Simplified Model Characterization and Control of an Unmanned Surface Vehicle

Abstract

This study presents the modeling and control of the unmanned surface vehicle (USV) SABALO. Two models were built, one based on a transfer function matrix and another based on state variables, and from these models, two control strategies were developed. The first strategy is based on independent Proportional-Integral/Proportional-Derivative (PI/PD) controllers complemented by a decoupling system, and the second strategy is based on state variable feedback. The two control strategies were evaluated and contrasted. Results demonstrated that the decoupler effectively eliminated variable interaction, enhancing stability in straight trajectories and directional changes. Meanwhile, state feedback control demonstrated markedly faster response times and superior precision, accompanied by higher energy consumption. The study concludes that both strategies are effective, but their suitability depends on the mission. The decoupler could be ideal for energy-efficient, long-duration operations, while state feedback could be appropriate for dynamic environments requiring rapid maneuvers.

1. Introduction

An Unmanned Surface Vehicle (USV) is an autonomous craft that operates on the surface of a body of water without any personnel onboard. They can be remotely operated by an operator positioned on land or another vessel. This capability ensures a primary line of defense and facilitates the inspection of other ships, all while keeping manned boats at a safe distance to minimize unnecessary risks. The evolution of USVs since their inception in the 1960s has significantly advanced maritime operations. Initially, USVs were designed to perform basic remote-controlled tasks; however, with the integration of cutting-edge technologies, they can perform military and scientific research missions, such as monitoring marine/river areas, conducting oceanographic measurements, measuring river water flow, mapping, and detecting riverbank lines [1,2,3,4,5]. As their applications have diversified and become more complex, the need for accurate modeling and control strategies has grown significantly.

Accurate dynamic modeling and parameter identification have become essential for enhancing the precision and robustness of USV control systems. A number of approaches have been proposed to enhance these aspects. For instance, Abrougui et al. developed a comprehensive mathematical model with three degrees of freedom (DOF), achieving effective parameter identification through sea trials and significantly improving the autopilot’s accuracy for waypoint tracking and route following [6]. Setiawan et al. presented a detailed dynamic modeling approach for USVs employing the nonlinear Levenberg–Marquardt method, which facilitated reliable parameter estimation validated through real-time experiments [7]. Furthermore, Zhang and Ren [8] introduced a non-parametric multi-output Gaussian process (MOGP) learning method, capable of effectively capturing complex nonlinear interactions across multiple DOFs, thus providing enhanced robustness and computational efficiency. In a similar vein, Xu et al. [9] employed a combination of least square support vector machines (LS-SVMs) and a cuckoo search optimization algorithm to achieve accurate parameter identification, underscoring the efficacy of this approach in comparison to traditional methods. Moreover, Zhong et al. [10] employed a nonlinear multi-innovation least-squares (NMILS) algorithm, further improving the accuracy of parameter identification and ensuring robust control performance through finite-time sliding mode control (FTSMC), validating their methods in simulation environments. Ma et al. [11] proposed a multi-model predictive control strategy to handle path-following tasks under wide-range speed variations, enhancing control stability and adaptability in dynamic marine environments. These methodologies significantly contribute to the field; however, their application in real life is limited due to the complexity of some of the proposed models.

In addition, regardless of their purposes or missions, USVs require an effective maneuvering controller to ensure successful operation throughout their missions. Many works have been developed focusing on USV control, as shown in Table 1. The majority of these strategies rely on complex methodologies, including predictive control, adaptive control, artificial intelligence (AI), deep learning (DL), and nonlinear control strategies [12,13,14]. In recent years, machine learning techniques, including deep reinforcement learning (DRL) and neural network-based models, have become increasingly prevalent [13]. These techniques have demonstrated remarkable adaptability and self-learning capabilities in complex and dynamic environments. For instance, Zhao et al. [8] proposed a DRL-based path-following method, which showed improved convergence but required high computational resources to handle continuous action spaces. Similarly, Wang et al. [12] developed an optimal path-following controller for USVs using reinforcement learning, but its reliance on extensive training data limited its real-world adaptability.

The recent literature has also emphasized the importance of the accurate modeling and control of USVs. Several studies have proposed a novel cost control method guaranteed by network-based modeling and sampling for USVs subjected to stochastic cyber-attacks, addressing the robustness and performance degradation problems caused by network instabilities [15]. Furthermore, other research has introduced a hybrid adaptive dynamic programming approach for the optimal control of USV tracking, which has been shown to significantly improve trajectory tracking performance and minimize energy consumption [16]. Furthermore, an adaptive formation control strategy was proposed for USV obstacle avoidance with asymmetric input saturation, demonstrating improved maneuverability and operational safety in complex marine environments [17]. Furthermore, other authors have presented a multi-model predictive control strategy for USV trajectory tracking over various velocity variations. This approach demonstrated significant robustness and adaptability to dynamic marine conditions, effectively maintaining trajectory accuracy despite wide speed variations [11]. These novel approaches reflect the growing trend of integrating advanced modeling techniques with robust control strategies to address the challenges posed by environmental disturbances, cyber-attacks, and operational uncertainties.

Nevertheless, despite the advantages of machine learning-based methods, they present inherent limitations that hinder their practical applicability in real-time marine operations. A primary challenge is the necessity of substantial volumes of high-quality data for practical model training and generalization. Additionally, the parameter tuning process can be computationally expensive and sensitive to variations in environmental conditions, leading to performance degradation when encountering unfamiliar scenarios. Furthermore, the interpretability of neural network-based models is a matter of concern, as it complicates the implementation of safety-critical applications, where transparency and predictability are essential [11,15,16].

