ECE 380 PROBLEM SET 6: SOLUTIONS
1. P (s) =
1
s(s+1)2 + 13 )
, C(s) = K
Since the system is in the unity feedback configuration,
1
1
2
1 , (s) = s(s + 1) + 3 ) + K
2
s(s + 1) + 3 )
1
2
,
N (s) = 1, D(s) = s (s + 1) +
3
3
, n = 3, m = 0
open loop poles = 0
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
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
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 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)
ECE380: PROBLEM SET 4
1. Determine whether the following polynomials are stable or unstable (without computing their roots).
If the polynomial is unstable, determine how many roots are in the CRHP.
(a) D(s) = 3s2 2s + 1
(b) D(s) = s3 5s2 s 6
(c) D(s) = s4
ECE380: PROBLEM SET 9
1. Draw Nyquist plots by hand for the following systems that are placed in the standard unity feedback
configuration, with K = 1.
(a) L(s) =
(s1)(s10)
(s+1)(s+10)
(b) L(s) =
s2 +25s+100
s(s+1)2
(c) L(s) =
s2 1
s2
After you draw the N
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