Iterative learning boundary control for a class of gantry crane systems with time-varying disturbance

Abstract

This paper develops an iterative learning control (ILC) scheme for gantry crane systems subject to external disturbances. The system dynamics are modeled by partial differential equations (PDEs) with boundary conditions. To suppress vibrations and to compensate for unknown boundary disturbances, an ILC law incorporating boundary feedback control is proposed. The operator semigroup theory is employed to establish the well-posedness of the closed-loop system. Exponential stability under the proposed controller is rigorously proven using Lyapunov’s direct method. Comparative simulation results demonstrate the effectiveness and superior performance of the presented ILC scheme, validating its practical applicability.

Keywords

gantry crane iterative learning control partial differential equation distributed-parameter system operator semigroup boundary control

Introduction

Gantry cranes are essential material handling systems widely employed in manufacturing, construction, and marine operations. A typical gantry crane system consists of a motorized trolley, a flexible cable, and a suspended payload. The inherent flexibility of the cable induces significant payload vibrations during transportation, posing challenges for precise and safe operation. Consequently, substantial research efforts have focused on developing effective control strategies for gantry crane systems (Nguyen et al., 2024; Wang et al., 2017; Goubej and Helma, 2019; Moradi et al., 2009; Burul et al., 2010; Tumari et al., 2012; Ramli et al., 2018; Frikha et al., 2018; Cuong et al., 2021; Albalta and Yalçın, 2023; Wen et al., 2021; Khudhaira et al., 2021; Otto et al., 2023; Rigatos et al., 2024; Yang et al., 2025b). Proposed methodologies encompass optimal control (Nguyen et al., 2024; Wang et al., 2017; Goubej and Helma, 2019), adaptive control (Frikha et al., 2018; Cuong et al., 2021; Albalta and Yalçın, 2023), H_∞ control (Moradi et al., 2009; Burul et al., 2010; Tumari et al., 2012), backstepping control (Wen et al., 2021; Khudhaira et al., 2021; Wen et al., 2022), and neural network control (Yang et al., 2025a; Tang et al., 2024). A prevalent limitation in the existing literature is the reliance on the models described by ordinary differential equations (ODEs), due to the consideration based on rigid cables. However, practical gantry cranes typically employ flexible cables, exhibiting distributed flexibility effects. ODE-based models fail to accurately capture these dynamics. Accurate representation of such systems necessitates modeling via partial differential equations (PDEs).

Beyond gantry cranes, PDE-based models provide essential modeling frameworks for diverse distributed-parameter systems in practical engineering, including flexible marine risers (Ge et al., 2009; He et al., 2011) and Euler-Bernoulli beams (Ge et al., 2011; Ma et al., 2023b). Boundary control offers a practical methodology for governing such PDE-based systems, as it requires actuation only at physical boundaries. Consequently, this approach has motivated considerable research attention toward diverse control strategies (He and Ge, 2016; Ma and Lou, 2021; Thull et al., 2005; Aguilar, 2021; Stürzer et al., 2018; d’Andréa Novel et al., 1994; d’Andréa Novel and Coron, 2000; Wijnand et al., 2021; Wen et al., 2021, 2022). In He and Ge (2016), a cooperative controller incorporating adaptation laws is developed to address the control challenges encountered in gantry crane systems with constrained tension. Leveraging the integral-barrier Lyapunov functions method, this approach establishes uniform boundedness for closed-loop system. The authors in Ma and Lou (2021) present an adaptive control method for gantry crane systems subject to parametric uncertainties and external disturbances, where two boundary adaptive laws are proposed to ensure the uniform ultimate boundedness of gantry crane systems. While the strategies in He and Ge (2016) and Ma and Lou (2021) effectively achieve control objectives and ensure stability, they require the application of control forces simultaneously to both the trolley and the suspended payload which limits practical implementation. The authors in Thull et al. (2005) develop a passivity-based control scheme with trajectory planning for gantry cranes to suppress vibrations and position cargo. The author Aguilar (2021) proposes a state-feedback H_∞ control strategy to resist the disturbances and suppress the vibration of the gantry crane. While effective for control objectives, above works omit rigorous well-posedness analysis, which is a critical requirement for PDE systems. In Stürzer et al. (2018), the authors establish well-posedness through operator semigroup and spectral theory, and further analyze both asymptotic and exponential stability. The authors in d’Andréa Novel et al. (1994) establish well-posedness of the closed-loop system while achieving exponential error convergence. Building on this foundation, d’Andréa Novel and Coron (2000) improve the controller through novel feedback terms for targeted vibration suppression. Concurrently, Wen et al. (2022) designs dual boundary feedback controllers using backstepping-derived kernel functions to simultaneously dampen oscillations and estimate inaccessible states. However, the backstepping method also has its weaknesses. It is highly dependent on the selection of the target system, and the kernel function of the backstepping transformation usually requires solving a PDE to obtain, which often brings significant difficulties.

Given the operational prevalence of gantry crane systems in industrial settings, persistent external disturbances necessitate robust control solutions to ensure stability. Various disturbance-rejection methods have been developed (Thull et al., 2005; Aguilar, 2021; Ramli et al., 2018; Huang et al., 2024; Guo and Jin, 2013; Zhao et al., 2023), including sliding-mode control (Zhang and Wu, 2025; Yang et al., 2025b; Huang et al., 2024; Zhang et al., 2024), disturbance observer approaches (Guo and Jin, 2013; Zhao et al., 2023), fuzzy control (Aguiar et al., 2021; Bounemeur and Chemachema, 2021; Boulkroune et al., 2025; Bounemeur et al., 2018; Boulkroune et al., 2017; Bounemeur and Chemachema, 2023; Abdelhamid and Mohamed, 2025), and neural network (Ma et al., 2022; Ramli et al., 2018; Ma et al., 2023a; Wang and Cui, 2025). While the sliding-mode control effectively rejects disturbance using its upper bound. However, the upper bound of the disturbance may be unknown in practice. Similarly, the disturbance observer techniques face limitations due to stringent disturbance characterization requirements. Although neural networks possess excellent nonlinear approximation capabilities, their poor interpretability and limited generalization to out-of-distribution scenarios can compromise closed-loop stability. This prevents the provision of stability guarantees typically required in safety-critical applications. In Yang et al. (2025b), the authors propose an adaptive proportional-integral-derivative sliding-mode control framework for ODE-based 4-degree of freedom tower crane systems, where system uncertainties and unknown time-varying external disturbance are addressed in their work. The authors in Wang et al. (2025) investigated a control method under the Iterative Learning Control (ILC) scheme for the ODE-based rotary cranes with input saturation and uncertainties. However, using the ODE equation to describe the flexible structure would reduce the modeling accuracy and might bring some problems such as control spillover, which could lead to some safety hazards in practical applications. The authors in Zhang and Wu (2025) and Huang et al. (2024) respectively proposed two different forms of sliding-mode controllers for the PDE-based gantry crane system, but both of their controllers require a priori knowledge of the disturbance bound. The above methods have certain limitations in terms of disturbance suppression.

