Iterative Learning Control System Research Articles

Towards Transparent Control Systems: The Role of Explainable AI in Iterative Learning Control

This paper presents a novel approach to improving the performance and interpretability of Iterative Learning Control (ILC) systems through the integration of Explainable Artificial Intelligence (XAI) techniques. ILC is a powerful method used across various domains, including robotics, process control, and traffic management, where it iteratively refines control inputs based on past performance to minimize errors in system output. However, traditional ILC methods often operate as "black boxes," making it difficult for users to understand the decision-making process. To address this challenge, we incorporate XAI, specifically SHapley Additive exPlanations (SHAP), into the ILC framework to provide transparent and interpretable insights into the algorithm's behavior. The study begins by detailing the evolution of ILC, highlighting key advancements such as predictive optimal control and adaptive schemes, and then transitions into the methodology for integrating XAI into ILC. The integrated system was evaluated through extensive simulations, focusing on robotic arm trajectory tracking and traffic flow management scenarios. Results indicate that the XAI-enhanced ILC not only achieved rapid convergence and high control accuracy but also maintained robustness in the face of external disturbances. SHAP analyses revealed that parameters such as the proportional gain (Kp) and derivative gain (Kd) were critical in driving system performance, with detailed visualizations providing actionable insights for system refinement. A crucial metric for control precision was the root mean square error (RMSE), which was reduced to as low as 0.02 radians in the robotic arm case, indicating extremely precise tracking of the intended route. Similarly, the ILC algorithm effectively maintained the ideal traffic density within the predetermined bounds in the traffic management scenario, resulting in a 40% reduction in congestion compared to baseline control measures. The resilience of the ILC algorithm was also examined by introducing changes to the system model, external disturbances, and sensor noise. The algorithm demonstrated a high degree of stability and accuracy in the face of these disruptions. For instance, in the robotic arm case, adding noise to the sensor readings had a negligible effect on the algorithm's performance, increasing the RMSE by less than 5%. This integration of XAI into ILC addresses a significant gap in control system design by offering both high performance and transparency, particularly in safety critical applications. The findings suggest that future research could further enhance this approach by exploring additional XAI techniques and applying the integrated system to more complex, real-world scenarios.

Method for predicting the lost input of iterative learning control systems over AWGN channels

This paper researches the method for predicting the lost input of iterative learning control (ILC) systems over additive white Gaussian noise (AWGN) channels. It directly takes into account a proportional ILC controller, which guarantees the convergence of objects under ideal communication conditions. By employing prior information on ILC controllers and AWGN channels, a state space model is built. According to this model and the innovation analysis theory, a predictor is developed for actuators to calculate the input transmitted by ILC controllers but lost in AWGN channels. Compared with the iteration compensation in existing literature, the proposed prediction compensation simultaneously guarantees the convergence accuracy and speed of ILC systems over AWGN channels. Numerical simulation demonstrates the usefulness of the developed prediction method.

Almost sure and mean square convergence of iterative learning control system with state feedback under random packet dropouts

This paper investigates the almost sure and mean square convergence of a state feedback-based iterative learning control scheme for a class of discrete-time multi-input–multi-output linear systems with a direct feedback term in the presence of random data dropouts. Firstly, the realisability of the control system is considered. It is shown that the realisability of the control system is only determined by the direct feedback matrix, which has nothing to do with the input–output coupling matrix, and the direct feedback matrix to be full-row rank is enough to guarantee the system realisable. Secondly, the convergence of output tracking errors is analysed in both the almost sure and mean square senses, respectively. By the contraction mapping method combined with matrix decomposition and transformation techniques, it is strictly proved that the output sequence can converge to any given desired trajectory in the sense of almost sure and mean square, respectively, if and only if the general spectral radius condition is satisfied. Finally, an example is given to show the correctness of the theoretical analysis.

Optimal state filters for networked iterative learning control systems with data losses and noises

SummaryData losses and noises in both forward and feedback channels significantly impact the convergence of networked iterative learning control (ILC) systems. To address this issue, this article considers a class of linear time‐invariant objects controlled by proportional ILC controllers, an optimal state filter is then designed at the ILC controller side that aims to guarantee the convergence of the input transmitted by ILC controllers. First, two data transmission processes are introduced to account for the effects of data losses and noises. Second, a filtering model is established utilizing only the object information and the aforementioned data transmission processes. Third, the optimal state filter is designed on the basis of the orthogonal projection principle. This filtered state facilitates the acquisition of actual output errors, thus improving the convergence of the input transmitted by ILC controllers. Simulation results demonstrate the effectiveness of the proposed state filters.

