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(zhì)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(guān)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(yù)，而在于構(gòu)建健全的市場機(jī)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(yù)和門檻限制。在眾多市場作用機(jī)制中，市場導(dǎo)向的創(chuàng)新試錯機(jī)制最為關(guān)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(shí)需求做出正確反應(yīng)和科學(xué)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(yùn)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(guān)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(guān)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(guān)小美的母親也說：“他說他是總經(jīng)理他就是了?。磕阋膊粍觿幽X子想想？你們認(rèn)識才幾天啊？”

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(hù)工程規(guī)模大，且施工相當(dāng)困難。故結(jié)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(diào)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(fù)的缺點(diǎn)，充分發(fā)揮了路基抗災(zāi)能力相對較強(qiáng)的特點(diǎn)，并在隧道口延伸明洞，明洞上部設(shè)置必要的傾向河側(cè)的緩沖層，進(jìn)一步提高線路的抗災(zāi)能力，達(dá)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(jī)碼環(huán)階數(shù)為2，碼環(huán)相關(guān)間隔設(shè)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(xué)課程與教學(xué)論形成了一些分支并獨(dú)立成書后如果不開設(shè)選修課，就弱化了本教材的功能，造成學(xué)生認(rèn)知機(jī)構(gòu)的不完整。為此，把獨(dú)立出去的分支教材《化學(xué)教學(xué)論實(shí)驗(yàn)》《化學(xué)微格教學(xué)》《化學(xué)教學(xué)測量與評價》《化學(xué)學(xué)習(xí)論》等與《化學(xué)課程與教學(xué)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(shè)置了一些欄目。如:每一章前面用“思維導(dǎo)圖”進(jìn)行引導(dǎo)，使學(xué)生瞬間可瀏覽到一個清新直觀的網(wǎng)絡(luò)圖，概覽整章的知識脈絡(luò)，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(fèi)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(yàn)采用隨機(jī)區(qū)組設(shè)計(jì)，3次重復(fù)，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(jī)抽取10株進(jìn)行株高、葉片數(shù)、分蘗數(shù)的測量統(tǒng)計(jì)。測產(chǎn)時采用三點(diǎn)法，每小區(qū)隨機(jī)抽取3個點(diǎn)，每個點(diǎn)選取1 m2進(jìn)行測產(chǎn)。試驗(yàn)具體設(shè)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(shù)患者手術(shù)時間、麻醉時間比較 三組患者手術(shù)時間、麻醉時間比較差異無統(tǒng)計(jì)學(xué)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(xí)近平新時代中國特色社會主義思想，凝結(jié)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(xiàn)了中國智慧。當(dāng)前，世界正處于大發(fā)展大變革大調(diào)整的時期，全球治理危機(jī)頻仍、挑戰(zhàn)迭出。習(xí)近平新時代中國特色社會主義思想，順應(yīng)了“和平、發(fā)展、合作、共贏”的時代潮流，飽含了對關(guān)于人類前途命運(yùn)重大問題的深邃思考，倡導(dǎo)構(gòu)建人類命運(yùn)共同體，以中國發(fā)展的實(shí)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(dú)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(xiàn)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(zhì)量和經(jīng)濟(jì)狀況進(jìn)行描述性分析和Pearson相關(guān)分析。如表1所示。社會隔離均值為(1.578±0.598)分，生存質(zhì)量、經(jīng)濟(jì)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(zhì)量(r=-0.222，P<0.01)呈顯著負(fù)向相關(guān)性，與經(jīng)濟(jì)狀況(r=-0.090，P<0.05)呈顯著負(fù)向相關(guān)性。生存質(zhì)量和經(jīng)濟(jì)狀況呈顯著正向相關(guān)性(r=0.183，P<0.01)。

and

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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(jì)算復(fù)雜度.

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含多重時滯非線性系統(tǒng)的分?jǐn)?shù)階迭代學(xué)習(xí)控制的收斂性分析

李娜娜,蔣威

(安徽大學(xué)數(shù)學(xué)科學(xué)學(xué)院,安徽,合肥 230601)

本文研究了含多重時滯非線性系統(tǒng)的分?jǐn)?shù)階迭代學(xué)習(xí)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(xué)習(xí)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(yàn)證所提方法的有效性.

分?jǐn)?shù)階迭代學(xué)習(xí)控制,收斂,多重時滯

O231.2

2016-10-10.

國家自然科學(xué)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