Eigensystem Realization Algorithm (ERA)

Eigensystem realization algorithm with modal indicators including consistent mode indicator and modal participation factor

Eigensystem realization algorithm with modal indicators including consistent mode indicator and modal participation factor.
Example file is provided for the identification of 2DOF system subject to impulse excitation with added uncertainty (gaussian white noise) to the response.

function [Result]=ERA(Y,fs,ncols,nrows,inputs,cut,shift,EMAC_option)

Inputs :

Y: Free vibration output data in a form of Y=[Y1 Y2 ... Y_Ndata] Yi is Markov Parameter of size (outputs,inputs) and the total size is (outputs,inputs*Ndata) where outputs is the number of output channels, inputs is the number of inputs which equals to 1 unless free vibration data comes from Multi-reference channels NExT. Ndata is the length of the data samples
fs: Sampling frequency
ncols: The number of columns in hankel matrix (more than 2/3 of No. of data)
nrows: The number of rows in hankel matrix (more than 20 * number of modes)
inputs: The number of inputs which equals to 1 unless free vibration data comes from Multi-reference channels NExT
cut: cutoff value=2*no of modes
shift: Shift value in the final row and column blocks (Increase EMAC sensitivity) usually =10
EMAC_option: if this value equals to 1, EMAC will be independent of the number of columns (calculated only from observability matrix not from controllability)

Outputs :

Result: A structure consist of the below components
Parameters: NaFreq : Natural frequencies vector
DampRatio: Damping ratios vector
ModeShape: Mode shape matrix
Indicators: MAmC : Modal Amplitude Coherence
EMAC: Extended Modal Amplitude Coherence
MPC: Modal Phase Collinearity
CMI: Consistent Mode Indicator
partfac: Participation factor
Matrices A,B,C: Discrete A,B and C matrices

References:
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[1] R. Pappa, K. Elliott, and A. Schenk, “A consistent-mode indicator for the eigensystem realization algorithm,” Journal of Guidance Control and Dynamics (1993), 1993.

[2] R. S. Pappa, G. H. James, and D. C. Zimmerman, “Autonomous modal identification of the space shuttle tail rudder,” Journal of Spacecraft and Rockets, vol. 35, no. 2, pp. 163–169, 1998.

[3] Al Rumaithi, Ayad, "Characterization of Dynamic Structures Using Parametric and Non-parametric System Identification Methods" (2014). Electronic Theses and Dissertations. 1325.
https://stars.library.ucf.edu/etd/1325

[4] Al-Rumaithi, Ayad, Hae-Bum Yun, and Sami F. Masri. "A Comparative Study of Mode Decomposition to Relate Next-ERA, PCA, and ICA Modes." Model Validation and Uncertainty Quantification, Volume 3. Springer, Cham, 2015. 113-133.