Li Li and Faouzi Ghrib
Department of Civil and Environmental Engineering
University of Windsor
Modal Identification
Using a Genetic and Nelder

Mead
Approach
Modal
identification
means
the
determination
of
the
modal
parameters
of
structures
from
vibration
measurements
.
The
modal
parameters
are
natural
frequencies,
mode
shapes
and
damping
ratios
of
each
mode
.
Modal
parameters
are
important
because
they
describe
the
inherent
dynamic
properties
of
structure
.
They
are
the
eigenvalues
and
eigenvectors
of
dynamic
equations
.
What is Modal Identification?
These
modal
parameters
can
serve
as
input
to
Finite
Element
model
updating
(such
as
:
the
minimum
rank
perturbation
and
sensitivity
based
model
update),
and
proceed
to
subsequent
steps
like
damage
identification
and
health
monitoring
.
Why Modal Identification?
1.
Equation solving approaches
2.
Minimization approaches (have strong relation
with optimization problem)
3.
Correlation approaches
4.
Subspace approaches (based on the state space
innovation model)
According to Ljung (1999), a simple
classification of identification techniques is:
Parameter identification as an
optimization approach
However, among all these approaches, the most
widely used in civil engineering modal identification
are the correlation and subspace approaches, not
the seemly straightforward minimization
(optimization) approaches.
Parameter identification as an
optimization approach
1.
The Prediction Error Method (PEM)
2.
The Bayesian maximum a posteriori (MAP)
estimation
3.
Direct time domain least

mean square approach
Parameter identification as an
optimization approach
1. The Prediction Error Method (PEM):
This method uses an ARMAX model structure
(Autoregressive Moving Average with exogenous
excitation) of the form.
Parameter identification as an
optimization approach
1. The Prediction Error Method (PEM):
The parameter estimation of the ARMAX model
structure is by solving a minimization problem of
the residual function.
The minimization is a nonlinear problem due to the
nonlinear dependency of the model residual upon
the model parameter vector.
Parameter identification as an
optimization approach
2. The Bayesian maximum a posteriori (MAP)
estimation:
In the Bayesian approach, the modal parameters
are assumed to be random variables whose
particular realization we must estimate.
In this approach, we attempt to minimize the
Bayesian mean

square error (BMSE) defined as
Parameter identification as an
optimization approach
2. The Bayesian maximum a posteriori (MAP)
estimation:
The estimator that minimizes the BMSE is difficult
to get, thus a suboptimal option is adopted that
maximizes the conditional mean of the parameter
θ
given observations Y,
Time

domain nonlinear least square
problem
The equation of motion for a damped dynamic
system using finite element method can be
formulated as:
Here M,C,K ∈ R
n
×
n
denote the mass, damping and
stiffness matrix, whereas g(t) denotes the load
vector.
Time

domain nonlinear least square
problem
The modal matrix Φ = {φ
k
} is solution of K = λM
with respect to the eigenvalues. The modal
transformation Φ
T
. Φ transforms the mass and the
stiffness matrix into diagonal matrices
Time

domain nonlinear least square
problem
the objective function to be minimized is:
Where t
j
is the sampling time, and
θ
is the vector of
parameters to be identified. M is the number of
observations, usually this is a very large number.
Time

domain nonlinear least square
problem
the objective function can be written as:
The vector R=(r
1
, …, r
M
) is called the residual.
We see that:
The necessary conditions for optimality require that
Parameter identification as an
optimization approach
All these minimizations are nonlinear problems due
to the nonlinear dependency of the model residual
upon the modal parameter vector.
Non

convex and non

differentiable objective
functions in high

dimensional spaces.
The analytical gradient and Hessian are not
available.
Many local minimums.
The measured outputs are always contaminated by
noise!
Thus make it to possess many local
minimums.
Parameter identification as an optimization
approach
Due to these difficulties, global convergence
methods must be employed.
Two methods has been hybridized to solve the
optimization problem, the
genetic algorithm
and
the
Nelder

Mead
method.
Nelder

Mead simplex is a kind of
direct
method;
that means, it does not require the evaluation of
derivatives.
Solving the NLS problem
In modal identification application, the Nelder

Mead is much better than Levenberg

Marquardt
when the initial guess is not very close to the exact
solution, and the noise level is high.
This is partly due to the difficulty in computing
gradients; the Levenberg

