Effective Open-loop Transfer Function method for design of multi-loop PI/PID controller for interacting multivariable processes

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1 Internatonal Journal of Electronc and Computer Scence Enneern 6 Avalable Onlne at ISSN Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee A. Harha Ran, K. Rajeh, B. Har Krhna Department of Electronc and Electrcal Enneern Jawaharlal Nehru Technolocal Unverty, Anantapur, Inda Emal harharan@mal.com Abtract In term of control mult loop procee are very complcated when compared to nle loop procee due to the extence of nteracton between manpulated and controlled varable. Durn the lat everal decade th topc ha drawn lot of reearch attenton and many multvarable control approache have been propoed. Th Project propoe an Effectve Openloop Tranfer Functon (EOTF) method for PI/PID den of multloop control procee, by employn the concept of Internal Model Control (IMC) wth reduced EOTF and Dynamc Relatve Gan Array (DRGA). Wth th method a multloop control ytem decompoed nto a et of ndependent nle nput nle output (SISO) loop. The advantae of the propoed method are: () the overall control ytem performance better compared wth extn method, and () the control ytem robut. Th technque work wth atfactory performance even under nfcant model mmatche that commonly occur n the proce control ndutry. Wth the help of Effectve Openloop Tranfer Functon (EOTF) baed method and DRGA concept, dynamc nteracton are atfactorly taken n to account. Benchmark example of multvarable ndutral procee wth dfferent nteracton charactertc are ued n th paper to demontrate the mplcty and ectvene of the propoed den method. The propoed method mulated un MATLAB/SIMULINK. Keyword multloop, PI/PID, Effectve openloop tranfer functon (EOTF), Internal model control (IMC), Dyanamc relatve an array (DRGA). I. INTRODUCTION The PID controller by far the mot common control alorthm. Mot feedback loop are controlled by th alorthm or mnor varaton. F.how the typcal tructure of a PID control ytem. Mot chemcal procee are bacally multnput/multoutput (MIMO) ytem. Depte the development of advanced multvarable controller, the multloop PI/PID control un multple Snlenput/nleoutput (SISO) PI/PID controller reman the tandard for controlln MIMO ytem wth modet nteracton becaue of t mple and falure tolerant tructure and adequate Performance [,]. However, due to proce and loop nteracton, the den and tunn of multloop controller much more dffcult compared wth that of nleloop controller. Snce the controller nteract wth each other, the tunn of one loop cannot be done ndependently. Applyn the tunn method for a SISO ytem to multloop ytem often lead to poor performance and tablty. Much reearch ha been focued on how to cently take loop nteracton nto account n the multloop controller den. Many method have been propoed, ncludn the detunn method, equental loop clon (SLC) method, relay autotunn method, and ndependent loop method. The bet lo modulu tunn (BLT) method propoed by Luyben [3] a typcal example of the detunn method, wheren each ndvdual controller frt dened baed on the Zeler Nchol (Z N) tunn rule [4] by norn proce nteracton from other loop. Then, the nteracton are taken nto account by detunn each controller untl the multvarable Nyqut tablty atfed. The attractvene of th method due to the mplcty n mplementaton and comprehenblene for control enneer. However, a dadvantae that the controller ettn are made more conervatve. The ndependent loop method ued to urmount the retrcton of the relay autotunn. A dcued by a number of author [ 5], the ndependent loop method ha a potental advantae n that the falure tolerance of the overall control ytem automatcally uaranteed, wheren each controller ndependently dened baed on the correpondn openloop and cloedloop tranfer functon, thereby atfyn the nequalty contrant on the proce nteracton []. A potental dadvantae of the method When the nteracton amon dfferent loop are modet a ISSN 77956/VN65

