STATE VARIANCE CONSTRAINED FUZZY CONTROL VIA OBSERVER-BASED FUZZY CONTROLLERS

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1 Jounal of Maine Science and echnology, Vol. 4, No., pp (6) 49 SAE VARIANCE CONSRAINED FUZZY CONROL VIA OBSERVER-BASED FUZZY CONROLLERS Wen-Je Chang*, Yi-Lin Yeh**, and Yu-eh Meng*** Key wods: takagi-sugeno fuzzy models, state vaiance constaints, linea matix inequalities (LMI). ABSRAC In this pape, a state vaiance constained design method is developed to deal with the obseve-based fuzzy contol poblem fo continuous akagi-sugeno (-S) fuzzy stochastic models. he stability poblem consideed in this pape is a standad feasibility poblem with seveal Linea Matix Inequalities (LMIs), which can be solved numeically using the inteio-point method. he pupose of this pape aims at finding obseve-based fuzzy contolles such that the closed-loop system is asymptotically stable and the state vaiance pefomance constaints ae satisfied. Finally, a numeical example is included to demonstate the effectiveness of the poposed contolle design method. INRODUCION Pape Submitted /9/5, Accepted 7/8/5. Autho fo Coespondence: Wen-Je Chang. wjchang@mail.ntou.edu.tw. *Pofesso, Depatment of Maine Engineeing, National aiwan Ocean Univesity, Keelung, aiwan, R.O.C. **Maste, Depatment of Maine Engineeing, National aiwan Ocean Univesity, Keelung, aiwan, R.O.C. ***Student, Depatment of Maine Engineeing, National aiwan Ocean Univesity, Keelung, aiwan, R.O.C. he fuzzy logic has been applied extensively in the aeas of industial system and consume poducts. It has emeged in tun to tackle the contol poblems whee the plants ae pooly modeled by mathematics. Howeve, the heuistics-based [4] appoach in the ealy days of fuzzy logic contol lack the fomula and system design methodology that guaantees the basic equiements such as stability and acceptable pefomance. he theoetic eseaches on the issue ae conducted actively by many theoists since the -S fuzzy model [9] is emeged. he -S fuzzy model is descibed by a set of IF- HEN ules. hese ules locally epesent linea inputoutput elations of nonlinea systems. he stability analysis and contolle design fo -S type fuzzy systems ae discussed in [-4,, 9, 5, 6]. he contolle design is caied out based on the fuzzy model via socalled Paallel Distibuted Compensation (PDC) concept [-4,, 5, 6]. Based on PDC concept [-4,, 5, 6], Lyapunov theoy is the main tool fo designes to deal with the stability analysis and synthesis of -S fuzzy models. he advantage of the PDC concept is to design linea feedback gains fo each local linea model and let the oveall system input can be blended by these linea feedback gains. In contol theoy, the pefomance constaint usually plays an impotant objective in addition to the stability equiement. Fo example, the state vaiance constaints [5, 7-9] ae impotant pefomance equiements fo the stochastic contol systems. In this pape, we not only conside the stability of nonlinea systems, but also conside the state vaiance constaints. It is shown that this state vaiance constained poblem can be tansfomed into a LMI poblem. In [, 5, 6], the stability and stabilization of the -S fuzzy system is detemined by solving a set of LMIs. It is well known that the LMI poblem can be solved numeically using an inteio-point method []. Fo pactical systems, the states of a system ae often not eadily available. Unde such cicumstances, some papes [, 3, 4] deal with the fuzzy obseve design poblem fo the -S fuzzy models. It allows the designes to find a common positive definite matix, which can be solved by LMI method fo the closed-loop systems. his pape focuses on the fuzzy contol poblem of continuous -S fuzzy stochastic systems unde the assumption that the complete state vecto cannot be measued. Accoding to the fuzzy obseve design concept and fuzzy contol theoy, an obseve-based fuzzy contolle design methodology is developed to achieve specified state vaiance constaints fo the -S fuzzy models. o validate the effectiveness of the poposed obseve-based fuzzy contolle design method, a numeical example is taken fo a -S fuzzy model subject to the individual state vaiance constaints.

