Research Article Variance and Passivity Constrained Fuzzy Control for Nonlinear Ship Steering Systems with State Multiplicative Noises

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1 Matheatical Pobles in Engineeing Volue 13 Aticle ID pages Reseach Aticle Vaiance and Passivity Constained Fuzzy Contol fo Nonlinea Ship Steeing Systes with State Multiplicative Noises Wen-Je Chang and Bo-Jyun Huang Depatent of Maine Engineeing National Taiwan Ocean Univesity Keelung Taiwan Coespondence should be addessed to Wen-Je Chang; wjchang@ail.ntou.edu.tw Received 7 Januay 13; Accepted 18 Febuay 13 Acadeic Edito: Chang-Hua Lien Copyight 13 W.-J. Chang and B.-J. Huang. This is an open access aticle distibuted unde the Ceative Coons Attibution License which peits unesticted use distibution and epoduction in any ediu povided the oiginal wok is popely cited. The vaiance and passivity constained fuzzy contol poble fo the nonlinea ship steeing systes with state ultiplicative noises is investigated. The continuous-tie Takagi-Sugeno fuzzy odel is used to epesent the nonlinea ship steeing systes with state ultiplicative noises. In ode to siultaneously achieve vaiance passivity and stability pefoances soe sufficient conditions ae deived based on the Lyapunov theoy. Eploying the atix tansfoation technique these sufficient conditions can be expessed intes of linea atix inequalities. By solving the coesponding linea atix inequality conditions a paallel distibutedcopensationbasedfuzzycontollecanbeobtainedtoguaanteethestabilityoftheclosed-loopnonlineashipsteeing systes subject to vaiance and passivity pefoance constaints. Finally a nueical siulation exaple is povided to illustate the usefulness and applicability of the poposed ultiple pefoance constained fuzzy contol ethod. 1. Intoduction In the liteatue the poble of contolling suface ships in aneuveing situations has been eceiving oe and oe attention fo the opeational safety and envionental viewpoints [1 8]. Fo ship steeing the stochastic distubancesaewaveswindgustsandobsevationeosand the deteinistic distubances ae cuent ean wind and cuvatue of the pespecified tack. In ode to deal with the coplex distubances the adaptive contol appoach was developed to solve the ship steeing contol poble [1 4]. In [5] an output feedback contol poble was studied by using the vectoial obseve backstepping technique. Applying the sliding ode contol a ultivaiable pefoanceconstained contolle was developed in [6] fothedynaic ship positioning systes. In [1 6]the dynaic ship steeing systes wee appoxiated by the lineaized odels that ay have sacificed soe nonlinea chaacteistics. In ode to eseve the chaacteistics of oiginal nonlinea systes the authos of [7 8] used the Takagi-Sugeno (T-S) fuzzy odel to epesent the nonlinea dynaic ship steeing systes. Without loss of geneality the T-S fuzzy odel [7 8] can appoxiate a wide class of nonlinea systes by ixing seveal linea subsystes with IF-THEN ules. In contol engineeing it is always equied to develop soe ethodologies fo designing contolles to achieve ultiple pefoance equieents. In addition to the stability pefoance constaint the individual state vaiance constaint and passivity constaint ae usually consideed fo the contol poble of linea and nonlinea systes. The individual state vaiance pefoance equieents of engineeing systes ae usually expessed as uppe bounds on the steady-state vaiances fo the stochastic contol systes. Cuent contol design techniques such as Linea Quadatic Gaussian (LQG) and