Finite-time output regulation method for a class of uncertain nonlinear systems
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- Ἰεζάβελ Λαιμός
- 5 χρόνια πριν
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1 Electri c Machies ad Cotrol Vol. No. 0 Oct :,,, ;, : ; ; ; ; DOI: /j. emc TP 7 A X 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 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 : ( 05A ) ; ( YQ05087) ; ( , ) ; ( 05KTSCX059) ; ( C540) ; ( 04KQNCX08) : ( 98 ),,,, ; ( 980 ),,, ; ( 970 ),,,, : ( 963 ),,,, : ( 966 ),,,, :
2 Ω ρ i x x x i Δf i ρ i x x x i i 3 x R 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 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 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
3 0 τ w = σ w L Sw σ w L q - Sw σ w T Γ = 0 T ψ = 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 β z 3d z - 3 τ 3 z 3d τ 3 z3d + z 3d = z - 3 z 3d 0 = z 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 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 β z- d τ z d τ zd + z d = z - z d 0 = z 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 -
4 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 y = - y τ - z - = - y τ 37 + ~ f z + K z α s α s + s ρ 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 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 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 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 s i y i + τ i + + η i + y i s i K i Sig s i α = - β - s i +
5 - s i s i + + y i + - s s i β - s - y i + τ i + + η i + y i + s i + - s i s i + + y i 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 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
6 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 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 β - s i - β s i + s i K i Sig s i α + - s i s i + + y i + - s i - s s i K i Sig s i α s i - s 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 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 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
7 4 0 ψ = [ ] G = k = 3 Q = 5 [ ] - 0 J = 5 [ ] 5 H = 0 [ ] 4 [ ] T - = zd = - z τ d - z - = - τ z d + z + w - w + K Sig s α + γ s 64 z = x - w ξ = ξ - s 65 ξ = - ξ - k s s = x + w + w - σ d u u = - ξ + 5ξ - k Sig s α - 5s + zd 66 Lyapuov FTOR 7 ASOR u u = - k s - ξ + 5ξ - 5s + zd 67 k = 0 α = 0. 7 e u 7 Fig. Output regulatio error 5 Fig. Cotrol iput. J JIANG Zhogpig HUANG Jie. Stabilizatio ad Output Regulatio by Noliear Feedback a Brief Overview J. Acta Automatica Siica DING Z. Output regulatio of ucertai oliear systems with oliear exosystems J. IEEE Trasactios o Automatic Cotrol LU M HUANG J. A class of oliear iteral models for global robust output regulatio problem J. Iteratioal Joural of Robust ad Noliear Cotrol 04 /a - /a. 4. J ZHANG Xuxi CHENG Daizha. Robust output regulatio of a class of o-miimum phase oliear systems J. Joural of Syste Sciebce ad Mathematical Sciebce J CHEN Zuoxia JI Haibo. Output regulatio for a class of oliear systems with adaptive iteral model J. Cotrol ad Decisio
8 0 5 J WU Jubi SU Weizhou. Performace limits i output regulatio of a ucertai oliear system uder disturbaces J. Cotrol Theory & Applicatios method for a class of ucertai oliear systems J. Electric Machies ad Cotrol LI Shihua SUN Haibi YANG Ju YU Xighuo. Cotiuous fiite-time output regulatio for disturbed systems uder mismatchig coditio J. IEEE Trasactios o Automatic Cotrol BHAT S P BERNSTEIN D S. Cotiuous fiite-time stabilizatio of the traslatioal ad rotatioal double itegrators J. IEEE Trasactios o Automatic Cotrol WU S Radice G Gao Y Su Z. Quaterio-based fiite time cotrol for spacecraft attitude trackig J. Acta Astroautica J. 0 9 JI Haibo CHEN Zuoxia Robust adaptive output regulatio for oliear systems with dyamic ucertaities J. Joural of Uiversity of Sciece ad Techology of Chia 008 CHEN Hua WANG Chaoli YANG Fag XU Wei-dog. Fi ite-time saturated stabilizatio of oholoomic mobile robots based o visual servoig J. Cotrol Theory & Applicatios J DING Shihog LI Shihua LUO Sheg. Guidace law desig based o cotiuous fiite-time cotrol techique J. Joural of Astroautics J J LIU Huixia DING Shihog LI Shihua et al. Fiite-time cotrol MENG Guizhi MA Kemao. Dyamic surface output regulatio of PMSM positio servo system J. Electric Machies ad Cotrol HONG Yiguag XU Yagsheg HUANG Jie. Fiite-time cotrol for robot maipulators J. Systems & Cotrol Letters SU Yuxi SWEVERS Ja. Fiite-time trackig cotrol for robot maipulators with actuator saturatio J. Robotics ad Computer-Itegrated Maufacturig LIU Haitao ZHANG Tie. Adaptive eural etwork fiite-time cotrol for ucertai robotic maipulators J. Joural of Itelliget & Robotic Systems J BHAT S P BERNSTEIN D S. Fiite-time stability of homogeeous systems C / /Proceedigs of the America Cotrol Coferece ( : )
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