Design of Robust PI Controller for Counter-Current Tubular Heat Exchangers

Transcription

1 Deign of Robut PI Controller for Counter-Current Tubular Heat Excanger Jana Závacká Monika Bakošová Intitute of Information Engineering Automation Matematic Faculty of Cemical Food Tecnology STU in Bratilava Radlinkéo Bratilava Slovaa jana.zavacka@tuba.k Abtract: Te paper preent an approac for robut PI controller deign for a ytem affected by parametric uncertainty. Te metod i baed on plotting te tability boundary locu in te plane of controller parameter tat i called (k p k i )-plane. Deigned robut PI controller i implemented for control of two counter-current tubular eat excanger in erie wit uncertain parameter in wic keroene a a product of ditillation in a refinery a to be cooled by water. Te controlled variable i te temperature of te outlet tream of te keroene from te econd eat excanger te control input i te volumetric flow rate of te inlet tream of te cold water in te econd eat excanger. Simulation reult of robut PI control of eat excanger are alo preented. Keyword: interval uncertainty robut control robut PI controller tubular eat excanger Introduction Heat excanger belong to te tard equipment in te cemical food indutrie. In te food indutry variou model ave been uggeted for te prediction of fouling tickne milk outlet temperature in a eat excanger (Nema Datta 006). Control of te outlet temperature of a counter-current tubular eat excanger by te predictive functional control i own in Abaoui et al A boundary geometric control trategy i propoed to control te internal fluid temperature at te outlet of a counter-current eat excanger by manipulating te inlet external fluid temperature (Maidi et al. 009). Ti paper preent a metod for deign of robut PI controller for two counter-current tubular eat excanger in erie. Te metod i baed on plotting te tability boundary locu in te (k p k i )-plane. Te parameter of a tabilizing PI controller are determined from te tability region (Söylemez et al. 003). Te robut PI controller tabilize a controlled ytem wit interval parametric uncertaintie wen te tability region i found for ufficient number of Karitonov plant (Barmi 994). Two counter-current tubular eat excanger in erie are conidered a a controlled ytem wit two uncertain parameter in wic keroene a a product of te ditillation in a refinery a to be cooled. Keroene flow in te inner tube te cooling water i in te ell of te eat excanger. Deigned robut PI controller wa implemented for control of counter-current tubular eat excanger ti paper preent imulation reult. Teoretical Robut PI Controller Deign Conider te ingle-input ingle-output (SISO) control ytem own in Fig. Here w i te et point e i te control error u i te control input y i te controlled output. Te controlled proce i decribed by te tranfer function Nb ^ Gba ^ Da ^ () were b a are vector of uncertain parameter N D are polynomial in wit coefficient wic depend on b a. C() i te tranfer function of a PI controller in te form Fig.. Control ytem. Acta Cimica Slovaca Vol. 6 No. 03 pp DOI: 0.478/ac

