe value of the error is determined according to the accuracy value,
Figure BDA0002476946280000117
represents Li(τ) derivative with time;
defining a differential approximation matrix D, D ∈ RN×(N+1)Then, there are:
Figure BDA0002476946280000118
in the formula, DH1Is that
Figure BDA0002476946280000119
In a simplified writing method of the present invention,
Figure BDA00024769462800001110
denotes τkDerivative of the ith Lagrangian difference basis function at node, τkRepresenting the kth time difference value node, wherein an angle code k represents a label for distinguishing different time difference value nodes;
the differential equation can be rewritten to the form of an approximate matrix as follows:
Figure BDA00024769462800001111
in the formula, X (τ)k) Indicating state variables at time node τkTo the precise value, U (τ)k) Representing the controlled variable at a time node taukAn accurate value is determined;
the differential equations are converted into a number of algebraic constraint problems.
Terminal state X (τ)f) Can be obtained by Gauss multiplication, and the initial value X (tau)0)＝X(-1)
Figure BDA0002476946280000121
In the formula, wkRepresenting gaussian pseudo-spectral coefficients;
similarly, the objective function can be expressed approximately as:
Figure BDA0002476946280000122
in the formula, x (1) represents a state quantity at a discrete time 1;
the path constraint c [ ] and the boundary constraint ψ [ ] are:
c[X(τk),U(τk),τk]≤0 (21)
ψ[X(τ0),X(τf)]＝0 (22)
in the formula, X (τ)0) Denotes an initial time state quantity, X (τ)f) Indicating the state quantity at the end time, τfIndicating an end time;
step 3, predicting and controlling the nonlinear model of the tether deployment process
And introducing nonlinear model prediction tracking control to perform tracking control on the optimal unfolding track of the tether. Firstly, tracking control is carried out by taking the optimal track of the dumbbell model in the step 2 as a reference track, and the feasibility of the control method is verified. And then, on the basis, observation errors are considered, disturbance in a white noise form is applied to the internal angle of the surface in the expansion process, and the applicability of the nonlinear model prediction tracking control is further verified. And finally, tracking control is carried out by taking the optimal track of the bead point model as a reference track, and verification of closed-loop control aiming at the accurate model is completed.
The principle of the invention is as follows:
the essence of step 2 of the invention is an open loop control, namely, constraint is input and path planning is completed before the system is started, and no state information is fed back in the unfolding process. Once a problem occurs in the operation process of the system, compensation cannot be performed. Under the action of modeling and observation errors, random external disturbance and other factors, an actual control system almost always deviates from a pre-designed open-loop orbit. Therefore, it is necessary to introduce closed-loop control based on step 2, transmit control information to the controlled object through system operation, and feed back the state information of the controlled object to the input to modify the operation process so that the output of the system meets the expected requirement.
Given a controlled process, it is measured at discrete times tnThe state quantity x (n) (n is a natural number). By "controlled" is meant that in each time domain a control input u (n) can be selected which will affect the future state of the system. The purpose of the overall tracking control is to determine the control input u (n) such that the state quantity x (n) can follow the reference trajectory x as much as possiblerefThis means that if the current state is far from the reference trajectory, it is desirable for the control system to approach the reference trajectory, and similarly if the current state is consistent with the reference trajectory, it is desirable for it to remain at the current positiondAnd U (n) ∈ U ═ RmX is a state quantity, R is a real number set, and d and m are labels for distinguishing the two sets; make the reference track constant and equal to x*＝0，xref(n)＝x*0 (n ≧ 0). Based on such a constant reference trajectory, the tracking problem is reduced to a stability problem.
Since it is desirable to pair x (n) and the reference trajectory xrefThe current deviation of (n) reacts, so the control input U (n) is in a feedback state, i.e. in the form of U (n) ═ μ (X (n)), the state quantity X ∈ X is mapped to in the control force set UMap μ.
