Page 1 June 5, 2013

Giuseppe Abbiati email: giuseppe.abbiati@ing.unitn.it

Department of Civil, Environment and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123, Trento, Italy.

A numerical tool for the tuning of nonlinear state space models

Abbiati G, Bursi OS, Cazzador E, Mei Z

SERIES Concluding Workshop - Joint with US-NEES JRC, Ispra, May 28-30, 2013

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1. The author gratefully acknowledges the financial supports from the European Union through the SERIES project (Grant number: 227887).

2. The author gratefully acknowledges the financial supports of the University of Trento for Lab. activities

SERIES: Seismic Engineering Research Infrastructures for European Synergies

Acknowledgments

Motivation of the research

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Lightweight state space models result to be very attractive for twofold purposes: • the characterization of PSs

• the modeling of complex NSs

A plenty of well-known differential models accounting for hysteresis, strength and stiffness degradation, pinching, hardening and softening behaviors can be easily assembled to carefully describe both NSs and PSs. In this perspective, a robust method for parameter tuning acts as a fulcrum for a framework for model management. The choice of the time-frequency technique was made because of its robustness.

• The Short-Time-Fourier-Transform

• Implementation of the identification tool in the MatLAB environment

• Case Study #1: Nonlinear identification of a steel-concrete frame structure

• Case Study #2: Model reduction of nonlinear hysteretic piers

• Conclusions

Outline

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, e i tX x t w t dt

The Short-Time-Fourier-Transform Analytical definition

Time-frequency domain representation Time domain representation

w t

,X

x t

w[ra

d/s

]

t[s]

time localization frequency localization

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In the present implementation the compact support Hanning window w(t) was considered:

1 cos 2,

2 2 2

0,2 2

t L L Lt

w tL L

t t

L/2 L/2

The Short-Time-Fourier-Transform The Hanning window

2

2

, e

, e

i t

L

i t

L

X x t w t dt

X x t w t dt

Hanning windows lengths of 2÷4 times the main period of the system being identified are suggested

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Identification of the parameters of the state space model Initial Value Problem associated to the parameter identification

, ,s s tx g x p

Simulated Xs,j signals are generated by means of a parameterized state space model g characterized by a parameters vector p:

, /2

, , , [ / 2, / 2]

/ 2k

s s k k

s k e L

t t L L

L

x g x p

x x

The associated Initial Value Problem (IVP) is defined over the generic k-th window time span as follow:

The initial value of each state coordinate Xs,j is selected from the relevant measured signals Xe,j , when available. If it is not the case, they are picked up from simulated signals of windows k-1-th considering identified parameters pid,k-1.

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Model parameters p minimize the error function between time-frequency

representations of experimental Xe,j and simulated Xs,j signals, respectively for

each k-th time window:

Identification of the parameters of the state space model Penalty function based on STFTs of measured and simulated signals

, , 2

,

, 2

, , ,

( ) arg min

,

e j k s j k

k j id k k

j

e j k

X X d

Q Q

X d

p

p

p p p

Where j refers to the j-th channel/state coordinate, whilst αj is a weighting

factor. The optimal window length allows contemporary for:

The well-conditioning of the optimization problem

The time localization of parameters p.

1.0s and 2.0s lengths were considered for Case Studies #1 and #2, respectively.

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Identification of the parameters of the state space model The MatLAB pattern search algorithm setting

OPTIONS = psoptimset('TolX',1e-5,...

'TolFun',1e-5,...

'Display','iter',...

'InitialMeshSize',2,...

'ScaleMesh','on',...

'MaxIter',1000,...

'MaxFunEvals',10000,...

'SearchMethod',@GPSPositiveBasisNp1,...

'MaxMeshSize',2,...

'CompleteSearch','off',...

'CompletePoll','off',...

'Vectorized','off‘,...

'UseParallel','always’);

Parameters were normalized toward average starting values estimated through

engineering sense. A pattern search solver was used and relevant setting follows:

Performances improved thanks to the Parallel Computing Toolbox which allows

for concurrent function evaluations, up to the number of CPU cores.

Global Optimization toolbox Parallel Computing toolbox

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Identification of the parameter of the state space model State space model definition

• The Model variable contains model data

• The Load variable keeps loadings corresponding to the actual time window

function dydt = Pier3DoFs_GBW(t,y,Model,Load)

% Bouc-Wen model

A = Model.Ave(1) * Model.Par(1); % stiffness

B = Model.Ave(2) * Model.Par(2); % beta

G = Model.Ave(3) * Model.Par(3); % gamma

N = Model.Ave(4) * Model.Par(4); % exp

f = interp1(Load.Time,Load.Load,t); % external force at time t

…

end

Simulated signals are generated by the ode15s MatLAB stiff solver because of its

robustness.

