Subject description - BE3M35ORR

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BE3M35ORR Optimal and Robust Control
Roles:PV Extent of teaching:2P+2C
Department:13135 Language of teaching:EN
Guarantors:Hurák Z. Completion:Z,ZK
Lecturers:Hurák Z. Credits:6
Tutors:Gurtner M., Hurák Z. Semester:L

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This advanced course will be focused on design methods for optimal and robust control. Major emphasis will be put on practical computational skills and realistically complex problem assignments.

Study targets:

Design advanced feedback controllers for realistically complex systems, while using existing specialized software.


The unifying concept is that of minimization of some optimization criterion. The properties of the resulting controller depend upon which criterion is minimized. Minimizing the popular integral-of-square-of criterion seeks a trade-off between a regulation error and a control effort. The modern theory introduces the concept of a system norm. Minimizing the H2 norm generalizes the classical LQ/LQG control. Minimizing the Hinf norm gives a controller which is robust (insensitive) to inaccuracies in the mathematical model of the system. The mu-synthesis is then an extension of Hinf methodology for systems with structured uncertainty. Hence robust control can be viewed as an offspring of the powerful paradigm of optimal control. The presented optimization-based control design can be solved either offline, or online. In the latter case the optimization can be done by invoking some nonlinear programming solver in every sampling period. This is the essence of model predictive control, which will be briefly introduced in this course. Also included in this course will be methods for time optimal and suboptimal control, which have already been found useful in applications with stringent timing requirements. In addition, semidefinite optimization and linear matrix inequalities will be introduced as these constitute a very flexible framework both for analysis and for numerical computation in robust control. Finally, computational methods for reduction of model and controller order will be covered in the course.

Course outlines:

1. Motivation for optimal and robust control; Introduction to optimization: optimization without and with constraints of equality and inequality types (Lagrange multipliers, KKT conditions)
2. Intro to algorithms for numerical optimization: steepest descent, Newton, quasi-Newton, projected gradient, ...
3. Optimal control for a discrete-time LTI systems – direct approach: discrete-time LQ-optimal control on a finite time horizon, receding horizon control (aka model predictive control, MPC).
4. Optimal control for a discrete-time LTI system – indirect approach: LQ-optimal control, finite and infinite-time horizons, discrete-time algebraic Riccati equation (DARE).
5. Dynamic programing in discrete and continuous time: Bellmans principle of optimality, HJB equation, application to derivation of LQ-optimal control problem.
6. Optimal control for a continuous-time system – indirect approach: introduction to calculus of variations, differential Riccati equations, continuous-time LQ-optimal control (regulation and tracking).
7. Optimal control for a continuous-time system with free final time and constraints on the control variable: Pontryagin's principle of maximum, time-optimal control.
8. Numerical methods for optimal control for continuous-time systems: direct and indirect, shooting, multiple shooting, collocation.
9. Some extensions of LQ-optimal control: LQG-optimal control (augmentation of an LQ-optimal state feedback with Kalman filter); robustification of an LQG controller using an LTR method; H2 optimal control as a generalization of LQ/LQG-optimal control.
10. (Models of) uncertainty and robustness; analysis of robust stability and robust performance.
11. Design of a robust controller by minimizing the Hinf norm of the system: mixed sensitivity minimization, general Hinf optimal control problem, robust Hinf loopshaping, mu-synthesis.
12. Analysis of achievable control performance.
13. Reduction of the order of the system and the controller.
14. Semidefinite programming and linear matrix inequalities in control design.

Exercises outline:

Some exercises (mainly those at the beginning of the semester) will be dedicated to solving some computational problems together with the instructor and other students. In the second half (or so) of the semester, exercises will also be used by the students to work on the assigned (laboratory) projects.


Compulsory • Sigurd Skogestad a Ian Postlethwaite. Multivariable Feedback Control – Analysis and Design. 2nd ed., Wiley, 2005. Some 15 copies reserved for students of this course in the university library. • For topics not covered in Skogestad's book, lecture notes have been created by the lecturer and made available to the students through the course Moodle page. In addition, some other resources will be referenced/linked when needed such as papers, online texts. Majority of topics/lectures are prepared in the form of videos uploaded on Youtube (AA4CC channel, Optimal and robust control playlist). Recommended • Kirk, Donald E. 2004. Optimal Control Theory: An Introduction. Dover Publications. Available online through the university library. But also affordable in print. • Gros, Sébastien, a Moritz Diehl. 2020. Numerical Optimal Control. Draft. KU Leuven. Freely available online at • Rawlings, James B., David Q. Mayne, a Moritz M. Diehl. 2017. Model Predictive Control: Theory, Computation, and Design. 2nd ed. Madison, Wisconsin: Nob Hill Publishing, LLC. Freely available at • Anderson, Brian D. O., a John B. Moore. 2007. Optimal Control: Linear Quadratic Methods. Dover Publications. 10 copies in the library. • Borrelli, Francesco, Alberto Bemporad, a Manfred Morari. 2017. Predictive Control for Linear and Hybrid Systems. Cambridge, New York: Cambridge University Press. The authors made an electronic version freely available at


(Informally recommended) prerequisites for successful passing of this course is a good background in the following areas and topics:
1.) basics of dynamic systems and feedback control: feedback control, stability, magnitude and phase margins, PID control, frequency methods for control design.
2.) linear (matrix) algebra: linear equations and their numerical solution using LU, Cholesky and QR matrix decompositions, eigenvalues, eigenvectors, positive (semi)definite matrix, singular value decomposition, conditioning of a matrix.
3.) complex functions of complex variables: analytic function, z-transform and Laplace transform and their regions of convergence, Fourier transform.
4.) random processes: random process, white noise, correlation, covariance, (auto)correlation function, spectral density.


Taught together with the "Czech" B3M35ORR course. In English only .


Optimization, Robustness, Optimal control, LQ-optimal control, LQG-optimal control, Model predictive control, Robust control, Dynamic programming, Calculus of Variations, Model order reduction, Semidefinite programming, Linear matrix inequalities.

Subject is included into these academic programs:

Program Branch Role Recommended semester
MEKYR_2021 Common courses PV 2

Page updated 16.7.2024 17:51:31, semester: Z,L/2023-4, Z/2024-5, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)