Li Nana,Jiang Wei

(School of Mathematical Science,Anhui University,Hefei230601,China)

Convergence analysis of fractional iterative learning controlnonlinear systems with multiple state delays

Li Nana,Jiang Wei

(School of Mathematical Science,Anhui University,Hefei230601,China)

This paper presents fractional-order iterative learning control(FOILC) nonlinear system with multiple state delays.It involves external disturbances and output measurement noises which obtain some new and interesting criteria to guarantee the convergence of the tracking error in the sense of the λ-norm.The convergence of the system outputs to the desired trajectory is ensured in the absence of disturbances and output measurement noises.We estimate the upper bound of the tracking error.Finally,the validity of the proposed method are veri fi ed by an example.

fractional-order iterative learning control(FOILC),convergence, multiple time delays

1 Introduction

The formal concept of iterative learning control(ILC)was published in 1978 by Uchiyama (in Japanese)and in 1984 by Arimoto et al.Iterative learning control(ILC),which belongs to the intelligent control methodology,is an approach for improving the transient performanceof systems that operate repetitively over a fi xed time interval[1-2].In recent years,the adaptiveness and robustness of linear or nonlinear ILC schemes become very popular topics,which extended the applications of ILC to more complicated problems[4,6-7,9].Some other interesting conclusions and surveys can be found in[8,11].Moreover,in the past three years,the applications of the ILC technique to medical treatments and engineering are getting more and more popular[12-13].

優(yōu)質的營商環(huán)境往往與旺盛的市場活力、富于創(chuàng)新的市場氛圍相關聯(lián)。大連營商環(huán)境短板問題主要表現(xiàn)在市場活力不足，激發(fā)市場活力不能靠人為干預，而在于構建健全的市場機制，讓 “創(chuàng)新什么、怎么創(chuàng)新、生產(chǎn)什么、怎么銷售”等決策真正“市場說了算”，最大程度減少行政干預和門檻限制。在眾多市場作用機制中，市場導向的創(chuàng)新試錯機制最為關鍵，它決定了創(chuàng)新的生成模式及市場活動，即依靠大眾創(chuàng)新，并一視同仁提供公平市場競爭環(huán)境?，F(xiàn)在的問題是過度依賴行政手段、行政指令、行政化價值觀來影響市場的創(chuàng)新活動，不能按照市場真實需求做出正確反應和科學決策。主要有三大表現(xiàn)：

The FOILC is a relatively new topic in ILC,many fractional-order ILC problems were presented aiming at enhancing the performance of ILC scheme for linear or nonlinear systems [18,20].Allow for the application of fractional-order ILC scheme to the control of complex systems and the cancelation of the dependence of ILC scheme to the system orders,time delays and uncertainties are inevitable in most industrial processes.There are few research results on iterative learning control for systems with time-delay[9-10]as well as for systems with disturbance[5-6].Also robustness of the learning control is established in the presence of initial function error.However,there is little consideration on the FOILC design of nonlinear multiple time-delays system and its convergence property.

不談收入，只談賓利全新歐陸GT，那自然是件再愜意不過的事了。而如果將全新歐陸GT的車頂去掉，相信所收獲的又會是另一片天地。我在全新歐陸GT敞篷版在上海的全球同步首發(fā)儀式現(xiàn)場目睹了這輛“超豪華運動旅行敞篷座駕”亮相，并試著在充斥著賓利車主的人群中，去端詳它的與眾不同。

關小美極力解釋：“他在北京有不錯的前途！可他寧愿放棄北京的高薪來到這個小城鎮(zhèn)，這樣有情有義的男人到哪里去找？”關云飛反駁說：“這正說明他做事太沖動，很不成熟。將來萬一你和他處不好，你會吃大虧的……”關小美的母親也說：“他說他是總經(jīng)理他就是了啊？你也不動動腦子想想？你們認識才幾天??？”

