Performance Analysis and Comparison of Different Tuning Strategies of PI Controller in Conical Tank

Transcription

1 Indian Journal of Science and Technology, Vol 9(11), DOI: /ijs/2016/v9i11/89299, March 2016 ISSN (Prin) : ISSN (Online) : Performance Analysis and Comparison of Differen Tuning Sraegies of PI Conroller in Conical Tank E. Kesavan *, A. Agalya, P. Palpandian and S. Manoharan Deparmen of Elecronics and Insrumenaion Engineering, Karpagam College of Engineering, Myleripalayam Village, Ohakkal Mandapam Pos, Coimbaore , Tamil Nadu, India; kesavanehiraj2013@gmail.com, agalya.ice@gmail.com, palfeb28@gmail.com, manoish07@gmail.com Absrac Background/Objecives: An adapive Proporional Inegral (PI) conroller for a conical ank liquid level process is simulaed in MATLAB plaform. Mehod/Saisical Analysis: The Scheduled PI and Inernal model conrol (IMC) based PI performance are compared in erms of seling ime rise ime, overshoo, Inegral of Squared Error (ISE), Inegral of Absolue Error (IAE), Inegral Time Absolue Error (ITAE) and disurbance rejecion. For IMC based PI uning, he exac model of he process obained by on line esimaion of process gain and ime consan. Findings: The IMC based uning procedure ensures good performance by Proporional and Inegral (PI) conroller. The Ziegler Nichols (ZN) uning rules and esimaion based uning are used for scheduled Proporional and Inegral (PI) and inernal model conrol based PI conroller. Applicaions/Improvemen: Derivaive conroller can be added o his PI conroller for disillaion process. Keywords: Disurbance Rejecion, IMC based PI, Inegral of Square Error, Scheduled PI 1. Inroducion A sysem performance descripion by mahemaical equaion is highly difficul in nonlinear sysems. Mos of he process in indusries are non linear whose mahemaical expression canno provide general soluions. The nonlinear processes are very less predicable han linear process. Tanks requiremen in he indusries are more where ransferring and soring he liquids and fluids plays lively role 1. The primary problem in indusries is aking a perfec conrol acion in order o have desired ank level. The conical ank is well known nonlinear sysem which ensures he sele free a he boom of he ank. The normal PI conrollers are o ake perfec conrol acion only for linear process 2-4. The adapive conrollers are perfec for nonlinear process o reach he se poin wihou overshoo 1,5. Gain scheduled PI conroller provides beer performance for nonlinear liquid process of level mainain in conical and spherical ank 6. Bu gain scheduling is he ime consuming process. Because of he approximaion is always behind he process. In scheduling nonlinear sysem consider as many linear sysems and design he PI conroller for differen region (linear region) 7. Someimes his scheduled PI conroller less effecive due o he approximaion for linearizaion. To overcome his problem on line uning of parameers of he conroller was done based on gain and ime consan of he process 8. So ha online esimaions of gain and ime consan is needed o une he conroller parameers. Esimaion of ime consan and gain can be done by he presen and pas values of he inpu and oupu of he process. Model based 9 inernal model conrol is one of he bes uning algorihm for uning he parameers of he PI conroller Comparison of esimaion based PI wih scheduled PI (Figure 5, Figure 7 and Figure 9) shows ha he esimaion based PI provides igh conrol acion and ensures less ime consuming during he uning of PI conroller (Secion 2.2 and 2.4). Here ZN uning rules and IMC based uning were used o une he scheduled and esimaion based PI conroller. * Auhor for correspondence

