Investigation of P and PD Controllers Performance in Control Systems with Steady-State Error Compensation

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1 ELECRONICS AND ELECRICAL ENGINEERING ISSN No. 5(2) ELEKRONIKA IR ELEKROECHNIKA 25 AUOMAION ROBOICS AUOMAIZAVIMAS ROBOECHNIKA Invsigaion of P and PD Conrollrs Prforman in Conrol Sysms wih Sady-Sa Error Compnsaion D. Lvisausas. orius Pross Conrol Dparmn Kaunas Univrsiy of hnology Sudnų sr Kaunas Lihuania phon: mails: donaas.lvisausas@u.l omas.orius@u.l hp://dx.doi.org/.5755/j Inroduion A oporional-ingral-drivaiv (PID) onrollr is h mos ommon onrollr in indusrial oss onrol sysms []. h hr-rm onrol aion as ino aoun h sn rror (P) aumulaion of pas rrors (I) and diion of fuur rrors (D) and h wighd sum of hs aions dpnds on h wighing offiins ha ar adjusd o h opimum valus for h dsird rspons of onrolld oss. Pariular appliaions rquir only on (P I) or wo (PI PD) onrol aions o ovid h rquird auray of onrol. Choosing of onrol law dpnds on rquirmns o a onrol sysm. On of frqun rquirmns is absn of sady-sa rror bwn h s-poin and h onrolld oupu. h sady-sa rror is an inbuil hararisi of fd-ba onrol sysms wih P and PD onrollrs and h mos ommon way of h rror liminaion is o add an ingral rm (I) o h onrol law. Alhough h ingral rm of PI and PID onrol laws allows auomai liminaion of h rror i nds o inras h osillaory or rolling bhavior of h oss rspons. h ohr disadvanag rlad o h ingral rm is an ingral windup oblm ha ours a larg s-poin hangs and h oss sauraion [ 3]. h alrnaiv way of h sady-sa rror liminaion is o add an adjusabl rs rm o h onrol signal or o add a orriv rm o h inpu of sysm wih P or PD onrollr. As h abov onrollrs dmonsra br sabiliy opris appliaion of his apoah an b worhwhil. In his wor w invsiga prformans of P and PD onrollrs in h sysm wih sady-sa rror ompnsaion by adding a orriv rm o h sysm inpu. h onrol sysm rsponss ar valuad wih h ingral of im-wighd absolu rror (IAE) [2]. h rsuls ar ompard wih hos of ordinary PI and PID onrol sysms. Prforman of P onrollr in h sysm wih sady-sa rror ompnsaion and h sysm adapaion o im-varying opraing ondiions is invsigad via ompur simulaion of h s-poin onrol sysm of dissolvd oxygn onnraion (DOC) in bah opraing mod bioraor. P and PD onrol sysms wih sady-sa rror ompnsaion Sruur of fd-ba P (PD) onrol sysm wih sady-sa rror ompnsaion is shown in Fig.. Fig.. P (PD) onrol sysm wih sady-sa rror ompnsaion K Δy is ransfr offiin bwn h s- In Fig. poin and h orriv rm; G s is ransfr funion G s and of onrolld oss; P G s s PD ar ransfr funions of P and PD d onrollrs rspivly; y and y s ar h oss oupu and h s-poin rspivly; y is orriv rm; and ar h oss and h onrollr gain offiins rspivly; d is diffrniaion im onsan s is Lapla opraor. Wih P or PD onrollr a sady-sa opraing ondiions h orriv rm y a h inpu of fd-ba onrol sysm ha ompnsas dviaion of oss oupu from h s-poin saisfis h ondiion 63

