Sharif University of Technology

Department of Electrical Engineering

Parametric Robust Control

Spring 2011 (1389-90)

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Course Information

Course Description:

This is a graduate course for students who passed Linear Control and Modern Control Systems. The outline of the course is:
  1. A Global Overview: robustness problems, historical perspective, Kharitonov's THM, uncertainty structure, mathematical programming approach, the value set concept, family paradigm, robustness analysis, robustness margin, robust synthesis, conclusion.
  2. Notation for Uncertain Systems: notations for uncertain parameters, uncertainty bounding sets and norms, notation for families, real versus complex.
  3. Case Study: The Fiat Engine, and Some Microgrid Examples: modeling, control, transfer function matrix for the engine, uncertain parameters in the model, closed-loop polynomial
  4. Robust Stability with a Single Parameter: stability and robust stability, basic definitions and examples, generalized root locus, invariant degree concept, eigenvalue criteria for robustness, introduction to robust D-stability
  5. Kharitonov's Theorem: interval polynomial family, Kharitonov polynomials, Kharitonov's theorem, graphical interpretation, overbounding via interval polynomials, extensions with degree dropping, frequency sweeping function, Kharitonov's THM with complex coeeficients
  6. The Value Set Concept: the value set, the zero exclusion condition, zero exclusion for robust D-stability, boundary sweeping functions
  7. Polytopes of Polynomials: affine linear uncertainty structure, introduction to polytopes of polynomials, polygonal value sets.
  8. The Edge Theorem: lack of extreme points results for polytopes, the edge THM, Fiat Example.
  9. Distinguished Edges: parallelotopes, parpolygons, setup with an interval plant, the Thirty-Two Edge THM.
  10. The Sixteen Plant Theorem: Sixteen distinguished plants, the Sixteen Plant THM, Controller synthesis technique.
  11. Multilinear Uncertainty Structure: multilinear and polynomic uncertainty, interval matrix family, lack of extreme point and edge results, the mapping THM, geometric interpretation, value set interpretation
  12. Spherical Polynomial Families: boxes versus spheres, Soh-Berger-Dabke THM, value set for spherical polynomial family, overbounding via spherical family, spherical set, Barmish and Tempo THM, affine linear uncertainty structures, testing function for robust stability.

Text Book:

B. R. Barmish, New Tools for Robustness of Linear Systems, MacMillan, New York, 1994.


  1. S. P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control - The Parametric Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1995.
  2. S. P. Bhattacharyya, A. Datta, L. H. Keel, Linear Control Theory: Structure, Robustness, and Optimization, CRC Press, Taylor & Francis Group, 1 Edition, January 30, 2009.