CDS 140b: Control of Bifurcations and Limit Cycles

This page contains my lecture outline and notes for a set of lectures that I will be giving in CDS 140b Introduction to Dynamics, in Winter 2008. This page is mainly intended as a place for me to keep my notes, but might be useful as a reference for the lecture (the final lecture notes will be posted on the CDS 140 web page).

Goals

• Describe how bifurcations and limit cycles arise in engineering applications
• Review some tools for characterizing bifurcations and limit cycles
• Show how feedback can be used for design of (nonlinear) dynamics

Lecture 1: Introduction and review

Outline Introduction Dynamic behavior: eq points, bifurcations, limit cycles Applications: compression systems, combustion, cavity flows Brief review of stability and bifurcations Review of bifurcations: pitchfork, Hopf, double Hopf Example: rotating stall and surge Normal forms for bifurcations Stabilization to an equilibrium point Nonlinear reachability: Lie algebra rank condition Review of Lyapunov-based stabilization: CLFs, Sontag's formula Looking forward Strongly nonlinear systems Actuator limits Project ideas

Lecture Materials Lecture presentation Reading W. K. Lord, D. G. MacMartin and T. G. Tillman, Flow Control Opportunities in Gas Turbine Engines. AIAA, 2000. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 2003. Chapter 19 - Normal Forms. E.D. Sontag. A "universal" construction of Artstein's theorem on nonlinear stabilization. Systems Control Letters, 13(2):117-123, 1989.

Notes

• This lecture is intended to serve as an introduction to the material that will be covered in the set of lectures, including motivating applications
• Need to figure out how to cover project ideas in this lecture, since students who want to work on the project would need to get started soon (so this can't wait until the last lecture)
• Not sure how much normal form material is going to be required. Look through Liaw/Abed + Wang/Murray to get a sense. I also need to find a good source for this
• The review of Lyapunov-based stabilization is intended to show simple techniques for control of bifurcations in a nonlinear setting. Plan to cover Sontag's formula, which students often don't see in other contexts.
• Can use UTRC presentations for the introductory material

Lecture 2: Control of bifurcations

Outline Bifurcation control through nonlinear feedback Effects of actuation limits Example: compression system instabilities

Lecture Materials Lecture presentation Reading Liaw and Abed Wang and Murray, Bifurcation with actuation limits

Notes

• This lecture will be based on the second VKI lecture, but need to add in more complete derivations of the results
• Work through Liaw and Abed paper in some detail (blackboard), then cover Yong Wang's results using powerpoint

Lecture 3: Control of limit cycles

Outline Motivating examples: combustion and cavity instabilities Control oriented modeling techniques Controlling the magnitude of limit cycles Describing function analysis (intro)

Lecture Materials Reading Khibnik et al (combustion instabilities) Rowley et al (cavity flow instabilities)

Notes

• Current plan for this lecture is to cover Clancy Rowley's thesis results as the main topic, but with results from combustion instabilities integrated in
• An important element of this lecture is the use of gray box modeling techniques: talk about how to pull out nonlinearities in useful ways
• Describing functions would be nice to use here, but not sure if someone has already done this (so that the results are worked out). Most likely, I'll just introduce the idea of describing functions here and then use lecture 4 to cover the full details

Lecture 4: Describing function analysis

Outline Describing functions Open problems: robustness

Lecture Materials Lecture notes Reading Describing function notes (AM08 supplement)

Notes

• Current plan is to use some of the notes from CDS 221 from the year when we covered describing functions. I need to dig these notes up and digitize them.
• If there is time, I'd like to convert this material into some supplemental notes that can be posted on the AM08 wiki page.