The current idea of model predictive control (linear or non-linear) is to use a process model to predict and optimize future system behavior. The following model is used for illustration:
x+＝f(x,u) (23)
where f X × U → X is a known and generally non-linear mapping that will follow state X at the next instant in time+Assigned to state x and control force u. For any given control sequence u (0),.. u (N-1) and number of prediction intervals N ≧ 2, starting from the current state x (N), a predicted trajectory x is constructed by iteration (4-1)uThe method comprises the following steps:
xu(0)＝x(n)，xu(k+1)＝f(xu(k),u(k))，k＝0,...,N-1. (24)
in this way, a system x (n + k) at time t may be obtainedn+kPredicted state x of timeu(k) In that respect Thus, a discrete time t is obtained which depends on the selected control sequence u (0)n,...,tn+NThe system above predicts the state.
By using the optimal control to select u (0), u (N-1), let xuAs close to x ═ 0 as possible. For this purpose, functions are used
Figure BDA0002476946280000144
To measure xu(k) And x is 0. In the model predictive control, not only are deviations of the system state trajectory from the reference trajectory allowed, but the distance of the value u (k) from the reference control u can be controlled if necessary, where u is likewise 0. For this reason, a quadratic function is often chosen to be expressed as follows:
Figure BDA0002476946280000143
wherein | · | | represents the conventional euclidean norm, λ is a controlled weighting parameter, λ ≧ 0; if no control input is required, let λ be 0. The optimal control problem can be expressed as:
Figure BDA0002476946280000141
the discretized model of the dumbbell model is:
Figure BDA0002476946280000142
in the formula, TsWhich represents a discrete time interval of time,
Figure BDA0002476946280000145
representing a first derivative of the equation of state;
the discretized model of the bead point model is:
Figure BDA0002476946280000151
in the formula, x1～x6Respectively, the tether length, tether release speed, first segment internal angle speed, second segment internal angle, and second segment internal angle speed.
Example (b):
the optimal trajectory of the dumbbell model is as follows:
analyzing the formula (1) to obtain
Figure BDA0002476946280000152
Let X1＝[x1x2x3x4]The target is unfolded on a circular orbit, and the earth orbit angular velocity omega is 1.1 × 10-3rad/s, the final rope length of release is 1000m, the star mass m2500kg, tether density ρ 1.85 × 10-3g/m3. Setting an initial state and a final state according to the actual task expansion requirement
X1(t0)＝[0 0 10 1](29)
X1(tf)＝[0 0 1000 0](30)
The control force of the tether is provided by the tension when the tether is tensioned, so the tension can only be positive, the lower limit of the tension is set to ensure that the tether is always in a tensioned state, and the upper limit of the tension is required to be set in consideration of the problem of the tension limit of the tether
10-5≤T≤100 (31)
Setting the tether deployment convergence time to
0≤tf≤10000 (32)
The deployment process is constrained as follows
Figure BDA0002476946280000161
The oscillation suppression during the deployment of the tether is studied, so that the objective function can be designed as follows by giving priority to preventing the tether from generating an excessive oscillation angle due to the constant change of the deployment rate during the deployment
Figure BDA0002476946280000162
From the constraints described above, calculations using GPOPS-II can be made to obtain information about X1＝[x1x2x3x4]The system is at tfThe unfolding is completed when the time is 9987s, and the change curves of various parameters of the system in the unfolding process are shown in figures 3-7. As can be seen in fig. 3-7, the system allows the tether to release freely during the initial stages of deployment, with the system face angle increasing with deployment, and the tether providing significant tension to slow the tether as the tether face angle approaches the upper constraint limit. It is readily apparent that each time the tether in-plane angle is again reduced to zero, the system will resume deployment, and so on until the tether completes the entire deployment process. It can be seen that the tether after being deployed to a predetermined length is controlled for a period of time, and finally the in-plane angle, release speed and in-plane angular speed are all controlled to 0, so that the accuracy requirement is met, and the system reaches an equilibrium state when the tension is maintained at 1.816N. When the tension is 0 in the unfolding process, other parameters all meet the constraint condition, namely the system is considered not to generate winding and other phenomena at the moment. The convergence of the interior angle of the plane is completed by the whole optimal trajectory for unfolding, and the unfolding process is the same as expected, so that the optimal trajectory for unfolding completely meets the objective fact.