2 5 Jounal of Maine Science and echnology, Vol. 4, No. (6) OBSERVER-BASED -S FUZZY SYSEMS Studies on univesal appoximation of fuzzy systems have achieved geat pogess in the past few yeas. In eality, geneal fuzzy systems ae mainly of two classes: Mamdani fuzzy systems [4] and -S fuzzy systems [9]. he main diffeence between these two classes lies in thei consequent of the fuzzy ules. In this pape, a -S type fuzzy model is used to constuct a nonlinea stochastic system as follows. Plant Rule i: IF x (t) is M i... and x nx (t) is M inx, HEN x(t )=A i x(x)+b i u(t )+D i v(t ), y(t) = C i x(t) + E i µ(t), i =,,...,, () whee x(t) R n x is the state vecto; u(t) R n u is the contol input vecto; and y(t) R n y is the contol output vecto in i-th ule. he v(t) R n v and µ(t) R n µ ae stationay zeo-mean mutually independent white noise pocesses with covaiance V > and Ω >, espectively. he matices, A i R n x n x, Bi R n x n u, Ci R n y n x, Di R n x n v, and Ei R n y n µ ae constant; i =,,..., and is the numbe of IF-HEN ules. he M ij ae fuzzy sets and it is assumed that B i is full-column ank. Besides, the pais (A i, B i ) and (A i, C i ) ae contollable and obsevable, espectively. he state and output equations fo the system can be epesented in tem of the ules () as x(t )= h i (t )A i x(t )+ h i (t )B i u(t )+ h i (t )D i v(t ), () y(t )= h i (t )C i x(t )+ h i (t )E i µ(t ). (3) n x whee h i (t) = ω i (t)/ i ω i (t ), ω i (t) = = j Π M ij(x j (t)) and = M ij (x j (t)) is the gade of membeship of x j (t) in M ij (t); ω i (t) is the weight of the i-th ule. In some nonlinea systems, the system states usually cannot be completely measued. heefoe, the designes need to design the fuzzy obseves to estimate the states fo the fuzzy system in ode to implement the fuzzy contolle. In [4], the authos consideed the socalled sepaation popety fo a contolle and an obseve design fo the linea stochastic systems. he fuzzy obseves equie to satisfy the condition x(t) x(t ) when t, whee x(t ) denotes the estimated state vecto of the fuzzy obseve. In this pape, the fuzzy obseve is descibed as follows: Obseve Rule i: IF x (t) is M i... and x nx (t) is M inx, HEN x(t )=A ix(t )+B i u(t )+K i (y(t ) y(t )), y(t )=C i x(t ), i =,,...,, (4) whee K i R n x n y ae obseve gain matices and x(t ) R n x is the state vecto of obseve. he y(t) and y(t ) ae the output of the fuzzy system and the fuzzy obseve, espectively. hen, the final estimated state and output of the fuzzy obseve ae chaacteized as follows. x(t )= h i (t )A i x(t )+ h i (t )B i u(t ) + j = + j = h i (t )h j (t )K i C j [x(t ) x(t )] h i (t )h j (t )K i E j µ(t ), (5) y(t )= h i (t )C i x(t ), (6) he same weight h i (t) of i-th ule of the fuzzy system () and (3) is used fo the fuzzy obseve (5) and (6). he desied paametes of the fuzzy obseve ae the gain matices K i in each ule. In this pape, the concept of PDC [, 3, 7,,, 3, 5, 6] is used to synthesize obseve-based fuzzy contol laws fo the nonlinea system, which is epesented by continuous -S type fuzzy stochastic model (). he basic idea of the PDC appoach is to design the feedback gains fo each ule in the fuzzy models. Linea contol design techniques can be used to design these linea contolles fo each ule. Hence, the nonlinea system contolle can be blended by local linea fuzzy contolles shaing the same fuzzy sets with the continuous -S type fuzzy stochastic model (). By using the obseved states fom the fuzzy obseve, the obsevebased fuzzy contolle becomes Obseve-based Fuzzy Contolle Rule i: IF x (t) is M i... and x nx (t) is M inx HEN u(t )=G i x(t ),, i =,,...,, (7) whee i =,,..., and is the numbe of IF-HEN ule. he oveall obseve-based fuzzy contolle becomes u(t )= h i (t )G i x(t ). (8)

