POTENTIAL ENERGY BASED STABILITY ANALYSIS OF FUZZY LINGUISTIC SYSTEMS. 1. Introduction

Transcription

1 Iranan Journal of Fuzzy Systems Vol. 2, No. 1, (2005) pp POTENTIAL ENERGY BASED STABILITY ANALYSIS OF FUZZY LINGUISTIC SYSTEMS A. A. SURATGAR AND S. K. Y. NIKRAVESH ABSTRACT. Ths paper presents the basc concepts of stablty n fuzzy lngustc models. The authors have proposed a crteron for BIBO stablty analyss of fuzzy lngustc models assocated to lnear tme nvarant systems [25]-[28]. Ths paper presents the basc concepts of stablty n the general nonlnear and lnear systems. Ths stablty analyss method s verfed usng a benchmark system analyss. 1. Introducton For many real-world systems and processes, a mathematcal descrpton n the form of dfferental/dfference uatons or other conventonal model s ether nfeasble or mpractcable due to complexty nvolved, the tme consderaton, and the ntrnsc nature of nformaton ncompleteness. For fndng a practcal way studes have been done to analyze fuzzy control systems [32]-[12]. Tanaka and et al [30] proposed a method for stablty analyss of TSK model by fndng a common Lyapunov functon, Percup and et al [19] used the center of manfold theory for fuzzy system stablty analyss. The authors of ths paper proposed a suffcent condton for stablty of TSK fuzzy model [22], Farnawata [14] and Lnder [4] separately worked on robust stablty controller desgn. Many other efforts, whch have been done n the TSK stablty analyss, are found n [21]. They use classcal approaches for stablty analyss of fuzzy systems. Unfortunately these approaches conflct wth the smplcty dea, whch was the man am of Zadeh when presentng hs fuzzy system approach. For smplcty n system analyss and humanty nterface, a lngustc model s offered [35]. Some authors have done some ncomplete researches on ths area [8]-[34]. Recently Margalot and Langholz [15]-[17] have proposed some approaches for lngustc nonlnear systems. The basc dea s usng crsp uaton of Lyapunov for the stablty analyss n ther approaches. Furuhash et. al [8]-[34] have proposed a defnton for ulbrum n fuzzy lngustc model. In ther defnton some thngs seems not to be qute rght [23]. Untl now the stablty analyss of lngustc systems s an open problem [12]. Here we wll extend a stablty analyss method [24] to fuzzy lngustc systems assocated wth a class of appled nonlnear systems. Secton 2 proposes some prelmnary defntons n computaton wth word. In secton 3 a theorem for stablty analyss of crsp system s proposed. Secton 4 presents the lngustc stablty concept and ncludes a benchmark problem. Fnally secton 5 Receved: Ths manuscrpt was recommended by the permanent commttee of the 2003 Conference on Intellgent Systems, Mashhad, Iran. A bref verson of ths paper was earler ncluded n the proceedngs of the CIS2003. Key words and phrases : Fuzzy modelng, Stablty analyss, Necessary and suffcent condton for stablty, Potental energy

2 66 A. A. Suratgar and S. K. Y. Nkravesh ncludes a concluson. 2. Defntons n Lngustc Calculus In ths secton we propose some defntons. These defntons are used n the next sectons n order to defne ulbrum and stablty concepts. Defnton 2.1. Center of Lngustc Value: consder, as an example, the followng lngustc value membershp functons: FIGURE 1. Gaussan membershp functons Lng. The X cent s the center of lngustc value LngM f and only f (1) µ ( x ) = µ ( X ) The lngustc values are { 1, Lng2, Lng3, Lng4,... } max x lngm Note 1. If there s an fnte nterval (.g. [ a, b] ) where µ lngm = 1, then the mean pont of nterval s defned as center of lngustc value. Note 2. If there s an nfnte nterval (.g. [ a, ) ) where µ = 1, then the fnte lngm boundary of nterval s defned as center of lngustc value. Defnton 2.2. Lngustc Neghborhood: Consder a set of fuzzy sets as (2) Set = { A, B, C,... } Where A, B, C,... are the fuzzy sets. Sort the center of lngustc values of these fuzzy sets ncreasngly. Assocate the sorted suence wth N 1 m. Thus the frst term s 1 and ts assocated fuzzy set s labeled lng 1, the second term s 2 and ts assocated fuzzy set s labeled lng 2, cent lngm 1 Nm s the frst m terms of natural number suence.