Thus, accurate modeling integrated with robust control strategies is essential to achieving reliable and efficient USV operations. The combination of modeling and control facilitates trajectory optimization and addresses environmental disturbances and operational stability challenges. Although these strategies improve trajectory-tracking accuracy, they often suffer from high implementation complexity, which makes them impractical for real-time marine operations. In contrast, our approach optimizes USV trajectory tracking by leveraging a more computationally efficient strategy while maintaining adaptability to environmental disturbances. Our method balances accuracy and real-time feasibility by reducing the dependency on extensive offline training and computationally intensive deep learning architectures, ensuring robust performance without incurring excessive computational costs. To achieve this, the proposed work develops control strategies based on simple models that effectively capture the dynamic behavior of USVs. Two models with two inputs and two outputs were constructed as follows: one based on a transfer function matrix and another using state variables. From these, two control strategies were implemented. The first utilizes independent Proportional-Integral/Proportional-Derivative (PI/PD) controllers with a decoupling system to mitigate possible inter-loop interactions. The second strategy employs state feedback control with an integrated state estimator to address sensor limitations and enhance overall system observability.

The paper is organized as follows: Section Table 1 contextualizes the study within the existing body of knowledge. Section 2 provides detailed insights into the USV’s design and hardware specifications. Section 3 details aspects such as the input and output variables of the system, the experiments carried out to generate the data, the identified models, and the results of the validations of such models. Section 4 evaluates two control strategies, an independent loop control with a decoupler and state feedback control, supported by performance metrics and simulations. Finally, the investigation concludes with Section 5 and Section 6, summarizing the findings and highlighting the most significant outcomes.

Table 1. Summary of related works on USV modeling and control strategies.

Title	Authors	Methods	Results	Contributions	Advantages and Disadvantages
Adaptive Control Scheme for USV Trajectory-Tracking Under Complex Environmental Disturbances via Deep Reinforcement Learning	Yuan Zhou et al. [13]	Deep Reinforcement Learning (TD3) for trajectory tracking	Improved tracking under dynamic conditions; enhanced learning stability	Robust DRL-based control for USVs, superior to MPC	A: High adaptability to disturbances; D: High computational cost
Network-Based Modeling and Sampling Guaranteed Cost Control for Unmanned Surface Vehicle Systems Under Stochastic Cyber-Attacks	Kui Ding and Quanxin Zhu [15]	DStochastic composite system with Lyapunov functions	Stabilizes USVs under cyber-attacks with controlled cost	Introduces a cost-effective control method for cyber-attack scenarios	A: Ensures system stability under uncertain conditions; D: Relies on precise system modeling
Multi-Model Predictive Control Strategy for Path-Following of Unmanned Surface Vehicles in Wide-Range Speed Variations	Yingkai Ma et al. [11]	Multi-model predictive control (MPC) with LMI-based robust control and adaptive LOS guidance	Improved path-following accuracy and robustness under speed variations	Enhanced USV maneuverability and control efficiency in dynamic speed conditions	A: Higher efficiency in dynamic speed conditions; D: Computational complexity and need for precise models
Investigation of Wind Effects on UAV Adaptive PID Based MPC Control System	A. S. Martinez Leon et al. [18]	Adaptive PID-based MPC control strategy	Improved UAV stabilization and positioning accuracy in the presence of wind disturbances	Enhanced remote surveillance missions along coastlines; insights into wind effects on UAV control	A: Effective for coastal surveillance missions; D: Limited to UAV applications, not directly applicable to USVs
Adaptive Formation Control for Obstacle Avoidance in USVs	Hu Yancai, Liu Yang, et al. [17]	RBF neural networks and APF method for formation control	Effective obstacle avoidance while maintaining formation	Provides robust formation control in dynamic environments	A: Handles complex obstacle avoidance; D: May struggle with numerous obstacles or extreme conditions
Safe Autonomy for Uncrewed Surface Vehicles Using Adaptive Control and Reachability Analysis	Karan Mahesh et al. [19]	Model Reference Adaptive Control (MRAC) with Moving Horizon Estimator (MHE) for real-time disturbance estimation	45–81% reduction in position error compared to PID; enhanced real-time safety certification	Improved USV stability in dynamic environments and robust real-time safety verification	A: Improved stability and safety verification in dynamic environments; D: High implementation complexity and computational requirements
Evolution of Algorithms and Applications for Unmanned Surface Vehicles in the Context of Small Craft: A Systematic Review	Luis Castano-Londono et al. [20]	Systematic literature review using PRISMA and bibliometric analysis	Identified 387 studies on USV applications and algorithm trends	Provided a comprehensive mapping of USV research, focusing on small craft applications	A: Comprehensive mapping of USV research; D: Does not propose new methods, only a review
Hybrid Adaptive Dynamic Programming for Optimal Tracking Control of USVs	Shan Xue, Ning Zhao, et al. [16]	Integration of IRL and DED mechanisms	Reduced communication overhead and improved tracking	Reduces model dependency in tracking control	A: Efficient learning with minimal data transmission; D: Requires significant computational resources
A Model Predictive Control Approach for USV Autonomous Cruising via Disturbance Learning	Maotong Cheng et al. [14]	Learning-based MPC, augmented by LSTM residual model	Improved course-keeping under disturbances; LSTM eliminates model mismatch	Improved hydrodynamics modeling for USVs through MPC and LSTM	A: Enhanced hydrodynamic modeling; D: Dependency on historical data for LSTM training
Disturbance Estimation and Rejection in an Underwater Autonomous Vehicle	A. Thomas et al. [21]	Integral observer for disturbance estimation; PID for rejection	Accurate disturbance estimation and effective disturbance rejection in underwater environments	Effective disturbance estimation and rejection for underwater vehicles using integral observer	A: Effective disturbance handling; D: Limited to underwater vehicles, not directly applicable to USVs
A Comparison of Intelligent Models for Collision Avoidance Path Planning on Environmentally Propelled Unmanned Surface Vehicles	Carlos Barrera et al. [22]	AI-based path planning using ANN, SVM, Random Forest, and Multiple Logistic Regression	ANN achieved the best accuracy and lowest error in trajectory optimization	Demonstrated AI effectiveness in collision avoidance and compliance with COLREGs	A: Demonstrated effectiveness of AI in collision avoidance; D: Requires significant computational resources for training
DRL-Based Motion Control for Unmanned Surface Vehicles with Environmental Disturbances	Xiangyu Wu et al. [23]	TD3 algorithm (DRL) combined with PI controller	Improved stability and efficiency under wind and wave disturbances	Superior motion planning for USVs using TD3-PI under environmental conditions	A: Superior motion planning under environmental conditions; D: Requires extensive training and tuning
USV Course Control Strategy based on the Disturbance Observer Under Disturbances of Winds and Waves	Xiuren Yue [24]	Nonlinear disturbance observer; composite control strategy (feedforward/feedback)	Strong anti-disturbance ability, accurate course tracking under continuous disturbances	Improved anti-disturbance strategy and course tracking for USVs	A: Improved anti-disturbance strategy; D: Limited to specific disturbance types
Heading Control System Design for a Micro-USV Based on an Adaptive Expert S-PID Algorithm	Runlong Miao et al. [25]	Adaptive expert S-PID algorithm tested in pool and lake environments using STM32-ARM and LabWindows/CVI	Achieved 2–3° heading error vs. 5–6° with traditional PID, with improved stability and responsiveness	Robust framework enhancing precision and reliability in micro-USV control	A: Robust framework for micro-USV control; D: Limited scalability to larger USVs

