Iterative Learning Control (ILC)

Updated 1 April 2026

Iterative Learning Control (ILC) is a control methodology that refines control inputs by learning from accumulated tracking errors in repetitive tasks.
It employs learning and robustness filters to guarantee convergence under both asymptotic and monotonic criteria for precise trajectory tracking.
ILC is widely applied in industrial robotics, precision motion systems, and advanced control paradigms to manage model uncertainties and nonminimum-phase behaviors.

Iterative Learning Control (ILC) is a class of control methodologies designed for systems that perform the same finite-duration task repeatedly. ILC exploits repeatability: after each execution (“trial”), the control input is updated based on accumulated tracking errors, enabling convergence to high-precision trajectory tracking even in systems where conventional feedback alone cannot achieve this performance. Originally developed for industrial robotics and precision motion systems, ILC has evolved into a rigorously analyzed, highly structured framework encompassing MIMO systems, nonlinear plants, constrained optimization, and integration with learning-based approaches (Oomen, 2020).

1. Fundamental Concepts and Theoretical Foundations

ILC addresses the problem of achieving high-accuracy output tracking for finite-horizon, repetitive systems. The canonical setup is a plant that, on each trial $k$ , is reset to a known initial condition and executes a prescribed trajectory under input $u_k$ ; performance is measured via tracking error $e_k$ compared to a desired reference $r$ .

The basic ILC update is

$u_{k+1} = u_k + L\,e_k,$

where $L$ is a learning operator (often a filter). More generally, an extended two-filter formulation is prevalent: $u_{k+1} = Q\,(u_k + L\,e_k),$ with $Q$ a robustness or regularization filter, and $L$ a learning filter. The lifted formulation—stacking time samples into vectors—captures signal evolution over a trial, with the core iteration

$e_{k+1} = (I - P\,L)\,e_k,$

where $u_k$ 0 represents the plant in “lifted” form as a lower-triangular Toeplitz matrix of Markov parameters, and $u_k$ 1 can be structured for causality, preview, or noncausal action (Koscielniak, 2023).

Two central convergence notions are employed (Koscielniak, 2022, Koscielniak, 2022):

Asymptotic Convergence (AC): $u_k$ 2, with $u_k$ 3 the spectral radius, ensures all error components decay asymptotically. However, AC does not preclude large learning transients.
Monotonic Convergence (MC): $u_k$ 4, i.e., the largest singular value is less than unity, guarantees the norm of the error strictly decreases each iteration. MC is strictly stronger than AC and eliminates large transient overshoot.

The z-domain (frequency) criterion provides a scalable approach (complexity $u_k$ 5 versus $u_k$ 6 for matrix-based tests for length- $u_k$ 7 trials): for a causal operator,

$u_k$ 8

guarantees monotonic $u_k$ 9-norm convergence (Koscielniak, 2023).

ILC must account for “learning transients,” which arise because for band-diagonal (causal) ILC matrices, the operator has repeated eigenvalues, resulting in polynomially modulated decaying terms in the error sequence, with transients potentially lasting $e_k$ 0 trials (Koscielniak, 2022, Koscielniak, 2023). Recent work has revealed classes of soliton-like (traveling-wave) solutions that persist even when all classic convergence criteria are satisfied (Koscielniak, 2022).

2. Trackability and Realizability in ILC

A foundational contribution is the formalization of trackability: for a continuous-time MIMO plant described in Laplace domain as

$e_k$ 1

a reference trajectory $e_k$ 2 is trackable if there exists an input $e_k$ 3 such that

$e_k$ 4

Trackability is a strictly weaker property than realizability, as it does not require uniqueness of $e_k$ 5; in over-actuated systems ( $e_k$ 6), infinitely many input functions can generate the same output trajectory (Meng et al., 2022).

Rigorous necessary and sufficient algebraic trackability conditions are provided:

Under-actuated ( $e_k$ 7): Trackability is equivalent to two conditions: initial-state alignment ( $e_k$ 8) and

$e_k$ 9

This leads to a unique input solution.

Over-actuated ( $r$ 0): Every $r$ 1 with appropriate initial alignment is trackable; the solution set is infinite-dimensional.

A key result links trackability to perfect tracking in feedback-based ILC: convergence to $r$ 2 is possible if and only if $r$ 3 is trackable, and the final asymptotic input matches the set of inputs defined by the trackability condition (Meng et al., 2022).

