Linear Matrix Inequalities Based on Linear Quadratic Regulator for the H2/Hinf Control of the Inverter Split Air Conditioner Temperature

Transcription

1 Purdue Univerity Purdue e-pub International Refrigeration and Air Conditioning Conferene Shool of Mehanial Engineering Linear Matrix Inequalitie Baed on Linear Quadrati Regulator for the H/Hinf Control of the Inverter Split Air Conditioner eperature Yaubin Yang Indutrial ehnology Reearh Intitute Min-Der Wu Indutrial ehnology Reearh Intitute Yu-Choung Chang Indutrial ehnology Reearh Intitute Follow thi and additional work at: Yang, Yaubin; Wu, Min-Der; and Chang, Yu-Choung, "Linear Matrix Inequalitie Baed on Linear Quadrati Regulator for the H/ Hinf Control of the Inverter Split Air Conditioner eperature" (). International Refrigeration and Air Conditioning Conferene. Paper hi douent ha been ade available through Purdue e-pub, a ervie of the Purdue Univerity Librarie. Pleae ontat epub@purdue.edu for additional inforation. Coplete proeeding ay be aquired in print and on CD-ROM diretly fro the Ray W. Herrik Laboratorie at Herrik/Event/orderlit.htl

2 84, Page Linear Matrix Inequalitie Baed on Linear Quadrati Regulator for the H/Hinf Control of the Inverter Split Air Conditioner eperature YauBin YANG *, Min-Der WU, Yu-Choung CHANG 3 Indutrial ehnology Reearh Intitute, Energy & Environent Reearh Lab., Chutung, Hhihu, aiwan ( , , yaubinyang@itri.org.tw) Indutrial ehnology Reearh Intitute, Energy & Environent Reearh Lab., Chutung, Hhihu, aiwan ( , , MinDerWu@itri.org.tw) 3 Indutrial ehnology Reearh Intitute, Energy & Environent Reearh Lab., Chutung, Hhihu, aiwan ( , , yuhoung@itri.org.tw) * Correponding Author ABSRAC In the ulti-evaporator air onditioner yte, the opreor peed and the expanion value opening degree affet oupling the evaporator wall teperature in two-phae region and the uperheat on the evaporator tube. hi reearh ued yte identifiation to get the inner & outer loop tranfer funtion between the air-onditioner and it environent. he anti-windup ethod avoided the aturation phenoenon generated fro integral ontroller. he linear atrix inequalitie (LMI) baed on linear quadrati regulator (LQR) ixed H and Hinf ontrol algorith oputed the optial ontrol gain fro the input and tate perforane indexe. In the target teperature ahieveent exaple for the inverter plit air onditioner, thi propoed ethod ontrolled the air onditioner with two evaporator teperature effiiently through thoe two optial feedbak gain. In addition, the reult how a novel path for energy aving by the propoed LMI ethod.. INRODUCION For the inverter plit air onditioner, the indoor teperature wa ontrolled by the evaporator teperature and the uperheat. And the evaporator teperature and the uperheat were ontrolled by the opreor peed and the everal eletroni expanion valve opening degree. he ontrol for ulti-input ulti-output yte (MIMO) wa well-known uh ore diffiult than that for ingle input ingle output yte (SISO) beaue of the oupling effet epeially for plit air onditioner unit. Lin and Yeh (7) ued deoupled ontrol hee for teperature ontrol of ulti-evaporator air onditioning yte baed on yte odeling and identifiation. Yakubovih (96) provided the firt olution for atrix inequalitie. he ultiple-input ultiple-output linear atrix inequality algorith got ore attration in reent year (Palhare et al., 996; De Oliveira et al, ). hi propoed tudy baed on LMI algorith ahieved the target teperature fro evaporator teperature and uperheat adjutent by ontrolling opreor peed and eletroni expanion valve opening degree. hi tudy invetigated the teperature ontrol for air onditioner baed on linear atrix inequalitie algorith. he paper i organized a follow. Setion preent brief review for inverter air-onditioning yte. Setion 3 preent the tranfer funtion fro yte indenifiation. Setion 4 preent the anti-windup and etion how LMI forulation. Setion 6 preent everal nuerial reult for the propoed approahe. International Refrigeration and Air Conditioning Conferene at Purdue, July -,

