Lecture Non Linear Systems

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

allowed. Show all your work and simplify your answers. Credit will be given

for partial answers. Answers that are not simplified may not receive full credit.

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

your final answer should be in Cartesian coordinates.

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

⇥