Operational modal analysis of the International Space Station via Fast and Relaxed Vector Fitting

Marina Cózar Alcázar Department of Aerospace Engineering, Universidad Carlos III de Madrid [email protected] R.Chakraborty, H. Jain, G.-S. Seo, A review of active probing-based system identification techniques with applications in power systems, International Journal of Electrical Power & Energy Systems 140 (2022) 108008. 1 L.Ljung, System Identification: Theory for the User, 2nd Edition, Prentice Hall, Upper Saddle River, NJ,1999 2 P.Avitabile, Experimental modal analysis, Sound and vibration 35 (1) (2001) 20–31. 3 Janson, M., 2013, “A Systematic Search for Trojan Planets in the Kepler Data”, The Astrophysical Journal, vol. 774, no. 2, Art. no. 156, doi:10.1088/0004-637X/774/2/156. 4 C. R. Farrar, S. W. Doebling, D. A. Nix, Vibration–based structural damage identification, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 359 (1778)(2001) 131–149. 5 M. Civera, G. Calamai, L. Z. 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Rep.NASA/TM–2018–220013, NASA Glenn ResearchCenter, accessed: 2025-06-11 (2018).URL https://ntrs.nasa.gov/api/citations/20180002013/downloads/20180002013.pdf 20 International Space Station PIMS Acceleration Data File Description Document, Tech. Rep. PIMS-ISS-101, NASA Glenn Research Center, accessed: May 16, 2025 (2001). URL https://gipoc.grc.nasa.gov/pims/ reports/PIMS-ISS-101_revBaseline.pdf 21 Preliminary stage: Initial poles Stage 1: Pole identification Stage 2: Residue identification resonance peaks The following figure is the output of NExT and, therefore, the input for FRVF Method ωn difference ζn difference MAC NExT-ERA 0.05% 1.75% 0.762 NExT-FRVF 0.05% 2.30% 0.960 No weight Weak inverse weight Strong inverse weight noise = x x cases under study resonance peaks Benchmark method ωn difference MAC NExT-ERA 1.34% 0.891 SSI-CVA 0.82% 0.823 Operative structure Data preprocessing System Identification Damage analysis What is OMA? OMA estimates modal parameters using only structural responses, making it suitable for systems where applying controlled inputs is impractical due to [6],[7],[8]: Its size Its complexity to fit installations Its difficulty to respond to artificial excitations e.g. Domain Available data Time Output-only Frequency Input-output What is System Identification? SI includes the methods to model the behaviour of a physical system [1] using: Measured data Mathematical model Validation criterion [2] SI techniques are usually classified according to these categories: ERA FRVF b e n c h m a r k m e t h o d m e t h o d t o d e v e l o p ! (adapted from [3]). NExT algorithm Fast Fourier Transform (FFT) FRVF algorithm β should cover the frequency range α = β/100 s m a l l t o a v o i d i l l c o n d i t i o n i n g once unknowns are resolved new poles ! new residues! This process is repeated until RMSE is below the threshold It is a least-squares (LS) iterative process! unknowns natural frequencies NExT-ERA and NExTFRVF closely match analytical results! NExT-FRVF and NExT-ERA Results The FRVF approximation (red line) matches the FRF obtained from NExT, indicating an almost perfect fit [11] and suggesting very promising results! Modes identified by the FRVF approximation are displayed in the stabilisation diagrams: 1 Default FRVF configuration: no noise, non weighted, 5 iterations, fs=3600 Hz and model order from 8 to 100 1 spurious poles physical modes The following table lists the maximum deviation of the modal parameters with respect to analytical results between all modes for each method. Maximum errors correspond to Mode 8, the less excited mode! Mode 7 found Mode 7 found Mode 7 and 8 not found Weight effect Weighting tunes the FRVF approximation [18], adjusting the signal importance at different frequencies in the LS problem It gives greater weight to the weaker parts of the signal It is an intermediate option to avoid overemphasising noisy region The configuration for the parametric analysis is the default one with noise=2% 2 Experimental case study The performance of the NExT-FRVF algorithm is verified using real operational data from the ISS, which is provided by Space Acceleration Measurement System (SAMS) sensors [20]. Experimental model Although the ISS centre of mass experiences near-zero gravity, its components are mechanically connected and vibrate individually. These vibrations propagate through joints or acoustically through the air. Sensors system Acceleration data from the station is obtained through measurements of the SAMS sensors, whose locations are shown in the figure [21]. Therefore, the analysis focuses on local modes of that specific part of the station. T h e s e d a t a a r e p u b l i c ! Y o u c a n c h e c k i t i n t h e N A S A w e b s i t e a s s o c i a t e d t o t h i s Q R c o d e ! 