3 . BRIEF REVIEW FOR INVERER AIR-CONDIIONING SYSEM 84, Page he air-onditioner ontain four ajor part: outdoor unit ontain the opreor, ondener, eletroni expanion valve and outdoor fan; indoor unit ontain the evaporator and the indoor fan. It operation yle trutural funtion are a follow: (a) the opreor: the opreor with the peranent agnet bruhle otor opree the lowpreure low-teperature gaeou refrigerant into high-preure high-teperature gaeou refrigerant. he refrigerant flow an be ontrolled by the opreor peed, and it peed i et to ~ 9 Hz; (b) the ondener: the outdoor fan reove heat fro the ondener to the outdoor atophere, o the high-preure high-teperature gaeou refrigerant beoe the high-preure iddle-teperature liquid refrigerant; () the eletroni expanion valve: the eletroni expanion valve onvert the high-preure iddle-teperature liquid refrigerant into the lowpreure low-teperature liquid refrigerant. he level of lower teperature and preure an be ontrolled by the eletroni expanion valve opening, and the opening degree i et to ~ Pule; (d) the evaporator: the lowpreure low-teperature liquid refrigerant through the inhaled higher indoor teperature air evaporate into lowpreure low-iddle-teperature gaeou refrigerant. In the eantie, thi inhaled relatively higher indoor teperature air ool down and diharge bak into the roo, o that the indoor abient teperature an go down into the required teperature. he low-preure low-iddle-teperature gaeou refrigerant at the evaporator export re-enter the opreor. One whole air-onditioner yle i opleted by thoe proedure. he entrane and export of the evaporator are in the two-phae oexitene of low-teperature liquid and lowiddle-teperature gaeou refrigerant. he refrigerant fro the evaporator export to the opreor entrane an be reheated by the abient teperature, and thi reheat proedure ake low-iddle-teperature gaeou refrigerant overheat. he low-preure low-teperature liquid refrigerant through the evaporator in iotheral tate beoe lowpreure low-teperature gaeou refrigerant working region, a liquid-ga two-phae oexitene aturation zone. If the refrigerant i heating up ontinually, then the refrigerant working region will leave the aturation zone and enter the overheated gaeou zone. he inreaed teperature whih the evaporated low-preure low-teperature refrigerant leave the aturated zone and keep heating up on the working region before entering the opreor i defined a the uperheat. h r _ etk e ok v K K f f / ˆ ˆ hk v ˆ r h _ etk e ik v u K K / u k Wall hk G ( ) H ( ) u rk. : Outer Loop... : Inner Loop Figure : Inverter air-onditioner yte ontrol flow hart he reation rate of the inner loop and the outer loop wa different, o thi tudy diued two feedbak loop. he inner loop defined the relationhip between the opreor and the ultiple evaporator, and the outer loop defined the relationhip between the ultiple evaporator and the environent. he notation ubript k ean fro to k and the nuber of ultiple evaporator i k. For exaple, the evaporator teperature ean and e e two evaporator in thi paper. International Refrigeration and Air Conditioning Conferene at Purdue, July -,