ILC offers a promising alternative for rejecting boundary disturbances with minimal model dependency. Unlike conventional approaches, ILC compensates for unknown boundary disturbances through iterative refinement of control inputs using historical error data (Wang et al., 2025; Liu et al., 2019). This model-agnostic approach demonstrates particular efficacy in flexible structures, as validated in flexible structure applications (Meng and He, 2020).

Motivated by the aforementioned challenges, an ILC scheme is proposed to mitigate the effects of unknown time-varying disturbances. The contributions of this work can be outlined below.

(1) A novel ILC framework incorporating boundary feedback control is developed to simultaneously achieve payload positioning and vibration suppression in the presence of unknown disturbances.

(2) The controller eliminates the restrictive assumption of a known disturbance bound, thereby enhancing practical robustness and applicability.

(3) Rigorous theoretical analysis via operator semigroup theory establishes the well-posedness of the closed-loop system. Exponential stability is rigorously proven using Lyapunov’s direct method.

(4) The proposed approach is validated through comparative simulations and hardware experiments, demonstrating its effectiveness over conventional methods.

The remaining sections of this paper are outlined as follows: In Dynamic model and preliminaries Section, the model of the gantry crane is established. Main result Section describes the design of ILC. Besides, the well-posedness and stability of closed-loop system are analyzed. Numerical simulations and physical experiments Section validates the efficacy of the proposed controllers through numerical simulations and experimental analysis. Finally, Conclusion Section gives a conclusion of this paper.

The notations in Table 1 are used throughout this paper.

Table 1.

Notations.

Notations
$R$	Set of real number
$R^{+}$	Set of positive real number
$L^{2} (0, L)$	Class of lebesgue measurable functions satisfying $\int_{0}^{L} \| f (x) \|^{2} d x <$ ∞
$L_{p} ([0, τ]; H)$	Class of lebesgue measurable functions satisfying $\int_{0}^{τ} \| f (t) \|_{H}^{p} d t < \infty$
$L_{\infty} [0, \infty)$	Class of bounded measurable functions on [0, ∞)
$H^{m} (0, L)$	Sobolev space $W^{m, 2} (0, L)$ as $D^{α} u \in L^{2} (0, L)$ for 0 ≤ α ≤ m
$H^{*}$	Dual space of $H$
Ran(A)	Range of the operator A
D(A)	Domain of the operator A

Dynamic model and preliminaries

Dynamics of gantry crane system

Figure 1 illustrates the gantry crane system structure. The cable displacement at position x and time t during the k-th iteration is denoted by w_k (x, t), while u_k(t) represents the boundary control force applied to the trolley, and d_k(t) denotes the time-varying boundary disturbance acting on the trolley. Key system parameters include: trolley mass M, payload mass m, cable length L, cable linear density ρ, and gravitational acceleration g. The iteration index $k \in N$ corresponds to the learning cycle period. $\dot{(\cdot)} = \partial (\cdot) / \partial t$ and (⋅)′ = ∂(⋅)/∂x are used throughout this paper.

Before presenting the model of gantry crane, the following assumption is given.

Assumption 1

Rahn and Christopher (2001):

• The cable is completely flexible and non-stretchable.

• Transversal and angular displacements are negligible.

• Load mass acceleration is negligible compared to gravitational acceleration g.

Taking the impact of time-varying boundary disturbance d_k(t) into consideration, and employing Hamilton’s principle along with Assumption 1, the following model is used to describe gantry crane system:

\begin{align} {[T (x) w_{k}^{'} (x, t)]}^{'} - ρ {\ddot{w}}_{k} (x, t) = 0, \forall (x, t) \in (0, L) \times (0, t_{T}) \end{align}

(1)

\begin{align} T (0) w_{k}^{'} (0, t) - M {\ddot{w}}_{k} (0, t) + u_{k} (t) - d_{k} (t) = 0, \end{align}

(2)

\begin{align} w_{k}^{'} (L, t) = 0 . \end{align}

(3)

where (1) is governing equation, and equations (2)-(3) are boundary conditions for x = 0 and x = L respectively. t_T ∈ (0, ∞) is the time interval for each iteration. T(x) denotes the tension of cable at point x = P (Rahn and Christopher, 2001).

T (x) = g [m + ρ (L - x)] .

(4)

Remark 1

The terminal state at each iteration serves as the initial condition for the subsequent iteration, satisfying w_k(⋅, t_T) = w_k+1(⋅, 0).

Preliminaries

This subsection introduces some lemmas and an assumption to the disturbance to help us make the analysis of the system.

Since the magnitude of disturbance energy is finite, the following assumption is made.

Assumption 2

The unknown time-varying boundary disturbance satisfies $d_{k} (t) \in L_{\infty} (0, t_{T}), \forall k \in N$ .

Lemma 1

Zhang et al. (2024): For any $a (x) \in H^{1} (0, L)$ , the following inequality can be obtained

\int_{0}^{L} a^{2} (x) d x \leq 2 L a^{2} (0) + 2 L^{2} \int_{0}^{L} a_{x}^{2} (x) d x .

(5)

Lemma 2

Zhang et al. (2024): For any $s (x), z (x) \in H^{1} (0, L)$ , the following inequality holds

\begin{align} |z (0) s (0)| & \leq \int_{0}^{L} [|z_{x} (x)| |s (x)| + |z (x)| |s_{x} (x)|] d x \\ + \frac{1}{L} \int_{0}^{L} |z (x)| |s (x)| d x, \end{align}

(6)

The same applies to $|z (L) s (L)|$ .

Main result

Controller design

To achieve precise payload positioning and effective vibration suppression in the gantry crane system governed by (1)–(3) while rejecting unknown time-varying disturbances d_k(t), an ILC control scheme is designed. The block diagram of the control scheme is shown as Figure 2.

