Longman 11/09/05
EEME E3601
Handout Number 10
ROOT LOCUS
Three Basic Equations
A. Standard Form of the Characteristic Polynomial
The root locus rules can help one see how the roots of a characteristic polynomial change
as one varies one parameter, call it
Homework 10
EEME E6601
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem B-7-1 page 561 5th Edition
(Problem B-8-1 page 612 4th Edition)
Problem B-7-24 page 564 5th Edition
(Problem B-8-27 page 615 4th Edition)
Note: You can use the straight line approx
Homework 8
EEME E6601
Problem 1
Work Problem B-7-1 page 561 5th Edition of textbook (this is same as Problem B-8-1
page 612 of 4th Edition).
Problem 2
In the last lecture we made the straight line approximation of the magnitude and the phase
Bode plots of
Homework 9
EEME E6601
Problem 1
attached.
Work Problem B-7-15 on Page 563 of the 5th edition (copy of the page is
Problem 2
Work Problem B-7-16 on Page 563 of the 5th edition (see attached)
Problem 3
Work Problem B-7-17 on Page 563 of the 5th edition (see
EEME E4601
Homework 7
Spring 2012
Problem 1
Given the matrix
2 1 1
A 1 3 2
1 1 2
a) Determine the eigenvalues and corresponding eigenvectors.
b) Write the matrix in diagonal form.
c) Calculate A5 , making use of diagonal matrix from b).
Problem 2
Given
EEME E4601 Homework 4
Problem 1
k 1
g (kT ) S (k 1 n)T )u (nT )
n 0
Show that
Zcfw_g (kT ) z 1Zcfw_u(nT )Zcfw_S (kT )
Problem 2
(a)
G(s) 1/ (s 1)
i)
ii)
iii)
Find differential equation relating yd(t) to y(t).
Find the unit step response S(t) of the system
HW#1 Problem
EEME E4601
Give general solution of following homogeneous different equations in terms of real valued
function.
Do the same for the following difference equations.
6. y(k+2) + 3y(k+1) + 2y(k) = 0
7. y(k+2) + 4y(k+1) + 4y(k) = 0
8. y(k+2) + y(
EEME E4601
Homework 3
Spring 2012
Problem 1
Find Y(z) in terms of V(z) and Yd(z).
Problem 2
Create closed loop difference equation.
Problem 3
Find the general solution of homogeneous equation yH(k) as a function of K for all K0. Give the
solution in terms
EEME E4601
Homework 6
Spring 2012
Use the Jurys stability test to answer Problem 1. Solve Problem 3 in two ways, use the Jurys
test, and also solve using a bilinear transformation and Routh criterion. Note the Routh test rules
are in an appendix of the te
LAPLACE TRANSFORMS USED TO SOLVE DIFFERENTIAL EQUATIONS
EEME E6601
Columbia University
Prof. Richard Longman
Definition
The Laplace transform F ( s ) or L[ f (t )] of a function f (t ) is given by
F (s) =
#
"
0
e! st f (t )dt
where s is a complex number s
Homework 5
EEME E6601
Professor Longman
Problem 5
In class we developed the separation theorem or certainty equivalence principle that
allows one to design the controller and the observer separately. In particular, one can
place the poles of the controlle
General Information
EEME E6601
Fall 2011
Professor Richard Longman
Office hours: 2:30 to 3:30PM Monday and Wednesday
Office: 232 Mudd
Phone: (212)854-2959
E-Mail: RWL4@columbia.edu
You can try to ask me questions outside my office hours. Be sure to make i
Longman
EEME E6601
Frequency Response
(2) Response to commands. The transfer function going from command to output
G1 ( s)G2 ( s)
Y ( s) =
YC ( s)
1 + G1( s)G2 ( s) H ( s)
is used to study how well the control system can follow commands of interest. If
Prof. Longman
EEME E6601
Homework 1
Problem 3
Problem 4
Problem 5
Refer to the handout on Laplace Transforms Used to Solve Differential Equations. Use
Laplace transforms to solve the following differential equation
d2y
dy
+ 4 + 3y = 1
2
dt
dt
for the foll
a
g@* r >otf r < ) o r
>$
ocfw_*$ao,@e@o&6f;@6q2@>o@o6gQ6$ofi$>rqAa><ocfw_Sr@r$Toq@ qa<06 seocfw_g>E<tg@(t.r>
@rigllalf,tl@6@pe*ef cfw_aqotpdFo+6cfw_ra
0
?
0
RQrQB@a
0
>?@
rrqoo>
LfN+
E tn e 66ol
prabler'
7
9oltt-ho't
1"
Q@
lY,^'tdp Shi t Shi yurade ZOa+@
Professor Longman
EEME E6601
Homework 2
In Homework 1, you were asked to manipulate these equations to eliminate the
intermediate variable u, and create a differential equation for output y in terms of the
input e, and do it in two ways, in the time domai
6eo
Qroa I
F;lt
L
fiunde.
S'olqion
icfw_ t^/ 3
Shi lotfirztttt
( ys t+w@u>tunbia.
(5')
lht
LN
p
6rti cul^r
firrtt) g
Fu,rt (- t1
I
ro
f, r.r,l q
s
mi sS z-ng part
bUE JJ
ea
.at
t,e
(hr+ 0, eot
t'e o'
4.5t'"n
Po*
1
twt
)n
(ht'+8t+c )eot
t
tur(Ltt t
?o'
cus
where u (t ) = 1 .
Problem 5
Find the exponential of matrix At by the Laplace transform method (see file
6601LaplaceTransforms.pdf)
e At = L!1[( sI ! A)!1 ]
;
"0 1%
A=$
'
# !2 !3&
Problem 8
Suppose that A,! B,! C are all n ! n matrices, and that A(t ) = B
Prof. Longman
EEME E6601
Homework 3
Background Information
Recall that when one has both a forcing function from the command and a forcing
function from the disturbance, you can find the particular solution y pd (t ) associated with
the command yd (t ) w
Homework 6
EEME E6601
Problem 1: Work Problem B-5-21 on page 267 of Ogata 5th Edition.
Problem 2: For the system given in Problem B-5-22, determine the range of gains K for
which the system has a settling time of 2 seconds or less.
Problem 3: Work Problem