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1 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< \ -S;? P A" ( 6M [ X& 5 +; \ ; D7 ] +; I$ X & >?# ; U (6 3 I q +W R% RBH % a >M 5 د! "#$#% & '( a.khan9@ms.tabrzu.a.r >" *+ + LN6 H+ +ˆ, ghaem@tabrzu.a.r >" *+ "3Q G+ +!" I" " mbadamh@tabrzu.a.r >" *+ "3Q G+ + 3, + *ق (393//4 :[l/ m" 393//9 :, m" ) ISSN: htt://nsee.sut.a.r +V -S E -;3 K, D -;3V K, :3J -S K, E + -;3 K, E A+< [ V V +/.3VV EV:+ VDV F! F KE + +9) ),.3VV h 3 : }+3" Z(" +9), -;3 K, :3J 3 : }+3" k3" +3VCH B3V) V VEV K, 3? +(" 3 : }+3" +9), -;3 K, E + }K3" K+5 [ B4 D! -S K, E 3 vc + 3< V 3VV V,D V\ KV, X3V/H }" F +/ >HO +.+ X( (non-pdc) K+3.O.+ +/ +! F (LMIs)!"! B3) +/ k+ 3<!" -;3 K, -S E +9), +/ >HO K, X3/H }"!"! (LMIs) Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

2 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 5 Sahand Unversty ofehnology Journal of Nonlnear Systems n Eletral Engneerng Vol.3, No., Summer 5 ISSN: htt://nsee.sut.a.r Stablty Analyss and Controller Desgn for Interval ye- -S Fuzzy Systems Based on Interval ye- -S Fuzzy Observer Under non-parallel Dstrbuted Comensaton and Fuzzy Lyaunov Funton Aroah Arman Khan, Sehraneh Ghaem, and Mohammad Al Badamhzadeh 3 Faulty of Eletral and Comuter Engneerng, Unversty of abrz, abrz, Iran, a.khan9@ms.tabrzu.a.r Corresondng Author, Assstant Prof., Faulty of Eletrall and Comuter Engneerng, Unversty of abrz, abrz, Iran, ghaem@tabrzu.a.r 3 Assoate Prof., Faulty of Eletral and Comuter Engneerng, Unversty of abrz, abrz, Iran, mbadamh@tabrzu.a.r ABSRAC Keywords Interval tye- -S fuzzy model, Stablty analyss, Fuzzy Lyaunov funton, Lnear matr nequaltes (LMIs) In ths aer, we nvestgate the desgn method for nterval tye- (I) -S fuzzy ontroller based on I -S fuzzy observer for nonlnear systems along wth unertanty arameters. In order to analyze the stablty and synthess the ontrol methods onvenently, an I ( S) fuzzy model s aled through reresentng the dynam of nonlnear systems and dynam of observer. Unertanty arameters are atured by I membersh funton haraterzed by the lower and uer membersh funtons. In ths aer, for I fuzzy ontroller, the membersh funtons and number of rules an be freely hosen dfferent from the I S fuzzy model and I -S fuzzy observer. hs method s known non- Parallel Dstrbuted Comensaton. o redue the onservatveness of stablty analyss, a fuzzy Lyaunov funton anddate s aled. he stablty ondtons n term of lnear matr nequltes (LMIs) are obtaned. Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