18th August 2015 07:00 a.m. UTC 1 2 3 Signal processing Reference data belongs to: This data has been processed in 3 steps: Obtain a common time interval between data sets Interpolate data on common time vector to correct SAMS sensors lag Express accelerations in a common reference system [21]. Y i n d i c a t e s t h e y a w a n g l e , f o r t h i s w o r k ! e n o u g h r e s o l u t i o n P t h e p i t c h a n g l e , a n d R t h e r o l l a n g l e [0, 0.005, 0.01, 0.015, 0.02] output standard deviation random array NExT-FRVF identifies modes more accurately but NExT-ERA identifies a higher number of modes Noise effect output = output + noise Adding noise could lead to the impossibility to reach the method convergence by generating spurious poles that fit noise instead of the signal higher errors mode not found N E x T - E R A Convergence is reached by third iteration Iterations effect By increasing iterations spurious modes are reduced and mode identification is more robust N E x T - F R V F Sampling frequency effect fs = 3000Hz fs = 5000Hz aliasing effects spurious modes wider frequency range without aliasing higher time resolution higher visibility of noise NExT-FRVF identifies more accurately almost all modes in all noise cases What did we do? An output-only System Identification (SI) method combining the Natural Excitation Technique (NExT) with Fast Relaxed Vector Fitting (FRVF) for Operational Modal Analysis (OMA). Validation is performed on a numerical I-beam and real acceleration data from the International Space Station (ISS). Results benchmarked against the standard NExT–ERA approach show that the proposed method is robust, accurate under noise, and suitable for vibration-based Structural Health Monitoring (SHM). Signal fitting: FRVF Vector Fitting (VF) fits frequency-domain responses via rational functions. Developed from 1960s work on transfer functions [13], [14], VF was formalised by Gustavsen and Semlyen (1999) for electrical systems and later applied to mechanical and civil structures because of the analogy between disciplines [15], [16]. In this method, the function used to describe the dynamic behaviour of the system and the one to be fitted is the FRF, H(ω). To do this, Partial Fraction approach is chosen: where: Cn are residues pn are poles n o n l i n e a r w i t h r e s p e c t t o i t s p a r a m e t e r s b u t l i n e a r w i t h r e s p e c t t o t h e F R F Iterative process residues poles compensation terms C n a n d p n c a n b e r e a l o r d a n d e o n l y t a k e r e a l v a l u e s ! c o m p l e x c o n j u g a t e p a i r s , Relaxed VF vs Original VF Fast VF vs Original VF update It gives more flexibility for pole relocation! Applies a QR decomposition to large sparse LS matrices [17] Less computational cost! Numerical case study We test the NExT-FRVF algorithm on the following structure: Numerical model: aluminum beam Simulation parameters: F = 1N fs = 3600 Hz Ts = 30 s F Quasi-steady Vibratory Transient This study focuses on the vibratory regime: The ISS microgravity environment is characterised by three different components: Fitting configuration? Lesson learnt! The configuration used for all fittings is chosen based on the experience gained from the numerical case: fs = 10 Hz Model order = [20, 100] weight = 1 iterations = 5 Experimental results validation In the experimental part, NExT-FRVF results are benchmarked against NExT-ERA and SSI-CVA methods, since analytical data is not available. The following table shows the maximum deviation of both methods with respect to NExT–FRVF In general, ωn deviation remains below 1%, and MAC exceeds 0.98 For now, the selected modes are validated primarily by temporal consistency and agreement with NExT-ERA and SSI-CVA modes, but do they have physical meaning? To assess physical relevance, the 3D direction of SAMS motion is examined Sensors 121f03 and 121f04 move together! Their motion is constrained by the ISS structure, leading to smaller accelerations and synchronized