Predicting and controlling a dumbbell model nonlinear model:
the above-described optimal trajectory is used as a reference trajectory, and equation (27) is used as a study target to simulate the target on a circular orbit with an earth orbit angular velocity Ω of 1.1 × 10-3rad/s, the final rope length of release is 1000m, the sub-star mass m2500kg, tether density ρ 1.85 × 10-3g/m3. The initial state of the state variable is
x0＝[0 0 10 1](35)
The deployment process is constrained as follows
Figure BDA0002476946280000171
The control input limit condition is
10-5≤T≤100 (37)
The Q and R matrices are respectively expressed as follows
Figure BDA0002476946280000172
Let model predict time domain N be 10, time interval TsThe discrete time point number k is 9987, and the variation curve of each parameter of the system in the simulation process is shown in fig. 8-11. In fig. 8 to 11, blue represents an optimal trajectory obtained by the gaussian pseudo-spectral method, and red represents a closed-loop control trajectory obtained by the nonlinear model predictive control tracking. It can be easily seen from fig. 8 that the rope length unwinding trajectory obtained by the model predictive control performs better tracking on the reference trajectory. The internal angle in fig. 9 effectively completes the tracking of the target track in the whole unfolding process, the maximum value of the internal angle is basically the same as the maximum value of the internal angle of the optimal track in the unfolding process, and convergence is completed at last to meet the control requirement. In fig. 10 and 11, the closed-loop control trajectories of the tension and the tether release speed both have higher overlap ratio with the reference trajectory, which indicates that the control effect is good. The whole closed-loop unfolding process is in accordance with expectation, and the feasibility of the nonlinear model predictive control is proved.
The model prediction control research object is a discretization model, and the formula (2) is7) Applying white noise p with power of 1W to the internal angle of the middle plane, and discretizing a state equation x+The following were used:
Figure BDA0002476946280000181
the other parameters are unchanged, and the change curves of the parameters of the system in the simulation process are shown in fig. 12-15. In fig. 12 to 15, blue represents an optimal trajectory obtained by a gaussian pseudo-spectrum method, and red represents a closed-loop control trajectory obtained by prediction control tracking of a nonlinear model. It can be seen from fig. 13 that the inner angle of the tether face always performs effective tracking in the presence of interference, and the tracking effect is good. As can be seen from fig. 14, in the case of considering the disturbance, the control force tether tension performs optimal control at each moment in response to the existence of gaussian white noise, so that the control force variation trend also conforms to the reference trajectory while the parameters such as the internal angle of the plane and the like are controlled. It can be seen that under the condition of disturbance, the model prediction control can still well complete the tracking control of the reference track. The feasibility of nonlinear model predictive control in the actual tether deployment process is demonstrated.
And (3) performing nonlinear model prediction control on the bead point model:
assuming that the target is spread on a circular orbit, the earth orbit angular velocity omega is 1.1 × 10-3rad/s, the final rope length of release is 2000m, the star mass is m2500kg, tether density ρ 1.85 × 10-3g/m3. The initial state of the state variable is
X2(t0)＝(100 1 0 0 0 0) (40)
The deployment process is constrained as follows
Figure BDA0002476946280000191
The control input limit condition is
10-5≤T≤100 (42)
The Q and R matrices are respectively expressed as follows
Figure BDA0002476946280000192
Let model predict time domain N be 10, time interval TsThe discrete time point number k is 7054, and the variation curve of each parameter of the system in the simulation process is shown in fig. 16-fig. 20 below. In fig. 16 to 20, blue represents an optimal trajectory obtained by gaussian pseudo-spectroscopy, and red represents a closed-loop control trajectory obtained by nonlinear model predictive control tracking. It can be seen that in this simulation, the rope length, the internal face angle, the rope release speed and the rope tension are kept in good tracking state in the whole process, and the convergence is completed at the moment of the end of the deployment. The simulation meets the expected indexes, and the effectiveness of the nonlinear model predictive control is proved.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical solution according to the technical idea proposed by the present invention falls within the protection scope of the claims of the present invention.

