CN113110560A - Satellite formation reconstruction model prediction control algorithm based on Chebyshev inequality - Google Patents

Satellite formation reconstruction model prediction control algorithm based on Chebyshev inequality Download PDF

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CN113110560A
CN113110560A CN202110564334.1A CN202110564334A CN113110560A CN 113110560 A CN113110560 A CN 113110560A CN 202110564334 A CN202110564334 A CN 202110564334A CN 113110560 A CN113110560 A CN 113110560A
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CN113110560B (en
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李彬
季袁冬
张凯
江秀强
周旭艳
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Sichuan University
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Abstract

The invention discloses a satellite formation reconstruction model predictive control algorithm based on Chebyshev inequality, which comprises the steps of firstly constructing a mathematical model of a system, then reconstructing the mathematical model by setting an opportunity constraint condition of a time-varying affine feedback structure and a control variable, and further obtaining a corresponding reconstructed mathematical model, compared with the traditional control mode, the invention utilizes the randomness and the statistical characteristic of uncertain factors, under the framework, an objective function exists in an expected form, probability uncertainty description is utilized to define opportunity constraint, the opportunity constraint allows the system to utilize the random characteristic of uncertainty, allows the system to violate the constraint under a given probability, and allows the system to seek the balance between realizing a control target and ensuring the probability constraint to meet due to uncertainty; compared with the prior art, the algorithm can well solve the problem of satellite formation control with constraint under the condition of interference.

Description

Satellite formation reconstruction model prediction control algorithm based on Chebyshev inequality
Technical Field
The invention relates to the technical field of satellite control, in particular to a satellite formation reconstruction model prediction control algorithm based on a Chebyshev inequality.
Background
Satellite technology also encounters many challenges in view of the current general development of space and space technology, which will result in the weight and size of individual satellites becoming heavier, their structure and function becoming more complex, development, launch costs and mission risks increasing significantly therewith [1 ]. In addition, many complex tasks cannot be performed independently by a single satellite. Therefore, the formation flying of the microsatellites can not only replace the function of a large satellite, but also break through the size limitation of the traditional large satellite from the design idea, and can realize the task which can not be completed by some large satellites [2 ]. When a plurality of small satellite formation groups are adopted to execute tasks, when a certain satellite in the formation groups breaks down, the formation modes can be changed to continue to finish the tasks, so that the risk of breaking down is reduced. Meanwhile, the small-sized satellite formation is used for replacing a single large-sized satellite, so that the method has a plurality of potential advantages of low-cost design, high reliability, low risk, high performance, short development period, high system viability, flexible emission speed and the like, the application field and performance of the traditional satellite are expanded, and the method has a wide space application prospect. By utilizing the regular spatial position distribution of each sub-satellite in the satellite formation, the tasks of three-dimensional observation and imaging, accurate positioning and navigation, atmospheric detection and weather forecast, astronomical observation, geophysical detection and the like of the ground target can be realized.
The satellite formation cooperative control technology can effectively improve the capability of processing uncertainty, enhance the quick response capability and reduce the difficulty of entering the space, and is one of the most important directions of the international space technology development.
At present, the mainstream formation control method at home and abroad mainly comprises the following steps: sliding mode control, robust control, optimal control, adaptive control, PD-fuzzy association control, LQR control, Model Predictive Control (MPC) and the like. For example: rongpengji [3] proposes a PD-fuzzy association control method, selects relative position error and speed error as fuzzy linguistic variables, introduces Proportional Differential (PD) control aiming at large position error, and acts by a PD controller when the relative position error reaches a certain threshold value. Simulation results show that PD control is suitable for large position errors, fuzzy control is suitable for small errors, and two control methods are fully utilized; ulybyshev [4] provides a Linear Quadratic Regulator (LQR) control method for long-term formation flight in a plane based on a linear Hill equation, adopts a nonlinear accurate equation, adds random influences such as atmospheric resistance, solar motion and the like, a thruster, measurement errors and the like, and simulates formation of two-star low-orbit satellites; qiguo Yan and Sparks [5] and the like propose a periodic pulse control method based on discrete LQR.
