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

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].

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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.

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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;ϕd(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.

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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

本文所有采集数据均来源于HRO观测站，大口径天线系统在B1频段增益约为78dB，采集时长为10s，数据比特位数为14，软件接收机码环阶数为2，码环相关间隔设置为0.2码片.

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:

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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.

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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.

试验采用随机区组设计，3次重复，小区面积50 m2。栽培密度为133 395株/hm2，株行距均为25 cm×30 cm。每小区随机抽取10株进行株高、叶片数、分蘖数的测量统计。测产时采用三点法，每小区随机抽取3个点，每个点选取1 m2进行测产。试验具体设置如表1所示。

Then

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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 三组行上腹部手术患者手术时间、麻醉时间比较 三组患者手术时间、麻醉时间比较差异无统计学意义(P>0.05)，见表2。

(c2)ϕk(t)=ϕd(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:

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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.

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首先对社会隔离、生存质量和经济状况进行描述性分析和Pearson相关分析。如表1所示。社会隔离均值为(1.578±0.598)分，生存质量、经济状况均值分别为(3.894±0.514)分和(2.270±0.599)分。社会隔离与生存质量(r=-0.222，P<0.01)呈显著负向相关性，与经济状况(r=-0.090，P<0.05)呈显著负向相关性。生存质量和经济状况呈显著正向相关性(r=0.183，P<0.01)。

and

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其中，Ndeep(l,x)为搜索深度，其定义如式(4)所示，且1≤ Ndeep(l,x) ≤Nr；C为更新半径，且C2=2Nr σ2，为常数.Rx-SD检测依靠式(3)来更新搜索半径，减小了接收端的搜索范围，降低了接收端的计算复杂度.

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含多重时滞非线性系统的分数阶迭代学习控制的收敛性分析

李娜娜,蒋威

(安徽大学数学科学学院,安徽,合肥 230601)

本文研究了含多重时滞非线性系统的分数阶迭代学习控制(FOILC).它包含了外部干扰和输出控制噪音,通过引入λ－范数,获得在开环和闭环迭代学习控制作用下,系统控制输入以及跟踪误差收敛的充分条件.最后,通过数值仿真验证所提方法的有效性.

分数阶迭代学习控制,收敛,多重时滞

O231.2

2016-10-10.

国家自然科学基金(11371027,11471015).

李娜娜(1990-),硕士生,研究方向：泛函微分方程理论方向.

A

1008-5513(2017)01-0069-13

10.3969/j.issn.1008-5513.2017.01.008

2010 MSC:93C15,93C40,34A37