2 ys y ys. () + From quaion () i follows ha h orriv rm is rlad wih h s-poin valu by a rlaionship whr Δy y K y (2) s Δy K. (3) Simulaion of h onrol sysms prforman Prformans of P and PD onrol sysms wih sady-sa rror ompnsaion and h ordinary PI and PID onrol sysms wr invsigad via numrial simulaion implmnd in MALAB/Simulin nvironmn. In h simulaion xprimns h onrolld oss was modld by a 3 rd ordr ransfr funion wih im dlay G s s s s xp τ s 2 3 (4) whr is oss gain 2 3 ar im onsans and τ is im dlay. In h simulaions h gain offiin was s and svral ss of h modl paramr 2 3 τ valus wr usd. h varian valus of h paramrs ar givn in abl. h P PD PI and PID onrollrs in simulaion xprimns wr und for ah s of modl paramrs o minimiz h IAE ririon. h onrollr paramrs wr opimizd in 2 sps: a h s sp h oss modl (4) was apoximad by a firs ordr plus im dlay (FOPD) modl and h IAE ririon uning ruls dvlopd for h FOPD modl [2] wr applid o drmin h firs apoah valus of h onrollr paramrs; a h 2 nd sp h firs apoah paramr.2 valus wr imovd using h Quadrai Programming algorihm (funion fminon ) from h MALAB Opimizaion oolbox library. h IAE ririon valus wr alulad for h onrol sysm rsponss o a uni sp s-poin hang. In Fig. 2 h simulaion rsuls ar snd for h modl paramrs s No givn in abl. h sp hang in s-poin has bn inrodud a im. Valus of h onrol sysms prforman ririon (IAE) alulad for all invsigad varians ar ompard in Fig. 3. abl. Pross modl paramrs usd in simulaion xprimns.2 Varian No Modl paramrs y().6 y() s-poin PI IAE=8547 P IAE= s-poin PID IAE=2776 PD IAE= a) b) Fig. 2. Simulad rsponss of h onrol sysms o a uni sp s-poin hang: a P onrol sysm wih sady-sa rror ompnsaion vrsus ordinary PI onrol sysm; b PD onrol sysm wih sady-sa rror ompnsaion vrsus ordinary PID onrol sysm 64

3 2 P PI 35 3 PD PID 8 25 IAE 6 4 IAE Varian numbr a) b) Varian numbr Fig. 3. Comparison of h IAE ririon valus alulad from h simulaion xprimns: a P wih sady-sa rror ompnsaion vrsus PI; b PD wih sady-sa rror ompnsaion vrsus PID Rfrring o h valuad valus of h IAE ririon w find ha h P onrollr wih h sady-sa rror ompnsaion is abou 5% br han h PI onrollr wih rsp o man valus (a normalizd sandard dviaion 4.9% ) and h PD onrollr is abou 3% br han h PID onrollr ( 4.8% ). Saisial ss on omparison of 2 sampl mans [4] ov ha h obsrvd diffrn in h IAE ririon valus ar signifian wih a 5 % lvl of signifian. Simulaion of h onrol sysms prforman for onrolling h dissolvd oxygn onnraion in bioraor Dissolvd oxygn onnraion (DOC) is an imporan hnologial paramr of arobi frmnaion osss ha is o b auraly onrolld a a pariular lvl. h onrolld oss is nonlinar and nonsaionary hrfor adapaion of fd-ba onrollr o im-varying opraing ondiions is rquird o nsur an aura s-poin onrol ovr nir bah frmnaion yl. On of dvlopd apoahs o adapaion of h DOC onrollr paramrs is basd on adapiv ransfr funion ha is drivd from h oss sa modl and updad on-lin wih h masurd valus of oss variabls [5]. h ransfr funion paramr valus ar dirly inrodud in h onrollr uning ruls ha ar usd for ralulaion h onrollr paramrs a ah im disrizaion sp. In his wor h abov onrollr adapaion hniqu is applid for adapaion of PI onrollr in an ordinary fd-ba onrol sysm and adapaion of P onrollr and h ransfr offiin bwn h spoin and h orriv rm in h onrol sysm wih sady-sa rror ompnsaion. In h simulaion xprimns h onrolld oss is modld by a s of quaions [5 6]: dc β γ α u q Csa C OUR (5) d dc C C (6) d whr C is DOC mmol L - ; C is signal of dissolvd oxygn (DO) lrod %; C sa is sauraion valu of DOC mmol L - Csa.2 H H is Hnry s onsan L mmol - ; u is sirring spd (onrol aion) s - ; q is air supply ra L s - ; OUR is volumri oxygn upa ra mmol L - s - ; is im s; is im onsan of DO lrod s; is oporionaliy offiin H.2 ; α β γ ar modl paramrs. h modl quaion (5) rsns mass balan on oxygn in frmnaion broh and quaion (6) rsns firs ordr dynamis of DO lrod. Paramrs of h modl quaions ar an from h rangs rpord in [6]: H.796 α.5 β 2. γ.2. In h viiniy of sady-sa opraing ondiions a im poin h oss dynamis an b rsnd by linar diffrnial quaions drivd by linarizaion of h sa quaions (5) (6): dc β γ α uq C d β- γ αβu q C sa C u (7) dc C C d (8) whr C C and u ar small dviaions of C C and u from h oss sa poin a im. By applying h Lapla ransformaion o quaions (7) (8) h onrolld oss dynamis an b rsnd by a 2 nd ordr ransfr funion: G C u s C u C u s s (9) 65

