Observer-Based Robust Control for Dynamic Positioning in Float-Over Installation of Offshore Converter Stations

Abstract

With the development of offshore wind power progressing towards larger-scale and deeper-water projects, the float-over installation of offshore converter stations has become a mainstream solution due to its high carrying capacity, efficiency and cost-effectiveness. This study addresses the dynamic positioning (DP) challenges during this operation, where traditional PID controllers often struggle with performance under complex environmental loads. An Observer-Based Robust Controller (OBRC) is proposed and integrated with a constant parameter time-domain model (CPTDM) to simulate the DP process of a novel T-U barge. Time-domain simulations for both standby and entry phases were conducted under various wave directions and periods. The results demonstrate that the OBRC significantly outperforms the conventional PID controller in maintaining positioning accuracy. The findings provide critical insights into motion responses and control strategies, offering valuable guidance for the design and safe operation of future float-over installations.

1. Introduction

The global offshore wind industry is rapidly advancing, with a clear trend towards larger-scale projects in deeper waters. This march toward “large-scale” and “deep-water” development poses substantial challenges for offshore power transmission. High-voltage direct current (HVDC) technology is widely used in the field of offshore wind power because of its low cost, good control performance and low energy loss [1,2]. Therefore, the offshore booster station and converter station which converts an alternating current emitted by a wind farm into a high-voltage direct current, came into being.

Understanding nonlinear dynamic responses is fundamental to securing the safety of float-over installations. The prediction of these responses relies on model tests and numerical simulation. Given the substantial time and financial investment required for physical model tests, they are chiefly employed to verify numerical analyses. In contrast, the controllability, computational efficiency, and cost-effectiveness of numerical simulation make it the central technique for float-over installation analysis. This numerical work is primarily carried out in two major streams: the application of commercial marine software and the use of self-developed programs.

Dynamic positioning technology in the field of marine engineering is one of its enduring research hotspots. Zhang [3] used the PID control model to simulate the dynamic response of HYSY278 when loading the upper block at sea during the standby stage and the entry stage. The trial and error method and the control variable method were used to tune the PID parameters. The positioning ability, motion response and thruster thrust of the barge under different PID parameters were explored. Finally, the optimized PID parameters were obtained and compared with the test results. Xu et al. [4] evaluated the probability of continuous operation in the East China Sea using dynamic positioning barges and pointed out that dynamic positioning barges have stronger applicability to harsh sea conditions. He et al. [5] established a dynamic positioning model based on a nonlinear observer and a PID controller to simulate the nonlinear motion of the ship entering stage. The model considers the effects of wind, waves and current forces and also considers the coupling between the jacket fender facilities of the barge. Ye et al. [6,7] constructed an observer-based robust controller to simulate the dynamic response of the floating crane during operation, considering the influence of uncertain factors in the modeling process, including mooring load, fluid damping force and external disturbances, such as sling force, wind wave and current environmental load. Liu et al. [8] performed a comprehensive nonlinear dynamic analysis of a dynamically positioned (DP) crane vessel, revealing its nonlinear dynamic characteristics and stability performance under PID control. The superior performance of the proposed OBRC compared to the conventional PID controller is evident in the presented results. It is insightful to further contextualize this performance within the landscape of modern DP control strategies. As extensively reviewed by Gao and Li [9], advanced DP control has evolved into multi-functional composite forms to handle the complex and intertwined challenges of the marine environment. These include adaptive control (e.g., using Neural Networks or Fuzzy Logic), disturbance observer-based control, model predictive control (MPC) and robust control techniques. Among these, MPC is particularly notable for its ability to explicitly handle system constraints and optimize performance over a future horizon [6,10]. While MPC can theoretically yield near-optimal performance, its computational demand can be significant, especially for high-fidelity nonlinear models, posing challenges for real-time implementation on existing DP system hardware.

Most commercial marine software utilizes the time-domain model based on the Cummins equation [11]. In the calculation process, each time step needs to integrate the convolution term in the Cummins equation, which is quite time-consuming and causes the accumulation of accuracy. Since the convolution term is a linear time-invariant system, it can be replaced by other constant coefficient models that are also linear systems, such as the state-space model (SSM) and transfer function (TF), and the transfer function can also be easily converted into a state-space model. The state-space model itself is a time-domain formulation, which has led to its widespread adoption in place of the computationally intensive convolution term. Its high computational efficiency and relevance to control analysis make it particularly suitable for marine structures. For example, Duarte et al.state-spacestate-spacestate-space [12] demonstrated that using the state-space model for radiation damping calculations reduced the required time by about 75% compared to the convolution approach.state-space Taghipour et al. [13] further illustrated the high efficiency of the state-space model at small time steps. When using a time step of 0.001 s, the calculation speed of the state-space model can reach 80 times the calculation speed of the convolution integral. Chen et al. [14] also applied the state-space model to replace the convolution term in the Cummins equation for time-domain modeling of wave-induced impact oscillators, which laid a foundation for efficient analysis of nonlinear impacts (e.g., LMU impacts) in float-over installations. Due to its high computational efficiency, the state-space model is widely used in the field of marine engineering.

Building upon the state-space modeling technique, Hu et al. [15] developed a three-degree-of-freedom (sway–heave–roll) collision model that integrated the DSU, LMU, and nonlinear sway fenders. The modeling challenge arises from the intrinsic complexity of float-over installation, which features strong hydrodynamic and mechanical coupling effects transmitted through the LMU, DSU, fender, and mooring systems. Consequently, when utilizing commercial marine software for dynamic response analysis, scholars frequently adopt simplified models. Typical simplifications include neglecting the finite strength of LMUs and DSUs, idealizing the geometry of the LMU jack and receiver, omitting the mooring and dynamic positioning systems, or artificially constraining the floating body’s degrees of freedom state-space.

The docking slow collision system includes the DSU and LMU, as shown in Figure 1. The DSU is located on the deck of the floating barge, which buffers the vertical collision load between the upper block and the floating barge during the weight transfer of the upper block. The LMU is located at the top of the jacket leg and buffers the horizontal and vertical collision loads generated between the upper block and the jacket during the weight transfer of the upper block.

