DSCC2012-MOVIC

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1 ASME 5th Annual Dynamc Systms and Control Confrnc jont wth th JSME th Moton and Vbraton Confrnc DSCC-MOVIC Octobr 7-9,, Fort Laudrdal, Florda, USA DSCC-MOVIC-886 LMI CONROL DESIGN FOR NONLINEAR VAPOR COMPRESSION CYCLE SYSEMS Bn L, Nra Jan, Andrw G. Allyn Dpt. of Mchancal Scnc and Engnrng Unvrsty of Illnos Urbana-Champagn, IL 68 ABSRAC o ffctvly control vapor comprsson cycl (VCC) systms whos dynamcs ar hghly nonlnar, t s ncssary to dvlop plant modls and control laws for dffrnt opratng rgons. hs papr prsnts a frst-prncpls modlng framwork that capturs four opraton mods ovr th opratng nvlop to construct an nvarant-ordr swtchd systm. o synthsz a mult-nput mult-output (MIMO) control systm, th Lnar Quadratc Rgulator (LQR) tchnqu s framd as a control optmzaton problm wth Lnar Matrx Inqualty (LMI) constrants whch can b smultanously solvd for th st of consdrd lnar systms. Stablty and prformanc charactrstcs of th controlld systm ar guarantd usng a common quadratc Lyapunov functon. Smulaton rsults n a cas study show that th LMI-basd controllr can mantan systm opraton at optmal st-ponts wth mod swtchng ovr a wd opratng nvlop. NOMENCLAURE Varabl h Spcfc Enthalpy ṁ Mass flow rat P Prssur Q Hat transfr rat mpratur W Powr ω Comprssor spd η Isntropc ffcncy γ Man vod fracton ζ Fracton of hat xchangr lngth covrd by zon Subscrpt c Condnsr Evaporator n Inlt out Outlt rf Rfrgrant w Hat xchangr wall c, c, c 3 Suprhatd, two-phas, subcoold zon n condnsr, wo-phas, suprhatd zon n vaporator INRODUCION h prmary goal of any ar condtonng or rfrgraton (AC/R) applcaton s to mov nrgy from on locaton to anothr. An dal vapor comprsson cycl (VCC) systm, as shown n Fg., s a thrmodynamc systm drvn by th phas charactrstcs of th rfrgrant that s flowng through t. VCC systms transfr hat by mans of rpatng phas transtons n th rfrgrant whch can lad to th changng of th flud charactrstcs n ach of th hat xchangrs (condnsr and vaporator). h mxtur of lqud and vapor rfrgrant dstrbutd n th hat xchangrs maks th phas transton procss hghly nonlnar and complx [-3]. hrfor, t s ncssary to dvlop control stratgs whch account for systm nonlnarts and mantan dsrabl prformanc ovr th ntr opratng nvlop. Advancd control of VCC systms n a varty of AC/R applcatons has bn wdly studd n th ltratur [4-8]. In partcular, mult-nput mult-output (MIMO) control dsgn s attractv du to th strong couplng n VCC systms. H t al. [4] proposd a Lnar Quadratc Gaussan (LQG) MIMO controllr wth gan schdulng to adapt to changng opratng condtons. Rasmussn and Allyn [5] prsntd a Youla paramtrzaton framwork for th gnraton of a local modl and local controllr ntwork (LMN/LCN) and dmonstratd th ffctvnss of MIMO gan schdulng control stratgs on prformanc rgulaton of th nonlnar systm. A LQG controllr basd on a frst-prncpls smulaton modl was dvlopd n [8], and th controllr prformanc was valuatd for opratng condtons 3% away from th nomnal opratng pont [9]. Copyrght by ASME

