Simplifying PID Control. Optimizing Plnt Performnce. Performnce Monitoring Fundmentls: Demystifying Performnce Assessment Techniques Roert C. Rice, PhD Rchelle R. Jyringi Dougls J. Cooper, PhD Control Sttion, Inc. Deprtment of Chemicl Engineering Control Sttion, Inc. One Technology Dr. University of Connecticut One Technology Drive Tollnd, CT 684 Storrs, CT 669-3 Tollnd, CT 684 o.rice@controlsttion.com rchelle@engr.uconn.edu doug.cooper@controlsttion.com ABSTRACT Rel-time performnce monitoring to identify poorly or under-performing loops hs ecome n integrl prt of preventtive mintennce. Among others, rising energy costs nd incresing demnd for improved product qulity re driving forces. Automtic process control solutions tht incorporte rel-time monitoring nd performnce nlysis re fulfilling this mrket need. While mny softwre solutions disply performnce metrics, however, it is importnt to understnd the purpose nd limittions of the vrious performnce ssessment techniques since ech metric signifies very specific informtion out the nture of the process. This pper reviews performnce mesures from simple sttistics to complicted modelsed performnce criteri. By understnding the underlying concepts of the vrious techniques, reders will gin n understnding of the proper use of performnce criteri. Bsic lgorithms for computing performnce mesures re presented using exmple dt sets. An evlution of techniques with tips nd suggestions provides reders with guidnce for interpreting the results. INTRODUCTION Over the pst two decdes, process control performnce monitoring softwre hs ecome n importnt tool in the control engineer s toolox. Still, the numer of performnce tests nd sttistics tht cn e clculted for ny given control loop cn e overwhelming. The prolem with controller performnce monitoring is not the lck of techniques nd methods. Rther, the prolem is the lck of guidnce s to how to turn sttistics into meningful nd ctionle informtion tht cn e pplied to improve performnce. The performnce nlysis techniques discussed in this pper re seprted into three sections. The first section detils methods for identifying process chrcteristics using tches of existing dt. The second section outlines methods used for rel-time or dynmic nlysis of streming process dt. These re vitl techniques for the timely identifiction nd interprettion of chnging process ehvior nd deteriorting loop performnce. The third section outlines techniques tht id in the identifiction of intercting control loops. The techniques presented in this pper use Microsoft Excel to clculte corresponding performnce mesures. Reders my otin complimentry copy of the Excel worksheet y contcting Bo Rice vi emil t o.rice@controlsttion.com. Performnce Monitoring Fundmentls Pge 1 of 19
P r o c e s s V r i l e / S e t p o i n t P r o c e s s V r i l e / S e t p o i n t Simplifying PID Control. Optimizing Plnt Performnce. IDENTIFYING PROCESS CHARACTERISTICS Set Point Anlysis There re numer of techniques for nlyzing closed loop process dt tht is collected during Set Point response experiment. These techniques permit n orderly comprison of process response shpes nd chrcteristics. When nlyzing Set Point response, the criteri used to descrie how well the process responds to the chnge cn include Pek Overshoot Rtio, Decy Rte, Set Point Crossing, Rise nd Settling. These criteri cn e used oth s specifictions for commissioning of control loops s well s for documenting chnges in performnce due to the djustment of the controller or process prmeters. Figure 1 elow shows closed loop response to Set Point chnge. To clculte the Set Point criteri mentioned ove, we ssign the following definitions: A B C = Size of the Set Point step = Size of the first pek ove the new Set Point or stedy stte = Size of the second pek ove the new stedy stte 4. 3. 8 3. 6 3. 4 3. 3.. 8 A B C. 6. 4. 4. 3. 8 3. 6 3. 4 3. 3.. 8. 6. 4. 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19 T im e ( m in s ) 5% of y(t) y(t) 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19 T im e ( m in s ) t t settle t pek rise Figure 1 - Process response to Set Point chnge with lels indicting response fetures Performnce Monitoring Fundmentls Pge of 19
