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Laplace Transforms. 1. Standard notation in dynamics and control (shorthand notation) 2. Converts mathematics to algebraic operations 3. Advantageous for block diagram analysis. Chapter 3. Laplace Transform. Example 1:. Chapter 3. Usually define f(0) = 0 (e.g., the error).

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Laplace Transforms

1.Standard notation in dynamics and control

(shorthand notation)

2. Converts mathematics to algebraic operations

3. Advantageous for block diagram analysis

Chapter 3


Laplace Transform

Example 1:

Chapter 3

Usually define f(0) = 0 (e.g., the error)


Other Transforms

etc. for

Chapter 3



Example 3.1

Solve the ODE,

First, take L of both sides of (3-26),

Chapter 3


Take L-1,

From Table 3.1 (line 11),



system at rest (s.s.)

Chapter 3

Step 1Take L.T. (note zero initial conditions)



Step 2a. Factor denominator of Y(s)

Chapter 3

Step 2b. Use partial fraction decomposition

Multiply by s, set s = 0


For a2, multiply by (s+1), set s=-1 (same procedure

for a3, a4)

Step 3.Take inverse of L.T.

Chapter 3

(check original ODE)

You can use this method on any order of ODE,

limited only by factoring of denominator polynomial

(characteristic equation)

Must use modified procedure for repeated roots,

imaginary roots


Laplace transforms can be used in process control for:

1. Solution of differential equations (linear)

2. Analysis of linear control systems

(frequency response)

3. Prediction of transient response for

different inputs

Chapter 3


Factoring the denominator polynomial


Chapter 3

Transforms to e-t/3, e-t

Real roots = no oscillation



Transforms to

Complex roots = oscillation

Chapter 3

From Table 3.1, line 17 and 18


Chapter 3

Let h→0, f(t) = δ(t) (Dirac delta) L(δ) = 1

Use L’Hopital’s theorem


If h = 1, rectangular pulse input


Difference of two step inputs S(t) – S(t-1)

(S(t-1) is step starting at t = h = 1)

By Laplace transform

Chapter 3

Can be generalized to steps of different magnitudes

(a1, a2).


One other useful feature of the Laplace transform

is that one can analyze the denominator of the transform

to determine its dynamic behavior. For example, if

the denominator can be factored into (s+2)(s+1).

Using the partial fraction technique

Chapter 3

The step response of the process will have exponential terms

e-2t and e-t, which indicates y(t) approaches zero. However, if

We know that the system is unstable and has a transient

response involving e2t and e-t. e2t is unbounded for

large time. We shall use this concept later in the analysis

of feedback system stability.


Other applications of L( ):

A. Final value theorem


Example 3: step response

Chapter 3

offset (steady state error) is a.

  • Time-shift theorem
  • y(t)=0 t < θ

C. Initial value theorem

Chapter 3

by initial value theorem

by final value theorem


Chapter 3

Previous chapter

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