The PID and 2DOF control of the integral system-influence of the 2DOF parameters and practical implementation

O iginal Pape Measu emen and Con ol 2022, Vol. 55(1-2) 94–101 © The Au ho (s) 2022 A icle euse guidelines: sagepub.com/jou nals-pe missions DOI: 10.1177/00202940221076961 jou nals.sagepub.com/home/mac The PID and 2DOF con ol o he in eg al sys em - influence o he 2DOF pa ame e s and p ac ical implemen a ion Radek Gu as 1 , Radek S ambe sky 1 and Mi osla Mahdal 1  Abs ac The a icle deals wi h he issue o using he Two Deg ee o F eedom (2DOF) PID con olle o con ol an in eg al sys em and in es iga es by he simula ion and expe imen al measu emen wha influence i has on he cou se o he con ol p ocess compa ed o s anda d PID con olle . The con olled plan is ep esen ed by he DC elec ic mo o wi h wo m gea and i s ou pu sha o a ion angle. The a icle s udies he e ec o he added pa ame e s o he 2DOF con olle on he dynamics o he closed-loop con ol. The influence o hese pa ame e s is hen e alua ed using he quali y o eedback con ol c i e ia ITAE. The pape s udies how he o e shoo o he con olled a iable du ing he se poin s ep is elimina ed using 2DOF con ol heo y. The o e shoo is caused due o an agg essi e uning o he con olle o elimina e he dis u bance e ec on he con olled a iable o he in eg al plan s wi h dead zones. Keywo ds in eg al sys em, 2DOF PID con olle , con ol sys em, DC mo o Da e ecei ed: 9 Sep embe 2021; accep ed: 8 Janua y 2022 In oduc ion In he indus ial en i onmen oday, he posi ion con ol o a ious de ices is a c ucial ask. We a e inc easingly en- coun e ing equi emen s o imp o e he accu acy and e fi- ciency o p oduc ion p ocesses, whe e i is necessa y o eplace he posi ioning o he limi swi ches wi h eedback con ol o achie e op imal condi ions o he echnology. This ask hus e y o en leads o in eg al sys em con ol, o en wi h nonlinea p ope ies, as desc ibed in he pape . 1 The ypical app oach o he eedback con ol is using he PD con olle . 2–4 The e is no pe manen con ol de ia ion o linea in eg al sys ems such as pneuma ic, elec ic, o hy- d aulic sys ems unless he e a e non-linea i ies like s a ic ic ion. PID con ol can lead o emo ing he pe manen con ol de ia ion in such sys ems. 5–7 Howe e , he PID con ol may no always be ideal o his ask, leading o o e shoo s and unwan ed slip and slide e ec s. The e o e, i is o en beneficial o use o he algo i hms ha can handle he p oblem be e . One such op ion is o use he PID con olle algo i hm wi h wo deg ees o eedom (2DOF). 8–13 Two Deg ee o F eedom PID con olle s add se poin weigh s o he p opo ional and de i a i e e m o he al- go i hm o ensu e ha he e ec o he dis u bance is quickly elimina ed while supp essing o e shoo when acking he se poin . 8,9 Thedeg eeo eedomo acon olle isdefined as he numbe o closed-loop ans e unc ions ha can be uned independen ly, 14–18 which p o ides addi ional op ions o uning he con olle ega ding change o se poin o dis u bance alue. Ac i e dis u bance ejec ion con ol, which uses he ex- ended s a e obse e 19–21 o he Dis u bance obse e , 22,23 can be used as an al e na i e. Fo bo h o hese me hods, we a e equi ed o c ea e he in e se model o he plan . The simpli- fica ion o he 2DOF con olle is ha he fil e is applied o he se poin signal and depends on al eady known PID pa ame e s only. Due o ha , he implemen a ion o he 2DOF is also possible wi hou he need o de ailed knowledge abou he plan . 