Data-driven TID controller for double-pendulum overhead crane system with distributed-mass payload

Abstract

This study proposes a data-driven tilt–integral–derivative (TID) controller for a nonlinear multi-input, multi-output (MIMO) double-pendulum overhead crane system with a distributed-mass payload (DPOC-DMP), in which the controller parameters are tuned using the Safe Experimentation Dynamics Algorithm (SEDA). The main advantage of the proposed data-driven TID controller is its ability to enhance control accuracy and robustness without reliance on an explicit mathematical model. By incorporating a tilt action into the integral–derivative structure, the proposed TID controller significantly improves transient behaviour and suppresses load sway in nonlinear crane dynamics. The tuning of the TID controller parameters is performed using SEDA, which facilitates stable convergence via random but safeguarded perturbations. The effectiveness of the proposed TID controller was evaluated through simulations by analysing tracking errors, sway suppression, control effort, and robustness under external disturbances. Furthermore, the performance of the proposed controller was compared with that of a conventional PID controller. The results demonstrated that the proposed data-driven TID controller achieved superior performance in terms of objective function value, total tracking error, total control input energy, sway suppression capability, and faster recovery under external disturbance conditions. The simulation results show improved transient performance with reduced overshoot and enhanced sway suppression, together with lower control input energy compared with the PID controller. Quantitatively, the proposed method achieved a lower objective function value (310.6951 vs. 326.4870), reduced total tracking error (0.1619 vs. 0.1720), and lower control input energy (3.9647 vs. 4.3902), while maintaining faster recovery under disturbance conditions. In particular, the proposed TID framework improves control accuracy by 2.82% compared with the traditional PID controller, demonstrating its effectiveness in managing complex, nonlinear MIMO crane systems.

Keywords

tilt–integral–derivative (TID) control data-driven control nonlinear MIMO overhead crane Safe Experimentation Dynamics Algorithm (SEDA)load sway suppression

Introduction

In recent years, the efficient management of crane systems has attracted considerable interest due to their pivotal role in container shipping and logistics, particularly in the loading and unloading cycle between ocean freighters and port terminals. The primary objectives include reducing freight costs, simplifying loading procedures, decreasing energy consumption, minimizing mechanical complexity, and expanding handling capacity through technological progress.¹ Compared with bridge structures, controlling cranes is significantly more challenging due to their underactuated nature, strongly coupled dynamics, and nonholonomic constraints. These traits render precise positioning difficult, even with the installation of supplementary devices, such as column-mounted dampers.^2,3 A particular challenge is payload swing and bounce during load transfer, which has prompted both industry practitioners and control engineers to develop accurate controllers capable of moving loads faster while suppressing oscillations and residual motion. Beyond crane systems, differential equations are extensively analysed in applied mathematics to model diverse phenomena, such as fractional-order dynamics in population systems,⁴ the stability of nonlinear biochemical reactions,⁵ and the evolution of complex networks under stochastic perturbations.⁶ While these studies offer rigorous theoretical frameworks for various domains, the complex mechanical coupling and underactuated nature of cranes necessitate specialized data-driven control to bridge the gap between abstract mathematical modelling and practical performance.

In line with recent trends, control strategies have been developed to improve positioning accuracy and suppress swing in crane systems. A comparative study of a backstepping controller was conducted by Y. Li et al.,⁷ which ensured fast trolley positioning and effective swing suppression in underactuated bridge crane systems by decomposing the nonlinear dynamics into manageable subsystems. A variable-parameter, time-varying sliding mode controller was introduced by Wang et al.⁸ for distributed-mass double-pendulum bridge cranes. The controller dynamically adjusted its sliding surface to improve robustness, suppressed load swing to within 5°, and eliminated chattering under uncertain operating conditions. In the same year, an H∞ output-feedback controller was introduced by C. Li et al.⁹ to position and suppress payload swing precisely in a nonlinear overhead crane system. This was accomplished by modelling nonlinearities with a Takagi–Sugeno fuzzy approach, estimating unmeasured states through a fuzzy observer, and applying a “virtual desired-variable synthesis” method to transform the tracking problem into a stabilization task while attenuating external disturbances.

Subsequently, a finite-time sliding mode controller was designed and introduced by Nguyen et al.¹⁰ to stabilize a fuzzy overhead crane carrying copper electrode plates. The controller achieved finite-time convergence and maintained minimal swing under disturbances such as friction and air resistance by linearizing the coupled nonlinear dynamics. In the same year, L. Yang & Ouyang¹¹ introduced a precision-positioning adaptive controller to achieve accurate trolley and beam positioning. The controller employed adaptive update laws to handle uncertainties such as varying payload mass, rope length, and friction errors without requiring model linearization or swing angle feedback. While the aforementioned control methods have shown promising results in managing single-input multi-output (SIMO) crane systems, their effectiveness remains limited in more complex multi-input, multi-output (MIMO) crane systems. A SIMO crane system refers to a control setup where a single input signal, such as voltage or force, is used to influence multiple output variables of a crane, such as the horizontal position of the trolley and the swing angle of the payload.¹² This configuration is typically suitable for simplified crane models with minimal actuator interactions. Therefore, existing SIMO control schemes may no longer be effective in addressing the challenges posed by MIMO crane systems.

Acknowledging the limitations of conventional SIMO-based control strategies, researchers have increasingly shifted towards multi-input, multi-output (MIMO) frameworks that account for additional control channels and provide a more realistic representation of actual operations. In response to the increased complexity of MIMO crane systems, advanced methods like Linear Quadratic Regulator (LQR)¹³ and energy-based nonlinear controllers¹⁴ have been explored to enhance coupling and reduce oscillations. Further developments include adaptive fractional-order sliding mode control for fault tolerance¹⁵ and adaptive fuzzy control to manage dead zones and strong nonlinearities.¹⁶ Expanding on these advancements, recent literature has introduced robust mechanisms to address environmental disturbances and mechanical constraints. Al Saaideh et al.¹⁷ proposed a backstepping controller integrated with an extended high-gain observer for offshore cranes, demonstrating superior robustness against unknown wave-induced roll/heave motions while maintaining precise trajectory tracking. For 7-DOF overhead cranes, G. Li et al.¹⁸ developed an adaptive nonlinear controller that utilizes an energy-like function to handle friction uncertainties and actuator overshoots, achieving better effectiveness and robustness compared to existing state-of-the-art methods. Addressing generalizability, Zhai et al.¹⁹ designed an adaptive neural network unified control framework based on modified normal forms, which improves transient performance and mitigates the chattering problem inherent in traditional sliding mode controllers while guaranteeing stability under persistent disturbances. Additionally, Y. Yang et al.²⁰ introduced a barrier function-based adaptive composite sliding mode control that integrates actuated and underactuated state errors on a single surface, enhancing coupling and reducing energy consumption through simplified structural complexity. To further optimise operational safety under input constraints, Y. Yang et al.²¹ developed a model-free PID-SMC method using a time-varying scale function, which allows for predetermined steady-state and transient performance without requiring prior knowledge of uncertainty upper bounds. In summary, while traditional linear and energy-based mechanisms focus on basic stability, these recent adaptive and composite strategies prioritize superior robustness against unmodelled disturbances and environmental uncertainties while significantly reducing the energy required for oscillation suppression through more efficient coupling mechanisms.

Although these MIMO methods have improved realism and robustness, they still present two major gaps. First, most designed MIMO controllers are model-based, requiring accurate mathematical models to achieve good performance. As a result, any modelling errors or oversimplifications can significantly degrade controller effectiveness, making these approaches highly sensitive to model accuracy. Second, most existing crane control strategies have been developed for point-mass payloads (PMPs), which are simpler to model and control. In contrast, relatively few studies have addressed distributed mass payloads (DMP), which introduce additional nonlinearities and coupling effects. This gap highlights the need to investigate the efficacy of advanced model-free MIMO controllers for crane systems operating with DMPs, which require more adaptive and robust control strategies.