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Case Study #1: Nonlinear identification of a steel-concrete frame Description of the case study

Plan view of the prototype

structure tested at JRC

Moment-resisting frames

The full scale steel–concrete composite

structure was constructed of three identical

moment-resisting frames, arranged at a

spacing of 3.0 m.

The structure was subjected to PsD tests

at 4 PGA levels at JRC.

Structure prototype

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1 1 1 1 1 ,1 2 2 2 2 2 ,2[ , , , , , , , , , , , ]s sA s A s p

Case Study #1: Nonlinear identification of a steel-concrete frame State space model for the identification

A modified Bouc–Wen hysteretic model, capable of taking into account both

degradation in stiffness and slip for a 2-DoFs chain-like system was adopted.

2-DoFs chain-like model

of the frame

S-DoF oscillator based on a

Bouc-Wen spring and a slip spring in series

n = 1 was assumed

Degrading trend of the first floor stiffness

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Penalty function defined STFT

characterized by 1.0s length time windows

Case Study #1: Nonlinear identification of a steel-concrete frame Identified parameters

Identified parameters on 20s length 100 Hz sampled acceleration signals

Bursi, O. S., Ceravolo, R., Erlicher, S., & Fragonara, L. Z. (2012). Identification of the hysteretic behaviour of a partial-strength steel – concrete moment-resisting frame structure subject to pseudodynamic tests. doi:10.1002/eqe

, ,2 2

1

, 2

, , ,1

( )2

,

e j k s j k

k

j

e j k

A A d

Q

A d

p

p

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Case Study #2: Model reduction of nonlinear hysteretic piers Description of the Rio Torto viaduct case study

Pier #9 Pier #11

Substructuring scheme (blue – NS, red – PS)

Mock-up 1:2.5 scale models

State space models of reduced piers (NSs) were tuned with the present TF tool with respect to a refined OpenSEES fiber based Reference Model (RM). Goal: Obtaining a good energy dissipation and transversal displacement matching with the OpenSEES RM for each pier.

The bridge will be subjected to PsD tests on the using the reaction wall of the JRC

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Case Study #2: Model reduction of nonlinear hysteretic piers Refined OpenSEES fiber based FE model of the bridge

Hysteretic loops of at Ultimate Limit State

Pier #9 Pier #11

1. Kent-Scott-Park model for concrete (Concrete01)

2. Menegotto-Pinto model for rebars (Steel02)

3. Nonlinear shear behaviour of transverse beam (hysteretic)

Detail of deck-pier connection

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Case Study #2: Model reduction of nonlinear hysteretic piers Linear substructuring of piers applying the Guyan method

No out-of-plane displacements of piers were considered

3-DoFs pier plane superelement

FE pier structural scheme

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1 1

2 2

3 3

1

1 11 12 13 1 12 13 1 1

2 21 22 23 2 21 22 23 2

3 31 32 33 3 31 32 33 3

1

0

0

0

(

u v

u v

u v

v m m m f k k u r

v m m m f k k k u

v m m m f k k k u

r A sgn

1 1 1 1( ) ) | |nv r r v

Loads applied to each single pier were recorded from OpenSEES TH analyses

Case Study #2: Model reduction of nonlinear hysteretic piers State space model of hysteretic piers

Bouc-Wen spring

[ , ]A p

n = 1 and = 0 were assumed

Transversal displacement and velocity

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,1 ,1 2

,1 2

,1 ,1 2

,1 2

, , ,1

( ) ...2

,

, , ,1

2,

e k s k

k

e k

e k s k

e k

V V d

Q

V d

U U d

U d

p

p

p

STFT characterized by 1.0 s length time windows were considered.

Case Study #2: Model reduction of nonlinear hysteretic piers Definition of the penalty function

Transversal velocity

Transversal displacement

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Case Study #2: Model reduction of nonlinear hysteretic piers Preliminary identification session

A [

N/m

] U

1 d

sp. [m

]

Time [s]

Pier #9 at Serviceability Limit State

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Case Study #2: Model reduction of nonlinear hysteretic piers Improved state restoring force accounting for softening behaviour

1 1 1 1 12

1

( ( ) ) | |1

nAr sgn v r r v

u

U

1 d

sp. [m

] E

1 [

J]

Time [s]

Pier #9 at Serviceability Limit State

--- OpenSEES --- Reduced model

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Conclusions

• With regard to HSDS, state space models result attractive for both the characterization of PSs and the modelling of complex NSs.

• A numerical tool implemented in MatLAB and devoted to the tuning of state space models is presented.

• The time-frequency approach is selected because of its robustness.

• The effective capabilities of the proposed software are presented throughout two application case studies.

Page 22 June 5, 2013

Giuseppe Abbiati email: giuseppe.abbiati@unitn.it Phone: +39-0461-282571

Thank you for your attention! Questions?