ELEC ENG 4CL4 Control System Design
Solutions to Homework Assignment #4
1. The nominal model for a plant is given by:
Go ( s ) =
1
.
( s + 4 )( s + 2 )
Assume that this plant has to be controlled in a one-d.o.f. feedback loop such that the closedloop pole
ELEC ENG 4CL4 Control System Design
Solutions to Homework Assignment #3
1. Consider the following two Nyquist diagrams labelled #1 and #2:
Nyquist Diagram #1
Nyquist Diagram #2
4
from to 0
1
3
0.8
2
0.4
from to 0
Imaginary Axis
Imaginary Axis
0.6
0.2
=
EE 4CL4 Control System Design
Solutions to Homework Assignment #3
1. A discrete-time system with input u[k] and output y[k] is described by the difference equation:
y[k ] 0.8 y[k 1] + 0.15 y [k 2] = 0.2u[k i ] .
a. Build a state space model for i = 0.
b.
EE 4CL4 Control System Design
Solutions to Homework Assignment #1
1. Explain how a high-gain feedback loop can be used to produce an implicit inverse of the plant
model.
(20 pts)
From Fig. 2.7 of Goodwin et al., u = h r z = h r f u . Passing each side of
ELEC ENG 4CL4 Control System Design
Solutions to Homework Assignment #5
1. Consider the following recursive equation describing the relationship between the input u [ k ]
and the output y [ k ] in a discrete-time (sampled-data) system:
y [ k ] + 0.5 y [ k
Electrical Engineering 4CL4: Control System Design
Solutions to Midterm Exam 2004
1. A closed-loop controller is preferable to an open-loop controller for most control problems.
a. Explain why this is the case.
b. List under what conditions an open-loop c
EE 4CL4: Control System Design
T. Kirubarajan
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Ontario, Canada
These viewgraphs are based on the text
Linear System: Theory and Design by Chi-Tsong Chen
Oxford University Press
EE 4CL4: Linear Systems (4)
T. Kirubarajan
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Ontario, Canada
These viewgraphs are based on the text
Linear System: Theory and Design by Chi-Tsong Chen
Oxford University Press, 1
EE 4CL4: Control Systems Design (3)
T. Kirubarajan
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Ontario, Canada
These viewgraphs are based on the text
Linear System: Theory and Design by Chi-Tsong Chen
Oxford University
EE 4CL4: Control Systems Design (2)
T. Kirubarajan
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Ontario, Canada
These viewgraphs are based on the text
Linear System: Theory and Design by Chi-Tsong Chen
Oxford University
EE 4CL4 Control System Design
Solutions to Midterm Exam 2003
1. The figure below shows a simple pendulum system in which a cord is wrapped around a fixed
cylinder. The motion of the system that results is described by the differential equation:
(l + R ) +
Electrical Engineering EE 4CL4
Solutions to Final Exam 2003
1. The figure below shows the Bode diagram for the open-loop transfer function C(s)Go(s) of a
system to be placed in a one-d.o.f. unity-feedback loop.
Bode Diagram
Magnitude (dB)
10
0
Mg
10
20
30
ELEC ENG 4CL4 Control System Design
Solutions to Homework Assignment #1
1. Consider a system that obeys the differential equation:
d 2x
dx
+ 2 + cos x = 0 .
2
dt
dt
a. Linearize this equation around the operating point x = 4 .
b. Derive a state-space repr
EE 4CL4 Control System Design
Solutions to Homework Assignment #7
1. Consider a plant with the nominal model:
Go (s ) =
e 0.5 s (s + 5)
.
(s + 1)(s + 3)
Build a Smith predictor such that the dominant closed-loop poles are located at s = 2 j0.5
and the con
EE 4CL4 Control System Design
Solutions to Homework Assignment #6
1. For the system with the open-loop frequency response generating the Nyquist plot shown in
Fig. 1, estimate the:
a. stability gain margin,
b. stability phase margin, and
c. sensitivity pe
EE 4CL4 Control System Design
Solutions to Homework Assignment #8
1. Consider a feedback control loop where:
Go (s )C (s ) =
9
.
s (s + 4 )
a. Verify Lemma 9.1 on page 242 of Goodwin et al. for this open-loop transfer function.
Note that if the integral c
EE 4CL4 Control System Design
Solutions to Homework Assignment #5
1. Determine the open-loop transfer function of the system generating the root locus plot shown
in Figure 1.
(25 pts)
Root Locus Plot
20
15
Imaginary Axis
10
5
0
5
10
15
20
5
4
3
Real Axis
EE 4CL4 Control System Design
Solutions to Homework Assignment #10
1. The output y(t) of a continuous-time system having a unit step input u(t) is sampled every 1
second. The expression for the sampled sequence cfw_y[k] is given by:
y[k ] = 0.5 0.5(0.6)
k
EE 4CL4 Control System Design
Solutions to Homework Assignment #2
1. Consider an electronic amplifier with input voltage vi(t) and output voltage vo(t). Assume that:
vo (t ) = 8vi (t ) + 2 .
(1)
a. Show that the amplifier does not strictly satisfy the pri
EE 4CL4 Control System Design
Solutions to Homework Assignment #9
1. Compute the Z-transform for the discrete sequences that result from sampling the following
signals a 1 Hz:
a. e 0.1t cos(0.5t + 4 ) .
b. t 2 e 0.25t .
(25 pts)
If the sampling frequency
EE 4CL4 Control System Design
Solutions to Homework Assignment #4
1. The input-output model for a system is given by:
d 2 y (t )
d y (t )
&
y
and y (t ) =
where &(t ) =
.
2
dt
dt
&(t ) + 7 y (t ) + 12 y (t ) = 3u (t ) ,
&
y
a. Determine the system transfe
ELEC ENG 4CL4 Control System Design
Solutions to Homework Assignment #2
1. A system has the transfer function:
H (s) =
s+ 1
2
.
s2 + 2s + 1
a. Will the step response of this system exhibit undershoot or overshoot? Explain your
answer in terms of the locat