Impact of the Sampling Period on the Design of Digital PID Controllers

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1 Impact of th Sampling Priod on th Dsign of Digital PID Controllrs Dimitris Tsamatsoulis Halyps Building Matrials S.A. 17 th klm Nat. Road Athns Korinth, Aspropyrgos, Grc Abstract Th impact of th sampling priod on th paramtrization of a digital PID controllr in th frquncy domain is attmptd using thr diffrnt digital approximations of th intgral action. Th controllr is implmntd in th industrial procss of rgulation of th cmnt sulphats in th cmnt mill outlt. Th maximum snsitivity, M s, has bn utilizd as a main robustnss critrion. For th sam M s, proportional and diffrntial gain, a ris of th sampling priod lads to a dcras of th intgral gain k i for all th thr approximations. For th sam sampling priod, th function btwn proportional and intgral gain diffrs for th thr approximations studid. If th dsign satisfis two critria simultanously, maximum snsitivity and phas margin in th currnt study, thn th prmissibl PID gains zon bcoms narrowr. Kywords digital PID; sampling; frquncy; cmnt; robustnss I. INTRODUCTION Digital controllrs ar implmntd nowadays in numrous applications in ordr to provid fdback control on a continuous tim procss, whos outputs ar sampld in discrt tim instancs. As it has bn noticd by Astrom t al. [1] in th industrial procss control th PID typ controllrs constitut a prcntag mor than 95% of th installd ons. Thus, th study of th discrt PID controllrs rmains a challnging fild of rsarch du to its larg fild of application. Various tchniqus hav bn dvlopd for th dsign of digital controllrs. Th first attmpt to analys and dsign discrt tim control systms was dscribd in a papr by Kalman [2]. Madsn t al. [3] rfrs two main approachs for dsigning a digital controllr: Th dirct sampld data approach and a scond on calld digital rdsign. Th rdsign mthod involvs an initial dvlopmnt of an analog controllr mting th systm rquirmnts. Thn this controllr is transformd to a digital on through a sris of rdsign procdurs. Sborg t al. [4] dscribs two basic mthodologis for digital controllr dsign: Th dirct synthsis ( Dahlin s mthod, Vogl Edgar algorithm, intrnal modl control) and minimum varianc control. Das t al. [5] has bn utilizd th continuous and discrt tim Linar Quadratic Rgulator (LQR) thory in dsigning of optimal analog and discrt PID controllrs. Yousfzadh t al. [6] aftr an initial dsign in th frquncy domain of an analog PID, thn applis th pol-zro mapping mthod to tun its digital form. Prtz t al. [7] rfrs that th most popular approach in dsigning digital compnsators is th frquncy domain basd mthod. Okuyama [8] has bn prsntd a dsigning problm of discrt-tim PID algorithm solvd in th frquncy domain. During th procdur, a modifid Nichols diagram is applid. Papadopoulos t al. [9] utilizd also th frquncy domain to dscrib a gnral procss modl. Thn th transition to th z domain in conjunction with a symmtrical optimum critrion has bn implmntd to gnrat tuning ruls of digital PID controllrs for intgrating procsss. This study aims at invstigating of th paramtrization of a digital PID controllr in th frquncy domain by applying it to an actual industrial procss of high importanc as concrns th quality of th manufacturd goods: Th rgulation of th cmnt sulphats in th cmnt mill outlt. Du to th significanc of th sampling priod in th controllr dsign, an xtnsiv sarch has bn attmptd to b carrid out. II. PROCESS MODEL Th SO 3 control and rgulation is prformd by sampling cmnt in th mill outlt, masuring th sulphats and changing th gypsum prcntag in th cmnt composition. Gypsum is th main sourc of SO 3. Anothr sourc is th clinkr which constituts th main cmnt componnt and th fly ash also in th cas of pozzolanic cmnts. Fig.1. Block diagram of th SO 3 rgulation fdback loop. 135

