41 10 Vol. 41, No ACTA AUTOMATICA SINICA October, ,, (Least square support vector machines, LS-SVM)., LS-SVM,,,, ;,,, ;,. DOI,,,,,

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1 4 0 Vol. 4, No ACTA AUTOMATICA SINICA October, 05,, Least square support vector machnes, LS-SVM)., LS-SVM,,,, ;,,, ;,. DOI,,,,,.., 05, 40): /j.aas.05.c507 Data-based Approxmate Soluton for a Class of Affne Nonlnear Systems wth Partally Unknown Functons ZHANG Guo-Shan WANG Yan-Hao Abstract A new method named onlne unbased least square support vector machnes LS-SVMs) s proposed by usng onlne samplng data to fnd the approxmate soluton of a class of partally unknown affne nonlnear systems wthn the nfnte nterval. Frstly, we elmnate the bas of LS-SVMs by ntroducng a parameter to avod redundant computaton, meanwhle, we gve the data ponts whch are closer to the current moment more weghts by ntroducng a weght functon to the optmzed target, thus the computatonal accuracy s mproved. Secondly, the method of sldng tme wndow s employed to acheve the approxmate soluton for affne nonlnear systems wthn the nfnte nterval and meet the requrement of real-tme solvng. Fnally, smulatons of numercal examples demonstrate the effcency and superorty of the proposed method. Key words Affne nonlnear system, data-drven control, least square support vector machnes LS-SVM), sldng tme wndow, machne learnng Ctaton Zhang Guo-Shan, Wang Yan-Hao. Data-based approxmate soluton for a class of affne nonlnear systems wth partally unknown functons. Acta Automatca Snca, 05, 40): ,., [ ] [3] [4].. [5] [6], Taylor Runge-Kutta [7].,,.,. [8] Adoman Manuscrpt receved May, 05; accepted July 7, , 64730) Supported by Natonal Natural Scence Foundaton of Chna , 64730) Recommended by Assocate Edtor HOU Zhong-Sheng School of Electrcal Engneerng and Automaton, Tanjn Unversty, Tanjn 30007,,. [9],,,..,.,,, [0], []. [] Least square support vector machnes, LS-SVM),.,.,, [3 4],

2 [5],,,,.,,,.. LS-SVM) SVM),.,. LS-SVM ξ, ) SVM ξ, SVM,,. SVM LS-SVM, SVM,.. LS-SVM,.. LS-SVM. yx) = w T ϕx) + b ), w R h, b R, ϕ ) : R m R h, R m ) R h ). ϕ ),, [],,. {x, y }, =,,, N, x R m, y R, w, b, ).,, LS-SVM : mn Jw, ξ) = wt w + γ ξt ξ s. t. y = w T ϕx ) + b + ξ, =,,, N ), γ R +,, ) mn wt w) mn ξt ξ),., : Lw, b, ξ, α) = Jw, ξ) N [ ] 3) α w T ϕx ) + b + ξ y Karush-Kuhn-Tucker KKT) [6], : = 0 ξ = α ξ γ w = 0 w = N α ϕx ) = 0 N α = 0 b = 0 w T ϕx ) + b + ξ y = 0 α w, ξ : [ ] [ ] [ ] Ω + I N γ N α y = T N 0 b 0 4), N = [,,, ] T R N, α = [α, α,, α N ] T R N, y = [y, y,, y N ] T R N, I N. Ω R N N, j : Ω j = Ω, j) = Kx, y j ) = [ϕx )] T ϕx j ). yx) = N α Kx, x) + b 5), y,. Mercer [6], ϕ ) [7]., [8] : ϕu) T ϕv) = Ku, v), : m n n+m u n v m 6) [ϕ n) u)] T ϕ m) v) = m n [ϕu) T ϕv)] = m n [Ku, v)] = m+n Ku, v) u n v m 7)