Simplifying PID Control. Optimizing Plnt Performnce. As shown in Figure 1, the time when the mesured process vrile first crosses the new Set Point nd the time t which it reches its first pek re used to descrie controller performnce. This performnce metric is clled Set Point Crossing nd it provides insight into the reltive speed with which the process responds to chnge. Another populr mesurement is Settling. Settling descries the time required for the mesured process vrile to first enter nd then remin within nd whose width is computed t specified rnge of the totl chnge in y(t). In our exmple, rnge of + 5% is shown. Additionl criteri re summrized in Tle 1 elow. Populr vlues include 1% Pek Overshoot Rtio nd 5% decy rtio. It is importnt to note tht these criteri re not independent. A process with lrge decy rtio will likely hve long Settling wheres process with long Rise will likely hve long pek time. The cceptility of these metrics is sujective nd will e closely tied to your process nd overll control ojective. Criteri Interprettion Clcultion Pek Overshoot Rtio The POR is the mount y which the (POR) = B/A (POR) process vrile surpsses Set Point. An ggressive controller cn increse the mount of overshoot ssocited with Set Point chnge. Decy Rte A lrge Decy Rte is ssocited with n ggressive controller, nd visile oscilltions re present in the Set Point response. The smller the Decy Rte, the fster the oscilltions will e dmpened. Decy Rtio = C/B Pek & Rise Settling These mesurements guge the time response to chnge in the Set Point. A lrge pek nd Rise could e the result of sluggish controller. The Settling is the time for the process vrile to enter nd then remin within nd. spent outside the desired level generlly reltes to undesirle product. Therefore, short Settling is sought. Rise = t rise Pek = t pek Settling = t settle Tle 1 - Interprettion of Set Point Response Criteri Other closed loop performnce metrics include the integrl of error indexes which focus on devition from Set Point. The Integrl Squred Error (ISE) is very ggressive ecuse squring the error term provides greter punishment for lrge error. The Integrl Asolute Error (ITAE) is the most conservtive of the error indexes; the multipliction y time gives greter weighting to error tht occurs fter longer pssge of time. The Integrl Asolute Error (IAE) is moderte in comprison to these two. Additionl indexes cn e derived depending on the system requirements. Integrl Squred Error (ITSE) comines the time weighting with the exggerted punishment for lrger error. Performnce Monitoring Fundmentls Pge 3 of 19
Integrl of Error Set Point Bump Criteri Process Vrile / Set Point PV / SP Process Vrile / Set Point Simplifying PID Control. Optimizing Plnt Performnce. The formul for clculting the Integrted Error indexes re listed elow. IAE T e() t dt (1) ISE T e () t dt () T ITAE t e() t dt (3) T te () t dt ITSE (4) Often the ove indexes re used s criteri in controller tuning. Typiclly, users will choose one of the ove metrics nd define optiml control s the tunings tht chieve the minimum vlue of the index. Figure shows the process vrile s response to Set Point chnge under vrious controller tunings rnging from poor/unstle to conservtive. The results re summrized in Tle. Poor Tuning Aggressive Tuning Convsertive Conservtive Tuning Tuning 141.4 141. 141.4 141. 141. 141 141 14.8 14.6 14.4 14. 14 139.8 3 8 13 18 3 141 14.8 14.6 14.4 14. 14 139.8 38 43 48 53 58 14.8 14.6 14.4 14. 14 139.8 77 8 87 9 97 c Figure - Set Point Response of ) poorly, ) ggressively, c) conservtively tuned PI controller Poorly Tuned Aggressively Tuned Conservtively Tuned POR 33% 1% Decy Rte 44% 4% Rise. min.9 min 13.7 min Pek.4 min 4.1 min 13.7 min Settling 1.8 min 6.5 min 14.7 min IAE 3.1.8 5.49 ISE 1.4 1.3 3.17 ITAE 17.69 8.64 5.4 ITSE.97 1.1 7.85 Tle - Results of Set Point Response Criteri, nd Integrl of Error clcultions for Figure Performnce Monitoring Fundmentls Pge 4 of 19
