Unfalsified Adaptive Control with Online Optimization to a ph Neutralization Process

Transcription

1 Research Artcle Unfalsfed Adatve Control wth Onlne Otmzaton to a H Neutralzaton Process anet Wonghong Deartment of Electrcal Engneerng, Faculty of Engneerng, Kasetsart Unversty, S Racha Camus, S Racha, Chonbur 030, haland Abstract hs wor resents a data-drven adatve controller wth an onlne otmzaton. he roosed algorthm comutes a real-tme controller arameter vector for a PID controller structure usng only lant nut-outut measurement data. he controller can be adated by the algorthm to handle a change of set-onts wth a mnmum nowledge about a lant. Fnally, we demonstrate the erformance of the algorthm usng a H neutralzaton rocess wth an adataton of PID controllers. Keywords: Data-drven Control; Adatve Control; Unfalsfed Control; Onlne Otmzaton; Automatc PID Controller unng; H Neutralzaton Process. Introducton Data-drven ndustral rocess control has been actvely researched n recent years []. In many ndustral alcatons, dervng a model from the frst rncles that s both smle and relable for controller desgn s both dffcult and tme-consumng. Nowadays, the rogress of data-acquston technology s easy and straghtforward to collect a large amount of measurements from ndustral rocesses. In [], they clamed that the utlzaton of measurement data as an alternatve to hyscal nowledge to desgn smle fxed-order controllers, e.g. a PID controller, has become somethng of ncreasng nterest throughout the years because t s cheaer and less tmeconsumng. he motvaton for ths aer has arsen from the roblem of a tradtonal model-based aroach. If we can derve an accurate model of a lant, t s suffcent to desgn a good erformng controller, e.g. aerosace alcatons, t s straghtforward Corresondence : tanet@eng.src.u.ac.th to desgn such a controller. However, modelbased aroaches may not be enough to coe wth a stuaton when a model s dffcult to obtan or even unavalable (worst case stuaton) [3]. For ths reason, a data-based methodology should be alternatvely consdered to solve ths ssue. hs s the case when we deal wth ndustral alcatons, e.g. a H neutralzaton rocess whch the modelng ste may be hard to derve. Unfalsfed control s one of the datadrven technques [3]. he man attractve ont of ths aroach s that no lant model s requred to desgn an adatve controller. Usng ths concet, an adatve controller s mlemented by means of swtchng among many canddate controllers n a redefned set. he lant nut sgnal and the lant outut sgnal are observed whle one canddate controller s actve n the feedbac loo, and they are used for the unfalsfcaton rocedure to decde on whch canddate

2 Vol.0, No.4, October-December 05 controller n the set should be swtched on as the next actve controller. he very frst verson of the unfalsfed control concet was orgnally roosed by [4]. It was motvated by a lmtaton of tradtonal model-based theores. he unfalsfed control nfluenced the adatve control theory because t can classfy the erformance of a gven set of canddate controllers usng only the observed lant nut-outut data to meet a gven closed-loo secfcaton. he ey ssue s the comutaton of a fcttous reference sgnal and a fcttous error sgnal of a nonactve canddate controller whle another controller s actve. In [4], they clamed that usng ths aroach based on a gven cost functon, the observed lant nut-outut data are suffcent to falsfy canddates from a set of controllers that need not actually be n the feedbac loo to be falsfed. hs s an mortant ssue because the swtchng of an actve controller n the set of unfalsfed controllers can gve a better transent resonse than other adatve controller algorthms avalable at that tme, e.g. [5], [6], [7] whch requre all canddate controllers to be actve one by one n the feedbac loo to chec whether or not they are sutable. hs mles that t can tae a long tme to end u wth the last swtchng to a stablzng controller f the number of canddate controllers s large. Further develoments by [8] and [9] led to a concet of a cost-detectable cost functon. Unfalsfed adatve controllers are not always safe usng the algorthm roosed n [4]. Safe means that the closed-loo stablty s always guaranteed. One examle of a cost-detectable cost functon was roosed n [8]. he cost-detectablty means that nstablty of all canddate controllers can be detected by a cost functon. Moreover, the swtchng method of an actve controller roosed n [4] was modfed n [9] to be mlemented wth a cost-detectable cost functon to acheve a safe algorthm. A swtchng algorthm always comares the hammasat Internatonal Journal of Scence and echnology erformance of the actve controller wth that of all non-actve controllers. If the actve controller erforms badly, a mnmum-cost non-actve controller n a redefned set s swtched to. Otherwse, the actve controller s always n the feedbac loo. Such an algorthm was roosed by [0], so called - hysteress swtchng algorthm. he cost functon roosed by [9] was reorted by [], [], and [3] that there was a severe roblem. It wll only detect nstablty of a controller n the set of canddate controllers f ths controller s actve. In [], they roved that the roblem orgnated from the ole-zero cancelaton n the transform of the fcttous error sgnal. hs leads to falure of the cost functon of [9] f t s used to the canddate controllers as shown by an examle n []. A new fcttous error sgnal to solve ths roblem was also roosed by []. In [], they also develoed a new cost-detectable cost functon that can detect nstablty of non-actve canddate controllers. hs cost functon leads to a ossblty to otmze a redefned set of canddate controllers. An evolutonary algorthm s used as an otmzaton technque because t s easy to handle and manulate a set of canddate controllers. he smulatons were erformed usng a thrdorder stable lnear system wth a tme delay as an unnown lant. Usng ths method, an otmal erformance could be acheved after the evolutonary algorthm was executed once. Further develoments by [4] resented a way to deal wth a nonlnear rocess model and measurement errors at outut. he effectveness of ndustral automatc tunng for the adataton of lnear sngle-loo unfalsfed control algorthm was demonstrated va a nonlnear non-mnmum hase contnuous strred-tan reactor model. Followng on the concet resented n [4], ths algorthm conssts of 3 modules:. Fcttous reference sgnal comutng module. Cost comutng module and 8