3 W.J. Chang et al.: State Vaiance Constained Fuzzy Contol Via Obseve-Based Fuzzy Contolles 5 By substituting (8) into () and (5), state and obseve equations of the fuzzy system can be descibed as follows. x(t )= h i (t )A i x(t )+ x = j = h i (t )h j (t )B i G j x(t ) + h i (t )D i v(t ), (9) j = + j = + j = h i (t )h j (t )(A i + B i G j )x(t ) h i (t )h j (t )K i C j [x(t ) x(t )] h i (t )h j (t )K i E j µ(t ). () Let x(t )=x(t ) x(t ), R ij = (A i + B i G j )+(A j + B j G i ) and R ij = B ig j + B j G i,, then (9) can be ewitten as x(t )= i h i (t )h j (t )(A i + B i G i )x(t = )+ h i (t )h j (t )R ij x(t ) h i (t )h j (t )B i G i x(t )+ h i (t )h j (t )R ij x(t ) + h i (t )D i v(t ) () he obseve eo dynamics becomes x(t )= i h i h j (t )(A i K i C i )x(t = )+ h i (t )h j (t )H ij x(t ) h i (t )h j (t )K i E i µ(t )+ h i (t )h j (t )H ij µ(t ) i h i (t )D i v(t ), () = whee H ij = (A i K i C j )+(A j K j C i ) and H ij = K i E j + K j E i. Augmenting () and () yields: whee χ(t )= k = h i (t )h i (t )h k (t )[L i χ(t )+N ik v (t )] + h i (t )h j (t )[L ij χ(t )+N ij v (t )], (3) χ(t )= x(t ) v(t ), v (t )= x(t ) µ(t ), L i = A i + B i G i B i G i, L A i K i C ij = R ij R ij, i H ij N ik = D k D k K i E i, N ij = H ij. If L i is a stable matix, the state covaiance matix X i of each subsystem of (3) can be defined by [4, 3, 3, 4] X i = lim E [χ(t )χ(t t ) ] (4) Let the common covaiance matix fo (3) be X such that X = X i = X aa X ab, i =,,...,, (5) X bb X ab and X = X >, then X satisfies the following Lyapunov equation fo each ule [4, 3, 3-4]: L i X + XL i + N i ΦN i = (6) whee Φ = V Ω. A -S type fuzzy obseve and an obseve-based fuzzy contolle ae used to constuct a nonlinea stochastic system as above. Based on the common state covaiance matix defined in (5), efeences [5-7] povided the conditions and solutions fo the optimal filte gains K i as follows: K i = X bb C i (E i ΩE i ), (7) (A i + B i G i )X aa + X aa (A i + B i G i ) B i G i X bb X bb (B i G i ) + D i VD i = (8) A i X bb + X bb (A i K i C i ) + D i VD i = (9) whee X aa >, X bb >, X ab = X bb and (X aa X bb ) > ae defined in (5). Note that the above conditions ae necessay and sufficient [7] fo the existence of optimal estimatos since they satisfy the Wiene-Hopf equation [5]. Fom the esults of [6-7], it can be found that the optimal filte gain K i = X bb C i (E i ΩE i ) leads to the fact that the steady state eo between the system state x(t) and the estimated state x(t ) conveges to zeo

4 5 Jounal of Maine Science and echnology, Vol. 4, No. (6) when t. Note that the state covaiance matix X defined in (5) satisfies the following Lyapunov equation: (A i + B i G i )(X aa X bb ) + (X aa X bb )(A i + B i G i ) + X bb C i (E i ΩE i ) C i X bb = () Besides, the assumption X ab = X bb implies that the estimate x and the eo x ae othogonal, i.e., Exx =. Without loss of geneality, this assumption has been applied in the design of optimal filte fo the continuous-time systems [6, 8]. Fom (7), if continuous -S type fuzzy model () is coupted only by state noise without measuement noise (i.e., Ω = ), then the optimal filte gain K i does