A Method of Performance Assessment of PID Controller with Actuator Saturation

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1 Inrnaional Confrnc on Mcharonic, Elcronic, Indurial and Conrol Enginring (MEIC 5 A Mhod of rformanc Amn of ID Conrollr wih Acuaor Sauraion Xia Hao Faculy of Elcronic and Elcrical Enginring Dalian Univriy of Tchnology Dalian, China xifihxi@6.com Yu Mingli Faculy of Elcronic and Elcrical Enginring Dalian Univriy of Tchnology Dalian, China yumingli@mail.dlu.du.cn Abrac Th uro of conrol loo rformanc amn i o find a ror mhod o maur h diffrnc bwn h acual rformanc and h rdicd rformanc of h indurial conrol loo and furhr idnify roblm and mak adjumn. Thrfor, a ror bai o valua h corrondingly rformanc i vry imoran. Acuaor auraion i a common hnomnon in h indurial roducion. In ordr o ablih a rformanc indx accuraly, h acuaor auraion i nd o b akn ino accoun. Thi ar roo a mhod o a h racking rformanc of ID conrollr undr acuaor auraion. Th oimal conrollr aramr can b xrd a h funcion of auraion lvl bad on curv fiing. Th lowr bound of h ron rformanc ha bn ablihd. By comaring wih h lowr bound hrough inrnal modl conrol (IMC rincil rgardl of auraion, i can b hown ha h rood mhod i mor rliabl. Th validiy of h nw rformanc indx i illurad by imulaion. Kyword-oin racking; inrnal modl conrol; ID conrollr; acuaor auraion; rformanc amn I. INTRODUCTION Th xid rformanc roblm of indurial conrol loo cau h rourc wa and vn afy roblm. Th uro of conrol loo rformanc amn i o find a ror mhod o maur h diffrnc bwn h acual rformanc and h rdicd rformanc of h indurial conrol loo and furhr idnify roblm and mak adjumn. Thrfor, a ror bai o valua h corrondingly rformanc i vry imoran []. In ordr o ablih a rformanc indx accuraly, h hnomnon,.g. acuaor auraion, fricion conrol valv and conrol valv juming, i nd o b akn ino accoun. Th rformanc indx ar ablihd o indica h rformanc of a conrol ym, and h mo common indx ar random rformanc and drminiic rformanc indx. Random rformanc indx rflc h diffrnc bwn h currn conrol rformanc and h minimum varianc conrol (MVC. Harri [] xlord h famou minimum varianc conrol and Dborough [3] alid i in univaria fdforward- fdback ym. Ko [4] alid i in cacad conrol ym o olv h roblm of random rformanc valuaion. Drminiic rformanc amn mhod can valua h rformanc of diurbanc rjcing or oin racking rgardl of h limiaion of h conrollr rucur. Swanda [5] found ha h adjuing im and ingraion aboluly rror (IAE can b h indx o a h rformanc of oin racking. Marin [6] rcommndd ha ing h ingraion of im and abolu rror a h indx o valua h rformanc of ID conrollr. Thr ar many xid mhod [7] o un and dign h conrollr, uch a IMC mhod [8], Ziglr-Nichol mhod [9], Cohn-Coon mhod and h dirc ynhi mhod []. Howvr, h mhod ignord h iuaion whn h acuaor aurad. Th conrol loo will hav a oor rformanc whn h conrollr acuaor ay in h auraion and on-loo a. Alhough hr ar many comnaion mhod o olv h acuaor auraion roblm [], mo of hm ar no ay o analyz h ym rformanc or h conrollr dign i vry comlicad. In hi ar, a nw mhod i rood o valua h oin racking rformanc of conrollr loo wih acuaor auraion. Thi ar analyzd h IMC uning mhod of ID conrollr, ablihd h limiaion of h rood mhod and found h rlaionhi bwn h auraion and conrollr aramr. Comard wih h indx of IMC rincil by imulaion, h nw mhod i mor accura han IMC mhod. II. INSTRUCTION OF IMC-ID CONTROLLER A. IMC Tuning Mhod of ID Conrollr r( - ( C( u( Figur. Th fdback conrol loo of SISO y( Th SISO fdback loo wa hown in Fig.. ID conrollr C( i : C( K + Td T i 5. Th auhor - ublihd by Alani r 84

