The Eigensystem realization algorithm (ERA) is a

system identification The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design of experiments for efficiently generating informative data f ...

technique popular in

civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...

, in particular in

structural health monitoring Structural health monitoring (SHM) involves the observation and analysis of a system over time using periodically sampled response measurements to monitor changes to the material and geometric properties of engineering structures such as bridges a ...

. ERA can be used as a

modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. Examples would include measuring the vibration of a car's body when it is attached to a shaker, or the noise pattern in a room when excited by a loudspeake ...

technique and generates a

system realization In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of ( time-varying) matrices (t),B(t),C(t),D(t)/math> such tha ...

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 ''Galileo'' was an American robotic space probe that studied the planet Jupiter and its moons, as well as the asteroids Gaspra and Ida. Named after the Italian astronomer Galileo Galilei, it consisted of an orbiter and an entry probe. It was ...

, turbines,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. civil structures and many other type of systems.

Uses in structural engineering

In structural engineering the ERA is used to identify

natural frequencies The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...

, mode shapes and damping ratios. The ERA is commonly used in conjunction with the Natural Excitation Technique (NExT) to identify modal parameters from ambient vibration. The technique has been applied to buildings, bridges, and many other type of structural systems. In the area of structural health monitoring the ERA and other modal identification techniques play an important role in developing a model of the structure from experimental data. The state space representation, or the modal parameters are used for further analysis and identify possible damage in structures.

Algorithm

It is recommended to review the concepts of State-space representation and

vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, suc ...

before studying the ERA. Given pulse response data form the Hankel matrix :

H(k-1) = \beginY(k) & Y(k+1) & \cdots & Y(k+p) \\ Y(k+1) & \ddots & & \vdots \\ \vdots & & & \\ Y(k+r) & \cdots & & Y(k+p+r) \end

where

Y(k)

is the

m \times n

pulse response at time step

k

. Next, perform a

singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is r ...

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 Q_n

. Then the discrete time system realization can be given by: :

\hat = D_n^ P_n^T H(1) Q_n D_n^

\hat = D_n^ Q_n^T E_m

\hat = E_n^T P_n D_n^

To generate the system states

\Lambda = \hat \hat

where

\hat

is the matrix of

eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

for

\hat

Example

Consider a single degree of freedom (SDOF) system with stiffness

k

, mass

m

, and damping

c

. The

equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...

for this SDOF is :

m\ddot(t)+c\dot(t)+kx(t)=p(t)

where

x

is the displacement of the mass and

t

is time. The continuous state-space representation of this system is :

\dot=As(t) + Bu(t)

y = Cs(t) + Du(t)

where

s

represent the states of the system corresponding to the displacement

x

and velocity

\dot

of the SDOF. Note that the states are usually denoted by

x

. However, here

x

is used for the SDOF displacement.

References

{{Reflist Systems theory

Uses in structural engineering

Algorithm

Example

See also

References