Simplifying PID Control. Optimizing Plnt Performnce. Disturnce Anlysis A disturnce is defined s nything other thn the controller output signl tht ffects the mesured process vrile. In n intercting plnt environment, ech control loop cn hve mny different disturnces tht impct performnce. By understnding the type of disturnce nd its impct on the control loop, engineers, opertors nd technicins cn more esily identify the cuse nd work towrds n pproprite solution. Auto-correltion is method tht is used to determine how dt in time series re relted [1]. By compring current process mesurement ptterns with those exhiited in the pst, the nture of disturnces nd how they ffect system cn e nlyzed. The eqution for clculting the uto-correltion reltionship is: rk ( ) (5) i [( y( i) y)( y( i k) y)] i ( y( i) y) Where: y = mesured process dt y = the Set Point or the series verge if there is n offset k = time dely in smples i = smple numer (or smple time) Auto-correltion vlues will lwys rnge etween negtive one nd one. If dt is rndom, the vlues will e pproximtely zero for ll time. Any vlue tht is significntly non-zero will indicte tht the dt is non-rndom. A strong uto-correltion will hve n initil vlue ner one or negtive one nd the trend will e liner, nd this demonstrtes ptter where ech mesurement dicttes the next. A moderte uto-correltion is one in which the plot egins elow one (or ove negtive one) nd decreses mgnitude towrds zero ut displys noise. An uto-correltion of closed loop dt cn lso give n estimte of the response time for n isolted disturnce. Another performnce sttistic tht cn prove useful with the identifiction of trends in dt is the Power Spectrum. Power Spectrum is clculted y computing the discrete Fourier trnsform of the process dt. A Fourier trnsform is mthemticl expression of the dt represented y series of two-dimensionl sine wves, nd the Power Spectrum is computed y squring the complex coefficients determined y those sine wves. The Power Spectrum shows the frequency t which chnge is occurring nd the mgnitude of the chnge [9]. The shpes nd heights of ech pek on Power Spectrum plots provide relevnt informtion out the system nd its performnce. Specificlly, the shpe of the Power Spectrum curve yields informtion out the nture of the disturnces y displying its frequency. Similrly, n increse in pek heights compred to historicl dt indictes tht the process hs greter devition from Set Point or from its historicl men. Low powers nd low frequencies re most desirle, s they re ssocited with smll devitions from Set Point nd lower verge vlues. Performnce Monitoring Fundmentls Pge 5 of 19
Simplifying PID Control. Optimizing Plnt Performnce. Figures 3-6 provided on the following pges show four different scenrios in which the utocorreltion nd Power Spectrum cn e useful in understnding the nture of the disturnce impcting the system. Only the single pulsed disturnce shown in Figure 3 is noticele from csul evlution of the process dt. By using the uto-correltion nd Power Spectrum tools, however, one cn identify chrcteristics for ll four disturnces. In Figure 3, the process is upset with single pulsed disturnce. The uto-correltion shown in Figure 3c shows n initil pek where the process is responding to the step up then the negtive pek occurs pproximtely 1 minutes lter when the disturnce steps ck down. This is chrcteristic of n isolted disturnce. If second pulse hd occurred, nother similr pttern would e expected to pper on the uto-correltion plot. The Power Spectrum of the process dt is shown in Figure 3d. Since the frequency of chnge corresponds to the frequency of the disturnce, n isolted disturnce is locted t pproximtely zero frequency on the Power Spectrum plot. There is no other disturnce occurring t ny other frequency, so the power quickly drops off nd the remining vlues re close to zero. Figure 4 shows the process with no disturnce impcting the system. Neither the utocorreltion nor the Power Spectrum contins ny ovious peks. In fct, oth trends show rndom vlues close to zero. This indictes the control loop is undistured nd performing well. The oscillting disturnce depicted in Figure 5 yields n oscillting uto-correltion. The Power Spectrum nlysis shows tht the oscillting disturnce is single cycle sine wve since there is one strong dominnt pek t the wve s frequency. If second disturnce ws cting on the system second pek would