Table 1. Summary of related works on USV modeling and control strategies.

Title	Authors	Methods	Results	Contributions	Advantages and Disadvantages
Adaptive Control Scheme for USV Trajectory-Tracking Under Complex Environmental Disturbances via Deep Reinforcement Learning	Yuan Zhou et al. [13]	Deep Reinforcement Learning (TD3) for trajectory tracking	Improved tracking under dynamic conditions; enhanced learning stability	Robust DRL-based control for USVs, superior to MPC	A: High adaptability to disturbances; D: High computational cost
Network-Based Modeling and Sampling Guaranteed Cost Control for Unmanned Surface Vehicle Systems Under Stochastic Cyber-Attacks	Kui Ding and Quanxin Zhu [15]	DStochastic composite system with Lyapunov functions	Stabilizes USVs under cyber-attacks with controlled cost	Introduces a cost-effective control method for cyber-attack scenarios	A: Ensures system stability under uncertain conditions; D: Relies on precise system modeling
Multi-Model Predictive Control Strategy for Path-Following of Unmanned Surface Vehicles in Wide-Range Speed Variations	Yingkai Ma et al. [11]	Multi-model predictive control (MPC) with LMI-based robust control and adaptive LOS guidance	Improved path-following accuracy and robustness under speed variations	Enhanced USV maneuverability and control efficiency in dynamic speed conditions	A: Higher efficiency in dynamic speed conditions; D: Computational complexity and need for precise models
Investigation of Wind Effects on UAV Adaptive PID Based MPC Control System	A. S. Martinez Leon et al. [18]	Adaptive PID-based MPC control strategy	Improved UAV stabilization and positioning accuracy in the presence of wind disturbances	Enhanced remote surveillance missions along coastlines; insights into wind effects on UAV control	A: Effective for coastal surveillance missions; D: Limited to UAV applications, not directly applicable to USVs
Adaptive Formation Control for Obstacle Avoidance in USVs	Hu Yancai, Liu Yang, et al. [17]	RBF neural networks and APF method for formation control	Effective obstacle avoidance while maintaining formation	Provides robust formation control in dynamic environments	A: Handles complex obstacle avoidance; D: May struggle with numerous obstacles or extreme conditions
Safe Autonomy for Uncrewed Surface Vehicles Using Adaptive Control and Reachability Analysis	Karan Mahesh et al. [19]	Model Reference Adaptive Control (MRAC) with Moving Horizon Estimator (MHE) for real-time disturbance estimation	45–81% reduction in position error compared to PID; enhanced real-time safety certification	Improved USV stability in dynamic environments and robust real-time safety verification	A: Improved stability and safety verification in dynamic environments; D: High implementation complexity and computational requirements
Evolution of Algorithms and Applications for Unmanned Surface Vehicles in the Context of Small Craft: A Systematic Review	Luis Castano-Londono et al. [20]	Systematic literature review using PRISMA and bibliometric analysis	Identified 387 studies on USV applications and algorithm trends	Provided a comprehensive mapping of USV research, focusing on small craft applications	A: Comprehensive mapping of USV research; D: Does not propose new methods, only a review
Hybrid Adaptive Dynamic Programming for Optimal Tracking Control of USVs	Shan Xue, Ning Zhao, et al. [16]	Integration of IRL and DED mechanisms	Reduced communication overhead and improved tracking	Reduces model dependency in tracking control	A: Efficient learning with minimal data transmission; D: Requires significant computational resources
A Model Predictive Control Approach for USV Autonomous Cruising via Disturbance Learning	Maotong Cheng et al. [14]	Learning-based MPC, augmented by LSTM residual model	Improved course-keeping under disturbances; LSTM eliminates model mismatch	Improved hydrodynamics modeling for USVs through MPC and LSTM	A: Enhanced hydrodynamic modeling; D: Dependency on historical data for LSTM training
Disturbance Estimation and Rejection in an Underwater Autonomous Vehicle	A. Thomas et al. [21]	Integral observer for disturbance estimation; PID for rejection	Accurate disturbance estimation and effective disturbance rejection in underwater environments	Effective disturbance estimation and rejection for underwater vehicles using integral observer	A: Effective disturbance handling; D: Limited to underwater vehicles, not directly applicable to USVs
A Comparison of Intelligent Models for Collision Avoidance Path Planning on Environmentally Propelled Unmanned Surface Vehicles	Carlos Barrera et al. [22]	AI-based path planning using ANN, SVM, Random Forest, and Multiple Logistic Regression	ANN achieved the best accuracy and lowest error in trajectory optimization	Demonstrated AI effectiveness in collision avoidance and compliance with COLREGs	A: Demonstrated effectiveness of AI in collision avoidance; D: Requires significant computational resources for training
DRL-Based Motion Control for Unmanned Surface Vehicles with Environmental Disturbances	Xiangyu Wu et al. [23]	TD3 algorithm (DRL) combined with PI controller	Improved stability and efficiency under wind and wave disturbances	Superior motion planning for USVs using TD3-PI under environmental conditions	A: Superior motion planning under environmental conditions; D: Requires extensive training and tuning
USV Course Control Strategy based on the Disturbance Observer Under Disturbances of Winds and Waves	Xiuren Yue [24]	Nonlinear disturbance observer; composite control strategy (feedforward/feedback)	Strong anti-disturbance ability, accurate course tracking under continuous disturbances	Improved anti-disturbance strategy and course tracking for USVs	A: Improved anti-disturbance strategy; D: Limited to specific disturbance types
Heading Control System Design for a Micro-USV Based on an Adaptive Expert S-PID Algorithm	Runlong Miao et al. [25]	Adaptive expert S-PID algorithm tested in pool and lake environments using STM32-ARM and LabWindows/CVI	Achieved 2–3° heading error vs. 5–6° with traditional PID, with improved stability and responsiveness	Robust framework enhancing precision and reliability in micro-USV control	A: Robust framework for micro-USV control; D: Limited scalability to larger USVs