The controller is designed as follows:

\begin{align} u_{k} (t) & = ξ_{1} w_{k} (0, t) + ξ_{2} {\dot{w}}_{k} (0, t) + ξ_{3} w_{k}^{'} (0, t) + ξ_{4} {\dot{w}}_{k}^{'} (0, t) \\ + {\hat{δ}}_{k - 1} (t) s i g n (φ_{k} (t)), \end{align}

(7)

where φ_k(t) is an auxiliary function defined as

φ_{k} (t) = k_{1} T (0) w_{k}^{'} (0, t) - k_{2} w_{k} (0, t) - k_{3} M {\dot{w}}_{k} (0, t),

(8)

and the signum function is given by

sign (x) ≔ \{\begin{cases} 1 & x > 0, \\ 0 & x = 0, \\ - 1 & x < 0 . \end{cases}

The iterative term ${\hat{δ}}_{k} (t)$ compensates for external boundary disturbances and is designed as

{\hat{δ}}_{k} (t) = {\hat{δ}}_{k - 1} (t) + η φ_{k} (t) s i g n (φ_{k} (t)), {\hat{δ}}_{- 1} (t) = 0

(9)

The parameters ξ₁, ξ₂, ξ₃, ξ₄ are

\begin{align} ξ_{1} = - \frac{k_{2} k_{4}}{k_{3}} - \frac{β}{2 k_{1} k_{3}}, \end{align}

(10)

\begin{align} ξ_{2} = - \frac{k_{2}}{k_{3}} - k_{4} M - \frac{1}{k_{3}} + \frac{M β}{2 k_{1} k_{2}}, \end{align}

(11)

\begin{align} ξ_{3} = - T (0) + \frac{k_{1} k_{4}}{k_{3}} T (0) + \frac{β T (0)}{2 k_{2} k_{3}}, \end{align}

(12)

\begin{align} ξ_{4} = \frac{k_{1}}{k_{3}} T (0) . \end{align}

(13)

k₁, k₂, k₃, k₄, and β satisfy the following inequalities

\{\begin{cases} \frac{k_{1}}{2} - \frac{α_{4} ρ L T (0)}{2} - \frac{β ρ L^{2}}{T (L)} > 0, \\ k_{1} - α_{4} L - β > 0, \\ \frac{k_{2}}{2} - β ρ L > 0, \\ k_{1} α_{1} - \frac{α_{4} L T (0)}{2} > 0, \\ k_{2} α_{2} > 0 \\ k_{3} α_{3} M - \frac{α_{4} ρ L T (0)}{2} > 0, \\ β + \frac{α_{4} T (L)}{2} > 0, \\ \frac{α_{4} T (L)}{2} - β > 0, \\ k_{4} > 0, \end{cases}

(14)

where α₁, α₂, and α₃ are

α_{1} = \frac{β T (0)}{2 k_{2}}, α_{2} = \frac{β}{2 k_{1}}, α_{3} = 1 - \frac{k_{3} M β}{2 k_{1} k_{2}} .

(15)

Well-posedness analysis

The Hilbert space $H$ for well-posedness analysis is defined as

H = H^{1} (0, L) \times L^{2} (0, L) \times R^{2} : w^{'} (L) = 0 .

(16)

Applying controller (7) yields the closed-loop system. To formulate it as an abstract Cauchy problem in semilinear differential form, define the linear operator $A : D (A) \to H$ :

\begin{align} A P & = [z (x), \frac{{[T (x) w^{'} (x)]}^{'}}{ρ}, z (0), \\ - {k_{4} ψ - [α_{1} w^{'} (0) - α_{2} w (0) - α_{3} z (0)]]}^{T}, \end{align}

(17)

where

P = {[w (x), z (x), ϕ, ψ]}^{T} .

(18)

The domain of operator A is

\begin{align} D (A) & = \{{[w (x), z (x), ϕ, ψ]}^{T} \\ \in H^{2} (0, L) \times H^{1} (0, L) \times R^{2}, w_{x} (L) = 0\} . \end{align}

(19)

The closed-loop system then becomes an evolution equation in $H$ :

\frac{d P_{k} (t)}{d t} = A P_{k} (t) + D P_{k} (t), P_{k} (0) = P_{k 0},

(20)

where

P_{k} (t) = {[w_{k} (x, t), {\dot{w}}_{k} (x, t), w_{k} (0, t), φ_{k} (t)]}^{T},

(21)

and

D P_{k} (t) = {[0, 0, 0, k_{3} (d_{k} (t) - {\hat{δ}}_{k - 1} (t) s i g n (ψ))]}^{T} .

(22)

The inner product on $H$ is

\begin{align} {⟨P_{1}, P_{2}⟩}_{H} = k_{1} \int_{0}^{L} T (x) w_{1}^{'} (x) w_{2}^{'} (x) d x \\ + k_{1} \int_{0}^{L} ρ z_{1} (x) z_{2} (x) d x \\ + β ρ \int_{0}^{L} w_{1} (x) z_{2} (x) d x + β ρ \int_{0}^{L} w_{2} (x) z_{1} (x) d x \\ + α_{4} ρ \int_{0}^{L} (x - L) T (x) w_{1}^{'} (x) z_{2} (x) d x \\ + α_{4} ρ \int_{0}^{L} (x - L) T (x) w_{2}^{'} (x) z_{1} (x) d x \\ + k_{2} ϕ_{1} ϕ_{2} + ψ_{1} ψ_{2}, \end{align}

(23)

where

\begin{array}{l} P_{1} = {[w_{1} (x), z_{1} (x), ϕ_{1}, ψ_{1}]}^{T}, \\ P_{2} = {[w_{2} (x), z_{2} (x), ϕ_{2}, ψ_{2}]}^{T}, \end{array}

k₁, k₂, α₄, β, γ₃, γ₄ are positive constants satisfying the following inequalities

\{\begin{cases} k_{1} T (L) - γ_{3} α_{4} ρ L T^{2} (0) - 2 γ_{4} β ρ L^{2} > 0, \\ k_{2} - 2 γ_{4} β ρ L > 0, \\ k_{1} - \frac{α_{4} L}{γ_{3}} - \frac{β}{γ_{4}} > 0 . \end{cases}

(24)

Lemma 3

The norm ${‖\cdot‖}_{H}$ induced by (23) is equivalent to the usual norm $‖\cdot‖$ on $H^{1} (0, L) \times L^{2} (0, L) \times R^{2}$ .

Proof

The proof is deferred to Appendix A. □

Theorem 1

Suppose that Assumptions 1-2 hold. The closed-loop system governed by equations (1)–(3) with controller (7) is well-posed for every iteration period. Specifically, for each $k \in N$ , there exists a unique mild solution defined for t ∈ [0, t_T].

