Research Article The Effect of Initial State Error for Nonlinear Systems with Delay via Iterative Learning Control

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1 Hindawi Publishing Corporaion Advances in Mahemaical Physics Volume 216, Aricle ID , 6 pages hp://dx.doi.org/1.1155/216/ Research Aricle The Effec of Iniial Sae Error for Nonlinear Sysems wih Delay via Ieraive Learning Conrol Zhang Qunli Deparmen of Mahemaics, Heze Universiy, Heze, Shandong 27415, China Correspondence should be addressed o Zhang Qunli; qunli-zhang@126.com Received 6 February 216; Acceped 23 March 216 Academic Edior: Ming Mei Copyrigh 216 Zhang Qunli. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. An ieraive learning conrol problem for nonlinear sysems wih delays is sudied in deail in his paper. By inroducing he λ-norm and being inspired by rearded Gronwall-like inequaliy, he novel sufficien condiions for robus convergence of he racking error, whose iniial saes are no zero, wih ime delays are obained. Finally, simulaion example is given o illusrae he effeciveness of he proposed mehod. 1. Inroducion Since ieraive learning conrol is proposed by Arimoo e al. in 1984 (see [1], his feedforward conrol approach has become a major research area in recen years. Ieraive learning conrol, which belongs o he inelligen conrol mehodology, is an approach for fully uilizing he previous conrol informaion and improving he ransien performance of sudied sysems ha is suiable for repeiive movemens. Is goal is o achieve full range of racking asks on finie inerval (see [2 14]. In pracice, he conrol problem of sysems wih delays has always been an ineresing research, since ime delay can be ofen encounered in a wide range, such as aircraf sysems, urboje engines, microwave oscillaors, nuclear reacors, and chemical processes. The exisence of ime delay in a sysem may degrade he conrol performance and even a wors may become a source of insabiliy. Sabilizaion problem of conrol sysems wih delay has received much aenion for several decades and some research resuls have been repored in he lieraure (see [3, 11, 13, 15 23]. However, only a few resuls are available for nonlinear sysems, combining wih he ieraive learning conrol iems, wih ime delays [11, 24 26]. In his paper, under he case ha he kh ieraive sae vecor x k ( is differen from he (k+1h ieraive sae vecor x k+1 (, hais,x k+1 ( x k ( =,heieraivelearning conroller of nonlinear ime-delayed sysems is designed by using λ-norm and rearded Gronwall-like inequaliy. Before ending his secion, i is worh poining ou he main conribuions of his paper as follows. (1 The ieraive learning conrol problem for nonlinear sysems wih delays is invesigaed. Tha is, we consider he sysem k ( =f(, x k ( +g(, x k ( τ +u k (, (1 which is differen from he menioned sysem k ( =f(, x k ( +u k ( (2 in pas lieraure. (2 BasedonreardedGronwall-likeinequaliyandhe convergence of racking error e k ( = y d ( y k (, he pracical oupu y k ( is deerminae by he previous ieraive learning conrol informaion x k (, u k (, φ k,h (. 2. Preliminaries Throughou his paper, he 2-norm for he n-dimensional vecor x=(x 1,x 2,...,x n T is defined as x = ( n i=1 x2 i 1/2, while he λ-norm for a funcion is defined as λ = sup [,T] {e λ },wherehesuperscript represens he ranspose and λ>. I and represen he ideniy marix and a zero marix, respecively.

