Delay Dependent Robust Stability of T-S Fuzzy. Systems with Additive Time Varying Delays

Transcription

1 Appled Maemacal Scences, Vol. 6,, no., - Delay Dependen Robus Sably of -S Fuzzy Sysems w Addve me Varyng Delays Idrss Sad LESSI. Deparmen of Pyscs, Faculy of Scences B.P. 796 Fès-Alas Sad_drss9@yaoo.fr El Houssane ssr LESSI. Deparmen of Pyscs, Faculy of Scences B.P. 796 Fès-Alas el_ssr@yaoo.fr Absrac s paper presens delay dependen sably condons of -S fuzzy sysems w addve me varyng delays. e approac s based on consrucng a new Lyapunov-Krasovsk funconal, and Fnsler s lemma. e perurbaons consdered are norm bounded and e resuls are expressed n erms of LMIs. Numercal examples are provded o sow e effecveness of e presen ecnque, compared o some recen resuls. Keywords: Addve me varyng delays, -S fuzzy sysem, Lner marx nequaly (LMI), Robus sably. Inroducon Durng e pas wo decades e sably analyss for akag-sugeno (-S) fuzzy sysems [3] as been suded exensvely. Los of sably crera of -S fuzzy sysems ave been expressed n lnear marx nequaly LMIs va dfferen approaces [6, 4, 6]. ese fuzzy sysems are descrbed by a famly of fuzzy IF HEN rules. However, all e aforemenoned meods are proposed for medelay free S fuzzy sysems. In pracce me delay ofen appears n many praccal sysems suc as cemcal processes, meallurgcal process, long ransmsson lnes n pneumac, mecancs, and communcaons neworks, ec. [,9]. Snce me delay, s usually a source of nsably and degradaon of

2 Idrss Sad and El Houssane ssr sysems performance, e analyss and syness ssues of fuzzy sysems w me delay as receved more aenon n recen years [3, 5, 6, 9]. Some approaces developed for general delay sysems ave been appled o deal w fuzzy sysems w me delays, e.g., e Lyapunov Krasovsk funconal approac [, 3], L e al. [3]. Applyng e model ransformaon also called Moon s nequaly [8] for boundng cross erms, Guan and Cen [4] ave suded e delay-dependen robus sably and guaraneed cos conrol of e me-delay fuzzy sysems. Recenly a free wegng marx approac as been employed n [5, 6,, 5]. In Lu e al. [7], e problem of sably for unceran -S fuzzy sysems w me varyng delay as been suded by employng a furer mproved free wegng marx meod. e free wegng marx approac as been sown o be less conservave an e prevous approaces. In e leraure e fuzzy sysems w me-varyng delay ave been modelled as a sysem w a sngle delay erm n e sae. Recenly n [8,, ], was noed a n neworked conrolled sysem, successve delays w dfferen properes are nroduced n e ransmsson of sgnals beween dfferen pons roug dfferen segmens of neworks. us s approprae o consder dfferen medelays τ ( ) and τ ( ) n e same sae were, τ ( ) s e me-delay nduced from sensor o conroller and τ ( ) s e delay nduced from conroller o e acuaor. In s work, movaed by e above dea, we derve a new and mproved delaydependen condon for asympoc sably of -S fuzzy sysem w wo addve delay componens. e condon s exended o cover sysems w norm bounded unceranes. e suffcen condons for asympoc sably and robus sably analyss are derved by usng Lyapunov-Krasovsk funconal meod and makng use of mproved ecnque and Fnsler s lemma. By solvng a se of LMIs, e upper bounds of e me delays can be obaned. We provde wo llusrave examples o sow a e new sably condons proposed n s paper are less conservave. n n n Lemma [7]: Consder a vecor χ R, a symmerc marx Q R and m n marx Β R, suc a rank ( Β ) < n. e followng saemens are equvalen:. χ Q χ <, χ suc a Β χ, χ. Β Q Β <. μ R : Q μβ Β< v. n m F R : Q + FΒ+Β F < Were Β denoes a bass for e null-space of Β n n Lemma [4]: for any consan marx M M R, M >, scalar n γ η >, vecor funcon ω:, [ γ] R suc a e negraons n e followng are well defned, en:

