# CDS 101/110 - Dynamic Behavior

From MurrayWiki

CDS 101/110a | Schedule | Recitations | FAQ | AM08 (errata) |

## Contents |

## Overview

The learning objectives for this week are:

- Students should be able to use a phase portraits to describe the behavior of dynamical systems and determine the stability of an equilibrium point
- Students should be able to find equilibrium points for a nonlinear system and determine whether they are stable using linearizations (all) and Lyapunov functions (CDS 110/210)
- Students should be able to explain the difference between stability, asymptotic stability, and global stability

**Monday:** Qualitative Analysis and Stability (Slides, MP3)

- This lecture provides an introduction to stability of (nonlinear) control systems. Formal definitions of stability are given and phase portraits are introduced to help visualize the concepts. Local and global behavior of nonlinear systems is discussed, using a damped pendulum and the predator-prey problem as examples.

- Lecture handout
- MATLAB code: phaseplot.m, boxgrid.m, L2_1_stability.m, oscillator.m, invpend.m, predprey.m

**Wednesday:** Stability Analysis using Lyapunov Functions (Notes, MP3)

- Lyapunov functions are introduced as a method of proving stability for nonlinear systems. Simple examples are used to explain the concepts.

**Friday:** Recitations

## Reading

- K. J. Åström and R. M. Murray,, Princeton University Press, 2008..
- CDS 101: Read sections 4.1-4.3 [30 min]
- CDS 110: Read sections 4.1-4.4, up to Krasolvski-Lasalle (p 118) [60 min]
- CDS 210: Review sections 4.1-4.3, read sections 4.4-4.5 [60 min]

## Homework

## FAQ

**Monday**

- Can a Lyapunov function be used to show/prove a system is unstable at an equilibrium point similar to the way it can be used to show asymp stability?
- I'm a 101/110 student. Can I sit in on the 210 lectures on Fridays without being enrolled in 210?
- In slide 12, what is special about the red cycle as opposed to the blue curves?
- In slide 9, why did you only linearize around the downward equilibrium point?
- What is the significance of having eigenvalues that are 0? I think I heard you say "in that case you don't know anything". Does that mean you cannot determine if the system is stable or asymptotically stable?
- You mentioned "aggressive" dynamics a couple of times. Please define. How/when is it useful?
- You said spiraling out to a bounded circle is unstable, but doesn't that still satisfy the epsilon-delta condition of stability? Or are there limits on epsilon?

**Wednesday**

**Homework**