3 Potental Energy Based Stablty Analyss of Fuzzy Lngustc Systems 67 The + ζ Z neghborhood of lngr s shown by Ω N (3) lngr = { lngj R ζ j R + ζ } L.. ( ) ζ Example 2.1. Consder fuzzy sets wth the followng membershp functons. FIGURE 2. A sample of membershp functons Then L. N. (4) Ω ( lng3) = { lng2, lng3, lng4} ζ = 1 Defnton 2.3. Lngustc Deleted Neghborhood: The Lngustc Neghborhood of lngr whch does not contan lngr, s called Lngustc Deleted Neghborhood of lngr. It s shown as Ω, N (5) lngr { lngj R j R j r} L. D... ( ) = ζ + ζ ζ Defnton 2.4. Fuzzy ulbrum subset: Consder a rule base that descrbes the dynamc relaton of the state varables of a system wthout exogenous nput as follows: (6) R : If x( k) s LngH then x( k + 1) s LngE = 1,..., m If there s a rule n whch Lng H and Lng E are lngustcally ual, then Lng H Lng E Lng s called a fuzzy ulbrum subset.

4 68 A. A. Suratgar and S. K. Y. Nkravesh Defnton 2.5. Equlbrum halo: Consder P = { lngp, lngp, lngp,.., lngp N } as a set of all fuzzy ulbrum subsets. Then lngp s called ulbrum halo f for ntal condton x ( k 0 ) = lngp the followng rule f x(k) s lngp then x(k+1) s lngp. s fred wth hghest degree of frng, among the other rules, for all the k > k0. 3. A Necessary and Suffcent Condton for Stablty Analyss of a Class of Applcable Mechancal Systems [24] Ths secton revews some defntons about the potental energy and conservatve forces n the pure mechancal systems. We have observed that the work done aganst a gravtatonal or an elastc force depends only on the net change of poston and not on the partcular path followed n reachng the new poston. The Force felds wth ths characterstc are called conservatve force felds whch posses an mportant mathematcal property and wll be llustrated as follows. Consder a force feld as shown n fg.1, where the force F s a functon of ts coordnates. FIGURE 3. The sample path for descrbng the potental energy The work done by the force F durng the dsplacement dr of ts pont of applcaton s (7) du = F. dr The total work done along ts path form 1 to 2 s 2 2 (8) U = F. dr = ( F dx + F dy F dz) x y The ntegral F. dr s a lne ntegral dependent, n general, upon the partcular path followed between any two ponts 1 and 2 n space. If, however, F. dr s an exact dfferental dv of some scalar functon V of the coordnates, then z

5 Potental Energy Based Stablty Analyss of Fuzzy Lngustc Systems 69 2 (9) U = dv = ( V V ) V V 1 whch depends only on the end ponts of the moton and whch s thus ndependent of the path followed. The mnus sgn n front of dv s arbtrary but s chosen to agree wth the customary desgnaton of the sgn of potental energy changes n the earth s gravty feld. If V exst, the dfferental change n V becomes (10) dv = dx + dy + dz y z Comparng t wth (11) dv = F. dr = F dx + F dy F dz 2 1 x y + z yelds (12) F x = and So the force may also be wrtten as the vector F y = and y F z = z (13) F = V where the symbol stands for the vector operator del whch s defned as follows (14) = + j + k y z The quantty V s known as the potental functon, and the expresson V s known as the gradent of the potental functon. When the force components are dfferentable from a potental functon as descrbed above, the force s sad to be conservatve, and the work done by the force between any two ponts s ndependent of path followed. In the followng we propose the necessary and suffcent condtons for asymptotc stablty of some appled mechancal systems. Theorem 3.1. Consder a mechancal system whch s perturbed wth a force and starts to move at ntal tme. There s no exogenous nput. Suppose all the forces are conservatve forces except the frctonal torque, and x s system s solated ulbrum pont. The ulbrum state x s asymptotc stable f and only f there s a symmetrc deleted neghborhood Ω of x such that (15) and where the x Ω de potental dx de x potental : > 0 dx x ( x ) = 0 =1,2,3 E potental s the potental energy of the system.

