modelling the position control of a segment of the e-elt ... · integrated approach is illustrated...
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ACTUATOR 2012, 13th International Conference on New Actuators, Bremen, Germany, 18–20 June 2012574
P 11
Modelling the Position Control of a Segment of the
E-ELT Using OOFELIE:: Multiphysics Integrated
FEM-based Approach Ph. Nachtergaelea, L. Gamonalb, O. Brülsb
a Open Engineering, Angleur, Belgium b University of Liège, Department of Aerospace and Mechanical Engineering (LTAS), Liège, Belgium
Abstract:
This paper presents the extension of a multiphysics software solution allowing to perform integrated simulation
of multiphysics controlled systems. This enhancement relies on an innovative formulation of time integration
schemes allowing to take into account simultaneously, in an integrated FEM-based approach, the non linear
structural response of a system and the controller dynamics. Interest and feasibility of this unified approach is
illustrated through the Modelling of the position control of a segment of the primary mirror of the E-ELT
(European Extremely Large Telescope), a highly representative application of complex multiphysics controlled
systems.
Keywords: Multiphysics, Control, Integrated Simulation, E-ELT, Time integration, Non Linear, OOFELIE
Introduction
Today, many technical systems rely on intimate
interactions between a multiphysics device and a
control system. A virtual prototype should thus be
able to describe these interactions with high
reliability and numerical accuracy in a user-friendly
environment. This calls for an integration of two
different simulation concepts. On one hand, modern
finite element simulation tools, such as
OOFELIE::Multiphysics, allow the integrated
simulation of large, strongly coupled multiphysics
models in a wide range of applications such as
electromagnetic, pyro-piezoelectric and electrostatic
sensors and actuators or fluid-structure interaction
(FSI). On the other hand, functional simulation
packages, such as Matlab/Simulink, are widely used
for the analysis of the dynamic response of control
systems.
In order to model the interactions between a detailed
physical model and a control system model, a link
needs to be established between these two
approaches. If the detailed model is linear (or
linearized), its behaviour is represented by constant
matrices that might be exported as a full or as a
reduced-order model to the control system
simulation environment. Alternatively, some
packages allow a co-simulation between the two
software packages [4], so that the equations of
motion of each subsystem are solved separately,
usually in an uncoupled or weakly coupled way, and
information between the two codes are exchanged at
some specific communication times. Special care
should then be taken to ensure the stability and an
acceptable level of accuracy.
In contrast, a fully integrated approach is proposed
in this work, relying on new functionalities
implemented in the finite element solution for the
modular Modelling of the control system. The
control system is thus described using the block
diagram language, a language very familiar to the
control engineer, and the strongly coupled equations
of motion are constructed and solved monolithically
in a unique environment, hence improving
convergence robustness. This approach relies on an
innovative time integration scheme with
unconditional linear stability for the coupled
problem [2] and global second-order accuracy [1].
From a user perspective, all properties of the system
are thus specified in a unique simulation
environment and there is no need to exchange
models from one package to the other, which might
be an error-prone process, or to implement a co-
simulation interface. In this paper, this fully
integrated approach is illustrated with the simulation
of the thermo-mechanical behaviour of a segment of
the E-ELT and its control system.
The European Extremely Large Telescope
The E-ELT will be the largest optical/near-infrared
telescope in the world and will thus gather much
more light than the largest optical telescopes existing
today. It will be able to correct for the atmospheric
distortions (i.e., fully adaptive and diffraction-
limited) from the start, providing images 16 times
sharper than those from the Hubble Space Telescope.
The E-ELT is a 40-m class, fully steerable telescope,
with integrated wavefront control. The optics are
mounted on an altitude azimuth telescope main
structure, with two massive cradles for the elevation
motions and azimuth tracks. The main structure
weighs approximately 2800 tons (Fig. 1).
ACTUATOR 2012, 13th International Conference on New Actuators, Bremen, Germany, 18–20 June 2012 575
Fig. 1: E-ELT main structure [5]
The primary mirror is composed of 984 hexagonal
segments. The motion of each segment is controlled
individually using 3 position actuators which act
perpendicularly to the mirror surface. Edge-sensors
are used to measure the relative motion between
adjacent segments. An important difficulty for the
control design comes from the highly distributed
nature of the actuators and sensors.
