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CONTROL SYSTEMS ENGINEERING
Sixth Edition
International Student Version
Norman S. Nise
California State Polytechnic University, Pomona
WILEY
John Wiley Er Sons, Inc. 326 Chapter 6 Stability
3. What would happen to
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Control Systems
Engineering CONTROL SYSTEMS ENGINEERING !
Sixth Edition
International Student Version
Norman S. Nise
California State Polytechnic University, Pomona
WILEY
John Wiley & Sons, Inc. Probl
(i)
(s«~a)3+o 3 ‘
s+a
(s+a)3+(o2 '
a. Using the frequency shift theorem and the Laplace transform of sin rot. F(s) =
b. Using the frequency shift theorem and the Laplace transform of cos (0t. F(s) =
7
l l A l . . l - - t-
c. Usmg the integration theorem.
EEE 3310: Control Engineering
Homework #3
Due: 31 Oct. 2014
Please carefully read Chapter 4 of a textbook and solve the following problems.
1. Solve problem 4.2 (10 point)
2. Solve problem 4.15 (10 point)
3. Solve problem 4.20 (15 point)
4. Solve problem
EEE 3310: Control Engineering
Homework #2
Due: 29 Oct. 2014
Please carefully read Chapter 3 of textbook and solve following problems:
1.
2.
3.
4.
5.
6.
Solve problem 3.2
Solve problem 3.9
Solve problem 3.10
Solve problem 3.14
Solve problem 3.15
Solve prob
EEE 3310: Control Engineering
Homework #4
Due: 26 Nov. 2014
Please carefully read Chapter 6 of a textbook and solve the following problems.
1. Solve problem 6.1 (15 point)
2. Solve problem 6.6 (15 point)
3. Solve problem 6.32 (20 point)
4. Solve problem 6
EEE 3310: Control Engineering
Homework #1
Due: 8 Oct. 2014
Please carefully read Chapter 2 of textbook and solve following problems:
1.
2.
3.
4.
5.
6.
Solve problem 2.2
Solve problem 2.7
Solve problem 2.9
Solve problem 2.18
Solve problem 2.21
Solve proble
2- 3
16.9
11
10
09
7.
Let F .
(a) Compute the outward flux of F through the sphere .
[6pts]
(b) Compute the outward flux of F through the ellipsoid .
[6pts]
8.
Let F and be the boundary surface of the solid region
enclosed by the paraboloid and the p
2- 3
16.8
11
10
09
6. Use Stokes' theorem to evaluate
F where
F
and is the triangle with vertices and , oriented counter
clockwise as viewed from above. [7pts]
2- 3
16.7
10
09
5. Let be the surface obtained by rotating the curve , about
the -axis, where and is continuous.
(a) Find the vector equation of the surface . [3pts]
(b) Show that the area of is
.
[6pts]
08
7. [10pts] Let be the surface obtained by rot
2- 3
16.5
11
09
4.
(a) Show that di v , where is a scalar function defined on
R .
Assume that has continuous second order partial derivatives on .
[5pts]
(b) Use the result of (a) to show that if is harmonic (that is, ) on , and
if on the boundary curve
2- 3
16.4
11
09
2. (a) Let be the region bounded by a positively oriented, piecewise-smooth,
simple closed curve in the plane.
given by
Prove that the area of the region is
.
[6pts]
(b) Use the formula in (a) to find the area under one arch of the cyc
2- 3
16.3
11
10
09
1.
(a)
Let F .
Determine whether
or
not
the vector
field
conservative, find a function such that F .
(b) Find
F where :
[6pts]
sin
F
is conservative.
If
[6pts]
s i n s i n cos cos
it
is
2- 2
15.8
11
10
09
8. (a) Evaluate
.
[5pts]
(b) Evaluate
.
[5pts]
08
7.[8pts] Use spherical coordinates to find the volume of the solid that lies under
the plane and above the cone .
8. [8pts] The volume of the closed ball
in
is , by evaluat
2- 2
15.6
11
09
7. Evaluate the following integrals.
(a)
cos
cos
ar ctan
(b)
[6pts]
[6pts]
(c)
, where and denotes the greatest
integer which is not larger than . [6pts]
(d)
[6pts]
10.
Determine whether the statement is true or false
2- 2
15.3
11
10
09
7. Evaluate the following integrals.
(a)
cos
cos
ar ctan
(b)
[6pts]
[6pts]
(c)
, where and denotes the greatest
integer which is not larger than . [6pts]
(d)
10.
[6pts]
Determine whether the statement is true or fa
2
14.2
11
10
09
8. Find the limit, if it exists, and prove your statement using
definition of limit, or show that the limit does not exist.
(a)
(b)
08
lim
[7pts]
lim
[7pts]
8. Find the limit, if it exists, and prove your statement using
definition
2
13.3~13.4
08
3.[10pts] Show that is perpendicular to the binormal vector and the
unit tangent vector , and deduce from them that
for some number (called the torsion of curve). (Here is the
principal unit normal vector.)
4. [7pts] Reparametrize the curv
2
13.2~13.3
13.2
10
13.3
11
10
09
4. Let be the unit tangent, unit normal, and binormal
vector, respectively on
a smooth space curve and be a
point of the curve.
(a) Show that the vector at lies in the plane determined by
and ,
constants.
that is,
where
2- 1
12.5
11
10
09
2. Find the volume of the tetrahedron with sides
, , and .
[5pts]
3. Find parametric equations for the line through the point that
is parallel to the plane and perpendicular to the line
, , . [5pts]
08
1.[7pts] Let and be the lines wi
2- 1
12.4
11
10
09
1.(a) Show that ,
where and are noncoplanar vectors, and
(Hint: , )
[7pts]
(b) Show that for any 3-dimensional nonzero vectors and , is
orthogonal to .
[5pts]
11.1 Sequences
(Limit Laws for sequences)
If
1.
and
are convergent sequences and
is a constant, then
(Definition 1)
lim lim lim
A sequence
has the limit ,
lim lim
lim
lim
lim lim lim
if we can make the terms as close to as
we like by taking s
10.1 Curve Defined by Parametric
.
Equations.
()
.
1. Parametric Equations
3 (parameter; )
(parametric equation; )
.
: initial point
: terminal point
,
,
(Ex 2)
(parametric curve; ) .
What curve is represented by the following
parametr