6 70 A. A. Suratgar and S. K. Y. Nkravesh Note1. Wthout loss n generalty x s chosen as the orgn of coordnates and x s supposed to be the reference of system s potental energy,.e. (16) E ( ) = 0 potental x Note2. In Newtonan dynamc systems the set of two numbers, the poston and velocty at tme t 0, s qualfed to be called the state of the system at tme t 0 [2]. In the other word [ x y z x& y& z& ] T s a state vector. Ths concept s used n the theorem s proof. Note 3. It s supposed that E potental s drectonal dervable on Ω. Note 4. It s supposed that the perturbed system at ts ntal condton has potental energy but does not have knetc energy. Proof. see [24]. 4. A Necessary and Suffcent Condton for Stablty Analyss of a Class of Lngustc Fuzzy Models Ths secton uses Theorem 3.1. and proposes a necessary and suffcent condton for stablty of lngustc models assocated wth a class of lnear and/or nonlnear mechancal systems. Theorem 4.1. Consder a system wthout exogenous nputs and all appled forces (except the frctons forces) are conservatve forces. An ulbrum halo s asymptotc stable f and only f t has a symmetrc lngustc deleted neghborhood such that the potental energy of system s greater or ual to ts potental energy. Proof. In the prevous secton we saw the necessary and suffcent condton for asymptotc stablty of an ulbrum pont of the gven system. The followng are uvalent. 1- Ω x de potental ; x Ω : > 0. x dx 2-A local mnmum of E potental (x) s E x ). potental ( Consder an arbtrary drecton as x. If the mnmum potental energy s at x then n a gven neghborhood of x we have: If 0 = x > x then

7 Potental Energy Based Stablty Analyss of Fuzzy Lngustc Systems 71 If 0 = x < x then E potental ( x ) < E potental ( x ) E ( x ) < E ( x ) potental potental Thus (17) (18) Therefore x > x x < x E E potental potental x> x = x< x 0 > 0 = 0 < 0 E potental x = 0 > 0 x E potental x = 0 > 0 x It means that f a local mnmum of E potental (x) s E x ), then potental ( On the other hand, f Ω Ω x de potental ; x Ω : > 0. x dx x de potental ; x Ω : > 0 x dx t s obvous that a local mnmum of E potental (x) s E x ). potental ( Hence Theorem 3.1. can be restated as follows. The ulbrum pont of the system gven n Theorem1 s asymptotc stable f and only f a local mnmum of potental energy s n x. However crsp model concepts can be extended n the fuzzy model. The ulbrum pont n a crsp model s assocated wth the ulbrum halo n fuzzy model; the crsp neghborhood s assocated wth a lngustc neghborhood and so on. Usng the above dea t s concluded that In the system gven n Theorem 3.1., an ulbrum halo s asymptotc stable f and only f there s a symmetrc lngustc deleted neghborhood of ulbrum halo such that potental energy of the system there s greater than or ual to the potental energy of ulbrum halo. Example 3.1. Consder a pendulum system.

8 72 A. A. Suratgar and S. K. Y. Nkravesh The rules of ulbrum subsets s as follows FIGURE 4. A pendulum Lng19 Lng 4 Lng13 Lng14 Lng11 Lng10 Lng9 Lng8 X ( k) =, Lng10 Lng11 Lng9 Lng1 Lng16 Lng7 Lng6 Lng12 Lng10 Lng10 Lng10 Lng10 Lng11 Lng11 Lng11 Lng10 Z ( k) =, Lng10 Lng10 Lng10 Lng10 Lng10 Lng10 Lng10 Lng10 where the Z and X are the coordnates of the system. The ulbrum halo of system wll be obtaned as X(k)= Lng10 and Z(k) = Lng10 The system s potental energy rule base of ulbrum halo s as follows: If X(k) s Lng10 and Z(k) s Lng10 then E s Lng10. Potental