Consider a class of fractional nonlinear multiple time-delays systems with disturbance as follows:

where k denotes the kth repetitive operation of the system and t is the time in the operation interval[0,T];xk(t)∈Rn,uk(t)∈Rr,and yk∈Rmare the states,control input and output of the system,respectively;wk(t)∈Rn,vk(t)∈Rmare bounded external disturbance and measurement noise;τ1,…,τm1,η1,…,ηm2are positive constant time delays;?d(t)is the initial piecewise continuous function of the system;Functions

[3]Sun M X,Wang D.Sampled-data iterative learning control for nonlinear systems with arbitrary relative degree[J].Automatica,2001,37:283-289.

are piecewise continuous in t;g:Rn×[0,t]?→Rmis di ff erential for all x and t with partial derivatives gxand gt.

The problem on designing an ILC for uncertain plants with time delays has not been fully investigated,and only a limited number of the results are available so far.In[14],theauthor has discussed the general systems with multiple time-delays.In this paper,I take this results to FOILC.We obtain some sufficient conditions to guarantee the system outputs,states and system input to converge to desired trajectories with bounded tracking errors.When the external disturbances and measurement noises decay to zero,the bounds of the tracking errors also decay to zero.It is shown that the multiple time delays in state variables do not a ff ect the ILC convergence property signi fi cantly.

(3) 工程處治方案。由于橋隧相連部位的山體高陡，采用坡面主動防護工程規(guī)模大，且施工相當困難。故結合隧道口河流高階地較為寬闊的有利條件，在不影響河流行洪的條件下，決定對崩塌落石影響區(qū)的橋梁調整為路基與隧道相接。這樣避免了橋梁一旦受到危巖破壞而很難修復的缺點，充分發(fā)揮了路基抗災能力相對較強的特點，并在隧道口延伸明洞，明洞上部設置必要的傾向河側的緩沖層，進一步提高線路的抗災能力，達到對崩塌危巖的有效處治。

2 Preliminaries

De fi nition 2.1The fractional integral of order γ with lower limit zero for a function f∈L1([0,∞))?→R can be written as

In this paper,we denote l∞norm by∥·∥,λ?norm by∥·∥λ.The following de fi nitions on norms are used in the following discussions.

De fi nition 2.2The Riemann-Liouville derivative of order γ with the lower limit zero for a function f:[0,∞)?→R can be written as

本文所有采集數(shù)據(jù)均來源于HRO觀測站，大口徑天線系統(tǒng)在B1頻段增益約為78dB，采集時長為10s，數(shù)據(jù)比特位數(shù)為14，軟件接收機碼環(huán)階數(shù)為2，碼環(huán)相關間隔設置為0.2碼片.

De fi nition 2.3The Caputo derivative of order γ for a function f:[0,∞)?→R can be written as

Lemma 3.1Let dibe a sequence of real number which converges to the limit d∞as i?→∞,suppose that aiis a sequence of real number such that

Lemma 2.4[19]Suppose β＞0,m(t)is a nonnegative function locally integrable on J and n(t)is a nonnegative,nondecreasing continuous function de fi ned on n(t)≤M,t∈J,and suppose y(t)is nonnegative and locally integrable on J with

Then

Remark 2.5Under the hypothesis of Lemma 2.4,let m(t)be a nondecreasing function on J.Then we have

3 System respresentation and assumptions

Given a desired output bounded trajectory yd(t)and an appropriate initial condition xd(t),t∈[?m,0],there exists a unique control input ud(t),such that when uk(t)=ud(t),the system has a unique bounded desired state xd(t)satisfying:

It is well known that if f and h satisfy uniformly Lipschitz conditions respectively then one can use standard methods to derive that the problem(2)has a unique solution x∈PC(J,R)(See[1])which is given by integral equation:

