5 Feedback

Linear systems have two basic types, namely open or closed loop.

Open loop Here the input $u$ is subject to linear operations and produces an output $y$. For example

$$y(t)=atu(t)+u^{\prime \prime }(t).$$

Closed loop or feedback systems Here we take the input, subject it to linear operations, and also take the output, feed it back into the system, apply linear operations to the output and then add the modified input and the modified output. For example

$$y^{\prime \prime }+b{y}^{\prime }+cy=u$$

can be written as

$$y=-c\int \int y-b\int y+\int \int u.$$