MULTIVARIABLE FUZZY CONTROL WITH ITS APPLICATIONS IN MULTI EVAPORATOR REFRIGERATION SYSTEMS

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1 MULTIVARIABLE FUZZY CONTROL WITH I APPLICATIONS IN MULTI EVAPORATOR REFRIGERATION SYSTEMS LIAO QIANFANG Schoo of Eectrca and Eectronc Engneerng A thess submtted to the Nanyang Technoogca Unversty n parta fufment of the requrement for the degree of Doctor of Phosophy 25

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3 Acknowedgement Frst and foremost I woud ke to express my sncere grattude to my supervsors Prof. Ca Wenjan and Prof. Wang Youy for ther patent supervson tremendous support and nvauabe gudance throughout the course of my research work. Ther nsghtfu comments and thoughtfu dscussons greaty nspre me. It s a bg honor and peasure to have them as my advsors. I woud ke to partcuary thank Prof. L Shaoyuan and Prof. L Nng of Shangha Jao Tong Unversty for ther patent supervson and constructve gudance durng my graduate study and encouragng me to pursue a hgher degree. And they st gve me ther precous support durng my Ph.D study whch I am so gratefu for. I want to thank Dr. Yan Ja and Dr. Dng Xudong for ther contnuous gudance and hep throughout the duraton of my research. I aso thank my coeagues cassmates and frends n Process Instrumentaton Laboratory: Xu D Yang Chen Lu Changxa Yn Xaohong Wu Bngje Wu Qong Cu Can Zhao Le Wang Xn L Chao Chen Can Hu Shen Chen Haoran J Yongfeng etc. and the aboratory supervsor Mr. Yock for ther hep and support durng the course of my study. I woud ke to acknowedge Schoo of Eectrca and Eectronc Engneerng Nanyang Technoogca Unversty and Energy Research NTU (ERI@N) for provdng the fnanca support research factes and opportuntes for ths study. Last but not east I want to express my deepest thanks to a members of my beoved famy: my father my mother my sster and my brother. They gve me steadfast ove support and understandng and stand by me through thck and thn. I coudn t reaze my dreams wthout them. From the bottom of my heart I ove them aways and forever.

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5 Tabe of Contents Summary... III Fgure st... V Tabe st... VI Chapter. Introducton.... Overvew of mut evaporator refrgeraton system contro....2 Motvatons and objectves Major contrbutons of the thess Organzaton of the thess... Chapter 2. T S fuzzy modeng for MIMO processes Introducton Type T S fuzzy modeng Type 2 T S fuzzy modeng Smuaton Summary Chapter 3. Interacton anayss and oop parng methods based on T S fuzzy modes for MIMO processes Introducton Loop parng crtera Cacuatons based on Type and Type 2 T S fuzzy modes Smuaton Summary... 4 Chapter 4. Effectve T S fuzzy mode for decentrazed contro of MIMO processes Introducton EM Controer desgn based on EMs Smuaton Summary Chapter 5. Sparse contro based on T S fuzzy modes for MIMO processes Introducton Sparse contro structure seecton Independent desgn based on EMs Smuaton Summary Chapter 6. Appcatons n MER systems I

6 6. Introducton Expermenta resuts of MER system contro Contro structures for the MER system EMs for the MER system Decentrazed and sparse contros for the MER system Comparsons of Type and Type 2 EM based contros Summary Chapter 7. Concusons and future work Concusons Future work... 2 References... 5 Author s pubcatons... Appendx A... Appendx B... 4 Appendx C... 9 II

7 Summary Mut evaporator refrgeraton (MER) system s a cost effectve devce for a budng wth dfferent coong requrements because t s comprsed of a snge compressor and a snge condenser but mutpe evaporators each operatng at a specfed temperature to smutaneousy cater to dfferent coong oads such that the effcency can be mproved and the economc cost can be reduced. Compared wth a snge evaporator refrgeraton system an MER system s more dffcut to contro due to ts compex structure wth cross coupng effects among dfferent evaporators. Ths thess presents a seres of nove Type and Type 2 Takag Sugeno (T S) fuzzy mode based studes for mutvarabe process contro to manpuate MER systems. The methodooges and the contrbutons are summarzed as foows:. Decentrazed contro s predomnant n mutvarabe process contro appcatons because of ts smpcty and effectveness. The frst task for devsng decentrazed contro s to determne a contro confguraton where the pared nput output oops have mnmum cross coupng effects. In ths thess by defnng the steady state gan and normazed ntegrated error on a T S fuzzy mode a reatve normazed gan array (RNGA) based oop parng crteron s presented to anayze the nteractons and par nputs and outputs to determne the contro oop confguraton for decentrazed contro of an MER system. Smpe cacuaton procedures based on both Type and Type 2 T S fuzzy modes are provded. Compared wth the exstng fuzzy mode based oop parng approaches usng ony steady state gan the proposed method empoys both steady and dynamc nformaton of the process to par the contro oops such that a more proper contro confguraton can be gven for manpuatng an MER system. 2. When devsng a decentrazed controer after contro confguraton determned the oca controer desgn for one oop requres the nformaton of other oops snce there are nteractons among the contro oops. A manner caed effectve T S fuzzy mode (EM) s presented to descrbe the nteractng effects on a certan oop caused by other cosed oops for controer desgn of an MER system. III

8 The EM of a certan oop s derved by ncorporatng the quantfed nteractons provded by RNGA based parng crteron nto the coeffcents of ts ndvdua T S fuzzy mode. Smpe cacuaton methods to obtan Type and Type 2 EMs are gven. Wth the EMs of pared oops a mut nput mut output (MIMO) process can be approxmatey consdered as mutpe ndependent snge nput snge output (SISO) processes and then the decentrazed controer desgn can be greaty factated by near SISO contro agorthms. Compared wth exstng T S fuzzy mode based decentrazed contro addng extra terms to the ndvdua fuzzy modes to express the nteractng effects EM s a practca way that can reduce the compexty n both modeng and controer desgn. 3. To further mprove the contro performance n handng the strong nteractons of an MER system a gudene to devse sparse contro based on both Type and Type 2 T S fuzzy modes s presented whch ncudes frst a method n terms of RNGA based parng crteron s ntroduced to anayze the nteractons between pared and unpared oops and then seect a sparse contro structure by addng severa unpared eements to the pared structure; second based on the EMs of seected oops an ndependent controer desgn approach s gven that an MIMO process can be approxmatey regarded as a group of non nteractng SISO processes and then the near SISO contro agorthms can be apped to manpuate nonnear cosed couped MER systems. Sparse contro can acheve mproved performance over decentrazed contro when cost fewer cacuatons than fu dmensona contro. 4. The appcatons of the proposed Type and Type 2 T S fuzzy mode based methods n an expermenta MER system wth three evaporators to cater to the coong requrements for ar condtonng pershabe food storage and freezng are presented. The expermenta resuts vadate the practcabty and effectveness of the proposed contro structure seectons and EM methods for decentrazed and sparse contro. And the comparatve resuts of Type and Type 2 T S fuzzy systems n terms of robustness and computatona cost are gven. IV

9 Fgure st Fgure. Schematc of an MER system... 2 Fgure 2. A n n MIMO process... 3 Fgure 2.2 Type 2 fuzzy sets... 8 Fgure 2.3 The errors of Type and Type 2 fuzzy modes Fgure 3. Typca waveforms for processes... 3 Fgure 3.2 AN MIMO process wth dsturbances Fgure 4. Interactng case wth λ and γ > Fgure 4.2 Interactng case wth λ and γ Fgure 4.3 Interactng case wth λ > and γ >... 5 Fgure 4.4 Interactng case wth λ > and γ... 5 Fgure 4.5 Decentrazed contro system based on EMs for a 2 2 process Fgure 4.6 A fuzzy mode based cosed oop contro system for an MIMO process Fgure 4.7 Decentrazed contros for the MIMO process n Eq. (4.34)... 6 Fgure 4.8 Decentrazed contros for the process changed as Eq. (4.35) Fgure 4.9 Decentrazed contros for the process changed as Eq. (4.36) Fgure 4. ETF based decentrazed contro for the process changed as Eq. (4.37) Fgure 4. EM based decentrazed contros for process changed as Eq. (4.37) Fgure 5. A cosed oop MIMO contro system Fgure 5.2 A cosed oop SISO contro system Fgure 5.3 Sparse contros for the MIMO process n Eq. (4.34)... 8 Fgure 5.4 The comparsons of sparse and decentrazed contros for the MIMO process n Eq. (4.34)... 8 Fgure 5.5 Sparse contros for the process changed as Eq. (4.36) Fgure 5.6 The comparsons of sparse and decentrazed contros for the process changed as Eq. (4.36) Fgure 5.7 Sparse contros for the process changed as Eq. (4.37) Fgure 5.8 The comparsons of sparse and decentrazed contros for the process changed as Eq. (4.37) Fgure 5.9 ETF based sparse contro for the process changed as Eq. (5.2) Fgure 5. EM based sparse contros for the process changed as Eq. (5.2) Fgure 6. An expermenta MER system Fgure 6.2 Schematc dagram of the MER system Fgure 6.3 Pressure (P) enthapy (h) chart of the MER system Fgure 6.4 Step responses of the three outputs of the MER system... 9 Fgure 6.5 Type EM based contros for the MER system Fgure 6.6 Type 2 EM based contros for the MER system Fgure 6.7 The manpuaton varabes of the MER system Fgure 6.8 Comparsons between Type and Type 2 EM based contros V

10 Tabe st Tabe 2. Type 2 T S fuzzy modes... 8 Tabe 2.2 The centers of fuzzy custers for Eq. (2.22) Tabe 2.3 The consequent parameters of Type and Type 2 modes for Eq. (2.22) Tabe 2.4 The RMSEs of Type and Type 2 modes for Eq. (2.22) Tabe 3. The comparsons of MAEs... 4 Tabe 4. Typca gan and phase margn pars Tabe 4.2 The IAEs of Type and Type 2 EM based decentrazed contros for the process n Eq. (4.34) Tabe 4.3 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.35) Tabe 4.4 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.36) Tabe 4.5 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.37) Tabe 5. Coeffcent cacuatons of EMs for unpared oops n dfferent cases Tabe 5.2 The IAEs of Type and Type 2 EM based sparse contros for the process n Eq. (4.34)... 8 Tabe 5.3 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (4.36)... 8 Tabe 5.4 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (4.37) Tabe 5.5 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (5.2) Tabe 6. The IAEs of Type and Type 2 EM based contros for the MER system Tabe 6.2 Comparson of the contro varabe cacuatng tme based on a Type and a Type 2 EM Tabe A. The centers of fuzzy custers for the process n Eq. (3.27)... Tabe A.2 The consequent parameters of Type and Type 2 T S fuzzy modes for the process n Eq. (3.27)... 2 Tabe B. The centers of fuzzy custers for the process n Eq. (4.34)... 4 Tabe B.2 The consequent parameters of Type and Type 2 T S fuzzy modes for the process n Eq. (4.34)... 5 Tabe C. The centers of fuzzy custers for the MER system... 9 Tabe C.2 The consequent parameters of Type and Type 2 T S fuzzy modes for the MER system... 2 VI

11 Chapter. Introducton. Overvew of mut evaporator refrgeraton system contro Refrgeraton systems move heat from one physca ocaton to another. Because of ths remarkabe property t has been wdey used n modern socety and pays an mportant roe n ar condtonng for comfort food producton and dstrbuton chemca and ndustry processes and speca appcatons []. The eectrcty consumpton of refrgeraton systems n a budng accounts for reatvey arge proporton of the tota energy usage. In countres and regons wth md cmates the usage of refrgeraton systems costs up to 3% of the tota eectrc energy consumpton. In a tropca zone such as Sngapore wth an average annua ambent temperature of 29.4ºC and an average annua ambent reatve humdty of 85% the eectrcty consumpton for refrgeraton systems can be much hgher. Accordng to Budng & Constructon Authorty (BCA) s Green Mark program [2] more than 52% of the tota eectrc energy consumpton of Sngapore s used for runnng ar condtonng systems whch costs a arge amount of money. Therefore even a sma mprovement of effcency n runnng refrgeraton systems coud save consderabe energy and greaty reduce the expendture n energy consumpton. In a genera budng such as a househod estate a commerca compex or an offce budng typcay there are three coong demands for ar condtonng pershabe food storage and freezng respectvey. A convenent manner to meet dfferent demands s to use dfferent refrgeraton systems that cater to dfferent coong oads ndependenty. However ths approach may not be economcay feasbe due to the hgh nta cost space requrements and redundant expendtures n operaton and mantenance. Another aternatve s to adopt a mut evaporator refrgeraton (MER) system constructed by a snge compressor and a snge condenser wth three evaporators each operatng at a specfc evaporatng temperature. Compared wth usng mutpe ndependent snge evaporator refrgeraton systems an MER system can satsfy dfferent coong requrements smutaneousy so that the effcency n

12 runnng refrgeraton systems can be mproved and then the economc cost can be reduced. The schematc of an MER system wth three evaporators s shown n Fg... Fgure. Schematc of an MER system The workng prncpe of ths MER system s: crcuatng refrgerant as a saturated vapor enters the compressor and s compressed nto a superheated vapor wth hgher pressure and temperature. Then the refrgerant s cooed and condensed nto qud phase n the condenser. Afterwards the sub cooed qud refrgerant wth a hgh pressure s dvded nto three fows that go through three expanson vaves where ther pressures are reduced abrupty. And then the three cod qud refrgerant fows enter the evaporators where they evaporate at dfferent temperatures to absorb the heat of ambent envronments and become saturated vapors wth dfferent pressures. To reduce the refrgerant vapor pressure from the evaporators wth hgher temperatures to that from the one wth owest temperature a pressure reguaton devce s used to mantan the requred pressure dfferences. After pressure reguaton the three fows wth the same pressure are merged nto one and the refrgerant s routed back nto the compressor to compete the refrgeraton cyce. Compared wth the conventona snge evaporator refrgeraton cyce an MER system as shown n Fg.. s more dffcut to reguate and contro snce t has a more 2

13 compex structure that conssts of three cyces whch nevtaby nteract wth each other and can ntroduce strong cross coupng effects. In the terature to date ony a mted number of studes can be found regardng controer desgns for MER systems. For exampe [3] presented an approach to controng the evaporatng temperatures and superheat vaues of a mut unt ar condtonng system that uses feedback nearzaton to compensate for the nonnearty n the system dynamcs wth a proportona ntegra (PI) controer desgn. [4] deveoped a method to produce a frst order near transfer functon matrx to descrbe a trpe evaporator ar condtoner for a cascade controer desgn and [5] used the same dentfcaton method to obtan a frst order transfer functon matrx for the fow dstrbuton contro n a dua evaporator ar condtoner. Severa mode predctve contro methods for mut evaporator ar condtoners were gven n [6-8]. A capacty contro approach for a mut evaporator ar condtonng system was deveoped and vadated by expermenta tests n [9]. In [] an optma contro strategy was presented to determne the maxmum coeffcent of performance (COP) for a dua evaporator refrgeraton system wth dfferent cod storage rooms where the settng temperatures were 5 and 23 respectvey. In addton methods usng fuzzy ogca reasonng to derve controers for mut evaporator systems were proposed n [ 2]. It s noted that the overwhemng majorty of exstng contro methods for MER systems are deveoped for mut evaporator ar condtoners where a the evaporators work at smar temperatures (ar condtonng) whe very few are avaabe for an MER system where the evaporators operate at dfferent temperatures to satsfy dfferent coong oads (ar condtonng food storage and freezng)..2 Motvatons and objectves An MER system wth evaporators workng at dfferent temperatures s an effectve devce to save the energy cost for a budng that has dfferent coong requrements. And a proper contro method s necessary and mportant for runnng such an MER system. The ack of ths type of contro method as ntroduced n the ast secton has motvated the author to do the reevant studes and deveop approprate and practca 3

