Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #6
Due: March 10, Thursday
1. In this problem you will study the linear system x = Ax, and show that for any
constant which is greater than the real parts of all eigenvalues of A, there exists a
> 0
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #2
Due: February 9, Tuesday
1. Khalil, Problem 2.20.
2. Make a bifurcation plot for each of the following two models:
a)
x = + x ln(1 + x)
b)
x = x ln(1 + x).
Identify the bifurcation values for the p
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #4
Due: February 23, Tuesday
1. Khalil, Problem 4.3.
2. Consider a system described by the second order differential equation:
y + h(y)y + g(y) = 0
where yg(y) > 0 for all y 6= 0 and h(y) > 0 for all
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #5
Due: March 1, Tuesday
1. Consider the system
x 1 = x2
x 2 = g(k1 x1 + k2 x2 ),
k1 , k2 > 0,
where the nonlinearity g() is such that
g(y)y > 0 y 6= 0
lim
Z y
y 0
g(z)dz = +.
a) Using an appropriat
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2014
Midterm
March 13, 2014
The duration is 80 minutes. Each problem is worth 25 points. Closed book/notes; one
formula sheet allowed.
1. Consider the following system defined on the nonnegative quadrant:
1
x 1 = x
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #9
Due: April 28, Thursday
1. Suppose a dynamical system is dissipative with respect to supply rates s1 (u, y) and
s2 (u, y). Show that it is also dissipative with respect to the supply rate s(u, y) :
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2012
Midterm
March 8, 2012
The duration is 80 minutes. Each problem is worth 25 points. Closed book/notes; two
formula sheets allowed.
1. Name three phenomena that can occur only in nonlinear systems. Give an examp
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2010
Midterm
March 11, 2010
The duration is 80 minutes. Each of the ve questions weights 20 points. Closed
book/notes; two formula sheets allowed.
1. Consider the system
1 = 21 2
2 = 1 3 .
Find the equilibria and
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
EE222  Spring16  Lecture 1 Notes1
Licensed under a Creative Commons
AttributionNonCommercialShareAlike
4.0 International License.
1
Murat Arcak
January 19 2016
Nonlinear Systems
x = Ax + Bu x = f ( x, u)
(1)
Analysis:
x = f ( x )
x = f (t, x )
f : Rn
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
EE222  Spring16  Lecture 23 Notes1
Licensed under a Creative Commons
AttributionNonCommercialShareAlike
4.0 International License.
1
Murat Arcak
April 21 2016
Stability of Interconnected Systems
Consider the interconnected system in Figure 1 where eac
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
EE222  Spring16  Lecture 9 Notes1
Licensed under a Creative Commons
AttributionNonCommercialShareAlike
4.0 International License.
1
Murat Arcak
February 16 2016
LaSalleKrasovskii Invariance Principle
Applicable to timeinvariant systems.
Allows us
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #8
Due: April 21, Thursday
1. Khalil, Problem 13.2.
2. The dynamics of the translational oscillator with rotating actuator (TORA) depicted
below are described by:
x 1 = x2
x1 + x24 sin x3
cos x3
x 2
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #7
Due: April 7, Thursday
1. Given the linear system x = Ax + Bu with initial condition x(0) = 0, we would like
R
to check if the trajectories x(t) generated by unit energy inputs ( 0 uT (t)u(t)dt 1)
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #1
Due: February 2, Tuesday
1. Khalil, Problem 1.18.
2. Duffings equation, x + x x + x3 = cos(t), exhibits chaotic behavior for its parameters
in certain ranges. Simulate this equation for (, , ) = (0
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2012
Midterm
March 8, 2012
The duration is 80 minutes. Each problem is worth 25 points. Closed book/notes; two
formula sheets allowed.
1. Name three phenomena that can occur only in nonlinear systems. Give an examp
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
M. Arcak
EE 222 NONLINEAR SYSTEMS
Spring 2016
Homework #3
Due: February 16, Tuesday
1. Design a second order system with a stable focus and a stable limit cycle. You may
use polar coordinates, but the final system equations should be given in the Cartesia
Non linear systems  analysis, stability, and control
EE 222

Spring 2016
EE222  Spring16  Lecture 12 Notes1
Licensed under a Creative Commons
AttributionNonCommercialShareAlike
4.0 International License.
1
Murat Arcak
February 25 2016
Linear TimeVarying Systems
Khalil Section 4.6, Sastry Section 5.7
x (t) = (t, t0 ) x (t0