Marquardt requires the
gradients, but Nelder

Mead does not.
And the calculation of gradient in the modal
identification is not easy.
Solving the NLS problem
Using
fourth

order
Runge

Kutta
method
to
integrate
the
system’s
response
under
a
impulsive
load
.
The
simulated
system
is
treated
as
the
“true
system”,
and
the
simulated
acceleration
outputs
are
taken
as
the
measurements
.
Noise
is
added
to
simulated
outputs
to
mimic
the
measurement
noise
which
is
unavoidable
in
real
engineering
.
In
most
civil
engineering
structures,
acceleration
is
the
only
response
that
can
be
measured
well
.
Accelerometers
are
more
accurate
and
cheaper
than
other
measurement

meters
.
Example

1: a SDOF spring

mass system
The
exact
value
of
stiffness
to
the
“true
system”
is
10
,
and
damping
ratio
is
0
.
05
.
The
mass
is
taken
as
1
,
there
is
no
loss
of
generality
here,
since
we
can
always
transform
the
mass
of
a
system
to
be
unity
using
modal
transform
.
There
are
only
two
parameters
to
be
estimated,
the
stiffness
and
the
damping
ratio
.
Below
is
the
identification
results
using
Nelder

Mead
method
.
The iteration of Nelder

Mead terminates when the diameter of
the simplex is smaller than the tolerance. The tolerance is set to
be 0.001.
Example

1: a SDOF spring

mass system
Initial guess= [8 9 11; 0.02 0.03 0.06]
tol = 0.001, 50% noise
Example

1: a SDOF spring

mass system
Example

1: a SDOF spring

mass system
Observations
:
The initial guess is crucial in the performance of nonlinear LS
identification.
With a good initial guess, this method still performs well even
under a large portion of noise.
Damping are more susceptible to noise. If noise is large, damping
estimates are poor, but stiffness estimate can be good.
Initial guess of damping is more crucial if the noise level is high.
Example

2: a MDOF truss structure
An electrical transmission tower
Example

2: a MDOF truss structure
An electrical transmission tower
The exact frequency and damping of first mode
:
Frequency =
17.366
, damping ratio =
0.03
a bad initial guess: 50 51 52 ; 0.03 0.04 0.03
Identified:
237.3501, 1.8068
a good initial guess: 15 17 18 ; 0.03 0.04 0.03
Identified:
17.3655, 0.0300
Example

2: a MDOF truss structure
An electrical transmission tower
The convergence to a local minimum and the global minimum
we can identify several several modes at the same time by using a
larger simplex.
Provide good initial guesses:
16.0000 17.0000 18.0000 15.0000 15.5000
50.0000 51.0000 52.0000 51.5000 53.0000
0.0300 0.0400 0.0300 0.0400 0.0340
0.0500 0.0450 0.0510 0.0440 0.0460
Example

2: a MDOF truss structure
An electrical transmission tower
The identified first two modal properties:
17.3490 17.3490 17.3490 17.3489 17.3490
51.7733 51.7732 51.7731 51.7730 51.7730
0.0303 0.0303 0.0303 0.0303 0.0303
0.5856 0.5854 0.5854 0.5855 0.5855
The exact solutions:
17.366
,
0.03
50.213, 0.05
Example

2: a MDOF truss structure
An electrical transmission tower
How can we get the good
Initial guess!
Initial guess!!
Initial guess!!!
Now the goal is:
Use
a
“global
method”
to
localize
a
‘‘promising
area’’
likely
to
contain
a
global
minimum
;
it
is
necessary
to
well
‘‘explore’’
the
whole
search
domain
.
When
a
promising
area
is
detected,
a
“local
convergence
method”
must
be
used
to
‘‘exploit’’
this
area
and
obtain
the
optimum
as
accurately
and
quickly
as
possible
.
Problem
the global optimization of multi

minima functions
The
ideal
of
combine
global
method
and
local
convergence
method
is
not
new
.
Jer

Nan Juang et al proposed an OKID

LS approach. Where they
start nonlinear LS iterations from OKID (observer/Kalman
identification) algorithm results. OKID is very effective in practice
and it is one state

of

the

art method in system identification of
modal structures; however it gives poor, usually overestimate or
underestimate damping ratio in lightly damped structures.
Using LS as post

processor of OKID, Juang got good damping
results.
Problem
the global optimization of multi

minima functions
Damping is always the difficult
part in identification of modal
properties, and we see the
nonlinear Least

