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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1.Standard notation in dynamics and control
2. Converts mathematics to algebraic operations
3. Advantageous for block diagram analysis
Usually define f(0) = 0 (e.g., the error)
Solve the ODE,
First, take L of both sides of (3-26),
From Table 3.1 (line 11),
system at rest (s.s.)
Step 1Take L.T. (note zero initial conditions)
Step 2a. Factor denominator of Y(s)
Step 2b. Use partial fraction decomposition
Multiply by s, set s = 0
for a3, a4)
Step 3.Take inverse of L.T.
(check original ODE)
You can use this method on any order of ODE,
limited only by factoring of denominator polynomial
Must use modified procedure for repeated roots,
1. Solution of differential equations (linear)
2. Analysis of linear control systems
3. Prediction of transient response for
Transforms to e-t/3, e-t
Real roots = no oscillation
Complex roots = oscillation
From Table 3.1, line 17 and 18
Let h→0, f(t) = δ(t) (Dirac delta) L(δ) = 1
Use L’Hopital’s theorem
If h = 1, rectangular pulse input
(S(t-1) is step starting at t = h = 1)
By Laplace transform
Can be generalized to steps of different magnitudes
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
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.
A. Final value theorem
Example 3: step response
offset (steady state error) is a.
by initial value theorem
by final value theorem