0 07 0 Electri c Machies ad Cotrol Vol. No. 0 Oct. 07 54088 :,,, ;, : ; ; ; ; DOI: 0. 5938 /j. emc. 07. 0. 05 TP 7 A 007-449X 07 0-008- 08 Fiite-time output regulatio method for a class of ucertai oliear systems LIU Hai-tao TIAN Xue-hog YU Guo-ya WANG Gui LIU Hua-lao School of Mechaical ad Power Egieerig Guagdog Ocea Uiversity Zhajiag 54088 Chia Abstract To solve a output regulatio problem of a class of ucertai oliear systems drive by a liear eutrally stable exosystem a fiite-time coverget output regulator desig method is proposed. Accordig to the desig the iteral model priciple was combied with fiite-time stability theorem. Firstly based o ecessary coditios of output regulatio problem the output regulatio problem of the give plat ad exosystem was trasferred to a robust stabilizatio problem for the so-called augmeted system. Secodly a robust fiite-time stable cotroller was desiged by usig the dyamic surface techique which guarateed all the closed-loop sigals to be bouded ad regulatio errors to coverge to a ay give rage i fiite time. Simulatio results show that the proposed regulator is effective ad feasible. Keywords oliear systems fiite-time stability output regulatio iteral model dyamic surface 05-07 - 6 : ( 05A03030307) ; ( YQ05087) ; ( 537500, 440604) ; ( 05KTSCX059) ; ( C540) ; ( 04KQNCX08) : ( 98 ),,,, ; ( 980 ),,, ; ( 970 ),,,, : ( 963 ),,,, : ( 966 ),,,, :
0 09 0 Ω ρ i x x x i Δf i ρ i x x x i i 3 x R - 7 8 u t 0 π 4 π w = R w 9 π i+ w = π i - d i - f i π π i - 0 Δf i π π i w t i - 3 4-6 σ w = π - d - f π w - Δf π w t 4 4 z = x - π w zi = z i + + ~ f i z z z i + Δ ~ f i z z z i i - z = u - σ w + ~ f z + Δ ~ f z e = z 5 x i = x i + + d i w + f i x - i + Δf i x - i w t x = u + d w + f x + Δf x w t Δ ~ f i = Δf i z + π z i + π i - Δf i π π i ~ y = x f i = f i z + π z i + π i - f i π π i e = y - R w i 6 x i = x x x i T x = x x x T R u y g i C Δf i R m 3 7 7 q a 0 a R w d i w a q - w Ω R m L q Swσ w = a 0 σ w + a L Sw σ w + + a q - L Sw σ w 8 L w = Sw i z z i Δ ~ f i z z i i z z i 7
0 τ w = σ w L Sw σ w L q - Sw σ w T Γ = 0 T ψ = 0 0 0 a 0 a a q - ξ = G + HJ ξ + φ φ 3 z - = - ~ f 5 - z - K - Sig s - α - γ - s - + z-d 6 Step s = z - z d z d = 0 s = z - z d = z + ~ f + Δ ~ f 3 z - z - = - f ~ - K Sig s α - γ s 4 K > 0 0 < α < γ = ρ + + β > 0 > 0 β > 0 z - τ τ zd + z d = z - z d 0 = z - 0 Step 5 s = z - z d 6 s = z 3 + ~ f i + Δ ~ f i - z d 7 z - 3 = - f ~ - K Sig s α - γ s + zd 8 γ = ρ + 3 + β z 3d z - 3 τ 3 z 3d τ 3 z3d + z 3d = z - 3 z 3d 0 = z - 3 0 Stepi i 9 s i = z i - z id 0 si = z i + + ~ f i z z z i + 9 Δ ~ f i z z z i - zid 3 σ w = Γτ τ = ψτ Hurwitz G R q q H z - R q i + = - ~ f i - K i Sig s i α - γ i s i + zid G H G ψ ψ Γ Sylvester γ i = ρ i + 3 + β z- i + Tψ - GT = HΓ T J τ i + z i + d = ΓT - R q 7 τ i + zi +d + z i +d = z - i + z i +d 0 = z - i + 0 η = G + HJ η 0 3 σ w = Jη Step s = z - z d 4 s = u - σ w + ~ f + Δ ~ f - z d 5 γ - = ρ - + 3 + β z- d τ z d τ zd + z d = z - z d 0 = z - 0 7 ζ = ξ - η - Hs 0 ζ = ξ - η - Hs = G + HJ ξ + φ - G + HJ η - Hs 8 φ = - Hk s - G + HJ Hs 9 ξ = G + HJ ξ - Hk s - G + HJ Hs 5 ζ = G + HJ ζ - Hk s - Hs = G + HJ ζ - Hk s - 30 H u - Jη + f ~ + Δ f ~ - zd 3 u = Jξ - JHs - K Sig s α -
