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.

Algorithm

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 ${\hat {A}}$.