5B5766 Robust Control Theory, 6 p, Fall 2003

Instructors

Ulf Jönsson (ulfj@math.kth.se), room 3711, Lindstedtsv. 25, phone. 790 84 50.
Chung-Yao Kao (cykao@math.kth.se), room 3725 Lindstedtsv. 25, phone. 790 6298.

Examiner

Ulf Jönsson (ulfj@math.kth.se), room 3711, Lindstedtsv. 25, phone. 790 84 50.

Course start

The course starts on September 15 at 10.15 in room 3721, Lindstedtsv. 25. There will be one or two lectures each week on

Examination

Course description

This is a course on modern robust control theory. Robust control has been one of the most active areas within the systems and control field since the late 1970s. It deals with uncertain systems. The motivation for using uncertain system models is that most physical systems are too complicated to be exactly represented by a single tractable mathematical model. Instead various types of uncertain linear operators are introduced in the model to capture the effects of nonlinear terms, high order unmodeled dynamics, and time-variations due to changing operating conditions. The uncertain system model thus consists of a nominal linear part and an (often highly structured) uncertainty in feedback interconnection. This can be viewed as a family of systems where each instance of the uncertainty corresponds to one system in the family. The main problem is thus to find analysis and synthesis methods that guarantee stability and satisfactory performance for all systems in the family. Much of the effort in robust control has been focused on finding computationally efficient methods to solve this problem either exactly or otherwise with as little conservatism as possible.

Several significant achievements were made during the the first phase of the robust control era which ended in the late 80s. Some examples are the scaling techniques for analysis of systems with structured uncertainty, a model order reduction technique with a rigorous bound on the approximation error, and several alternative solutions to the $\Hinf$ optimal control problem. These results are now supported by commercial software packages such as Mutools and are routinely used in practical applications.

The development continued in the 90s, now with a strong emphasis on convex optimization and computational complexity issues. A strong driver was the development of efficient interior point algorithms for the solution of Linear Matrix Inequalities (LMI). It was established that the solution to many problems in systems and control can be formulated as LMIs and thus be considered tractable. This started an intensive activity that lead to several significant contributions, such as LMI techniques for $\Hinf$ synthesis, gain scheduling, various multiobjective control problems, and model reduction of uncertain systems.

Another driver was the influence from techniques for quadratic optimization, which was introduced to systems theory by Yakubovich and his colleagues during the 70s. A useful contribution in this direction was the generalization of the S-procedure losslessness result into a robust control setting. This result has been instrumental in showing non-conservatism of certain scaling techniques for robust stability analysis.

We will cover several of the main directions of robust control. The emphasis of the course will be on the latest developments based on convex optimization at the same time as we convey the basic principles at core of the subject.

Course material

Preliminary course content

Last modified: September 12, 2003 by Ulf Jönsson, ulfj@math.kth.se.