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 = For A”= I arid N = this cwndition is sufficient and tlie t for N > the following conditions must also be tcsted To hrrnulatc the necessary iitrrl sufficient conclition, the Hvru12tz deternununts for / I = , , * , N are set up: ( 16.18) (16.19) - _ I - i, a I-lurwitz polynomial if all Hurwih deterniinarits arc positivc A{, > 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)