Uploaded November 2022 | Updated September 2026, 1 week ago
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.
Playlist: youtube.com/playlist?list=PLMrJAkhIeNNTYaOnVI3QpH7jgULnAmvPA
Course Website: http://faculty.washington.edu/sbrunton/me564/
@eigensteve on Twitter
eigensteve.com
databookuw.com
This video was produced at the University of Washington
%%% CHAPTERS %%%
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)
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.
Playlist: youtube.com/playlist?list=PLMrJAkhIeNNTYaOnVI3QpH7jgULnAmvPA
Course Website: http://faculty.washington.edu/sbrunton/me564/
@eigensteve on Twitter
eigensteve.com
databookuw.com
This video was produced at the University of Washington
%%% CHAPTERS %%%
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)