2 FIGURE. SCHEMAIC OF AN IDEAL VCC SYSEM. Although t has bn shown that VCC systms xhbt nonlnar bhavor across thr ntr opratng nvlop, a partcular hat xchangr formulaton wth a fxd numbr of flud zons s normally assumd for VCC systm control dsgn n th opn ltratur. It has also bn dntfd that hghly nonlnar dynamcs volv whn th systm loss ts condnsr subcoold zon or vaporator suprhatd zon [3]. hs typs of rfrgrant transtons can occur du to varyng systm boundary condtons. hs papr gos byond prvous fforts to dvlop a control framwork that s capabl of handlng th transtons nsd th flud zons. Motvatd by rcnt work on dynamc modlng of rfrgrant phas transtons [], ths papr prsnts a swtchd modl ntwork, thrby nablng modl-basd MIMO control dsgn for th nonlnar VCC systm. h VCC systm can b rprsntd by a st of four dffrnt opratng mods that span th normal opraton nvlop. h ky contrbuton of ths work s th formulaton of an nvarant-ordr frst-prncpls swtchd modl systm amnabl to swtchd systm controllr dsgn across th normal opratng nvlop of a typcal VCC systm. h rst of ths papr s organzd as follows. Scton ntroducs th swtchd modl ntwork n trms of th dffrnt mods of rfrgrant dstrbuton durng opraton. A frstprncpls dynamc modl for ach opraton mod s dscrbd n Scton 3 along wth th lnarzd modl valdaton. Psudo-stat dynamc quatons ar usd to mantan th sam ordr n ach mod of th swtchd modl ntwork. Scton 4 prsnts a Lnar Matrx Inqualty (LMI) rprsntaton of th Lnar Quadratc Rgulator (LQR) control dsgn. A modl-basd swtchng control schm wth st-pont optmzaton s prsntd n Scton 5. Smulaton rsults dmonstrat th advantags of th swtchd modl framwork on nonlnar VCC systm control across ts opratng nvlop. A concluson scton summarzs th man pont of ths papr. FIGURE. FOUR OPERAION MODES OF HE VCC SYSEM. SYSEM OPERAION For ths work, a movng-boundary modl formulaton [] s utlzd to dscrb all hat xchangr dynamcs. As dscussd n [, ], th rfrgrant phas transtons n th condnsr swtch back and forth btwn 3-zon (suprhatd, two-phas and Copyrght by ASME

3 subcoold) and -zon (suprhatd and two-phas) modl rprsntatons undr dffrnt opratng condtons. hr also xst two dffrnt rfrgrant dstrbuton confguratons n th vaporator modl, whch ar -zon (two-phas and suprhatd) and -zon (two-phas) modl rprsntatons. hrfor, th VCC systm conssts of four dffrnt opraton mods across ts opratng nvlop as dpctd n Fg.. Exognous sgnals, such as a dsrd rfrnc trajctory or systm dsturbancs, and/or ndognous schdulng varabls, such as masurd prssur or tmpratur sgnals [5] can drv th VCC systm to swtch btwn dffrnt opraton mods. h swtchd opraton mod framwork s summarzd n Fg. 3. opnng, condnsr ar flow rat, and vaporator ar flow rat (s Fg. ). h output varabls ar condnsr prssur, condnsr hat rjcton rat, vaporator prssur, and vaporator coolng capacty, dnotd as y = Pc Q c P Q. h condnsr and vaporator ar nlt tmpraturs ar rgardd as dsturbancs: w = ar, c ar,. h matrcs A, B, C and F ar obtand from th modl lnarzaton, and =,,3,4 dnots th opraton mod numbr. Not that on lnar modl s drvd for ach mod, and ths papr focuss on th swtchng opraton btwn dffrnt mods. Howvr, multpl lnarzd modls can also b dvlopd for a sngl opraton mod bcaus th dynamcs wthn ach mod ar thmslvs nonlnar. 4 δ x = A δ x + B δu + Fδ w δ y = C δ x () 3 h dynamcs of th lnarzd modl ar compard aganst thos of th nonlnar modl for ach opraton mod. On valdaton of th lnarzd modl for mod opraton (s Fg. ) s gvn hr. Fgur 4 shows th systm nput sgnals, and Fgur 5 compars th dynamc rspons of condnsr prssur, vaporator prssur, and vaporator ar outlt tmpratur, of th lnarzd and nonlnar modls. h dynamc rsponss match wll, thrby valdatng th lnarzaton. FIGURE 3. SCHEMAIC OF HE SWICHED OPERAION MODE NEWORK. Comprssor Spd (RPM) Valv Opnng (%) SYSEM MODELING AND LINEARIZAION As mntond arlr, th VCC systm dynamcs ar hghly nonlnar. In ordr to apply lnar control tchnqus on th VCC systm, a