pper. The continuously pulsed rndom disturnce depicted in Figure 6 is difficult to identify since the mgnitude of the disturnce is within the rnge of noise. The disturnce is not impcting the system t regulr frequency ecuse the length of time ssocited with the disturnce pulses is not constnt. Therefore the Power Spectrum does not show ny significnt peks outside the rnge of the noise. The uto-correltion gives n indiction of disturnce tht is not ssocited with the process noise ecuse there is strong pek t 5 minutes. Also, there re slight clusters ove nd elow the x-xis, especilly close to zero. These clusters re not s regulr s the oscillting disturnce. In this sitution comprison to historicl dt nd fmilirity with the process is vitl. Performnce Monitoring Fundmentls Pge 6 of 19
Correltion.15 1.8 3.45 5.1 6.75 8.4 1.1 11.7 13.4 15 16.7 18.3 1.6 3.3 4.9 6.6 8. 9.9 31.5 33. 34.8 36.5 Power P r o c e s s V r i l e / S e t P o i n t D i s t u r n c e Correltion.15 1.8 3.45 5.1 6.75 8.4 1.1 11.7 13.4 15 16.7 18.3 1.6 3.3 4.9 6.6 8. 9.9 31.5 33. 34.8 36.5 Power P r o c e s s V r i l e / S e t P o i n t D i s t u r n c e Simplifying PID Control. Optimizing Plnt Performnce. S i n g l e P u l s e d D i s t u r n c e D i s t u r n c e P r o f i l e 4. 5 4. 4 4. 3 4. 4. 1 3. 1 5. 9 5. 7 5 4. 5 5 3. 9. 3 5 3. 8 3. 7. 1 5 3. 6 1. 9 5 3. 5 1 3 4 5 T i m e ( m i n ) Autocorreltion - Single Pulsed 1. 7 5 1 3 4 5 T i m e ( m i n ) Power Spectrum - Single Pulsed.5.1.4.3 c.1 d..8.1.6 -.1 -. -.3 -.4 -.5 Lg (Min).4..1..3.4.5.6 Frequency Figure 3 - For process sujected to pulsed disturnce here re the ) process vrile response ) disturnce profile c) uto-correltion nd d) Power Spectrum plots D i s t u r n c e N o t C h n g i n g D i s t u r n c e P r o f i l e 4. 5 4. 4 4. 3 3. 1 5. 9 5 4. 4. 1 4 3. 9 3. 8 3. 7 3. 6 3. 5 18 19 1 3 T i m e ( m i n ). 7 5. 5 5. 3 5. 1 5 1. 9 5 1. 7 5 18 19 1 3 T i m e ( m i n ) Autocorreltion - Disturnce Not Chnging Power Spectrum - Disturnce Not Chnging.5.1.4.3..1 c.1.8 d.6 -.1 -. -.3 -.4 -.5 Lg (min).4..1..3.4.5.6 Frequency Figure 4 - For n unchnging process here re the ) process vrile response ) disturnce profile c) uto-correltion nd d) Power Spectrum plots Performnce Monitoring Fundmentls Pge 7 of 19
Correltion.15 1.8 3.45 5.1 6.75 8.4 1.1 11.7 13.4 15 16.7 18.3 1.6 3.3 4.9 6.6 8. 9.9 31.5 33. 34.8 36.5 Power P r o c e s s V r i l e / S e t P o i n t D i s t u r n c e Correltion.15 1.8 3.45 5.1 6.75 8.4 1.1 11.7 13.4 15 16.7 18.3 1.6 3.3 4.9 6.6 8. 9.9 31.5 33. 34.8 36.5 Power P r o c e s s V r i l e / S e t P o i n t D i s t u r n c e Simplifying PID Control. Optimizing Plnt Performnce. O s c i l l t i n g D i s t u r n c e D i s t u r n c e P r o f i l e 4. 5 4. 4 4. 3 4. 4. 1 3. 1 5. 9 5. 7 5 4. 5 5 3. 9. 3 5 3. 8. 1 5 3. 7 3. 6 1. 9 5 3. 5 1 1 4 14 1 3 4 1 4 4 1 5 4 T i m e ( m i n ) 1. 7 5 1 1 4 14 1 3 4 1 4 4 1 5 4 T i m e ( m i n ) Autocorreltion - Oscillting Disturnce Power Spectrum - Oscillting Disturnce.5.4.3 c.1.1 d..8.1.6 -.1 -..4 -.3. -.4 -.5 Lg (min).1..3.4.5.6 Frequency Figure 5 - For process sujected to n oscillting disturnce here re the ) process vrile response ) disturnce profile c) uto-correltion nd d) Power Spectrum plots D i s t u r n c e P r o f i l e C o n t i n u o u s l y P u l s e d R n d o m D i s t u r n c e 4. 5 4. 4 4. 3 4. 4. 1 4 3. 9 3. 8 3. 7 3. 6 3. 5 6 7 8 9 1 T i m e ( m i n ) Autocorreltion - Continuously Pulsed Rndom Disturnce 3. 1 5. 9 5. 7 5. 5 5. 3 5. 1 5 1. 9 5 1. 7 5 6 7 8 9 1 T i m e ( m i n ) Power Spectrum - Continuously Pulsed Rndom Disturnce.5.4.3 c.1.1 d..1.8.6 -.1 -..4 -.3 -.4 -.5 Lg (min)..1..3.4.5.6 Frequency Figure 6 - For process sujected to continuously pulsed rndom disturnce here re the ) process vrile response ) disturnce profile c) uto-correltion nd d) Power Spectrum plots Performnce Monitoring Fundmentls Pge 8 of 19