4 84, Page 3 he indoor teperature rk were generated by the evaporator teperature and the uperheat hk. he outer loop feedbak error e ok between the indoor etting teperature and the indoor teperature r _ etk rk were ultiplied by the ontroller feedbak gain K to get the evaporator teperature ˆ and the uperheat ˆhk. he high uperheat f ake the opreor daage eaily, o the uperheat value ˆhk ut be greater than uperheat etting value. he outer feedbak gain K need the anti-windup to avoid integrator aturation. h_ etk f he evaporator teperature and the uperheat hk were generated by new opreor peed and new eletroni expanion valve. he inner loop feedbak error e ik between ( ˆ, ˆ ) fro outer loop and ( h_ etk, ) hk fro inner loop were ultiplied by the ontroller feedbak gain K to get opreor peed and eletroni expanion valve opening. he inner feedbak gain K need the anti-windup to avoid integrator aturation. he high opreor peed ake the whole yte daage eaily, o the opreor peed ut be le than new opreor peed. hoe proedure baed on the two feedbak ontrol gain ake the plit type air onditioner ahieve the target teperature. Indoor teperature hange were uh lower than the outdoor unit. he propoed ethod handled the eparately and ade indoor teperature ontrol effiieny. In the ultiple-input ultiple-output inverter air-onditioning yte, the ooling fan peed of indoor unit and outdoor unit wa et to be ontant; the opreor peed and eletroni expanion valve opening were et to the ontrollable input; the evaporator teperature and the uperheat hk were et to the output. he yte ontrol flow hart wa hown in figure. 3. Syte Identifiation 3. RANSFER FUNCION FROM SYSEM IDENIFICAION he evaporator i the exhange plae between air-onditioner teperature and indoor teperature.. D. He et al (99) etablihed equivalent vapor opreion yle dynai paraeter, and diued evaporator odel through the two-phae region and the uperheat zone. he outer and inner loop are the ultiple-input ultipleoutput paraeter yte, and the yte identifiation tranfer funtion are hown in Figure. In Figure (a), the opreor peed and the eletroni expanion valve opening degree are input, and the evaporator teperature and the uperheat hk whih affet heat aborptive apaity in the evaporator are output. In Figure (b), the evaporator teperature and the uperheat hk are input, and the indoor teperature rk are output. Sine the yte identifiation i for MIMO yte, the pae tate for yte an be expreed a e () ( k ) ( k ) k y G G h v k k G () hk z H Hy ; r ( ) k ( k ) (k ) H () (a ) (b ) Figure : MIMO paraeter blok diagra th he tranfer funtion etiation G of data by the leat quare ethod how, Gˆ ( ) y () he ( ) th data an be expreed a y G, o the tranfer funtion etiation G i hk rk h k () International Refrigeration and Air Conditioning Conferene at Purdue, July -,

5 84, Page 4 ˆ G ( ) y ( x x) y (3) After alulation and rearrangeent, the reurive leat quare equation i Gˆ Gˆ ( ) x I x ( ) x y x Gˆ (4) he tranfer funtion etiation H ha iilar equation 4 for. (a) (b) () (d) 8 v h e v 7 h e Pule Speed (Hz) 6 eperature C eperature C ie (Minute) ie (Minute) 3 4 ie (Minute) 3 4 ie (Minute) Figure 3: yte identifiation for inner loop (a) (b) 3 () 3 (d) e h r r eperature C e eperature C h eperature C eperature C ie (Minute) ie (Minute) 4 ie (Minute) 4 ie (Minute) Figure 4: yte identifiation for outer loop For the inner loop, the opreor peed, the firt eletroni expanion valve opening and the eond eletroni expanion valve opening were fixed at Hz, 4 Pule and Pule repetively. Firt, the firt eletroni expanion valve opening degree hanged fro to 8 Pule. Seond, the eond eletroni expanion valve opening degree hanged fro 8 to 4 Pule. Finally, the opreor peed hanged fro 4 to 6 Hz. Eah period took 4. inute, and there were. inute between two period. he other two paraeter reained ontant while one paraeter varied with tie. For the outer loop yte identifiation, the firt evaporator reain. C and uperheat hanged fro to C. he eond evaporator reained C and uperheat hanged fro 3 to C. Eah period took inute. he area of firt roo wa around 48 and the area of eond roo wa around 6. he height of the roo wa around.7. Figure howed the MIMO paraeter blok diagra for outer and inner loop, figure 3 wa the yte identifiation for inner loop and figure 4 wa the yte identifiation for outer loop. he yte identifiation by reurive leat quare ethod wa the firt-order low-pa filter hown in equation G( ) H ( ) () ANI-WINDUP International Refrigeration and Air Conditioning Conferene at Purdue, July -,