In satellite control, state constraints and control constraints are common, and if these constraints are ignored, poor control performance of a real system may result. However, existing methods of satellite formation control are hardly able to handle control problems that contain constraints. The greatest advantage of model predictive control is its ability to handle constraints resulting from its prediction of the future dynamic behavior of the system, which can be represented graphically as quadratic programming or online solved nonlinear programming problems by adding constraints to future input, output or state variables.
In deep air, the satellite formation control will also take into account the effects of disturbances, however, standard MPC algorithm design does not take into account the disturbances and uncertainties of the actual system. Although the classical robust MPC algorithm can handle disturbances and uncertainties, its idea is based on the min-max method [6], i.e. minimizing the objective function in case disturbances and uncertainties reach the worst. However, the interference or uncertainty in the actual system is random or fuzzy, and it is obvious that the method has strong conservative property, which will result in the degradation of the system performance.
Reference documents:
[1] hosonger, admire, Liulin, satellite constellation and formation flight problem review [ J ] astronomy progress, 2003,21(3): 231-.
[2] The method for controlling the dynamics and the relative position keeping of the flying orbit of the formation of the microsatellite from the Chinemen, the Liujianfeng and the microsatellite [ J ] modern defense technology 2003,31(6):33-37.
[3]Pengji Wang,Di Yang.PD-fuzzy formation control for spacecraft formation flying in elliptical orbits.Aerospace Science and Technology.2003,7(7), 561–566.
[4]Ulybyshev Y.Long-term formation keeping of satellite constellation using linear-quadratic controller.Journal of Guidance,Control,and Dynamics.1998,21(1), 109–115.
[5]Schaub H,Vadali S R,Junkins J L,et al.Spacecraft formation flying control using mean orbit elements[J].Journal of the Astronautical Sciences,2000, 48(1):69-87.
[6]Kothare M V,Balakrishnan V,Morari M.Robust constrained model predictive control using linear matrix inequalities.Automatica.1996,32(10), 1361-1379.
Disclosure of Invention
Aiming at the defects of large limitation and poor accuracy in the prior art, the invention provides a satellite formation reconstruction model prediction control algorithm based on the Chebyshev inequality, which has state feedback, can process a constraint system and a disturbance or uncertainty system by utilizing the randomness and the statistical characteristic of uncertain factors, and can realize fuel optimization.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a satellite formation reconstruction model predictive control algorithm based on Chebyshev inequality comprises the following steps:
s1, establishing a mathematical model of the satellite formation dynamic system under random disturbance based on the circular orbit or near circular orbit condition;
s2, setting a time-varying affine feedback structure, adding opportunity constraint conditions of control variables, and describing the system model in the step S1 as an SMPC formation control problem solved by utilizing stochastic model predictive control;
s3, reconstructing the chance constraint conditions of the control variables in the step S2 into a linear expression form through a Kantanli inequality;
s4, reconstructing the SMPC formation control problem in the step S2 into an easily-solved optimal control problem according to the relationship between the mean value and the variance of the respectively-set state quantity and the control quantity based on the reconstructed opportunity constraint condition;
and S5, solving the optimal control problem in the step S4 by adopting a CVX tool box in MATLAB and carrying out simulation processing.
Specifically, the expression of the mathematical model under random disturbance established in step S1 is:
xt+1＝Axt+But+Gωt
wherein the content of the first and second substances,
Figure BDA0003080332490000031
in order to set the system matrix,
Figure BDA0003080332490000041
representing a state variable indicating the relative position of the target satellite in the LVLH coordinate system,
Figure DEST_PATH_FDA0003080332480000013
represents a control variable that is affected by thrust considerations,
Figure BDA0003080332490000043
representing external random disturbance, the distribution information of which is unknown, only the mean and variance of the distribution are known, and t is the current time.
