CN111781827B - Satellite formation control method based on neural network and sliding mode control - Google Patents

Satellite formation control method based on neural network and sliding mode control Download PDF

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CN111781827B
CN111781827B CN202010488772.XA CN202010488772A CN111781827B CN 111781827 B CN111781827 B CN 111781827B CN 202010488772 A CN202010488772 A CN 202010488772A CN 111781827 B CN111781827 B CN 111781827B
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高志峰
单越超
钱默抒
蒋国平
林金星
张孝波
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a satellite formation control method based on a neural network and sliding mode control, which comprises the following steps: establishing a dynamic model of the rigid-body spacecraft; converting the dynamic model into a second-order mathematical model; determining a control error and an error limited function of the attitude quaternion according to a second-order mathematical model; accumulating control errors of attitude quaternions of associated following satellites in a satellite formation system to obtain a system lumped error; and performing online compensation on external interference by using a radial basis function neural network, defining a sliding mode surface according to the concentrated error, and obtaining a control law of the distributed controller of the satellite formation system based on the sliding mode surface. According to the method, under the condition that the interference exists in the operating environment of the satellite formation system, the interference can be quickly estimated and compensated online in real time, so that the satellite formation system keeps the expected formation attitude to fly.

Description

Satellite formation control method based on neural network and sliding mode control
Technical Field
The invention relates to aerospace flight control, in particular to a satellite formation control method based on a neural network and sliding mode control.
Background
The multi-agent system is a complex network for generating complex clustering behaviors through simple information interaction among individuals with the same or different sensing and communication capacities, and is widely applied to the fields of sensor networks, mobile robot systems, unmanned aerial vehicle systems, satellite systems and the like.
Existing multi-agent systems generally have three typical control configurations: centralized, distributed, and decentralized. The centralized type is to regard the multi-agent system as an integral design controller, and the method has simpler controller design but higher cost and is not suitable for the multi-agent system with larger scale. The distributed controller design is to design the controller for each following satellite in the multi-agent system independently, and the method is simple, the cost is lower compared with the centralized method, but the coupling of the system is strongly restricted. The distributed controller design utilizes information of individuals and adjacent individuals, which is much lower in cost than the centralized design, easy to implement and applicable to large-scale multi-agent systems. Most controller designs for multi-agent systems now employ a distributed architecture.
In the formation control problem, it is desirable to make the relevant state of each single body reach the same through a multi-agent system, and if each single body is fault-free and interference-free, the distributed controller is designed to be simpler; if one or more individual units in the multi-agent system are subjected to external compound interference, the influence of the external compound interference is considered when designing the distributed controller. In the existing system, the interference is usually compensated online in real time by using an interference sensor for processing external interference on a single spacecraft, but due to the coupling phenomenon in a multi-agent system, the accuracy of interference information obtained by using an interference observer method for the multi-agent system is not ideal.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to provide a satellite formation control method based on a neural network and sliding mode control, and the defect that the existing satellite formation control method is poor in external interference resistance is overcome.
The technical scheme is as follows: the invention provides a satellite formation control method based on a neural network and sliding mode control, which comprises the following steps:
(1) establishing a dynamic model of the angular velocity and attitude quaternion of the rigid-body spacecraft;
(2) converting the dynamic model into a general uncertain non-linear multi-input multi-output second-order mathematical model;
(3) determining a control error and an error limited function of the attitude quaternion according to a second-order mathematical model;
(4) accumulating control errors of attitude quaternions of associated following satellites in a satellite formation system to obtain a system lumped error;
(5) designing a sliding mode surface according to the centralized error;
(6) and on the basis of the sliding mode surface, performing online compensation on external interference by using the radial basis function neural network to obtain the self-adaptive law of the radial basis function neural network parameters and the control law of the distributed controller of the satellite formation system.
