# Mini-Projects

- Due No due date
- Points 30
- Submitting a file upload

## Suggestions for Projects

*Your time budget for this should be around 2 days of work. Your delivery is a 10 min presentation (e.g. powerpoint), to be done before summer vacations.*

*Below you find a rough description of suggestions. You can also suggest a topic yourself. Inform us about your choice before you start. We have more material and hints that we will provide. *

### Zipf's law

Study Zipf's law and give a heuristic motivation and analysis, see The Zipf Mystery

and zipf_explanation.pdf

### Nonlinear prediction and filtering: The Kushner/ Zakai equations

There is a generalization of the Fokker-Planck equations to the situation where you have a nonlinear SDE with nonlinear measurements. Assume the state of the system evolves according to

and a noisy measurement of the system state is available:

The estimation of x given z now results in a non-Gaussian probability distribution p(x). The Kushner equation describes a PDE for p(x). There is an alternative equation, which is even simpler, called the Zakai equation that gives an unnormalized version of p(x). The task is to describe these equation, it would be nice with an understandable proof and intuition on a small example. There is also relations to recent research on so called Feedback Particle Filters, that would be interesting to mention, but it might be out-of-scope

### Event-based control by Lebesgue sampling

Twenty years ago we wrote a paper about a small toy problem where you sample a signal whenever it passes a certain threshold. The information is hence not provided equidistantly in time. We showed that this method can give a more "efficient" use of measurements. You have all the tools to understand this paper and explain its content. You should also interview Marcus Thelander about what has happened after the paper was written and what he is doing research on (ask about Mirkin's recent work for instance).

### Jump Linear Systems and Johan Nilsson's thesis

Jump Linear Systems is a model extension where the state space matrices (A,B,C,D) change at random instances in time. Probability of the changes are given by the state of a finite state machine. This was used in JN's thesis to analyse a situation where random fluctuations in computation and communication delays impact control performance. The LQG problem was solved in his thesis by a generalization of the HJB equations, giving a system of coupled Riccati equations.

### Stochastic control of Level Up-crossings and Anders Hansson's thesis

The thesis by Anders Hansson, studies the problem of minimizing up-crossings of a stochastic control system. Describe the main result and illustrate with a simple example.

### Partially Observable Markov Decision Processes (POMDPs)

The general framework of Markov decision processes with incomplete information was described in a paper by Karl Johan Åström in 1965 in the case of a discrete state space. Summarize the contribution of this paper. Relate to the area of dual control theory.

### Stochastic differential equations on manifolds (and improved Gibb's sampling)

A recent article illustrates how to use theory from stochastic differential equations and Brownian motion evolving on a manifold to improve efficiency of monte carlo sampling methods. It's a slightly more mathematically advanced problem. You should understand most of the words in the abstract of the paper already if you should choose this project.

### Stochastic Differential Equations in Julia

Try out some available package(s) for SDEs in Julia and evaluate its quality and performance.