4 C u C u sa u αuq β C C β γ (). () Assuming quasi-sady-sa ondiions for DOC ( dc d ) h dynami paramrs C u and C u an b simad on-lin using masurd valu of OUR and s-poin valu of DOC ( C C ): s For adapaion of PI onrollr paramrs h Ziglr&Nihols uning ruls [2] ar applid:.9 C u C u C u 3.33 (8) i C u. (9) h adapiv P onrol wih sady-sa rror ompnsaion is implmnd by h onrol sysm snd in Fig. 5. C u q C β γ s βα OUR C u β (2) Cs. (3) OUR In h adapiv onrol sysm h updad ransfr funion modl (9) is furhr rdud o a FOPD modl G * C u C xp u C u C u s (4) whr C u C u C u C u ar rsulan im onsan and rsulan im dlay of h onrolld oss a im rspivly. h modl paramrs C u and C u ar updad by fiing h FOPD modl (4) a ah sampling im o a simulad sp rspons of h modl (9) using h Smih s apoximaion hniqu [2]. h updad FOPD modl is dirly applid for onrollr paramrs on-lin adapaion using onrollr uning ruls dvlopd for simpl dynami modls. h adapiv PI onrol is implmnd by h onrol sysm snd in Fig. 4. and Fig. 5. DOC adapiv onrol sysm wih P onrollr and sadysa rror ompnsaion h disr P onrol algorihm ralizd in h onrol sysm (Fig. 5) is u (2) in whih adapaion of h onrollr gain is ralizd by h Ziglr&Nihols uning rul [2] h oss gain C u. (2) C u C u is simad from h modl quaion (5) a sady-sa ondiions ( dc d ): C u s dc d γ αq Cs Cs OUR β. (22) Fig. 4. DOC adapiv onrol sysm wih PI onrollr In h onrol sysm (Fig. 4) h vloiy form of disr PI onrol algorihm is usd: u u u (5) u (6). (7) i C C s h ransfr offiin bwn h s-poin and h K C is simad from h alulad orriv rm valus and K C. (23) Prformans of h onrol sysms snd in Fig. 4 and Fig. 5 wr invsigad via ompur simulaions ralizd in MALAB/Simulin nvironmn. h simulaion rsuls ar shown in Fig

5 OUR().24 () a) b) K C () C s () C() s-poin PI onrollr P onrollr ) d) Fig. 6. Simulad prformans of h DOC adapiv onrol sysms a s-poin sp hangs and im-varying oxygn upa ra: OUR (a) adapaion of P onrollr gain (b) adapaion of ransfr offiin bwn s-poin and orriv rm () DOC onrolld a s poin (d) In h simulaion xprimns disurbans of h DOC s-poin sp hangs undr im-varying oxygn upa ra wr applid. Sp hangs of h s-poin from % o 2 % from 2 % o 4 % and from 4 % o 3 % wr inrodud a im poins 5 s 2 s and 35 s rspivly. im-varying rajory of OUR snd in Fig. 6 (a) is hosn o simula los o ralisi opraing ondiions a bah frmnaions. Adapaion of P onrollr gain and h ransfr offiin bwn h s-poin and h orriv rm o im-varying opraing ondiions is dmonsrad in Fig. 6 (b) and Fig. 6 () rspivly. Rsponss of h onrolld DOC o h s-poin hangs ar givn in Fig. 6 (d). h solid lin dmonsras rsponss of h adapiv P onrol sysm wih sady-sa rror ompnsaion h dod lin rsponss of h adapiv PI onrol sysm. h simulaion rsuls dmonsra ha h DOC adapiv P onrol sysm wih sady-sa rror ompnsaion ouprforms h adapiv PI onrol sysm in rsp of ris im sling im and ingral of absolu valu of h rror. Conlusions Prformans of P and PD onrollrs in onrol