Further, the kinetic model is represented as:
Figure BDA0002520240900000021
Figure BDA0002520240900000022
Figure BDA0002520240900000023
wherein, Ji∈R3×3An inertia matrix of the ith following satellite of the rigid body spacecraft; omegai∈R3Is the angular velocity relative to the rigid body frame; i is3An identity matrix of three rows and three columns, ui∈R3Is a control input of the rigid-body spacecraft;
Figure BDA0002520240900000024
is the attitude quaternion of the rigid spacecraft; q. q.s0iAnd q isviSatisfy the relation
Figure BDA0002520240900000025
di∈R3The external disturbance moment vector of the ith following satellite of the rigid body spacecraft is unknown;
the index x represents the diagonally symmetric matrix, expressed as:
Figure BDA0002520240900000026
further, the second order mathematical model is represented as:
Figure BDA0002520240900000027
wherein x is1i＝qviIs a quaternion of the spacecraft;fi(x) And gi(x) Are nonlinear terms, and are respectively expressed as:
Figure BDA0002520240900000028
Figure BDA0002520240900000031
Tdiis a non-linear term with interference, expressed as:
Figure BDA0002520240900000032
further, the control error e of the ith following satellitei∈R3Specifically, it is represented as:
ei＝qvi-qdi＝x1i-xdi(7)
Figure BDA0002520240900000033
Figure BDA0002520240900000034
wherein xdi＝qdiIs the attitude of the pilot satellite, i.e. the expected attitude;
the error limited function for the ith following satellite is specifically expressed as:
Figure BDA0002520240900000035
wherein the content of the first and second substances,
Figure BDA0002520240900000036
and has | ei(0)|＜ρi(0)，ρi(t)＜ei(t)。
Further, the lumped error of the ith following satellite is expressed as:
Figure BDA0002520240900000037
wherein, aijIs an element in adjacency matrix a; the ith following satellite can receive the message of the jth following satellite, aijIt equals 1, otherwise it equals 0; biIs an element in the matrix B, B when the following satellite is associated with the pilot satelliteiEqual to 1, otherwise equal to 0; lijIs an element in the matrix L, L ═ D-a; wherein the D matrix is a diagonal matrix, the value of which corresponds to the accumulated value of all elements of each row of the A matrix;
the lumped error of the satellite formation system is expressed as:
E＝Mξ (12)
wherein
Figure BDA0002520240900000038
n refers to the number of following satellites.
Further, the designed slip form surface is expressed as:
Figure BDA0002520240900000041
wherein C ∈ R3×3Is a strictly positive parametric gain matrix.
Further, the radial basis function neural network is represented as:
Figure BDA0002520240900000042
wherein the content of the first and second substances,
Figure BDA0002520240900000043
is the output of the ith radial basis function neural network,
Figure BDA0002520240900000044
is the weight matrix of the radial basis function neural network, N is the number of neurons of the neural network;
Figure BDA0002520240900000045
is an activation function of the neural network; epsilondiIs the approximation error of the bounded function, satisfies
Figure BDA0002520240900000046
The adaptive law of the radial basis function neural network parameters is expressed as:
Figure BDA0002520240900000047
Figure BDA0002520240900000048
wherein eta iswi、ησiIs a predetermined constant, ηwi>0,ησi>0。
Further, the control law of each following satellite in the satellite formation system is represented as:
Figure BDA0002520240900000049
wherein the content of the first and second substances,
Figure DEST_PATH_GDA00026622374500000410
has the advantages that: compared with the prior art, the satellite formation control method enables the satellite system to keep formation flying under the condition of external interference; the adopted radial basis function neural network has high convergence speed, the self-adaptive law of the weight matrix is easy to process, and the calculated amount is reduced; in addition, the sliding mode variables of the sliding mode surface adopted by the method adopt lumped errors, the limited function does not limit the system type, the application range is wide, and a complicated change process is not needed, so that the strange phenomenon is avoided.
Drawings
Fig. 1 is a schematic flow chart of a satellite formation control method according to the present application;
FIG. 2 is a topological diagram of a multi-satellite formation of a satellite system of the present application;
FIG. 3 is a diagram of the attitude of each satellite under external disturbance in simulation verification;
FIG. 4 is a velocity plot for each satellite under external interference in simulation verification;
fig. 5 is a variation trend graph of the attitude component of each satellite in simulation verification.
Detailed Description
The invention is further described below with reference to the following figures and examples:
the application discloses a satellite formation control method based on a neural network and sliding mode control, as shown in fig. 1, the method comprises the following steps:
s101, establishing a dynamic model of the angular velocity and attitude quaternion of the rigid-body spacecraft.