3 Delay dependen robus sably 3 η η η η ω ( β)m ω( β)d β ω( β)dβ M ω( β)dβ Lemma 3 []: Le Q Q, H, E, and F sasfyng F F I are appropraely dmensoned marces, e followng nequaly : Q + HFE + E F H < Is rue, f and only f e followng nequaly olds for any marx Y >, Q + HY H + E YE <.. Sysem descrpon Consder a S fuzzy me-varyng delay sysem, wc can be descrbed by a S fuzzy model, composed of a se of fuzzy mplcaons, and eac mplcaon s expressed by a lnear sysem model. e rule of e S fuzzy model s descrbed by followng IF HEN form: Plan Rule : IF z s W and and z g s W g HEN x& (A + ΔA )x + (A d + ΔA d )x( ) () x φ, [,],,,...,r were z, z,, z g are e premse varables, and W j, j,,..., g n are fuzzy ses, x R s e sae varable, r s e number of f-en rules, φ s a vecor-valued nal condon, and s e me-varyng delays sasfyng, & d,, & d, + and d d + d () e paramerc unceranes ΔA and Δ A d are me-varyng marces w approprae dmensons, wc can be descrbed as : Δ A ΔA D F E,,,..., r (3) [ ] [ ] d E d Were D, E, E d are known consan real marces w approprae dmensons and F are unknown real me-varyng marces w Lebesgue measurable elemens bounded by: F F I,,,..., r (4) By usng e cener-average deffuzzfer, produc nference and sngleon fuzzfer, e global dynamcs of -Z fuzzy sysem () can be expressed as

4 4 Idrss Sad and El Houssane ssr r x& µ (z)[(a + ΔA )x + (A d + ΔA d )x( )] (5) Were, r µ (z) ω (z) / ω (z), ω ( z) W j (z j) And Wj (z j ) s e membersp value of z j n µ (z) are µ (z), µ (z). r g j W j, some basc properes of 3. Man resuls In s secon, we wll oban e sably crera for -S fuzzy me-varyng delay sysems w wo addve me varyng delay based on a new Lyapunov- Krasovsk funconal approac. Frs e followng nomnal sysem of sysem (5) wll be consdered: x& A x dx( ) (6) x φ, [,] Were r (z) A A µ and A d µ (z) A d r eorem : e sysem descrbed by (6) and sasfyng condons () s asympocally sable f ere exs symmerc posve defne marces P, Q, Q, R, R and any appropraely dmensoned marces, F, F, F, suc a R -R > and e followng LMIs are feasble for,,..., r Were, Φ Φ Q F A d F P F F Φ Q Φ < (7) Φ 33 F df Φ 44 Q + R + F A FO

5 Delay dependen robus sably 5 Φ Q Q ( d)(r R Φ Φ 33 Q ( d d )R F A d A df 44 Q + Q F F + + ) Proof: Defne e followng Lyapunov Krasovsk funconal V(x ) V (x ) + V (x ) + V3 (x ) (8) Were, V (x ) x Px (9) V (x ) x& (s)q x(s)dsd & θ + x& (s)q x(s)dsd & θ () + θ + θ V (x ) x (s)r x(s)ds + x (s)r x(s)ds () 3 Compung e me dervave of (9)-() one oban, V & (x ) x Px () & [ x& Q x & x& ( +θ)q x( & +θ)d ] θ+ [ x& Q x & x& ( +θ)q x( & +θ)d ] θ V & (x ) x& Qx & x& (s)qx(s)ds & + x& Qx & x& (s)q x(s) & ds (3) For any symmerc posve defne marces Q and Q e followng nequales always old, see []. x & x & (s)q x(s)ds & (s)q x(s)ds & x& x& (s)q x(s)ds & (s)q x(s)ds & Were + Applyng e above nequales o e negral erms n (3) one oban, V & (x ) x& Q x & By usng lemma we oban: x& (s)q x(s)ds & + x& Q x & x& (s)q x(s)ds & (4)

6 6 Idrss Sad and El Houssane ssr V& (x ) x& Q x & + x& Q x & [ x x( ] Q [ x x( ] [ x( ) x( ] Q [ x( ) x( ] [ x Q x x Q x( + x ( Q x( ] x& Qx & + x& Qx & [ x ( )Qx( ) x ( )Qx( + x ( Qx( ] (5) V & 3(x ) x Rx ( & )x ( )Rx( ) + ( & )x ( )R x( ) ( & & )x ( )R x( ) x R x ( d)x ( )(R R )x( ) (6) ( d d )x ( )R x( ) Were we assume a R >R. Le ξ ( x ( ) x ( ( )) x ( ( )) x ( ) ) ( ) & akng accoun of (), (5) and (6), and leng Ψ Q + R Q Q Q ( d )(R R ) Q Q ( d d )R Q + Q P (7) We oban: Now le [ A A I] B ~ V(x & ) ξ Ψξ (8) d F F F and [ ] F. en, snce r µ ( z ) we can verfy a B ~ ξ, ξ. Snce condon (7) olds, follows a e marces FB ~ B ~ Ψ + + F < and erefore by lemma we ave ξ Ψξ < wc mples a V& (x ) <. s complees e proof.

