A Spatio-Temporal Fuzzy Logic System for Process Control

Transcription

1 Proceedings of the 2009 IEEE Internationa Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October A Satio-Temora Fuy Logic System for Process Contro Han-Xiong Li Schoo of Mechanica & Eectrica Eng Centra South University, China; Det of MEEM, City University of Hong Kong; Abstract A nove aication of tye-2 fuy system is resented by deveoing a satio-temora fuy ogic controer (FLC) for the distributed arameter system (DPS). The nove difference to the tye-2 fuy system is to ay the secondary MF for a different hysica variabe sace domain. Using a number of sensors ocated on the satia domain, the 3D fuy membershi function can be obtained that contains satio-temora information. The 3D inference wi incude satia T-norm oeration and the traditiona rue inference. The tye-reduction wi become the satia reduction before the traditiona defuification. This saitio-temora FLC is successfuy aied to a cataytic reaction rod to demonstrate its effectiveness and otentia to a wide range of engineering aications. I. INTRODUCTION Many of industria rocesses are inherenty distributed arameter systems (DPS) that their states, contros, oututs and rocess arameters may vary temoray and satiay [1], such as the therma, fuid fow and chemica reactor rocess. The traditiona aroach, which ouary eists in many engineering aications, is to ignore the distributed nature of the system and derive the simified mode based on the assumtion of umed arameters. Thus the cassica contro theory, which reies on the accurate mathematica mode of the rocess, can be aied. As an inteigent method, the fuy ogic controer (FLC) is widey used in industria rocesses due to its inherent robustness [2-4]. Though some traditiona FLCs have been used for DPS contro [5-8], however, these traditiona fuy aroaches are not inherenty designed for the satiay distributed dynamic rocess because of their two-dimensiona (2D) feature. The tye-2 fuy system introduces a secondary membershi function (MF) for deaing with more come rocesses [9-11]. Unfortunatey, the common ractice is to ay both rimary and secondary MF to describe the same variabe, which has imited its otentia aications. Recenty, a nove three-dimensiona (3D) FLC, based on the concet of the tye-2 fuy set has been deveoed for satiay distributed dynamic systems [12-13]. In difference to the tye-2 fuy system, it aies the secondary MF to a different hysica domain sace, whie the rimary MF sti Xiao-Gang Duan Schoo of Mechanica & Eectrica Eng Centra South University China for the temora variabe. Thus, a satio-temora fuy system is deveoed for the rea aication in the rocess industry. Since the satia information is difficut to obtain for buiding the secondary MF, the actua imementation of this 3D MF reies on the sensors ocated on the sace domain, which wi rovide the satia information. This 3D MF is actuay a virtua tye that consists of a grou of 2D MF. After the 3D fuification, the 3D MF wi be formed with the he of sensors. The 3D inference wi incude the satia T-norm oeration and the rue inference. Finay, the satia reduction needs to be carried out, which is equivaent to the tye-reduction in tye-2 fuy systems, before the traditiona defuification. II.THREE-DIMENSIONAL FUZZY LOGIC CONTROLLER The noninear distributed arameter system discussed above with one actuator ( 1 ) is considered as shown in Fig. 1, where oint measurement sensors are ocated at 1, 2,, in the one-dimensiona sace domain and an actuator u acts on the distributed rocess u Fig. 1: Sketch of a DPS with a distributed actuator u and oint measurement sensors. For controing such a satio-temora rocess, a cometey new fuy contro method shoud be deveoed with the inherenty designed features for the satio-temora rocess. One of the essentia eements of this tye of fuy system is the 3D fuy set used for modeing the 3D uncertainty. A. Three-dimensiona fuy set A 3D fuy set is introduced in Fig.2 by deveoing a third dimension for satia information from the traditiona fuy set [24] /09/$ IEEE 799