not exist. In next section, the obseve-based fuzzy contol poblem is solved such that the state vaiance constaints ae satisfied. OBSERVER-BASED FUZZY CONROL WIH SAE VARIANCE CONSRAINS In this section, the Lyapunov appoach is used to discuss and analyze the stability conditions fo the -S fuzzy models. As well as, the individual state vaiance constaints ae also consideed in the obseve-based fuzzy contolle design pocess. A vaiance constained fuzzy contol methodology fo continuous -S fuzzy stochastic systems has been developed in [6]. Howeve, it did not conside the obseved-state feedback contol technique. o offe a lucid pesentation of the obsevebased fuzzy contol theoy fo continuous -S fuzzy model (), we conside the stability analysis fo the - S fuzzy model () by applying the fuzzy obseves. he stability conditions of the obseve-based fuzzy contol poblem ae stated by using the optimal estimations defined in (7-9) via the following theoem. heoem Fo a -S fuzzy model (), which is diven by (8) and (), the closed-loop system is possessed by an optimal filte gain K i = X bb C i (E i ΩE i ) with X aa >, X bb > and X ab = X bb, which satisfy equations (7-9). If thee exist common positive definite matices X aa and X aa X bb > satisfying the following conditions A i X bb + X bb A i X bb C i (E i ΩE i ) C i X bb + D i VD i = () (A i + B i G i )( X aa X bb )+(X aa X bb )(A i + B i G i ) + X bb C i (E i ΩE i ) C X bb < () R ij ( X aa X bb )+(X aa X bb )R ij <, (3) then the equilibium of the closed-loop continuous -S fuzzy stochastic contol system () is asymptotically stable in the lage and X aa > X aa. Poof: Fom the statements of above section, it is clea that the matix K i pefoms an optimal filte gain if and only if thee exist matices X aa > and X bb > such that (7-9) ae all satisfied with X ab = X bb defined in (5). Substituting (7) into (9) and eaanging yields A i X bb + X bb A i + D i VD i = X bb C i (E i ΩE i ) C i X bb, (4) which is equivalent to (). Subtacting () fom (), one has (A i + B i G i )( X aa X aa )+(X aa X aa )(A i + B i G i ) < Using the simila method yields (5) R ij ( X aa X aa )+(X aa X aa )R ij < (6) It is easy to find that the optimal filte gain K i defined in (7-9) leads to the fact x(t ) when t. Fom heoem 3 of [], one can deduce that if thee exist a common positive definite eo state covaiance matix ( X aa X bb )> satisfying () and (3), then the equilibium of continuous fuzzy contol system () is asymptotically stable in the lage due to X bb C i (E i ΩE i ) C i X bb and x(t ). In the case of the matices (A i + B i G i ) and R ij being stable, one can obtain that X aa X aa > fom (5) and (6). Hence, it can be concluded that if conditions (-3) ae all satisfied with X aa X aa > and ( X aa X bb )>, then the equilibium of the closed-loop nominal continuous -S fuzzy stochastic contol system () is asymptotically stable in the lage and X aa is the uppe bound of matix X aa, i.e., X aa > X aa. # Applying the esults of heoem, the common state covaiance matix X has an uppe bound matix. It can be defined as X, i.e., X = X aa X ab, i =,,...,, (7) X bb X ab