2 Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee 7 mple multloop PI/PID controller normally adequate, however, the decentralzed controller den method may fal to ve acceptable repone f there ext evere loop nteracton. A a reult, many ndutral decentralzed controller are tuned looely to enure ytem tablty, whch caue ncent operaton and hher enery cot. For th, the ndependent loop method for IMC type multloop controller [4, 5] ued to reduce the conervatm. Fure. Confuraton of Control Sytem II. PROPOSED ALGORITHM A. Effectve openloop tranfer functon wth dynamc relatve an: Recently, everal reearcher have ntroduced the concept of an ectve openloop tranfer functon (EOTF) to take nto account the loop nteracton n the novel den of a multloop controller [6 ]. The advantae of the EOTF nvolve reduced modeln requrement and eae of mplementaton whle the potental dadvantae reducton n achevable control performance due to retrcted controller tructure []. Effectve openloop tranfer functon method[6] for multloop PI/PID controller den method for multvarable procee nclude three tep: () Un th concept, the den of a multloop controller can be reaonably converted to the den of a nleloop controller; () un the nformaton obtaned n the frt tep to we can fnd the reduced order EOTF to obtan the ectve relatve an and dynamc relatve an(drga) of a ven tranfer functon matrx for cloed loop ytem; and (3) Once a reduced EOTF obtaned, any PID tunn method for a SISO ytem can be appled for the den of each ndvdual PID controller. In th tudy, the IMCPID tunn rule ueted by Lee et al. [4] were choen. Conder the openloop table multloop ytem n F.3., where r, u, and y are the etpont, manpulated, and controlled varable vector, where r, u, and y are dcarded from r, u, and y, repectvely. Let the EOTF of loop be defned a the tranfer functon relatn u wth y where loop open whle all other loop are cloed. F. 3. how the block daram for the concept of the EOTF of loop. The EOTF dffer from the ornal openloop tranfer functon (OTF) by tranmon nteracton throuh a path ncludn other loop. It clear that the EOTF correpond to the actual open loop tranfer functon under multloop tuaton and thu, tunn of the controller of loop hould be done baed on the EOTF,, rather than the ornal OTF,. G() ( ) ( )... n ( ) () ( ) ( )... n ( ) n ( ) n ( )... nn ( ) The den of a multloop controller can be reaonably converted to the den of a nleloop controller. On the ba of tructure decompoton, the multloop control ytem completely eparated nto equvalent ndvdual SISO loop, and thu the ect of the proce and controller on the loop nteracton and ubequent ytem properte, uch a rht half plane (RHP) zero and pole, nterty, and tablty, are elucdated [6]. In an alternate manner, on and Ca [9] ueted that the EOTF provde both an and phae nformaton for multloop controller den n four way. The advantae of the EOTF nvolve reduced modeln requrement and eae of mplementaton. Furthermore, a hown n F.3., a multloop MIMO ytem can be condered a et of n ndependent SISO ytem wth correpondn EOTF. From the block daram n F. 3. wth r, u obtaned by ~ ~ c u G c y G c ( u + G u ) () ISSN 77956/VN65

3 IJECSE, Volume, Number A. Harha Ran et al. 8 Fure. Block daram for the concept of the ectve openloop tranfer functon n an n n multloop ytem: loop open whle all other loop are cloed. Fure 3. Multloop ytem and equvalent ndependent SISO ytem wth the correpondn EOTF. Where G ~ c denote multloop controller matrx n whch c dropped form G ~ c. G denote a tranfer functon matrx where both the th row and column are removed from G. The term r and c denote the th row and column vector of matrx G where dcarded, repectvely. Rearrann Eqn. () yeld c u Gc ( I + G Gc ) u (3) Therefore, the relaton between y and u wrtten a r r c y u + u [ Gc ( I + G Gc ) ] u (4) The complcaton of the dynamc nteracton clear from Eqn. (3.4).The openloop dynamc between y and u depend not only up on the nle tranfer functon,, but alo on the