MATH 551: Linear Dynamic Systems
Assignment 1
Due: September 12, 2016
1. (Product Spaces) Let X1 , X2 , . . . , Xk be vector spaces over the same field F. The
Cartesian Product
k
Y
X = X1 X2 . . . Xk =
Xi
i=1
is defined to be the set of all vectors of the
MATH 551: Linear Dynamic Systems
Assignment 2
Due: September 21, 2016
1. Let A be a real matrix, not necessarily square. Of course both AA0 and A0 A are square
matrices but they are not necessarily of the same size, where A0 denotes the transpose
of A. A
ESE 551: Linear Dynamic Systems
Assignment 3
Due: October 12, 2016
1. Let A and X be n n matrices. The linear transformation L : Rnn Rnn is defined
by
LX = A0 X + XA.
Compute the adjoint of L.
solution: Define an inner product h, i : Rnn Rnn R by hX, Y i
ESE 551: Linear Dynamic Systems
Assignment 4
Due: October 24, 2016
1 1 3
1 4 3
3 3 4
1. Is the matrix
positive definite or not?
2. Given the control system
x1
d
x2 =
dt x3
x4
0
1
0
0
x1
0
0
1
0 x2
0
0
0
1 x3
1 2 1 1
x4
0
0
+ u,
9
3
y = x1 +
ESE 551: Linear Dynamic Systems
Assignment 3
Due: October 12, 2016
1. Let A and X be n n matrices. The linear transformation L : Rnn Rnn is defined
by
LX = A X + XA.
Compute the adjoint of L.
2. Consider a continuous linear mapping (a bounded operator)
L
ESE 551: Linear Dynamic Systems
Assignment 4
Due: October 24, 2016
1. Is the matrix
1 1 3
1 4 3
3 3 4
positive definite or not?
Solution. If the matrix above, denoted by A, is positive definite, then det(A) > 0.
However, det(A) = 15 < 0, thus A is not p
ESE 551: Linear Dynamic Systems
Assignment 2
Due: September 21, 2016
1. Let A be a real matrix, not necessarily square. Of course both AA and A A are square
matrices but they are not necessarily of the same size, where A denotes the transpose
of A. A mome
ESE 551: Linear Dynamic Systems
Assignment 1
Due: September 12, 2016
1. (Product Spaces) Let X1 , X2 , . . . , Xk be vector spaces over the same field F. The
Cartesian Product
k
Y
X = X1 X2 . . . Xk =
Xi
i=1
is defined to be the set of all vectors of the
Confirming Pages
9.4
3.
4.
Temperature Measurement
Law of intermediate junction metals. As illustrated in Figure 9.39, if a
third metal is introduced within a junction creating two new junctions (A-C
and C-B), the measured voltage will not be affected as
Confirming Pages
3.3
Junction Diode
Zener Regulation Performance
83
EXAMPLE 3.2
We wish to determine the regulation performance of the zener diode circuit shown in
Figure 3.8 for a voltage source Vin whose value ranges between 20 and 30 V. For the zener
d
Confirming Pages
A.1
Systems of Units
527
CLASS DISCUSSION ITEM A.2
Common Use of SI Prefixes
For each of the prefixes listed in Table A.4, think of or research an example of a measurable physical quantity for which the prefix is commonly used to express
Confirming Pages
10.7 Selecting a Motor
torque
motor
torque-speed
curve
467
operating
point
operating
speed
load
line
speed
Figure 10.33 Motor operating speed.
