On interval tracking performance evaluation and practical varying sampling ILC

Transcription

1 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, On interval tracking performance evaluation and practical varying sampling ILC Yun Xu, Dong Shen and Youqing Wang College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, P.R. China ABSTRACT This paperconsiders the evaluation of interval tracking error for sampled control performance and an associated sampling technique to enhance the tracking performance. The upper bounds of the tracking error profile of arbitrary sample interval for both the linear system and nonlinear system are first given. A practical sampled-data iterative learning control with varying sampling rates is proposed to ensure a prior given tolerant tracking error. In this control strategy, the inter-sample behaviour is checked to determine which intervals are not satisfactory when the given tracking performance at-sample time instants is satisfied, and then the sampling frequency for such intervals is increased. Both at-sample and inter-sample tracking performance are satisfied after enough learning iterations. Two examples are simulated to demonstrate the effectiveness of the proposed sampling strategy. ARTICLE HISTORY Received 1 August 216 Accepted 17 December 216 KEYWORDS Iterative learning control; sampled-data control; tolerant tracking performance; practical varying sampling; inter-sample error bound 1. Introduction The idea of iterative learning control (ILC), which involves the use of tracking information from the previous iterations to construct new input and can improve tracking performance for repetitive systems, was first proposed in Arimoto, Kawamura, and Miyazaki (1984). The principle of ILC mimics the human ability using the past experience and knowledge to complete a certain task repeatedly. A classic example is shooting a basketball from a fixed point. A basketball player learns from previous experiments and constantly revises his/her shooting angle and intensity, thereby ensuring that he/she could hit the basket successfully after enough attempts. ILC is simple and effective as it requires less system information and is a typical data-driven approach to handle nonlinearity, strong coupling, modelling difficulty and high-precision tracking control problems. As a consequence, ILC has attracted much attention of researchers (Ahn, Chen, & Moore, 27; Bristow, Tharayil,& Alleyne, 26; Shen & Wang, 214),andiswidelyusedinindustrialproduction (Bifaretti, Tomei, & Verrelli, 211; Ji,Hou,&Zhang,216; Ouyang, Zhang, & Gupta, 26; Xu, Chu, & Rogers, 214;Zhao, Lin, Xi, & Guo, 215).ItisworthnotingthatILCisafeedforward type control method, which is advantageous to ensure the convergence along the iteration axis. Thus, it is also capable of combining with other control techniques for further performances. For example, in Ouyang et al. (26), a hybrid approach is provided for robot manipulators with integrating ILC and switching method to ensure a fast convergence rate. ILC was first proposed in a continuous form called continuous ILC. It saves the continuous signals of the input, the output and the desired trajectory into a memory, and generates the continuous input by using intact continuous information. However, in practice, saving all the continuous history information requires considerable storage. Therefore, discretetime ILC was developed to address this issue. As the name suggests, discrete-time ILC involves design and analysis in discrete form. This approach significantly reduces the amount of storage andcalculation,aswellasensuresthecontroleffect.itssimple and convenient design has attracted the interest of researchers suchasthosefocusingonstochasticsystems(oh&lee,215), quantised problem (Shen & Xu, 216; Xu& Shen, 216), data dropouts problem (Shen & Wang, 215a, 215b) and random iteration length problem (Shen, Zhang, Wang, & Chien, 216). However, the controlled plant is often a continuous-time system rather than a discrete system. Thus, scholars further consider so-called sampled-data iterative learning control (SDILC). SDILC designs discrete-time ILC law for continuous-time systemsbasedonthesampleddata(abidi&xu,211;chien,1997; Oomen, Wijdeven, & Bosgra, 27; Sun& Wang, 21). The results on SDILC are rather limited and most existing papers focusedonconvergenceatthesamplinginstants,whiletheintersampleperformanceislessevaluated.notethattheconvergence at the sampling instants can be treated as that of the discretetime ILC, which is easy to guarantee; however, the inter-sample convergence is much harder to achieve because the control signal is constant rather than continuously varying for the specified interval. In fact, fairly few results can be found on the intersample tracking performance in the ILC field. While the issues of the existing publications related to SDILC have covered delays, initial shifts, bounded disturbances and other traditional ILC issues, in-depth study on the analysis and synthesis of SDILC stillrequiresmoreefforts.inthefollowing,wegiveabriefliterature review on SDILC in the light of kernel issues. Chien et al. focusedontheeffectofcombiningthecurrent feedback mechanism with ILC under bounded disturbances or noises. Specifically, for linear systems, Chien and Tai (24) and Chien, Hung, and Chi (214) introduced the sampled tracking error of the current batch into ILC algorithm, where the bounded convergence at sampling instants was established. Chien and Ma (213) showed that the convergence rate was increased when the feedback controller was incorporated with CONTACT Dong Shen shendong@mail.buct.edu.cn 217 Informa UK Limited, trading as Taylor & Francis Group

