Dual-Mode Data-Driven Iterative Learning Control: Applications in Precision Manufacturing and Intelligent Transportation Systems

1  Introduction

Iterative Learning Control (ILC), as an optimization method for repetitive dynamic systems, enhances tracking accuracy through iterative adjustment of control inputs, demonstrating significant potential in both precision manufacturing and intelligent transportation systems (ITS) [1–3]. In manufacturing applications, ILC has been effectively employed to improve positioning accuracy, surface quality, and production throughput [4]. Within transportation domains, its data-driven characteristics enable continuous improvement of autonomous vehicle trajectory tracking [5,6], platoon coordination, and traffic signal timing optimization [7] in dynamic environments.

As an important branch of modern control theory, the theoretical framework of ILC is established upon six fundamental postulates [8].

1.   The reference signal r of each iteration is fixed.

2.   The initial state xk(0) is reset to the same value for each iteration.

3.   The length of time T of each iteration is fixed.

4.   The system dynamics operator G is fixed for each iteration.

5.   The input signal update follows the law uk+1=F(uk,ek).

6.   The system dynamics operator G is invertible.

The aforementioned postulates establish the complete theoretical architecture for iterative learning control, with the corresponding framework diagram presented in Fig. 1.

Figure 1: Typical closed-loop structure of ILC

•   The error information acquisition link provides the necessary feedback basis for the system.

•   The control law updating link is responsible for the dynamic adjustment of the algorithm parameters.

•   The input signal correction link completes the optimization of the control command generation.

Through proper algorithm design, this approach theoretically guarantees the convergence of system tracking errors to zero. With demonstrated advantages in both rapid convergence and asymptotic optimization performance [9,10], it exhibits significant application value in control engineering.

In terms of historical development, the initial P-type ILC algorithm features a simple structure but suffers from limited noise robustness and slow convergence rate. The D-type ILC formulation enhances high-frequency noise suppression capability while exhibiting sensitivity to measurement quantization noise due to its derivative operation. The subsequently developed PD-type ILC synergistically combines these approaches, achieving an optimal balance between robustness and convergence performance through gain scheduling. Recently, advanced ILC architectures integrating adaptive control and robust control techniques have demonstrated significant improvements in tracking accuracy, closed-loop stability, and convergence rate [11–13].

The successful application of ILC in industrial robotics provides an effective solution to the accuracy bottleneck problem in repetitive trajectory tracking [14,15]. In precision manufacturing systems, it significantly improves machining accuracy and productivity, thereby promoting the transformation and upgrading of manufacturing industries [16,17]. For motion control platforms, ILC establishes a novel methodology for achieving high-speed and high-precision control objectives [18–20]. The successful applications of ILC not only validate its theoretical foundations but also demonstrate its practical effectiveness for sustainable development. Through continuous technological advancements, ILC has achieved significant breakthroughs in both theoretical system construction and engineering implementations. Current research focuses on three primary directions: the evolution of learning algorithms, improvements in analytical methodologies, and integration with emerging technologies such as precision manufacturing. Furthermore, scholars are making substantial contributions to cutting-edge research areas including frequency-domain analysis, two-dimensional system theory, and energy-based control methods [21–25].

ILC exhibits distinctive advantages in precision manufacturing applications due to its model-independent nature [26,27]. The data-driven characteristic of ILC is particularly well-suited to address the complex challenges inherent in precision manufacturing processes [28]. Precision manufacturing systems involve the coupling of multiple physical fields, including complex phenomena such as thermal-force-fluid coupling, and it is extremely challenging to establish accurate mathematical models [29,30]. ILC effectively addresses modeling challenges by directly optimizing process parameters through iterative refinement of measured part morphology data [31]. In metal additive manufacturing, ILC enables autonomous adjustment of critical parameters including laser power and scanning speed based on layer-by-layer dimensional analysis, eliminating dependence on complex melt pool dynamics models [32].

In the face of repetitive disturbances, ILC demonstrates excellent compensation capabilities [33]. Manufacturing processes exhibit multiple periodic disturbances, including fluctuations in the powder feeding system, thermal accumulation effects, and inter-layer temperature variations [34,35]. Through iteration, ILC automatically identifies these recurring disturbance patterns and compensates for them in subsequent printing cycles. For instance, in polymer 3D printing, ILC adapts to extrusion changes caused by nozzle temperature fluctuations, maintaining consistent layer deposition thickness [36]. Progressive layer deposition enables iterative enhancement of ILC precision, yielding asymptotically diminishing disturbance influence [37,38].

ILC and precision manufacturing technology exhibit inherent synergy, primarily manifested in the strong alignment between their process characteristics and control requirements [39]. The cyclic characteristics inherent in precision manufacturing operations exhibit particular compatibility with the recursive learning paradigm implemented by ILC.

•   Firstly, its cyclic machining characteristics provide opportunities for ILC to achieve continuous optimization [40,41].

•   Secondly, the repetitive trajectory of equipment actuators represents the ideal control target for ILC.

•   Thirdly, multi-physics phenomena in machining processes exhibit complex nonlinear dynamics that can be effectively regulated through ILC-based control systems.

Furthermore, the complex nonlinear dynamic characteristics formed by the heat-force coupling, material removal and other multi-physical fields involved in the machining process can be effectively controlled by the data-driven characteristics of ILC [42–44].

It is noteworthy that the data-driven advantages of ILC extend beyond manufacturing applications, demonstrating significant value in intelligent transportation systems characterized by more complex dynamic behaviors. Building on ILC, researchers have integrated it with fuzzy logic to develop an adaptive data-driven traffic signal control strategy. VISSIM simulation results confirm that this approach significantly outperforms conventional fixed-time and actuated control strategies in terms of both intersection capacity improvement and traffic flow adaptation [45]. Requiring minimal prior knowledge, this method effectively handles stochastic disturbances in transportation systems through synergistic integration with alternating response-based urban control frameworks [46].

