Optimality and Robustness


Recommended prerequisite for participation in the module

The module builds on knowledge obtained through the course Multivariable Control.

Content, progress and pedagogy of the module

The aim of this module is to obtain qualifications in formulation and solution of control problems where the objective can be formulated as an optimization problem in which the trajectories of inputs, state variables and outputs are included in an objective function and can be constrained. The formulation will include a model which describes the dynamic behavior of the physical plant with given control inputs and disturbances. Models describing disturbances and references can be included to describe predictive problems. A further aim is to provide methods to analyze robustness of closed loop stability and performance when discrepancy between the physical plant and the model is bounded by specified uncertainty bounds and to study dimensioning methods, which aim to ensure robustness of stability and performance given specified uncertainty bounds.

Learning objectives


  • Must have an understanding of basic concepts within optimal control, such as linear models, quadratic performance, dynamic programming, Riccati equations etc.
  • Must have an understanding of the use of observers to estimate states in a linear dynamical system
  • Must have insight into the stability properties of optimal controllers
  • Must have insight into the stability properties of finite horizon control, and how to ensure stability
  • Must have knowledge about performance specifications that are not quadratic
  • Must have knowledge of additive and multiplicative model uncertainty
  • Must have insight into the small gain theorem and its applications in robust control
  • Must have insight into robust stability and robust performance


  • Must be able to formulate linear control problems using models of disturbances and references combined with a quadratic performance function and solve them using appropriate software tools, e.g. Matlab
  • Must be able to introduce integral states in control laws to eliminate steady state errors
  • Must be able to design observers while taking closed-loop stability into account
  • Must be able to utilize quadratic programming to solve predictive control problems with constraints.
  • Must be able to use software tools such as Matlab to solve constrained optimization problems
  • Must be able to formulate the standard robustness problem as a two-input-two-output problem and solve it using appropriate methods
  • Must be able to assess the limitations model uncertainty sets impose on the achievable performance for systems described by linear models
  • Must be able to use singular value plots and the H infinity norm of appropriate transfer function to assess robustness
  • Must be able to perform H infinity norm optimization as a method to tune controllers


  • Must be able to formulate and solve optimal control problems with references and disturbances
  • Must understand the implications of disturbances and uncertainties in the context of linear dynamical systems, and be able to address these via robust control design

Type of instruction

As described in ยง 17.



Name of examOptimality and Robustness
Type of exam
Written or oral exam
AssessmentPassed/Not Passed
Type of gradingInternal examination
Criteria of assessmentThe criteria of assessment are stated in the Examination Policies and Procedures

Facts about the module

Danish titleOptimal- og robust regulering
Module codeESNCAK2K2
Module typeCourse
Duration1 semester
Language of instructionEnglish
Empty-place SchemeYes
Location of the lectureCampus Aalborg
Responsible for the module


Education ownerMaster of Science (MSc) in Engineering (Control and Automation)
Study BoardStudy Board of Electronics and IT
DepartmentDepartment of Electronic Systems
FacultyThe Technical Faculty of IT and Design