Specifically, the kinetic model is represented as:
Figure BDA0002520240900000051
Figure BDA0002520240900000052
Figure BDA0002520240900000053
wherein, Ji∈R3×3An inertia matrix of the ith following satellite of the rigid body spacecraft; omegai∈R3Is the angular velocity relative to the rigid body frame; u. ofi∈R3Is a control input of the rigid-body spacecraft;
Figure BDA0002520240900000054
is the attitude quaternion of the rigid spacecraft; q. q.s0iAnd q isviSatisfy the relation
Figure BDA0002520240900000055
di∈R3The external disturbance moment vector of the ith following satellite of the rigid body spacecraft is unknown external disturbance.
Sign (sign)×Represents a diagonally symmetric matrix, represented as:
Figure BDA0002520240900000056
s102, the dynamic model is converted into a general uncertain nonlinear Multiple Input Multiple Output (MIMO) second-order mathematical model.
In particular, using x1i＝qviThe second order mathematical model is represented as follows:
Figure BDA0002520240900000057
Figure BDA0002520240900000058
wherein x is1iIs a quaternion for the spacecraft.
To simplify the above expression, the above non-linear term is expressed as fi(x) And gi(x) Instead of this.
Figure BDA0002520240900000061
Figure BDA0002520240900000062
Figure BDA0002520240900000063
The above formula can be simplified to a second order model as follows:
Figure BDA0002520240900000064
wherein T isdiIs a non-linear term with interference.
S103, determining a control error and an error limited function of the attitude quaternion according to a second-order mathematical model.
In particular, the control error e of the ith following satellitei∈R3Specifically, it is represented as:
ei＝qvi-qdi＝x1i-xdi (8)
Figure BDA0002520240900000065
Figure BDA0002520240900000066
the error dynamics equation (10) can be restated as a set representing the set of all following satellites in the satellite formation system.
Figure BDA0002520240900000067
Wherein the content of the first and second substances,
Figure BDA0002520240900000068
g＝diag(g1,g2,…,gn),
Figure BDA0002520240900000069
the error limited function for the ith following satellite is specifically expressed as:
Figure BDA00025202409000000610
wherein the content of the first and second substances,
Figure BDA00025202409000000611
and has | ei(0)|＜ρi(0)，ρi(t)＜ei(t), combined formula (12) has | [ xi ]i(t)∣＜1，∣ξi(t)Tξi(t)∣＜1。
And S104, accumulating the control errors of the attitude quaternion of the associated following satellites in the satellite formation system to obtain a system lumped error.
Specifically, in the satellite formation system, a controller is designed by using a distributed control method, and each following satellite only uses information of the following satellite adjacent to the following satellite. The lumped error for the ith following satellite is expressed as:
Figure BDA0002520240900000071
wherein, aijIs an element in adjacency matrix a; the ith following satellite can receive the message of the jth following satellite, aijIt equals 1, otherwise it equals 0; biIs an element in the matrix B, B when the following satellite is associated with the pilot satelliteiEqual to 1, otherwise equal to 0; lijIs an element in the matrix L, L ═ D-a; where the D matrix is a diagonal matrix whose values correspond to the accumulated values of all elements of each row of the a matrix. The lumped error of the satellite formation system is expressed as:
E＝Mξ (14)
wherein
Figure BDA0002520240900000072
n refers to the number of following satellites. Directed graphs because of graph theory representation
Figure BDA0002520240900000073
Contains at least one spanning tree, so that M is guaranteed to be invertible.
S105, designing a sliding mode surface according to the centralized error, wherein the sliding mode surface is expressed as:
Figure BDA0002520240900000079
wherein C ∈ R3×3Is a strictly positive parametric gain matrix.
S106, based on the sliding mode surface, the radial basis function neural network is used for carrying out online compensation on external interference, and the self-adaptive law of the radial basis function neural network parameters and the control law of the distributed controller of the satellite formation system are obtained.