Claims (3)

1. A tether oscillation suppression control method for a tether satellite system is characterized by comprising the following steps:
step 1, establishing a dynamic model of a tethered satellite system;
firstly, neglecting the influence of attitude motion of a satellite and tether flexibility, establishing a dumbbell model of a two-body tether satellite system on a circular orbit through a second Lagrange equation, and expanding the tether model by utilizing a tether multi-body dynamics modeling method to obtain a bead point model; deducing to obtain the unfolding mode of the bead point model based on the assumption that the number of tether segments is unchanged when the bead point model is unfolded;
step 2, planning an optimal path of the tether deployment process;
firstly, taking an internal angle of a surface as an objective function to obtain an optimal track unfolding state of the object; respectively taking time optimization, time optimization and tension simultaneous optimization as objective functions to perform expansion trajectory planning, and comparing the system performance and stability under each objective function; selecting a corresponding target function according to the comparison conclusion, and performing optimal track planning on the bead point model;
step 3, introducing nonlinear model prediction to perform tracking control on the tether track;
firstly, tracking control is carried out by taking the optimal track of the dumbbell model as a reference track, and the feasibility of the control method is verified; on the basis, the observation error is synthesized, the inner angle of the surface in the expansion process is disturbed in a white noise mode, and the applicability of the nonlinear model prediction tracking control is verified; and finally, performing tracking control by taking the optimal track of the bead point model as a reference track to finish the verification of closed-loop control aiming at the accurate model.
2. The tethered satellite system tether deployment oscillation suppression control method of claim 1, wherein the specific method for establishing the tethered satellite system dynamic model in step 1 is as follows:
the dumbbell model for the tethered satellite system comprises the following steps:
Figure FDA0002476946270000011
wherein θ represents an in-plane angle, θ 'represents an in-plane angular velocity, θ "represents an in-plane angular acceleration, ξ represents an original length of the dimensionless tether, ξ' represents a release velocity of the dimensionless tether, ξ" represents an acceleration of the dimensionless tether,
Figure FDA0002476946270000021
denotes the face external angle, T denotes tether tension, m denotes the total system mass, m1Denotes the mass of the mother and the star, m2The mass of the sub-star is represented,
Figure FDA0002476946270000022
representing the current release length of a tether, and omega representing the orbital angular velocity of the tethered satellite system;
the tethered satellite system operates undisturbed at radius R0On a circular track, the track inclination angle is
Figure FDA0002476946270000023
Orbital angular velocity of tethered satellite system
Figure FDA0002476946270000024
Wherein the gravitational constant is mue＝3.986005×1014m3/s2When in modeling, the main star and the sub-star are regarded as mass points, the tether is kept in a straight line state in the releasing process, the length is l, ξ represents the original length of the dimensionless tether,
Figure FDA0002476946270000025
lcrepresents the tether maximum release length;
the tethered satellite bead point model is:
Figure FDA0002476946270000026
Figure FDA0002476946270000027
Figure FDA0002476946270000031
in the formula, the corner code N represents the number of nodes, N is 1,2, …, N-1, and N represents the total number of the current nodes; lnThe length of the nth segment of tether is shown,
Figure FDA0002476946270000032
the release speed of the nth segment of tether is shown,
Figure FDA0002476946270000033
representing the acceleration of the nth segment of the tether,
Figure FDA0002476946270000034
