A Unified Approach for Sensitivity Design of PID Controllers in the Frequency Domain

Transcription

1 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns A Unf Aroach for nstvty Dsgn of PID Controllrs n th Frquncy Doman TOORAN EMAMI JOHN M WATIN Dartmnt of Elctrcal Engnrng an Comutr cnc Wchta tat Unvrsty 85 Farmount Wchta ansas UNITED TATE OF AMERICA txmam@wchtau jwatkns@org Abstract: - In ths ar a grahcal tchnqu s ntrouc for fnng all contnuous-tm an scrt-tm roortonal ntgral rvatv (PID controllrs that satsfy an H snstvty constrant of an arbtrary orr transfr functon wth tm lay Ths roblms can b solv by fnng all achvabl PID controllrs that smultanously stablz th clos-loo charactrstc olynomal an satsfy constrants fn by a st of rlat comlx olynomals A ky avantag of ths rocur s that t only ns on th frquncy rsons of th systm Th lta orator s us to scrb th controllrs bcaus t not only osssss numrcal rorts suror to th scrt-tm shft orator but also convrgs to th contnuous-tm controllr as th samlng ro aroachs zro A unf aroach allows us to us th sam rocur for scrt-tm an contnuoustm H snstvty sgn of PID controllrs y-wors: - nstvty sgn frquncy rsons lta oman PID controllrs an tm lay Introucton Bcaus of th xtnsv us of roortonal ntgral rvatv (PID controllrs n nustry thr has bn a sgnfcant ffort to trmn th st of all PID controllrs that mt crtan sgn goals As th ntnt of ths rsarch s to vlo sgn mthos that can b al n nustry ths mthos shoul ossss svral ky attrbuts Frst thy shoul b alcabl to a broa st of lants In orr for th mthos to b alcabl n th rocss control nustry t s artcularly mortant that thy hanl tm-lays Ially th sgn mthos woul b sml to unrstan an asy to mlmnt Mthos that n only on th frquncy rsons of th systm lmnat th n for a lant mol whch may not b avalabl n som alcatons Fnally sgns on rctly n th gtal oman allow for asy comutr mlmntaton Not surrsngly most of th arly work n ths ara sought to fn all contnuous-tm PID controllrs that stablz th nomnal lant mol Much of th arly work n ths ara was on by Bhattacharyya an collagus an assum knowlg of th systm transfr functon mol [] [] Many of ths rsults n on gnralzatons of th Hrmt-Bhlr thorm [] Thy vlo rsults bas on thorms by Pontryagn an a gnralz Nyqust crtron [] Th mtho ntrouc by Tan n [5] brok th numrator an nomnator of th lant transfr functon nto vn an o arts In [6] [7] an [8] a nw mtho whch not nvolv comlx mathmatcal rvatons was us to solv th roblm of stablzng an arbtrary orr transfr functon whn only th frquncy rsons of th lant transfr functon was known Byon stablty nvstgators hav also look at rformanc an robustnss Th authors n [5] an [7] foun rgons whr th controllrs wr guarant to mt crtan gan an has margn rqurmnts PID controllrs that also satsfy gan crossovr has crossovr an banwth rqurmnts for oubl ntgrator systms wth lay wr foun n [9] In [] an [] th aramtrs of PID controllr wr trmn usng a mtahurstc algorthm mtho In [] th mtahurstc algorthm mtho was us to ajust th PID aramtrs to mt th rformanc rqurmnt for a ourng task In [] th authors us a fractonal PID controllr to mt th rformanc rqurmnt for an actv magntc barng systm In ths ar an aatv gntc algorthm was us to trmn th PID controllr IN: Issu 5 Volum May 9