2. Experimental Platform

The experimental platform used in this study is a prototype USV called SABALO, built by the IN-NOVA Group for the Colombian Navy, shown in Figure 1. The “Almirante Padilla” Naval Cadet School developed the control system. The USV is a monohull platform equipped with the following:

• High-performance embedded controller, CompactRIO-9075.
• Global Position System (GPS) module, VK2828U7G5LF.
• Inertial Measurement Unit (IMU), UM7-LT.
• Ultrasonic speed sensor, CS4500.
• Electric water propeller thrust, a TP100 Inrunner brushless motor.
• Steering nozzle, HS-7954.
• Telemetry systems, based on an Xbee-PRO SX module.

The propeller thrust is driven by an electric motor connected to a power driver. The power the motor delivers is given by a PWM (Pulse Width Modulation) signal with a period of 10 ms. The pulse width varies from 0% to 100%, with an accuracy of 0.1%, so the engine delivers from 0% to 100% of the power. The thrust angle is driven by a servo motor connected to the nozzle at the thruster outlet. The servomotor angle is given by a PWM signal with a period of 10ms and pulse width that varies from 1 ms to 2 ms, with an accuracy of 1 μs. This leads to a nozzle angle that ranges from −45º to 45º. The GPS and IMU signals give the course and speed measurements. These sensors deliver data via RS232 serial port at a rate of 115,200 bps every 100 ms in NMEA (National Marine Electronics Association) format, with a resolution of less than 0.5º for heading and less than 0.1 m/s for speed. The speed delivered by the ultrasonic sensor is compared with the speed offered by the GPS to estimate the speed of the water current in which the boat is sailing. The physical characteristics of the USV are summarized in

Table 2. USV parameters.

Parameter	Value
Length	2462 mm
Beam	896 mm
Draft	15 mm
Weight	117 kg
Speed	9 kn

.

3. Unmanned Surface Vehicle Models

From the perspective of control system design, the USV SABALO is a 2 × 2 system. This includes two inputs or manipulated variables and two outputs or controlled variables, as is shown in

Table 3. System inputs and outputs variables.

Type	Variable	Symbol	Range
Input	Propeller Thrust	$m_1$	0∼100 %
	Propeller Angle	$m_2$	−45°∼45°
Output	USV Speed	$c_1$	0∼9 kn
	USV Heading	$c_2$	0°∼360°

.

3.1. System Inputs and Outputs

The system inputs are the jet thrust, $m_1$, and the propeller angle, $m_2$. The system outputs are the USV speed, $c_1$, and the USV heading, $c_2$. Table 3 also shows the ranges of variation of these variables. The propeller thrust is determined by a pulse-width modulation signal, which enables the engine driver to regulate the power output. The propeller angle is set by a servomotor, which adjusts the nozzle position at the thruster outlet, allowing for the precise direction of the propeller stream. The speed and heading of the USV are determined by GPS and inertial measurement unit (IMU) measurements. These measurements are delivered digitally at a frequency of 10 Hz.

Therefore, it is possible to build a model composed of four transfer functions $G_{ij}\left(s\right)$, relating the ith controlled variable with the jth manipulated variable, as shown in Figure 2.$G_{11}\left(s\right)$ and $G_{12}\left(s\right)$ represent the effects of propeller thrust and propeller angle, respectively, on USV speed, while $G_{21}\left(s\right)$ and $G_{22}\left(s\right)$ represent the effects of propeller thrust and propeller angle, respectively, on USV Heading.

3.2. Experimental Data Generation

Generating the data needed to build the vessel model consists of two stages. First, a constant thrust force is applied through the propeller ($m_1$) and a steady speed is reached; Figure 1b. Once the USV speed has stabilized, a yaw torque is applied by changing the propeller angle ($m_2$), making the vessel spin in circles, Figure 1c. It is worth mentioning that the tests were carried out in a 25 m × 50 m pool. Figure 3 shows the vessel’s behavior in the described maneuver.

3.3. USV Proposed Models

3.3.1. Transfer Function Model

Note that transfer function $G_{21}\left(s\right)=0$ indicates that $m_1$ does not affect $c_2$. The above can be seen in Figure 3, where it can be observed that the step change in $m_1$ does not cause any effect on $c_2$. It is only until a change is induced in $m_2$ that $c_2$ begins to change. The effects of $m_1$ and $m_2$ on $c_1$, $G_{11}\left(s\right)$ and $G_{12}\left(s\right)$, respectively, can be approximated by a First Order Plus Dead Time (FOPDT) model such as that in (1), where K, $\tau$, and $t_0$ are the process’ gain, constant time, and dead time, respectively.

$$G\left(s\right)={\displaystyle \frac{K{e}^{-t_{0}s}}{\tau s+1}}$$

(1)

The model in (1) is an important transfer function used to approximate the response of higher-order processes. The so-called Fit3 method presented in [26] (see Figure 4(Left)) can be used to calculate the parameters of a FOPDT model from the step response of the process as follows.

$$K={\displaystyle \frac{\Delta c_s}{\Delta m}},\tau =1.5(t_2-t_1),t_0=t_2-\tau$$

(2)

On the other hand, note the integrating behavior of $c_2$—in this case, an Integrating First Order Plus Dead Time (IFOPDT) model (see (3))—can be used to approximate the effects of $m_2$ on $c_2$, that is $G_{22}\left(s\right)$.

