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 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
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,
The exercise sessions are given on
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: firstname.lastname@example.org.
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
The course will be given from overhead projector sheets. Copies of these sheets
will be handed out.
- The LQ-theory is covered for example in R.W. Brockett, Finite
Dimensional Linear Systems, Wiley, 1970, sections 21 and 23. We will, roughly, follow
There are many other sources where this material is covered, e.g.,
H. Zwart, Optimal control, Course notes for subject 156062,
Department of Applied Mathematics, University of Twente, 1998, see sections
2.3, 2.4, 3.3.
H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley, 1972.
- For the part on Realization theory, notes will be handed out.
- For the LQG-theory we follow J.C. Willems, Recursive filtering,
Statistica Neerlandica, pages 1 - 39, 1978. An off-print of this
article will be handed out. This material is, of course, covered in many
other places, for example, in the book by Kwakernaak and Sivan.
- For the -control, we follow the recent manuscript by
H.L. Trentelman and J.C. Willems, Dissipative differential systems and
the state space control problem, submitted for publication. A
copy of this manuscript will
be handed out.
- Exercise sets will be handed out, and a selection of them will be covered
during the exercise sessions.
- All this material will be posted on
http://www.math.rug.nl/~willems, see under ``Teaching''.
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
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
- 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. interpretation.
- control: Induced norms. The -norm
as the -induced norm. The double Riccati equation solution to
the -control problem. Robustness and the small loop gain theorem.