6 84, Page he integrator redue the yte teady-tate error but the integrator aturation aue the windup phenoenon. In equation 6, the paraeter u ha three poible output orreponding to the ontroller input u. u for u u in in u for u u u u (6) in ax u for u u ax ax In figure, the differene between u and u ultiplie by oeffiient, then add the error ignal e i the input of integrator. hi anti-windup an uppre integrator aturation phenoenon aued by integrator. In general, the bigger value, the fater aturation onvergent rate. A variable-truture (withing) for proportional integral antiwindup ethod wa introdued (VSPI) (Hodel and Hall, ), i.e. u with u Kxr i in linear operation, and u ( u u ) with u Kxr i in aturation.. LMI forulation. Direte tie tate pae equation with anti-wind up for inverter plit air onditioner yte he yte direte tie dynai equation an be expreed a tate pae for, ( k ) A ( k) Bu ( k) B u ( k) Bw( k); z( k) E ( k) Eu ( k) Gw( k) (7) Where u i ontrol input, w i diturbane input, z i the referene output, u with u Kxr i in linear operation, and u ( u u ) with u Kxr i in aturation.. Hinf Forulation he ontrol i the loed-loop yte to iniize Hinf rank fro the diturbane input w to the referene output z. A. J. Connolly pointed out the deigned ontroller fro the entire ontrol with Hinf etiation an effetively redue the tranient repone tie and the teady-tate error (Connolly, 99). If the tranfer funtion of the yte ha bounded in equation 8, then z z z H up up (8) R R ( ) Where i the bounded and J w z z J i perforane index In the Lyapunov tability theory, the yte i aid to be ayptoti table if there exit a yetri atrix P P in equation 7. It i aued that funtion V( k) i x ( k) Px( k ), and feedbak openator uk ( ) i Kx( k). he direte Lyapunov equation an be expreed a follow, V w ( k) w( k) z ( k) z( k) Where V V( k ) V( k) (9) Fro Equation 9, in the linear operation, A PA P E E * w B PA ( I B PB ) w () r B PA B PB B PB r w ( I B PB ) B P( Br A) Where A A B K, B B and E E E K () In the aturation ituation, International Refrigeration and Air Conditioning Conferene at Purdue, July -,

7 84, Page 6 A PA P E E * * * B PA B PB * * u u w BPA BPB ( ribpb) * w r BPA BPB BPB BPB r w ( I B PB ) B P(( Bu B r) A) Where A A B K, B B B and E E (3) ().3 H Forulation Chooe perforane index J, o that J ( xqx uru) Where Q E E, and R E E (4) In linear operation, A PA P E E K E E K * r BPAEEK BPB EE r () In aturation ituation, A PA P E E * * u B PA B PB * u (6) r BPA BPB BPB r.4 Linear atrix inequality expreion he two iportant onept were hown here before deviating linear atrix inequalitie. Q S he firt i the Shur opleent: M S R R, QSR S (7) he eond i the invere atrix expreion: ( A BCD) A A B( C DA B) DA (8) For the onvenient oputation, et P Y, and Z KY : here exity Y uh that Hinf nor LMI in linear operation an be expreed a: Y * * * * * * * AY B Y * * (9) B I * EY I here exity Y here exitw W uh that Hinf nor LMI in aturation an be expreed a: Y * * * * * E E * * * * * * * AY B B Y * * B I * EY I and Y Y uh that H nor LMI in linear operation an be expreed a: () International Refrigeration and Air Conditioning Conferene at Purdue, July -,

8 84, Page 7 Y * * * W B E E * * he optiization proble in( W ) ubjet to, Y, WY, B Y AY B Y * EY EKY I here exitw W and Y Y uh that H nor LMI in aturation an be expreed a: Y * * * * E E * * * W B he optiization proble in( W ) ubjet to, Y, * * WY, B Y AY B B Y * EY I () (). NUMERICAL RESULS eperature C (a) r r r - et eperature C (b) h h - et h Speed (Hz) () Pule 4 3 (d) v v eperature C ie (Minute) (a) r r r - et ie (Minute) eperature C 4 6 ie (Minute) 4 ie (Minute) Figure repone oparion for the firt roo (b) h h h - et 4 6 ie (Minute) Speed(Hz) () 4 ie (Minute) Figure 6 repone oparion for the eond roo Pule 4 ie (Minute) 4 3 (d) 4 ie (Minute) v v An inverter plit air-onditioner operating ondition wa hown here. he outdoor teperature, firt indoor roo teperature, the eond indoor roo teperature, the opreor peed, firt eletroni expanion valve opening and the eond eletroni expanion valve opening were et to 3 C, 9 C, 8. C, ~ 9 Hz (6 rp ~ 7 rp), ~ Pule, and ~ Pule, repetively. he indoor target teperature wa deterined to at 6 C. he experiental ahine are plit type with R4A refrigerant, and the ooling apaitie are 3.kW and.kw. he opreor peed and eletroni expanion valve opening were the input; two-phae wall teperature and uperheat were the output. hi tudy wa baed on ultiple-input ultiple output (MIMO) yte, o the evaporator teperature and uperheat were ro-oupling ontrolled by opreor peed and eletroni expanion valve opening at the ae tie. hoe MIMO iulated reult baed on ixed H/Hinf LMI and traditional interation hee were hown in figure, 6. he indoor teperature reult in figure, 6(a) wa dereaed gradually before they reahed the teady tate. he traditional iteration hee wa for validation purpoe. heir teady tate error teperature for two indoor roo were aeptable, ay le than. C. he dah line i for propoed LMI ethod and the olid line i for traditional iteration ethod. In figure, 6(), the highet opreor peed for LMI wa 7.6% le than that for International Refrigeration and Air Conditioning Conferene at Purdue, July -,

