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Graduate school in Systems, Optimization, Controland Networks (SOCN) |
Graduate school courses scheduled in the academic year 2006-2007
"OPTIMIZATION METHODS FOR VARIATIONAL DATA
ASSIMILATION"
Lecturers : S. Gratton (CERFACS, France) and Ph. L. Toint (F.U.N.D.P.,
Namur, Belgium)
Spring 2007
"DESIGN METHODS FOR CONTROL SYSTEMS"
Lecturers : M. Steinbuch (Eindhoven University of Technology, The Netherlands) and G. Meinsma (University of Twente, The Netherlands)
"MECHATRONICS : DYNAMICS OF ELECTROMECHANICAL AND PIEZOELECTRIC SYSTEMS, WITH APPLICATIONS IN VIBRATION CONTROL"
Lecturer : A. Preumont (Université Libre de Bruxelles, Belgium)"AN INTRODUCTION TO MONTE CARLO, MARKOV CHAIN MONTE CARLO (MCMC), AND SEQUENTIAL MONTE CARLO METHODS (SMC)"
Lecturer : Anuj Srivastava (Florida State University, U.S.A.)
Detailed contents
1. VARIATIONAL PROBLEMS IN ENGINEERING AND COMPUTERS SCIENCE
The calculus of variations is concerned with finding extrema and can therefore be seen as a branch of optimisation. The techniques and problems, however, differ in an important aspect, the space over which one optimises is usually infinite dimensional, e.g. a space of curves lying in a vector space or even on a manifold, fulfilling certain boundary and smoothness conditions. The techniques and ideas from finite dimensional calculus therefore have to be generalised appropriately. Nevertheless, it is often possible to apply certain rules to derive the so-called Euler-Lagrange equations in a straightforward way. To find a solution for the Euler-Lagrange equation can be a real challenge, even numerically.
1. Introduction and Mathematical Preliminaries
2. Simplest Examples
3. The First Variation and the Euler-Lagrange Equation
4. More Advanced Examples from Mechanics, Robotics and Control
5. Second Order Conditions
6. Numerical Aspects
Overview :
This course is on the basic principles of the calculus of variations and how to solve variational problems frequently appearing in engineering and computer science. We will develop the basic ideas from scratch based on examples from classical mechanics, path planning in robotics and interpolation problems, optimal control, and aspects of geometric control. There exists a vast literature on the subject. Examples and applications to be discussed can be found in particular in :
J. L. Troutman, Variational Calculus and Optimal Control, Springer, 1996
B. van Brunt, The Calculus of Variations, Springer, 2004
V. Jurdjevic, Geometric Control Theory, Cambridge, 1997
M. Giaquinta and S Hildebrandt, Calculus of Variations I/II, Springer 1996
M. I. Zelikin, Control Theory and Optimization I, Springer, 2000
A. Agrachev and Y Sachkov, Control Theory from the Geometric Viewpoint, Springer,
2004
A. Bloch, Nonholonomic Mechanics and Control,
Springer, 2003
P. Crouch and F Silva Leite, The Dynamic Interpolation Problem: On Riemannian
Manifolds, Lie
Groups, and Symmetric Spaces, J. of Dynamical and Control systems, Vol. 1,
No. 2, 1995, 177-202.
Prerequisites :
Participants are assumed to have a modest background in linear algebra and calculus in several variables. Basic knowledge in optimisation, linear systems and some control is advantageous.
Schedule : 9h15 - 12h15
This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve
2. OPTIMIZATION
METHODS FOR VARIATIONAL DATA ASSIMILATION
Lecturers : S. Gratton (CERFACS, France) and Ph. L. Toint (F.U.N.D.P., Namur, Belgium)
This course is an advanced and comprehensive presentation of most minimization methods used in variational data assimilation.
The mathematical problem to be solved is a very large nonlinear least-squares problem. The solution time has to be tightly controlled to comply with operational requirements. These two constraints (problem size, controlled solution time) have stimulated theoretical developments as well as the emergence of complex solvers combining various techniques. The course will address the following theoretical questions of interest for Data Assimilation methods : truncation of iterative methods based on algebraic stopping criteria, preconditioning issues, sufficient convergence conditions for inexact Krylov methods, use of multigrid techniques in optimization. Mathematical justifications will be presented for these techniques and numerical illustrations will be provided on academic examples.