This study focuses on a single segment of the
primary mirror of the E-ELT. It aims at analyzing
the interactions between the complex structure of the
segment and the controller response.
Modelling a Segment of the E-ELT
The model has been elaborated based on a CAD file
defining the segment geometry (Fig. 2) and on
technical reports from ESO that describe the
behaviour and physical properties of the segment
components as well as their interactions.
Fig. 2: Initial CAD of the segment
This initial geometry has been firstly adapted,
through the proper CAD healing and simplification
procedures, to remove the fine details that are
negligible regarding the expected accuracy of the
simulations. Then, each structural component of the
segment has been meshed and their thermo-
mechanical properties have been defined. Finally,
the whole segment model has been obtained by
assembling the components together using
appropriate gluings and elements.
The resulting finite element (FEM) model, illustrated in
Fig. 3, involves about 157000 nodes, 525000 elements
and 468000 degrees of freedom (DOF).
Fig. 3: FEM model of the segment
In a first step, constant material properties have been
defined. However, temperature dependency could be
taken into account in a straightforward way.
Reduced Models and User Elements
Model order reduction (MOR) means algorithms to
automatically process numerical FEM models,
characterize their relevant properties and generate
corresponding reduced models that involve very few
variables but remain representative of the initial full-
scale model behaviours.
In OOFELIE::Multiphysics, a SuperElement Model
(SEM) corresponds to an open data structure initially
dedicated to handle such reduced models. However,
this concept has been extended and generalized in an
innovative way so that a SEM now also allows the
complete and explicit definition of compact models
using a large panel of methods such as analytical
definitions, the use of experimental data or even user
programming. This multiphysics data structure can
handle both linear and non linear models and it
proposes multiple interfacing capabilities, as
illustrated in Fig. 4.
Fig. 4: SEM interfacing capabilities
ACTUATOR 2012, 13th International Conference on New Actuators, Bremen, Germany, 18–20 June 2012576
MOR techniques can advantageously be applied in
the context of this study. Indeed, as the controller
interacts with a limited number of DOF of the
segment model (sensing and actuators driving
points), a thermo-mechanical reduced model of the
segment can be generated [3] using these interaction
points as interface nodes. This allows to connect the
controller model to the segment model regardless of
the nature of this last one, that can be either the full
3D FEM model or its reduced version (SEM).
Fig. 5: Simulation using a SEM of the segment
This approach allows to use a linear reduced model
during the early stages of the controller model
design (Fig. 5) and then perform integrated
simulations with non linear 3D FEM model
afterwards (Fig. 6), without any adaptation.
Modelling the Position Actuators (PACT)
Multiple methods can be considered to model the
actuators connected to the segment.
The raw approach would be to insert in the segment
model a full 3D FEM model of the actuators.
However, this could lead to a huge model and would
require to take simultaneously into account multiple
physical fields, hence increasing the model
complexity.
An alternate approach consists in generating a
compact model of the actuator that represents
correctly their internal behaviour but involves few
interface DOF interacting with the segment model.
Such model could be obtained by applying MOR
techniques on 3D FEM model of the actuators or by
directly introducing in a user element the equations
of the actuator.
The adoption of compact models being more elegant
and computationally efficient than the full 3D FEM
method, equivalent SEM have been used to model
the position and segment shape actuators of the
system.
In a first step, a simple linear model of each actuator
has been considered that combines a constant
mechanical stiffness with piezoelectric coupling
term.
The segment deformation resulting from the
excitation of one of the three position actuator is
illustrated in Fig. 6.
Fig. 6: Effect of a single PACT actuation
Modelling the Segment PACT Controller
The M1 controller aims at compensating
disturbances on the global shape of the mirror based
on a feedback of the edge sensors displacements. In
the model, this system is described in the finite
element simulation package as a block diagram
model in state space form (Fig. 7).