2 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns aramtrs that otmz a mult-objctv cost functon In [] constran ol assgnmnt was us for sgn of PD controllrs for a oubl ntgrator lant mol wth tm lays or tm constant As ths controllrs must b mlmnt on ral systms sgn mthos that al wth robustnss ar of artcular mortanc In [] an [5] ak an collagus look at ffrnt mthos for H controllr sgn of PID controllrs Ho us a gnralzaton of th Hrmt-Bhlr thorm for H PID sgn [6] Tantars l an Bhattacharyya look at sgn of frst-orr controllrs n [7] Unfortunatly ths mthos that alt wth robustnss not work rctly wth systms wth tm-lay whch ar rvalnt n th rocss control nustry In [8] l an Bhattacharyya allow for tm-lays n th nomnal mol whn thy nvstgat th stablty roblm for lants wth no ols or zros on th jω axs an a known tm lay All of th mthos n []-[8] ar bas n contnuous-tm systms As mor an mor controllrs ar mlmnt as gtal comnsators sgn mtho that work rctly n th gtal oman bcom mor mortant Unfortunatly most of th work n ths ara has concntrat on sgn of contnuous-tm PID controllrs In [9] th lta orator was us to obtan a unf aroach for fnng stablty bounars of PID controllrs for arbtrary orr transfr functons wth tm lay n th frquncy oman Th lta orator was us to scrb controllrs n th scrt-tm bcaus t not only rovs numrcal rorts suror to th scrttm shft orator but also convrgs to th contnuous-tm as th samlng ro aroachs zro [] [] In [] uchomsk us th lta orator to sgn robustly stabl PID controllrs for low orr known systm transfr functons In [] [] [5] [6] an [7] th authors of ths ar vlo tchnqus for fnng all achvabl contnuous-tm PID controllrs that smultanously stablz th clos-loo systm an satsfy an H snstvty comlmntary snstvty wght snstvty robust stablty constrant or robust rformanc constrant In ths ar th goal s to fn a unf aroach for contnuous-tm an scrt-tm snstvty sgn of PID controllrs Ths mtho s alcabl for sngl-nut-sngl-outut (IO ror transfr functons of any orr wth tm lay A unf aroach usng th lta orator allows us to us th sam rocur for scrt-tm an contnuous-tm H snstvty sgn As ths work buls uon th straght-forwar vlomnt n [9] t os not rqur th lant transfr functon mol but only th frquncy rsons of th systm If th lant transfr functon s known w can aly th sam rocurs by frst comutng th frquncy rsons Th rmanr of ths ar s organz as follows Th sgn mthoology s ntrouc n cton A numrcal xaml that monstrats th alcaton of ths mtho s rsnt n cton Fnally th rsults of ths ar ar summarz n cton Dsgn Mthoology A IO contnuous-tm lant transfr functon wth tm lay τ s fn as - s G ( s = G( s τ ( Th quvalnt mol n th lta oman whn th outut of lant s saml an a zro-orr hol s lac at th nut can b foun from [] as G ( ( T L T s G s = ( + whr T s th samlng ro T s th gnralz transform an as fn n [] s gvn by s T = = st T T ( Consr th IO systm shown n Fg whr G ( s th lant an G ( s th PID controllr c Th rfrnc nut sgnal an th rror sgnal ar r an z rsctvly Th outut of th controll lant s y Th PID controllr s fn as IN: Issu 5 Volum May 9

3 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns whr G ( = + + c + T an ( ar th roortonal ntgral an rvatv gans rsctvly Th rror sgnal that w want to mnmz s fn as functon n (8 can b wrttn n trms of ts magntu an has angl as ( ( j β β ω (9 If (9 hols thn for ach valu of β z = r y (5 jθ ( β ( y + Gc ( G ( Fg Block agram of snstvty functon Th transfr functons n Fg can b xrss n th frquncy oman Th lant transfr functon can b wrttn n trms of ts ral an magnary arts as G ( β = R ( β + ji ( β (6 m jω T = whr β = jωt Th PID controllr T T s fn n th frquncy oman as β G ( β = + + c β + T β Th trmnstc valus of an (7 for whch th clos-loo charactrstc olynomal s Hurwtz stabl hav bn foun n [9] (wth a small ffrnc n th aramtrzaton of th PID controllrs In ths ar th roblm s to fn all PID controllrs that stablz th systm an satsfy th snstvty constrant whr r ( β (8 ( β = + G ( β G ( β s th snstvty functon an s a ostv ral scalar Th comlx c z must b tru for som θ [ π whr θ = ( β Consquntly all PID controllrs that satsfy (8 must l at th ntrscton of all controllrs that satsfy ( for all θ [ π [] To accomlsh ths for ach valu of θ [ π w wll fn all PID controllrs on th bounary of ( It s asy to show from ( that all th PID controllrs on th bounary must satsfy P( ωθ T = ( j whr P( ωθ T = + G ( β G ( β θ c Not that ( rucs to th frquncy rsons of th stanar clos-loo charactrstc olynomal as ubsttutng (6 (7 an jθ = cos θ + j sn θ nto ( an solvng for th ral an magnary arts yls an X + X + X = Y R R R R ( X + X + X = Y ( I I I I whr X = ωr ( β R ( cos( ωt + ( ωt T X = ωr ( β + I ( β R m snc IN: Issu 5 Volum May 9