Further, in step S1, the compact mode of state, control and perturbation is defined as follows:
Figure BDA0003080332490000044
Figure BDA0003080332490000045
Figure BDA0003080332490000046
the mathematical model established in step S1 is rewritten into the following expression:
xt＝Gxxxt+Gxuut+Gxωωt
wherein the content of the first and second substances,
Figure BDA0003080332490000047
representing the corresponding known system matrix after the conversion.
Further, the time-varying affine feedback structure in step S2 is:
Figure BDA0003080332490000048
Figure BDA0003080332490000049
Figure BDA00030803324900000410
wherein the content of the first and second substances,
Figure BDA00030803324900000411
is a feedback matrix obtained by the computation of the Riccati equation,
Figure BDA00030803324900000412
representing defined control variables utIs determined by the average value of (a) of (b),
Figure BDA00030803324900000414
state variable x representing a definitiontAverage value of (1), then state xtThe average value of (a) is:
Figure BDA00030803324900000413
further, the opportunity constraint conditions of the control variables in the step S2 are:
Pr[Ρ][hTut+k≤d]≥1-β,k＝1,2,…,N-1
wherein, Pr[Ρ][·]Representing the probability under the distribution of the random perturbation P,
Figure BDA0003080332490000051
is a constant vector, d ∈ R is a given constant, { · R }TRepresenting the transpose operator, β ∈ (0,1) is the probability of a constraint violation,
P＝{Ρ:E[Ρ][ωt]＝μ0,E[Ρ][(ωt-μ0)(ωt-μ0)T]＝Σ}
E[Ρ]represents the statistical expectation under P distribution, μ0Indicating expectation, Σ denotes variance.
Further, the system model expression of the SMPC formation control problem obtained in step S2 is as follows:
Figure BDA0003080332490000052
subject to:xt＝Gxxxt+Gxuut+Gxωωt
Figure BDA0003080332490000053
Figure BDA0003080332490000054
Pr[Ρ][hTut+k≤d]≥1-β,k＝1,2,…,N-1
wherein Q, R ', Q' are corresponding weights,
Figure BDA0003080332490000055
is the desired position, J denotes the objective function, xi(k) Value, u, representing the state quantity of the ith satellite at time ki(k) Indicating the ith satellite is at kThe values of the control variables, i and j, respectively represent the ith satellite and the jth satellite.
Further, the linearized expression after the chance constraint condition reconstruction in step S3 is in the form of:
Figure BDA0003080332490000056
wherein epsilon belongs to (0,1) as a set parameter, sigmauThe variance of the control amount u is indicated.
Further, the process of setting the mean and variance of the state quantity and the control quantity, respectively, in step S4 is:
respectively setting state quantity and control quantity:
Figure BDA0003080332490000057
then there are:
Figure BDA0003080332490000058
δut＝K(Gxxδxt+Gxωwt)
wherein, I represents an identity matrix,
Figure BDA0003080332490000061
make it
Figure BDA0003080332490000068
Figure BDA0003080332490000062
Then there are:
Figure BDA0003080332490000063
Figure BDA0003080332490000064
order to
Figure BDA0003080332490000065
In summary, the following are provided:
Σz＝E(δztδzt T)＝(A+BK)(GxxΣtGxx T+GxωΣtGx T)(A+BK)T
wherein
Figure BDA0003080332490000066
Further, in the step S4, the objective function of the mathematical model of the SMPC formation control problem in the step S2 is reconstructed into the following form:
Figure BDA0003080332490000067
wherein the weight Qm＝diag(Q,Q,…,QNR ', …, R'), tr (·) denotes the traces of the matrix; an optimal control problem is obtained by combining the corresponding constraints.
Compared with the prior art, the invention has the following beneficial effects:
compared with the traditional control mode, the method utilizes the randomness and the statistical characteristics of uncertain factors, under the framework, an objective function exists in a desired form, probability uncertainty description is utilized to define opportunity constraint, the opportunity constraint allows systematic utilization of random characteristics of uncertainty, allows the system to violate the constraint under a given probability, and allows systematic seeking of trade-off between realization of a control target and guarantee of probability constraint satisfaction due to uncertainty.