1．教材體系的新變化。(1)凸顯教材的多功能性。由于化學課程與教學論形成了一些分支并獨立成書后如果不開設選修課，就弱化了本教材的功能，造成學生認知機構的不完整。為此，把獨立出去的分支教材《化學教學論實驗》《化學微格教學》《化學教學測量與評價》《化學學習論》等與《化學課程與教學論》再回歸為一本，發(fā)揮教材的多功能性。(2)教材體例的新變化。在每一章都設置了一些欄目。如:每一章前面用“思維導圖”進行引導，使學生瞬間可瀏覽到一個清新直觀的網(wǎng)絡圖，概覽整章的知識脈絡，增加對教材知識的概括性理解。

In this paper,we introduce the following assumptions:

Assumption 1The functions f,h,and gxare uniformly bounded.In the sequel,we use bf,bhand bgxto denote the upper bounds for f,h and gx.Furthermore,the input-output coupling matrix gxh is full column rank.

Assumption 2The system disturbance wk(t)and output measurement noise vk(t)are bounded by positive constants bwand bv1,respectively,on[0,T](i.e.,∥wk(t)∥≤bw,∥vk(t)∥≤bv1),andcDαtνk(t)is also bounded by positive constant bv2.

老鱖魚一看刀子掉地上了，撲上又和我廝打，他一臉菜色，營養(yǎng)不良，根本沒啥力氣，我沒費大勁，就一腳把他踢翻。他爬起來，還要跟我打，我又一腳把他踢倒。

Assumption 3The function f,g and gx,gtand h are uniformly globally Lipschitz in x on[0,T].g is di ff erentiable for all x and t with partial derivatives gxand gt.There exist positive constants kf;kh;kg;kgx;kgtsuch that:

Our objective is to design an iterative scheme to generate the control input uk(t)such that the system output yk(t)converges to yd(t)and the control input uk(t)converges to ud(t)as k goes to in fi nity for all t within the time interval[0,T].

Remark 1Compared to the[14],this paper expended iterative learning control to the fractional nonlinear system with multiple time-delay.It veri fi ed that the iterative learning control and delays are irrelevant.

where

denotes the tracking error at kth repetition;P:Rm×[0,T]?→Rr×mand Γ:Rm×[0,T]?→Rr×mare learning gain matrices with norm bounds bpand bc,respectively.The control input uk(t)is piecewise continuous for all k on t∈[0,T].

For the sake of brevity,the following notations will be used:

We introduce the following lemma which will be used in the following section.

試驗采用隨機區(qū)組設計，3次重復，小區(qū)面積50 m2。栽培密度為133 395株/hm2，株行距均為25 cm×30 cm。每小區(qū)隨機抽取10株進行株高、葉片數(shù)、分蘗數(shù)的測量統(tǒng)計。測產(chǎn)時采用三點法，每小區(qū)隨機抽取3個點，每個點選取1 m2進行測產(chǎn)。試驗具體設置如表1所示。

Then

魏金枝：《柔石傳略》，丁景唐、瞿光熙編《左聯(lián)五烈士研究資料編目》，上海文藝出版1981版，第218頁。

provided the right side is point-wise de fi ned on[0,∞),where Γ(·)is the gamma function.

where

is a vector,

is a matrix and h(t)(t∈[0,T])is a real function.

4 Main results

(c1)∥I?Γ(,)gxk(,)h(·,·,…,·)∥≤ρ＜1,and the initial condition.

Theorem 4.1Consider the fractional nonlinear multiple time-delays system(1)with ILC updating law(3)and satisfying the assumptions(1),(2),and(3).If

2.1 三組行上腹部手術患者手術時間、麻醉時間比較 三組患者手術時間、麻醉時間比較差異無統(tǒng)計學意義(P>0.05)，見表2。

(c2)?k(t)=?d(t),t∈[?m,0]are satis fi es,and if the external disturbance and the output measurement noise approach to zero,then the bounds of the tracking error∥ud?uk∥,∥xd?xk∥, and∥yd?yk∥approach to zero when k→∞for all t∈[0,T],that is,the control input uk(t) convergences to ud(t),the state xk(t)to xd(t),and the output yk(t)to yd(t).When there exists external disturbance and measurement noise,the tracking error∥yd?yk∥is estimated as:

To realize above control objective,we consider a updating law:

（四）從世界意義看，習近平新時代中國特色社會主義思想，凝結著對人類發(fā)展重大問題的深邃思考，為解決人類面臨的共同問題貢獻了中國智慧。當前，世界正處于大發(fā)展大變革大調整的時期，全球治理危機頻仍、挑戰(zhàn)迭出。習近平新時代中國特色社會主義思想，順應了“和平、發(fā)展、合作、共贏”的時代潮流，飽含了對關于人類前途命運重大問題的深邃思考，倡導構建人類命運共同體，以中國發(fā)展的實踐拓展發(fā)展中國家走向現(xiàn)代化的途徑，給世界上那些既希望加快發(fā)展又希望保持自身獨立性的國家和民族提供了全新選擇，為解決人類普遍面臨的問題貢獻了中國智慧和中國方案。

Remark 2We can observe that the bounds of the FOILC fi nal tracking errors are directly a ff ected by the disturbance and measurement noise,moreover,it can be observed that when the bounds of disturbance and measurement noise approach to zero,through the FOILC repetitive operations,the bound of tracking error∥ek(t)∥λapproaches to zero asymptotically.Therefore,iterative learning control method itself can not reject or compensate the uncertainties or disturbances for the multiple time-delay systems.

5 Numerical simulations

To demonstrate the e ff ectiveness of the proposed control scheme,the following nonlinear multiple time-delays systems with external disturbances and output measurement is presented for numerical simulations:

where t∈[0,2].Let us suppose that x(t)=[0.5 0.5]T;for t∈[?0.8,0]is the initial state function,and the desired trajectory is given by yd(t)=1.The objective of the control is to track the trajectory yd(t),over the time interval t∈[0,2].The simple choices of the control gains P and Γ in iterative learning control rule(3)are P=Γ=0.1.We apply the proposed FOILC rule(3)to the nonlinear system.By calculation,we can show the tracking performance of the FOILC system output at the interval t∈[0,2]when the FOILC rule(3)is iterativelyexecuted at di ff erent times.If the tolerance bound is chosen to be equal to 0.015,we fi nd that the control objective is achieved at the 11th iteration.Our simulation results illustrate that the proposed ILC rule in the form of(3)is robust with respect to small disturbance and measurement noises.

Fig.1 The solid line represents the desired trajectory yd(t),and the dashed,the dotted, and the dashed-dotted lines represent the FOILC system output yk(t)when the FOILC rule has executed 3,8,11 times,respectively.

[1]Podlubny I.Fractional Di ff erential Equations[M].New York:Academie Press,1999.

[2]Hilfer R.Application of Fractional Calculus in Physics[M].Singapore:World Science Publishing,2000.

首先對社會隔離、生存質量和經(jīng)濟狀況進行描述性分析和Pearson相關分析。如表1所示。社會隔離均值為(1.578±0.598)分，生存質量、經(jīng)濟狀況均值分別為(3.894±0.514)分和(2.270±0.599)分。社會隔離與生存質量(r=-0.222，P<0.01)呈顯著負向相關性，與經(jīng)濟狀況(r=-0.090，P<0.05)呈顯著負向相關性。生存質量和經(jīng)濟狀況呈顯著正向相關性(r=0.183，P<0.01)。

and

[4]Xu J X.A new fuzzy logic learning control scheme for repetitive trajectory tracking problems[J].Fuzzy Sets and Systems,2003,133:57-75.

[5]Park K H,Bien Z H,Wang D H.Design of an iterative learning controller for a class of linear dynamic systems with time delay[J].IEE Proceedings Control Theory and Applications,1998,145:507-512.