In railway control applications, reference [47] proposed a D-type ILC scheme incorporating overspeed protection, subsequently developing a multi-train coordination strategy that achieves precise maintenance of safe headway distances. To address nonlinear parametric uncertainties and multiple unknown state delays in the system, reference [48] designed an adaptive ILC method for high-speed trains that successfully ensures accurate tracking of desired displacement and velocity trajectories. This study innovatively employs spatial state differentiator technology to transform the nonlinear train operation model from the time domain to the spatial domain, thereby constructing a constrained spatial adaptive controller [49]. Furthermore, reference [50] provides a systematic examination of the key technical challenges and future development prospects of deep reinforcement learning in intelligent transportation applications. Beyond specific railway applications, the development of intelligent transportation systems requires integrated solutions across multiple technical dimensions.

To support the development of efficient, secure, and intelligent future transportation systems, a series of studies have proposed targeted solutions from a vehicle-road-cloud cooperative perspective [51]. Reference [52] employs non-orthogonal multiple access and deep reinforcement learning to optimize inter-vehicle communication resource allocation, ensuring information timeliness while reducing energy consumption. Reference [53] introduces an intelligent task offloading mechanism that leverages edge server resources to enhance the energy efficiency and training performance of federated learning. Reference [54] utilizes multi-agent deep reinforcement learning to coordinate digital twin maintenance and real-time task processing in cloud servers, thereby improving resource utilization efficiency. Together, these studies establish a multi-level optimization framework spanning terminal communications, edge computing, and cloud scheduling. While these communication and computing frameworks provide the infrastructure foundation, the core control algorithms operating within this infrastructure require simultaneous advancements in precision and robustness.

The dual-mode framework proposed in this work simultaneously incorporates the capability of iterative learning to asymptotically eliminate repetitive errors and the ability of online adaptation to promptly suppress unexpected disturbances. Precision manufacturing and intelligent transportation systems are adopted as two highly representative application scenarios to validate the effectiveness and superiority of this unified framework across different domains.

2  Fundamental Theory of ILC

This section focuses on the core theory of ILC. It first elaborates on the principle of its core update law, and then analyzes the stability conditions of the update law in time and frequency domains. Based on the special characteristics of ILC, this section strategically relaxes certain constraints, thereby lowering control energy usage without sacrificing system performance.

2.1 Model Free ILC Design Framework

ILC primarily deals with discrete repetitive processes as its core subject. Such systems operate periodically over a finite time duration, with each run referred to as an “iteration”. Their dynamic behavior is characterized by evolution in both the time domain and the iteration domain: the system state evolves along the time axis, while control performance improves with successive iterations. By leveraging this repetitive nature, ILC updates the current control input using information from past iterations, thereby achieving asymptotic convergence of the tracking error over the iteration domain.

As the fundamental ILC algorithm, P-type ILC operates by scaling the tracking error in the k-th iteration with a proportional gain, which then modifies the control input for the (k+1)-th iteration to iteratively minimize errors [55].

Consider the dynamics of a discrete-time linear system at the k-th iteration

{xk(t+1)=Axk(t)+Buk(t),yk(t)=Cxk(t),(1)

where t∈{0,1,⋯,T} is time step, T is task period, uk(t),yk(t),xk(t) are control inputs, outputs, and states, respectively; and yd(t) is desired output [56].

The control law of P-type ILC can be expressed in [57] as

uk+1(t)=uk(t)+Kp⋅ek(t+1),(2)

where uk(t) denotes the control input of the k-th iteration at time t; ek(t)=yd(t)−yk(t) denotes the tracking error of the k-th iteration; yd(t) is the desired output; yk(t) is the actual output; Kp is the learning gain matrix, which indicates the strength of error correction.

The P-type ILC law in discrete-time systems takes the following form

uk+1(i)=uk(i)+Kp⋅ek(i+1),(3)

where i denotes a discrete time step [58].

Combining the advantages of P-type ILC and sliding mode control, the work in [59] designed a P-type closed-loop sliding mode iterative learning controller with forgetting factor, as shown in Table 1. This controller first guides the system from an arbitrary starting point to the sliding surface, then ensures rapid convergence to the origin along this surface, effectively eliminating the influence of varying initial conditions across iterations.

The D-type iterative learning algorithm utilizes error derivative components to update control inputs, leveraging the error change rate from previous iterations. Ideal for repetitive motion applications, this technique improves upon basic P-type ILC by adding dynamic error compensation through differentiation, resulting in quicker convergence and better tracking performance [60–62].

The update formula for the control input in [63] as

uk+1(t)=uk(t)+Kde˙k(t),(4)

where Kd represents the differential gain matrix (design parameter) and e˙k(t) is differential term of error, usually approximated by a difference

e˙k(t)≈ek(t)−ek(t−1)Δt,(5)

where Δt is sampling interval.

To enhance tracking precision and system robustness, reference [64] develops a combined open/closed-loop D-type iterative learning control scheme for non-canonical nonlinear systems, integrating both historical error data and instantaneous feedback information. For a comparison of the three methods, please refer to Table 2.

PD-type ILC synergistically combines proportional and derivative feedback mechanisms. Building upon conventional P-type ILC through the incorporation of error differentiation, this approach substantially enhances both transient response characteristics and disturbance rejection performance. The underlying principle operates through dual compensation: the proportional component eliminates steady-state errors to guarantee precision, while the derivative term anticipates error variations to mitigate overshooting and dampen oscillations [68,69]. For a comparison of the three methods, please refer to Table 3.

For the PD-type update law, the iterative update formula for the control inputs is given in [70]

uk+1(t)=uk(t)+Kpek(t)+Kde˙k(t),(6)

where Kp and Kd are proportional and differential gain matrices, respectively, and e˙k(t) is error differentiation term.