Meanwhile, although the classical robust MPC algorithm can handle disturbances and uncertainties, its idea is based on the min-max method, i.e. minimizing the objective function in case disturbances and uncertainties reach the worst. However, in the actual operating environment, the interference or uncertainty factors to the entire satellite system are random or fuzzy, and therefore the method is too conservative in dealing with the operating conditions, while the method of the present invention uses probabilistic uncertainty descriptions to define opportunistic constraints that allow systematic use of the random nature of uncertainty, allow the system to violate constraints at a given probability, allow the system to systematically seek a trade-off between achieving control objectives and ensuring that probabilistic constraints are met due to uncertainty, and therefore, are more flexible than conventional approaches, can handle both constrained and perturbed or uncertain systems, and can achieve fuel optimization.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention.
Fig. 2 is a schematic diagram illustrating an optimal trajectory simulation under SMPC in an embodiment of the present invention.
FIG. 3 is a diagram illustrating an optimal trajectory simulation under MPC in an embodiment of the present invention.
Fig. 4 is a diagram illustrating a simulation of the real-time distance between two satellites under SMPC according to an embodiment of the present invention.
FIG. 5 is a diagram illustrating a simulation of the real-time distance between two satellites under MPC in accordance with an embodiment of the present invention.
Detailed Description
The invention will be further illustrated by the following specific embodiments:
at present, the common research at home and abroad is the control of formation of satellites with reference orbits being circular orbits or near circular orbits, the research on the formation control of elliptical orbits is not mature, a Clohessy-Wiltshire equation is generally utilized to construct a relative motion dynamics model of circular orbit formation satellites, a C-W equation can only be used for designing the formation shape of a target satellite running on the circular orbits or the near circular orbits, and the C-W equation has certain limitation in the design of the formation shape of the elliptical orbits. Based on the above, the present invention can describe the relative motion dynamics of the satellite when the reference orbit is the circular orbit by using the following differential equation:
Figure BDA0003080332490000071
Figure BDA0003080332490000081
Figure BDA0003080332490000082
wherein η ═ x y z]TIs the relative position of the target satellite in the LVLH coordinate system,
Figure BDA0003080332490000083
is the angular velocity of the target satellite orbit, μ is the gravitational constant of the earth, and r is the orbital radius of the target satellite.
Assuming that the distance between two satellites is much smaller than the orbit radius of the target satellite,
namely, it is
Figure BDA0003080332490000084
Furthermore, considering the influence of the disturbance ω and the thrust u, the following differential equation can be converted into the well-known C-W equation:
Figure BDA0003080332490000085
Figure BDA0003080332490000086
Figure BDA0003080332490000087
order to
Figure BDA0003080332490000088
u＝[ux uy uz]，ω＝[ωx ωy ωz]The above equation is written in the form of a state space:
Figure BDA0003080332490000089
wherein A isd,BdConsidering the sampling time Δ T for a corresponding constant matrix, the discrete system is then represented as follows: x (t +1) ═ ax (t) + bu (t) + ω (t)
Considering that the thrust of the satellite in each direction in formation maintenance is within a certain range, the thrust in each direction in the satellite formation maintenance research is constrained as follows:
||ui||≤umax
aiming at the problem of the flight of the formation of the satellite, the satellite can overcome disturbance to reach an expected position and complete the conversion of an initial formation and a target formation under the condition of considering the existence of the external disturbance. In addition, it is also considered that a certain distance is maintained between every two satellites even if the formation satellites do not reach the desired positions, so as to avoid collision. In order to improve the service life of the satellite, the problem of fuel consumption needs to be fully considered. Therefore, in order to improve the control precision and reduce the formation cost, the error between the actual state and the target state, the energy consumption and the error between the real-time state difference of every two satellites and the target state difference of every two satellites can be used as the performance indexes of the reconstruction of the satellite formation, namely:
Figure BDA0003080332490000091
in the formula, E [. cndot]Representing the desirability of finding, N is the number of predicted steps, N is the number of queued satellites,
Figure BDA0003080332490000092
indicating the expected position of the ith satellite,
Figure BDA0003080332490000093
representing the target distance difference between the ith satellite and the jth satellite, Q, R ', Q' are semi-positive definite matrixes respectively representing that the satellites reach the expected positions and the fuelThe optimal weighting factor keeping a certain distance can be adjusted according to the actual conditions of the simulation environment. It is desirable that the satellites reach the desired position without collision between two satellites, so that Q and Q' can be adjusted to be the same.