sysms wih sady-sa rror ompnsaion ar invsigad and ompard wih hos of PI and PID onrollrs in ordinary fd-ba onrol sysms. h invsigaion is arrid ou by ompur simulaion of h onrol sysm prformans in onrolling osss wih various dynami paramrs. h onrollrs wr und o minimiz h IAE ririon ha was usd as a masur of onrol qualiy. h simulaion rsuls dmonsra ha h P and PD onrollrs in onrol sysms wih sady-sa rror ompnsaion ouprform h PI and PID onrollrs in ordinary fd-ba onrol sysms rspivly. Saisial ss ov ha h obsrvd diffrns in h IAE ririon valus ar signifian. Prforman of P onrollr in h sysm wih sady-sa rror ompnsaion and adapaion of onrollr gain was invsigad by onrolling h dissolvd oxygn onnraion in bah opraing mod bioraor a s-poin sp hangs and prmann hang in oss dynamis. h simulaion rsuls show ha h sysm wih adapiv P onrollr and sady-sa rror 67

6 ompnsaion ovids shorr ris and sling ims ompard wih hos of adapiv PI onrollr. h invsigaion rsuls ov ha appliaion of P and PD onrol sysms wih h sady-sa rror ompnsaion is worhwhil. Rfrns. Asrom K. J. Hagglund. PID Conrollrs: hory Dsign and uning 2nd d. Insrumn Soiy of Amria Rsarh riangl Par NC p. 2. Smih C. A. Corripio A. B. Prinipls and Prai of Auomai Pross Conrol 3rd d. John Wily & Sons In p. 3. Krjar O. Spia I. Frishr R. Implmnaion of Full Faurd PID Rgulaor in Miroonrollrs // Elronis and Elrial Enginring. Kaunas: hnologija 2. No. 7(3). P DOI:.5755/j Shillr J. J. Srinivasan R. Spigl A. Murray R. Shaum's Oulins: Probabiliy and Saisis 3rd d. MGraw Hill Profssional Publishing p. 5. Lvišausas D. An Algorihm for Adapiv Conrol of Dissolvd Oxygn Connraion in Bah Culur // Biohnology hniqus. Kluwr Aadmi Publishrs 995. Vol. 9. No. 2. P Nilsn J. Villadsn J. Lidn G. Bioraion Enginring Prinipls 2nd d. Kluwr Aadmi/Plnum Publishrs p. Rivd 22 5 Apd afr rvision D. Lvisausas. orius. Invsigaion of P and PD Conrollrs Prforman in Conrol Sysms wih Sady-Sa Error Compnsaion // Elronis and Elrial Enginring. Kaunas: hnologija 22. No. 5(2). P In h papr prforman of P and PD onrollrs in onrol sysms wih sady-sa rror ompnsaion is invsigad and ompard wih a prforman of ordinary PI and PID onrol sysms. h invsigaion is arrid ou by ompur simulaion of h onrol sysms in onrolling h osss wih various dynami paramrs. Prforman of P onrollr is also invsigad by opraing in h onrol sysm wih sady-sa rror ompnsaion and h onrol sysm adapaion o im-varying opraing ondiions. h invsigaion rsuls show ha ha h P and PD onrollrs in h sysms wih sady-sa rror ompnsaion ouprform h ordinary PI and PID onrol sysms rspivly. Ill. 6 bibl. 6 abl. (in English; absras in English and Lihuanian). D. Lvišausas. orius. P ir PI rguliaorių viimo yrimas valdymo sismos su ompnsuojama sain palaida // Elronia ir lrohnia. Kaunas: hnologija 22. Nr. 5(2). P Sraipsnyj iriamas P ir PD rguliaorių viimas valdymo sismos su ompnsuojama sain palaida. Šių sismų viimas palyginas su įasų PI ir PID valdymo sismų viimu. yrimas alias saimninio modliavimo būdu valdan osus su įvairiomis dinaminėmis savybėmis. aip pa iširas P rguliaoriaus viimas valdymo sismoj su ompnsuojama sain palaida isiaiančioj i laiui bėgan inančių oso dinaminių savybių. yrimo rzulaai rodo ad P ir PD rguliaoriai sismos su ompnsuojama sain palaida lidžia pasii grsnius valdymo oybės rodilius palygini su įasinėmis PI ir PID valdymo sismomis. Il. 6 bibl. 6 ln. (anglų alba; sanrauos anglų ir liuvių.). 68