In addition to the aforementioned control schemes, PID controllers have been widely applied in control systems because of their simplicity, practicality, ease of implementation, and effectiveness, and several variants have been adopted for crane operations.²² For example, Choi et al.²³ introduced a neural network-based two-degree-of-freedom PID (2-DOF PID) controller for gantry cranes, aiming to achieve precise trolley placement and minimize container swing. By employing a neural network self-tuner that adaptively adjusts its parameters $(K_{p}, K_{i}, K_{d}, α_{p}, β_{p}, α_{θ}, β_{θ})$ online, the controller provided robustness against external disturbances and payload variations. Later, Jaafar & Mohamed²⁴ proposed an efficient PID controller for a double-pendulum crane system, with its parameters tuned using a Particle Swarm Optimization (PSO) algorithm. By employing two fitness functions that considered horizontal distance sway with and without payload parameters, this controller achieved accurate positioning while simultaneously minimizing hook and payload oscillations.

More recently, Elhabib et al.²⁵ designed a fractional-order PID (FOPID) controller with parameters optimised using a genetic algorithm in MATLAB. The controller demonstrated high precision in trolley positioning, a significant reduction in sway oscillations in double-pendulum cranes, and superior performance compared with classical PID controllers, while maintaining robustness across varying payload masses. Although these PID variants have demonstrated considerable success in improving performance and robustness, their effectiveness remains limited when applied to highly nonlinear, strongly coupled MIMO crane systems. In particular, very few studies have investigated controllers capable of simultaneously addressing trolley motion, hoist variations, and swing suppression under complex crane dynamics with DMP.

To address this research gap, the Tilt–Integral–Derivative (TID) controller is considered in this work, as it provides finer frequency-response shaping and additional tuning flexibility, offering strong potential to overcome the limitations of existing PID approaches in crane applications. The TID controller is a class of fractional-order controller that uniquely features a “tilted” proportional component, distinguishing it from a standard PID controller. Its advantages include enhanced capability to modify closed-loop system parameters and enhance overall robustness. In addition, it provides superior disturbance rejection compared with FOPID controllers, with the tilt component offering greater resilience while simplifying the tuning process.^26,27 These features have enabled improved transient and steady-state performance as demonstrated in many applications such as load frequency control (LFC) of multi-area interconnected power systems,²⁶ magnetic levitation (Maglev) control,²⁷ combined frequency and voltage regulation in interconnected microgrids²⁸ and speed control and operation of brushless direct current (BLDC) motor drives for electric vehicles (EVs),²⁹ suggesting strong potential for MIMO crane system control.

In addition to active control strategies, passive anti-vibration structures have also been widely investigated for shock and vibration suppression in engineering systems. Among these approaches, X-structure-based mechanisms have attracted considerable attention due to their beneficial nonlinear stiffness characteristics and inherent vibration isolation capability without requiring external energy input.³⁰ Recent studies have demonstrated the effectiveness of X-structured mechanisms in various engineering applications, including bistable electromagnetic wave energy converters,³¹ nonlinear inerter-based wave energy harvesters,³² and multi-degree-of-freedom vibration isolation platforms.³³ Although these passive approaches provide reliable vibration attenuation with simple implementation, their performance is generally limited by fixed mechanical characteristics and lack of adaptability to varying operating conditions. In contrast, the active data-driven control approach proposed in this study offers greater flexibility and performance optimisation for nonlinear MIMO overhead crane systems with distributed mass payload.

Based on the identified gaps, this paper introduces a data-driven control framework based on interconnected TID controllers, designed for a MIMO double-pendulum overhead crane with distributed mass payload (DPOC-DMP). The motivation for this approach is to overcome the limitations of model-based controllers,^34,35 which heavily depend on model accuracy, and to address the lack of studies involving distributed mass payloads (DMPs).³⁶ Unlike most existing crane control approaches that rely on accurate mathematical models or simplified payload representations, the proposed method employs a fully data-driven tuning mechanism that directly utilizes system response information for controller optimisation. This enables improved robustness against modelling uncertainties and nonlinear coupling effects compared with model-based controllers reported in the literature. Furthermore, while previous studies have primarily focused on SIMO crane configurations or point-mass payload models, the present work demonstrates improved tracking accuracy, sway suppression, and control efficiency in a nonlinear MIMO crane system with distributed mass payload. The obtained results show that the proposed data-driven TID controller achieves a lower objective function value and reduced control input energy compared with the conventional PID controller, highlighting its superior performance for complex crane dynamics. To achieve this, the Safe Experimentation Dynamics Algorithm (SEDA) is employed within the MATLAB/Simulink environment to fine-tune the five parameters of each TID controller. SEDA was selected because of its stochastic search capability to navigate complex, multimodal solution spaces and thereby avoid premature convergence when tuning TID gains. As described by Shukor et al.,³⁷ its safe-experimentation update rule, simple tuning coefficients, and ability to retain the best parameters ensure a good exploration and exploitation balance, making it well-suited for data-driven control of the nonlinear MIMO DPOC-DMP system.

The evaluation compares the proposed TID controller with a conventional PID controller under the same benchmark setup, ensuring fairness in simulation parameters. Assessment criteria include convergence behaviour, integral of square error, integral of square input, overall performance index derived from the objective function, and time-domain response specifications, with additional evaluations under external disturbance scenarios. The simulation results demonstrated that the optimised data-driven TID controller achieves superior positioning accuracy for both trolley and hoist in the MIMO DPOC-DMP system. The controller also significantly suppresses hook and payload sway compared to the conventional PID controller, thereby enhancing robustness and ensuring operational safety under complex DMP dynamics. Based on these findings, the main contribution of this work is summarized as follows:

(i) A novel data-driven TID control framework has been developed for the double-pendulum overhead crane with distributed mass payload (DPOC-DMP), representing one of the first reported attempts to address such a complex crane configuration.

(ii) This work presents one of the first applications of the SEDA to tune a data-driven TID controller, providing reliable optimisation and ensuring effective convergence of controller parameters.

(iii) The proposed data-driven TID controller significantly outperforms conventional PID controllers in stabilizing DMP crane dynamics, reducing error, suppressing oscillations, and enhancing robustness, thereby demonstrating the practical advantage of data-driven control over traditional model-based approaches.

The remaining sections of this paper are organized as follows: the next section presents the mathematical modelling of the DPOC-DMP, outlining the nonlinear dynamic equations that form the basis for controller development. The subsequent section introduces the proposed data-driven TID control framework, beginning with the formulation of the TID controller structure for the nonlinear MIMO crane system, followed by an overview of SEDA, and concluding with a detailed procedure for implementing SEDA to tune the TID controller parameters. This is followed by a section that evaluates the performance of the proposed data-driven TID controller through a series of simulation studies, including benchmarking against a conventional PID controller, assessing tracking accuracy, sway suppression capability, control effort efficiency, transient response characteristics, and robustness under external disturbances. Finally, the last section summarizes the findings of this work and provides concluding remarks.

Modelling of DPOC-DMP system

In this section, the mathematical modelling of the DPOC-DMP system is presented based on the formulation of Bello et al.,³⁶ providing dynamic equations that serve as the foundation for the proposed controller design. The dynamic equations for the DPOC-DMP are developed using the Euler-Lagrange method to account for its highly nonlinear, underactuated, and strongly coupled nature. While many studies simplify payloads as point masses, research by Bello et al.³⁶ emphasizes that incorporating distributed mass is crucial for accurately capturing the complex interactions and multiple oscillation modes seen in real-world crane operations. This validated dynamic model serves as the foundation for control strategies aimed at simultaneous trolley positioning and payload sway suppression. Figure 1 illustrates the schematic of the DPOC-DMP. The system consists of a trolley of mass $m_{t}$ moving along the horizontal $x$ -axis, the position of the trolley, $x$ , the payload length, $l_{p}$ , the gravitational coefficient, g, a hook of mass $m_{h}$ suspended by a cable of length $l_{1}$ , and a distributed payload of mass $m_{p}$ attached to a rigging cable of length $l_{2}$ . The hook swing angle is represented by $θ$ , while the payload sway angle is denoted by $φ$ . Finally, the driving force acting on the trolley, the damping coefficients of trolley, the applied hoisting force required to lift the payload, and the damping coefficient of hoisting are denoted by $F_{x}$ , $f_{x}$ , $F_{l}$ and $f_{l}$ , respectively.

