Many practical systems follow the same operation mode where they repeatedly complete a given task in a finite time interval. For instance, the industrial production process generally consists of successive batches of production tasks; that is, the system completes a production batch following a given procedure within the desired time interval and then repeats it again and again. For such systems that can be clearly divided into successive operation batches, if the operation time lengths of each batch are identical and the operation circumstances of different batches are similar, then we can fully utilize the operation data and experience to adjust the action strategy for the next batch. This basic concept of "learning" motivates the proposal and developments of iterative learning control (ILC), which is now an important branch of intelligent control [1]. In other words, ILC is a typical control strategy mimicking the learning process of the human being, of which the pivotal idea is to continuously learn the inherent repetitive factors of system operation processes based on various data from completed batches such that the tracking performance is gradually improved. This control strategy imposes little requirement on system information and thus is typically a datadriven control methodology, which can effectively deal with the traditional control challenges such as highnonlinearity, strong coupling, modeling difficulty, and tracking of high precision.
After developments over three decades, ILC has resulted in a number of valuable results in both theory and applications; for details, see survey papers and special issues [2][7]. We note that the invariance of system dynamics including identical tracking reference, identical operation length, and identical initial state is a basic requirement of ILC, for which the proposed update law can reduce the invariance and improve tracking performance. Recently, much effort has been devoted to relax this requirement. For example, in [8], [9], attempts have been made for the nonrepetitive uncertain system to take into account essential limitations of ILC dealing with nonrepetitive factors. The case of nonrepetitive parameters was also explored in a recent paper [10] among others. Moreover, scholars are working on novel analysis and synthesis approaches other than the conventional contraction mapping method, which imposes some restrictive conditions on the systems. The repetitive process based approach has shown its effectiveness in [11][14], and ILC can be easily turned into a repetitive process whose dynamics and control problems have been well investigated. Various stability criteria have been studied in [11][14] for different problems which can be applied to derive fruitful results of ILC by suitable transformation. We note that the 2D system based approach [15] and frequency based approach [16] are both important synthesis methods for deriving performanceguaranteed controller design of ILC. In addition, it should be pointed out that, along with fast developments in theoretical analysis, the applications of ILC have been greatly enlarged such as robotics [17], [18], dualmode flyback inverter [19], and stroke rehabilitation systems [20]. In sum, ILC has gained significant progress for both theoretical analysis and practical applications in the past decades.
In order to achieve excellent control performance, most ILC literature depends on the acquisition and utilization of full system information and operation data. That is, the data employed by the learning algorithms are assumed to have infiniteprecision. To this end, we have to increase the quantity and precision of sensors for complex systems to acquire more accurate information, increase the network bandwidth to transmit mass data, and increase the number of servers and improve the computation ability to guarantee good execution of complex algorithms. All of these inevitably increase the system burden and control cost. On the one hand, due to various uncertainties, the practical systems would suffer data dropout and loss during the operation, which results in additional difficulty in acquiring complete information. On the other hand, if we could efficiently reduce the acquisition and computation of mass data, provided that the tracking precision and control performance is decreased, we can not only reduce the cost of hardware and software, but also increase operational efficiency and system robustness. In consideration of the above two aspects, it is of great theoretical and practical significance to design datadriven ILC algorithms with incomplete information such that a high quality of control performance is achieved. We note that the influence of incomplete information on the tracking performance of datadriven ILC is essentially a robustness problem of ILC. It is worth pointing out that such robustness problem is different from the traditional modelbased robustness problem. That is, the former emphasizes the perspective of data, which focuses on the inherent restriction between the incomplete information and control performance, whereas the latter emphasizes the perspective of the model, which concentrates on the robustness with respect to the unmodeled dynamics.
In practical applications, there are various factors that can lead to the incomplete information problem, including both objective and subjective factors. To make our expression clear to follow, we classify the incomplete information scenarios into two categories: passive incomplete information and active incomplete information. Passive incomplete information refers to incomplete data and information caused by practical system limitations during data collection, storage, transmission, and processing, such as sensor/actuator saturation, data dropouts, communication delay, packet disordering, and limited transmission bandwidth. This incomplete information problem is common in networked control systems that are widely employed in engineering implementations due to their high flexibility and robustness. Active incomplete information refers to incomplete data and information caused by manmade reduction of data quantity and quality on the premise that the specified control objective is satisfied, such as sampling and quantization. By sampling, we acquire the operation data of a continuoustime system with a specified frequency only and skip the information between adjacent sampling time instants. By quantizing, we transform a value interval as an integer within a finite or infinite candidate set, which is common in the conversion from analog signal to digital signal. Clearly, both sampling and quantization can reduce the mass of data, which reduces the burden in acquiring, storing, and transmitting and increases the system operating efficiency. Therefore, it is of great importance to investigate how incomplete information influences control performance as well as determine how large the influence is and how to overcome the influence.
We note that the control design and analysis with both passive and active incomplete information have obtained many results in traditional control methodologies, especially in the field of networked control systems. However, ILC differs from traditional control methodologies in that it considers dualevolution along both the time axis and iteration axis. The kernel dynamics lie in the iterationaxis, which is essentially different from the timeaxisbased evolution of traditional system dynamics. Consequently, the results in networked control systems cannot be extended to ILC directly. Indeed, in ILC field, related results are very few and there are many open problems. Moreover, for learning control with incomplete information, it is most important to consider the data robustness of incomplete information and the associated overall design of the control systems; that is, it is important to understand the inherent restriction between incomplete information and control performance in a novel framework.
This paper is devoted to providing a survey of ILC with incomplete information, where we address the recent progress on ILC with passive incomplete information such as data dropouts, communication delays, and iterationvarying length, as well as with active incomplete information such as sampling and quantization. We will give a research framework for various incomplete information problems from the perspective of design and analysis techniques. Moreover, we provide a primary discussion on the data robustness and related topics in ILC with incomplete information. It is expected that the survey can help the reader to grasp the overall view of this topic and comprehend the fundamental techniques. The structure of the overview is shown in Fig. 1. We note that, to some extent, terminal ILC and pointtopoint ILC can be regarded as a type of incomplete information. The methods for this issue have been well reviewed in [5] and thus will not be repeated here.