Proof 1

The well-posedness of the closed-loop system follows from the existence of a C₀-semigroup on $H$ . Using Definition 5.1.4 in Curtain and Zwart (2020), we show that operator A generates a C₀-semigroup of contractions on $H$ . The verification procedure comprises:

Step 1: We prove operator A is dissipative in $H$ by showing ${⟨ A P, P ⟩}_{H} \leq 0$ for all $P = {[w (x), z (x), ϕ, ψ]}^{T} \in D (A)$ .

\begin{align} {⟨A P, P⟩}_{H} \\ = - k_{2} α_{2} ϕ^{2} - T (0) [k_{1} α_{1} - \frac{α L T (0)}{2}] {w^{'}}^{2} (0) - k_{4} ψ^{2} \\ - [k_{3} α_{3} M - \frac{α_{4} ρ L T (0)}{2}] z^{2} (0) - β \int_{0}^{L} T (x) {w^{'}}^{2} (x) d x \\ - \frac{α_{4}}{2} \int_{0}^{L} T^{2} (x) {w^{'}}^{2} (x) d x - \frac{α_{4} ρ}{2} \int_{0}^{L} ρ g (L - x) z^{2} (x) d x \\ - \frac{α_{4} ρ}{2} \int_{0}^{L} T (x) z^{2} (x) d x + β ρ \int_{0}^{L} z^{2} (x) d x \\ \leq - k_{2} α_{2} ϕ^{2} - T (0) [k_{1} α_{1} - \frac{α_{4} L T (0)}{2}] {w^{'}}^{2} (0) - k_{4} ψ^{2} \\ - [k_{3} α_{3} M - \frac{α_{4} ρ L T (0)}{2}] z^{2} (0) - β \int_{0}^{L} T (x) {w^{'}}^{2} (x) d x \\ - \frac{α_{4}}{2} \int_{0}^{L} T^{2} (x) {w^{'}}^{2} (x) d x \\ - [\frac{α_{4} T (L)}{2} - β] \int_{0}^{L} ρ z^{2} (x) d x, \end{align}

(25)

where integration by parts is employed and parameters satisfying

\{\begin{cases} α_{1} k_{1} - \frac{α_{4} L T (0)}{2} > 0, \\ α_{2} k_{2} > 0, \\ α_{3} k_{3} M - \frac{α_{4} ρ L T (0)}{2} > 0, \\ \frac{α_{4} T (L)}{2} - β > 0, \\ α_{4} > 0, \\ β > 0, \\ k_{5} > 0 . \end{cases}

(26)

Step 2: This step aims to show that there exists a λ₀ > 0 such that (λ₀I − A) is surjective, that is, $R a n (λ_{0} I - A) = H$ . To do this, it suffices to demonstrate that for any $Q (x) = {[f (x), g (x), \tilde{ϕ}, \tilde{ψ}]}^{T} \in H$ , the unique weak solution P ∈ D(A) exists satisfying

(I - A) P = Q,

(27)

where I is the identity operator on

H

(27) can be written as

\{\begin{cases} w (x) - z (x) = f (x), \\ z (x) - \frac{{[T (x) w^{'} (x)]}^{'}}{ρ} = g (x), \\ ϕ - z (0) = \tilde{ϕ}, \\ (1 + k_{4}) ψ + [α_{1} w^{'} (0) - α_{2} w (0) - α_{3} z (0)] = \tilde{ψ} . \\ w^{'} (L) = 0 . \end{cases}

(28)

Rearranging (28) yields

\begin{align} w (x) - \frac{{[T (x) w^{'} (x)]}^{'}}{ρ} = f (x) + g (x) \end{align}

(29)

\begin{align} w^{'} (0) - χ_{1} w (0) = χ_{2} \tilde{ψ} - χ_{3} \tilde{ϕ} \end{align}

(30)

with coefficients

\begin{array}{l} χ_{1} & = \frac{[(1 + k_{4}) (k_{2} + k_{3} M) + α_{2} + α_{3}]}{[(1 + k_{4}) k_{1} T (0) + α_{1}]}, \\ χ_{2} & = \frac{1}{[(1 + k_{4}) k_{1} T (0) + α_{1}]}, \\ χ_{3} & = \frac{(1 + k_{4}) k_{3} M + α_{3}}{(1 + k_{4}) k_{1} T (0) + α_{1}} . \end{array}

Subsequently, choosing a test function

v (x) \in H^{1} (0, L)

, multiplying both sides of equation (29) and integrating over x from 0 to L yields

\begin{align} \int_{0}^{L} [w (x) v (x) + \frac{T (x)}{ρ} w^{'} (x) v^{'} (x)] d x + \frac{T (0)}{ρ} w^{'} (0) v (0) \\ = \int_{0}^{L} f (x) v (x) d x + \int_{0}^{L} g (x) v (x) d x . \end{align}

(31)

Then combining (30) and (31) yields

\begin{align} \int_{0}^{L} [w (x) v (x) + \frac{T (x)}{ρ} w^{'} (x) v^{'} (x)] d x + \frac{T (0)}{ρ} χ_{1} w (0) v (0) \\ = \int_{0}^{L} f (x) v (x) d x + \int_{0}^{L} g (x) v (x) d x \\ + \frac{T (0) v (0) χ_{3}}{ρ} \tilde{ϕ} - \frac{T (0) v (0) χ_{2}}{ρ} \tilde{ψ} . \end{align}

(32)

Since the left side of (32) is a bi-linear form, denoting $a (y (x), v (x)) = \int_{0}^{L} [w (x) v (x) + T (x) / ρ w^{'} (x) v^{'} (x)] d x + T (0) / ρ χ_{1} w (0) v (0)$ . To prove the existence and the uniqueness of the solution by the Lax-Milgram theorem (Lax, 1954), it remains to verify the continuity and coerciveness of a (y(x), v(x)) on $H^{1} (0, L)$ . By utilizing the Hölder inequality and Lemma 2, it can be obtained

\begin{align} |a (y (x), v (x))| \\ \leq \int_{0}^{L} |y (x)| |v (x)| d x + \frac{T (0)}{ρ} \int_{0}^{L} |y^{'} (x)| |v^{'} (x)| d x \\ + \frac{T (0)}{ρ} χ_{1} |y (0) v (0)| \\ \leq (1 + \frac{T (0) χ_{1}}{ρ L}) {‖y (x)‖}_{L^{2}} {‖v (x)‖}_{L^{2}} \\ + \frac{T (0) χ_{1}}{ρ} {‖y^{'} (x)‖}_{L^{2}} {‖v (x)‖}_{L^{2}} \\ + \frac{T (0) χ_{1}}{ρ} {‖y (x)‖}_{L^{2}} {‖v^{'} (x)‖}_{L^{2}} \\ + \frac{T (0)}{ρ} {‖y^{'} (x)‖}_{L^{2}} {‖v^{'} (x)‖}_{L^{2}} \\ \leq c_{α} {‖y (x)‖}_{H^{1}} {‖v (x)‖}_{H^{1}}, \end{align}

(33)

where

c_{α} = 1 + \frac{T (0) χ_{1}}{ρ L} + \frac{T (0)}{ρ} + \frac{2 T (0) χ_{1}}{ρ} .