14 approaches to manpuatng ths type of MER system. The conventona and cassca contro methods are deveoped based on essentay quatatve and quanttatve technques that requre more or ess accurate mathematca modeng [3]. However for an MER system as shown n Fg.. because of ts compex structure strong nonnearty and the nfuence of dsturbances and nteractons an accurate mathematca mode s generay dffcut to derve. On the other hand fuzzy contro can be carred out wthout requrng accurate mathematca modes snce a fuzzy mode can be but based on data sampes human experence or both [4-7]. Moreover fuzzy contro has proved to be an effectve robust and powerfu too to manpuate compex defned or even non anaytc processes [3 8]. Thus fuzzy contro has been recommended as an aternatve to conventona contro approaches [3 8]. Currenty two fuzzy mode structures are popuar n research and appcaton one s Mamdan fuzzy mode [9] and the other s the Takag Sugeno (T S) fuzzy mode [2]. Both of them consst of a group of IF (antecedent) THEN (consequent) fuzzy rues. The dfference between them s that Mamdan fuzzy mode uses fuzzy sets as ts consequents whe T S uses rea vaued dynamc near poynomas nstead. A T S fuzzy mode can descrbe a goba nonnear process by a group of oca near poynomas whch are smoothy merged by fuzzy membershp functons. And theoretca proof has been gven that a T S fuzzy mode s a unversa approxmator for any smooth nonnear systems wth arbtrary degree of accuracy n any convex compact area [5-7]. Compared to the Mamdan fuzzy mode T S fuzzy mode can greaty reduce the number of fuzzy rues when modeng hgher order systems and subsequenty s ess prone to the curse of dmensonaty [8]. Furthermore t provdes a patform to deveop methodooges that combne ntegent human reasonng and cassca mathematca approaches for system anayss and controer desgn. Therefore author has been motvated to study nove and effcacous methods based on T S fuzzy modes to manpuate MER systems. Reguatng an MER system as shown n Fg.. requres mut nput mut output 4

15 (MIMO) process contro because t generay needs to adjust severa varabes such as the vave openngs fan speeds and compressor power to cater to dfferent coong oads. Compared to snge nput snge output (SISO) counterparts the MIMO process controers are much more dffcut to reaze because of the exstence of nteractons between process nputs and outputs. At present a number of T S fuzzy mode based contro methods for MIMO processes are avaabe [2-32]. Accordng to the contro structure the exstng methods can be cassfed nto two groups: centrazed contro and decentrazed contro. Centrazed contro can hande the nteractons f t s we desgned snce a fu dmensona contro structure s empoyed. However the fu dmensona contro works wth acceptabe computatona cost ony n manpuatng ow dmensona processes. For arge scae processes t w ead to greaty ncreased compexty for controer desgn because the cacuatons of hgh dmensona matrces are nvoved and may resut n dffcut or even mpossbe to mpement contro strateges. Decentrazed contro coud be much smper than centrazed contro snce t uses a dagona contro structure nstead of fu dmensona contro structure. In contemporary ndustra contro practces decentrazed contro s the predomnant approach to reguatng MIMO processes because of ts smpcty n desgn tunng and mpementaton and mantenance wth ess cost [33-35]. However usng the mted contro structure may gve deterorated cosed oop performance because of the nteractons caused by the exstence of non zero off dagona eements n the matrx descrbng nput output reatonshps for an MIMO process [35]. Therefore the prmary step for decentrazed contro desgn s to par the nputs and outputs to determne a contro confguraton wth mnmum cross coupng effects such that the burden n handng the nteractons can be reduced as much as possbe. Currenty very few pubshed studes can be found wth respect to nteracton anayss and contro structure seecton based on fuzzy modes. Among the few [36] proposed a method to utze steady state gans cacuated based on fuzzy bass functon networks modes to evauate nteractons for an MIMO process. However for an MER system whch s a dynamc process usng ony steady state gans to measure the nteractng effects may gve naccurate resuts for contro 5

16 structure seecton snce no dynamc nformaton s taken nto account; subsequenty the cost for controers to cope wth nteractons w ncrease and the contro performance may be deterorated. Therefore the author was motvated to nvestgate a T S fuzzy mode based method consderng both steady state gans and dynamc nformaton to anayze the nteractons and seect a proper contro confguraton for an MER system that ensures the mnmzed nteractng effects for a decentrazed contro system. When devsng a decentrazed controer for an MER system after the contro confguraton has been determned the performance of one contro oop cannot be assessed wthout the nformaton of other oops snce they nteract wth each other [37]. In the exstng T S fuzzy mode based decentrazed contro aws for a certan dagona contro oop extra terms that characterze the nteractng effects are added to ts ndvdua open oop mode to express the nteractng resuts. A smpe exampe s gven as foows: R : IF u s C n j j j j THEN y a u f ( u ) (.) where R denotes the th fuzzy rue u s the th nput C s the th fuzzy custer y s the th output. y a u s the mode of ndvdua open oop y u n th fuzzy rue f ( u ) s an extra term that denotes the nteractng effects caused by u j j j and n fj( uj) s the sum of extra terms that descrbe the nteractons from other j j oops. The oca controer for contro oop y x of a decentrazed contro system s desgned based on the mode wth extra terms as n Eq. (.) to hande or emnate the nfuence of nteractons. However usng extra terms n controer desgn for an MER system whch s used to smutaneousy cater for dfferent coong demands may not be feasbe because: An MER system can form a arge scae process snce t has penty of varabes to be chosen as the nputs and the outputs. For a arge scae process the number of 6

17 extra terms n the mode woud be arge. The dentfcaton of these extra terms may drastcay ncrease the cost and compexty n process modeng. An MER system generay has ntrcate structure and consderabe unmeasured physca changes. The nteractons among the contro oops may be very dffcut or even mpossbe to drecty gauge or evauate whch causes obstaces to obtan the extra terms. An MER system s nonnear. For a nonnear MIMO process dfferent workng condtons may requre dfferent contro confguratons and then resut n changed nteractng effects whch ead to chaenges n fndng sutabe extra terms to descrbe the varant nteractons. In ght of these ssues the author was motvated to nvestgate an aternatve method nstead of usng extra terms to descrbe the nteractons for decentrazed controer desgn of MER systems. For an MIMO process where the nteractons among the oops are weak or modest decentrazed contro usng the smpest contro structure can generay work wth satsfactory resuts. However strong coupng effects may exst n an MER system that decentrazed contro woud gve degraded performance due to ts mted contro structure. In ths case a more compcated contro structure shoud be empoyed whe generay not necessary gong to fu dmensona structure whch eads to the study of sparse contro. Based on the dagona decentrazed contro structure by addng severa off dagona controers sparse contro ncudng bock dagona contro and tranguar contro has the abty to acheve better performance than decentrazed contro [38]. To the best of author s knowedge no study regardng sparse contro structure seecton and sparse contro desgn based on fuzzy mode has been proposed whch motvated the author to deveop a fuzzy mode based crteron for anayzng the nteractons between dagona and off dagona eements to seect a proper sparse contro structure and then devse sparse fuzzy controers for an MER system wth cosey couped effects. The study n ths thess ams to deveop a seres of T S fuzzy mode based practca 7

18 methods for MER system contro. Accordng to the motvatons descrbed prevousy the objectves of ths thess are summarzed as foows: Based on T S fuzzy modes study a method that empoys both steady and dynamc nformaton to evauate the nteractons and then par the oops to determne the decentrazed contro confguraton wth mnmum cross coupng effects for an MER system. Investgate an aternatve method whch s smpe feasbe and effectve n MER system contro appcatons to express the nteractng effects based on T S fuzzy modes wthout usng extra terms for decentrazed contro desgn. Deveop a crteron to determne whether an MIMO process of an MER system has strong nner nteractons that decentrazed contro may not fuy hande. Afterwards seect a proper sparse contro structure by addng severa off dagona controers to the dagona contro oops and then desgn sparse controers based on T S fuzzy modes for the MER system. Appy the methods to contro an expermenta MER system to vadate ther practcabty and effectveness..3 Major contrbutons of the thess In ths thess a seres of nove and practca T S fuzzy mode based methods for MER system contro are presented. And these methods are studed based on both Type and Type 2 T S fuzzy modes. The major contrbutons are descrbed as foows:. Loop parng based on T S fuzzy modes For an MIMO process of an MER system a T S fuzzy mode based oop parng crteron s proposed that utzng steady state gans and normazed ntegrated errors of ndvdua oops to measure the nteractons and then determne the decentrazed contro confguraton. Normazed ntegrated error accounts for the response speed that can be used to represent the dynamc property of a process. Compared wth the exstng fuzzy mode based oop parng methods usng ony steady state gan the 8

19 proposed method can gve a more proper contro structure for an MER system snce both steady and dynamc nformaton of the process are consdered to par the contro oops. Moreover smpe procedures to cacuate steady state gan and normazed ntegrated error based on both Type and Type 2 T S fuzzy modes are gven. 2. Effectve T S fuzzy mode (EM) for decentrazed contro In order to assst decentrazed controer desgns for MER systems n handng the nteractons a nove manner caed effectve T S fuzzy mode (EM) s presented as an aternatve to the mode wth extra terms as n Eq. (.). For a certan contro oop n a process of an MER system an EM s derved by ncorporatng the nteractng effects quantfed by the oop parng crteron nto the coeffcents of ts ndvdua T S fuzzy mode thus the EM has same structure but revsed coeffcents from that of ts ndvdua fuzzy mode. Smpe approaches to cacuatng the coeffcents of both Type and Type 2 EMs are provded. Based on the EMs of pared oops an MIMO process can be approxmatey regarded as mutpe non nteractng SISO processes for decentrazed controer desgn. Compared wth the exstng decentrazed fuzzy contro methods addng extra terms to the ndvdua mode to characterze nteractng resuts EM method s a feasbe way that can greaty reduce the cost n process modeng and the foowng controer desgn. Moreover the oca mode of an EM has the same near structure as that of ts ndvdua T S fuzzy mode thus near SISO contro technques can be apped to desgn decentrazed contro through parae dstrbuted compensaton (PDC) [5] for the MER systems. 3. Sparse fuzzy contro In terms of the proposed oop parng method a crteron s gven to determne whether a process of an MER system has strong coupng effects among oops that decentrazed contro may not fuy hande and then to seect an approprate sparse contro structure based on T S fuzzy modes. Afterwards by vrtue of EMs of the seected oops an ndependent desgn approach s gven that the sparse controer 9

20 desgn for an MIMO process can be converted to a group of ndependent snge oop controer desgns usng near contro agorthms. Sparse contro s abe to acheve greaty mproved performance by empoyng a tte more compex contro structure than decentrazed contro and save the cost by utzng smper contro structure when compared to fu dmensona contro. Furthermore ths study provdes a patform to utze mature and deveoped near SISO contro schemes to manpuate nonnear and strong couped MER systems. 4. Appcatons The proposed T S fuzzy mode based methods are apped to an expermenta MER system wth three evaporators for ar condtonng pershabe food storage and freezng to demonstrate ther practcabty and effectveness. The expermenta resuts prove that the proposed methods are feasbe n usng near SISO contro agorthm to reguate ths MER system so that the outputs can track ther reference vaues. And the comparsons of decentrazed and sparse contro are gven to demonstrate that sparse contro s abe to acheve more satsfactory resuts when the nteracton anayss ndcates that the coupng effects among part of oops are strong that severa off dagona controers shoud be added to dagona decentrazed contro structure. And the comparsons between Type and Type 2 T S fuzzy contro systems are gven to anayze the performances of these two types of fuzzy mode n terms of robustness and computatona cost..4 Organzaton of the thess The rest of ths thess s organzed as foows: Chapter 2 ntroduces fuzzy modeng for an MIMO process that a T S fuzzy mode s but for each ndvdua oop and then a T S fuzzy mode matrx can be formed for the overa process by coectng a these ndvdua open oop modes. The detaed Type and Type 2 T S fuzzy modeng methods are ntroduced and a smuaton exampe s gven to demonstrate and compare ther errors. Chapter 3 descrbes a T S fuzzy mode based oop parng method to determne the

21 contro confguraton wth mnmum cross coupng effects for decentrazed contro. And the cacuaton procedures based on both Type and Type 2 T S fuzzy modes are presented. A smuaton exampe s used to show and compare the resuts obtaned from these two types of T S fuzzy mode. Chapter 4 presents an EM method to assst decentrazed contro n handng nteractons. Smpe cacuatons to obtan both Type and Type 2 EMs are provded. The approach to devsng controers based on Type and Type 2 EMs usng near SISO contro agorthms s gven. A nonnear process s used as an exampe to demonstrate and compare the performances of Type and Type 2 EM based decentrazed controers and ther effectve transfer functon (ETF) based counterpart. Chapter 5 provdes a gudene for devsng sparse contro for a cosey couped MIMO process ncudng a crteron of sparse contro structure seecton and an ndependent desgn method based on EMs to factate sparse controer desgn n deang wth the nteractons. The nonnear process n Chapter 4 s used as the exampe to compare the performances among the decentrazed and sparse controers desgned based on ETF Type and Type 2 EMs. Chapter 6 presents the appcatons of the proposed methods of Chapter 2 5 on an expermenta MER system. The performances of decentrazed and sparse contro based on Type and Type 2 EMs are compared and anayzed. Chapter 7 concudes the thess and provdes some possbe future research drectons.

22 Chapter 2. T S fuzzy modeng for MIMO processes 2. Introducton In an MER system penty of varabes can be seected as nputs and outputs to form an MIMO process such as vave openngs fan speeds and compressor power. These varabes can be utzed to reguate refrgerant fow rates or the temperatures of the evaporators to satsfy dfferent coong oads. In ths thess t s assumed that the processes to dea wth are open oop stabe nonsnguar at steady state condtons and square ( n n) whch can generay be expressed by Fg. 2. where y s ( n) are outputs and u s ( j n) are nputs. Modeng and dentfcaton are key steps j n devsng contro systems for an MIMO process snce an accurate mode can provde usefu nformaton for deveopng and testng dfferent types of advanced approaches. Ths chapter descrbes T S fuzzy modeng methods. In a process as shown n Fg. 2. there are n! nput output confguratons for a cosed oop contro system. And when a oops open there are 2 n ndvdua snge open oop processes (denoted by oop y u j n ). The ndvdua oop propertes are mportant for the foowng j studes n ths thess such as nteracton anayss oop parng EM method etc. To competey refect the steady and dynamc nformaton of ndvdua oops a nove modeng manner s adopted n ths thess where a SISO T S fuzzy mode s but for each ndvdua oop. When a oops open the ndvdua open oop modes are aways dentfabe wth proper persstent exctatons snce there are no nteractons from cosed oops to affect the dentfcaton resuts [39]. Denote the T S fuzzy mode for the ndvdua open oop y uj as f for the whoe process a fuzzy mode matrx can be formed by coectng a these ndvdua modes as: f f 2 f n f f f F (2.) n f nn f f f n n2 nn 2

23 Consequenty the nformaton of ndvdua oops can be obtaned from F to carry out nteracton anayss contro structure seecton and controer desgn for the MIMO process. Fgure 2. A n n MIMO process T S fuzzy mode can be constructed based on the nput output data pars samped from the orgna MIMO process. Generay the constructon ncudes two parts: one s the dentfcaton for antecedents whch s to dvde the data nto severa fuzzy custers. The other s the dentfcaton for consequents whch s to dentfy the parameters of the oca near poynomas. In ths chapter the methods to construct both Type and Type 2 T S fuzzy modes are presented. Type 2 fuzzy theory was proposed n [4] as an extenson of Type (tradtona) fuzzy theory [4]. Compared wth Type fuzzy set where the fuzzy membershps are crsp Type 2 fuzzy set has the fuzzy membershps that are themseves fuzzy [42]. Wth ths ncreased fuzzness Type 2 fuzzy set can descrbe eves of uncertanty vagueness and mprecson wth whch Type fuzzy set strugges [43] and provdes addtona degrees of freedom for desgn that make t possbe to drecty descrbe the nfuence of uncertantes [44]. A competed theory of Type 2 Mamdan fuzzy ogc systems was deveoped by [ ] at the end of ast century. Soon after Type 2 T S fuzzy system was ntroduced and dscussed n 999 [47] whch provded addtona choce and gave the chance to combne Type 2 fuzzy theory and conventona mathematca approaches for the systematc method deveopment. At the end of ths chapter smuaton resuts are provded to show and compare the errors of Type and Type 2 T S fuzzy modes. 2.2 Type T S fuzzy modeng For oop y uj n a n n MIMO process when ts ndvdua open oop mode 3