3 0 : 747, 7). : 0 [Ku, v)] = ϕu)t ϕv)) u 0[Ku, v)] = ϕu)t ϕv)) v 0 [Ku, v)] = ϕu)t ϕv)) u. = ϕ u) T ϕv) = ϕu) T ϕ v) = ϕ u) T ϕv) b, 4). [5] λ, ω, b,,. : [ ϖ T = w T, b ] [ ] ϕx), φx) = 8) λ λ ) ) mn yx) = ϖ T φx) 9) Jϖ, e) = ϖt ϖ + et e s. t. y = ϖ T φx ) + e, =,,, N 0).3 [],.,,...3. [3 4],. : K N + K, N, {x, y } R m R, = K N + ), K N + ),, K.,. x K+, y K+ ), x K N+, y K N+ ). {x, y } R m R, = K N +), K N +3),, K +,, K +.,,,,,.. 4) Ω + I ) N α = y ) γ yx) = N α Kx, x) ), Ω R N N, Ω j = Kx, x j ) = φx ) T φx j ) = ϕx ) T ϕx j ) + λ. 7) : Ω m n x, y) = m+n Kx, y) n x m y : = [φ n) x)] T φ m) y) Ω 0 0x, y) = [φx)] T φy) = [ϕx)] T ϕy) + λ Ω 0x, y) = [φx)] T φ y) = [ϕx)] T ϕ y) Ω x, y) = [φ x)] T φ y) = [ϕ x)] T ϕ y) 3) Fg..3. Schematc of the sldng tme wndow.,,.., 0 40,, 0 Hz, 0. ;,

4 748 4 LS-SVM ;,.. : {ẋx = fx) + gx)ut) y = hx) 4), x = [x, x,, x m ] T R m, ut) R p. : fx), gx)ut). : f x) u t) fx) =., ut) =. f m x) u p t) g x) g p x) gx) =..... g m x) g mp x) ˆxt), gx)ut)., fx),. gx)ut) = gux, t) = gu.. LS-SVM K, x t [t nf, t sup ],, xt ) = [x t ), x t ),, x m t )] T, = K N + ), K N + ),, K, N., fx), gx)ut)., LS-SVM. ˆx l t) = w T l φt) + b l = w T l Φt) 5) [ ], φt) w T l = [w T l, b l /λ], Φt) =, l = λ,,, m. fx) gx)ut) : ϕx w T w T ) b m ˆfx) =. ϕx.... ) w T m w T. + b. = mm ϕx m ) b m [w T, b ] λm [wt m, b ] λm..... [w T m, bm ] λm [wt mm, bm [ϕx ), λ] T. = [ w T lk] m m [Ψx)] m, [ϕx m ), λ] T λm ] l =,,, m; k =,,, m 6) ĝx)ûut) = ĝûux, t) = ˆx ˆfˆx) 7) [t nf, t sup ], w l,. : mn N [ ˆx ˆfx) ] ) gux, t) t ) s. t. ˆxt ) = xt ) + et ) 8), N, ˆxt ), xt ), ˆfx), gux, t), et ).,. θ 0 < θ ), [3] h) = θ + ) θ, =,,, N 9) N, N., θ = 0.. ˆxt ) ˆfx) 5) 6), gux, t)., h = h K + N), K, N

5 0 : 749. : mn s. t. w T l w l + c l= k= l= w T lk w lk + ε =K N+ l= h e l) + =K N+ l= h ξl) + gu T l gu l + τ h e l) l= =K N+ l= w T l Φ t ) = w T lkψx k) + gu l x, t ) + e l k= w T l Φt ) = x l + ξ l f l x ) = w T lkψx k) + e l k= 0), ξ l, e l ˆx l t ) x l t ), ˆft ) f l t ) ), c, ε, τ. : L w l, w lk, e l, e l, ξl, αl, β l, γ l, gu l ) = w T l w l + c h e l) + l= k= l= w T lk w lk + ε =K N+ l= =K N+ l= h ξ l + gu T l gu l + τ h e l) l= =K N+ l= ) f l x ) e l w lk Ψx k) l= =K N+ l= =K N+ l= =K N+ γ l β l w T l Φt ) α l k= x l ξl) + αlgu l x, t )) ) w T l Φ t ) e l w lk Ψx k) k= ), α l, β l, γ l, = K N + ), K N +),, K; l =,,, m. KKT,, : = 0 e e l = α l l c h = 0 ξ ξl l = β l ε h = 0 e e l = γ l l τ h = 0 w l = αl + γl ) Ψx w k) l =K N+ = 0 gux, t ) = αl gu l = 0 w lk = αl + γl ) Ψx w k) lk =K N+ = 0 w T l Φt ) x l ξl = 0 β l γ l = 0 f l x ) = m w l kψx k) + e l k= = 0 w T αl l Φ t ) e l Ψx k) gu l x, t ) = 0, k= l, k =,,, m; = K N + ), K N + ),, K ) w l, w lk, e l, e l, ξl, : R H S α l 0 H T 0 γ l = f l 3) S 0 Q β l, l =,, m, α l = [αl, αl,, αl N ] T, γ l = [γl, γl,, γl N ] T, β l = [βl, βl,, βl N ] T, j = Ω t, t j ) + m Ω 0 k= 0 x k, xk) j, j = Ω 0 t, t j ),κ j = Ω 0 0t, t j ), Λ = m ) Ω 0 0 x k, x j k : R = k= x l + h c N..... N NN + h c Λ Λ N H =..... Λ N Λ NN N S =..... N NN

