A NEW PROCEDURE FOR ROBUST CONTROL DESIGN OF COOLING MACHINES BASED ON VAPOR COMPRESSSION

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Chastity Stewart
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1 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG A NEW PROCEDURE FOR ROBUST CONTROL DESIGN OF COOLING MACHINES BASED ON VAPOR COMPRESSSION Joe Mara Galvez Department of Mechancal Engneerng Federal Unverty of Mna Gera Av. Antono Carlo 667, Pampulha, Belo Horzonte, MG, Brazl Emal: jmgalvez@ufmg.br Abtract. Coolng and ar-condtonng ytem are often poorly effcent concernng energy conumpton. Ther operaton cycle often nclude a thermotat that wtche on and off a motor-compreor gear. Recently a great amount of reource are beng appled n the reearch on non-pollutng flud, on the degnng of new equpment and on the development of new controllng technque. The beneft of the mprovement on coolng ytem control technque nclude, bede the energy avng, a) the accurate and contnuou and unform temperature control n the target envronment, b) the hortenng of the uperheatng regon, c) a lower number of tart-top by tme unt ncreang n th way the equpment lfetme, d) fater coolng repone, e) maller and lghter compreor, etc. However, the multaneou control of the freezng power and uperheatng degree a challengng tak. Bede the model uncertante, the trong cro-couplng nteracton that ext between the ytem nput and output and the tme delay n the control loop are ome of the problem that mght deterorate the controller deal performance. Th paper preent a robut control degn procedure a an alternatve to more complex and expenve MIMO controller. The objectve to ndependently control the comfort temperature and the uperheatng. The propoed technque allow robut control charactertc to be ncluded naturally n the controller degn. Keyword: Robut Control, Model Uncertante, Coolng Sytem.. Introducton The mprovement of power conumpton effcency of ndutral devce one of the man ue for the ncomng century. In the frt century of the ndutral age, the world populaton ha vrtually exploded, nature ha been almot devatated and energy reource have been depleted. In pte of that, the human lvng comfort ha become eental for mot of the whole world populaton, even for the thrd world people; and becaue of that energy per-capta conumpton hould contnuouly ncreae n the future. It a fact that the next decade are gong to tetfy a contnuou and trenuou earch for new devce and technologe to ave energy reource. The energy conumpton by heatng and coolng ytem n commercal and ndutral buldng correpond to 5% of the world energy conumpton (Imbab, 99). Heatng and coolng ytem are hgh-energy conumpton procee (Arguello-Serrano, 999) whoe operaton n commercal and ndutral buldng tll are neffcent. It already known that the oluton for an effcent operaton of heatng and coolng ytem rele on the proper choce and degn of an automatc control ytem. Low cot controller uch a On/Off control and PID control are currently ued a the tandard controller n the heatng, ventlaton and ar condtonng (HVAC) ndutry. However, ther low energy effcency caue an extra-undered energy burnng. An mportant fact that caue the low energy effcency that mot control degn are only capable of dealng wth contant thermal load, whch not the general cae; n practce, thermal load are tme varyng. The temperature enor locaton another dffcult n the control feld of heatng and coolng devce. The natural poton for the temperature enor cloe to, or even nde, the target envronment; n practce, to avod the ncluon of tme delay n the control loop, the temperature enor uually located cloe to the heat ource (nk). Furthermore, conventonal ngle-nput ngle-output (SISO) control of coolng machne not capable of controllng the freezng power and uper-heatng ndependently (due to the I/O cro couplng). Several control tratege to deal wth the control problem of tme varyng procee, tme delay and I/O cro couplng have been propoed by the control communty. Among them, robut control, adaptve control and ntellgent control are the mot mportant. A drawback of thee ophtcated alternatve that they are uually expenve and requred advanced computatonal reource. To face tme-varyng thermal load, tme delay and I/O cro couplng, new-low cot multnput mult-output (MIMO) control tratege mut be explored. Th paper ntroduce a MIMO control cheme that permt the ndependent control of the output varable of coolng machne baed on the vapor compreon cycle. Fgure how the chematc dagram of a coolng machne of th type.. The MIMO Feedback Control Output feedback ha been the ndutral tandard for control purpoe not only to hape the plant repone, fulfllng performance pecfcaton, but alo to deal wth output dturbance and model uncertante.

