LAPLACE TRANSFORMS
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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 (steady-state 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

Root-locus Analysis
Based on characteristic eqn of closed-loop transfer function

Plot location of roots of this eqn

Same as poles of closed-loop transfer function

Parameter (gain) varied from 0 to

Multiple parameters are ok

Vary one-by-one

Plot a root “contour” (usually for 2-3 params)

Quickly get approximate results

Range of parameters that gives desired response