24 f s a Type T S fuzzy system the IF THEN fuzzy rues of f can be expressed as: R : IF x ( k) s C THEN y ( k) a u ( k ) a u ( k ) a u ( k p) j j p j b y ( k ) b y ( k q) q (2.2) where 2 M M s the number of fuzzy rues n f ; x ( k) s a vector that s constructed of the past nputs and outputs: x ( k) [ u ( k ) u ( k ) u ( k p) y ( k ) y ( k q)] j j j p and q are ntegers that p and q / T s the tme deay n oop y uj and T s the sampng nterva; C represents the th fuzzy custer; y ( k ) s the output of th rue; a ( r p) and b ( s q) are the r coeffcents of the near poynoma n th rue. The tota output y ( k ) s the weghted sum of oca outputs as: s M ( x ( k)) y ( k) M ( ) ( ( )) ( ) M x ( x ( k)) y k k y k (2.3) where ( x ( )) s the fuzzy membershp functon of x ( k) n fuzzy custer k C that ( x ( k)) and M ( x ( k)). The nput output data pars of oop y uj can be denoted as z ( k) [ x ( k) y ( k)] k N where N s the number of data pars samped from ndvdua oop y uj. These data pars are dvded nto M fuzzy custers: 2 M C C C then f s correspondngy characterzed by M fuzzy rues: 2 M R R R. A method s ntroduced here to construct Type T S fuzzy mode 4

25 as n Eq. (2.2). In ths method for antecedents Gustafson Kesse (G K) custerng agorthm [48] whch empoys the adaptve dstance norm to detect custers of dfferent geometrca shapes s used to cassfy the data pars. And for consequents east square method s adopted to dentfy the parameters of the oca near poynomas. The detaed steps are gven as foows:. Randomy ntaze a fuzzy membershp matrx U for the data sampes U ( z ()) ( z (2)) ( z ( N )) ( ()) ( (2)) ( ( N )) ( ( k)) z z z z M N (2.4) M M M ( z ()) ( z (2)) ( z ( N )) where ( ( k)) z denotes the ntazed fuzzy membershp vaue of z ( k) [ x ( k) y ( k)] n th custer and M ( z ( k)) for k N. 2. Based on U the center of the th custer denoted by z c ( M ) can be cacuated by z c N 2 z ( k) ( z ( k)) k N 2 ( z ( k)) k (2.5) where [ zc x c yc ] x c s the center of nput vectors and y s the center of outputs n th custer. 3. Based on the centers z c ( M ) the fuzzy covarance matrx for the th c custer denoted by F can be computed by F N 2 T ( z ( k)) z ( k) zc z ( k) zc k N k ( z ( k)) 2 (2.6) 4. Each fuzzy custer has ts own norm nducng matrx denoted by A whch can 5

26 be obtaned by pq /( 2) A det F F (2.7) where det denotes the vaue of the determnant. 5. Based on A the adaptve dstance between ( ) z k and z c denoted by D( z ( k) z ) can be yeded c T c c c D( z ( k) z ) z ( k) z A z ( k) z (2.8) 6. Based on the adaptve dstances D( z ( k) z c ) ( M ) the updated fuzzy membershp of z ( k) n th custer C can be cacuated by s f a D( z ( k) zc ) M D( z ( k) zc ) s M s s D( z ( k) zc ) s ( z ( k)) f any D( z ( k) zc ) s M s (2.9) f D( z ( k) z ) c then the updated fuzzy membershp matrx can be obtaned as: U ( z ( k)). M N 7. Gven a termnaton toerance f U U t means the data cassfcaton s satsfactory; otherwse et U U repeat Step Assgn z ( k) to the custer n whch t has the argest fuzzy membershp such that the data sampes are dvded nto M custers. For each custer usng east square agorthm to dentfy the coeffcents of poynoma n consequents ( a ( r p) and b ( s q) ). The T S fuzzy modeng s competed. r s When gven an new nput x ( k) to the f ts membershp ( x ( k)) n th fuzzy rue can be cacuated by 6

27 s f a D( x ( k) xc ) M D( x ( k) xc ) s M s s D( x ( k) xc ) s ( x ( k)) f any D( x ( k) xc ) s M s f D( x ( k) xc ) where D( ( k) c ) ( k) c ( k) c T (2.) x x x x x x. And then the output of the T S fuzzy mode can be obtaned by Eq. (2.3). Remark 2.: The operatng condton range that a T S fuzzy mode can cover s determned by the range that the data samped from the process. To guarantee the performance of fuzzy mode based controer the data pars shoud be samped from the entre operatng range of the process. And the seecton of fuzzy rue number depends on the process propertes the operatng range and the precson requrement etc. The mnma number that can satsfy the demand for accuracy shoud be chosen such that the computatona compexty n the foowng fuzzy mode based studes can be mnmzed. 2.3 Type 2 T S fuzzy modeng When there are a arge number of uncertantes that the Type fuzzy mode may not be abe to fuy hande Type 2 fuzzy mode can be used to descrbe the process. Type 2 fuzzy set was proposed based on a concept of Words mean dfferent thngs to dfferent peope [4]. Its exampes are gven n Fg The fuzzy membershp grade of an eement n a Type 2 fuzzy set can be consdered as a Type fuzzy set that ncudes prmary and secondary membershps. In the eft sde of Fg. 2.2 the perpendcuar wdth of the footprnt denotes the range of prmary membershps and the coor depth of the footprnt characterzes the vaues of secondary membershps: ghter coor means smaer vaue and darker means arger. Part (a) of Fg. 2.2 s an exampe of genera Type 2 fuzzy set. When a the secondary membershp vaues are whch means the prmary membershps are eements of an nterva Type fuzzy set 7

28 (can be caed nterva set whch s actuay a cassca set) as shown n Part (b) of Fg. 2.2 t becomes nterva Type 2 fuzzy set [45]. It s noted that because of the ncreased fuzzness the dentfcaton or the controer desgn usng Type 2 fuzzy mode requre much more computatona cost than that usng Type fuzzy mode. In order to reduce the cost the majorty of exstng Type 2 fuzzy methods for controer desgns are deveoped based on nterva Type 2 fuzzy mode because the computatons assocated wth t are very manageabe. Fgure 2.2 Type 2 fuzzy sets For the Type 2 T S fuzzy mode severa resuts can be found [ ] to demonstrate that t offers a sgnfcant mprovement on ts Type counterpart n terms of robustness under the nfuence of uncertanty. There are three dfferent structures of Type 2 T S fuzzy mode proposed n [47] as shown n Tabe 2. a of whch can descrbe processes wth nexact nformaton better than Type T S fuzzy mode [47]. Tabe 2. Type 2 T S fuzzy modes Type 2 T S fuzzy modes Mode I Mode II Mode III Fuzzy sets n Type 2 Type 2 Type Antecedents fuzzy sets fuzzy sets fuzzy sets Coeffcents n Type Crsp Type Consequents fuzzy sets numbers fuzzy sets 8

29 As descrbed n Tabe 2. Mode I can descrbe the process wth uncertantes more competey than Mode II and Mode III snce t has addtona degree of fuzzness n both antecedents and consequents. In ths secton a modeng method to construct Type 2 T S fuzzy Mode I usng nterva Type 2 sets as the antecedents and nterva sets as the parameters n consequents s ntroduced. Snce Mode II and Mode III have a part (antecedent or consequent) same as that of Type fuzzy mode combne ths Type 2 T S fuzzy Mode I constructon method wth the Type T S fuzzy modeng approach ntroduced n Secton 2.2 t s easy to obtan Type 2 T S fuzzy Mode II or Mode III for the process. For oop y uj n a n n MIMO process when ts ndvdua open oop mode f s a Type 2 T S fuzzy system the IF THEN fuzzy rues of f can be expressed as R : IF x ( k) s C THEN y ( k) a u ( k ) a u ( k ) a u ( k p) j j p j b y ( k ) b y ( k q) q (2.) where C ( M ) denotes the th nterva Type 2 fuzzy set the fuzzy membershp of x ( k) n C s an nterva that can be denoted as ( ( )) ( ( )) x k b x k rb( x ( k)) where b( x ( k)) and rb( x ( k)) are the eft and the rght bounds. a ( r p) and b s ( s q) are aso r ntervas wth eft and rght bounds as a = r a b r a rb r ( r p) and b = s b b s b rb s ( s q ). And the output of th fuzzy rue s y ( k) y ( k) y ( k) that can be cacuated by [24] b rb y b( k) a b u j ( k ) a b u j ( k ) a b p u j ( k p) b b y ( k ) b b q y ( k q) y rb( k) a rb u j ( k ) a rb u j ( k ) a rb p u j ( k p) b rb y rb q ( k ) b y ( k q) (2.2) 9

30 Based on these M fuzzy rues a type reduced set can be obtaned [24 47] as y ( k) f ( x ( k)) y b( k) y rb( k) (2.3) An approach to cacuatng y b( k ) and y ( k ) s gven n [47] whch s an teratve process. In ths thess a smper way as foows s adopted for savng computatons as rb y y b rb ( k) ( k) M M b b M b rb rb M ( x ( k)) y ( k) rb ( x ( k)) ( x ( k)) y ( k) ( x ( k)) (2.4) Based on Eq. (2.4) the crsp output can be derved by defuzzfyng the type reduced set y ( k ) as y ( k) y ( k) y ( k) b rb (2.5) 2 It shoud be noted that dfferent from the fuzzy membershps n Type fuzzy mode M b( x ( k)) and M rb ( x ( k)) n Type 2 fuzzy mode may not equa. Type 2 T S fuzzy modeng method s an extenson of the Type fuzzy modeng method. The steps are presented as foows [24]. For antecedents: Based on the data sampes z ( k) [ x ( k) y ( k)] ( k N ) M fuzzy custer centers can be determned by G K agorthm as ntroduced n Secton 2.2. And then a crsp fuzzy membershp ( z ( )) can be cacuated by Eq. k (2.9) for each sampe n th custer whch s utzed as the center of Type 2 fuzzy membershp ( z ( )) for z ( k) as k b( z ( k)) rb( z ( k)) ( z ( k)) (2.6) 2 2

31 where where b( z ( k)) ( z ( k)) rb( z ( k)) ( z ( k)) denotes the radus of the nterva ( z ( )). The vaue of k (2.7) represents the degree of nfuence from uncertantes. It can be determned by expert experence or cacuated from the data sampes as foows. Frsty assgn z ( k) ( k N ) to the custer n whch t has the argest fuzzy membershp to dvde data sampes nto M custers. In each custer choose R ( R ) groups of data sampes. The seecton crteron s that the dstances among the data n a group shoud be smaer than a gven mt denoted by whch s z ( k) z ( s) where ( ) z k and z () s are dfferent sampes n a group. Secondy n each group defne the maxmum membershp vaue as max( ) and r the mnmum as mn( ) r R. And then r can be approxmatey evauated by the foowng equaton. R max( ) mn( ) r r (2.8) R 2 r Afterwards the nterva ( z ( )) of z ( k) can be obtaned by ( z ( )) and accordng to Eq. (2.7). k k For consequents: In order to dentfy the bounds of a r ( r p) and b ( s q) the fuctuatng range of output y caused by the uncertantes denoted s by y ( ) needs to be assessed. One way to determne y s based on the y expert experence another way s based on the data sampes as foows. Frsty n each custer choose R ( R ) groups of data sampes. The seecton crteron s that the dstances among the nput vectors of the sampes n a group are not 2

32 arger than a gven mt denoted by 2 whch means 2 x ( k) x ( s) where x ( k) and x () s are dfferent samped nput vectors n a group. Secondy n each group defne the maxmum vaue of the output as max( y ) and mnmum as mn( y ) r R and then the fuctuatng range of the output r r denoted by y can be evauated as max( ) mn( ) y y r r y max r R 2 y max y M (2.9) After obtan the vaue of y two data pars can be obtan from each sampe z ( k) [ x ( k) y ( k)] as z z ( k) [ x ( k) y ( k)] where y ( k) y ( k) y b b b ( k) [ x ( k) y ( k)] where y ( k) y ( k) y rb rb rb (2.2) Consequenty n each custer the coeffcents n the two near poynomas of Eq. (2.2) can be dentfed by east square agorthm based on z ( k ) and z ( k ) respectvey such that the foowng equatons wth ntervas as ts coeffcents can be acqured for each fuzzy rue. b rb y ( k) y b( k) y rb( k) a b a rb u j ( k ) a b a rb u j ( k ) a b p a rb p u j ( k p) b b b rb y ( k ) b b q b rb q y ( k q) (2.2) When gven a new nput x ( k) to the Type 2 T S fuzzy mode f the center of ts fuzzy membershp n each fuzzy custer can be cacuated by Eq. (2.) and the eft and the rght bounds can be determned wth ( M ) by Eq. (2.7). Afterwards usng Eq. (2.2) (2.4) and (2.5) to obtan the crsp output y ( k ). Remark 2.2: The vaue of a fuzzy membershp shoud be n the nterva [ ] whch 22

33 means the eft and the rght bounds of ( z ( )) shoud satsfy k ( z ( k)) ( z ( k)). Therefore when the vaues of b rb ( M ) are obtaned f ( z ( k)) then et ( z ( )) ; If ( z ( k)) then et ( z ( )). rb k b k Remark 2.3: Wth the ncreased fuzzness n the parameters the Type 2 fuzzy mode has stronger abty to descrbe uncertantes whe the computatona cost of ts modeng procedure and the foowng controer desgn s ncreased accordngy even though usng nterva Type 2 fuzzy mode. Therefore when the degree of uncertanty s n the range that tradtona fuzzy modes can fuy hande usng Type fuzzy mode nstead of Type 2 fuzzy mode s abe to reduce the compexty n both process modeng and controer desgn. 2.4 Smuaton Snce the modeng method n ths thess s to construct T S fuzzy modes for ndvdua oops whch are SISO processes n ths secton the ntroduced Type and Type 2 T S fuzzy modeng methods are apped on a SISO nonnear process to demonstrate and compare ther accuraces. Consder the foowng nonnear functon 2 dy 36 y 2 u ( u) y dt 5u 2 (2.22) where the tme deay s.sec the sampng nterva s T.sec thus / T. And the range of nput s [ 2]. Suppose there are dsturbances wth random vaues mted n [.8.8] on the output choose the vaues of p and q n Eq. (2.2) and Eq. (2.) as p and q and then the samped data pars are n the z x where x ( k) u( k ) y( k ) form of ( k) [ ( k) y( k)]. Coect 2 data pars from Eq. (2.22) and set the number of fuzzy rues as 5. Based on these samped data pars a Type and a Type 2 T S fuzzy mode for ths nonnear process can be constructed usng the modeng methods ntroduced n ths chapter. The dentfed 23

34 centers of fuzzy custers denoted by z [ x ( k) y ( k)] where c c c x c uc ( k ) yc( k ) ( 5 ) are gven n Tabe 2.2. The radus of the nterva ( x ( k)) for Type 2 fuzzy mode s.2 ( 5 ). The dentfed parameters of the consequents n Type and Type 2 T S fuzzy modes are presented n Tabe 2.3. The errors of two types of T S modes are ustrated n Fg. 2.3 and root mean square errors (RMSE) are empoyed to characterze the accuraces of the modes whch are presented n Tabe 2.4. As can be seen n Fg. 2.3 and Tabe 2.4 Type 2 T S fuzzy mode can acheve hgher degree of accuracy than Type T S fuzzy mode when the process s under the nfuence of dsturbance. 3 error of Type- mode sampe No. 3 error of Type-2 mode sampe No. Fgure 2.3 The errors of Type and Type 2 fuzzy modes Centers of Tabe 2.2 The centers of fuzzy custers for Eq. (2.22) No. of fuzzy custers custers C R (=) R 2 (=2) R 3 (=3) R 4 (=4) R 5 (=5) u c(k τ) y c(k ) y c(k)

