Fuzzy Gain Scheduling of PID Controller for a MIMO Process

Transcription

1 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 Fuzzy Gan Scheulng of PID Controller for a MIMO Process N anagasaba Department of Instrumentaton Engneerng, Annamala Unversty, Chambaram, Tamlnau, Ina N Jaya, Ph D Department of Instrumentaton Engneerng, Annamala Unversty, Chambaram, Tamlnau, Ina ABSTRACT Ths paper escrbes the evelopment of a fuzzy gan scheulng scheme of PID controllers for three tank process Ths paper presents the controllers for three tank mult loop system usng fuzzy gan scheulng The applcaton of fuzzy logc controller (FLC) appears to be encouragng n the sense that t s robust n sturbance rejecton uner varous contons The controller esgne by FLC technque s base on the choce of Fuzzy rules an Reasonng s use to etermne the controller parameters base on the error sgnal an ts frst fference Smulaton results show that better control performance can be acheve n comparson wth conventonal-pi controllers The smulaton result of the process s carre out by usng MATLAB smulnk software eywors FLC, three tank, mult-loop 1 INTRODUCTION Most of the processes n power plants, refnery process, arcrafts an chemcal nustres are multvarable or multnput mult-output (MIMO) an control of these MIMO [3]processes are more complcate than SISO processes The methoology use to esgn a controller for the SISO process cannot be apple for MIMO process because of the nteracton exhbts between the loops Many methos have been presente n the lterature for control of MIMO process Proportonal-ntegral-ervatve (PID) or Proportonal-Integral (PI) base controllers are use very commonly to control three tank systems Usually two types of control schemes are avalable to control MIMO processes The frst s ecentralze control scheme or multloop control scheme, where sngle loop controllers are use (the controller matrx s a agonal one) The secon scheme s a [3] full multvarable controller known as the centralze controller Mult loop controllers o not explctly conser the ecouplng of the nter-loop nteractons unlke full multvarable controllers The applcaton of knowlege-base systems n process control s growng, especally n the fel of fuzzy control In fuzzy control, lngustc escrptons of human expertse n controllng a process are represente as fuzzy rules or relatons Ths knowlege base s use by an nference mechansm, n conjuncton wth some knowlege of the states of the process n orer to etermne control actons Although they o not have an apparent structure of PID controllers, fuzzy logc controllers may be consere nonlnear PID controllers whose parameters can be etermne on-lne base on the error sgnal an ther tme ervatve or fference In ths paper, a rule-base scheme for gan scheulng of PID controllers s esgne for MIMO process The new scheme utlzes fuzzy rules an reasonng to etermne the controller parameters an the PID controller generates the control sgnal It s emonstrate n ths paper that human expertse on PID gan scheulng can be represente n fuzzy rules Furthermore, better control performance can be expecte n the propose metho than that of the PID controllers wth fxe parameters 2 THREE TAN PROCESS The three-tank system consere for stuy [5] s shown n Fg 1 The controlle varables are the level of the tank h1 an tank h3 In flow of tank1 (fn1) an n flow of tank3 (fn3) are chosen as manpulate varables to control the level The open loop response of the three tank process s shown n Fg2 The steay state operatng ata of the Three-tank system s gven n Table1The materal balance equaton for the above three-tank [1] system s gven by h1 fn1 Az1 2g h1 h2 (1) t s1 S1 h2 AZ1 AZ3 2g(h1 h2) 2g(h2 h3) (2) t S2 S2 h3 fn3 AZ3 AZ2 2g(h2 h3) t s3 S3 S3 Where, fn1 - Inflow of tank-1 fn3 - Inflow of tank-3 (S1-S3)- Area of the tank h1, h2 an h3-level of the tank 2g h3 (3) 13