2 2 Y. XU ET AL. the feedforward ILC. Similar results for nonlinear systems were given in Chien (1997, 2) and Chien, Wang, and Chi (214). Sun et al. contributed a series of papers on affine nonlinear systems with arbitrary relative degree (Sun & Wang, 2, 21) andinitialshifts(sun,li,&zhu,213; Zhu,He,& Sun, 26). For nonlinear systems with a well-defined relative degree, Sun, Wang, and Wang (24) used lower order differentiations of tracking error with the order less than the relative degree to generate an input sequence. Moreover, Sun and Wang (2, 21), Sun et al. (24, 213) and Zhu et al.(26) showed bounded convergence under bounded disturbances and bounded initial shifts. The initial rectifying mechanism for initial shifts was discussed in Sun et al. (213) and Zhu et al.(26) under the SDILC framework. In addition, Sun et al. (213) proposed a novel varying-order algorithm differing from other papers. Xu et al. conducted a relatively basic and comprehensive study on the design and analysis of SDILC, featured with the frequency domain method (Abidi & Xu, 211; Huang,Xu, Venkataramanan, & Huynh, 214; Xu, Abidi, Niu, & Huang, 212). It was found that a monotonic convergence condition can be derived more easily in the frequency domain than in the time domain. Criteria for the selections of learning type and sampling time were presented in Abidi and Xu (211). An experiment on a piezoelectric motor was detailed in Xu et al. (212). The closedloop feedback controller was included in SDILC design, and this controller outperformed well-tuned open-loop and PI control algorithms (Huang et al., 214). Traditional ILC problems have also been attempted for SDILC such as time delays (Fan, He, & Liu, 29), optimal control (Zhou, Tan, Oetomo, & Freeman, 213) and singular systems (Sun, Fang, & Han, 22). However, it should be pointed out that these attempts are just at the first step and many efforts are desiderated for their perfection. From the above papers, we find that most existing research focusedontheconvergenceatthesampleinstants,buttheperformance in sampling intervals is less evaluated. However, when considering the sampled control of the continuous-time system, perfect tracking means not only good at-sample performance but also satisfactory inter-sample behaviour. This is one major difference between sampled-data control and discrete-time control.specifically,intheilcfordiscrete-timesystems,thecontrol objective is to ensure the precise tracking at the specified time instants, while the optimal sampled-data ILC should pay attention to both the at-sample and inter-sample tracking performance simultaneously. To solve this problem, Oomen et al. presented an optimal multirate ILC criterion under the closed-loop multirate ILC set-up and provided an experiment on a wafer stage system to illustrate better inter-sample behaviour (Oomen et al., 27; Oomen,Wijdeven, &Bosgra,29, 211). However, the topic is far away from complete as many related issues such as the design and analysis of irregular sampled-data ILC are still open. This is the first motivation of this paper. That is, we aim to provide an evaluation of the inter-sample tracking performance. Moreover, it is natural that high sampling rate results in rich data, which further leads to precise tracking performance. Therefore, in some applications, it is straightforward to use the fastest sampling rate up to the limit of the hardware. However, such simple mechanism may lead to a great waste of sampling cost and computation burden for some practical systems, especially for slow varying systems such as chemical reaction processes. Specifically, some reaction process is quite slow and thus one operation would run for a long period. The fast sampling for such kind of slow varying systems yields an excessive cost in sampling, storage and computation. In fact, the sampling rate is usually slow to save cost. Nevertheless, slow sampling may result in that the inter-sample behaviour is not so good as expected. Thus, it is of great interest to consider the trade-off between sampling cost and tracking performance. This is the second motivation of this paper. That is, we aim to propose an irregular sampling technique to balance the conflict between sampling cost and tracking performance. This paper contributes to an evaluation of the inter-sample tracking error and a varying sampling technique to construct an irregular sampled-data ILC for ensuring the prior given tolerant control performance. Specifically, the technical contributions of this paper consist of two parts. First, for high accuracy tracking of continuous-time systems, the upper bounds of inter-sample tracking errors are first evaluated provided that the zero-error at-sample tracking has been precisely achieved, which implies that a smaller sampling period corresponds to a smaller intersample tracking error. The cases for both linear and nonlinear systems are discussed separately, and the evaluation for linear systems is tighter than that for nonlinear systems as more uncertainties are involved for the latter case. Second, to balance the conflict between sampling cost and tracking performance, a practical varying sampling technique is introduced on the basis of the above observations. Specifically, we start from a slow sampling rate and increase the rate for part of the intervals according to the inter-sample tracking performance. The associated SDILC is proposed to ensure the good/tolerant tracking performance for both at-sample and inter-sample cases. Two illustrative examples are also detailed to show the effectiveness oftheproposedstrategy.itisworthpointingoutthatanalternative method for increasing the tracking performance is to compute the virtual sampled data for the points located in the sampleintervalandusethesevirtualsampleddataforinput updating. However, such method would need the system information for computing the virtual information. Moreover, if system noises and/or uncertainties are involved, the computation errors should be well addressed. Our method in this paper is to adjust the actual sampling rate. Therefore, the available data aremoreaccurateandtheproposedsdilcisadata-driven method. The rest of the paper is organised as follows. Section 2 provides the problem formulation. Sections 3 and 4 derive the expression of maximum interval tracking error for both linear and nonlinear systems, respectively. Section 5 details a varying sampling rate strategy and its associated SDILC. Two practical systems illustrate the effectiveness of the proposed method in Section6. Section7 concludes this paper. Notations. R denotes the set of real numbers, and R n denotes the n-dimensional space. N denotes the set of non-negative integers. For a vector x, without further specifications, x denotes the Euclidean norm and for a matrix M, M is the induced norm.