35 Tabe 2.3 The consequent parameters of Type and Type 2 modes for Eq. (2.22) No. of Type mode Type 2 mode Fuzzy rues a b a b R (=) R 2 (=2) R 3 (=3) R 4 (=4) R 5 (=5) a rb b b b rb Tabe 2.4 The RMSEs of Type and Type 2 modes for Eq. (2.22) Type of T S fuzzy mode RSME Type.753 Type Summary In order to competey express the propertes of ndvdua oops n an MIMO process n ths chapter a SISO T S fuzzy mode was estabshed for each ndvdua oop to form a fuzzy mode matrx for an MIMO process. Both Type and Type 2 T S fuzzy modeng methods were ntroduced. Compared wth a Type fuzzy mode the Type 2 fuzzy mode wth ncreased fuzzness has stronger capabty to descrbe the uncertantes and acheve hgher accuracy whch has been proved by the smuaton resuts. Whe the computatona costs for the mode constructon and the foowng studes based on Type 2 fuzzy mode are aso ncreased. Whch type of T S fuzzy mode shoud be chosen depends on the degree of uncertantes and the requrement of accuracy n the rea appcatons. Based on the Type and the Type 2 T S fuzzy mode matrces usefu nformaton can be derved to carry out the studes n ths thess. In the next chapter the nteractons anayss and oop parng to determne the optma decentrazed contro confguraton based on both Type and Type 2 T S fuzzy mode matrces for an MIMO process are presented. 25

36 Chapter 3. Interacton anayss and oop parng methods based on T S fuzzy modes for MIMO processes 3. Introducton Athough consderabe sophstcated technques have been proposed for mutvarabe contro decentrazed contro s more prevaent n ndustra appcatons because t possesses outstandng advantages such as smpcty n desgn and mpementaton fewer parameters for tunng and ow cost n mantenance [33 34]. How to cope wth the nteractons among the oops n an MIMO process s the man probem for decentrazed contro to sove. The prmary step for decentrazed controer desgn s to par the nputs and the outputs to obtan a contro confguraton where the pared oops have mnmum crossng couped effects such that they can mosty resembe a group of SISO processes whch are ndependent to each other subsequenty the controer desgn and tunng can be argey factated by SISO contro methods [35 5]. Currenty a number of approaches for nteracton anayss and oop parng are avaabe to determne the decentrazed contro structure. One of the most popuar methods s the reatve gan array (RGA) based crteron that was proposed by [52] n 966. Generay RGA based parng rues are used n conjuncton wth the Nedernsk ndex (NI) [53] to guarantee system stabty. The RGA NI based crteron ony empoys the open oop steady state gan matrx of an MIMO process to evauate the nteractng effects and determne contro confguraton. It s very smpe n cacuaton and the scang s ndependent because of ts rato nature [54]. However snce no dynamc propertes of the process are consdered usng ony steady state gan may resut n ncorrect parng resuts consequenty the contro performance may be degraded. In order to overcome ths mt severa oop parng methods that usng both steady state gan and dynamc propertes have been deveoped ater. Such as the Dynamc RGA (DRGA) methods [55-59] whch empoy the transfer functon matrx nstead of the steady state gan matrx for RGA cacuaton. And the effectve reatve 26

37 gan array (ERGA) based method [5] whch adopts steady state gan and bandwdth of the process transfer functon eement for nteracton measurement. And the reatve normazed gan array (RNGA) based crteron [35] uses steady state gan matrx and normazed ntegrated error matrx of the process for oop parng. Among these methods RNGA based oop parng crteron has promnent advantages: t gves a comprehensve descrpton for dynamc nteractons among ndvdua oops wthout requrng the specfcaton of the controers and t can provde unque oop parng resuts. Moreover t s very smpe for researchers and engneers to understand and make parng decsons n theoretc studes and rea appcatons [35]. These crtera are a proposed and can be easy mpemented based on transfer functons whe cannot be drecty apped to a fuzzy mode that conssts of IF THEN rues and s ntrnscay nonnear. Mutvarabe fuzzy contro requres the fuzzy mode based method for nteracton anayss and oop parng. To the best of author s knowedge very few pubshed academc papers can be found n ths area. Among the few [36] has proposed a method to anayze the nteractons among oops based on fuzzy bass functon networks mode usng RGA. As descrbed before RGA based oop parng rues may provde naccurate resuts snce no dynamc nformaton of the process s ncuded. Another probem n [36] s that t used a fuzzy modeng manner that n MISO fuzzy modes were constructed for an n n MIMO process from whch the characterstcs of the ndvdua oops may not be obtaned. Furthermore sngeton rather than poynoma was adopted to be the consequents n the fuzzy rues whch may not have the capabty to competey refect the propertes of the process. In ths chapter based on the T S fuzzy mode matrx F where a SISO mode s constructed for each oop of an MIMO process as descrbed n Chapter 2 the defntons of steady state gan matrx and normazed ntegrated error matrx are gven to present the RNGA based oop parng rues. And the cacuaton procedures based on both Type and Type 2 T S fuzzy modes are provded. The proposed method s smpe n computaton and gves an aternatve to decde the contro 27

38 confguraton for an MIMO process where the exact mathematca modes are dffcut to derve. Compared wth the exstng fuzzy mode based parng approaches the proposed method usng RNGA crteron s abe to gve more approprate contro confguratons snce both steady and dynamc propertes of the oops are consdered. A smuaton exampe s empoyed to compare the resuts obtaned from T S fuzzy modes to that from accurate mathematca functons to demonstrate the smpcty accuracy and effectveness of the proposed method. And the comparson between Type and Type 2 T S fuzzy modes s aso provded to prove that Type 2 T S fuzzy mode can acheve more accurate resuts under the nfuence of uncertantes. 3.2 Loop parng crtera Accordng to the defnton proposed n [52] the reatve gan for a oop n an MIMO process descrbed by T S fuzzy mode can be defned as ( y / uj ) ur j (3.) ( y / u ) j yr where denotes the reatve gan of oop y uj ( y / uj ) u r j s the steady state gan of ndvdua oop y uj and ( y / uj ) y r s the process gan of the same oop when a other oops cose. In order to cacuate from f operatng pont must be gven due to the nonnear nature of fuzzy mode. Dfferent an operatng ponts may resut n dfferent reatve gans. Denote an operatng pont x for oop y uj of an n n MIMO process as x [ u ( k ) u ( k p) y ( k ) y ( k q)] (3.2) j j where j n. Denote the steady state gan of f based on x as k. In ths thess the step response of the process s utzed to defne the vaue of steady state gan whch means k s the gan n output of the process when the nput s a unt step sgna. Coect k of each oop the steady state gan matrx of 28

39 F denoted by K can be formed as k k2 k n k k k K (3.3) n k nn k k k n n2 nn As the defnton n Eq. (3.) the reatve gan for f s the rato of two gans frst the steady state gan of the soated oop y uj k second the steady state gan n the same oop when a other oops are cosed denoted by k ˆ. k kˆ (3.4) Consequenty the RGA for F denoted by s expressed as 2 n (3.5) n nn n n2 nn whch can be cacuated ony usng the steady state gans of ndvdua oops as [52] K K (3.6) T where ndcates eement by eement product and K s the transpose of the T nverse of K. Smar to the parng rues for an MIMO process descrbed by transfer functon matrx the RGA parng rues for an MIMO process descrbed by T S fuzzy mode matrx are that nputs and outputs n a decentrazed fuzzy contro system shoud be pared as foows. () The pared RGA eements shoud be postve () The pared RGA eements shoud be cosest to () Large RGA eements shoud be avoded. 29

40 By pacng the pared oops at the dagona poston n K through coumn swap the Nedernsk ndex (NI) [53] for T S fuzzy mode denoted by N can be defned as where det K s the determnant of det K N (3.7) n k K after coumn swap and k s ( n) are the dagona (pared) eements. NI provdes a necessary condton for a stabe pared system that s: f the NI s negatve the processes w be unstabe for a possbe (any) vaues of controer parameters (.e. t w be structuray monotonc unstabe ) [35]. Therefore an addtona rue for RGA based oop parng s (v) N for the pared structure. The man advantage of RGA NI based oop parng crteron s that t s smpe n cacuaton as t ony depends on the steady state gan matrx whch s generay easy to compute. However a potenta weakness of ths crteron s the same fact that t ony uses the steady state gans whch based on the assumpton of perfect oop contro to determne oop parng and no dynamc nformaton of the process s consdered. In order to make more accurate and effectve evauaton of contro oop nteractons the normazed ntegrated error whch can be used to represent dynamc property of a process s ncuded nto the assessment of nteractng effects. The concept of normazed ntegrated error s ntroduced as foows. Normazed Integrated Error: Normazed ntegrated error s the normazed sum of the errors between the steady state vaue and the outputs of a process. Same as that of the defnton of k the normazed ntegrated error of y uj based on the T S fuzzy mode denoted by e s defned from the step response of y uj as e y( ) y( r T) T (3.8) k r where T s the sampng nterva y( ) y( k) s the steady state output vaue k 3

41 when the nput s a unt step sgna. And y ( r T) s the output at the rth sampe tme of f. Normazed ntegrated error s drecty reated to the process dynamcs snce t refects the response speed of the y to u. Smaer absoute vaue of e j ndcates that ths oop has fast response to the partcuar nput whe arger one ndcates the oop has sower response [35]. Note that e of a rea process s generay not equa to thus n ths thess t s assumed that e. Two typca exampes are gven n Fg. 3. where the shaded area determnes the vaue of e. Fgure 3. Typca waveforms for processes For the overa process the normazed ntegrated error matrx of F denoted by E s expressed as e e 2 e n e e e E (3.9) n e nn e e e n n2 nn So far two mportant factors from f have been defned for nteracton anayss: 3

42 () Steady state gan k : t refects the effect from u j on the gan of y when the process reaches the steady state condton; () Normazed ntegrated error e : t ndcates the response speed of y to u j. In order to combne both steady state gan and normazed ntegrated error for nteracton measurement and oop parng the normazed gan of f k rato of steady state gan to normazed ntegrated error s utzed: N as the k N k (3.) e Eq. (3.) provdes a tota effects of u j to y. For the overa process the normazed gan matrx of F denoted by K N can be expressed as kn kn 2 kn n kn 2 kn 22 k N 2n KN k N Κ nn E (3.) kn n kn n2 kn nn where ndcates the eement by eement dvson. Denote the normazed gan of f when a other oops cose as k ˆN kˆ ˆ ˆ N k / e and e ˆ s the normazed ntegrated error of oop y uj when a other oops cose. And then the reatve normazed gan can be defned as k kˆ N (3.2) N And then the RNGA denoted by s 2 n (3.3) n nn n n2 nn 32

43 whch can be cacuated by ony usng the nformaton of ndvdua oops as [35 6 6] K K (3.4) T N N The RNGA can be taken as a compement to RGA NI based parng crteron such that the dynamc property of the process can be ncuded nto the contro structure decson. The updated parng rues for an MIMO process are the foowng. () A pared RGA and RNGA eements shoud be postve () The pared RNGA eements shoud be cosest to () (v) Large RNGA eements shoud be avoded NI shoud be postve. RNGA based oop parng crteron provdes mportant nsghts nto the ssue of contro structure seecton because t takes RNGA RGA and NI nto consderaton. The sgnfcances of ths parng method are: The parng crteron s mpemented on T S fuzzy mode constructed from data sampes whch means the decson for contro confguraton can be made based on data sampes wthout exacty knowng other nformaton such as the nterna structure or parameters of the system. Therefore t provdes a smpe and feasbe way to evauate the nteractons among the oops and then determne an approprate oop parng structure. Compared wth the exstng oop parng method based on fuzzy modes n [36] the proposed method s based on the T S fuzzy mode matrx whch can provde ndvdua oop propertes of an MIMO process and usng RNGA crteron that empoys both steady state and dynamc nformaton to measure the nteractons such that t s abe to provde more accurate resuts for contro oop confguraton. 3.3 Cacuatons based on Type and Type 2 T S fuzzy modes In ths secton the cacuatons of RNGA oop parng crteron based on both Type and Type 2 T S fuzzy modes are presented. 33

44 . Cacuaton based on Type T S fuzzy mode In the vcnty of the gven operatng pont x as n Eq. (3.2) nonnear Type T S fuzzy mode of Eq. (2.2) can be approxmatey represented by a near functon by ettng ( x ( k)) ( x ) ( M ) as y ( k) f ( x ( k)) a u ( k ) a u ( k p) b y ( k ) b y ( k q) j p j q (3.5) M where a ( x ) a ( r p) and b ( x ) b ( s q). r r M s s Through Z transform [62] Eq. (3.5) can be converted to a dscrete transfer functon as a a z a z G z z ( ) ( ) p Y () z p () 2 q U j z b z b 2 z b qz (3.6) where G () z denotes the nearzed functon of oop y uj n Z doman Y( z) Z [ y] and U j( z) Z [ u ]. Based on G () z the steady state gan k of j oop y uj at the operatng pont x can be easy computed accordng to fna vaue theorem as a a z a z k G z z p p ( ) z 2 q ( b z b2 z b qz ) z (3.7) a a a p ( b b b ) 2 q Note that usng fna vaue theorem of Z transform to cacuate steady state gan can greaty reduce the computatona compexty when compared to the method proposed n [36]. Accordng to the defnton of normazed ntegrated error n Eq. (3.8) the nput s a unt step sgna that can be expressed by 34

45 U ( z) Z[ u ] j j 2 r r z z z z z z r (3.8) Substtute Eq. (3.8) nto Eq. (3.6) the output Y ( z) Z [ y ] becomes a a z a z z Y z G z U z z ( b z b z b z ) z p p ( ) ( ) j ( ) 2 q 2 q (3.9) And accordng to the defnton n Z transform Y () z can be expressed as Y z y T y T z y T z y r T z 2 r ( ) ( ) ( ) (2 ) ( ) r y ( r T) z r (3.2) Accordng to Eq. (3.2) Eq. (3.8) can be rewrtten as y( ) y( r T ) ( y ( ) y ( r T )) r r e T T k k r y ( ) ( y ( T ) y ( T ) y ( r T ) ) r T k y z z y T y T z y r T z k r r ( ) ( ) ( ( ) ( ) ( ) ) r y ( ) z Y ( z) r y( ) z Y( z) T T k k z k z z T z (3.2) For a stabe process when the nput s a unt step sgna t s easy to earn that y ( ) k thus Eq. (3.2) can be revsed by e z Y () z T z k z (3.22) Substtutng Eq. (3.7) and (3.9) nto Eq. (3.22) gves that 35

46 p z ( b b q ) a a z a pz z e z T q z a a a p ( b z b qz ) z p ( b b q ) a a z a pz z T T q a a a p ( bz b qz ) z p p q r s w a r ( z ) a wb s z z r w s q a a a p b z b qz z T T z z ws q p p q r mn( w s) r ( ) w s sgn r w s a z a b z z w s a a a p b z b qz mn( ws ) r sgn w s r z w s z q a a a p b z b qz z z z z T T z p r p q ws z z a a b z w s T T (3.23) It s obvous that ( z ) / ( z ) z z z r 2 r and 2 ( r ) z z z z r therefore t foows that z z r z z r and z w s z ws Then Eq. (3.23) can be arranged to obtan p p q r w s sgn r w s a a a p b b q ra a b w s w s e T T (3.24) Consderng Eq. (3.7) and Eq. (3.24) the steady state gan and normazed ntegrated error of each oop can be easy cacuated from f ( j n ). And then K and E can be formed to cacuate and N. Fnay the contro confguraton can be determned accordng to the rues of RNGA based oop parng crteron. 36