0 ~ f - γ + k s - tah + zd 3 γ = ρ - + β zd d z = z- - z d 33 τ 3 3 ζ = G + HJ ζ - Hk s - Hs = G + HJ ζ - Hk s H u - Jη + f ~ + Δ f ~ - zd = - ( ) G + HJ ζ - H Jζ + Δf ~ - γ s - tah 7 34 Δ ~ f - tah - tah y = z d - z - = z d + f ~ + K Sig s α + γ s - xd 35 y i+ = z i+d - z - i+ = z i+d + f ~ i + K i Sig s i α + γ i s i - z s = Jζ - K Sig s α - γ + k s + Δf ~ - tah id 36 Youg's s i s i Δf i s i ρ i +s i ρ i / + / > 0 s = s + y - K Sig s α - γ s + Δf ~ y i + η - y i +η - /ω + ω / ω > 0 43 si = s i+ + y i+ - K i Sig s i α - γ i s i + Δf ~ i 4 37 40 y = - y τ - z - = - y τ 37 + ~ f z + K z α s α s + s ρ + 3 + β + s ρ ρ z 38 z y + y τ η s s y K 39 i y + = - y i + + τ i + i i s ρ i + 3 + β + s i ρ i i ρ i z j z j f ~ i z j z j + K i α s i α si + - z id = - y i + + η τ i + s s i + y y i z id zid i + η i + s s i + y y i z id zid = i K i α s i α s i + s i ρ i + 3 + β + s iρ i i i j = y j z ρ i z j z j z id 40 f ~ i z j z j + - z id 4 = s y i j = s j + Ω i p 0 R Ω Ω i R +3 η i + Ω Ω i η - V- Step Lyapuov = - V - s i s i = - s i + - y i + yi + + - y i + 4 - s i s i + + y i + - K i Sig s i α - γ i s i + Δ ~ f i + - - y i + ( + η τ i + y i + i + ) - s i s i + + y i + - K i Sig s i α - βs i - s - - 3 - s i + - + - y i + τ i + + η i + y i + - - s i K i Sig s i α = - β - s i +
- s i s i + + y i + - s - 3 - s i + - + - - β - s - y i + τ i + + η i + y i + s i + - s i s i + + y i + - - 3 - s i + - + - ω - - - β - s i - β - - η- τ i + ω y i + y i + + - + - ω - - - η- τ i + ω - β - y i + σ - β( - s i + - y i + ) - - - η- τ i + ω - β - y i + σ - βv - l 0 44 σ = - ( + ω ) 45 > 0 46 = mi - η- τ i + ω - β - Step Lyapuov V = V - + s + ζt Qζ 47 Q G T Q + QG = d > - ki k > 0 槡 θ T t T s + ζ ω 36 37 - θ V = V- + s s + ζtqζ + Qζ ζt - V + s s - kζ T ζ + s + ζ T QH + s Jζ 48 s l s + 6l ζ T QH l ζ T ζ + QH 4l s Jζ l s + J ζ T ζ 49 V = V- + s s + ζtqζ + Qζ ζt Jζ - K Sig s α - γ + k s + σ - βv - + s Δf ~ - tah - kζ T ζ + s + ζ T QH σ - βv - - βs - s K Sig s α - k s - kζ T ζ + s + ζ T QH + s Jζ σ - βv - - βs - k s - kζ T ζ + l s + 6l + l ζ T ζ + QH + l 4l s + J ζ T ζ σ - βv - k - l - l s - ( k - l - J - β Q ) ζ T ζ + + QH 6l 4l k - l - l 0 k - l - J - β Q 0 σ = σ + ( + QH 6l 4l ) 50 5 V = - βv + σ 5 0 V θ + V t 0 - θ e -ct θ = σ β Π = s R s ω - 53 ω - s = z = e Vs Lyapuov V s = s i si = = s i 54
0 3 s s + - s i s i + + y i + - K i Sig s i α - γ i s i + Δ f ~ i s Jζ - K Sig s α - γ + k s + Δ ~ f + - s i s i + + y i + - K i Sig s i α - γ i s i + Δ f ~ i - s K Sig s α + s Jζ - k s - βs + - s i s i + + y i + - K i Sig s i α - s - 3 - s i - β - s i + - s K Sig s α - k - l s + J ζ T ζ + - s i s i + + y i + - K i Sig s i α - s - β - 3 - - β - s i - β s i + s i K i Sig s i α + - s i s i + + y i + - s i - s - 3 - s i K i Sig s i α s i - s - 3 - s i K i Sig s i α s i + - s i + + J ζ T ζ s i + s i + + y i + - + + J ζ T ζ + + - y i + + J ζ T ζ - s i K i Sig s i α + δ 55 δ = + - y i + + J ζ T ζ 56 μ = + α / < μ < K mi = mi K i ad K ~ V s = μ K mi Vs - K i s i +α - K mi + δ s +α / i + - K ~ s μ i + δ δ - K ~ V μ s + δ 57 V s s s = p > λ + δ /p μ = p Vs - λv μ s K ~ V s p V s 0 p t 0V s t p 8 37 T s K ~ - μ V s s 0 -μ 58 3 Vs - λv μ s 0 V s t s s 4 46 τ i + < η - ω 59 + β + ω 3 s + + s + y + y p 0 s + + s p p p 0 3 30 3 4 x = x + w + w + x = u - w + Δ x t y = x e = y - w x w = w w = - w 60 6 Δ x t 0. 60 6 3 π w = w π w = - w - w 6 σ w = - w w - Δ x t Δ x t = 0. si t τ w = σ w L sw σ w 59 w = τ ψτ w 63
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