3 hammasat Internatonal Journal of Scence and echnology Vol.0, No.4, October-December Suervsor module. hese modules lead to an automatc controller tunng wthout hyscal nowledge about a lant. We wll demonstrate how controller arameter can be adated va a PID controller structure. hs aer s organzed as follows: In secton, a data-based adatve control based on unfalsfed control s resented. hen, n secton 3, one alcaton to a H neutralzaton rocess s demonstrated wth an evoluton of PID controllers to handle changes of set-onts. Fnally, conclusons and future wor are gven.. Materals and Methods ˆ ( n d e (t ) C ( r (t ) Intellgent algorthm ˆ ( Adatve PID controller + - P d( u ( y( Real Process Fg.. Data-based Adatve Control System. We consder a SISO system n the contnuous tme-doman as shown n Fg.. A mang P, Cˆ( ) : L e L that ( e r u(, y( s called an adatve control system. We denote r L e as the reference sgnal or the external exctaton. We assume that the real rocess P n Fg. s comletely unnown. A blac-box mang : L e L that transforms P e transforms u y( s called an unnown lant P. Only nformaton that we need to measure s the rocess outut y (. he rocess nut u ( s generated from one actve PID controller by usng the controller arameter vector θ ˆ ( whch s comuted by the roosed algorthm. hs algorthm requres only a data vector sgnal d( whch conssts of the reference sgnal r (, the control sgnal u( and the observed outut sgnal y ( ; r( d u( y( As shown n Fg., we have three mortant comonents to buld such an algorthm they are as follows:. Fcttous reference sgnal comutng module. Cost comutng module and 3. Suervsor module. We wll dscuss them later n detals. ˆ Suervsor Module,...,,... m An adatve set of m canddate controllers Cost comutng Module J ( t ),,..., m Fcttous Reference Sgnal Comutng ~ r t Module ( ),,..., m Fg.. Structure of ntellgent algorthm. r( d( y( u( A set of canddate controllers s defned as θ,..., θ,..., θ ( where θ s a controller arameter vector. Snce we am ths algorthm to deal wth a nonlnear rocess, ( can be changed over tme n order to adat to new rocess dynamcs due to a change of set-onts.. Fcttous Reference Sgnal Comutng Module he orgnal fcttous reference sgnal for a canddate controller C ( s defned [4] n Lalace doman as R ~ ( ) s C ( U( Y( () and the orgnal fcttous error sgnal for a canddate controller C ( can be comuted from ~ ~ E ( R ( Y( C ( U( () m 8

