Mini Projects (2.5 hp)

Yde's presentation: LMI_synthesis.pdf Download LMI_synthesis.pdf

Mini-project can be done as an extra course worth 2.5 hp (roughly 1.5 week of work).

Suggested date for presentation is Monday Feb 17 2025 at 13.15

List of suggested projects

Q-parametrization (+ structural constraints ?)

The Q-parametrization, also called Youla-Kucera parametrization, is a cornerstone to optimization-based controller design. It can be used to transform the (nonlinear and nonconvex) problem of controller design to a linear and convex optimization problem: All stable closed loop systems can be parametrised as $G_{cl}(s) = G_0(s) + G_1(s)Q(s)G_2(s)$ , where $Q(s)$ can be chosen as any stable system. The book chapter by Megretski presents a state-space version (which IMHO is the best way to gain intuition) but there are also alternatives with fractional MIMO transfer functions. There are also some generalizations to cases where one wants to respect structural constraints on the controller. Make your own choice of which material to study and present. Some suggested references:
Megretski Chapter 7- Q parametrization Download Megretski Chapter 7- Q parametrization
Wikipedia, Youla-Kucera parametrization Links to an external site.
Lessard-Sall 2012, Decentralized two-player LQG problem Download Lessard-Sall 2012, Decentralized two-player LQG problem

Nevanlinna-Pick Interpolation (+ Reproducing Kernel Hilbert Spaces ?)

A stabilized closed loop system must satisfy some interpolation constraints, given by the open loop systems zeros and poles in the RHPL. The sensitivity function must for instance satisfy $S(z) = 1$ when z is an open loop zero and $S(p) = 0$ when p is a open loop pole. One way to find H-infinity optimal controllers is to use so called "Nevanlinna-Pick Interpolation Links to an external site.". This theory is mathematically beautiful, and not that hard to grasp, at least for the SISO case when everything is scalar. There are many good papers and books.

Some references:

RKHS_interpolation_NevanlinnaPick.pdf Download RKHS_interpolation_NevanlinnaPick.pdf

RKHSMScthesis2012.pdf Download RKHSMScthesis2012.pdf

PickMScthesisCarlsson2016.pdf Download PickMScthesisCarlsson2016.pdf

https://users.metu.edu.tr/baver/MS.Thesis.pdf Links to an external site. (see e.g. pp 66-68 below)

It would be nice if the report used Handin 3 as example and illustrates what happens when one mimimizes $M_s$ or $M_t$ . Furthermore, can you figure out what happens if one changes stability domain to $\Omega_\alpha :=\{s: \;\mathrm{Re}(s) \leq -\alpha\}$ , i.e. requires all closed loop poles to have distance $\alpha$ .

To fully understand the theory behind Pick interpolation it helps to also learn a little about Reproducing Kernel Hilbert Spaces, a topic which is used also in machine learning (the kernel trick etc), visualisation of high-dimensional data and other areas Links to an external site.. This could be added if two persons want to work with this topic in a joint project.

Polyak Heavy Ball and Tannenbaum Gain Margin Optimization

This paper by Polyak describes an interesting link between algorithm design in optimization theory and control synthesis in robust control. A suitable project for anyone interested in this field.

- https://www.arxiv.org/pdf/2409.19882 Links to an external site. (Oct 2024)

Integral Action — "Disturbance Observer" vs "Explicit Form"

In the course we mentioned different ways of introducing integral action. The task is to deepen the understanding of the pros and cons of these approaches. Remember windup, bumpless transfer to manual control, modeling errors, MIMO. Remark: A control Alumni is interested in this problem, related to autonomous driving.

References

- Hagander+Åkesson ECC 2003 Download Hagander+Åkesson ECC 2003
- https://lucris.lub.lu.se/ws/portalfiles/portal/4720262/8627312.pdf (See Ch 4.6 in MPC Tools manual)
- https://lucris.lub.lu.se/ws/portalfiles/portal/4796996/8840415.pdf (See Ch 5 in this Lic)
- inputdisturbanceobserver.pdf Download inputdisturbanceobserver.pdf

MPC (and MPC_Tools)

Summarize MPC and some MPC-Tools matlab toolbox Links to an external site.. What exists in python/julia? Illustrate on some example from the course.

LQG design using polynomial methods

LQG design using polynomial methods is presented in the book Computer Controlled Systems by Åström and Wittenmark. This was the course book at the department for a long time in our advanced control course. Present how the discrete time LQG controller can be found this way. Illustrate on some of the examples in the course.

L1-optimization and SDP

Sometimes it is motivated to use peak-norms ( $L_\infty$ ) to measure size of disturbances and output signals. To find a controller that minimizes the so induced system norm is called the $L_1$ -optimization problem.

This old presentation by Polyak Download presentation by Polyak gives an introduction. Here is another reference from Encyclopdia of Systems and Control Links to an external site.

Root-locus and step-size control

Summarize the properties of root-locus and describe how it can be efficiently calculated. Note that it is a good idea to develop a method for automatic step-size control, since parts of the root-locus is numerically much more sensitive than others. Describe how this can be done.

ControlSystems.jl (or Python?)

Illustrate and summarize the status of the Julia Toolbox for control systems, mostly developed at our department. What development, and possible changes, would you propose? It would be nice to use examples from the course when illustrating things.

Package and document nice code for GangOfFours and Nichols plots

(Tikz? See Thelander report 2016 as example)

Your own suggestion

The best alternative. But describe your idea to BoB first before you start working on it and get his ok.