Contents :
1. Description of Data Assimilation problems and of the underlying statistical assumptions : Bayesian estimation. Presentation of the academic examples used along the lectures.
2. Mathematical difficulty of the problem as viewed from perturbation theory. Partial condition theory based on differential calculus.
3. Basic methods for nonlinear least-squares problems. Convergence properties and algebraic stopping criteria. Preconditioning techniques. Convergence theory for inexact Krylov solvers.
4. Introduction to multigrid and trust-region techniques. Convergence of a multilevel trust-region algorithm to first order critical points. Practical implementation : smoother, grid-transfer operators and use of approximate Hessians.
Prerequisites :
Basic knowledge of matrix algebra and optimization.
Supporting material :
Notes will be distributed during the course.
Related references :
R. Conn, N. Gould and Ph. L. Toint. Trust-Region Methods, SIAM, Philadelphia,
USA, 2000.
L. Giraud, S. Gratton. On the sensitivity of some spectral preconditioners.
SIAM J. Matrix Analysis and Applications, accepted for publication.
S. Gratton, A. Sartenaer and Ph.L. Toint. A numerical exploration of recursive
multiscale unconstrained optimization. In Oberwolfach Reports 2005: Optimization
and Applications F. Jarre, C. Lemarchal, J. Zowei, editors (University of Hamburg,
Optimization and Applications, Oberwolfach).
S. Gratton, A. Sartenaer and Ph.L. Toint. Second-order convergence properties
of trust-region methods using incomplete curvature information, with an application
to multigrid optimization. Journal of Computational Mathematics, accepted for
publication.
A.S. Lawless, S. Gratton, and N.K. Nichols. An investigation of incremental
4d-var using non-tangent linear models. Quart. J. Royal Met. Soc. , 131,459-476,
2005.
Dates : November 27-29 and December 01-04-06-08, 2006
Schedule : 9h15 - 12 h15
This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve
Lecturers : M. Steinbuch (Eindhoven University of Technology, The Netherlands) and G. Meinsma (University of Twente, The Netherlands)
Objective :
The course presents "classical," "modern" and "post modern" notions about linear control system design. First the basic principles, potentials, advantages, pitfalls and limitations of feedback control are presented. An effort is made to explain the fundamental design aspects of stability, performance and robustness.
Next, various well-known classical single-loop control system design methods,
including Quantitative Feedback Theory, are reviewed and their strengths and
weaknesses are analyzed. The course includes a survey of design aspects that
are characteristic for multivariable systems, such as interaction, decoupling
and input-output pairing. Further LQ, LQG and some of their extensions are reviewed.
Their potential for single- and multi-loop design is examined.
After a thorough presentation of structured and unstructured uncertainty, model
design methods based on H-infinity-optimization (in particular, the mixed sensitivity
problem and McFarlane-Glover's loopshaping problem) and mu-synthesis are presented.
Contents :
1. Introduction to feedback theory.
Basic feedback theory, closed-loop stability, stability robustness, loop shaping,
limits of performance.
2. Classical control system design.
Design goals and classical performance criteria, integral control, frequency
response analysis, compensator design, classical methods for compensator design.
Quantitative Feedback Theory.
3. Multivariable Control
Multivariable poles and zeros, interaction, interaction measures, decoupling,
input-output pairing, servo compensators.
4. LQ, LQG and Control System Design
LQ basic theory, some extensions of LQ theory, design by LQ theory, LQG basic
theory, asymptotic analysis, design by LQG theory, optimization, examples and
applications
5. Uncertainty models and robustness
Parametric robustness analysis, the basic perturbation model, the small-gain
theorem, stability robustness of the basic perturbation model, stability robustness
of feedback systems, numerator-denominator perturbations, structured singular
value robustness analysis, combined performance and stability robustness.
6. H-infinity optimization and mu-synthesis
The mixed sensitivity problem, loop shaping, the standard H-infinity control
problem, state space solution, optimal and suboptimal solutions, integral control
and HF roll-off, mu-synthesis, application of mu-synthesis.
Lecture notes :
A full set of lecture notes will be made available on the course website.
Prerequisites :
Basic undergraduate courses in sytems and control. Some familiarity with MATLAB is helpful for doing the homework exercices.