Robust model-based predictive iterative learning control for systems with non-repetitive disturbances

Iterative Learning Control (ILC) is commonly used for batch processes. However, it may face difficulties when dealing with non-repetitive disturbances and inconsistent initial states. In situations with non-repetitive disturbances, the output may disobey constraints and negatively impact tracking performance when using existing predictive ILC algorithms. This paper introduces a new model-predictive ILC incorporating feed-forward and feedback mechanisms. This new approach evaluates and attenuates the impact of non-repetitive disturbances on the output. As a result, constraints are guaranteed, and tracking performance is preserved and improved, even in the presence of non-repetitive disturbances. Furthermore, if the desired trajectory is unattainable, the proposed ILC can robustly track an optimal trajectory while still guaranteeing constraints. The convergence is proven rigorously. Finally, two examples are provided to demonstrate the effectiveness of this new approach.

Monotonically convergent iterative learning control by time varying learning gain revisited

This note considers the problem whether we can design a time-varying learning gain to ensure the monotone convergence of system output tracking errors (SOTEs) in the sense of unweighted 1/2/∞-norm for iterative learning control systems. Firstly, it points that the iterative learning control update law with the exponentially decaying learning factor considered in Moore et al. (2005) cannot ensure the monotonic convergence of SOTEs in the sense of unweighted 1-norm, 2-norm or ∞-norm. Secondly, it is strictly proven that there exists a time-varying learning gain to ensure the monotone convergence of SOTEs in the sense of unweighted ∞-norm and there is no time-varying learning gain to ensure the monotone convergence of SOTEs in the sense of unweighted 1-norm. Finally, this paper presents a sufficient condition under which there exists a time-varying learning gain to ensure the monotone convergence of SOTEs in the sense of unweighted 2-norm.

Input predictors for networked iterative learning control systems with data dropouts and time delays

Hold-up compensation decelerates the convergence of iterative learning control (ILC) systems with data dropouts and time delays. Only depending on the prior knowledge of both ILC controllers and transmission channels, this paper develops a predictor to calculate the input not received on time due to data dropouts and time delays. First, a controller adopting the proportional learning strategy is considered directly, which is appropriate for objects in ideal communication conditions. After that, two data-receiving equations are given to describe the effect of data dropouts and one-step time delays. Finally, a predictor is designed according to the innovation analysis approach. Since the prediction uses all historical input at the identical time index in previous iterations, the predicted input is more approximate to the one not received on time than the input held up for compensation. Simulation results show the object with prediction compensation tracks the expected trajectory faster than that with input-hold compensation.

Data-based trackability criteria and control design for disturbed learning systems

This paper aims to establish a data-based control design framework for linear iterative learning control (ILC) systems subject to disturbances. Through the proper selection of the test inputs, some output data are collected to exploit two trackability criteria of the desired reference for ILC systems such that the implementability of the perfect tracking objective is ensured, where the persistent excitation condition may not be imposed. Furthermore, two classes of data-based ILC updating laws are presented by incorporating the design idea of the observer, for which the collected input and output data are only leveraged. It is shown that given any trackable desired reference, the perfect tracking objective can be realized for ILC systems under the proposed data-based ILC updating laws in the absence of any model information.

Enhancing Iterative Learning Control With Fractional Power Update Law

The P-type update law has been the mainstream technique used in iterative learning control (ILC) systems, which resembles linear feedback control with asymptotical convergence. In recent years, finite-time control strategies such as terminal sliding mode control have been shown to be effective in ramping up convergence speed by introducing fractional power with feedback. In this paper, we show that such mechanism can equally ramp up the learning speed in ILC systems. We first propose a fractional power update rule for ILC of single-input-single-output linear systems. A nonlinear error dynamics is constructed along the iteration axis to illustrate the evolutionary converging process. Using the nonlinear mapping approach, fast convergence towards the limit cycles of tracking errors inherently existing in ILC systems is proven. The limit cycles are shown to be tunable to determine the steady states. Numerical simulations are provided to verify the theoretical results.

Observer-Based Distributed Methods for Learning Control Systems

In learning systems, high operation precision is often a desirable objective for the algorithm design. Though centralized algorithms are generally adopted, they are subjected to restrictive hypotheses on the learning systems. To overcome this challenging problem, we aim to propose some distributed learning algorithms that focus specifically on achieving the perfect tracking tasks for iterative learning control (ILC) systems. By noting the equivalent relation between the perfect tracking problem of ILC systems and the solving problem of linear algebraic equations (LAEs), we first present an observer-based distributed learning algorithm to solve LAEs, where a multiagent system is constructed with every agent being only required to access some partial information for LAEs. The distributed learning algorithm benefits from integrating both observer-based design and consensus-based design ideas such that for any solvable LAE, all agents can agree on a common solution of it under any initial conditions of agents, regardless of whether it has a unique solution or not. Then, with the distributed learning algorithm for LAEs, we further develop two classes of distributed learning control algorithms for ILC systems, which establish the perfect tracking objective in the presence of the trackable desired output even without using the basic relative degree condition that is generally imposed for conventional ILC.

