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
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 app
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
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 ,
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)
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) + 2
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
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.
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 (
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.
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: [email protected]
You can try to ask
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
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 differe
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 o
6eo
Qroa I
F;lt
L
fiunde.
S'olqion
icfw_ t^/ 3
Shi lotfirztttt
( ys [email protected]>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'
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 th
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 s
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 sett