Aim: This course covers the so-called Linear-Quadratic-Gaussian (LQG) control problem. With the Riccati equation, the Kalman filter, the separation theorem, and , the tex2html_wrap_inline54 problem as its main building blocks, this theory forms the focal point of the impressive development in control theory since 1960. The course will cover the high-lights only, we will not dwell into details.

Lectures: The lectures are given on Tuesday 21/3, 28/3, 4/4, 11/4, 18/4, 25/4, 9/5, 16/5, 23/5, from 9.15 - 11.00, in room WSN 31, and on Thursday 23/3, 30/3, 6/4, 13/4, 20/4, 27/4, 11/5, 18/5, 25/5, from 11.15 - 12.00, in room RC 0059. Lecturer: J.C. Willems, IWI 323,
email: J.C.Willems@math.rug.nl.

The exercise sessions are given on Wednesday 22/3, 29/3, 5/4, 12/4, 19/4, 26/4, 10/5, 17/5, 24/5, from 9.15 - 12.00, in room WSN 25B, under the direction of Shiva Shankar, IWI 330, email: shankar@math.rug.nl.

Exercise Set 1
Exercise Set 2
Exercise Set 3
Exercise Set 4

Take Home Exam 1
Take Home Exam 2
Take Home Exam 2: Hints and Corrections

Course material: The course will be given from overhead projector sheets. Copies of these sheets will be handed out. Printed sources:

Examination: The course counts for 4 study-points. The course mark will be based on two take-home examinations and one MATLAB simulation set. The first take-home exam is due on May 9. The second take-home exam and the simulation set are both due on June 1. The take-home examinations should be made strictly personal. Violations will not be tolerated! The simulation set may be done in groups of 2. We wish to warn you that the take-home exams will be non-trivial so, please, count on one week on intensive concentration for each of them.


Linear state space systems (review): Controllability. Observability. Stability. Pole placement by state feedback. Observers. Stabilization by output feedback. This material, already covered in the pre-requisite course Introduction to System Theory, will be reviewed without proofs.

Linear Quadratic (LQ) control: State feedback control. The finite horizon LQ problem. The Riccati differential equation. The infinite horizon problem. The algebraic Riccati equation (ARE). Solvability and properties of its solution.

Realization theory: Convolutions. State space systems. Hankel matrix. Realizability. Minimality and equivalent realizations.

Stochastic linear systems: Brownian motion and Wiener integrals. Linear systems driven by a Wiener process. Propagation of the mean and the covariance. Stationarity. Stochastic realization theory: Markov representations of stationary Guassian processes.

Kalman filtering: Linear least squares estimation. Smoothing, filtering, and prediction. Recursive filtering. The Kalman filter. Infinite-time case. Wiener filtering.

The linear quadratic Gaussian (LQG) problem: Output feedback control. The certainty equivalence principle. The separation theorem. Solution of the LQG-problem. tex2html_wrap_inline60 interpretation.

tex2html_wrap_inline54 control: Induced norms. The tex2html_wrap_inline54 -norm as the tex2html_wrap_inline66 -induced norm. The double Riccati equation solution to the tex2html_wrap_inline54 -control problem. Robustness and the small loop gain theorem.