## Chapter 21. Block-Diagram Algebra and the Feedback Equation

In Chapter 20 we saw that the dynamic behavior of a system is given as the solution to a differential equation. We also saw how the Laplace transform could be used to repackage all the dynamic information contained in a linear, time-invariant differential equation into a simple function (the *transfer function*). In this chapter, we will show how the dynamic behavior of a *combination* of systems can be found from the transfer functions of the individual elements.

## Composite Systems

In Chapter 20, we saw that, in the frequency domain, a
system’s dynamic response *y*(*s*) to an external input
*u*(*s*) is given by the product of the system’s
transfer function *H*(*s*) and the input^{[22]}

*y*(

*s*) =

*H*(

*s*)

*u*(

*s*)

We can express this equation as a *block diagram*, where the system (described by its transfer function *H*) transforms the input *u* to the output *y*:

Obviously, we can combine several such systems in series, with the output of one serving as input to the next:

The output of this composite system is the product of its components:

*y*(

*s*) =

*H*(

*s*)

*G*(

*s*)

*u*(

*s*)

This follows simply because the output of the first element is *x*(*s*) = *G*(*s*) *u*(*s*) and because the output of the second component, acting on the output of the first, is *y*(*s*) = *H*(*s*) *x*(*s*). Therefore, the transfer function ...

Get *Feedback Control for Computer Systems* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.