H design ethods do not see to give a diect solution to this kind of design poble since they lack a convenient avenue fo iposing design objectives stated in tes of uppe bounds on the vaiance values. The covaiance contol theoy developed in [9] povided a oe diect ethodology fo achieving the individual vaiance constaints than the LQG contol theoy. The idea of covaiance contol theoy has been widely applied in solving ultiobjective contol pobles as well as filteing pobles see [6 1 13] fo instance. In addition to the individual

2 Matheatical Pobles in Engineeing x 3 (t) x 6 (t) 1 x 1 (t) x 5 (t) x (t) Figue 1: The coodinate of ship steeing syste. x 1 (t) Rule 1 1 Rule 4 Rule Rule x 3 (t) Figue : The ebeship function of state x 3 (t) Tie (s) Figue 3: Responses of state x 1 (t). 1 vaiance constaint the passivity constaint is also consideed in this pape. In the liteatue the passivity concept [14 1] is pesented to issue stability analysis and synthesis of linea o nonlinea systes. It is well known that the passive theoy uses the behavios of enegy dissipative between input andoutputofsystes.theostipotantdefinitionsof the passive theoy ae stoage function [16 ] andpowe supply [16 1] fo decision the popety of passivity. As fo a stoage function in passive theoy it easues the aount of enegy stoed in the intenal of the syste; hence it is natually descibed by using intenal vaiables of syste state vaiables. To the authos knowledge so fa the vaiance and passivity constained stabilization poble of nonlinea stochastic ship steeing systes with ultiplicative noises has not been investigated. It is also an open poble to discuss the vaiance and passivity constained fuzzy contolle design fo the stochastic T-S fuzzy odels with ultiplicative noises. Fo this eason this pape ties to develop an applicable fuzzy contol ethodology fo achieving the individual vaiance constaints and passivity constaints siultaneously. The contol poble fo the systes with ultiplicative noise has ecently eceived a geat deal of attention. Such odels ae found in any physical systes such as aeospace engineeing systes [] signal pocessing systes [3] and biological oveent systes [4 5]. Diffeent fo the taditional additive noise ultiplicative noise is oe pactical since it allows the statistical desciption of the noise to be unknown apiobut depend on the contol and state solution. So fa any eseaches have been woking on vaious kinds of investigation of filteing and contol fo the contol systes with ultiplicative noise. Thus thee have been seveal appoaches fo dealing with this poble including the linea atix inequality (LMI) appoach [3] the Riccati diffeence equation appoach [5] the gae theoetic appoach [6] and the fuzzy contol appoach [19 7 9]. Since this odel eflects oe ealistic popeties in x (t) Tie (s) Figue 4: Responses of state x (t). engineeing a coplete theoy which includes contol and estiation is wothy to be developed. The contibution of this pape is to develop a ethodology to design a fuzzy contolle such that the stability constaint individual vaiance constaint and passivity constaint ae siultaneously achieved fo the nonlinea stochastic ship steeing systes with ultiplicative noise. The T-S fuzzy odel is used to epesent the nonlinea stochastic ship steeing systes with ultiplicative noise. AccodingtotheT-Sfuzzyodelwithultiplicativenoise thee ae few appoaches investigated to siultaneously achieve ultiple pefoance equieents such as stability constaint individual vaiance constaint and passivity constaint. Theefoe the ultiobjective fuzzy contol ethodology pesented in this pape is wothy of the attention of contol enginees. Accoding to the T-S fuzzy odel [3 31] a paallel distibuted copensation (PDC) [31] based fuzzy contolle is designed via the Lyapunov theoy in this pape. Subject to the individual vaiance constaint and passivity constaint new sufficient conditions ae deived by assigning

