Week 2

Question 1

Transfer Function

Answer 1

The transfer function of a LTI system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, under the assumption that all initial conditions are zero.

Question 2

The response of the system to three types of inputs

Answer 2

Impulse inputs
Step inputs
Ramp inputs

Question 3

For a unit-step input

Answer 3

Question 4

Substituting into the first-order input-output relationship

Answer 4

Question 5

Unit-impulse

Answer 5

Question 6

Unit-ramp

Answer 6

Question 7

Second-order systems are those with dynamics that can be described by an equation of the form

Answer 7

Question 8

The system is characterised by two parameters

Answer 8

ωn, the undamped natural frequency, describes the unforcedoscillatory behaviour of the system, in the absence of damping
ζ, the damping ratio, describes the level of damping in the system

Question 9

The value of ζ determines the form (shape) of the response

Answer 9

0 < ζ < 1 : system is underdamped, transient response is oscillatory and decaying
ζ > 1 : system is overdamped, transient response decays without oscillation
ζ = 1 : system is critically damped, transient response is on boundary between being underdamped and overdamped (no oscillation)