3 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 3 Problem statement: Given a desired trajectory y d (t), and a tolerant bound of the tracking error ϵ for the continuous-time system, the control object is to assign a proper sampling strategyandaproperilclawsothatthegeneratedinputsequence u k (j ) could ensure that the maximal tracking error during the wholetimeintervalisnolargerthanthegivenboundϵ after sufficient learning iterations. That is, max t y d (t) y k (t) ϵ, t [, T], k K where K is a sufficiently large integer. Figure 1. Block diagram of SDILC. 2. Problem formulation Consider the following single-input single-output nonlinear continuous-time system: ẋ k (t) = f (x k (t), u k (t)), y k (t) = g(x k (t)), where k denotes different iterations, t denotes the time axis, t [, T], and T is the iteration length. u k (t) R p, y k (t) R q and x k (t) R n are the input, output and state, respectively. The functionsf( ), g( ) are smooth in their domain of definition. As a special case of Equation (1), the linear system is formulated as follows: ẋ k (t) = Ax k (t) + Bu k (t), y k (t) = Cx k (t), where A, B and C are system matrices with appropriate dimensions. The continuous-time system can be well handled if the continuous differential of tracking error is used for update law. However, it is hard to obtain the continuous differential signal in many practical applications. Meanwhile, when computers are involved in the control design, it is reasonable to generate the control signals in discrete-time form. Thus, the sampled-data controlisapromisingwaytofillthegap.toperformthisimplementation, a sampler and a holder are introduced to achieve the A/D and D/A transformation. The block diagram of SDILC is given in Figure 1, wherea sampler is implemented at the output side to generate the sampled output, the learning controller produces the discrete input for the next iteration using the stored discrete input values and sampled outputs as well as the reference trajectory, and a holder is adopted to regain the continuous input signal for the controlled system. In this paper, the sampling period is denoted by.without loss of any generality, it is assumed that T/ is an integer and denote N T/ as the number of sampling instants. Then, only the output at the time instant j, wherej is a positive integer, is sampled for input updating. Therefore, we could only obtain the input signal at j, j N. To generate the continuous control signal, the zero-order holder is adopted as follows: (1) (2) u(t) = u( j ), t [ j, j + ). (3) Remark 2.1: Thecontrolobjectiveistoguaranteeatolerant bound of tracking errors after enough learning iterations. Thus, it is an objective in the sense of limitation. In other words, an appropriate sampling strategy should be designed such that lim k e k (t) ϵ for arbitrary t valued in [, T] ratherthan only at the sampled time instants. However, in practical applications, verifying the limiting performance is difficult. Consequently, we could set a prior acceptable iteration number as the sufficient learning iterations. In addition, the tracking error at sampling time instants would affect the inter-sample behaviour, while in practical application, it is difficult to ensure the zeroerror tracking performance in finite iterations. Thus, given the upper bound of inter-sample tracking bound ϵ, wesetthetol- erant error at sampling time instants to be far smaller than the given bound so that perfect tracking at sampling instants is approximately achieved after finite learning iterations. Details are illustrated in Section 6. Remark 2.2: Even if the tracking errors at sampling time instants are infinitesimal or approach zero, some intervals may exist where the tracking performance is not satisfactory, especially when the sampling frequency is slow. To guarantee the desired tracking results, the sampling frequency needs to be adjusted in those time intervals. This motivates us to consider the nonuniform sampling, i.e. different sampling rates exist in an iteration. Such sampling mechanism can efficiently balance the conflict between sampling cost and tracking performance. Details are shown in Section 5. The following assumptions are required: Assumption 2.1: The nonlinear functions f(x) and g(x) satisfy the globally Lipschitz condition in the definition domain, i.e. f(x 1,y 1 ) f(x 2,y 2 ) f 1 x 1 x 2 +f 2 y 1 y 2 and g(x 1 ) g(x 2 ) g x 1 x 2. Assumption 2.2: Theinitialstateispreciselyresetforalliterations, i.e. x k () = x d (). Assumption 2.3: The desired trajectory y d (j ) and y c (t) is realisable, where y d (j ) denotesthesampledoutputvaluesofy c (t). That is, for an appropriate initial state x d (), a unique input series u d (j ), j N,calledstepsignals,existssuchthattheoutputof(1) or (2) equals y d (j ) at sampling times instants. In addition, for an appropriate initial state x d (), thereexistsauniqueinputu c (t), called continuous signals, such that the output of (1) or (2) equals y c (t) for all t [,T]. Remark 2.3: We give some brief remarks on the assumptions, which are all basic conditions and widely used in most ILC papers. Assumption 2.1 gives the globally Lipschitz condition of the nonlinear functions, which is essential in ensuring the stability of the original continuous-time system. It has been illustratedinchapter7ofxuandtan(216) thatthesystem