4 Vol.0, No.4, October-December 05 Snce we use a PID controller structure d s C ( ( ), thus we s s obtan ~ R ( ( d n n n n s s U( Y( ) s ( ) s n Now we can comute the fcttous reference sgnals and the orgnal fcttous error sgnals for all m canddate controllers to be the nuts for the cost comutng module.. Cost Comutng Module he orgnal cost functon s defned n [9], ~ ~ e u( J (3) ~ r he roblem of (3) was roven n [] that unstable dynamcs of non-actve destablzng controller are not excted usng (3) and thus t cannot be used for a controller arameter otmzaton. However, the cost comutng module uses the modfed cost functon as defned n [] e u J (4) r( where e ( t ) s the new fcttous error sgnal and u ( t ) s the new fcttous control nut sgnal and s a ostve small number. We consder ~ ~ ~ ~ ~ E ( ) ( ) ( ) ( ) ~ ( ) s E ( R s S s R s S s C ( P( R ( Although the senstvty functon S( s unnown because P( s unnown, we stll can estmate t usng the deconvoluton n tme ~ doman. ~ ( ~ ( ) ~ ( e s r s t r e~ and s a deconvoluton oerator. A ractcal way to comute was shown n []. hen we can comute e ~ s r( and n hammasat Internatonal Journal of Scence and echnology u c e where c L C (. Fnally, the comuted cost values for all m-canddate controllers wll be sent to the suervsor module..3 Suervsor Module he suervsor wll erform two functons:. Selecton of actve controller Snce we wor wth an unnown lant, we arbtrarly choose the ntal set of canddate controllers and choose one controller n that set to be actve frst. If the rocess s stable by tself, we should start wth a low gan controller (small ) and then the suervsor wll select the best avalable controller from the ntal set. hs crteron s based on the wor of [3], hysteress swtchng algorthm. hs algorthm scans erformances of all canddate controllers. If the actve controller erform oorly, t wll swtch to the best avalable controller n that set.. Adataton of a whole set of canddate controllers (onlne otmzaton) he suervsor has to montor whether or not the rocess outut has almost settled to the current set-ont. If t does so, the suervsor wll actvate an evolutonary algorthm (EA) to obtan an otmzed controller for ths setont and use t to generate a new set of canddate controllers ( for the next setont. In ths wor, we use an evoluton strategy (ES) [5] as the EA because t s easy to handle and manulate the controller arameter vector. Evoluton Strategy It s an otmzaton roblem for a ftness functon f θ ) : ( 83

5 hammasat Internatonal Journal of Scence and echnology Vol.0, No.4, October-December 05 θ mn f ( θ ) θ denotes an n-dmensonal object arameter vector, e.g. a PID controller arameter vector n a search sace of canddate controllers, PID EA K n d and θ denotes an otmal controller arameter vector. Reresentaton he ES wors wth a oulaton P (q) of the sze and. denotes the number of arent ndvduals and denotes the number of offsrng ndvduals. q denotes the number of generatons. An a ndvdual conssts of a controller arameter vector θ and an nternal (self-adataton) n- dmensonal strategy arameter vector s, and ts ftness value f ( θ ) : a θ, s, f ( θ )). ( Note that s s not nvolved n the comutaton of the ftness of the ndvdual but t s transmtted to the offsrng. he ndvduals a.e. arent ndvduals construct a oulaton, a,,,..., and offsrng ndvduals a ˆ,,,...,. At generaton, the oulaton of the arent ndvduals and the offsrng ndvduals can be defned as P ( q) a ( q), a( q),..., a ( q) P ( q) a ˆ ( q), aˆ ( q),..., aˆ ( q). Recombnaton A basc recombnaton n the ES uses two arent ndvduals to create one chld ndvdual. o obtan chldren ndvduals, the recombnaton s erformed tmes. here are two recombnaton varants: ) dscrete recombnaton and ) ntermedate recombnaton. We assume that two arent vector x and y are unformly randomly chosen from P (q) to roduce a chld vector z : x or y : dscrete z x y : ntermedate where,,..., n (n = 3 for PID controller structure). he former s used for the controller arameter vectors and the latter s used for the strategy arameter vectors. 84 Mutaton Mutaton s very mortant for the ES because t s the source of genetc varatons. After the recombnaton, each chld ndvdual s mutated to an offsrng ndvdual. Mathematcally, each controller arameter vector θ s mutated usng a normal dstrbuton ( 0, sˆ ): ˆ ˆ (0, ˆ s s ). Accordng to the self-adataton mechansm of the ES, each strategy arameter s s modfed log-normally: s ˆ ex( (0,)) s where s an external arameter. Usually t s nversely roortonal to the square root of the roblem sze. hus an n offsrng ndvdual s defned as aˆ (ˆ θ,ˆ s, f (ˆ θ )) l l where l,,..., l. Selecton l l here exst two dfferent selecton for the evoluton strategy, (, ) and ( ) selectons. he dfference between both selectons s defned by the set of ndvduals nvolved n the selecton. he former selects the best ndvduals out of the offsrng, j

