Parametric Identification of Nonlinear Devices Using Soft Computing

Prof. Giuseppe Carlo MARANO
Technical University of BARI, Italy

parametric identification of nonlinear devices for seismic protection using soft computing techniques



BASE ISOLATION
Isolators



NTC/08 - EN 15129


(
mx ( t ) + g x , x, t , Θ = f ( t )
&& & )



Experimental vs analytical responce force

f exp ( t )

(
f a x, x, t ,ϑ
& )


f exp ( t ) Design vector

x (t )
ϑ
x (t )
& minimize
tend

∫ abs ( f ( t ) − f ( x, x, t,ϑ ) ) dt
exp
& e

OF ϑ = ( ) tstart
tend

∫ abs ( f ( t ) ) dt
tstart
exp



tend

∫ abs ( f ( t ) ) dt
tstart
exp

tend

∫ abs ( f ( t ) − f ( x, x, t,ϑ ) ) dt
tstart
exp
& e



To be identified

&& ( t ) − µ ( 1 − y 2 ( t ) ) y ( t ) + y ( t ) = sin ( ω f t )
y &



Searching for a more reliable mathematical models of the investigated
systems…
Mathematical model of Experimental set-up
the system

Simulated system Measured system
response response

Features from the Features from the
simulated response measured response

Minimize the Evaluate
difference correlation

New set of system   Reliable model
parameters



Non-classical algorithms: they
deal with socially, phisically and/or
biologically inspired paradigms
(Perry et al., 2006)

In this field, the most adopted is
soft computing algorhitm


genetic algorhitms (GA)
GA’s are based on Darwin’s theory of evolution

Reproduction Competition

surviving Selection

Evolutionary computing evolved in the 1960’s. GA’s were
created by John Holland in the mid-70’s.

Nuove prospettive del monitoraggio strutturale Giuseppe
Carlo Marano Politecnico di Bari

GA scheme
PRIMA GENERAZIONE
Generazione dopo generazione, la
popolazione evolve verso una
soluzione ottima.

GENITORE 1 GENITORE 2
SECONDA GENERAZIONE

Gli algoritmi genetici GENITORE 1 GENITORE 2
TERZA GENERAZIONE
vengono utilizzati per risolvere
una varietà di problemi per cui i
normali metodi di ottimizzazione
risultano poco appropriati (discontinuità, non
differenziabilità, forti non linearità etc.)
GENITORE 1 GENITORE 2


Particle Swarm Optimization



Test
Load Velocity
Test Type stroke Cycle
(kN) (mm/s)
(±mm)
7No.
1 20 92 (20%) 3
50
Constitutive law test
2 750 20 230 (50%) 3
3 750 20 460 (100%) 3
4 Damping efficiency test 750 17 460 (100%) 10



M is the effective mass
 Cα is the damping coefficient

My + Cα y = p
&& & sgn[·] is the signum function
α is the damping law
exponent
K1 is the elastic stiffness
My + Cα y α = p
&& & p is the time-varying force
M is the effective mass
Cα is the damping coefficient

My + Cα sgn [ y ] y + K1 y = p
α sgn[·] is the signum function
&& & & α is the damping law exponent
K2 and K0 are two constants
My + Cα sgn [ y ] y + ( K 2 y 2 + K1 y + K 0 ) = p
α
&& & & M is the effective mass
 C1 is the internal damping coefficient
 Cα is the damping coefficient

My + C1 y + Cα sgn [ y ] y + K1 y = p
α sgn[·] is the signum function
&& & & & α is the damping law exponent
p is the time-varying force



Non-classical Identification methods
Algorithm Short description
DEA01 A DEA whose mutation operator is given by Eq.(1) and with binomial crossover as in Eq.(6)

DEA02 A DEA whose mutation operator is given by Eq. (2) and with binomial crossover as in Eq.(6)



DEA05 A DEA whose mutation operator is given by Eq. (5) and with binomial crossover as in Eq.(6)

A DEA with adaptive mutation – as in Eq.(8) – and a free-parameter crossover given by Eq.
DEA06
(10)
A PSOA whose velocity model is Eq.(11), with inertia weight as in Eq.(13), social and
PSOA01
cognitive factors as in Eq.(14)
A PSOA in which the velocity updating rule (based on the use of the constriction factor) is
PSOA02
given by Eq.(15)
A PSOA based on the use of chaotic maps (so-called chaotic PSOA) for both inertia weight
PSOA03
and acceleration factors
PSOA04 A PSOA with passive congregation in which the velocity updating rule is given by Eq.(19)
A modified multi-species real-coded genetic algorithm with specialized operators for each
MGAR
subpopulation, see [17] and [18]



Objective Function results obtained using a linear viscous

Test Mean Max Min Std
Test 1 0.324322 0.324322 0.324322 0
Test 2 0.363997 0.363997 0.363997 2.8E-16
Test 3 0.272685 0.272685 0.272685 1.68E-16
Test 4 0.297829 0.297829 0.297829 1.68E-16
Objective Function results obtained using a generalized viscous
Test 1 0.254494 0.254494 0.254494 4.26E-14
Test 2 0.332256 0.332257 0.332256 1.39E-07
Test 3 0.264244 0.26426 0.264243 2.99E-06
Test 4 0.28234 0.28234 0.28234 2.45E-09