2 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG Tradtonally, the ndutral control communty ha reled on the ntrnc robutne of output feedback controller to face the control degn problem for SISO plant. A dverty of controller tunng algorthm ha been uccefully developed and appled to SISO ndutral plant. Behnd th ucce there ha alway been a property that ext for all phycal ytem, the domnance of the low-frequency pole n the ytem tme repone. Th fact ha been the background of nearly every robut control degn technque. Conderng th concept n the controller degn, there no need for olvng the modelng problem a rgorouly a t could be requred wthout the pole domnance property. Several attempt have been made to extend the SISO degn technque to the MIMO cae. In th context, the uually trong nput-output cro-couplng extng n MIMO ytem become a mportant a the model uncertante due to the ze (order) of large-cale ytem. Wth ome excepton, the ucce of MIMO control degn alo depend on the pole domnance property. In recent year, the reearch ha been focued n new uncouplng technque. It worth to menton the poneer contrbuton from Brtol (966), Kouvartak (979), Mee (98), McAvoy, (983) and Grodder and Morar (986). Some charactertc of the ue of thee technque are: The degn procedure uually carred out n the frequency doman. Model uncertante are ealy repreented n the frequency doman (partcularly, non-tructural uncertante. Low frequency model are, n general, accurate enough for control degn n th envronment. The tandard PI controller (that repond for more de 9% of the ndutral controller degned to perform n the low frequency range. Output dturbance are uually low frequency gnal. 3. The MIMO Control A Bref Revew Th ecton preent a bref revew of the bac concept on multvarable control ytem. The followng baed on the book from Macejowk (989) and Skogetad et al (996). The ytem output, y(, gven by y( = T ( P( r( S( d( T ( m( () where r( the reference nput, d( repreent the dturbance and m( the meaurement noe. the functon S( known a the output entvty functon and defned a ( = [ I G( ] S () and the ytem cloed loop tranfer functon (or complementary entvty), T(, then gven by T ( = S( G( (3) the nput entvty functon defned a ( = [ I G( ] S () and t correpondng complementary functon a T ( = G( S ( (5) a multplcatve model for the plant uncertanty can be wrtten a G( = G ( [ I W ( ] (6) o Hence, the followng crtera to ae the ytem performance and tablty can be etablhed: a) The crteron for nomnal performance defned by S( Wp ( < (7) where W p ( a performance weghtng matrx and ha the form [] I Wp ( = wp ( (8)

3 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG thu, the nomnal performance crteron can be re-wrtten a σ [ S( ] < (9) w ( p where σ [.] the greatet ngular value of [.] b) The crteron for robut performance (non-tructured uncertanty) gven by ( W ( S ( ) σ ( W ( T ( γ σ ) () p where γ = mn(plant-controller condton number). c) The crteron for robut tablty (non-tructured uncertanty) defned by T( W ( < () where W () an uncertanty weghtng matrx and ha the form [] I W ( = w ( () d) The crteron for robut tablty can be re-wrtten a σ [ T ( ] < (3) w ( e) The robutne analy baed on tructured ngular value wa ntroduced by Doyle et al (98). In th cae, a robut performance condton for tructured uncertanty can be etablhed a ( () < ω µ Q (a) where, the matrx Q() defned a Q( and = Q Q ( ( Q Q ( ( = Wp ( So ( W ( So ( W p ( So ( Go ( W ( So ( Go ( (b) ( I Go ( ) So ( = ) (c) where w p ( and w I ( are defned n the frequency doman. Alo, a robut tablty condton for tructured uncertanty can be wrtten a ( Q ( )) < ω µ (5) Equaton from (7) to (5) are appled n Secton 6 to ae the cloed loop ytem robutne and to valdate the controller degn.. The Coolng Sytem Th paper concerned wth the control of a ytem conttuted by an expanon valve, an evaporator and a compreor a hown n Fg.. The ytem nput are the expanon valve openng poton, whch defne the ma flow rate (MFR) and the compreor peed, whch control the volume flow rate (VFR). The ytem output are the uper heatng, Τ, and the freezng power, Q, (Fg.). Ideally, the openng of the expanon valve and the velocty of the varable-peed compreor would be ued to regulate the uper heatng and to control the generaton of freezng power (Fg.3 wth G ( = G ( = ). Unfortunately, th not the cae. Actually, each of the output a functon of both nput (the valve openng poton and the compreor velocty) a hown n Fg.3. Th mean that G ( and G ( can not be neglected n practce.