Zhu et al. [16] constructed a three-degree-of-freedom constant parameter time-domain model (CPTDM) for the float-over installation of gravity platforms, considering the hydrodynamic coupling between the barge and the platform foundation. They pointed out that in the float-over installation system, with the relative motion between the upper block and the barge and the foundation, the separation-collision-reseparation phenomenon will occur between the DSU and the LMU, and a strong nonlinear effect will occur at this time. In order to more accurately capture the strong nonlinearity of the system when the impact and separation occur at the junction of LMU and DSU, the strong nonlinearity of the system is more effectively captured. Zou et al. [17] combined the open source mooring load calculation program MoorDyn with CPTDM to construct a six-degree-of-freedom CPTDM for jacket float-over installation and pointed out the difference in nonlinear response characteristics between three-degree-of-freedom and six-degree-of-freedom systems.

In the simulation of dynamic positioning of float-over installation, most scholars use the PID model which is convenient for modeling, but the method of setting parameters is often result-driven, and it is necessary to determine the optimization parameters through trial calculation or control variables, which lacks the consideration of external environment and barge’s own factors. In addition, when calculating the motion response of the dynamic positioning barge, the dynamic positioning system will bring some nonlinear effects to the barge. It is difficult to accurately capture the nonlinear dynamic response of the system by using the traditional convolution integral method. Therefore, it is necessary to develop a more efficient and accurate time-domain algorithm. On the one hand, efficient time-domain solution can save a lot of computing resources. On the other hand, accurate time-domain model can capture more nonlinear characteristics in the system and provide more accurate motion response feedback for the controller.

While the proposed CPTDM framework does not directly simulate the internal stress of LMU/DSU components, it provides a high-fidelity prediction of the global motion responses and impact loads (e.g., forces at fender and LMU locations) under complex environmental conditions. These precise load predictions are essential input data for subsequent dedicated finite element analysis (FEA) to evaluate the bearing capacity and detailed mechanical response of the LMU, DSU and other structural units, ensuring their design can withstand the operational loads encountered during the installation.

The OBRC control strategy proposed in this paper is combined with the floating structure, mooring system and wave energy suppression technology verified by Chen et al. [18] the study of deep-sea aquaculture platform, which is expected to provide a more robust and intelligent technical path for the floating installation of large-scale wind power structures in the deep sea in the future.

This study aims to investigate whether the proposed Observer-Based Robust Controller (OBRC) can significantly enhance the dynamic positioning performance and operational safety of a float-over barge during the installation of offshore converter stations under complex environmental conditions. To address the highly nonlinear and time-varying nature of the coupled barge-environment system, high-fidelity time-domain numerical simulations based on a constant parameter time-domain model (CPTDM) are essential for obtaining reliable and comprehensive dynamic response results. A series of environmental conditions, including various wave directions and periods, are selected for time-domain simulations covering both the standby and entry operational phases. Comparative analyses are then conducted on key aspects, including positioning accuracy, motion responses (such as accelerations and angular displacements) and load effects under different control strategies and wave scenarios. Ultimately, the feasibility and advantages of the OBRC-based control strategy are evaluated based on the integrated analysis of these results. The study is guided by the following research questions:

• Can the proposed Observer-Based Robust Controller (OBRC) provide superior dynamic positioning performance compared to the conventional PID controller for a float-over barge under complex environmental conditions?
• How do different wave directions and periods affect the key motion responses and positioning accuracy of the T-U barge during the critical standby phase?
• What are the comparative dynamic response characteristics between the bow entry and stern entry installation methods under various wave directions?

Section 2 establishes the theoretical foundation, introducing the novel T-U barge design and detailing the constant parameter time-domain model (CPTDM) for efficient simulation, alongside the observer-based robust control (OBRC) strategy with guaranteed stability. Section 3 describes the implementation of the coupled OBRC-CPTDM numerical framework in MATLAB/Simulink and defines the specific environmental conditions and controller parameters for the standby and entry phase analyses. Section 4 then presents a comprehensive discussion of the time-domain results, systematically validating the superior performance of the OBRC over a conventional PID controller and evaluating the dynamic response of the barge under various wave conditions for both operational phases.

2. Materials and Methods

2.1. Numerical Modeling of New T-U Barge

In this chapter, a new type of T-U barge for offshore large-scale wind power structure installation, referred to as T-U barge, will be proposed. This type of barge combines the characteristics of a T-type barge and U-type barge. It is narrowed in the bow to form a T-type bow and slotted in the stern to form a U-type stern. After completing the three-dimensional modeling of the barge, hydrodynamic calculation and analysis are carried out. The viscous correction of T-U barge is carried out, including rolling damping correction and artificial damping correction within the moonpool. For hydrodynamic coefficient calculation of structures involving heave motion (a key degree of freedom in float-over installation), Zhang and Ishihara [19] studied the hydrodynamic coefficients of multiple heave plates using large eddy simulations (LES) with the volume of fluid (VOF) method, providing insights into the effects of geometric parameters (e.g., spacing ratio, aspect ratio) on heave-related hydrodynamics—this supports the rationality of viscous correction and geometric parameter optimization for the T-U barge. The effect of damping correction is evaluated by the hydrodynamic coefficient, frequency-domain motion Response Amplitude Operator (RAO), impulse response function and frequency-domain–time-domain verification, and a complete hydrodynamic model of T-U barge is constructed.

This versatile barge is engineered for the concurrent installation and disassembly of offshore wind turbines, converter stations, and booster stations. Its T-shaped bow facilitates the float-over installation of large structures, such as 10,000 t converter stations [20,21], onto pre-slotted jacket platforms. Conversely, the U-shaped stern enables the hoisting or floating installation of smaller, non-slotted structures, including kiloton-scale booster stations and individual wind turbines (fixed or floating) [22,23]. Furthermore, the U-shaped groove can be utilized for launching foundations, such as jackets and semi-submersible bases. To accommodate this wide range of applications, the T-U barge is designed with an operational capacity for structures weighing from 1000 t–15,000 t. Crucially, its hydrodynamic performance is designed to be comparable to, if not superior to, that of traditional flat-top and T-barges. The specific case study in this paper involves a converter station located in the northern Yellow Sea, northeast of the Shandong Peninsula, at a water depth of 58.4 m.