The satellite formation problem model is established as follows:
Figure BDA0003080332490000094
subjectto：xt+1＝Axt+But+Gωt
||ui||≤umax
i,j∈{n}
wherein J represents an objective function, xi(k) Value, u, representing the state quantity of the ith satellite at time ki(k) A value representing the control quantity of the ith satellite at time k.
Aiming at a random disturbance system, a random model predictive control Strategy (SMPC) algorithm with state feedback is designed, the algorithm aims to utilize the randomness and the statistical characteristics of uncertain factors, under the framework, an objective function exists in a desired form, opportunity constraints are defined by probability uncertainty description, the opportunity constraints allow systematic utilization of random characteristics of uncertainty and allow a system to violate constraints at a given probability, and the SMPC allows systematic seeking of a trade-off between realizing a control target and ensuring that the probability constraints are met due to uncertainty. For the treatment of opportunity constraint, a state feedback control mechanism is designed, soft constraint is converted into hard constraint treatment according to the Kantaili inequality, and the method does not need to know the distribution function of random disturbance and only needs to know the values of a mean value matrix and a covariance matrix. The target function is reconstructed by means of the mean value and the variance, so that the SMPC problem is reconstructed into an optimal control problem which is easy to solve.
Embodiment mode 1
The embodiment is taken as a basic embodiment of the invention, the prediction control algorithm of the reconstruction model of the satellite formation based on the Chebyshev inequality is applied to the flight of the satellite formation, and comprises the following steps:
first, a system model under a random perturbation system is described as follows:
xt+1＝Axt+But+Gωt
wherein the content of the first and second substances,
Figure BDA0003080332490000101
in order to set the system matrix,
Figure DEST_PATH_FDA0003080332480000012
representing a state variable indicating the relative position of the target satellite in the LVLH coordinate system,
Figure 303112DEST_PATH_FDA0003080332480000013
represents a control variable that is affected by thrust considerations,
Figure BDA0003080332490000104
representing external random disturbance, the distribution information of which is unknown, only the mean and variance of the distribution are known, and t is the current time.
The compact modes of simultaneous state, control and perturbation are defined as follows:
Figure BDA0003080332490000105
Figure BDA0003080332490000106
Figure BDA0003080332490000107
the system model can be rewritten as follows:
xt＝Gxxxt+Gxuut+Gxωωt
wherein the content of the first and second substances,
Figure BDA0003080332490000108
representing the corresponding known system matrix after the conversion.