Dates : March 19-21-23-26-28-30, 2007.
Schedule :
9h15 -12h15
March 19 , room 00.03, Celestijnenlaan 200S
March 21- 26-28-30 , room 00.62, ESAT
March 23, room 01.60, ESAT
This
course will take place at the Katholieke Universiteit Leuven,
Department Elektrotechniek ESAT, Kasteelpark
Arenberg 1, 3001 Heverlee
Contents :
1) Lagrangian dynamics of mechanical systems:
Kinetic state functions, Virtual work principle, D'Alembert's principle, Hamilton's
principle, Lagrange's equation, gyroscopic systems, Jacobi integral, Rayleigh-Ritz
method.
2) Lagrangian dynamic of electrical network:
Constitutive equations of circuits elements, Hamilton's principle (charge formulation
and flux linkage formulation), Lagrange's equation (charge formulation and flux
linkage formulation)
3) Lagrangian dynamics of electromechanical systems:
Constitutive equations for transducers (movable-plate capacitor, movable-core
inductor, moving-coil transducer)
Hamilton's principle and Lagrange's equation (charge formulation and flux linkage
formulation)
Examples: loudspeaker, microphone, electrodynamic isolator, geophone, magnetic
suspension.
Control examples: sky-hook damper, magnetic levitation.
Self-sensing.
4) Piezoelectric systems:
Constitutive equations of piezoelectric transducers, electromechanical coupling
factor, structure with a piezoelectric transducer.
Piezoelectric material (constitutive equations, coenergy density function).
Hamilton's principle.
Rosen's transformer.
5) Piezoelectric laminates:
Beam actuator, Hamilton's principle, piezoelectric loads, laminar sensor, modal
actuator and sensor.
Beam with collocated actuator-sensor pair, pole-zero pattern, modal truncation.
Kirchhoff plate, multi-layer laminate.
6) Active and passive damping with piezoelectric transducers:
Active strut, Integral Force Feedback, voltage vs. current control, modal coordinates.
Damping via resistive shunting and inductive shunting.
Self-sensing.
ATTENTION !!!
Dates : This course will not be given on the previously announced dates at KUL.
It will be rescheduled in academic year 2007-2008.
Overview :
This course is geared towards introducing basic techniques from computational statistics, for use in engineering problems. In particular, we will focus on random sampling, Monte Carlo estimation, and stochastic optimization techniques, in problems where complex probability distributions result from nonlinear systems and non-Gaussian noise. As an example, we will demonstrate the use of these random sampling techniques in nonlinear filtering. Similarly, we will also apply these tools to develop stochastic optimization techniques.
Specific Aims :
(i) To introduce basic ideas behind random sampling and Monte Carlo estimation.
(ii) To demonstrate computational approaches for solving complex estimation
and optimization problems that are difficult to solve analytically.
(iii) To demonstrate Kalman filtering and nonlinear filtering from a Bayesian
perspective, and to introduce SMC methods for nonlinear filtering.
Contents :
1 Monte Carlo Methods:
Motivation, classical Monte Carlo techniques, simulation of random variables,
error reduction Techniques
2 Introduction to Stochastic Processes:
Stochastic processes: definition, Poisson process, random walk, and standard
Wiener-process; simulations of stochastic processes.
3. Markov chain Monte Carlo methods:
Markov processes, stationarity, homogeneous Markov chains; Metropolis-Hastings
algorithm: general algorithm, examples,
Markov random fields and image analysis.
4. Sequential Monte Carlo Methods (also called particle filtering or ensemble
Kalman Filtering)
Kalman filter for Gauss-Markov systems (from a Bayesian perspective), importance
sampling, sequential Monte Carlo algorithm, applications to hidden Markov models
5. Stochastic Optimization
Gradient process, stochastic gradient process (Langevin's diffusion process),
Metropolis- adjusted stochastic gradient process, simulated annealing.
Prerequisites :
Linear algebra, advanced calculus, introduction to probability theory
Supporting material :
A printed version of all class notes will be handed out in the first lecture. As an additional reference, the textbook "Monte Carlo Statistical Methods" by C. P. Robert and G. Casella (2004) can also be used although it will not be necessary.
Evaluation procedure :
The students will be assigned some homework problems to be implemented in matlab and will be evaluated on the basis of their cumulative homework performances.