2 2 (, Fig. 2: A three-dimensiona fuy set. B. Configuration of 3D FLC Since the 3D fuy set is inherenty defined to eress the satia information, it woud be usefu to deveo fuy contro for the distributed arameter rocess that has satio-temora nature. Theoreticay, the 3D fuy set or 3D goba fuy MF is the assemby of 2D traditiona fuy sets at every satia ocation. Practicay, this 3D fuy MF can be aroimatey constructed by 2D fuy MF at each sensing ocation. The more sensors are used in the sace domain, the better 3D goba fuy MF coud be obtained. As there is ony one contro outut and no recise or quantitative information that can be used for design, the feasibe action is to contro the overa behavior of the satia domain instead of accuratey maniuating each satia ocation. Thus, a centraied rue base might be more aroriate, which can aso avoid the eonentia eosion of rues when sensors increase. 3D FLC t Cris Inut 3D Fuification Satia T-norm 3D fuy inut rue Inference Rues 3D Inference Fig. 3: Basic structure of 3D FLC The new FLC wi have the same basic structure as the traditiona one, which is comosed of fuification, inference and defuification as shown in Fig. 3. Due to its unique 3D nature, some detaied oerations of this new FLC wi be different from the traditiona one as shown in Fig. 3 and Fig. 4 for satia information rocessing and comressing. 1) Fuification Simiar to the traditiona FLC, there are two different fuifications: singeton fuifier and non-singeton fuifier [14]. If finite sensors are used, this 3D fuification can be considered as the assemby of the traditiona 2D fuification at each sensing ocation. Therefore, for discrete 3D fuy outut Satia Reduction 3D Deuification Cris outut Traditiona Deuification t measurement sensors ocated at 1, 2,, shown in Fig.1, ( i ) [ 1( i ),, J ( i )] is defined as J cris satia inut variabes in sace domain Z { 1,, }, where ( i) X IR ( 1,, J ) denotes the cris inut at the measurement ocation i for the satia inut variabe ( ), X denotes the domain of ( i). The fuification for each cris satia inut variabe ( ) is uniformy eressed as one 3D fuy inut A X in the discrete form as foows A ( (, /( (, X1 Z 1( X X 1 1 X2 Z 2( X X A ( (, /( (, 2 2 A ( (, /( (, XJ Z J ( X X J J J J 2) Rue Inference a) Rue Base Using the 3D fuy set, the th rue in the rue base can be eressed as foows where R : if ( ) is C 1 and and J ( is 1 C J then u ( is G. (1) R denotes the th rue, 1, 2,, N ; ( ) denotes satia inut variabe, 1,, J ; A denotes 3D fuy set; u( denotes the contro action at satia ocation, u( U IR, and G denotes a outut 3D fuy set; N is the number of fuy rues. b) Inference The inference engine of the 3D FLC is eected to transform a 3D fuy inut into a traditiona fuy outut. Thus, the inference engine is the same as traditiona inference oeration. The ony difference is that this 3D fuy inference is at each satia ocation. The inference rocess is about the oeration of 3D fuy set incuding union, intersection and comement oeration. Considering the fuy rue eressed as in (1), the rue resents a fuy reation R : C1 CJ G, 1,2,, N. Thus, a traditiona fuy set is generated via combining the 3D fuy inut and the fuy reation reresented by rues. Satia T-norm Oeration This first oeration in the inference is to transform the 3D fuy inut A X into a 3D set W, Through the su-star comosition oeration on the inut set and antecedent set, W is denoted by W I C D I C D with the grade of the MF derived as 800

3 3 (, ece, ) W this 3D set into a 2D set that can aroimatey I C D su ee, cece[ (, ece, ) (, ece, )] describe the overa imact of the satia distribution with I C D resect to the outut u (. su ee, cece[ E( e, ) CE( ce, ) ( e, ) ( ce, )] C D {su ee[ E( e, ) ( e, )]} {su [ (, ) (, )]} C cece CE ce ce ( u, d D (4) Fig. 4 gives a demonstration of satia information fusion in d the case of two cris inuts from the sace domain Z, i.e. where denotes a continuous ane curve and ( [ 1 (, 2( ]. d denotes the arc ength of. 1 2 Fig. 4: Satia T-norm oeration 3D Inference Oeration The imication oeration is eecuted through the foowing equation ( u, (, (, ( u, C1 where Mamdani imication is used; stands for a t-norm; ( u, is the membershi grade of the consequent set of the 3-D goba fuy MF fired rue R ; is the outut 3D fuy set of the fired rue R. Then, CJ G is defined by the union of reations 1,, N R ( u, ( u, (2) where N is the number of rues in the rue base. When an inut is aied, the comosition of the fuy set W to which beongs (2) can be written as ( u, (, ( u, (3) 3) 3D Defuification Satia Reduction C X 1 C X C X 2 T-norm W W 1,, N The set shows an aroimate fuy satia distribution for each outut u (, in which hysica information such as the sensor ocations and their reative distance between each other are aroimatey embedded. The satia reduction oeration is to comress the satia distribution information ( u, into 2D information (u). The 3D set coud be simy regarded as a 2D satia MF on the ane (, for each outut u (. Thus the centroid oeration in (4) might be a good otion to comress W 2-D fuy satia MF at ( Satia fusion G Traditiona Defuification Oeration After the inference, a traditiona fuy set is roduced. Then, the same defuification used in the traditiona FLC wi be taken to roduce a cris outut. Since there is no standard defuifier in theory, the center-of-sets [15] is chosen as the defuifier in this aer due to its sime comutation. In center-of-sets defuification, the consequent set of each fired rue R ( 1, 2,, N ) is reaced by a singeton situated at its centroid, whose amitude equas the firing eve, then the resut of defuification is the centroid of these singetons. The outut eression is given as foows N N u c 1 1 where c U is the centroid of the consequent set of the fired rue R ( 1, 2,, N ), which reresents the consequent set G in (1), and N is the number of fire rues, N N. C. Design of 3D FLC for satio-temora rocesses If ony error and error in change are used as the cris satia inuts for 3D FLC, then a 3D two-term FLC as shown in Fig. 5 wi be designed for a distributed arameter system. E( and R( are the scaed satia inuts for 3D two-term FLC, where E ( denotes the scaed satia error, R( denotes the scaed satia error in change, E( Ke( e( IR, R( Kd ( e ( IR, K e ( and K d ( denote the scaing factors of actua error e ( and error in change e ( de( / dt, resectivey. Then, a sime 2D rue base is generated with J 2 in (1). The 3D FLC roery utiies the satia information ony with a sime rue base, taking fu advantages of the traditiona fuy contro described in subsection B of section II. Currenty, there is no mature method that coud guide design of 3D FLC for the DPS. Without the we-deveoed aroach, a these have to be considered using the ractica knowedge of the rocess when designing for the distributed arameter rocess [16]. In genera, design rocedure of 3D FLC can be summaried as foows. 1) Proery design the overa behavior measurement of the satia domain according to contro requirement. 2) Proery aocate measurement sensors in satia domain according to the rocess nature and overa 801