The new T-U barge’s overall dimensions are based on the HYSY228 from the China National Offshore Oil Corporation, which has a float-over installation capacity of approximately 18,000 t. The modified design incorporates a slotted stern, which reduces the vessel’s load capacity and consequently affects the arrangement of its internal ballast tanks. Preliminary estimates indicate a resultant float-over installation capacity of 14,000 t–16,000 t for the T-U barge, aligning with the target capacity range previously established.

In addition, the slot width of the T-U barge stern is designed to be 36 m. First of all, considering that the size of the kiloton step-up station or converter station is 15 m–30 m [24,25], the width of the slot should not be too narrow and a certain margin should be left to install buffer equipment such as fenders. Secondly, in order to improve the applicability of the barge to jacket foundations of different sizes and reduce the influence of the barge on the foundation design, the slot width is as close as possible to the width of the bow. When the size of the jacket is less than 36 m and the inlet slot cannot be designed, the U-shaped stern can be used for installation.

On the other hand, in order to ensure high initial stability of the barge, the load capacity and the structural strength of the stern, the width of the stern is widened by 1.75 m on the basis of the ‘HYSY228’ so that the maximum width of the stern part reached 58 m. The dimensions of the new T-U barge are shown in Figure 2. In this chapter, the barge in the standby phase is taken as the research object, that is, the upper block is loaded on the DSF of the barge and equipped with fixed facilities, and the upper block is rigidly connected with the barge. Therefore, in the analyses and calculations in this chapter, it was necessary to input the mass parameters of the barge itself and the upper block into the hydrodynamic model at the same time. The specific parameters of the barge and the upper block are as shown in

Table 1. Quality parameters of new T-U barge.

Parameter	Unit	Value
Mass displacement, $\mathsf{\Delta }$	[t]	52,507
The center of gravity position in the direction of the captain, $L_{OA}$	[m]	86.42
Height of center of gravity, $V_{CG}$	[m]	−2
$R_{xx}$	[m]	15
$R_{yy}$	[m]	50
$R_{zz}$	[m]	50

Note: The origin of the center of gravity in the direction of the captain is at the stern of the ship, and the plateau point of the center of gravity is at the still water surface.

and

Table 2. Quality parameters of upper block.

Parameter	Unit	Value
$L_{Deck}$	[m]	62
$B_{Deck}$	[m]	51
$H_{Deck}$	[m]	30
Mass displacement, $\mathsf{\Delta }$	[t]	13,000
The center of gravity position in the direction of the captain, $L_{OA}$	[m]	120
Height of center of gravity, $V_{CG}$	[m]	40.54
$R_{xx}$	[m]	29.54
$R_{yy}$	[m]	16.93
$R_{zz}$	[m]	29.06

Note: The coordinate system used to determine the center of gravity is the same as in Table 1.

. In the table, the origin of the center of gravity in the direction of the captain is at the midline of the stern, and the plateau point of the center of gravity is at the still water surface. The numerical model of the novel T-U barge was developed in this study. Its overall scale is based on the ‘HYSY228’ vessel, and it is a conceptual design with no physical manufacturer involved.

2.2. Time-Domain Model with Constant Coefficients Based on State-Space

Under the assumption of potential flow theory, only the first-order hydrodynamic force is considered, and the motion equation of the floating structure without speed can be expressed by the Cummins equation [11]:

$$\left[M+A\right]\ddot{\eta }(t)+{\displaystyle {\int }_0^{t}K}(t-\tau )\dot{\eta }(\tau )d\tau +C\eta (t)=f^{\mathrm{exc}}(t)$$

(1)

In the formula: $M$ is the mass matrix of the offshore floating structure; $A$ is the additional mass matrix; $\mathrm{K}\left(\mathrm{t}\right)$ is the delay function matrix; C is the coefficient moment of hydrostatic restoring force array; $\eta (t)$, $\dot{\eta }(t)$ and $\ddot{\eta }(t)$ are the displacement, velocity and acceleration vectors of the floating structure, respectively; and $f^{exc}(t)$ is the excitation force acting on the floating structure.

The added mass $A$ and the delay function $\mathrm{K}\left(\mathrm{t}\right)$ in the Cummins equation are the quantities related to the hydrodynamic parameters of the offshore floating structure. In order to explore the mathematical relationship between them, Fourier transform is performed on Equation (1):

$$\tilde{\eta }(j\omega )={\{-{\omega }^2\left[M+A(\omega )\right]+j\omega B(\omega )+C\}}^{-1}{\tilde{f}}^{exc}(j\omega )$$

(2)

In the formula: $\tilde{\eta }(j\omega )$ is the Fourier transform of $\eta (t)$; $A(\omega )$ is the additional mass varying with frequency; $B(\omega )$ is the wave damping; ${\tilde{f}}^{exc}(j\omega )$ is the Fourier transform of the external load $f^{exc}(t)$; and $\omega$ is the angular frequency.

