Wiley signals and systems e book TLFe BO 398

wc.11 as causality, another important criterion that must be met for a system t o
be implemented is stabilitj. II a system is stable, and the input signal i s bounded,
then the output signal cannot grow beyond limit. This condition niust be fulfilled,
if continuous systerris are to be rcdised based on the fundamental pliysical principle
of conservation of energy, for example, electrical, optical, rnechanical, hydraulic or
pneumatic systcms. For a discrete system the signal ainplilude must stay within
certain limit5 as defiiied by the perrnittcd numerical range of- the coinpiiter being
used. The first thing we will do in this chapter is investigate the connections
between stability, the frequency respoiisc and the impulse response foi gmcral LTIsystems. Then we will restrict ourselves to caim-11systems and coiisidei stability
tests using the pol~-xerodiagram. Fiiially we will disciiss sonic typical uses of
ms with feedback. We will again be dealing with continuous arid discrete
systems a1 the same time.

A5

There me various possible ways of definirig the stability of a system. Among them,
BlRO-stability is particularly useful for LTI-systems. First of all we will introduce
the concepts boimded functzori arid bounded series.
21:

Bounded function, bounded series

A functzon as s a d to De bounded when 22s muynstude
for nll lime t:
[z(t)j< Ml < 00,Slt.

2s

less than

U

fixed

IimPt

(16.1)

~ ~ o ~ ~ e s ~ othe
~ dconditzon
a n ~ l ~ ,

lz[k]l < M2 < w, ' d k .
zs s,uficien,t for

n senes to be bounded.

With these definitions we can define stability:

(16.2)