Recent advances in PD-type ILC for T-S fuzzy nonlinear systems have addressed key implementation challenges. Reference [71] introduced a finite-frequency domain design to enhance robustness, while reference [72] developed a variable-gain scheme optimizing the convergence-noise trade-off. For non-uniform trial lengths, reference [73] proposed a recursive update mechanism that efficiently constructs complete learning sequences, outperforming conventional zero-padding and search methods in both data utilization and storage efficiency.

Despite benefiting from the proportional term’s accuracy and the derivative term’s predictive correction, PD-type ILC is prone to failure in environments with significant noise or non-repetitive disturbances. The derivative component exacerbates high-frequency noise, leading to control signal chattering that degrades performance and threatens stability. The common solution of adding noise filters introduces its own problems, namely phase lag and the challenge of filter parameter tuning.

2.1.1 Model-Based ILC Design Framework

Model-based ILC can significantly improve the learning efficiency and control accuracy by combining information from system dynamics models [75]. The core idea is to use the model to predict the error evolution dynamics and design a higher-order learning law. In this section, two typical methods, inverse model ILC and optimal control ILC, are introduced in detail, including algorithm design, theoretical analysis, and engineering applications.

The inverse model ILC is implemented as proposed in [76], where a dynamic inverse model of the system is constructed to directly compute control inputs for compensating the tracking error

uk+1(t)=uk(t)+G−1ek(t),(7)

where G−1 denotes inverse of system model G. If system is linear and invariant, G can be expressed as a transfer function matrix, for nonlinear systems, the inverse model needs to be approximated by linearization or numerical methods.

Optimal control ILC transforms an iterative learning problem into a dynamic optimization problem by minimizing a composite objective function containing the error energy and the control energy to solve the optimal input update law [77]. Its generalized form can be formulated as

uk+1=arg⁡minu⁡(‖ek+1‖Q2+‖u−uk‖R2),(8)

where Q and R are the weighting matrices for error and control inputs, respectively, and ek+1=yd−Guk+1 needs to be predicted using the system model.

The performance of inverse-model ILC is critically dependent on model accuracy, as minor mismatches can cause instability, restricting its use in practical scenarios. Conversely, optimal control ILC suffers from the empirical tuning of its weighting matrices and high computational complexity, hindering its real-time implementation.

2.1.2 Data-Driven ILC Design Framework

By integrating ILC with model predictive techniques for repetitive batch processes, the iterative learning model predictive control approach enables progressive enhancement of tracking accuracy while effectively rejecting real-time disturbances [78].

Reference [79] introduced an indirect data-driven ILC strategy for nonlinear repetitive systems, focusing on enhancing PID feedback control performance through reference trajectory adjustment, eliminating dependence on precise system modeling. Subsequently, reference [80] developed an advanced data-driven higher-order optimal ILC algorithm to significantly decrease computational burden.

The higher order learning control law is

uk(t)=uk−1(t)+∑m=1Mαm(θ^k(t)⋅ek−m(t))(9)

where uk(t) is the control input at k-th iteration moment t, αm is higher-order factor, θ^k(t) is the estimated gradient vector, and ek−m(t) indicates the tracking error from the (k−m)-th iteration. Furthermore, a unified data-driven framework is proposed for various practical scenarios, including optimal ILC, optimal point-to-point ILC, and optimal terminal ILC [81].

The mathematical model describing a typical nonlinear batch process [82] is given by

xk(t+1)=f(xk(t),uk(t))+w(t),(10)

where k is the batch index, t∈{0,1,…,Tk−1} is the discrete time index, and Tk is the time horizon. The state xk(t)∈X⊆Rnx and input uk(t)∈U⊆Rnu are constrained by physical sets X and U. The function f:X×U→X is Lipschitz continuous in both arguments. The disturbance w(t) is iteration-invariant with ‖w(t)‖≤ ϵ.

A prior-knowledge migration mechanism based on deep neural networks was proposed in [82], and its relationship with the ILMPC controller is shown in Fig. 2.

Figure 2: ILMPC architecture based on a priori knowledge migration mechanism

The unified framework of dual-mode data-driven ILC integrates iterative learning control with online adaptation. Its core concept lies in leveraging ILC on the iteration axis to learn and eliminate repetitive disturbances, while online feedback strategies like adaptive control are utilized on the time axis to suppress non-repetitive disturbances and model uncertainties in real time.

This paper presents a general update law for the control signal, formulating the integration of the two modes within a unified mathematical framework.

uk+1(t)=ℒ(uk(t),ek(t))⏟ILC Mode +𝒜(xk(t),ek(t))⏟Adaptive Mode .(11)

In this framework, ℒ(⋅) signifies the ILC update law, which is based on historical batch data, while 𝒜(⋅) corresponds to the adaptive law utilizing the real-time state xk(t) and error ek(t) of the current batch. The ILC module refines the baseline control signal iteratively, and the adaptive module compensates for dynamic variations in real-time, thereby allowing the system to converge progressively over batches while remaining robust to uncertainties during each operation.

The inherent gap between the ideal conditions of theoretical assumptions and the pressing demands of practical applications serves as the primary driving force behind the evolution of ILC theory. To successfully deploy ILC in the complex dynamic scenarios discussed in Section 4, such as precision manufacturing and intelligent transportation, it is essential to relax these classical assumptions and develop more robust and adaptive ILC frameworks. The following section will explore a range of advanced ILC strategies designed for this purpose.

3  ILC for Broader Application Scenarios

The theoretical framework of ILC is based on six postulates, which tend to have large limitations in practical applications [83]. To enhance the applicability of ILC, some scholars in recent years have proposed a new idea that ILC is inherently applicable to any control scenario with repetitive characteristics [84,85]. Guided by this idea, researchers have attempted to gain greater flexibility at the algorithm design level by gradually removing single or multiple traditional assumptions [86–88]. However, the sixth postulate, system invertibility, cannot be readily relaxed since it serves as a necessary and sufficient condition for guaranteeing the existence of a unique ideal control input that achieves perfect tracking. Without this assumption, the theoretical basis for asymptotic zero-error convergence would be fundamentally undermined. Therefore, this section focuses only on the first five postulates. Among the six postulates, the sixth is universally applicable across system types, while the first five form the core focus of this section.