3 Matheatical Pobles in Engineeing 3.15 Rule i: x 3 (t) if z 1 (t) is M i1 and z q (t) is M iq then x (t) =[A i + N ei V ei (t)]x(t) + B i u (t) + D i w (t) y (t) = C i x (t) + H i w (t) (1a) (1b) x 4 (t) Tie (s) Figue 5: Responses of state x 3 (t) Tie (s) Figue 6: Responses of state x 4 (t). whee z 1 (t)...z q (t) ae the peise vaiables that ay be functions of the state vaiables M iq is the fuzzy set q is the peisevaiablenube is the nube of fuzzy ules x(t) R n x is the state vecto u(t) R n u is the input vecto y(t) R n y istheoutputvectoandthepocesses w(t) R n w and V ei (t) R ae utually independent zeo-ean Gaussian white noise with intensities W (W > ) and 1 espectively. Refeing to [33] it is assued that E{w(t)} = E{V ei (t)} = E{x(t)w(t)} = E{x(t)}E{w(t)} = E{x(t)V ei (t)} = E{x(t)}E{V ei (t)} = and E{w(t)V ei (t)} = E{w(t)}E{V ei (t)} =.TheaticesA i R n x n x B i R n x n u D i R n x n w C i R n y n x H i R n y n w andn ei R n x n x ae constant. Without loss of geneality it is assued that the peise vaiables of the pevious T-S fuzzy odel ae easuable. Given the pai (x(t) u(t)) the oveall fuzzy odel can be descibed as follows: x (t) = y (t) = h i (z (t)) {[A i + N ei V ei (t)]x(t) + B i u (t) + D i w (t)} h i (z (t)) {C i x (t) + H i w (t)} (a) (b) a coon uppe bound atix fo the state covaiance atices of all fuzzy ules. By using the LMI technique [3] to solve the pevious sufficient conditions a PDC-based fuzzycontollecanbeobtainedtoguaanteetheindividual vaiance constaint and passivity constaint siultaneously. Finally a nueical exaple fo contolling a nonlinea stochastic ship steeing syste with ultiplicative noise is given to deonstate the applicability and effectiveness of the poposed fuzzy contolle design appoach.. Syste Desciption and Poble Stateent Conside a nonlinea stochastic syste which is constucted by a continuous-tie T-S fuzzy odel with ultiplicative noise as follows. whee h i (z(t)) = ω i (z(t))/ ω i(z(t)) ω i (z(t)) = q j=1 M ij (z j (t)) h i (z(t)) and h i(z(t)) = 1. ApplyingtheconceptofPDCthefuzzycontolleis designed to shae the sae IF pat of the T-S fuzzy odel (1a) and (1b).Thepoposedfuzzycontollecan be epesented as follows. Rule i: o if z 1 (t) is M i1 and and z q (t) is M iq then u (t) = F i x (t) u (t) = (3) h i (z (t)) { F i x (t)}. (4)

4 4 Matheatical Pobles in Engineeing Substituting (4) into(a) the closed-loop T-S fuzzy odel canbeobtainedasfollows: x (t) = h i (z (t)) h j (z (t)) j=1 [(A i B i F j )x(t) + D i w (t) + N ei V ei (t) x (t)]. Consideing each subsyste of the T-S fuzzy odel (5) the steady-state covaiance atix of the state vecto x(t) has the following fo: (5) X i = X T i > (6) whee X i = li t E[x(t)x T (t)] and X i is the unique solution of the following Lyapunov equation fo each ule [34]: (A i B i F i ) X i + X i (A i B i F i ) T (7) + D i WD T i + N ei X i N T ei =... The state vaiance constaint consideed in this pape is defined as follows: [X i ] kk σ k (8) whee [ ] kk denotes the kth diagonal eleent of atix [ ] and σ k k = 1...n x denote the oot-ean-squaed constaints fo the vaiance of syste states. In addition to