2 Th variation of th SO 3 contnt of th raw matrials rprsnts disturbancs that hav to b liminatd. Th block diagram and th transfr functions of th SO 3 rgulation procss ar shown in Figur 1. Th blocks rprsnt th subsqunt sub-procsss: GP = mixing of matrials insid th milling installation; GS = cmnt sampling; GM = SO 3 masurmnt; GC = control law. Th signals of th fdback loop ar dnotd with th following symbols: %G, %CL, %FA = th prcntag of gypsum, clinkr and fly ash rspctivly. %SO3 = th cmnt sulphats. %SO3_S, %SO3_M = SO 3 of th sampld and masurd cmnt corrspondingly. %SO3_T = sulphats targt. = %SO3_T - %SO3_M is th rror of th rgulation. d = th fdrs disturbancs. n = nois of th SO 3 masurmnt. Th schm shown in Figur 1 compriss all th basic componnts of th fdback control loop. A. Transfr Functions of th Control Loop To modl th control loop, th transfr functions and thir Laplac transform has bn considrd. Th dtrmination of th dynamics btwn th gypsum contnt of th matrial fd to th mill and th SO 3 of th cmnt in th mill circuit outlt has bn achivd with a stp rspons tst in an actual closd circuit cmnt mill. A first ordr with tim dlay (FOTD) modl dscribs this dynamics which constituts th procss transfr function GP. Th Laplac transform of this modl is givn by quation (1): GP = %SO3 %G = k g xp ( T D s) 1 + T 0 s (1) Whr k g = th dynamics gain (%SO 3 /%Gypsum), T D, T 0 = th dlay tim and first ordr tim constant rspctivly (min). During th sampling priod an avrag sampl is collctd. Thus, th variabls %SO3 and %SO3_S in th tim domain ar connctd with th rlation (2): Whr k p, k i, k d ar th proportional, intgral and drivativ gain of th controllr rspctivly. To driv an xprssion for th digital form of th controllr, th analog formula is usd in th instancs t = k and t = (k-1) whr k is an intgr. By subtracting th two quations th digital xprssion of PID is drivd, dscribd by (6) in tim domain. Its Laplac transform is givn by (7). For comparison rasons th Laplac form of th analog PID is providd also by th formula (8). %G k %G k 1 = k p k k 1 + k i k k + k 2 2 k (6) = k k i p + 1 xp s 1 xp s (7) = k p + k i s s (8) Probably th bst critrion of robustnss is th snsitivity function dtrmind by th Laplac quation (9): S = GC GP GS GM (9) Th variabl 1/M s can b intrprtd as th shortst distanc btwn th opn loop Nyquist curv and th critical point (- 1,0) shown in th Figur 2. In th sam figur othr systm proprtis, charactrizing th systm stability ar also shown: - Th gain margin, g m - Th gain crossovr frquncy, ω gc - Th phas margin, φ m - Th snsitivity crossovr frquncy, ω sc - Th maximum snsitivity crossovr frquncy, ω mc. t+ %SO3_S t = 1 GS = t %SO3 dt Th Laplac transform of (2) is givn by (3): (2) %SO3 _S %SO3 = 1 s 1 s (3) Whr = th sampling priod (min). Th sampl transfr from th sampling point to th quality laboratory, th masurmnt procdur of SO3, th calculation of th nw gypsum contnt and its transfr to th mill fdr lasts T M minuts. This dlay is xprssd by (4). GM = %SO3 _M %SO3 _S = T M s (4) Th PID controllr has as input th masurd valus of SO 3, %SO3_M and producs th nw prcntag of gypsum, %G. Its analog implmntation is dscribd by (5): %G t = k p + k i dt 0 t d dt (5) Figur 2. Maximum snsitivity, phas margin and crossovr frquncis 136

3 Analog PID, k d =0, Ts=1 min Digital PID, k d =0, Ts=1 min Digital PID, k d =0, Ts=10 min Digital PID, k d =0, Ts=20 min Figur 3. Maximum snsitivity as function of k p, k i and for analog and digital PID implmntation III. RESULTS AND DISCUSSION To invstigat th prformanc of th digital PID in th rgulation of th givn procss, for diffrnt valus of k d in %G/%SO3 min, th subsqunt ara of th othr two paramtrs has bn slctd: 0 k p 1 in %G/%SO3, 0.01 k i in %G/%SO3/min. Th procss paramtrs ar th following: k g =0.42 %G/%SO3, T 0 =14 min, T D =8 min, T M =20 min. Th sampling priod is a variabl undr invstigation. A. Comparisons of Analog and Digital PID Initially th prformancs of analog and digital controllrs ar compard using M s as a critrion. For k d =0, th maximum snsitivity as a function of k p, k i for analog and digital controllrs is shown in Figur 3. For fast sampling frquncy, =1 min, th prformanc of both implmntations is th sam. As th sampling priod augmnts for th sam pair (k p, k i ) th M s of digital controllr incrass too. Th abov rsult is vry significant: Th tuning of a digital PID by using its analog implmntation and fast sampling priod, lads to an inaccurat M s if th actual sampling priod applid in th procss is high nough. To study th impact of th sampling priod on th digital controllr prformanc for various (k p, k i, k d ) sts th nxt procdur has bn followd: (i) Two sts (k p, k i ) 1 =(0.5, 0.031) and (k p, k i ) 2 =(0.2, 0.04) hav bn slctd; (ii) a fast sampling priod =1 min and thn priods from 10 to 120 min with a stp of 10 min; (iii) drivativ gains from 0 to 5 with a stp of 1; (iv) Th M s valus hav bn computd. Th rsults ar shown in Figurs 4 and 5. As incrass, M s incrass too, by tnding in an asymptotic valu for ach k d. Th impact of k d is not in on dirction: for lowr an incras of k d causs a dcras of M s, whil for high valus of, th rlation btwn k d and M s is th rvrs. Figur 4. Maximum snsitivity as function of, k d for (k p,k i)=(0.5, 0.031) 137

4 For small sampling priod =1min- for th sam proportional gain, whn k d incrass, th intgral gain incrass too. Th rvrs happns to =30 min For highr sampling priods, thr is ngligibl impact of th diffrntial gain on th function btwn k p, k i. Figur 5. Maximum snsitivity as function of, k d for (k p,k i)=(0.2, 0.04) B. Digital PID Dsign Satisfying On Critrion To invstigat th impact of th sampling priod on th digital PID dsign, th maximum snsitivity has bn slctd as th critrion. Th rquirmnt placd in th dsign is 1.45 M s PID sts satisfying this constraint hav bn dtrmind for sampling priods ranging from 1 min to 120 min. Th PID rgions for k d = 0, 2, 4 ar shown in Figurs 6, 7, 8 rspctivly. Figur 8. Plots of (k p, k i) for M s є [1.45,1.55], various and k d=4. Figur 6. Plots of (k p, k i) for M s є [1.45,1.55], various and k d=0. C. Digital PID Dsign Satisfying Two Critria In this cas two critria of prformanc and robustnss hav bn combind: (a) Th maximum snsitivity M s and (b) th phas margin φ m. Th dsign is applid for k d =0, 1.45 M s 1.55 and 60 φ m 70. As in th prvious cas, varid from 30 min to 120 min. Th dsign is prformd for ach critrion sparatly and th corrsponding rgions (k p, k i ) ar found. Th common rgion satisfis both critria. Th rsults ar shown in Figurs 9 to 11. From ths Figurs it is concludd that for ach th common ara satisfying both critria is much narrowr compard with th on satisfying ach critrion. Additionally this common sgmnt is locatd to small valus of k p. Gnrally an incras of rsults in a dcras of both k p and k i. Figur 7. Plots of (k p, k i) for M s є [1.45,1.55], various and k d=2. From ths Figurs th following conclusions can b xtractd: Th magnitud of has strong function on th sts (k p, k i, k d ) driving M s є [1.45, 1.55]. Gnrally, whn incrass, for th sam k p, k d, th intgral gain is moving to lowr valus. Figur 9. Plots of (k p, k i) for = 30 min and k d=0. 138

5 k + k 1 %G k %G k 1 = k p k k 1 + k i T 2 s T k + k 2 2 k (11) s = k p + k i 1 + xp s 2 1 xp s 1 xp s (12) Figur 10. Plots of (k p, k i) for = 60 min and k d=0. Using ths two implmntations th controllr has bn dsignd satisfying th maximum snsitivity critrion, for 1.45 M s 1.55, k d =0 and ranging from 1 min to 120 min. Th rsults ar shown in Figurs 12, 13 and th comparison has to b mad with th ons shown in Figur 6. Figur 11. Plots of (k p, k i) for = 120 min and k d=0. D. Othr Digital Implmntations of th PID Controllr Th digital implmntation of th analog PID dscribd by [6] includs backward diffrncs of both intgral and drivativ actions. Thr ar svral othr digital implmntations as Astrom t al. analyzd [1]. Two of thm hav bn invstigatd as concrns th impact of th sampling priod on th controllr robustnss and prformanc: (i) forward diffrncs (ii) Tustin approximation which is also known as bilinar or trapzoidal approximation. Both implmntations hav bn applid to th intgral trm of th PID. Th drivativ action continud to b approximatd by backward diffrnc. Th digital PID with th forward diffrnc of th intgral