Optimal and Robust Control

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Code Completion Credits Range Language
BE3M35ORR Z,ZK 6 2P+2C English
Zdeněk Hurák (guarantor)
Zdeněk Hurák (guarantor), Martin Gurtner
Department of Control Engineering

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.


(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.

Syllabus of lectures:

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.

Syllabus of tutorials:

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.

Study Objective:

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

Study materials:


• 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).


• 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 https://www.syscop.de/teaching/ss2017/numerical-optimal-control.

• 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 http://www.nobhillpublishing.com/mpc-paperback/index-mpc.html.

• 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 http://cse.lab.imtlucca.it/~bemporad/publications/papers/BBMbook.pdf.

Further information:
Course webpage is on https://moodle.fel.cvut.cz/course/BE3M35ORR
Time-table for winter semester 2021/2022:
Time-table is not available yet
Time-table for summer semester 2021/2022:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2022-08-09
For updated information see http://bilakniha.cvut.cz/en/predmet4679706.html