$$G\left(s\right)={\displaystyle \frac{K{e}^{-t_{0}s}}{s(\tau s+1)}}$$

(3)

The methodology proposed in [27] (see Figure 4(Right)) can be used for the characterization of the integrating processes by solving the equation system composed of (2) and (4) (given $t_1^f$, $t_2^f$ and ${\tau }_f=1.5(t_2^f-t_1^f)$).

$$\begin{array}{cc}\hfill -e^{{\displaystyle \frac{-(t_1-t_0)}{\tau }}}& =-{\displaystyle \frac{\tau e^{\frac{-(t_1^f-t_0)}{\tau }}}{\tau -{\tau }_f}}+{\displaystyle \frac{{\tau }_f{e}^{\frac{-(t_1^f-t_0)}{{\tau }_f}}}{\tau -{\tau }_f}}\hfill \\ \hfill -e^{{\displaystyle \frac{-(t_2-t_0)}{\tau }}}& =-{\displaystyle \frac{\tau e^{\frac{-(t_2^f-t_0)}{\tau }}}{\tau -{\tau }_f}}+{\displaystyle \frac{{\tau }_f{e}^{\frac{-(t_2^f-t_0)}{{\tau }_f}}}{\tau -{\tau }_f}}\hfill \end{array}$$

(4)

The transfer function matrix model obtained is shown in (5).

$$\begin{array}{c}\hfill \left[\begin{array}{c}{c}_1\\ c_2\end{array}\right]=\left[\begin{array}{cc}{G}_{11}\left(s\right)& G_{12}\left(s\right)\\ G_{21}\left(s\right)& G_{22}\left(s\right)\end{array}\right]\left[\begin{array}{c}{m}_1\\ m_2\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{K_{11}{e}^{-t_{0_{11}}s}}{{\tau }_{11}s+1}}& {\displaystyle \frac{K_{12}{e}^{-t_{0_{12}}s}}{{\tau }_{12}s+1}}\\ {\displaystyle \frac{K_{21}{e}^{-t_{0_{21}}s}}{{\tau }_{21}s+1}}& {\displaystyle \frac{K_{22}{e}^{-t_{0_{22}}s}}{s({\tau }_{22}s+1)}}\end{array}\right]\left[\begin{array}{c}{m}_1\\ m_2\end{array}\right]\\ \hfill =\left[\begin{array}{cc}{\displaystyle \frac{0.07254e^{-1.5s}}{3s+1}}& {\displaystyle \frac{-0.06431e^{-3s}}{2.16s+1}}\\ 0& {\displaystyle \frac{0.7e^{-s}}{s(s+1)}}\end{array}\right]\left[\begin{array}{c}{m}_1\\ m_2\end{array}\right]\end{array}$$

(5)

Figure 5 compares the actual and the model responses. The $R^2$ statistic obtained was $0.9778$ and $0.9992$ for $c_1$ and $c_2$, respectively. The transfer function matrix model approximates the dynamic response of the actual system quite well, with maximum estimation errors of $1.56\%$ and $0.13\%$ for speed and heading, respectively.

3.3.2. State Space Model

The state of a dynamical system is the smallest set of variables (called state variables) such that knowledge of these variables at $t=t_0$, together with knowledge of the input to $t\ge t_0$, completely determines the system’s behavior at any $t\ge t_0$. State-space models use state variables to describe a system using a set of first-order differential Equations (6),

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =Ax\left(t\right)+Bu\left(t\right)\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right)\hfill \end{array}$$

(6)

where $x\left(t\right)$ is the state vector, $u\left(t\right)$ is the input vector, $y\left(t\right)$ is the output vector, A is the state matrix, B is the input matrix, and C is the output matrix.

A state-space model was obtained for the USV using the N4SID algorithm (Numerical Subspace State Space System Identification). The N4SID algorithm is used to identify state-space models of dynamic systems from input–output data [28]. Some of the advantages of this algorithm are the following [29]:

• No iterative optimization: N4SID is non-iterative and therefore fast.
• Numerically robust: Uses Singular Value Decomposition, which is stable and reliable.
• Works well with noise: Especially with short or noisy data records.

The steps followed in N4SID are explained in [30]. Below is a summary of them.

• Step 1: Construct block Hankel matrices. For a chosen block size i, construct the block Hankel matrices from the data as follows:

  $$U_p=\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_j\\ u_2& u_3& \cdots & u_{j+1}\\ ⋮& ⋮& \ddots & ⋮\\ u_i& u_{i+1}& \cdots & u_{j+i-1}\end{array}\right],Y_p=\left[\begin{array}{cccc}{y}_1& y_2& \cdots & y_j\\ y_2& y_3& \cdots & y_{j+1}\\ ⋮& ⋮& \ddots & ⋮\\ y_i& y_{i+1}& \cdots & y_{j+i-1}\end{array}\right]$$

  (7)
  Define the future input–output Hankel matrices $U_f,Y_f$. The number of columns $j=N-2i+1$.
• Step 2: Toeplitz matrix. Compute the oblique projection of future outputs $Y_f$ onto the row space of past data $[U_p;Y_p]$, orthogonal to the space covered by $U_f$, as follows:

  $$\mathcal{P}={\mathrm{Proj}}_{[U_p;Y_p]\perp U_f}\left(Y_f\right)$$

  (8)
• Step 3: Singular Value Decomposition (SVD). Apply SVD to the projection matrix:

  $$\mathcal{P}=U\Sigma V^{\top }$$

  (9)
  The system order n is determined by inspecting the singular values in $\Sigma$. Retain the dominant n components as follows:

  $${\Sigma }_1=\mathrm{diag}({\sigma }_1,\cdots ,{\sigma }_n),U_1=U(:,1:n)$$

  (10)
  Then, estimate the extended observability matrix:

  $${\mathcal{O}}_i\approx U_1{\Sigma }_1^{1/2}$$

  (11)
• Step 4: Estimate the state sequence. Estimate the state sequence from the following:

  $${\widehat{X}}_f={\Sigma }_1^{1/2}{V}_1^{\top }$$

  (12)

  where $V_1=V(:,1:n)$.
• Step 5: Estimate the system matrices. With ${\widehat{X}}_k$ estimated, solve the following equations via least squares:

  $$\begin{array}{c}\hfill {\widehat{X}}_{k+1}=A{\widehat{X}}_k+B{u}_k\\ \hfill y_k=C{\widehat{X}}_k+D{u}_k\end{array}$$

  (13)
  This yields the state-space matrices $A,B,C,$ and D.