trm is dscribd by (9) in tim domain. Its Laplac transform is providd by (10): %G k %G k 1 = k p k k 1 + k i k 1 k + k 2 2 k (9) = k p + k i xp s 1 xp s 1 xp s (10) Th corrsponding xprssions of th Tustin approximation ar givn by (11), (12). Figur 12. Plots of (k p, k i) for forward diffrnc of intgral part Figur 13. Plots of (k p, k i) for Tustin approximation of intgral part Th following conclusions can b xtractd from Figurs 6, 12, 13. For high frquncy sampling all th thr approximations giv similar (k p, k i ) zons of qual M s Th gnral trnd is that an incras of, causs a dcras k i for th sam k p. Th bhavior of th thr approximations diffrs significantly, i.. if a PID has bn paramtrizd using 139

6 a crtain digital form, th robustnss dos not rmain th sam in cas th sam gains ar applid to anothr digital PID implmntation. For 30 min, th function btwn k p, k i is not th sam for th thr approximations. In th backward approximations a ris of k p, causs a dcras of k i. Th rvrs function appars in th forward approximation. Th trapzoidal rul shows an intrmdiat bhaviour, btwn th othr two. CONCUSIONS An invstigation of th impact of th sampling priod on th dsign and paramtrization of a digital PID controllr has bn attmptd using th frquncy domain. An industrial procss of high importanc as concrns th quality of th products has bn chosn: Th rgulation of th cmnt sulphats in th cmnt mill outlt. Thr diffrnt approximations hav bn utilizd for th intgral action of th controllr: Th backward, forward and Tustin approximations. In cas th dsign should satisfy on robustnss constraint, th maximum snsitivity, M s, has bn chosn as such. In this cas for th sam M s and diffrntial gain, a ris of th sampling priod lads to a dcras of th intgral gain for all th thr approximations. An incras of th diffrntial gain has not strong impact on th (k p, k i ) rgion satisfying th M s rquirmnt whn th backward diffrnc is utilizd and Ts 60 min. Th function btwn k p and k i, for th sam sampling priod diffrs for th thr approximations studid. Thus th way th PID is implmntd digitally plays an important rol too. If th dsign satisfis two critria simultanously, maximum snsitivity and phas margin in th currnt study, thn th (k p, k i ) zon bcoms mor narrow. An incras of, has not a svr influnc on th k i valus but causs a slight drop of th proportional gains. A furthr futur study using th z-transform and procss simulations can xtnd th rsults of th currnt study and gnraliz thm to a varity of othr procsss. REFERENCES [1] K. Astrom and T. Hagglund, Advancd PID Control, Instrumntation, Systms and Automatic Socity: Rsarch Triangl Park, 2006, pp. 1, pp [2] R.E.Kalman, Nonlinar Aspcts of Sampld Data Control Systms, Proc. Of th Symposium on Non-linar Circuit Analysis, vol.vi, 1956, pp [3] J. M. Madsn, L. Shih and S. Guo, Stat-spac Digital PID Controllr Dsign for Multivariabl Analog Systms with Multipl Tim Dlays, Asian Journal of Control, vol. 8, pp , Jun [4] D. E. Sborg, T. F. Edgar and D.A. Mllichamp, Procss Dynamics and Control, 2nd d., Jon Wily & Suns, 2004, pp [5] S. Das, I. Pan, K. Haldr, S. Das and A. Gupta, LQR Basd Improvd Discrt PID Controllr Dsign via Optimum Slction of Wighting Matrics Using Fractional Ordr Intgral Prformanc Indx, Applid Mathmatical Modlling, vol. 37, pp , 15 March [6] V. Yousfzadh, N. Wang, Z. Popovic and D. Maksimovic, A Digitally Controlld DC/DC Convrtr for an RF Powr Amplifir, IEEE Transactions on Powr Elctronics, vol. 21, pp , January [7] M. M. Prtz and S.Bn-Yaakov, Invstigation of Tim Domain Dsign of Digital Controllrs for PWM Convrtrs, IEEE 24th Convntion of Elctrical and Elctronics Enginrs, Isral, 2006, pp [8] Y. Okuyama, Discrtizd PID Control on an Intgr Grid, Socity of Instrumnt and Control Enginrs Annual Confrnc, Tokyo, 2008, pp [9] K. G. Papadopoulos, N. D. Tslpis and N. I. Margaris, Analytical Tuning Ruls of Digital PID Controllrs for Intgrating Procsss via th Symmtrical Optimum Critrion, Procdings of th th Mditrranan Confrnc on Control & Automation (MED), Barclona, Spain, 2012, pp