J. Acad. Idus. Res. Vol. 1(7) December 01 393 RESEARCH ARTICLE ISSN: 78-513 Dead Time comesators for Stable Processes Varu Sharma ad Veea Sharma Electrical Egieerig Deartmet, Natioal Istitute of Techology, Hamirur-177005, Himachal Pradesh, Idia veeaaresh@gmail.com; +91 19754536(O) Abstract Dead time is ofte reset i cotrol systems as comutatioal or iformatioal delay but i most cases it is very small ad is eglected. Dead time is widely foud i the rocess idustries whe trasortig materials or eergy. Geerally stable rocesses are rereseted by first-order-lus-dead-time or secod order-lus-dead-time models for aalysis. The roblem of cotrol desig for rocesses with dead time is quite crucial ad log-stadig. The advet of the Smith Predictor (SP) rovided the idustrial cotrol commuity with aother tool to tackle the cotrol of rocesses where the resece of dead time was imairig closed-loo erformace. I this aer, aalysis of stable rocesses with dead time is doe. Here PI cotroller ad Smith Predictor are used as dead time comesators. The comarative study with resect to referece trackig ad disturbace rejectio of PI cotroller ad Smith Predictor for the cosidered rocesses has bee covered i this aer. Keywords: Process dead time, iformatioal delay, Smith Predictor, PI cotroller, stable rocesses. Itroductio All the feedback systems are geerally rereseted by the liear lumed arameters mathematical model. This is valid so log as the time take for eergy trasmissio is egligible i.e. the outut begis to aear immediately o alicatio of iut. This is ot quite true of trasmissio chael-lies, ies belts, heat exchagers, coveyors etc. I such cases a defiite time elases after alicatio of iut before the outut begis to aear. This tye of ure time lag is kow as trasortatio lag or dead-time. Dead times or time delays, are foud i may rocesses i idustry. Dead time ca arise i a cotrol loo for a umber of reasos (Nagrath ad Goal, 1997; Normey-Rico ad Camacho, 007). Dead time issues ca be addressed by a simle chage i desig of the rocess. The rocess lat is desiged such that the sesors are located close to the actio. For ay rocess larger the dead time, the quality erformace becomes more difficult to achieve. The major difficulties i cotrollig dead-time rocesses are as (Normey-Rico ad Camacho, 007): (i) the effect of the disturbaces is ot felt util a cosiderable time has elased; (ii) the effect of the cotrol actio takes some time to be felt i the cotrolled variable; (iii) the cotrol actio that is alied based o the actual error tries to correct a situatio that origiated some time before. Delay is uavoidable i may cotrol systems. Most of the classical methods such as root locus ad Nyquist criterio that aalyze the cotrol system caot deal with delay. Moreover, systems with delay have ifiite dimesios which make it imossible to exress the system i state sace. But for aalysis ad to uderstad the dyamic behavior of the systems, mathematical modelig of rocesses is imortat. I case of dead time rocesses geerally FOPDT ad secod order lus dead time (SOPDT) models are formed for aalysis without losig the characteristics of the rocess (Abdel Fattab et al., 004; Rahmat ad Sahazati, 008). It is also doe because aalysis of higher order rocesses becomes quite difficult. Geerally rocess idetificatio methods are used to reduce higher order rocesses ito FOPDT ad SOPDT models. The FOPDT model is rereseted by K Ls P(s) e (1) 1 Ts Where K, T ad L are real umbers. T > 0 is the equivalet time costat of the lat ad K is the static gai. L > 0 is the equivalet dead time (Morari ad Zafiriou, 1989; Coughaow, 1991; Seborg, 004; Sigh, 009). Whe it is desirable to rereset a smoother ste resose i the first art the trasiets or a oscillatory ste resose, a secod-order rocess with a dead time is used. Ls Ls K e K e P(s) () (1 T 1s)(1 Ts) s s 1 Where, K,T,T,, ad L are real umbers. As i the 1 FOPDT model K is the static gai ad L > 0 the equivalet dead time. T 1 > 0 ad T > 0 are time costats of the lat i the case of a o-oscillatory resose while the damig coefficiet, (0,1) ad the atural frequecy > 0 are used whe the rocess exhibits a oscillatory ste resose. Youth Educatio ad Research Trust (YERT) Varu Sharma & Veea Sharma, 01