Specifically, the radial basis function neural network is represented as:
Figure BDA0002520240900000074
wherein the content of the first and second substances,
Figure BDA0002520240900000075
is the output of the ith radial basis function neural network,
Figure BDA0002520240900000076
is the weight matrix of the radial basis function neural network, N is the number of neurons of the neural network;
Figure BDA0002520240900000077
is an activation function of the neural network; epsilondiIs the approximation error of the bounded function, satisfies
Figure BDA0002520240900000078
Solving the self-adaptive law of the parameters of the radial basis function neural network by a sliding mode control method, which is expressed as follows:
Figure BDA0002520240900000081
Figure BDA0002520240900000082
wherein eta iswi、ησiIs a predetermined constant, ηwi>0,ησi>0。
And further obtaining a control law expression of each following satellite in the satellite formation system by using the sliding mode surface as follows:
Figure BDA0002520240900000083
wherein the content of the first and second substances,
Figure BDA0002520240900000084
in order to verify the stability of the sliding mode controller provided by the application, the invention utilizes the Lyapunov stability analysis theory to verify the effectiveness of the controller provided by the invention, and the method comprises the following steps:
defining the Lyapunov function:
V＝VS+VW+Vσ (21)
wherein
Figure BDA0002520240900000085
A Lyapunov function representing a sliding mode surface;
Figure BDA0002520240900000086
a Lyapunov function representing a neural network;
Figure BDA0002520240900000087
representing the stability of the neural network estimation error; when the stability of these error variables is verified, it can be verified that the control law of equation (17) is valid.
The lyapunov function of formula (21) is derived to obtain the following form:
Figure BDA0002520240900000088
wherein the content of the first and second substances,
Figure BDA0002520240900000089
the following simple inequality can also be obtained:
Figure BDA00025202409000000810
Figure BDA0002520240900000091
substituting the adaptive law and the controller control law into equation (21), and substituting equations (23) and (24) into equation (22), the following inequalities can be obtained:
Figure BDA0002520240900000092
wherein the content of the first and second substances,
Figure BDA0002520240900000093
by selecting an appropriate controller gain, ηwi＝2Ci,ησi＝2CiWherein C isi＝min[C,ηw,ησ]The inequality (22) can be expressed in the form:
Figure BDA0002520240900000094
the final consistent bounding is obtained by inequality (23),
Figure BDA0002520240900000095
and the system model utilized by the invention is stable, and the control method of the invention is effective.
The invention utilizes Matlab2018 software to carry out simulation verification on the control method of the invention:
as shown in fig. 2, two following satellites can receive the information of the pilot satellite, instead of only one following satellite in the existing production.
(1) Selecting parameters of a satellite attitude control system:
Figure BDA0002520240900000096
(2) initial parameter selection:
Figure BDA0002520240900000097
[ωi1,ωi2,ωi3]＝[0,0,0]
(3) external interference setting:
assuming that interference occurs on the satellites 1 and 3, in order to simulate various uncertain factors in the space, we choose as much interference as possible, highlighting the superiority of the control algorithm of the present invention.
d＝5*sin(t)
(4) The invention selects the parameters of a controller and the self-designed parameters of a radial basis function neural network:
Ω＝{x∣‖x‖≤0.5,x∈R3}
C＝2*diag{1,1,1}
r＝0.1,χi＝200,ηwi＝20,ηi＝150,ησi＝15
as shown in fig. 3, the attitude of each of the 4 following satellites reaches the attitude position of the pilot satellite in about 5 seconds, and compared with the existing results, the attitude of the pilot satellite needs about 10 seconds to reach the expected attitude, which illustrates the superiority of the controller in the present invention. In which the following satellites 1 and 3 are subject to external interference, are handled very quickly by the controller of the invention, but the time for the following satellites 2, 4 to reach the desired attitude is somewhat slower due to the time lag with which the following satellites 2, 4 receive information.
As shown in fig. 4, when the four following satellites reach the attitude of the pilot satellite, the angular velocities corresponding to the four following satellites also tend to be zero, which indicates that the four following satellites do not need to adjust their own attitudes any more. The rationality of the controller of the present invention was demonstrated.