4 4 Y. XU ET AL. may have finite escape time so that the tracking problem cannot be solved. This example implies the necessity of Assumption 2.1.Moreover,Assumption 2.2 is a basic conditionto guarantee the space and time repetition of the system. While relaxations of Assumption 2.2 have been discussed in many papers, the objective of this paper is to propose some fundamental results for the error evaluation. Thus, we use Assumption 2.2 to avoid additional influences of initial error on the inter-sample tracking errors. Finally, Assumption 2.3 provides the existence condition of input solution to the desired reference so that the following SDILC procedures are well defined. 3. Tracking error bound for linear system case The tracking error in sampling intervals under the step input signal u d (j ), t [j, j + ), for linear systems defined by Equation (2)isanalysedinthissection.Thefollowingtheorem presents the upper bound of the tracking error in a sampling interval. Theorem 3.1: If the linear system (2) samples uniformly with a period and works under the desired step input signals u d (j ), then for any sampling interval [j,j + ),themaximumvalue of the tracking error norm profile e(j + t ) satisfies e( j +t ) C e A t B μ( j )t < C e A B μ( j ), where μ(j ) = max t u c (j + t) u d (j ), t [, ], provided that the tracking errors at sampling time instants are zero. Here, the maximum tracking error within the sampling interval [j,j + ) isassumedtobeattimej + t. Proof: In a sampling period [j, j + ), the inter-sample trackingerrorwiththeuseofstepsignalscanbeexpressedas follows: e( j + t) = y c ( j + t) y d ( j + t) = Cx c ( j + t) Cx d ( j + t) [ j +t ] = C e At x c ( j ) + e A( j +t τ) Bu c (τ )dτ [ C e At x d ( j ) + j +t = C = C j t j j +t j (4) ] e A( j +t τ) Bu d ( j )dτ e A( j +t τ) B [ u c (τ ) u d ( j ) ] dτ e A(t τ) B [ u c ( j + τ) u d ( j ) ] dτ, (5) where x c (j ) = x d (j ) isduetothepriorconditionthatthe zero-error tracking at sampling time instants has been achieved. According to the mean value theorem, Equation (5) can be written as = e( j + t) t Ce A(t τ) B[u c ( j + τ) u d ( j )]dτ = Ce A(t t ) B[u c ( j + t ) u d ( j )]t, (6) where t t. Taking norms to both sides of Equation (6) yields e( j + t) C e A t t B u c ( j + t ) u d ( j ) t < C e A t B μ( j )t, (7) where μ(j ) = max t u c (j + t) u d (j ), t [, ]. Obviously, the value of C e A t B μ(j )t monotonically increases as t increases. Therefore, the above inequality renders the following inequality: e( j + t) < C e A t B μ( j )t < C e A B μ( j ). (8) For the absolute value of tracking error, i.e. e(j + t),atleast onemaximumvalueexistsin(j, j + ) duetothecondition e(j ) =, e(j + ) = and the interval length is finite. Assume that j + t is the time when the maximum is achieved, then e( j + t ) C e A t B μ( j )t < C e A B μ( j ). This completes the proof. Remark 3.1: It has been proved in Sun and Wang (21) that the desired discrete input sequence can ensure asymptotic zero-error tracking at sampling instants. Thus, in order to give the essential bound of the inter-sample tracking error, we assume that all tracking errors at the sampling time instants are zero. Then, we have e( j + ) = C ea(t τ) B(u c ( j + τ) u d ( j ))dτ =. Suppose that u d (j ) is always larger (or smaller) than the upper (or lower) bound of u c (j + t), the exact tracking at sampling point j + is not realisable, i.e. e(j + ). This contradiction implies that min τ u c (j + τ) < u d (j ) < max τ u c (j + τ). In addition, u c (j + τ) iscontinuousintheinterval τ, thus, according to the mean theorem, there must exist at least one time instant t,< t <,suchthatu d (j ) = u c (j + t ). Note that in practical implementations, it is difficult to ensure the zero-error tracking within finite iterations, thus we would use a sufficient small threshold instead of zero to suspend the algorithm (see Section 5 for details). 4. Tracking error bound for nonlinear system case The estimation of the inter-sample tracking error bound for nonlinear systems is analysed in this section. It should pointed out that, comparing with the linear system case, more uncertainties are involved for the nonlinear system case, thus the estimation for nonlinear systems would be rougher than that for linear systems. This conjecture is verified in the following. Theorem 4.1: If the nonlinear system (1) samples uniformly with asmallperiod and works under the desired step input signals u d (j ), then for any sampling interval [j, j + ), the maximum value of the tracking error norm profile e(j + t ) (9)

5 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 5 satisfies e( j + t ) g f 2μ( j )t < g f 2μ( j ) 1 f 1 t 1 f 1 (1) where μ(j ) = max t u c (j + t) u d (j ),t [, ],provided that the tracking error at sampling time instants are zero. Here, the maximum tracking error within the sampling interval [j, j + ) isassumedtobeattimej + t. Proof: In a sampling period [j, j + ), the interval tracking errorwiththeuseofstepsignalscanbeexpressedasfollows: e( j + t) = g(x c ( j + t)) g(x d ( j + t)), (11) and by Assumption 2.1 e( j + t) g x c ( j + t) x d ( j + t). (12) In a sampling period [j, j + ), the inter-sample system states can be written as t x( j + t) = x( j ) + f (x( j + τ),u( j + τ))dτ. (13) According to the mean value theorem, the difference of states x c (j + t)andx d (j + t)isgivenby x c ( j + t) x d ( j + t) = t ( f (x c ( j + τ),u c ( j + τ)) f (x d ( j + τ),u d ( j )))dτ = [ f (x c ( j + t ), u c ( j + t )) f (x d ( j + t ), u d ( j + t )) ] t (14) where t t. Takingnormstobothsidesofthelastequalityandapplying the Lipschitz conditions, we obtain and Define x c ( j + t) x d ( j + t) [ f 1 x c ( j + t ) x d ( j + t ) + f 2 u c ( j + t ) u d ( j + t ) ] t. (15) λ = max x c ( j + t) x d ( j + t), t [, ], (16) t μ = max u c ( j + t) u d ( j ), t [, ]. (17) t The expression (15)holdsforallt in the interval [, ), which implies λ ( f 1 λ + f 2 μ)t. (18) If the sampling interval is small enough such that 1 f 1 >, then by rearrangingequation (18), we obtain λ f 2μt 1 f 1 t. (19) Using definitions (16)and(17), we can rewrite Equation (12)as e( j + t) g f 2μt 1 f 1 t. (2) Assume that the maximum appears at time j + t within the sampling period [j, j + ), we have e( j + t ) g f 2μt 1 f 1 