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

Lecture Materials Lecture presentation Handwritten notes on stabilization Reading R. M. Murray, R. L. Behnken, S. Yeung and Y. Wang, Compression System Dynamics: Control and Applications. RTO AVI/VKI Special Course on Active Control of Engine Dynamics, 2001. E.D. Sontag, A "universal" construction of Artstein's theorem on nonlinear stabilization. Systems Control Letters, 13(2):117-123, 1989. W. K. Lord, D. G. MacMartin and T. G. Tillman, Flow Control Opportunities in Gas Turbine Engines. AIAA Fluids 2000. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 2003.

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

Lecture Materials Lecture notes Compression systems slides Reading Y. Wang, S. Yeung, R. M. Murray, Bifurcation Control of Rotating Stall with Actuator Magnitude and Rate Limits. Automatica 38(4):597-610 and 611-625, 2002. Y. Wang and R. M. Murray, Effects of magnitude saturation in control of bifurcations. IFAC World Congress, 1999. E. H. Abed and J. H. Fu, Local feedback stabilization and bifurcation control, I. Hopf bifurcation and II. Stationary bifurcation. Systems and Control Letters, 1987. D-C. Liaw and E. H. Abed, Active Stabilization of Rotating Stall: A Bifurcation-Theoretic Approach. Automatica, 1994.

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

Lecture Materials Lecture notes Combustion example (slides 7-9, from L1) Reading Y. Wang and R. M. Murray, A geometric perspective on bifurcation control. Conference on Decision and Control, 2000. R. M. Murray, C. A. Jacobson, R. Casas, A. I. Khibnik, C. R. Johnson Jr, R. bitmead, A. A. Peracchio and W. M. Proscia, System Identification for Limit Cycling Systems: A Case Study for Combustion Instabilities. American Control Conference, 1998.

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

Lecture Materials Lecture notes Combustion example Reading K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, 2008. Chapter 9 - Frequency Domain Analysis. (Section 9.5, Describing functions) A. I. Mees, Describing Functions: Ten Years On. IMA Journal of Applied Mathematics, 32:221-233, 1984. A. Banaszuk, P. G. Mehta, C. A. Jacobson, A. I. Khibnik, Limits of Achievable Performance of Controlled Combustion Processes. IEEE T. Control Systems Technology, 14(5):881-895, 2006. A. Gelb and W. E. Vander Velde. Multiple-Input Describing Functions and Nonlinear System Design. McGraw Hill, 1968 (online version available here)

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.

Homework and Project Ideas

• Media:cds140b_bifctrl-hw.pdf Homework set