Introduction to Time-Delay Systems

Lecturer: Leonid Mirkin


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)


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 eπ (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.


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.


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


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