Furthermore,

a (y (x), y (x))

also satisfies

\begin{align} a (y (x), y (x)) \\ = \int_{0}^{L} [y^{2} (x) + \frac{T (x)}{ρ} y^{2} (x)] d x + \frac{T (0) χ_{1}}{ρ} y^{2} (0) \\ \geq \int_{0}^{L} [y^{2} (x) + {y^{'}}^{2} (x)] d x \\ \geq c_{β} {‖y (x)‖}_{H^{1}}^{2}, \end{align}

(34)

where

c_{β} = \min \{1, T (L) / ρ\}

The continuity and coerciveness are demonstrated through equations (33) and (34) respectively. Since the right-hand side of (31) defines a bounded linear functional, the Lax-Milgram Theorem (Lax, 1954) guarantees the existence and uniqueness of a weak solution w ∈ H¹ (0, L) to equations (29)-(30). Consequently, (27) admits a unique weak solution $P = {[w (x), z (x), ϕ, ψ]}^{T} \in D (A)$ , established in accordance with the theoretical framework presented in Chapters 5 and 6 of Banks (2012).

Step 3: Having established that A is a maximal monotone operator, Corollary 2.4.3 in Cazenave and Haraux (1998) implies D(A) is dense in $H$ . Thus, by the Lumer-Phillips theorem (Banks, 2012), A generates a C₀-semigroup of contractions on $H$ .

Finally, by the definition of ${\hat{δ}}_{k} (t)$ in (9), it is evident that sign (φ_k(t)) is bounded with ${\hat{δ}}_{k - 1} (t)$ continuous on t ∈ [0, t_T]. The Assumption 2 indicates that $d_{k} (t) \in L_{\infty} (0, t_{T})$ , implying that d_k(t) belongs to $L_{p} (0, t_{T})$ , 1 < p ≤ ∞, by Theorem 2.14 in Adams and Fournier (2003). Consequently, $D_{k} \in L_{p} ([0, t_{T}]; H)$ with 1 < p ≤ ∞. According to Definition 5.1.4 in Curtain and Zwart (2020), there is a unique mild solution to the closed-loop system for each iteration period t ∈ [0, t_T). □

Stability analysis

This subsection establishes exponential stability for the closed-loop system under the proposed iterative learning controller (7).

Theorem 2

Suppose that Assumptions 1-2 hold. The closed-loop system (1)-(3) under the designed iterative learning controller (7) is exponentially stable if the following conditions are satisfied

\{\begin{cases} \frac{k_{1}}{2} - \frac{α_{4} ρ L T (0)}{2} - \frac{β ρ L^{2}}{T (L)} > 0, \\ k_{1} - α_{4} L - β > 0, \\ \frac{k_{2}}{2} - β ρ L > 0, \\ α_{1} k_{1} T (0) - \frac{α_{4} L T^{2} (0)}{2} > 0, \\ α_{2} k_{2} > 0, \\ α_{3} k_{3} M - \frac{α_{4} ρ L T (0)}{2} > 0, \\ \frac{α_{4} T (L)}{2} - β > 0, \\ k_{4} > 0, \\ β > 0, \\ α_{4} > 0 . \end{cases}

(35)

Proof 2

Consider the following Lyapunov candidate function

V_{k} (t) = V_{1, k} (t) + V_{2, k} (t) + V_{3, k} (t) + V_{4, k} (t),

(36)

where

\begin{align} V_{1, k} (t) & = \frac{k_{1}}{2} \int_{0}^{L} T (x) {w_{k}^{'}}^{2} (x, t) d x + \frac{k_{1}}{2} \int_{0}^{L} ρ {\dot{w}}_{k}^{2} (x, t) d x \\ + \frac{k_{2}}{2} {w_{k}}^{2} (0, t), \end{align}

(37)

\begin{align} V_{2, k} (t) & = α_{4} ρ \int_{0}^{L} (x - L) T (x) w_{k}^{'} (x, t) {\dot{w}}_{k} (x, t) d x, \end{align}

(38)

\begin{align} V_{3, k} (t) & = β ρ \int_{0}^{L} w_{k} (x, t) {\dot{w}}_{k} (x, t) d x, \end{align}

(39)

\begin{align} V_{4, k} (t) & = \frac{φ_{k}^{2} (t)}{2} . \end{align}

(40)

Step 1: The Lyapunov function can be reformulated as the norm, that is, $V_{k} (t) = 1 / 2 {‖P_{k} (x, t)‖}_{H}^{2}$ which is positive definite when condition (24) holds. Furthermore, it satisfies the bounds:

\begin{align} 0 & \leq (1 - η_{1}) [V_{1, k} (t) + V_{4, k} (t)] \leq V_{k} (t) \\ \leq (1 + η_{1}) [V_{1, k} (t) + V_{4, k} (t)], \end{align}

(41)

where

η_{1} = m a x \{α_{4} ρ L T (0) / k_{1} + 2 β ρ L^{2} / k_{1} T (L), α_{4} L + β / k_{1}, 2 β ρ L / k_{2}\}

Step 2: We prove ${\dot{V}}_{k} (t) \leq - λ V_{k} (t)$ to establish exponential convergence. The proof is deferred to Appendix B.

It follows from ${\dot{V}}_{k} (t) \leq - λ V_{k} (t)$ that

V_{k} (t) \leq V_{k} (0) e^{- λ t} .

(42)

Then combining (41), it can obtain

\frac{k_{1} (1 - η_{1})}{2} \int_{0}^{L} T (x) {w_{k}^{'}}^{2} (x, t) d x \leq V_{k} (0) e^{- λ t} .

(43)

Applying Cauchy-Schwarz inequality yields

{[\int_{0}^{x} w_{k}^{'} (u, t) d u]}^{2} \leq L \int_{0}^{L} {w_{k}^{'}}^{2} (x, t) d x,

(44)

where x ∈ [0, L]. Combining (43) and (44)

|w_{k} (x, t) - w_{k} (0, t)| \leq \sqrt{\frac{2 L V_{k} (0)}{T (L) k_{1} (1 - η_{1})} e^{- λ t}} .