Fig. 7: Controller block diagram
The control scheme is based on a singular value
decomposition of the sensor actuator interaction
matrix J which defines the kinematic relation
between the actuator displacements a and the
induced motion at the edge sensors y:
y J a=
TJ U V=
For any edge sensor error y, the feedback control
scheme is based on a decomposition of the
measurements in modal space using the matrix UT, a
modal integral control ( ) ( /( ))H s diag k s b= + ,
where k is the integral gain and b is a leakage term,
and a back-transformation to the coordinates of the
actuators using the matrix V, so that the required
corrections are defined as
1 ( ) T
a V H s U y=
This strategy is general and applies to the whole M1-
control, however, in this work, it is particularized to
ACTUATOR 2012, 13th International Conference on New Actuators, Bremen, Germany, 18–20 June 2012 577
a single-segment control where all neighbour
segments are considered as fixed.
Integrated Simulation Approach
Based on a representation of the control system
dynamics in state space form within an integrated
finite element approach allows the formulation of
the set of strongly coupled equations of motion
( , ) ( )
( )
( , , , , )
( , , , , )
ext
q
ext
q
M q g q q K g t L y
C K K q t
x f q q x t
y g q q x t
+ + = +
+ + =
=
=
The first equation represents the mechanical
equilibrium with possible geometric nonlinearities,
the second is the thermal equilibrium, the third is the
control state equation, and the fourth is the control
output equation. q is the vector of generalized
displacements, is the vector of temperatures, x is
the vector of states, y is the vector of outputs and Ly
represents the actuator forces.
A monolithic and implicit time integration is used to
solve the coupled equations for all the variables in
one shot. More precisely, this method is based on an
extension of the generalized- method for systems
of first- and second-order differential equations [1,
2]. The solution is thus computed in a numerically
stable way and second-order accuracy of the final
results can be guaranteed.
Application and Numerical Results
The integrated simulation tool has been exploited for
the analysis of an E-ELT segment. It is assumed that
the neighbour segments are fixed in space so that the
edge sensors actually measure absolute vertical
displacements. The position actuators are considered
as ideal and their internal stiffness is not taken into
account here. The controller is based on the values k
= 8 (integral gain) and b = 0 (no leakage term). A
vertical disturbance force is applied on the segment
at the level of the first edge sensor. The force
follows a step function activated at t = 0 s, with an
amplitude of 10 N.
Fig. 8 shows the response of the first three edge
sensors (ES) and of the three position actuators
(PACT). Transient vibrations are observed at the
beginning of the simulation and are then damped out
due to the presence of structural damping in the
system. Then, the integral control system manages to
reduce significantly the amplitude of the
displacements at the edge sensors.
In conclusion, the results are physically consistent so
that this study demonstrates the ability of the
proposed integrated simulation method to analyse
control-structure interactions problems in industrial
applications using OOFELIE::Multiphysics.
Fig. 8: Simulation results for the E-ELT segment
Acknowledgements
This research work was carried out under grant
number 6020 (Multi- ) from the Walloon Region
which is gratefully acknowledged.
The authors would like to thank Mr. B. Bauvir
(ESO) and its co-workers for kindly providing a
large amount of technical data about the E-ELT
segment.
References
[1] O. Brüls and M. Arnold. The generalized-alpha
scheme as a linear multistep integrator: Toward a
general mechatronic simulator. ASME Journal of
Computational and Nonlinear Dynamics,
3(4):041007, 10 pages, 2008.
[2] O. Brüls and J.-C. Golinval. On the numerical
damping of time integrators for coupled
mechatronic systems. Computer Methods in
Applied Mechanics and Engineering, 197(6-
8):577-588, 2008.
[3] P. Nachtergaele, D. Rixen, A. Steenhoek,
Efficient weakly coupled projection basis or the
reduction of thermo-mechanical models. Journal
of Computational and Applied Mathematics
(JCAM), 234(7):2272-2278, 2010.
[4] O. Vaculin, M. Valasek, and W.R. Krüger.
Overview of coupling of multibody and control
engineering tools. Vehicle System Dynamics,
41:415-429, 2004.
[5] www.eso.org
( , ) ( )
( )
( , , , , )
( , , , , )
ext
q
ext
q
M q g q q K g t L y
C K K q t
x f q q x t
y g q q x t
&& &
&
& &
&
The first equation represents the mechanical