proce and controller term n all other loop. Th alo mple that n prncple, tunn of one controller cannot be done ndependently and depend on other controller. However, th complcated relaton due to the dynamc nteracton can be reaonably mplfed by ntroducn everal aumpton. Aumn that the controller nclude nteral acton to avod offet and that the cloedloop dynamc, by properly tuned controller, are uffcently fater than the openloop dynamc, then a perfect control approxmaton, G G I + G G I can be condered at frequence lower than the croover frequency. c ( c ) Therefore, Eqn. (3.4) reaonably mplfed a follow r c y [ ( G ) r c G Gc ( I + G Gc ) ] u [ ( G ) ] u u (5) Note that the EOTF of loop,, cont of a proce dynamc term only and doe not nclude knowlede of other controller. Furthermore, the EOTF can be compactly expreed n term of the DRGA, a follow A (6) ISSN 77956/VN65

4 Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee 9 Where A denote the th daonal element of the DRGA and calculated by T A [ G ( G ) ] (7) Where the ymbol denote the element by element multplcaton (Hadamard or Schur product) and the upercrpt T denate the tranpoe of a matrx. The phycal meann of DRGA clearly ndcated from Eqn. (6): the th daonal element of the DRGA mple a rato of the openloop tranfer functon to the ectve openloop tranfer functon of loop, under the aumpton of perfect control of other loop. B. Reduced EOTF for controller den: One of the mot common approache for controller den to ue a reducedorder model that mplfe the proce dynamc. Snce the EOTF lkely to how a complcated dynamc model form, the approach un a reducedorder model enerally requred. Any conventonal model reducton technque can be appled for th purpoe. In th ecton, a mple model reducton technque appled to approxmate the EOTF to a reducedorder model, uch a the frtorder plu dead tme (FOPDT) and the econdorder plu dead tme (SOPDT) model. A twonput, twooutput (TITO) multdelay proce one of the mot commonly encountered multvarable procee n the proce ndutry [9 3]. A lare number of prevou tude focued on denn multloop control ytem of TITO procee. For a ytem, the eneral table quare tranfer functon matrx repreented a ( ) G S (8) The DRGA obtaned from Eqn(6) ( ) ( ) A ( ) A ( ) ( ) (9) ( ) ( ) ( ) ( ) For a 3 3 ytem, the dervaton of EOTF ubtantated n a mlar manner a that of the ytem. The daonal element of the DRGA are etablhed from (6) a follow ( ) A () Thu the EOTF for the frt, econd and thrd loop are conttuted a ( ) ( ) () A een from the equaton above, the reultn EOTF are uually too complcate to be drectly utlzed for the controller den. To overcome th dffculty, the EOTF have to be mplfed to loworder model, uch a FOPDT and SOPDT. A lot of model reducton technque are avalable, ncludn, but not lmted to, the leat quare alorthm [3 34], polynomal approxmaton, Lauerre expanon and the Gauan frequency doman approach. Any technque can be appled toward fttn the EOTF nto a loworder model. In th work, for the purpoe of evaluatn the propoed EOTF, a mple model reducton technque wa propoed baed on the cocent matchn method. Expandn n a Maclaurn ere n ve ( ) a + b + c + d + e + O( ) () Where the cocent of the polynomal are 4 a (), d ( ), d ( ) b, 3 d d ( ) ( ) c d d e 3 4 d 6 4 d d ISSN 77956/VN65

5 IJECSE, Volume, Number A. Harha Ran et al. Snce the FOPDT model mot wdely ued for modeln monotonc table dynamc, ven t mplcty wth reaonable performance, the FOPDT dynamc a a reducedorder model mut be condered frt. θ r _ ke (3) + Expandn the reduced EOTF ven by Eqn. (3..6) n Maclaurn ere n alo ve r _ 3 k k( θ + ) + k θ + ( θ + ) + O( ) (4) Table Relaton between proce parameter and polynomal cocent for typcal proce model MODELS θ ke + k a ; ( ) ke θ ± a + ( )( ) + + k a ; RELATIONS ( θ ) ; b + a θ c ( ) + θ + a ( m ) ; b a a [ ( m )] + a c a ; d 3( ) + m a a c 4 5( ) + m a a Where θ + +, θ, 3 θ θ, + ( θ + ) + + ( θ + ) 6 θ 3 + ( θ + ) + ( θ + ) +, 3 θ θ θ θ θ θ θ + + ( θ ) + θ + + θ Where k,, and θ hould