t. (21) ToindicatethesecondinequalityinEquation(1), the monotonic property of the function h(t) = gf 2μt 1 f 1, t [, ] needs to t be examined. Note that d dt h(t) = which implies that function h(t) = gf 2μt 1 f 1 t is tenable. This com- increasing and therefore g f 2μt 1 f 1 t pletes the proof. gf 2μ >, (22) (1 f 1 t) 2 < g f 2μ 1 f 1 is monotonically Remark 4.1: To ensure the validity of Equation (19), the denominator part should be larger than zero, i.e. 1 f 1 t > (ort < 1/f 1 ). To satisfy this condition, should be small enough to satisfy < 1 f 1. It is worth mentioning that it is a sufficient requirement of sample interval rather than a necessary condition. However, note that we will keep increasing the sampling rate if the inter-sample tracking error is not satisfactory, thus the length of sample intervals decrease so that there always exists a suitable guaranteeing the validity of Equation (19). Remark 4.2: Different analysis methods are applied for linear and nonlinear systems. For liner systems, the state equation is solvable and analytical as shown in Equation (5), and then the tracking error can be directly expressed by system matrix A, B and C asshowninequation(6). But the situation is not valid for nonlinear systems. In fact, the state analytical solution is nearly impossible to obtain. Therefore, the difference of states x c (j + t)andx d (j + t)hastobederivedfromthemeanvaluetheorem as given in Equation (14). Moreover, the relationship between system outputs and states is nonlinear, leading to that norm of the difference e(j + t) is bounded by parameters f and g.in short, the nonlinear system formulation has more uncertainty in establishing the estimated error bound. Remark 4.3: Since the nonlinear system formulation introduces more uncertainties in evaluating the estimation of the error bound, it is expected that the estimation bound for the linear system is tighter than that for the nonlinear system. To show this, we restrict the estimation (1) to the linear system, then it is evident that f 1 = A, f 2 = B,andg = C for the linear case. In such situation, the distinctions of the upper bounds for linear and nonlinear systems are e A in Equation (23) and 1 1 A

6 6 Y. XU ET AL. in Equation (1), respectively. Note that canbeselectedtobe smallenoughtoensurethepositivenessof1 A according to Remark 4.1. Define an auxiliary function q(x) = e x (1 x). It is evident that q() = 1and q(x) = xe x.thus q(x) < forx > x x, implying that q(x) is monotonically decreasing for x >. This fact further yields that q(x) < q() = 1forx >, which implies e x < 1 for < x < 1. Consequently, the estimation for linear 1 x systems is tighter than that for nonlinear systems. Remark 4.4: Both linear and nonlinear cases show that a small sample period corresponds to small intersample tracking errors. The validity of the linear case is evident. Here, we verify it for the nonlinear case, that is, we show the last term of Equation (1)is monotonically increasing with respect to. Tothisend,,we denote ē( ) = g f 2μ( j ) 1 f 1, t [, ], as a function of.then, the derivative of this function is given as follows: dē( ) d = gf 2μ( j ) (1 f 1 ) 2 + gf 2μ ( j ) 1 f 1, (23) dμ( j ) where μ ( j ) = d.roughlyspeaking,μ (j ) isusually non-negative because the maximum error μ(j ) is nondecreasing along with the increase of interval length, which implies that gf 2 μ ( j ) 1 f 1. Therefore, the derivative dē( ) is positive. Consequently, the monotonic increase property with respect to the d interval length is also valid for the nonlinear system case. 5. Varying sampling rate strategy With sufficient learning iterations and a proper ILC design, the asymptotical convergence at sampling time instants is easy to achieve, while controlling the inter-sample tracking behaviour is difficult. It is impossible to achieve zero-error convergence in the sampling intervals unless the controlled plant works under continuous input signal u c (t). Therefore, what we can do is to reduce tracking errors in sampling intervals and guarantee it within the given bound. Sections 3 and 4 indicate that a high sampling rate corresponds to a small inter-sample tracking error bound. Through increasing the sampling rate, the maximal inter-sample tracking error would show a decreasing tendency. This observation is also used to derive the so-called sampling rate-dependent technique (Saab & Toukhtarian, 215). Consequently, a simple idea for our control objective is to increase the sampling rate so that the interval length is decreased and the inter-sample behaviour is improved. However, increasing sampling rate is not always practical because an excessively high sampling rate usually leads to a waste of computation time, energy and storage space. On the contrary, lower rate sampling means less calculation cost, but lower rate sampling may result in a larger inter-sample tracking error no matter how many learning iterations are performed. To balance the tracking performance and computation cost, a varying sampling rate strategy is presented in this section. By varying sampling rate strategy, we mean that the sampling rate varies during different time sections of the entire iteration. To be specific, we first set a low sampling rate and the entire operation is divided into several time intervals. Then, after enough learning iterations, we check the inter-sample behaviour to determine whether the maximal tracking error is larger than the given bound. If the maximal tracking error of an appointed intervalissmallerthanthegivenbound,thenthesamplingrate does not need to be increased for this interval further. Otherwise, we increase the sampling rate to divide the interval into several subintervals to further improve the performance. A high sampling rate is expected to be applied to the necessary sections only