# 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
1. Introduction and applications [30 m, slides]
2. Brief review of stability and bifurcations [20 m, slides]
• Review of bifurcations: pitchfork, Hopf, sub/super critical
• Example: rotating stall and surge
3. Stabilization to an equilibrium point [20 m, board]
• Review of Lyapunov-based stabilization: CLFs, Sontag's formula
• Strongly nonlinear systems (if time)
4. Looking forward [10 m, slides]
• Actuator limits
• Nonequilibrium behavior
• Project ideas

### 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 Looks like I don't need normal forms at all, so I'm skipping 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.
• Use VKI presentations for the introductory material

## Lecture 2: Control of bifurcations

##### Outline
1. Analysis of the Moore-Greitzer model [20 m; board]
• Review of the model (form + polynomials)
• Nonlinear analysis (standard CDS 140a tools)
2. Control of bifurcations [20 m; board]
• Normal form for control of bifurcations
• Abed and Fu: bifurcation controllability (if time)
3. Effects of actuation limits [20 m; board]
• Magnitude limits
• Rate limits
4. Implementation and applications [20 m; slides]
• Actuation mechanisms: bleed valves, air injection
• Experimental results from Caltech rig

### 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
• Finish up with implementation results, including the story of IHPTET program + Caltech experimental results

## Lecture 3: Control of limit cycles

##### Outline
1. Motivating example: combustion instabilities [20 m; board]
• Review of the essential physics
• Culick model + dynamical systems analysis
2. Bifurcation control of limit cycles [15 m; board]
• Extension of Abed and Fu/Wang & M results to Hopf
• Geometric version of results
3. Harmonic balance [20 m; board]
• State space derivation, with time delays
4. Implementation and applications [20 m; slides]
• Actuation mechanisms: speakers versus fuel modulation
• UTRC experimental results + other applications

### 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 Results are a bit too linear; use combustion instabilities as well, with cavity flow as example application
• 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 (via harmonic balance) here and then use lecture 4 to cover the full details

## Lecture 4: Describing function analysis

##### Outline
1. Describing functions [20 m; board]
• Review of the basic approach (non-rigorous)
• Examples: input saturation, relay control
• Stability of limit cycles
2. Theory [30 m; board]
• Review of Mees formulation (from CDS 221)
3. Random input describing functions [20 m; board]
• Description of method
• Application: combustion instabilities
4. Wrap up [10 m; slides]
• Summary of results from lectures
• Discussion of open problems in nonlinear control