In the USV Sabalo case, the obtained state-space model was $x\left(t\right)={\left[\begin{array}{cc}{x}_1\left(t\right)& x_2\left(t\right)\end{array}\right]}^T$, $u\left(t\right)={\left[\begin{array}{cc}{m}_1\left(t\right)& m_2\left(t\right)\end{array}\right]}^T$, $y\left(t\right)={\left[\begin{array}{cc}{c}_1\left(t\right)& c_2\left(t\right)\end{array}\right]}^T$, and

$$\begin{array}{cc}\hfill A& =\left[\begin{array}{cc}1.2426\times {10}^{-3}& 3.3967\times {10}^{-3}\\ -1.0803\times {10}^{-1}& -2.1629\times {10}^{-1}\end{array}\right]\hfill \\ \hfill B& =\left[\begin{array}{cc}-1.7388\times {10}^{-5}& -6.7773\times {10}^{-5}\\ 1.3788\times {10}^{-3}& -1.3431\times {10}^{-3}\end{array}\right]\hfill \\ \hfill C& =\left[\begin{array}{cc}4.0251& 11.3239\\ -8540.2908& 4.7127\end{array}\right].\hfill \end{array}$$

(14)

Therefore, we have two state variables ($n=2$), two input variables ($q=2$), and two output variables ($p=2$). Note that the state variables $x_1\left(t\right)$ and $x_2\left(t\right)$ are abstract; therefore, they cannot be quantified directly. Figure 6 compares the actual and the state space model responses. The statistic $R^2$ obtained was $0.9271$ and $0.9985$ for $c_1$ and $c_2$, respectively. The state-space model approximates the dynamic response of the actual system quite well, with maximum estimation errors of $18.7\%$ and $0.3\%$ for speed and heading, respectively.

4. Unmanned Surface Vehicle Control

Two control strategies were implemented for the vessel as follows: one based on two independent control loops with decoupler blocks, and the other on state variable feedback with a state estimator. Both alternatives are presented below.

4.1. Control Strategy 1—Independent Control Loops with Decoupling of Interactions

Figure 7 presents the block diagram for the proposed strategy. The decoupled block $D_{12}\left(s\right)$ is installed to cancel the effect of $m_2$ on $c_1$.

$G_{c1}\left(s\right)$ is PI controller tuned with the lambda tuning method [31], while $G_{c2}\left(s\right)$ is a PD controller tuned with the approach proposed in [27] for the integrating process. The parameters of each controller are presented in

Table 4. PI and PD controller parameters.

Controller	Parameter	Value
PI	$K_{c_1}={\displaystyle \frac{{\tau }_{11}}{2K_{11}{t}_{0_{11}}}}$	13.7861
	${\tau }_{i_1}={\tau }_{11}$	3
PD	$K_{c_2}={\displaystyle \frac{{\tau }_{22}}{2K_{22}{t}_{0_{22}}}}$	0.7143
	${\tau }_{d_2}={\tau }_{22}$	1

.

On the other hand, from the block diagram algebra, we can derive

$${\displaystyle \frac{C_1\left(s\right)}{M_2\left(s\right)}}=G_{11}\left(s\right)D_{12}\left(s\right)+G_{12}\left(s\right)=0.$$

(15)

By solving the decoupler transfer function from (15) and replacing $G_{11}\left(s\right)$ and $G_{12}\left(s\right)$ from (5), we obtain

$$D_{12}\left(s\right)=-{\displaystyle \frac{G_{12}\left(s\right)}{G_{11}\left(s\right)}}={\displaystyle \frac{(1.231s+0.4104)}{(s+0.463)}}{e}^{-1.5s}$$

(16)

The experimental tests were conducted in a controlled environment, as follows: a pool with calm water, a surface free from obstacles, and with the vessel floating horizontally. Setpoints for speed and heading were selected so the vessel remained well away from the pool’s edges to protect it from potential collisions. Figure 8 shows the response of control systems without a decoupler ($c_1$ and $c_2$) and with decoupler ($c_{d1}$ and $c_{d2}$). The systems are excited with an increase of 3 kn in $c_1^{set}$ and a subsequent 45° increase in the $c_2^{set}$, the setpoints for speed and heading, respectively.

In the case of the control system without a decoupler (blue continuous line), the interaction effect is observed in $c_1$ once the change in $c_2^{set}$ is generated, that is, the oscillation in the speed at t = 10 s. However, when the decoupler is included (red dashed line), it is observed that the interaction effect is wholly canceled, $c_{d1}$. It is essential to highlight that the limits of the manipulated variables are not exceeded.

4.2. Control Strategy 2—State Feedback Control

In this case, the state-space realization in (6) was used to develop a control strategy based on state feedback, as shown in Figure 9. The state feedback gain, $K_C$, was calculated as follows:

• Step 1. Check controllability: Construct the controllability matrix.

  $$\mathcal{C}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]$$

  (17)
  If $\mathrm{rank}\left(\mathcal{C}\right)=n$, the system is controllable, and you may proceed. Otherwise, full pole placement is not possible.
• Step 2. Calculate the characteristic polynomial of A.

  $$\Delta \left(s\right)=\det (sI-A)=s^n+{\alpha }_{n-1}{s}^{n-1}+\cdots +{\alpha }_{1}s+{\alpha }_0$$

  (18)
• Step 3. Choose the desired closed-loop poles: Select n desired eigenvalues $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\}$ based on the desired transient response and stability. Compute the desired characteristic polynomial.

  $$\begin{array}{c}\hfill {\Delta }_{\mathrm{des}}\left(s\right)=(s-{\lambda }_1)(s-{\lambda }_2)\cdots (s-{\lambda }_n)\\ \hfill {\Delta }_{\mathrm{des}}\left(s\right)=s^n+{\overline{\alpha }}_{n-1}{s}^{n-1}+\cdots +{\overline{\alpha }}_{1}s+{\overline{\alpha }}_0\end{array}$$

