Advanced Scence and Technoogy Letters Vo.83 (ISA 205), pp.60-65 http://dx.do.org/0.4257/ast.205.83.2 Research on Compex etworks Contro Based on Fuzzy Integra Sdng Theory Dongsheng Yang, Bngqng L, 2, He Jang, Xngyu Lu 2, Yhe Wang 3 Coege of Informaton Scence and Engneerng ortheastern Unversty, Laonng Provnce, Chna 2 State Grd Laonng Eectrc Power Suppy Co., Ltd., Laonng Provnce, Chna 3 State Grd Shenyang Eectrc Power Suppy Co., Ltd., Laonng Provnce, Chna Abstract. A fuzzy ntegra sdng mode contro method based on the fnte tme stabe theory and ntegra sdng mode contro theory s proposed to sove a cass of compex network probems. The compex networks are expressed by T-S fuzzy modes wth bounded approxmaton errors by usng the approxmaton capabty of T-S fuzzy modes. The paper proposes fuzzy dynamc ntegra sdng mode contro scheme and the scheme s desgned for the compex networks based on ther T-S fuzzy approxmaton modes. Keywords: Fuzzy, compex networks, ntegra sdng mode, nonnear system.. Introducton Sdng mode contro s aso caed varabe structure contro. It s under the acton of the sdng mode controer, and controed system comes nto hgh frequency swtched state to force t move contnuousy on the sdng surface. Eventuay t w converge to the desred sdng mode aong the sdng mode surface so as to acheve the goa of contro. Sdng mode contro s a very effectve method n deang wth uncertan hgh-order nonnear dynamc system []. It gves expresson to the uncertan nonnear system s smaer error, externa dsturbance s strong robustness and the smpe agorthm desgn. But t can produce hgh frequency chatterng of sdng mode n the actua sdng mode contro system due to the nfuence of the non-dea effects. Wth the deveopment of compex networks n recent years, compex network appears n varous knds of actua system. A compex network composes of many nodes and every node connects wth each other. It s hgh dmenson, strong coupng and uncertanty, so the tradtona snge node contro theory s dffcut to sove mutpe nodes compex network system's anayss and desgn probems. In the past ten years, many researches are based on the anayss of the T - S fuzzy mode n order to study the ssue of compex nonnear systems and compex networks. A seres of fuzzy ntegra sdng mode contro scheme of the nonnear system can be put forward wth ths mode, because t provdes the advantage of tradtona contro theory and technoogy. ISS: 2287-233 ASTL Copyrght 205 SERSC
Advanced Scence and Technoogy Letters Vo.83 (ISA 205) Ths paper puts forward a new knd of fuzzy dynamc ntegra sdng mode contro method to reaze the stabty of the nonnear system. The key s that the ntegra sdng surface depends on both of the system state vector x and contro nput vector u. The sdng moton can be guaranteed by sovng LMI matrx, and the ntegra sdng surface aong wth the dynamc sdng-mode controer [2] can be obtaned. 2 Probem descrptons and the premnary knowedge The th chd node of the compex network can be expressed n formua () : dx () t f( x (), t u ()) t () Consder a cass of compex network composed of chd nodes S,, 2,. Each rue of chd node S can use T - S fuzzy mode to descrbe: ζ s M and and ζ g s M g ;Then If dx () t F x () t G w () t G u () t Ω () t (2) 2 Ω () t H x () t (3) j j j j z () t K x () t Lu () t (4) y() t K x() t L w(), t,2,,2 r (5) y y F, G, G 2, K, L represent the rue node matrx of the th chd node. Hj represents the nterconnecton matrx of the th chd node and the jth chd node. n m x R, u R, w, z, y respectvey represent the system state, contro, dsturbance, measurabe output and the output vector. If adoptng the snge pont burred, the product of reasonng and center average souton of the bur, the above formua can be represented as: r η ζq 2 j j j j (6) dx () t ( ())( t F x () t G w () t G u () t H x ()) t r η ζq z () t ( ())( t K x() t Lu ()) t (7) r η ζq y y y () t ( ())( t K x () t L w ()) t (8) Copyrght 205 SERSC 6
