Course detail

Robust and algebraic control

FEKT-MPA-RALAcad. year: 2023/2024

The course is focused on application of algebraic theory for control circuit’s synthesis. It consists of algebraic theory, the controller designs using polynomial methods, structured and unstructured uncertainties of dynamic systems and introduction into robust control.

Language of instruction


Number of ECTS credits


Mode of study

Not applicable.

Offered to foreign students

Of all faculties

Entry knowledge

The subject knowledge on the Bachelor´s degree level is requested.

Rules for evaluation and completion of the course

Exercises. Individual project. Max. 30 points.
Exam. Max 70 points.
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.


To introduce to the students universal tool for solving tasks of automatic control and to became familiar with robust control.
After passing the course, student should be able to
- solve algebraic equations and understand algebraic theory
- utilize basic algebraic methods for controller designs
- explain the relationship between sensitivity function and modulus stability margin
- describe the possibilities of sensitivity function shaping and use them for robust controller design
- determine stability of interval polynomials
- utilize parametric and non-parametric uncertainties in the environment of MATLAB Simulink
- design the controller using H infinity mixed sensitivity method

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Havlena, V., Štecha, J.: Moderní teorie řízení, Skriptum ČVUT, Praha 2000 (EN)
Doyle, Francis, Tannenbaum: Feedback Control Theory, Macmillan Publishing (EN)
Scherer, Weiland: Linear matrix inequalities in control. DISC, 2000 (EN)

Recommended reading

Not applicable.


Classification of course in study plans

  • Programme MPA-CAN Master's, any year of study, summer semester, elective
  • Programme MPAD-CAN Master's, any year of study, summer semester, elective
  • Programme MPC-KAM Master's, any year of study, summer semester, compulsory-optional

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Introduction into problematic.
2. Algebraic theory. Solution of polynomial equation, general solution, special solutions, solvability condition.
3. Application of algebraic methods to simple controller designs. Pole placement method, exact model matching problem, the group of stabilizing controllers.
4. Sensitivity function shaping design. Sensitivity function and modulus margin, sensitivity function template, additional polynomials in controller and in its design.
5. Time optimal discrete control. Feedforward control,
6. Quadratically optimal discrete control, 1DOF, 2DOF, finite and stable time optimum control with nonzero initial conditions.
7. Stochastic control. Minimum variance control, the evaluation of MVC controllers, generalized minimum variance control.
8. Interval polynomials. Zero exclusion principle, value sets, Mikhailov-Leonard stability criteria, Kharitonov polynomials.
9. Introduction to robust control. Robustness, signal and system norms.
10. H infinity control. Linear fractional transformation, mixed sensitivity design, gamma iteration.
11. H2 control, comparison with LQ control, H2 optimal state controller design, H2 optimal state observer design, duality of both approaches.
12. Uncertainties description. Classification of uncertainties, affine/polytopic uncertainties, gain scheduling controller design. Additive, multiplicative and feedback uncertainties. Small gain theorem, D-K iteration.
13. Linear Matrix Inequalities (LMI), quadratic form LJ and its conversion to LMI, LQG using LMI, H infinity using LMI.

Fundamentals seminar

14 hours, compulsory

Teacher / Lecturer


7. The work on the project. Design of the controller using sensitivity function shaping.
8. Computation with parametric uncertainties. Interval uncertainties. Conversion to structured uncertainties.
9. H infinity controller design. Loop shaping design method.
10. H infinity controller design. Mixed sensitivity design.
11. H infinity controller design , inverted pendulum example.
12. Robust controller design for MIMO systems.
13. Reserve, conclusions.

Exercise in computer lab

12 hours, compulsory

Teacher / Lecturer


1. Becoming familiar with functions from Symbolic Math Toolbox in MATLAB.
2. Basic notations in algebraic methods.
3. Preparation of function for the computation of the general and particular solution of polynomial equation.
4. Stabilizing controller design, modal controller, EMMP problem solution.
5. Design of time optimal controllers for one degree of freedom control structure.
6. Design of time optimal controllers for two degrees of freedom control structure.