2 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 Table 1 Steay state operatng parameter Three tank System h1, h2, h3 n m 07,05,03 fn1 an fn3 n ml/sec 100 Outflow coeffcent (Az1, Az2, Az3) Area of tank (S1-S3)nm Acceleraton ue to gravty n m/sec e-5,3057e- 5,2307e-5 Fg1: Schematc agram of three tank process L E R T h 3 DESIGN OF PID CONTROLLER The transfer functon of a PID controller has the followng form: Gc (s) p s (4) s Where p,, an are the proportonal, ntegral, an ervatve gans, respectvely Another useful equvalent form of the PID controller s G c (s) p (1 1/(T s T s) (5) - Where T = p / an T = / p T an T are known as the ntegral an ervatve tme constants, respectvely The screte-tme equvalent expresson for PID [12] control use n ths paper s gven as n U () = p T s e() 1 Δ T s Here, u(k) s the control sgnal, s the error between the reference an the process output, T s s the samplng pero for the controller, an Δ Δ e(k 1) the parameters of the PID controllers p, an or p, T an T can be manpulate to prouce varous response curves from a gven process Fnng optmum ajustments of a controller for a gven process s not trval In the followng secton, an onlne gan scheulng scheme of the PID controller base on fuzzy rules s presente The man objectve of ths process s to control the level of Tank1 an Tank3 The fuzzy gan scheulng scheme s esgne base on the thorough knowlege of the three tank process Also the PI controller parameters for the nteractng three tank process are obtane by utlzng rect synthess metho The controller settngs for loop1 c1 =124137, T 1 =13365 an for loop2 c2 =14589, T 2 =13155 The close loop response of the three tank process for a set pont change n tank1 from ts operatng value of 07m to 1m s shown n Fg3 an ts corresponng effect of nteracton n tank3 Fg 4 shows the close loop response of tank3 for a set pont change of03 m to10 from ts operatng value an ts corresponng effect of nteracton n tank1 an ther values are tabulate n Table 5 In accumulaton of that a loa value of 02 m s ae an subtracte n the process uner the 07m for tank1operatng pont an the servo an regulatory responses are plotte n Fg 5 to 6 The servo trackng response of tank-1 s shown n Fg7 Fg 2: Open loop Response of the three tank process 14

3 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 Fg 3: Close loop response of Tank 1 for Conventonal PI controller Fg 6: Servo an Regulatory response of Tank 3 for Conventonal PI controller Fg 4: Close loop response of Tank 3 for Conventonal PI controller Fg 7: Servo trackng response of Tank 1for Conventonal PI controller 4 FUZZYGAIN SCHEDULING Fg8 shows the PID control system wth a fuzzy gan scheuler The approach taken here s to explot fuzzy rules an reasonng to generate controller parameters It s assume that p, are n prescrbe ranges [ p,mn, max ] an [ mn, max ], respectvely Fuzzy rules an reasonng Fg 5: Servo an Regulatory response of Tank 1 for Conventonal PI controller Input Control PID controller sgnal + - Process Outp ut Fg 8: PID control system wth a fuzzy fan scheuler The approprate ranges are etermne expermentally an wll be gven n equaton (12) For convenence, p an are normalze nto the range between zero an one by the followng lnear transformaton: p / p pmn pmax pmn 15