3.1 Remove Postulate 1

In adaptive trajectory tracking for robots, strict time intervals are unnecessary. Thus, reference [89] relaxes time constraints by dynamically optimizing tracking time points (e.g., pick and place times), enabling flexible time allocation. A two-stage optimization framework is then proposed under this relaxed time scheme.

To satisfy the tracking requirements at a finite number of intermediate time points and sub-regions, with the rest of the time free for optimization, the generalized ILC algorithm is designed using a project-by-projection approach in the work of [90]. The iterative update law is

uk+1=PΩ(uk+Ge∗(I+GeGe∗)−1eke),(12)

where

•   uk: Control input vector at iteration k.

•   Ge: Extended system dynamics model (input-to-output mapping).

•   Ge∗: Adjoint operator of Ge.

•   eke=re−yke: Extended tracking error (re: reference, yke: output).

•   PΩ: Projection operator enforcing input constraints Ω.

•   I: Identity matrix.

To be applicable to spatial path tracking without time constraints, such as laser cutting as well as AM. The work [91] presents an ILC formulation that encodes path tracking requirements via Pr projection operations and Φr constraint satisfaction, yielding an optimization problem formulation.

minu⁡f(u,y)s.t.y=Gu,Pry=Prr~,y∈Φr,u∈Ω.(13)

The study in [92] develops an adaptive ILC scheme with data-driven quantization, requiring only time-specific data sampling while relaxing non-essential trajectory constraints.

Design of quantized adaptive learning control law

uk(t)=uk−1(t)+ρwe,k−1Φ^k(t)ek−1(t+1)λ+|Φ^k(t)|2−ρΦ^k(t)∑i=0t−1Φ^k(i)Δuk(i)λ+|Φ^k(t)|2,(14)

where we,k−1 is the weight factor for quantization error, Φ^k(t) is the system parameter estimated by the adaptive updating law, and ρ and λ are the design parameters to ensure convergence.

To reduce point-to-point tracking error, reference [93] proposed an enhanced gradient algorithm with an adaptive gain mechanism for tracking error

uk+1=uk+rk−1HTΨTek,(15)

where rk is the adaptive gain parameter, ek=yr−Ψyk is tracking error, Ψ is the selection matrix, and H is the system lift matrix.

Meanwhile literature [94] proposes a spatial ILC method based on 2D convolution. The updating law can be expressed in the spatial and frequency domains respectively as

fj+1(x,y)=(lf∗fj)(x,y)+(le∗ej)(x,y),(16)

where lf and le denote the filter impulse responses, fj and ej denote the 2D functions of the inputs and errors, respectively, with j=0,1,... denotes iteration index.

3.2 Remove the Postulate 2

The traditional ILC assumes identical initial states for each batch, but in practice, initial states are often unknown and vary due to perturbations, preventing error convergence. Literature [95] addressed this by proposing an initial state learning law that combines with point-to-point iterative learning control (P2PILC) to jointly update control inputs and initial states. The learning law dynamically adjusts the initial state of next batch xk+1(t1) based on the first tracking error ek(t1)

xk+1(t1)=xk(t1)+BΓ(ΦG)Tek(t1),(17)

where the adjustment is weighted by the control matrix B, learning gain Γ, and system characteristics ΦG.

Gradient-based P2PILC input update law

uk+1=uk+Γ(ΦG)Tek,(18)

where Γ is the learning gain matrix, ΦG represents the extracted form of the system matrix, and this formulation ensures optimization of tracking errors only at specified time points.

In the work of [96], an interactive adaptive ILC is constructed by simultaneously removing the constraints of iterative parameter changes and initial state from the system, and the parameter estimation update formula

ψ^k+1i=ψ^ki+ski(yk(i)−y^k(i))u¯ki,(19)

where ski is search step and y^k(i) is an estimate of the system output. Control input update law

uk+1=uk+Γk+1G^k+1ek,(20)

where Γk+1 is adaptive learning gain and ek denotes the tracking error. The adaptive mechanism relaxes traditional ILC constraints to make it applicable to a wider range of non-repetitive systems.

The work in [97] proposes a finite-time extended state observer that eliminates the conventional requirement for identical initial conditions in iterative learning control systems. The observer dynamics are given by

z^1,k=Az^1,k+Buk+L1(z¯y,k+ϕ1(z¯y,k)),z^2,k=h^k+L2(z¯y,k+ϕ2(z¯y,k)),(21)

where z^1,k and z^2,k are state estimates, z¯y,k=yk−Cz^1,k is output estimation error, ϕ1(z¯y,k) and ϕ2(z¯y,k) are nonlinear functions, and L1 and L2 are observer gains.

3.3 Remove Postulate 3

In practice, systems are often subject to uncertainties and disturbances that make strict repeatability difficult to guarantee. Removing this constraint can significantly enhance system adaptability and flexibility, improve the robustness and reliability of the control system, and extend ILC applicability to non-repeatable scenarios.

The article [98] proposes an adaptive ILC method for 2D nonlinear MIMO parameter systems. The method addresses three major non-repeatable uncertainties faced by traditional ILC in practical applications.

•   Stochastic initial shifts: The initial state fluctuates randomly around the desired value.

•   Non-repetitive references: Reference trajectories may change in each iteration.

•   Non-uniform trial lengths: Batch lengths vary randomly in each iteration.

An adaptive control law uk(i,j) and parameter update law Φ^k(i,j) are designed as

uk(i,j)=Φ^k(i,j)ψk(i,j),(22)

Φ^k(i,j)=Φ^k−1(i,j)+ηek−1∗(i+1,j+1)ψk−1(i,j)Tμ+‖ψk−1(i,j)‖2,(23)

where η is the step size and μ is the weight factor. In framework of the fixed model, external non-repeatability is handled by adaptive laws and random variables to extend the applicability scenarios of ILC.