lnarzd modl of th systm dynamcs s ndd. In th cas of th swtchd modl ntwork, th nonlnar dynamc modl for ach ndvdual opraton mod s lnarzd. h ntrstd radr s rfrrd to [, ] for th nonlnar modl drvatons and to [] for th lnarzaton procdur. h dynamc stats of th swtchd modl ar dscrbd n Eq. (). Frst-ordr psudo-stat quatons [] ar addd n ach opraton mod to nsur th contnuty of stats and mantan a 3 th -ordr swtchd modl of th systm. x = [ ζ P h ζ P h γ γ ] () c c c3 cw cw cw3 w w c h stat-spac rprsntaton s gvn n Eq. (), whr th vctors δ x, δ u, δ w and δ y ar th dynamc dvatons from th normal opratng pont. h nput varabls, u = [ u u u u ] 3 4, rprsnt comprssor spd, valv Condnsr Ar Flow Rat (kg/s) m(s) Evaporator Ar Flow Rat (kg/s) FIGURE 4. SYSEM INPU SIGNALS FOR LINEARIZED MODEL VALIDAION; DASHED LINE CORRESPONDS O HE RIGH Y-AXIS. LMI-BASED CONROL DESIGN hs scton prsnts an LMI rprsntaton of LQR control dsgn usng th swtchd modl framwork as shown n Fg. 3. In ordr to guarant zro stady-stat trackng rror, th orgnal plant modl n Eq. () s augmntd wth ntgratd rror stats btwn th dsrd rfrnc and th actual plant output as shown n Eq. (3) whr y s s th output vctor valuatd 3 Copyrght by ASME

4 Condnsr Prssur (kpa) Evaporator Prssur (kpa) Evaporator Ar Outlt mp. ( C) Nonlnar Lnar m (s) FIGURE 5. LINEARIZED MODEL VALIDAION. at a stady stat opratng pont about whch th systm was lnarzd. as whr s ( ) dt ( δ ) z = y r = y + y r dt (3) h augmntd stat-spac modl rprsntaton s rwrttn s ( ) x = A x + B δu + F δ w + E y r (4) Usng LMI tchnqus [3], th LQR problm can b rformulatd [4,5] as a convx optmzaton problm ovr P and M subjct to and mn r( QP) + r( N ) (7) P, M, ˆ X A P + PA B M M B M P M R < P Q / N R M / > M R P P >. Hnc, th LMI-basd LQR dsgn problm can b summarzd as th mnmzaton of Eq. (7) undr th constrants gvn by Eqs. (8) and (9). h optmal LQR controllr s prsntd n Eq. (). h man advantag of ths formulaton s that th soluton provds a common Lyapunov matrx P - to nsur systm stablty durng mod swtchng opratons and achv dsrd prformanc of th closd-loop systm. h ntrstd radr s rfrrd to [4, 5] for a dtald drvaton of th LMI-basd LQR dsgn problm. δu = K x K = M P (8) (9) () x = δ x z A B A = = B C F F = = E I (5) Snc som stat varabls n Eq. () cannot b masurd drctly (.g., normalzd two-phas zon lngth n th condnsr and vaporator), a standard Kalman fltr [6] s dvlopd basd on th lnarzd plant modl n Eq. () for opraton mod, as gvn blow whr δ x ˆ s an stmat of th stat varabl δ x. () ( ) δ xˆ = A δ xˆ + B δu + Fδ w + L δ y C δ xˆ m h optmal LQR controllr s obtand by fndng th statfdback gan K that mnmzs th prformanc ndx h obsrvr gan matrx L s computd from J ( δ u R δ u ) = x Qx + dt. (6) whr S L = SC V () 3 3 R s th soluton of th Rccat quaton 4 Copyrght by ASME

5 S A + A S S C V C S + W =. (3) h nos covaranc matrcs W and V n Eq. (3) ar slctd as th dntty matrcs, I and I rspctvly, n ths papr. Wth th stat obsrvr, th MIMO fdback control dagram for ach opraton mod s prsntd n Fg. 6. δ ˆx whr and (,, ) m h h = η η = f ( P, P c, ω) (6) W = f ( m ) (7) rf n c out comprssor (5) W fan, c ar, c W fan, = f3( m ar, ). (8) FIGURE 6. BLOCK DIAGRAM OF MIMO FEEDBACK CONROL FOR OPERAION MODE. h functons f, f, and f 3 ar mprcal nonlnar functons that ar spcfc to th partcular VCC systm that s bng optmzd. h vctor of optmzaton varabls s CASE SUDY Buldng upon th control framwork dscussd abov, a modl-basd swtchng control cas study s prsntd hr