(45)

Similarly, it can obtain

|w_{k} (0, t)| \leq \sqrt{\frac{2 V_{k} (0)}{k_{2} (1 - η_{1})} e^{- λ t}} .

(46)

Applying inequality

|a| - |b| \leq |a - b|

to (45) yields

|w_{k} (x, t)| \leq \sqrt{\frac{2 L V_{k} (0)}{T (L) k_{1} (1 - η_{1})} e^{- λ t}} + \sqrt{\frac{2 V_{k} (0)}{k_{2} (1 - η_{1})} e^{- λ t}} .

(47)

It implies that the closed-loop system is exponentially stable. Moreover, the auxiliary term φ_k(t) satisfies

|φ_{k} (t)| \leq \sqrt{\frac{2 V_{k} (0)}{1 - η_{1}} e^{- λ t}} .

(48)

Furthermore,

V_{k + 1} (0) = V_{k} (t_{T}) \leq V_{k} (0) e^{- λ t_{T}} .

(49)

Let k = m + n, where $n \in N$ and k ≥ m, then, (49) can be rewritten as

V_{m + n} (0) \leq V_{m} (0) e^{- n λ t_{T}} .

(50)

Then it can obtain

\begin{align} {\hat{δ}}_{k} (t) = {\hat{δ}}_{m + n} (t) & = {\hat{δ}}_{m - 1} (t) + η \sum_{i = 0}^{n} |φ_{m + i} (t)| \\ \leq {\hat{δ}}_{m - 1} (t) + η \sum_{i = 0}^{n} \sqrt{\frac{2}{1 - η_{1}} V_{m + i} (t)} \\ \leq {\hat{δ}}_{m - 1} (t) + \sqrt{\frac{2 η^{2} V_{m} (0)}{1 - η_{1}}} \sum_{i = 0}^{n} e^{- \frac{λ t_{T}}{2} i} \\ = {\hat{δ}}_{m - 1} (t) + \sqrt{\frac{2 η^{2} V_{m} (0)}{1 - η_{1}}} \frac{1 - e^{- \frac{λ t_{T}}{2} n}}{1 - e^{- \frac{λ t_{T}}{2}}} . \end{align}

(51)

Letting t ∈ [0, t_T] and n → ∞, it can obtain that

\begin{align} \lim_{n \to \infty} {\hat{δ}}_{k} (t) & = {\hat{δ}}_{m - 1} (t) + \sqrt{\frac{2 η^{2} V_{m} (0)}{1 - η_{1}}} \frac{1 - e^{- \frac{λ t_{T}}{2} n}}{1 - e^{- \frac{λ t_{T}}{2}}} \\ \leq {\hat{δ}}_{m - 1} (t) + \sqrt{\frac{2 η^{2} V_{m} (0)}{1 - η_{1}}} \frac{1}{1 - e^{- \frac{λ t_{T}}{2}}} < \infty . \end{align}

(52)

which shows that

{\hat{δ}}_{k} (t)

is bounded with t_T ∈ (0, ∞). □

Numerical simulations and physical experiments

Numerical simulations

In this subsection, numerical simulations are provided to validate the effectiveness of proposed boundary controller (7) and iterative term ${\hat{δ}}_{k} (t)$ in (9). The time step and space step are chosen as △t = 1 × 10⁻⁴ s and △x = 1 × 10⁻² m. The tracking error e (0, t) = w (0, t) − w_d is defined as tracking error of trolley, where w_d represents the target location, which is set to w_d = 500 mm. Besides, initial conditions are set as w₀ (0, t) = 100 mm, ${\dot{w}}_{0} (0, t) = 0 m / s$ . System parameters are chosen as L = 1 m, ρ = 0.2 kg/m, M = 2.1 kg and m = 10 kg. The time-varying disturbance d_k(t) is modeled by $d_{k} (t) = 0.2 (1 + s i n (π t) + 3 s i n (3 π t) + 5 s i n (5 π t))$ .

Boundary controller (7) with iterative term (9): The parameters of controller (7) are set as k₁ = 1.2, k₂ = 5, k₃ = 1, k₄ = 5, and β = 1. The initial value of the iterative term ${\hat{δ}}_{- 1} (t)$ and the parameter η are set as ${\hat{δ}}_{- 1} (t) = 0$ and η = 1.5 in the iterative term (9). The iteration interval is 0.1 s with k = 100 iterations.

Figure 3 illustrates the trends of the iterative term and auxiliary function. These results reveal that the iterative term rapidly converges to the disturbance upper bound and stabilizes the system state.

Next, we compare the controller (7) with several existing controllers.

Case 1 (PD control)

Control problem of gantry crane can be solved using a PD type controller (Wen et al., 2021; Aguilar, 2021):

u (t) = - a_{1} e (0, t) - a_{2} e_{t} (0, t),

(53)

The parameters a₁ = 50 and a₂ = 45 were deliberately selected based on extensive experimentation, yielding optimal results across multiple tests. Comparative responses are shown in Figure 4. While this control scheme is simple, requiring only position and velocity feedback, the simulation results demonstrate that the payload requires more time to reach the target position than under the proposed controller (7).

Furthermore, external disturbances limit the effectiveness of vibration suppression. The initial phase exhibits a steep slope Figure 4(b), indicating significant transient vibrations. Notably, although these vibrations progressively attenuate over time, they fail to decay to zero.

Case 2 (boundary control strategy in Rahn and Christopher (2001))

The control strategy in Rahn and Christopher (2001) is as follows

u (t) = - b_{1} e (0, t) - b_{2} e_{t} (0, t) + b_{3} e_{x} (0, t) .

(54)

Controller gains are set to b₁ = 100, b₂ = 60 and b₃ = 800, determined through extensive parametric tuning to achieve satisfactory dynamic responses in the closed-loop system. The corresponding responses are shown in Figure 5. From Figure 5(a), it can be observed that the payload reaches the target position within 4 seconds. In comparison with controller 54, the proposed controller (7) achieves faster and smoother positioning. Furthermore, Figure 5(b) demonstrates effective vibration damping, evidencing superior vibration suppression relative to the controller 54. However, residual vibrations persist at the target position, and excessive transient vibrations remain during the initial phase. When subject to external disturbances, controller (7) outperforms this approach in both positioning accuracy and vibration suppression.

Case 3 (boundary control strategy in Wen et al. (2021))

The boundary control strategy is

u (t) = - d_{1} e (0, t) - d_{2} e_{t} (0, t) + d_{3} e_{x} (0, t) + d_{4} e_{x t} (0, t) .