2 Performance Analysis and Comparison of Differen Tuning Sraegies of PI Conroller in Conical Tank 2. Process Descripion The process conains a conical ank whose heigh and radius are 10cm and 7cm and oule valve coefficien is 0.2. The connecions of Inle valve (conrol valve, Figure 1) is from pump o ank and oule valve is from ank o reservoir. dv 1 2 = prh* ( fin - fou ) d Parameers for Modeling (4) The developmen of model for he process is needed he parameers are abulaed (Table 1) as analyical values. Table 1. Process parameers Values Maximum heigh of he ank 10 cm Maximum radius of he ank 7 cm Oule valve coefficien 0.2 Maximum inle flow rae 6.3cm 3 /sec Figure 1. Experimenal seup of liquid level process. Pump can suck he liquid from reservoir and gives o ank via conrol valve. Conrol valve ge conrol signal a he rae of 4-20mA from conroller and regulaes he flow of liquid depending upon he level in he conical ank. Level in he ank is measured by using a differenial ransducer and is ransmied o he inerfacing uni in he form of curren a he rae of 4-20mA. Conroller compares he se poin wih ransmied ou and produce conrol a signal based on he error. 2.1 Modeling for Conical Tank The rae of change of volume is given by Mahemaical equaion for liquid level sysem is derived from mass balance equaion. Toal mass of accumulaion Toal mass of inpu Toal mass of oupu = - ime ime ime (1) The mass balance equaion saes ha change in volume equal o difference beween inle and oule flow rae of he conical ank 1. dv (2) = f in - f ou d dv A ( fin fou ) d = * - (3) 2.1 Mehod of Gain Scheduling The model of he sysem creaed by using MATLAB sofware o obain non linear response of he process. The process response is shown in Figure 2. The parameers of he process can be obained by process reacion curve mehod for differen operaing regions. From he process parameers, Conroller parameers are deermined by ZN uning rules for various regions o design he scheduled PI wih he help of process parameers 1. Conroller parameers of process are proporional gain and Inegral gain which are obained from he below polynomial equaions. K p = *SP *SP *SP *SP (5) K i = *SP *SP *SP *SP (6) Figure 2. Open loop response of conical ank. 2 Vol 9 (11) March Indian Journal of Science and Technology

3 E. Kesavan, A. Agalya, P. Palpandian and S. Manoharan 2.3 Linear Process Parameer Esimaion ys () K us () - (7) + 1 response racking of conroller are more convenien by using IMC based PI 15. y(s)τs+y(s)=ku(s) (8) dy() + y() = Ku() (9) y ( )-y ( -1) + y() = ku() y ( -1) ku() + y () = æ ö ç + 1 è ø The digial form of he equaion (11) yn ( -1) ku( n) + yn ( ) = æ ö ç + 1 è ø (10) (11) (12) Le us consider he pas daa samples n (n-1) in above discree model yn ( -2) ku( n- 1) + (13) yn ( - 1) = æ ö ç + 1 è ø By solving he equaions (12) and (13), he gain and ime consan are esimaed as shown in (14) and (15) un ( )-Tn ( )( (-1)-yn ( )) Kn ( ) = (14) yn ( ) un ( ) unyn ( ) (-1)-ynun ( ) (-1) Tn ( ) = un ( )( yn ( -1) -yn ( -2))-un ( -1)( yn ( -1-yn ( )) (15) Where K(n)-gain of he model T(n)-Normalized ime consan The Esimaed values of gain K and ime consan can be calculaed shorly as much as possible o design perfec conroller IMC based Tuning Algorihm The normal PI conroller is he mos ransparen and simple conroller. IMC describes inernal model conrol (Figure 3) which plays crucial role for uning he parameer of PI conroller in order o have a se poin racking and disurbance rejecion 14. The frequen uning and desired Figure 3. Block diagram for IMC based PI design. The Process G P can be conrolled by he conroller Q c. The process deviaes from he model due o disurbance occurring of he process. Conroller Q c will be equal o he Inverse of he model if process is exacly same as he model 16. Q c =Inverse of G P * if G P = G P * The models of he ransfer funcion represened by wo par, hey are non inverible and inverible par. Inverible and non inverible par are good suff and bad suff porions. ygp*(s)= GP*(+)(s) GP*(-)(s) (16) GP*(s)=1.K*/[τ*(s)+1] (17) Here non inverible par will be cancelled and ge Q c *(s)=inv[g P *(-)(s)]= [τ*(s)+1]/k* (18) Q c (s)=q c *(s).f(s) = [τ*(s)+1]/[k*.(lem(s)+1] (19) Equivalen feedback conroller G c (s)=q c (s)/(1-q c (s)g P *(s)) =(τs+1)/(k p.lem.s (20) G c (s)=[kc.(τi.s+1)]/(τi.s) (21) The conroller parameers are Kc= τ/ K lem and τ i = τ 15 The equaion for PI is é 1 ù (22) KC e () + e () êë i ò úû 3. Resul and Discussion A Conical ank whose process parameers are varying Vol 9 (11) March Indian Journal of Science and Technology 3