5 W.J. Chang et al.: State Vaiance Constained Fuzzy Contol Via Obseve-Based Fuzzy Contolles 53 whee X is the uppe bound of common state covaiance matix X, i.e., X > X. Fom the stability conditions of heoem, one can obtain LMI stability conditions with espect to Z i = G i ( X aa X bb ), and they can be espected in the following theoem. heoem Fo a -S fuzzy model (), which is diven by (8) and (), the closed-loop system is possessed by an optimal filte gain K i = X bb C i (E i ΩE i ) with X aa >, X bb > and X ab = X bb, which satisfy equations (7-9). If thee exist common positive definite matices X aa and X aa X bb >, satisfying the following conditions, then the stability conditions (-3) of heoem ae achieved. whee σ ϕ denote the Root-Mean-Squaed (RMS) constaint fo the vaiances of system states. his poblem is efeed to as the state vaiance constained design using the obseve-based fuzzy contol. he pupose of this pape is to find common positive definite matices X aa and ( X aa X bb ) to satisfy stability conditions (-3) subject to the state vaiance constaint (3). Fo this pupose, this pape povides a fomula to find the obseve-based fuzzy contolles fo achieving individual state vaiance constaint (3). heoem 3 If stability conditions (), (8-9) and the following LMIs ae satisfied fo a given σ ϕ >. hen, the - S fuzzy model (), diven by (8) and (), is stable and the state vaiance constaint (3) is achieved. Θ i X bb C i C i X bb (E i ΩE i ) <, i =,,...,, (8) σ ϕ X aa I ϕ I ϕ X aa >, ϕ =,,..., n x (3) X aa A i X aa + X aa A i A i X bb X bb A i + A j X aa + X aa A j A j X bb X bb A j + B i Z j + B j Z i +(B i Z j + B j Z i )<, (9) whee Z i = G i ( X aa X bb ) and Θ i ae defined as Θ i = A i X aa + X aa A i A i X bb X bb A i + B i Z i + Z i B i (3) he poof of heoem can be obtained via Schu complement []. By Schu complement, one can get that () and (3) ae equivalent to (8) and (9). In heoem, the stability conditions of heoem ae tansfomed into LMIs. Based on the stability conditions of heoem, this pape not only conside to design stable obseve-based fuzzy contolles but also to achieve individual state vaiance constaints. he state vaiance pefomance constained design poblem is an application of uppe bound state covaiance contol appoach [5,, 3]. he pupose of this pape is to find the set of contolles G i, which satisfy the stability conditions of heoem, such that the uppe bound state covaiance matix X aa satisfies the following vaiance pefomance objectives: lim E [x ϕ(t )] = [X aa ] ϕϕ [ X aa ] ϕϕ σ t ϕ, ϕ =,,..., n x, (3) whee I ϕ = [ ] R n x denotes a ow vecto with the ϕth element is and othes ae. Poof: In ode to tansfom the pesent contol poblem into a LMI poblem, it is necessay to tansfom the state vaiance constain (3) into a LMI fom. Hence, the LMI condition (3) is diectly constucted fom the state vaiance constaint (3). It is noted that the obseve-based fuzzy contol poblem with state vaiance constaints can be dealt with by the LMI citeions of heoem 3. hese LMI citeions can be solved numeically using an inteiopoint method []. In next section, a numeical example is povided to veify the usefulness of the poposed obseve-based fuzzy contolle design method. A NUMERICAL EXAMPLE o illustate the poposed obseve-based fuzzy contol appoach, conside a two-ule ( = ) -S fuzzy model, which is descibed as follows. Plant Rule : IF x (t) is M HEN ẋ(t) = A x(t) + B u(t) + D v(t) y(t) = C x(t) + E µ(t) Plant Rule : IF x (t) is M HEN ẋ(t) = A x(t) + B u(t) + D v(t) y(t) = C x(t) + E µ(t) (33a) (33b)