the n-th tether face outer angle is shown,
Figure FDA0002476946270000035
represents the (n + 1) th tether face external angle,
Figure FDA0002476946270000036
represents the outer angle of the n-1 th tether face,
Figure FDA0002476946270000037
which is indicative of the out-of-plane angular velocity,
Figure FDA0002476946270000038
represents the out-of-plane angular velocity of the nth segment,
Figure FDA0002476946270000039
represents the n-th out-of-plane angular acceleration, θnDenotes the angle of the n-th face, θn+1Denotes the internal angle of the n +1 th segment, θn-1Indicates the internal angle of the n-1 th segment,
Figure FDA00024769462700000310
represents the angular velocity of the nth section of the face,
Figure FDA00024769462700000311
denotes the angular acceleration of the nth section, TnRepresenting the nth tension, Tn+1Denotes the n +1 th tension, Tn-1Showing the tension of the (n-1) th segment,
Figure FDA00024769462700000312
which represents the quality of the node n,
Figure FDA00024769462700000313
representing the quality of the node n +1,
Figure FDA00024769462700000314
representing forces other than gravitational tension, "-" represents the corresponding axis,
Figure FDA00024769462700000315
showing projection of the nth segment to the x-axisThe other force(s) on the first side,
Figure FDA00024769462700000316
other forces projected to the x-axis direction by the (n + 1) th segment, x represents the x-axis of the orbital coordinate system,
Figure FDA00024769462700000317
representing the other forces projected by the nth segment into the y-axis direction,
Figure FDA00024769462700000318
other forces projected to the y-axis direction by the (n + 1) -th segment, y represents the y-axis of the orbit coordinate system,
Figure FDA00024769462700000319
representing the other forces projected by the nth segment into the z-axis direction,
Figure FDA00024769462700000320
representing other forces projected to the z-axis direction by the n +1 th segment, wherein z represents the z-axis of the orbit coordinate system;
the centroid of the dominant star is the system centroid O2With a track radius of R0(ii) a The total number of the current nodes is N, and the serial numbers are sequentially increased from the subsatellite to the main star (B)1,B2,…,BN) Wherein the first node B1Position coincident with subsatellite centroid, end node BNThe position coincides with the centroid of the main star; the external force borne by the tether is intensively applied to the node during modeling, the node mass is determined by a centralized mass method according to the linear density of the tether, and the subsatellite mass is superposed to the first node B1The above step (1); with tether density ρ, the discrete node mass is given by:
Figure FDA0002476946270000041
in the formula (I), the compound is shown in the specification,
Figure FDA0002476946270000042
represents a node oneMass of (c), m2Denotes the subsatellite mass,/1Denotes the first strand length,/n-1Represents the length of the n-1 segment of rope;
let n be 2, n is the sum of n,
Figure FDA0002476946270000043
Fnwhen 0, the system equation is simplified as follows:
Figure FDA0002476946270000044
wherein the content of the first and second substances,
Figure FDA0002476946270000045
representing first tether deployment acceleration, θ1The internal angle of the first tether face is shown,
Figure FDA0002476946270000046
representing the in-plane angular velocity of the first segment of tether,
Figure FDA0002476946270000047
representing the angular acceleration, theta, in the plane of the first tether segment2Representing the internal angle of the second tether face,
Figure FDA0002476946270000048
representing the second segment tether in-plane angular velocity,
Figure FDA0002476946270000049
representing the second segment tether in-plane angular acceleration,
Figure FDA00024769462700000410
a quality of the node is represented as one,
Figure FDA00024769462700000411
representing node two quality, T1Indicating the tension on the first length of tether,
Figure FDA00024769462700000412
indicating the release rate of the first segment of tether.