[6]Zhang B L,Tang G Y,Zhen S.PD-type iterative learning control for nonlinear time-delay system with external disturbance[J].Journal of Systems Engineering and Electronics,2006,44:600-605.

[7]Li Y,Chen Y Q,Ahn H S.Fractional-order iterative learning control for fractional-order linear systems[J].Asian Journal of Control,2011,13:1-10.

其中，Ndeep(l,x)為搜索深度，其定義如式(4)所示，且1≤ Ndeep(l,x) ≤Nr；C為更新半徑，且C2=2Nr σ2，為常數(shù).Rx-SD檢測依靠式(3)來更新搜索半徑，減小了接收端的搜索范圍，降低了接收端的計算復雜度.

[8]Chen Y Q,Wen C Y.Iterative Learning Control)Convergence,Robustness and Applications[C].London: Springer,1999.

[9]Lan Y H,He L J.P-type iterative learning control of fractional order nonlinear time-delay systems[J].In: Proceedings of the 24th Chinese Control and Decision Conference,2012:1027-1031.

[10]Sun M X,Wang D.Initial condition issues on iterative learning control for nonlinear systems with time delay[J].International Journal of Systems Science,2001,32:1365-1375.

[11]Benchohra M,Henderson J,Ntouyas S K,et al.Existence results for fractional order functional di ff erential equations with in fi nite delay[J].Journal of Mathematical Analysis and Applications,2008,338:1340-1350.

[12]Li Y,Chen Y Q,Ahn H S.Fractional-order iterative learning control for fractional-order linear systems[C].In Proceedings of the Symposium on Learning Control at IEEE CDC 2009.China:Shanghai,2009.

[13]Deng W H.Smoothness and stability of the solutions for nonlinear fractional di ff erential equations[J].Nonlinear Anal.,2010,72:1768-1777.

[14]Ma F,Li C.Iterative learning control design of nonlinear multiple time-delay systems[J].Applied Mathematics and Computation,2011,218:4333-4340.

[15]Geng X G,Xue B.An extension of integrable peakon equations with cubic nonlinearity[J].Nonlinearity, 2009,22:1847-1856.

[16]Lazarevic M P.PDα-type iterative learning control for fractional LTI system[C].Proceedings of the 16th International Congress of Chemical and Process Engineering.2004:1503-1507.

[17]Kilbas A A,Srivastava H M,Trujillo J J.Theory and Applications of Fractional Di ff erential Equations[J].Amsterdam:Elsevier,2006.

[18]Bristow D A,Tharayil M,Alleyne A G.A survey of iterative learning control[J].IEEE Control Systems Magazine,2006,26:96-114.

[19]Jiang W.The controllability of fractional control systems with control delay[J].Computers and Mathematics with Applications,2012,64:3153-3159.

[20]Ye H,Gao J,Ding Y.A generalized Gronwall inequality and its application to a fractional di ff erential equation[J].Journal of Mathematical Analysis and Applications,2007,328:1075-1081.

含多重時滯非線性系統(tǒng)的分數(shù)階迭代學習控制的收斂性分析

李娜娜,蔣威

(安徽大學數(shù)學科學學院,安徽,合肥 230601)

本文研究了含多重時滯非線性系統(tǒng)的分數(shù)階迭代學習控制(FOILC).它包含了外部干擾和輸出控制噪音,通過引入λ－范數(shù),獲得在開環(huán)和閉環(huán)迭代學習控制作用下,系統(tǒng)控制輸入以及跟蹤誤差收斂的充分條件.最后,通過數(shù)值仿真驗證所提方法的有效性.

分數(shù)階迭代學習控制,收斂,多重時滯

O231.2

2016-10-10.

國家自然科學基金(11371027,11471015).

李娜娜(1990-),碩士生,研究方向：泛函微分方程理論方向.

A

1008-5513(2017)01-0069-13

10.3969/j.issn.1008-5513.2017.01.008

2010 MSC:93C15,93C40,34A37