to dmonstrat th ffctvnss of th proposd approach. h modl-basd control schm s shown n Fg. 7, whr th local control ntwork s composd of four ndvdual controllrs, and th swtchd modl ntwork s usd to rprsnt th dynamcs of th nonlnar VCC systm. hs scton s dvdd nto two parts. Frst, an optmzaton problm for dtrmnng optmal opraton st-ponts s dscrbd. Scond, for a partcular VCC systm, smulaton rsults dmonstrat th ffctvnss of th mod swtchng control on achvng th optmal st-ponts. FIGURE 7. MODEL-BASED SWICHING CONROL SCHEME. St-pont Optmzaton In ordr to dtrmn th optmal opraton st-ponts for th VCC systm for a gvn prformanc rqurmnt, a nonlnar optmzaton problm can b formulatd and solvd. On common ffcncy objctv s to mnmz th total powr consumpton n th VCC systm whl mtng th dsrd coolng capacty. In ths cas, th objctv functon s J = W = W + W + W (4) powr total comprssor fan, c fan, 4 3 whr v P P h m m m (9) = c n, c rf ar, c ar, v * ( J powr ) = arg mn. () v A numbr of dffrnt constrants must b mposd n ths optmzaton problm. h dsrd coolng capacty, C ds, s nforcd as a hard constrant on th systm as shown n Eq. () whr C ds s spcfd by th usr. (,, ) C = Q = m h h () ds rf out n Addtonal nqualty and qualty constrants ar rqurd for th optmzaton, as lstd n Eq. (). Not that rf,sat,c and rf,sat, ar th saturatd rfrgrant tmpraturs at P c and P, rspctvly. h frst and scond constrants n Eq. () nsur th corrct tmpratur gradnts at th outlt of th vaporator and condnsr, rspctvly. h thrd and fourth constrants ar modfd for ach ndvdual opraton mod (s Fg. ). For xampl, n opraton mod 3 or 4, th vaporator modl s a on-zon (two-phas) rprsntaon. hrfor, th vaporator suprhat, rf,out, rf,sat,, s xactly zro. In opraton mod or 3, th condnsr subcoold tmpratur, rf,sat,c rf,out,c, bcoms zro. hs may rsult n a dffrnt optmal soluton snc th dynamcs of th sysm ar dffrnt whn t s opratng n ths mod. As shown latr n ths papr, for a gvn coolng capacty, th optmzaton ylds a bst opraton mod wth rgards to th ffcncy mtrc J powr dfnd n Eq. (4). h ffth constrant nsurs that th frst law of thrmodynamcs s not volatd. 5 Copyrght by ASME

6 rf, out, ar, ar, c rf, out, c rf, sat, rf, out, rf, out, c rf, sat, c Q + W = Q total c M st M M sys sys 5% () h fnal constrant lstd n Eq. () nsurs that th total amount of rfrgrant mass s mantand n th systm, ndpndnt of th mod n whch th systm s opratng. M st s th amount of mass stmatd to b n th systm basd on a gvn st of valus for th optmzaton varabls, and M sys s th total mass spcfd by th usr. An rror of 5% s allowd du to th ffct of modlng uncrtants on th calculaton of M st. h dynamcs of th VCC systm ar prmarly drvn by th rfrgrant mass dstrbuton throughout th systm. Snc th VCC systm s closd, th total amount of mass rmans constant at all tms, and ths physcal constrant naturally lmts how th systm can oprat. h objctv functon s nonlnar; thrfor, th functon fmncon n th MALAB Optmzaton oolbox s usd to fnd a soluton to th optmzaton problm. Controllr Implmntaton h VCC systm paramtrs n [] ar appld for th swtchd modl ntwork formulaton and controllr dsgn. Va th st-pont optmzaton dscrbd abov (agan appld to th systm dscrbd n []), th bst opraton mod, wth rspct to mnmal powr consumpton, can b found for a partcular coolng capacty. A common ffcncy mtrc for VCC systms s th coffcnt of prformanc (COP) gvn by Q COP = W total (3) whr a hghr COP ndcats mor ffcnt opraton. Snc Q s constrand to b qual to th dsrd coolng capacty C ds (s