# System Stability

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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