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 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)