Modal frequency identification using the predictor subspace approach for a space solar power station

The system’s modal frequency parameters are determined by using a technique known as predictor-based subspace identification (PBSID), which is based on the dynamic properties of space solar power stations’ huge size and low frequency. First, the dynamic equation for the space solar power station with an Abacus configuration’s attitude-vibration coupling is developed. Additionally, the PBSID approach is used to build the relevant parameter matrix, and singular value decomposition (SVD) is chosen to evaluate the system’s structural frequency parameters. The right input signals are created via numerical simulation, and the best sensor placement technique yields the associated vibration response signals. The computing results then demonstrate that the modal frequency characteristics of the space solar power station may be successfully identified by the PBSID method, which is based on SVD. Additionally, the findings demonstrate the superior noise immunity capabilities of the PBSID algorithm when compared to the standard modal parameter identification approaches.


Introduction
Academics were very worried about Space Solar Power Stations (SSPSs) since they were a novel kind of massive space construction [1].Structural vibration resulting from the size of SSPSs, which may often surpass kilometers, would significantly affect the system's attitude motion.In this instance, onorbit identification technology has the potential to obtain the structure's modal parameters, including mode shape, damping ratio, and frequency, which would enable it to serve as a reference for controlling system parameter correction in addition to monitoring system operational performance.For the purpose of precisely estimating the structural vibration characteristics of the system, it is thus essential to evaluate the modal parameters of the SSPSs.
The dynamics issues of SSPSs have been investigated by many scholars.Khartov et al. [2] studied the precision control problem of SSPS in low orbit.Li and Deng [3] established a dynamic model of a multi-rotary-joint SSPS for the design and simulation of high-precision attitude control systems.A strategy for optimizing the arrangement of temperature and mass in the antenna module of SSPS was presented by Yang et al [4].The modal parameter identification of the SSPS system has received less attention than other aspects of the field, such as system stability analysis, ground/sun orientation and control, and orbit and attitude dynamics modeling [5].
Identifying the parameters of in-orbit spacecraft has been the subject of several investigations in the last few decades.The subspace identification method, least squares approach, and eigensystem IOP Publishing doi:10.1088/1742-6596/2764/1/012094 2 realization algorithm (ERA) are common time-domain identification techniques [6].The aforementioned studies mostly concentrate on the conventional rigid-flexible coupling spacecraft, which have flexible appendages and stiff body parts at the center.The size of these structures usually does not exceed a hundred meters.However, SSPSs can reach the kilometer level and are often considered as a fully flexible structure.The parameter identification research for such unconstrained flexible ultra-large space structures is rare.
A big SSPS identification challenge is the major topic of this study.We use a predictor-based subspace identification (PBSID) technique to determine the relevant system modal frequency characteristics.A planar SSPS dynamic model is created in this work, along with the PBSID method's computing steps.In numerical simulation, by designing proper in-orbit input signals and obtaining corresponding vibration response signals through the optimization sensor placement method, the SSPS's modal frequency characteristics are determined.The PBSID algorithm can successfully detect the structural frequency parameters, as shown by the simulation results.This technique offers greater noise immunity capabilities than the widely utilized ERA approach.

Dynamic model of the SSPS's attitude-vibration coupling with an Abacus arrangement
There are many in-orbit arrangements for the SSPSs that have been suggested, including multi-rotaryjoint, planar, tether, integrated symmetric concentrator, and solar sail tower.For convenience, this study selects the planar Abacus configuration proposed by NASA as the research object, as shown in Figure 1.Consequently, this type of SSPS can be simplified into a huge unconstrained flexible plate model.The following assumptions are made: (1) based on the Kirchhoff hypothesis, a small deformation case is used; (2) certain nonlinear variables in the dynamic equation's derivation may be disregarded if the attitude rotation is sluggish and the attitude angle change is minimal; (3) finding the system's structural modal parameters is the aim of this study; hence, there is no consideration of the consequences of orbital motion or gravity gradient.According to the previously mentioned assumptions, the dynamic equation that describes the connection between attitude and vibration in the SSPS may be written in the following linearized form [8] is input from thrusters that control the vehicle; is the modal stiffness matrix, where q  denotes the qth order frequency; ζ is the damping ratio; r L and e L are the matrices of input coefficients in the following way: ) is the vector that represents the position of the ith thruster in regard to the coordinate system with which the body is associated; v represents the matrix that represents the antisymmetric of the given vector, while the superscript "  " indicates the number of thrusters; ( ), ( ),..., ( ) ) is the matrix representing the geometry of the translational motion of the node associated with the ith thruster.
Equation ( 1) is rewritten to the structure of the subsequent generalized dynamic equation: is the vector of states, and M , E , and K are the matrices for mass, damping, and stiffness, respectively.
If we establish a novel state vector and select the planar structure's attitude, the system outputs are represented by the out-of-plane vibration signals.Then, we may further rewrite Equation ( 3) to the subsequent state-space representation: Where ( ) t y is the system output signal, and c A , c B , C , and D represent the n n  system, n r  input, m n  output, and m r  transition matrices, respectively, as follows: Where I is a unit matrix, and diag( , Φ is a mode shape matrix, which may be produced from finite element analysis.