Learning-Ability of Discrete-Time Iterative Learning Control Systems with Feedforward

This paper considers the learning-ability for discrete-time iterative learning control (ILC) systems with feedforward. More specifically, the relation between the output realizability and the feedforward matrix is first established. Then, the learning-ability of four ILC systems is considered. It is shown that the proportional type (P-type) update law can only ensure the fully asymptotic learning-ability. By only using the feedforward matrix, a more efficient point-wise P-type update law is developed, which can ensure the fully -step learning-ability, where is the trial length. In the case that the state is measurable and controllable, it is proven that the update law with current state feedback can ensure the fully monotone learning-ability and the fully 2-step learning-ability, respectively. In addition, by only using the output data at the previous trial, a full output feedback update law is proposed, which can respectively ensure the fully 2-step learning-ability and the fully monotonic learning-ability.

Data-Based Control for Learning Systems With Nonlinear Dynamics: A Case Study

This paper focuses on exploiting a data-based control design framework for a class of iterative learning control (ILC) systems with nonlinear dynamics. A test method is first presented for nonlinear ILC systems to generate some helpful output data under some specific test inputs. Then, a data-based P-type ILC updating law is developed by resorting to these input and output data, for which a simple data-based selection condition on the gain matrix is provided to accomplish the perfect tracking objective of ILC systems having the globally Lipschitz nonlinear dynamics. The proposed data-based P-type ILC updating law is applicable for nonlinear ILC systems without the common full rank requirement, where none of the specific model information is utilized. A simulation example is given to illustrate the validity of the data-based ILC design framework for nonlinear ILC systems.

Dynamic ILC for Linear Repetitive Processes Based on Different Relative Degrees

The current research on iterative learning control focuses on the condition where the system relative degree is equal to 1, while the condition where the system relative degree is equal to 0 or greater than 1 is not considered. Therefore, this paper studies the monotonic convergence of the corresponding dynamic iterative learning controller systematically for discrete linear repetitive processes with different relative degrees. First, a 2D discrete Roesser model of the iterative learning control system is presented by means of 2D systems theory. Then, the monotonic convergence condition of the controlled system is analyzed according to the stability theory of linear repetitive process. Furthermore, the sufficient conditions of the controller existence are given in linear matrix inequality format under different relative degrees, which guarantees the system dynamic performance. Finally, through comparison with static controllers under different relative degrees, the simulation results show that the designed schemes are effective and feasible.

Control Design for Iterative Methods in Solving Linear Algebraic Equations

In the interaction between control and mathematics, mathematical tools are fundamental for all the control methods, but it is unclear how control impacts mathematics. This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results. By proposing an iterative method that integrates a learning control mechanism, a class of tracking problems for iterative learning control (ILC) is explored for the problem solving of LAEs. A trackability property of ILC is newly developed, by which analysis and synthesis results are established to disclose the equivalence between the solvability of LAEs and the controllability of discrete control systems. Hence, LAEs can be solved by equivalently achieving the perfect tracking tasks of resulting ILC systems via the classic state feedback-based design and analysis methods. It is shown that the solutions for any solvable LAE can all be calculated with different selections of the initial input. Moreover, the presented ILC method is applicable to determining all the least squares solutions of any unsolvable LAE. In particular, a deadbeat design is incorporated to ILC such that the solving of LAEs can be completed within finite iteration steps. The trackability property is also generalized to conventional two-dimensional ILC systems, which creates feedback-based methods, instead of the common used contraction mapping-based methods, for the design and convergence analysis of ILC.

Lyapunov‐based iterative learning feedback control design for parabolic MIMO PDEs with time‐varying delays

This paper presents an iterative learning control approach to achieve precise tracking control for a class of repeatable parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) with time-varying delays over a finite time interval. Feedback control is utilised in the iterative learning control system to improve the convergence speed of the repeatable system. A Lyapunov–Krasovskii functional is constructed to deal with the time-varying delay problem caused by the model uncertainty of MIMO PDEs. Collocated piecewise control and piecewise measurement is considered based on the distributions of actuators and sensors to generate multiple inputs and multiple outputs. By using the mean value theorem for integrals, variants of the Poincaré–Wirtinger inequality in the 1D space, Cauchy–Schwartz inequality and Gronwall–Bellman lemma, sufficient conditions that guarantee the convergence of the iterative learning feedback control system are presented. Numerical simulation experiments and comparisons verify the effectiveness and advantages of the proposed method.