4 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns X Y X R R I ( ωt snc( ωt cos( ωt ( ωt sn R ( β = ω + I ( βsnc m = ω cos θ = ωi ( cos( ωt + snc( ωt ( ωt ( ωt ( ωt cos( ωt = I m m ( β T X R ( β ωi ( β R ( βsnc + X = ω sn snc I I ( β m + ω Y = sn θ I Ths s a thr-mnsonal systm n trms of th controllr aramtrs an Th bounary of ( can b foun n th ( lan for a fx valu of Aftr sttng ( an ( can b rwrttn as to th fx valu X X Y X R R R R X X = Y X I I I I ( olvng ( for all ω an θ [ π gvs th followng quatons: ( ωθ T = ( cos( ωt ( cos θ + R ( β sn ( ωt sn θ + ( cos( ωt sn θ + + I ( β m sn ( ωt ( cos θ + an ( ( ωt + G ( β cos (5 whr snc ( ωt cos( ωt + ( ( T + ( ( T ( ( β ( cos( ωt + ( ωθ T = ω + ω R ( β snc ω sn θ I ( β snc ω cos θ m G m (6 G ( β = R ( β + I ( β ttng ω = n ( w obtan X ( R X ( = I (7 an conclu that ( θ T s arbtrary an ( θ T = unlss I ( = R ( = m whch hols only whn G ( s has a zro at th orgn By lttng T n (5 an (6 th contnuoustm snstvty bounars ar foun as ( ( θ R ( ω cosθ + I ( ω sn ( ωθ an ω whr m = G ( jω ( ωθ ω = + ( R ( ω ( sn θ + I ( ω ( cosθ m G ( jω m (8 (9 G ( jω = R ( ω + I ( ω R ( ω an I ( m ω ar th ral an magnary arts of th contnuous-tm lant transfr functon rsctvly IN: Issu 5 Volum May 9

5 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns Th rocur can b rat n th ( lan Aftr sttng ( can b rwrttn as to a fx valu X X Y X R R R R X X = Y X I I I I ( an ( olvng ( for all ω θ [ π gvs th sam xrsson as (5 for ( ωθ T an th followng quaton for ( ωθ T ( cos( ωt + ( ωθ T = + ω snc ( ωt ( R ( β ( sn θ + I ( β ( cosθ m ( ( T ω G ( β snc ω ( At ω = must b qual to zro for a soluton to xst Furthrmor as I ( = for all ral lants m ( θ T s arbtrary an cos θ ( θ T = ( R ( Lttng T n (5 an ( gvs th sam xrsson as (8 for ( ωθ an for th contnuous-tm snstvty bounary of ( ωθ th followng xrsson: ( ωθ = + ω ( R ( ω ( sn θ + I ( ω ( cosθ m ω G ( jω ( Lastly th soluton s foun n th ( lan Aftr sttng to a fx valu of ( an ( ar rwrttn as X X Y X R R R R X X = Y X I I I I ( Although th coffcnt matrx s sngular a soluton wll xst n two cass Frst at ω = ( θ T s arbtrary an ( θ T = unlss I ( = R ( = whch hols only whn th m lant has a zro at th orgn In such a cas a PID comnsator shoul b avo as th PID ol cancls th zro at th orgn an th systm bcoms ntrnally unstabl A scon st of solutons occurs at any frquncy ω whr ( ω θ T (from(5 s qual to At ths frquncs ( ω θ T an ( ω θ T must satsfy th followng straght ln quaton ( cos( ω T + ω θ T = + ω snc ( ωt ( R ( β ( sn θ + I ( β ( cosθ m ω G ( β ( snc( ωt ( (5 Lttng T n (5 w can gt th followng xrsson for th contnuous-tm cas ( ω θ = + ω ( R ( ω ( sn θ + I ( ω ( cosθ m ω G ( jω (6 IN: Issu 5 Volum May 9

