Stability and Feeclback Svsteriis 16.2 2.. 2 Discrete Systems A rorrcspondirrg result tiolds for discrete systems: The poles of the system fiinction H z of a causal arid stable discr
Trang 1390 16 Stability and Feeclback Svsteriis
16.2 2 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 time- domain (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 ~esponse here 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 th e 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(*rilating the iridividiinl pole locatioris, they jnst determine whether all poles lie in the left half of the s-plane (or unit circle o l the z-plane) We will only br ronsidering one test for continuous and discrete systems becai~se the tests all have essentially the banie effect,