the pevious individual state vaiance constaints (8) the passivity constaint intoduced in the following definition is also consideed in this pape. Definition 1 (see [18]). The syste (5) with extenal distubance w(t) and output y(t) is called stictly inpuassive if thee exists a positive scala γ and syetic positive definite atix S = S T >such that E{ y T (x) Sw (t) dt} > E { γw T (t) w (t) dt} (9) fo all and w(t) =. ThepuposeofthispapeistofindfeedbackgainsF i that canbeusedtoconstuctthepdc-basedfuzzycontolle(3) o (4) such that the individual state vaiance constaint (8) and passivity constaint (9) ae all satisfied siultaneously. 3. Fuzzy Contolle Design fo T-S Fuzzy Models with Multiplicative Noises The sufficient conditions fo guaanteeing the stability individual vaiance constaint and passivity constaint of closedloop T-S fuzzy odel with ultiplicative noise ae deived in this section. By assigning a coon uppe bound atix of the state covaiance atices fo all fuzzy ules the sufficient conditionsaedeivedbasedonthelyapunovtheoyand passivity theoy. Accoding to the closed-loop T-S fuzzy odel (5) the sufficient conditions fo achieving the stability individual vaiance constaint and passivity constaint ae deived in the following theoe. Theoe. If thee exisositive definite atices X >and S >andfeedback gains F i and dissipative ate γ satisfying the following sufficient conditions then [X i ] kk σ k and the closedloop syste (5) is stictly inpuassive and asyptotically stable: [ (A i B i F i ) X + X(A i B i F i ) T + D i WD T i + N ei XN T ei XC T i S ] < (1) [ S T C i X γi H T i S ST H i ] ( G ) X + X( G T ) < fo i<j (11) X diag (σ 1...σ n x )< (1) whee G ij = A i B i F j and diag(σ 1...σ n x ) denotes a diagonal atix with the diagonal eleents σ 1 σ...and σ n x. Poof. To analyze the stability of the closed-loop T-S fuzzy syste (5) a Lyapunov function is chosen as V(x(t)) = x T (t)px(t) whee P = P T >. The expected value of diffeential of the Lyapunov function V(x(t)) along the tajectoies of (5) is given as follows: E{ V (x (t))} =E{ T (t) Px (t) +x T (t) Px (t)} =E{ h i (z (t)) {[G ii x (t) + D i w (t) + T N ei V ei (t) x (t)]

5 Matheatical Pobles in Engineeing 5 =E{ Px (t) +x T (t) P [G ii x (t) + D i w (t) + N ei V ei (t) x (t)]} + i<j h i (z (t)) h j (z (t)) x T (t) {( G T ) P + P ( G )} x (t)} h i (z (t)) {x T (t) [G T ii P + PG ii + (V T ei (t) NT ei P + PN eiv ei (t))]x (t) +w T (t) D T i Px (t)+xt (t) D i w (t) } + i<j h i (z (t)) h j (z (t)) x T (t) {( G T ) P + P ( G )}x (t)}. (13) Due to E{x(t)w(t)} = E{x(t)}E{w(t)} = and E{x(t)V ei (t)} = E{x(t)}E{V ei (t)} = onehas E{ V (x (t))} =E{ + h i (z (t)) {x T (t) [G T ii P + PG ii]x(t)} i<j h i (z (t)) h j (z (t)) x T (t) {( G T ) P + P ( G )} x (t)}. (14) It is obvious that if condition (1) is satisfied then one has (A i B i F i ) X + X(A i B i F i ) T + D i WD T i + N ei XN T ei < (15) (A i B i F i ) X + X(A i B i F i ) T. (16) x 5 (t) x 6 (t) Tie (s) Figue 7: Responses of state x 5 (t) Tie (s) Figue 8: Responses of state x 6 (t). Let P = X 1 and ultiplying each ite of the pevious inequality fo left-hand side and ight-hand side by Pthen one can obtain the following inequality: G T ii P + PG ii <. (17) Besides ultiplying both sides of (11)by Pone has ( G T ) P + P ( G ) <. (18) It is obvious that if the conditions (17) and(18) ae satisfied then E{ V(x(t))} < can be obtained fo (14). Thus one can conclude that if conditions (1)and(11) ae satisfied then the closed-loop syste (5)isasyptoticallystable.