2 2 Advances in Mahemaical Physics Lemma 1 (see [1, 27]. Consider sup [,T] (e λ x (τ dτ 1 λ x ( λ. (3 Lemma 2 (see [28] rearded Gronwall-like inequaliy. Consider such an inequaliy u ( a( + and suppose ha n i=1 b i ( f i (, s w i (u (s ds, b i ( < 1, (4 (1 all w i (i= 1,2,...,nare coninuous and nondecreasing funcions on [, + and are posiive on (, + such ha w 1 w 2 w n ; (2 a( is coninuously differeniable in and nonnegaive on [, 1,where, 1 are consans and < 1 ; (3 all b i : [, 1 [, 1 (i= 1,2,...,n are coninuously differeniable and nondecreasing such ha b i ( on [, 1 ; (4 all f i (, s, i= 1,2,...,n, are coninuous and nonnegaive funcions on [, 1 [, 1. Take he noaion W i (s, s fl s (dz/w s i (z for s>,where s >is a given consan. I is denoed by W i (s simply when here is no confusion. If u( is a coninuous and nonnegaive funcion on [,saisfying (4, hen u ( W 1 n b n ( [W n (r n (+ b n ( where r n ( is deermined recursively by r 1 ( fl a( + a (s ds, r i+1 ( fl W 1 i b n ( [W i (r i (+ b n ( T< 1 and T is he larges number such ha b n (T W i (r i (T+ b n ( max f n (τ, s ds τ<t max f n (τ, s ds], τ< T, max f n (τ, s ds], τ< i=1,2,...,n 1; + dz w i (z, i=1,2,...,n. s (5 (6 (7 3. Main Resuls Consider he following sysem wih ime delay: k ( =f(,x k (+g(,x k ( τ+u k (, u k+1 ( =u k ( +Me k ( ψ k,h ( (x k+1 ( x k (, y k ( =Cx k ( +Du k ( φ k,h (, φ k+1,h ( =φ k,h ( Dψ k,h ( (x k+1 ( x k (, where x k ( R n, y k ( R n,andu k ( R n are he sae vecor, oupu vecor, and inpu vecor, respecively. k is he number of ieraions, k {1,2,3,...} and [,T]. ψ k,h(θdθ = 1, ( h, M, C, D are real consan marices; e k ( = y d ( y k (, y d ( is a reference oupu. Supposehahereexisheboundedconsansl f >and l g >such ha f(,x k+1 ( f(,x k ( l f x k+1 ( x k (, g(,x k+1 ( τ g(,x k ( τ l g x k+1 ( τ x k ( τ. Theorem 3. For sysem (8 and a given reference y d (,ifhere exis marices M, C, D and funcions ψ k,h ( and φ k,h ( such ha (8 (9 I DM ρ<1, (1 where ρ is a consan, hen sysem (8 wih he ieraive learning conrol law can guaranee ha y k ( y d ( is bounded bu y k ( canno rack y d ( on [,h]and lim k y k ( = y d ( on [h,t]for arbirary iniial sae x k (. Proof. I is easy o know ha, for any [h,t], x k+1 ( x k ( = ( k+1 (θ k (θdθ+x k+1 ( x k ( = (f (θ, x k+1 (θ f (θ, x k (θ dθ + (g (θ, x k+1 (θ τ g (θ, x k (θ τ dθ + (u k+1 (θ u k (θdθ+x k+1 ( x k ( = (f (θ, x k+1 (θ f (θ, x k (θ dθ + (g (θ, x k+1 (θ τ g (θ, x k (θ τ dθ + Me k (θ dθ

3 Advances in Mahemaical Physics 3 (x k+1 ( x k (( ψ k,h (θ dθ 1 = (f (θ, x k+1 (θ f(θ,x k (θ dθ + (g (θ, x k+1 (θ τ g (θ, x k (θ τ dθ + Me k (θ dθ. So we obain from condiion (9 x k+1 ( x k ( l f x k+1 (θ x k (θ τ +l g x k+1 (θ x k (θ τ + M e k (θ. (11 (12 In his paper, we use Lemma 2. Taking =, b 1 ( =, b 2 ( = τ, a( = M e k(θ dθ, W 1 (s = l f s (dz/z = 1 l f lns, andw 2 (s = l g s (dz/z = l 1 g lns, henr 1 ( = M e k(θ dθ and r 2 ( = exp[ln( M e k(θ dθ + (l g/l f dθ] = e (l g/l f M e k(θ dθ.sowehave x k+1 ( x k ( exp[ln(e (l g/l f M e k(θ dθ + τ dθ] = τ e (1+l g/l f M e k(θ dθ: e k+1 ( =e k ( +y k ( y k+1 ( =e k ( +Cx k ( +Du k ( φ k,h ( Cx k+1 ( Du k+1 ( +φ k+1,h ( =e k ( C(x k+1 ( x k ( D(u k+1 ( u k ( +(φ k+1,h ( φ k,h ( =e k ( C(x k+1 ( x k ( DMe k ( +Dψ k,h ( (x k+1 ( x k ( +(φ k+1,h ( φ k,h ( = (I DM e k ( C(x k+1 ( x k (, e k+1 ( I DM e k ( + C x k+1 ( x k ( I DM e k ( + C e(1+l g/l f T M e k (θ. (13 (14 Using Lemma 1 and muliplying boh sides of he above inequaliy (14 by e λ and aking he λ-norm, we have e k+1 ( λ I DM e k ( λ + C M e(1+l g/l f T λ e k ( λ =( I DM + C M e(1+l g/l f T λ e k (. λ (15 Thus, condiion (1 can guaranee ( I DM + C M e (1+l g/l T f /λ < 1 by selecing λ sufficienly large, so we have lim k e k ( λ =for any [h,t].ifollowsfrom he equivalence of norms; we ge ha lim k e k ( =. For any [,h], x k+1 ( x k ( = (f (θ, x k+1 (θ f(θ,x k (θ dθ + (g (θ, x k+1 (θ τ g (θ, x k (θ τ dθ + Me k (θ dθ (x k+1 ( x k (( ψ k,h (θ dθ 1, x k+1 ( x k ( l f x k+1 (θ x k (θ τ +l g x k+1 (θ x k (θ τ + M e k (θ + x k+1 ( x k ( ψ k,h (θ dθ 1 l f x k+1 (θ x k (θ τ +l g x k+1 (θ x k (θ τ + M e k (θ + η, where η=sup [,h] ( x k+1 ( x k ( ψ k,h(θdθ 1. I is easy o know ha by using Lemma 2. (16 x k+1 ( x k ( e (1+l g/l f ( M e k (θ + η (17

4 4 Advances in Mahemaical Physics Then e k+1 ( I DM e k ( + C x k+1 ( x k ( I DM e k ( + C e (1+l g/l T f ( M e k (θ + C e (1+l g/l T f η. (18 Using Lemma 1 and muliplying boh sides of he above inequaliy by e λ and aking he λ-norm, we have e k+1 ( λ ( I DM + C M e(1+l g/l f T λ e k ( +η λ =σ e k ( λ +η, where σ = I DM + C M e (1+l g/l f T /λ: (19 e k+1 ( + η λ σ 1 σ( e k ( + η λ. (2 σ 1 Imiaing he above proof, he resul lim k e k ( λ =for any [h,t]is obained by selecing λ sufficienly large. So i is rue ha e k ( is bounded on [,h]. Remark 4. When he number of ieraions k and [h, T] [, T], he racking error saisfies ha e k ( on [,T]for arbirary iniial sae x k (. Sysem (8 is k ( =f(,x k (+g(,x k ( τ(+u k (, u k+1 ( =u k ( +M( e k ( ψ k,h ( (x k+1 ( x k (, y k ( =C( x k ( +D( u k ( φ k,h (, φ k+1,h ( =φ k,h ( D( ψ k,h ( (x k+1 ( x k (, (21 where he delay τ( saisfies < τ( γ < 1. SimilaroheproofofTheorem3,inequaliy(14is wrien as e k+1 ( I D( M ( e k ( + C ( M ( e (1+(1 γl g/l T f e k (θ, e k+1 ( λ ( I D( M ( + C ( M( e(1+(1 γl g/l f T λ e k (. λ Thenwehavehefollowing. (22 Theorem 5. For sysem (21 and a given reference y d (, if here exis marices M(, C(,andD( and funcions ψ k,h ( and φ k,h ( such ha C( M( is bounded and I D(M( ρ < 1, where ρ is a consan, hen sysem (21 wih he ieraive learning conrol law can guaranee ha y k ( y d ( is bounded bu y k ( canno rack y d ( on [,h]and lim k y k ( = y d ( on [h,t]for arbirary iniial sae x k (. When y k ( = s(, x k (, u k ( φ k,h ( and s(, x k (, u k ( saisfies <δ 1 I<s u = s (x k (,u k ( u k ( <δ 3 I<s x = s (x k (,u k ( x k ( i is easy o obain ha δ 2 I, δ 4 I, s(,x k+1 (,u k+1 ( s(,x k (,u k ( =s x (ζ (x k+1 ( x k ( +s u (ζ (u k+1 ( u k (, (23 (24 where ζ [x k (+ω(x k+1 ( x k (, u k (+ω(u k+1 ( u k (], ω (, 1. Consider he following sysem: k ( =f(,x k (+g(,x k ( τ+u k (, u k+1 ( =u k ( +Me k ( ψ k,h ( (x k+1 ( x k (, y k ( =s(,x k (,u k ( φ k,h (, φ k+1,h ( =φ k,h ( s u ψ k,h ( (x k+1 ( x k (. SimilaroheproofofTheorem3,wege e k+1 ( λ (25 ( I s um + s x M e(1+l g/l f T (26 λ e k (. λ So we have he following resul. Theorem 6. For sysem (25 and a given reference y d (, if condiions (9 are rue and here exis marix M and funcions ψ k,h ( and φ k,h ( such ha ψ k,h(θdθ = 1, >h, s x is bounded, max( I δ 1 M, I δ 2 M ρ < 1, whereρ is a consan, hen sysem (25 wih he ieraive learning conrol law can guaranee ha lim k y k ( = y d ( is bounded bu y k ( canno rack y d ( on [,h]and lim k y k ( = y d ( on [h,t]for arbirary iniial sae x k (.