4 Performance Analysis and Comparison of Differen Tuning Sraegies of PI Conroller in Conical Tank wih respec o ime is conrolled by gain scheduled PI and IMC based PI conroller. The Gain scheduled PI provides less efficien conrol acion han IMC based PI conroller. Here he conrol performance is obained for various desired response and compared boh conrollers for he se poin of 4cm, 6cm and 8cm. Performance of IMC based PI and parameer scheduled PI is compared for conical ank. The servo and regulaory response were simulaed using MATALB. Figure 7. Oupu response using Scheduled PI and IMC based PI for he se poin of 6 cm. Figure 4. Conroller oupu of he Scheduled PI and IMC based PI for he se poin of 4 cm. Figure 8. Conroller oupu of he Scheduled PI and IMC based PI for he se poin of 8 cm. Figure 5. Oupu response using Scheduled PI and IMC based PI for he se poin of 4 cm. Figure 9. Oupu response using Scheduled PI and IMC based PI for he se poin of 8 cm. 4 Figure 6. Conroller oupu of he Scheduled PI and IMC based PI for he se poin of 6 cm. Vol 9 (11) March Table 2. Comparison Table of Scheduled PI and IMC PI for he Se poin 4cm Conroller Se poin Seling ime Rise ime Scheduled PI 4cm 78s 17s IMC PI 4cm 20s 10s Conroller overshoo ISE IAE ITAE Scheduled PI 0.25cm IMC PI Nil Indian Journal of Science and Technology

5 E. Kesavan, A. Agalya, P. Palpandian and S. Manoharan The IMC based PI provides no overshoo and small oscillaion in oupu han Parameer scheduled PI conroller. The Scheduled PI conroller is designed for differen nominal operaing poin (from op o boom of he ank). The scheduled PI provides oscillaion in he range beween he nominal values (Figure 5, 7 and 9). This is mainly due o he cross secional variaions in enire region of he ank. Bu he IMC based uning overcomes his drawback effecively, which is no able o aain by scheduled PI conroller ((Figure 5, Figure 7 and Figure 9). Figure 4, 6 and 8 show he conroller oupu for boh conrolling echniques. The scheduled PI conroller has large seling ime, Rise ime, Overshoo, ISE (Inegral of square error), IAE (Inegral of absolue error) and ITAE (Inegral of ime absolue error) [Table 2]. (23) 2 ISE = ò e( ) d 0 2 ITAE = ò e( ) d 0 IAE = ò 0 e( ) d Figure 10. Posiive Disurbance rejecion performance using Scheduled PI and IMC based PI. Table 3. Process parameers (24) (25) Conroller Posiive disurbance Occurring Time Posiive disurbance Rejecion Time Scheduled 100 h s 175 h s PI IMC PI 100 h s 106 h s Conroller Negaive disurbance Occurring Time Negaive disurbance Rejecion Time Scheduled 200 h s 275 h s PI IMC PI 200 h s 206 h s Figure 11. Negaive Disurbance rejecion performance using Scheduled PI and IMC based PI. The Scheduled PI is less effecive conroller han IMC based PI conroller for rejecion of disurbance in he conical ank. The posiive and negaive disurbances are added o he process afer he seling ime a he 100 h second and 200 h second [Table 3]. The scheduled conroller akes more ime o reain he se poin of 4cm. Bu IMC based PI reain he Se poin as soon as possible by reducing he seling ime. 4. Conclusion In his work, he effecualiy of he IMC based PI and ineffecual of scheduled PI for nonlinear sysem are illusraed. The regulaory and servo performance for boh uning mehodology are simulaed using MATLAB. The online esimaion of gain and ime consan based uning of PI conroller provides beer conrol acion and disurbance rejecion for nonlinear process han Scheduled PI conroller. The ISE, IAE and ITAE are calculaed in order o analysis he regulaory response of he conroller (seling ime, rise ime, overshoo and oscillaion). The small value of ISE and IAE shows ha, here is a small overshoo and seling ime using IMC based uning han oher. The servo mechanism is analysed by adding he disurbance o he process and how quickly oupu is reained o he se poin of he process. The proposed conrol algorihm does no require any difficul mahemaical calculaion. This IMC based PI conroller uning ensures low cos of compuaion and opimal conrol acion even he disurbance is added o he process. 5. References 1. Bhuvaneshwari NS, Uma G, Rangaswamy TR. Adapive Vol 9 (11) March Indian Journal of Science and Technology 5