6 6 Matheatical Pobles in Engineeing Subtacting (7) fo(15) one has (A i B i F i )(X X i )+(X X i )(A i B i F i ) T (19) + N ei (X X i ) N T ei <. Due to the fact that conditions (1) and(11) ae satisfied one can obtain that the closed-loop syste (5) is stable and the closed-loop syste atix (A i B i F i ) is stable. In this case it can be concluded that X X i via inequality (19). Fo the condition (1)andX X i onehas [X i ] kk [X] kk σ k. () Fo achieving the attenuating pefoance the passivity theoy povides a useful and effective tool to design the contolle to achieve the enegy constaints fo the closedloop systes. Consideing the passivity constaint defined in Definition 1 one can define a pefoance function such as E{ (γw T (t) w (t) y T (x) Sw (t))dt} whee =E{ (γw T (t) w (t) y T (x) Sw (t) + V (x (t)))dt V (x (t)) } E{ (γw T (t) w (t) E{K (x w t)} K (x w t) y T (x) Sw (t) + V (x (t)))dt} (1) (γw T (t) w (t) y T (x) Sw (t) + V (x (t)))dt. () Using inequality (14) and Schu copleent [3] one has E {K (x w t)} =E{ T x (t) h i (z (t)) [ w (t) ] [ GT ii P + PG ii C T i S x (t) S T C i γi H T i S ][ ST H i w (t) ] + i<j h i (z (t)) h j (z (t)) x T (t) {( G T ) P + P ( G )} x (t)}. (3) Due to the fact D i WD T i and N eixn T ei >it is obvious that condition (1)iplies [ (A i B i F i ) X + X(A i B i F i ) T XC T i S S T C i X γi H T i S ST H i ]<. (4) Multiplying inequality (4) on the left-hand and ighthand sides by diag{p I} thenonecanobtainthefollowing inequality via P = X 1 : [ GT ii P + PG ii C T i S S T C i γi H T i S ST H i ]<. (5) If conditions (18) and (5) hold then one can obtain E{K(x w t)} <.Fo() the inequality E{K(x w t)} < iplies E { y T (x) Sw (t) dt} >E{ γw T (t) w (t) dt} (6) fo all nonzeo extenal distubance. Fo Definition 1 it can be thusconcludedthatifconditions(1) and(11) ae satisfied then the closed-loop syste (5)is stictly inputpassive. Theoe povides the sufficient conditions fo solving the state feedback gains such that the closed-loop syste (5) is asyptotically stable subject to individual state vaiance constaints and passivity constaint. The conditions deived in Theoe cannot be solved by the LMI technique [3] becausetheyaeofthebilineaatixinequalityfos. In ode to apply the LMI technique to solve the pesent poble the conditions of Theoe can be ewitten in the following coollay by using the atix tansfo technique.

7 Matheatical Pobles in Engineeing 7 Coollay 3. If thee exisositive definite atices X > and S > and dissipative ate γ and atices K i satisfying condition (1) andthefollowingsufficientconditionsthen [X i ] kk σ k and the closed-loop syste (5) is stictly input passiveandasyptoticallystable: A [ i X B i K i + XA i K T i BT i + D i WD T i + N ei XN T ei XC T i S ] < (7) [ S T C i X γi H T i S ST H i ] ( A ix B i K j + A j X B j K i ) + ( A T ix B i K j + A j X B j K i ) < fo i<j (8) whee K i = F i X. Poof. Conside conditions (1) and(11) of Theoe. By setting new vaiables K i = F i X conditions (1) and(11) can be ewitten as (7) and(8) espectively. It is obvious that conditions (1) and(7)-(8) ae of LMI fos fo the desied paaetes X and K i. Based onthe conditionsofcoollay3 the fuzzy contol gains can be solved via LMI technique [3] byusingmat- LAB LMI-Toolbox. In ode to deonstate the applicability and effectiveness of the poposed fuzzy contolle design appoach a nueical exaple fo the contolle design poble of a nonlinea ship steeing syste is consideed in the next section. 