PRESENTATION
In coincidence with the workshop MISC'99 to be held on 24-26 February 1999 in Girona, a pre-workshop tutorial course "Introduction to interval arithmetic and its applications" will be organized by the University of Girona. The aim of the course is to bring together researchers and doctorate students that work in systems and control engineering so as to give them the opportunity to have a knowledge of the method of intervals and to stimulate discussion of relevant problems and fertile lines of future research.
Three prominent speakers, Professor Jürgen Garloff, Professor Vladik Kreinovich and Professor Bob Barmish, will provide an interesting and comprehensive state-of-the-art tutorial on motivations, basic theories and applications of interval analysis to systems and control. The past two decades have witnessed steadily increasing recognition and appreciation of Interval Analysis in solving problems in systems and control engineering. For instance, problems like analysis and synthesis of robust controllers for uncertain plants or fuzzy interference have been stated from the interval point of view. Recent developments have generated a great deal of interests in emerging tools aimed at exploiting qualitative, semiqualitative and interval simulation and their application to fault detection and diagnosis and to system identification.

SCHEDULE
Monday afternoon 22 February 1999 15:30 - 19:00
Tuesday morning 23 February 1999 09:00 - 11:00 11:30 - 13:30
Tuesday afternoon 23 February 1999 15:30 - 19:00

PARTICIPATION FEE
The participation fee will be 15000 Spanish pesetas. Advanced registration is highly recommended (Registration form: PostScript, PDF)

CONTACT ADDRESS
Dr. Josep Vehí
Department of Electronics, Automatic Control and Computer Engineering
University of Girona, Campus Montilivi, Edifici P-II
E-17071 Girona, SPAIN
misc99@eia.udg.es

TOPICS OF THE COURSE:

INTRODUCTION TO INTERVAL COMPUTATIONS

1. Why intervals and why interval computations?
- Inexact knowledge of physical and technical constants; rounding errors; approximation formulas of function,
  derivative and integral. Sensitivity analysis and tolerance problem
2. Basic properties of interval arithmetic
3. Rounded interval arithmetic
4. Enclosures for the range of a function
- Naive interval computations; mean value form and other centered forms
5. Solution of systems of linear interval equations
- Linear interval equations; M-matrices and inverse positive matrices; preconditioning; interval Gauss elimination;
  interval Gauss-Seidel iteration .
6. Solution of systems of nonlinear interval equations
- Branch-and-bound methods (methods which do not use derivatives). Newton-type methods. Automatic
  differentiation; forward and backward. Optimal bisection. Constraint propagation approach
7. Optimization
- Branch-and-bound methods (methods which do not use derivatives). Newton-type methods. Optimization of
  non-differentiable functions (method of generalized derivatives)

APPLICATIONS OF INTERVALS

1. Error estimation for data processing - the basic application of interval computations:
- why data processing (indirect measurements); why it is important to estimate errors of data processing
- two opposite situations when we do not know probabilities, only intervals: cutting-edge fundamental science
  super-precise measurements); manufacturing applications (crude measurements)
- linearization and generalized (affine) interval computations
- cases when interval computations are necessary
- in general, interval computations are intractable; ways out: overestimation, Monte-Carlo methods
2. Additional problems:
- transforming an input-output relation into an explicit algorithm; experimental determination of the input-output
  relation (system identification); changing the world: synthesis, design, control, and decision making
3. Transforming an input-output relation into an explicit algorithm: real-life applications of solving systems of equations:
- systems of algebraic equations. Case study: computer graphics
- systems of differential equations. Case study: particle accelerators
- systems of integral equations. Case study: astrophysics
4. Interval-valued system identification:
- formulation of the problem. Typical situation: systems of linear interval equations: Case study: non-destructive
  evaluation of aerospace constructions
5. Synthesis, design, control, and decision making: real-life applications of interval optimization
- problems with finitely many alternatives: how to compare intervals
- problems in which an alternative can be characterized by finitely many real numbers: interval optimization
- case study: chemical spectral analysis; image processing in radioastronomy; detection of elementary particles
- problems in which alternatives can be characterized by functions, e.g., in control: economical planning with
  Leontieff's input-output model award-winning mobile robot; telemanipulators (MIT/Utah robotic arm)
- general introduction to robust control: Kharitonov's theorem; its applications
6. Beyond interval computations:
- intervals + probabilities
- intervals + expert knowledge
- first step: nested intervals and fuzzy logic
- second step: interval-valued degrees of belief
- case study: using expert knowledge to control a mobile robot
7. Probabilistic robustness with intervals