during the entire operation to reduce computation load and improve tracking performance. The detailed implementation of the proposed varying sampling rate strategy is given as follows. Practical verifications are given in the next section. Step 1 The sampling period is initiated based on working experience, and then N sampling instants are given, where N = [T/ ]. Step 2 Design a discrete or sampled ILC update law based on current sampling rate, where the control target is to achieve the zero-error tracking performance at the sampling time instants. Step 3 At each iteration, the maximal at-sample error is checked to determine whether the maximum error is smaller than pϵ or not. That is, we check whether max j e k (j ) pϵ is true or not. If true, then go to Step 4;otherwise,goto Step 2.Notethatp (, 1) is a prior given small enough constant. Step 4 For each time interval, check whether the maximum error in the interval is smaller than ϵ. Thatis,wecheck whether max t [j, j + ] e k (t) ϵ is true or not. If it is true, then no changes need to be made. If not, then the sampling rate is increased for the unsatisfactory intervals. A simple rate increase mechanism is to double the sampling rate for all unsatisfactory intervals, that is, we further divide the selected interval into two identical pieces. Step 5 If the validation in Step 4 is satisfied for all intervals, i.e. thecontrolobjectiveisachieved,thenterminatethealgorithm. Otherwise, go to Step 2. The tracking error could be sufficiently reduced as long as the sampling rate is sufficiently high. Therefore, through increasing the sampling rate at certain time intervals and learning sufficient iterations, it is expected to obtain a smaller tracking error than the given error bound during the entire running time [, T], i.e. y d (t) y k (t) ϵ, t [, T]. The rate adjusting might occur several times. Then, the collection of iterations with the same sampling distribution is called a stage in the following. As a result, the learning process usually comprises several stages before the tracking error becomes small enough. For an arbitrary tolerant error bound, the suitable input sequence based on samplingdatamaybeofthefollowingform: u k ( j 1 1 ), t [ j 1 1, j ) [, t 1 ), j 1 =, 1, 2,...,k 1 u k ( j 2 2 ), t [t 1 + j 2 2, t 1 + j ) u k (t) = [t 1, t 2 ), j 2 =, 1, 2,...,k 2. u k ( j m m ), t [t m 1 + j m m, t m 1 + j m m + m ) [t m 1, T], j m =, 1, 2,...,k m (24)

7 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 7 where i is the sampling period of the ith section, and m denotes the number of sections constituting operation length. Different sections can have different sampling rates, that is, some sections can be sampled roughly, while other sections are sampled densely. Remark 5.1: To illustrate the design of update law in Step 2,we consider the linear system (2) as an example here. The sampleddata dynamic equation can be described as x k ( j + ) = x k ( j ) + Ɣu k ( j ), y k ( j ) = Cx k ( j ). (25) The relative degree of the sampled-data system is assumed to be one, and the following P-type ILC is adopted to achieve zeroerror tracking performance at the sampling time instants: u k+1 ( j ) = u k ( j ) + Le k ( j + ), j N. (26) The convergence condition can be formulated as I LCƔ < 1, (27) where L isthelearninggainofilc.notethatthecontrolmatrix Ɣ depends on the sampling period. Thus, the convergence condition (27) and tracking performance are not always satisfied. Once Ɣ changes, the learning gain should be changed accordingly. Moreover, to guarantee the monotonic convergence, a new type of ILC can be designed based on the design technique in Abidi and Xu (211). For example, D-type ILC is suitable if the transfer function P(z) is second-order with a single integrator, and D 2 -type ILC is suitable for either a second-order or a thirdorder P(z) with two integrators. On the other hand, the matrix Ɣ determines the relative degree of the sampled-data system. There may exist one sampling frequency in which the relative degree of the sampled-data system is lager than one. Then, the convergence condition (27) becomesinvalid.insuchcase,the tracking error e k (j + ) shouldbereplacedbye k (j + r ), where r denotes the corresponding relative degree. The design and analysis of these algorithms have been presented in previous studies (Abidi & Xu, 211; Chien, 1997, 2; Chien& Ma,213; Chien & Tai, 24; Chien, Wang, & Chi, 214; Chien,Hung,& Chi,214; Huangetal.,214; Sun& Wang,2, 21; Sunetal., 24; Xuetal., 212; Xu, Huang, Venkataramanan, & Tuong, 213). Remark 5.2: ThetypeofILCalgorithmsandselectedlearning gains have significant influence on the convergence speed and tracking performance. That is, different ILC algorithms and/or different gains can lead to different learning iteration numbers and computation load. However, this subject is beyond the scope of this paper. Therefore, in Remark 5.1, we simply adopt the traditional P-type learning algorithm as an illustration. This ILC scheme is also used in the following simulations (see the next section for more details). One may argue that the traditional P-type learning law may result in poor transient performance. Consequently,moreeffortsarerequiredforfurtheranalysis,and many open problems remain for the design, analysis and optimisation of SDILC. Remark 5.3: To determine exactly whether the tracking error by a certain sampling strategy is satisfactory, sufficient trials should be prior performed with the ILC update law until the tracking errors at sampling instants approach zero. This condition is the reason why we choose p (, 1) small enough in Step 3,thatis, asmallp means the degree that the input sequence approaches sufficiently to u d (j ) when the condition is satisfied. The value of p canbevariableindifferentstages. Remark 5.4: Using the strategy presented above, we could always find a suitable sampling rate distribution to ensure that the maximal tracking error during the whole time interval is smaller than the given bound ϵ. On one hand, smaller sampling period makes smaller tracking error bound. On the other hand, the inequality g f 2μ( j ) 1 f 1 ε leads to ε gf 2 μ( j ) + f 1 ε. (28) From the inequality (28), we have the sufficient value of the sampling period that guarantees the tolerant tracking performance, while the actual generated sampling period may be larger than this value. 