4 4 behavior measurement. 3) Proery design 2D fuy MFs for each inut from the satia domain. These 2D fuy MFs wi form a 3D goba fuy MF on the entire satia domain. 4) Design and tune scaing gains for a inuts and outut for the overa behavior measurement, and the contro erformance. 5) Design contro rue base according to the contro requirement. This design is simiar to those in the traditiona FLC [17-18]. from y (, 0, the rocess wi move to another satiay non-uniform steady state with the eak in the midde of the rod without any contro. Therefore, the contro obective is to stabiie the rod temerature aong the rod ength. A B e ( E( K e ( e( R( s K d ( 3D fuy reasoning u Ku U A b 1 ( b 2 ( Fig. 6. Cataytic rod. B Fig. 5: Structure of 3D FLC system. III. APPLICATION A. Cataytic reaction rod Consider the robem of a cataytic reaction in a reactor as shown in Fig. 6 [1], where the reactor is fed with ure secies A and a ero-th order eothermic cataytic reaction of the form A B takes ace on a rod. The reaction is eothermic, so that a cooing medium that is in contact with the rod is used for cooing. Under the assumtions of constant density, constant heat caacity of the rod, constant conductivity of the rod, constant temerature at both ends of the rod, and ecess of secies A in the furnace, the mathematica mode describing the satio-temora evoution of the dimensioness rod temerature is given as the foowing araboic PDE: 2 y(, y(, e1 (, ) 2 T y t t (5) U b( U ( T (, T e subect to the boundary conditions and initia condition: y(, 0, 0 y(, 0, 0 (6) y(, y0(, 0 where 50 denotes a dimensioness heat of reaction; T 4 denotes a dimensioness activation energy; 2 denotes a dimensioness heat transfer coefficient; (, with [ 0, ] denotes the dimensioness temerature of the rod, which is satiay deendent; y ( ) denotes the initia temerature, which is satiay deendent; b ( U ( denotes 0 a satiay distributed cooing source with b( distribution and U ( as the maniuated inut. U as the In this case, there is one oint sources that are used to coo the reactor, i.e. b ( ( 0.5 ), where () denotes Dirac deta function. It is verified that the satiay uniform oerating steady-state y (, 0 is an unstabe one [1] as shown in Fig. 7. For any initia condition with a deviation Fig. 7. Unstabe rod temerature evoution from initia condition y ( sin( ). 0 B. Deign of 3D FLC For simicity and convenience, seven oint sensors are uniformy ocated aong the ength of the reactor with / 8 with 1,, 7 for coecting the satia distribution of the temerature y (,. Therefore, at each saming time k, seven satia temeratures are measured as y(, with 1,, 7. Two satia inuts for 3D FLC are error and error in change, i.e. e( { e( )} and e( { e( )}, where e( ) yd (, y(,, e( ) e(, e(, k 1), k and k 1 are the k th and k 1 th same time. Initiay, the scaing factor for each e ( ) is set to be 1, the scaing factor for each e( ) is set to be 1.27, and the scaing factor for u is set to be A the scaed inut variabes e ( ), e( ) and outut variabe u are normaied into [-1, 1]. Due to finite sensors used, the MFs for error and change in error are discrete. As described reviousy, the 3D goba fuy MF is the assemby of 2D fuy MFs at each sensing ocation. Each e ( ) and e( ) as we as the contro outut u are cassified into seven inguistic abes as ositive arge (PL), ositive midde (PM), ositive sma (PS), ero (O), negative sma (NS), negative midde (NM), and negative arge (NL). For convenience, the MF is chosen as trianguar shae. Then, the actua outut U ( is eressed as U ( K u( u 802