4 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG The trong cro-couplng nteracton among nput and output characterze th type of ytem. In th cae, the ytem dynamc can be defned by a matrx tranfer functon of the form: T( G( G ( MFR( = [ Y ( ] = [ G( ][ U ( Q ( G( G ( ] (6) VFR( Several lnear and non-lnear computatonal model for coolng ytem can be found n the techncal lterature (Koury, 998; Machado, 996; Rocha, 995; Outtagart, 99). The model dentfcaton procedure (Machado, 996) led to x matrx tranfer functon of the form: 5.6.9( 7 ) G ( G ( (5 ) (59.5 ) [ G ( ] = = (7) G ( ) ( ) 33.89( ).(63 ) G (5.65 )(67.79 ) (8 )(9 ) It hould be notce that the plant non-mnmal phae and alo non trctly proper. Q Condener Valve Bolng Zone Compreor Lqud Evaporator Vapor Q Super Heatng Zone Fgure. The Coolng Sytem. MFR( T( MFR( G( T( Coolng G( VFR( Sytem Q( VFR( G( G( Q( Fgure. The Open Loop Sytem. Fgure 3. The Coolng Sytem Cro Couplng. 5. The x MIMO Control Law. The propoed trategy bacally a frequency-doman procedure. In th cae, the MIMO controller degn carred out n two tep. Frt a MIMO pre-compenator, K (, degned to cale the ytem and reach dagonal domnance at low frequence and then a MIMO controller, K (, degned to meet performance pecfcaton. The advantage of th procedure t that for degn purpoe, K ( dagonal and can be treated a a multple SISO degn nce G(K ( trongly dagonal domnant at low frequence and dagonal at w =. Addtonally, exact modelng only requred at teady tate (w = ) or at mot at low frequence. The MIMO control law ha the form: [ U ] [ K ( ][ K ( ][ R( Y( ] = [ ] [ E( )] ( = (8a)

5 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG where [ R ( ] = T( Setpont T( T ( Error MFR( ; [ Y ( ] = ; [ E( ] = ; [ U ( ] = Q ( Setpont Q ( Q ( Error VFR( (8b) and [ K ( ] = [ ][ K ( ] = K ( K ( K ( (8c) for K ( = K (decouplng at w = ), the x MIMO PI controller become N K I K P D [ ] = [ K][ K ( ] = [ K ] = [ K] K I K N P D ( ( ( ( N D N D ( ( ( ( (9) It hould be notced that, all entre of have the general form of SISO PI controller, however, the engneer only ha to determne the dagonal element of K (. The propoed degnng technque lead to a cloed loop tranfer functon that can be approxmated (at low frequence by a dagonal matrx tranfer functon, Eq. (). T ( T ( T ( Q ( T ( T( Setpont T ( Q ( Setpont [ Y( ] [ R( ] () Becaue of that the ndependent control of uperheatng and freezng power tangble a t hown n the next ecton. Fgure how the MIMO controller tructure. Fgure 5 how the block dagram for the cloed loop ytem. E( MFR( T( Setpont - E( MFR( T( E( Control Algorthm VFR( Coolng Sytem Q( E( VFR( - Q( Setpont Fgure. The MIMO Controller Scheme. Fgure 5. The Cloed Loop Sytem. 6. Aeng The Controller Performance. In th work, the nomnal performance crteron wa pecfed a σ [ S( ] < = () w ( 5 p And the crteron for robut tablty wa choen a σ [ T( ] < = () w (. It clear that n order to fulfll robut tablty pecfcaton the plant mut be made trctly proper. Th can be acheved by ung a matrx low-pa flter of the form [ F ( ] = F ( (3) ) In th cae the fnal controller equaton become