be dentfed to approxmate a cloe a poble over control relevant frequency rane. When the EOTF dynamc too complcated to be properly expreed by a mple FOPDT model, reducton to the FOPDT model reult n nfeable and/or θ value that are neatve or complex. A uch, the FOPDT model not vald for approxmatn the EOTF dynamc and a uch, more eneral dynamc uch a a SOPDT model ha to be condered. It ha been hown from prevou work [8] that a SOPDT model wth a neatve/potve zero atfactorly repreent mot of the complcated proce dynamc for the control relevant purpoe. ( ± + ) r _ a ke θ ( + )( + ) The relaton between the polynomal cocent and the proce parameter for the SOPDT model are lted n Table. The SOPDT model parameter value of,, a, and θ can alo be obtaned from thoe relaton. ;, (5) ISSN 77956/VN65

6 Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee It mportant to predct when and/or θ of the FOPDT model become nfeable n the propoed model reducton method. Suppoe that the FOPDT model attempted whle the actual EOTF ha the SOPDT dynamc ven by Eqn. (5). For the cae of SOPDT wth a neatve zero, a > + or ( ) + + ( ) > a > + (6.a) For the cae of SOPTD wth a potve zero, a > + (6.b) Where θ + +. For example, n the propoed model reducton method, ether a lare overhoot or a tron nvere repone uch a: a > + caue nfeable and/or θ value n the FOPDT model. If the EOTF monotonc and ha no lead term, the propoed method alway ve feable and θ value. It hould be noted that feable parameter value do not uarantee accuracy of model reducton. Identfcaton performance by any reducedorder model requre fnal confrmaton by comparn the EOTF wth the reduced EOTF. III. DESIGN OF MULTILOOP PID CONTROL Once a reduced EOTF obtaned, any PID tunn method for a SISO ytem can be appled for the den of each ndvdual PID controller. In th tudy, the IMCPID tunn rule ueted by Lee et al. were choen. The IMCPID den approach commonly ued for the PID controller tunn n the proce ndutry becaue of t many advantae, ncludn mplcty, robut performance, and analytcal form. The overall procedure for drvn the tunn rule of loop a follow: r _ Frt, the reduced EOTF,, decompoed to r _ e ff P P, where P A M A and P M are the nonmnmum porton wth an allpa form and the mnmum phae porton, repectvely. The conventonal IMC flter, f, elected a: f ( ) ( ) m λ +, n whch λ a den parameter that provde the tradeoff between performance and robutne. It the dered cloedloop tme contant for the etpont trackn. The flter order m elected a a potve nteer o that the controller proper and realzable. Then, the deal feedback controller to yeld the dered cloed loop repone perfectly ven by P ( ) (7) q M c r m q + P A ( ) _ ( ) ( λ ) Where q the IMC controller and dened by: P f M Snce the above reultn controller doe not have a tandard PID controller form, t requred to approxmate the deal feedback controller G c to the equvalent PID controller form. Expandn n a Maclaurn ere n yeld c f ( ) ' f ''() f '''() 3 c f () f () + +! + 3! + L (8) The controller ven by Eqn. (7) nterpreted a the tandard PID controller by un the frt three term and truncatn the hher order term, ven by c ( ) K c + + D (9) l Where ISSN 77956/VN65

7 IJECSE, Volume, Number A. Harha Ran et al. ' ' f K c f (), () l, f ''() D f () ' f () The dervatve and nteral tme contant computed from Eqn. (4.3) could have neatve value when the reduced EOTF model ha a tron lead term. In th cae, a PID controller n ere wth the frtorder la flter tructure recommended for ue c ( ) K c + + D () l F + Where f '''() ' ' f '''() f () ' '' Kc f () + f () f '' l + '''() f () 3( f ()) ; f '' 3 f () () ; 3 f () f D '''() '' ' ; F '' f '''() f () 3 f () f () 3 f () Performance ndex: To evaluate cloedloop performance, the nteral abolute error (IAE) crteron condered, whch defned a: IAE e( t) dt () Where e( t) r( t) y( t). Robutne ndex: In th tudy, a wellknown method for robut tablty [, 6, 44, and 45] utlzed for a far comparon wth other comparatve method. The multple ource of