4 / mn max mn (6) Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl Small Bg In ths scheme, PID parameters are etermne base on the current error an ts frst fference e (k)the ntegral tme constant s etermne wth reference to the ervatve tme constant, e, T αt (7) 1 An the ntegral gan s thus obtane by 2 αt / α p / p (8) The parameters, p fuzzy rules of the form If e (k) s A an e (k) s B, then an are etermne by a set of p s C, s D, an = = 1, 2, m (9) Here, A, B, C an D are fuzzy sets on the corresponng supportng sets: s a constant The membershp functons (MF) of these fuzzy sets for e (k) an e (k) are shown n Fg9 In ths fgure, N represents negatve, P postve, ZO approxmately zero, S small, M meum, B bg Thus NM stans for negatve meum, PB for postve bg, an so on The fuzzy sets C an D may be ether Bg or Small an are characterze by the membershp functons shown n Fg10, where the grae of the membershp functons an the varable x (= p or ) have the followng relaton: 1 4μ SMALL (x) = ln x or x SMALL (μ) e for small, 4 1 4μ BIG (x) = ln (1 x) or x BIG (μ) 1e for Bg (10) 4 N B N M N S Z O P S P M Fg 9: Membershp functons for e (k) an e (k) P B X= e (k) an e (k) Fg 10: Membershp functons for p an The fuzzy rules n (12) may be extracte from operator s expertse Here the rules are erve expermentally base on the step response of the process The PID controller shoul have a large proportonal gan, a large ntegral gan, an a small ervatve gan Thus the proportonal gan ( p ) can be represente by a fuzzy set Bg an the ervatve gan by a fuzzy set Small The ntegral acton s etermne wth reference to the ervatve acton as n eqn (7) For the PID controller, takng a small or a small ntegral tme constant T wll result n a strong ntegral acton Whether the ntegral acton shoul be strong or weak s etermne n the scheme by comparson wth the well-known synthess tunng rule If s PB an Δ Small, an =2 s Zo, then p s Bg s Note that may also be consere as a fuzzy number whch has a sngleton membershp functon as shown n Fg5 For example, becomes 2 when s Small Aroun pont b 1 expect a small control sgnal to avo a large overshoot That s, the PID controlle shoul have a small proportonal gan, a large ervatve gan, an a small ntegral gan Thus the followng fuzzy rule s taken If 0 0 s ZO an Δ Bg, an =5 0 s NB, then p s Small s Thus a set of rules, as shown n Table 2, may be use to aapt the proportonal gan p The tunng rules for an s gven n Tables 3 an 4, respectvely, n the tables, B stans for Bg, an S for SmallThe truth value of the th rule n (12) s obtane by prouct of the MF values n the anteceent part of the rule μ Δ μ μ A B (11) Where A s the MF value of the fuzzy [15] set A gven a value of e (), an B the MF value of the fuzzy set B gven a value of e (k) Base on the values of p an for each rule are etermne from ther corresponng membershp functons The mplcaton process of a fuzzy p or 16

5 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 [20] rule s shown n Fg 11By usng the membershp functons n Fg9; we have the followng conton [18]: m μ 1 1 Then, the efuzzfcaton yels the followng: Here (12) m μ, 1 (12b) a m μ a 1 (12c) p s the value of for the the rule (see Fg 12) p corresponng to the grae s obtane n the same way Once p,, an are obtane, the PID controller Parameters are calculate from the followng equatons that are ue to (6) an (8) p ) p, mn (13a) ( p max (, max mn p, mn ), mn (13b) (13c) Base on an extensve smulaton stuy on varous processes, a rule of thumb for etermnng the range of p an the range of s gven as pmn mn 032 u, 008 u T u, p maz max 06 u 015 u T u (14) Where u an T u are, respectvely, the gan an the pero of oscllaton at the stablty lmt uner P-control [19] Note that there are other forms for the fuzzy tunng rules n (12) Some examples are as follows: 1) If e (k) s A an e (k) s B, then s E 2 ( p / ( ) 2) If e (k) s A an e (k) s B, then p s C, s D, an p s C T s D, an s E 3) If e (k) s A an e (k) s B then u(k) p0 ( 0 T s ) je (j) ( 0 /T s ) Δ Table 3 fuzzy tunng rules for S Small, MS: Meum Small M: Meum, B:Bg Fg 5: Sngleton membershp functons for Table 2 Fuzzy tunng rules for p NB NM NS ZO PS PM PB NB S S S S S S S NM B B S S S B B NS B B B S B B B ZO B B B B B B B PS B B B S B B B PM B B S S S B B PB S S S S S S S NB NM NS ZO PS PM PB NB B B B B B B B NM S B B B B B S NS S S B B B S S ZO S S S B S S S PS S S B B B S S PM S B B B B B S PB B B B B B B B 17

