Chapter 26. Topics Beyond This Book

The last few chapters have offered a fairly comprehensivse sketch of what could be called basic or elementary feedback theory. Of course, there is much more that could be said.

Discrete-Time Modeling and the z-Transform

The theory presented here assumes that time is a continuous variable. This is not true for digital systems, where time progresses in discrete steps. When applying the continuous-time theory to such processes, care must be taken that the step size is smaller (by at least a factor of 5–10) than the shortest time scale describing the dynamics of the system. If this condition is not satisfied, then the continuous-time theory can no longer be safely regarded as a good description of the discrete-time system.

There is an alternate version of the theory that is based directly on a discrete-time model and that is generally useful if one desires to treat discrete time evolution explicitly. In discrete time, system dynamics are expressed as difference equations (instead of differential equations) and one employs the z-transform (instead of the Laplace transform) to make the transition to the frequency domain.

Structurally, the resulting theory is very similar to the continuous-time version. One still calculates transfer functions and examines their poles and zeros, but of course many of the details are different. For instance, for a system to be stable, all of its poles must now lie inside the unit circle around the origin in the (complex) ...

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