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Fig. 1 Main structure of the overview 
The rest of this paper is arranged as follows. Section Ⅱ gives the basic formulation, design and analysis techniques, and primary convergence results of ILC. In Section Ⅲ, the recent progress on ILC with passive incomplete information is discussed, where the issues of random data dropouts, communication delays and limits, and iterationvarying lengths are elaborated, respectively. In Section Ⅳ, we proceed to review the progress on ILC with active incomplete information, where the sampling and quantization issues are emphasized. The data robustness and promising research directions are expounded in Section Ⅴ. Section Ⅵ concludes the paper with remarks.
Notations: Throughout the paper, we use
In this section, we provide the basic formulation of ILC as well as the primary design and analysis techniques. To this end, we first propose the essential principle of ILC. In particular, the fundamental idea of ILC is to improve the tracking performance for a given reference along the iteration axis. The main concept of networked ILC is shown in Fig. 2, where
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Fig. 2 Framework of networked ILC 
Now we proceed to the basic formulation of ILC according to the discretetime system. Consider the following discretetime linear timeinvariant system:
$ \begin{split} x_k(t+1)&=Ax_k(t)+Bu_k(t)\\ y_k(t)&=Cx_k(t) \end{split} $  (1) 
where
We denote the reference trajectory as
We assume the initial state to be reset to the desired one at each iteration, which is the wellknown identical initialization condition (i.i.c.). That is,
Note that the correction mechanism of ILC is to employ the tracking error information of previous iterations to adjust the input signal. To this end, denote the tracking error
$ \begin{align} u_{k+1}(t)=h(u_k(\cdot), \ldots, u_0(\cdot), e_k(\cdot), \ldots, e_0(\cdot)) \end{align} $  (2) 
where
$ \begin{align*} u_{k+1}(t)=h(u_k(\cdot), e_k(\cdot)). \end{align*} $ 
Additionally, the update law is usually linear for simplicity. A simple but common update law is as follows:
$ \begin{align}\label{Plaw} u_{k+1}(t)=u_k(t)+Ke_k(t+1) \end{align} $  (3) 
where
For system (1) and update law (3), a basic convergence condition on
$ \begin{align*} \ICBK\ < 1 \end{align*} $ 
where
For discretetime ILC, the lifting technique is a useful tool to transform the twoaxisbased evolution dynamics into oneaxisbased evolution dynamics. To see this point, considering system (1) and learning law (3) and noting that the iteration length is
$ \begin{align*} U_k&=[u_k^T(0), u_k^T(1), \ldots, u_k^T(N1)]^T\\ Y_k&=[y_k^T(1), y_k^T(2), \ldots, y_k^T(N)]^T \end{align*} $ 
as the lifted supervectors of input and output at the
$ {G}=\left[ \begin{array}{ccccc} CB &0 &0&\ldots&0\\ CAB&CB&0&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots \\ CA^{N1}B&CA^{N2}B&\ldots &\ldots&CB \end{array} \right] $ 
then we have
$ Y_k={G}U_k+{d} $ 
where
$ {d}=[(CAx_0)^T, (CA^2x_0)^T, \ldots, (CA^Nx_0)^T]^T. $ 
Similarly, we can define
$ U_{k+1}=U_k+{K}E_k $ 
where
$ \begin{align*} E_{k+1}= &Y_dY_{k+1}=Y_d{G}U_{k+1}{d}\\ = &Y_d{G}U_k{GK}E_k{d}\\ = &E_k{GK}E_k\\ = &(I{GK})E_k. \end{align*} $ 
Consequently, noting that
At the end of this section, we remark that the asymptotical tracking performance is derived according to the tracking error
In this section, we provide an indepth survey of ILC with passive incomplete information, where we concentrate on random incomplete information scenarios such as random data dropouts, communication delays and limits, and iterationvarying lengths. The common factor of these scenarios is that their information loss is due to practical conditions and environments. We note that other hardware limitations such as sensor/actuator saturation may also reduce the quality of data and information; however, they are omitted in this paper as they are generally deterministic.
A. Random Data DropoutsFrom Fig. 2 it is seen that the measured output and generated input are transmitted through networks. Due to data congestion, limited bandwidth, and linkage fault, the data packet may be lost during transmission. The data transmission has two alternative states: successful transmission and loss. Thus, the data dropout is usually described by a random binary variable, say
1) Random sequence model (RSM): For each time instant
2) Bernoulli variable model (BVM): The random variable
$ \begin{align} \label{Bernoulli} \mathbb{P}(\gamma_k(t)=1)=\overline{\gamma}, \quad \mathbb{P}(\gamma_k(t)=0)=1\overline{\gamma} \end{align} $  (4) 
where
3) Markov chain model (MCM): The random variable
$ \begin{align} \label{Markov} P=\left[\begin{array}{cc} P_{11}&P_{10} \\ P_{01}&P_{00} \end{array}\right] =\left[\begin{array}{cc} \mu&1\mu \\ 1\nu&\nu \end{array}\right] \end{align} $  (5) 
with
We first remark on the inherent connections among the above three models. Clearly, BVM is a special case of MCM as MCM would convert into BVM when
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Fig. 3 Data dropout models 
From the definition of RSM, we note that RSM only requires an upper bound of successive data dropouts along the iteration axis for every time instant
To clearly describe the average level of data dropouts along the iteration axis, we introduce a concept called data dropout rate (DDR), which is defined as
$ \begin{align} \pi=\left[\frac{1\nu}{2\mu\nu}, \frac{1\mu}{2\mu\nu}\right]. \end{align} $  (6) 
Then, DDR for MCM is
Taking the recent research literature into account, we observe that the progress can be reviewed from five perspectives: system types, data dropout models, dropout positions, update schemes, and analysis techniques, as is shown in Fig. 4. In the past decade, ILC under random data dropouts has been fully developed in all the perspectives; however, there are still open problems for further research.