9 Potental Energy Based Stablty Analyss of Fuzzy Lngustc Systems 73 Usng the system s potental energy rule base, there s a deleted neghborhood, wth potental energy greater than or ual to the ulbrum halo potental energy. If X(k) s Lng9 and Z(k) s Lng11, then E s Lng11. If X(k) s Lng9 and Z(k) s Lng10, then If X(k) s Lng11 and Z(k) s Lng10, then Potental E Potental s Lng10. E Potental s Lng10. If X(k) s Lng11 and Z(k) s Lng11, then E Potental s Lng11. Usng Theorem 4.1., the ulbrum halo of system s asymptotc stable. Ths analyss concdes wth the classc stablty analyss. 5. Concluson Ths paper has presented the basc concepts of stablty analyss for the fuzzy lngustc models; also ths paper presents the necessary and suffcent condton for stablty analyss of a class of appled mechancal nonlnear systems. Fnally the proposed stablty analyss method s verfed usng a benchmark problem. REFERENCES [1] P. Albertos, R. Stretzel and N. Mort, Control engneerng soluton, a practcal approach, IEE Press, (1997). [2] C. T. Chen, Introducton to lnear system theory, Prentce Hall, Englewood Clffs (1970) 83. [3] Y. Dng, H. Yng and S. Shao, Theoretcal analyss of a takag-sugeno fuzzy PI controller wth applcaton to tsssue hypertherma therapy, Proc. of IEEE on Computatonal Intellgence, Vol. 1 (1998) [4] S. S. Farnwata, A robust stablzng controller for a class of fuzzy systems, Proc. of IEEE Conf. of Decson and Control, Vol. 5 (1999) [5] T. Furuhash, H. Kakam, J. Peter and W. Pedrycz, A stablty analyss of fuzzy control system usng a generalzed fuzzy petr net model, Porc. of IEEE Internatonal Conference on Computatonal Intellgence, Vol. 1 (1998) [6] S. M. Guu and C. T. Pang, On asymptotc stablty of free fuzzy systems, IEEE Trans. On Fuzzy Systems., Vol. 7 (1999) [7] T. Hasegawa and T. Furuhash, Stablty analyss of fuzzy control systems smplfed as a dscrete system, Control and Cybernetcs, Vol. 27, (1998), No. 1 (1998) [8] T. Hasegawa, T. Furuhash and Y. Uchkawa, Stablty analyss of fuzzy control systems usng petr nets, Proc. of 5-th IEEE Int. Conf. On Fuzzy Systems, (1996). [9] X. He, H. Zhang and Z. Ben, Analyss on D stablty of fuzzy system, Porc. of IEEE World Congress on Computatonal Intellgence, (1998). [10] G. Kang, W. Lee and M. Sugeno, Stablty analyss of TSK fuzzy systems, Proc. of IEEE Internatonal Conference on Computatonal Intellgence, Vol. 1 (1998) [11] S. Kawamoto, K. Tada, A. Ishgame and T. Tanguch, An approach to stablty analyss of second order fuzzy systems, Proc. Frst IEEE Int. Conf. On Fuzzy Systems,(1992). [12] E. Km, A new approach to numercal stablty analyss of fuzzy control systems, IEEE Trans. On Syst. Man and Cyber., Part C, Vol. 31 (2001) [13] H. K. Lam, F. H. F. Leung and P. K. S. Tam, Stablty and robustness analyss and gan desgn for fuzzy control systems subject to parameter uncertantes, Proc. of 9-th Internatonal Conf. On Fuzzy Systems., Vol. 2 (2000) [14] P. Lnder and B. Shafa, Qualtatve robust fuzzy control wth applcaton to 1992 ACC Benchmark, IEEE Trans. On Fuzzy Systems, Vol. 7 (1999)