Is the motor torque-speed curve well matched to the load? If the load has
a well-defined torque
Confirming Pages
9.3
Stress and Strain Measurement
397
where Vex is the DC voltage applied to the bridge called the excitation voltage. Substituting these expressions into Equation 9.14 and rearranging gives
R2
R1
- = -R4
R3
(9.17)
If we know R2 and R3 ac
Confirming Pages
4.4 Bandwidth and Frequency Response
127
where i is the frequency component number. For example, for i = 2 corresponding to
frequency 20,
A' = (Aout Ain)2 A2 = 0.25A2
2
(4.24)
Applying Equation 4.23 to each frequency component results in
Confirming Pages
10.6 Stepper Motors
457
Table 10.1 Unipolar full-step phase sequence
Step
1
2
3
4
CW
1
ON
OFF
ON
OFF
2
ON
OFF
OFF
ON
CCW
3
OFF
ON
OFF
ON
4
OFF
ON
ON
OFF
Table 10.2 Unipolar half-step phase sequence
1
2
3
4
1
ON
OFF
ON
OFF
1.5
ON
OFF
OFF
O
Confirming Pages
8.2 Quantizing Theory
351
many more video demonstrations of various music, sound, and vibration concepts,
see Internet Link 8.2.
8.2
Video Demo
QUANTIZING THEORY
Now we look at the process required to change a sampled analog voltage into
Confirming Pages
Questions and Exercises
157
c. Assuming the signal is being measured by a measurement system with the frequency response curve shown in Question 4.10, plot the spectrum of
the resulting output of the system.
4.12. Determine and plot the f
Confirming Pages
5.6 Noninverting Amplifier
169
Vin
t
Vout
t
Figure 5.9 Illustration of inversion.
Therefore, the voltage gain of the amplifier is determined simply by the external
resistors RF and R, and it is always negative. The reason this circuit is
Confirming Pages
9.2
Position and Speed Measurement
383
CLASS DISCUSSION ITEM 9.3
LVDT Signal Filtering
Given the spectrum of a time-varying core displacement, what effect does the
choice of the primary excitation frequency have, and how should the low-p
Confirming Pages
7.4
Programming a PIC
269
PIC16F84 are listed in Table 7.2. Each command consists of a name called the
mnemonic and, where appropriate, a list of operands. Values must be provided for
each of these operands. The letters f, d, b, and k cor
Confirming Pages
7.10
Practical Considerations
335
' Turn off the diagnostic LED
High led
' Wait to ensure the PIC16F88 is initialized
PAUSE 500
' Initialize I/O signals
High enc_oe
' disable encoder output
Low enc_sel
' select the encoder counter high by
Confirming Pages
7.5
Picbasic Pro
279
my_byte.3 1 would set bit 3 in my_byte to 1 and my_array[9].7 0 would set the
MSB of the last element of my_array to 0. All bits within a variable can be set by
assigning a value or expression to the variable with an
Confirming Pages
5.11 Differentiator
of C is large at low frequencies. At very high frequencies (i.e., 1/R1C), the
output is attenuated to zero because the feedback loop is effectively a short. Lab
Exercise 6 explores and Video Demo 5.5 demonstrates how t
Confirming Pages
Questions and Exercises
6.13.1
249
IEEE Standard Digital Symbols
The IEEE standard symbols for illustrating digital input and output conditions are
shown in Figure 6.46. When drawing or reading digital schematics, it is important to
recog
Confirming Pages
4.10 Second-Order System
4.10.2
143
Frequency Response of a System
To determine the frequency response of a linear system, we apply a sinusoidal input
and determine the output response for different input frequencies. For the secondorder
Confirming Pages
2.9 Impedance Matching
49
source
Rs
+
+
Vs
RL (load resistance)
VL
Figure 2.35 Impedance matching.
To find the load resistance that maximizes this power, we set the derivative of the
power equal to 0 and solve for the load resistance:
2
d
Confirming Pages
Questions and Exercises
Section 3.5
115
Field-Effect Transistors
3.22. For the MOSFET used in the experiment of Figure 3.31, estimate the fully-on drainto-source resistance Ron from the values in the plot.
3.23. Answer Question 3.20 using
Rev. Confirming Pages
3.4 Bipolar Junction Transistor
97
current drops. This effect is called emitter degeneration. The reason for this is that
the base current adds to the collector current, causing an increase in the emitter current (IE IB IC), creating