Dates : May 2, 4, 9, 11, 14, 16, 2007
Schedule : 9h15 - 12h15
This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve
Other courses scheduled in the academic year 2006-2007
Overview :
The purpose of this course is to give an overview of the field of convex optimization, focussing both on the modelling and algorithmic aspects.
The course will start with an introduction to the notion of structured convex
optimization and the reasons why this class of problems is worth studying, both
from the theoretical (favourable algorithmic complexity) and practical points
of view (resolution of large-scale problems).
Various techniques allowing the formulation of optimization problems in a convex
way will be presented, emphasizing the roles of duality and conic formulations.
Convex problems can be solved by a class of methods called interior-point algorithms,
whose key concepts will be explained along with the corresponding algorithmic
complexity results. Finally, some noteworthy applications of convex optimization
will be described.
Lecture 1. Introduction :
convex sets, convex functions and convex optimization ; black-box vs. structured
optimization and complexity.
Lecture 2. Models :
Convex modeling of optimization problems ; linear, quadratic and semi definite
modeling
Lecture 3. Models :
Duality for convex optimization ; conic formulation ; convex separable optimization
Lecture 4. Methods :
interior-point methods for linear optimization ; complexity
Lecture 5. Methods :
interior-point methods for convex optimization ; self-concordant barrier functions
Lecture 6. Applications :
polynomial optimization, MAX-CUT, machine learning, etc.
References :
S. Wright, Primal-Dual Interior-Point Methods, SIAM, 1996.
J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization,
MPS/SIAM Series on Optimization 1, 2001.
A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis,
Algorithms, and Engineering Applications, MPS/SIAM Series on Optimization 2,
2001.
Y. Nesterov, Introductory lectures on convex optimization: a basic course, Applied
Optimization 87, Kluwer Academic Publishers, 2003.
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
Dates : Fridays 10/11 (9h15-12h15), 17/11 (9h15-12h15), 24/11 (9h15-12h15), 1/12 (14h-17h !), 8/12 (14h-17h !), 15/12 (9h15-12h15)
This course will take place at CORE, 34 voie du Roman Pays, B-1348, Louvain-la-Neuve (room b-135 except for 17/11 and 24/11 : room c-035).
Overview :
In this course we will examine the main results underlying the theory of integer and mixed integer programming. Examples will be chosen with the practical goal of understanding what happens in the existing commercial systems such as Cplex and Xpress-MP, and of learning how to improve problem formulations and/or use design alternative algorithms if and when these MIP systems fail.
Lecture 1. Background Theory: Polyhedra, Lattices, Hilbert Bases, Duality.
Lecture 2. Valid Inequalities for General IPs and MIPs
Lecture 3. Valid Inequalities and Convex Hulls for "Easy" Structured MIP Sets
Lecture 4. Valid Inequalities, Lifting and Separation for "Hard" Structured MIP Sets
Lecture 5. Branch-and-Cut and Applications
Lecture 6. Decomposition Algorithms: Branch-and-Cut-and-Price and Resource Decomposition
References :
Basic: Integer Programming, L.A. Wolsey, Wiley, New York 1998
More Advanced: Production Planning by Mixed Integer Programming, Y. Pochet and
L.A. Wolsey, Springer, New York 2006.
Integer and Combinatorial Optimization, G.L. Nemhauser and L.A. Wolsey, Wiley,
New York 1988 (paperback edition).
Dates : Wednesday afternoons from 14h-17h from 14/2/07 till 21/3/07
This course will take place at CORE, 34 voie du Roman Pays, B-1348, Louvain-la-Neuve ( room c-035).
This course is given in the framework of the Chaire Francqui at ULB
Overview :
Linear models are at the basis of many engineering activities. The aim of this course is to identify these models from experimental data. In real life, nonlinear distortions violate the ideal linear framework. Two solutions are discussed to extend the classic linear modelling approach. First the linear framework will be extended to include these distortions using best linear approximations and nonlinear noise sources. Alternatively, the nonlinear distortions will be explicitly modelled.
Specific Aims :
- To give an introduction to system identification theory: this offers a systematic
approach to the extraction of models from experimental data. Besides the plant
model, identification theory also provides a description of the uncertainty
bounds.