Literature [99] improves the traditional ILC method by making three main improvements.

•   No longer requiring the system dynamics to be repeated in each iteration.

•   Removing the restriction that the initial conditions are fixed.

•   Allowing the system parameters to vary over time.

To this end, the authors developed a neural network-enhanced adaptive ILC framework, demonstrating improved robustness and practicality. The work in [100] introduces an adaptive Kalman filtering-augmented ILC algorithm, which enables real-time compensation for time-varying dynamics through simultaneous online parameter estimation and covariance matrix adaptation, while maintaining rigorous convergence guarantees.

Literature [101] proposes a D-type iterative modified update law that eliminates the dependence of the system on strict repeatability and ensures convergence and tracking performance of the system under non-repetitive uncertainty

uk+1(t)=uk(t)+Γ[e˙k(t)+θ(t)ek(0)],(24)

where Γ is learning gain matrix, θ(t) is correction function, and ek(0) is used to compensate for the initial error.

3.4 Remove the Postulate 4

In practical systems like aircraft control with variable iteration times, the work in [102] introduced a feedback-assisted PD-type quantized ILC approach for discrete linear systems with random batch lengths and initial conditions. Two quantization schemes were developed

uk+1(t)=uk(t)+Γ1Q(ek∗(t+1))+Γ2Q(ek∗(t))+KQ(ek+1∗(t)),(25)

uk+1(t)=uk(t)+Γ1ek∗(t+1)+Γ2ek∗(t)+Kek+1∗(t),u¯k+1(t)=Q(uk+1(t)).(26)

Scheme 1 achieves zero-error convergence, while Scheme 2 achieves bounded convergence.

For discrete-time systems exhibiting nonuniform trial durations, the work in [103] transforms the variable-length problem into a projection problem within a multi-affine subspace using Hilbert space optimization theory, and designs an implementable control algorithm.

The error between desired and actual trajectories is mathematically expressed as

ek=Fk(yd−yk),(27)

where Fk is a trial length-dependent diagonal matrix that sets errors for missing data to zero.

In the Hilbert space, the following objective function is minimized

J(uk+1)=‖ek+1‖Q2+‖uk+1−uk‖R2,(28)

which simultaneously reduces tracking error and input variation, with weight matrices Q and R regulating performance.

The core problem of non-uniform length ILC is solved by the alternating projection method

•   The problem is transformed into a multisubspace projection and convergence is rigorously proved.

•   Propose a feedback-feedforward structure for causal realization to support input constraints.

The work in [104], on the other hand, transforms the ILC problem into a quadratic programming problem with constraints, as in the following equation, and solves it using the interior point method, which achieves efficient convergence in the presence of input constraints.

minuk+112uk+1THuk+1 + cTuk+1s.t.ζuuk+1 ≥ ζk+1,(29)

where H=2(GTM~QG+R), with M~ representing the expectation of Mk. Here, G is the system’s Toeplitz matrix, Q and R are positive definite weight matrices, and c=−2[(GTM~QG+R)uk+GTM~Qek]. The interior point method ensures efficient convergence under input constraints. Complementarily, literature [105] proposes an alternating projection-based ILC framework to enhance convergence rates.

3.5 Remove the Postulate 5

By removing the constraints of fixed iteration time and a specified update law, the ILC theoretical framework becomes more relevant to engineering needs and adaptable to a wide range of non-repetitive scenarios. To enhance industrial robotic path tracking accuracy, reference [106] developed a dual-phase iterative learning approach comprising sequential model error compensation and motion trajectory optimization. In the model correction phase, difference between actual system and nominal model is quantified by defining a model mismatch quantity Δτi

Δτi=τm.i−T(qm,i,q˙m,i,q¨m,i).(30)

By constructing an objective function containing three regularization constraints as shown in the following equation. Among them, the γ1 term is used to control the magnitude of the correction, the γ2 term ensures smooth transitions between iterations, and the γ3 term guarantees time-domain continuity. This structured regularization design maintains correction accuracy while preventing overfitting

αi∗=arg⁡minαi⁡‖Δτi−αi‖22+γ1‖α‖22+γ2‖Δiα‖22+γ3‖Δkα‖22.(31)

The nonlinear optimal control problem in trajectory planning is reformulated as a convex-concave optimization through the introduction of auxiliary variables a(s) and b(s), with the temporal domain transformed to the normalized interval s∈[0,1].

a(s)=s¨(t),b(s)=s˙2(t),b′(s)=2a(s).(32)

The composite cost function comprises two distinct components: the first term 1b(s) corresponds to time-optimality metric, while the second term is a regularization of rate of change of control quantity. The constraints systematically consider the dynamics of the system, actuator limits, and kinematic constraints to form a complete description of the optimization problem.

The article [107] achieves control energy minimization and high-precision path tracking by transforming the path tracking problem into a convex optimization problem as shown inthe following equation and incorporating the indirect reference update framework of ILC

minu~,a,b⁡∫01u~T(s)u~(s)b(s)ds+E^(T^,u^),(33)

where u~(s) denotes the spatial input signal, and a(s) and b(s) represent the spatial acceleration and velocity squared, respectively, corresponding to the energy consumption in the stationary phase. This equation transforms the time domain problem into a spatial domain convex optimization problem.

4  Case Studies: ILC in Precision Manufacturing and ITS

This section presents typical application examples of ILC in the manufacturing field and intelligent transportation, along with an in-depth discussion of the unique advantages and technical details of this method across different processes. Through four representative cases, the analysis demonstrates how ILC can effectively improve both accuracy and efficiency.

