· Can be used to find normal input reponses for linear systems
· It is most useful for finding an output response for a system given an arbitrary input function
· It is also the basis for other methods that come later for system analysis.
· A unit impulse function
· For a unit step function,
· If we look at an input signal (force here) we can break it into very small segments in time. As the time becomes small we can approximate it with a set of impulses.
Figure 7.1 An impulse as a brief duration (instant) pulse
· If we put an impulse into a system the output will be an impulse response.
Figure 7.2 Response of the system to a single pulse
· If we add all of the impulse responses together we will get a total system response. This operation is called convolution.
Figure 7.3 A set of pulses for a system gives summed responses to give the output
· Consider the unit impulse of a system with the given differential equation. Note: This method is only valid for trivial differential equations with only one homogeneous term. The preferred method is shown later.
· Note that the derivation of the unit impulse function assumed zero initial conditions, so the process of convolution must also assume systems start at rest and undeflected.
· The following example shows the use of the convolution integral to find a number of responses.
· The convolution integral can also be solved numerically. This is particularly useful for systems with arbitrary inputs.
· This can be applied to the previous example for a unit step input to find the system position at 10 seconds, with a 2 second time step.
· A Scilab program to perform the previous calculation numerically is shown below.
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