6 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns Examl In ths scton a numrcal xaml s us to monstrat th alcaton of ths mtho Consr th scon-orr lant transfr functon from [7] 5( s 6s G ( s = (7 ( s + ( s + 5 Th goal s to fn all scrt-tm PID controllrs that stablz th clos-loo systm an satsfy th H snstvty constrant n (8 for = whn th samlng ro s T = scons Usng ( th scrt-tm lta-oman quvalnt of th systm n (7 s gvn by Th Bo rsons of th snstvty functon s shown n Fg As can b sn ( β = 5 whch s lss than = Th sgn goal s mt for th scrt-tm systm Th clos loo st rsons of th systm wth th PID controllr n (9 s shown n Fg As can b sn th clos-loo st rsons of th scrt-tm systm s stabl an has no ovrshoot a sttlng tm of 85 scons an zro stay stat rror 5 tablty Bounary G 6 T ( ( = ( + T ( + 96( + 8 (8 5 nstvty Rgon Equatons (5 an (6 ar us n th ( lan for a fx valu of = Equaton (6 s us to fn th ral an magnary arts of (8 n th frquncy oman As scuss rvously th PID stablty bounary of th nomnal systm can b foun by sttng = n (5 an (6 All PID controllrs that satsfy th H snstvty constrant n (8 ar foun by sttng = n (5 an (6 for θ [ π an fnng th ntrscton of all rgons Th stablty bounary an th rgon that satsfs th H snstvty constrant ar shown n Fg Th ntrscton of all rgons ns th stablty bounary of th ( lan s th H snstvty rgon for th scrt-tm systm To vrfy th rsults an arbtrary controllr from ths rgon s chosn gvng us th scrt-tm PID controllr as G c ( = (9 X: 579 Y: Fg tablty bounary an snstvty rgon of scrttm systm n th ( lan Magntu (abs 8 6 Bo Dagram Frquncy (Ra/c (ra/sc ystm: Pak gan (abs: 5 At frquncy (ra/sc: 85 Fg Magntu of scrt-tm snstvty functon frquncy rsons IN: Issu 5 Volum May 9

7 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns Amltu t Rsons ystm: T ttlng Tm (sc: 85 st rsos n lan ystm: T Fnal Valu: Th scon mtho uss (5 an ( n th ( lan for a fx valu of = Th PID controllr s sgn to satsfy th wght snstvty constrant wth = Th rgon that satsfs th snstvty constrant an th stablty bounary s shown n Fg 6 Th ntrscton of all rgons ns th stablty bounary of th ( lan s th snstvty rgon for th scrt-tm systm Tm (sc Fg Clos loo st rsons of scrt-tm systm In orr to comar ths rsults wth th contnuous-tm PID controllr quatons (8 an (9 ar us n th ( lan for a fx valu of = Th stablty bounary an th rgon that satsfs th H snstvty constrant ar shown n Fg 5 Th ntrscton of all rgons ns th stablty bounary of th ( lan s th H snstvty rgon for th contnuous-tm systm Comarng Fg an Fg 5 w can s that whl th stablty bounary an th H snstvty rgons ar smlar for th scrt-tm an contnuous-tm controllrs th contnuous-tm rgons ar largr - - X: Y: 576 tablty Bounary nstvty Rgon Fg 6 tablty bounary an snstvty rgon of scrttm systm n th ( lan 5 tablty Bounary To vrfy th rsults an arbtrary controllr from ths rgon s chosn gvng us th PID controllr Gc 57 ( = + + ( + 5 X: 78 Y: 6 nstvty Rgon Fg 5 tablty bounary an snstvty rgon of contnuous-tm systm n th ( lan Th substtuton of (8 an ( nto (8 gvs ( β = As th magntu of th snstvty functon s lss than th sgn goal s mt for th scrt-tm systm In orr to comar ths rsults wth th contnuous-tm PID controllr quatons (8 an ( ar us n th ( lan for a fx valu of = Th stablty bounary an th rgon that satsfs th H snstvty constrant ar shown IN: Issu 5 Volum May 9