5 Advances in Mahemaical Physics 5 4. Numerical Example For furher illusraion, we consider he following sysem: k ( =.8 cos 2 (x k (.5 ( x k ( 1 +1 x k ( 1 1 +u k (, u k+1 ( =u k ( +.9e k ( ψ k,h ( (x k+1 ( x k (, y k ( = anh (2x k ( +.9u k ( φ k,h (, φ k+1 ( =φ k (.9ψ k,h ( (x k+1 ( x k (, (27 where M =.9 and ψ k,h ( = { π { 2.5 cos ( π, [,.5, 2.5 (28 {, [.5, 5], aking he reference y d ( = sin +1. From he above numerical example, i can be easily proved ha he condiions of Theorem 6 are saisfied. 5. Conclusion In his paper, considering he ieraive learning conrol problem for nonlinear sysems wih delays, he novel sufficien condiions for robus convergence of he racking error have been addressed. Compeing Ineress The auhor declares ha here are no compeing ineress. Acknowledgmens ThisworkissupporedbyheNaionalNauralScienceFoundaion of China ( , he Naural Science Foundaion of Shandong Province of China (ZR211AL18, ZR211AQ8, and a Projec of Shandong Province Higher Educaional Science and Technology Program (J13LI2, J11LA6. References [1] S. Arimoo, S. Kawamura, and F. Miyazaki, Beering operaion of Robos by learning, Roboic Sysems, vol.1,no.2, pp , [2] H.-S. Lee and Z. Bien, A noe on convergence propery of ieraive learning conroller wih respec o sup norm, Auomaica,vol.33,no.8,pp ,1997. [3] Y. Chen, Z. Gong, and C. Wen, Analysis of a high-order ieraive learning conrol algorihm for uncerain nonlinear sysems wih sae delays, Auomaica, vol.34,no.3,pp , [4] J.-X. Xu and Y. Tan, A composie energy funcion-based learning conrol approach for nonlinear sysems wih imevarying parameric uncerainies, IEEE Transacions on Auomaic Conrol, vol. 47, no. 11, pp , 22. [5] A. Tayebi, Adapive ieraive learning conrol for robo manipulaors, Auomaica,vol.4,no.7,pp ,24. [6]R.H.Chi,Z.S.Hou,andJ.X.Xu, AdapiveILCforaclass of discree-ime sysems wih ieraion-varying rajecory and random iniial condiion, Auomaica,vol.44,no.8,pp , 28. [7] C.-J. Chien, A combined adapive law for fuzzy ieraive learning conrol of nonlinear sysems wih varying conrol asks, IEEE Transacions on Fuzzy Sysems,vol.16,no.1,pp.4 51, 28. [8] Y.-C. Wang and C.-J. Chien, Decenralized adapive fuzzy neural ieraive learning conrol for nonaffine nonlinear inerconneced sysems, Asian Conrol, vol. 13, no. 1, pp , 211. [9]J.-X.Xu,D.Huang,V.Venkaaramanan,andT.C.T.Huynh, Exreme precise moion racking of piezoelecric posiioning sage using sampled-daa ieraive learning conrol, IEEE Transacions on Conrol Sysems Technology,vol.21,no.4,pp , 213. [1] Q. Lv, Y.-C. Fang, and X. Ren, Ieraive learning conrol for acceleraed inhibiion effec of iniial sae random error, Aca Auomaica Sinica,vol.4,no.7,pp ,214. [11] J.-M. Wei, Y.-A. Hu, and M.