2 Th conrolld objc i aumd a abilizaion roc, and i can b aroximad a FODT: K T or SODT modl: K T T Whr K, T and θ rrn roc gain, im conan and lag im conan. According o ( and (3, h aramr of ID conrollr und by IMC rincil ar xrd in (4(5: T K, Ti T, Td K( T T K, Ti T, Td K( T Finally, h ranfr funcion of clo-loo i: Gcl ( ( ( For ay analyi and bcau h dlayd imac on ym rformanc i imilar o h imac of oiiv ol, w dfind a [7], and hi will lad o crain rror from h acual ym. Rfrring o h(6, only h aramr λ influnc h aramr of ID conrollr and h conrol rformanc. Hnc, how o find h aramr λ i h mo imoran iu during h whol dign. B. Th Effc of Sauraion for IMC-/ID rformanc Thr ar many nonlinar hnomna nd o b concrnd, hi ar focu on h acuaor auraion which lad o h ranin valu of ym ron. Th conrol objc i: 5 Th I conrollr aramr ar K., Ti 5. Afr IMC uning, h aramr ar K, Ti. Th indx ar hown in Tab.I. TABLE I. DIFFERENT INDEX VALUE WITH AND WITHOUT IMC TUNING Conrollr Diffrn Indx Valu original 3.5 a IMC und In hi xaml, h IAE and ISE ar all mall and indica h good oin racking rformanc. If h valu of acuaor auraion i.8, h conrol ignal and ym ouu ar hown in Fig.. u( y( h original conrollr conrollr und by IMC h original conrollr conrollr und by IMC Figur. Th conrollr ouu and ym ron wih acuaor auraion(u=.8 TABLE II. DIFFERENT INDEX VALUE WITH ACTUATOR SATURATION Conrollr Diffrn Indx Valu original 3.5 a IMC und I can b n ha h IAE and ISE bcom biggr and h rformanc of IMC uning i ffcd by h acuaor auraion. If w ak h oimal indx calculad by IMC mhod wihou conidring h auraion a h andard indx, hr will b an rror for h acual oimal indx bcau h ffc of auraion. So i nd o find a mhod o lc h righ λ o avoid h auraion and choo h und conrol rformanc a. III. THE METHOD OF ID CONTROLLER DESIGNING UNDER SATURATION CONDITION A. Th mhod of ID conrollr digning undr auraion condiion In hi ar, vral conrol objc modl wr randomly lcd. W lcd I/ID conrol aramr rfr o ach diffrn conrol objc modl and imulad o obain conrol daa and rcordd h corrondingly imum ouu of conrollr. Finally w found h rlaionhi bwn aramr of I/ID conrollr and h imum ouu of conrollr. Sinc h oimum ingraion im can b obaind by IMC uning mhod, w go h aramr of T i and T d. A for h FODT modl, h mhod of curv fiing a following: 85

3 S i h raio bwn h im dlay θ and im conan T, which rang from. o commonly. In hi ar, h roc gain K i chon a,, 3,4 for 4 grou, im dlay θ i chon a,, 3,4 for 4 grou and h conan λ i chon a.t,.4t,.6t,.8t,t for 5 grou. Afr h daa wa collcd, w ablihd h rlaionhi bwn K and U by Curv Fiing in Malab, which wa hown in Tab.3. Whr R rrn h dgr of curv fiing, quard ummaion rror (SSE, adjud R and h roo man quar rror (RMSE rflc h goodn of curv fiing. In Tab. III, h valu of adjud R clo o, whil h SSE and RMSE clo o, which howd an accura fiing rul. Hr w wo cofficin A and B and dfind diffrn valu of cofficin in h rang in Tab. IV : coffici n valu TABLE III. RELATION COEFFICIENT VALUES Diffrn Valu A B SSE R Adj. R RMSE TABLE IV. COEFFICIENT VALUES OF A AND B UNDER THE DIFFERENT RANGE OF S Cofficin valu Diffrn Rang of S A.38S S R B.3S S+.78 R According o h S, h aramr of conrollr in auraion condiion wa finally calculad. A for.<s<.,h aramr wr hown a follow. K ( U B A A.38S.9865 B.3S.8 Whn.<S<. K ( U B A A.376S.6374 B.8 S.78 A for h SODT modl, S i h raio bwn h im dlay and im conant, who common valu rang from. o. V i h raio bwn T and T, R qual o V/S. Th roc gain K, im dlay θ wr chon a,, 3,4 for 4 grou. Afr h daa wa collcd, w ablihd h rlaionhi bwn S and U by Curv Fiing in Malab, which wa hown in Tab.V. Th xrion cofficin wa hown in Tab.VI. U AK B TABLE V. COEFFICIENT VALUES OF A AND B UNDER THE DIFFERENT RANGE OF S coffici n valu Diffrn Valu A B SSE R Adj. R RMSE Cofficin valu TABLE VI. EXRESSION COEFFICIENT Valu A.75R+.8S R.935,.995,.9944,.746 B -.55R-.6S+.869 R.759,.885, ,.5 In hi mhod, h cofficin wa dfind by h auraion dgr of h conrollr ouu. Th calculad cofficin wa influncd by roc gain, im conan and im dlay. In hi ar, h nw mhod wa rood on h bai of h IMC mhod, o w can dign h conrollr by IMC uning mhod dircly whn i i undr unaurad condiion. B. Th Lowr Bound of ID Conrollr rformanc undr Sauraion Th obaind ym clod-loo ranfr funcion from Fig. wa hown a follow. Gcl ( T T ( ( T 86

