Decentralized Control for Time-Delay Interconnected Systems Based on T-S Fuzzy Bilinear Model

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1 ensors & ransucers Vol. 75 Issue 7 July 24 pp ensors & ransucers 24 by IFA Publshng. L. hp:// Decenralze Conrol for me-delay Inerconnece ysems Base on - Fuzzy Blnear Moel * Guo ZHANG Yunwang GE Elecrcal Engneerng Deparmen Luoyang Insue of cence an echnology Luoyang 4723 P. R. Chna * E-mal: zhangguo6363@63.com Receve: 23 Aprl 24 /Accepe: 3 June 24 /Publshe: 3 July 24 Absrac: he paper presens he problem of ecenralze sae feeback conrol for nonlnear nerconnece sysems wh me-varyng elay n boh saes an npus whch s compose by a number of akag-ugeno (- fuzzy blnear subsysems wh nerconnecons. Base on he Lyapunov creron an he parallel srbue compensaon scheme he elay-epenen sablzaon suffcen conons are erve for he whole close-loop fuzzy nerconnece sysems. he corresponng ecenralze fuzzy conroller esgn s convere no a convex opmzaon problem wh lnear marx nequaly (LMI consrans. Fnally a smulaon example shows he effecveness of he propose approach. Copyrgh 24 IFA Publshng. L. eywors: Nonlnear nerconnece sysem Decenralze conrol Delay-epenen Lnear marx nequaly (LMI Fuzzy blnear moel.. Inroucon Large-scale nerconnece sysems can be foun n many real-lfe praccal applcaons such as elecrc power sysems nuclear reacors economc sysems process conrol sysems compuer neworks an urban raffc nework ec. []. On he oher han here are few sues concernng wh he sablzaon conrol for he nerconnece nonlnear sysems [2]. nce lnearzaon echnque an lnear robus conrol are use hese resuls are always conservave an only applcable o some specal nonlnear nerconnece sysems. Due o he physcal confguraon an hgh mensonaly of nerconnece sysems a cenralze conrol s neher economcally feasble nor even necessary [3]. herefore ecenralze scheme s preferre n conrol esgn of he large-scale nerconnece sysems [4]. In recen years - (akag- ugeno moelbase fuzzy conrol has arace we aenon essenally because he fuzzy moel s an effecve an flexble ool for conrol of nonlnear sysems [5-7]. In hs approach he - fuzzy moel subsues he consequen fuzzy ses n a fuzzy rule by a lnear moel. Local ynamcs n fferen saespace regons are represene by lnear moels an he overall moel of he sysem s represene as he fuzzy nerpolaon of hese lnear moels. Jus because of hs - fuzzy moel has been pa conserable aenon an s wely use o he conrol esgn of nonlnear nerconnece sysems. he problem of sablzaon of nonlnear nerconnece sysems was sue n [7] whle robus sablzaon of a class of mulple me-elay nonlnear nerconnece sysems was nvesgae. I s noe ha he above nonlnear nerconnece sysems are all base on - fuzzy lnear moel. 32 hp://

2 ensors & ransucers Vol. 75 Issue 7 July 24 pp I s known ha blnear moels can be escrbe many physcal sysems an ynamcal processes n engneerng fels [8]. here are wo man avanages of he blnear sysem. One s ha proves a beer approxmaon o a nonlnear sysem han a lnear one. Anoher s ha many real physcal processes may be appropraely moele as blnear sysems when he lnear moels are naequae. A goo example of a blnear sysem s he populaon of bologcal speces escrbe by θ / θv where v s he brh rae mnus eah rae an θ enoes he populaon. I s mpossble o approxmae he aforemenone equaon by a lnear moel [8]. Conserng he avanages of blnear sysems an - fuzzy