47 2. Cacuaton based on Type 2 T S fuzzy mode Compared wth the cacuaton procedure based on Type T S fuzzy mode the cacuaton based on Type 2 T S fuzzy mode requres one more step of defuzzfcaton snce the parameters of Type 2 T S fuzzy mode are of ncreased fuzzness. Based on the fuzzy rues n Eq. (2.) n the vcnty of the gven operatng pont x as n Eq. (3.2) the foowng type reduced set of output can be obtaned as y ( k) y b( k) y rb( k) a u ( k ) a u ( k ) a u ( k p) j j p j b y ( k ) b y ( k q). q (3.25) In Eq. (3.25) a r a b r a rb r ( r p ) and b s b b s b rb s ( s q) where a b r M b b r M ( x ) a ( x ) b a rb r M rb rb r M ( x ) a ( x ) rb and b b s M b b s M ( x ) b ( x ) b b rb s M rb rb s M ( x ) b ( x ) rb and y b( k) a b u j ( k ) a b u j ( k ) a b p u j( k p) and b y ( k ) b y ( k q) b b q y rb( k) a rb u j ( k ) a rb u j ( k ) a rb p u j ( k p). b y ( k ) b y ( k q) rb rb q And the foowng near equaton can be obtaned by defuzzfyng y ( k ) n Eq. (3.25) accordng to the method presented n Eq. (2.5) as y b( k) y rb( k) y ( k) 2 a u ( k ) a u ( k p) b y ( k ) b y ( k q) j p j q (3.26) 37

48 where a r a b r a rb r ( ) / 2 r p and b ( b b ) / 2 s b s rb s s q. Eq. (3.26) can be used to approxmatey represent a Type 2 T S fuzzy mode f n the area around x and t can be converted to a dscrete transfer functon as the form n Eq. (3.6) by Z transform. Consequenty the steady state gan k and normazed ntegrated error e of each oop can be easy cacuated by usng Eq. (3.7) and Eq. (3.24) and then and N can be cacuated based on K and E. Fnay the contro structure can be determned n accordance wth the rues of RNGA based oop parng crteron. Eq. (3.7) and Eq. (3.24) seem to be compcated but actuay very smpe n rea appcatons snce the number of p and q are generay not arge. For exampe when p and q the Type and the Type 2 T S fuzzy rues are n the form of and R : IF x ( k) s C THEN y ( k) a u ( k ) b y ( k ) j R : IF x ( k) s C THEN y ( k) a u ( k ) b y ( k ) j In ths case k and e can be smpy cacuated by: a b k e T T b b 3.4 Smuaton In ths secton both Type and Type 2 T S fuzzy mode matrces are constructed for an MIMO process wth bounded dsturbance and then the parng structures are determned based on these two types of fuzzy modes by usng RNGA based oop parng crteron. The comparsons are presented to demonstrate that both Type and Type 2 fuzzy modes can provde accurate parng structure and the resuts obtaned 38

49 from Type 2 modes are wth smaer errors than that from Type modes when the process s under the nfuence of uncertantes. Consder an n n process where n 3 as descrbed n [35] ts mathematca transfer functon matrx s as foows. 9s 5s 3s e 9e 3e s 7s s 4s 3s 35s 3s 2s 5s 5e 8e 7e G () s s 9s s 33s s 3s (3.27) 3s 7s s 6e 3e e s 5s s 4s 3s 25s Its accurate steady state gan matrx denoted by K k 33 and accurate normazed ntegrated error matrx denoted by E e are [35] 33 K and E And then RGA and RNGA 33 can be obtaned as and The oop parng structure determned by RNGA based crteron s y u / y u / y u where NI s N Fgure 3.2 AN MIMO process wth dsturbances Suppose there are dsturbances added to the process when coectng data sampes of ndvdua oops as shown n Fg. 3.2 and the dsturbances are random but bounded n 39

50 .3.3. Choose p q and sampng nterva T sec y j. For oop u sampe the data pars as z ( k) [ x ( k) y ( k)] where x ( k) [ u ( k ) y ( k )] N 3. The constructed Type and Type 2 T S j fuzzy modes are presented n Appendx A. The operatng ponts are x [ u ( k ) y ( k )] [ ] for j 23 from the Type fuzzy j modes the foowng resuts can be derved as K E Accordng to these resuts the parng structure can be determned by RNGA based crteron as: y u2 / y2 u3 / y3 u where NI s N.765. From the Type 2 fuzzy modes the foowng resuts can be obtaned as K E Accordng to these resuts the parng structure can be determned by RNGA based crteron as: y u2 / y2 u3 / y3 u where NI s N As demonstrated n the above resuts both Type and Type 2 T S fuzzy modes but based on the data wth nexact nformaton can gve accurate parng structures. In order to further compare the degrees of accuracy between these two types of fuzzy mode the mean absoute error (MAE) ndex s empoyed to show ther errors. Snce the RGA RNGA and NI are a cacuated from K and E we compare MAE of 4

51 K : n n 2 MAE / and MAE of j k k k n E : n n e e e / n MAE j 2 between Type and Type 2 T S fuzzy modes. The resuts are gven n Tabe 3.. Tabe 3. The comparsons of MAEs Type of fuzzy mode k MAE e MAE Type Type As shown n Tabe 3. both k MAE and e MAE of Type 2 fuzzy mode are ess than that of Type fuzzy mode whch demonstrates that Type 2 fuzzy mode can yed more accurate resuts than Type fuzzy mode under the nfuence of uncertantes. 3.5 Summary Ths chapter ntroduced RNGA based oop parng crteron for MIMO processes descrbed by T S fuzzy modes and the smpe cacuaton procedures based on both Type and Type 2 T S fuzzy modes were presented. The proposed method can anayze the nteractons between process nputs and outputs and then par the oops to determne the decentrazed contro confguraton wthout requrng accurate mathematca functons of an MIMO process. Compared wth the exstng fuzzy mode based approaches whch ony use steady state gan for oop parng ths method can provde a more proper contro structure snce both steady and dynamc nformaton of the process were consdered. The smuaton demonstrated that under the nfuence of uncertantes both Type and Type 2 fuzzy modes can gve accurate contro confguraton whe the errors of the cacuated resuts of Type 2 fuzzy modes were smaer than that of Type fuzzy modes. Snce ths parng crteron can quantfy the nteractons among the oops an effectve fuzzy mode can be formed by ncorporatng the nteractons nto the ndvdua fuzzy mode for a certan pared oop. Effectve fuzzy mode s a smpe and practca way wthout usng extra terms to express the nteractng resuts for decentrazed controer desgn whch w be eaborated n the next chapter. 4

52 Chapter 4. Effectve T S fuzzy mode for decentrazed contro of MIMO processes 4. Introducton In decentrazed controer desgn for an MIMO process the performance of one oop cannot be assessed wthout the nformaton of other oops snce controers nteract wth each other [37]. In the exstng T S fuzzy mode based decentrazed contro methods extra terms are added to the ndvdua open oop mode to characterze the nteractng effects as n Eq. (.). Snce the extra terms w ncrease the compexty n both modeng and controer desgn for an MIMO process of an MER system n ths chapter an aternatve caed effectve mode s presented to descrbe the nteractng effects nstead of usng extra terms. Effectve modes are used to descrbe a group of non nteractng equvaent SISO processes whch are but based on the contro oops to represent an MIMO process for decentrazed controer desgn. By vrtue of dfferent oop parng crtera recenty a number of effectve mode methods based on transfer functons caed effectve transfer functons (ETF) have been deveoped. [37] proposed a method usng dynamc reatve gan to derve ETFs for equvaent open oop processes wthout pror knowedge of controer dynamc propertes whch s smpe and effectve for ow dmensona MIMO system but conservatve for hgh dmensona system due to the nevtabe modeng errors n formuaton. [63] presented an ETF method n terms of effectve reatve gan array (ERGA) for MIMO processes severa exampes demonstrated ts smpcty and effectveness n both ow and hgh dmensona system. The dsadvantage of ths method s that the computaton of ERGA requres the vaue of crtca frequency of each ndvdua oop. Dfferent crtera for seectng crtca frequency ponts may resut n dfferent ERGAs and then causes uncertantes n parng structures and the ETFs. In order to overcome ths weakness the approaches usng RNGA to obtan ETFs for controer desgns were ater deveoped [38 64] whch can obtan unque resuts of contro structure and effectve modes for controer 42

53 desgns and acheve good performance wth ess computatona compexty. To the best of the author s knowedge no pubshed studes regardng oop parng crteron based effectve mode have been found n fuzzy area. Because of the ntrnsc nonnearty and speca structure of fuzzy modes the exstng ETF methods are not drecty appcabe to fuzzy contro systems. In order to factate decentrazed fuzzy contro n ths chapter based on the T S fuzzy mode matrces ntroduced n Chapter 2 and the RNGA parng crteron presented n Chapter 3 an effectve T S fuzzy mode (EM) method s proposed. For a pared oop n an MIMO process an EM can be obtaned by smpy scang the coeffcents of ts ndvdua open oop T S fuzzy mode accordng to the quantfed nteractons provded by RNGA crteron and the scaed coeffcents are further revsed to adapt dfferent nteractng cases. EM can refect the nteractng effects caused by a other cosed oops on both steady and dynamc propertes through a smpe and drect manner whe the ntegrty of contro system can be mantaned. The consequents of an EM aso have the same near structure as that of ts ndvdua T S fuzzy mode. Hence based on the EMs of pared oops the oca controers of a decentrazed contro system can be ndependenty desgned usng near SISO contro agorthms through parae dstrbuted compensaton (PDC) [5]. Compared wth the exstng decentrazed fuzzy contro approaches no extra terms to characterze nteractons are requred n the EM method whch can save consderabe costs n modeng and controer desgn. When the parng structure changes as the operatng condtons vary n a nonnear process the EMs can be qucky updated snce the cacuaton s smpe whch makes t appcabe n rea tme contros. Compared wth the ETF methods EM can work wthout knowng exact mathematca functons of the process and provdes a bass to deveop robust controers snce fuzzy system s powerfu to hande the uncertantes. In the smuaton secton the performances of decentrazed controers based on ETFs and EMs for an MIMO process wth and wthout parametrc uncertantes are presented and compared to demonstrate the superortes of the proposed EM method over ts ETF counterpart. And the comparsons between 43

54 Type and Type 2 EMs are aso gven to show that Type 2 fuzzy system can acheve more robust resuts under the nfuence of uncertanty. 4.2 EM Denote the EM for oop y uj n an n n MIMO process as f ˆ. At a certan operatng condton suppose the optma contro confguraton has been determned. Snce the contro oop when other oops cose has smar frequency propertes wth that when other oops open f the process s we pared [63 65] t s reasonabe to et the structure and part of parameters of f ˆ be same as that of f. For Type EM the fuzzy rues of f ˆ can be expressed as R : IF x ( k) s C THEN y ( k) aˆ u ( k ˆ ) aˆ u ( k ˆ ) aˆ u ( k ˆ p) j j p j b y ( k ) b y ( k q) q (4.) where a ˆ r ( r p M ) and ˆ are the dfferent parameters from ts orgna Type T S fuzzy mode as n Eq. (2.2). And for Type 2 EM the fuzzy rues of f ˆ can be expressed as R : IF x ( k) s C THEN y ( k) aˆ u ( k ˆ ) aˆ u ( k ˆ ) aˆ u ( k ˆ p) j j p j b y ( k ) b y ( k q) q (4.2) ˆ ˆ ˆ ( r p M where a r a b r a rb r parameters from ts orgna Type 2 T S fuzzy mode as n Eq. (2.). ) and ˆ are the dfferent The quantfed nteractons from RNGA based parng crteron can be utzed to cacuate the parameters of EMs. The nteractng effects on steady state gan can be derved by reatve gan whe the nteractng effects on dynamc property can be obtaned from both reatve gan and reatve normazed gan as eˆ (4.3) e 44

55 where defned as reatve normazed ntegrated error represents the normazed ntegrated error change of oop y uj when other oops cose. The reatve normazed ntegrated error array denoted by can be cacuated as Based on and gven as foows. (4.4) nn the approaches to obtan Type and Type 2 EMs are. Cacuatons to determne coeffcents of Type EM Accordng to Eq. (3.7) the steady state gan k ˆ at the gven operatng pont x of Type EM f n Eq. (4.) can be cacuated as ˆ kˆ aˆ aˆ aˆ p (4.5) ( b b b ) 2 q M where aˆ ˆ ( x ) a r p. Submttng Eq. (3.4) and Eq. (3.7) nto r r Eq. (4.5) the foowng equaton to can be obtaned. aˆ r a r (4.6) And then a ˆ n Eq. (4.) can be cacuated by r aˆ r a r (4.7) Accordng to Eq. (3.24) the normazed ntegrated error e ˆ of f ˆ can be cacuated by p p q ˆ ˆ r w s sgn r w s aˆ ˆ ˆ a a p b b q ra a b w s w s eˆ T ˆ T (4.8) By consderng Eq. (4.6) Eq. (4.8) can be rewrtten as 45

56 p p q a r a w r b s w s sgn w s r w s eˆ ˆ T T a a2 a p b b q p p q ra a b w s sgn w s r w s r w s a a a b p b q T T ˆ (4.9) Substtutng Eq. (3.24) and Eq. (4.3) nto Eq. (4.9) after arrangement gves the foowng equaton to cacuate ˆ. p p q r w s sgn r w s a a a p b b q ra a b w s w s ˆ ( ) (4.) Severa expermenta resuts demonstrate that for we pared MIMO processes the vaues of s of pared oops are cose to. Therefore the cacuaton of ˆ for pared oop n Eq. (4.) can be smpfed as (4.) ˆ Eq. (4.) s a practca formua to cacuate EMs. Even though t s ess accurate than Eq. (4.) a number of smuatng exampes prove that the contro performances are comparabe by these two formuas. Moreover Eq. (4.) s much more straghtforward expanabe and understandabe than Eq. (4.). Based on Eq. (4.7) and (4.) Type EMs can be obtaned. Whe the Type 2 EM can be obtaned by smar procedure wth one more step of defuzzfcaton. 2. Cacuatons to determne coeffcents of Type 2 EM: Based on the Type 2 T S fuzzy mode f for oop y uj the steady state gan k at the gven operatng pont x can be cacuated by Eq. (3.7) where a M M b( ) a b r rb( ) a rb r a b r a x x rb r r M M 2 2 b( x ) rb( x ) (4.2) 46

57 b M M b( ) b b s rb( ) b rb s b b s b x x rb s s M M 2 2 b( x ) rb( x ) (4.3) r p s q. The steady state gan k ˆ for the Type 2 EM f can be derved by an equaton wth the same form as Eq. (4.5) where aˆ M M ( ) ˆ ( ) ˆ ˆ ˆ b a b r rb a rb r a b r a x x rb r r M M 2 2 b( x ) rb( x ) ˆ r p. (4.4) Consderng Eq. (3.4) Eq. (3.7) Eq. (4.5) and Eqs (4.2) (4.4) gves that aˆ b r a b r and aˆ rb r a rb r (4.5) The eft and the rght bound of ˆ a ˆ ˆ r a b r a rb r n Eq. (4.2) can be cacuated as aˆ b r a b r and aˆ rb r a rb r (4.6) Smar to the method for determnng Type EM the tme deay ˆ of Type 2 EM can be cacuated by an equaton wth the same form as Eq. (4.) where the parameters a r ( r p) and b s ( s q) are determned by Eq. (4.2) and Eq. (4.3). And the cacuaton can aso be smpfed as Eq. (4.) snce the vaues of s of pared oops are cose to. By the above smpe cacuatng methods t s easy to obtan a Type and a Type 2 EM for a pared oop that can exhbt the nteractng effects on both steady state gan and dynamc property when a other oops cose. However t s requred that the contro system possesses ntegrty property whch means the overa contro system shoud reman stabe whether other contro oops are put n and/or removed [38 63]. And the ntegrty requres that when controng a certan oop after a other oops 47

58 removed the performance of the controer desgned based on the EM shoud be no more aggressve than that of the controer desgned based on the ndvdua mode [38]. Therefore the coeffcents n f ˆ must take dfferent vaues for dfferent nteractng cases. There are four dfferent nteractng cases that are cassfed by four combnatons of the vaues of and. How to revse the coeffcents n f to guarantee the ntegrty property of a contro system s dscussed beow. ˆ Fgure 4. Interactng case wth λ and γ > Case : and In ths case k ˆ / k and ˆ e / e whch means k k and eˆ e. An exampe of the Bode dagrams of f and f ˆ can be seen n part (a) of Fg. 4.. ˆ k ˆ k means that the steady state gan of oop y uj when other oops cose s not smaer than that of the ndvdua oop. Snce the nteractons from the other oops enarge the effect of u j on y the controer gans need reducng to guarantee the system stabty. In ths case for Type EM a ˆ ( r p M ) s determned by Eq. (4.7) and for Type 2 EM a ˆ b r and r a ˆ rb r ( r p M ) are determned by Eq. (4.6) such that the 48

