Module Description

Module: Optimal and Robust Control


Optimal and Robust ControlLecture2Summer Semester
Optimal and Robust ControlRecitation Section (small)2Summer Semester

Module Responsibility:

Prof. Herbert Werner

Admission Requirements:


Recommended Previous Knowledge:

  • Classical control (frequency response, root locus)
  • State space methods
  • Linear algebra, singular value decomposition

Educational Objectives:

Professional Competence

Theoretical Knowledge
  • Students can explain the significance of the matrix Riccati equation for the solution of LQ problems.
  • They can explain the duality between optimal state feedback and optimal state estimation.
  • They can explain how the H2 and H-infinity norms are used to represent stability and performance constraints.
  • They can explain how an LQG design problem can be formulated as special case of an H2 design problem.
  • They  can explain how model uncertainty can be represented in a way that lends itself to robust controller design
  • They can explain how - based on the small gain theorem - a robust controller can guarantee stability and performance for an uncertain plant.
  • They understand how analysis and synthesis conditions on feedback loops can be represented as linear matrix inequalities.
  • Students are capable of designing and tuning LQG controllers for multivariable plant models.
  • They are capable of representing a H2 or H-infinity design problem in the form of a generalized plant, and of using standard software tools for solving it.
  • They are capable of translating time and frequency domain specifications for control loops into constraints on closed-loop sensitivity functions, and of carrying out a mixed-sensitivity design.
  • They are capable of constructing an LFT uncertainty model for an uncertain system, and of designing a mixed-objective robust controller.
  • They are capable of formulating analysis and synthesis conditions as linear matrix inequalities (LMI), and of using standard LMI-solvers for solving them.
  • They can carry out all of the above using standard software tools (Matlab robust control toolbox).

Personal Competence

Social Competence

Students can work in small groups on specific problems to arrive at joint solutions.


Students are able to find required information in sources provided (lecture notes, literature, software documentation) and use it to solve given problems. 

ECTS-Credit Points Module:



Oral exam

Workload in Hours:

Independent Study Time: 124, Study Time in Lecture: 56

Course: Optimal and Robust Control


Herbert Werner




Summer Semester


  • Optimal regulator problem with finite time horizon, Riccati differential equation
  • Time-varying and steady state solutions, algebraic Riccati equation, Hamiltonian system
  • Kalman’s identity, phase margin of LQR controllers, spectral factorization
  • Optimal state estimation, Kalman filter, LQG control
  • Generalized plant, review of LQG control
  • Signal and system norms, computing H2 and H∞ norms
  • Singular value plots, input and output directions
  • Mixed sensitivity design, H∞ loop shaping, choice of weighting filters
  • Case study: design example flight control
  • Linear matrix inequalities, design specifications as LMI constraints (H2, H∞ and pole region)
  • Controller synthesis by solving LMI problems, multi-objective design
  • Robust control of uncertain systems, small gain theorem, representation of parameter uncertainty


  • Werner, H., Lecture Notes: "Optimale und Robuste Regelung"
  • Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan "Linear Matrix Inequalities in Systems and Control", SIAM, Philadelphia, PA, 1994
  • Skogestad, S. and I. Postlewhaite "Multivariable Feedback Control", John Wiley, Chichester, England, 1996
  • Strang, G. "Linear Algebra and its Applications", Harcourt Brace Jovanovic, Orlando, FA, 1988
  • Zhou, K. and J. Doyle "Essentials of Robust Control", Prentice Hall International, Upper Saddle River, NJ, 1998