4.1 Extrusion Speed and Trajectory Tracking ILC in FDM Printing

Ceramic paste extrusion faces line inhomogeneity due to viscosity variations and speed mismatches, along with environmental and material fluctuations [108]. Conventional open-loop control relies on trial-and-error parameter tuning, proving inefficient for dynamic disturbances. Current approaches use experimental data to train predictive models [109], but complex paste rheology leads to prediction errors, limiting linewidth control accuracy [110,111]. ILC compensates model errors by identifying perturbations and adjusting extrusion linewidth [112–114]. Its learning capability handles slurry nonlinearities, improving print quality [115], as shown in Fig. 3.

Figure 3: ILC structure diagram

In the extrusion of ceramic materials, there is a clear kinetic relationship between the screw extrusion speed Ve and screw rotational speed n

Ve=2whdcos⁡θD2n,(34)

where w denotes the screw channel width, h indicates the flight depth, d signifies the screw core diameter, n is the angular velocity of screw rotation, D represents the extrusion die orifice diameter, and θ denotes the screw helix. Eq. (34) describes the linear relationship between screw extrusion speed Ve and screw speed n, reflecting the mechanical transmission characteristics of the extrusion system. Increasing n or decreasing D increases Ve [115].

The deposition line width W vs. extrusion speed Ve and print speed Vxy is given by

Vxy=πd24hd+πh2⋅Ve,(35)

W=π2⋅Ve/Vxy+4+π−2π⋅d.(36)

Improved algorithms are proposed to improve control performance. Higher-order ILC with forgetting factor

uk+1(t)=λuk(t)+∑i=0pLiek−i(t+1),(37)

where, λ is the forgetting factor, which balances the new/old data weights and prevents overfitting, Li is learning gain matrix, and p denotes historical error order, which utilizes the historical error information to enhance the convergence speed. It has better results for dealing with nonlinear time-varying systems [116].

Adaptive gain adjustment rule, adjusted to cumulative error, online adaptation to process perturbations

Kp(k)=Kp0+γ∑i=1k|e(i)|,(38)

where Kp0 denotes the initial proportional gain and γ denotes the adaptive rate, which determines the adjustment rate [117].

Fig. 4 shows a vision feedback-based extrusion 3D printing process control system, which can be combined with ILC to achieve high-precision closed-loop quality control.

Figure 4: System configuration for closed-cycle quality regulation in 3D printing material deposition processes

The aforementioned study demonstrates the effectiveness of ILC in controlling the ceramic extrusion process—a SISO system—where the screw speed is adjusted to directly compensate for viscosity disturbances, thereby stabilizing the line width. However, the control challenges in additive manufacturing extend far beyond this scenario. When the process involves strong spatiotemporal coupling and interactions among multiple physical fields, such as thermal management in laser powder bed fusion of metals, the complexity of the problem increases significantly. Consequently, the control objective shifts from regulating a single geometric variable to coordinating multiple physical fields, placing greater demands on the control strategy.

4.2 Interlayer Temperature Control in Laser Selective Melting

The control of the temperature field in laser selective zone melting (SLM) process faces a key challenge [118]: the need to avoid overheating melt pool while ensuring that material is sufficiently melted [119]. To address this multi-objective control requirement, an innovative model-free ILC scheme has been proposed in the literature [120].

Examining the selective laser melting process depicted in Fig. 5: The laser beam performs raster scanning in the build region with a fixed spot diameter d and a scanning speed of s. The spacing between adjacent raster lines (i.e., the shaded row spacing) is maintained as h, corresponding to the center spacing of the melting tracks.

Figure 5: Illustration of a single-layer SLM process

Setting the laser center x(t) at time t, it is necessary to control temperature of the surrounding ring area of radius r to satisfy [120]

Tmelt<T(r)<Tvapor,(39)

where the temperature T(r) within an annular region of radius r around the laser center x(t) must satisfy

•   Lower bound Tmelt: Material melting temperature (ensures proper fusion).

•   Upper bound Tvapor: Material vaporization temperature (prevents excessive ablation).

For a predefined laser path x(⋅), the laser power time-varying function p(⋅) is optimized to achieve

minE(Tmax(⋅),Tmin(⋅),p(⋅))s.t.t(⋅)=g(p(⋅),x(⋅)),(40)

where E denotes the cost function that quantifies degree of deviation of temperature extrema from the target interval, and g denotes the operator that describes the nonlinear mapping of the laser power to the temperature field [121].

The structure of SLM equipment is shown in Fig. 6, which mainly consists of following components: energy beam, recoating blade, part, powder, build platform, and powder platform [122].

Figure 6: Diagram of SLM machine setup

To solve the difficulty of requiring an exact model for traditional gradient computation, a path inversion gradient estimation method was proposed in the literature [118]. The method first constructs the local linearization matrix of the time-varying system.

Gk=[g1(1)0⋯0g2(1)g2(2)⋯0⋮⋮⋱⋮gN(1)gN(2)⋯gN(N)],(41)

where gj(i) represents the impulse response of the input tj to the output ti at the moment.

The computation of the concomitant operator is realized by introducing the time reversal operator T (unit inverse order matrix) and the path backscan response matrix [123], as expressed by

Gk∗=TG←kT.(42)

Fig. 7 illustrates the single iteration flow of gradient descent-based model-free ILC algorithm.

Figure 7: Convergence plot of gradient-ILC for SLM thermal control

The case study on interlayer temperature control demonstrates the capability of ILC to handle complex, nonlinear distributed parameter systems that are difficult to model accurately. The control objective in this context extends to stabilizing the entire temperature field. Subsequent research shifts focus from manufacturing processes to motion systems, specifically high-speed train operation control. This domain involves high-order nonlinear complexities, including multi-agent coordination, state constraints, and large-range operation. Such challenges require extending ILC applications from physical field control to the cooperative optimization of multi-body motion systems.

