Eigensystem realization algorithm

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. The ERA was proposed by Juang and Pappa and has been used for system identification of aerospace structures such as the Galileo spacecraft, turbines, civil structures and many other type of systems.

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 ${\hat {A}}$.

See also

• Frequency domain decomposition
• Stochastic Subspace Identification
• ERA/DC