Li Li and Faouzi Ghrib

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23 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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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