3 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 5 53" -S E K, D + -;3 K, :3J :- -S E K, E + E + (PDC )K+3 }K3" K+5 X3/H +/ 3".+ f K ==6 -;3 K, :3J -S E K, D + -S E E! +/ >HO A+< O 3 A+< D+5 B3N D F E F!K + 3< E A+<.[7-] O (LMI 3 )!"! K+ +:3J B3N +/ w K, E! K, E K O + 35 HA -S E K, D 3 A -S E K, -S E! -S E P HA.[8] }"! HA }" D!K " + g++ [.+ + K, 3? +(" E -S E D +/.+,D \ +/ >HO + 3: X3/H }" F D-S E -S B3N E - 3\.O 353 +/ >HO 6 7" + 3: X3/H }" -+>,+ {: +,D \ K, X3/H }" ( X3/H }+3" ++ +/ >HO 6.[9] + A+< l/ X(+ z+ K, X3/H }" + *! 3 : }+3" LG X3/H \ 3\ ( E 3\ -;3 K, :3J ++.+,D \ [4 - ] -;3 K, :3J L? -;3 K, :3J 3 : }+3" 3! +/ (.[5] + g++ Bh= Bh= -;3 K, E 95 K+ 49 C.[5] O +9), -;3 K, :3J -;3 K, :3J K+ 3\ MA H=.[6] 3 K, :3J! E B==6" ( ++.3,D \ : KE [ [6].[8-6 4] + 7J+ -S E K+ 4 E -;3 9)+ + MA H=.+ g++ LN4 3< - ;3 K, :3J K G z+ +9), -;3 K, :3J -S E z+ K, D + E A+< >H O +/ >HO + K, X3/H \ (non-pdc) K+3 }K3" K + E -S E K, 3? +(" D -S E! -S E K, 3? +(" D! K, E! E + K, 3? \ +3" 3< E A+< K D.., +E f 3 : ( + D E 3 7= PDC G! E D D akag - sugeno Parallel Dstrbuted Comensaton 3 Lnear Matr Inequlty Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

4 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 53 >H D -S E! -S K, E 3 : +/ K, K f 3 : \ +3" O + 3 b+< 3< O!!? K+ 4 + O K+ +/ >HO K, K K Y 6 + K!? 3< D K++ L?! K+ HA }" 4D \ 3 :.3 3! B3N E D -S K, E! -S K, E E! +/ k+ +9), -;3 K, :3J! -S K, E -C g++ (LMI)!" -E 4 -C.3 g++ +9), -;3 K, :3J D -S K, E 3 -C.3 g++! -S E E! 5 -C.+ +9), -;3 K, :3J E S K 7 -C.+ g++ +/ >HO 6 -C.3 g++ E -S E D -S E.+ DJ O -C.3 [6] +; -\ ; W Rb? A" -S; - Z(" K B3N ψ K, 3? -;3 K, :3J -S E K,! F Rule : IF f () s M % and,...,f () s M% HEN & = A + B u =,,..., ψ ψ () :3< f ( ) K, u + +K, 3? w3 +9), - ;3 K, :3J F M% O `" HA `" f"" B, A O B R, A R n m n =,,..., ψ, =,,..., +٢ -;3 -S K, E. H E* u R m l! HA + R n l.! : K B3N 7+ 3? + 3 L U w (), w (), =,,..., h K 3? () :O w () = µ (f ()) µ (f ())... µ (f ()) L M M Mψ w () = µ (f ()) µ (f ())... µ (f ()) U M M Mψ ψ ψ (3) :3< Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

5 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< µ (f ()) [,] M µ (f ()) [,] M 54 (4) : ) >K + +lh. <3 3 : }+3" 3 : 5 f" µ (f ()) µ (f ()) M U M w () w () L :3 (5) K! J (6) :O K + B3N! -S K, E L U = = & = w ()v ()(A + B u) + w ()v ()(A + B u) = w % ()(A + B u) = w % () = w ()v () + w ()v () [,] L U w % () =, v () [,], v () [,] = (7) :O (8) :3<.3,D \.5 + h3(! }+3" v (), v () O [8] -;3 K, :3J O +; -\ ; W Rb? D7-S ; -3 7= w * \ K, 3? + +9), - ;3 K, D F (9) + B3N D w3 K, 3?. D G! J w :!G D (7)! -S E w3 K, K Rule : If f () s M % and,...,f () s M% hen & ˆ = A ˆ + B u + L (y y ˆ ), =,,..., ψ ψ (9) f ( ) K, u + 7+ K, 3? w3 +9), - ;3 K, :3J F }+3" K, u J+. [8] + e += F ψ =,,..., ψ, =,,..., M% O :3<! K! >J"!? D K++ L?!! K+ HA }" 3 : Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