Eq. ()), mnmzng W total s quvalnt to maxmzng COP. Fgur 8 shows a comparson of th rsults n trms of COP among opraton mods for dffrnt coolng capacty rqurmnts: hgh (.kw), mdum (.8kW), and low (.kw) capacty. h condnsr and vaporator ar nlt tmpraturs ar st to b 6 C and C, rspctvly, and th total amount of rfrgrant mass n th systm s spcfd as 35g. It can b sn that th mnmum powr consumpton s achvd n opraton mod to mt th hgh coolng capacty, n mod to mt th mdum coolng capacty, and n mod 3 to mt th low coolng capacty. abl summarzs th optmal st-ponts for th bst opraton mod at ach capacty lvl, along wth th dsrd systm nputs and rfrgrant mass dstrbuton n ach of th hat xchangrs. Hgh Capacty (. kw) Mdum Capacty (.8 kw) Low Capacty (. kw) 3 4 COP 3 4 FIGURE 8. PERFORMANCE COMPARISON BEWEEN DIFFEREN OPERAION MODES FOR EACH OF HREE DESIRED COOLING CAPACIIES. ABLE. SUMMARY OF OPIMIZAION RESULS Hgh capacty Mdum capacty Low capacty Opraton mod 3 Cond. prssur (kpa) Evap. prssur (kpa) Cond. subcoolng ( C).9 Evap. suprhat ( C) Comp. spd (rpm) 5 7 Valv opnng (%) Cond. ar flow (kg/s)..3. Evap. ar flow (kg/s).6..5 Mass n cond. (g) Mass n vap. (g) h LMI-basd optmal control problm s solvd to fnd th optmal stat fdback gan matrx K and obsrvr gan matrx L for ach opraton mod, (K, L ), (K, L ) and (K 3, L 3 ). In th LQR dsgn, th wghtng matrx Q s chosn as th dntty 7 7 matrx I, and th matrx R s gvn n Eq. (4) to mt th systm actuators saturaton lmts. R = (4) 6 Copyrght by ASME

7 h dsrd coolng capacty profl s gvn n Fg. 9, and Fg. shows th corrspondng opraton mod swtchng squnc. h controllr prformanc whl trackng th optmal st-ponts (condnsr prssur, condnsr hat rjcton rat, vaporator prssur, and vaporator coolng capacty) s prsntd n Fg.. It s shown that th dsgnd controllrs can mantan systm opraton at th optmal st-ponts wth mod swtchng. Coolng capacty (kw)..8. m (s) FIGURE 9. DESIRED COOLING CAPACIY PROFILE. Opraton mod Condnsr prssur (kpa) Condnsr hat rjcton (kw) Evaporator prssur (kpa) Evaporator capacty (kw) 3 4 m (s) FIGURE. MODE SWICHING SEQUENCE St-pont Smulatd output m (s) FIGURE. CONROLLER RACKING PERFORMANCE. Fgurs and 3 show th control nput sgnals to th VCC systm. It s vdnt by th larg changs n comprssor spd, valv opnng, and hat xchangr ar flow rats that th systm s bng ffctvly controlld ovr a vry larg opratng rgon. As llustratd, th proposd swtchng control schm drvs th systm from on opraton mod to anothr basd on th dtrmnd optmal st-ponts lstd n ab.. It should b notd that control tchnqus, such as bumplss transfr [7] or dynamc compnsator dsgn [8], can b appld to nsur smoothr control sgnals durng mod swtchng. Comprssor spd (RPM) Valv opnng (%) FIGURE. CONROLLED VCC SYSEM INPUS AND. Condnsr ar flow rat (kg/s) Evaporator ar flow rat (kg/s) 4.5 m (s) FIGURE 3. CONROLLED VCC SYSEM INPUS 3 AND 4. CONCLUSIONS hs papr prsnts a swtchd modl ntwork to rprsnt th nonlnar VCC systm dynamcs ovr th normal opratng nvlop. Four dffrnt opraton mods ar ntroducd to formulat a 3 th -ordr swtchd modl of th systm. h proposd LMI control dsgn framwork guarants th stablty and prformanc charactrstcs of th controlld closd-loop systm. A st-pont optmzaton problm s solvd to dtrmn th optmal st-ponts for achvng a dsrd coolng capacty profl for a partcular VCC systm. A modl-basd swtchng 7 Copyrght by ASME