5 5 where k and k 1 are the k th and k 1th same time. A inear contro rue base wi be used with the foowing format If e ) is PS and e ) is NS then u ) is G ( ( where e ) and e ) are scaed satia inut variabes, ( ( reresenting error and error in change resectivey; PS and NS denote 3D fuy sets, which are assembed by 2D fuy sets PS and NS at each sensing ocation; u ) is the contro action; G is a 3D fuy set. The rue weight is defauted as unity. Singeton fuifier is used for fuification, the minimum is used for the t-norm in the intersection oeration, the maimum is used for the t-conorm in the union oeration, and the Center-of-sets is used for defuification. C. Simuation comarison The simuation resuts shown in Fig. 8 and Fig. 9 indicate that the contro erformance of the roosed 3D FLC is better than that of the 2D FLC. ( ( Fig. 9. The traditiona FLC fais to stabiie the rocess I. CONCLUSIONS In this aer, a nove aication of tye-2 fuy system is resented by deveoing a satio-temora FLC for controing a cass of satio-temora rocesses. Simiar to the tye-2 FLC, this 3D FLC aso consists of fuification, rue inference, satia reduction and defuification. In difference to the tye-2 fuy system, the secondary MF is used to rocess satia variabe. With the he of sensors ocated on the satia domain, the 3D MF can be obtained. With the nove structure, a sime 2D rue base can sti be used for. Thus, rues wi not increase as sensors increase for measuring satia information, which minimies the comutation for rea word aications. Finay, the 3D FLC is successfuy aied to a cataytic reaction rod. ACKNOWLEDGMENT The work is artiay suorted by a grant from RGC of Hong Kong (CityU: ) and a grant from NSF China ( ). Fig. 8. Rod temerature evoution rofie and its contro signa from 3D FLC. REFERENCES [1] P. D. Christofides, Noninear and Robust Contro of Partia Differentia Equation Systems: Methods and Aications to Transort-Reaction Processes. Boston: Birkhäuser, [2] W. H. Ray, Advanced rocess contro. New York: McGraw-Hi, [3] M. J. Baas, The Gaerkin method and feedback contro of inear distributed arameter systems, J. Math. Ana. A., vo. 91, , [4] R. F. Curtain, Finite-dimensiona comensator design for araboic distributed systems with oint sensors and boundary inut, IEEE Trans. Autom. Contro, vo. AC-27, no. 1, , Feb [5] J. Lin and F. L. Lewis, Two-time scae fuy ogic controer of feibe ink robot arm, Fuy Sets and Systems, vo. 139, no. 1, , [6] D. I. Sagias, E. N. Sarafis, C. I. Siettos, and G.. Bafas, Design of a mode identification fuy adative controer and stabiity anaysis of noninear rocesses, Fuy Sets and Systems, vo. 121, no.1, , [7] P. Sooraksa, and G. R. Chen, Mathematica modeing and fuy contro of a feibe-ink robot arm, Math. Comut. Modeing, vo. 27, no. 6, , [8] M.-R. Akbaradeh-T, Fuy contro and evoutionary otimiation of come systems, Ph.D. Dissertation, The University of New Meico, USA, [9] J. M. Mende, Uncertain Rue-Based Fuy Logic Systems: Introduction and New Directions. Uer Sadde River, NJ: Prentice-Ha, [10] J. M. Mende and R. I. Bob John, Tye-2 Fuy Sets Made Sime, IEEE Trans. Fuy Syst., vo. 10, no. 2, [11] J. M. Mende and H. W. Wu, Tye-2 Fuistics for Nonsymmetric Interva Tye-2 Fuy sets: Forward Probems, IEEE Trans. Fuy Syst., vo. 15, no. 5,

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