P RACTICE M ID - TERM E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all you answers.
1. The following equation models the motion of a pendulum:
+ k sin =
where R is the angle of the pendulum with the vertical, R an applied torque, and k a pos
P RACTICE M ID - TERM E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all you answers.
1. Consider the nonlinear systems
y + y + y = u2 1.
(a) Compute a state-space representation for the system with input u and output y
(b) Linearize the system
P RACTICE M ID - TERM E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all your answers.
1. Consider the one-link robot in the following gure:
m
y
x
where denotes the angle of the link with the horizontal, the torque applied at the base, (x, y) th
P RACTICE F INAL E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all you answers.
1. For which values of a, b, and c does the transfer function
(s 1)(s 2)
s3 + as2 + bs + c
has a two-dimensional realization that is both controllable and observabl
P RACTICE F INAL E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all you answers.
1. Consider the matrix
A=
0
1
2
.
3
Compute A100 and eAt .
Hint: Diagonalize A.
2.
(a) Show that if J is a stable Jordan block then for every t 0 all the eigenvalue
P RACTICE F INAL E XAM
L INEAR S YSTEMS
Jo o P. Hespanha
a
Please explain all you answers.
1. Consider the following nonlinear system
x1 = x1 + u 1
x = x2 + u 2
2
x3 = x2 u 1 x1 u 2
y = x2 + x2 + x2
2
3
1
(a) Linearize the system around the equilibrium p
SOLUTIONS IN MOTION
Motoman's HP20 robot provides a powerful solution to meet your automation needs
MULTIPLE ROBOT CONTROL
CUTTING DISPENSING HANDLING MACHINE TENDING PACKAGING MATERIAL REMOVAL
MATERIAL HANDLING
HP20
Payload: 20 kg
PACKAGING
Versatile, H
UCSB
ME 17: Mathematics of Engineering
Spring 2011
Homework 4
Consider a domain [a, b] [c, d] and the heat equation in two spatial dimensions:
u
=D
t
2u 2u
+2
x2
y
+ S,
where D is the diusion constant, u is the temperature and S is the source term. A nume
UCSB Linear Systems I
ME243A/ECE230A, Fall 2011
Instructor: Francesco Bullo
This is the website for the UCSB course ME 243A / ECE 230A Linear Systems I, Fall 2011 (lecture times during
period: Th.22sep2011 through Fri.02dec2011). This information is avail
UCSB Introduction to Robotics
ME170A/ECE181A, Fall 2011
Instructor: Francesco Bullo
This is the website for the UCSB course ME 170A / ECE 181A Introduction to Robotics, Fall 2011 (lecture times
during period: Th.22sep2011 through Fri.02dec2011). This info
Chapter 4
The Conguration Space
Steven M. LaValle
University of Illinois
Copyright Steven M. LaValle 2006
Available for downloading at http:/planning.cs.uiuc.edu/
Published by Cambridge University Press
128
S. M. LaValle: Planning Algorithms
The brief tre