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)
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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