Iterative Learning Control (ILC) constitutes one of the most influential methodologies for improving control performance in systems that execute identical tasks repeatedly over a finite time horizon. Its origins can be traced to early conceptual work by Murray (1967) and Edwards (1974), but the method was formally introduced in the seminal publication of Uchiyama (1978), who demonstrated that control inputs can be systematically refined by exploiting information from previous task executions. The field gained international recognition following the influential work of Arimoto et al. (1984), which established the theoretical foundations of learning laws and demonstrated their effectiveness in robotic trajectory tracking. These contributions initiated a research direction that has since evolved into a mature and rapidly expanding discipline.

Over the past three decades, ILC has developed into a comprehensive framework with applications in robotics, industrial automation, precision manufacturing, and batch processes. Early research focused primarily on linear time-invariant systems and simple proportional-type learning laws. These classical formulations provided essential insights into convergence, monotonicity, and robustness, forming the basis for subsequent theoretical advancements. However, modern engineering systems increasingly exhibit nonlinear, time-varying, and high-dimensional dynamics, which has motivated the development of more sophisticated ILC architectures.

Contemporary ILC research addresses several key challenges:

• Nonlinear and time-varying dynamics:
  Traditional linear ILC methods often fail to guarantee convergence in systems with strong nonlinearities. As a result, nonlinear ILC frameworks—based on Lyapunov methods, contraction theory, or differential dynamic programming—have been proposed to extend applicability to complex robotic and manufacturing systems.
• Robustness to disturbances and model uncertainties:
  Robust ILC formulations incorporate H∞ methods, adaptive learning gains, or set-based approaches to ensure stability under bounded uncertainties. These methods are essential for real-world systems where disturbances vary across iterations.
• Constraints and limited actuation:
  Many practical systems operate under actuator saturation, state constraints, or safety limits. Optimization-based ILC, including model predictive ILC (MPILC), has emerged as a promising direction for handling such constraints while preserving learning performance.
• Sparse or noisy measurements:
  Classical ILC assumes full-state or full-output measurements across iterations. However, modern systems—such as soft robots or flexible manipulators—often rely on sparse sensing. This has led to the development of observer-based ILC, learning filters, and data-driven reconstruction techniques.
• Integration with machine learning and data-driven methods:
  Recent advances incorporate neural networks, Gaussian processes, Koopman operator theory, and reinforcement learning to model complex dynamics and improve learning efficiency. These hybrid approaches aim to overcome the limitations of purely model-based or purely data-driven strategies.

Despite these advances, several critical gaps remain in the current state of the art:

1. Limited applicability to highly nonlinear, compliant, or soft robotic systems:
   Soft manipulators exhibit strong nonlinearities, hysteresis, and unmodeled dynamics that challenge existing ILC frameworks. Current methods often require dense sensing or accurate models, which are rarely available in such systems.
2. Insufficient handling of sparse or indirect measurements:
   Many emerging robotic platforms rely on minimal sensing (e.g., end-effector tracking only). Existing ILC approaches struggle to reconstruct latent states or compensate for unobserved dynamics, limiting their performance.
3. Lack of unified frameworks combining physics-based modeling with data-driven learning:
   While physics-informed neural networks, Koopman-based models, and reinforcement learning have been explored independently, there is no established methodology that integrates these tools into a coherent ILC framework for nonlinear systems.
4. Challenges in ensuring stability and robustness in data-driven ILC:
   Data-driven models introduce approximation errors that may accumulate across iterations. Ensuring convergence and robustness in such hybrid systems remains an open research problem.
5. Limited experimental validation on advanced robotic platforms:
   Many state-of-the-art ILC methods remain validated only in simulation or on rigid robotic systems. There is a clear need for experimental studies on soft manipulators, pneumatic actuators, and other emerging platforms.

Given these gaps, the development of new learning-based control strategies that combine physics-informed modeling, data-driven system identification, and robust ILC principles is both timely and necessary. Such methods have the potential to significantly advance the capabilities of nonlinear, compliant, and sparsely observed systems—domains that are increasingly relevant in modern robotics, intelligent manufacturing, and autonomous systems.

The proposed research directly addresses these challenges by integrating advanced modeling techniques with iterative learning mechanisms, supported by experimental validation on complex robotic platforms. This aligns with current international research trends and responds to clear unmet needs in the state of the art.