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Robust Control

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  • Lecturer: Mokrane Boudaoud (mokrane.boudaoud@sorbonne-universite.fr)
  • Course code: MU5RBR30-ROB
  • Student workload: 12 h of lectures, 8 h of tutorials, 8 h of labs
  • Credits: 3 ECTS
  • Specialization tracks: ROBOT (elective) SI MeDH
  • Semester offered: S1 S2 S3 S4
  • Language of instruction: French English
  • Targeted audience: Eng. Sc. department Other : ENSAM
  • Localization : PMC Campus Other :

Course Overview

In control theory, robust-control methodologies aim to design a controller that guarantees robust stability and performance when the controlled process has nonlinear dynamics or is subject to parameter variations or external disturbances. This question is often formulated as an optimization problem. The objective of this course unit is to provide an overview of robust-control methodologies for linear systems with parametric uncertainty and disturbances, using the H∞ framework.

Mots-clés : System interconnections, closed-loop systems, H∞ norm, singular values, modeling of uncertain systems, small-gain theorem, standard H∞ problem, loop shaping, weighting functions.

Prerequisites

Students should have previously acquired the following prerequisites to follow this course:

  • Classical control theory and state-space representation

Intended Learning Outcomes

By the end of this course, students will be able to:

  1. Calculate the singular values of a transfer matrix.
  2. Calculate the norms of a vector, matrix, signal and system.
  3. Understand and apply the small-gain theorem.
  4. Model a closed-loop system with disturbance inputs.
  5. Understand the principle of frequency separation for closed-loop disturbance rejection.
  6. Model uncertainties.
  7. Analyze the stability of uncertain linear systems.
  8. Define the standard H∞ problem from a set of specifications.
  9. Model an augmented system with weighting functions.
  10. Define and calculate weighting functions.
  11. Calculate a standard H∞ controller.

Indicative Teaching Sequence and Methods

Week C/TD/TP* Content Preparation Learn.\ outc.
S1 C1 Part 1: Introduction to robust control 1.1 - Modern robust-control theory: an introduction 1.2 - Fundamentals of robust control 1.2.1 - Modeling and analysis of linear systems (Laplace transform, transfer function, state-space representation, controllability, observability) 1.2.2 - System interconnections (sum, product, linear fractional transformation) 1.2.3 - Closed-loop system configurations (feedback loops, pre-compensator, post-compensator, etc.) 1.2.4 - Characteristic functions (open-loop transfer functions, sensitivity function, complementary sensitivity function, etc.)
S2 C2 Part 2: Mathematical tools 2.1 - Norms of matrices, signals and systems 2.2 - Singular values 2.3 - Calculation of the H∞ norm 2.4 - Linear matrix inequalities AAV1, AAV2
S2 TD1 Exercises on C1 and C2
S3 C3 Part 3: Modeling disturbances and uncertainties 3.1 - Disturbance modeling and frequency-separation concepts for disturbance rejection 3.2 - Uncertainty modeling (additive representation, multiplicative representation, etc.) 3.3 - Robust stability (small-gain theorem) 3.4 - Passivity AAV3, AAV4, AAV5, AAV6
S3 TD2 Exercises on C3
S4 C4 Part 4: H∞ control 4.1 - Standard H∞ problem 4.2 - General synthesis and augmented scheme 4.3 - Solving the H∞ problem AAV7, AAV8, AAV9
S4 TD3 Exercises on C4
S5 C5 Part 5: Loop shaping and H∞ controller synthesis. 5.1 - Loop shaping 5.2 - Selection of weighting functions 5.3 - Example: control of a second-order system. AAV10, AAV11
S5 TD4 Exercises on C5
S6 C6 Part 6: LMI techniques for H∞ synthesis. 6.1 - LMI techniques for H∞ synthesis. 6.2 - Application example
S7 TP_part 1 AAV1-AAV11
S8 TP_part 2 AAV1-AAV11
  • C/TD/TP respectively corresponds to lectures, tutorials and lab sessions.

Indicative Assessment of Intended Learning Outcomes (1st session)

Week Individ./group In-person/remote Type of exam Evaluated outcomes Scale %
S9 Individual In-person Written exam AAV1-AAV11 70%
S10 Group In-person TP AAV1-AAV11 30%

2nde session

Session Individ./group In-person/remote Type of exam Evaluated outcomes Scale %

Bibliographic references

    1. G. Duc and S. Font, Commande H∞ et μ-analyse, Hermes Science Publications, Paris, 1999.
    1. Doyle john, bruce francis and allen tannenbaum. feedback control theory. ISBN-10 : 0023300116, 1991.
    1. K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, 1996.

Logo SDI Date of generation of this unit description: 06/10/2026 Logo SDI