Ogilvie [26] established the relationship between the added mass $A(\omega )$ and the wave damping $B(\omega )$ and the delay function $\mathrm{K}\left(\mathrm{t}\right)$ with the change in frequency through the study of hull motion prediction, namely

$$A(\omega )=A-{\displaystyle \frac{1}{\omega }}{{\displaystyle \int }}_0^{\infty }K(t)\sin (\omega t)\mathrm{d}t$$

(3)

$$B(\omega )={\displaystyle {\int }_0^{\infty }K}(t)\cos (\omega t)\mathrm{d}t$$

(4)

The Fourier transform of the delay function $\mathrm{K}\left(\mathrm{t}\right)$ can be expressed by $A(\omega )$ and $B(\omega )$:

$$\tilde{K}(j\omega )={\displaystyle {\int }_0^{\infty }K}(t)e^{-j\omega t}dt=B(\omega )+j\omega [A(\omega )-A(\infty )]$$

(5)

According to Equation (3), the additional mass in Equation (1) is

$$A=\underset{\omega \to \infty }{\lim }A(\omega )=A(\infty )$$

(6)

Thus, Equation (1) can be expressed as

$$\left[M+A(\infty )\right]\ddot{\eta }(t)+{{\displaystyle \int }}_0^{t}K(t-\tau )\dot{\eta }(\tau )d\tau +C\eta (t)=f^{\mathrm{exc}}(t)$$

(7)

To improve computational efficiency and stability in simulating multi-body hydrodynamic interactions, Zou et al. [27] developed a CPTDM that combines the damping lid method with state-space models, effectively addressing gap resonance phenomena in close-proximity floating systems such as FLNG-LNG side-by-side operations. To address the computational inefficiency and numerical instability of traditional Cummins-based models in multi-body floating systems, Zou et al. [18] developed a fully coupled time-domain model for catamaran float-over installations using a CPTDM with state-space realization. Their model incorporates impact buffering systems and nonlinear mooring dynamics, and was validated against AQWA using the Kikeh Spar benchmark case. Furthermore, Zhang et al. [28] extended this approach to hybrid floating wind-wave platforms (FWWPs), integrating surge-flap-type WECs with a semi-submersible FOWT. Their CPTDM-SSM framework yielded a computational efficiency increase of one order of magnitude over AQWA while maintaining high fidelity in motion and mooring response predictions.

Although the time-domain model described by the Cummins equation based on convolution integral has been widely used in the field of marine engineering, it still takes a lot of time to solve the convolution integral directly, and it is not easy to use when modeling the motion control system of marine engineering structures. The convolution term in the Cummins equation describes a linear time-invariant system, so it can be described by a state-space model. A typical state-space model has the following expression:

$$\begin{array}{c}\dot{z}(t)=A_{c}z(t)+B_{c}u(t)\\ y(t)=C_{c}z(t)+D_{c}u(t)\end{array}$$

(8)

In the formula, $z$ is the state quantity, $u$ is the input quantity, $y$ is the output quantity and $A_c\in {ℝ}^{N\times N}$, $B_c\in {ℝ}^{N\times 1}$, $C_c\in {ℝ}^{1\times N}$ and $D_c\in {ℝ}^{1\times 1}$ are the coefficient matrices.

We use the velocity term in the convolution integral as the input of the state-space and the velocity-related part of the radiation force as the output. For a single floating body system, the state-space of a certain degree of freedom can be written as

$$\begin{array}{c}{\dot{z}}_{ij}\left(t\right)=a_{ij}{z}_{ij}\left(t\right)+b_{ij}{\dot{x}}_j\left(t\right)\\ y_{ij}\left({\dot{x}}_j\left(t\right)\right)=c_{ij}{z}_{ij}\left(t\right)+d_{ij}{\dot{x}}_j\left(t\right),\hspace{1em}i,j=1,\dots ,6\end{array}$$

(9)

where $a_{ij}\in {ℝ}^{q\times q}$, $b_{ij}\in {ℝ}^{q\times 1}$, $c_{ij}\in {ℝ}^{1\times q}$ and $d_{ij}\in {ℝ}^{1\times 1}$, and $q$ is the order of state-space.

After replacing the convolution term with the state-space model, the time-domain equation can be expressed in the form of a matrix:

$$[M+A(\infty )]\ddot{x}(t)+B_L\dot{x}(t)+y\left(\dot{x}\left(t\right)\right)+Cx(t)=F^{exc}(t)+F^E(t)$$

(10)

All coefficients in Equation (10) are constant coefficients, and it is not necessary to integrate the impulse response function at each time step with the solution convolution integral, so the model is called the CPTDM.

2.3. Dynamic Positioning

To address the transient performance issues in dynamic positioning (DP) systems under varying environmental conditions, Brodtkorb et al. [29] proposed a hybrid observer-controller framework that switches between model-based and signal-based observers based on real-time performance monitoring. In contrast to observer-switching-based approaches, Hu and Du [30] proposed a robust nonlinear DP control law that explicitly incorporates thruster dynamics, input saturation and unknown time-varying disturbances using command-filtered backstepping combined with a disturbance observer and auxiliary dynamic system. The time-domain simulations and controller design were implemented using MATLAB (Version R2023a; MathWorks, Inc., Natick, MA, USA) and Simulink. The constant parameter time-domain model (CPTDM) and state-space model were realized through in-house scripts developed within this environment.

2.3.1. PID Control

PID control is a classical control method which is widely used in the field of marine engineering. One of the most important application scenarios is the ship’s dynamic positioning system. In order to realize the dynamic positioning of the ship, the PID controller calculates and outputs the control instructions through the error of the system, using the proportional, integral, and derivative to achieve the control goal.

$$\tau =K_P\tilde{\eta }+K_I{\displaystyle \int \tilde{\eta }}dt+K_D{\displaystyle \frac{d\tilde{\eta }}{dt}}$$

(11)

In the formula, $\tilde{\eta }$ is the system error, that is, the difference between the real-time position of the floating body and the target point (position and heading angle), $K_P$ is the proportional function, which can be understood as the stiffness of the DP system, and the output is proportional to the error of the system. The restoring force, $K_I$, is the integral coefficient, which can continuously provide the slowly varying restoring force for the system; $K_D$ is the differential coefficient, which can be understood as the damping of the DP system, which can alleviate the hysteresis effect of the control to a certain extent.

2.3.2. Observer-Based Robust Controller and Its Stability Analysis

In this paper, an observer-based robust controller (OBRC) [6,7] based on Simulink is introduced for dynamic positioning of float-over barges. The stability of the closed-loop dynamic system is verified by Lyapunov’s second method.