4. Constained Fuzzy Contolle Design fo Nonlinea Ship Steeing Systes Conside a nonlinea ship steeing syste with state ultiplicative noises. It is custoay to wite the dynaic equations using a coodinate fae fixed to the ship such as Figue 1. Its dynaic equation can be witten as follows [7]: 1 (t) = cos (x 3 (t))x 4 (t) sin (x 3 (t))x 5 (t) (t) = sin (x 3 (t)) x 4 (t) + cos (x 3 (t)) x 5 (t) 3 (t) =x 6 (t) +.3w (t) 4 (t) =.358x 1 (t).797x 4 (t) +.915u 1 (t) +.1V 1 (t) x 4 (t) +.5V (t) x 4 (t) (9a) (9b) (9c) (9d) +.8V 1 (t) x 5 (t) +.5V 1 (t) x 6 (t) +.6V (t) x 5 (t) +.1V (t) x 6 (t) (9f) whee x 1 (t) and x (t) ae the eath-fixed positions (x y) x 3 (t) is yaw angle of the ships x 4 (t) is suge x 5 (t) is sway x 6 (t) is yaw odes and u(t) = [u 1 (t) u (t) u 3 (t)] T is the contol and oent by the thuste syste. Besides the pocesses V 1 (t) V (t) andw(t) ae all utually independent zeo-ean Gaussian white noise with intensity 1. In this pape the pevious nonlinea ship steeing syste is odeled by a T-S fuzzy odel [31] by assuing that theyawanglechangesbetweenx 3 (t) ( π/ +π/). The ebeship function of state x 3 (t) is given in Figue. The T-S fuzzy odel fo nonlinea ship steeing systes contains the following ules. Rule 1: Rule : if x 3 (t) is about π ( π <x 3 <) then x (t) =[A 1 + N e1 V e1 (t)] x(t) + B 1 u (t) + D 1 w (t) y (t) = C 1 x (t) + H 1 w (t) (3a) 5 (t) =.8x (t).818x 5 (t).14x 6 (t) +.78u (t) u 3 (t) +.1V 1 (t) x 5 (t) +.7V 1 (t) x 6 (t) +.1V (t) x 5 (t) +.5V (t) x 6 (t) (9e) if x 3 (t) is about ( π <x 3 <) then x (t) =[A + N e V e (t)] (3b) 6 (t) =.394x (t).54x 5 (t).468x 6 (t) x(t) + B u (t) + D w (t) u (t) u 3 (t) y (t) = C x (t) + H w (t)

8 8 Matheatical Pobles in Engineeing Rule 3: Rule 4: if x 3 (t) is about ( < x 3 < π ) then x (t) =[A 3 + N e3 V e3 (t)] x(t) + B 3 u (t) + D 3 w (t) y (t) = C 3 x (t) + H 3 w (t). if x 3 (t) is about + π ( < x 3 < π ) then x (t) =[A 4 + N e4 V e4 (t)] whee x(t) + B 4 u (t) + D 4 w (t) y (t) = C 4 x (t) + H 4 w (t) (3c) (3d) β 1 1 β A 1 = [ ] [ ] 1 α α 1 A = [ ] [ ] 1 α α 1 A 3 = [ ] [ ] β 1 1 β A 4 = [ ] [ ].915 B 1 = B = B 3 = B 4 = [ ] [ ] T D 1 = D = D 3 = D 4 =.3 [ ] [ ] C 1 = C = C 3 = C 4 =[ 1 ] H 1 = H = H 3 = H 4 =1. N 11 = N 1 = N 13 = N 14 =.1 [.1.7] [.8.5] N 1 = N = N 3 = N 4 =.5 [.1.5] [.6.1 ] (3e) and α=sin( ) and β=cos(88 ). In geneal the PDC appoach is a popula T-S fuzzy contolle design ethod. In the PDC concept one needs to fist design a linea contolle fo each ule afte which the contolle of the entie nonlinea syste can be ade by blending the linea contolles of all ules. The fuzzy contolle poduced using PDC is epesented as follows. Rule 1: Rule : Rule 3: Rule 4: if x 3 (t) is about π then u (t) = F 1 x (t) ( π <x 3 <) if x 3 (t) is about ( π <x 3 <) then u (t) = F x (t) if x 3 (t) is about ( < x 3 < π ) then u (t) = F 3 x (t) if x 3 (t) is about π ( < x 3 < π ) then u (t) = F 4 x (t) (31a) (31b) (31c) (31d)

9 Matheatical Pobles in Engineeing 9 whee F i i= ae the desied feedback gains fo the fuzzy contolle design. Fo stating the fuzzy contolle design we select the supply ate γ = 1 S = I with copatible diension and assign the individual state vaiance constaints as σ 1 = 3 σ =3 σ 3 =. σ 4 = σ 5 =1andσ 6 =.8.Solvingthe sufficient conditions of Coollay 3 via LMI technique [3] the atix X and F i canbeobtainedasfollows: X = [ [ F 1 = [ ] F = [ ] [ ] [ ] F 3 = [ ] F 4 = [ ]. [ ] [ ] ] ] (3) The siulation esponses of states ae shown in Figues with initial condition x() = [1 1 ] T. Besides the extenal distubances V 1 (t) V (t) andw(t) ae chosen as zeo-ean white noises with vaiance one. Fo the siulation esults the effect of the extenal distubance w(t) on the poposed syste can be