Optimization in Control, Systems Analysis and Identification

Dr.-Ing. Stefan Streif
E-Mail: Stefan.Streif@nospam@@nospamovgu.de

Dipl.-Ing. Philipp Rumschinski
E-Mail: Philipp.Rumschinski@nospam@@nospamovgu.de

Optimization is an important and diverse tool in many research and application fields spanning from economics, logistics to control and systems theory. Since the mid 1990s it was realized that many optimization problems are instances of so-called LMIs (Linear Matrix Inequalities). These instances remain an active research area in particular in the control, systems analysis and identification community to this day. Many applications can be written as feasibility or optimization problems over LMIs or BMIs (Bilinear Matrix Inequalities). Applications include stability analysis of linear and nonlinear systems, nonlinear systems identification, design of experiments, robust controller design, model predictive control and fault diagnosis.

The main goals of this course are to introduce the participants to the possibilities of LMIs and to give them a sound foundation for further studies in more recent research areas such as relaxation based methods.

The first part of the course deals with the mathematical theory and process control applications of LMIs and BMIs. Software and algorithms for solving LMIs and BMIs are introduced. It is shown that many important process control applications can be rewritten as LMIs and BMIs. In-class exercises and computer exercises are an essential part of the course.

In the second part of the course, particular applications are worked out in mini-projects and/or homework assignments and the results are presented by the students.

At the end of the course, the students will be able to

• Reformulate various analysis and synthesis problems in control, systems analysis and identification in an abstract manner
• Apply and solve own problems using LMIs using state-of-the-art software and algorithms
• Read and understand publications dealing with LMIs and BMIs

Control techniques considered in the course:

• Lyapunov functions to ensure invariance, stability, performance, robust performance.
• Control criteria: Dissipativity, IQCs, H2-norm, H∞-norm, peak-to-peak norm.
• Frequency domain techniques: robustness analysis of a control system, IQCs and multipliers, relations to classical tests and to μ-theory.
• Controller Design: State-feedback and output-feedback synthesis algorithms for robust stability, nominal performance and robust performance, mixed control problems and parametrically-varying systems and control design.

• Control Engineering I (Regelungstechnik)
• Lineare Algebra courses (Mathematik I + II)
• Systems Theory (Systemtheorie)
• Nonlinear Control (Nichtlineare Regelungstechnik)

Important notes:

• Course material will be send out per mail
• Time and date of the course is not fixed yet and will be agreed upon in a poll of all registered participants

Course schedule (subject to changes and polls):

1st day	9am - 12am	Lectures
	1pm - 3pm	Exercises
2nd day	9am - 12am	Lectures
	1pm - 3pm	Exercises
3rd day	9am - 12am	Lectures
	1pm - 3pm	Computer exercises
4th day	9am - 12am	Presentation of mini projects
5th day	9am - 12am	Presentation of mini projects

Part I -- Introduction

• Motivation and overview

Part II -- Linear Matrix Inequalities Feasibility Problems

• Basics (e.g. linear and polynomial inequalities, convexity, ...)
• The Schur Complement (e.g. bounded real lemma, maximum singular value, ...)
• Applications (e.g. stability of linear, nonlinear and time-varying systems)
• Exercises

Part III -- Linear Matrix Inequalities Optimization Problems

• Basics (e.g. semidefinite programming, Farkas' lemma, ...)
• Applications (e.g. parameter estimation, optimal experimental design, robust control)
• Exercises

Part IV -- Algorithms and Software to solve Linear and Bilinear Matrix Inequalities

• Algorithms (elipsoid and interior-point algorithm)
• Software (YALMIP, SEDUMI, and others)
• Computer Exercises

Part V -- Bilinear Matrix Inequalities

• Basics (definitions, non-convex problems, relaxation methods, ...)
• Applications (e.g. set-based estimation)
• Computer Exercises

Part VI -- Mini-Projects and/or Homework Assignments

• Topics include: stability and nominal performance, controller synthesis, robust stability and robust performance (systems with parametric uncertainties), analysis of input-output behavior, and many more!

Course examination depends on the number of participants; however, mini-projects and/or homework assignments will be the considered.

• Linear Matrix Inequalities in Control
  Carsten Scherer and Siep Weiland
  Available online.