ECE380: PROBLEM SET 8
1. Consider the plant P (s) =
with controller C(s) = K.
1
s(s2 +2s+2) ,
and suppose that it is placed in the unity feedback configuration
(a) Using the rlocus command in MATLAB, plot the positive root locus of P (s) and find the (pos
ECE380: PROBLEM SET 6
For the following problems, you can use the polyval and roots commands in MATLAB to evaluate
polynomials and find their roots. For the first four problems, draw the root locus by hand, and perform
the breakaway point calculations. Yo
ECE 380 PROBLEM SET 7: SOLUTIONS
1. (a) P (s) =
s+5
s(s+2)
s+zlead
, C(s) = K s+p
where plead = 10.
lead
Assuming the system is in the unity feedback configuration,
H(s) =
P (s)C(s)
K(s + 5)(s + zlead )
=
1 + P (s)C(s)
s(s + 2)(s + 10) + K(s + 5)(s + zlea
ECE 380 PROBLEM SET 5: SOLUTIONS
1. (a) P (s)C(s) =
KP s+KI
.
s(s3 +5s2 +3s2)
Try (s) =
Since the system is in the unity feedback configuration,
P (s)C(s)
KP s + KI
.
= 4
3
1 + P (s)C(s)
s + 5s + 3s2 + (KP 2)s + KI
Applying the Routh-Hurwitz test to evalu
ECE 380 PROBLEM SET 3: SOLUTIONS
1
1. (a) Remembering i = C dv(t)
dt or v = C
v1 (t) and v2 (t) can be written as:
R
i(t)dt for capacitors and v = L di(t)
dt for inductors,
di(t)
1
v1 (t) = L
+ Ri(t) +
dt
C
1
i(t)dt , v2 (t) =
C
Z
Z
i(t)dt .
R
R
d
d2
i(t)
ECE380: PROBLEM SET 1
1. A review of complex numbers.
(a) Compute the magnitude and the phase of the following complex numbers, and write them in
polar form. Also, state which parts of the complex plane (OLHP, CLHP, ORHP, CRHP) each
number lies in.
i. 5j
ECE380: PROBLEM SET 2
1. Consider the system with transfer function
H(s) =
1
.
s+1
Assume that the initial condition is zero. Compute the response to the input u(t) = cos t by applying
the formula Y (s) = H(s)U (s), using partial fractions, and then calcu
ECE 380
PROBLEM SET 9: SOLUTIONS
1. Once the Bode plots are drawn for the systems, the contour L(C1 ) is obtained by drawing the
magnitude and phase in the complex plane. The contour L(C2 ) is just the reflection of the
contour L(C1 ) about the real axis.
ECE 380 PROBLEM SET 4: SOLUTIONS
1. (a) D(s) = 3s2 2s + 1
Since all the coefficients are not positive, D(s) is unstable. The number of poles in
the CRHP is determined by number of sign changes in the first column of the Routh
array.
s2
s1
s0
3
-2
1 = 0+2
ECE380 PROBLEM SET 2: SOLUTIONS
1. The Laplace transform of the input, u(t) = cos t, is U (s) = s2s+1 . Plugging it into the equation
Y (s) = H(s)U (s):
1
s
s
Y (s) =
=
.
2
s+1
s +1
(s + 1)(s + j)(s j)
We will perform a partial fraction expansion on Y (s)
ECE 380 PROBLEM SET 8: SOLUTIONS
1. Unity feedback configuration with P (s) =
1
s(s2 +2s+2)
, C(s) = K
(a) Using the rlocus command in MATLAB, the positive root locus of P (s) is shown below
with K = 4 marked
as the gain for which there is an imaginary a
ECE 380 Assignment 2
Analog Control Systems: Spring 2018
Department of Electrical and Computer Engineering
University of Waterloo
Problem 1 (Convolution Practice)
(a) For the function
f (t) =
0
1
0
if
if
if
t<0
0t<1
t1
compute f f , the convolution of f w
ECE 380 Assignment 1
Analog Control Systems: Spring 2018
Department of Electrical and Computer Engineering
University of Waterloo
Problem 1 (Complex Arithmetic)
(a) For each of the following complex numbers s, compute the magnitude |s| and phase s and
wri
ECE 380 Extra Resources (Section 002)
Analog Control Systems: Spring 2018
Instructor: J. W. Simpson-Porco
Department of Electrical and Computer Engineering
University of Waterloo
This document lists some pointers to extra material, organized by topic. In
Jeremie Benhamron
SE 380 Introduction to Feedback Control
Assignment 1 Solutions
1.
a)
i.
Im
5j
|5j| = 5
&
\5j =
2
or
90
5j = 5ej 2
= /2
5j is in the CLHP and CRHP.
R
ii.
Im
= arctan(4/3) +
R
3
3
3
p
( 3)2 + ( 4)2 = 5
4
1
\ 3 4j = tan
+ 4.069 or 233.
Jeremie Benhamron
SE 380 Introduction to Feedback Control
Assignment 2 Solutions
1.
f (t)
8
>
<0 if t < 0
f (t) = 1 if 0 t < 1
>
:
0 if t 0
2
1
0
1
2
t
a)
Convolution of x(t) with y(t) is denoted by x(t) y(t) and is defined as:
x(t) y(t) =
Z1
x( )y(t
)d
ECE380: PROBLEM SET 3
1. For the following RLC circuit:
(a) Determine the differential equation model relating the input voltage v1 (t) to the output voltage
v2 (t).
(b) Determine the transfer function H(s) =
V2 (s)
V1 (s) .
(c) Now, for the values R = 3,
ECE380: PROBLEM SET 7
You can use MATLAB to solve some of the problems. The angle and abs commands will be useful for
calculating phase and magnitude.
1. Consider the plant P (s) =
s+5
s(s+2) .
s+zlead
(a) Design a lead controller C(s) = K s+p
so that the
ECE 380: PROBLEM SET 5
1. Consider the unity feedback loop with P (s) =
1
s3 +5s2 +3s2 .
(a) Suppose we want to use the PI controller C(s) = KP + KsI . For what values of KP and KI
will the closed loop system be stable? Provide a quantitative plot of this
UNIVERSITY OF WATERLOO
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
E& CE 380 Analog Control Systems
Midterm Examination
Monday, February 24, 2003
INDICATE IF YOU ARE IN THE ELECTRICAL ENGINEERING OR COMPUTER
ENGINEERING STREAM ON YOUR ANSWER BOOKS