Figure 1.

Double-pendulum overhead crane with DMP.

The nonlinear dynamics of the DPOC-DMP system are derived using the Lagrangian formulation and have been adopted directly from Bello et al.³⁶ To capture the practical operating conditions of crane systems, the modelling explicitly incorporates the hoisting mechanism, where the suspension cable length $l_{1}$ is variable. The governing equations for the hoisting case are given as follows:

(m_{t} + m_{h} + m_{p}) \ddot{x} - (m_{h} + m_{p}) (2 {\dot{l}}_{1} \dot{θ} \cos θ + l_{1} \ddot{θ} \cos θ - l_{1} {\dot{θ}}^{2} \sin θ + {\ddot{l}}_{1} \sin θ) - m_{p} l_{2} (\ddot{φ} \cos φ - {\dot{φ}}^{2} \sin φ) = F_{x} - f_{x} \dot{x}

(1)

(m_{h} + m_{p}) (l_{1} \ddot{θ} - \ddot{x l_{1}} \cos θ + 2 l_{1} {\dot{l}}_{2} \dot{θ} + {g l}_{1} \sin θ) - m_{p} l_{1} l_{2} (\ddot{φ} \cos (θ - φ) + {\dot{φ}}^{2} \sin (θ - φ)) = 0

(2)

m_{p} l_{2} (\ddot{x} \cos φ - 2 l_{1} {\dot{l}}_{2} \dot{θ} \cos (θ - φ) - l_{1} \ddot{θ} \cos (θ - φ) + l_{1} {\dot{θ}}^{2} \sin (θ - φ)) + m_{p} ({l_{2}}^{2} + \frac{1}{12} {l_{p}}^{2}) \ddot{φ} - m_{p} l_{2} ({\ddot{l}}_{1} \sin (θ - φ) + g \sin (θ - φ)) = 0

(3)

(m_{h} + m_{p}) (- \ddot{x} \sin θ + {\ddot{l}}_{1} - l_{1} \dot{θ} + g (1 - \cos θ)) - m_{p} (\ddot{φ} l_{2} \sin (θ - φ) - \dot{φ} l_{2} \sin (θ - φ)) = F_{l} - f_{1} \dot{l_{1}}

(4)

These equations clearly highlight the highly nonlinear nature of the DPOC-DMP system, where the motion of the trolley is intrinsically coupled with the oscillatory dynamics of both the hook and the distributed-mass payload. An external force input applied to the trolley excites complex interactions between the hoist, hook, and payload, thereby presenting significant challenges for control design. Importantly, this dynamic model has been validated using an actual experimental crane setup, ensuring its reliability and practical relevance. The system parameters used in this study are summarized in Table 1.

Table 1.

DPOC-DMP parameters.

Variable	Symbol	Values
Mass of trolley	$m_{t}$	1.115 kg
Mass of hook	$m_{h}$	0.1 kg
Mass of payload	$m_{p}$	0.215 kg
Cable length between trolley and hook	$l_{1}$	0.2 m - 0.4 m
Cable length between hook and payload	$l_{2}$	0.2 m
Length of DMP	$l_{p}$	0.2 m
Viscous damping coefficients of trolley	$f_{x}$	82 Ns/m
Viscous damping coefficients of hoisting	$f_{l}$	75 Ns/m
Gravitational constant	$g$	9.81 m/s²

Data-driven TID control design based on SEDA

Building upon the nonlinear dynamics of the DPOC-DMP system derived in the previous section, this part introduces the proposed data-driven TID control framework. Firstly, the design of the TID controller for the MIMO DPOC-DMP is presented. Secondly, an overview of the SEDA algorithm is provided. Finally, the implementation of SEDA for data-driven tuning of the TID controller parameters is explained.

TID controller design

This section presents the design of the TID controller applied to the DPOC-DMP system. The Tilt-Integral-Derivative (TID) controller is a fractional-order control method that introduces a “tilted” proportional component to provide finer frequency-response shaping and increased tuning flexibility compared to conventional PID controllers. It has been shown to offer superior disturbance rejection and improved transient performance in various complex systems, such as magnetic levitation and interconnected power grids.^26,27 In the context of crane systems, the TID structure is employed to enhance control accuracy and resilience against nonlinear coupling effects without requiring an explicit mathematical model. The main control objective is to ensure accurate trolley and hoist positioning while effectively suppressing hook and payload sway, even in the presence of disturbances and uncertainties in the crane’s nonlinear MIMO dynamics. The general closed-loop configuration is illustrated in Figure 2, where $r (t) \in R^{q}$ is the reference input, $K (s) \in R^{p}$ is the TID controller, $u (t) \in R^{p}$ is the controller output and $y (t) \in R^{q}$ is the output measurement, with $p$ and $q$ indicating the number of inputs and outputs, respectively. To be precise, according to the mathematical model presented in Equations (1)–(4), the input vector of the DPOC-DMP system is defined as $u (t) = {[\begin{array}{l} F_{x} & F_{l} \end{array}]}^{T}$ , while the corresponding output vector is given by $y (t) = {[x l θ ϕ]}^{T}$ . The symbol G represents the DPOC-DMP crane system. In this context, $t_{s}$ denotes as the sampling interval such that time expressed as $t = 0, t_{s}, 2 t_{s}, 3 t_{s}, \dots, D t_{s}$ where D represents the total number of samples.

Figure 2.

The TID control system.

Next, the proposed TID controller is expressed in the form of a control gain matrix $K (s)$ , where each entry corresponds to an individual controller channel of the MIMO crane system. The structure is defined as

K (s) = [\begin{array}{l} m_{11} (s) & \dots & m_{1 q} (s) \\ ⋮ & ⋱ & ⋮ \\ m_{p 1} (s) & \dots & m_{p q} (s) \end{array}],

(5)

where

m_{i j} (s)

represents the transfer function of the TID controller corresponding to the i-th input and j-th output channel. Each element

m_{i j} (s)

incorporates the proportional, tilt, integral, derivative, and filtering terms, and is formulated as

m_{i j} (s) = K_{P i j} (\frac{1}{s^{\frac{1}{K_{T i j}}}} + \frac{1}{K_{I i j} s} + \frac{K_{D i j} s}{1 + (\frac{K_{D i j}}{N_{i j}}) s}),

(6)

where

K_{P i j} \in R

denotes the proportional gain,

K_{T i j} \in R

is corresponds to the tilt gain,

K_{I i j} \in R

is the integral gain and

K_{D i j} \in R

represents the derivative gain. The term

N_{i j} \in R

defines the filter coefficient used in the derivative action, mitigating high-frequency noise effects. Note that the addition of the tilt term sets the TID controller apart from the conventional PID structure. While PID relies only on proportional, integral, and derivative actions, the TID introduces a fractional-order tilt component

s^{\frac{1}{K_{T i j}}}

that provides an extra degree of freedom to the control design structure. This feature thus improves robustness, enhances disturbance rejection, and yields better transient response compared to PID, making the TID controller more suitable for nonlinear DPOC-DMP systems.

Overview of SEDA

In this section, the SEDA is introduced as the optimisation method employed for tuning the parameters of the proposed TID controller. Originally developed by Marden et al.,³⁸ the SEDA optimisation technique is a game-theoretic approach used to explore optimal design parameters by introducing random perturbations to selected elements of the design parameter vector. Specifically, only a subset of elements of the design parameter vector is perturbed during the searching process, while the remaining elements are maintained to the current best solutions obtained so far. Since only a limited subset of elements is perturbed based on a predefined probability, the SEDA’s search mechanism exhibits a more controlled exploitation behaviour compared to conventional local neighbourhood search methods, while maintaining superior optimisation accuracy. This controlled perturbation strategy prevents excessive parameter variations between iterations, thereby improving convergence stability and reducing the likelihood of performance degradation during the controller tuning process. Furthermore, the algorithm is favoured for its simplicity, low computational cost, and strong balance between exploration and exploitation.³⁷ To date, the SEDA has been demonstrated to be effective for various controller tuning problems. For example, it has been successfully efficient in tuning several advanced PID controllers such as Sigmoid-based Secretion Rate Neuroendocrine–Proportional–Integral–Derivative (SbSR-NEPID),³⁹ NEPID,⁴⁰ Multiple-node Hormone Regulation-NEPID (MnHR-NEPID),¹² and Brain Emotional Learning-based Intelligent Controller with PID (BELBIC-PID).⁴¹ These successful applications support the adoption of SEDA for optimising the TID controller parameters in the DPOC-DMP system in this research.