6 54 Jounal of Maine Science and echnology, Vol. 4, No. (6) whee A =.5.5, A = , X aa = G = [ ], (36) B =.5, B =.6, C = C = [5 ], D =.5.5, D =.5, E = E =, In Figue, the fuzzy sets fo Plant Rule and ae descibed by two tiangula membeship functions. In this example, it is assumed that the state vaiance constaints of this system have the following foms: [X aa ] 6, [X aa ] 3, [X aa ] 33 5 (34) Besides, the covaiance matices of zeo-mean white noises v(t) and µ(t) ae V = and Ω =, espectively. Befoe finding the solutions of common positive definite uppe bound covaiance matix X aa, the positive definite matix X bb can be obtained by solving the algebaic Riccati-like Eq. (). X bb = (35) o cay on, using LMI-toolbox of MALAB, one can find the solutions of common positive definite uppe bound covaiance matix X aa and feedback gains G i fom LMIs (8-9) and (3). -.5 Rule Rule Rule x (t).5 Fig.. he membeship function of x (t). G = [ ] (37) Applying X aa and X bb to constuct X, which is defined in (7), one has X = Subtacting X bb fom X aa, one can obtain X aa X bb = (38) >. (39) Substituting X bb into (7), the fuzzy obseve gains K i can be obtained as follows: K = [ ], K = [ ] (4) Since conditions (-3) of heoem ae all satisfied, one can conclude that the closed-loop continuous obseve-based -S fuzzy stochastic contol system () is asymptotically stable by applying the obseve gains (4) and the fuzzy contol gains (37). Besides, it also achieved the goal of the state vaiance pefomance constaints (34). In the simulation, the initial states ae given as [x () x () x 3 ()] = [ ]. Figue and Figue 3 show the input signals of v(t) and µ(t) fo the simulations. Applying fuzzy contol gains G and G in the simulation, Figue 4 shows the esponses of contol input. In addition, Figue 5, Figue 6 and Figue 7 show the esponses of tue states X(t) and the estimated state ^X(t) fo the contolled -S fuzzy model (33), espectively. Fom these simulation esults, the state vaiances of closedloop nonlinea system (33) ae calculated as follows:

7 W.J. Chang et al.: State Vaiance Constained Fuzzy Contol Via Obseve-Based Fuzzy Contolles 55 v (t) Fig.. he input signals of v(t). 8 x (t) and x^ (t) Plant state x (t) -.6 Obseve state x^ (t) Fig. 5. he esponses of tue state x (t) and estimated state x (t ) fo contolled fuzzy model (33). µ (t) u (t) Fig. 3. he input signals of µ(t) Fig. 4. he esponses of contol input u(t). 8 8 x (t) and x^ (t) Plant state x (t) Obseve state x^ (t) Fig. 6. he esponses of tue state x (t) and estimated state x (t ) fo contolled fuzzy model (33). x 3 (t) and x^ 3 (t) Plant state x 3 (t) -.4 Obseve state x^ 3 (t) Fig. 7. he esponses of tue state x 3 (t) and estimated state x 3 (t ) fo contolled fuzzy model (33).

8 56 Jounal of Maine Science and echnology, Vol. 4, No. (6) va(x (t)) =.75, va(x (t)) =.4 and va(x 3 (t)) =.58 (4) whee va(x (t )) denotes the vaiance of system state x (t ), =, and 3. It can be found that the closed-loop fuzzy system is stable and the vaiance constaints (34) ae satisfied. CONCLUSIONS he obseve-based fuzzy contolle design poblem has been solved in this pape subject to individual state vaiance constaints. he filteing optimal contol technique was used to design the obseves fo the continuous -S fuzzy stochastic models. he LMI conditions wee developed fo the existence of obseve-based fuzzy contolles, which allows designes to assign a specified uppe bound common state covaiance matix to the closed-loop continuous -S fuzzy models. he PDC concept was applied to design fuzzy contolles fo each ule and to solve the solutions of the obseve-based fuzzy contolles. he poposed appoach aimed at finding obseve-based