59 controer w be desgned based on f ˆ where the steady state gan s kˆ k /. eˆ e means that the response speed of oop y uj when other oops cose s sower than that of the ndvdua oop. The enarged normazed ntegrated error w reduce the phase margn and may cause nstabty. In ths case we et the tme deay ˆ n both Type and Type 2 EMs be determned by Eq. (4.) such that the controer w be desgned based on f ˆ where the normazed ntegrated error s e e. ˆ The Bode dagram of f ˆ wth revsed parameters n ths case can be seen n part (b) of Fg. 4.. Case 2: and Fgure 4.2 Interactng case wth λ and γ In ths case k ˆ / k and ˆ e / e whch means k k and eˆ e. An exampe of the Bode dagrams of f and f ˆ can be seen n part (a) of Fg ˆ k ˆ k same as that n Case. 49

60 eˆ e means that the response speed of oop y uj when other oops cose s not sower than that of the ndvdua oop. The reduced normazed ntegrated error w enarge the phase margn. However n consderaton of the contro system ntegrty the tme deay of EMs shoud be same as that of ther ndvdua T S fuzzy modes.e. ˆ such that the controer desgn w be based on the unchanged normazed ntegrated error: eˆ e. The Bode dagram of f ˆ wth revsed parameters n ths case can be seen n part (b) of Fg Case 3: and Fgure 4.3 Interactng case wth λ > and γ > In ths case k ˆ / k and ˆ e / e whch means k k and eˆ e. An exampe of the Bode dagrams of f and f ˆ can be seen n part (a) of Fg ˆ k ˆ k means that the steady state gan of oop y uj when other oops cose s smaer than that of the ndvdua oop. Even though the nteractons from other oops acts n opposton to the effect of u j on y the controer gan cannot be magnfed for better performance due to the contro system ntegrty 5

61 consderaton. Hence a ˆ of Type EM and a ˆ a ˆ of Type 2 r EM ( r p M ) shoud be same as that of ther ndvdua modes.e. a ˆ r a a ˆ b r a and a ˆ rb r a such that the controer r b r rb r b r rb r desgn w be based on the unchanged steady state gan: k ˆ k. eˆ e same as that n Case. The Bode dagram of f ˆ wth revsed parameters n ths case can be seen n part (b) of Fg Case 4: and Fgure 4.4 Interactng case wth λ > and γ In ths case k ˆ / k and ˆ e / e whch means k k and eˆ e. An exampe of the Bode dagrams of f and f ˆ can be seen n part (a) of Fg ˆ k ˆ k same as that n Case 3. e ˆ e same as that n Case 2. The Bode dagram of f ˆ wth revsed parameters n ths case can be seen n part (b) of Fg

62 After revsng the coeffcents n EMs of n pared contro oops can be obtaned to descrbe n non nteractng equvaent snge oops to represent an n n MIMO process. Afterwards each oca controer of a decentrazed contro system for ths process can be ndependenty desgned based on an EM through near SISO contro agorthms. An exampe for a two nput two output process s shown n Fg Fgure 4.5 Decentrazed contro system based on EMs for a 2 2 process The steps of usng the presented oop parng crteron based EM method to deveop a decentrazed controer for an MIMO process are gven n the foowng. ). At an operatng condton cacuate the steady state gan and the normazed ntegrated error from the T S fuzzy mode of each ndvdua oop to seect a contro confguraton for the overa process accordng to the RNGA based parng crteron. ). Accordng to the nformaton provded by the RNGA based crteron scae the ndvdua T S fuzzy mode coeffcents of pared contro oops to obtan EMs. Then each oca decentrazed controer can be ndependenty derved based on an EM usng near SISO contro approaches. ). When the operatng condton changes repeat step ) and step ) to update contro confguraton and EMs for contro varabe cacuatons. The sgnfcances of the proposed EM method are as foows. EM can descrbe the nteractng effects on both steady and dynamc propertes wthout usng extra terms or changng the near structure of oca T S fuzzy mode whch provdes a smpe but effectve manner to approxmate the nteractng resuts that can greaty reduce the compexty and cost of process modeng and the foowng controer desgns especay for arge scae MIMO 52

63 processes. The cacuatons are qute smpe such that n an MER system whch s a nonnear process the EM of updated contro confguraton for changed workng condton can be obtaned mmedatey. And t ony utzes the nformaton of ndvdua open oop modes whch are aways dentfabe wth proper persstent exctatons. Moreover fuzzy mode can be dentfed based on data sampes or human experence wthout requrng the knowedge of nner process nformaton. Therefore t s feasbe and effectve n compex MIMO process contros and can be apped n rea tme appcatons for manpuatng MER systems. Compared wth the exstng oop parng crtera based effectve mode methods that are a proposed based on near transfer functons dentfed at certan workng ponts the EM method provdes an aternatve when the accurate mathematca functons are unavaabe. And fuzzy mode s a goba descrpton for the process that can save the computatona cost and tme for new transfer functon dentfcaton when the workng ponts change n rea tme appcatons. Moreover ths method ays a foundaton to deveop robust controers snce fuzzy system s powerfu to hande uncertantes. The EM can refect proper steady and dynamc propertes n dfferent cases to guarantee the contro system ntegrty. Based on EMs of pared oops decentrazed controer desgn for an MER system can be regarded as mutpe ndependent snge oop controer desgns and greaty factated by near SISO contro agorthms. Furthermore ths method presents Type and Type 2 EMs under a unfed framework where the comparatve studes of Type and Type 2 T S fuzzy systems n terms of robustness and computatona cost can be carred out to quatatvey and quanttatvey anayze ther propertes and dfferences. 4.3 Controer desgn based on EMs Ths secton ntroduces how to desgn controers based on Type and Type 2 EMs. A cosed oop T S fuzzy mode based contro system for an n n MIMO process s shown n Fg. 4.6 where rv n denote the reference vaues; 53

64 contro varabes u ( j n) are determned through controer G c accordng to j d s ( n) whch are the dfferences between reference vaues and system outputs d rv y. Suppose the contro confguraton has been determned and the pared oops have been paced n the dagona postons through coumn swappng by vrtue of the EMs of the pared oops the near SISO contro technques can be empoyed to devse decentrazed controers for the MIMO process. Fgure 4.6 A fuzzy mode based cosed oop contro system for an MIMO process Theoretcay any near contro agorthms can be apped to desgn controers based on an EM through PDC [5]. In order to deveop a practca controer for a rea MIMO process the contro agorthms shoud be robust to the dsturbance and smpe n cacuaton and mpementaton. Currenty a number of contro agorthms have been proposed to cope wth the nteractons and dsturbance such as mode predctve contro (MPC) and the robust contro. They are more sutabe to be used n hgher eve of a contro strategy due to the computatona compexty. Whe at the ower eve the contro agorthms based on proportona ntegra (PI) or proportona ntegra dervatve (PID) controers are wdey used for reguatng contros [38]. Dfferent methods have been proposed to cacuate the gans of a PID controer such as Zeger Nchos wth detunng factor method proposed n [66] the bggest og moduus tunng agorthm presented n [67] the reatve gan based tunng approach gven n [65] and the sequenta oop tunng manner ntroduced by [68] etc.. 54

65 Recenty a PID controer tunng method based on gan and phase margns s proposed [69] whch can guarantee both robustness and performance. Severa exampes have demonstrated that t can acheve better contro performances for both set pont change and oad dsturbance than many other PID contro tunng methods n [63]. In ths secton ths gan and phase margns based agorthm s seected to present the controer desgns based on Type and Type 2 EMs. For smpcty the foowng Type and Type 2 T S fuzzy rues wth p and q 2 whch can be used to approxmate the majorty of ndvdua oops n the MIMO processes are utzed to present controer desgns. The Type T S fuzzy rue expressed as R : IF x ( k) s C THEN y ( k) a u ( k ) b y ( k ) b y ( k 2) j 2 (4.7) and the Type 2 T S fuzzy rue expressed as R : IF x ( k) s C THEN y ( k) a u ( k ) b y ( k ) b y ( k 2) j 2 (4.8) Based on Eq. (4.7) and Eq. (4.8) the procedures to devse controers are gven as foows.. Controer desgn based on Type EM For a certan pared oop y u ts Type EM f ˆ can be expressed as R : IF x ( k) s C THEN y ( k) aˆ u ( k ˆ ) b y ( k ) b y ( k 2) 2 (4.9) The near poynoma of th fuzzy rue as n Eq. (4.9) can be converted to a dscrete transfer functon form by through Z transform aˆ G z z Y () z () 2 U ( z) b z b 2z ˆ (4.2) And a standard dscrete PID controer for G () z n Eq. (4.2) usng backward dfferencng approxmaton of dervatve can be expressed by 55

66 K T K G ( z) K ( z ) U () z I D c P D ( z) z T (4.2) where K K and K are proportona ntegra and dervatve gans P I D respectvey and D ( z) Z [ d ]. Accordng to the gan and phase margns based PID tunng method [69] the parameters of the controer shoud be chosen to satsfy the foowng equaton. ˆ KT G ( z) G ( z) z z ˆ c (4.22) where K / (2 A T) A s the specfed gan margn whch s nterreated to ˆ m m phase margn m. Some typca gan phase margn pars are gven n Tabe 4.. Substtute Eq. (4.2) and Eq. (4.2) nto Eq. (4.22) to have ˆ ˆ ˆ a z KI T K D G ( z) Gc ( z) K 2 P ( z ) ( b z b2z ) z T K 2K K D D D 2 ˆ aˆ ( P I ) ( P ) z K K T K z z T T T 2 2 b z b z z KP 2 KD / T KD / T ˆ z z aˆ ( KP KI T KD / T ) z KP KI T KD / T KP KI T KD / T 2 b z b z z KT z z ˆ 2 Tabe 4. Typca gan and phase margn pars 2 m /4 /3 3 / 8 2 / 5 A m (4.23) In order to satsfy Eq. (4.23) the foowng equatons are utzed. 56

67 K T aˆ ( K K T K / T ) b b P I D K 2 K / T P D P I D K K T K / T K / T D 2 P I D K K T K / T Consequenty the proportona ntegra and dervatve gans can be derved by KT KP ( b 2 b2 ) ( b 2 b 2) aˆ ˆ ˆ 2a Am K KI ( b b2 ) ( b b2) aˆ ˆ ˆ 2a Am T 2 KT T KD ( b2 ) ( b2) aˆ ˆ ˆ 2a Am (4.24) Submttng Eq. (4.24) nto Eq. (4.2) to obtan G () z for th fuzzy rue as c KI T KD Gc ( z) KP ( z ) z T ( b 2 b ) ( b b ) ( b ) = 2aˆ A ˆ 2aˆ A ˆ z 2aˆ A ˆ ( ) z m m m b z b z 2 aˆ A ( z 2 2 ˆ m ) (4.25) And Eq. (4.25) gves the foowng equaton to derve a contro varabe n tme doman for th fuzzy rue as b b u k u k d k d k d k (4.26) 2 ( ) ( ) ( ) ( ) ( 2) 2aˆ ˆ ˆ ˆ ˆ ˆ Am 2a Am 2a Am Accordng to PDC the u ( k ) s a weghted sum of oca contro varabes u ( k ) ( M ) and shares the same membershp functons of f ˆ as M u ( k) x ( k) u ( k) (4.27) 2. Controer desgn based on Type 2 EM For a certan pared oop y u ts Type 2 EM f ˆ can be expressed as 57

68 R : IF x ( k) s C THEN y ( k) aˆ u ( k ˆ ) b y ( k ) b y ( k 2) j 2 (4.28) where y ( k) y ( k) y ( k) y b( k ) and y rb( k ) are expressed by b rb y ˆ ˆ b( k) a b u j ( k ) b b y ( k ) b b2 y ( k 2) y ˆ ˆ rb( k) a rb u j ( k ) b rb y ( k ) b rb2 y ( k 2) (4.29) Eq. (4.29) can be converted to two dscrete near functons through Z transform as ˆ a b G b( z) z 2 b b z b b2z ˆ a rb G rb( z) z 2 b rb z b rb2z ˆ ˆ (4.3) Accordng to the gan and phase margns based tunng method a contro varabe nterva for th rue: u ( k) u ( k) u. ( k) can be cacuated based on Eq. (4.3): b rb d ( k) b b d ( k ) b b2 d ( k 2) u b( k) u ( k ) 2aˆ ˆ b Am d ( k) b d ( k ) b d ( k 2) rb rb2 rb( ) ( ) 2aˆ ˆ rb Am u k u k (4.3) and the contro varabe nterva for oop y u can be obtaned through PDC as u ( k) u b( k) u rb( k) ( k) u ( k) ( k) u ( k) M M b x b rb x rb M M b x ( k) rb x ( k) (4.32) The crsp contro varabe u ( k ) s derved by defuzzfyng u ( k ) n Eq. (4.32) as u ( k) u ( k) u ( k) b rb (4.33) 2 Remark 4.: Accordng to the ntroduced controer desgn procedures n theory the computatona cost to obtan a contro varabe based on an nterva Type 2 EM doubes that based on a Type EM because of the ncreased fuzzness n the 58

69 coeffcents of the Type 2 EM. 4.4 Smuaton A three nput three output ( 3 3) nonnear process s gven as foows. 2 2 x x2 5x x2 6x2 x2 4x 5x2 8x x2 u x3 x4 3 x4 6x3 5x4 3x3 x3x4 x5 u2 2 x5 x6 4x7 2 x6 x7 5x5x6 x7 x7 4x5 23x6 x7 7x5x6 x7 u 3 y 5x 5x 6x 2x 4x 9x x y2 8x 2x2 3x3 4x5 6x6 2x7 y3 x x2 4x3 2x4.4x5.2x (4.34) where y s ( 23 ) and u j s ( j 23 ) are outputs and nputs respectvey x r s ( r 7 ) are state varabes. The tme deays n ths process are 2 2 (sec) and 3 (sec) for 23. Suppose there are dsturbances added to the process when coectng data sampes of ndvdua oops as shown n Fg. 3.2 and the dsturbances are random but bounded n.2.2. Choose the sampng nterva as T.sec. The Type and the Type 2 T S fuzzy modes as the forms of Eq. (4.7) and Eq. (4.8) are constructed based on the samped nput output data pars and the dentfed resuts are presented n Appendx B. Gven the operatng ponts as x u ( k ) y ( k ) y ( k 2) j for j 23 from the Type T S fuzzy mode matrx the foowng resuts can be obtaned as

70 Accordng to the RNGA based oop parng crteron the optma contro confguraton can be determned: y u3 / y2 u / y3 u2 where Based on and the reatve normazed ntegrated error array can be derved as For the pared oops and thus the Type EMs of oop y u3 and oop y2 u are determned by Case 3 and the Type EM of oop y3 u2 s determned by Case 4 as descrbed n Secton 4.2. From the Type 2 T S fuzzy mode matrx the foowng resuts can be obtaned Accordng to the RNGA based oop parng crteron the optma contro confguraton can be determned: y u3 / y2 u / y3 u2 whch s same as that obtaned from Type T S fuzzy mode matrx and Based on and reatve normazed ntegrated error array can be derved as the

71 For the pared oops and thus the Type 2 EMs of oop y u3 and oop y2 u are determned by Case 3 and the Type 2 EM of oop y3 u2 s determned by Case 4. Based on the EMs of pared oops by usng the contro agorthm ntroduced n Secton 4.3 the decentrazed fuzzy mode based controers can be derved. For comparson the same contro agorthm s empoyed to desgn decentrazed PID controers for ths MIMO process based on the ETFs n terms of RNGA crteron [38 64] and the transfer functons are obtaned by nearzng the MIMO processes of Eq. (4.34) at the gven ponts wthout dsturbances (the detas are n Appendx B). The gan and the phase margn are chosen as Am 3 and m /3 and reference vaues are rv.5 rv2 and rv3. Appy ETF and EM based controers to manpuate ths MIMO process ther performances are shown n Fg y.5 ETF based decentrazed contro.5 Type- EM based decentrazed contro Type-2 EM based decentrazed contro y y Tme(sec) Fgure 4.7 Decentrazed contros for the MIMO process n Eq. (4.34) As can be seen n Fg. 4.7 even though the fuzzy modes are but based on the data wth nexact nformaton the controers based on EMs can acheve better resuts 6

