An Integrated Design of Course-Keeping Control and Extended State Observers for Nonlinear USVs with Disturbances

Abstract

The integrated design problem of non-fragile controllers and extended state observers (ESOs) for nonlinear unmanned surface vehicles (USVs) under mismatched disturbances is addressed in this paper. First, an integrated model combining the USV system and the rudder system is developed, which includes a second-order underdamped system and a Norrbin nonlinear model incorporating uncertainties. Due to the coupling issues in the design of controllers and observers caused by parameter perturbations or other unmodeled dynamics, an integrated design method, which enables the simultaneous computation of controller gains, observer gains, and disturbance compensation gains, is proposed, effectively addressing these issues. Ultimately, the performance of the designed strategy is verified through a simulation, with the data used in the simulation derived from the real Qingshan USV.

1. Introduction

Marine vessels serve as essential tools for maritime operations and have played a pivotal role in marine exploration and resource development [1,2]. Consequently, marine vessels have long been a prominent focus of academic research, attracting widespread attention across multiple disciplines [3,4,5]. With rapid technological advancements, unmanned surface vehicles (USVs) have emerged as a transformative innovation in the field of marine engineering, receiving significant attention due to their immense potential in enhancing marine resource exploitation, strengthening environmental monitoring, and optimizing remote operational processes [3,4,5,6]. As indicated by their name, USVs are designed to operate autonomously, requiring neither remote control nor onboard operators, thus representing a novel paradigm in autonomous maritime operations and offering transformative solutions to traditional challenges in marine exploration. Compared to traditional vessels, USVs offer advantages such as compact size, superior maneuverability, and enhanced safety, which have facilitated their widespread adoption in both military and civilian applications [7].

Foundational research on USVs begins with the development of mathematical models to describe their dynamic behavior. Among these models, the Nomoto model proposed by Nomoto et al. [8] employs low-order linear differential equations to characterize the transverse rocking motion of a vessel, effectively capturing the influence of rudder angles on course dynamics under steady-state conditions. However, it fails to accurately describe the nonlinear dynamics induced by high-speed operations and complex environmental conditions. To address these limitations, Inoue et al. [9] introduced more sophisticated hydrodynamic analyses into the Nomoto framework, enabling a more comprehensive description of vessel motion. Building upon this, Kijima et al. [10] further incorporated nonlinear terms, significantly enhancing the model’s capability to capture ship dynamics at the cost of increased computational complexity. In contrast, the Norrbin model [11] introduces nonlinear terms directly into the Nomoto structure, providing a more detailed representation of yaw dynamics compared to the original Nomoto model. Overall, the Inoue and Kijima models focus on multi-degree-of-freedom hull motion coupling and are suitable for overall maneuvering analysis, whereas the Nomoto and Norrbin models are more appropriate for single-degree-of-freedom heading control analysis. Nevertheless, in practical ship heading control, rudder angles are not directly generated but must be achieved by controlling the rudder actuator. In the control of USVs, trajectory tracking is often achieved using modified backstepping methods [12]. Lei et al. [13] proposed a nonlinear networked predictive control strategy to address challenges posed by communication constraints—including network delays, packet loss, and packet disorder—as well as external environmental disturbances during trajectory tracking tasks. Furthermore, research on trajectory tracking is closely linked to studies on course-keeping control systems (CCS), as precise heading regulation forms the foundation for effective trajectory tracking. Therefore, CCS research is of great significance. However, most existing studies on CCS focus primarily on modeling the USV dynamics, whereas in practical applications, the controller must also provide precise rudder inputs. Thus, integrated modeling of USV and rudder systems becomes crucial, yet it remains insufficiently explored.

In the context of CCS, ensuring robustness against external disturbances and achieving fault tolerance in the event of equipment failures or component ingress are essential for maintaining long-term system stability and reliability. Thus, enhancing the robustness of CCS remains a critical research topic. Xiong et al. [14] designed a non-fragile fault-tolerant controller considering the inherent uncertainty of USVs and partial rudder failures; however, under strong environmental disturbances, its control performance showed slight deficiencies. Observers, which serve as one of the most effective tools for disturbance detection in CCS, have been widely applied in related studies. For instance, Qiu et al. [15] proposed a dual-loop robust composite control strategy by incorporating a finite-time uncertainty observer and an auxiliary dynamic system into a trajectory linearization control framework, effectively addressing uncertainties, disturbances, and rudder saturation. He et al. [16] developed a nonlinear disturbance observer combined with finite-time dynamic surface control to handle environmental perturbations, rudder dynamics, and model uncertainties in USVs. Wu et al. [17] introduced an active disturbance rejection technique using an extended state observer (ESO), which, by incorporating extended states, enables the simultaneous estimation of system states and external disturbances, significantly improving performance and applicability compared to conventional observers. Due to its simple structure, wide applicability, and strong robustness, the ESO has been extensively applied in marine environments, unmanned systems, and power systems.