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Fig. 4 The research framework of ILC with data dropouts 
1) Analysis Techniques: For smooth reading, we first review the analysis techniques and the related convergence results, especially the convergence meanings in consideration of the randomness of data dropouts besides optional stochastic noises. We review papers from the research groups in this issue to provide a basic outline of recent works.
Ahn et al. provided earlier attempts to the ILC for linear systems in the presence of data dropouts [21][23] using the Kalman filtering based technique, which was first proposed by Saab in [24]. The main difference among the contributions lies in the descriptions of data dropouts. In particular, the first paper [21] assumed that the whole output vector was considered as a packet, whereas this assumption was relaxed to the case that only partial information of an output vector may suffer loss problem in [22]. Moreover, in [23] both data dropouts and delayed control signals were taken into account. In [24], the input was derived by optimizing the input error covariance and thus the meansquare convergence of the input sequence was obtained. Therefore, [21][23] all contributed to a meansquare convergence.
Bu et al contributed different research angle for this problem in [25][29]. First, by using the exponential stability theory of asynchronous dynamical systems, which was given by Hassibi et al in [30], the convergence of both first and highorder update laws was established with an existence assumption of certain quadratic Lyapunov functions. Such a technique is not easy to extend to other systems and the authors used an expectationbased transform technique to derive the convergence for linear systems. In particular, in [26] the recursion of the tracking errors along the iteration axis, where the random data dropout variable was involved, eliminated the randomness by taking mathematical expectation to both sides. As a result, only the convergence in expectation sense was obtained. The techniques were then extended to nonlinear systems in [27] and an inequality of the input error rather than a recursion was obtained due to the nonlinearity. Moreover, in [28], a new
Liu and Ruan considered the problem using the traditional contraction mapping method in [32][34]. In [32], both linear and affine nonlinear systems were taken into account, where the data dropouts were assumed to occur at both the output and input sides. The recursion of the input error was first taken with an absolute operator and expectation operator, and then the convergence in expectation sense was derived using a technical lemma on contraction with respect to all previous iterations. As a result, the design condition for learning gains is fairly restrictive. A similar problem was also addressed in [33] following the same procedures of [32], where the difference between the two papers was the renewal of output information. When removing the data dropout at the input side, the results for both intermittent and successive update algorithms were also given in [34]. To recap, in these results, in order to allow a general successive data dropouts along the iteration axis, a restrictive convergence property for nonnegative sequences was derived and employed, which in turn may limit its applications.
Shen et al considered the random data dropouts for stochastic systems in [35][42], where the stochastic approximation was employed to derive the almostsure and meansquare convergence. First, Shen and Wang proposed the RSM for data dropouts in [35] for both linear and nonlinear systems with stochastic noises. The almostsure convergence was obtained by introducing a decreasing sequence to suppress the noise influence and improve the input signal. However, in [35], the control direction was assumed to be known prior, and this restriction was removed in [36], where a novel direction probing mechanism was employed. When considering the BVM, [37], [38] also addressed both intermittent and successive update schemes with a strict almostsure convergence analysis for linear and nonlinear systems, respectively. Note that stochastic noises are involved in the systems. Thus, the controller design and convergence analysis are distinct from the existing related literature. Detailed performance comparisons between the two types of algorithms and for related design parameters were also provided in [37], [38]. Moreover, the general data dropout case, i.e., both networks at the output and input sides suffering loss, was considered in [39][41] for deterministic linear systems, stochastic linear systems, and nonlinear systems, respectively. In these three papers, the data dropout was only described as a Bernoulli variable without any further restrictions on its successive dropouts. Note that the input fed to the plant and the one generated at the learning controller may be different due to the lossy network at the input side. Thus, the asynchronism between the two inputs should be well depicted. In fact, such asynchronism was modeled as a Markov chain and then the almostsure and meansquare convergence were established in the papers. The first attempt for data dropouts modeled by Markov chain was given in [42]. For both noisefree and stochastic linear systems, a unified framework was established for the design and analysis of ILC for three models, namely, RSM, BVM, and MCM. Both mean square and almost sure convergence of the input sequence to the desired input were strictly established. In short, the stochastic approximation technique is successfully applied to systems with stochastic noises and random data dropouts in the above papers.
There are scattered results on this topic such as in [43][47]. In [43], the authors contributed a detailed analysis of the effect of data dropouts. In particular, when only a single packet at the output side or the input side was dropped, the fundamental influence of data dropouts on tracking performance was carefully evaluated and revealed that neither a contraction nor expansion arose. This technique was then extended in [44] to study the general data dropout case; that is, networks at both output and input sides suffer data dropouts. In [45], both data dropouts and communication delays were jointly considered, where the expectation operator and the traditional contraction mapping technique with
To recap, the main techniques for addressing random data dropouts are done by either eliminating the randomness by taking mathematical expectation or projecting the problem into a traditional analysis framework for stochastic systems using Kalman filtering and stochastic approximation techniques. We should emphasize that the former method actually ignores the specific effect but considers the averaged performance of data dropouts.
2) System Types: Like the development processes of other control methodologies, the research results for linear systems are much more than that for nonlinear systems. We note that ILC focuses on evolution along the iteration axis, whereas the timeaxisbased dynamics is less significant due to finite operation length. Therefore, research for linear timeinvariant systems and linear timevarying systems have little distinction. Results with linear systems include [21], [23], [25], [26], [28], [29], [32], [33], [39], [42], [44], [45], most of which are the discretetime type.
