Wiley signals and systems e book TLFe BO 407

302

16. Stability and Feedback Systems

A polynomial i s a Hw-irirt~polpiianiral if all of i t s zeros have a negative real part.
‘The system is htablt. if its numerator polyrioinid Q ( s ) is a Hurwitx polynomial, and
Nurwil z’s stability criteria dctcrniiiie wliel,her this is the case. The test consists
OP

two parts:

A nwessary coiidit,ioiifor a Hurwitz polynoniial i s that all eoe%cicnts a , are
positive,
Q,, > 0,
n = 1,..,,(16.17)

v,

It can be only ht. fiilfilled if all orders sn in &(s) exist, as otherwise tlnert.
would be a coeficierit u n = 0.
For A”= I arid N = 2 this cwndition is sufficient and tlie t
for N > 2 the following conditions must also be tcsted.
To hrrnulatc the necessary iitrrl sufficient conclition, the Hvru12tz deternununts for / I = 1 , 2 , . * . , N are set up:

( 16.18)

(16.19)

-

_

I
-

i, a I-lurwitz polynomial if all Hurwih deterniinarits arc positivc.
A{, > 0 for p = 1,2,. . .N

(16.20)

That concliides the Iliirwitz test. lilit~iliustratc how it is carried out in Example 16.4. A rdatecl procedure Rotitli’s stability t r h t - can bc found in 123, 191.
~

16.2,3.2

Discrete Systems

Foi. dihcrete syste~ns,it must be tkterniiiitd whetlic~all poles lie>within t he unit
circle We will fhcrefore be using a procedure which maps the cornplex Z - ~ ~ R I I P
onto the corriplcx s-planc. whtre we can carrv out a stability test for continuous
systcms.
Tlic bzlzneur transform is a sullabl~way of doing tlris:

(16.21)