6 750 4 κ + κ N ε h Q =..... κ N κ NN + ε h Λ + Λ N τ h T =..... Λ N Λ NN + τ h, f l = [f l x ), f l x ),, f l x N )] T, x l = [x l, x l,, x N l ] T., : ˆx l t) = =K N+ α l Ω 0 t, t) + β l Ω 0 0 t, t) ), l =,, m 4), Ω. fx) gx)ut) : ˆf l x) = ĝû l x, t) = k= =K N+ k= =K N+ =K N+ ) ) α l + γ l Ω0 0 x k, x k α l Ω t, t) + β l Ω 0 t, t) ) + ) ) α l + γ l Ω0 0 x k, x k, l =,,, m 5) Radal bass functon, RBF),, : Kx, y) = exp ). x y) LS-SVM σ c, τ, ε σ., [9]. K +, K N +, K +, LS-SVM,.,,..3 : ) ; ) LS-SVM ; 3) ; 4) ), ; 5)., ;,, ; 6) 4)..4,,.,,,., LS-SVM., E,,. E,, ;,.,,.. Fg. 3 Flow chart of the algorthm,., 60, 3 8., E 0 4., 0 Hz,.

7 0 : 75, fx), gx)ut).. MSE = N xt ) ˆxt )) N 6), N, xt ), ˆxt ). [] : ) LS-SVM,, ; ),, ; 3),. [],, 3.. [, 0] : 5 6. [] 3. f t), gu x, t) f t), gu x, t) Fg. 3 3 Comparson of the system s approxmate soluton and exact soluton ẋt) = xt) x t) + xt) = + tanh )) t + log +, fx) = x, gx)ut) = x +. xt) 3,,. []. gx)ut) fx) ,,.. [, ]. ẋxt) = fx) + gx)ut) ) x + x, fx) = x x ) cost) sn t) t gx)ut) = t snt) t snt) x0) = [0 ] T. x t) = snt), x t) = + t. x t), x t) Fg. 4 Table 4 ft), gux, t) Comparson of ft), gux, t) approxmate soluton and exact soluton t [0, ] xt) MSE of xt) n terms of dfferent numbers of tranng ponts N) [] 0.69E 07.5E E 4.4E E E E E

8 75 4 Table t [0, 3] x t) MSE of x t) n terms of dfferent numbers of tranng ponts N) [] 0.057E 08.84E E E E.49E E E 4 3 Table 3 t [0, 3] x t) MSE of x t) n terms of dfferent numbers of tranng ponts 7 f t), gu x, t) Fg. 7 Comparson of f t), gu x, t) approxmate soluton and exact soluton N) [] E 3 5.9E E E E E E E 4 8 f t), gu x, t) Fg. 8 Comparson of f t), gu x, t) approxmate soluton and exact soluton 5 x t) E t) Fg. 5 Comparson of x t) approxmate soluton, exact soluton and error E t)., ẋx = fx, t) ẋx = fx, t) + gx, t)u,. 4 6 x t) E t) Fg. 6 Comparson of x t) approxmate soluton, exact soluton and error E t),,. 4 : ),,,.,,,. ),

Decentralzed nonlnear optmal predctve exctaton control for mult-machne power systems. Internatonal Journal of Electrcal Power & Energy Systems, 04, 55: 60 67 3 Q G Y, Chen Z Q, Yuan Z Z.

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2 : 237.,. [6 7] (Markov chan Monte Carlo, MCMC). MCMC, [8 9].,,, [0 ].,, : ),,,.,, ; 2),,.,.,. : ),.,,. ; 2),.,,. ; 3), EM, EM,.,, EM, EM. K M,.,. A

2 : 237.,. [6 7] (Markov chan Monte Carlo, MCMC). MCMC, [8 9].,,, [0 ].,, : ),,,.,, ; 2),,.,.,. : ),.,,. ; 2),.,,. ; 3), EM, EM,.,, EM, EM. K M,.,. A 38 2 Vol. 38, No. 2 202 2 ACTA AUTOMATICA SINICA February, 202.,.,, EM,.. DOI,,, 0.3724/SP.J.004.202.00236 Data Assocaton n Vsual Sensor Networks Based on Hgh-order Spato-temporal Model WAN Ju-Qng LIU