- To discuss the nonparametric and parametric identification of linear dynamic
systems: the discussions are made in the frequency domain, but many results
carry over to the time domain formulation. The equivalences and differences
between time-and frequency domain identification will be highlighted.
- To analyse the impact of nonlinear distortions on the linear framework: Are
there nonlinear distortions present? At what level? What is their nature? The
Volterra/Wiener theory is used as a framework.
- To include nonlinear distortions in the linear system identification theory.
- To model nonlinear distortions using different classes of nonlinear models.
During the course many illustrations with experimental results coming from different
application fields will be shown.
Outline :
February 7, 2007 at 16:30
Inaugural Lecture (1 Hour)
February 8, 2007 from 10:00 to 12:45
Frequency Response Function Measurements (2.5 hours)
Setup; random and periodic excitations; bias and variance analysis; leakage;
averaging techniques; experiment design; extension to multiple-input-multiple-
output systems;
February 15, 2007 from 10:00 to 12:45
Impact of Nonlinear Distortions on the Linear Framework (2.5 hours)
Nonlinear framework; coherent and stochastic nonlinear contributions; best
linear approximation; detection, qualification and quantification of nonlinear
distortions; optimized experiments.
February 22, 2007 from 10:00 to 12:45
System Identification (2.5 hours)
Introduction; consistency, efficiency, Cramer-Rao bound; (weighted) least
squares and maximum likelihood estimation; errors-in-variables framework;
model selection; model errors.
March 1, 2007 from 10:00 to 12:45
Identification of Linear Systems (2.5 hours)
Basic choices (including time- and frequency domain identification); errors-in-
variables and output error formulation; (non)parametric noise models; model
selection; impact of nonlinear distortions.
March 8, 2007 from 10:00 to 12:45
Identification of Nonlinear Systems (2.5 hours)
Basic problem; models: nonparametric, block structured, and state space
representation; initialization methods.
Dates : February 7, 8, 15, 22 and March 01, 8, 2007
This course will take place at Solbosch Campus of ULB, 50 Av. F.D. Roosevelt, B-1050 Brussels
For the inaugural lecture : Room UB5.132
All other lectures : Room UB4.132
Content :
1. Introduction to polytopes: fundamental notions, cyclic polytopes and other
basic
examples, permutahedron and associahedron, faces, polarity, rational and 0/1-polytopes.
2. Steinitz's Theorem (a graph is the graph of a 3-polytope iff it is planar
and 3-connected) :
Truemper's revision of Steinitz's original proof.
3. Gale transforms: various definitions, classification of polytopes with few
vertices,
nonrational polytopes, (oriented) matroids.
4. The Upper Bound Theorem: shellability, the Euler-Poincaré formula,
h-vectors and the
Dehn-Sommerville equations, maximizing the number of facets.
5. Topics following suggestions from the audience.
References :
Grünbaum, Branko. Convex polytopes. Second edition, prepared by Volker
Kaibel, Victor Klee
and Günter M. Ziegler. Graduate Texts in Mathematics, 221. Springer-Verlag,
New York, 2003.
Ziegler, Günter M. Lectures on polytopes. Graduate Texts in Mathematics,
152. Springer-Verlag, New York, 1995.
Dates :
Tuesday, March 13, from 2 p.m. to 5 p.m.
Tuesday, March 20, from 2 p.m. to 5 p.m.
Tuesday, March 27, from 2 p.m. to 5 p.m.
Tuesday April 17, from 2 p.m. to 5 p.m.
*Friday* April 27, from *1* p.m. to 4 p.m.
Room 2.NO.906, on the 9th floor of the Mathematics-Physics Building, Campus
de la Plaine, ULB
People who want to attend are invited to send an e-mail to: doignon@ulb.ac.be
This course will take place at ULB, Brussel.
Registration
You can register electronically by filling in the following form via the web
:
http://www.inma.ucl.ac.be/graduate/graduate_registration.html
If you have problems with this, please contact Lydia DE BOECK
The admission is free for doctoral students and participants from Belgian academic
institutions. Other participants are requested to pay a registration fee of
EURO 500,- per course but a waiver can be obtained under special conditions
(contact the secretariat).
Payment can be made by bank transfer to the account n°310-0959001-48 with
the mention "AUTO2866 ACTIVITES DIDACTIQUES"
Secretariat