Advanced Scence and Technoogy Letters Vo.83 (ISA 205) g ( q ( t)) Mq ( q ( t)), ( q ( t)) r q γ ζ ζ η ζ γ ( ζ ( t)) q γ ( ζ ( t)) q (9) ζ,, ζ s antecedent varabe, M ( ζ ( t)) s membershp of g q q M q. It s observed that γ ( ζ q ( t )) 0, η ( ζ ( t)) 0, q η ( ζq ( t)),,, J,, r, r s th chd node rue's amount. For any compex network, f( x ( t), u ( t)) s n the nterva of X r U and f (0, 0) 0.There exst s postve constant ε f. So the T-S fuzzy mode s descrbed as foows: r η ζq 2 j j j j (0) fˆ( x, u ) ( ())( t F x () t G w () t G u () t H x ()) t f( x, u ) fˆ ( x, u ) E( x, u ) r η ( ζq())(( t F F ) x() t G w() t ( G2 G2 ) u() t H j j j () r η ζq 2 T T T (2) Exu (, ) ( ( t))[ F, G ] [ X, U ] F, G ε (3) 2 f Accordng to (6),(9), (), the approxmaton error upper bound vaue can obtan sma enough by seectng enough arge vaues of fuzzy rues n the T-S fuzzy mode[3]. Through the anayss of the approxmaton error, one can get: r () η ( ζq ())(( ) () () ( 2 2 ) () j j j j x t t F F x t G w t G G u t H x (24) 3 The desgn of dynamc ntegra sdng mode contro For the provded nonnear mode(), (6), the foowng fuzzy ntegra sdng surface s desgned as foows: 62 Copyrght 205 SERSC
Advanced Scence and Technoogy Letters Vo.83 (ISA 205) r ( ) r ( ) ( ) ( ( )) ( ( ) ( )) t t x u x ( ( )) u S ( t A d t A e t η 0 ζq t A F x φ G2 u φ dφ η 0 ζq t A d( t) x( t) x(0) e( t) u( t) u(0) (35) A A E H respectvey descrbe m*n m*m m*n m*m matrx. x u Theorem: Consder the nonnear system n formua () and the fuzzy system n formua (6).One can use the foowng fuzzy controer: ζ s M and and ζ g s M g ;Then If - - - u x u x u () () () () j j() ( ω υ()) sgn( ()),,,,, j j u t E x t H u t G w t A A H x t A A t A s t J r (46) the fuzzy controer usng fuzzy rues can aso be descrbed as foows: r - - - u x u x u u () t η ( ζq ()) t E x () t H u () t G w () t A A H j xj () t A A ( ω υ ()) t A sgn( s ()) t j j (57) υ () t ε A x [ x T () t, u T ()] t T (68) f ω s a postve constant, the cosed-oop contro system trajectory can keep on the sdng surface amost snce the fnte tme by usng the above ntegra controer, Furthermore constructng Lyapunov functon[4] for the subsystem functon to prove: T Vt () s() ts () t (79) 2 r r x u x u η ζq 2 η ζq s () t A x () t A u () t ( ()) t A ( F x () t G u ()) t dt ( ()) t A ( E x () t H u ()) t dt (20) Then from(4), (7), (20)one has: T Vt () s() ts () t (2) 2 r r x u x u η ζq 2 η ζq s () t A x () t A u () t ( ()) t A ( F x () t G u ()) t dt ( ()) t A ( E x () t H u ()) t dt (22) Copyrght 205 SERSC 63
Advanced Scence and Technoogy Letters Vo.83 (ISA 205) Then from (8), (9), (22)one has: Vt () ω s() t ω 2 Vt () (23) Through the above formua one can prove that the system's trajectory[5] can keep on the sdng surface n the mted tme, and the above argument has gven the reevant proof. So defne other sdng surface equaton n order to descrbe the trajectory convenenty. r t 2 () { () η ( ())( ) ( ) } 0 ζq φ φ s t Af t t CF CE x d (24) ( ) ( ) T T f () t x t x 0, x () t [ x (), t u ()] t, C [,0 ] T I n n m, x u F [ F, G2 ], A [ A, A ], E [, E H ] One can draw the foowng two cosed oop contro system equaton: C [0, I ] T, 2 m n m r ˆ η ζq 2 j j j j x () t f( x, u ) ( ())( t F x () t G w () t G u () t H x ()) t (25) r ˆ η ζq j j j j u(t) g( x, u ) ( ())( t E x () t G w () t H u () t H x ()) t (26) The above two equatons are equvaent to: r 2 () η ( ζq ())[( ) ( ) j j () ()] j j x t t CF CE x t H x t G wt (27) If the desgn of cosed-oop contro equaton (27) s reasonabe, then the stabty of sdng mode controer can be guaranteed. 4 Concuson A nove dynamc ntegra sdng mode contro scheme has been deveoped for compex networks systems based on T-S fuzzy modes. It has been shown that the sdng mode can be acheved and mantaned amost surey snce the nta tme wth the ntegra sdng surface and a nove dynamc sdng-mode controer. And suffcent condtons to guarantee the stabty of the sdng moton are gven n terms of LMIs. 64 Copyrght 205 SERSC
Advanced Scence and Technoogy Letters Vo.83 (ISA 205) References. Sjac, D. D.: Decentrazed Contro of Compex Systems [M]. Boston: Academc Press, (99) 2. Ho, D. W. C. and u, Y.: Robust Fuzzy Desgn for onnear Uncertan Stochastc Systems va Sdng-mode Contro. IEEE Transactons on Fuzzy Systems, 5, 350 (2007) 3. Utkn, V. I.: Varabe Structure Systems wth Sdng Modes [J]. IEEE Transactons on Automatc Contro, 22, 22 (977) 4. Utkn, V. I., Gudner, J., Sh, J.: Sdng Mode Contro n Eectromechancay Systems [M]. London: Tayor & Francs, (999) 5. Xa, Y. Q., Chen, J., Lu, G. P.: Robust Adaptve Sdng Mode Contro for Uncertan Tmedeay Systems [J]. Internatona Journa of Adaptve Contro and Sgna Processng, 23, 863 (2009) Copyrght 205 SERSC 65