Giuseppe Abbiati

EACS MEETING 2012

EUROPEAN ASSOCIATION FOR CONTROL OF STRUCTURES

Genoa, Italy, 18-20 June 2012

Novel Partitioned Time Integration Schemesfor DAE Systems based on Generalized-alpha methods

2012/7/6Page 1

• Interfield partitioned algorithms for time-integrating heterogeneous subsystems applied to Hamilton form of the equation of motion: PLSRT-2 and IPLSRT-2 methods

• Interfield partitioned algorithms for time-integrating heterogeneous subsystems applied to Euler-Lagrange form of the equation of motion: the PM method

• The novel Partitioned Parallel Generalized-ρ (PPG-ρ) method for time-integrating heterogeneous subsystems applied to Hamilton form of the equation of motion

• Identification and control design of actuators in TT1 test rig

Outline

2012/7/6Page 2

PLSRT-2 and IPLSRT-2 methods

2012/7/6Page 3

PLSRT2 – Dt = 4 ms

IPLSRT2 – Dt = 4 ms

NS PS

0 5 10 15 20 25-0.015

-0.01

-0.005

0

0.005

0.01

0.015

ynye

0 5 10 15 20 25-0.015

-0.01

-0.005

0

0.005

0.01

0.015

ynye

Numerical and experimental simulations with PLSRT2 and IPLSRT2 algorithms done by Zhen Wang

2012/7/6Page 4

Spectral stability: the PLSRT2 method is conditionally stable

Parallel LSRT2 (PLSRT2)

Stability limits decline when b1

increases.

1

A B B

B A A

m kbm k

æ

æ

æ

æ

æ

æ

æ

æ

à

à

à

à

à

à

à

à

ò

ò

ò

ò

1122

10-5 10-4 10-3 10-2 10-1 Dt A10-1110-910-710-5

0.0010.110

e

ò là u° Bà u° Aæ uB

æ uA

2012/7/6Page 5

The PM method

2012/7/6Page 6

1 1 1 , 1 1

/ ( 1)/ ( 1)/ , ( 1)/ /

/ /

( , )

( , )

0

T

T

sub sub sub sub sub

sub sub

A A A A A A An n n ext n n

B B B B B B Bn j n n j n n j n ext n j n n j n

A A B Bn j n n j n

M u R u u F L Λ

M u R u u F L Λ

L u L u

The PM methodPegon and Magonette, 2002

Split-mass Single-Degree-of-Freedom (S-DoF) system:

Equations of motion:

2012/7/6Page 7

The interfield parallel method – Pegon-Magonette method – exploits 2tA in subdomain A. It allows the interfield parallelization.

Algorithms applied tothe Euler-Lagrange formof the equations ofmotion

A. Bonelli, O.S. Bursi, L. He, P. Pegon, G. Magonette. Convergence analysis of a parallel interfield method for heterogeneous simulation with dynamic substructuring. International Journal for Numerical Methods Engineering, 75 (7); 2008. p. 800-825.

The PM methodPegon and Magonette, 2002

2012/7/6Page 8

The Partitioned Parallel Generalized-ρ (PPG-ρ) method

2012/7/6Page 9

1 1 1 1

0

T

T

A A A A A An n n n

B B B B B Bn j ss n j ss n j ss n j ss n j ss

A A B Bn j ss n j ss

M y K y f G Λ

M y K y f G Λ s

G y G y

Equations of the coupled problem written in Hamiltonian form:

Crucial for the consistency is to apply to the subdmain B the residual on thebalance equation of subdomain A due to the interpolation of the state vector overthe smallest time step:

Partitioned Parallel Generalized- (PPG- ) methodLagrange multipliers inherited from the PM method (Pegon and Magonette, 2002)

, ,A A A A f A A fn j ss n j ss n j ss n j ss s G f M y K y

2012/7/6Page 10

asymptotic spectral radius modulus 0,1

31 1 1, ,2 1 1 2m f m f

1

1

1

1 1

m

f

f

n n m n n

n n f n n

n n f n n

n n n n n

y y y y

y y y y

u u u u

y y t y t y y

m f fn n ny y u y y u

1st order ODE

1st orderDAE

The Generalized- (G-) method(Jansen et al, 1999)