uncertanty are lumped nto a nle complex perturbaton (multplcatve nput/output form). Snce the output uncertanty often le retrctve than nput uncertanty n term of control performance [6], the robut tablty of multloop control ytem examned under output multplcaton uncertanty. For a proce wth an output uncertanty of [ I + ( )] G( ), the upper bound of the robut tablty ven a : ~ ~ ~ γ σ ( ) < / σ [( I + G( jω ) G c ( jω )) G( jω ) G c ( jω )] < σ [ I + ( G( jω ) G c ( jω )) ], ω Where γ repreent the deree of robut tablty, perturbaton a a multplcatve ~ output, and σ and σ maxmum and mnmum nular value, repectvely. It hould be noted that: G j ) Gc ( j ) nvertble. ( ω ω For a far comparon, all of the controller ben compared were dened to have the ame deree of robut tablty n term of the γ value throuhout all mulaton example. For the propoed control ytem, the γ value wa kept the ame a or larer than thoe of the other method. Note that a control ytem wth a larer γ value mple more robut tablty. A. Comparon of controller performance for Vnante and Luyben (VL) column method [3]: The performance of controller dened baed on Effectve openloop tranfer functon method compared wth mple tunn controller dened by J.Lee et al,lee et al and He et al Fure 4.(a), 4.(b), 4.(c) and 4.(d) compare the repone of y and y for unt tep chane n et pont y and y repectvely. It clear that ettln tme le for the Effectve openloop tranfer functon method than the mple tunn method. The reultn controller parameter, toether wth the performance ndce calculated un the abovementoned method, are ummarzed n Table. The propoed PI/PID controller provde a ood performance wth fat and wellbalanced repone n comparon wth thoe of the extn method. The ectvene of the propoed PI/PID controller alo confrmed by t mallet IAE value n Table. ISSN 77956/VN65

8 Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee 3 (a) Fure 4. Mult performance comparon of cloedloop performance to the equental tep chane n the etpont for the VL column for (a) frt output (b) econd output. (b) (a) Fure 5. performance comparon of (a) frt output for the repone of the VL column where loop cloed and loop open (b) econd output for the repone of the VL column where loop cloed and loop open. Table. Comparon of Controller parameter and reultn performance ndce for the VL column. (b) Tunn method Loop k c l Propoed method (PID) Propoed method (PI) D λ γ IAE Nomna l VL (+4%) VL (4%) t J. Lee et al M.Lee et al Hee et al B. Comparon of controller performance for Wood and Berry (WB) column model [48]: The performance of controller dened baed on Effectve openloop tranfer functon method compared wth mple tunn controller dened by Ho et al, Lee et al and Loh et al Fure 4.3(a) and 4.3(b) compare the repone of y and y for unt tep chane n et pont y and y repectvely. It clear that ettln tme le for the Effectve openloop tranfer functon method than the mple tunn method. The reultn controller parameter, toether wth the performance ndce calculated un the abovementoned method, are ummarzed n Table 4.. The propoed PI/PID controller provde a ood performance wth fat and well ISSN 77956/VN65

9 IJECSE, Volume, Number A. Harha Ran et al. 4 balanced repone n comparon wth thoe of the extn method. The ood performance of the propoed method readly apparent. The ectvene of the propoed PI/PID controller alo confrmed by t mallet IAE value n Table 4.. (a) Fure 6. Performance comparon of (a) frt output and (b) econd output for the repone cloedloop repone to the equental tep chane n the etpont for the WB column. (b) Table 3. Comparon of Controller parameter and reultn performance ndce for the WB column. Tunn method Propoed method (PID) Propoed method (PI) Loop Ho et al. Lee et al. Loh et al. k c l D..4 λ γ IAE t Nomnal WB (+4%) WB (4%) IV.CONCLUSION A method for den of PI/PID decentralzed controller for multnput multoutput ytem un Effectve Openloop Tranfer Functon (EOTF) method developed n th work. It appled to () Vnante and Luyben (VL) column model [3], () Wood and Berry (WB) column model [48]. A mulaton tudy wa carred out to evaluate the propoed approach. The reult are compared wth thoe of ome extn tunn method and decentralzed method of on [7]. The performance evaluated n term of IAE and robutne. Robutne tudy wa conducted by nertn a perturbaton uncertanty of ±4% n the proce an. The reult howed that the propoed control ytem hold robut tablty well n the plantmodel mmatch cae. It concluded that Effectve Openloop Tranfer Functon (EOTF) method for PI/PID decentralzed controller den of multnput multoutput ytem ve better reult compared wth other decoupln method, and decentralzed method. V. REFERENCE [] P.J. Campo, M. Morar, Achevable cloedloop properte of ytem under decentralzed control: condton nvolvn the teadytate an IEEE Tran. Automat. Control 39 (994) ISSN 77956/VN65

10 Effectve Openloop Tranfer Functon method for den of multloop PI/PID controller for nteractn multvarable procee 5 [] W.L. Luyben, Smple method for tunn SISO controller n multvarable ytem, Ind. En. Chem. Proce De. Dev. 5 (986) [3] J.G. Zeler, N.B. Nchol, Optmum ettn for automatc controller Tran. ASME 64 (94) [4] D.Q. Mayne, The den of lnear multvarable ytem Automatca 9 (973) 7. [5] M. Hovd, S. Skoetad, Sequental den of decentralzed controller Automatca3 (994) [6] K.J. Åtröm, T. Hälund, Automatc Tunn of PID Controller Intrument Socety of Amerca, Reearch Tranle Park, NC, 988 [7] S.H. Shen, C.C. Yu, Ue of relayfeedback tet for automatc tunn of multvarable Sytem, AIChE J. 4 (994) [8] P. Grodder, M. Morar, Interacton meaure for ytem under decentralzed Control, Automatca (986) [9] S. Skoetad, M. Morar, Robut performance of decentralzed control ytem by ndependent den, Automatca 5 (989) 9 5. [] M. Hovd, S. Skoetad, Improved ndependent den of robut decentralzed Controller, J. Proce Control 3 () (993) [] J. Jun, J.Y. Cho, J. Lee, One parameter method for a multloop control ytem Den, Ind. En. Chem. Re. 38 (999) [] Z.. Zhu, Structural analy and tablty condton of decentralzed control Sytem, Ind. En. Chem. Re. 35 (996) [3] H.P. Huan, J.C. Jen, C.H. Chan, W. Pan, A drect method for multloop PI/PID controller den, J. Proce Control 3 (3) [4] Cu, E.W. Jacoben, Performance lmtaton n decentralzed control, J. Proce Control () [5] E.H. Brtol, On a new meaure of nteracton for multvarable proce control, IEEE Tran. Automat. Control (966) [6] L.S. Tun, T.F. Edar, Analy of controloutput nteracton n dynamc ytem AIChE J. 7 (98) [7] S. Skoetad, I. Polethwate, Multvarable Feedback Control, John Wley and Son, New York, 996. [8] Q. on, W.J. Ca, M.J. He, A practcal loop parn crteron for multvarable Proce, J. Proce Control 5 (5) [9] E.H. Brtol, Recent reult on nteracton n multvarable proce control, n: Proceedn of the 7t Annual AIChE Meetn, Houton, [] L.S. Tun, T.F. Edar, Analy of controloutput nteracton n dynamc ytem AIChE J. 7 (98) [] S. Skoetad, I. Polethwate, Multvarable Feedback Control, John Wley and Son, New York, 996. [] Q. on, W.J. Ca, M.J. He, A practcal loop parn crteron for multvarable Proce, J. Proce Control 5 (5) [3] M.J. He, W.J. Ca, W. N, L.H. e, RNGA baed control ytem confuraton for multvarable procee, J. Proce Control 9 (9) [4] F. Shnkey, Proce Control Sytem, McGrawHll, New York, 979. [5] W.L. Luyben, Proce Modeln, Smulaton and Control for Chemcal Enneer, McGrawHll, New York, 99. [6] V. Strejc, Leat quare parameter etmaton, Automatca 6 (5) (98) [7] R. Pntelon, P. Gullaume, Y. Rolan, J. Schouken, H.V. Hamme, Parametrc dentfcaton of tranfer functon n the frequency doman a urvey, IEEE Tran. Automat. Control 39 () (994) [8] B. Nnne, Interal contrant on the accuracy of leat quare etmaton, Automatca 3 () (996) [9] P.J. Gawthrop, M.T. Nhtla, Identfcaton of tmedelay un a polynomal dentfcaton method, Syt. Control Lett. 5 (985) [3] R. Malt, S.B. Ekonolo, J. Raot, S.I.S.O. Dynamc, MIMO ytem approxmaton baed on optmal Lauerre method, IEEE Tran. Automat. Control 43 (9) (998) ISSN 77956/VN65