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