Formally, the optimisation problem is expressed as

\min_{p \in R^{n}} f (p),

(7)

which minimizes the objective function

f

with respect to the design parameter vector

p \in R^{n}

. SEDA performs iterative updates on the parameters to find the optimal configuration. The update law of SEDA is expressed as:

p_{i} (k + 1) = {\begin{cases} h ({\bar{p}}_{i} - K_{g} {r v}_{2}), if {r v}_{1} \leq E, \\ {\bar{p}}_{i}, if {r v}_{1} > E, \end{cases}

(8)

where

k = 1, 2, \dots, k_{\max}

is the iteration index,

p_{i} \in R

denotes the

i

-th element of

p \in R^{n}

{\bar{p}}_{i} \in R

denotes the

i

-th elements of

\bar{p} \in R^{n}

and

\bar{p}

is the design vector that refers to the obtained best solution so far. The term

k_{\max}

specifies the maximum number of iterations, while

K_{g}

is a scalar that controls the interval size of the random search step. The parameter

E

represents the probability of performing a random perturbation. The symbols

{r v}_{1} \in R

and

{r v}_{2} \in R

are random numbers which are independently generated and uniformly distributed in

[0, 1]

and design parameter boundary

[p_{\min}, p_{\max}]

, respectively. Furthermore, the mapping function

h ({\bar{p}}_{i} - K_{g} {r v}_{2})

in Equation (8) is defined as

h (\cdot) = {\begin{cases} p_{\max}, if {\bar{p}}_{i} - K_{g} {r v}_{2} < p_{\max}, \\ {\bar{p}}_{i} - K_{g} {r v}_{2}, if p_{\min} \leq {\bar{p}}_{i} - K_{g} {r v}_{2} \leq p_{\max}, \\ p_{\min}, if {\bar{p}}_{i} - K_{g} {r v}_{2} > p_{\min} . \end{cases}

(9)

In the above equations, the design parameter boundary

[{p_{\min}, p}_{\max}]

are chosen according to preliminary design knowledge. The scalar

K_{g}

regulates the step size of the random perturbations, influencing how aggressively the algorithm explores the design space. A larger

K_{g}

promotes exploration by allowing wider parameter jumps, while a smaller

K_{g}

encourages exploitation around the current best solution. The probability factor

E

further controls the likelihood of accepting new random steps, thereby providing a balance between exploration of new regions and exploitation of existing promising solutions. More specifically, SEDA operates by applying random perturbations to selected elements of the current best parameter vector

\overset{ˉ}{p}

, and then deciding whether to keep or update them according to a predefined probability coefficient

E

.The procedure of SEDA can be summarized as follows:

Step 1: Initialize the values of $p_{\max}$ , $p_{\min}$ , $K_{g}$ and $E$ . Set $k = 0$ and assign the initial design parameter as $p (0)$ . Evaluate the corresponding objective function $f (p (0))$ . Store $\bar{p} = p (0)$ and $\bar{f} = f (p (0))$ as the current best solution.

Step 2: If $(p (k)) < \bar{f}$ , update $\bar{p} = p (k)$ and $\bar{f} = f (p (k))$ . Otherwise, go to Step 3.

Step 3: Generate random numbers ${r v}_{1}$ and ${r v}_{2}$ , and update $p (k)$ according to Equation (8).

Step 4: Evaluate the new objective function $f (p (k + 1))$ .

Step 5: If the termination criterion is satisfied, stop and record the optimal solution

p_{o p t} ≔ \arg \min_{p ϵ {p (0), p (1), \dots, p (k + 1)}} f (p) .

Otherwise, increase $k = k + 1$ and repeat Step 2.

Implementation of SEDA for TID controller

This section outlines how SEDA is employed to adjust the TID controller parameters for the DPOC-DMP system. The primary objective of this tuning is to obtain optimal controller parameters that minimize the overall system error while ensuring smooth control effort and stable payload transfer of nonlinear DPOC-DMP systems. The algorithm structure and operation are adopted from Marden et al.,³⁸ who originally developed the SEDA for data-driven control optimisation.

Specifically, the SEDA is employed to fine-tune all elements of TID control parameters $(K_{P i j}, K_{T i j}, K_{I i j}, K_{D i j} and N_{i j})$ that are grouped respectively in vector gains $K_{P}, K_{T}, K_{I}, K_{D} and N$ based on the control performance of the DPOC-DMP systems, which can be obtained through its input and output data. To enhance and provide a much faster tuning process, the parameter search within the defined tuning range is accelerated by applying a logarithmic scale to the design parameter $p$ , such that the TID controller parameters are formulated as:

ψ = [K_{P}, K_{T}, K_{I}, K_{D}, N] \in R^{n},

(10)

where each element of

ψ

is defined as

ψ_{i} = 10^{p_{i}} (i = 1, 2, \dots, n)

. Subsequently, the control system’s performance is quantified using the indices

{\bar{e}}_{j} ∶ = \int_{t_{0}}^{t_{f}} {| r_{j} (t) - y_{j} (t) |}^{2} d t,

(11)

{\bar{u}}_{i} ∶ = \int_{t_{0}}^{t_{f}} {| u_{i} (t) |}^{2} d t,

(12)

where

r_{j} (t)

y_{j} (t)

, and

u_{i} (t)

correspond to the

j

-th and

i

-th elements of vectors

r (t)

y (t)

, and

u (t)

, respectively. The time interval

[t_{0}, t_{f}]

represents the evaluation period of the system performance, where

t_{0} \in {0} \cup R_{+}

and

t_{f} \in R_{+}

. Then, the objective function is expressed as follows

J (K_{P}, K_{T}, K_{I}, K_{D}, N) = \sum_{j = 1}^{q} w_{1 j} {\bar{e}}_{j} + \sum_{i = 1}^{p} w_{2 i} {\bar{u}}_{i},

(13)

where the value of parameter

K_{P} ≔ {[K_{P 11}, K_{P 12}, \dots, K_{P p q}]}^{T}

K_{T} ≔ {[K_{T 11}, K_{T 12}, \dots, K_{T p q}]}^{T}

K_{I} ≔ {[K_{I 11}, K_{I 12}, \dots, K_{I p q}]}^{T}

K_{D} ≔ {[K_{D 11}, K_{D 12}, \dots, K_{D p q}]}^{T}

N ≔ {[N_{11}, N_{12}, \dots, N_{p q}]}^{T}

. The symbol

{\bar{e}}_{j}

represents the integral square error, and

{\bar{u}}_{i}

denotes the integral square input, each weighted by coefficients

w_{1 j}

and

w_{2 i}

, respectively. The optimisation process aims to minimize an objective function J, which combines both tracking accuracy and control corresponds to

{\bar{e}}_{j}

and

{\bar{u}}_{i}

, respectively. Finally, the procedure for the data-driven TID controller design based on the SEDA algorithm is described as follows:

Step 1: Initialize the parameters, $f (p) = J (K_{P}, K_{T}, K_{I}, K_{D}, N)$ and represent each design parameter as $p_{i} = \log ψ_{i}$ . Then, determine the maximum number of iterations $k_{\max}$ .

Step 2: Apply the SEDA algorithm for minimizing the objective function $f (p) .$

Step 3: Once $k_{\max}$ iterations are completed, obtain the optimal output $p_{opt} = \overset{`}{p} (k_{\max})$ . Then, calculate the optimal parameters as $ψ_{opt} = {[10^{p 1 opt} 10^{p 2 opt} \dots 10^{p n opt}]}^{T}$ and apply them to the TID controller within the DPOC-DMP control system shown in Figure 1.

