Handout Number 10
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
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
Work Problem B-7-1 page 561 5th Edition of textbook (this is same as Problem B-8-1
page 612 of 4th Edition).
In the last lecture we made the straight line approximation of the magnitude and the phase
Bode plots of
Work Problem B-7-15 on Page 563 of the 5th edition (copy of the page is
Work Problem B-7-16 on Page 563 of the 5th edition (see attached)
Work Problem B-7-17 on Page 563 of the 5th edition (see
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).
EEME E4601 Homework 4
g (kT ) S (k 1 n)T )u (nT )
Zcfw_g (kT ) z 1Zcfw_u(nT )Zcfw_S (kT )
G(s) 1/ (s 1)
Find differential equation relating yd(t) to y(t).
Find the unit step response S(t) of the system
Give general solution of following homogeneous different equations in terms of real valued
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(
Find Y(z) in terms of V(z) and Yd(z).
Create closed loop difference equation.
Find the general solution of homogeneous equation yH(k) as a function of K for all K0. Give the
solution in terms
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
Prof. Richard Longman
The Laplace transform F ( s ) or L[ f (t )] of a function f (t ) is given by
F (s) =
e! st f (t )dt
where s is a complex number s
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
Professor Richard Longman
Office hours: 2:30 to 3:30PM Monday and Wednesday
Office: 232 Mudd
You can try to ask me questions outside my office hours. Be sure to make i
(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
Refer to the handout on Laplace Transforms Used to Solve Differential Equations. Use
Laplace transforms to solve the following differential equation
+ 4 + 3y = 1
for the foll
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>
E tn e 66ol
lY,^'tdp Shi t Shi yurade ZOa+@
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
icfw_ t^/ 3
( ys t+w@u>tunbia.
Fu,rt (- t1
f, r.r,l q
mi sS z-ng part
(hr+ 0, eot
where u (t ) = 1 .
Find the exponential of matrix At by the Laplace transform method (see file
e At = L!1[( sI ! A)!1 ]
# !2 !3&
Suppose that A,! B,! C are all n ! n matrices, and that A(t ) = B
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
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