citicized as follows: E{ yt (x) Sw (t) dt} E{ γwt (t) w (t) dt} =.4. (33) The atio value of (33) is bigge than deteined dissipation ate γ=1; one can find that condition (9) ofdefinition 1 is satisfied. Besides the vaiances of states x 1 (t) x (t) x 3 (t) x 4 (t) x 5 (t) andx 6 (t) ae and espectively.itisobvious that the individual state vaiance constaints ae all satisfied. Theefoe the nonlinea ship steeing syste (9a) (9b) (9c) (9d) (9e) and (9f) contolled by the poposed fuzzy contolle (31a) (31b) (31c) and (31d)is asyptotically stable subject to individual state vaiance constaints and stictly inpuassivity constaint. 5. Conclusions The pefoance-constained fuzzy contolle design poble fo the nonlinea ship steeing syste has been studied in this pape. The nonlinea ship steeing syste was odeled byastochastict-sfuzzyodelwithstateultiplicative noises. This pape also consideed the individual state vaiance constaint and passivity constaint. Based on the PDC concept the poposed fuzzy contolle design appoach was caied out by solving the LMI stability conditions. On the othe hand these LMI stability conditions can be solved by the convex optial pogaing algoith. In the nueical exaple a nonlinea ship steeing syste has been intoduced to illustate the usefulness and effectiveness of the poposed fuzzy contol ethodology. Acknowledgents The authos would like to expess thei sincee gatitude to anonyous eviewes who gave the soe constuctive coents citiciss and suggestions. This wok was suppoted by the National Science Council of the Republic of China unde Contact NSC11-1-E Refeences [1] P. T. K. Fung and M. J. Gible Dynaic ship positioning using a self-tuning kalan filte IEEE Tansactions on Autoatic Contolvol.AC-8no.3pp [] J. V. Aneonge Adaptive steeing of ships-a odel efeence appoach Autoaticavol.no.1pp [3]R.E.ReidA.K.TugcuandB.C.Meas Theuseofwave filte design in kalan filte state estiation of the autoatic steeing poble of a tanke in a seaway IEEE Tansactions on Autoatic Contolvol.AC-9no.7pp [4] J.M.GodhavnT.I.FossenandS.P.Bege Non-lineaand adaptive backstepping designs fo tacking contol of ships Intenational Jounal of Adaptive Contol and Signal Pocessing vol. 1 no. 8 pp [5] T. I. Fossen and A. Govlen Nonlinea output feedback contol of dynaically positioned ships using vectoial obseve backstepping IEEE Tansactions on Contol Systes Technologyvol. 6 no. 1 pp [6]W.J.ChangandK.Y.Chang Multivaiablepefoanceconstained sliding ode contol fo ship Yaw-otion systes with petubations Intenational Jounal of Adaptive Contol and Signal Pocessingvol.14no.4pp [7] H.Y.ChungS.M.WuandW.J.Chang Regionalfuzzycontol fo nonlinea ship steeing systes Intenational Jounal of

10 1 Matheatical Pobles in Engineeing Innovative Coputing Infoation and Contolvol.4no.7pp [8] W.J.ChangH.J.LiangandC.C.Ku Fuzzycontolledesign subject to actuato satuation fo dynaic ship positioning systes with ultiplicative noises Poceedings of the Institution of Mechanical Enginees. Pat I: Jounal of Systes and Contol Engineeingvol.4no.6pp [9] A. Hotz and R. E. Skelton Covaiance contol theoy Intenational Jounal of Contolvol.46no.1pp [1] W. J. Chang and H. Y. Chung A study of H no and vaiance-constained design using dynaic output feedback fo linea discete systes Intenational Jounal of Contol vol. 57 no. pp [11] W. J. Chang and J. N. Lin Robust dynaic contolle design with ulti-constaints fo linea petubed discete systes Asia Jounal of Contolvol.5no.4pp [1] Y. Hung and F. Yang Robust H filteing with eo vaiance constaints fo uncetain discete tie-vaying systes with uncetainty