6 Vol.0, No.4, October-December 05 hammasat Internatonal Journal of Scence and echnology whle the latter selects the best ndvduals out of the unon of arents and offsrng to form the next oulaton. [6] onted out that has to be chosen larger than one and a rato of s otmal accordng to the 7 acceleratng effect of self-adataton. ermnaton Convergence of the ES s defned n the sense of robablstc convergence lm Pr( f ( θ ( q)) f ) q where f s a global mnmum. In ractce, f the locaton of the global mnmum f f ( θ ) s reached wth an arbtrary small, the termnaton crteron s lm Pr( f ( θ ( q)) f ( θ ) ). q Remar: after termnaton of ES, there s a roblem,.e. a dversty of canddate controllers s lost. Before t wll be used for the next oeratng ont, we needs to dversfy t as defned: θ ( t ) θ,,, n d...,,,,,,... F F F F F j n...,,,,,,... n n n F n F F F n d...,,,,,,... F d F d d F d F d where F s a constant. 3. Results and Dscusson We consder the nonlnear rocess model for neutralzaton rocesses n a contnuous strred tan reactor (CSR) [7] as shown n Fg. 3. he rocess dynamcs can be descrbed as Fg.3. H neutralzaton rocess. ( I ( x ))( u I ( x ) F) x (4) V ( I ( x )/ x ) R x ( x x) (5) y x (6) where y x s the measured outut. In ths wor, we assume a erfect measurement. he feasblty of the roosed nonlnear control scheme for neutralzaton rocess where acetc acd (CH 3 COOH) s neutralzed by sodum hydroxde (NaOH) n a CSR wth a volume of V R 5.5L. he stochometry of ths reacton s descrbed by CH3COOH NaOH CH3COONa HO he H-value of the reactor effluent has to be adjusted wthout steady state offset n a range between H 5 and H 0 n the resence of varyng rocess streams and devatons from the standard ttraton stream s c B, n 0. 0 mol/l. he nomnal concentraton of acetc acd n the rocess stream s c A, n mol/l and the nomnal flow rate of the rocess stream s F 0 L/h. Due to the ncomlete dssocaton of acetc n water and ts reacton wth sodum acetate, the system behaves le a buffer soluton for low H-values. For ths reacton system the nverse of the standard ttraton curve s exlctly gven by 85

7 hammasat Internatonal Journal of Scence and echnology Vol.0, No.4, October-December 05 0 I ( H) H 0 c H B, n K c W A, n H 0 0 /( 0 H K W ( Ks H) 4 where K s and K 0 mol /L for W room temerature. Because of the low concentraton of the reactants and the small reacton enthaly of the neutralzaton rocess, we can assume constant temerature wthn the reactor. he flow rate of erstaltc um whch sules the ttratng stream s lmted to the range 0 u 45L / h. he samlng erod s of second and the addtonal tme delay 0 seconds for H sensor. 3. Smulaton Setu he smulatons are carred out under the followng assumtons: ) he H neutralzaton rocess model s used as the unnown lant P. ) Oeraton scheme: H 7( ) : 0s t 300s; H 9.5( ) : 300s t 600s; H 6( 3) : 600s t 900s; r j H 8.5( 4) : 900s t 00s; H 7.5( 5) : 00s t 500s; H 6.5( 6) : 500s t 800s. Note that we frst move the rocess to the standard oeratng ont y 7 from the orgn (start-u). hen we move to the second oeratng after t 300s etc. 3) he PID controller structure s set u as follows a) he ntal controller set of PID controllers conssts of 7 canddate controllers: θ ( 0),,, t θ n d,5,0, 0 n,5,0 d 0.0,,0 We choose the frst actve controller as ˆ (0),,0.0. ) 86 b) he search sace for PID canddate controllers s PID EA K n d 000, ,000 [0.0000,000]. c) We defne the dversty for 7 canddate controllers as θ ( t ) θ,,, n d,, F F F j n,, n F n F n d,, F d d F d where F s chosen. 4) 0. 0 for the swtchng of an actve controller. 5) 0. 0 to smoothen the dervatve term 6) he cost functon s used: e u, t j t j t j, t j J rj t j, t j where s a control weghtng. 7) he controller tunng onts t ) are ( j every 00s after a change of setonts. 8) he ( ) selecton s used for ES. 3. Smulaton Results We frst run the algorthm wth the low gan controller θ ˆ(0),,0.0 that we choose from the ntal controller set to start u the rocess to the frst oeratng ont. he suervsor wll ee ths controller u to 00 seconds because there s no a better controller avalable. At tme 00 seconds, the suervsor actvates the ES as shown n Fg. 5. Note that after the termnaton, the ES returns almost the same controller (Uer). hus we need to dversfy t before t uses for the second set-ont (Lower). hs wll be et as the new set of canddate controllers for the second oeratng ont. he otmzed controller

