# Lecture Non Linear Systems

Autor:   •  February 24, 2019  •  Course Note  •  680 Words (3 Pages)  •  17 Views

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EE M242A, Winter 2017 — Midterm Exam

Feb. 8, Wednesday, 4pm–6pm

Name: ID:

Instructions:

Closed book, no calculators allowed. Two letter-sized sheets of notes are

Write only on the FRONT of each page.

I am aware of and will abide by UCLA’s policy on academic integrity during

this exam. This means I will not use or give to others unauthorized materials,

information, or study aids during this exam. I recognize that “unauthorized

materials” includes other students’ exam papers.

Signature:

1

Problem 1 (20 points). Design a second-order (i.e., two-state) systemwith

a stable limit cycle and an unstable focus. You may use polar coordinates, but

Solution. One possibility:

r˙ = −r(r − 1)

✓˙ = 1

where

x = r cos ✓

y = r sin ✓.

Then

x˙ = r˙ cos ✓ − r sin ✓ = −(r − 1)r cos ✓ − r sin ✓ = −x(

p

x2 + y2 − 1) − y

y˙ = r˙ sin ✓ + r cos ✓ = −(r − 1)r sin ✓ + r cos ✓ = −y(

p

x2 + y2 − 1) + x.

2

Problem 2 (20 points). Consider the system:

x˙ 1 = x2

˙ x2 = −x1 + x2(2 − 3x21

− 2x22

).

(a) Show that the set {(x1, x2) : x21

+ x22

 1} is a positively invariant set.

(b) Show that the system has a periodic orbit.

Solution.

(a) Consider the set {(x1, x2) : x21

+ x22

...