Simplifying PID Control. Optimizing Plnt Performnce. REAL-TIME PERFORMANCE MONITORING With growing ccess to plnt-wide process dt, rel-time control loop monitoring hs ecome incresingly populr. Mny employ the Hrris Index, sed on minimum vrince control principles, s the preferred strtegy. At the hert of every performnce monitoring system is the ility to identify prolem within the control loop s soon s possile. Presented in this pper is comprison of the Hrris Index to simpler strtegies for monitoring controller performnce. Descriptive sttistics re seprted into three ctegories: mesures of centrl tendency, mesures of spred, nd mesures of shpe. The men is the most common mesure of centrl tendency, providing insight into the loction of process center or verge stte of opertion. In contrst, the mesures of spred provide informtion out the degree to which individul vlues re clustered or to which they devite from the men vlue in distriution. The minimum nd mximum re the simplest mesures of spred nd give only rnge of vlues. The Vrince nd Stndrd Devition re other populr mesures of spred tht provide more useful numericl vlue sed upon their devition from the men. Lstly, mesures of shpe re used to descrie the distriution of dt vlues. The skewness of these vlues refers to the degree of symmetry present in the dt set. Ech of these descriptive sttistics cn provide insight into how the control loop is functioning. These sttistics re most commonly clculted for the process vrile, controller output, nd controller error. Shown in Eqution 6, the Hrris Index is vlue sed on the comprtive performnce of current control to minimum vrince control, MVC. In clculting the minimum vrince, n utoregressive moving verge model is fit to the process dt. This is predictive model tht represents the ction minimum vrince controller would tke. If the disturnces tht ffected the process cn e predicted y MVC, then the current controller is performing poorly in comprison. If the disturnces re determined to e rndom, then the controller is performing s well s MVC. The Hrris Index is difficult to clculte. It is importnt to note tht under MVC the uto-correltion of the dt is zero fter the initil process dely. Therefore, uto-correltion cn e used to ssess whether or not the system is displying minimum vrince. Eqution 6 - The Hrris Index is computed s [4]: I H (6) y mv Where: I H = the Hrris Index = the Vrince of the process dt y mv = the minimum Vrince When the process displys minimum vrince, the Hrris Index is one. To estlish seline, the Hrris Index should e clculted while the system is in pek performnce nd then used for comprison purposes ginst future vlues. The Hrris Index is useful for ssessing the output vrince due to stochstic disturnces. It cnnot give specific informtion out Set Point chnges, known disturnce vriles, Settling, decy rtio or stility [1]. Performnce Monitoring Fundmentls Pge 9 of 19
Simplifying PID Control. Optimizing Plnt Performnce. The reliility of the Hrris Index depends on the strength of the model nd the estimtion of the process ded time. The prmeters for the model need to e determined using the Box nd Jenkins method [1], prior knowledge, or tril nd error. Poor model selection or erroneous ded time estimtions will result in misleding vlues of the Hrris Index. An dditionl performnce metric introduced in this pper is Stndrd Vrition. The Stndrd Vrition is normlized mesure of devition of the mesured process vrile from the Set Point of process. It is detiled in Eqution 7 s shown elow. PV SP n 1 Stndrd Vrition 1% Averge( PV ) Where: PV : Mesured Process Vrile SP : Set Point n : Numer of Dt Points (7) Using this method smller Stndrd Vrition will represent less devition from Set Point. Some fctors tht cn impct the Stndrd Vrition include the numer of Set Point chnges s well s the numer of disturnces tht impct the process. The Stndrd Vrition cn e used to guge performnce improvement reltive to retuning loop. If the vlue for Stndrd Vrition is smller fter retuning, then performnce hs een improved. It should e noted tht when compring efore nd fter performnce index, the dt needs to e collected for sufficient period of time such tht the numer of disturnces impcting the system re pproximtely equl. Two lterntive pproches for computing performnce include oth moving nd sttic clcultions. Moving clcultions re computed on moving suset of the complete dt set. Sttic clcultions, however, re computed sed on the performnce mesurements of the entire dt set. The results of