There are some papers for nonlinear systems such as [27], [32][34], [41], [43]. However, we note that nonlinear systems are generally of the affine type. This is because affine nonlinear systems separate the evolution influence of the previous state and the current input with respect to time instants. Moreover, the nonlinear functions are assumed to be globally Lipschitz. That is, for a nonlinear function
In addition, stochastic noises are also included in systems in several papers including [22], [35][38], [40]. Specifically, in [22], [35], [37], [40] both random systems disturbances and measurement noises are assumed for linear systems, whereas in [36], [38] only measurement noises are considered as the involved systems are nonlinear. For systems with stochastic noises, the techniques of stochastic control would play an important role in the design and analysis. We also remark that a few results on special systems are reported such as singular systems [46] and multiagent systems [47]. It is worth pointing out that the ILC problem for special types of systems under data dropouts have few reports.
3) Data Dropout Models: As we have clarified at the beginning of the section, there are three models of random data dropouts, namely, RSM, BVM, and MCM. The most popular model is BVM, where data dropouts have a clear probability distribution and good independence. Most ILC papers adopt this model, including [21][23], [25][29], [32][34], [37][41], [44][46]. However, a major issue in BVM is the treatment of successive data dropouts where several limitations are imposed in the existing literature. In particular, the data dropout is independent for different time instants and different iterations in BVM. Thus, it is natural that adjacent data packets may be dropped simultaneously. In many existing papers, in order to provide a specified data compensation, additional requirements are imposed. For instance, in [27], [43], the dropped packet was compensated for with a packet onetimeinstant back within the same iteration. Consequently, a limitation arises where packets at adjacent time instants are not allowed to drop within the same iteration. In [44][46] the lost packet was compensated for with the packet at the same time instant, but oneiteration back. Consequently, there is no simultaneous data dropout at the same time instant across any two adjacent iterations under this condition. Indeed, a more suitable compensation mechanism for the lost packet is to employ the packet at the same time instant from the latest available iteration. In other words, say we find a packet,
There are quite a few papers on other models. In [35], [36] the RSM was used for data dropouts. In this case, the statistical property of data dropouts is removed and thus can vary along the iteration axis. In other words, the distinct difference with RSM is the removal of steady distribution assumptions on data dropouts. In [42], a unified framework was proposed for all the three models where MCM was first studied in the ILC field. Moreover, the authors of [43] carefully analyzed the effect of single packet loss. For the multiple packet loss case, a general discussion was given instead of strict analysis and description. The authors claimed that the data dropout level should be far smaller than 100
4) Dropout Positions: As is seen from Fig. 2, there are two networks connecting the plant and the learning controller, which are separated into different sites. One is at the measurement side to transmit the output information back to the learning controller. The other is at the actuator side to transmit the generated input signal to the plant for the next operation process. To facilitate convergence analysis, most papers only assume data dropouts at the measurement side, while the network at the actuator side is assumed to work well, as in [21], [22], [25], [26], [28], [29], [35][38]. Although some papers claimed that their results can be extended to the general case that both networks suffer packet loss, it is actually not a trivial extension.
In particular, when the network at the actuator side is assumed to work well, i.e., all generated input signals can be successfully transmitted to the plant, the computed control generated by the learning controller and the actual control fed to the plant are always the same. Thus, the input used in the update algorithm is always equal to the actual control. However, when the network at the actuator side is lossy, the computed control may be lost during the transmission and then the plant has to compensate for it with other available signals. Consequently, the actual control may differ from the computed control. In other words, there exists an additional asynchronism between the computed control and the actual control. This random asynchronism imposes extra difficulty in addressing the data dropout problem since it is hard to separate from evolution dynamics as an individual variable. As a matter of fact, it has been proven in [39][41] that such asynchronism can be described by a Markov chain when modeling the dropouts by BVM, which paves a novel way to establish the convergence. Other papers considering the general data dropout position problem include [27], [32][34] where the randomness of the data dropout at the actuator side is eliminated by taking mathematical expectation for recursions of both input errors and tracking errors.
5) Update Schemes: There are two major update schemes which can be referred to when designing the update algorithms. One is eventtriggering and the other one is iterationtriggering. We provide a brief explanation of the schemes by taking the algorithms in the learning controller as an example. The principle of the first update scheme is as follows: if the output information is successfully transmitted, then the learning controller employs such information to generate a new input signal; otherwise, the learning controller would stop updating until the corresponding output information is successfully transmitted in the subsequent iterations. In other words, when the corresponding packet is lost, it is replaced by 0. Clearly, this updating scheme is eventtriggering. We call it an intermittent update scheme (IUS). The principle of the other update scheme is as follows: if the output information is successfully transmitted, then the learning controller employs such information to generate the input, which is same as the previous update scheme; if the output information is lost during transmission, then the learning controller would employ the iterationlatest available output information for generating the input, which is different from the previous scheme. This update scheme keeps working for all iterations no matter whether the information is lost or not, so it is iterationtriggering. We call it a successive update scheme (SUS).
When considering an unreliable network at the measurement side, it has been shown that both IUS and SUS work well for the learning controller, as shown in [37], [38]. It is worth pointing out that a SUS outperforms an IUS when the DDR is large, as it continuously improves the tracking performance. When considering the unreliable network at the actuator side, it is clear that the IUS scheme is not applicable. In other words, the computed control packet which is lost cannot be simply replaced by 0 as it would greatly damage the tracking performance. That is, the lost input signal must be compensated for with a suitable packet to maintain the operation process of the plant. Clearly, the simple compensation mechanism is to employ the latest available input from the previous iteration. In such case, we may regard it as a SUS. As a matter of fact, such mechanism for the input has been reported in [32][34], [39][41]. From another viewpoint, we could regard an IUS as a noncompensation type and a SUS as a simple compensation type. Generally, a sufficient compensation for the dropped data can effectively improve the tracking performance. Thus the specific compensation mechanism is of great significance according to particular problems, but related results are very few.