(55)

The parameters are set to d₁ = 110, d₂ = 90, d₃ = 600, and d₄ = 50, with system dynamics shown in Figure 6. In comparison with controller (55), controller (7) achieves comparable positioning speed while providing effective vibration suppression. It can be seen from Figure 6(b) that the initial vibration amplitude of the system under controller (55) is substantial, and persistent oscillations occur due to external disturbances. In contrast, the proposed iterative learning controller (7) exhibits smoother transient behavior than (55) and maintains the slope within ±5 × 10⁻³ rad, confirming superior vibration suppression.

Case 4 (sliding-mode boundary control strategy in Huang et al. (2024))

The sliding surface is

\begin{align} s (t) & = \int_{0}^{t} T (0) e_{x} (0, τ) d τ - g_{1} \int_{0}^{t} e (0, τ) d τ \\ - M e_{t} (0, t) - g_{2} e (0, t), \end{align}

(56)

and the sliding-mode controller is

u (t) = - g_{1} e (0, t) - g_{2} e_{t} (0, t) + (g_{3} + \bar{d}) sign [s (t)],

(57)

The controller gains are set to g₁ = 25, g₂ = 30, g₃ = 5 and $\bar{d} = 9$ , consistent with the values reported in Huang et al. (2024). From Figure 7, it can be observed that the proposed iterative learning controller (7) demonstrates significant advantages in the presence of external disturbances. Figure 7(a) illustrates the transient dynamics at point x = L, showing that the payload reaches the target position within 2 seconds with smooth positioning. Furthermore, Figure 7(b) demonstrates that the slope is maintained within ±5 × 10⁻³ rad and converges to zero within 2 s, confirming effective vibration suppression. Since controller (57) lacks angle feedback, small residual vibrations persist at the target position. Additionally, while controller (57) requires a priori knowledge of the disturbance bound, the proposed controller does not, thereby enhancing practical applicability.

Case 5 (sliding-mode boundary control strategy in Zhang and Wu (2025))

The sliding surface is

s (t) = h_{1} T (0) e_{x} (0, t) - h_{2} e (0, t) - h_{3} M e_{t} (0, t),

(58)

and the sliding-mode controller is

\begin{align} u (t) & = - \frac{h_{1}}{h_{3}} T (0) e_{x t} (0, t) - T (0) e_{x} (0, t) - \frac{h_{2}}{h_{3}} e_{t} (0, t) \\ + h_{4} s (t) + (h_{5} + \bar{d}) sign [s (t)], \end{align}

(59)

The controller gains are set to h₁ = 1, h₂ = 20, h₃ = 12, h₄ = 10, h₅ = 3 and $\bar{d} = 9$ in accordance with the original literature (Zhang and Wu, 2025). From the results presented in Figure 8, it can be observed that the proposed controller (7) outperforms controller (59) in both positioning speed and vibration suppression. Furthermore, controller (59), similar to controller (57), requires prior knowledge of the disturbance bound.

The average computational time for a single run of each controller during simulation is presented as follows:

• Case 1 in Aguilar (2021): 1.6905 × 10⁻⁷ s

• Case 2 in Rahn and Christopher (2001): 1.8488 × 10⁻⁷ s

• Case 3 in Wen et al. (2021): 2.0268 × 10⁻⁷ s

• Case 4 in Huang et al. (2024): 2.1455 × 10⁻⁷ s

• Case 5 in Zhang and Wu (2025): 2.6501 × 10⁻⁷ s

• Controller (7): 3.4081 × 10⁻⁷ s

The results indicate that the Case 1 in Aguilar (2021) controller exhibits the shortest execution time due to its simple structure. As the architectural complexity of the controllers increases, the corresponding computational cost also increases accordingly. The proposed ILC controller (7) requires the longest execution time among the controllers compared. However, even the longest execution time of 3.4081 × 10⁻⁷ s is negligible and does not compromise the feasibility of real-time implementation, which will be further validated through the subsequent experimental section.

Physical experiments

In this subsection, physical experiments are executed using a self-constructed experimental platform to demonstrate the practical viability of the proposed iterative learning controller (7) in practical applications. The experimental platform is depicted in Figure 9. The system parameters of experimental platform are as follows: M = 2.1 kg, m = 9.4 kg, L = 1 m and ρ = 0.2 kg/m.

The control algorithm is implemented on an STM32F103 embedded platform, with a Panasonic MSMF012L1U2M servo motor driven by a MADLT05SF driver. The system operates at a 20 ms control period. Initial conditions are set to w_k (x, 0) = 100 mm and ${\dot{w}}_{k} (x, 0) = 0 m/s$ , with a target position w_d = 500 mm. The parameters of controller (7) are: k₁ = 1.2, k₂ = 35, k₃ = 10, k₄ = 100, β = 1 and η = 0.25. A total of k = 135 iterations are selected and the time interval t_T = 0.1 s. Additionally, the dynamic frictions, identified as the main disturbance, are approximated with $\bar{d} = 17.5 N$ . Comparative experiments against controllers (53)-(55) and (57) from Case 1-Case 5 (Aguilar, 2021; Rahn and Christopher, 2001; Wen et al., 2021; Huang et al., 2024; Zhang and Wu, 2025) demonstrate the proposed ILC’s performance. Feedback gains for controllers are chosen as follows:

• Case 1 (53): a₁ = 625, a₂ = 150.

• Case 2 (54): b₁ = 480, b₂ = 170, b₃ = 100.

• Case 3 (55): d₁ = 450, d₂ = 200, d₃ = 700, d₄ = 20.

• Case 4 (57): $g_{1} = 140, g_{2} = 30, g_{3} = 17, \bar{d} = 17.5$

• Case 5 (59): $h_{1} = 0.86, h_{2} = 20, h_{3} = 9.5, h_{4} = 25, h_{5} = 8, \bar{d} = 17.5$

Figures 10a and 10(b) depict the position w_k (0, t) of the trolley and the cable slope $w_{k}^{'} (0, t)$ , respectively. Figure 11 shows the evolution of the iterative term.

As shown in Figure 10(a), the proposed controller achieves positioning speed comparable to the other controllers while exhibiting smoother trolley motion. Furthermore, Table2 demonstrates that the overshoot can be maintained within 0.0449 rad, thereby avoiding excessive overshoot. Additionally, the results presented in Figure 10(b) and Table2 confirm that vibrations are effectively suppressed. The system under the controller in Aguilar (2021) exhibits significant initial vibrations and fails to suppress persistent oscillations. Controllers in Rahn and Christopher (2001) and Wen et al. (2021) attenuate excessive vibrations but fail to eliminate persistent oscillations. Controller in Huang et al. (2024) exhibits slower vibration suppression during the initial stage due to the absence of angle feedback, with effective vibration suppression achieved only after 6 seconds.