Optimal input filtering for networked iterative learning control systems with packet dropouts and channel noises in both sides

AbstractThis article is concerned with the convergence performance of wireless networked iterative learning control (ILC) systems with data dropouts and channel noises in both sensor‐to‐controller and controller‐to‐actuator channels. In order to improve the convergence performance of such ILC systems, an optimal input filter is developed at the actuator side to estimate controller updated input with the effect of these network uncertainties. Specifically, a filtering model is constructed only by using the learning process of P‐type controllers and the transmission process of both measured output data and updated input data. On the basis of this model and the orthogonality principle, the filter in the sense of linear minimum variance is designed at the actuator side so that the controller updated input could be estimated in iteration domain. The convergence performance of filtering error covariance matrix is analyzed theoretically. Moreover, because the filter design does not employ plant information, the tracking performance of any plant with P‐type learning controllers can be improved by driving with the filtered input. Simulation results show that the proposed filtering method is effective.

가속센서 데이터 기반 반복 학습 제어 적용 디스플레이 패널 이송 로봇 경로 오차 보상에 대한 연구

Transfer robots for large-sized panels used in the display industry need to compensate for path error and reduce vibration. The iterative learning control (ILC) technique can simply compensate for the uncertainty of a control system in a repetitive motion. This study introduces an ILC compensation system applied with an accelerometer to a display panel transfer robot control system. The ILC technique was used to reduce the path error and vibration induced the flexibility of the large size robot. This method was applied to a robot system without the system model of the mechanical and measurement elements. To improve the iterative learning performance through the accelerometer, the ILC is configured by applying an acceleration element and time shift method to the PD-Offline ILC algorithm. In addition, based on the characteristics of repetitive motion, the ILC derives an acceleration data-based position estimation value. In this study, the ILC system and a large-sized panel transfer robot were implemented in MATLAB-Simulink with RECURDYN. The path errors and vibration level of the robot with a suggested ILC of 20 repeated learnings were reduced by more than 90%.

Iterative Learning Control for Discrete-Time Systems With Full Learnability.

This article considers iterative learning control (ILC) for a class of discrete-time systems with full learnability and unknown system dynamics. First, we give a framework to analyze the learnability of the control system and build the relationship between the learnability of the control system and the input-output coupling matrix (IOCM). The control system has full learnability if and only if the IOCM is full-row rank and the control system has no learnability almost everywhere if and only if the rank of the IOCM is less than the dimension of system output. Second, by using the repetitiveness of the control system, some data-based learning schemes are developed. It is shown that we can obtain all the needed information on system dynamics through the developed learning schemes if the control system is controllable. Third, by the dynamic characteristics of system outputs of the ILC system along the iteration direction, we show how to use the available information of system dynamics to design the iterative learning gain matrix and the current state feedback gain matrix. And we strictly prove that the iterative learning scheme with the current state feedback mechanism can guarantee the monotone convergence of the ILC process if the IOCM is full-row rank. Finally, a numerical example is provided to validate the effectiveness of the proposed iterative learning scheme with the current state feedback mechanism.

Performance Improvement of Low-Cost Iterative Learning-Based Fuzzy Control Systems for Tower Crane Systems

This paper is dedicated to the memory of Prof. Ioan Dzitac, one of the fathers of this journal and its founding Editor-in-Chief till 2021. The paper addresses the performance improvement of three Single Input-Single Output (SISO) fuzzy control systems that control separately the positions of interest of tower crane systems, namely the cart position, the arm angular position and the payload position. Three separate low-cost SISO fuzzy controllers are employed in terms of first order discrete-time intelligent Proportional-Integral (PI) controllers with Takagi-Sugeno-Kang Proportional-Derivative (PD) fuzzy terms. Iterative Learning Control (ILC) system structures with PD learning functions are involved in the current iteration SISO ILC structures. Optimization problems are defined in order to tune the parameters of the learning functions. The objective functions are defined as the sums of squared control errors, and they are solved in the iteration domain using the recent metaheuristic Slime Mould Algorithm (SMA). The experimental results prove the performance improvement of the SISO control systems after ten iterations of SMA.

Iterative Learning Control Design for Nonlinear Systems with Reference Trajectory Switching

This paper continues the development of iterative learning control for systems performing the same finite duration task repeatedly. Each execution is known as a trial, and the finite duration of each trial is known as the trial length. In many designs the reference trajectory is not changed during the generation of the trials, but there are applications where changing, or switching, this trajectory is required. Design in this case has been addressed based on a linear model for the dynamics, and there has been some results on design for examples where a nonlinear model of the dynamics has to be used. The development of this latter case based on the use of the recently developed vector Lyapunov function-based stability theory for nonlinear repetitive processes combined with feedback linearization is the subject of this paper. Moreover, the case when the reference trajectory is required to switch after a number of trials have been completed is considered and a switching rule is developed to avoid to avoid the increase in the tracking error after switching occurs.