CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER NICHOLS METHOD

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1 Journal of ELECTRICAL ENGINEERING, VOL 54, NO 7-8, 3, CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER NICHOLS METHOD Mikuláš Huba Kaarína Žáková The high pracical ignificance of he I T d -plan approximaion ha been hown by he perhap mo frequenly ued mehod for conroller uning by Ziegler and Nichol, modified laer for ampled daa yem by Takahai e al Thi paper exend previou work by analying alo conrol rucure ha include I T and more general I T T d -yem All hee approximaion are ued for uning of he P and PI conroller deigned for ingle inegraor yem wih conrained inpu K e y w o r d : P (PD) conroller, conroller uning, Ziegler-Nichol mehod INTRODUCTION One of he mo widepread uning rule wa inroduced by [3] already a he beginning of 4 in he la cenury They have preened wo differen mehod of he conroller eing While in he fir one, he conroller parameer are deermined on he bai of he ranien repone meauremen, in he ulimae eniiviy mehod i i neceary o bring he yem o he abiliy boundary by raiing he gain of a P conroller Ziegler and Nichol have deigned conroller o guaranee he quarer ampliude raio (quarer ampliude damping) Thi crierion wa alo followed by eg [9] and [] Ziegler- Nichol mehod creae a aring poin for number of oher uning rule SETTING OF THE PROPORTIONAL ACTION For he ake of impliciy, Ziegler and Nichol have no analyzed he ype of ime lag involved in he conrol loop In order o how he poible conequence, he behavior of an inegraor yem combined wih wo baic paraiic delay: he dead ime and he ime conan delay will be analyzed A imilar approach wa ued previouly by [8] I T d yem Opimal eing of P conroller Le u conider he inegral + dead ime yem F () = K S e T d () ha i going o be conrolled by an analog P conroller The uggeed deign of he opimal conroller eing ha correpond o he muliple dominan pole follow a requiremen of monoonic ranien repone a he preence of poible maximal value of conroller gain Therefore, i eing can be found by looking for a conroller gain exreme I mean he opimal gain of he P conroller K C ha o aify he condiion dk C d! = () whereby K C can be expreed from he characeriic equaion of he yem cloed loop: K C K S e T d + = (3) Solving () and (3) lead o he opimal conroller gain and dominan real pole K Cop = ek S T d (4) op = /T d (5) Thee formulae are known for long ime ee eg [8], [4] or [] The derived gain can be ued for eing of P conroller in he rucure wih conrained conrol ignal (Fig ), oo The block Plan and Delay ogeher repreen any proce ha can be approximaed by I T d model The dominan order dynamic preened by a ingle inegraor i uppoed o be deployed in he feedforward pah The ranpor delay can be preen a any place wihin he feedback Criical conroller gain For comparion, in addiion o he opimal gain of he P conroller here wa analyically derived alo a gain a he abiliy margin For hi one a erm criical conroller gain wa inroduced To derive he criical cloed loop gain we can ubiue = jω (6) Deparmen of Auomaion and Conrol, Faculy of Elecrical Engineering and Informaion Technology, Slovak Univeriy of Technology, Ilkovičova 3, 8 9 Brailava, Slovakia [huba, kaarinazakova]@elfubak ISSN c 3 FEI STU

2 Journal of ELECTRICAL ENGINEERING VOL 54, NO 7-8,3 89 K cop Plan Delay K c K T + Fig P conroller wih conrained oupu Fig Conrol rucure wih I T yem ino (3) In hi way, here arie a e of wo equaion K C K S co(t d ω) = K C K S in(t d ω) = ω (7) whoe oluion lead o he criical conroller gain A rivial oluion of (7) ω = lead o he minimum criical value K Ccr = (8) The econd oluion give he maximum criical value of he abiliy area K Ccr = The correponding period of ocillaion i I T yem Opimal conroller gain In he cae of I T -yem π K S T d (9) P u = π ω = 4T d () F () = K S (T + ) () whoe dynamic i deployed in he conrol rucure according o Fig, he opimal conroller gain can be derived from he requiremen of double real pole of he cloed loop characeriic equaion Thi