6 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 55 O B+u" 6 K+ k=, + 35 DK++ L?! K+ HA K, u D+ DK++ L? HA K+ k=, 3 : \ KE 3\ HA + R n l.! `" HA `" f"" (٩) D L B R, A R n n n n +/. H E* u 4 + m l! HA B3N D 5! 5 f"" ŷ ˆ :3 Z("K y = y ˆ = = = w % ()C w % ()C ˆ () -S E! -S E 5 `" C `" + D -S E! -S E 5 () + B3N 7" 3 w % (), O D :(. C = C =... = C = C ().O K + B3N D -S K, E J ˆ& = w % ()(A ˆ + B u + L (y y) ˆ = () +; -\ ; W Rb? IPP -S ; -4.3 g++ () D 3 A (7) -S K, E +/ 3\ +9), -;3 K, E ++ : [8] K+ K, 3? 7= w!? k=, K + B3N E w3 K, 3? Rule : IF g () s N % AND... AND g () s N % HEN u = G ˆ ψ ψ (3) +9), -;3 K, :3J % N : \. HA F, =,,..., ψ, g () m n G R, =,,..., K, u K, + 3< w3 Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

7 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< M () m (),, = m () ψ m () = µ (g ) = ψ = N% m () = µ (g ) N% µ (g ) µ (g ) N% N% 56 (4) (5) (6) (7) 3 : 5 7+ h 3 : 5 f"" m () (7) " (3) k+.[ 8] K, + m () µ N % (g ()) µ N % K h K 7+ (g +("! DK++ L?!! K+ HA }" E w3 3 : }+3" K, u B3N E 3?. D! K, E w3 :3 Z(" K + B3N HA F, u = % = m ()G ˆ (8) : [8] O β ()m () + β ()m () m () =, m () = % % = ( β k ()m k () + βk ()m k ()) k= (9) ) β + β = ( ) ( ) + Z(" -/ K+ u β [ ], β [ ] O. [8] 3,D \.5 + h3( IPP -S D 7 -S a A" -S? P A"-5! HA +/ E* +/ X K + B3N E* Z(" D A+<. +! HA D ( L 4) ) 3 +/ E* D+ 3< e = ˆ () : [] O K B3N + () () w D E J () when t () ˆ when t () :[8] \ () (8) () (7) + 53" w % () = m % () = w % ()m% () = = = = = () Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

8 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 57 :O K + B3N (8) E () D (7) -S K, E E! F = = = = { } & = w % ()m% () (A + B G ) B G e { } e & = w % ()m% () (A L C)e (3) 3< 3 +/ (3) E! +3D L G HA F, +/ X O J +! HA D L 4)! HA m %, w% K+ m % (), w % () 5 D ( 6M-6,D \ K + B3N f""! =9A! 9 `" HA u (3) + A + BG BG a =, Q e = A LC :3!3K K + B3) :3 (4) (3) + 3< non- a = % a = = & w % m Q G -;3 K, :3J K, X3/H }" Z(" Y< K+! =9A! +/ >HO + (5) :* \ + K + B3N K, X3/H }".O! =9A! + +/ w PDC V = % % a a = = w m P P = P >, R n n (6) :3< : +? K H( (5) + 53" [ a M + λ & a M] & a w % m % Q a = = ==A : F λ > +/ >HO = G `" M R n n O K! l/ Y 3/ (+3",D \ \ () +!" 53" :3 +3" [-9]! HA Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