J. Acad. Idus. Res. Vol. 1(7) December 01 394 Dead time comesators PI Cotroller: Whe dead time is very small ad for slow variatios of the outut sigal PID cotrol is a better choice but whe dead time is log eough the cotrol erformace obtaied with a roortioal-itegralderivative (PID) cotroller is limited. Predictive cotrol is required to cotrol a rocess with a log dead time efficietly. Therefore, if a PID cotroller is alied o this kid of roblems, the derivative art is mostly switched off ad oly a PI cotroller without redictio is used (Nagrath ad Goal, 1997). I a itegral error comesatio scheme, the outut resose deeds i some maer uo the itegral of the actuatig sigal. This tye of comesatio is itroduced by usig a cotroller which roduces a outut sigal cosistig of two terms, oe roortioal to the actuatig sigal ad the other is roortioal to its itegral. Such a cotroller is called roortioal lus itegral cotroller. A PI cotroller is a secial case of the PID cotroller i which the derivative (D) of the error is ot used. The most famous tuig method for PI cotrollers is the Ziegler-Nicholas rule (ZN). It was develoed usig simulatios with differet systems where the equivalet dead time L ad time costat satisfy the coditio i.e. L 1 or called lag domiat systems. The ZN settigs T are bechmarks agaist which the erformaces of other cotroller settigs are comared i may studies. The Smith Predictor: The Smith redictor iveted by Smith i 1957 is a tye of redictive cotroller for systems with ure time delay (Palmor, 1996). Figure 1 shows this ideal situatio i a geeral case, where the cotroller C(s) is tued usig oly G(s) ad the real outut y (t) is the outut of G(s), y1 delayed L uits of time. y(t) = y1(t L). I this situatio the dead time has o effect o the closed loo trasiets, as the closed-loo trasfer fuctio is Ls Y(s) C(s)G(s)e R(s) 1 C(s)G(s) Fig. 1. Ideal cotrol for dead time rocesses. (3) The real imlemetatio of this solutio is, i geeral, ot ossible i ractice maily because the sesor caot be istalled i the desired ositio ad/or the rocess dead time is ot caused by mass trasortatio. A simle solutio for this roblem ca be obtaied usig the idea of redictio ad will be alied here to a stable rocess. If a dead-time-free model G (s) of the lat P(s) = G(s)e Ls is cosidered, it is ossible to feed the outut of this model to the cotroller as show i figure. Fig.. Oe loo redictor. I this structure if G (s) = G(s) the rimary cotroller C(s) ca be tued cosiderig oly G(s) ad the obtaied closed- loo erformace is the same as i the ideal case as i equatio (3).The equivalet cotroller for this system is C eq C(s) (s) (4) 1 C(s)G (s) This oerates i a oe-loo maer. This strategy is kow as oe-loo redictor based cotrol ad it is clear that it caot be used i ractice because the cotroller does ot see the effect of the disturbaces ad also model mismatches are ot take ito accout ad, therefore, all the beeficial roerties of feedback disaeared. A better solutio for this roblem was roosed by Smith based o a closed-loo redictor structure of the oe-loo-stable rocess. I this strategy, the redictio at time t is comuted by the use of a model of the lat without dead time G (s) ad, i order to correct the modelig errors, the differece betwee the outut of the rocess ad the model (icludig the dead time P (s) = G (s)e Ls is fed back, as ca be see i figure 3. Fig. 3. The Smith redictor structure. Youth Educatio ad Research Trust (YERT) Varu Sharma & Veea Sharma, 01
J. Acad. Idus. Res. Vol. 1(7) December 01 395 With this structure, if there are o modelig errors or disturbaces, the error betwee the curret rocess outut ad the model outut e (t) will be ull ad the cotroller ca be tued as if the lat had o dead time. Thus, i the omial case this structure gives the same erformace as the ideal solutio. To cosider the modelig errors, the differece betwee the outut of the rocess ad the model icludig dead time is added to the oe-loo redictio, as ca be see i the scheme of figure 3. If there are o modelig errors or disturbaces, the error betwee the curret rocess outut ad the model outut will be ull ad the redictor outut sigal y(t) will be the dead-time-free outut of the lat. A simle solutio to this roblem is to use a filter F r (s) with uitary static gai F r (0)=1. The filter should be desiged to atteuate oscillatios i the lat outut esecially at the frequecy where the ucertaity errors are imortat. This ca be doe by low ass filter that icreases the robustess of the cotroller. Results ad discussio Here simulatio results of three rocesses such as stirred tak heat exchager; electric ove temerature cotrol ad couled tak rocess are show ad discussed. PI cotroller ad Smith Predictor are used to cotrol these rocesses ad how the erformace of these cotrollers is iflueced by the variatio i dead time is also discussed i this sectio. Stirred tak heat exchager: The FOPDT model of stirred tak heat exchager (Morari ad Zafiriou, 1989; Coughaow, 1991) rocess is cosidered as 0.0396s e G( s) 0.0s 1 Here i this FOPDT model of stirred tak heat exchager the dead time is very small ad the tuig arameters are take as K 0.01 ad T i 0.1, which are chose usig Cohe ad Coo tuig rule. For Smith Predictor, the tuig arameters are cosidered same as above for PI cotroller. Actually the cotrol algorithm i a Smith 1 Predictor is usually a PI cotroller. Here, F 0.0s 1 is used as a filter to