Square provides
one promising approach to fine

tuning it.
LS

to optimize the estimate of damping
GA is a paradigm that mimic the Darwinian theory
of natural selection.
As natural selection works solely
By and for the good of each being,
All corporeal and mental endowments
Will tend to progress toward perfection.
Charles Darwin, Origin of Species
the global optimization of multi

minima functions
–
using Genetic Algorithm
Basic steps of genetic algorithm (Davis, 1991)
Step1: establish a base population of chromosomes.
Step2: determine the fitness value of each chromosome.
Step3: create new chromosomes by mating the current generation.
(reproduction, crossover, mutation)
Step4: delete old members of the population.
Goto step2, continue until the predetermined condition is achieved.
GA is efficient to explore a wide search space and detect a
promising ‘‘valley’’, it is a self

start method, no need for
good initial guess, it is also robust against noise; but it is
slow in fine

tuning, takes too much time to search for the
bottom of this valley.
Slow convergence of GAs before providing an accurate
solution is a well

known drawback, closely related to their
lack of exploiting any local information.
the global optimization of multi

minima functions
–
using Genetic Algorithm
Various combinations of GA and some hill

climbing algorithm have been proposed in the
literature.
the global optimization of multi

minima functions
–
using Genetic Algorithm and a local

convergence algorithm
Assume parameters (chromosomes)

> simulate response

> calculate the fitness.
Binary encoding of the modal parameters (number of
chromosome bits controls the resolution of identified
parameters).
The fitness function is simply the inverse of the least mean
square error of output accelerations.
Genetic algorithm
For modal identifications
Two kinds of combinations:
Sequential combination and nested combinations.
For
nested combinations
, the GA loop forms the outer
loop, and the LS forms an inner loop; the LS modify the
offspring to seek better solutions with a specified number
of iterations.
For
sequential combinations
, simply use LS to do fine

tuning after GA.
the global optimization of multi

minima functions
–
using Genetic Algorithm and a local

convergence algorithm
Example

1: a SDOF spring

mass system
Use
Genetic
Algorithm
to
solve
this
problem
:
The initial population is randomly distributed between a pre

specified lower bound and upper bound of the parameters to be
identified.
The range for the initial population of k is chosen as 1~100
The range for the initial population of c is chosen as 0.01~0.20
They are very wide ranges, we can safely assume the true values
must be within this range.
A 20% level Gaussian white noise is added to the measurement.
Use Genetic Algorithm to solve example

1
Evolution parameters:
population size = 50
crossover rate = 0.8
mutation rate = 0.15
after 100 generations, the b
est member:
k(1) = 10.306 c(2) = 0.067 fitness = 298.011
k(1) = 10.306 c(2) = 0.051 fitness = 299.078
k(1) = 10.306 c(2) = 0.049 fitness = 299.096
Exact values (k=10, c=0.05)
As we can see, this is a very good initial guess for Nelder

Mead to
start with.
Solving modal identification problems as a nonlinear
Least Square optimization problem is a promising
direction.
Global convergent methods, such as the GA algorithm,
can be used to search an good initial guess for local
convergent methods to do fine

tuning.
With good initial iterations, the Nelder

Mead direct
method works well in modal identification, at least for
simple structural models;
Even the damping ratios can be accurately estimated. (this
is especially remarkable and useful)
Conclusions
The
competence
of
minimization
modal
identification
approaches
rely
on
the
advance
of
optimization
techniques
.
Conclusions
Rachid
Chelouah,
Patrick
Siarry,
Genetic
and
Nelder
–
Mead
algorithms
hybridized
for
a
more
accurate
global
optimization
of
continuous
multiminima
functions,
European
Journal
of
Operational
Research,
148
(
2003
)
335
–
348
.
Z
.
Michalewicz,
Genetic
Algorithms
+
Data
Structures
=
Evolution
Programs
,
Springer

Verlag,
Heidelberg,
1996
.
C.G. Koh, Y.F. Chen, C.Y. Liaw,
A hybrid computational strategy for identification of
structural parameters,
computers and structures, 81 (2003) 107
–
117.
Jer

Nan Juang,
Optimized system identification
, research report, NASA/TM

1999

209711
Joanna Iwaniec, Tadeusz Uhl
, the application of the nonlinear least squares frequency
domain method to estimation of the modal model parameters, 2002
References
Thank you
！
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