9 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< & w & % = m% = = = 58 (8) + K+ n n n n S R, Y R `" Z(". m %, w% K, K K Y m &%, w& % O l= l= w&% S =, =,,..., l a a m&% Y =, =,,..., l a a :3 +3" (8) (9) :O K B3) V & (9) (7) (6) + 53" & % a a % & a a % & a a = = = = = = V & = w m% P + w m% P + w m% P % % & a a & l a l a & l a l a = = l= l= = w m ( P + w P + m P ) % % & a a & l a l a & l a l a = = l= l= = w m ( P + w (P + S ) + m (P + Y ) ) w% m % ( a P & a w& l a l + a + & l a l + a = = l= l= = + + [ a M + λ & a M] & a w% m% Q a = = (P S ) m (P Y ) ) % % & a a & l a l a & l a l a = = l= l= = w m ( P + w (P + S ) + m (P + Y ) + M & MQ + λ & M & λ & MQ ) a a a a a a a a (3) P + S >, P + Y >, =,,...,, =,,..., :, : ' V m & % w & % 9 >P );D = w &% σ, m& % φ (3) :3!3K K B3N (3) + φ σ O % & a a l a l a l a l a = = l= l= V & w m % ( P + σ (P + S ) + φ (P + Y ) + M & MQ + λ & M & λ & MQ ) a a a a a a a a (3).3!3K K B3N (3)! + (7) + Z(" M `" K+ 4 + Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

10 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 59 % & a a l a l a = = l= V & w m % ( MM P M M + σ MM (P + D )M M + φ MM (P + F )M M + MM MM M & l= l a l a a a MM MQ M M + λ & MM MM M & a a a a λ & MM MQ M M ) a a (33) :3!3K K B3N (33) + % & a a l a l a = = l= V & w m % ((M ) M P M (M ) + σ (M ) M (P + S )M (M ) & l a l a a a l= + φ (M ) M (P + Y)M (M ) + (M ) M MM (M ) (M ) M MQ M (M ) + λ(m & )M MM (M & ) a a a a λ(m & ) M MQ M (M )) a a (34) : 4) + + (7) +, [!D (34) + K+ K + G+ 53" (M ) M MM (M & ) (M ) M MQ M (M ) a a a a + λ(m & )M MM (M & ) λ (M & ) M MQ M (M ) = a a a a (35) :3 Z(" f (+ K `" u W = M, G = N W, L = W X, V = WP W, Ε = WS W, Ι = WY W, l l & l l a a l= l= ϒ = σ (V + Ε ), Γ = φ (V + Ι ), ζ = W, γ = W ; =,,...,, e `" `" W W W = dag[w, W ] O + (35) + J. HA F, D `" ( f `" m n n m N R,X R QW W ζ ζ γ = λqw λw γ :3 K + B3N +3" (36) QW λqw W = λw : +? K + (36) + 53" (37) Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

11 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 6 `" + Š^ X< f"" F = dag[i, W,I, W ] n n D = dag[i, W, I, W ] n n `" vm :O K + (37) + AW BG W BG W W A + W LC W ζ ζ γ = A λ W λbg W λbg λw γ λ W A + λw LC λw (38) Ξ = AW BN BG W A + XC :@,D \ :3 K + B3N +3" + (34) + ζ ϒ + Γ + Ξ + Ξ V + W + λξ ζ γ V W (W W ) + + λξ λ + γ V & w% m % = = ζ = wm % % ζ γ Θ = = γ (39) :O Θ = Ω = = = w m Ω ϒ + Γ + Ξ + Ξ V + W + λξ V + W + λξ λ (W + W ) >HO s.3 +? V& < w Ω < D+ 3. V& <! =9A! +/ +.+ O K? A+< +/!. φ σ 3< m&% φ w &% σ + 3 ==A : F λ >,: ]H () - ;3 K, D (7)! K, E K+ LG" (3) +9), -;3-S K, E E :3,D \ K `" D+ + +/! =9A B3) (8) -;3 K, E S = S R, Y = Y R, V = V R, Ε = Ε R n n n n n n n n n n m n n m n n Ι = Ι R, N R, X R, W = W R, W = W W = W R, W = W R, ( =,,...,, =,,..,),( ) n n n n W Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