72 wth smaer overshoots compared to that based on the ETFs nearzed from exact mathematca modes. And n ths case the performances of Type and Type 2 EM based decentrazed contros are comparabe. In a bd to ceary demonstrate ther dfference the Integra of the Absoute vaue of the Error (IAE) a performance crteron gven by [7] s empoyed to refect the error accumuaton of a contro resut whch s defned as IAE( y ) rv y ( k) T k where T s the sampng nterva. The IAEs of three outputs under the Type and the Type 2 EM based decentrazed contros n ths case s gven n Tabe 4.2. Tabe 4.2 The IAEs of Type and Type 2 EM based decentrazed contros for the process n Eq. (4.34) Type of fuzzy mode IAE( ) IAE( ) IAE( ) y y 2 y 3 Type Type As shown n Tabe 4.2 the IAEs of Type and Type 2 EM based contros whch denote the vaues of ther error accumuatons are very cose. In order to test the robustness of these controers we suppose that n the equaton group for the process of Eq. (4.34) the thrd equaton x3 x4 s changed as that n Eq. (4.35) due to the uncertantes x x.5u (4.35) 3 4 Appy prevousy desgned controers to the process wth changed equaton as n Eq. (4.35) the contro performances are gven n Fg. 4.8 and the comparson of IAEs between Type and Type 2 contro s presented n Tabe 4.3. As ustrated n Fg. 4.8 both Type and Type 2 EM based controers can outperform ETF based controer. And t can be seen that outputs under Type 2 EM based contro are of a tte smaer overshoots than that under Type EM based contro whch s proved by the vaues of IAEs presented n Tabe 4.3 that IAEs of Type 2 contro are smaer. 62

73 2 y ETF based decentrazed contro Type- EM based decentrazed contro Type-2 EM based decentrazed contro y y Tme(sec) Fgure 4.8 Decentrazed contros for the process changed as Eq. (4.35) Tabe 4.3 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.35) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type When the changed parameter n the process s further enarged as x x 2.3u (4.36) 3 4 the performances of the prevous desgned decentrazed controers apped to the process wth changed equaton as n Eq. (4.36) are shown n Fg. 4.9 and the IAE comparson between Type and Type 2 contro s gven n Tabe 4.4. As ustrated n Fg. 4.9 n ths case the outputs under ETF based controer are oscatng and cannot merge wth reference vaues whe the outputs under EM based controers can successfuy hande the parametrc uncertantes and make the outputs reach ther references staby and the IAEs of Type 2 contro are st smaer than that of Type contro. Compared wth the performances n Fg. 4.8 and Tabe 4.3 the dfference 63

74 between Type and Type 2 EM based contro s more apparent n Fg. 4.9 and Tabe 4.4 that Type 2 EM based controer acheves better resuts wth smaer amptude of oscaton and error accumuatons. y ETF based decentrazed contro Type- EM based decentrazed contro Type-2 EM based decentrazed contro y y Tme(sec) Fgure 4.9 Decentrazed contros for the process changed as Eq. (4.36) Tabe 4.4 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.36) Type of fuzzy mode IAE( ) IAE( ) IAE( ) y y 2 y 3 Type Type In order to ceary demonstrate the robustness of Type and Type 2 EM based controers we contnue to enarge the changed parameter as x x 2.5u (4.37) 3 4 In ths case the process under ETF based decentrazed controer becomes unstabe and dvergent as n Fg. 4. whe the process under both Type and Type 2 EM based controers s st stabe as shown n Fg. 4.. And the IAEs are presented n Tabe 4.5. From Fg. 4. and Tabe 4.5 t s very cear and easy to fgure out that 64

75 Type 2 EM based decentrazed controer can provde much mproved performance n terms of oscatng amptude and settng tme when compared wth Type EM based controer. 2 5 y y2 y3 5 Outputs Tme(sec) Fgure 4. ETF based decentrazed contro for the process changed as Eq. (4.37) Fgs prove that both Type and Type 2 EM based decentrazed controers can acheve much better performances than ther ETF counterpart. And the IAEs n Tabes demonstrate that when arge uncertanty appears Type 2 EM based controer can gve more robust resuts than Type EM based one. 3 2 Type- EM based decentrazed contro Type-2 EM based decentrazed contro y y y Tme(sec) Fgure 4. EM based decentrazed contros for process changed as Eq. (4.37) 65

76 Tabe 4.5 The IAEs of Type and Type 2 EM based decentrazed contros for the process changed as Eq. (4.37) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type Remark 4.2: The smuaton resuts demonstrate that under the uncertanty wth the degree that Type fuzzy mode can hande the decentrazed controers based on Type and Type 2 EMs perform comparaby. When the degree of uncertanty ncreases Type 2 fuzzy mode s abe to gve better performance than Type fuzzy mode n terms of robustness. Snce the computatona cost and compexty based on Type 2 fuzzy mode wth respect to modeng oop parng and controer desgn are more than that based on Type fuzzy mode n order to acheve satsfactory performance whe mnmze the cost whch type of fuzzy mode shoud be utzed depends on the degree of uncertanty the expectaton of modeng accuracy and the requrement of controer robustness. 4.5 Summary In ths chapter both Type and Type 2 EMs are proposed to descrbe the nteractons among the oops to factate decentrazed contro for MIMO processes. For a certan oop n an MIMO process an EM s obtaned by scang the parameters of ts ndvdua open oop mode accordng to the quantfed nteractng effects provded by RNGA based parng crteron and the scaed parameters are further revsed to adapt dfferent cases such that the ntegrty of contro system can be ensured. The proposed EM method can express the nteractng resuts on both steady and dynamc propertes of a oop caused by other oops. Based on the EMs of pared oops each oca controer of a decentrazed contro system can be ndependenty desgned by usng near SISO contro schemes. Compared wth the exstng decentrazed fuzzy contro methods addng extra terms n the ndvdua modes to characterze the coupng effects EM provdes an easer and more 66

77 practca way to express the nteractons that can reduce compextes n both modeng and controer desgn. Compared wth the exstng ETF methods EM method gves an aternatve when mathematca functons are unavaabe and offers a bass to deveop robust controers for the process snce fuzzy system s strong n handng uncertantes. Smuaton resuts demonstrate that EM based controers outperform ther ETF based counterpart and they can acheve the goas even f the orgna process parameters change sgnfcanty. It s aso proved that as the degree of uncertanty ncreases Type 2 fuzzy system can gve more satsfactory performance than Type fuzzy system. Snce the EM can embody the nteractons from other oops more nterestng contro technques can be deveoped based on t. In the foowng chapter a sparse contro method based on EM for an MIMO process w be presented. 67

78 Chapter 5. Sparse contro based on T S fuzzy modes for MIMO processes 5. Introducton For an MIMO process where the nteractng effects among the oops are weak or modest prevousy proposed decentrazed contro can generay work wth satsfactory performance. However for an MER system wth cosey couped effects t may gve degraded performance due to the mted contro structure. On the other hand fu dmensona contro such as centrazed contro or fu decoupng contro generay provde mproved performance n manpuatng cosey couped MIMO processes when compared wth decentrazed contro. However fu dmensona contro may resut n drastcay ncreased computatona cost and compexty especay for arge scae processes makng the contro agorthms dffcut to mpement. To deveop an effcent controer that can hande strong coupng yet reman vabe n appcatons sparse contro ncudng bock dagona contro and tranguar contro whch can make a compromse between decentrazed contro and fu dmensona contro n structure have been deveoped. By addng severa off dagona controers to the dagona decentrazed contro structure sparse contro f t s we desgned s abe to acheve better performance for cosey couped MIMO processes compared to decentrazed contro whe can be much smper n cacuaton and appcaton than fu dmensona contro. In [7] an approach usng Observabty and Controabty Graman to form a Partcpaton Matrx (PM) for nteracton measurement and contro structure seecton was deveoped and n [72] t was further dscussed. Ths work was ony used to support near systems unt recenty. It was extended to bnear and nonnear systems n [73 74]. These studes have not presented cear crtera to unquey determne a sparse contro structure for an MIMO process whe a cear crteron n terms of RNGA based oop parng crteron was proposed n [38]. Moreover [38] presented an ndependent desgn method usng 68

79 ETFs so that the sparse controer desgn for an MIMO process becomes mutpe ndependent SISO controer desgns. Severa exampes demonstrated ts smpcty and effectveness when compared to decentrazed contro and fu dmensona contro. However these methods are proposed based on knowng mathematca functons of an MIMO process whch mpes that ther mpementatons may be mted when accurate mathematca functons cannot be derved. Sparse contro based on a fuzzy mode can be a souton. To the best of the author s knowedge no studes concernng sparse contro structure seecton and sparse controer desgn based on fuzzy mode have been deveoped. In a bd to f the gap n ths chapter the study of [38] s extended to fuzzy area to present a gudene for sparse contro system desgn based on both Type and Type 2 T S fuzzy modes wth smpe and manageabe cacuatons. For an MIMO process the nteractons between process nputs and outputs are anayzed n terms of RNGA oop parng crteron to determne whether the coupng effects are strong enough that the MIMO process shoud choose sparse contro over decentrazed contro and a cear crteron s presented to unquey determne a sparse contro structure. By empoyng EMs an ndependent desgn method s ntroduced so that the sparse controer desgn for an MIMO process can be converted to mutpe ndependent snge oop controer desgns. The proposed gudene gves an aternatve to manpuate cosey couped mutvarabe processes where mathematca modes are unavaabe and t ays a bass to devse robust fuzzy controers. Moreover the comparatve studes of Type and Type 2 T S fuzzy modes can be carred out under ths gudene. In the smuaton secton the performances of ETF Type and Type 2 EM based sparse contros are provded and compared wth each other as we as ther decentrazed contro counterparts to demonstrate the effectveness of the proposed method. Smar to the smuaton n Chapter 4 parametrc uncertantes are ntroduced to the process to show the robustness of these controers. 5.2 Sparse contro structure seecton Accordng to RNGA based oop parng crteron ntroduced n Chapter 3 the 69

80 decentrazed contro confguraton for an MIMO process can be determned. By pacng the pared oops n the dagona poston through coumn swappng a decentrazed controer for the MIMO process n a cosed oop contro system as n Fg. 4.6 can be expressed as Gc Gc2 G c (5.) Gcn Decentrazed contro utzes the smpest contro structure to manpuate an MIMO process and t can work wth satsfactory resuts when the nteractons among the oops are not strong when for cosey couped processes ts mted structure may strugge wth the nteractons. On the other hand a centrazed controer usng fu dmensona structure to emnate nteractng effects can be expressed as nn Gc Gc2 Gc n Gc2 Gc22 G c2n G c (5.2) G G G c n c n2 c nn nn The dsadvantage of the fu dmensona controer s that the computatona cost w be greaty ncreased when compared wth decentrazed contro especay for arge scae processes where the cacuatons of hgh dmensona matrces w be nvoved that may cause dffcutes n the mpementaton. To make a compromse between decentrazed contro structure as n Eq. (5.) and centrazed contro structure as n Eq. (5.2) sparse contro wth the foowng structure can be empoyed. Gc 2 Gc2 ngcn 2 Gc2 Gc22 2 ng c2n G c (5.3) n Gc n n2 Gc n2 Gc nn where s a seecton ndex wth vaue of or. When t means the off dagona controer G c s added to the contro structure and when t 7

81 means no controer n the poston. If for j n ( j) Eq. (5.3) becomes a decentrazed contro as n Eq. (5.) and f for j n ( j) Eq. (5.3) becomes a centrazed controer as n Eq. (5.2). For sparse contro the vaues of s ( j n j) are not the same. Compared wth decentrazed contro sparse contro can acheve mproved performance n handng the nfuence of nteractons whe ts compexty s ess than that of centrazed contro. One of the key steps for sparse contro deveopment s to determne the vaues of s ( j n j) n Eq. (5.3) to seect a proper contro structure whch can be acheved by vrtue of RNGA based oop parng crteron. By swappng the coumn of fuzzy mode matrx F RGA RNGA and reatve normazed ntegrated error array to pace the pared eements on the dagona postons an nteracton ndex denoted by can be defned as nn k / kˆ k / kˆ eˆ / e k / kˆ k / kˆ eˆ / e N N N N (5.4) ( j ) can be utzed to anayze the steady and the dynamc nteractons between pared and unpared eements to determne the vaue of. Two extreme cases of the vaues of are anayzed as foows [38].. s very sma. A very sma vaue of means ether / s very sma or / s very sma. A very sma vaue of / mpes k s much smaer than k thus the steady state effect on y from u j s very tte when compared wth that from u. Whe a very sma vaue of / 7

82 ndcates e s much smaer than e thus the response speed of oop y u s very rapd when compared wth that of oop y u whch can be j consdered as a hgh frequency nterference and can be effectvey ftered out by the pared oop wth reatve sow response speed. 2. s very arge. A very arge vaue of means ether / s very arge or / s very arge. A very arge vaue of / mpes k s much arger than k. If oop y uj s ncuded n the controer the contro system w be qute senstve to modeng errors whch means a sma modeng error or a sma change n the contro varabe u j w be magnfed through ths oop and resut n a very arge error n the output y and then the contro woud be very hard to acheve for such a oop. On the other hand a very arge vaue of / ndcates e s much arger than e thus oop y uj presents qute sow reactons and can be regarded as a constant dsturbance that can be easy handed by the pared oop controer. Accordng to the above anayss the off dagona oops wth the vaue of nether too sma nor too arge shoud be consdered nto controer desgn. As n Fg. 4.6 where a cosed oop contro system for an MIMO process s shown t s easy to earn that n the controer matrx G c c j G s the controer usng d to determne contro varabe u j whch mpes c j G s based on y to cacuate u j. Therefore n G c the controer c j G reated to oop y uj s at the transposed poston of f n F. When the vaue of s modest j to add controer G c j to G c such that oop j y u can be ncuded nto controer desgn. An exampe of determnng sparse contro structure for G c based on seected eements n F s 72

83 gven as foows (wth * n the postons of seected eements). * * * * * * * * F G c. * * * * * * To deveop an effectve contro structure that factate the controer desgn whe savng the computatona cost as much as possbe the foowng crteron to determne the vaue of to seect a proper sparse contro structure can be used [38]..5 j 8 for j (5.5) otherwse Remark 5.: Snce the RGA and RNGA for a 2 2 process s symmetrc the contro type for 2 2 process s ether centrazed structure or decentrazed structure. Whereas for an n n process wth n 2 sparse contro can be apped. In the practca appcatons t s rare to fnd a process sutabe for centrazed contro or fu decoupng contro therefore decentrazed contro of sparse contro methods are generay more approprate for MIMO contro system desgns [38]. 5.3 Independent desgn based on EMs Fgure 5. A cosed oop MIMO contro system The fuzzy mode matrx for an MIMO process s a dscrete system that can be denoted by F ( z ) f ( z ) nn. As the cosed oop contro system shown n Fg. 5. a genera controer desgn pattern s to satsfy T F ( z ) Gc ( z ) I (5.6) z 73