Figure 3 illustrates a detailed flow diagram that integrates the tuning strategy employed for the TID controller using the SEDA optimisation method. As shown, the diagram is composed of two primary blocks. The first block represents the implementation of the TID controller for the MIMO DPOC–DMP system, while the second block highlights the operation of the SEDA-based optimisation process. The initial block focuses on reducing the fitness function $J (ψ)$ based on the system’s measured input $u (t)$ and output $y (t)$ . The second block employs the SEDA algorithm to randomly perturb each design parameter $p_{i}$ according to the perturbation law determined by $r v_{1}$ and $r v_{2}$ . The perturbed parameters are then translated into the TID controller parameters through $ψ_{i} = 10^{p_{i}}$ , for $i = 1, 2, \dots, n$ , establishing the current TID parameter set that is applied back to the first block. This reciprocal interaction between both blocks is repeated iteratively until the maximum number of iterations $k_{\max}$ is reached, resulting in the optimal design parameters $p_{opt}$ that yields improved tracking performance and reduced control effort.

Figure 3.

Block diagram of implementation for TID controller of MIMO DPOC-DMP system.

Remark 3.1: It is important to emphasize that the TID–SEDA framework proposed in this study is a model-free, data-driven approach. Unlike traditional control methods that rely on precise mathematical plant models, SEDA optimises the TID parameters using only input and output data. Consequently, this methodology is designed for seamless transition to experimental platforms, as it can be applied to an actual crane rig just as effectively as a simulation, provided that the operational data is available. While this study focuses on the rigorous theoretical and numerical validation of the algorithm, the model-independent nature of the framework ensures its readiness for real-world experimental implementation without further structural modification.

Numerical example

In this study, the proposed data-driven TID controller, tuned using the SEDA, was evaluated to verify its capability to improve control performance of the MIMO DPOC-DMP system. The evaluation was conducted on a nonlinear MIMO crane benchmark within the MATLAB/Simulink environment under a data-driven control framework. The proposed TID controller is implemented for the DPOC-DMP system, and its parameters were optimised using the SEDA-based tuning technique presented in the previous section. The control performance of the proposed SEDA-tuned TID controller was compared with that of a conventional PID controller, which is widely adopted as a benchmark controller for crane control applications. To provide a more comprehensive evaluation of robustness, supplementary tests were conducted in the presence of external disturbances, in which both controllers were assessed within the same simulation framework using the previously determined optimal parameters. Specifically, the following criteria were used to evaluate the effectiveness of the proposed TID controller:

(i) The objective fitness function $J (K_{P}, K_{T}, K_{I}, K_{D}, N)$ , together with the integral of square error $\sum_{j = 1}^{q} {\bar{e}}_{j}$ , and integral of square input $\sum_{i = 1}^{p} {\bar{u}}_{i}$ were used to assess the overall tracking accuracy and control effort.

(ii) The percentage improvement in control performance $J (ψ)$ between the proposed TID controller and the conventional PID controller was determined using:

% J_{Δ} (ψ) = \frac{| J {(ψ)}_{T I D} - J {(ψ)}_{P I D} |}{J {(ψ)}_{P I D}} \times 100 % .

(14)

(iii) The time-domain responses of the MIMO DPOC-DMP crane system obtained using the proposed SEDA-tuned TID controller were analysed and compared with those produced by the conventional PID controller. Key transient response metrics, namely rise time, $T_{r}$ , settling time, $T_{s}$ and percentage overshoot, %OS, were employed to evaluate the system’s performance in response to a unit step reference input in the displacement of the trolley $y_{1} (t)$ . Specifically, settling time is defined as the duration required for the response to remain within 2 % of its final value, while rise time represents the time taken for the response to transition from 10 % to 90 % of the final value. In addition, the evaluation also considers the cable length between the trolley and hook $y_{2} (t)$ , the hook sway angle $y_{3} (t)$ , the payload sway angle $y_{4} (t)$ as well as the corresponding control inputs, namely the trolley force $u_{1} (t)$ and hoist force $u_{2} (t)$ .

(iv) The robustness of the proposed SEDA-tuned TID controller was further assessed by introducing external disturbances into the MIMO DPOC-DMP crane system. The controller’s ability to maintain stable tracking performance and suppress oscillations under disturbance conditions was compared with that of the conventional PID controller.

Next, the MIMO DPOC-DMP system illustrated in Figure 1 was examined. It consists of two control inputs $p = 2$ and four outputs $q = 4$ . The inputs are defined as $u_{1} (t) = F_{x} (t)$ for the trolley driving force and $u_{2} (t) = F_{l} (t)$ for the hoisting force. The system outputs are given by $y_{1} (t) = x (t)$ , $y_{2} (t) = l_{1} (t)$ , $y_{3} (t) = θ (t)$ , and $y_{4} (t) = φ (t)$ , corresponding to the trolley displacement, cable length between trolley and hook, hook sway angle, and payload sway angle. The desired reference trajectory for evaluating the proposed data-driven TID controller is defined as

r (t) = {[\begin{array}{l} 0.5 & 0.4 & 0 & 0 \end{array}]}^{T} \forall t .

(15)

The structure of the TID controller, denoted as

K (s)

, is defined as

K (s) = [\begin{array}{c} K_{11} (s) & 0 & K_{13} (s) & K_{14} (s) \\ 0 & K_{22} (s) & 0 & 0 \end{array}],

(16)

where each element

K_{i j} (s)

represents the corresponding TID control channel between the input–output pairs.

The dimension of the TID controller design parameter vector is

n = 20

, expressed as

ψ : = [K_{P}, K_{T}, K_{I}, K_{D}, N] \in R^{20} .

The objective is to determine the optimal parameter set

ψ

that minimizes the performance index

J (ψ)

. The weighting coefficients are selected as

w_{11} = 1500

w_{12} = 1250

w_{13} = 1000

w_{14} = 1000

w_{21} = 20

and

w_{22} = 10

. The simulation is run until

t_{f} = 20

, and the maximum number of iterations for the SEDA optimisation is fixed at

k_{\max} = 500 .

The SEDA parameters are selected based on preliminary tuning trials, with

K_{g} = 0.022

E = 0.66

, and the design parameter bounds are defined as

p_{\min} = - 2.0

and

p_{\max} = 3.0

. The initial parameter vector

p (0)

is generated within the prescribed bounds based on preliminary tuning and prior knowledge of the system characteristics to ensure feasible convergence while respecting the physical and control constraints of the system. All parameter values and tuning coefficients employed in the simulation study are summarized in Table 2.

Table 2.

Design parameters.

$ψ$	TID	$p (0)$	$10^{p (0)}$	$p_{o p t}$	$10^{p_{o p t}}$
$ψ_{1}$	$K_{P 11}$	$0.2$	$1.5849$	$0.6494$	$4.4607$
$ψ_{2}$	$K_{T 11}$	$1.9$	79.4328	$2.6894$	$489.1569$
$ψ_{3}$	$K_{I 11}$	$3.0$	1000.0000	$0.8915$	$7.7888$
$ψ_{4}$	$K_{D 11}$	$0.7$	$5.0119$	$- 0.2932$	$0.5091$
$ψ_{5}$	$N_{11}$	$0.8$	$6.3096$	$0.8777$	$7.5454$
$ψ_{6}$	$K_{P 22}$	$1.5$	$31.6228$	$0.3141$	$2.0609$
$ψ_{7}$	$K_{T 22}$	$1.8$	63.0967	$- 1.1639$	$0.0686$
$ψ_{8}$	$K_{I 22}$	$0.6$	$3.9811$	$- 0.6245$	$0.2374$
$ψ_{9}$	$K_{D 22}$	$- 0.5$	$0.3162$	$0.8254$	$6.6889$
$ψ_{10}$	$N_{22}$	$- 0.4$	$0.3981$	$1.7188$	$52.3316$
$ψ_{11}$	$K_{P 33}$	$1.2$	$15.8489$	$1.2482$	$17.7109$
$ψ_{12}$	$K_{T 33}$	$2.5$	$316.2278$	$- 0.3786$	$0.4182$
$ψ_{13}$	$K_{I 33}$	$2.0$	$251.1886$	$0.1469$	$1.4024$
$ψ_{14}$	$K_{D 33}$	$1.5$	$31.6228$	$- 0.6864$	$0.2059$
$ψ_{15}$	$N_{33}$	$0.2$	$1.5849$	$- 0.1531$	$0.7028$
$ψ_{16}$	$K_{P 44}$	$0.4$	$2.5119$	$- 0.7862$	$0.1636$
$ψ_{17}$	$K_{T 44}$	$2.6$	398.1072	$1.3656$	$23.2073$
$ψ_{18}$	$K_{I 44}$	$- 1.0$	$0.1000$	$1.4768$	$29.9771$
$ψ_{19}$	$K_{D 44}$	$0.2$	$1.5849$	$1.7410$	$55.0779$
$ψ_{20}$	$N_{44}$	$- 0.1$	$0.7943$	$2.6782$	$476.6767$

Remark: To ensure a fair comparison, both the proposed TID controller and the conventional PID controller were evaluated under identical simulation conditions, including identical weighting coefficients, simulation duration, and disturbance profiles. This approach ensures consistency and fairness across all performance evaluations.