8 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns n Fg 7 Th ntrscton of all rgons ns th stablty bounary of th ( lan s th H snstvty rgon for th contnuous-tm systm Comarng Fg 6 an Fg 7 w can s that whl th stablty bounary an th H snstvty rgons ar smlar for th scrt-tm an contnuous-tm controllrs th contnuous-tm rgons ar largr tablty Bounary 5-5 Fasbl - - X: 8 Y: 69 nstvty Rgon Fg 7 tablty bounary an snstvty rgon of contnuous-tm systm n th ( lan Th thr mtho s al n th ( lan for a fx valu of = 5 Plots of ( T T ( ωθ (from (5 for varous valus of θ [ π ar shown n Fg 8 for th scrt-tm systm For ach curv th ω ar th frquncs at whch ( ωθ T = = 5 Each ω s substtut nto (5 to fn th rqur bounars In aton w hav th bounary at ( θ T = for th scrt-tm systm w Fg 8 Plots of ( ωθ T vrsus ω us to fn valus of ω for scrt-tm systm Th rgon that satsfs th snstvty constrant an th stablty bounary s shown n Fg 9 Th ntrscton of all rgons ns th stablty bounary of th ( lan s th H snstvty rgon for th scrt-tm systm To vrfy th rsults an arbtrary controllr from ths rgon s chosn gvng us th scrt-tm PID controllr G c 69 ( = 5+ + ( + Th substtuton of (8 an ( nto (8 gvs ( β = 9 As th magntu of snstvty functon s lss than th sgn goal s mt for th scrt-tm systm IN: Issu 5 Volum May 9

9 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns tablty Bounary nstvty Rgon 5 5 X: Y: Fasbl Fg 9 tablty bounary an snstvty rgon of scrttm systm n th ( lan In orr to comar ths rsults wth th contnuous-tm PID controllr lots of ( ωθ an ( ωθ (from (8 for varous valus of θ [ π ar shown n Fg For ach curv th ω ar th frquncs at whch ( ωθ = = 5 Each ω s substtut nto (6 to fn th rqur bounars In aton w hav th bounary at ( contnuous-tm systm Th stablty bounary an th rgon that satsfs th H snstvty constrant ar shown n Fg Th ntrscton of all rgons ns th stablty bounary of th ( lan s th H snstvty rgon for th contnuous-tm systm Comarng Fg 9 an Fg w can s that whl th stablty bounary an th H snstvty rgons ar smlar for th scrt-tm an contnuous-tm controllrs th contnuous-tm rgons ar largr w Fg Plots of ( ωθ vrsus ω us to fn valus of ω for contnuous -tm systm - - X: 569 Y: 776 tablty Bounary nstvty Rgon Fg tablty bounary an snstvty rgon of contnuous-tm systm n th ( lan Conclusons A grahcal tchnqu was ntrouc for fnng all achvabl contnuous-tm or scrt-tm PID controllrs that satsfy an H snstvty constrant for an arbtrary-orr transfr functon wth tm lay Ths mtho s sml to unrstan an rqurs only th frquncy rsons of th lant A numrcal xaml wth a tm lay was rsnt to monstrat th alcaton of ths mtho It was shown that th contnuous-tm an scrt-tm IN: Issu 5 Volum May 9

