# Introduction to Time-Delay Systems

### Lecturer: Leonid Mirkin

### Syllabus

The course is about system-theoretic and control aspects of the effect of time delays on linear control systems. The following subjects will be covered:

- mathematical models of time-delay systems
- stability and roots of characteristic quasi-polynomials
- stability analysis (delay sweeping, Lyapunov extensions)
- stabilization methods and dead-time compensation (Smith predictor and its modifications, finite-spectrum assignment)
- implementation of dead-time compensators
- robustness to uncertainty in time delays
- optimal control: some problem-oriented ideas (if time permits)

### Formalities

The course is open for all PhD students and gives **7.5 ECTS credits**. Knowledge of basic linear finite-dimensional systems theory (transfer functions, frequency response, convolution, state-space realizations, stability, system norms, LQG/H2, etcetera) is expected.

**Grading:** based entirely on homework solutions. There will be 6 (or, maybe, 5) homework assignments, graded on a scale of 0 to 100. To pass the course, the grade of each assignment must be at least e^{π} (implying that the submission is compulsory) and the average grade of 5 (or 4) best solutions must be at least 67. Solutions should be submitted electronically in PDF (*typeset*, not a scan of a handwritten text) to
.

**Grades** can be seen here.

### Schedule

The classes will be held on Wednesdays in Seminarierummet (M2498), starting **October 24**. Tutorial sessions are scheduled to the 10:15-12:00 slot, lectures—to the 15:15-17:00 slot. During the first week, both slots will be filled with lectures.

**Special schedules**: the 15:15-17:00 classes of November 28 and December 5 are moved to the 8:15-10:00 schedule. On November 28 the location is still Seminarierummet, whereas all lectures of December 5 (including 10:15-12:00) will be held in Konferensrummet, 1167B.

### Lectures

- Introduction; time delays in frequency domain; rational approximations; state space; modal properties
- Stability analysis: Nyquist criterion, delay sweeping; Lyapunov’s methods
- Problem-oriented stabilization methods: Smith controller, finite spectrum assignment
- Dead-time compensation: loop shifting, Youla parametrisation, etcetera
- Implementation of dead-time compensators
- Topics in optimal control of systems with delays and preview
- Robustness to delay uncertainties

### Homework

- Assignment 1, submission deadline: Oct 31, 10:00am
- Assignment 2, submission deadline: Nov 7, 10:00am
- Assignment 3, submission deadline: Nov 14, 10:00am
- Assignment 4, submission deadline: Nov 28, 8:00am
- Assignment 5, submission deadline: Dec 5, 8:00am
- Assignment 6, submission deadline: Dec 21, 10:00am