conrol he blnear fuzzy sysem base on he - fuzzy moel wh blnear rule consequence was arace he neres of researchers [9-2]. he - FB may be suable for some classes of nonlnear plans. he robus sablzaon for connuous-me fuzzy blnear sysem (FB s sue n [9] hen he resul was exene o he FB wh me-elay only n he sae []. However he paper [] s only consere he elay n he sae an he ervaves of meelay ( < s requre. o far he ecenralze conrol of nonlnear nerconnece sysems base - blnear moel has no been scusse. In hs paper he purpose s o evelop a sae feeback conroller an oban some suffcen conons for nonlnear nerconnece sysems whch s concerne abou he me-varyng elay boh n sae an npu. For objecve - fuzzy blnear moel s employe for he nonlnear nerconnece sysems. hen base on he parallel srbue compensaon (PDC scheme he robus sablzaon conons can be esablshe an moreover he ecenralze conroller esgn proceure can cas as solvng a se of lnear marx nequales (LMIs. 2. ysem Descrpon an Assumpons Conser a me-elay nerconnece sysem Ω compose of subsysems Ω Each fuzzy rule of he subsysem Ω can be represene by a - fuzzy blnear moel as follows m m m R f ξ ( s F an... ξv ( s F v hen x ( Amx ( Bmu( Nmx( u( A x ( B u ( N m m m x( u( C ( j j jmxj x ( φ ( [- τ ] m 2... r ( where r s he number of he fuzzy rules. ξ j ( an F m j 2... v are some measurable premse j n varables an fuzzy ses. x ( R u ( R are he sae vecor an conrol npu respecvely. A A N N R B B R n n n m m m m m m enoe he sysem marces wh approprae n nj mensons. Cjm R represens he nerconnecon marx beween he h an he j h subsysems. s a me-varyng fferenable funcon ha sasfes τ where τ s a real posve consan as he upper boun of he mevaryng elay. I s also assume ha α < an a s a known consan. By usng sngleon fuzzfer prouc nferre an weghe efuzzfer he sysem can be expresse by he followng globe moel: r m ξ m m m x ( h ( ([ A x ( B u ( N x ( u ( A x ( N x ( m m m u ( B u ( C x ( ] m j j jm j where h ωm( ξ ( ( ξ ( ( ( m ω m ξ m r v m j mj (2 ω ( ξ ( μ ( ξ (. μ ( ξ ( s he grae of membershp of ξ ( n j mj m F j. In hs paper s assume ha ωm( ξ ( an r ω ( ( m m ξ > for all. hen we have he followng conons hm( ξ ( an r h ( ξ ( m m for all. In he consequen we use abbrevaon hm x ( u ( o replace h ( ξ ( x ( u ( respecvely for m convenence. Base on PDC he fuzzy conroller shares he same premse pars wh (; ha s he h fuzzy conroller s formulae as follow m m f ξ ( s F an... ξv ( s F v ρmx( hen u ( ( x mmx ρ snθ ρ cos θ x ( m m m From (3 he u ( can be wren as: ρmx( u ( x mmx (4 ρ snϕ ρ cos ϕ x ( m m m 33

3 ensors & ransucers Vol. 75 Issue 7 July 24 pp n where m R s he local conroller gan o be eermne an ρ > s he scalar o be assgne. mx ( sn θm cos θ m x mmx x mmx mx ( sn ϕm cos ϕ m x x x x m m m m 2... ; m 2... r. he overall fuzzy conrol law can be represene by r ( h sn mρ θ m m r h mρ m θmmx r ( sn m mρ ϕ m r m mρ ϕ m m u u h cos ( h cos x ( (5 By subsung (5 no (2 he h close-loop subsysem can be represene as r ( [ m n Λ mn ( mn Λ mn j j jm j x h h x (6 x ( C x (] where Λ A ρ snθ N ρ cos θ B mn m n m n m n Λ mn Am ρ snϕnnm ρ cosϕnbm n. he objecve of he paper s o esgn ecenralze fuzzy sae feeback conrollers (5 such ha he close-loop sysems (6 s ecenralze robus sably. 