requiremen correpond o whereby T + + K C K S = () K Cop = 4K S T (3) op = T (4) The opimal conroller gain (3) can alo be applied in he rucure hown in Fig Here, he block Plan repreen he dominan ingle inegraor dynamic and he block Delay repreen he idenified ime conan T Criical conroller gain The criical conroller gain for hi yem i going o infiniy, ince all coefficien of he characeriic polynomial are poiive and according o Rouh-Schur e of abiliy i mean ha he yem i able 3 I T T d yem 3 Opimal conroller gain K Ccr = (5) The la inegral model ha will be conidered i an inegral yem wih boh dead ime and ime conan F () = K S (T + ) e T d (6) The opimal conrol gain can be deermined from he characeriic equaion (T + ) + K C K S e T d = (7) Afer aking a derivaive of (7) according o he complex variable and expreing T + K C K S T d e T d! = (8) K C K S T d e T d = (T + ) (9) we ge he double real pole value by mean of a imple ubiuion (9) ino (8):, = T d + T ± Td + 4T () T T d Subiuing he value of correponding o he ign + ino (9) we ge he opimal gain of he proporional conroller K Cop = T + Td + 4T T +T d T K S Td e d +4T T Afer denoing he oal ime lag L = T d + T ()

3 9 M Huba K Žáková: CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER-NICHOLS METHOD 5 5 y() 3 Criical conroller gain The derivaion of he criical cloed loop gain come ou from he characeriic equaion (7) Subiuion = jω lead o he pair of equaion u() K C K S co (ωt d ) = ω T K C K S in (ωt d ) = ω Unforunaely, i i no poible o compue he value of criical gain analyically and herefore only a graphical repreenaion of K Ccr correponding o ε = i inroduced in Fig 3 Thi i cloe o ha correponding o ε Fig 3 Opimal and criical P conroller gain Solid line: I T T d yem wih ε = ; dahed line: I T d yem ( ε ); doed line: I T yem ( ε = ) 3 SETTING OF THE PROPORTIONAL AND INTEGRAL ACTIONS K c v ^ i v i T f + K K (T f +) Delay Up o now, only he deign of he P conroller for he ingle inegraor ha been conidered The addiional ime lag were compenaed by decreaing he opimal conroller gain Due o hi, an increae of ime delay caue an increae of he poible permanen conrol error One poibiliy i o compenae he permanen conrol error by he feedforward conrol The oher poibiliy i o ue a winduple inegral acion The baic conrol loop ( order yem + delay + P conroller) can be exended by block ha erve for reconrucion and compenaion of an inpu yem diurbance v i I creae he inegral acion of he conroller Fig 4 Reconrucion and compenaion of he inpu yem diurbance (winduple P I conroller) and he raio here follow ε = T d T () T f + K c T f + v i K K c (T f +) K Delay Then T d = Lε ε +, T = L ε + (3) K Cop = + ε + 4 K S ε L ε + e +ε ε +4 (4) The graphical dependence of he normed gain K op = K Cop K S on he oal lag L for ε = i preened in Fig 3 I i inereing o noe ha hi opimal value lie beween he value correponding o he limi iuaion wih ε = and ε Thi concluion regarding I T T d yem can be generalized alo for oher value ε (, ) The dependence of he opimal and criical gain on he raio T d /T can be found eg in [8] Fig 5 Equivalen rucure for he proporional band o he cheme in Fig 4 The new conrol rucure (ee alo [5], [7] and [6]) can be een in Fig 4 The included filer erve for a reducion of poible noie in he rucure For impliciy, i i conneced wih reconrucion block In he cae when only a linear par of auraion i conidered, he whole conrol rucure can be modified o he form ha i preened in Fig 5 I i eviden ha he conrol correpond o P I conroller wih he ranfer funcion ( C() = K C + ) (5) T f