10 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns sgns can b unrstoo unr a common framwork through th lta orator 5 Acknowlgmnts Ths work was suort n art by rt Arosystms Inc Bong Intgrat Dfns ystms an th Grauat chool at Wchta tat Unvrsty W woul lk to acknowlg th fnancal suort of all th sourcs that ma ths rsarch ossbl Rfrncs: [] P Bhattacharyya H Challat an L H l Robust Control: Th Paramtrc Aroach Ur al Rvr NJ: Prntc-Hall 995 [] G J lva A Datta an P Bhattacharyya PID Controllrs for Tm-Dlay ystms Boston: Brkhäusr 5 [] W Ho A Datta an P Bhattacharyya Gnralzatons of th Hrmt-Bhlr thorm Lnar Algbra an ts Alcatons Vol [] H Xu A Datta an P Bhattacharyya PID stablzaton of LTI lants wth tm-lay Procngs of th n IEEE Confrnc on Dcson an Control Mau Hawa 8- [5] N Tan Comutaton of stablzng PI an PID controllrs for rocsss wth tm lay IE Transactons Vol 5 - [6] ujolžć an J M Watkns tablzaton of an arbtrary orr transfr functon wth tm lay usng PI an PD controllrs Procngs of Amrcan Control Confrnc Mnnaols Mnnsota 6 7- [7] ujolžć an J M Watkns tablzaton of an arbtrary orr transfr functon wth tm lay usng PID controllr Procngs of 5th IEEE Confrnc on Dcson an Control an Dgo CA 6 [8] M ak Prorts of stablzng PID gan st n aramtr sac IEEE Transactons on Automatc Control Vol 5 No [9] J Watkns an G Pr Invstgatng th ffcts of cross-lnk lays on saccraft formaton control Journal of th Astronautcal cnc Vol 5 No 5 8- [] M P Tzamtz F N oumbouls an M G karts On th molng an controllr sgn for th outut has of ourng rocss WEA Transactons on ystm an Control Vol No 9 - [] F N oumbouls an MP Tzamtz A mtahurstcs aroach for controllr sgn of multvarabl rocss th IEEE Confrnc on Emrgng Tchnologs an Factory Automaton Patras Grc 7 9- [] L Y Chang an H C Chn Tunng of fractonal PID controllrs usng aatv gntc algorthm for actv magntc barng systm WEA Transactons on ystm an Control Vol 8 No [] Žáková On ty of controllr sgn for lay oubl ntgrator systm WEA Transactons on ystm an Control Vol No [] M ak an Amoto PID controllr otmzaton for H control by lnar rogrammng Intrnatonal Journal of Robust an Nonlnar Control : 8-99 [5] M ak Fx structur PID controllr sgn for stanar H control roblm Automatca Vol 6 9- [6] M T Ho ynthss of H PID controllrs: a aramtrc aroach Automatca Vol [7] R N Tantars L H l an P Bhattacharyya H Dsgn wth frst orr controllrs IEEE Transactons on Automatc Control Vol 5 No [8] L H l an P Bhattacharyya Controllr synthss fr of analytcal mols: thr trm controllrs IEEE Transactons on Automatc Control Vol 5 No [9] T L J M Watkns T Emam an ujolžć A unf aroach for stablzaton of arbtrary orr contnuous-tm an scrt-tm transfr functons wth tm lay usng a PID controllr Procngs of 6 th IEEE Confrnc on Dcson an Control Nw Orlans LA 7-5 [] R H Mlton an G C Goown Dgtal Control an Estmaton A Unf Aroach Englwoo Clffs NJ: Prntc-Hall 99 IN: Issu 5 Volum May 9

11 WEA TRANACTION on YTEM an CONTROL Tooran Emam John M Watkns [] T Emam J M Watkns R T O Brn A unf rocur for contnuous-tm an scrt-tm root-locus an Bo sgn Procngs of th Amrcan Control Confrnc Nw York NY [] P uchomsk Robust sgn n lta oman for IO lants: PI an PID controllrs ystm Analyss Molng mulaton Vol 9-69 [] T Emam an J M Watkns nstvty sgn of PID controllrs for arbtrary orr transfr functons wth tm-lay al to a DC motor wth communcaton lay Procngs of IEEE Mult Confrnc on ystms an Control an Antono Txas 8 [] T Emam an J M Watkns Comlmntary snstvty sgn of PID controllr for arbtraryorr transfr functons wth tm-lays Procngs of 8 AME Dynamc ystms an Control Confrnc Ann Arbor Mchgan 8 [5] T Emam an J M Watkns Wght snstvty sgn of PID controllr for arbtraryorr transfr functon wth tm lay Procngs of th Elvnth IATED Intrnatonal Confrnc on Intllgnt ystms an Control Orlano Flora 8-5 [6] T Emam an J M Watkns Robust stablty sgn of PID controllrs for arbtrary-orr transfr functons wth uncrtan tm lay Procngs of th st outhastrn ymosum on ystm Thory Tullahoma Tnnss [7] T Emam an J M Watkns Robust rformanc charactrzaton of PID controllrs n th frquncy oman Procngs of th 8 th Intrnatonal Confrnc on Alcatons of Elctrcal Engnrng Houston Txas May 9-7 IN: Issu 5 Volum May 9