As shown in FIG. 5, the postures of the four following satellites are put together for comparison, so that the four following satellites can be more intuitively seen to almost simultaneously arrive at the expected position, and the effectiveness of the satellite formation control method is proved. Where the following satellites 2, 4 are somewhat behind due to the time lag of the communication.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

Claims (5)

1. A satellite formation control method based on a neural network and sliding mode control is characterized by comprising the following steps:
(1) establishing a dynamic model of the angular velocity and attitude quaternion of the rigid-body spacecraft;
(2) converting the dynamic model into a generally uncertain non-linear multiple-input multiple-output second-order mathematical model, wherein the second-order mathematical model is expressed as:
Figure FDA0003553750060000011
wherein x is1i＝qviIs a quaternion of the spacecraft; f. ofi(x) And gi(x) Are nonlinear terms, and are respectively expressed as:
Figure FDA0003553750060000012
Figure FDA0003553750060000013
Tdiis a non-linear term with interference, expressed as:
Figure FDA0003553750060000014
(3) determining the control error and error limited function of the attitude quaternion and the control error e of the ith following satellite according to the second-order mathematical modeli∈R3Specifically, it is represented as:
ei＝qvi-qdi＝x1i-xdi (7)
Figure FDA0003553750060000015
Figure FDA0003553750060000016
wherein xdi＝qdiIs the attitude of the pilot satellite, i.e. the expected attitude;
the error limited function for the ith following satellite is specifically expressed as:
Figure FDA0003553750060000017
wherein the content of the first and second substances,
Figure FDA0003553750060000018
and has | ei(0)|<ρi(0)，ρi(t)<ei(t)；
(4) Accumulating control errors of attitude quaternions of associated following satellites in a satellite formation system to obtain a system lumped error;
(5) designing a sliding mode surface according to the centralized error;
(6) based on the sliding mode surface, performing online compensation on external interference by using a radial basis function neural network to obtain a self-adaptive law of parameters of the radial basis function neural network and a control law of a distributed controller of a satellite formation system, wherein the radial basis function neural network is expressed as follows:
Figure FDA0003553750060000021
wherein the content of the first and second substances,
Figure FDA0003553750060000022
is the output of the ith radial basis function neural network,
Figure FDA0003553750060000023
is the weight matrix of the radial basis function neural network, N is the number of neurons of the neural network;
Figure FDA0003553750060000024
is an activation function of the neural network; epsilondiIs the approximation error of the bounded function, satisfies
Figure FDA0003553750060000025
The adaptive law of the radial basis function neural network parameters is expressed as:
Figure FDA0003553750060000026
Figure FDA0003553750060000027
wherein eta iswi、ησiIs a predetermined constant, ηwi>0,ησi>0。
2. The method of claim 1, wherein the kinetic model is represented as:
Figure FDA0003553750060000028
wherein, Ji∈R3×3An inertia matrix of the ith following satellite of the rigid body spacecraft; omegai∈R3Is the angular velocity relative to the rigid body frame; i is3An identity matrix of three rows and three columns, ui∈R3Is a control input of the rigid-body spacecraft;
Figure FDA0003553750060000029
is the attitude quaternion of the rigid spacecraft; q. q.s0iAnd q isviSatisfy the relation
Figure FDA00035537500600000210
di∈R3The external disturbance moment vector of the ith following satellite of the rigid body spacecraft is unknown;
sign (sign)×Represents a diagonally symmetric matrix, represented as:
Figure FDA00035537500600000211
3. the method of claim 2, wherein the lumped error for the ith following satellite is expressed as:
Figure FDA0003553750060000031
wherein, aijIs an element in adjacency matrix a; the ith following satellite can receive the message of the jth following satellite, aijIt equals 1, otherwise it equals 0; biIs an element in the matrix B, B when the following satellite is associated with the pilot satelliteiEqual to 1, otherwise equal to 0; lijIs an element in the matrix L, L ═ D-a; wherein the D matrix is a diagonal matrix, the value of which corresponds to the accumulated value of all elements of each row of the A matrix;
the lumped error of the satellite formation system is expressed as:
E＝Mξ (12)
wherein
Figure FDA0003553750060000032
n refers to the number of following satellites.
4. A method according to claim 3, wherein the designed slip-form surface is expressed as:
Figure FDA0003553750060000033
wherein C ∈ R3×3Is a strictly positive parametric gain matrix.
5. The method of claim 4, wherein the control law of each following satellite in the satellite formation system is expressed as:
Figure FDA0003553750060000034
wherein the content of the first and second substances,
Figure FDA0003553750060000035
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