6 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG [ K F ] [ F( ][ K ( ][ K ( ] = [ F( ][ )] ( = () The practcal conequence of Eq. () that wth the ncluon of a flter n the control loop the compreor peed change mother than wthout t. Th wll caue an mproved uper heatng tranent profle. Several technque for multvarable loop hapng can be found n the lterature (Macejowk, 989, Skogetad, 996, Ho & Xu, 998). In th cae, a robut performance wa acheved ung the followng control law [ K K ( ] = = F ( F K K ( ) ( ) ( ) Fnally, the cloed loop tranfer functon ha the form (5) Y ( = T ( R( = [ I G( ] G( K ( R( (6) F 7. Expermental Reult. Smulaton reult are preented here to llutrate the controller performance. Fgure 6 to 9 graphcally dplay the controller degnng procedure. Fgure 6 preent the ytem open loop repone. It how the uper-heatng (quadrant II) and freezng power (quadrant IV) tme repone. It can be oberved (n quadrant I and III) the trong effect of the I/O cro couplng (n the deal cae, the tme repone hown n quadrant I and III hould reman at zero for all tme or at leat return to zero at teady tate). Fgure 7 how the pre-fltered ytem, G( F(, open loop repone. It hould be noted that the ncluon of the pre-flter F( low down the repone of G (. Fgure 8 preent the effect of the decouplng pre-compenator K (. It how the uper-heatng (quadrant II) and freezng power (quadrant IV) open loop tme repone to a unt tep. It can be oberved (n quadrant I and III) that the teady tate effect of the I/O cro couplng were elmnated by the propoed control law. Fnally, Fgure 9 preent the ytem cloed loop performance. Fgure to preent the effect of the pre-flter and the decouplng pre-compenator n the frequency doman. Fgure how the non-trctly proper charactertc of the G ( tranfer functon (t alo non-mnmal phae a hown by Eq. (7)). Fgure preent the effect of the pre-flterng on G (. Fgure how the ytem decouplng n the frequency doman due to the ncluon of K (. Fgure 3 and dplay the robutne analy addreed by Eq. (7) to (5). 6 G( Repone 6 G( Repone Tme 8 G( Repone 8 G( Repone Tme Fgure 6. Plant Open Loop Step Repone [G(].

7 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG 6 G(F( Repone 6 G(F( Repone G(F( Repone 8 G(F( Repone Fgure 7. Pre-Fltered Plant Open Loop Step Repone - [G( F(]. 3 G(F( Repone.5 G(F( Repone G(F( Repone G(F( Repone Fgure 8. Pre-Compenated Pre-Fltered Plant Open Loop Step Repone - [G(F(K (].. G(F( Repone.5 G(F( Repone G(F( Repone. G(F( Repone Fgure 9. Cloed Loop Step Repone - T(.

8 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG G( Repone 9.5 G( Repone G( Repone G( Repone Fgure. Plant Open Loop Repone - [G(]. G (F( Repone G(F( Repone G (F( Repone 5 G(F( Repone Fgure. Pre-Fltered Plant Open Loop Repone - [G( F(]. G(F( Repone G(F( Repone G(F( Repone 5 G(F( Repone Fgure. Pre-Compenated Pre-Fltered Plant Open Loop Repone - [G( F( K (].

9 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG Weghtng Functon (/W & /Wp) Controller Repone Open Loop Prncpal Gan Nomnal Performance Condton Fgure 3. The Degn Procedure..35 Robut Performance Condton (NSU) Robut Stablty Condton (NSU) µ(q) - Robut Performance Condton (SU).5 µ(q) - Robut Stablty Condton (SU) Fgure. Controller Robutne Valdaton.