3. The tethered satellite system tether deployment oscillation suppression control method of claim 1, wherein the specific method for step 2 planning the optimal deployment of the tethered satellite system tether is as follows:
the state equation of the controlled object is expressed as:
Figure FDA00024769462700000413
wherein t represents a continuous time, x (t) represents a state quantity at time t, and u (t) represents a control quantity at time t;
the objective function for optimizing the performance index is expressed as follows:
Figure FDA0002476946270000051
where φ represents an optimization target, x (t)f) Represents tfState quantity of time, tfRepresenting the end time, and representing the end time node L by an angle code f to represent a state equation of the optimization object;
the path constraint c [ ] and the boundary constraint ψ [ ] are expressed as follows:
Figure FDA0002476946270000052
gaussian pseudospectral method with discrete time region tau ∈ < -1,1 [ ]]The inner Gaussian point is a matching point, and the time t ∈ [ t ] is continued0,tf]Conversion to τ is as follows:
Figure FDA0002476946270000053
in the formula, t0Denotes a start time, τ denotes a discrete time region;
differential equation of
Figure FDA0002476946270000054
And the objective function J is expressed as:
Figure FDA0002476946270000055
Figure FDA0002476946270000056
wherein x (τ) represents a τ time state quantity, u (τ) represents a τ time control quantity, and x (1) represents a 1 time state quantity;
approximating the system state variables by N +1 Lagrange polynomials to obtain:
Figure FDA0002476946270000057
wherein X (τ) represents an N-order difference approximation function of X (τ), i represents a label for distinguishing a difference node or a basis function, and X (τ)i) Representing the exact value of the state variable at the node of the time difference, τiRepresenting a time difference node, Li(τ) represents a Lagrangian difference basis function;
Figure FDA0002476946270000061
Figure FDA0002476946270000062
in the formula, τjRepresenting time difference value nodes, wherein the corner codes j represent labels for distinguishing different time difference value nodes;
approximating the control variable with N lagrangians to obtain:
Figure FDA0002476946270000063
wherein the content of the first and second substances,u (τ) represents an N-order difference approximation function of U (τ), U (τ)i) Indicating control variable at time node τiThe value of the accuracy of the measurement is determined,
Figure FDA0002476946270000064
representing a Lagrangian difference basis function;
Figure FDA0002476946270000065
Figure FDA0002476946270000066
derivation of equation (11) yields:
Figure FDA0002476946270000067
in the formula (I), the compound is shown in the specification,
Figure FDA0002476946270000068
denotes the time derivative of X (τ), X (τ)i) Indicating state variables at time node τiThe value of the accuracy of the measurement is determined,
Figure FDA0002476946270000069
represents Li(τ) derivative with time;
defining a differential approximation matrix D, D ∈ RN×(N+1)Then, there are:
Figure FDA00024769462700000610
in the formula (I), the compound is shown in the specification,
Figure FDA00024769462700000611
is that
Figure FDA00024769462700000612
In a simplified writing method of the present invention,
Figure FDA00024769462700000613
denotes τkDerivative of the ith Lagrangian difference basis function at node, τkRepresenting the kth time difference value node, wherein an angle code k represents a label for distinguishing different time difference value nodes;
the differential equation is rewritten to the form of an approximate matrix as follows:
Figure FDA0002476946270000071
in the formula, X (τ)k) Indicating state variables at time node τkTo the precise value, U (τ)k) Indicating control variable at time node τkAn accurate value is determined;
terminal state X (τ)f) Obtained by Gauss's product, the initial value X (tau)0)＝X(-1)
Figure FDA0002476946270000072
In the formula, wkRepresenting gaussian pseudo-spectral coefficients;
similarly, the objective function is approximately expressed as:
Figure FDA0002476946270000073
in the formula, x (1) represents a state quantity at a discrete time 1;
the path constraint c [ ] and the boundary constraint ψ [ ] are:
c[X(τk),U(τk),τk]≤0 (21)
ψ[X(τ0),X(τf)]＝0 (22)
in the formula, X (τ)0) Denotes an initial time state quantity, X (τ)f) Indicating the state quantity at the end time, τfIndicating the end time.
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