Nevertheless, most existing studies assume that the controller and observer designs satisfy the separation principle. In practice, however, due to parameter uncertainties and controller perturbations, unavoidable coupling effects exist between the controller and the observer, leading to mutual influence. To address this issue, Do et al. [18] proposed an integrated design of a PI observer and a state feedback controller for linear parameter-varying (LPV) systems, effectively mitigating the coupling problems associated with separate designs, as validated through simulations on a vehicle suspension platform. Rodrigues et al. [19] proposed an integrated fault-tolerant control strategy for LPV systems subject to actuator faults by designing an adaptive polytopic observer combined with a controller to reduce the adverse effects of faults on observation performance. In studies of uncertain switched systems under external disturbances, Menhour et al. [20] developed an integrated $H_{\infty }$ controller based on a PI observer, fully considering system uncertainties during observer design. Although some existing research has partially addressed the coupling between controller and observer, systematic studies on the integrated design of controllers and observers under uncertainties in the context of ship systems remain insufficient.

This paper investigates a novel integrated $H_{\infty }$ control strategy incorporating an ESO for nonlinear USVs with uncertainty, controller ingestion, and coupling between the controller and observer. The main contributions can be summarized as follows:

(1)

To make the CCS more realistic, an integrated model is constructed that encompasses the characteristics of both the unmanned surface vehicle (USV) system and the rudder system. In the modeling process, the Norrbin model with parameter uncertainties is used to describe the USV system, while a second-order underdamped model is employed to represent the rudder system.

(2)

A unified CCS strategy is proposed to address the perturbations in the controller, system parameter uncertainties, and the coupling between the controller and the extended state observer (ESO). This strategy aims to enhance the robustness and performance of the system by effectively managing the challenges posed by these factors.

Section 2.1 introduces a composite control strategy based on the ESO. Section 2.2, Section 2.3 and Section 2.4 first analyze the characteristics of the USV system and its rudder system, and based on these characteristics, an integrated CCS model is constructed. In Section 3.1, the specific design process of the extended observer is presented. Section 3.2 introduces an integrated control method combining non-fragile control with ESO, providing a detailed proof of the approach. This method addresses the coupling problem between the controller and the observer and considers system uncertainties. Additionally, sufficient conditions for calculating the observer gain, controller gain, and disturbance compensation gain are derived using the Linear Matrix Inequality (LMI) method. In Section 4.1, a real ship test is conducted, where system parameters are obtained by model matching using data measured during actual navigation. Section 4.2 validates the performance of the proposed method through simulation experiments. Finally, Section 5 summarizes the main findings of the paper and suggests directions for future research.

2. Problem Formulation and Preliminaries

2.1. Structure of CCS

As illustrated in Figure 1, the CCS is composed of a PID controller, an interference observer, a rudder, and an unmanned surface vehicle. The rudder input signal is as follows:

$$\begin{array}{c}\alpha \left(t\right)=(K_x+\Delta K_x)e\left(t-\tau \left(t\right)\right)+K_d\widehat{d}\left(t\right),\end{array}$$

(1)

$\alpha \left(t\right)$ comprises the PID controller output signal $(K_x+\Delta K_x)e\left(t-\tau \left(t\right)\right)$ influenced by time delay and controller perturbations, along with the disturbance compensation signal $K_d\widehat{d}\left(t\right)$. In this context, $\delta \left(t\right)$ denotes the rudder angle, $\widehat{d}\left(t\right)$ denotes the estimated disturbance, $\psi \left(t\right)$ represents the actual heading angle, and ${\psi }_d$ denotes the desired heading angle.

To address the inherent coupling between the controller and the observer in practical scenarios, as well as the model uncertainties in the CCS, we propose an integrated design approach that simultaneously considers the controller gain $K_x$, observer gain L, and disturbance compensation gain $K_d$.