Figure 4 illustrates the convergence of the objective function $J$ for the TID controller optimised using the SEDA method over $k_{\max} = 500$ iterations. This result demonstrates that the proposed method successfully minimises the objective function and yielded the optimal TID parameters $p_{opt}$ listed in Table 2. The results also indicate that the output responses of the crane system, namely $y_{1} (t)$ , $y_{2} (t)$ , $y_{3} (t)$ , and $y_{4} (t)$ , under TID control are illustrated in Figures 5–8, respectively, whereas the control inputs $u_{1} (t)$ and $u_{2} (t)$ are depicted in Figures 9 and 10. The responses were evaluated against those obtained by the conventional PID controller for reference. For the trolley displacement $y_{1} (t)$ , the TID controller exhibited a significantly lower percentage overshoot of only 0.0021% compared to 1.8597% for the PID controller, indicating superior overshoot suppression performance. Although the PID controller achieved a slightly faster rise time of 1.5851 s versus 1.6626 s and a settling time of 3.6900 s versus 5.3580 s, the TID controller provided a more accurate and stable transient response with negligible overshoot, which is desirable for precise trolley positioning. For the cable length response $y_{2} (t)$ , both controllers produced a smooth response with a slightly higher overshoot of 0.020 % for the TID controller compared to 0.0043% for the PID controller, while maintaining reduced oscillatory behaviour that helped suppress excitation and ensured safer hoisting operations.

Figure 4.

The convergence curve of the objective function $J (K_{P}, K_{T}, K_{I}, K_{D}, N)$ .

Figure 5.

Trolley displacement $y_{1} (t)$ responses.

Figure 6.

Cable length between trolley and hook $y_{2} (t)$ responses.

Figure 7.

Hook sway angle $y_{3} (t)$ responses.

Figure 8.

Payload sway angle $y_{4} (t)$ responses.

Figure 9.

The trolley force, $u_{1} (t)$ responses.

Figure 10.

The hoist force, $u_{2} (t)$ responses.

The hook sway angle $y_{3} (t)$ response demonstrated that the TID controller produced less maximum sway with a value of 0.1205 radian in comparison with the PID controller with a value of 0.1330 radian, reflecting improved hook sway suppression performance, and achieved a shorter settling time of 5.47 s compared to 8.01 s for the PID controller. Similarly, the payload sway angle $y_{4} (t)$ responses demonstrated that the TID controller yielded a slightly smaller maximum sway of 0.2808 radian compared to 0.2875 radian for the PID controller, and it also settles faster with a settling time of 2.54 s compared to 7.73 s for the PID controller.

In terms of control effort, the TID controller required significantly lower control input

u_{1} (t)

with a maximum trolley force of 2.8400 N compared to the PID controller, which reached 4.3706 N, while the minimum input trolley force of the TID controller decreased slightly to –0.0304 N, whereas the PID controller remained at zero, indicating smoother actuation behaviour. Meanwhile, for

u_{2} (t)

, both controllers exhibit low hoisting input force, indicating that minimum control input energy is required to maintain the cable length between the trolley and hook. Specifically, the TID controller exhibited a maximum input of 0.0028 N in comparison with the PID controller, which attained a value of

2.78 \times 10^{- 6}

N, while both controllers maintained zero minimum input, confirming that the proposed TID controller achieved improved regulation with reduced overall control effort. These results are summarized in Table 3.

Table 3.

The results of the time responses analysis of the MIMO DPOC-DMP system.

Response	Performance parameter	TID	PID
$y_{1} (t)$	Rise time, $T_{r}$ (s)	$1.6626$	$1.5851$
	Settling time, $T_{s}$ (s)	$5.3580$	$3.6900$
	Percentage overshoot, %OS (%)	$0.0021$	$1.8597$
$y_{2} (t)$	Percentage overshoot, %OS (%)	$0.0207$	$0.0043$
$y_{3} (t)$	Maximum sway angle (radian)	$0.1205$	$0.1330$
$y_{3} (t)$	Settling time, $T_{s}$ (s)	5.47	$8.01$
$y_{4} (t)$	Maximum sway angle (radian)	$0.2808$	$0.2875$
$y_{4} (t)$	Settling time, $T_{s}$ (s)	2.54	7.73
$u_{1} (t)$	Maximum trolley input force (N)	$2.8400$	$4.3706$
$u_{1} (t)$	Minimum trolley input force (N)	$- 0.0304$	$0$
$u_{2} (t)$	Maximum hoisting input force (N)	$0.0028$	${2.78 \times 10}^{‐ 6}$
$u_{2} (t)$	Minimum hoisting input force (N)	$0$	$0$

Furthermore, Table 4 presents the numerical comparison between the TID and PID controllers in terms of the objective function

J

, integral square error

\sum_{j = 1}^{q} {\bar{e}}_{j}

and integral square input

\sum_{i = 1}^{p} {\bar{u}}_{i}

, which further supports the results reported in Table 3. It can be observed that the TID controller achieved a lower objective function value of

J = 310.6951,

compared to

J = 326.4870

obtained by the PID controller, indicating improved overall control performance. In addition, the total tracking error

\sum_{j = 1}^{q} {\bar{e}}_{j}

is reduced from

0.1720

to 0.1619, demonstrating the superior tracking capability of the TID controller. Similarly, the integral square input

\sum_{i = 1}^{p} {\bar{u}}_{i}

is decreased from

4.3902

to 3.9647, implying that the TID controller achieved this performance with lower control effort. These results confirmed that the proposed data-driven TID–SEDA framework not only improves tracking accuracy but also enhanced control efficiency by reducing unnecessary actuator activity and producing smoother control actions. In addition, the proposed TID controller achieved a control accuracy improvement

J_{Δ}

of 2.82% compared to the conventional PID controller, further confirming the effectiveness of the SEDA algorithm in fine-tuning the TID parameters and improving the overall closed-loop performance.

Table 4.

Numerical result of DPOC-DMP system.

Controller	PID	TID
$J$	326.4870	310.6951
$\sum_{j = 1}^{q} {\bar{e}}_{j}$	0.1720	0.1619
$\sum_{i = 1}^{p} {\bar{u}}_{i}$	4.3902	3.9647

On the other hand, the robustness of the proposed data-driven TID controller, tuned via the SEDA, is further investigated under the presence of an unpredictable external disturbance in comparison with the standard PID controller. The robustness of the TID was evaluated when a disturbance $d (t) = {[0 0 0 d_{y 4} (t)]}^{T}$ was introduced into the output of the MIMO DPOC-DMP system. After the disturbance settings illustrated in Figure 11, an external disturbance was introduced to the payload sway angle $y_{4} (t)$ based on the equation presented below

d_{y_{4}} (t) = {\begin{cases} 0, & 0 < t < 3, \\ - 0.1, & 3 \leq t < 3.5, \\ 0.1, & 3.5 \leq t < 4, \\ 0, & t \geq 4 . \end{cases}

(17)

Figure 11.

Disturbance signal $d_{y 4} (t)$ .

