Wiley signals and systems e book TLFe BO 405

16. Stability and Feeclback Svsteriis

390

16.22 . 2

Discrete Systems

A rorrcspondirrg result tiolds for discrete systems:

The poles of the system fiinction H ( z ) of a causal arid stable discrete
LTI-s,ybtern lie within the unit circle of the x-plane.

IIere we can also see the connection betweerl the locsation of tlie poles in Che
complex frequency plane arid the system's characteristic frequencies in the timedomain (sec Exercise 16.5). The stability conditions can alsn be illustrated to aid
understauding, as in Figure 13.6. The internal term of the system ~esponsehere
only decays if thr corresponding poles lie within the unit circle. Again, the initial
states have rio influence if one waits long enough.

Example 16.3
Figarc. 16 2 shows four pole-zero tliagrnnis of both contix~uous(left) a i d discrete
(right) system. We can bee inixnediately fiom tlie locatioix of the poles tlmt only
the first and third conlirruous system (likewise discrete systeixi) are stable. If,
hom7ever, biliiteral impulsc responses are permitted - systems which are not causal
- the first three continuous systems and tlie first, second and fourth disciete system
are stable. The ROC must be chosen so t h a t it lnrludes the imaginary axis of the
s-plane, <)I' the unit circle of the a-plane.
po1t.s lie directly on thc iniaginary
axis or unit circle, t h i s t,ecornes irnlmssible, and the system will always be ematabl~.

Stability Criteria

If the trijnsfer fiinction of an LTI-system in rat,ional fonn is determined from a
differential or cliffercwe ecpat,ion, only the numerator and denominator coefficients
arc: obtained at. first. In order to check the stability with the location of the poles,
the zeros of the denominator polynomial must be determined. This can be given
in closed form for polynomials up to the third degree, but for higher-order systems
iterative procedures arc necessary. Once the digital computers needed to carry out
these procedurcs m w c unmailablc, but modern computers can do this easily. To
avoid having to perform the nurnerical senrch for the zeros by linnd, a series of easy
t o perform stability tests were developed, Instead of cal(*rilatingthe iridividiinl pole
locatioris, they jnst determine whether all poles lie in the left half of the s-plane
(or unit circle ol the z-plane). We will only br ronsidering one test for continuous
and discrete systems becai~sethe tests all have essentially the banie effect,