2DOF con olle can be uned manually o by quan i a i e me hods as 9,24 based on educing he c i e ia desc ibed in Chap e 6. o his pape . Ano he op ion is also he combina ion o he 2DOF con olle wi h he dis u bance obse e . 25 The same echnique can also be used wi h o he ypes o eedback C ea i e Commons CC BY: This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 License (h ps://c ea i ecommons.o g/licenses/by/4.0/) which pe mi s any use, ep oduc ion and dis ibu ion o he wo k wi hou u he pe mission p o ided he o iginal wo k is a ibu ed as specified on he SAGE and Open Access pages (h ps://us.sagepub.com/en- us/nam/open-access-a -sage). 1 Depa men o Con ol Sys ems and Ins umen a ion, VSB –Technical Uni e si y o Os a a, Os a a, Czech Republic Co esponding au ho : Radek Gu as, Depa men o Con ol Sys ems and Ins umen a ion, VSB –Technical Uni e si y o Os a a, 17. lis opadu 15/2172, Os a a 708 00, Czech Republic. Email: adek.gu as@ sb.cz loop con ol as Fuzzy con ol, which o e us he op ion o design he esponse p ope ies based on expe expe ience, S a e con olle , which allow us o design he con ol sys em based on he ull s a e eedback, o F ac ionalPIDcon olwhe e he de i a ion and in eg a ion e ms o PID con olle a e exchanged wi h he ac ional-o de in eg a ion and de i a ion e m which leads o ackle he p oblem o dead zones nonlinea i ies in in eg al sys ems. 26 Fo bo h me hods, he addi ional fil e can be added o al e na e he se poin signal. Howe e , he quan i a i e design o he fil e o non-PID con ol is hen based on Me a- heu is ic op imiza ion algo i hms. 27 The implemen a ion o he 2DOF fil e was published be o e in e . 28, whe e adi ional 2DOF is modified o 2DOF wi h an in e nal model o ime- delay p ocesses. The combina ion o he 2DOF and F ac ional PID con olle design was hen published in e . 29.In hescope o his pape , we will ocus on he adi ional 2DOF PID Con olle , whe e he design, implemen a ion, and compa ison be ween he nume ical simula ions and ac ual single-chip compu e implemen a ion. Desc ip ion o he sys em The in eg al sys em (see Figu e 1) is a DC mo o wi h a wo m gea , whe e he con olled a iable is he angle o o a ion o he ou pu sha . A esis ance senso p o ides he measu emen in a po en iome ic se up. The A duino Nano mic ocon olle was used as a model con ol sys em o coope a e wi h an ex e nal AD con e e and an adap ing elec onic boa d. The ac ion a iable is ealized ia a DC mo o d i e (Pololu Simple High-Powe Mo o Con olle 18 15, see Figu e 2 and Figu e 3) using a PWM signal and H-b idge di ec ion swi ching as a esponse o he con olle ou pu . The MOSFET swi ching ca ie equency o he d i e is 25 kHz. All he pa ame e s o he d i e a e configu able h ough he USB po and dedica ed so wa e. The ou pu signal is an analog alue in he ange o 0–5V p ocessed by a 12-bi AD con e e . The ange o he angle o o a ion o he ou pu sha is app oxima ely 0–230°. By i s physical na u e, he sys em i sel exhibi s nonlin- ea i ies, some o which a e compensa ed by he algo i hm, such as he ini ial insensi i i y o he mo o o he inpu signal caused by ic ion and nonlinea exci a ion o he mo o coils. This beha io is ela i ely supp essed by he o se o he inpu signal o p e iously expe imen ally measu ed alues. O he nonlinea i ies o he sys em we e neglec ed o his wo k and conside ed in he iden ifica ion o he sys em by selec ing he mos sui able sys em pa ame e s. A duino nano was chosen as a