remove dead time estimatio errors. Actually the filter which is used to remove dead time dead time estimatio errors is i the form 1 1 F where 0.5 ad T L 1 st f 1 s L f where L is the dead time (Normey-Rico ad Camacho, 007). From figure 4, it is clear that Smith Predictor rovides much faster resose as comared to PI cotroller ad also Smith Predictor rejects the disturbace earlier as comared to PI cotroller. Fig. 4. Ste resose, PI v/s Smith Predictor. I the above aalysis, the iteral model matched the rocess model P ( s ) exactly but i ractical situatios the iteral model is oly a aroximatio of the true rocess dyamics. Electric ove temerature cotrol system: The FOPDT model of electric ove temerature system (Sigh, 009) is G ( s ) 7 0 s 1.6 3 e 1 3 4 8 0 s This system has a log dead time. Now whe a PI cotroller is alied o this system with K 3.5 ad T i 773.63 usig Cohe ad Coo tuig rule, the result obtaied is show i figure 5. Fig. 5. Ste resose with K 3.5 ad T 773.63. i Fig. 6. Ste resose, PI v/s Smith Predictor. G ( s) e L s Youth Educatio ad Research Trust (YERT) Varu Sharma & Veea Sharma, 01
J. Acad. Idus. Res. Vol. 1(7) December 01 396 From figure 6, it is clear that Smith Predictor rovides much faster resose as comared to PI cotroller ad also Smith Predictor rejects the disturbace earlier as comared to PI cotroller. Fig. 8. Ste resose with differet time delays. Couled tak rocess: The results reseted above are for FOPDT models ad ow a SOPDT model of a couled tak rocess (Rahmat ad Rozali, 008) with a small delay is cosidered as G( s) 0.4s 0.0331e s 0.0315s 0.048 Whe a PI cotroller is alied o the above system described by the equatio with K 0.0315 ad T i 3 0. 1 which are chose usig Zeigler-Nicholas tuig method, the result obtaied is show below i figure 7. Fig. 9. Ste resose, PI v/s Smith Predictor. Fig. 7. Ste resose with K 0.0315 ad T 30.1. i Figure 8 shows how ste resose is affected by icreasig the delay i dead time art. Here four differet values of dead time are used. For aalysis uroses, SOPDT model with log delay time is used. Therefore the cosidered model is G ( s) s 600s 0.0331e 0.0315s 0.048 Figure 9, shows ste resose, deictig comariso betwee PI cotroller ad Smith Predictor. It is clear that Smith Predictor rovides much faster resose as comared to PI cotroller ad also Smith Predictor rejects the disturbace earlier as comared to PI cotroller. So, it is imortat to uderstad how robust the Smith Predictor to ucertaity o the rocess dyamics ad dead time (Saravaakumar, 006). Smith redictor is able to cotrol the rocess havig larger value of dead time i a much faster ad accurate way as comared to covetioal PI cotroller. Coclusio PI cotroller ad Smith Predictor are good dead time comesators for dead time rocesses. The cotrol algorithm i a Smith Predictor is a PI cotroller ad it also uses the idea of redictio. Whe a comariso is made betwee the erformace of PI cotroller ad Smith Predictor for log dead time rocesses, better results are obtaied with Smith Predictor. Smith Predictor elimiates the effect of the dead time i the set oit resose. A good trade- off betwee robustess ad erformace ca be obtaied by aroriate tuig of rimary cotroller. Whe the rocess exhibits itegral dyamics, the classical Smith Predictor fails to rovide a ull steady state error i the resece of a costat load disturbace. Refereces 1. Abdel Fattab, H.A., Gesraba, A.M. ad Haafy, A.A. 004. Cotrol of Itegratig Dead Time Processes with Log Time Delay. America Cotrol Coferece Bosto, Massachusetts.. 4964-4970.. Coughaow, D.R. 1991. Process Systems Aalysis ad Cotrol. McGraw-Hill Iteratioal Editio, secod editio. Youth Educatio ad Research Trust (YERT) Varu Sharma & Veea Sharma, 01
J. Acad. Idus. Res. Vol. 1(7) December 01 397 3. Morari, M. ad Zafiriou, E. 1989. Robust Process Cotrol. Pretice Hall, Eglewood Cliffs, New Jersey. 4. Nagrath, I.J. ad Goal, M. 1997. Cotrol System Egieerig. New Age Iteratioal (P) Limited, secod editio. 5. Normey-Rico, J.E. ad Camacho, E.F. 007. Cotrol of Dead Time Processes. Sriger-Verlag Lodo Limited. 6. Palmor, Z.J. 1996. The Cotrol Hadbook, Time Delay Comesatio: Smith Predictor ad its Modificatios. CRC Press ad IEEE Press. 7. Rahmat, M.F. ad Rozali, S.M. 008. Modellig ad cotroller desig for a couled tak liquid level system: aalysis ad comariso. J. Tekologi. 48(D): 113-141. 8. Saravaakumar, G., Wahidha Bau, R.S.D. ad George, V.I. 006. Robustess ad Performace of Modified Smith Predictors for rocesses with loger dead-times. ACSE J. 6(3): 41-46. 9. Seborg, D.E., Edger, T.F. ad Mellicham, D.A. 004. Process Dyamics ad Cotrol. Joh Wiley & Sos, secod editio, USA. 10. Sigh, S.K. 009. Process Cotrol Cocets, Dyamics ad Alicatios. PHI Learig Pvt. Ltd. New Delhi. Youth Educatio ad Research Trust (YERT) Varu Sharma & Veea Sharma, 01