12 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 6 :3 O K LMI 3< V > V + Ε >, V + Ι > ϒ + Γ + Ξ + Ξ V + W + λξ < ( =,,...,, =,,..,),( ) V + W + λξ λ (W + W ) :O AW BN BG ϒ = σ l (V l + Ε ), Γ = φ l (V l + Ι ), Ξ = l= l= W A + XC :O K B3N L D G HA F, B3)@+ G = N W, =,,..., L = W X, =,,...,?$ h-7 E/ }M3 u" +K " O * \ + K + +/ + z3g( E/! :[6] DK++ L? θ && θ = & g sn( θ) aml θ sn( θ) / a os( θ)u 4L / 3 aml os ( θ) (4) E3< L a = / (m + M ) M E/ 75 m P5 v g E/ }M3 u" +K : K B3N! +/ B+u" 6. 3"3 f!a ++ + θ u O E/ m [m m ] = [ 3] kg mn ma M [M M ] = [8 6] kg mn ma L = m = θ 5 π / 5 π / (rad) = θ& [ 5 5] (rad / s) ] :K B3N HA \ :O K + B3N HA Bh( & g aml os( ) sn( a os( ) ) u = ( ) + & 4L / 3 am 4L / 3 am Los ( ) Los ( ) (4) Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

13 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 6 :3,D \ K + B3N K, u g am L os( ) sn( ) a os( ) f () ( ) f ( ) Los ( ) =, = 4L / 3 amlos ( ) 4L / 3 am :O O (4)!-S E (4) Bh( 53" (4) (4) (9) () k+ 53" A = A =, A3 A 4, B B 3, B B4 fmn = = fma = = f = = mn f ma f =.78, f = 8.48, f =.765, f =.6 mn ma mn ma C = C = C = C = C = [ ] 3 4 (43) + 4 +! K! >J" +! (?+ HA B+u" 6 K+ k=, (43) + f=" 3< DK++ L?! E+ HA K+ " 3 : }+3" Z(" + HA E5 B3N D! E K, 3? h 3 : }+3".O (?+! K+.3,D \ K z3g( E/ -S E w3 3 : }+3" : E5 h 3 : 3 : }+3" µ = µ =.3e M% M% µ = µ = e 3 4 M% M% µ = µ = e 3 M% M% 4 M% M%..5 µ = µ =.5e.5.5 µ = µ = e M% M% 3 4 M% M% 3 M% M% 4 M% M%. µ = µ =.3e µ = µ =.5e µ = µ = e }" +3CH B3N E w3 K, u J+ 3,D \ K, 3? > E + 3 : }+3". O +? k+ 3< 3,D \ DK++ L? HA u : g () e.35 K B3N = = K, u 53" E h m ( ) = µ ( ) = m ( ) = µ = e N% N%.35 m ( ) = µ ( ) = m ( ) = µ = µ ( ) N% N% N% β = β =.5 :3 +3" E! E K, K h A O + w% w% m% m% w %& = = &, m& % = = & t t Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

14 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 63 K B3N E K, E! K, E K Y }" k=, m% w% G+ 53" :O w w m% m% w & % % % & m& % & = =, = = t t & = (w% [ ] w [ ] w 3[ ] w 4[ ] ) + % + + % % = (w% + w% + w% + w% ) = w% = 3 4 = 4 :O E! E K, K K Y h A HA u B+u" 6 53" J :3 6 K B3N E w& % σ = 3.99, w& % σ = w& % σ = 7.8, w& % σ = m& % φ = 4.5, m& % φ = 4.5 :O K B3) D HA F,? K+ 4 + G = [ ], G =[ ] L = L =,L3 L4.6 = = :K Z9 C H+ k+ 6" + H E*! HA m/ K s () : () : X() = [.3 ] ˆX() = [ ] X() = [.3 ] ˆX() = [ ] m = m mn = kg, M = M mn = 8kg m = m ma = 3kg, M = M ma = 6kg (( ) Z9 C 75 Z9 C 75 3 A + z3g( E/! 3 E K. F 3 + +! HA 3 D +/ Z9 C H+ k+ m M Z9 C 75 H+ k+ + + HA m/ () LG. 7= D E 3 D +,+ =" HA m/ 3 O 53" Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