sui able mic ocon olle - based con ol sys em o con ol his plan . I s main ad an- ages a e he possibili y o as and simple p og amming, easy po abili y o code o o he simila boa ds, he possibili y o p og amming om he MATLAB/Simulink en i onmen while main aining a low-cos concep , andeasyimplemen a ionin o exis ing ha dwa e, unlike e.g., gene al PIC MCU. 30 The desc ibed block diag am o he con ol sys em can be seen in Figu e 2.Figu e 3 shows he elec onics im- plemen a ion o he expe imen . Theo y o he 2DOF PID con olle Ou con ol sys em aims o calcula e he co ec con ol signal o he inpu o he plan desc ibed abo e. Fo he con ol, we a e going o use he linea eedback con olle . All he signals will be p esen ed in he complex domain using he Laplace ans o m, whose defini ion is: XðsÞ¼L xð Þg ¼ Z ∞ 0 xð Þes d (1) Whe e he ime ( ) domain signal x( ) is ans e ed in o he complex (s) domain X(s). Figu e 1. DC mo o as in eg al plan . Figu e 2. Block diag am o he con ol sys em. Gu as e al. 95 The p ima y me hod o con olling he plan s a e Y(s) is he P opo ional-In eg al-De i a i e (PID) (based on P o- po ional, In eg al, and De i a i e e ms) con olle . The con ol loop diag am can be seen in Figu e 4. This me hod uses he eedback om he con ol signal Y(s), which is hen compa ed wi h he wan ed (se poin ) signal W(s). The e o signal can be calcula ed as: EðsÞ¼WðsÞYðsÞ(2) Using he con olle desc ibed by he ans e unc ion: GRðsÞ¼UðsÞ EðsÞ(3) whe e U(s) is he con ol ac ion signal which is he inpu o he con olled plan . The con ol plan ans e unc ion de- sc ibes he dynamics o he plan : GSðsÞ¼YðsÞ UðsÞVðsÞ¼0 ¼YðsÞ VðsÞUðsÞ¼0 (4) whe e he V(s) is he dis u bance. Usually, du ing he design phase o he con olle design, we se he expec a ion o he closed con ol loop o be: The ans e unc ion o he se poin o he ou pu is equal o one. GwyðsÞ¼YðsÞ WðsÞVðsÞ¼0 →1 (5) The ans e unc ion o he dis u bance o he ou pu is ze o. G yðsÞ¼YðsÞ VðsÞWðsÞ¼0 →0 (6) Due o he physical p ope y o he mechanical sys em, his expec a ion canno be implemen ed as e e y s a e con ain some dynamics, and he sys em canno c ea e he co esponding con ol signal agains he u u e unknown dis u bance V(s). The di e ence be ween he s anda d PID con olle and 2DOF PID con olle will be p esen ed on he plan , which was desc ibed abo e, and he expe imen al iden ifica ion (desc ibed in he nex chap e ) lead o he ans e unc ion: GsðsÞ¼ k1 sðT1sþ1Þ(7) whe e k 1 is he plan gain, and T 1 is he ime cons an due o ine ia. Fi s , le us desc ibe he p ope y o he PID con olle whose ans e unc ion is: GRðsÞ¼kp1þ1 TIsþTDs(8) whe e k P is he con olle gain, and i is equal o he p opo ional (posi ion) eedback, T I is he in eg al cons an o he con olle , and he k P /T I is equal o he in eg a ion o he con ol e o whose unc ion is o ob ain Y(s) = W(s) in he s eady s a e. The las e m is he de i a i e e m con aining he con olle ’s de i a i e cons an T D ,whe ek P T D is equal o he de i a ion ( eloci y) eedback. The 2DOF con olle has a simila s uc u e. The only di e ence is he fil e wi h he wo cons an s band c. Thanks o hese pa ame e s, we can con ol he influence o he se poin a iable compa ed o he ou pu a iable. The 2DOF con olle can be desc ibed wi h he equa ion: UðsÞ¼KP8 > < > : bWðsÞYðsÞþ 1 TIs½WðsÞYðsÞ… …þTDs½cWðsÞYðsÞ 9 > = > ; (9) I he b=1andc= 1, hen he 2DOF con olle is equi alen o he common PID con olle . I he b=0andc=0 hen hefil e is ully ac i e. We can di ide his con olle in o wo blocks- he common PID con olle and a sepa a e fil e . A final diag am can be seen in Figu e 5. Then he ans e unc ion o he fil e is: GFðsÞ¼cTITDs2þbTIsþ1 TITDs2þTIsþ1(10) The ans e unc ion o he se poin o he ou pu is equal o: Figu e 4. Closed loop con ol sys em diag am. 9 Figu e 3. Pic u e o he con ol sys em. Figu e 5. Closed loop con ol sys em wi h inpu fil e . 