M23 – PARIS‐SACLAY 09/05/2016 – 13/05/2016

Time-delay, sampled-data and PDE systems

Emilia Fridman
School of Electrical Engineering,
Tel Aviv University, Israel emilia/


Time-delay appears naturally in many control systems. It is frequently a source of instability although, in some systems, it may have a stabilizing effect. A time-delay approach to sampled-data control, which models the closed-loop system as continuous-time with delayed input/output, has become popular in networked control systems (where the plant and the controller exchange data via communication network). The beginning of the 21st century can be characterized as the "time-delay boom" leading to numerous important results. The emphasis of the course is on the Lyapunov-based analysis and design for time-delay, sampled-data and networked control systems. An LMI approach to some classes of PDEs will be discussed.

The course is designed around the text book:
E. Fridman, Introduction to Time-Delay Systems: Analysis and Control. Birkhauser, 2014.


Models of systems with time-delay: sampled-data and networked-control systems, traffic flow and drilling pipe models.
2. Basic theory. Classification of time-delay systems, existence of solutions. Solution of linear non-homogenous equations. Controllability, observability of linear systems.
3. Stability and performance analysis. Direct Lyapunov approach: Krasovskii and Razumikhim methods. An LMI approach to stability and performance. The small gain theorem approach.
4. Control design: predictor-based control, LQR problem. LMI approach to robust stabilization and H1 control. Systems with saturated actuators.
5. Discrete-time delay systems.
6. Sampled-data and networked control systems: a time-delay approach.
7. PDEs: an LMI approach to analysis and design.