6 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 Table 4 fuzzy tunng rules for k NB NM NS ZO PS PM PB NB NM NS ZO PS PM PB e(k ) A B A B e e Fg12: Implcaton process of a fuzzy rule Where po, 0, 0 are constant Although these rules have fferent forms, they are equvalent to each other uner certan contons It seems relatvely easer to set the fuzzy tunng rules by (11) The fuzzy gan scheulng scheme has been teste on a varety of processes The mult loop fuzzy gan scheulng PID controller parameters of the process calculate as follows Frst the error an ts frst fference are calculate from the sample process output Then the values of p an for each rule are etermne by the fuzzy p, j p reasonng process, as shown n Fg 6 Fnally, the PID parameters are obtane by usng equaton (12), (13) an (14) 5 SIMULATION RESULTS AND DISCUSSIONS The performances of the two controllers are evaluate usng performance nces namely settlng tme (t s ), Integral Square error (ISE) an Integral Absolute Error (IAE) The close loop response of the three tank process for a set pont change n tank1 from ts operatng value of 07m to 1m s shown n Fg13 an ts corresponng effect of nteracton n tank3fg 14 shows the close loop response of tank3 for a set pont change of 03 m to10 m from ts operatng value an ts corresponng effect of nteracton n tank1 an ther values are tabulate n Table 5Wth same operatng contons, a smulatons runs are carre out for rect synthess base three tank process for comparatve stuy The performance nces are calculate n terms of settlng tme, Integral Square Error (ISE) an Integral Absolute Error (IAE) an values are charte n Table 5 From the table t s observe that ntellgent fuzzy control system gves satsfactory performance than the PI controller In accumulaton of that a loa value of 02 m s ae n the process uner the 07m for tank1operatng pont an the servo an regulatory responses are plotte n Fg 15 to 16 The servo trackng response of tank-1 s shown n Fg17 The performance nces are calculate n terms of settlng tme, Integral Square Error (ISE) an Integral Absolute Error (IAE) an values are charte n Table 5 From the table t s observe that fuzzy gan scheulng control system gves satsfactory performance than the PI controller an clearly ncates that the control augmente the control system s conserably reuce the effect of loa sturbance n the process varable compare to the PI controller Fg 13: Close loop response of Tank1 for Fuzzy logc controller Fg 14: Close loop response of Tank3 for Fuzzy logc controller 18

7 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl 2014 Table 5 Performance evaluaton of the controller Controllers parameters Types of controllers Loop - 1 Loop 2 T S (sec) ISE IAE T S (sec) ISE IAE Fuzzy PID Controller e e-007 Fg 15: Servo an regulatory response of Tank1for Fuzzy logc controller Conventon al - PI Controller Fg 16: Servo an regulatory response of Tank3 for Fuzzy logc controller Fg 17: Servo trackng response of Tank1 for Fuzzy logc controller 19