Autoaticavol.39no.7pp [13] S. Baoand and H. Khaloozadeh On the closed-fo odel fo state covaiance assignenoble IET Contol Theoy & Applicationsvol.4no.9pp [14] C.I.BynesA.IsidoiandJ.C.Willes Passivityfeedback equivalence and the global stabilization of iniu phase nonlinea systes IEEE Tansactions on Autoatic Contol vol. 36 no. 11 pp [15] L. Xie M. Fu and H. Li Passivity analysis and passification fo uncetain signal pocessing systes IEEE Tansactions on Signal Pocessing vol. 46 no. 9 pp [16] R.LozanoB.BogliatoO.EgelandandB.MaschkeDissipative Systes Analysis and Contol Theoy and ApplicationSpinge LondonUSA. [17] W.-J.ChangC.C.KuandW.Chang Analysisandsynthesisof discete nonlinea passive systes via affine T-S fuzzy odels Intenational Jounal of Systes Sciencevol.39no.8pp [18] W.-J.ChangC.-C.KuandP.-H.Huang Robustfuzzycontol fo uncetain stochastic tie-delay Takagi-Sugeno fuzzy odels fo achieving passivity Fuzzy Sets and Systesvol.161no. 15 pp [19] W. J. Chang S. S. Jheng and C. C. Ku Fuzzy contol with obust and passive popeties fo discete-tie Takagi-Sugeno fuzzy systes with ultiplicative noises Poceedings of the Institution of Mechanical Enginees. Pat I: Jounal of Systes and Contol Engineeingvol.6no.4pp [] O. Kaneko and T. Fujii On discete tie nonnegative stoage functions and state functions in Poceedings of the 39th IEEE Confence on Decision and Contol vol. 4 pp Decebe. [1] J. Mohseni E. Yaz and K. Olejniczak State-dependent LMI contol of discete-tie nonlinea systes in Poceedings of the 37th IEEE Confeence on Decision and Contol (CDC 98) pp Decebe [] S. M. Joshi On optial contol of linea systes in the pesence of ultiplicative noise IEEE Tansactions on Aeospace and Electonic Systesvol.AES-1no.1pp [3] E. Geshon U. Shaked and I. Yaesh H contol and filteing of discete-tie stochastic systes with ultiplicative noise Autoaticavol.37no.3pp [4] C. M. Hais and D. M. Wolpet Signal-dependent noise deteines oto planning Natue vol. 394 no pp [5] E. Todoov Stochastic optial contol and estiation ethods adapted to the noise chaacteistics of the sensoioto syste Neual Coputationvol.17no.5pp [6] Y. A. Phillis Estiation and contol of systes with unknown covaiance and ultiplicative noise IEEE Tansactions on Autoatic Contol vol. 34 no. 1 pp [7] W. J. Chang W. Y. Wu and C. C. Ku H constained fuzzy contol via state obseve feedback fo discete-tie Takagi-Sugeno fuzzy systes with ultiplicative noises ISA Tansactionsvol.5no.1pp [8] W. J. Chang L. Z. Liu and C. C. Ku Passive fuzzy contolle design via obseve feedback fo stochastic Takagi-Sugeno fuzzy odels with ultiplicative noises Intenational Jounal of Contol Autoation and Systesvol.9no.3pp [9] H. T. Yau C. C. Wang C. T. Hsieh and C. C. Cho Nonlinea analysis and contol of the uncetain ico-electo-echanical syste by using a fuzzy sliding ode contol design Coputes and Matheatics with Applicationsvol.61no.8pp [3] L. Huang K. Wang P. Shi and H. R. Kaii A novel identification ethod fo genealized T-S fuzzy systes Matheatical Pobles in Engineeing vol.1aticleid893871pages 1. [31] K. Tanaka and H.O. Wang Fuzzy Contol Systes Design and Analysis: A Linea Matix Inequality Appoach JohnWiley& Sons 1. [3]S.BoydL.ElGhaouiE.FeonandV.BalakishnanLinea Matix Inequalities in Syste and Contol Theoyvol.15SIAM Philadelphia Pa USA [33] G. Eli S. Ui and Y. Isaac H Contol and Estiation of State- Multiplicative Linea Systes Spinge London UK 5. [34] H. Y. Chung and W. J. Chang Constained vaiance design fo bilinea stochastic continuous systes IEE Poceedings Dvol. 138 no. pp

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