  (19)
• Step 4. Calculate the feedback gains for the equivalent controllable canonical form.

  $${\overline{K}}_C=\left[\begin{array}{cccc}{\overline{\alpha }}_{n-1}-{\alpha }_{n-1}& \cdots & {\overline{\alpha }}_1-{\alpha }_1& {\overline{\alpha }}_0-{\alpha }_0\end{array}\right]$$

  (20)
• Step 5. Calculate the equivalence transformation.

  $$S:=P^{-1}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]\left[\begin{array}{ccccc}1& {\alpha }_{n-1}& \cdots & {\alpha }_1& {\alpha }_0\\ 0& 1& {\alpha }_{n-1}& \cdots & {\alpha }_1\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 1& {\alpha }_{n-1}\\ 0& 0& 0& 0& 1\end{array}\right]$$

  (21)
• Step 6. Calculate feedback gains $K_C={\overline{K}}_{C}P={\overline{K}}_C{S}^{-1}$.

Since the state variables in the state-space model in (14) are abstract and cannot be directly quantified, it was necessary to integrate a state observer into the control strategy, as shown in Figure 9. The output $\widehat{x}\left(t\right)$ of the observer is governed by (22). Calculating the estimator gain, L, by transposing matrix A and replacing $C^{T\prime }$ with matrix B in the above procedure to calculate $K_C$ is possible.

$$\dot{\widehat{x}}\left(t\right)=(A-LC)\widehat{x}\left(t\right)+Bm\left(t\right)+Lc\left(t\right)$$

(22)

The state feedback control strategy in Figure 9 includes a pre-filter gain $K_F$ to guarantee that the controlled variable is commanded to track a setpoint asymptotically (setpoint tracking).

Let $c^{set}$ denote a desired constant value for the output $c\left(t\right)$. The control objective is to design a control law, which may depend on $x\left(t\right)$ and $c^{set}$, so that the closed-loop control system is stable and that the tracking error $e\left(t\right)=c^{set}-c\left(t\right)$ tends to zero as $t\to \infty$. Since $c^{set}\ne 0$, the steady-state value of $x\left(t\right)$, $x_{ss}$ cannot be 0. Assume that a control law has been chosen so that both $x\left(t\right)$ and $m\left(t\right)$ converge to a steady-state value as $t\to \infty$. Therefore, in a steady state, (6) can be written in matrix form as in (23).

$$\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]\left[\begin{array}{c}{x}_{ss}\\ m_{ss}\end{array}\right]=\left[\begin{array}{c}0\\ I_q\end{array}\right]c^{set}$$

(23)

where 0 is a zero matrix of size $n\times q$, $x_{ss}={lim}_{t\to \infty }x\left(t\right)$ and $m_{ss}={lim}_{t\to \infty }m\left(t\right)$ are steady-state values of the state and manipulated variables, respectively, with $I_q$ being a $q\times q$ identity matrix.

If $\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]$ is nonsingular, (23) can be solved uniquely for ${\left[\begin{array}{cc}{x}_{ss}& m_{ss}\end{array}\right]}^T$, as in (24).

$$\left[\begin{array}{c}{x}_{ss}\\ m_{ss}\end{array}\right]={\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ I_q\end{array}\right]c^{set}$$

(24)

Furthermore, by considering the state feedback control law $m=m_{ss}-K_C(x-x_{ss})$, which can be rewritten as follows

$$\begin{array}{cc}\hfill m& =m_{ss}-K_C(x-x_{ss})\hfill \\ \hfill & =\left[\begin{array}{cc}{K}_C& I_p\end{array}\right]\left[\begin{array}{c}{x}_{ss}\\ m_{ss}\end{array}\right]-K_{C}x\hfill \\ \hfill & =\left[\begin{array}{cc}{K}_C& I_q\end{array}\right]{\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ I_q\end{array}\right]c^{set}-K_{C}x\hfill \\ \hfill & =K_F{c}^{set}-K_{C}x\hfill \end{array}$$

(25)

we obtain that

$$K_F=\left[\begin{array}{cc}{K}_C& I_q\end{array}\right]{\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ I_q\end{array}\right].$$

(26)

It is worth mentioning that the previous result guarantees setpoint tracking for constant step reference inputs. For high-order reference inputs, integral actions must be included in the control strategy [32]. Equation (26) was used to calculated the $K_C$ in (27).

The gains of the state feedback $K_C$, state observer L, and pre-filter $K_F$ are shown in (27). Closed-loop poles were chosen at $-0.8$ and $-0.9$, and observer poles were chosen five times faster than closed-loop poles.

$$\begin{array}{cc}\hfill K_C& =\left[\begin{array}{cc}-2.8823\times {10}^4& 241.6017\\ -2.9511\times {10}^4& -112.1038\end{array}\right]\hfill \\ \hfill L& =\left[\begin{array}{cc}9.0898\times {10}^{-4}& -1.5\times {10}^{-3}\\ 0.2899& 1.4929\times {10}^{-4}\end{array}\right]\hfill \\ \hfill K_F& =\left[\begin{array}{cc}34.4597& 3.3822\\ -10.6382& 3.4507\end{array}\right]\hfill \end{array}$$

(27)

Experimental tests were conducted under the same conditions described above. Figure 10 shows the response of the state feedback control, where $c_{1_{SFB}}$ and $c_{2_{SFB}}$ are the controlled variables obtained with the state feedback control strategy. The system was excited with an increase of 3 kn in $c_1^{set}$ and a subsequent increase of 45° in $c_2^{set}$, as in the previous case.

The Integral Absolute Error (IAE) index, $I_{IAE}={\int }_0^{t_f}\left|e\left(t\right)\right|dt$, was used to evaluate the performance of the control strategies. The IAE serves as a performance metric in control systems, quantifying the cumulative magnitude of tracking errors over time without regard to their sign. Lower IAE values signify improved tracking accuracy. This metric is commonly used in controller optimization to ensure a smoother system response, minimizing control effort.

Table 5. Performance indices.