Figures 12–15 illustrate the responses of the trolley displacement $y_{1} (t)$ , cable length between trolley and hook $y_{2} (t)$ , hook sway angle $y_{3} (t)$ , and payload sway angle $y_{4} (t)$ , respectively, under the disturbance condition. From the trolley displacement $y_{1} (t)$ in Figure 12, it was observed that the proposed TID controller exhibited smaller transient deviation and returned to the reference trajectory of 0.5 m more rapidly after the disturbance compared to the PID controller, indicating superior disturbance rejection and recovery capability. For the cable length between trolley and hook $y_{2} (t)$ , both controllers maintained stable behaviour during the disturbance event. However, the TID controller produced smoother transient behaviour with negligible oscillatory effects, demonstrating improved decoupling and robustness of the hoisting dynamics. Furthermore, the responses of the hook sway angle $y_{3} (t)$ in Figure 14 clearly showed that the TID controller produced lower peak deviation during the disturbance and achieved a faster settling time back to the equilibrium position compared to the PID controller. This indicated that the proposed TID controller was more effective in suppressing sway oscillations induced by sudden external excitation. Similarly, improvement was observed in the payload sway angle response $y_{4} (t)$ , where the TID controller exhibited reduced oscillation amplitude and more rapid damping following the disturbance, confirming enhanced sway suppression capability for the distributed mass payload system. In summary, the disturbance study demonstrated that the proposed data-driven TID framework outperformed the conventional PID controller in terms of robustness, recovery speed, sway suppression, and overall stability under unexpected external disturbances. These results confirmed the effectiveness of the proposed tuning approach in maintaining safe and reliable crane operation under uncertain and dynamically changing conditions.

Figure 12.

Trolley displacement with disturbance $y_{1} (t)$ responses.

Figure 13.

Cable length between trolley and hook with disturbance $y_{2} (t)$ responses.

Figure 14.

Hook sway angle with disturbance $y_{3} (t)$ responses.

Figure 15.

Payload sway angle with disturbance $y_{4} (t)$ responses.

The obtained results highlight the capability of the proposed data-driven interconnected TID controller in handling the nonlinear MIMO overhead crane system with distributed mass payload. Compared with the conventional PID controller under identical conditions, the proposed approach achieves improved tracking accuracy, reduced sway angles, and lower control input energy, as reflected by the smaller objective function value. These improvements demonstrate the effectiveness of the data-driven tuning approach in capturing the system dynamics without requiring an accurate mathematical model. The proposed control strategy shows strong potential for practical implementation in industrial overhead crane operations such as manufacturing lines, automated warehouses, construction material handling, and port container transportation, where accurate positioning and effective sway suppression are essential for operational safety and efficiency. In addition, the data-driven tuning framework is particularly suitable for real-world crane systems where model uncertainties, parameter variations, and nonlinear coupling effects make conventional model-based controller design difficult. Overall, the results confirm that the proposed data-driven interconnected TID controller provides a practical and superior alternative to conventional model-based and PID control approaches for nonlinear MIMO overhead crane systems with distributed mass payload.

Conclusion

This paper presented a data-driven TID controller for the nonlinear MIMO DPOC–DMP crane system, tuned using the SEDA algorithm. Simulation results demonstrated that the optimised TID controller achieved a lower objective function value (310.6951) compared to the PID controller (326.4870), corresponding to an improvement of 2.82% in overall control accuracy. The proposed controller also reduced the total tracking error from 0.1720 to 0.1619 and decreased the total control input energy from 4.3902 to 3.9647, indicating improved control efficiency. In addition, the TID controller provided better suppression of hook and payload sway angles with smoother transient responses and faster settling under both nominal and disturbance conditions. These results confirm that the proposed data-driven TID–SEDA framework is effective in enhancing tracking performance, reducing oscillations, and improving robustness for nonlinear MIMO crane systems with distributed-mass payloads. Future work will focus on extending the proposed framework to a multi-objective optimisation setting to simultaneously balance multiple performance criteria such as tracking accuracy, control effort, and robustness. In addition, experimental validation on a real crane testbed will be pursued to further assess the practical feasibility of the proposed approach. The versatility of the proposed TID–SEDA framework allows for potential extension to a broad range of nonlinear and coupled MIMO systems where high-precision oscillation damping and robust positioning are critical. In electric drive systems, the framework could optimise speed and torque control in electric vehicles to handle varying load conditions and improve energy efficiency. For robotic manipulators, specifically those used in medical surgeries or high-speed manufacturing, the SEDA-tuned TID controller can enhance path-tracking accuracy while suppressing mechanical vibrations. Furthermore, in aerial vehicles, such as quadrotors or delivery drones, the strategy can be applied to maintain stable flight attitudes and payload stability under unpredictable wind gusts and aerodynamic disturbances. Future investigations will focus on adapting the safe-experimentation mechanism of SEDA to these real-time operational constraints to ensure both performance and safety across these diverse platforms.

Footnotes

Acknowledgement

The highest gratitude is especially extended to the Universiti Malaysia Pahang Al-Sultan Abdullah for the financial assistance provided under the MTUN Strategic Collaboration Research Grant, RDU243501. Also special thanks to the Ministry of Higher Education for the financial assistance provided under Fundamental Research Grant Scheme (FRGS) No. FRGS/1/2024/TK07/UMP/03/1 (University reference RDU240101).

ORCID iDs

Mohd Zaidi Mohd Tumari

Mohd Helmi Suid

Mohd Ashraf Ahmad

Mohammad Osman Tokhi

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 Universiti Malaysia Pahang Al-Sultan Abdullah (RDU243501) and Malaysia Ministry of Higher Education (FRGS/1/2024/TK07/UMP/03/1).

Declaration of conflicting interests

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

References

Ramli

Mohamed

Abdullahi

, et al. Control strategies for crane systems: A comprehensive review. Mechanical Systems and Signal Processing 2017; 95: 1–23. https://doi.org/10.1016/j.ymssp.2017.03.015

Kataria

Jangid

. Seismic protection of the horizontally curved bridge with semi-active variable stiffness damper and isolation system. Advances in Structural Engineering 2016; 19(7): 1100–1117. https://doi.org/10.1177/1369433216634477

Madhekar

Jangid

. Variable dampers for earthquake protection of benchmark highway bridges. Smart Materials and Structures 2009; 18(11): 115011. https://doi.org/10.1088/0964-1726/18/11/115011

Ghosh

. Fractional order modeling of ecological and epidemiological systems: Ambiguities and challenges. The Journal of Analysis 2025; 33(1): 341–366. https://doi.org/10.1007/s41478-024-00836-y

You

Guo

. Pattern formation for a reversible biochemical reaction model with cross-diffusion and Michalis saturation. Journal of Mathematical Chemistry 2025; 63(3): 888–910. https://doi.org/10.1007/s10910-025-01705-0

Huang

Wang

. Assessing the robustness of physical networks under attack uncertainty. Reliability Engineering & System Safety 2025; 262: 111231. https://doi.org/10.1016/j.ress.2025.111231

Zhou

Zhu

. A backstepping controller design for underactuated crane system. In: 2018 30th Chinese Control and Decision Conference (CCDC), Shenyang, China, 9-11 June 2018, pp. 5707–5712. https://doi.org/10.1109/CCDC.2018.8407619

Wang

Tan

Zhang

, et al. A time-varying sliding mode control method for distributed-mass double pendulum bridge crane with variable parameters. IEEE Access 2021; 9: 67140–67152. https://doi.org/10.1109/ACCESS.2021.3079303

Xia

Wang

. H∞ output-feedback anti-swing control for a nonlinear overhead crane system with disturbances based on T-S fuzzy model. IEEE Access 2021; 9: 136128–136137. https://doi.org/10.1109/ACCESS.2021.3115948

10.

Nguyen

Yang

, et al. Design and implementation of finite time sliding mode controller for fuzzy overhead crane system. ISA Transactions 2022; 124: 468–479. https://doi.org/10.1016/j.isatra.2019.11.037

11.

Yang

Ouyang

. Precision-positioning adaptive controller for swing elimination in three-dimensional overhead cranes with distributed mass beams. ISA Transactions 2022; 127: 419–430. https://doi.org/10.1016/j.isatra.2021.08.035

12.

Mohd Tumari

Ahmad

Suid

, et al. An improved marine predators algorithm tuned data-driven multiple-node hormone regulation neuroendocrine-PID controller for multi-input–multi-output gantry crane system. Journal of Low Frequency Noise, Vibration and Active Control 2023; 42(4): 1666–1698. https://doi.org/10.1177/14613484231183938

13.