15 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 64 Z9 C 75 H+ k+ + + HA () LG.+ + HA.+ L 4) ()- ((=[.3 ])) (rad) and estmaton of (rad) ()- estmaton of ((=[.3 ])) ()- ((=[-.3 ])) ()- estmaton of ((=[-.3 ])) tme(se) m = M = 8 = + m/ :LG.5.5 ()-error between & estmaton of (()=[.3 ]) ()error between & estmaton of (()=[-.3 ]) error tme(se) m = 3 O 53" Z9 C 75 H+ k+ + + H+ k+ + + M = 8 : (4) LG.+ + HA m/ (3) LG HA 3 D.+ L 4) Z9 C 75 3 ()- (()=[.3 )] (rad/s) & estmaton of (rad/s) - - ()- estmaton of (()=[.3 ]) ()- (()=[-.3 ]) ()- estmaton of (()=[-.3 ]) tme(se) m = M = 8 = + m/ : 3 LG Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

16 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 65.5 ()- error between & estmaton of (()=[.3 ]) ()- error between & estmaton of (()=[-.3 ]) error tme(se) m = M = Z9 C H+ k+ m : 4 LG M = 8 = + H E* (5) LG 5 5 ()- ontrol sgnal for ()=[.3 ] ()- ontrol sgnal for ()=[-.3 ] u tme(se) m = M = 8 + u H E* :5 LG M = 6 = + K (Z9 C 75) ( L= D E 3 7= H+ k+ + + HA m/ (6) LG.+ 7J+ Z9 C H+ k+ m = 3 +,+ =" HA m/ 3 O 53" Z9 C H+ k+ + + HA (7) LG.+ + HA 3 D.+ L 4) Z9 C 75 3 O 53" Z9 C 75 H+ k+ + + H+ k+ + + HA (9) LG.+ + HA m/ (8) LG HA 3 D.+ L 4) Z9 C 75 Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

17 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 66.5 ()- (()=[.3 ]) (rad) and estmaton of (rad) ()-estmaton of (()=[.3 ]) ()- (()=[-.3 ]) ()-estmaton of (()=[-.3 ]) tme(se) º =» = µ¹ + µ m/ :6 LG.5.5 ()-error between & estmaton of (()=[.3 ]) ()error between & estmaton of (()=[-.3 ]) error tme(se) m = 3 M = 6 :7 LG 3 ()- (()=[.3 ]) (rad/s) and estmaton of (rad/s) - - ()-estmaton of (()=[.3 ]) ()- (()=[-.3 ]) ()-estmaton of (()=[-.3 ]) tme(se) m = 3 M = 6 = + m/ : 8 LG Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

18 K+3 }K3" K+5 K, X3/H }" G +9), ;3 K, D 3 A +9),- ;3-SK, E E! +/ >HO A+< 67.5 ()- error between & estmaton of (()=[.3 ]) ()- error between & estmaton of (()=[-.3 ]) error m = 3 M = 6 tme(se) :9 LG 5 5 ()- ontrol sgnal for ()=[.3 ] ()- ontrol sgnal for ()=[-.3 ] u tme(se) m = 3 M = 6 + u H E* :LG :3 L: K B3N HA K+ G+ 53" + e = ˆ ˆ e& = & & e = ˆ e& = & ˆ& :3 +3" h k " e = e& + ˆ ˆ G+ O Y f!a ++ B3N G+ 53" + +3 D> += : 6H K+ O 3 4) 9 K HA K+ J.3 d w3 D K, KE +9), -;3 K, Y -S K, E K K / K, E D KE z+ `Q. +/ Journal of Nonlnear Systems n Elet. Eng. Vol. 3, No., Summer5 394!" 3

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