We have classified the above literature on ILC under data dropouts in Table Ⅰ from the mentioned five perspectives. From this table, it can be seen that the data dropout problem has been deeply investigated from all perspectives. However, we note that the research for MCM and its generalization is promising.
Besides random data dropouts, there are many other random factors caused by limited communication capacity. Communication delay is one of them, which has been witnessed to somewhat progress in the past decade. In earlier attempts [23], [45], the timedelay within an iteration was discussed. Such a delay was assumed to occur for the input signal and modeled by a random matrix according to the lifted system in [23]. The Kalmanfilteringbased stability analysis technique was applied to derive an iterationstability of the proposed update law. In [45] the onestep delay was addressed such that the packet could be transmitted on schedule or onestep later. A Bernoulli random variable was used to describe a random delay, of which the randomness was eliminated by taking expectation in the convergence analysis.
The Bernoulli model was then employed in [49], [50] for describing the random oneiteration communication delay, where the communication delay was assumed to occur at both the output and input sides. That is, the output signal for updating the input may come from either the current or previous iteration, and obeys a simple Bernoulli distribution. Technically, the oneiteration delay provides a certain deterministic property of the communication delay, which allows us to construct a finiteiteration contraction along the iteration axis. Indeed, in [49] the error of the
The successive iterationbased communication delay was considered in [51]. In particular, a largescale system consisting of several subsystems was considered in the paper, where the communication between different subsystems suffered random and possibly asynchronous communication delays due to potentially different work efficiency among subsystems. The communication delay was modeled similarly to the RSM given in the last subsection and decentralized ILC algorithms were constructed based on available information. However, due to random successive communication delays, the memory was assumed to have enough capacity such that the arrived data can be well stored. An extreme case for the memory size is that only the data of one iteration can be accommodated by the memory. Clearly, it is the minimum buffer capacity to ensure the learning process. Such a case was studied in [52], where multiple communication constraints were considered for networked nonlinear systems, including data dropouts, communication delays, and packet disordering. In that paper, a RSM was employed to describe the combined effect of the multiple communication constraints. Both an IUS and a SUS were applied to construct the learning algorithms. Compared with [50], the restrictions on occurrence probability of communication delays were removed and successive communication delays were allowed in the progress. However, we would like to remark that the research on ILC with communication delays has gained little attention from scholars compared with that on ILC with data dropouts. The randomness of uncertain communication delay may lead to a mismatch of the input and tracking error in the update law (for example, (3)). It is vital to figure out the effect of this mismatch in convergence analysis and provide a data compensation mechanism in control synthesis.
C. IterationVarying LengthsIn Section ⅢA, the data dropout is considered independently for different time instants, whereas in practical applications, the data may be dropped dependently along the time axis. In other words, the data dropouts at the former time instants would have a direct influence on those at the later time instants within the same iteration. For example, if one data packet is dropped due to a linkage fault at some time instant, then the following data of the iteration may be all dropped. That is, to the learning controller, the iteration ends early. It results in a typical problem, called the iterationvarying length problem. This problem has been encountered in certain biomedical application systems. For example, while applying ILC in a functional electrical stimulation (FES) for upper limb movement and gait assistance, it has been seen that the operation processes end early for at least the first few passes due to safety considerations because the output significantly deviates from the desired trajectory [53]. The FESinduced foot motion and the associated variablelengthtrial problem are detailed in [54] and [55], which clearly demonstrate the violation of the identicaltriallength assumption typically used in ILC. Another example can be seen in the analysis of humanoid and biped walking robots, which feature periodic or quasiperiodic gaits [56]. For analysis, these gaits are divided into phases that are defined by the time at which the foot strikes the ground, and the duration of the resulting phases are usually not the same from iteration to iteration. A third example can be found in [57], where the trajectorytracking problem for a labscale gantry crane was investigated. In this example, the output was constrained to be within a small neighborhood of the desired reference, because the iteration would end if the output drifted outside the specified boundary, thereby resulting in the varyinglength iteration problem. Whether caused by the communication limits or by the safety consideration, iterationvarying length problem always results in incomplete information problem for the learning process.
There were some early research attempts to provide a suitable design and analysis framework for the iterationvarying length problem that contributed to the groundwork for subsequent investigations [53][57]. For example, based on experimental verifications and primary convergence analysis that were given in [53][55], a systematic proof of the monotonic convergence in different norm senses were further elaborated in [58]. In particular, necessary and sufficient conditions for monotonic convergence were derived strictly by carefully analyzing the path property of the proposed algorithm. Moreover, other issues including the controller design guidelines and influence of disturbances were also discussed. However, no specific formulation of iterationvarying length was imposed in this framework as it concerned the contraction between adjacent iterations.
The first random model of iterationvarying length was proposed in [59] for discretetime systems and then extended to continuoustime systems in [60]. In the model, a binary random variable was used to represent the occurrence of the output at each time instant and each iteration; that is, the random variable is equal to 1 if the output appears and 0 otherwise (similar to the model of data dropout). The variable was then multiplied with the tracking error denoting the actual information of the update process. To compensate for the lost information, an iterationaverage operator for averaging all historical data was introduced to the ILC algorithm in [59], whereas in [60], this average operator was replaced by a movingiterationaverage operator to reduce the influence of very old data. Both operators provide good compensation as shown by the theoretical analysis and simulations. Moreover, a lifted framework of ILC of a discretetime linear system was provided in [61] to avoid the conservatism of the conventional
Stronger convergence results were given in [62] and [63] for linear and nonlinear discretetime systems, respectively. In particular, the classical Ptype ILC algorithm was employed for the discretetime linear system in [62], where the possible iteration length has finite cases. Next, the evolution of liftederrorvectors along the iteration axis was transformed into a random switching system with finite switching states. Consequently, the authors established recursive computation formulas of such vectors' statistics (i.e., the mathematical expectations and covariances). The convergence in the mathematical expectation, mean square, and almost sure senses were derived simultaneously. In [63] the affine nonlinear system was considered. It is clear that the lifting techniques cannot be applied to such types of systems. As a result, a technical lemma on the commutativity of the expectation operator and the absolutevalue operator was first created for paving a novel way to derive the strong convergence. A recent work [64] proposed two improved ILC schemes to fully utilize the iterationmovingaverage operator. Specifically, a searching mechanism was introduced to collect useful information while avoiding redundant tracking information from the past, so a faster convergence speed was expected. In these contributions, the probability distribution of the random length is not required prior.