84 where I s an dentty matrx. The controer matrx can then be obtaned by T Gc ( z ) F ( z ) (5.7) z Gven the nonnear nature and speca structure wth IF THEN rues of a T S fuzzy mode t s qute dffcut or even mpossbe to drecty cacuate the nverse matrx F ( z ) whch causes an obstace n devsng the contro system G c( z ). In ths secton an ndependent controer desgn method based on EMs for sparse contro s ntroduced to provde a smpe and feasbe manner to sove ths probem. The cacuatng method to obtan EM ntroduced n Chapter 4 can be used for pared oops whe may not be appcabe to other oops snce ther s and s may be negatve and s may not be cose to. Hence the EM cacuatng method gven n Chapter 4 s revsed for dervng the EMs of unpared eements as shown n Tabe 5.. Tabe 5. Coeffcent cacuatons of EMs for unpared oops n dfferent cases Interactng Vaues of coeffcents Cases ˆ a ˆ r b r a and ˆ a rb r ˆ. By Eq. (4.7) By Eq. (4.6) By Eq. (4.) 2. By Eq. (4.7) By Eq. (4.6) ˆ 3. sgn( ) a r aˆ sgn( ) a aˆ sgn( ) b r b r rb r a rb r By Eq. (4.) 4. sgn( ) a r aˆ sgn( ) a aˆ sgn( ) b r b r rb r a rb r ˆ Subsequenty based on the fuzzy mode matrx F for an n n MIMO process we can obtan an EM matrx F ˆ expressed as 74

85 fˆ ˆ ˆ f 2 f n ˆ ˆ ˆ ˆ ˆ f 2 f 22 f 2n F f nn (5.8) fˆ ˆ ˆ n f n2 f nn where F ˆ s a dscrete system denoted by ˆ ˆ F ( z ) f ( z ) nn. For oop y u j the nteractons from other cosed oops can be measured usng dynamc reatve gan defned as [ ] f ( z ) ( ) ( ) ( ) (5.9) ˆ z f z f z fˆ ( z ) Let ˆ ˆ F ( z ) f ( z ) nn accordng to Eq. (5.9) for the overa process the dynamc RGA can be derved as ( z ) ( z ) ( z ) ˆ F F ( z ) (5.) nn Accordng to the cacuaton of as n Eq. (3.6) the dynamc RGA can be cacuated as (5.) ( ) ( ) T z F z F ( z ) Comparng Eq. (5.) wth Eq. (5.) an mportant reatonshp can be reveaed as ˆ ˆ ˆ f f 2 f n ˆ ˆ ˆ T ( ) ˆ f f f n F z F ( z ) (5.2) ˆ ˆ ˆ f n f 2 n f nn By consderng Eq. (5.2) Eq. (5.7) can be rewrtten as G T ˆ T ˆ T c( z ) F ( z ) ( ) z z F z (5.3) Let the error functon for Eq. (5.3) be defned as T ˆ T J( z ) J ( z ) Gc ( z ) F ( z ) (5.4) nn z and an ndex functon J mn for mnmzng the error functon J ( z ) n Eq. (5.4) 75

86 can be defned as the mnmum vaue of the sum of the absoute error between each eement n c( z G ) and z T T z F ˆ ( ) / ( ) as n n n n ˆ mn J z Gc z f j T z j j J = mn ( ) mn ( ) / ( ) (5.5) Form Eq. (5.5) the mnmzaton of s determned by J ( z ) requres the controer G z where c ( ) G ( z ) fˆ T / ( z ) G ( z ) fˆ T / ( z ) (5.6) c j c j In Eq. (5.6) G ( z ) fˆ can be regarded as the forward functon of a c j cosed oop SISO contro system as shown n Fg Fgure 5.2 A cosed oop SISO contro system And the controer G z that satsfes Eq. (5.6) s the objectve of a genera c ( ) controer desgn pattern for SISO processes smar to that n Eq. (5.6) for MIMO processes. Therefore based on EMs each eement of G c can be ndependenty desgned usng SISO contro agorthms to reaze a mutvarabe contro system and avod drecty cacuatng nverse matrx F ( z ). The steps of usng the proposed method to devse sparse contro for an MIMO process are brefy gven as foows. ). At an operatng condton determne pared oops usng RNGA based crteron and then seect severa unpared eements accordng to Eq. (5.5). ). Accordng to the nformaton provded by the RNGA based crteron scae the ndvdua T S fuzzy mode coeffcents of each seected oop to obtan EMs. 76

87 Afterwards ndependenty devse a SISO controer for each EM through near contro agorthms to form a sparse contro system for the whoe process. ). When the operatng condton changes repeat step ) and step ) to update contro structure and the EMs for sparse contro varabe cacuatons. The sgnfcances of the proposed gudene ncudng contro structure seecton and EM based ndependent desgn for sparse contro ncude the foowng. The gudene s mpemented based on T S fuzzy modes that can be but from data sampes whch provdes an aternatve where the accurate mathematca modes of the processes are unavaabe. It offers a bass to deveop robust fuzzy controers. Moreover ony the nformaton of the dentfabe ndvdua open oop modes s utzed. Therefore ths method s effectve and feasbe n rea MER contro appcatons. The gudene provdes a cear reasonabe and ow computatona cost manner to seect the off dagona oops for sparse controer desgn. And when the workng condton changes the contro structure and the correspondng EMs can be rapdy updated whch makes t sutabe to be apped n rea tme contro systems. The sparse controer for an MIMO process can be acheved by ndependenty desgnng a SISO controer for each seected oop based on the EM whch can greaty reduce the computatona compexty. Furthermore snce near contro agorthms can be apped to deveop contro systems based on EMs ths method provdes a way to empoy the smpe and mature near SISO contro agorthms to manpuate compex and cosey couped nonnear MER systems. Ths method can be apped on both Type and Type 2 T S fuzzy modes to devse sparse contro under a unform frame for an MIMO process whch provdes a patform to carry out the comparatve studes of these two types of T S fuzzy mode to demonstrate and anayze ther propertes and dfferences. 77

88 5.4 Smuaton Consder the 3 3 nonnear MIMO process n Eq. (4.34) gven the operatng ponts as u ( k ) y ( k ) y ( k 2) x j for j 23 based on the Type T S fuzzy mode matrx K E and can be cacuated. Pace the pared oops n the dagona postons through coumn swap we can have the foowng arrays Based on derved as the nteracton ndex for Type T S fuzzy mode matrx can be (5.7) Accordng to Eq. (5.5) two off dagona oops (ther s n of Eq. (5.7) are n bod) shoud be seected and added to the controer desgn. Based on Type 2 T S fuzzy mode matrx K E and can be computed. Pacng the pared eements of these matrces n the dagona postons through coumn swap yeds

89 Based on obtaned as the nteracton ndex for Type 2 T S fuzzy mode matrx can be (5.8) The two off dagona oops seected based on Type 2 T S fuzzy modes are the same as that seected based on Type T S fuzzy modes (ther s n of Eq. (5.8) are n bod). And these two eements are oop y2 u3 and oop y u n the orgna process. Therefore the seected oops for controer desgn n F of the orgna MIMO process n Eq. (4.34) can be expressed as (set to the unseected eements) f f 3 F f 2 f 23 (5.9) f 32 And then the controer matrx s Gc Gc2 G c Gc23 (5.2) Gc3 Gc32 Based on the EMs of the seected oops each nonzero eement n G c of Eq. (5.2) can be desgned ndependenty usng near SISO contro agorthms. Same as the decentrazed contro smuaton n Secton 4.4 n ths secton the gan and phase margns based contro agorthm ntroduced n Secton 4.3 usng Am 3 and m /3 s empoyed to devse both Type and Type 2 EM based sparse 79

90 contro systems. And the sparse controer based on ETFs n terms of RNGA parng crteron [38] s aso cacuated for comparson (the detas are presented n Appendx B). Set reference vaues same as that n the smuaton of Chapter 4 whch are rv.5 rv2 and rv3 the performances of these sparse contro systems when manpuatng the MIMO process n Eq. (4.34) wthout uncertanty are shown n Fg The IAEs of three outputs under Type and Type 2 EM based sparse contros n ths case are presented n Tabe 5.2. And the comparsons between these sparse controers and ther decentrazed counterparts n ths case are gven n Fg y.5 ETF based sparse contro Type- EM based sparse contro Type-2 EM based sparse contro y y Tme(sec) Fgure 5.3 Sparse contros for the MIMO process n Eq. (4.34) Tabe 5.2 The IAEs of Type and Type 2 EM based sparse contros for the process n Eq. (4.34) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type As can be seen n Fg. 5.3 and Tabe 5.2 Type and Type 2 EM based sparse controers acheve comparabe performances wth cose IAE vaues. And both of them can obtan better resuts than ETF based sparse controer. And as the comparsons 8

91 ustrated n Fg. 5.4 sparse controers can acheve smaer overshoots n the outputs than ther decentrazed controer counterparts. ETF based contro Type- EM based contro Type-2 EM based contro y Sparse contro Decentrazed contro y y Tme(sec) 2 3 Fgure 5.4 The comparsons of sparse and decentrazed contros for the MIMO process n Eq. (4.34) To test the robustness of these sparse controers smar as that n Secton 4.4 parametrc uncertanty s ntroduced to the process and then these sparse contro systems desgned based on orgna process are apped to manpuate the changed process. Consder the process wth changed equaton as n Eq. (4.36) where x x 2.3u the performances of ETF and EM based sparse controers n ths 3 4 case are shown n Fg The IAEs of Type and Type 2 EM based sparse contros are gven n Tabe 5.3. And the comparsons between sparse contro and decentrazed contro n ths case are gven n Fg Tabe 5.3 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (4.36) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type

92 2 y ETF based sparse contro Type- EM based sparse contro Type-2 EM based sparse contro y y Tme(sec) Fgure 5.5 Sparse contros for the process changed as Eq. (4.36) y ETF based contro - 5 Type- EM based contro Sparse contro Decentrazed contro - 5 Type-2 EM based contro y y Tme(sec) 5 Fgure 5.6 The comparsons of sparse and decentrazed contros for the process changed as Eq. (4.36) From Fg. 5.5 t can be seen that EM based contros can gve better resuts when compared wth ETF based contro. And Type 2 EM based sparse contro can acheve a tte smaer amptude of oscaton than Type EM based sparse 82

93 contro whch can be proved by the vaues of IAEs that the error accumuatons of the outputs under the Type 2 EM based contro are smaer than that under the Type EM based contro. The comparsons n Fg. 5.6 demonstrate that ETF based sparse contro can make the outputs reach the references staby n ths case whe ts decentrazed contro cannot and both Type and Type 2 EM based sparse contros outperform ther decentrazed contro counterparts wth respect to the amptude of oscaton and settng tme. When the parametrc uncertanty s further enarged as n Eq. (4.37) where x x 2.5u the performances of ETF and EM based sparse controers and the 3 4 comparsons of sparse and decentrazed contro are ustrated n Fg. 5.7 and Fg y ETF based sparse contro Type- EM based sparse contro Type-2 EM based sparse contro y y Tme(sec) Fgure 5.7 Sparse contros for the process changed as Eq. (4.37) Tabe 5.4 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (4.37) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type

94 3 ETF based contro Type- EM based contro 3 Type-2 EM based contro y Sparse contro Decentrazed contro y y Tme(sec) Fgure 5.8 The comparsons of sparse and decentrazed contros for the process changed as Eq. (4.37) respectvey. And the IAEs of Type and Type 2 EM based sparse contros are presented n Tabe 5.4. In ths case EM based sparse contros can st provde smaer oscatons than ETF based sparse contro. The superorty of Type 2 EM based sparse contro over Type EM based sparse contro wth respect to oscaton amptudes and error accumuatons as shown n Fg. 5.7 and Tabe 5.4 s more apparent than that n Fg. 5.5 and Tabe 5.3. Accordng to the comparson n Fg. 5.8 three sparse contro systems can acheve much better performance than ther decentrazed contro counterparts. To ceary demonstrate the robustness of these sparse controers the parametrc uncertanty can be further enarged as foows x x 2.95u (5.2) 3 4 In ths case the process under ETF based sparse contro becomes unstabe and dvergent as n Fg. 5.9 whe the process under both Type and Type 2 EM based sparse controers s st stabe as shown n Fg. 5.. And ther IAEs are presented n Tabe 5.5. From Fg. 5. t can be seen that as the degree of parametrc uncertanty 84

95 ncreases the advantage of Type 2 fuzzy system over Type fuzzy system n terms of robustness become more evdent that the process outputs under Type 2 EM based sparse contro have smaer oscatons and much smaer settng tme than that under Type EM based sparse contro. And the IAEs n Tabe 5.5 demonstrate that the error accumuatons of the outputs under Type 2 EM based sparse contro are much smaer than that under Type EM based sparse contro n ths case. Tabe 5.5 The IAEs of Type and Type 2 EM based sparse contros for the process changed as Eq. (5.2) Type of fuzzy mode IAE( ) y IAE( y 2) y 3 IAE( ) Type Type y y2 y3 Outputs Tme(sec) Fgure 5.9 ETF based sparse contro for the process changed as Eq. (5.2) Remark 5.2: The smuaton resuts prove that by addng severa off dagona controers to the contro structure sparse contro has the capabty to acheve much better resuts than decentrazed contro that t can be apped to manpuate the process wth strong nteractng effects and uncertantes where decentrazed contro s not abe to provde satsfactory resuts. The sparse controers based on Type and Type 2 fuzzy modes perform comparaby when the process s under the uncertanty wth the degree that Type fuzzy mode can hande. When arger uncertantes appear t s 85

96 proved that Type 2 fuzzy mode can gve more robust performance than Type fuzzy mode. Snce the computatona cost and compexty based on Type 2 fuzzy system n respect of modeng oop parng sparse contro structure seecton and controer desgn are more than that based on Type fuzzy mode n order to use mnmum cost to acheve satsfactory contro resuts n the appcatons the degree of uncertanty the expectaton of modeng accuracy and the requrement of controer robustness shoud be we consdered when makng decson of the seecton between Type and Type 2 fuzzy systems. 3 2 y Type- EM based sparse contro Type-2 EM based sparse contro y y Tme(sec) Fgure 5. EM based sparse contros for the process changed as Eq. (5.2) 5.5 Summary In a bd to mprove the contro performance for strong coupng MIMO processes of MER systems a gudene to devse T S fuzzy mode based sparse contro was presented n ths chapter. Through anayzng the nteractons usng RNGA for an MIMO process a seectng crteron that addng severa off dagona controers to the dagona contro structure to form a sparse contro structure s gven. Afterwards an ndependent desgn method was ntroduced that the sparse contro system desgn for an MIMO process can be transformed to a group of ndependent SISO controers n 86

97 terms of EMs. Ths gudene paves the way to utze near SISO contro agorthms to manpuate nonnear and strongy couped MIMO processes and robust controer can be derved snce fuzzy system s powerfu to hande the uncertantes. Moreover the comparatve research of Type and Type 2 fuzzy systems can be aunched to anayze and study ther propertes. The smuaton resuts demonstrate that sparse contro outperforms ts decentrazed contro counterpart. And compared wth the ETF based sparse contro both Type and Type 2 EM based sparse contros can obtan better performances. As the degree of uncertanty ncreases contro system based on Type 2 EMs s abe to acheve more robust and satsfactory resuts than that based on Type EMs. In the next chapter the proposed methods n ths thess are apped to an expermenta MER system to demonstraton ther practcabty and effectveness n MER systems. 87

98 Chapter 6. Appcatons n MER systems 6. Introducton In ths chapter the proposed Type and Type 2 T S fuzzy mode based methods are apped to evauate the nteractons seect contro structures and desgn decentrazed and sparse controers for an expermenta MER system. The photo of ths system s shown n Fg. 6.. Its schematc dagram and the correspondng pressure (P) enthapy (h) chart are gven n Fg. 6.2 and Fg Ths MER system utzes a snge compressor (COMP) and a snge condenser (COND) but three evaporators (EVAP EVAP2 and EVAP3) each workng at a specfed evaporatng temperatures to smutaneousy cater for three dfferent coong requrements: ar condtonng food storage and freezng. In ths expermenta MER system the pressure reguaton (PR) s composed of throttes and R34a s used as the refrgerant. For EVAP water s used as heat transfer fud to convey the coong to meet the ar condtonng requrements. And for EVAP 2 (food storage) and EVAP3 (freezng) whose evaporatng temperatures are ow that water may be frozen ethyene gyco souton s used nstead. Fgure 6. An expermenta MER system As shown n Fg. 6.2 and Fg. 6.3 the workng process of ths MER system can be ntroduced by the foowng steps. ). The refrgerant as a saturated vapor wth a ow pressure at state enters the 88

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