Eigensystem realization algorithm

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The Eigensystem realization algorithm (ERA) is a system identification technique popular in civil engineering, in particular in structural health monitoring. ERA can be used as a modal analysis technique and generates a system realization using the time domain response (multi-)input and (multi-)output data.[1] The ERA was proposed by Juang and Pappa [2] and has been used for system identification of aerospace structures such as the Galileo spacecraft,[3] turbines,[4] civil structures [5][6] and many other type of systems.


Given pulse response data form the Hankel matrix

H(k-1) = \begin{bmatrix}Y(k) & Y(k+1) & \cdots & Y(k+p) \\ Y(k+1) & \ddots & & \vdots \\ \vdots & & & \\ Y(k+r) & \cdots & & Y(k+p+r) \end{bmatrix}

where Y(k) is the m \times n pulse response at time step k. Next, perform a singular value decomposition of H(0), i.e. H(0) = PDQ^T. Then choose only the rows and columns corresponding to physical modes to form the matrices D_n, P_n, \text{ and } Q_n. Then the discrete time system realization can be given by:

\hat{A} = D_n^{-\frac{1}{2}} P_n^T H(1) Q_n D_n^{-\frac{1}{2}}
\hat{B} = D_n^{\frac{1}{2}} Q_n^T E_m
\hat{C} = E_n^T P_n D_n^{\frac{1}{2}}

To generate the system states \Lambda = \hat{C} \hat{\Phi} where \hat{\Phi} is the matrix of eigenvectors for \hat{A}.[5]

See also[edit]


  1. ^ Marlon D. Hill. "An Experimental Verification of the Eigensystem Realization Algorithm for Vibration Parameter Identification" (pdf). Retrieved August 24, 2011. 
  2. ^ Juang, J.-N.; Pappa, R. S. (1985). "An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction". Journal of Guidance, Control, and Dynamics 8 (5). 
  3. ^ Pappa, Richard S., and J-N. Juang. "Galileo spacecraft modal identification using an eigensystem realization algorithm." Structures, Structural Dynamics and Materials Conference, 25 th, Palm Springs, CA. 1984.
  4. ^ Sanchez-Gasca, J. J. "Computation of turbine-generator subsynchronous torsional modes from measured data using the eigensystem realization algorithm." Power Engineering Society Winter Meeting, 2001. IEEE. Vol. 3. IEEE, 2001.
  5. ^ a b Juan Martin Caicedo; Shirley J. Dyke; Erik A. Johnson (2004). "Natural Excitation Technique and Eigensystem Realization Algorithm for Phase I of the IASC-ASCE Benchmark Problem: Simulated Data". Journal of Engineering Mechanics 130 (1). doi:10.1061/(asce)0733-9399(2004)130:1(49). 
  6. ^ Brownjohn, James Mark William; Moyo, Pilate; Omenzetter, Piotr; Lu, Yong (2003). "Assessment of Highway Bridge Upgrading by Dynamic Testing and Finite-Element Model Updating". Journal of Bridge Engineering 8 (3): 162–172. doi:10.1061/(ASCE)1084-0702(2003)8:3(162). ISSN 1084-0702.