9 84, Page 8 the traditional iteration ethod. he lower opreor peed the lower eletri urrent uage. In figure, 6(d), for the eletroni expanion valve opening, the LMI ethod wa alo le than the tradition iteration ethod. he le opening the little ore eletri urrent uage, but the effet of opening on urrent wa le than that of opreor peed on urrent aording to epirial invetigation. he propoed LMI ethod provided a path for energy aving. 6. CONCLUSIONS hi paper preented a new approah for inverter plit air onditioner. he inner and outer loop LMI eparation hee were firt addreed here. he ethod didn t need to opute only one tranfer funtion repreent whole yte. hi propoed ethod ould deal with the two loop eparately in the ae way, o the inner loop gain and outer loop gain were granted. he reult howed that ultivariable teperature ontrol yte by linear atrix inequitie wa ontrollable and onvergent. he ontrol logi entioned a follow: two indoor teperature were ontrolled by two evaporator teperature and two uperheat. hoe uperheat were ontrolled iediately by the opreor peed and two eletroni expanion valve opening degree. Coparing the indoor teperature hange with the rk r _ etk uperheat etting value ould ontrol the indoor teperature effetively. h_ etk In the beginning of ontrol, the roo teperature wa uh larger than it target value, and uperheat wa at high tate. he ontrol yte redued uperheat by the high opreor peed and high eletroni expanion valve opening degree. When the roo teperature wa loe to etting value, the uperheat wa at low tate. he opreor peed and eletroni expanion valve opening had alo dropped to low peed and low opening. he yte wa operating under the teady tate; the indoor teperature wa table under the ontrol etting. Coparing with the teady tate uperheat, the high uperheat affeted high peed opreor. he yte inreaed energy ue. he low uperheat affeted low peed opreor. he yte dereaed energy ue. Figure, 6(b) howed the LMI ethod wa better energy aving than traditional iteration ethod. Until the indoor teperature etting value reahed and the opreor peed and eletroni expanion valve opening alo beae teady tate. he LMI forulation guaranteed the oputation tie wa uh fater than the traditional iteration tie. In addition, the opreor peed for LMI hee wa le than that for iteration hee, o thi ethod wa alo a novel energy aving approah, hown in figure, 6(). REFERENCES Connolly, A. J., Green, M., Chiharo, J. F., and Bitead, R. R., 99, he deign of LQG and ontroller for ue inative vibration ontrol and narrow band diturbane rejetion, Pro. 34th Conf. on Deiion and Control, Vol. 3, pp De Oliveira, M. C., Geroel, J. C., and Bernuou, J.,, Extended and Nor Charaterization and Controller Paraeterization for Direte-ie Syte, Int. J. Control, vol. 7, no. 9, pp He,. D., Liu, S., and Aada, H. H., 99, Modeling of Copreion Cyle for Advaned Control in HVAC Syte, Pro. Aerian Control Conf., pp Hodel, A. S., and Hall, C. E.,, Variable-Struture PID ontrol to Prevent Integrator Windup, IEEE ranation on Indutrial Eletroni, vol. 48, no., pp Lin, J. L., and Yeh,. J., 7, Modeling, Identifiation and Control of Air-onditioning Syte, Int. J. of Refrigeration, Vol. 3, No., pp. 9-. Palhare, R. M., Rao, D. C. W. and Pere, L. D., 996, Alternate LMI Charaterization of and Central Direte- ie Controller, Pro. of the 3th Conf. on Deiion and Control, Koba, Japan, pp Yakubovih,V.A., 96, he olution to ertain atrix inequalitie in autoati ontrol, Dokl. Akad. Nauk USSR 43 (96), pp ACKNOWLEDGEMEN he author gratefully aknowledge the finanial upport provided by the Bureau of Energy, Minitry of Eonoi Affair, aiwan, R. O. C. International Refrigeration and Air Conditioning Conferene at Purdue, July -,