The time-varying affine feedback structure is set as follows:
Figure BDA0003080332490000109
Figure BDA0003080332490000111
wherein the content of the first and second substances,
Figure BDA0003080332490000112
is a feedback matrix obtained by the computation of the Riccati equation,
Figure BDA0003080332490000113
representing defined control variables utIs determined by the average value of (a) of (b),
Figure BDA00030803324900001110
state variable x representing a definitiontAverage value of (1), then state xtThe average value of (a) is:
Figure BDA0003080332490000114
the formation problem can be converted to the following form:
Figure BDA0003080332490000115
subjectto：xt＝Gxxxt+Gxuut+Gxωwt
Figure BDA0003080332490000116
Figure BDA0003080332490000117
hTu≤d
i,j∈{n}
since the above equation is not in a convex form, it is not easy to calculate, and it must be converted into a convex form that can be calculated online. The preparation distribution of random disturbance is often unknown, so that the convex optimization reconstruction is carried out on the problem by using a distributed random model prediction control theory. And defining a disturbance distribution set only by knowing the mean value and the variance of the disturbance and not knowing the specific condition of the disturbance distribution. The perturbation information may be expressed as follows:
P＝{Ρ:E[Ρ][ωt]＝μ0,E[Ρ][(ωt-μ0)(ωt-μ0)T]＝Σ}
in the formula, E[Ρ]{. denotes the mathematical expectation under P distribution,
Figure BDA0003080332490000118
μ0is the mean of the perturbation, Σ is the variance of the perturbation,
Figure BDA0003080332490000119
represents the kronecker product, and defines E [ omega ]t]＝μ0，Σ[ωt]Σ, i.e. the mean and variance of the disturbance is known.
Opportunistic constraints consider the case where a decision may not satisfy the constraint and use a rule: the decision is allowed to satisfy the constraint to some extent and the probability that the decision satisfies the constraint is not less than a certain confidence level. The use of probabilistic constraints in the constraint process allows the hard constraints to be violated within a specified confidence interval, resulting in more efficient control.
The problem of uncertain linearity systems with opportunistic constraints is a very common problem. The unknown disturbance may be unbounded, it may not satisfy the hard constraints of the input, and therefore an opportunity constraint needs to be imposed to measure uncertainty.
The opportunity to set the control variables is constrained as follows:
Pr[Ρ][hTut+k≤d]≥1-β,k＝1,2,…,N-1
the uncertainty in the probability distribution is processed by adopting a distribution robust method, and the joint input opportunity constraint of the distribution robust is defined as follows:
Figure BDA0003080332490000121
wherein, Pr[Ρ][·]Representing the probability under the distribution of the random perturbation P,
Figure BDA00030803324900001210
is a constant vector, d ∈ R is a given constant, { · R }TRepresenting a transpose operator, β ∈ (0,1) being the probability of constraint violation;
adapting the distributed robust joint opportunity constraint to a more compact form:
Figure BDA0003080332490000122
then the SMPC formation control problem can be described as follows for a randomly perturbed system:
Figure BDA0003080332490000123
subjectto：xt＝Gxxxt+Gxuut+Gxωwt
Figure BDA0003080332490000124
Figure BDA0003080332490000125
Figure BDA0003080332490000126
the joint opportunity constraint is transformed as follows:
Figure BDA0003080332490000127
Figure BDA0003080332490000128
available according to the bang feroni inequality:
Figure BDA0003080332490000129
based on the above two-step process, the single opportunity constraint then transforms to the form:
Figure BDA0003080332490000131
convex optimization processing is carried out on the single chance constraint of the formula by utilizing the Chebyshev-Kartley inequality, and the mean value of y is taken as
Figure BDA0003080332490000139
And the variance is a random variable of Y, and for any alpha epsilon R is more than or equal to 0, the following steps are provided:
Figure BDA0003080332490000132
for the
Figure BDA0003080332490000133
Assuming any δ u ≧ 0, there are:
Figure BDA0003080332490000134
the following transformation is performed and then transformed into a convex optimization approximation as follows:
Figure BDA0003080332490000135
Figure BDA0003080332490000136
in the formula, sigmauThe variance of the control amount u is indicated.
The opportunity constraint is reconstructed as a deterministic representation as follows:
Figure BDA0003080332490000137
setting δ u ═ ε d, where ε ∈ (0,1) is an additional design parameter, and the deterministic constraint is linearized at the cost of an additional slightly tightened constraint, the standard linearization process allows restating the above equation as follows:
Figure BDA0003080332490000138
then, the objective function part is processed, converted into a more easily-calculated form, and split and reconstructed, so that the following form can be obtained:
Figure BDA0003080332490000141
wherein Q ism＝diag(Q,Q,…,QNR ', …, R'), tr (·) represents the traces of the matrix.