4.3 Trajectory Tracking of High-Speed Train Based on Adaptive ILC

The control system of high-speed trains (HST) [124] must balance safety, energy efficiency, and punctuality. Traditional methods (e.g., PID, model predictive control) rely on precise modeling and struggle to handle time-varying parameters and complex environmental disturbances. In recent years, data-driven and intelligent control methods have significantly improved control performance by leveraging historical data and iterative optimization [125].

Reference [46] designed an adaptive ILC, with its control law formulated as

ui,j=bj−1γεi,j+θ^i,jξj,θ^i,j=θ^i−1,j+βεi,jξjT,(43)

where εi,j denotes the local error, ξj incorporates known state functions, θ^i,j represents the parameter estimate, γ stands for the learning gain, and β refers to the adaptive gain matrix.

The synchronization of all carriages tracking the reference trajectory is ensured through the consensus error ei,j=vr−vi,j+c1(sr−si,j), as illustrated in Fig. 8. Fig. 9 demonstrates the convergence of maximum tracking error with respect to iteration numbers.

Figure 8: Tracking effect of the 100th iteration

Figure 9: Convergence of maximum tracking error with number of iterations

Reference [49] pioneered the solution of state constraints within a spatial iterative learning framework, offering novel insights for nonlinear system control. A constrained spatial adaptive ILC (CSAILC) scheme was designed as illustrated in Fig. 10.

Figure 10: Constrained spatial adaptive iterative learning control

Fig. 10 illustrates the CSAILC framework embedded within the hierarchical architecture of a train control system. Its functional modules operate collaboratively as follows: the ATS (Automatic Train Supervision) layer generates operational plans and target commands; the ATO (Automatic Train Operation) module acts as the core actuator, receiving optimized outputs from the ILC algorithm to regulate the traction and conventional braking systems for precise speed tracking; the ATP (Automatic Train Protection) module continuously monitors the system state to ensure all operations remain within the safety envelope, triggering emergency braking immediately upon constraint violation, thereby providing essential safety assurance for the entire learning process. Ground equipment and onboard units exchange data via train-ground communication. The adaptive ILC algorithm developed in this study operates within this framework, dynamically adjusting learning parameters online to achieve high-performance tracking in the iterative domain while strictly adhering to all safety constraints.

The control law is designed to

Fi(s)=F^i(s)+αi|ei|−θ^iT(s)δζi(s)+Kβi|ei(s)|.(44)

The parameter update law is designed to

θ^i=Pθ[θ^i−1]−pβiδζi.(45)

Through the aforementioned design, the convergence performance shown in Fig. 8 can ultimately be achieved.

The HST case demonstrates the effectiveness of ILC in multi-body dynamical systems with state constraints and coordination requirements. Furthermore, the applicability of ILC can be extended to larger-scale urban traffic systems. Highway traffic flow regulation presents challenges such as high stochasticity, interactions among heterogeneous agents (vehicles), and macroscopic periodic characteristics, with system models exhibiting significant uncertainty. This application marks the expansion of ILC from the aforementioned equipment-level control and fleet coordination to the optimized regulation of city-level networked systems.

4.4 ILC for Freeway Traffic Flow Regulation

Traffic congestion in freeway systems remains one of the major challenges in modern urban transportation [126]. Conventional control methods (e.g., ramp metering and variable speed limits) rely heavily on precise traffic modeling, yet real-world traffic systems exhibit strong nonlinearities and uncertainties, resulting in excessive model dependency and limited control efficacy [127].

Reference [128] proposed an ILC-based approach for freeway traffic density control, achieving traffic flow optimization through coordinated ramp metering and speed signaling regulation.

Design of ramp metering control

un+1(k)=un(k)+βen(k+1),en(k+1)=yd(k+1)−yn(k+1).(46)

The speed marker control is designed to

wn+1(k)=wn(k)+βen(k+2).(47)

Reference [129] proposed a hybrid strategy (ILC+TUC) that enhances robustness by integrating the iterative learning capability of ILC with the real-time responsiveness of feedback control. The hybrid strategy is formulated as

un(k)=unf(k)+γen(k).(48)

Fig. 11 illustrates the hybrid strategy, which leverages both offline learning and online adaptation to exploit the periodic characteristics of traffic flow while enhancing system robustness.

Figure 11: Hybrid controller block diagram

Table 4 provides a systematic comparison of the differentiated applications of ILC in precision manufacturing and ITSs across four dimensions. Precision manufacturing emphasizes micro/nano-scale trajectory tracking and multi-physical field disturbance suppression, often employing model-assisted and adaptive gain methods. In contrast, intelligent transportation systems focus on multi-agent coordination and communication topology optimization, predominantly utilizing distributed and hybrid feedback architectures. This comparison clarifies the distinct technical pathways of ILC across different domains and offers a theoretical basis for cross-disciplinary methodological integration.

Due to their distinct control objects and performance objectives, each case study adopts a specifically tailored ILC technical approach. The particular ILC framework applied in each case is summarized in Table 5.

The four case studies collectively highlight key commonalities. All processes exhibit distinct cyclic or repetitive characteristics. Each case involves complex dynamics that resist accurate modeling, and each employs historical operational data to compensate for model uncertainties and unknown disturbances. This approach supports high-performance tracking or regulation. Nevertheless, the implementations emphasize distinct aspects. Ceramic extrusion printing focuses on precise closed-loop geometric control. Laser powder bed fusion addresses spatiotemporal temperature optimization and inverse gradient estimation. HST control requires multi-vehicle coordination under state constraints, while highway traffic management emphasizes macroscopic optimization and hybrid integration of control strategies. These distinctions reflect the flexibility of ILC in adapting to diverse applications. They also demonstrate its consistent methodological core in addressing varied control challenges.

5  Experimental Verification

The experimental platform primarily consists of the five components shown in Fig. 12. This experiment involves collaborative operation between two key subsystems: the laser galvanometer system and the 2D precision translation stage. There are usually two mirrors inside the galvanometer, one is the X-axis mirror and the other is the Y-axis mirror. These two mirrors can be rotated at high speed to achieve rapid deflection of the laser beam. By precisely controlling the rotation angle of the mirrors, the pointing of the laser beam can be precisely controlled to realize fine laser processing.