8 Internatonal Journal of Computer Applcatons ( ) Volume 91 No10, Aprl CONCLUSION In ths paper Mult-loop fuzzy gan scheulng controller s esgne for three tank process an compare wth the controller esgne by Synthess tunng metho through smulaton The propose gan scheulng scheme uses fuzzy rules an reasonng to calculate the PID controller parameters Human knowlege an experence n control system esgn s explote n the tunng of a PID controller The scheme has been teste on varous set ponts n smulaton, an satsfactory results are obtane The result of the fuzzy gan scheulng control s analyze an clearly shows that potental avantages usng fuzzy control for a three tank process The control strateges have a goo set pont trackng wthout any offset wth reasonable settlng tme The comparson of above these controllers, reveals that fuzzy gan scheulng control s superor resultng n smoother controller output wthout oscllatons whch woul ncrease the actuator lfe 7 ACNOWLEDGEMENTS The author(s) wsh to thank Prof DrSPNatarajan, Hea of the Department of Instrumentaton Engneerng, an Annamala Unversty, Ina for hs kn permsson to carry out ths Research work 8 REFERENCES [1] SAbraham Lncon, DSvakumar JPrakash State an Fault Parameter Estmaton Apple to Three-Tank Bench Mark Relyng On Augmente State alman Flter, ICGSTACSE Journal, Volume 7, Issue 1, May 2007 [2] M Zhaung an DP Atherton (1994), PID controller esgn for TITO system, IEEP processes control Theory Appl, Vol 141 no2pp [3] S Skogesta,I Postlehwate (1996 ), Multvarable Feeback Control Analyss an Desgn, John Wley & Sons Chchester [4] Astrom an T Hagglun (1995), PID Controllers: Theory Desgn an Tunng, Instrument Socety of Amerca, 2n Eton [5] L ovács: Classcal an Moern Multvarable Control Desgnng Methos of the Three Tank System, Peroca Poltechnca Transactons on Automatc Control an Computer Scence, Vol 48/62, 2003, pp [6] J J Buckley an H Yng Fuzzy controller theory: Lmt theorems for lnear fuzzy control rules Automatca, 25(3): , March 1989 [7] L Anerson, GL Blankenshp, an L G Lebow, A rule-base aaptve PID controller, n Proc 27 th IEEE Conf Decson, Control, 1988, pp [8] P J Gawthrop an PE Nomkos, Automatc tunng of commercal PID controllers for sngle-loop an multloop applcatons, IEEE Control Syst Mag Vol 10, pp 34-42, 1990 [9] J Gertler an HS Chang An nstablty ncator for expert control, IEEE Control Syst Mag, vol6, pp 14-17, 1986 [10] CC Hang, The choce of controller zeros, IEEE Control Syst Mag, vol9, pp 72-75, 1989 [11] CC Hang, J Astrom, an W Ho, Refnements of the Zegler Nchols tunng formula, Proc IEE, Pt D, vol 138, pp , 1991 [12] T Iwasak an A Morta, Fuzzy atuo tunng for PID controller wth moel classfcaton, n Proc NAFIPS 90 Toronto, Canaa, June 6-8, 1990, pp [13] T tamor, A metho of control system esgn base upon partal knowlege about controlle processes, Trans SICE Japan, vol15 pp , 1979 (n Japanese) [14] BC uo, Automatc Control Systems, 5 th e Englewoo Clffs, NJ: Prentce-Hall, 1987 [15] CC Lee, Fuzzy logc n control system: Fuzzy logc controller, Part I, IEEE Trans Systs, Man, Cybern, vol SMC-20, pp , 1990 [16] Fuzzy logc n control systems: Fuzzy logc controller, Part II, IEEE Trans Syst Man Cybern, vol SMC-20, pp [17] CG Nesler, Experences n applyng aaptve control to thermal processes n bulngs, n Proc Amer Control Conf Boston, MA, 1985 Pp [18] TJ Procyk an EH Maman, A lngustc self organzng process controller, Automatca, vol, 15, pp 15-30, 1979 [19] M Sugeno, e, Inustral Applcatons of Fuzzy Control Amsteram, The Netherlans: North-Hollan, 1985 [20] T Takag an M Sugeno, Fuzzy entfcaton of systems an ts applcatons to moelng an control, IEEE Trans Syst Man Cybern, vol SMC-15, pp , 1985 [21] Venkata Ramesh Eara, BAmarenra Rey, Srkanth Monang an MVmala, Analytcal Structures for Fuzzy PID Controllers an Applcatons, Internatonal Journal of Electrcal Engneerng & Technology (IJEET), Volume 1, Issue 1, 2010, pp 1-17, ISSN Prnt : , ISSN Onlne: IJCA TM : wwwjcaonlneorg 20