Non linear systems  analysis, stability, and control
EE 222

Spring 2013
E5222 Fromm Set it/
SOL/\le 
Gabe, HofmaV'Vb
Problem 1: Bowing of a violin string with different models of stiction.
From the equation of motion, Li 2 (101: Fb(:t)). Let $1 : x and x2 = x'. Then the state model
is
f1($1,$2):$1 2:32 (1)
f2($1,$2) = 5
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE 222 Spring 2017  Discussion 2
Datong Paul Zhou, [email protected]
EE 222 Discussion 2
February 10, 2017
Periodic Orbits
Problem 1
Consider the system
x 1 = x1 + x31 + x1 x22
x 2 = x2 + x32 + x21 x2
Find its equilibria and sketch the system in t
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 6
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 3/2; Due 3/9
Problem 1. Consider the nonlinear equations
x1
1+
x1 + x2
1
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 1
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 1/19; Due 1/26
Problem 1: Bowing of a violin string with different model
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE 222 Spring 2017  Discussion 1
Datong Paul Zhou, [email protected]
EE 222 Discussion 1
February 3, 2017
Linear Systems
Consider the system
x = Ax,
Characterize (1) for the following cases:
"
#
a 0
A=
0 b
x R2 , A R22 .
"
#
a 1
A=
0 a
(1)
"
A=
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE 222 Spring 2017  Discussion 3
Datong Paul Zhou, [email protected]
EE 222 Discussion 3
February 17, 2017
Problem 1
Given the system x = Ax, x R2 with
"
a b
A=
c d
#
and a, b, c, d R, find conditions on the elements of A for which there are no pe
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 4
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 2/16; Due 2/23
Problem 1: A Bifurcation study of a 3D Pendulum. Conside
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 3
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 2/9; Due 2/16
Problem 1. Prove that each of the following systems has no
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 2
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 1/31; Due 2/9
Now that you are familiar with generating phase portraits,
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE 222 Spring 2017  Discussion 4
Datong Paul Zhou, [email protected]
EE 222 Discussion 4
February 24, 2017
Problem 1
Consider the two linear systems
x = A1 x
and x = A2 x
with
"
#
1
10
A1 =
100 1
"
1 100
A2 =
10 1
#
What can you say about the stab
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
http:/inst.eecs.berkeley.edu/ee222/
Course Outline
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley
Spring 2013
Lecture Informatio
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
3) d)
Example of initial conditions that the Jacobian linearized control scheme works:
x0 = [1, 0, 1]
Example of initial conditions that the Jacobian linearized control scheme fails:
x0 = [100, 50, 1.98]
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
Phase portrait of zero dynamics
Epsilon = 0.1
Simulating a control law
Simulating a control law that tracks level flight (constant altitude) zR = 1, and a
constant horizontal velocity xR = v = 0.17.
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Midterm Test
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2011
Issued 3/10
This is a one hour and 20 minute test.
Your solutions will be graded
Non linear systems  analysis, stability, and control
EE 222

Spring 2013
EE222 Nonlinear Systems: Analysis, Stability, and Control
Problem Set 5
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Spring 2017
Issued 2/23; Due 3/2
Problem 1: Describing function determination.
(a) Consider