the moving suset clcultion re grphed with the performnce mesure plotted long the verticl xis nd time long the horizontl xis. By using moving suset in lieu of complete tch clcultion, it is possile to identify the point in time when loop performnce egins to chnge. This in turn signls where, or t wht point in time, to egin n investigtion into cuses. Becuse of the ility to identify rel-time chnges to performnce, the moving suset method is recommended for control loop monitoring. Figure 7 shows the process vrile nd controller output trces for time-vrint process. A time-vrint process is system whose dynmic ehvior chnges with time. This chnge in ehvior cn e the result of degrding vlve performnce, het exchnger surfce fouling, ctlysts dectivting, or even fluctuting wether conditions. In the exmple shown, the system is under PI control nd the tuning vlues re constnt during the process trnsition. By using moving clcultion, the time t which the process egn to shift ws clerly identifile. Performnce Monitoring Fundmentls Pge 1 of 19
Controller Output Process Vrile / Set Point Simplifying PID Control. Optimizing Plnt Performnce. 5.6 Vrint Process Under PI Control 5.4 5. 5 49.8 49.6 5 49.4 49.9 49.8 4 6 8 1 1 49.7 49.6 49.5 49.4 49.3 49. 49.1 49 4 6 8 1 1 Figure 7 Process Dt nd Controller Output Visul inspection of the process trends shown in Figure 7 does not indicte significnt chnge in controller performnce. However, the plots tht follow sed on methods just discussed revel tht something in the process indeed hs chnged. By detecting this chnge efore it hs significnt impct on controller performnce, solutions including updting controller tunings cn e considered efore lrms re triggered. Figure 8 shows side-y-side comprison of the Hrris Index, Stndrd Devition, Vrince nd Stndrd Vrition of moving suset of dt for the time-vrint process. The dotted line in ech trce represents the pre-defined seline vlue. Since no two processes re like, ech process should hve its own seline or cceptle performnce limit s determined y mesurements collected under norml operting conditions when the system is understood to e running under good control. If the vlue for ny performnce criteri moves outside its performnce limit for specified mount of time, then tht system hs drifted to wrning sitution. All four methods show tht the process egins to drift from its seline vlue t out 55 minutes. Performnce Monitoring Fundmentls Pge 11 of 19
Vrince Stndrd Vrition Averge StDev Simplifying PID Control. Optimizing Plnt Performnce. Hrris Index of Percent Error Stndrd Devition of Percent Error 1.8 1.6 1.4 1. 1. 1.18 1.16 1.14 1.1 1.1 1.8 1.6 4 6 8 1 3.35E-3 3.5E-3 3.15E-3 3.5E-3.95E-3.85E-3 4 6 8 1 Moving Vrince of Percent Error Stndrd Vrition 1.11E-5 1.6E-5 1.1E-5 9.6E-6 9.1E-6 8.6E-6 8.1E-6 4 6 8 1 c.74.69.64.59.54.49.44.39.34.9 4 6 8 1 d Figure 8 - ) Hrris Index, ) Stndrd Devition, c) Vrince, nd d) Stndrd Vrition re clculted y the moving suset method. The time when the process model egins to chnge is pprent in ll plots. IDENTIFYING INTERACTING PROCESSES Intercting processes cn e troulesome in ny mnufcturing process. By identifying which systems interct, the disturnces cn e countercted rther thn perpetuted throughout the system. Even if n upstrem disturnce cnnot e eliminted, y identifying the source, feed-forwrd controller cn e used to mitigte the impct of disturnces nd improve downstrem loop performnce. Cross-correltion nlyzes the reltionship etween two dt series. By clculting set of correltion vlues t incresing time delys, picture develops tht shows how the dt series re relted through time. The cross-correltion is clculted s: rk ( ) i i [( y ( i) y )( y ( i k) y )] ( y ( i) y ) ( y ( i k) y ) i (8) Where: y nd y = process dt y nd y = the Set Point vlues (or the series verges) k = time dely in smples i = smple numer (or smple time) Performnce Monitoring Fundmentls Pge 1 of 19