Firstly, the three-degree-of-freedom control equation of DP system is defined:

$$\dot{\eta }=J\nu$$

(12)

$$M\dot{\nu }=-D\nu -F\eta +\tau +d_s$$

(13)

$$J=\left[\begin{array}{ccc}\cos (\psi )& -\sin (\psi )& 0\\ \sin (\psi )& \cos (\psi )& 0\\ 0& 0& 1\end{array}\right]$$

(14)

In the formula, $\eta$ is the motion of the floating body in the global coordinate system, $J$ is the 3 × 3 Euler rotation matrix, $\psi$ is the yaw angle of the floating body, $\nu$ is the motion speed of the floating body in the local coordinate system, $D$ is the positive definite damping matrix, $F$ is the positive definite stiffness matrix, $\tau$ is the control input and $d_s$ is the external disturbance, including the wind and flow load; the left multiplication matrix M⁻¹ on both sides of Equation (13) is moved to the right to obtain

$$\dot{\nu }=-A_1\eta -A_2\nu +B\tau +d$$

(15)

Thus, an observer-based robust controller (OBRC) can be constructed:

$$\dot{\hat{\eta }}=-K\hat{\eta }+K_1\tilde{\eta }+J\hat{\nu }$$

(16)

$$\dot{\hat{\nu }}=-{\hat{A}}_1\hat{\eta }-{\hat{A}}_2\hat{\nu }+B\tau +K_2(t)\tilde{\eta }$$

(17)

$$\tau =B^{-1}\{({\hat{A}}_1-J^T)\hat{\eta }-K_2(t)\tilde{\eta }+({\hat{A}}_2-\rho -{\rho }_1(t))\hat{\nu }\}$$

(18)

In the formula, $\hat{\eta }$ and $\hat{\nu }$ are the observed values of $\eta$ and $\nu$, and $\tilde{\eta }$ and $\tilde{\nu }$ are the observed errors of $\eta$ and $\nu$. The other gains in the controller, including the parameters of $K$, $K_1$, ${\hat{A}}_2$, $\rho$ and ${\rho }_1$, are designed as follows:

$${\lambda }_{\min }(K_1)>\left|\right|(1/2\beta )\mathsf{\Delta }{A}_1^T{H}^{-1}\mathsf{\Delta }{A}_1\left|\right|,$$

(19)

$${\lambda }_{\min }({\hat{A}}_2)>\left|\right|(3\beta /2)H\left|\right|$$

(20)

$${\lambda }_{\min }(K)>\left|\right|(1/2\beta )\mathsf{\Delta }{A}_1^T{H}^{-1}\mathsf{\Delta }{A}_1\left|\right|$$

(21)

$$\rho >\left|\right|(1/2\beta )\mathsf{\Delta }{A}_2^T{H}^{-1}\mathsf{\Delta }{A}_2\left|\right|+\left|\right|\mathsf{\Delta }d\left|\right|$$

(22)

$${\rho }_1(t)=\alpha \left|\right|(K_1|+K)|\left|\right|\left|\hat{\eta }\right|\left|\right|\left|\tilde{\eta }\right||$$

(23)

$$K_2(t)=-{\hat{A}}_1+J^T(t)$$

(24)

The detailed Lyapunov stability analysis for the closed-loop system has been moved to Appendix A to enhance the flow of the main text.

3. Dynamic Positioning Floating Barge Standby and Ship Dynamic Response Analysis

3.1. Numerical Model

Firstly, the time-domain model is constructed in Matlab. Based on the Cummins equation, the state-space model is used to replace the convolution integral term in the original equation to calculate the radiation force of the floating body in the wave, and the six-degree-of-freedom constant coefficient time-domain model of the floating body in the wave is established. In the time-domain model, the fourth-order Runge–Kutta method is used to realize the time step. The second-order wave force, wind load and current load calculated by the full Quadratic Transfer Function (QTF) method are input into the time-domain model as external loads.

Then, the OBRC proposed by Ye et al. [7] is modeled in Simulink. Then CPTDM is added to the Simulink model through the Matlab Function module in Simulink, and a two-way coupling is established with the OBRC to construct a closed-loop system including a motion solver and a controller, which is called the OBRC-CPTDM model. For fully coupled dynamic analysis of complex marine systems (e.g., integrating hydrodynamics, control and mooring), Chen et al. [31] constructed an aero-hydro-net-mooring time-domain model by coupling AQWA and FAST, which provides a reference for the coupled modeling logic of the OBRC-CPTDM framework—especially in handling multi-physical field interactions.

To solve the motion of the floating body in the DP state, it is necessary to input the OBRC-CPTDM model to input the target point and the initial conditions of the time-domain model, that is, the displacement and velocity of the floating body at the moment. Subsequently, the CPTDM performs time stepping, outputs the floating body motion response and inputs the deviation $\tilde{\eta }$ for the OBRC, and the OBRC outputs the DP force $\tau$ as the external load input of the CPTDM in the next time step. The overall process of OBRC-CPTDM is shown in Figure 3. For the verification and comparison of CPTDM’s own performance with AQWA, FAST and other software, we can refer to the study carried out by Zou et al. [17].

In this paper, the controller gains $K$, $K_1$, $\rho$ of OBRC are:

$$\begin{array}{l}K=9.077I\\ K_1=9.077I\\ \rho =0.0853\end{array}$$

(25)

where I is the unit matrix. In addition, the other constants in the controller are $\alpha =2$ and $\beta =1$.

The simulation environmental conditions were based on the metocean criteria for a reference offshore converter station located in the Bohai Sea region (water depth: 58.4 m). A current speed of 0.6 m/s and a wind speed of 10 m/s were adopted. The significant wave height and its corresponding spectral peak period were determined following the CCS Analysis Guide for Transportation and Floatover Installation of Large Marine Structures (2020) [20]. JONSWAP spectrum was used for wave spectrum, with peak factor $\gamma =1.4$ [32]. In addition, in this section, the performance of OBRC under the standby condition and the entry condition of the float-over installation are simulated, respectively. Under the standby condition, the OBRC and PID controllers are compared and studied.