Table 2.

Performance comparisons of different controllers.

Controllers	$\max_{t \in [0,5]} \|w^{'} (0, t)\|$	$\max_{t \in (5,10]} \|w^{'} (0, t)\|$
Case 1 in Aguilar (2021)	0.1957	0.1607
Case 2 in Rahn and Christopher (2001)	0.0948	0.0272
Case 3 in Wen et al. (2021)	0.0691	0.0145
Case 4 in Huang et al. (2024)	0.0816	0.0041
Case 5 in Zhang and Wu (2025)	0.0518	0.0012
Controller (7)	0.0449	0.0035

Controller in Zhang and Wu (2025) exhibits strong similarity to the proposed controller (7) in the experimental results. This similarity arises because both controllers employ feedback of four boundary states and incorporate signum function terms, which confer robustness against disturbances, while sharing similar control architectures. However, the proposed controller (7) eliminates the requirement for prior knowledge of the disturbance bound, thereby outperforming the methods in Huang et al. (2024) and Zhang and Wu (2025).

The proposed iterative learning controller significantly enhances overall dynamic performance. Moreover, the maximum cable vibration magnitude settles to ±0.0035 rad within 5 s. Furthermore, Figure 11 demonstrates the convergence of ${\hat{δ}}_{k} (t)$ to the estimated disturbance bound $\bar{d}$ . This convergence enables robust disturbance rejection of d_k(t) and complete suppression of cable vibrations.

The execution time of the controller was measured on an STM32F103C8T6 microcontroller, with the CPU clock frequency set to 72 MHz:

• Case 1 in Aguilar (2021): 8.4167 × 10⁻⁶ s

• Case 2 in Rahn and Christopher (2001): 9.6528 × 10⁻⁶ s

• Case 3 in Wen et al. (2021): 1.5444 × 10⁻⁵ s

• Case 4 in Huang et al. (2024): 1.9375 × 10⁻⁵ s

• Case 5 in Zhang and Wu (2025): 2.8542 × 10⁻⁵ s

• Controller (7): 3.2208 × 10⁻⁵ s

The results are consistent with those obtained from the simulations. As the architectural complexity of the controller increases, the corresponding execution time increases proportionally. Controller (7), which exhibits the longest execution time, requires 3.2208 × 10⁻⁵ s for computation. This computational overhead is negligible relative to the 20 ms control period employed in the system and poses no practical impact on real-time implementation. Notably, if the computational implementation is migrated from floating-point to fixed-point arithmetic, the computational cost can be substantially reduced.

Remark 2

Different parameters have an impact on the performance of the system under controller (7). Through numerical simulation and physical experiment, we can summarize the effects of parameter selection as follows:

• Parameter ξ₁ is the position feedback gain of the controller. Larger ξ₁ leads to the trolley reaching the desired position faster, but it brings larger payload swing. If ξ₁ is too small, the payload needs more time to the specified position.

• Parameter ξ₂ is the velocity feedback gain of the controller. Larger ξ₂ makes the trolley reach the desired position slower, while a smaller ξ₂ leads to a larger payload swing.

• Parameters ξ₃ and ξ₄ are the cable slope and cable swing velocity feedback gains of the controller. Larger ξ₃ and ξ₄ can achieve better vibration suppression and faster payload stabilization, but it takes more time to reach the target. Smaller ξ₃ and ξ₄ can cause larger payload swing.

• Parameter η is the learning rate of the iterative term. Larger η can make the controller suppress disturbances more quickly. However, if it is set too large, it will cause greater chattering.

Remark 3

Under actual working conditions, there may be sensor noise, which may lead to a reduction in control effect and even cause divergence. The following are some suggestions for working conditions with sensor noise:

• Sensor selection: The position and speed feedback of the trolley are usually provided by the servo motor and driver. Choosing an absolute multi-turn encoder can avoid drift and jitter problems caused by noise. It is recommended to choose an absolute single-turn encoder for the angle sensor of the cable.

• The data read from the encoder needs to undergo some algorithmic filtering, such as using a Kalman filter or a median filter.

• Appropriately reducing the parameter gain can lower the sensitivity to noise signals, but this will simultaneously reduce the control speed of positioning and vibration suppression.

Remark 4

The proposed ILC framework is potentially applicable to a broader class of distributed-parameter systems governed by PDEs, including but not limited to Euler-Bernoulli beam systems and reaction-diffusion systems. However, application to each specific system requires rigorous theoretical analysis employing similar techniques to those in this work: establishing well-posedness of the closed-loop system, and determining the feasible parameter ranges that ensure closed-loop stability. Different system dynamics and boundary conditions may necessitate problem-specific adaptations of the control design and corresponding theoretical verification.

Conclusion

An ILC strategy has been proposed to address vibration control of gantry crane systems subject to boundary time-varying disturbances. The existence of a closed-loop solution is formally established as an abstract Cauchy problem. Through the application of the operator semigroup theory, the closed-loop system has been demonstrated to be well-posed. Lyapunov’s direct method is employed to rigorously prove the exponential stability and convergence of the closed-loop system. Numerical simulations and physical experiments demonstrate the superior performance and practical effectiveness of the proposed ILC strategy in simultaneous payload positioning and vibration suppression.

Our future research will explore the following directions to enhance the practical applicability and robustness of the proposed control framework.

(i) Sensor faults and measurement delays: The current design assumes perfect state measurements. Future work will address control design under sensor faults and communication delays by incorporating fault-tolerant mechanisms and developing predictive observers that can mitigate the effects of imperfect information.

(ii) System parameter uncertainty: The proposed controller design relies on nominal system parameters. Extending the framework to account for parametric uncertainties in system dynamics, material properties, and boundary conditions is an important direction. Adaptive ILC control scheme that can online estimate and compensate for parameter variations will be developed.

(iii) Actuator nonlinearities and saturation: Practical actuators exhibit nonlinear behavior and saturation constraints. Future studies will investigate ILC strategies that explicitly account for actuator saturation, hysteresis, and dead-zones.

(iv) Data-driven and learning-based approaches: To further enhance adaptability in complex and uncertain environments, neural network-based control schemes combined with adaptive ILC will be one of our future works.

Footnotes

ORCID iDs

Xin Huang

Wei Wu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX22_2303) and the China Scholarship Council (No. 202306790049).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

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