Mechanical Model: Generalized viscous- linear elastic
Test 1
0.162356 0.163188 0.162077 0.000298
Test 2
0.203976 0.204116 0.203949 3.45E-05
Test 3
0.153384 0.153388 0.153384 7.23E-07
Test 4
0.127699 0.127699 0.127699 1.41E-12

Mechanical Model: Generalized viscous- quadratic elastic
Test 1
0.173636 0.254494 0.158448 0.022962
Test 2
0.208454 0.21712 0.203949 0.006284
Test 3
0.160706 0.26426 0.153025 0.026845
Test 4
0.12752 0.127699 0.126207 0.00049



Mechanical Model: Linear viscous
Test Type N.1 Test Type N.2 Test Type N.3 Test Type N.4
Parameters
v=92mm/s v=230mm/s v=460mm/s v=460mm/s
M (mean) - [kg] 0 0 0 0
M (max) - [kg] 0 0 0 0
M (min) - [kg] 0 0 0 0
C (mean) - [kN/
6.308518 9.955068 2.950677 3.599261
(mm/s)]
C (max) - [kN/
6.308518234 9.955068455 2.95067697 3.599260974
(mm/s)]
C (min) - [kN/
6.308518234 9.955068455 2.95067697 3.599260974
(mm/s)]

C (std) - [kN/(mm/s)] 3.32E-14 0 1.93E-15 3.15E-14



Mechanical Model: Fractional viscous

Test Type N.1 Test Type N.2 Test Type N.3 Test Type N.4
Parameters
v=92mm/s v=230mm/s v=460mm/s v=460mm/s
M (mean) - [kg] 1.75E-14 1.45E-11 0 0
M (max) - [kg] 8.74059E-13 7.26404E-10 0 0
M (min) - [kg] 0 0 0 0
M (std) - [kg] 1.24E-13 1.03E-10 0 0

C (mean) - [kN/(mm/s) ^ α] 321.4664 101.8108 20.93332 60.02495

C (max) - [kN/(mm/s)] 321.4663828 102.5398101 22.44238445 60.02544199
C (min) - [kN/(mm/s)] 321.4663828 101.058709 20.75427774 60.01439848

C (std) - [kN/(mm/s)^ α ] 1.05E-10 0.254748 0.284589 0.001661

α (mean) 0.121515 0.456479 0.647184 0.472998
α (max) 0.121514934 0.458176548 0.648755563 0.473033897
α (min) 0.121514934 0.454813372 0.634798957 0.472996579
α (std) 6.82E-14 0.00058 0.002352 5.61E-06



(a) (b)

(c) (d)



Maxwell
C K OF
Test 1 7.3107 139.2401 0.2882
Test 2 4.0123 259.4377 0.1784
Test 3 3.3335 205.6275 0.2869
Generalized Maxwell
C K   OF
132.0147 267.8712 0.33 1.0006 0.1269
Test 1 31
122.1587 358.5821 0.33 1.0060 0.1240
Test 2 60
119.2544 277.8710 0.33 1.0017 0.1535
Test 3 33
Generalized Voight
C K   OF
24.1856 0.7847 0.6932 2.0000 0.2300
Test 1

Test 2 1.0213 1.1889 1.2407 2.0000 0.1816
Test 3 4.7018 52.2115 0.9249 0.4756 0.3079
Voight
C K OF
Test 1 6.4963 12.6587 0.2877
Test 2 3.5944 15.5999 0.2049
Test 3 3.0240 12.8350 0.3074


50 mm 70 mm 104 mm



f BW ( t ) = kα x (t ) + ( 1 − α ) kz (t )
& & & (
z ( t ) = x ( t ) − β x (t ) z (t )
η −1
z (t ) − γ x (t ) z (t )
&
η
)



400 - 1220 KN vertical load

Test k α β γ η OF

50 mm 2.0812 0.4174 0.0078 -0.0065 1.8350 0.0961

70 mm 2.0522 0.3216 0.0018 -0.0015 2.0297 0.0783

140 mm 3.8513 0.2241 0.0276 -0.0202 1.4126 0.0710

Test k α β γ η OF

50 mm
2.579205 0.408575 0.14214 -0.11051 1.064357 0.09611
70 mm
2.880737 0.361206 0.030589 -0.02775 1.765703 0.078308
140 mm
3.926685 0.219169 0.041263 -0.02879 1.270448 0.07104



f BW ( t ) = kα x (t ) + ( 1 − α ) kz (t )
z (t ) = x(t ) − β x(t ) z (t )
& & & ( η −1
z (t ) − x(t ) z (t )
&
η
)



Model K    
BW 5 b1 2.0577 0.4237 0.0018 -0.0016 2.3154
parametrs
BW 4 1.3588 0.0697 5.8117e- 2.7033
parametrs b3 005
BW 5 2.0577 0.4237 0.0018 -0.0018 2.3154
parametrs b2
“forced “


Parametric Identification of Nonlinear Devices Using Soft Computing

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