3 96 Measu emen and Con ol 55(1-2) GWY ðsÞ¼GSðsÞGRðsÞGFðsÞ 1þGSðsÞGRðsÞ(11) GWY ðsÞ¼ k1kpðcTDTIs2þbTIsþ1Þ T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp (12) F om his equa ion, we can see ha using he con en ional PID con olle , we ob ain wo complex conjuga ed ze os. Thei e ec can be canceled using he 2DOF con olle when b¼0 and c¼0. I we se band c o di e en non-ze o o non-one, he ze o can be al e ed. I we use he subs i u ion: GPðsÞ¼ k1kp T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp (13) which is he ans e unc ion o he p opo ional sys em, we ob ain he ans e unc ion om he se poin o he ou pu : GWY ðsÞ¼GPðsÞþGPðsÞbTIsþGPðsÞcTDTIs2(14) The fi s e m does no depend on any o he bo cpa- ame e s, he second e m is he fi s de i a ion o he fi s e m, and he influence can be con olled by he pa ame e b,and he hi d e m is he second de i a ion o he fi s e m and can be con olled by he pa ame e c.I bo cis nega i e, i can lead o he unde shoo . bpa ame e can be used o make he sys em esponse o he s ep se poin change mo e agg essi e. Pa ame e cwill ha e only a minimal influence on he dynamics o he sys em esponse o he s ep se poin change. Visible di e ences will be no iceable igh a e he s ep se poin change. The ans e unc ion o he dis u bance o he ou pu is equal o: G yðsÞ¼ GSðsÞ 1þGSðsÞGRðsÞ(15) G yðsÞ¼ TIk1s T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp (16) As you can see, in he s eady s a e G yðsÞ¼0. Also, his ans e unc ion is no a ec ed by he 2DOF pa ame e s b and c. The 2DOF PID con olle used o he expe imen al e ifica ion in his pape is done by single chip compu e implemen a ion. 31 Expe imen al iden ifica ion o he in eg al plan By di ec iden ifica ion wi h he s ep inpu signal, da a de- sc ibing he p ope ies o he con olled sys em we e ob ained. The measu emen was pe o med by applying an inpu signal o 50% o he maximum ange o he sys em inpu , and he s ep esponse o he sys em o his signal was eco ded (see Figu e 6). In he figu e, he alue o he ou pu a iable is no malized o he maximal angle o he o a ion ange. The same no maliza ion is also used in he ollowing pa ag aphs. By p ocessing he inpu da a and no malizing he signals, he sys em’s ans e unc ion was ob ained in he o m o an in eg a ion sys em wi h fi s -o de ine ia. GSðsÞ¼ 1 0:3039sð0:0603sþ1Þ¼3:2906 sð0:0603sþ1Þ(17) Closed loop con ol A con ol sys em was designed o he plan (sys em) de- sc ibed by (17) based on he p e ious heo y. The PID con olle pa ame e s we e uned using he nume ic op i- miza ion me hod, and hey a e lis ed in Table 1. The expe imen al measu emen was pe o med as a esponse o a s ep change o se poin om he beginning o he measu emen . A ime = 15 s, he e was a s ep change o dis u bance in on o he plan . The exac p ocess was nume ically simula ed in he MATLAB /Simulink en i- onmen , whe e he iden ifiedsys emwascon olledbya 2DOF con olle wi h ac ion a iable limi a ion (see Figu e 7). The pa ame e s (weigh s) o he inpu fil e we e selec ed om he whole ange 0–1 and measu ed and simula ed wi h he s ep o 0.2. The dis u bance is simula ed by he so wa e d op o ac ion a iable ol age on he inpu o he plan . The esul s o he measu emen s and simu- la ions a e plo ed in Figu e 8. 