Control Strategy	IAE Index
1	${\mathrm{IAE}}_{c_1}$ = 12.39
	${\mathrm{IAE}}_{c_2}$ = 136.9
2	${\mathrm{IAE}}_{c_1}$ = 7.943
	${\mathrm{IAE}}_{c_2}$ = 37.91

shows the results obtained. Regarding control strategy 1, with control strategy 2, 35.89% and 72.31% reductions are obtained in the IAE indices for speed and heading, respectively.

5. Discussion

Transfer function matrix and state-space models were used to model the vessel. In both cases, the models’ responses, Figure 5 and Figure 6, are very close to the behavior of the actual system. Although the literature contains a wide range of complex models and parameter identification techniques applicable to USVs [8,15,33,34], whose results show satisfactory consistency between the predictions of the proposed models and the test data, the models proposed here show similar results in terms of estimation errors but stand out for their simplicity. The high value of the speed estimation error observed initially in the state-space model, $18.7\%$, is because the model induces an apparent effect of $m_1$ on $c_2$; however, estimation errors below $1.65\%$ are observed from that point onward. This initial peak estimation error does not occur in the model based on the transfer function because $G_{21}\left(s\right)=0$ is set.

Some advantages associated with the simplicity of the models used are as follows:

• Interpretability: Simple models are often more transparent and intuitive, making understanding the system’s behavior easier.
• Efficiency: Simpler models typically require fewer computations, making them ideal for real-time applications or iterative simulations.
• Lower hardware demands.
• Reduced data requirements: With fewer parameters, you need less experimental or observational data to calibrate the model.
• Lower risk of overfitting: Simple models are more robust, especially in noisy or limited datasets.
• Control-friendly: Many classic control strategies assume linear or low-order models.
• Observability and controllability: These properties are more straightforward to verify and maintain in simple systems.
• Closed-form solutions: Simpler models often allow for analytical solutions or approximations, revealing key insights into the dynamics.
• Sensitivity analysis: Easier to assess how parameters influence outcomes.
• Model verification: Simpler structures facilitate the identification of errors or inconsistencies in the model assumptions.
• Baseline comparison: Simple models provide a reference for testing the added value of complexity.
• Iterative improvement: Based on performance gaps, you can progressively add complexity as needed.

Regarding the performance of the control strategies, we can again state that simple control techniques show promising results. Figure 8 shows the benefits of including the decoupler block, achieving the complete elimination of the effect of the angle of the propeller ($m_2$) on the vessel’s speed ($c_1$). Generally, this strategy shows good response times without exceeding the limits of the manipulated variables (actuator saturation). Figure 10 allows us to compare the performance of both control strategies. In this case, better response times are observed with the state feedback strategy, reflected in the better performance indices (Table 5). However, as expected, the shorter response times imply greater energy consumption, reflected in the higher values in the manipulated variables. Additionally, the small effect of the propeller’s thrust ($m_1$) on the vessel’s heading ($c_2$) is observed, which does not occur in the previous strategy. However, the response speeds achieved make this strategy attractive for combat maneuvers.

Some advantages of classical control techniques, such as those used in this work, are as follows:

• Simple structure and intuitive understanding: Easy to implement with minimal code or analog components. Tuning can be performed using basic heuristics or a trial-and-error approach. Little specialist knowledge is needed; it is widely taught and used.
• Required system knowledge and modeling: It can often work with no explicit or straightforward model (e.g., gain and time constant). Design focuses on measured behavior (e.g., step response and frequency response).
• Very low computational load: It can run on inexpensive hardware or analog circuits. It is suitable for high sampling rates and fast dynamics.
• Real-world applicability: It is ideal for single-variable or loosely coupled systems, and these approaches are widely used in typical engineering applications due to their quick deployment and well-understood nature.

6. Conclusions

This work presents an unmanned surface vehicle’s modeling and control design, focusing on dynamic characterization and the evaluation of control strategies. Two modeling approaches, transfer function and state-space representations, were developed using experimental data collected under controlled test conditions. The resulting models closely approximate the system’s behavior with high statistical accuracy ($R^2>0.9$), validating their suitability for control design. Note that, although the model fit was excellent for the tested scenario, as evidenced by the high correlation coefficients presented, this level of accuracy cannot be guaranteed under different operating conditions, such as significant variations in propeller thrust or angle, without further validation or model recalibration. Accordingly, the reported correlation coefficients should not be interpreted as a universal indicator of model accuracy. Instead, they reflect the fit quality within the specific context and conditions under which the experimental data were collected and the models were derived.

While the literature presents a wide range of complex modeling and identification techniques for USVs, which also yield good agreement between predicted and measured dynamics, the models proposed here are distinguished by their simplicity. Despite this, they offer comparable performance in terms of estimation error. In the state-space model, an initial peak estimation error is observed due to an apparent induced interaction between propeller thrust and heading; however, subsequent errors remain consistently low. This effect is absent in the transfer function model.

Building on these models, two classical control strategies were designed and analyzed. Independent loop control with a decoupling block successfully eliminates the effect of propeller angle on the vessel speed. This strategy offers acceptable response times while keeping manipulated variables within operational limits, making it suitable for long-duration and energy-conscious missions such as humanitarian or environmental operations. On the other hand, we have a state feedback control, which provides faster dynamic responses, particularly advantageous for applications requiring agility and rapid maneuvering (e.g., tactical or evasive actions). The results confirm the improved performance indices compared to the first strategy. However, this performance comes at the cost of increased control effort, reflected in the higher actuator usage and energy consumption.

The successful application of classical control techniques further underscores their continued relevance and effectiveness in modern autonomous systems. Some of their key benefits include the following: ease of implementation and tuning with minimal computational resources, compatibility with simple models or empirical data, and low-cost deployment on embedded systems or resource-constrained platforms.

Experimental validation in real-world marine environments can only fully assess the robustness of the proposed models and control strategies. These tests, planned as part of future work, will provide a more comprehensive evaluation of the models’ predictive capabilities and control systems’ performance under the influence of environmental disturbances and varying operating regimes. This will also offer valuable insights into the potential refinements needed for deployment in operational scenarios.