The state-space equation in an innovative form
Firstly, taking into account how noise affects the system, Equation ( 4) can be further discretized to the following innovative form [9]: Where A and B are the discrete forms of matrices c A and c B , respectively; k is the sampling interval; K and ( ) k e are the Kalman gain matrix and innovation noise vector, respectively.We define a vector and thus, Equation ( 5) can be further written as: ( 1) ( ) Where  

Computation of system state variable
When it comes to the PBSID method, we specify the parameters and f as the duration of the previous window and the future window, respectively.If it is assumed that the system matrix  A is asymptotically stable, Equation ( 6) may thus take the form of: , ., , Where , T [ ( ), ( 1),..., ( )] ,..., ( )] ,..., ( )] ,..., ( )] If the length f of the future window measures the same as the length p of the window that came before it, i.e. p f  , then obtaining the values of matrices p CΞ and D may be accomplished by solving the least squares problem, which is discussed in the following paragraphs.We define a generalized observability matrix in the following form: The matrix product of p Γ and p Ξ may be represented as: Based on the expression form of p Ξ in Equations ( 7) and ( 8), the matrix p CΞ calculated by Equation ( 8) can be further expressed as: (13) Then, the system's order may be ascertained, and the state variable can also be calculated by using the following equation:

Estimation of system modal frequencies
The state variable , p p X may be acquired by SVD.To find the system output matrix C , the following least squares problem has to be solved: (16) Subsequently, the matrices A , B , and K may be derived by solving the given equation:  is sampling time.Therefore, the jth order modal frequency j  can be expressed as: Re( ) Im( ) The real and imaginary components of the eigenvalue cj  are denoted by the letters

Numerical simulation
An SSPS with the Abacus configuration may be thought of as a thin, rectangular plate with free boundary conditions, to keep things simple in the discussion.Table 1 contains the model's geometry and mass parameters, whereas the relevant definitions of the thruster positions and body coordinate system are shown in Figure 2. Table 1.The SSPS model's geometric and mass attributes.    2 and 3 provide the relative error findings between the identified values and theoretical values of the first through seventh modal frequencies.Most modal frequencies can be identified by using the PBSID method, according to the findings in Tables 2 and 3.The approach yields a greater identification accuracy than the ERA method when the SNR drops below 20 dB.The results indicate that compared with the classical ERA method, the PBSID algorithm has higher noise immunity ability.However, it is worth noting that certain orders of the frequencies are still not effectively identified in Tables 2 and 3.One reasonable explanation is that the optimization of actuators is not included in this analysis; instead, it just addresses the best locations for sensors.As a result, it is challenging to fully excite and get structural modal information.

Conclusion
The determination of frequency characteristics for space solar power stations is the main topic of this work.A planar Abacus configuration SSPS with a dynamic attitude-vibration coupling model is constructed.Furthermore, the PBSID algorithm is used to determine the system's frequency settings based on the outcomes of optimizing sensor location by using the EI approach.Under various SNRs, the frequency parameters' identification accuracy is compared to the conventional ERA approach.The following is the resultant conclusion: At high signal-to-noise ratios (SNRs), there is no discernible difference between the identification accuracy of the PBSID and standard ERA approaches.However, the related data show that the identification error of the ERA technique is much larger than that of the PBSID algorithm at low SNRs, such as SNR=20 dB in Table 3, and that noise interferes with the effective identification of certain orders of frequencies.According to the findings, the PBSID algorithm outperforms the traditional ERA approach in terms of noise immunity.

Figure 1 .
Figure 1.Space solar power station with planar Abacus configuration [7].The following assumptions are made: (1) based on the Kirchhoff hypothesis, a small deformation case is used; (2) certain nonlinear variables in the dynamic equation's derivation may be disregarded if the attitude rotation is sluggish and the attitude angle change is minimal; (3) finding the system's structural modal parameters is the aim of this study; hence, there is no consideration of the consequences of orbital motion or gravity gradient.According to the previously mentioned assumptions, the dynamic equation that describes the connection between attitude and vibration in the SSPS may be written in the following linearized form[8]: -inertia in the Directions of X and Y 4.5  10 15 kgꞏm 2 Moment of Inertia in the Z-axis 9.0  10 15 kgꞏm 2

Figure 2 .Figure 3 .
Figure 2. Simplified model and the location of thrusters.As the system input, an impulse excitation signal is created to mimic the thruster's excitation signal.Using Thrusters 1 and 3 as examples, we can see the relevant input signals in Figure 3. Furthermore, determining the location of vibration sensors is essential to precisely measure the structure's modal frequency.The locations of the vibration sensors in this research, which are based on the effective independence (EI) technique, are shown in Figure 4, with a selection of 20 sensors.According to Figure 4's findings, the four corners and associated boundaries make up the majority of the sensor placement places determined by the EI approach.

Figure 4 .Figure 5 .
Figure 4.The location of vibration sensors by the EI method.In Figure5, attitude angle response data in three directions are shown by using the proposed impulse excitation signal.Based on all three orientations, the findings indicate that the attitude angle changes by a very modest amount-no more than five degrees.Figure6shows the response results from vibration displacement and corresponding velocity for Sensor 1 in the Z-direction.

Figure 6 .
Figure 6.Displacement and velocity response results for Sensor 1 in the Z-direction.

Table 2 .
Identification Results of the First 7 Orders Vibration Frequencies (SNR=50 dB).

Table 3 .
Identification Results of the First 7 Orders Vibration Frequencies (SNR=20 dB).