3. Man Resuls Before proceeng wh he followng heorems we nrouce he followng lemmas whch wll be use n our resuls. Lemma [8]: Gven any marces M an N wh approprae mensons such haε > we have M N N M εm M ε N N. he followng heorem gves he suffcen conons for he exsence of he fuzzy conroller for he nerconnece sysem (6. heorem For gven scalars ρ > a > ε > ε 2 > 2... he nerconnec sysem (6 s ecenralze elayepenen asympocally sable f here exs marces P > R > 2... an m 2... ; m 2... r such ha he followng nequaly (7 s sasfe. Φ mm < 2... ; m 2... r (7a Φ mn Φ nm < 2... ; m< n r (7b φmn PA m where Φ mn φ 2mn 2 φ A P PA ( ε ε ρ PP mn m m 2 ε NmNm ε ( Bm n ( Bmn PC j j jmc jmp I R 2mn 2 NmNm 2 Bmn Bmn ( φ ε ε ( ( ( α R. Proof: ake he Lyapunov funcon canae as s s ( ( [ ( ( x ( ( ] ( srx ss V V x Px (8 he me ervaves of V ( along he rajecory of he sysem (6 s gven by V hh x Px x Px r ( m n[ ( ( ( ( mn x ( Rx ( ( x( Rx ( ] r hh[ (( m n x mnp P mn x ( mn x( Λ mn Px ( x ( PΛ mnx( x ( ( ( j CjmPx x P j j s C ( ( ( j j jmxj x Rx ( α x( Rx ( ] Λ Λ Applyng Lemma we ge Λ P PΛ A P PA ( ρ snθ mn mn m m n Nm P P ( ρ sn θnnm ( ρcos θnbmn P P ( ρ cos θnbmn A P PA ε ρ PP ε N N 2 m m m m ε ( Bmn ( Bmn ( Λ mn ( ( Λmn ( x ( AmPx ( x ( PA mx ( 2 x ( ε2ρ PPx ( x( ε2 NmNmx ( x ( ε 2 ( Bmn x Px x P x ( Bmn x ( (9 ( Conserng he followng nequaly an applyng Lemma yel 34

4 ensors & ransucers Vol. 75 Issue 7 July 24 pp r h m[ x j ( CjmPx ( x ( m j j s P C ( ] jmxj j j r h [ m x ( C mpx (... x ( m C mpx ( x ( C mpx (... x ( CmPx ( x ( PC mx(... x ( PC mx ( x ( PC mx (... x ( PC mx( ] r h [ ( ( m m x P C j j jmcjmpx x j ( xj( ] j j r h [ ( ( m x P m C j j jmcjmpx ( x ( x( ] ubsung (9 - ( no (8 resuls n V hh x r ( m nη ( Φ mn ( η ( m n r 2 [ h ( mη mmη ( Φ m r h m n mhnη Φ mn Φ nm η < ( ( ( ] ( (2 where η ( x ( x(. herefore s noe ha (7 mples V ( < so he nerconnece sysem (6 s asympocally sable. hus we complee he proof. he marx nequaly (7 leas o blnear marx nequaly (BMI opmzaon a non-convex programmng problem. Non-convexy mples he exsence of local mnma an he BMI problems are NP-har. In he followng heorem we wll erve a suffcen conon such ha he marx nequaly (7 can be ransforme no an LMI problem. heorem 2 For gven scalars ρ > a > ε2 > ε3 > 2... he nerconnec sysem (6 s ecenralze asympocally sable f here exs marces Z > R > 2... an Gm 2... ; m 2... r such ha he marx nequaly (3 s sasfe. Moreover he feeback gans are gven by G Z 2... ; m 2... r. m m * * * * * * ZA ZA R < ( NZ mn 77 * ( BG mn ; m< n r m n m n 2( α * * * * * 2I 2Z * * * * s ( NZ mn - 55 * * * ( BG mn 66 * * where m m m j j jm jm 2 ( ε ε2 ρ I R Z A A Z C C (3b NmZ BmGn ( NZ mn ( BG mn NnZ BnG m NmZ BmGn ( NZ mn ( BG mn NnZ BnG m ag{ ε I ε I} ag{ ε I ε I} Proof: leng P Z R PRP an nong M m mz. hen pre-mulplyng an posmulplyng ag{ P P I I I I I } o (3a resuls n P P * * * * * * * * * < B I m AmP ( α R * * * * * I I s * * * * Nm ε I Bmm ε I * * Nm ε 2I * m m ε ; m 2... r (4a Applyng he chur complemen o (4 a resuls n he conon (7a. mlar he (3 b s equvalen o (7 b. Accorng o heorem he nerconnece sysem (6 s asympocally sable. hus he proof s complee. 4. mulaon Example In hs secon he propose approach s apple o he followng example o verfy s effecveness. We conser a fuzzy blnear nerconnece sysem whch s compose of wo subsysems as follows m * * * * * * ZA m ( α R * * * * * I Z * * * * s NmZ ε I * * * < BmGm ε I * * NmZ ε 2I * BmGm ε 2I 2... ; m 2... r (3a ubsysem : R : f x s F hen x ( A x ( A x ( N x ( u ( N x ( u ( B u ( B u ( C x (;