6. Numerical experiments Two examples are given in this section to illustrate the effectiveness of our results. One is a linear system case and the other one is a nonlinear system case Linear system case Consider a piezomotor stage studied in Abidi and Xu (211). The driver and motor can be molded approximately as ẋ 1 (t) = x 2 (t), ẋ 2 (t) = k f ν M x 2(t) + k f M (t), y(t) = x 1 (t), (29) where x 1 is the motion position, x 2 is the motion velocity, M = 1kgisthemovingmass,k fν = 144 N is the velocity damping factor and k f = 6 N/V is the force constant. The desired trajectory is given as y d (t) =.3 +.3sin (2πt π/2) and the iteration length is.5 s. Here, the tolerant tracking error is set as ϵ = For simplicity and effectiveness, the traditional P-type ILC is adopted in the following simulation. A detailed convergence analysis is given in Abidi and Xu (211). According to the implementation in Section 5,we first sample five instants uniformly during the entire running time. For the first stage, the learning gain is selected as 25. The parameter p is set to.5. The performance at sampling instants is displayed in Figure 2,wherethelinedenotesthemaximalerroralongtheiteration axis. The maximal tracking error at the sampling instants approximates zero at the ninth iteration for the first stage. Then, we check the inter-sample behaviour, which is observed to be poor (as can be seen in Figure 3). Thus, the sampling rate is then increased. As a result, the maximal error leaps at the 1th iteration. The similar leap occurs at the 34th iteration where

8 8 Y. XU ET AL. maximal tracking error at sampling instants stage 1 stage 2 stage 3 pε iteration Figure 2. Maximal tracking error at sampling instants of linear system. system outputs desired trajectory 9th iteration: 5 ponits 33rd iteration: 1 points 6th iteration: 18 points t/ms Figure 4. Output profiles at different stages tracking error 1.2 x th iteration :5 ponits 33rd iteration :1 points 6th iteration :18 points error bound t/ms Figure 3. Tracking performance of linear system using the varying sampling technique. thesamplingratedistributionsareincreasedagainforseveral intervals. Figure 2 shows the maximal tracking error at sampling instants of the three learning stages according to different samplingratesanddistributions.specifically,thedashedline,dotand-dash line, solid line and dotted line denote stages 1, 2, 3, and error bound multiplied by parameter p, respectively. In addition, the given tracking condition on the sampling instants is satisfied at the 9th, 33rd and 6th iterations according to different stages. The parameters p for stages 2 and 3 are set to.5 and.85, respectively. The learning gains for stages 2 and 3 are set to 25 and 6, respectively. Figure3 shows the tracking error profiles along the time axis atthelastiterationofeachstage.specifically,thedashedline, dot-and-dash line, solid line and dotted line are the tracking error profiles of the 9th, 33rd and 6th iterations, and the given tolerant tracking error, respectively. From Figure 3, we can see that all the maximal inter-sample errors of each interval at the ninth iteration exceed the given bound ϵ. Therefore, all intervals are divided into two pieces, i.e. the sampling rate is doubled in stage 2. That is, the sampling period is.5 s, and 1 sampling instants are chosen in stage 2. After learning of another 24 iterations, i.e. at the 33rd iteration, the tracking error profile is displayed by the dot-and-dash line in Figure 3. The tracking error profiles of the fifth and sixth sampling intervals are smaller than ϵ. Thus,theinputsofthese two intervals no longer need to be updated in the following iterations. However, the tracking error profiles of the left intervals are yet unsatisfactory. Thus, the sampling rates in these intervals are doubled again. As a result, the sampling instants increase to 18 in stage 3. After another 27 iterations, the maximal tracking error profile of the entire iteration does not exceed the given bound ϵ at the 6th iteration. ThesystemoutputsareshowninFigure 4, where the dashed line,dot-and-dashlineandsolidlinedenotethesystemoutputat the 9th, 33rd and 6th iterations, respectively. The dotted line is the desired trajectory. Figure 4 shows that the output at the ninth iteration does not track the desired trajectory well, whereas the output at the 6th iteration almost coincides with the desired trajectory. These findings show the effectiveness of the proposed strategy Nonlinear system case A DC-motor driving a single rigid link through a gear is used as an example for the nonlinear system case. The dynamics is the same as that given in Wang (1998), (( J m + J ) ( l θ n 2 m + B m + B ) l θ n 2 m + Mgl ( ) n sin θm = u, n (3) andthelinkanglepositionisrelatedtothemotorangleas θ l = θ m /n, (31) where θ m, J m, B m and θ l, J l, B l are the motor and link angles, inertia and damping coefficients, respectively; n is the gear ratio; u is the motor torque; M isthelumpedmass;andl is the center of mass from the axis of motion. These parameters are given as J m =.3, B m =.3, J l =.44, B l =.25, M =.5, g = 9.8, n = 1.6, l =.15. θ m and θ m are chosen as state variables. The output is θ l,and let the desired trajectory be y d (t) = 1 6 πt πt3, t [, 3].