-M. Sun, Observer-based adapive ieraive learning conrol for a class of nonlinear ime delay sysems wih inpu sauraion, Mahemaical Problems in Engineering, vol. 215, Aricle ID , 19 pages, 215. [12] J. Li and J. Li, Ieraive learning conrol approach for a kind of heerogeneous muli-agen sysems wih disribued iniial sae learning, Applied Mahemaics and Compuaion, vol. 265, pp , 215. [13] L. Yan and J. Wei, Fracional order nonlinear sysems wih delay in ieraive learning conrol, Applied Mahemaics and Compuaion,vol.257,pp ,215. [14] H. Cai, Y. Huang, J. Du, T. Tang, D. Zuo, and J. Li, Ieraive learning conrol wih exended sae observer for elescope sysem, Mahemaical Problems in Engineering, vol. 215, Aricle ID 7151, 8 pages, 215. [15]S.S.Ge,F.Hong,andT.H.Lee, Robusadapiveconrolof nonlinear sysems wih unknown ime delays, Auomaica, vol. 41,no.7,pp ,25. [16] C. C. Hua, G. Feng, and X. P. Guan, Robus conroller design of a class of nonlinear ime delay sysems via backsepping mehod, Auomaica,vol.44, no.2,pp , 28. [17] X. D. Ye, Adapive sabilizaion of ime-delay feedforward nonlinear sysems, Auomaica, vol. 47, no. 5, pp , 211. [18] J. Na, Adapive prescribed performance conrol of nonlinear sysems wih unknown dead zone, Inernaional Adapive Conrol and Signal Processing, vol.27,no.5,pp , 213. [19] Z.-Y. Sun and Y.-G. Liu, Adapive conrol design for a class of uncerain high-order nonlinear sysems wih ime delay, Asian Conrol,vol.17,no.2,pp ,215. [2] Q.-L. Zhang, Synchronizaion of muli-chaoic sysems via ring impulsive conrol, Conrol Theory & Applicaions, vol. 27, no. 2, pp , 21. [21] Q. Zhang, Synchronizaion of muli-chaoic sysems wih ring and chain inermien connecions, Applied Mechanics and Maerials,vol ,pp ,213. [22]Q.Zhang, AclassofvecorLyapunovfuncionsforsabiliy analysis of nonlinear impulsive differenial sysems, Mahemaical Problems in Engineering, vol. 214, Aricle ID 64912, 9 pages, 214.

6 6 Advances in Mahemaical Physics [23] Q. Zhang, Marix measure wih applicaion in quanized synchronizaion analysis of complex neworks wih delayed ime via he general inermien conrol, Annals of Applied Mahemaics,vol.31,no.1,pp ,215. [24] W. S. Chen and L. Zhang, Adapive Ieraive learning conrol for nonlinearly parameerized sysems wih unknown imevarying delays, Inernaional Conrol, Auomaion and Sysems,vol.8,no.2,pp ,21. [25] J.-M. Wei, Y.-A. Hu, and M.-M. Sun, A new AILC for a class of nonlinearly parameerized sysems wih unknown delays and inpu dead-zone, JournalofAppliedMahemaics, vol. 214, Aricle ID 23818, 14 pages, 214. [26] J.-M. Wei, Y.-A. Hu, and M.-M. Sun, Adapive ieraive learning conrol for a class of nonlinear ime-varying sysems wih unknown delays and inpu dead-zone, IEEE/CAA Auomaica Sinica,vol.1,no.3,pp ,214. [27] M.-X. Sun and B.-J. Huang, Ieraive Learning Conrol, Naional Defence Indusry Press, Beijing, China, [28] R. P. Agarwal, S. Deng, and W. Zhang, Generalizaion of a rearded Gronwall-like inequaliy and is applicaions, Applied Mahemaics and Compuaion, vol.165,no.3,pp , 25.

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