The user-controlled algorithmc damping is tuned by means of the parameter: spurious modes damping

and noise rejection

where:

2012/7/6Page 11

1 1

1 1

1 1

1 1

1 1

1 1

1

1

n m n m n f n f

n n f n f n m m

n n n n

n n n n

t t

t t

v v y y

v y y v

y y y y

y y v v

2

2

n n t

n n t

O

O

v y

d y

MG-α solutionwith collocated values

Exact solutionReal values

The Modified Generalized- (MG-) method

The balance equation must be written at the end of the time step.

For that purpose collocated quantities are introduced as extension of the state

vector (Bruls et al)

1 1 1n n n My Ky f

2012/7/6Page 12

1

1

1 1 1

1 1 1

11

2 1 1 1

1

1 1

1

fmn n n n

m m

fn n n

m

fn n n

m

n f n f n mn

m

fmn n n n

m

t t

t

t

t t

y y v y

y M K f Ky

y y y

y y vv

y y v y

m

The MG- methodThe linear-implicit algorithm

N+1 Predictor

N+1 Solution

N+1 Collocated quantity

N+2 Predictor

2012/7/6Page 13

The MG- methodSpectral properties

Spectral radius Phase angle

Algorithmic damping Frequency error

2012/7/6Page 14 t Dimensionless frequency:

Split-mass S-DoFs system used for the simulations:

The Partitioned Parallel Generalized-ρ (PPG-ρ) methodNumerical estimation of the order of accuracy

1, 1, 0A B A B A Bm m m k k k c c

0 0,10B A B

A B A

m kbm k

The tangent stiffness is assumed to be time invariant and equal to the initial one.

Linearly implicit MG- Linearly implicit MG-

2012/7/6Page 15

PPG- methodNumerical estimation of the order of accuracy

1, 1ss

Displacements Velocities Collocatedquantities

Lagrangemultipliers

2012/7/6Page 16

PPG- methodNumerical estimation of the order of accuracy

Order of accuracy for the investigated couples of infinity spectral radius Rooand sub-stepping ss:

• With sub-stepping the algorthm is 1st order accurate

• Without sub-stepping the algorithm is 2nd order accurate

2012/7/6Page 17

PPG- methodNumerical simulation on a split-mass 3-DoFs stiff system

1 2 32,85Hz, 6,91Hz and 36,03Hzf f f

3 1 2 , f f f Stiff system

2,00 ms, 0, 20 ms, 10A B A Bt t ss t t

Simulation parameters:

2012/7/6Page 18

PPG- methodSplit-mass 3-DoFs stiff system simulation

1,00 0,60 w/o numerical damping:

- All the frequencies are preservedwith numerical damping:

- The lowest frequencies are preserved- The highest frequency damped out

1st DoF 1st DoF

3rd DoF 3rd DoF

Time-frequency analysis of the displacement responses

2012/7/6Page 19

Identification and control design of actuators

in TT1 test rig

2012/7/6Page 20

Sensors

Parker – Velocity loop

DSpace - Disp. loop

Displacement [m]Load [N]Accel. [m/s2]

Plant – Actuator

The Test Rig TT1- hardware upgrade -

2012/7/6Page 21

)()(

uLyL

UYGM

)()(

1 ,

,

ddInnerPM

InnerPMP dL

yLDY

KGKG

G

Control scheme of the plant (Actuator)

Velocity to displacement open-loop transfer function GM

Displacement to displacement closed-loop transfer function GP

Motor 1 / sy [mm]

Speed loop

v [mm/s]u [daV]KP,Inner

dd [mm]

Inner displacement loop GM

y [mm]

Motor 1 / sy [mm]

Speed loop

v [mm/s]u [daV]KP,Inner

dd [mm]

Inner displacement loop

y [mm]

GP

+-

+-

+-

+-

, 0.015P InnerK

2012/7/6Page 22

The displacement-to-displacement transfer function was identified by from SISO data using the Prediction Error Minimization (PEM) method

