Linear Systems of Differential Equations with Forcing: Convolution and the Dirac Delta Function

This video derives the fully general solution to a matrix system of linear differential equation with forcing in terms of a convolution integral. We start off simple, by breaking the problem down into simple sub-problems. One of these sub-problems is deriving the response of the system to an impulsive delta-function input (the so-called impulse response). This involves introducing the Dirac delta function. Next we show how a generic input forcing function u(t) may be seen as a sequence of infinitesimal delta functions, allowing us to derive the convolution integral solution.

This approach is a cornerstone of control theory and linear systems analysis.

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This video was produced at the University of Washington

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0:00 Overview
1:14 Case 1: Initial condition response with no forcing
5:19 The Dirac delta function
11:30 Case 2: Impulse response for delta function input
20:40 Case 3: Impulse response with an initial condition
28:55 Convolution integral for arbitrary forcing u(t)

Видео Linear Systems of Differential Equations with Forcing: Convolution and the Dirac Delta Function канала Steve Brunton