2 C () k p kp () Te tak i to find te parameter k p k i of te PI controller () tat tabilize te ytem in Figure. PI Controller Deign Decompoing te numerator te denominator polynomial of () into teir even ( e ) odd ( o ) part ubtituting j were i te frequency (Tan Kaya 003 Závacká et al. 008) give Ne^- ~ j~ No^- ~ Gj ( ~ ) De^- ~ j~ Do^- ~ (3) Te cloed-loop caracteritic equation i in te form D ( j~ ) 6kN i e^-~ -kp~ No^-~ -~ Do ^- jk 6 p~ Ne^- ~ ~ No^- ~ ~ De ^- 0 (4) Equating te real imaginary part of Δ(j ) to zero give k p ( N o ( )) k i (N e ( )) D o ( ) (5) After defining k p (N e ( )) k i (N o ( )) D e ( ) (6) (5) (6) can be written a F( ) N o ( ) G( ) N e ( ) H( ) D o ( ) (7) I( ) N e ( ) J( ) N o ( ) K( ) D e ( ) k p F( ) k i G( ) H( ) k p I( ) k i J( ) K( ) (8) From (8) parameter of te PI controller () are k p H^~ J^~ - K^~ G^~ F^~ J^~ - G^~ I^~ K^~ F^~ - H^~ I^~ F^~ J^~ - G^~ I^~ (9) (0) Solving tee two equation imultaneouly for 0 te et of parameter k p k i i obtained. Ten it i poible to plot te dependence of k i on k p wic i te tability boundary l(k p k i ω) in te (k p k i )-plane. Te tability boundary divide te parameter plane into table untable region. Te table region i found by te coice of teting point inide te region. Te metod i very fat effective owever frequency rating become important. An efficient approac to avoid frequency rating can be obtained by uing te Nyquit plot. It i only neceary to find real value of tat atify Im[G(j )] 0 () Stabilization of a plant wit interval parametric uncertainty Conider a feedback ytem (Figure ) wit te PI controller () te interval plant m m- Nb ^ b m bm - g b0 Gba ^ n n- Da ^ a n an - g a0 () were b i [b i b i ] I 0 m a j [a j a j ] j 0 n. Let te Karitonov polynomial aociated wit N( b) D( a) are (Barmi 994): N^ b b0 b b b3 f N^ b b0 b b b3 f N3^ b b0 b b b3 f N4^ b b0 b b b3 f D^ b a0 a a a3 f D^ b a0 a a a3 f D3^ b a0 a a a3 f D4^ b a0 a a a3 f (3) (4) By tang all combination of te N i ( b) D j ( a) for i j 3 4 te following family of ixteen Karitonov plant can be obtained Ni ^ b GK^ Gij ^ Dj ^ a (5) were i j 3 4; K 6. Te controller C() () robutly tabilize interval plant G( b a) if only if it tabilize eac of te ixteen Karitonov plant G K (). Define te et S ij (C()G ij ()) wic contain all value of te parameter of te controller C() wic tabilize G ij (). Te et S ij can be found uing te approac decribed in te previou ection. Ten te et of all te tabilizing value of parameter of a PI controller wic tabilize te interval plant () can be written a S(C()G K ()) S(C()G ()) S(C()G ()) S(C()G 44 ()) (6) Experimental Proce Decription Two counter-current tubular eat excanger in erie (Figure ) repreent te controlled proce 36 Závacká J. et al. Deign of Robut PI Controller for Counter-Current Tubular Heat Excanger

3 Fig.. Two counter-current tubular eat excanger (HE) in erie. in wic keroene a a product of te ditillation in a refinery a to be cooled. Keroene flow in te inner tube te cooling water in te ell of te eat excanger. Te tube of te eat excanger are made from teel. Te controlled variable i te temperature of te outlet tream of te keroene from te nd eat excanger te control input i te volumetric flow rate of te inlet tream of te cold water in te nd eat excanger. Two uncertain parameter are alo conidered in te eat excanger. Te eat-tranfer coefficient cange a te flow rate of te cooling medium cange te denity of te keroene depend on te temperature in te eat excanger. Tecnological parameter teady tate value of te eat excanger are ummarized in te Table were n i te number of eat excanger tube l i te lengt of te eat excanger d in i te inner diameter of te tube d out i te outer diameter of te tube d in i te inner diameter of te eat excanger A i te total eat tranfer area V i te volume c p i te termal capacity i te denity T in i te input temperature q i te volumetric flow rate. Te ubcript refer to te water keroene repectively. Te upercript ()-() denote te aociated eat excanger property. Te upercript denote te teady tate value. Tab.. Tecnological parameter teady tate value of eat excanger. Parameter Value Parameter Value n 40 q [m 3 min ] 0.89 l [m] 6 T () [ C] 78.0 d in [m] T () [ C] 43.4 d out [m] T () [ C].3 d in [m] T () [ C] 66.6 A [m ] 6.6 V [m 3 ] V [m 3 ] c p [J kg K ] c p [J kg K ] [kg m 3 ] T in [ C] 0.0 T in [ C] 80.0 q [m 3 min ] Te implified dynamic model of te firt eat excanger i decribed by two linear differential equation water: keroene: were ^ qtcp T AU Dj dt qtcp T Vtcp T ^0 T qtcptin- AUDj dt qtcpt Vtcp T ^0 T ^Tln - T ^ ^T - T ^ D j (7) (8) (9) Te implified dynamic model of te econd eat excanger i decribed imilarly water: keroene: were qtcptin AU Dj ^ ^ dt qtcp T Vtcp ^ ^ T ^0 T qtcpt- AU Dj ^ ^ dt qtcpt Vtcp ^ ^ T ^0 T ^ T ^ t T ^ t T ^ ^ - ^ ^ - Tin D j (0) () () Minimal maximal nominal value of uncertain parameter are in Table were U i te eat tranfer coefficient i te denity of keroene. Te nominal value of tee parameter are te mean value of te interval. Control of te Counter-Current Tubular Heat Excanger Te controlled variable i te temperature of te outlet tream of te keroene from te nd eat excanger te control input i te volumetric flow Závacká J. et al. Deign of Robut PI Controller for Counter-Current Tubular Heat Excanger 37