3.2. Time-Domain Simulation of Dynamic Positioning Float-Over Barge in Standby Phase

3.2.1. Standby Phase: Model Setup

Under the standby condition, the OBRC model is compared with the PID model, and the positioning ability of OBRC in different wave directions and different periods is preliminarily evaluated. The mass property of the barge used in this section is the parameter when the barge and the upper block are taken as a whole and the upper block has been towed to the installation position, which is given in

Table 3. The mass properties of the barge-block system during the standby phase of the bow entry condition.

Parameter	Unit	Value
Mass displacement, $\mathsf{\Delta }$	[t]	52,507
The center of gravity position in the direction of the captain, $L_{CG}$	[m]	86.422
Height of center of gravity, $V_{CG}$	[m]	8.560
$R_{xx}$	[m]	26.91
$R_{yy}$	[m]	48.10
$R_{zz}$	[m]	45.96

. The marine environmental conditions simulated in this section are shown in

Table 4. Time-domain simulation environment parameters of standby condition.

Controller	Wave Incidence Angle $\mathit{\beta }$ [°]	Significant Wave Height [m]	Period [s]	Wind Speed [m/s]	Current Speed [m/s]
OBRC	0	2	5, 6, 7, 8	15	1
	45
	90
	135
	180
PID	0	2	7	15	1
	45
	90
	135
	180

.

The environmental load parameters (significant wave height, spectral peak period, wind speed, and current speed) were selected based on the operational limits for float-over installation and the metocean data specific to the project site located in the northern Yellow Sea. The significant wave height of 2 m and wind speed of 15 m/s represented challenging but operationally feasible conditions. The wave periods (5–10 s) covered the range most critical for motion response. The current speed was set to a typical value of 1 m/s and are consistent with the guidelines provided by the China Classification Society (CCS) [33,34] for the analysis of marine operations.

3.2.2. Comparison Between OBRC Control Model and Classical PID Control Model

When using the PID controller, it is first necessary to determine the three gain values of the proportional coefficient $K_P$, the integral coefficient $K_I$ and the differential coefficient $K_D$. Referring to the PID parameters of the float-over barge HYSY278 proposed by Zhang [3] and after debugging according to the characteristics of the T-U barge, the proportional coefficient $K_P=0.001$, integral coefficient $K_I=0.00001$ and differential coefficient $K_D=0.08$ were taken. The target position was set at the barge’s center of gravity.

The time history curve of the offset radius of the T-U barge under different wave incident angles and different controllers is shown in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. It can be seen from the diagram that under the control of the PID model, the barge oscillates around the target point, and its oscillation period is determined by the proportional coefficient $K_P$ It can be considered that the equivalent natural period of the PID model in this oscillation period is about 200 s. When the external load is a constant load, the object controlled by the PID model will converge to the target point.

The convergence rate is governed by the derivative gain $K_D$, and the system behaves analogously to a damped spring–mass system. However, in this study, the complete quadratic transfer function (QTF) was adopted for calculating second-order wave forces, in place of the mean slow-drift approximation. In addition, wind and current loads were updated in real time according to the instantaneous velocity of the barge. Under such time-varying environmental conditions, the PID-controlled barge exhibited sustained oscillations around the target position.

By contrast, the observer-based robust controller (OBRC) achieved significantly better positioning performance under the same time-varying loads. With OBRC, the barge maintained a steady position sufficiently close to the target, confining the tracking error within a bounded range and effectively suppressing oscillatory motion. The governing equation of the OBRC model was obtained based on the hydrodynamic characteristics of the barge. The additional mass, linear damping and pre-estimated external load of the barge in the surge, sway and yaw directions are fully considered. Therefore, the OBRC model can more accurately calculate the DP force required by the barge at each time step under the coupling of wind, wave and current.

3.2.3. Sensitivity Analysis of Wave Direction and Period of OBRC Control Model

In order to fully explore the positioning ability of the OBRC model under different wave directions and different periods of wave loads, five wave directions, 0°, 45°, 90°, 135° and 180°, were taken, respectively, and each wave direction considered six spectral peak periods, as shown in Table 4.

For large offshore converter stations housing precision electrical equipment, strict monitoring of motion parameters—including inclination and acceleration—is essential during float-over installation [20]. This practice prevents excessive dynamic responses that could compromise structural integrity and air tightness, thereby ensuring the safety of sensitive equipment within the station. For large converter stations, the maximum acceleration in the translational direction should not exceed 0.1 g, and the roll angle and pitch angle should not exceed 5.0° and 2.5°, respectively. In the standby phase, the upper block is loaded on the barge and is towed to the installation position of the bow to wait for installation, so the above indicators must be attended to in the standby phase to ensure the safety of the upper block. In the standby phase, the barge and the upper block are still in a rigid connection state, which can be considered as a whole. Therefore, the following will be based on the barge’s indicators of the safety of the upper block in the standby phase.

Chen et al. [35] further developed a coupled heave–roll–pitch impact model to analyze the complex vertical interactions between the deck, barge and substructure during float-over installation, highlighting that multi-degree-of-freedom motions (e.g., roll, pitch) significantly affect impact loads and motion responses—this supports the necessity of sensitivity analysis for wave direction/period on roll/pitch of the T-U barge.

Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 present polar plots of the maximum surge, sway, and heave accelerations, maximum roll and pitch amplitudes, and maximum offset radius of the barge during the standby phase under various wave directions and spectral peak periods. As shown in Figure 9, the maximum surge acceleration under following sea conditions is notably higher than under other wave directions; nevertheless, the overall values remain well below the operational limit of 0.1 g. Figure 10 illustrates that the maximum sway acceleration occurs under beam sea conditions. Aside from head and following seas, sway accelerations show limited variation across wave directions and do not exceed the 0.1 g threshold. Regarding heave motion (Figure 11), the heave acceleration increases with wave period and is generally greater than that under head sea conditions. At a spectral peak period Tp = 10 s, the heave acceleration reaches approximately 0.8 m/s², approaching the specified limit of 0.1 g.