Respectively setting state quantity and control quantity: order to
Figure BDA0003080332490000142
Then there are:
Figure BDA0003080332490000143
δut＝K(Gxxδxt+Gxωwt)
wherein, I represents an identity matrix,
Figure BDA0003080332490000144
make it
Figure BDA00030803324900001411
Figure BDA0003080332490000145
Then there are:
Figure BDA0003080332490000146
Figure BDA0003080332490000147
is provided with
Figure BDA0003080332490000148
In summary, the following are provided:
Figure BDA0003080332490000149
wherein
Figure BDA00030803324900001410
Order to
Figure BDA0003080332490000151
The objective function of the mathematical model of the SMPC formation control problem is reconstructed to the form:
Figure BDA0003080332490000152
in conclusion, the SMPC formation control problem can be reconstructed into an optimal control problem which is easy to solve:
Figure BDA0003080332490000153
subject to:xt＝Gxxxt+Gxuut+Gxωωt
Figure BDA0003080332490000154
Figure BDA0003080332490000155
Figure BDA0003080332490000156
finally, a CVX tool box in MATLAB is used for processing the problem, disturbance is set to be Gaussian distribution, and simulation results show that the SMPC algorithm with state feedback designed by the invention is suitable for linear discrete systems which are possibly affected by unbounded disturbance and probability constraint, and the existing disturbance control strategies based on model prediction control and robust control are improved.
As shown in fig. 2 to 5, simulation results of formation of samsung under the SMPC strategy with disturbance set to gaussian distribution are shown, and S1, S2, and S3 respectively represent three satellites. The formation physical problem is described as: the three stars fly from the initial linear shape to finally form an isosceles triangle formation, and during formation, the problem that the collision is avoided when the member satellites still keep a certain shape to fly even if the member satellites do not reach the expected positions and the fuel is optimal is also considered. The simulation compares the SMPC control strategy with the MPC strategy, and can obtain the shape that the SMPC and MPC strategies complete formation almost simultaneously, but because no interference is considered in the design of the MPC strategy, the formation control under the SMPC strategy saves more fuel, as shown in the fuel consumption comparison in table 1 below, the validity of the algorithm is verified.
MPC(Kg) SMPC(Kg)
Gaussian disturbance 86.3145 57.5360
TABLE 1 comparison of fuel consumption
The above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention, but all changes that can be made by applying the principles of the present invention and performing non-inventive work on the basis of the principles shall fall within the scope of the present invention.

Claims (9)

1. A prediction control algorithm of a satellite formation reconstruction model based on a Chebyshev inequality is characterized in that: the method comprises the following steps:
s1, establishing a mathematical model of the satellite formation dynamic system under random disturbance based on the circular orbit or near circular orbit condition;
s2, setting a time-varying affine feedback structure, adding opportunity constraint conditions of control variables, and describing the system model in the step S1 as an SMPC formation control problem solved by utilizing stochastic model predictive control;
s3, reconstructing the chance constraint conditions of the control variables in the step S2 into a linear expression form through a Kantanli inequality;
s4, reconstructing the SMPC formation control problem in the step S2 into an easily-solved optimal control problem according to the relationship between the mean value and the variance of the respectively-set state quantity and the control quantity based on the reconstructed opportunity constraint condition;
and S5, solving the optimal control problem in the step S4 by adopting a CVX tool box in MATLAB and carrying out simulation processing.
2. The satellite formation reconstruction model predictive control algorithm of claim 1, wherein: the expression of the mathematical model under random disturbance established in step S1 is:
xt+1＝Axt+But+Gωt
wherein the content of the first and second substances,
Figure FDA0003080332480000011
in order to set the system matrix,
Figure FDA0003080332480000012
the state variable is represented by a number of variables,
Figure FDA0003080332480000013
the control variable is represented by a number of control variables,
Figure FDA0003080332480000014
representing an external random disturbance, and t is the current time.