In addition, some extensions have also been reported in the existing literature. Nonlinear stochastic systems were investigated in [65], where the bounded disturbances were included. The averageoperatorbased scheme similar to [59] was improved by collecting all available information. Nevertheless, we note that a Gaussian distribution of the variable iteration length was assumed, which limits the possible application range. In [66], the authors extended the method to discretetime linear systems with a vector relative degree. Thus, we need to carefully select the output data for the learning algorithms. In addition, the variable length issue was extended to stochastic impulse differential equations in [67] and fractional order systems in [68]. The sampleddata control for continuoustime nonlinear systems was proposed in [69], where both the generic PDtype and a modified PDtype scheme were employed with suitable design conditions of the learning matrices. We remark that the convergence analyses derived in these papers were primarily based on the mature contraction mapping method.
In short, as a special case of passive incomplete information, the iterationvarying length problem has gained some progress. However, the existing literature has witnessed the following limitations. First, most papers considered discretetime systems so that the possible length has finite outcomes. Second, the systems are limited to be linear or globally Lipschitz nonlinear. Third, the averageoperatorbased design of ILC controller is widely studied, which motivates us to consider how to efficiently use the available information. Novel analysis techniques are also of great interest to replace the conventional contractionmapping method. Additionally, the randomly iterationvarying length problem can be regarded as a special case of the data dropout problem; that is, the former is a timeaxisbased successive dropout case (from the actual ending time instant to the desired ending time instant). Therefore, the results in ILC with data dropouts can be applied to deal with the varying length problem and vice versa.
Ⅳ. ILC WITH ACTIVE INCOMPLETE INFORMATIONIn the previous section, we reviewed recent progress on ILC with passive incomplete information. In the section, we proceed to review the progress on ILC with active incomplete information. In other words, we collect the papers where information quality is intentionally reduced. Two major reduction actions are considered, namely, sampleddata ILC and quantized ILC. The former case indicates that only the signal at assigned time instants, rather than the whole time interval, are available, and the latter case indicates that only the assigned values rather than the precise values are available. By sampling and quantization, we can heavily reduce the amount of the data.
A. SampledData ILCIn this subsection, we present a review of sampleddata ILC from the perspective of research issues. Before that, we first formulate the problem of sampleddata ILC, as shown in Fig. 5. Let
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Fig. 5 The research framework of sampleddata ILC 
There are two primary problems associated with sampleddata ILC: the behavior at the sampling instants and how the interval performance (between sampling instants) is. To be specific, the former aims to construct suitable learning algorithms to guarantee convergence at the sampling instants, and the latter focuses on quantitative analysis of the tracking performance between different sampling instants and possible solutions to reduce the tracking errors in the sampling interval. Generally, the former problem is similar to discretetime ILC as they share the same design and analysis techniques. However, the latter problem indeed makes sampleddata ILC different from the traditional discretetime systems.
Considering the system models, both linear and affine nonlinear systems without disturbances attract the most attention, and both linear and affine nonlinear systems with bounded disturbances have been under investigation, while the other systems are of little consideration. The reference classification is given in Table Ⅱ. These papers are mainly written by several research groups with different special interests. Therefore, we review the publications by the research interests/groups. In each category, four perspectives of the publications are explored, i.e., the system model, the update scheme, the convergence result, and the analysis techniques.
1) FrequencyBased SampledData ILC: The frequencybased design and analysis of sampleddata ILC are presented in [70][73], where the kernel issue focuses on the fundamental analysis and synthesis of sampleddata theory in ILC.
Reference [70] presented a framework for the design and analysis of sampleddata ILC in both time and frequency domains. For a fundamental framework, the LTI system was adopted, while Ptype, Dtype, D
2) Bounded Convergence Under Bounded Disturbances: A series of papers on the bounded set convergence at the sampling time instants are contributed for linear and nonlinear systems with bounded disturbances [74][79]. In these papers, bounded system disturbances
In an early paper [74], the conventional Ptype update law was employed using the available sampling information for affine nonlinear systems. The convergence was conducted based on the wellknown
It is noted that different update algorithms are investigated by Chien and his coworkers including Ptype, Dtype, and feedback of current error. This research mainly focuses on bounded convergence to some given set by letting the sampling period be small enough under bounded disturbances.
3) SampledData ILC With Arbitrary Relative Degree: An indepth study on sampleddata ILC for nonlinear systems with arbitrary relative degree was carried out in [80][84]. The relative degree is a description of the inputoutput relationship, which reflects the minimum effect order between the input and its corresponding output. For continuoustime systems, the relative degree is defined by the Lie derivative of the output with respect to the input; for discretetime systems, it is defined by the function composition. However, for sampleddata control, the integral should be included to define the relative degree. Consider the following SISO affine nonlinear systems as an example,
$ \begin{split} \dot x_k(t)&=f(x_k(t))+b(x_k(t))u_k(t)\\ y_k(t)&=g(x_k(t)) \end{split} $  (7) 
where
$ \begin{align*} &\int_{j\Delta_T}^{(j+1)\Delta_T}L_bg(x(t_1))dt_1=0, \\ &\int_{j\Delta_T}^{(j+1)\Delta_T}\int_{j\Delta_T}^{t_1}\cdots\int_{j\Delta_T}^{t_i}L_bL_f^ig(x(t_{i+1}))dt_{i+1}\cdots dt_1=0, \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad 1\leq i\leq \eta2, \\ &\int_{j\Delta_T}^{(j+1)\Delta_T}\int_{j\Delta_T}^{t_1}\cdots\int_{j\Delta_T}^{t_{\eta1}}L_b L_f^{\eta1}g(x(t_{\eta}))dt_{\eta}\cdots dt_1\neq 0. \end{align*} $ 
Roughly speaking, a relative degree larger than 1 indicates that the direct inputoutput coupling matrix is zero. In such a case, it is interesting to ask whether the conventional Ptype update scheme guarantees the convergence. Such a problem was resolved in [80][82]. In particular, it was shown in [80], [81] that the basic Ptype scheme based on the available sampled data can ensure a zeroerror tracking for the sampling time instants. It was then extended to a general case called sampleddata ILC with lowerorder differentiations for general nonlinear systems in [82], where the authors used lowerorder to indicate that the derivative in the learning controller was less than the relative degree.