10 74 A. A. Suratgar and S. K. Y. Nkravesh [15] M. Margalot and G. Langholz, New approaches to fuzzy modellng and control desgn, World Scentfc Press, (2000). [16] M. Margalot and G. Langholz, Adaptve fuzzy controller desgn va fuzzy Lyapunov synthess, IEEE Conf. (1998). [17] M. Margalot and G. Langholz, Fuzzy control of a benchmark problem: a computng wth words approach, IEEE Conf. (2001). [18] W. Pedrycz and F. Gomde, A new generalzed fuzzy petr net model, IEEE Trans. On Fuzzy Systems, Vol. 2 (1994) [19] R. E. Precup, S. Pretl and S. Solyom, Center manfold theory approach to the stablty analyss of fuzzy control systems, EUFIT, (1999), Dortmund. [20] J. J. E. Slotne and W. L, Appled nonlnear control, Prentce Hall [21] M. Sugeno, On stablty fuzzy systems expressed by fuzzy rules wth sngleton consuents, IEEE Trans. On Fuzzy Systems., Vol. 7 (1999) [22] A. A. Suratgar and S. K. Nkravesh, A new suffcent condton for stablty of fuzzy systems, ICEE 2002, Tabrz, Iran, (2002). [23] A. A. Suratgar and S. K. Nkravesh, Comment on: stablty analyss of fuzzy control systems smplfed as a dscrete system, Control and Cybernetcs Journal, Submtted. [24] A. A. Suratgar and S. K. Nkravesh, Necessary and suffcent condtons for asymptotc stablty of a class of appled nonlnear dynamcal systems, IEEE Crcut and System Conf., Sharjah, (2003), (to appear). [25] A. A. Suratgar and S. K. Nkravesh, Two new approaches for lngustc fuzzy modelng and ntroducton to ther stablty analyss, IEEE World Congress Computatonal Intellgence, USA, (2002). [26] A. A. Suratgar and S. K. Nkravesh, Two new approaches for lngustc fuzzy modelng and ts stablty, IEEE Fuzzy System and Knowledge Dscovery FSKD 02, Sngapour, (2002). [27] A. A. Suratgar and S. K. Nkravesh, Two approaches for lngustc fuzzy modelng and ts stablty, Daneshvar Journal (to be appeared), (2002) (n persan). [28] A. A. Suratgar and S. K. Nkravesh, Varaton model: the concept and stablty analyss, IEEE World Congress Computatonal ntellgence, USA, (2003). [29] Kazuo Tanaka, Stablty and stablzablty of fuzzy-neural-lnear control systems, IEEE Trans. On Fuzzy Systems., Vol. 3 (1995) [30] K. Tanaka, T. Ikeda and H. O. Wang, A LMI approach to fuzzy controller desgns based on relaxed stablty condton, Proc. of the Sxth IEEE Internatonal Conf. on Fuzzy Systems., Vol. 1 (1997) [31] K. Tanaka and M. Sugeno, Stablty analyss and desgn of fuzzy control systems, Fuzzy Sets Systems, Vol. 45 (1992) [32] M. A. L.Thathachar and P. Vswanath, On stabltyof fuzzy systems, IEEE Trans. On Fuzzy Systems., Vol. 5 (1997) [33] L. X. Wang, Fuzzy systems as nonlnear dynamc systems dentfers, part II: stablty analyss and smulaton, Proc. of 31-th Conf. of Decson and Control, (1992 ). [34] H. Yamamoto and T. Furuhash, A new suffcent condton for stable fuzzy control system and ts desgn method, IEEE Trans. On Fuzzy Systems, Vol. 9 (2001) [35] L. A. Zadeh, Outlne of a new approach to the analyss of complex systems and decson processes, IEEE Transacton on Systems Man and Cybernetcs, Vol. SMC-3, No. 1 (1973) AMIR ABOLFAZL SURATGAR*, DEPARTMENT OF ELECTRICAL ENGINEERING, ARAK UNIVERSITY, ARAK, IRAN E-mal address: a_a_suratgar@yahoo.com SYED KAMALEDIN NIKRAVESH, DEPARTMENT OF ELECTRICAL ENGINEERING, AMIRKABIR UNIVERSITY OF TECHNOLOGY, TEHRAN, IRAN E-mal address: nkravesh@aut.ac.r *CORRESPONDING AUTHOR