# ENEE 769D: Control of Bifurcations and Chaos with Applications

## Course Goals:

If a nonlinear system leaves its stable operating regime through a steady change of system parameters, the original steady state gives way to new operating modes through a process known as bifurcation. Often in control practice, loss of stability is considered a failure of the controller and no analysis of the outcomes is pursued. In this course, we pursue the new subject of control design for (nonlinear) systems undergoing bifurcations and chaos, emphasizing applications from a variety of areas. Bifurcations most often occur when systems are driven into regimes of high performance, but less secure, operation. It is not uncommon for bifurcations to occur in sequences leading to chaotic behavior, and we will also address control of chaos in the course. Applications for which bifurcations and chaos have been observed include models of aircraft flight dynamics, jet engine stall, electric power system voltage collapse, chaotic behavior of lasers, electrical circuits, power electronic systems, heart dynamics, economic systems, and communication networks. We will pursue old and new applications in the course, and some of the students may pursue a measure of research into possible uncharted applications.

In the first part of the course we will focus on the fundamental notions and results of elementary bifurcation theory and chaos. We will cover elementary bifurcations of equilibria and limit cycles and the notion of chaos, giving the basic mathematical results and emphasizing their interpretation from a systems perspective. We will illustrate the results using selected applications from electrical and mechanical systems.

In the second part of the course we will consider control problems for systems undergoing bifurcation. We will show the role of linear control techniques and their limitations, and proceed to address the limitations by employing the bifurcation results studied in the first part. We will study two-stage control designs combining linear and nonlinear feedback. We will study control design techniques that maintain nominal steady states while controlling bifurcations and chaos, even in the presence of model uncertainty. We will use the applications studied earlier to illustrate the control methods.

In the third part of the course we will venture into a recent aspect of the subject, namely system monitoring for early detection of impending instability. We will consider techniques that employ probe signals and feedback to constantly monitor a system's margin of stability with and without availability of an accurate system model. We will also consider detection of the nature of an impending bifurcation in the same context.

To help students develop their own perspective of this young and growing subject area, early in the semester students will be given references from the research literature on applications related to bifurcation control. Students will be asked to select a topic from these or other publications that most interests them and to prepare a report on the topic.

## Course Prerequisite(s):

ENEE663 and ENEE661, or permission of instructor

## Topics Prerequisite(s):

The course assumes familiarity with linear control concepts and with the main stability results in nonlinear ordinary differential equations.

None

## Reference(s):

Class notes (to be distributed).
Several texts on bifurcation theory (TBA).
Review papers that will be copied and distributed.

## Core Topics:

• Bifurcation analysis
• Bifurcation control
• Nonlinear modeling for the applications considered

## Optional Topics:

Nonlinear phenomena in communication networks