6 Performance Analysis and Comparison of Differen Tuning Sraegies of PI Conroller in Conical Tank and opimal conrol of an nonlinear process using inelligen conrollers. Applied Sof Compuing. 2009; 9(1): Anandanaarajan R, Chidambaram M, Jayasingh T. Limiaion of a PI conroller for a firs order nonlinear process wih dead ime. ISA Transacions. 2006; 45(2): Babu SRR, Deepa S, Johivel S. A closed loop conrol of quadraic boos converer using PID conrolller. IJE Transacion B: Applicaions Nov; 27(11): Asrom KJ, Hagglund T. The fuure of PID conroller. Conrol Engineering Pracice. 2001; 9(11): Tan KK, Haung S, Ferdous S. Robus self-uning PID conroller for nonlinear sysems. Journal of Process Conrol. 2002; 12(7): Kuamr DD, Meenakshipriya M. Design and implemenaion of nonlinear sysem using gain scheduled PI conroller. Procedia Engineering. 2002; 38: Hizal NA. Gain scheduling adapive model conrol. Journal of Engineering and Environmenal Science. 1999; 23: Schei TS. Auomaic uning of PID conrollers based on ransfer funcion esimaion. Auomaica. 1994; 30(12): Morari M, Zafiriou E. Prenice-Hall: Englewood Cliffs, NJ: Robus Process Conrol Volanova R. IMC based robus PID design uning guidelines and auomaic uning. Journal of Process Conrol. 2008; 18(1): Tan W, Liu J, Chen T, Marquez HJ. Comparison some well known PID uning formulas. Compuer and Chemical Engineering. 2006; 30(9): Arpuha Vijayaselvi J, Rahakirushnan TK, Sundaram S. Performance assessmen of PID and IMC uning mehods for a mixing process wih ime delay. ISA Transacion. 2007; 47(3): Kesavan E, Rakesh Kumar S. PLC based adapive PID conrol of NON linear liquid ank sysem using online esimaion of linear parameers by difference equaions. Inernaional Journal of Engineering and Technology. 2013; 5(2): Arbogas JE, Cooper DJ. Exension of IMC uning correlaion for non-self regulaing (inegraing) process. ISA Transacions. 2007; 46(3): Li D, Zeng F, Jin Q, Pan L. Applicaion of an IMC based PID conroller uning sraegy in amospheric and vacuum disillaion uni. Non Linear Analysis Real World Applicaion. 2009; 10(5): Porwal A, Vyas V. Rourkela: Naional Insiue of Technology: Inernal model conrol and IMC based PID conroller NOMENCLATURE e() Error signal Kc Conroller gain y(s) Process oupu dv/d Change in volume u(s) Process inpu dh/d Change in heigh K Process gain A Area of he ank * Model parameers SP Sepoin f in Inle flow, cm 3 /s (lph) u(n) Presen inpu f ou Oule flow, cm 3 /s (lph) y(n) Prese oupu G p (s) Process T(n) Normalized ime consan G p *(s) Model of he process Greek Symbols G d (s) Disurbance τ Time consan H Maximum heigh, cm Abbreviaions h Heigh a liquid level of he conical ank, cm P Proporional K v Oule valve coefficien I Inegral K p Proporional gain IMC Inernal model conrol K i Inegral gain ZN Ziegler Nichols K(n) Gain of he model ISE Inegral squared error Conroller of he process IAE Inegral absolue error G c (s) R Maximum radius, cm ITAE Inegral of ime absolue error r Radius a liquid level of he conical ank, cm DPT Differenial pressure ransmier 6 Vol 9 (11) March Indian Journal of Science and Technology