Figure 12: Precision collaborative machining experimental platform

To verify the practical effectiveness of ILC in applications, this study designed and conducted a series of experiments to systematically evaluate system performance regarding tracking capability, fault tolerance, and spatial adaptability. The experiments firstly verified the trajectory tracking ability of ILC. In the test, the system is required to track a predefined complex trajectory, as shown in Fig. 13. The results show that after several iterations of learning, ILC can significantly improve tracking accuracy. The initial tracking deviation is substantial, yet progressive refinement through iterative cycles drives asymptotic convergence toward negligible error levels. This shows that the ILC can effectively optimize the control inputs through the “learning-correction” mechanism to achieve high-precision tracking.

Figure 13: Tracking effect for different number of iterations: (a) 4th tracking; (b) 6th tracking; (c) 8th tracking; (d) 20th tracking

To evaluate ILC stability under perturbation conditions, the experiment simulated common disturbance scenarios in additive manufacturing processes. Test results (Fig. 14) demonstrate that ILC rapidly adapts to perturbations and progressively corrects errors through subsequent iterations. The method exhibits enhanced robustness, particularly in compensating for periodic disturbances.

Figure 14: Tracking effect of different number of iterations after interference: (a) 31st tracking; (b) 52nd tracking; (c) 53rd tracking; (d) 100th tracking

To address spatial path tracking requirements in precision manufacturing, experiments further evaluated ILC performance in three-dimensional space. Complex spatial trajectories (Fig. 15) were designed to verify ILC adaptability in multi-degree-of-freedom systems. Experimental results demonstrate that ILC not only achieves precise spatial path tracking but also effectively responds to geometric variation challenges through dynamic control parameter adjustment.

Figure 15: Performance of ILC in three-dimensional space

6  Conclusions and Future Work

This paper systematically reviews the research progress and application achievements of ILC in precision manufacturing and ITS. The results demonstrate that ILC, through its unique iterative optimization mechanism, effectively addresses challenges such as nonlinear dynamics, multi-physics coupling, and periodic disturbances in precision manufacturing, significantly improving the accuracy of extrusion linewidth control in additive manufacturing and interlayer temperature regulation in selective laser melting. In the ITS domain, the integration of ILC with techniques such as fuzzy logic and reinforcement learning enhances the dynamic adaptability of autonomous vehicle trajectory tracking, train cooperative control, and traffic signal optimization. Theoretically, by relaxing traditional assumptions, ILC’s applicability has been extended to non-repetitive scenarios, while advanced variants like PD-type ILC and adaptive ILC further improve system robustness and convergence performance. Experimental validation confirms that ILC exhibits excellent performance in trajectory tracking, disturbance compensation, and spatial path control.

While the effectiveness of ILC is well recognized, it is essential to objectively acknowledge the limitations of current approaches, particularly in complex application scenarios such as multi-axis coordinated systems. Although ILC shows considerable potential in these applications, the existing methods still exhibit several constraints. Firstly, most current ILC designs rely on the assumption of approximately decoupled dynamics across axes, which may lead to degraded performance in strongly coupled and highly nonlinear cooperative motions. Secondly, many algorithms exhibit sensitivity to significant variations in system parameters or environmental disturbances between iterations, indicating that robustness remains an area requiring further improvement. Moreover, current research predominantly focuses on set-point tracking or the replication of predefined trajectories, while capabilities for online real-time trajectory adjustment or responding to unexpected obstacles remain underdeveloped. These limitations highlight clear directions for future in-depth investigation.

Building upon the research foundation and existing limitations identified in this study, future work will advance along four concrete directions. First, multi-axis cooperative control algorithms will be developed. These algorithms will integrate nominal dynamics feedforward with data-driven strategies, focusing on online estimation and compensation of coupling disturbances using iterative-domain disturbance observers. Second, robust adaptive gain scheduling strategies will be designed for time-varying operational conditions. These strategies will dynamically adjust learning gains based on error norms while guaranteeing convergence through Lyapunov-based methods. Third, a hybrid ILC architecture will be constructed to incorporate real-time sensory feedback. This architecture will address frequency-band coordination and stability between the iterative learning loop and the online feedback control loop. Fourth, systematic validation will be conducted on a multi-degree-of-freedom precision motion platform. A multi-objective optimization evaluation framework will be established to quantitatively analyze the Pareto front of various algorithms in terms of convergence speed, precision, and robustness. This effort will facilitate the transition of the methodology toward practical applications.

Acknowledgement: Not applicable.

Funding Statement: This research was funded by the Wuxi Young Scientific and Technological Talent Support Initiative, project number: TJXD-2024-203 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China, grant number: 24KJB470027.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Lei Wang, Menghan Wei, Ziwei Huangfu and Shunjie Zhu; methodology, Lei Wang, Xuejian Ge and Zhengquan Li; software, Menghan Wei and Shunjie Zhu; validation, Lei Wang, Menghan Wei and Ziwei Huangfu; formal analysis, Lei Wang, Xuejian Ge and Zhengquan Li; investigation, Menghan Wei, Ziwei Huangfu and Shunjie Zhu; resources, Lei Wang, Xuejian Ge and Zhengquan Li; data curation, Menghan Wei, Ziwei Huangfu and Shunjie Zhu; writing—original draft preparation, Lei Wang; writing—review and editing, Lei Wang, Xuejian Ge and Zhengquan Li; visualization, Lei Wang, Menghan Wei; supervision, Xuejian Ge and Zhengquan Li; project administration, Lei Wang and Xuejian Ge; funding acquisition, Lei Wang and Xuejian Ge. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Not applicable, as this is a narrative review based on existing literature.

Ethics Approval: This study did not involve any human or animal subjects, and therefore, ethical approval was not required.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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