RE: Laplace Transforms

LAPLACE TRANSFORMS
LAPLACE TRANSFORMS1.ppt (Size: 639.5 KB / Downloads: 21)
Definition
Transforms  a mathematical conversion from one way of thinking to another to make a problem easier to solve
Definition  Partial fractions are several fractions whose sum equals a given fraction
Purpose  Working with transforms requires breaking complex fractions into simpler fractions to allow use of tables of transforms
Different terms of 1st degree
To separate a fraction into partial fractions when its denominator can be divided into different terms of first degree, assume an unknown numerator for each fraction
Repeated terms of 1st degree (1 of 2)
When the factors of the denominator are of the first degree but some are repeated, assume unknown numerators for each factor
If a term is present twice, make the fractions the corresponding term and its second power
If a term is present three times, make the fractions the term and its second and third powers
Effect of Control Actions
Proportional Action
Adjustable gain (amplifier)
Integral Action
Eliminates bias (steadystate error)
Can cause oscillations
Derivative Action (“rate control”)
Effective in transient periods
Provides faster response (higher sensitivity)
Never used alone
Basic Controllers
Proportional control is often used by itself
Integral and differential control are typically used in combination with at least proportional control
eg, Proportional Integral (PI) controller:
Summary of Basic Control
Proportional control
Multiply e(t) by a constant
PI control
Multiply e(t) and its integral by separate constants
Avoids bias for step
PD control
Multiply e(t) and its derivative by separate constants
Adjust more rapidly to changes
PID control
Multiply e(t), its derivative and its integral by separate constants
Reduce bias and react quickly
Rootlocus Analysis
Based on characteristic eqn of closedloop transfer function
Plot location of roots of this eqn
Same as poles of closedloop transfer function
Parameter (gain) varied from 0 to
Multiple parameters are ok
Vary onebyone
Plot a root “contour” (usually for 23 params)
Quickly get approximate results
Range of parameters that gives desired response 