Simplifying PID Control. Optimizing Plnt Performnce. Cross-correltion vlues re lwys etween negtive one nd one. Positive vlues indicte tht process A directly ffects process B, so tht n incresed devition from verge in process A cuses n incresed devition in B. Negtive vlues indicte n inverse reltionship such tht n incresed devition in process A cuses decresed devition in process B. If there is no reltionship etween the dt sets, then the cross-correltion vlues will e close to zero. In ddition to reveling the level of interction etween control loops, cross-correltion cn lso e used to determine exctly how much time elpses efore the downstrem process will e impcted. At the point when there is gretest impct on the downstrem loop, there will e pek in the cross-correltion trend. Additionlly, cross-correltion is used to identify when disturnces re eing cused y recycle strem. If recycle strem occurs within single control loop, n uto-correltion cn e used to identify how the recycle influences the system. Power Spectrum is lso employed to identify nd nlyze intercting loops. Intercting loops re ffected y the sme events nd therefore hve Power Spectrum peks t the sme frequencies. Power Spectrum cnnot identify how long it tkes for chnge in one system to rech nother like cross-correltion cn, ut it cn e more useful when there re mny processes seprting the suspected intercting loops. Cross-correltion cn e muddled when there re mny processes with vrying reltionships, ut the Power Spectrum is more sensitive. If processes re ffected y events occurring t the sme frequencies, Power Spectrum will identify the interction. Controller Output 1 Tnk Level 1 Disturnce 1 Tnk Level Figure 9 - The intercting tnks process used to demonstrte Power Spectrum nd crosscorreltion To explore the ilities of cross-correltion nd Power Spectrum to identify intercting loops, consider the rry of tnks shown in Figure 9. The two upper tnks ech drin into the two lower tnks. Two controllers connect lower tnks to the upper tnks. If the level controllers on the ottom tnks re put into utomtic, disturnce in one of the lower tnks will ffect ll four tnks. If the level controllers re left in mnul, the tnk connectivity is roken nd disturnce impct remins locl to the prticulr tnk ffected. Now consider the system of tnks when they re operted in mnul mode. Figures 1, 1, nd 1c show the process dt when step to controller output 1 increses the flow to upper tnk 1. Figures 11 nd 11 show the cross-correltion of controller output 1 nd the levels in lower tnks 1 nd, respectively. The lrge peks on the grphs signify strong correltion for oth nd tht the mximum effect tkes pproximtely 15 minutes to impct Performnce Monitoring Fundmentls Pge 13 of 19
Simplifying PID Control. Optimizing Plnt Performnce. tnk level 1 nd 3 minutes to impct tnk level. Figures 1, 1, nd 1c show process dt collected during step disturnce in lower tnk 1. From the uto-correltion plots shown in Figures 13 nd 13, it is cler the disturnce hs n lmost instntneous negtive effect on the level in tnk 1 nd no effect on tnk. Figure 14 shows the Power Spectrum of controller output 1, tnk level 1, nd tnk level scled so they cn e displyed on the sme grph. All three systems shre peks t the sme frequencies nd this indictes they re intercting. Figure 14 shows the reltionship etween disturnce in tnk 1 nd the levels in tnks 1 nd. Disturnce 1 shres similr peks with tnk level 1, indicting they re intercting. Tnk level hs unique Power Spectrum, indicting it is responding to different stimuli. c Controller Output 1 Tnk Level 1 Tnk Level 66 65.5 65 64.5 64 63.5 63 6.5 6 61.5 61 5 1 15 5 3 4 3.5 3.5 5 1 15 5 3 4 3.5 3.5 5 1 15 5 3 Figure 1 Process dt during step chnge of controller output 1 ) controller output 1 ) tnk level 1 c) tnk level Performnce Monitoring Fundmentls Pge 14 of 19
1 13 4 36 48 59 71 83 95 16 118 13 141 153 165 176 188 1 1 14 6 39 51 64 77 89 1 114 17 14 15 165 177 19 3 15 Simplifying PID Control. Optimizing Plnt Performnce. Cross-correltion of Controller Output 1 nd Tnk Level 1.35.5.15.5 -.5 -.15 -.5 -.35 Cross-correltion of Controller Output 1 nd Tnk Level.35.5.15.5 -.5 -.15 -.5 -.35 Figure 11 - Cross-correltion digrms of the reltionship etween controller output 1 nd ) tnk level 1 ) tnk level c Disturnce 1 Tnk Level 1 Tnk Level 1.6 1.5 4 4 1.4 1.3 3.5 3.5 1. 3 3 1.1 1.5.5.9 39 41 43 45 47 49 51 53 55 39 41 43 45 47 49 51 53 55 39 41 43 45 47 49 51 53 55 Figure 1 Process dt during pulse disturnce in tnk 1 ) disturnce 1 ) tnk level 1 c) tnk level Performnce Monitoring Fundmentls Pge 15 of 19