fuzzy contolles such that the closed-loop system is asymptotically stable and the state vaiance constaints ae achieved, simultaneously. ACKNOWLEDGEMENS his wok was suppoted by the National Science Council of the Republic of China unde contact NSC93-8-E-9-. REFERENCES. Boyd, S., Ghaoui, L.E., Feon, E., and Balakishnan, V., Linea Matix Inequalities in Systems and Contol heoy, SIAM, Philadelphia, PA (994).. Chang, W.J., Model-Based Fuzzy Contolle Design with Common Obsevability Gamian Assignment, Jounal of Dynamic Systems Measuement and Contol- ASME, Vol. 3, No., pp. 3-6 (). 3. Chang, W.J., Fuzzy Contolle Design via the Invese Solution of Lyapunov Equations, Jounal of Dynamic Systems Measuement and Contol- ASME, Vol. 5, No., pp (3). 4. Chang, W.J. and Chung, H.Y., A Study of Nom and Vaiance Constained Design Using Dynamic Output Feedback fo Linea Discete Systems, Intenational Jounal of Contol, Vol. 57, No., pp (993). 5. Chang, W.J. and Sun, C.C., Constained Fuzzy Contolle Design of Discete akagi-sugeno Fuzzy Models, Fuzzy Sets and Systems, Vol. 33, No., pp (3). 6. Chang, W.J. and Wu, S.M., State Vaiance Constained Fuzzy Contolle Design fo Nonlinea ORA systems with Minimizing Contol Input Enegy, Poceedings of IEEE Intenational Confeence on Robot and Automation, aipei, aiwan, pp (3). 7. Chung, H.Y. and Chang, W.J., Constained Vaiance Design fo Bilinea Stochastic Continuous Systems, IEE Poceedings-Pat D: Contol heoy and Applications, Vol. 38, No., pp (99). 8. Chung, H.Y. and Chang, W.J., Covaiance Contol with Vaiance Constaints fo Continuous Petubed Stochastic Systems, Systems & Contol Lettes, Vol. 9, No. 5, pp (99). 9. Collins, J. E.G. and Skelton, R.E., Constained Vaiance Design Using State Covaiance Assignment, Poceedings of Ameican Contol Confeence, Seattle, WA, pp (986).. Hong, S.K. and Langai, R., Robust Fuzzy Contol of a Magnetic Beaing System Subject to Hamonic Distubances, IEEE ansactions on Contol Systems echnology, Vol. 8, No., pp ().. Jun, Y., Masahio, N., Hitoshi, K., and Ichikawa, A., Design of Output Feedback Contolles fo akagi- Sugeno Fuzzy Systems, Fuzzy Sets and Systems, Vol., No., pp ().. Kwakenaad, H. and Sivan, R., Linea Optimal Contol Systems, John Wiley and Sons, New Yok (97). 3. Li, N., Li, S.Y., and Xi, Y.G., Stability Analysis of - S Fuzzy System Based on Obseves, Intenational Jounal of Fuzzy Systems, Vol. 5, No., pp. -3 (3). 4. Mamdani, E.H., Applications of Fuzzy Algoithms fo Contol of Simple Dynamic Plant, Poceedings of IEEE Institution of Electical Engineeing and Contol Science, Vol., No., pp (974). 5. Maybeck, P.S., Stochastic Models, Estimation, and Contol, Academic Pess, New Yok (979). 6. Phillis, Y.A., Contolle Design of Systems with Multiplicative Noise, IEEE ansactions on Automatic Contol, Vol. 3, No., pp. 7-9 (985). 7. Phillis, Y.A., A Smoothing Algoithm fo Systems with Multiplicative Noise, IEEE ansactions on Automatic Contol, Vol. 33, No. 4, pp (988). 8. Phillis, Y.A., Estimation and Contol of Systems with Unknown Covaiance and Multiplicative Noise, IEEE ansactions on Automatic Contol, Vol. 34, No., pp (989). 9. akagi,. and Sugeno, M., Fuzzy Identification of Systems and Its Applications to Modeling and Contol, IEEE ansactions on Systems Man and Cybenetics, Vol. 5, No., pp. 6-3 (985).. anaka, K., Ikeda,., and Wang, H.O., A Unified Appoach to Contolling Chaos via an LMI-Based Fuzzy

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