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Ecole nationale supérieure de l'Énergie, l'Eau et l'Environnement
Engineering school in energy, water and environment
Our engineering & master degrees
Our engineering & master degrees

> Studies > E3-STU-COURSES

Non Linear and Robust Controls - 5EUS5CNL

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  • Number of hours

    • Lectures : 40.0
    • Tutorials : ?
    • Laboratory works : 20.0
    • Projects : ?
    • Internship : ?
    ECTS : 5.0


Ability for design and analysis of Hinfinity controlllers, robustness analysis, and the limits of the linearization, the analytical tools for nonlinear stability, and the basic principles of feedback control nonlinear state.

Contact Olivier SENAME



Industrial examples (automotive and electromechanical applications).
1 Tools
Hinf norm: how to dene the gain of a MIMO system ?
Singular values of a transfer matrix, introduction to H2 and H1 norms.
Example of a mass/srping/damper system.
Internal stability: Notion of well-posedness, Small Gain theorem
2 Performance analysis
Denition of the sensitivity functions
frequency-domain performance indices (sensitivity functions, stability and robustness margins, bandwidth, SISO and MIMO cases)
3 Hinf control design
Performance Specications: selection of weighting functions.
Loop-shaping Mixed sensitivity problem.
Solving the Hinf control problem: Obtaining the General control conguration
Hinf controller structure (state feedback, dynamic output feedback)
Problem solution using Riccati equations or LMIs -Bounded Real Lemma)
Illustrative examples
4 Uncertainty and robustness
Representing uncertainties: unmodelled dynamics, frequency forms, unstructured uncertainties
Parametric uncertainties, LFT forms, structured uncertainties
Robust stability analysis: M structure, small gain theorem
Robust stability for unstructured uncertainties.
Robust performance analysis: A simplied Hinf criterion,
Introduction to mu-analysis - structured uncertainties

5 Introduction to LMIs
What is an Linear Matrix Inequality ? Brief optimisa-
tion background, Denition
Stability issue: From Lyapunov equation to LMIs
Control design: problem formulation: Example on
State feedback
6 Short introduction to LPV systems
Denition of Linear Parameter Varying systems, stability issue, control design


Linear Systems, Transfer and state space approach, frequency and time-domain analysis


Semester 9 - The exam may be taken in french or in english FR EN

Project reports
Individual evaluation in Matlab tutorials

CC 50% + CT 50%

Additional Information

Semester 9 - This course is given in english only EN

Curriculum->Master's Degree in Engineering ASI->Semester 9


1. D. Alazard, C. Cumer, P. Apkarian, M. Gauvrit, and G. Ferreres. Robustesse et commande optimale. Cpadues Editions, 1999.
2. J.C. Doyle, B.A. Francis, and A.R. Tannenbaum. Feedback control theory. Macmillan Publishing Company, New York, 1992.
3. G. Duc and S. Font. Commande H1 et -analyse: des outils pour la robustesse. Herms, France, 1999.
4. G.C. Goodwin, S.F. Graebe, and M.E. Salgado. Control System Design. Prentice Hall, New Jersey, 2001.
5. Scherer, C. and Wieland, S. (2004). Linear Matrix inequalities in Control. lecture support, DELFT University.
6. S. Skogestad and I. Postlethwaite. Multivariable Feedback Control: analysis and design. John Wiley and Sons, 2005.
7. K. Zhou. Essentials of Robust Control. Prentice Hall, New Jersey, 1998.
8. A. Isidori, Nonlinear control systems, 3rd Ed., Springer, 1995.
9. H. Khalil, Nonlinear systems, 3rd Ed., Prentice Hall, 2002.

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Date of update February 8, 2017

Grenoble INP Institut d'ingénierie Univ. Grenoble Alpes