Eigensystem realization algorithm

The Eigensystem realization algorithm (ERA) is a system identification technique popular in civil engineering, in particular in structural health monitoring. ERA is a modal analysis technique which generates a system realization using the frequency response given (multi-)input and (multi-)output data.^[1]

Algorithm

Given pulse response data form the Hankel matrix

$H(k-1) = \begin{bmatrix}Y(k) & Y(k%2B1) & \cdots & Y(k%2Bp) \\ Y(k%2B1) & \ddots & & \vdots \\ \vdots & & & \\ Y(k%2Br) & \cdots & & Y(k%2Bp%2Br) \end{bmatrix}$

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 $\hat{A}$ .^[2]

References

^ Marlon D. Hill. "An Experimental Verification of the Eigensystem Realization Algorithm for Vibration Parameter Identification" (pdf). http://mceer.buffalo.edu/publications/resaccom/04-sp06/05Hill.pdf. Retrieved August 24, 2011.
^ 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).