9 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 9 maximal tracking error at sampling instants stage 1 stage 2 pε iteration Figure 5. Maximal tracking error at sampling instants of nonlinear system. systerm outputs desired trajectory 16th iteration: 15 points 52nd iteration: 28 points x t x 1 4 Figure 7. Output profiles at different stages tracking error 6 x th iteration:15 points 52nd iteration:28 points error bound t x 1 4 Figure 6. Tracking performance of nonlinear system using the varying sampling technique. Thetrackingerrorboundissetasthemillesimalofmax y d (t), i.e. ϵ = In the simulation, the classical P-type ILC is adopted, and the learning gain is set to 2. The parameter p is set as p =.45. We first divide the entire operation into 15 intervals. By using the varying sampling technique, the total learning process consists of two stages, where 15 and 28 sampling instants exist for the former stage (stage 1) and the latter stage (stage 2), respectively. The sampling positions are automatically selected by the algorithm itself similar to the linear system case. The performance at sampling instants is displayed in Figure 5 along the iteration axis, where the dot-and-dash line, the solid line and the dotted line denote the maximal tracking error profiles of stages 1 and 2, and error bound multiplied by parameter p, respectively. The at-sample tracking performance is satisfied at the 16th and the 52nd iterations for the two stages, respectively. Figure 6 shows the entire tracking performance along the time axis, where the dot-and-dash line, the solid line and the dotted line denote the maximal tracking error profiles at the 16th and the 52nd iterations, and the given tolerant tracking error, respectively. From Figure 6, one can see that two intervals exist, whose inter-sample tracking error profiles are smaller than the given bound ϵ at the 16th iteration. As a result, the input signals for these intervals are no longer updated in the following iterations. The other intervals are unsatisfactory in the first stage, and thesamplingrateisdoubledfortheseintervals,therebyproducing 28 sampling instants in the second stage. After another 36 learning iterations, the entire tracking error profile is acceptable at the 52nd iteration. Similar to the linear system case, we show the tracking performance of the 16th and 52nd iterations in Figure 7,wherethedotand-dashline,thesolidlineandthedottedlinedenotetheoutput profiles at the 16th and the 52nd iterations, and the desired trajectory, respectively. The tracking at the 52nd iteration is very good. In short, the proposed algorithm is also effective for nonlinear systems. 7. Concluding remarks In this paper, the inter-sample errors are analysed, and their upper bounds are given. On the basis of these upper bounds, a practical SDILC with varying sampling rates is proposed. The system samples quickly if a large inter-sample tracking error exists in last stage, whereas the sampling frequency is slow if theinter-sampletrackingerrorisacceptable.forthispurpose, a maximal/tolerant tracking error is given first as the control objective. The algorithm starts with a low sampling rate, and then it improves the tracking performance at sampling instants to be of sufficient precision by using the sampled data based on the learning mechanism. The inter-sample behaviour is then checked to determine which intervals are not satisfactory. The sampling rate is increased for these intervals. Repeating these steps ensures that the whole tracking objective could be well achieved. Two examples demonstrate the effectiveness of our strategy. For the further research, it is of great interest to deep the theoretical analysis. Disclosure statement No potential conflict of interest was reported by the authors.

10 1 Y. XU ET AL. Funding This work is supported by National Natural Science Foundation of China [grant number ], [grant number ]; Beijing Natural Science Foundation [grant number 41524]. Notes on contributors Yun Xu received her B.S. degree in Automation from beijing Institute of Petrochemical Technology, China, in 214. Now she is pursuing a M.S. degree at College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, China. Her research interests include sampleddata iterative learning control and adaptive iterative learning control. Dong Shen received his B.S. degree in Mathematics from Shandong University, Jinan, China, in 25. He received his Ph.D. degree in Mathematics from the Academy of Mathematics and System Science, Chinese Academy of Sciences (CAS), Beijing, China, in 21. From 21 to 212, he was a Post-DoctoralFellowwiththeInstituteofAutomation, CAS. From 216 to 217, He was a visiting scholar at National University of Singapore, Singapore. Since 212, he has been an associate professor with College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, China. His current research interests include iterative learning controls, stochastic control and optimization. He has published more than 4 refereed journal and conference papers. He is author of Stochastic Iterative Learning Control (Science Press, 216, in Chinese), co-author of Iterative Learning Control for Multi-Agent Systems Coordination (Wiley, 216), and co-editor of Service Science, Management and Engineering: Theory and Applications (Academic Press and Zhejiang University Press, 212). Dr Shen received IEEE CSS Beijing Chapter Young Author Prize in 214 and Wentsun Wu Artificial Intelligence Science and Technology Progress Award in 212. Youqing Wang received his B.S. degree from Shandong University, Jinan, Shandong, China, in 23, andhisph.d.degreeincontrolscienceandengineering from Tsinghua University, Beijing, China, in 28. He worked as a research assistant in the Department of Chemical Engineering, Hong Kong University of Science and Technology, from February 26 to August 27. From February 28 to February 21, he worked as a senior investigator in the Department of Chemical Engineering, University of California, Santa Barbara, USA. From August 215 to November 215, he was a visiting professor in Department of Chemical and Materials Engineering, University of Alberta, Canada. Currently, he is a full professor in Shandong University of Science and Technology and also Beijing University of Chemical Technology. His research interests include fault-tolerant control, state monitoring, modelling and control of biomedical processes (e.g. artificial pancreas system), and iterative learning control. He is an (associate) editor of Multidimensional Systems and Signal Processing and Canadian Journal of Chemical Engineering. He holds membership of two IFAC Technical Committees (TC6.1 and TC8.2). 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