7 Delay dependen robus sably 7 eorem : e unceran sysem (5) sasfyng condons () s robusly sable f ere exs symmerc posve defne marces P, Q, Q, R, R, Y and any appropraely dmensoned marces, F, F, F, suc a R -R > and e followng LMIs are feasble for,,r. Φ + E YE Q Φ F Ad F + E YEd Q Φ33 + E d YEd Were Φ, Φ, Φ 33 and Φ 44 are defned n (7). P F F F d F Φ44 F D < (9) F D F D Y Proof: Replacng A and A d by A + DF E and A d + DF E d n (7), respecvely, e correspondng formula of (7) for sysem (5) can be rewren as follows: Φ + HF E + E F H () < Were [ D F D F D F ] d. Accordng o Lemma 3, () s rue If ere exs Y >, suc a e followng nequaly olds: H and E [ E E ] Φ + HY H + E YE < () By Scur complemen, () s equvalen o (9). s complees e proof. Remark. o e bes of our knowledge, all e resuls sudyng -S fuzzy sysems w me delay consder sysems w sngle delay erm as: IF z s W and and z g s W g HEN x& (A + ΔA )x + (Ad + ΔAd )x( ) x φ, [,],,,...,r Were and & d, and ere s no resuls dealng w addve me varyng delay. Remark. For me delay sysems w sngle delay erm, free wegng marces approac as been used n [5, 6, 5, 7, 5] and less conservave resuls ave been esablsed compared w Moon s nequaly approac employed n [4, ]. In [9] sably crera for -s Fuzzy sysems w delay

8 8 Idrss Sad and El Houssane ssr ave been developed by employng neer free wegng marces nor model ransformaon and derved less conservave resuls an ose n e above references. In s paper, we use a new Lyapunov Krasovsk funconal and our meod s based on Fnsler s Lemma. Comparng eorem w corollary of [9], concernng e sably of sysem (6) w + and, e numbers of varables requred n eorem and corollary are 5n(n ) 3n + + and 6n + n respecvely. I can be seen a eorem n(n ) requres less number of varable a s. Consequenly, w our resuls e compuaonal demand on searcng for e soluon of sably condons can be allevaed. s advanage can be revealed especally for sysems w large dmenson n. A second advanage of our approac s a we expec a reduced conservasm. s s llusraed n e examples 4. Numercal examples In s secon, we am o demonsrae e effecveness of e proposed approac presened n s paper by eorem and eorem. Example : Consder a sysem w e followng rules: Rule : If z s W, en x& A x dx( ) If z s W, en x& A x dx( ) And e membersp funcons for rule and rule are μ (z), μ ( z) μ(z) + exp( z) Were,.5 A, A d.9, A, A d. Applyng eorem, we fx dfferen values of and searc for e correspondng upper bounds of. Hence we fx dfferen values of and searc for e correspondng upper bounds of. In order o compare w e leraure

9 Delay dependen robus sably 9 resuls, snce o our knowledge, ere s no resuls dealng w fuzzy sysems w addve delays, we le + suc a. en we apply e leraure condons. From our resuls we compue as n (). e resuls are summarzed n able. able : Upper bound for nvaran delays Meod Upper bound Wang e al.[].597 an and Peng.597 [5] Corollary Cen e al. [5].597 Fan e al. [7].597 s paper upper bound for gven upper bound for gven eorem of s paper,,5,,3,5,897,667,8,54,449,33 We can see a e maxmum allowable upper bound, + obaned by eorem s greaer an e bound obaned by e resuls of [, 5, 5, 7]. Our approac leads o less resrcve condon aloug compung + may be conservave, n fac e delays and may ave sarply dfferen properes and wen + reaces s maxmum, we do no necessarly ave bo and reac er maxmum a e same me. Example : Consder e followng unceran fuzzy sysem w wo addve me varyng delay: x& were, [(A + ΔA )x + (A + ΔA )x( ( ] μ (z) d d )).5.6 A, A d., A, A d E, E d.5, E.3, E d D.5

10 Idrss Sad and El Houssane ssr μ (z ) + exp( 3(z /.5 π / )) + exp( 3(z /.5 π / )) μ z ) (z ), ( μ Employng eorem of s paper, we calculae e upper bound of or of, wen e oer s known. e upper bound of s obaned by summng e wo delay bounds and. In order o compare w e leraure resuls, we consder e above sysem as an unceran fuzzy sysems w a sngle delay erm,.e., + sasfyng. en we apply e leraure condons. e resuls are presened n able. able : Upper bound for nvaran delay Meod Upper bound Cen e al. [5].43 Fan e al. [7].439 upper bound for gven upper bound for gven eorem of s paper e resuls guaranee e sably of unceran fuzzy sysem for nvaran delays. I s sown a e upper bound obaned by eorem s beer en ose obaned by sngle delay approac n [5, 7]. 5. Concluson We ave esablsed delay dependen condons for asympoc sably of -S fuzzy sysems w wo addve me varyng delays. e resuls are exended o cover e class of unceran -S fuzzy sysems. e perurbaons consdered are assumed o be norm-bounded. e LMIs proposed ave been obaned by ulzng a new Lyapunov Krasovsk funconal and Fnsler s lemma. e less conservaveness of e resuls s sown by wo numercal examples n wc we obaned a large delay upper bound as sown n able and able. References [] Y. Y. Cao and P. M. Frank. Sably analyss and syness of non lnear me delay sysems w akag-sugeno fuzzy models. Fuzzy ses and sysems 4 () 3-9.

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