6,7 Acco ding o he esul s, he influence o he bpa ame e is significan ly g ea e han he cpa ame e when acking he se poin in he o m o he s ep. P o ed also by he simula ion, he bpa ame e supp esses he e ec o p opo ional e m and elimina es he o e shoo o he con ol p ocess while he esponse o a dis u bance s ep emains unchanged. Figu e 6. DC mo o ou pu sha angle o he o a ion s ep esponse. Table 1. 2DOF PID Con olle unable pa ame e s. k P T I T D bc 0.2012 2.577 0.056 0–10–1 Gu as e al. 97 The e ec o he indi idual e ms acco ding o (13) and (14) a e shown in Figu e 9. We spli he h ee e ms in o sepa a e cou ses ha gi e he o e all con ol sys em esponse using he same basic PID unable pa ame e s and b= 0.4 and c= 0.8. Quali y o eedback con ol Mul iple me hods can be used o e alua e he quali y o he eedback con ol. As i was desc ibed ea lie , one o he expec a ions o he closed-loop con ol is ha he ans e unc ion o he se poin o he ou pu is equal o one. Tha also means ha he e o should be ze o. Fo physical sys ems, we wan he sys em e o o minimize as as as possible. As an example, we can choose one o hese c i e ia o e alua ing he se poin s ep change esponse cou se: ·In eg al o Absolu e E o (IAE) IIAEðsÞ¼Z ∞ 0 jeð Þjd (18) ·In eg al o Squa ed E o (ISE) IISEðsÞ¼Z ∞ 0 e2ð Þd (19) ·In eg al o Time mul iplied by Absolu e E o (ITAE) IITAEðsÞ¼Z ∞ 0 jeð Þjd (20) whe e a lowe numbe means a as e con ol p ocess. All hese me hods can be used o bo h non-oscilla ing and oscilla ing p ocesses. The disad an age o IAE and ITAE is ha i canno be calcula ed as a de i a i e o a ze o-c ossing o absolu e alue is no defined. The e o e, hese me hods can be calcula ed only nume ically. The ISE is defined, bu he oscilla ing p ocesses ha e a lowe e o han non- oscilla ing p ocesses. These me hods a e usually used o he quan i a i e me hods o calcula ing he con ol a iables as published in 9,24 . We calcula ed he ITAE in his a icle. Nume ical simula ion is hen compa ed o he measu emen s. The calcula ed esul s o di e en pa ame e s band ccan be seen in ables: measu emen s in Table 2 (ITAE) and Figu e 7. MATLAB/Simulink nume ical simula ion schema ic. Figu e 8. Closed loop con ol ( ed –simula ed, g ay - measu ed). Figu e 9. Compa ison o he influence o he 2DOF e ms (b= 0.4, c= 0.8). Table 2. Quali y o eedback con ol acco ding o ITAE c i e ia. b/c 0 0.2 0.4 0.6 0.8 1 0 11.46 11.54 11.58 11.36 11.41 11.35 0.2 10.37 10.36 10.52 10.26 10.48 10.45 0.4 9.35 9.37 9.35 9.45 9.47 9.31 0.6 8.36 8.41 8.39 8.19 8.40 8.45 0.8 7.33 7.36 7.28 7.33 7.44 7.31 1 7.09 7.07 7.12 7.16 7.08 7.40 98 Measu emen and Con ol 55(1-2) nume ical simula ions in Table 3 (ITAE). Bo h he ables a e plo ed in he su ace g aph in Figu e 10 and Figu e 11 (simula ed). As we can see om he esul s, he di e ences be ween he simula ion and measu emen s con aining small nonlinea i ies a e neglec able. The pa ame e chas only a iny influence on he quali y, while lowe ing he bpa ame e slows down he e- sponse. The esponse o he dis u bance is no a ec ed by he 2DOF con olle pa ame e s as s a ed abo e. Using an inpu 2DOF fil e , supp ession o he o e shoo o he con ol p ocess cou se is achie ed, while wi hou i , he con ol p ocess cou se shows an o e shoo o app oxima ely 7% a a gi en se