5B5715 Linear Stochastic Systems, spring 2007

Archived News

March 27

Eigth class. Finished presentation of chapter 7. Next week there is no class because of Easter, so the next lecture will be given on April 3. The last two lectures will be on chapter 8.

March 20

Seventh class. Presentation of parts of chapter 7. Introduced the perpendicularly intersecting property, scattering pair representation of splitting subspaces and Markovian splitting subspaces. The example on determining a minimal Markov splitting subspace gave rise to a good question, that shows how easy it is to make mistakes when working with subspaces. See example 7.3 among the exercises which illustrate this.

March 13

Sixth class. Finished presentation of chapter 6, and started presentation of chapter 7. The special predictor state spaces were defined and the relation to steady state Kalman filtering was explained. Then the geometric description of states in terms of splitting subspaces was introduced. This will be an important concept and it will be further analyzed in the following lectures. Seeing this for the first time, it might be a bit difficult to grasp. The abstraction level increases a bit in this part of the theory. Even if you do not understand all the details I think that it could be useful for the most of you to understand a bit of this geometric description and the underlying structure.

March 6

Fifth class. Presentation of chapter 6: Linear stochastic systems. The coordinate free representation of the state, observability, constructibility, and minimality. Rational spectral factorization, the positive real lemma and the Riccati inequality.

February 27

Fourth class. Presentation of chapter 6: Linear stochastic systems. Reminded about deterministic realization theory. Derived the backward model and showed how to determine a state space representation of it.

February 20

Third class. Presentation of chapter 4: Innovations, Wold decomposition and Spectral Factorization. We studied acausal and causal Wienerfiltering. Orthonormalizability and causal equivalence was connected to the existence of spectral factors. Wolds theorem was stated and the idea of the proof was outlined. Inner-outer factorization was left for self-studies. The next class we will then be ready for chapter 6 and rational stochastic realization for forward and backward models. I regard chapters 6 and 7 as the core of the course, so we will spend two classes each on them.

February 13

Second class. Presentation of chapter 3: Spectral representation of stationary processes. First hour: Definitions of stochastic integrals and measures. Second hour: Frequency representations of the covariance function and the processes themselves. Relation between the shift operator on stochastic variables and the multiplication with M_e in the L^2 space. Finally, representation of processes spanned by white noise. Please read the rest of chapter 3.5 carefully yourselves.
Next class is about Wiener filtering, Wold decomposition and the innovation process, so then the fun begins.

February 6

First class. 13.15-15.00, in room 3733 at Lindstedtsvägen 25.
Presentation of chapter 2: Geometry of second-order random processes. Stationary stochastic process were described and the Markov property was introduced. Please read the rest of the chapter where some more concepts are introduced, such as the definition of stochastic systems and causality, and will be discussed in more detail later in the course.

December 1:st

Homepage is born
Since the course is currently under construction, this is a great opportunity to give your opinions, please send your suggestions to: <penqvist@math.kth.se>