Another important issue is the initial rectifying problem [83], [84]. In other words, the initial state is shifted from its desired value. These papers propose an effective rectifying mechanism such that the actual output would be shifted back to the desired one after some time interval. In [83], the fixed initial shift was considered and the proposed initial rectifying action was able to drive the system output to the desired trajectory within a specified error bound. Then the initial shift was extended to an arbitrarily varying case and a socalled varyingorder sampleddata ILC was designed and analyzed. In all the studies, the convergence analysis was established with the help of a technical lemma, which is an extension of the contraction mapping principle.
4) Interval Performance of SampledData ILC: It is observed that in papers such as [74][84], only the performance at the sampling instants is considered while the intersample behavior is seldom discussed. However, achieving good performance at the sampling instants (atsample) can be at the expense of poor intersample behavior [85]. However, guaranteeing acceptable intersample tracking performance is a difficult problem for sampleddata ILC. Early attempts are given in [86], [87].
In [86], the multirate ILC approach was proposed to balance the atsample performance and the intersample behavior, where the key idea was to generate a command signal at a low sampling rate by using fast sampled measurements. The details of multirate systems and multirate ILC were given to enable an optimal sampleddata ILC in the paper. Further, the authors developed an ILC framework for sampleddata systems by incorporating the system identification and a loworder optimal ILC controller in [87], as an ongoing study of [86]. The proposed system identification procedure delivers a model that encompasses the intersample in a multirate setting for the closedloop system so that the resulting model could be used for the optimal ILC synthesis. As a consequence, the computational burden is much less than common optimizationbased algorithms for large systems.
In short, there still lack more indepth studies on the intersample behavior of sampleddata ILC including novel design and analysis technique for improving the tracking performance between different sampling instants.
5) Scattered Contributions: Reference [88] presented a limiting property of the inverse of sampleddata systems. To be specific, for a continuoustime system with a relative degree of one or two, the inverse of the corresponding sampleddata system can approximate the inverse of the original continuoustime system independently of the stability of the zeros as the sampling period
Timedelay was introduced into the affine nonlinear model in [89] with other settings similar to [74], [75]. The PDtype update scheme was employed with a bounded convergence analysis; however, the differential signal is not suitable for sampleddata implementation.
The sampleddata ILC for singular systems was addressed in [90] using a Ptype learning algorithm and
Based on the above reviews, we have several remarks. First of all, much attention is paid to LTI and affine nonlinear systems with/without bounded disturbances, whereas there has been little progress with timevarying systems, general nonlinear systems, and stochastic systems. Moreover, most papers contribute to the atsample performance, while the intersample behavior is seldom considered. However, good atsample tracking performance does not necessarily imply acceptable intersample behavior. Furthermore, the traditional contraction mapping method and its extensions are the main technique for convergence analysis, which restricts the research range of systems and problems. Last but not least, the implementation of sampleddata ILC in practical applications is of great significance, but few publications are found on this direction [92]. Therefore, a systematic framework of sampleddata ILC is yet blank and much effort should be made by considering the above aspects of sampleddata ILC. Meanwhile, a sampleddata control methodology is usually combined with the quantized technique to further reduce the data amount, where the latter is reviewed in the next subsection.
B. Quantized ILCTo reduce the communication burden, another effective method is to introduce a quantization mechanism. That is, we first quantize the measured signal and then transmit the signal. In fact, the quantization method has been deeply studied in the networked control field; however, few papers have been reported on quantized ILC.
An early attempt on the quantized ILC was given in [93], where the output measurements were quantized by a logarithmic quantizer and then fed to the controller for updating ILC law. By using the sector bound technique and conventional contraction mapping method, it was shown that the tracking error converged to a small range whose upper bound depended on the quantization density. Meanwhile, the tracking error also depended on the target value, which can be seen from the expression of the upper bound. That is, the larger the output measurement is, the larger the final tracking error upper bound is. To achieve zeroerror tracking performance, an alternative framework was proposed in [94], where the desired reference was first transmitted to the local plant to generate a tracking error and then the tracking error was quantized by a logarithm quantizer and transmitted. In other words, the tracking error, rather than the output signal, was quantized. This scheme can guarantee the zeroerror convergence with the inherent principle of the logarithmic quantizer. The extension to stochastic systems was addressed in [95], where a detailed comparison of the tracking index was provided by considering both stochastic noises and quantization error. It can be seen from the simulations that the ultimate index value is completely generated by the stochastic noises, indicating that the quantization error is eliminated asymptotically. The extension of the above quantization methods to input quantization case was provided in [96] with similar conclusions of [93], [94]. Similar idea of quantizing the measured error was also used in [97], [98] for dealing with discretetime and continuoustime multiagent systems, respectively. We remark that the logarithm quantizer should have infinitesimal precision near zero, which is hard to implement in applications. Thus, it is important to propose new quantization mechanisms to improve the tracking performance.
