## Chapter 7. Theory Preview

There exists a beautiful and rather deep theory of feedback systems, which we will sketch in Part IV. Yet for most of the applications that we are interested in, this theory is not strictly required. Moreover, the theory makes several assumptions that are not necessarily fulfilled by computer systems and is therefore not even fully applicable.

Nevertheless, the theoretical description yields several useful terms and concepts that are pervasive in all of control theory. In this chapter, we will summarize the most important of those ideas so that we can use them in the sequel. At this point, we will skip most motivation and justification—if you want to know more, please refer to Part IV.

## Frequency Representation

The classical theory of feedback systems is based on a mathematical operation (the *Laplace transform*) that allows us to express any function of *time* *t* as a function of the (complex) *frequency* *s*. The two representations are completely equivalent, and we can freely transform back and forth between the time domain and the frequency domain.

The Laplace transform is not universally applicable. It applies only to systems whose time evolution is described by linear, time-invariant differential equations. Many systems in the physical world are in this category; in particular this is true for many of the mechanical, electrical, and thermal assemblies for which classical feedback theory was originally developed.

## The Transfer Function

In the frequency representation, ...

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