**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:

- 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 this exposition.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.

**I.**-
**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. **II.**-
**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. **III.**-
**Realization theory**: Convolutions. State space systems. Hankel matrix. Realizability. Minimality and equivalent realizations. **IV.**-
**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. **V.**-
**Kalman filtering**: Linear least squares estimation. Smoothing, filtering, and prediction. Recursive filtering. The Kalman filter. Infinite-time case. Wiener filtering. **VI.**-
**The linear quadratic Gaussian (LQG) problem**: Output feedback control. The certainty equivalence principle. The separation theorem. Solution of the LQG-problem. interpretation. **VII.**-
**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.