4 Journal of ELECTRICAL ENGINEERING VOL 54, NO 7-8,3 9 y() 5 u() which are derived by approximaing he compued complex value by he real par or by he module, i cloe o he required one Then T f re = + 3 T d = 94Td, T f mod = T d + 4 e = 3555T d, (3) Fig 6 Tranien repone and conrol ignal for I T d yem wih he winduple P I conroller according o Fig 4 ( K S = 3, T d = 5 ) 3 Srucure wih ranpor delay The conroller deign follow he conrol rucure in Fig 4 where he block of delay repreen he ranpor delay in he loop Ariing from he characeriic polynomial of he cloed loop chp() = T f + ( + K S K C T f )e Td + K S K C e T d (6) he eing of conroller and filer can be deermined from he requiremen of i dominan riple real pole I can be achieved by olving a following e of equaion Hence where chp() = ; d d chp() = ; d chp() = (7) d op = +, (8) T d + I(5 7)τ K Cop = e +, (9) K S T d τ = (4 + )e 7 The ime conan of he filer i T f op = Iτ T d (3) However, he achieved oluion are complex number and he conroller and he filer a well are uppoed o work wih real parameer Experimenally, i can be hown ha he loop dynamic ha correpond o parameer, K C re = e + 3 =, K S T d K S T d 4 K C mod = e = 8 K S T d K S T d (3) For comparion, he conroller gain of he P conroller wa K Cop = 367/KS T d (4), eg approximaely 6 ime bigger ha he gain K Cre of he P I conroller In Fig 6 i i poible o compare ranien repone wih he correponding conrol ignal for boh ype of approximaion Simulaion are done for I T d yem conrolled by P I conroller according o he rucure in Fig 4 I can be een ha here i no a big difference beween boh approximaion and herefore he impler approximaion by he real par will be ued The correcne of he choen approximaion i alo obervable from Fig 7 Le u follow he approximaion by he real par Firly, he imulaion wa accomplihed wih he conroller parameer ha were e according o formulae given by (3) and (3) Then, he imulaion wa compared wih imulaion where conroller parameer were increaed or decreaed by % The qualiy of hee modified ranien repone and he correponding conrol ignal i lighly going down Finally, i i o noe ha reuled formulae (3) and (3) remind of formulae achieved by Ziegler and Nichol Their form i he ame, hey lighly differ only in conan 5 y() u() Fig 7 Tranien repone and conrol ignal for I T d yem conrolled by PI conroller Solid line: conroller parameer according o (3) and (3); dahed line: conroller parameer decreaed by %; doed line conroller parameer increaed by %

5 9 M Huba K Žáková: CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER-NICHOLS METHOD 3 Srucure wih ime conan delay The conroller deign follow he conrol rucure in Fig 4 where he block of delay repreen he ime conan delay in he loop The opimal eing of he conroller can be found from he cloed loop characeriic polynomial chp() = T T f 3 +T f ++(+K C K S T f )+K C K S (33) Following (7) one ge he opimal conroller eing K Cop = 3 ± I 3 8K S T T f op = 3T (3 ± I 3) ha correpond o he dominan riple pole op = 3T (34) Afer approximaion of he complex oluion by he real par or by he module one ge he following P I conroller eing 667 K C re = = 6K S T K S T 3 9 K C mod = = 9K S T K S T (35) 4 EXAMPLE Le u conider he yem decribed by he ranfer funcion F () = ( + ) 3 (38) Firly, he conroller deign i baed on he I T d, I T and I T T d approximaion Experimen how ha uch inegraor baed approximaion model hould concenrae mainly on he iniial par of he ep repone [4] Following hi requiremen one can receive model ranfer funcion 57 I T d e 745 I T 49 (95 + ) I T T d 3 (9 + ) e 5 and T f re = 9 T = 45T T fmod = 3 (36) 3T = 596T The mehod decribed in [], [3] and [3] ariing from I T d approximaion are compared wih he propoed P I conroller All conroller parameer can be found below For comparion he P conroller gain for he I T d yem wa K Cop = 5/K S T The graphical preenaion of he achieved oluion can be found in Fig 9 33 Srucure wih ranpor and ime conan delay Following he conrol rucure (Fig 4) wih a ranpor and a ime conan delay (I T T d yem) one ge he characeriic polynomial chp() = T T f 3 +T f +[(+K C K S T f )+K C K S ]e T d (37) A i wa menioned already before, he eing of conroller and filer can be deermined from he requiremen of i dominan riple pole ha can be found from equaion (7) Unforunaely he analyical oluion i no available The conroller parameer can be compued only numerically a depending on he oal lag L () and he raio ε () For example for ε = 5 one ge K C op = 93/K S L and T f op = 377L and for ε = i can be found ha K C op = 9/K S L and T f op = 37L The conroller parameer for ε = are illuraed in Fig 9 and here i hold K C op = 7/K S L and T f op = 3386L Conroller parameer K C T I (T f ) I T d I T I T T d Tyreu-Luyben Fruehauf e al Ziegler-Nichol The conrol ignal wa limied o U min = 5 and U max = 5 and a = 3 here wa conidered an inpu diurbance wih he value v i = 5 The imulaion reul are preened in Fig 8 The propoed P I conroller baed on I T d approximaion wa able o reach he required value wihou an overhoo The non preened imulaion following I T and I T T d approximaion had a oo long eling ime whereby he eling ime of ranien repone baed on I T