behaviour, a clear sign of physically meaningful modes. Why are the modes physical? Numerical model was defined and validated Strong agreement between NExT-FRVF and NExT-ERA under noiseless conditions Modal identification becoming more challenging as noise increases can be mitigated by increasing the sampling frequency Rising number of iterations beyond three did notably improve accuracy Algorithm was validated with ISS SAMS acceleration data from NASA. Identified modes show temporal consistency, match benchmark methods (NExT–ERA, SSI–CVA), and demonstrate clear physical relevance Reduction of computational cost Implementation of machine learning to set SI methods configuration automatically Generate an automated clustering function Test the method on damaged structure to check if the variations in the modal parameters are detected Implement the algorithm in a real SHM system Summary of results Future works How are they related? Ambient excitation Sensor data Filtered signal Modal parameters Structural assessment What is Structural Health Monitoring? Vibration-based SHM tracks modal parameters over time or under varying conditions to detect changes that may locate, and evaluate potential damage [4]. This is possible since main modal parameters (ωₙ, ζₙ, φₙ) are directly related to mass (M), stiffness (K) and damping (C) [5]. In this way, maintenance is optimised and safety is ensured. (retrieved from [13]) These structures are excited by ambient forces (e.g. wind, traffic) that cannot be neglected or isolated they are modelled as [9]: W h i t e G a u s s i a n N o i s e ( W G N ) O p e r a t i o n a l M o d a l A n a l y s i s o f t h e I n t e r n a t i o n a l S p a c e S t a t i o n v i a F a s t a n d R e l a x e d V e c t o r F i t t i n g From System Identification to Structural Health Monitoring The basics of Operational Modal Analysis (OMA) Understanding the methodology From equations to simulations When models meet measurements Conclusions Acceleration environment In this case, mode identification addresses natural frequencies between 0.01Hz and 2.5Hz, where primary excitations arise from vehicle structural modes (truss and solar arrays), together with crew activities (wake and sleep cycles). Modes identification The FRF to be approximated by FRVF is shown on the right for the first case. natural frequencies To demonstrate the robustness of the processing and fitting procedure, the consistency of identified modes over time was analysed between the four cases shown above: Colour matrices summarise frequency deviations and MAC values of Cases 2 to 4 relative to Case 1 (considered the reference one for this analysis). Case 1: Reference case, 18th August 2015, 07:00:00 Case 2: 10 minutes later Case 3: 90 minutes later Case 4: One day later Most of the deviations in ωn are within 1%, and the MAC values are generally around 0.9 These deviation values fall within the limits accepted by NASA [19] The modes identified across all cases are considered consistent Ambient excitation I n p r a c t i c e W G N ! Impulse Response Functions (IRFs) Frequency Response Functions (FRFs) Modal parameters T h i s s t e p i s n o t n e c e s s a r y f o r t i m e - d o m a i n S I m e t h o d s Ambient excitations are processed with techniques such as NExT as follows[11]: Data processing: NExT Cross-correlation functions approximate free-decay vibration responses. Then, these IRFs are obtained through correlation functions between signals, as expressed in the following equation: where: x(t) and y(t) are outputs at different locations Ts is the observation time τ is the time lag Rxy represents similarity between signals and yields the IRF [12] Initially applied to output-only scenarios by Laufer et al. for wind turbines [10], NExT has since been used in many applications with unknown or unmeasurable excitation. Gabriele Dessena Department of Aerospace Engineering, Universidad Carlos III de Madrid [email protected] Marco Civera Department of Structural, Geotechnical, and Building Engineering, Politecnico di Torino [email protected] Oscar E. Bonilla-Manrique Electronic Technology Department, Universidad Carlos III de Madrid [email protected] This work was supported by the Madrid Government (Comunidad de Madrid – Spain) under the Multiannual Agreement with the Universidad Carlos III de Madrid (IA_aCTRl-CM-UC3M). Scan here to access the online version of this poster! 2