10 Proceedng of COBEM 5 Copyrght 5 by ABCM November 6-, 5, Ouro Preto, MG 8. Fnal Comment. Tradtonally, clacal on-off controller for coolng machne have hown to be neffcent for energy avng purpoe (Rocha, 995; Machado, 996). Varable compreor peed operaton ha recently emerged a the oluton for the energy conumpton mnmzaton problem. The earchng for an nexpenve compreor peed controller currently on the focu of the attenton of the control communty and although ome fne reult can be found n the techncal lterature the fnal oluton tll under nvetgaton. Th paper hown that ndependent control of uperheatng and freezng power n a coolng ytem baed on vapor compreon a feable tak. A MIMO controller for a coolng machne wa degned and mplemented n computer mulaton. The potental of the propoed MIMO controller for avng energy and keepng comfort wa verfed through mulaton. The man feature of the propoed technque are: a) A long a the controller degned to work n the frequency range n whch the plant dagonal domnant, the degn can be accomplhed n a SISO envronment and th uually the cae of ndutral PI controller that are degned to work n low frequency. b) The mplementaton of the control law can be done a two ndependent ngle loop control. c) Due to the plant pre-compenator, model accuracy n general requred only at low frequence. 9. Reference Arguello-Serrano, B. & Vélez-Reye, A., (999), Nonlnear Control of a Heatng, Ventlatng, and Ar Condtonng Sytem wth Thermal Load Etmaton, IEEE Tranacton on Control Sytem Technology, vol. 7, pp Brtol, E.H., (966), On a New Meaure of Interacton for Multvarable Proce Control, IEEE Tranacton on Automatc Control, Vol., pp Doyle & G. Sten, (98), "Multvarable Feedback Degn: Concept for a Clacal/Modern Synthe", IEEE Tranacton on Automatc Control, Vol. 6, pp. -6. Doyle, J.C., (98), Analy of feedback ytem wth tructured uncertante, Proceedng of the IEE, Part D, vol 9, pp -5. Grodder, P. and Morar, M., (986), Interacton Meaurement for Sytem under Decentralzed Control, Automatca, vol., pp Ho, W.K. and Xu, W., (998), Multvarable PID Controller Degn Baed on the Drect Nyqut Array Method, Proceedng of the Amercan Control Conference, Phladelpha, Pennylvana, pp Imbab, M.S., (99), Computer Valdaton of Scale Model Tet for Buldng Energy Smulaton, Int. J. Energy Re., vol., pp Kouvartak, B., (979), "Theory and Practce of the Charactertc-Locu Degn Method", IEE Proceedng, Vol. 6, pp Koury, R., (998), Modelo Numérco de uma Máquna Frgorífca de Compreão de Vapor, Tee de Doutorado, UNICAMP, Campna, SP, Bral. Machado, L., (996), Modele de Smulaton et Etude Expermentale d un Evaporateur de Machne Frgorfque en Regme Trantore, Thée de Doctorat, L Inttut Natonal de Scence Applquee de Lyon, France. Macejowk, J.M., (989), Multvarable Feedback Degn, Addon-Weley Publhng Company. McAvoy, T. J., (983), Interacton Analy - Prncple and Applcaton, Intrument Socety of Amerca. Mee, A.I., (98), Achevng Dagonal Domnance, Sytem and Control Letter, vol., pp Outtagart, A., (99), Comportement Dynamque d un Evaporateur de Machne Frgorfque Almente par un Detendeur Electronque, Thée de Doctorat, L Inttut Natonal de Scence Applquee de Lyon, France. Rocha, R., (995), Contrbução ao Etudo de um Stema de Refrgeração com Compreor Operando a Velocdade Varável, Dertação de Metrado, PPGEE-UFMG, Bral. Skogetad, S. and Potlethwate, I., (996), Multvarable Feedback Control - Analy and Degn, John Wley & Son.. Reponblty Notce The author the only reponble for the prnted materal ncluded n th paper.

A PLC BASED MIMO PID CONTROLLER FOR MULTIVARIABLE INDUSTRIAL PROCESSES

A PLC BASED MIMO PID CONTROLLER FOR MULTIVARIABLE INDUSTRIAL PROCESSES ABCM Sympoium Serie in Mechatronic - Vol. 3 - pp.87-96 Copyright c 8 by ABCM A PLC BASE MIMO PI CONOLLE FO MULIVAIABLE INUSIAL POCESSES Joé Maria Galvez, jmgalvez@ufmg.br epartment of Mechanical Engineering