As shown in Figure 12, roll motion amplitudes remain minimal in head and following seas (0° and 180°). For beam seas, the roll response shows no significant difference for Tp ≤ 8 s, but increases markedly at Tp = 9 s and further at Tp = 10 s. In quartering seas (45° and 135°), the roll magnitude grows with increasing spectral peak period, with the stern-quartering wave direction (45°) consistently producing the largest response amplitudes across all periods.

Figure 13 shows the amplitude of the pitch motion of the barge, which increases with the increase in the period in the same wave direction. Due to the two special structures of the T-shaped bow and the U-shaped stern of the T-U barge, there is still a certain pitch amplitude under the action of transverse waves. In the same period, the amplitude of the pitch motions of the barge under the action of bow oblique waves and stern oblique waves are significantly larger than those of other wave directions. When Tp = 10 s, the amplitude of pitch motion reaches 1.9° under the action of stern oblique waves and reaches 1.6° under the action of bow oblique waves, which is close to the maximum pitch angle limit of 2.5°.

The maximum offset radius of the barge is shown in Figure 14, and the overall performance increases with the increase in the spectral peak period of the wave, and in each wave direction, the offset radius is the largest in the transverse wave, followed by the bow oblique wave, the stern oblique wave again, and finally the following wave and the head wave.

The maximum offset occurs at 90° (beam sea) and 45° (quartering sea) wave directions due to the interplay of wave drift forces and the barge’s hydrodynamic characteristics. In a 90° beam sea, the wave-induced second-order drift force in the sway direction is typically greatest. The DP system must work constantly to counteract this strong lateral forcing, leading to larger positioning errors. For 45° quartering seas, the wave drift force has significant components in both surge and sway directions. Furthermore, these waves create a strong yaw moment, challenging the heading control of the DP system. The coupling between surge, sway and yaw motions under these directional waves makes it particularly difficult for the controller to cancel all motions simultaneously, resulting in a large net offset.

3.3. Time-Domain Simulation of Dynamic Positioning Float-Over Barge Entering Ship Condition

3.3.1. Entry Phase: Model and Fender Setup

This section compares the dynamic response of the dynamically positioned float-over barge during different entry directions and analyzes the sensitivity of both installation methods to incident wave direction. Furthermore, after the barge enters the jacket, swaying fenders are employed to buffer the motion, thereby mitigating excessive collision forces between the barge and the jacket structure. The nonlinear stiffness characteristics of the fenders are modeled using a polynomial approximation [36], as shown in Figure 15. The mass properties of the barge–upper block system under both entry conditions are provided in

Table 5. Quality properties of barge-block system in standby phase of stern entry condition.

Parameter	Unit	Value
Mass displacement, $\mathsf{\Delta }$	[t]	52,507
The center of gravity position in the direction of the captain, $L_{CG}$	[m]	86.410
Height of center of gravity, $V_{CG}$	[m]	2.875
$R_{xx}$	[m]	22.822
$R_{yy}$	[m]	56.645
$R_{zz}$	[m]	55.200

, while the environmental parameters used in the simulations are summarized in

Table 6. Environmental load parameter setting of dynamic positioning entering ship condition.

Working Condition	CONTROLLER	Incidence Angle of Wave $\mathit{\beta }$ [°]	Significant Wave Height [m]	Period [s]	Wind Speed [m/s]	Current Speed [m/s]
Bow into the ship	OBRC	0	1.5	7	10	0.6
		45
		90
		135
		180
Stern into the ship	OBRC	0	1.5	7	10	0.6
		45
		90
		135
		180

. The specific layout of the fender system is illustrated in Figure 16 and detailed in

Table 7. Fender setting for bow entry condition.

Fender Number	Jacket X Coordinate [m]	Jacket Y Coordinate [m]	Jacket Z Coordinate [m]	Barge X Coordinate [m]	Barge Y Coordinate [m]	Barge Z Coordinate [m]
Swaying fender 1	−31	−16.6	0	−31	−16	0
Swaying fender 2	−31	16.6	0	−31	16	0
Swaying fender 3	0	−16.6	0	0	−16	0
Swaying fender 4	0	16.6	0	0	16	0
Swaying fender 5	31	−16.6	0	31	−16	0
Swaying fender 6	31	16.6	0	31	16	0
Swaying fender 7	−31.5	−25.5	0	−32	−25.5	0
Swaying fender 8	−31.5	25.5	0	−32	25.5	0

, with fender coordinates provided in Appendix A.

Due to the differing loaded positions of the upper block, the mass parameters of the barge vary between the bow-entry and stern-entry conditions. Wind loads on the upper block and current loads on the barge are considered, with the corresponding coefficients determined according to the China Classification Society (CCS) guidelines [33]. When the upper block is positioned forward or aft, the resulting wind load induces a yaw moment on the barge–block system, introducing operational challenges. This effect constitutes one of the primary distinctions between bow-entry and stern-entry procedures.

The entry phase simulation begins with the barge preparing to enter the jacket and concludes when the barge is fully inserted and the Leg Mating Unit (LMU) tip aligns with the receiver. The total entry distance is set to 80 m for both bow and stern entry, with a constant entry speed of 0.02 m/s. The total simulation time is 5000 s, including a 600 s system initialization period and 4000 s of dynamic entry simulation. Only surge motion is permitted during this process. The time history of the x-coordinate of the prescribed entry path is shown in Figure 17.

3.3.2. Time-Domain Simulation Results of Ship Entry Conditions

The time history curve of the error between the surge direction and the target path in the process of entering the ship under the two entry conditions is shown in Figure 18. It can be seen from the diagram that, when waves are incident from 0°, 90° and 180°, the error between the actual position and the set path of the barge in the surge direction under the conditions of bow entry and stern entry fluctuates around 0. Under the action of stern oblique waves (45°) and bow oblique waves (135°), the error between the actual position and the set path of the barge in the surge direction is stable at a non-zero value. Under the action of bow oblique waves, the error between the actual position and the path of the barge in the two working conditions is stable at a negative value, and the error under the action of stern oblique wave is stable at a positive value.