4 Auming h ing valu i a ignal, w finally obaind h ron from Fig.. Y ( Gcl ( R( [( T ] Tranformd by Lalac: u( h original conrollr conrollr und by IMC conrollr und by rood mhod, y ( ( /( T, Dviaion ignal:, ( r( y( ( T, Th indx of ID conrollr rformanc: ( T IAE ( d d d T / ( T ISE ( d d d.5 ( T / IV. SIMULATION In hi cion, w dicu h influnc of auraion hnomnon on ym and h ffcivn of h rood mhod of ID conrollr amn in auraion iuaion. Manwhil, h rul wr comard wih IMC uning mhod. Simulaion : Th aramr of I conrollr in mixur roc ar K. and Ti 6.6. Idnifid FODT modl i hown a follow: Th auraion dgr in hi ym wa a.3, and h conrollr aramr wr K. and T.6, which obaind by h i mhod in hi ar. Th conrollr ouu and h ym ron wa hown in Fig.3. Th diffrn indx valu wih acuaor auraion wa hown in Tab.V. Tabl VII. DIFFERENT INDEX VALUE WITH ACTUATOR SATURATION Conrollr Diffrn Indx Valu rim ym y( h original conrollr conrollr und by IMC conrollr und by rood mhod Figur 3. Th conrollr ouu and ym ron wih acuaor auraion i.3 Simulaion : Th aramr of ID conrollr in mixur roc ar K.5, Ti 3. and Td.8. Idnifid FODT modl i hown a follow: Th auraion dgr in hi ym wa a.95, and h conrollr ouu and h ym ron wa hown in Fig.4. Th diffrn indx valu wih acuaor auraion wa hown in Tab.VIII. Tabl VIII. DIFFERENT INDEX VALUE WITH ACTUATOR SATURATION Conrollr Diffrn Indx Valu rim ym IMC Mhod in hi ar Thorical valu IMC Mhod in hi ar Thorical valu

5 .5 h original conrollr conrollr und by IMC conrollr und by rood mhod ACKNOWLEDGMENT Th auhor would lik o xr arciaion for h financial uor by h Naional Naural Scinc Foundaion of China ( u( y( h original conrollr conrollr und by IMC conrollr und by rood mhod Figur 4. Th conrollr ouu and ym ron wih acuaor auraion i.95 V. CONCLUSION Acuaor auraion i a common hnomnon in h indurial roducion. Th corrc valuaion conrol loo rformanc in auraion iuaion can avoid h micalculad lo ffcivly. For h FODT modl and SODT modl, w rood a nw conrollr dign mhod in auraion iuaion and ablihd h lowr bound of rformanc indx. Finally w found h nw mhod wa mor accura o valua h rformanc of conrollr loo han h IMC mhod rgardl of h auraion. REFERENCE [] Z. Yu, J. Wang, B. Huang, Z. Bi, rformanc Amn of ID Conrol Loo Subjc o Soin Chang, J of roc Conrol, vol. (8,,. 64-7, doi:.6/j.jrocon..6.. [] T.J. Harri, Amn of Conrol Loo rformanc, Can. J. Chm.Engng., vol.67(, 989, , doi:.6/cinc [3] L. Dborough, T.J. Harri, rformanc Amn Maur for Univaria Fdback Conrol, Th Canadian Journal of Chmial Enginring, vol. 7, 99, doi:./cjc [4] B. Ko and T.F. Edgar, rformanc Amn of Cacad Conrol Loo, AIChE Journal, vol. 46,, doi:./aic [5] A.. Swanda, D.E. Sborg, Conrollr rformanc Amn Bad on Soin Ron Daa, rocding of h Amrican Conrol Confrnc. San Digo, Amrican Conrol Confrnc, IEEE r, 999, , doi:.9/acc [6] F.G. Marin, Tuning ID Conrollr Uing h ITAE Cririon, In J Eng Educa, vol. (3, 5,. -7. doi: X/9 $3.+.. [7] S. Skogad, Siml Analyic Rul for Modl Rducion and ID Conrollr Tuning, J. roc Conrol, vol. 3, 3, doi:.6/s959-54(6-8. doi:.6/s959-54(6-8. [8] D.E. Rivra, M. Morari, S. Skogad. Inrnal Modl Conrol: ID Conrollr Dign, Ind. Eng. Chm. roc D. Dv, vol. 5, 986, doi:./i3a4. [9] K. Arom, T. Hagglund, Rviiing h Ziglr Nichol S Ron Mhod for ID Conrol, roc Conrol, vol. 4, 4, , doi:.6/j.jrocon.4.. [] A. Yaman, John H, Exrimnal Sudy of Inrnal Modl Conrol, Ind. Eng. Chm. roc D. Dv, vol. 5, 986,. -8, doi:./i3a6. [] D.E. Sborg, T.F. Edgar, D.A. Mllicham, roc Dynamic and Conrol, Hobokn, Wily r, 4. 88