In [99], a uniform quantizer was used with an additional scaling mechanism implemented between the plant and controller. In this case, the measured signal is first scaled by prior scaling functions and then quantized by the uniform quantizer; then, at the controller, the received signal is converted using the scaling functions again to obtain a wellapproximation of the original signal. Such process is called the encoding and decoding mechanism. In fact, the scaling functions play a role to enhance quantization precision. In [47], another quantization method called
In sum, quantized ILC is still in its first stage compared with more fruitful results using conventional quantized control. Two valuable research directions should be highlighted for this issue. The first one is to provide an estimation on the relationship between quantized data and the tracking performance. The other one is to investigate effective soft mechanisms for data acquiring, transforming, transmitting, and recovering to eliminate or reduce the effect of quantized data.
Ⅴ. DATA ROBUSTNESS AND PROMISING DIRECTIONSAs has been explained in the previous sections, ILC requires little information on the system matrices. In other words, the design of learning controller mainly depends on the input and tracking information of the previous iterations. Thus, it is a typical datadriven method [101]. From this viewpoint, the ILC problem under incomplete information essentially is a data robustness problem. That is, the inherent control objective is to investigate how the control schemes perform according to different levels of data loss. Generally, if the designed learning control scheme can behave well even if most data is lost due to various restriction conditions, we say the scheme has good data robustness; if the designed learning control scheme is very sensitive to the data loss, we say the scheme has poor data robustness. However, we should note that the concept of data robustness is still unclear [101], and therefore, the research on ILC under incomplete information would settle a fundamental cognition and may guide us to find a direction in establishing the data robustness for datadriven control.
In the traditional control theory, robust control indicates an approach to controller design for dealing with model and/or parameter uncertainty. We define the robustness of this framework as the property of maintaining certain control performance when the uncertain parameters or disturbances vary within some set (typically compact). Therefore, the traditional control robustness is defined with respect to the system itself. While considering the datadriven control, the system information is excluded. Thus, it is not suitable to follow the above definition of control robustness. As a matter of fact, the robustness for datadriven control should be coined with respect to the information/data itself. Particularly, the inherent relationship between the incomplete information/data and the control performance would explicitly describe the robustness issue. Along this line, we would like to share the following points. First, the average data loss can approximate
In ILC with incomplete information, the emphasis should be put on the robustness significance contained in the lost information and related control system design. In other words, we should concentrate on the indepth understanding of the restriction and tradeoff between the information and tracking indices of ILC (such as tracking precision, convergence speed, control energy, and data amount). Based on this relation, we can evaluate the key factors of improving the tracking performance when losing partial data. In this respect, we highlight the following possible prospective research topics.
1) A good solution for data dropout problem can be extended to many other types of incomplete information environments; thus, it deserves more deep investigations on the essential points, for examples, the quantitative influence of data dropouts on tracking performance, novel compensation mechanisms of the lost data with respect to specified objectives, and the controller design and analysis under general data dropout environments.
2) When considering communication channels, many open problems are waiting for profound exploration and exploitation on various communication constraints such as random communication delay and multiple delays, random and/or unrecognized packet disordering, very limited communication bandwidth, insufficient memory storage, and multichannel transmission and fusion problem. Moreover, the combined effect of multiple communication constraints is also of interest.
3) Sampling is an effective and economic treatment of continuoustime systems using computer technology, whereas the specific involvement of sampling techniques is not so clear for applications. It yet lacks an explicit answer to the many practical requirements such as the lowest sampling frequency, the specific sampling pattern (uniform or nonuniform), the inherent relation between the sampling pattern and the control performance. Moreover, it is also important to develop suitable sampling framework to satisfy the tradeoff between minimum data amount and optimal tracking performance.
4) Quantized ILC is in its embryonic stage as only tentative convergence results for the common quantizer are provided, whereas the essential performance improvements based on finite precision quantizer are not investigated. The kernel issue is to deal with the inevitable quantization error, find out the tracking limitation using quantized data, search suitable treatments for eliminating or reducing the effect of quantization, and establish the analysis and synthesis framework for quantized ILC.
5) In the existing literature, the passive incomplete information is generally formulated by random variables and techniques in stochastic control are applied to derive the performance analysis, whereas the active incomplete information is usually described as a certain loss variable and the bounded convergence analysis for conventional ILC is achieved. Since the ILC problem can be well conducted as a repetitive process [102], it is expected the repetitive process based approach can provide a meaningful solution framework to the ILC with incomplete information.
When investigating the data robustness issue of ILC, we should pay special attention to the triple shown in Fig. 6: (incomplete) information, index, and control. The incomplete information not only includes both passive and active types, but also includes a mixture of both. The indices contain tracking precision, convergence speed, input energy, etc. The control part includes algorithms design and analysis as well as the experimental verification of the theoretical results. Based on this triple, we have a triple of key points in investigation: restrictive relationship, control system, and synthesis/analysis. In particular, the restrictive relationship between the incomplete information and control indices plays a fundamental role. With an indepth understanding of the relationship, one can implement the specific realization of the control system and then establish the synthesis and analysis framework for the specific problems.
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Fig. 6 The research triple of ILC with incomplete information 
In this paper, we have surveyed the recent progress on ILC with incomplete information, which is caused by practical conditions, or passive incomplete information, and manmade treatments, or active incomplete information. For passive incomplete information, the random loss conditions such as data dropouts, communication delay and constraints, and iterationvarying lengths are given much attention. For active incomplete information, we focus on the sampleddata ILC and quantized ILC, both of which considerably reduce the amount of data required for acquiring and processing. Based on this survey, it is observed that ILC with incomplete information is actually a case of the data robustness problem. For such a problem, two issues should be given sufficient concern: the first is to evaluate the influence of incomplete information on control performance, and the second is to design a suitable synthesis and analysis framework. It is expected that this survey will give the reader a better understanding of ILC with incomplete information and provide useful guidelines for further research to perfect the framework.
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