In order to measure the fluctuation of the deviation between the actual position of the barge in the dynamic path and the set path, the root mean squared error (RMSE) is introduced. The statistical values of the deviation radius of the actual position of the barge relative to the set path and the deviation of the heading angle under two working conditions are given in Figure 19. It can be seen from Figure 19a that the maximum offset radius of the barge in the process of entering the ship is much larger than the other four wave directions in the 90° wave direction. In 0°, 45°, 90° and 180° waves, the offset radius of the stern is larger than that of the bow, and only the offset radius of 135° waves is smaller. In contrast, as shown in Figure 19b, the RMSE of the offset radius is greatest under 90° wave incidence, followed by 45° and 135° directions, and smallest under 0° and 180° directions. No significant differences are observed in the RMSE among the 0°, 90°, and 180° wave conditions. At 45°, the offset radius RMSE of the stern entry condition is larger than that of the bow entry condition. On the contrary, at 135°, the offset radius RMSE of the stern entry condition is smaller than that of the bow entry condition.

On the other hand, in Figure 19c, it can be seen that the maximum heading angle of the ship’s bow entry is largest under the 45° wave direction, followed by 90° and 135° again, and there is almost no response under the 0° and 180° wave directions. When the ship enters the stern, the 90° wave direction is the largest, followed by 45° and 135° again. In the 90° wave direction, the maximum amplitude of the heading angle when the stern enters the ship is larger than that when the bow enters the ship, while in the 45° and 135° wave directions, it is the opposite, and it is smaller when the stern enters the ship. The RMSE of the heading angle is shown in Figure 19d. Under both conditions, the RMSE of the 45° wave direction is the largest, and it is much larger than under other wave directions and is followed by the 90° wave direction, 135° again, and the 0° and 180° wave directions, which are the smallest. The heading angle RMSE when the bow enters the ship at 45° and 90° are larger than those when the stern enters the ship, and vice versa, at 135°.

When the ship enters the ship’s bow, the wind force acting on the platform is on the side of the bow, 33.4 m away from the barge’s center of gravity. When the ship enters the ship’s stern, the wind force acting on the platform is on the side of the stern, 86.4 m away from the barge’s center of gravity. However, the size of the platform is larger under the condition of bow entry, so there is a larger wind area. Therefore, the yaw moment transmitted by the upper block to the barge will change with the wind direction, which causes the statistical value of the barge’s heading angle and offset radius to be high and low under the two working conditions.

4. Conclusions

In this study, an Observer-Based Robust Control (OBRC) strategy integrated with a constant parameter time-domain model (CPTDM) was proposed to enhance the dynamic positioning performance during float-over installation of offshore converter stations. Through time-domain numerical simulations, a systematic comparison was performed between the proposed OBRC and a conventional PID controller under various environmental conditions. The dynamic responses of the T-U barge during both the standby phase and the entry phase were thoroughly investigated. However, the present study has some limitations. The control strategy focuses on the OBRC approach, and comparative analyses with other advanced control methods, such as adaptive control or different robust architectures, are not included. Furthermore, the implementation challenges related to integrating the OBRC into existing dynamic positioning systems warrant further practical investigation.

The OBRC strategy provides strong robustness guarantees against model uncertainties and time-varying disturbances through its Lyapunov-based design, while avoiding the high online computational burden typically associated with nonlinear model predictive control (MPC). The results demonstrate that the OBRC achieves a favorable balance between performance, robustness and computational efficiency, rendering it a practically viable advanced control solution for float-over installation, especially when predicting all complex interaction dynamics for MPC is challenging.

A summary of the conclusions is given below:

1. Numerical simulations conducted in MATLAB/Simulink incorporating the dynamic positioning system within a closed-loop OBRC-CPTDM framework show that the OBRC exhibits superior dynamic positioning capability compared to the conventional PID controller. The OBRC maintains high positioning accuracy even under the coupled influences of wind, wave and current loads, whereas the PID controller results in persistent oscillations around the target point.

2. Evaluation of the dynamic response of the T-U barge during the standby phase indicates that, under specific wave conditions, particularly when the peak wave period Tp = 10 s, hazardous motion responses occur. For example, under 90° beam seas, the maximum heave acceleration reaches 0.8 m/s², and under 45° stern-quartering seas, the maximum pitch amplitude reaches 1.9°, which identifies these as critical wave directions.

Analysis during the entry phase comparing bow entry and stern entry methods reveals that the dynamic response characteristics are influenced by the loaded position of the platform, which affects the wind-induced yaw moment. Neither entry method is absolutely superior; however, both bow entry and stern entry operations encounter the greatest challenges—manifested as the largest offset radii and heading errors—under quartering (45° and 135°) and beam (90°) sea conditions, which is consistent with findings from the standby phase.

Despite its advantages, the practical implementation of the OBRC on offshore vessels faces several challenges. The controller’s performance is highly dependent on the accuracy of the vessel’s hydrodynamic model and the precisely identified parameters for the observer and control gains, which may require extensive system identification efforts. Additionally, although computationally more efficient than nonlinear MPC, the OBRC still imposes a higher computational load than a standard PID controller due to the state and disturbance observer calculations, necessitating more powerful processing hardware for real-time execution and potentially increasing the cost and complexity of retrofitting existing DP systems. Finally, integration of such an advanced controller into the layered architecture of a commercial DP system demands significant engineering effort and validation.

Future research should focus on several key areas:

(1)

Implementing hardware-in-the-loop (HIL) tests to validate the controller’s performance with real-time thruster dynamics and system delays.

(2)

Investigating fault-tolerant control strategies to handle thruster failures, which are critical for operational safety.

(3)

Extending the coupled model to include the nonlinear contact dynamics during the final mating phase, enabling a holistic simulation of the entire installation process.

The insights and framework provided by this study pave the way for more robust and intelligent DP solutions, ultimately enhancing the safety and efficiency of installing critical offshore renewable energy infrastructure.