3. Design Methodologies and Robustness

ILC algorithm design spans a broad spectrum from simple FIR learning filters to advanced optimization-based, data-driven, and machine learning architectures:

Frequency-Domain ILC: Fast learning is enabled by noncausal model inversion ( $r$ 4 approximating the inverse of the plant’s sensitivity), while robustness to plant uncertainty/noise is enforced via a contraction mapping theorem—i.e., the condition

$r$ 5

(Oomen, 2020). $r$ 6 is shaped for stability and noise/sensitivity roll-off as required.

Multivariable ILC: In systems with cross-coupling and interaction, centralized multivariable ILC using centralized $r$ 7, $r$ 8 matrices is essential for guaranteed convergence. Robust stability can be addressed via structured singular value ( $r$ 9) analysis or Gershgorin-type bounds, which provide rigorous frequency-domain guarantees under model uncertainties and unmodeled interaction (Blanken et al., 2018).
Constrained and Uncertain Systems: Incorporating explicit state/input constraints and handling plant/model uncertainty can be achieved via optimization-based ILC (OB-ILC), which formulates each update as a convex quadratic program, leveraging real output measurements for gradient estimation and using robust constraint tightening to guarantee feasibility (Liao-McPherson et al., 2022). Recent extensions embed ILC within tube-based MPC frameworks, enhancing disturbance rejection and ensuring constraint satisfaction under bounded stochastic noise and plant/model uncertainty (Zuliani et al., 25 Mar 2025).
Basis Function and Sparsity Approaches: To limit computational complexity and enhance generalization, feedforward signals can be parameterized in selected basis functions, with automatic, data-driven sparse subset selection (using LASSO/LARS) to trade off model complexity versus tracking error, especially for variable/parametric references (Ickenroth et al., 9 May 2025).
Data-Driven and Learning-Enhanced ILC: Nonparametric (Willems’ Fundamental Lemma) data-driven models can avoid explicit plant identification, enabling rapid and robust learning, and allowing acceleration strategies such as Nesterov momentum and hybrid schemes to empirically improve convergence rates (Wang et al., 2023). Deep learning architectures, such as the TAIL-ILC framework, enable generalization to unseen, nonrepetitive references by implicit imitation of classical ILC experts on low-dimensional latent representations (Vinjarapu et al., 2023).
Predictive and Gaussian Process ILC: Incorporation of quasi-periodic Gaussian processes (QPGP) allows modeling and prediction of inter-trial disturbances and drifts, facilitating anticipatory, predictive ILC updates that accelerate convergence and maintain computational efficiency for long-run and real-time applications (Nigam et al., 20 Feb 2026).

4. Extensions: Nonlinear, Cross-Coupled, and Adaptive ILC

Nonlinear/Nonminumum-Phase Systems: For discrete-time nonlinear systems with nonminimum-phase characteristics, stable inversion methodologies can be embedded to construct bounded, well-posed feedforward corrections. The invert-linearize ILC (ILILC) framework combines stable nonlinear inversion with Newton/Kantorovich-based iterative error minimization to guarantee quadratic convergence and avoid divergence in ill-conditioned problems (Spiegel et al., 2021).
Contour-Tracking and Cross-Coupled ILC: In contour-tracking contexts (e.g., industrial printers), cross-coupled norm-optimal ILC directly penalizes spatial (contour) error, employing iteration-varying, time-varying coupling/weighting matrices and closed-form lifted solutions or low-order linear quadratic tracking (LQT) formulations that scale efficiently for long trial horizons (Aarnoudse et al., 2022).
Adaptive, Multi-System, and Disturbance Observer Integration: ILC augmented with disturbance observers (DOBs) can achieve robustness to persistent and non-repetitive disturbances even across families of systems with differing linearized dynamics. Learning signals are adaptively filtered to account for model mismatch, and filter design incorporates explicit stability margins based on maximum plant mismatch norms (Modi et al., 2024).
Integration with Reinforcement Learning and MPC: ILC’s stability/convergence properties make it well-suited to inform or initialize deep RL agents for batch process optimization, acting as an “expert” during offline pretraining and safe online adaptation via hierarchical blending schemes. Two-layer cascaded architectures (batch-to-batch and within-batch ILC), each with Kalman filtering, can simultaneously ensure constraint satisfaction, disturbance rejection, and RL policy safety (Lin et al., 16 Mar 2026).