6 Journal of ELECTRICAL ENGINEERING VOL 54, NO 7-8, log(kc) - - log(l) Fig 8 Tranien repone and conrol ignal for he given example Seing of P I conroller i done according o approximaion by I T d model (olid line), Tyreu-Luyben mehod (normal dahed line), Fruehauf-Chien-Laurien mehod (hor dahed line) log(kc) log(l) log(ki) - - log(l) Fig 9 Opimal parameer of he P I conroller ( K i = /T f ) Solid line: I T T d yem ( ε = ); dahed line: I T d yem; doed line: I T yem; dah-and-do line: Ziegler-Nichol eing model wa longer han a he ranien repone following I T T d approximaion 5 DISCUSSION The paper wa dedicaed o he conroller deign following he conrol rucure wih he order dynamic in he feedforward pah and poible addiional paraiic delay in he feedback On he bai of uch diribuion of he dynamical elemen, P conroller following he dominan pole deign wa derived The deign procedure alo ake ino accoun alway preen limiaion in he conrol ignal The diurbance reconrucion and compenaion i olved by inroducing an I acion o he conrol rucure I lead o he eablihmen of P I conroller In conra o he quarer ampliude damping inroduced by Ziegler and Nichol, he conroller eing correpond o he double (riple) real dominan pole and guaranee he fae poible ranien wihou overhoo The comparion of he opimal P conroller gain (4) and (3) wih ha one given by he Ziegler-Nichol ranien repone mehod how ha he approximaion by he inegral+dead ime yem or inegral+ime conan yem yield he gain K Cop = 368/K S L (L = T d ) and K Cop = 5/K S L (L = T ) which are 7 or 4 ime maller han he value recommended by Ziegler and Nichol Thi i caued by he differen performance crieria: The double real pole correpond o ranien wihou overhoo, while Ziegler and Nichol ried o e he gain a high a poible in order o reduce he offe and influence of poible load change However, hi alo creae a cloed loop yem ha i very poorly damped and ha ha very poor abiliy margin Then, a alo menioned in [], he cloed loop gain i ypically 3 ime higher I i o noe ha analyical oluion for opimal and criical gain of P conroller were dicued for I T and I T d model already by Oldenbourg and Saroriu, [8] For he P I conroller i i poible o compare parameer K I = /T i ha for he approximaion by I T d model wih he dead ime L = T d give K Iop = 343/L (39) For I T model where L = T one ge K Iop = /L (4) The above compued parameer are very cloe o he value deermined by Ziegler and Nichol ha i K I = 3/L A far a he eing of he proporional gain of he P I conroller i conidered, Ziegler and Nichol propoed o decreae i value in comparion o he pure P conroller Thi wa due o he increaed inclinaion of he loop o ocillaion in he preence of he I par of he conroller The endency of he proporional parameer decreae can alo be een a he preened conroller deign Following he previou analyi, o accep only one ype of he dominan ime lag could lead o no ufficienly precie reul A broad cla of dynamical yem can be approximaed by a model conaining boh ype of he dominan ime lag (6) Then, new relaion for he opimal and he criical conroller gain depending on he raio T d /T can be found The previou analyi how ha he P conroller opimal gain compued on he bai of I T T d model lie beween value ha were compued for above menioned I T d and I T model For example, for T d /T = 5 he opimal gain i K Cop = 97/K S L