Displacement demand dd :

chirp sine up to 30Hz with 4mm

amplitude

Identification of the plant (Actuator)

2012/7/6Page 23

Two models for the transfer function are identified in order to design the control system:

sTsTK

DYG

pp

p

dP

21

2, 11~

sTsTsTK

DYG

ppp

p

dP

321

3, 111~

Kp = 0.98407 Tp1 = 0.053294 [s] Tp2 = 0.015493 [s]

Kp = 0.99132 Tp1 = 0.059422 [s] Tp2 = 0.0064085 [s] Tp3 = 0.0064086 [s]

• 2-poles model:

• 3-poles model:

Identification of the plant (Actuator)

2012/7/6Page 24

With respect to highest frequencies, the 3-poles model GP,3P reproduces with more accuracy the Empirical Estimated Transfer Function (EETF) of the

actuatore as shown by the Bode plots:

Identification of the plant (Actuator)

2012/7/6Page 25

Controller design

• Non model-based controllers:

• PID

• PD

• Model-based controllers (based on 2 and 3 poles TF):

• Internal Model Control (IMC)

• LQG

2012/7/6Page 26

1F PG s G s PG s

r yd

MG s,P InnerK

3

11 314FG s

s

Internal Model Control (IMC)

The saturation on the velocity command was taken into account also for the model prediction

The scheme is valid for both the 2 and 3 pole models of the plant

2012/7/6Page 27

LQG control

x A x B uy C x D u

LQG controllers based on the state space realisation of both the 2 and 3 poles model GM,2 and GM,3 are devised:

C,A B

+

-KPLLQRK

x̂

x yu

r

where: LQRK

KPL

Linear Quadratic Regulator static gain

Kalman observer 2012/7/6Page 28

Estimation of the performance of the controllers without specimen

The delay ranges between 8 ms (LQG-3P) and 17 ms (PID)

2012/7/6Page 29

Load cells

Electro-mechanic actuators

Laser displacement sensors

0, 0d

Set-up of the specimen for the test

Estimation of the performance of the controllers with specimen

The same displacement demand d is sent to both the actuators attached to the 200 kg mass specimen

2012/7/6Page 30

Estimation of the performance of the controllers with specimen

Resonance frequency due to the specimen arises at about 11 Hz

The controllers show a good response up to 10 Hz

2012/7/6Page 31

5 mm RMS, 5 Hz Band Limited White Noise (BLWN) displacement demand signal

Estimation of the Delay and the Error of the controllers

1/2

2

11/2

2

1

1

E 1001

N

i ii

N

ii

r yN

rN

1*

0max ( ) max

N m

i m ii

m m

r yt r y m t

N m

r: displacement demandy: displacement of the actuatorN: number of samplesm: shift samplest: sampling time

Error [%]: Delay of the controller [ms]:

RMS and Delay estimations were carried out in the cases with and without delay compensation

for each devised controller

2012/7/6Page 32

Forward prediction entails noise amplification:

hybrid simulations will help us to select the optimal controller

Without delay compensation

With delay compensation

Delay[ms]

Error[%]

Error[%]

PD 16.3 30.6 2.9PID 16.8 29.8 2.5LQG p2 7.7 15.5 4.7LQG p3 8.3 17.3 5.8IMC p2 8.6 25.5 3.2IMC p3 11.7 23.3 5.4

Estimation of the Delay and the Error of the controllers

Delay compensation done by means of 2nd order Lagrange polynomial extrapolation

2012/7/6Page 33

Conclusions

• The novel Partitioned Parallel Generalized-ρ (PPG-ρ) method with user-controlled algorithmic damping is proposed

• Its stability and accuracy properties are investigated by means of numerical simulations

• In order to perform hybrid simulations on the Test Rig TT1 with the proposed PPG-ρ :

• A significat noise reduction on sensors is achieved thanks to the hardware upgrade

• Identification of the actuator and design of controllers are carried out

2012/7/6Page 34

Thank you for the attention

2012/7/6Page 35