# 769I: Topics in Robust Control

## Course Goals:

Mathematical models of systems never exactly describe the physical systems they represent. While the control systems designer works with models, it is paramount that the controllers produced work well on the actual physical systems. One paradigm to achieve this is known as robust control. As a field of control theory, it is mathematically sophisticated. While some seminal work was done by pioneers in the 60s and 70s, most of the theory was developed in the past two decades. This course gives but a glimpse at some aspects of this theory, with emphasis on issues related to the ``structured singular value''.

## Course Prerequisite(s):

ENEE 663, or equivalent.

## Topics Prerequisite(s):

This course assumes mastery of the basic tools of linear algebra, and an understanding of multivariable system theory both in frequency domain and in state space.

## Textbook(s)

No textbook will be used. Notes will be distributed. A good reference for much of the course is:
• K. Zhou and J.C. Doyle, Essentials of Robust Control, Prentice Hall, 1998.
• ## Reference(s):

• K. Zhou, J.C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, 1995.
• B.R. Barmish, New Tools for Robustness of Linear Systems, MacMillan, 1994.
• J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, MacMillan, 1992.
• M. Green and D.J.N. Limebeer, Linear Robust Control, Prentice Hall, 1995.
• ## Core Topics:

1. Preliminaries
2. Parametric Families of Polynomials or Matrices
• Value set
• Kharithonov's theorem
• Polytopes of polynomials: Edge Theorem
• Rantzer's growth condition
• Guardian maps
3. Stability and Performance of Feedback Systems
• H2 and H-infinity
• Computing H2 and H-infinity norms
• Well-posedness and internal stability
• Weighted H2 and H-infinity performance
4. Linear Matrix Inequalities
• Definition and numerical solution
• Elementary applications in robust control
5. Model Uncertainty and Robustness
• Small Gain Theorem
• Robust stability
• Sufficient conditions for robust performance
6. Structured Robust Stability and Performance
• Linear fractional transformations
• Structured singular value: definition and bounds
• Computing the ``mu-norm''
• Structured robust stability and performance analysis: Small mu Theorem
• Glance at H-infinity synthesis and mu synthesis
7. Robustness Under Mixed Dynamic and Parametric Uncertainty
• ``Real mu'' and ``mixed mu'': Definition and elementary properties
• Necessary and sufficient condition: Small mu Theorem
• Convex upper bound: linear matrix inequality
• Connection with passivity theorem and absolute stability.

Lectures

## Grading Method:

### Last Updated:

29 November 1998, Andre Tits

khodary@eng.umd.edu