Gutierrez

Collado

. An LQR controller in the obstacle avoidance of a two-wires hammerhead crane. Neurocomputing 2017; 233: 61–71. https://doi.org/10.1016/j.neucom.2016.08.104

14.

Ouyang

Zhao

Zhang

. Enhanced-coupling nonlinear controller design for load swing suppression in three-dimensional overhead cranes with double-pendulum effect. ISA Transactions 2021; 115: 242–251. https://doi.org/10.1016/j.isatra.2021.01.009

15.

Van Trieu

Cuong

Dong

, et al. Adaptive fractional-order fast terminal sliding mode with fault-tolerant control for underactuated mechanical systems: Application to tower cranes. Automation in Construction 2021; 123: 103533. https://doi.org/10.1016/j.autcon.2020.103533

16.

Yang

Sun

Fang

. Adaptive fuzzy control for a class of MIMO underactuated systems with plant uncertainties and actuator deadzones: Design and experiments. IEEE Transactions on Cybernetics 2022; 52(8): 7561–7572. https://doi.org/10.1109/TCYB.2021.3050475

17.

Al Saaideh

Al-Solihat

Al-Rawashdeh

, et al. Decoupling and tracking control for offshore crane system effect by unknown roll/heave wave motions disturbances. In: 2024 American Control Conference (ACC), Toronto, Canada, 8-12 July 2024, pp. 1–6. https://doi.org/10.23919/ACC60939.2024.10644341

18.

. Adaptive anti-swing control for 7-DOF overhead crane with double spherical pendulum and varying cable length. IEEE Transactions on Automation Science and Engineering 2024; 21(4): 5240–5251. https://doi.org/10.1109/TASE.2023.3298192

19.

Zhai

Yang

, et al. Adaptive neural network unified control for general MIMO underactuated mechatronic systems with disturbances via modified normal forms. IEEE Transactions on Automation Science and Engineering 2025; 22(4): 19377–19390. https://doi.org/10.1109/TASE.2024.3411441

20.

Yang

Zhou

, et al. Barrier function-based adaptive composite sliding mode control for a class of MIMO underactuated systems subject to disturbances. IEEE Transactions on Industrial Informatics 2024; 20(9): 11429–11440. https://doi.org/10.1109/TII.2024.3396112

21.

Yang

Zhang

Zhou

, et al. Given-performance PID-SMC for 4-DOF tower crane systems under input constraints. IEEE Transactions on Automation Science and Engineering 2025; 22(3): 11444–11456. https://doi.org/10.1109/TASE.2024.3364239

22.

Milanesi

Mirandola

Visioli

. A comparison between PID and PIDA controllers. In: 2022 IEEE 27th International Conference on Emerging Technologies and Factory Automation (ETFA), Stuttgart, Germany, 6-9 September 2022, pp. 1–4. https://doi.org/10.1109/ETFA52439.2022.9921724

23.

Choi

Kim

Lee

, et al. A study on gantry crane control using neural network two degree of PID controller. In: ISIE 2001. 2001 IEEE International Symposium on Industrial Electronics Proceedings, Pusan, South Korea, 12-16 June 2001, pp. 1954–1959. https://doi.org/10.1109/ISIE.2001.932001

24.

Jaafar

Mohamed

. PSO-tuned PID controller for a nonlinear double-pendulum crane system. Communications in Computer and Information Science 2017; 752: 198–208. https://doi.org/10.1007/978-981-10-6502-6_18

25.

Elhabib

Wahid

Mohamed

, et al. Optimal control of double pendulum crane using FOPID and genetic algorithm. Communications in Computer and Information Science 2024; 1912: 419–431. https://doi.org/10.1007/978-981-99-7243-2_34

26.

Ahmed

Magdy

Khamies

, et al. Modified TID controller for load frequency control of a two-area interconnected diverse-unit power system. International Journal of Electrical Power and Energy Systems 2022; 135: 107528. https://doi.org/10.1016/j.ijepes.2021.107528

27.

Sain

Swain

Mishra

. TID and I-TD controller design for magnetic levitation system using genetic algorithm. Perspectives in Science 2016; 8: 370–373. https://doi.org/10.1016/j.pisc.2016.04.078

28.

Moschos

Parisses

. Combined frequency and voltage control of two-area multi-source interconnected microgrids via the 2DOF-TIDμ controller. E-Prime - Advances in Electrical Engineering, Electronics and Energy 2023; 5: 100268. https://doi.org/10.1016/j.prime.2023.100268

29.

Sayed

El-Zohri

Ahmed

, et al. Application of tilt integral derivative for efficient speed control and operation of BLDC motor drive for electric vehicles. Fractal and Fractional 2024; 8(1): 61. https://doi.org/10.3390/fractalfract8010061

30.

Jing

. The X-structure/mechanism approach to beneficial nonlinear design in engineering. Applied Mathematics and Mechanics 2022; 43(7): 979–1000. https://doi.org/10.1007/s10483-022-2862-6

31.

Jing

. A bistable X-structured electromagnetic wave energy converter with a novel mechanical-motion-rectifier: Design, analysis, and experimental tests. Energy Conversion and Management 2021; 244: 114466. https://doi.org/10.1016/j.enconman.2021.114466

32.

Wang

. Dynamic modeling and analysis of improved wave energy harvesters with novel X-structured nonlinear inerter mechanism. Journal of Vibration and Control 2025, Advance online publication. https://doi.org/10.1177/10775463251317668

33.

Jiang

Zang

Xiong

, et al. Design and vibration reduction performance analysis of a 6-DOF nonlinear vibration isolation platform by employing bio-inspired multi-joint X-structure. Nonlinear Dynamics 2025; 113(16): 21119–21150. https://doi.org/10.1007/s11071-025-11256-3

34.

Acosta

JÁ

Ortega

Astolfi

, et al. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Transactions on Automatic Control 2005; 50(12): 1936–1945. https://doi.org/10.1109/TAC.2005.860292

35.

Mohammadi

Maggiore

Consolini

. Dynamic virtual holonomic constraints for stabilization of closed orbits in underactuated mechanical systems. Automatica 2018; 94: 112–124. https://doi.org/10.1016/j.automatica.2018.04.023

36.

Bello

Mohamed

Efe

, et al. Modelling and dynamic characterisation of a double-pendulum overhead crane carrying a distributed-mass payload. Simulation Modelling Practice and Theory 2024; 134: 102953. https://doi.org/10.1016/j.simpat.2024.102953

37.

Shukor

NSA

Ahmad

Tumari

MZM

. Data-driven PID tuning based on safe experimentation dynamics for control of liquid slosh. In: Proceedings of the 2017 IEEE 8th Control and System Graduate Research Colloquium (ICSGRC), Shah Alam, Malaysia, 4-5 August 2017, pp. 62–66. https://doi.org/10.1109/ICSGRC.2017.807056

38.

Marden

Young

Arslan

, et al. Payoff-based dynamics for multiplayer weakly acyclic games. SIAM Journal on Control and Optimization 2009; 48(1): 373–396. https://doi.org/10.1137/070680199

39.

Ghazali

Ahmad

Raja Ismail

RMT

, et al. An improved neuroendocrine–proportional–integral–derivative controller with sigmoid-based secretion rate for nonlinear multi-input–multi-output crane systems. Journal of Low Frequency Noise, Vibration and Active Control 2020; 39(4): 1172–1186. https://doi.org/10.1177/1461348419867524

40.

Ghazali

Ahmad

Raja Ismail

RMT

. Adaptive safe experimentation dynamics for data-driven neuroendocrine-PID control of MIMO systems. IETE Journal of Research 2019; 68(3): 1611–1624. https://doi.org/10.1080/03772063.2019.1656556

41.

Saat

Ahmad

Ghazali

. Data-driven brain emotional learning-based intelligent controller-PID control of MIMO systems based on a modified safe experimentation dynamics algorithm. International Journal of Cognitive Computing in Engineering 2025; 6: 74–99. https://doi.org/10.1016/j.ijcce.2024.11.005