3. The satellite formation reconstruction model predictive control algorithm of claim 2, wherein: in step S1, the compact mode of state, control and perturbation is defined as follows:
Figure FDA0003080332480000015
Figure FDA0003080332480000016
Figure FDA0003080332480000017
the mathematical model established in step S1 is rewritten into the following expression:
xt＝Gxxxt+Gxuut+Gxωωt
wherein the content of the first and second substances,
Figure FDA0003080332480000021
4. the satellite formation reconstruction model predictive control algorithm of claim 3, wherein: the time-varying affine feedback structure in step S2 is:
Figure FDA0003080332480000022
Figure FDA0003080332480000023
Figure FDA0003080332480000024
wherein the content of the first and second substances,
Figure FDA0003080332480000025
is a feedback matrix obtained by the computation of the Riccati equation,
Figure FDA0003080332480000026
representing defined control variables utIs determined by the average value of (a) of (b),
Figure FDA0003080332480000027
state variable x representing a definitiontAverage value of (1), then state xtThe average value of (a) is:
Figure FDA0003080332480000028
5. the satellite formation reconstruction model predictive control algorithm of claim 4, wherein: the opportunity constraint conditions of the control variables in step S2 are:
Pr[Ρ][hTut+k≤d]≥1-β,k＝1,2,…,N-1
wherein, Pr[Ρ][·]Representing the probability under the distribution of the random perturbation P,
Figure FDA0003080332480000029
is a constant vector, d ∈ R is a given constant, { · R }TRepresenting the transpose operator, β ∈ (0,1) is the probability of a constraint violation,
P＝{Ρ:E[Ρ][ωt]＝μ0,E[Ρ][(ωt-μ0)(ωt-μ0)T]＝Σ}
E[Ρ]represents the statistical expectation under P distribution, μ0Indicating expectation, Σ denotes variance.
6. The satellite formation reconstruction model predictive control algorithm of claim 5, wherein: the system model expression of the SMPC formation control problem obtained in step S2 is as follows:
Figure FDA0003080332480000031
subject to:xt＝Gxxxt+Gxuut+Gxωωt
Figure FDA0003080332480000032
Figure FDA0003080332480000033
Pr[Ρ][hTut+k≤d]≥1-β,k＝1,2,…,N-1
wherein Q, R ', Q' are corresponding weights,
Figure FDA0003080332480000034
is the desired position, J denotes the objective function, xi(k) Value, u, representing the state quantity of the ith satellite at time ki(k) And the values of the control quantity of the ith satellite at the time k are shown, and i and j respectively show the ith satellite and the jth satellite.
7. The satellite formation reconstruction model predictive control algorithm of claim 6, wherein: the linearized expression form after the chance constraint condition reconstruction in the step S3 is:
Figure FDA0003080332480000035
wherein epsilon belongs to (0,1) as a set parameter, sigmauThe variance of the control amount u is indicated.
8. The satellite formation reconstruction model predictive control algorithm of claim 7, wherein: the process of setting the mean and variance of the state quantity and the control quantity, respectively, in step S4 is:
respectively setting state quantity and control quantity:
Figure FDA0003080332480000036
then there are:
Figure FDA0003080332480000037
δut＝K(Gxxδxt+Gxωwt)
wherein, I represents an identity matrix,
Figure FDA0003080332480000038
make it
Figure FDA0003080332480000039
Figure FDA00030803324800000310
Then there are:
Figure FDA00030803324800000311
Figure FDA0003080332480000041
order to
Figure FDA0003080332480000042
In summary, the following are provided:
Figure FDA0003080332480000043
wherein
Figure FDA0003080332480000044
9. The satellite formation reconstruction model predictive control algorithm of claim 8, wherein: in step S4, the objective function of the mathematical model of the SMPC formation control problem in step S2 is reconstructed as follows:
Figure FDA0003080332480000045
wherein the weight Qm＝diag(Q,Q,…,QNR ', …, R'), tr (·) denotes the traces of the matrix; an optimal control problem is obtained by combining the corresponding constraints.
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