5 ensors & ransucers Vol. 75 Issue 7 July 24 pp R : f x s F 2 2 hen x ( A x ( A x ( 2 2 N x ( u ( N x ( u ( 2 2 B u ( B u ( C x (; ubsysem 2: R : f x s F hen x ( A x ( A x ( N x u( N x ( u ( B u ( B u ( C x (; R : f x s F hen x ( A x ( A x ( N x ( u ( N x ( u ( B u ( B u ( C x (; where A A -3-9 A A22 ; N N -3 - N2 N22 ; B B B2 B22 ; A A A2 2 2 ; 24 A 4 7 N N 2 N2 N22 ; B B 2 B2 B22 ; C2 C22 - C2 C22. he membershp funcons are chosen as μ ( x μ ( x μ ( x 2 F F F exp( 2 x An μ ( x μ ( x μ ( x 2 F 2 2 F F2 exp( 2 x2 By leng ρ.75 ρ2.63 ε ε2.3 ε2 ε22.75 a a2 solvng LMIs (3 gves he followng feasble soluon: ; ; ; Inal conon s assume o be x x he smulaon resuls are shown n Fg. Fg. 2 an Fg. 3. Fg. an Fg. 2 show he sae responses of wo subsysems an conrol rajecory s shown n Fg. 3. I can be seen ha wh he ecenralze fuzzy conrol law he close-loop sysem s asympocally sable. he smulaon resuls show ha he fuzzy conroller propose n hs paper s effecve me(sec Fg.. ae responses of subsysem me(sec Fg. 2. ae responses of subsysem 2. x x 2 x 2 x 22 36

6 ensors & ransucers Vol. 75 Issue 7 July 24 pp me(sec 5. Concluson Fg. 3. Conrol rajecory. In hs paper a - fuzzy blnear moel s propose o suy he conrol problems for me-elay nonlnear nerconnece sysems usng fuzzy ecenralze conrol. Base on he Lyapunov creron he suffcen conons for elayepenen sablzaon of he nerconnece sysem are presene. he ecenralze conrollers esgnng problems can be formulae as a convex opmzaon problem wh LMI consrans. A smulaon example s nclue o show he effecveness of he propose approach. Acknowlegemens hs work s suppore by he Naonal Hgh echnology Research an Developmen Program of Chna (863 Program No. 22AA32. u u 2 References []. W.. Chan an C. A. Desoer Egenvalue assgnmen an sablzaon of nerconnece sysems usng local feeback IEEE ransacons on Auomac Conrol Vol. 25 Issue 7 98 pp [2]. E. J. Davson he ecenralze sablzaon an conrol of unknown nonlnear me varyng sysems Auomaca Issue 974 pp [3]. M. Jamsh Large-scale sysems: Moelng an conrol Elsever New York 983. [4]. Z. Hu Decenralze sablzaon of large scale nerconnece-sysems wh elays IEEE ransacons on Auomac Conrol Vol. Issue pp [5].. Yuan H. X. L an J. Cao Robus sablzaon of he srbue parameer sysem wh me elay va fuzzy conrol IEEE ransacons on Fuzzy ysems Vol. 3 Issue 6 28 pp [6]. C.. Pang an Y. Y. Lur On he sably of akag- ugeno fuzzy sysems wh me-varyng unceranes IEEE ransacons on Fuzzy ysems Vol. Issue 6 28 pp [7]. R. J. Wang Nonlnear ecenralze sae feeback conroller for unceran fuzzy me-elay nerconnece sysems Fuzzy es an ysems Vol. Issue 5 25 pp [8]. R. R. Mohler Blnear conrol processes Acaemc New York 973. [9].. H.. L an. H. sa - fuzzy blnear moel an fuzzy conroller esgn for a class of nonlnear sysems IEEE ransacons on Fuzzy ysems Vol. 3 Issue 5 27 pp []. J. M. L an G. Zhang Non-fragle guaranee cos conrol of - fuzzy me-varyngelay sysems wh local blnear moels Iranan Journal of Fuzzy ysems Vol. 2 Issue 9 22 pp [].. H.. L. H. sa Robus H fuzzy conrol for a class of unceran scree fuzzy blnear sysems IEEE ransacons on ysem Man an Cybernecs Vol. 38 Issue 2 28 pp [2] R. J. Wang W. W. Ln an W. J. Wang ablzably of lnear quarac sae feeback for unceran fuzzy me-elay sysems IEEE ransacons on ysem Man an Cybernecs Vol. 34 Issue 2 24 pp Copyrgh Inernaonal Frequency ensor Assocaon (IFA Publshng. L. All rghs reserve. (hp:// 37