8 Vol.0, No.4, October-December 05 wll be used as the new actve controller durng tme 00 seconds to tme 300 seconds. In addton, the suervsor wll assgn t as the frst actve controller for the second set-ont and allowng the swtchng between canddate controllers occurs n the set. he same rocedure s reeated agan at tme 400 seconds, at tme 700 seconds, at tme 000 seconds, at tme 300 seconds, and fnally at tme 600 seconds. hs leads to an automatc PID controller tunng as shown n Fg.. A good outut erformance can be acheved as shown n Fg.. hammasat Internatonal Journal of Scence and echnology 4. Concluson We roosed the ntellgent algorthm based on only measurement data. hs algorthm can be consdered as an economcal method that can deal wth a stuaton when a lant model s dffcult to derve or even unavalable to desgn a controller. hs s a case that we encounter ndustral roblems. In ths aer, we demonstrated that the algorthm can wor well wth a H neutralzaton rocess model whch the rocess dynamcs are hghly nonlnear. In the future, we wll nvestgate for a real sngle-nut sngle outut H neutralzaton rocess, e.g. a H control for hydrooncs. 87

9 hammasat Internatonal Journal of Scence and echnology Vol.0, No.4, October-December 05 Loss of Dversty after ES termnaton d n x 0-5 Dversty.5 d n Fg.5. ES actvaton at 00 seconds to generate the new set of canddate controller. Fg.6. Adatve PID controller. 88

10 Vol.0, No.4, October-December 05 hammasat Internatonal Journal of Scence and echnology Fg.7. Outut erformance. 5. References [] S. Yn, X. L, H. Gao, and O. Kayna, Data-Based echnques Focused on Modern Industry: An Overvew, IEEE ransactons on Industral Electroncs, Vol.6, , 05. [] S. Formentn, K.V. Heusden, and A. Karm, A Comarson of Modelbased and Data-drven Controller unng. Int. J. Adat. Control Sgnal Process, 00:-6, 0. [3] Z.S. Hou and Z. Wang, From Model- Based Control to Data-drven Control: Survey, Classfcaton and Persectve, Informaton Scences, Vol.53,. 3-35, 03. [4] M.G. Safonov and. sao, he Unfalsfed Control Concet and Learnng, IEEE ransactons on Automatc Control, Vol.4, No.6, , 997. [5] M. Fu and B.R. Barmsh, Adatve Stablzaton of Lnear Systems va Swtchng Control, IEEE ransactons on Automatc Control, Vol.3, , 986. [6] D.E. Mller and E.J. Davson, An Adatve Controller whch Provdes an Arbtrary Good ransent and Steadystate Resonse, IEEE ransactons on Automatc Control, Vol.36, No , 99. [7] S.R. Weller and G.C. Goodwn, Hysteress Swtchng Adatve Control of Lnear Multvarable Systems, IEEE ransactons on Automatc Control, Vol.39, , 994. [8] M. Stafanovc, R. Wang and M.G. Safonov, Stablty and Convergence n Adatve Systems, Proc. Amercan Control Conf., Boston,.93-98, 004. [9] R. Wang, A. Paul, M. Stefanovc and M.G. Safonov, Cost-detectablty and Stablty of Adatve Control Systems. Proc. IEEE CDC-ECC, Sevlle, ,

11 hammasat Internatonal Journal of Scence and echnology Vol.0, No.4, October-December 05 [0] A.S. Morse and D.Q. Mayne and G.C. Goodwn, Alcaton of Hysteress Swtchng n Parameter Adatve Control, IEEE ransactons on Automatc Control, Vol.37, , 99. [] A. Dehghan, B.D.O. Anderson, and A. Lanzon, Unfalsfed Adatve Control: A New Controller Imlementaton and Some Remars, Proc. Euroean Control Conf. Kos, , 007. [] C. Manuell, S.G. Cheong, E. Mosca, and M.G. Safonov, Stablty of Unfalsfed Adatve Control wth Non SCLI Controllers and Related Performance under Dfferent Pror Knowledge, Proc. Euroean Control Conf., Kos, , 007. [3]. Wonghong and S. Engell, Automatc Controller unng va Unfalsfed Control. Journal of Process Control, Vol., , 0. [4] A. Eben and J. Smth, Introducton to Evoluton Comutng. Srnger- Verlag, Berln, 003. [5] H.P. Schwefel, Collectve Phenomena n Evolutonary Systems. Prernts of the 3 st Annual Meetng of the Internatonal Socety for General System Research, volume, Budaest, [6] K.U. Klatt and S. Engell, Nonlnear Control of Neutralzaton Process by Gan-schedulng rajectory Control. Industral & Engneerng Chemstry Research, Vol.35, ,