ing. Conclusion The pape shows ha using he 2DOF con olle , we ob ain mo e flexibili y in designing he final desi ed sys em e- sponse. The heo y was implemen ed in o he A duino- based con ol sys em expe imen ally iden ified and con- olled by he 2DOF PID con olle . The ans e unc ion o se poin acking and he e- ac ion o he ou pu o he dis u bance was calcula ed o he in eg al sys em wi h he fi s -o de ine ia. The closed- loop con ol using he ypical PID con olle con ains wo ze os, which o he in eg al sys em makes he esponse mo e agg essi e om he beginning a e he se poin s ep change. Tha leads o as elimina ion o he dis u bance change bu causes an o e shoo . This beha io can be supp essed using he 2DOF PID con olle , whe e hese ze os can be canceled by se ing he band cpa ame e s o lowe alues. Using he bo cpa ame e s, we can also design he dynamics o se poin acking wi hou a ec ing he beha io o he closed-loop con ol sys em o he dis u bances. The measu emen and simula ion p o ed ha he o e all influence o he bpa ame e is conside ably mo e dominan when acking he se poin alue. Acco ding o he c i e ia used o e alua e he quali y o he con ol p ocess, you can conclude ha he se ling ime is no significan ly ex ended (i we conside a su ficien ly na ow ole ance band). Howe e , he se poin s ep unc ion e- sponse o e shoo is supp essed, which can be e y im- po an o some dynamic sys ems. Depending on he se ings o he 2DOF fil e pa ame e s, we can hen achie e an o e shoo - ee cou se while main aining he same se ling ime. In he case o he pa icula sys em used, his se ing is a ound he alue 0.8–0.9 o pa ame e b. Decla a ion o conflic ing in e es s The au ho (s) decla ed no po en ial conflic s o in e es wi h espec o he esea ch, au ho ship, and/o publica ion o his a icle. Funding The au ho (s) disclosed eceip o he ollowing financial suppo o he esea ch, au ho ship, and/o publica ion o his a icle: This wo k was suppo ed by he Eu opean Regional De elopmen Fund in he Resea ch Cen e o Ad anced Mecha onic Sys ems p ojec , CZ.02.1.01/0.0/0.0/16_019/0000867 wi hin he Ope a- ional P og amme Resea ch, De elopmen and Educa ion and he p ojec SP2022/60 Applied Resea ch in he A ea o Machines and P ocess Con ol suppo ed by he Minis y o Educa ion, You h and Spo s. ORCID iDs Radek Gu as h ps://o cid.o g/0000-0002-8704-1395 Mi osla Mahdal h ps://o cid.o g/0000-0002-9720-2201 Table 3. Quali y o eedback con ol acco ding o ITAE c i e ia (simula ed). b/c 0 0.2 0.4 0.6 0.8 1 0 11.09 11.10 11.10 11.10 11.10 11.10 0.2 10.11 10.12 10.12 10.12 10.12 10.12 0.4 9.13 9.14 9.14 9.14 9.14 9.14 0.6 8.15 8.16 8.16 8.16 8.16 8.16 0.8 7.17 7.18 7.18 7.18 7.18 7.18 1 7.09 7.08 7.08 7.08 7.08 7.08 Figu e 10. Resul s o he ITAE c i e ion o di e en band c pa ame e s. Figu e 11. Resul s o he ITAE c i e ion o di e en band c pa ame e s (simula ed). Gu as e al. 99 Re e ences 1. Gu as R and Mahdal M Nonlinea in eg al sys em con ol [online]. In: 2020 21 h In e na ional Ca pa hian Con ol Con e ence (ICCC), 2020, Koˇ sice, Slo ak Republic, 2020-10- 27, pp. 1–4. [ci . 2021-7-15]. DOI: 10.1109/ICCC49264.2020. 9257237 2. Kaya I. A PI-PD con olle design o con ol o uns able and in eg a ing p ocesses. ISA T ans 2003; 42(1): 111–121, [ci . 2021-9-9]. DOI: 10.1016/S0019-0578(07)60118-9. 3. Simhachalam D, Dey C and Mudi RK. An au o- uning PD con olle o DC se o posi ion con ol sys em. 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