7 94 M Huba K Žáková: CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER-NICHOLS METHOD (L = T d + T ), for T d /T =, K Cop = 3/K S L and for T d /T =, K Cop = 346/K S L The gain are 337, 33 and 89 ime maller han he value recommended by Ziegler and Nichol The concluion can be confirmed by he graphical dependence in Fig 3 ha i drawn for ε = Similar reul were alo achieved for P I conroller eing A i can be een from Fig 9 boh conroller parameer (K Cop and K Iop ) compued on he bai of I T T d model are again iuaed beween value correponding o I T d and I T model In he figure he opimal value for eleced model are compared wih he value deermined by Ziegler and Nichol The analyically compued parameer K I ha follow I T T d model i pracically equivalen o he value ha wa found by Ziegler and Nichol The coincidence i reached for ε = T d /T = 3 However, i i neceary o menion ha Ziegler and Nichol ued he claical rucure of P I conroller, wherea he P I conroller inroduced here aroe on he bai of reconrucion and compenaion of he inpu yem diurbance Such modificaion of conrol rucure enure maller overhoo of he ranien repone han he parallel P I conroller doe One of he main advanage of hi concep of he winduple P I conroller and i uning i ha hey can be generalized alo for he higher order approximaive model, which allow o achieve higher conrol dynamic [5] HUBA, M KULHA, P : Linear and Nonlinear (Sauraing) Srucure of PID Conrol, Summer Coure BEST Compued Aided Conrol Deign, Brailava 994 [6] HUBA, M KULHA, P : Sauraing Conrol for he Dominan Fir Order Plan, Proceeding of he IFAC Workhop Moion Conrol, Munich 995, pp 97 4 [7] HUBA, M KULHA, P BISTAK : Synhee der Syeme mi Begrenzungen: Dynamik der PI- und PID- Regler, Elekroechniche Kolloquium FAU Erlangen 995 (in German) [8] OLDENBOURG, R C SATORIUS, H : Selbäiger Regelnugen, Auflage, R Oldenbourg Verlag, München, 95 [9] PAAR, E A : Indurial Conrol Handbook, Vol 3, BSP Profeional Book 989 [] TAKAHASHI, Y CHAN, C S AUSLANDER, D M : Parameereinellung Bei Linearen DDC-Algorihmen, Regelungechnik und Proze-Daenverarbeiung 9 (97), [] TYREUS, B D LUYBEN, W L : Tuning PI Conroller for Inegraor/Dead Time Procee, Indurial Engineering Chemiry Reearch 3 (99), [] VITECKOVA, M VITECEK, A SMUTNY, L : Conroller Tuning for Conrolled Plan wih Time Delay, Preprin of IFAC Workhop on Digial Conrol (Pa, Preen and Fuure of PID Conrol), Terraa (Spain), pp [3] ZIEGLER, J G NICHOLS, N B : Opimum Seing for Auomaic Conroller, Tranacion of he ASME (94), [4] ZAKOVA, K : Deign of Conroller for Time-Delayed Syem, Slovak Univeriy of Technology, Faculy of Elecrical Engineering and Informaion Technology, Brailava 3 Received 8 Augu 3 Acknowledgemen The work ha been parially uppored by he Slovak Scienific Gran Agency, Gran No VG-45 Reference [] ASTROM, K J HAGGLUND, T : PID Conroller: Theory, Deign, and Tuning, nd ed, Inrumen Sociey of America, 995 [] CORRIPIO, A B : Tuning of Indurial Conrol Syem, Inrumen Sociey of America, Reearch Triangle Park, Norh Carolina [3] FREUHAUF, P S CHIEN, I L LAURITSEN, M D : Simplified IMC-PID Tuning Rule, ISA/93 Advance in Inrumenaion and Conrol 48 (993), [4] HUBA, M : Uing microcompuer a pecial conrol and meauring device (Použiie mikropočíačov ako špeciálnych riadiacich a meracích prírojov), EF SVŠT (in Slovak) Mikuláš Huba (Doc, Ing, CSc) born 95 in Ružomberok, graduaed from he Slovak Univeriy of Technology in Technical Cyberneic in 974 From 989 an aociaed profeor a he Deparmen of Conrol and Auomaion, Faculy of Elecrical Engineering and Informaion Technology, Slovak Univeriy of Technology, appoined from 996 alo by leading he Local Cenre of Diance Educaion Acive in he field of nonlinear conrol and modern learning echnologie, like Inerne baed ineracive educaion, compuer aided/baed conrol deign, mulimedia, ec Kaarína Žáková (Ing, PhD) graduaed from he Slovak Univeriy of Technology in Technical Cyberneic in 99 Since 99 he i wih he Deparmen of Auomaion and Conrol, Faculy of Elecrical Engineering and Informaion Technology, Slovak Univeriy of Technology Her curren reearch aciviie are conneced wih ynhei of nonlinear yem, compuer aided conrol deign and e-learning