4 Tab.. Uncertain parameter in eat excanger. Parameter Minimal value Nominal value Maximal value U [J min m K ] [kg m 3 ] rate of te inlet tream of te cold water in te nd eat excanger. For controller deign te matematical model of te counter-current tubular eat excanger in erie wit two uncertain parameter (Table ) wa obtained in te form of a tranfer function b b b0 Gba ^ 4 3 (3) a 4 a 3 a a a0 are able to aure te robut tability of te feedback cloed loop wit te controlled ytem (3) are found by te metod wic i baed on plotting te tability boundary in te (k p k i )-plane. Te region i found a te interection of te tability region obtained for 6 Karitonov plant own in Figure 3. were coefficient in te numerator te denominator lie in following interval: b i [b i b i ] i 0 a j [a j a j ] j Te value of te uncertain parameter are own in Table 3. Tab. 3. Uncertain parameter of tranfer function (3). Parameter Minimal value Maximal value b b b a 4 a a a a Ten combining of polynomial (3) (4) ixteen Karitonov plant (5) are created te approac decribed in te teoretical ection i ued for te robut PI controller deign. Uing (5) (6) for (3) give a 3 4 (a b k i b k p ) b 0 k i 0 (4) a 4 4 (a b k p ) a 0 b 0 k p b k i 0 (4) From (4) (5) PI controller parameter k p k i are calculated in dependence on te frequency according to (6) (7) 6 ^ab 3 - ab 4 ~ ^ab -ab -ab 3 0~ b 4 b ~ ^ - bb 0 ~ b0 ^ab 0- ab 0 ~ b 4 b ~ ^ - bb 0 ~ b0 k p 4 a3 ~ - ^a bk i~ bk 0 i b ~ 4 (6) (7) Te region of all PI controller parameter wic Fig. 3. Stability region for (0;.740). Te robut controller parameter k p k i wic tabilize all ixteen Karitonov ytem are coen from te table region te deigned PI controller i in te form 0. C ^ kp (8) Figure 4a preent imulation reult obtained during control of te model of te eat excanger wit te deigned controller (8). Te etpoint cange diturbance are in Table 4. Te diturbance were repreented by te tep cange of te temperature of te inlet tream of water keroene T in T in te volumetric flow rate of te keroene q. Figure 4b ow te control input. Concluion Te deign of te robut PI controller for control of te counter-current tubular eat excanger in erie wit two uncertain parameter i decribed in te paper. Te approac for deign of robut PI controller i baed on finding te tability boundary in te plane of controller parameter. 38 Závacká J. et al. Deign of Robut PI Controller for Counter-Current Tubular Heat Excanger

5 Fig. 4. a) Control repone of te temperature T () b) control input q generated by te robut PI controller (8). Tab. 4. Simulation condition. Setpoint cange Diturbance t [min] () T [ C] t [min] T in [ C] t [min] T in [ C] t [min] q [m 3 min ] Preented imulation reult confirmed tat te deigned robut PI controller i able to aure te etpoint tracng te diturbance rejection. Acknowledgement Te autor gratefully acknowledge te contribution of te Scientific Grant Agency of te Slovak Republic under te grant /0973/. Reference Barmi B (994) New Tool for Robutne of Linear Sytem. Macmillan Publiing Company New York. Abaoui MA Vernière-Haimi L Seguin D Abdelgani- Idrii MA (007) Applied Termal Engineering 7: Maidi A Diaf M Courriou J-P (009) Journal of Proce Control 9: Nema PK Datta AK (006) International Journal of Heat Ma Tranfer 49: Söylemez M Munro N Ba H (003) Automatica 39: 6. Tan N Kaya I (003) In Meditteranean Conference on Control Automation Rode Greece. Závacká J Bakošová M Vaneková K (008) In 8 t International Congre of Cemical Proce Engineering Prag Czec Republic. Závacká J. et al. Deign of Robut PI Controller for Counter-Current Tubular Heat Excanger 39