Power Power 1 13 4 36 48 59 71 83 95 16 118 13 141 153 165 176 188 1 1 13 4 36 48 59 71 83 95 16 118 13 141 153 165 176 188 1 Simplifying PID Control. Optimizing Plnt Performnce. Cross-correltion of Disturnce 1 nd Tnk Level 1 Cross-correltion of Disturnce 1 nd Tnk Level.35.35.5.5.15.15.5.5 -.5 -.5 -.15 -.15 -.5 -.5 -.35 -.35 Figure 13 - Cross-correltion digrms of the reltionship etween disturnce 1 nd ) tnk level 1 ) tnk level Impct of Controller Ouput, CO1, on Tnk 1 nd Tnk Level Impct of Disturnce, D1, On Tnk 1 nd Tnk Level 3E-4 E-4 Tnk 1 Level, PV1 Controller Output, CO1 [1^-5] Tnk Level, PV [1^1] 4.E-4 3.5E-4 3.E-4 Tnk 1 Level, PV1 Disturnce, D1 [1^-] Tnk Level, PV [1^4] E-4.5E-4.E-4 1E-4 1.5E-4 5E-5 1.E-4 5.E-5 E+.5.1.15..5.3.35.4.45.5 Frequency.E+.5.1.15..5.3.35.4.45.5 Frequency Figure 14 - Power Spectrums of ) controller output 1 nd tnk levels 1 nd nd ) disturnce 1 nd tnk levels 1 nd. The spectrums hve een scled so tht they cn e view on the sme grph. Performnce Monitoring Fundmentls Pge 16 of 19
Simplifying PID Control. Optimizing Plnt Performnce. CONCLUSIONS Performnce mesures re n integrl prt of optimizing nd mintining system performnce. Industry nd cdemi re constntly deriving new methods for performnce ssessment, ut the methods re only useful when they cn e fully understood nd used properly. It is importnt to understnd the theory, purpose nd limittions of the mesures efore relying on their informtion. In mny cses, the performnce ssessment methods only identify the strt of prolem, not the source. By understnding the sic principles nd disturnces tht impct your system, engineers will know wht to expect during norml opertion nd will e le to identify more quickly wht is norml opertion. This pper ddressed wide vriety of commonly used performnce ssessment techniques in n ttempt to demystify them for etter ppliction in monitoring. The techniques detiled in this pper for tckling rel-time process monitoring re twofold. First one cn identify when process strts to drift wy from seline opertion nd towrds triggering n lrm. Once prolem is identified, the use of uto-correltion, cross-correltion, nd Power Spectrum cn e used for detect the root-cuse. Performnce Monitoring Fundmentls Pge 17 of 19
Simplifying PID Control. Optimizing Plnt Performnce. REFERENCES 1. Box G.E.P., Jenkins, G.M., (197), Series Anlysis forecsting nd control, Holden-Dy, Inc. Sn Frncisco, CA. Burch, R. (4). Monitoring nd Optimizing PID Loop Performnce. ISA Annul Meeting, Houston, TX. 3. Desorough, L. nd R. Miller (1). Incresing Customer Vlue of Industril Control Performnce Monitoring--Honeywell's Experience. 6th Annul Interntionl Chemicl Process Control Meeting, Tucson, AZ. 4. Hrris, T. J. (1989). "Assessment of Control Loop Performnce." Cndin Journl of Chemicl Engineering 67: 856-861. 5. Hoo, K. A., M. J. Piovoso, et l. (3). "Process nd controller performnce monitoring: overview with industril pplictions." Int. J. Adpt. Control Signl Processing 17: 635-66. 6. Horch, A. nd A. J. Isksson (1999). "A modified index for control performnce ssessment." Journl of Process Control 9: 475-483. 7. Hung, H.-P. nd J.-C. Jeng (). "Monitoring nd Assessment of Control Performnce for Single Loop Systems." Ind. Eng. Chem. Res. 41: 197-139. 8. Ptwrdhn, R. S., S. L. Shh, et l. (). "Assessing the Performnce of Model Predictive Controllers." The Cndin Journl of Chemicl Engineering 8: 954-966. 9. Press W.H., et l(1986), Numericl Recipes: The Art of Scientific Computing, Cmridge University Press, New York, NY 1. Qin, S. J.(1998), Control Performnce Monitoring review nd ssessment, Computers nd Chemicl Engineering, 3, 173-186 11. Thornhill, N. F.(1998), Performnce Assessment nd Dignosis of Refinery Control Loops, AIChe Symposioum Series N o 3, 94, 373-379 Performnce Monitoring Fundmentls Pge 18 of 19
Simplifying PID Control. Optimizing Plnt Performnce. For more informtion, plese contct us t: Control Sttion, Inc. One Technology Drive Tollnd, CT 684 877-LOOP-PRO (877-566-7776) www.controlsttion.com Performnce Monitoring Fundmentls Pge 19 of 19