8 controllr s succssfully dsgnd to rgulat th systm prformanc across th nonlnar opratng nvlop. Futur work wll nvolv comparson studs btwn th proposd modl and control framwork and convntonal VCC systm control approachs as wll as fndng optmal squncs for mod swtchng n th prsnc of dsturbancs and othr xognous sgnals. ACKNOWLEDGMENS hs work was supportd n part by th sponsorng compans of th Ar-Condtonng and Rfrgraton Cntr at th Unvrsty of Illnos at Urbana-Champagn and n part by an award from th Dpartmnt of Enrgy (DOE) Offc of Scnc Graduat Fllowshp Program (DOE SCGF). REFERENCES [] Chng,., 6. Nonlnar Obsrvr Dsgn usng Contracton hory wth Applcaton to Hat Exchangrs havng Varyng Phas ranston. PhD thss, MI. S also URL dspac.mt.du/handl/7./3677 [] Rasmussn, B.P., 5. Dynamc lng and Advancd Control of Ar Condtonng and Rfrgraton Systms. PhD thss, Unvrsty of Illnos at Urbana-Champagn. S also URL [3] Catano, J.E., Zhang,.J., Pls, Y., Jnsn, M.K., and Wn, J..,. "Exprmntal Idntfcaton of Evaporator Dynamcs for Vapor Comprsson Rfrgraton Cycl durng Phas ranston". In hrmal and hrmomchancal Phnomna n Elctronc Systms, th IEEE Intrsocty Confrnc, pp. -5. [4] H, X.D., Lu, S., Asada, H., and Itoh, H., 998. "Multvarabl Control of Vapor Comprsson Systms". HVAC&R Rsarch, vol. 4, pp [5] Rasmussn, B.P., and Allyn, A.G.,. "Gan Schduld Control of an Ar Condtonng Systm usng th Youla Paramtrzaton". IEEE ransactons on Control Systms chnology, vol. 8, pp [6] Q, Q., and Dng. S.M., 9. "Multvarabl Control of Indoor Ar mpratur and Humdty n a Drct Expanson (DX) Ar Condtonng (A/C) Systm". Buldng and Envronmnt, vol. 44, pp [7] Andrson, M., Buhnr, M., Young, P., Httl, D., Andrson, C., u, J.L., and Hodgson, D., 8. "Mmo Robust Control for HVAC Systms". IEEE ransactons on Control Systms chnology, vol. 6, pp [8] Schurt, L.C., Hrms, C.J.L., and Nto, A.., 9. "A l-drvn Multvarabl Controllr for Vapor Comprsson Rfrgraton Systms". Intrnatonal Journal of Rfrgraton, vol. 3, pp [9] Schurt, L.C., Hrms, C.J.L., and Nto, A..,. "Assssmnt of th Controllng Envlop of a l-basd Multvarabl Controllr for Vapor Comprsson Rfrgraton Systms". Appld hrmal Engnrng, vol. 3, pp [] L, B., and Allyn, A.G.,. "A Dynamc l of a Vapor Comprsson Cycl wth Shut-down and Start-up Opratons". Intrnatonal Journal of Rfrgraton, vol. 33, pp [] McKnly,.L., and Allyn, A.G., 8. "An Advancd Nonlnar Swtchd Hat Exchangr l for Vapor Comprsson Cycls usng th Movng-boundary Mthod". Intrnatonal Journal of Rfrgraton, vol. 3, pp [] Rasmussn, B.P., and Allyn, A.G., 4. "Control-orntd lng of ranscrtcal Vapor Comprsson Systms". ASME Journal of Dynamc Systms, Masurmnt, and Control, vol. 6, pp. 54. [3] Boyd, S., El Ghaou, L., Fron, E., and Balakrshnan, V., 994. Lnar Matrx Inqualts n Systm and Control hory, sr. Studs n Appld and Numrcal Mathmatcs, vol. 5. Phladlpha, PA:SIAM. [4] Ko, H.S., Jatskvch, J., Dumont, G., and Yoon, G.G., 8. "An Advancd LMI-basd-LQR Dsgn for Voltag Control of Grd-connctd Wnd Farm". Elctrc Powr Systms Rsarch, vol. 78, pp [5] Olalla, C., Lyva, R., El Aroud, A., and Qunnc, I., 9. "Robust LQR Control for PWM Convrtrs: an LMI Approach.". IEEE ransactons on Industral Elctroncs, vol. 56, pp [6] Skogstad, S., and Postlthwat, I., 996. Multvarabl Fdback Control, John Wly & Sons, Nw York. [7] Grab, S.F., and Ahln, A., 996. Dynamc ransfr among Altrnatv Controllrs and Its Rlaton to Antwndup Controllr Dsgn. IEEE ransactons on Control Systms chnology, vol. 4(), pp [8] Ln, J.L., and Yh,.J., 9. Swtchng Control of Dual-vaporator Ar-condtonng Systms. Enrgy Convrson and Managmnt, vol. 5, pp Copyrght by ASME