Slides - College of Engineering, Purdue University

ECE 382—Feedback Systems Analysis and Design
˙
Stan Zak
School of Electrical and Computer Engineering
Purdue University
[email protected]
August 27, 2014
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
1 / 11
Today’s class
The first homework assignment posted
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
The The Laplace transform defintion
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
The The Laplace transform defintion
Examples of taking the Laplace transform
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
The The Laplace transform defintion
Examples of taking the Laplace transform
Some properties of the Laplace transform
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
The The Laplace transform defintion
Examples of taking the Laplace transform
Some properties of the Laplace transform
The initial-value theorem
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
Today’s class
The first homework assignment posted
Today’s topic—The Laplace transform review
The The Laplace transform defintion
Examples of taking the Laplace transform
Some properties of the Laplace transform
The initial-value theorem
The final-value theorem
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
2 / 11
The Laplace transform
Definition:
The Laplace transform is an operator that transforms a function of time,
f (t), into a new function of complex variable, F (s), where s = σ + jω,
Z ∞
F (s) = L(f (t)) =
f (t)e −st dt
0−
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
3 / 11
The Laplace transform is a linear operator
Schematic representation of the Laplace transform operator
f (t) -
˙
Stan Zak
(Purdue University)
L(·)
F (s) = L(f-(t))
ECE 382: Second class of Fall 2014
August 27, 2014
4 / 11
The Laplace transform is a linear operator
Schematic representation of the Laplace transform operator
f (t) -
L(·)
F (s) = L(f-(t))
The operator L denotes that the time function f (t) has been
transformed to its Laplace transform, denoted F (s)
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
4 / 11
The inverse Laplace transform
L−1 (F (s)) = f (t)
F (s)-
˙
Stan Zak
(Purdue University)
L−1 (·)
f (t) = L−1
(F (s))
-
ECE 382: Second class of Fall 2014
August 27, 2014
5 / 11
The inverse Laplace transform
L−1 (F (s)) = f (t)
F (s)-
L−1 (·)
f (t) = L−1
(F (s))
-
We will use only the one-sided Laplace transform
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
5 / 11
Consequences of using the one-sided Laplace transform
Because we are using the one-sided Laplace transform, we define all
functions, whose Laplace transforms we compute, to be zero for
t < 0−
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
6 / 11
Consequences of using the one-sided Laplace transform
Because we are using the one-sided Laplace transform, we define all
functions, whose Laplace transforms we compute, to be zero for
t < 0−
The definition of the unit step function, u(t),
1 if t ≥ 0
u(t) =
0 if t < 0
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
6 / 11
Consequences of using the one-sided Laplace transform
Because we are using the one-sided Laplace transform, we define all
functions, whose Laplace transforms we compute, to be zero for
t < 0−
The definition of the unit step function, u(t),
1 if t ≥ 0
u(t) =
0 if t < 0
The unit step function is also called the Heaviside function
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
6 / 11
Solving differential equations using the Laplace transform
Need the Laplace transforms of derivatives
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
7 / 11
Solving differential equations using the Laplace transform
Need the Laplace transforms of derivatives
L dfdt(t) = sF (s) − f (0− )
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
7 / 11
Solving differential equations using the Laplace transform
Need the Laplace transforms of derivatives
L dfdt(t) = sF (s) − f (0− )
Differentiating in the time domain corresponds to multiplying F (s) by
s and then subtracting the initial value of f (t)
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
7 / 11
Solving differential equations using the Laplace transform
Need the Laplace transforms of derivatives
L dfdt(t) = sF (s) − f (0− )
Differentiating in the time domain corresponds to multiplying F (s) by
s and then subtracting the initial value of f (t)
2 df (0− )
L dfdt(t)
= s 2 F (s) − sf (0− ) − dt
2
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
7 / 11
Solving differential equations using the Laplace transform
Need the Laplace transforms of derivatives
L dfdt(t) = sF (s) − f (0− )
Differentiating in the time domain corresponds to multiplying F (s) by
s and then subtracting the initial value of f (t)
2 df (0− )
L dfdt(t)
= s 2 F (s) − sf (0− ) − dt
2
n =
L dfdt(t)
n
df (0− )
d n−2 f (0− )
d n−1 f (0− )
s n F (s) − s n−1 f (0− ) − s n−2 dt − · · · − s dt n−2 − dt n−1
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
7 / 11
The impulse function
The impulse function, denoted δ(t), also called the Dirac function, is
a signal of infinite amplitude, zero duration, and unity area
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
8 / 11
The impulse function
The impulse function, denoted δ(t), also called the Dirac function, is
a signal of infinite amplitude, zero duration, and unity area
We can construct an impulse function as the limit of pulse functions
pεi (t) =
1
(u(t) − u(t − εi ))
εi
as εi → 0
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
8 / 11
Generating δ(t) as the limit of pulse functions
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
9 / 11
The Initial-Value Theorem
Theorem:
Let F (s) = L(f (t)) be a strictly proper rational function. Then,
lim sF (s) = f 0+
s→∞
˙
Stan Zak
(Purdue University)
ECE 382: Second class of Fall 2014
August 27, 2014
10 / 11
The Final-Value Theorem
Theorem:
Suppose that F (s) = L(f (t)) has poles only in the open left-half complex
plane, with the possible exception of a single-order pole at s = 0. Then,
lim sF (s) = lim f (t)
s→0
˙
Stan Zak
(Purdue University)
t→∞
ECE 382: Second class of Fall 2014
August 27, 2014
11 / 11

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