Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 ROUTH-HURWITZ CRITERION The Routh-Hurwitz riterio i lytil proedure for determiig if ll the root of polyomil hve egtive rel prt. The riterio give the umer of root with poitive rel prt. The firt tep i to otrut rry lled the Routh rry follow. For Q the Routh rry i m l k k where, d d,, d The Routh-Hurwitz riterio tte tht:
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity The umer of polyomil root i the right-hlf ple i equl to the umer of ig hge i the firt olum of the Routh rry. Exmple Coider the followig polyomil: Q Uig the Routh rry, determie the umer of root of the polyomil i the right-hlf ple. The Routh rry i, where, d. The umer of ig hge i zero. Therefore ll root re i the left-hlf ple. Exmple Coider the followig polyomil: Q Uig the Routh rry, determie the umer of root of the polyomil i the right-hlf ple. The Routh rry i Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity, where, d. There re two ig hge i the firt olum, -, d -, whih idite two root i the right-hlf ple. The root of the polyomil re -,. j.9 Note: Routh-Hurwitz riterio doe ot determie the ext lotio of the root of the polyomil doe ot determie the root vlue. Speil Ce Thi e our whe the firt elemet i row i zero ut ot ll the elemet i the me row re zero. I thi e, proeedig with the tdrd Routh-rry otritio proedure will ue diviio y zero. To void thi prolem, the zero i repled y very mll vlue. The the uully rry otrutio proedure otiue. At the ed, the ig i the firt olum re determied y llowig to pproh zero. Thi lo e doe durig the lultio. Exmple Coider the followig polyomil: Q Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Uig the Routh rry, determie the umer of root of the polyomil i the right-hlf ple. The Routh rry i lulted where d I the preedig lultio, the limit were tke t oveiet poit i the lultio rther th witig util the rry w omplete. Thi implifie the lultio d the fil form of the rry, d the fil reult i the me. Ipetig the firt olum of the rry, we otie tht regrdle of the ig of, there i lwy ig hge from to. Thi olude tht ytem tht fll uder thi e re ALWAYS UNSTABLE. Speil Ce Thi i the e whe ll the elemet i row of the Routh rry re zero. Thi will reult i premture termitio of the rry. For exmple, oider the polyomil Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity The Routh rry i Q ll zero row The rry ot e ompleted eue of the zero i the row. Aother exmple i The Routh rry i Q ll zero row The previou exmple how tht the preee of ll zero row idite tht the polyomil oti eve polyomil ftor. Thi eve polyomil i lled the Auxiliry polyomil. The oeffiiet of thi uxiliry polyomil re the elemet of the row diretly ove the row of zero. The order of the uxiliry polyomil i the order of the row tht oti it oeffiiet. I the previou exmple, i row of zero, whih idite tht row oti the oeffiiet of the uxiliry polyomil Q. Q Polyomil of thi e e lyzed i two wy. Firt, the uxiliry polyomil e ftored out. The the uxiliry polyomil d the remiig polyomil e lyzed eprtely. For exmple Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity The Routh rry i Q ll zero row The uxiliry polyomil i i the row, d i Now ftor Q from Q Q The Q However, diffiultie re eoutered whe workig with higher order polyomil. For exmple, let Q Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 77
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 The Routh rry i zero row ll The uxiliry polyomil i i the row, d i Q The eod pproh i to reple the row of zero y the oeffiiet of the derivtive of the uxiliry polyomil Q d d d otiue 7
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 79 Exmple Determie the umer of root i the right-hlf ple of the polyomil Q The Routh rry i Q d d Q The rry how tht there re o right-hlf ple root. However, ivetigtio how tht the uxiliry polyomil h imgiry xi root. Auxiliry Polyomil Auxiliry polyomil i the Routh-Hurwitz lyi re lwy eve ordered. The root of eve polyomil ome i pir ymmetril with repet to oth the rel d the imgiry xe. For exmple Re Im j Q
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Q Im Re Q,, j j Im Re Hee, the preee of uxiliry polyomil idite either utle ytem or mrgilly tle ytem t the et. Routh-Hurwitz i the Deig of Cotrol Sytem The Routh-Hurwitz riterio e ued to perform imple otrol ytem deig. It i motly ued to vlidte the tility of otroller deig or to et limit for otrol prmeter. Exmple Ue proportiol feedk otrol ytem to redued the tedy-tte error of the followig ytem to le th % of ott iput. Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud G p The trfer futio of proportiol otroller i G. The tedy-tte error i e G where G G G p The 9. e To vlidte the tility of the ytem for the required, we will hek the root of the hrteriti equtio of the ytem trfer futio H G G T Applyig the Routh-Hurwitz riterio to the hrteriti equtio of the ytem, the Routh rry i R + - Y
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity for tle ytem for tle ytem Therefore, for tle ytem, the otroller ott mut e. Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud