This chapter is concerned with the various techniques available for the analysis of the stability of discrete-time systems.

Suppose we have a closed-loop system transfer function

where 1 + *GH*(*z*) =0 is also known as the characteristic equation. The stability of the system depends on the location of the poles of the closed-loop transfer function, or the roots of the characteristic equation *D*(*z*) = 0. It was shown in Chapter 7 that the left-hand side of the *s*-plane, where a continuous system is stable, maps into the interior of the unit circle in the *z*-plane. Thus, we can say that a system in the *z*-plane will be stable if all the roots of the characteristic equation, *D*(*z*) = 0, lie inside the unit circle.

There are several methods available to check for the stability of a discrete-time system:

- Factorize
*D*(*z*) = 0 and find the positions of its roots, and hence the position of the closed-loop poles. - Determine the system stability without finding the poles of the closed-loop system, such as Jury's test.
- Transform the problem into the
*s*-plane and analyse the system stability using the well-established*s*-plane techniques, such as frequency response analysis or the Routh–Hurwitz criterion. - Use the root-locus graphical technique in the
*z*-plane to determine the positions of the system poles.

The various techniques described in this section will be illustrated with examples. ...

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