5. Practical Implementation, Convergence Properties, and Performance

Finite-Time and Rapid Convergence: FIR filter and circulant-matrix methods, which approximate the plant’s steady-state frequency response inverse, can deliver extremely rapid, sometimes finite-step, convergence. Practical implementation involves remedial corner (edge) gain adjustment in the learning gain matrix via steepest-descent optimization of the maximum singular value, ensuring monotonic norm decay (Liu et al., 2021, Liu et al., 2021). For nonminimum-phase plants, deletion of early rows/columns in learning matrices cures ill-conditioning and stabilizes convergence.
Performance Metrics and Experimental Validation: Empirical studies (industrial printers, robotic motion platforms, spacecraft sensors) validate theoretical guarantees, routinely demonstrating one to two orders-of-magnitude error reduction within tens of iterations. Comparative studies show that model-based conventional controllers can become destabilized by modeling errors, while ILC suboptimality gaps remain linear in model mismatch, often outperforming certainty-equivalent optimal control by orders of magnitude in ill-specified environments (Vemula et al., 2021).
Challenges and Best Practices: Major restrictions on ILC’s practical impact include sensitivity to non-repetitive disturbances, variable references, unmodeled cross-coupling, computational cost for long-horizon MIMO systems, and the requirement for experiment resetting. Workflow recommendations involve: (1) a-priori analysis using error decomposition and model uncertainty quantification; (2) frequency response measurement for robust filter synthesis; (3) explicit verification of monotonic convergence conditions (matrix or frequency-domain); (4) conservative filter/gain selection; and (5) periodic performance and safety monitoring during deployment (Oomen, 2020).

6. Contemporary Extensions and Open Problems

Research is actively extending ILC into domains previously considered intractable:

Task-Flexible and Generalizing ILC: Traditional ILC cannot generalize to nonrepetitive references. Recent deep learning-based imitation frameworks (e.g., TAIL-ILC) produce one-shot feedforward signals for unseen references, efficiently mimicking classical ILC solutions over a broad class of motion profiles (Vinjarapu et al., 2023).
Learning-Based Model Synthesis: Kernel-based (GP) models and nonparametric identification bypass parametric model-building, rapidly adapting to position-dependent or time-varying disturbances (Oomen, 2020, Nigam et al., 20 Feb 2026).
Wave Phenomena and Soliton Solutions: Wave-like (soliton) solutions, partially explained by Jordan-block (repeated eigenvalue) phenomena, introduce persistent spatially localized error packets that standard AC/MC criteria do not predict. Design adjustments, such as regularization via robust lowpass filters, can suppress solitons—at the expense of residual steady-state error (Koscielniak, 2023, Koscielniak, 2022).
Integrated Predictive and Constraint Handling: Combining ILC with predictive control (MPC) strategies brings true input/output constraint management and robustification against both additive noise and parametric/modeled/mistuned dynamics (Zuliani et al., 25 Mar 2025).

7. Tables: Key Design Conditions in ILC

Property	Sufficient Condition	Reference(s)
Asymptotic Conv.	$u_{k+1} = u_k + L\,e_k,$ 0	(Koscielniak, 2022)
Monotonic Conv.	$u_{k+1} = u_k + L\,e_k,$ 1	(Koscielniak, 2022)
Freq. Domain Conv.	$u_{k+1} = u_k + L\,e_k,$ 2	(Koscielniak, 2023)
Robust SISO ILC	$u_{k+1} = u_k + L\,e_k,$ 3	(Blanken et al., 2018)
Multivariable Robust	$u_{k+1} = u_k + L\,e_k,$ 4	(Blanken et al., 2018)
Basis Selection	Regularized: $u_{k+1} = u_k + L\,e_k,$ 5	(Ickenroth et al., 9 May 2025)
Trackability	$u_{k+1} = u_k + L\,e_k,$ 6 (with algebraic conditions)	(Meng et al., 2022)

Each entry corresponds to results validated in the cited sources.

ILC provides a unified, rigorous methodology for high-precision trajectory tracking in repetitive systems, with a continually expanding theoretical and algorithmic toolkit enabling application to nonlinear, constrained, multivariable, and learning-driven contexts. Ongoing developments leverage advances in optimization, machine learning, and statistical modeling to mitigate classical limitations and unlock robust, scalable deployment in next-generation precision motion, manufacturing, and process-control systems.