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 (1){xk(t+1)=Axk(t)+Buk(t),yk(t)=Cxk(t),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 (2)uk+1(t)=uk(t)+Kp⋅ek(t+1),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 (3)uk+1(i)=uk(i)+Kp⋅ek(i+1),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 (4)uk+1(t)=uk(t)+Kde˙k(t),where Kd represents the differential gain matrix (design parameter) and e˙k(t) is differential term of error, usually approximated by a difference (5)e˙k(t)≈ek(t)−ek(t−1)Δt,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] (6)uk+1(t)=uk(t)+Kpek(t)+Kde˙k(t),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.

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 (12)uk+1=PΩ(uk+Ge∗(I+GeGe∗)−1eke),where

uk: Control input vector at iteration k.

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

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 (14)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,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 (15)uk+1=uk+rk−1HTΨTek,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 (16)fj+1(x,y)=(lf∗fj)(x,y)+(le∗ej)(x,y),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.

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) (17)xk+1(t1)=xk(t1)+BΓ(ΦG)Tek(t1),where the adjustment is weighted by the control matrix B, learning gain Γ, and system characteristics ΦG.

Gradient-based P2PILC input update law (18)uk+1=uk+Γ(ΦG)Tek,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 (19)ψ^k+1i=ψ^ki+ski(yk(i)−y^k(i))u¯ki,where ski is search step and y^k(i) is an estimate of the system output. Control input update law (20)uk+1=uk+Γk+1G^k+1ek,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 (21)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)),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.

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.

An adaptive control law uk(i,j) and parameter update law Φ^k(i,j) are designed as (22)uk(i,j)=Φ^k(i,j)ψk(i,j), (23)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,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.

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 (24)uk+1(t)=uk(t)+Γ[e˙k(t)+θ(t)ek(0)],where Γ is learning gain matrix, θ(t) is correction function, and ek(0) is used to compensate for the initial error.

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 (25)uk+1(t)=uk(t)+Γ1Q(ek∗(t+1))+Γ2Q(ek∗(t))+KQ(ek+1∗(t)), (26)uk+1(t)=uk(t)+Γ1ek∗(t+1)+Γ2ek∗(t)+Kek+1∗(t),u¯k+1(t)=Q(uk+1(t)).

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 (27)ek=Fk(yd−yk),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 (28)J(uk+1)=‖ek+1‖Q2+‖uk+1−uk‖R2,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.

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

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 (30)Δτi=τm.i−T(qm,i,q˙m,i,q¨m,i).

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

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

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

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.

In the extrusion of ceramic materials, there is a clear kinetic relationship between the screw extrusion speed Ve and screw rotational speed n (34)Ve=2whdcos⁡θD2n,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 (35)Vxy=πd24hd+πh2⋅Ve, (36)W=π2⋅Ve/Vxy+4+π−2π⋅d.

Improved algorithms are proposed to improve control performance. Higher-order ILC with forgetting factor (37)uk+1(t)=λuk(t)+∑i=0pLiek−i(t+1),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 (38)Kp(k)=Kp0+γ∑i=1k|e(i)|,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.

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.

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.

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

For a predefined laser path x(⋅), the laser power time-varying function p(⋅) is optimized to achieve (40)minE(Tmax(⋅),Tmin(⋅),p(⋅))s.t.t(⋅)=g(p(⋅),x(⋅)),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].

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. (41)Gk=[g1(1)0⋯0g2(1)g2(2)⋯0⋮⋮⋱⋮gN(1)gN(2)⋯gN(N)],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 (42)Gk∗=TG←kT.

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

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.

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

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.

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

The parameter update law is designed to (45)θ^i=Pθ[θ^i−1]−pβiδζi.

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.

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

The speed marker control is designed to (47)wn+1(k)=wn(k)+βen(k+2).

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 (48)un(k)=unf(k)+γen(k).

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.

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.