2.2. Characteristics of USV

For the CCS of the USVs, we focus on the link between the heading and the rudder angle, and we assume that the USVs are perfectly symmetric and establish a moving coordinate system at its center of gravity. The energy unbounded environmental perturbation is assumed to be $\omega \left(t\right)$, and we can obtain a simplified model of the yaw rate $r\left(t\right)$ and the rudder angle $\delta \left(t\right)$ as follows [21]:

$$\begin{array}{c}{T}_1\dot{r}\left(t\right)+r\left(t\right)+=T_3\delta \left(t\right)+\omega \left(t\right),\end{array}$$

(2)

where $T_1$ and $T_3$ are USV time constant. Due to the interaction between the rudder angle and other dynamic factors (such as wind, waves, and currents), the system response exhibits nonlinear characteristics. The nonlinear effects can be simplified to yield the following model [22]:

$$\begin{array}{c}{T}_1\dot{r}\left(t\right)+r\left(t\right)+T_2{r}^3\left(t\right)=T_3\delta \left(t\right)+\omega \left(t\right),\end{array}$$

(3)

where $T_2$ is the newly introduced constant, and the nonlinear influence is embodied by $T_2{r}^3$, which still conforms to the Norrbin nonlinear mode [11]. Define $r\left(t\right)=\dot{\psi }\left(t\right)$. Due to unavoidable parameter uncertainties, the following uncertainty Norrbin model is applied to characterize the USV system based on Equation (3):

$$\begin{array}{c}\left(T_1+\Delta T_1\right)\ddot{\psi }\left(t\right)+\dot{\psi }\left(t\right)+T_2{\dot{\psi }}^3\left(t\right)=\left(T_3+\Delta T_3\right)\delta \left(t\right)+\omega \left(t\right).\end{array}$$

(4)

2.3. Dynamics and Modeling of the Rudder Actuator

In conventional vehicles, there is typically a delay between the rudder input signal and the rudder angle output, and this rudder characteristic is generally modeled as a first-order inertial link, particularly in large vehicles. However, the rudder of USVs has characteristics of high speed and high accuracy, making the first-order inertial system insufficient to accurately describe the rudder’s state. Instead, it is more appropriate to use a second-order linear underdamped model to represent the rudder system of USVs [15,22].

$$\begin{array}{c}\ddot{\delta }\left(t\right)+2\zeta {\omega }_n\dot{\delta }\left(t\right)+{\omega }_n^2\delta \left(t\right)=K_n{\omega }_n^2\alpha \left(t\right),\end{array}$$

(5)

where $\zeta$ is the damping ratio, ${\omega }_n$ is the natural frequency, $K_n$ is the magnification factor, $u\left(t\right)$ is the command rudder angle, $\left|\delta \left(t\right)\right|\le {35}^{\circ },\left|\dot{\delta }\left(t\right)\right|\le 3^{\circ }/s$. Define $T_4=2\zeta {\omega }_n$, $T_5={\omega }_n^2$, $T_6=K_n{\omega }_n^2$, the rudder system can be rewritten as

$$\begin{array}{c}\ddot{\delta }\left(t\right)+T_4\dot{\delta }\left(t\right)+T_5\delta \left(t\right)=T_6\alpha \left(t\right).\end{array}$$

(6)

2.4. Integrated Model of CCS

Combining Equations (4) and (6) and taking state variables, we have $e_1\left(t\right)=\int {\psi }_d-\psi \left(t\right)dt$, $e_2\left(t\right)={\psi }_d-\psi \left(t\right)$, $e_3\left(t\right)=\dot{\psi }\left(t\right)$, $e_4\left(t\right)=\delta \left(t\right)$, $e_5\left(t\right)=\dot{\delta }\left(t\right)$. The nonlinear CCS model can be obtained:

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{e}}_1\left(t\right)=e_2\left(t\right)\hfill \\ {\dot{e}}_2\left(t\right)=-e_3\left(t\right)\hfill \\ {\dot{e}}_3\left(t\right)=\left(-{\displaystyle \frac{1}{T_1}}+{\displaystyle \frac{\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}}\right)e_3\left(t\right)+\left(-{\displaystyle \frac{T_2}{T_1}}+{\displaystyle \frac{\Delta T_1{T}_2}{T_1\left(T_1+\Delta T_1\right)}}\right)e_3^3\left(t\right)\hfill \\ +\left({\displaystyle \frac{T_3}{T_1}}+{\displaystyle \frac{T_1\Delta T_3-T_3\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}}\right)e_4\left(t\right)+\left({\displaystyle \frac{1}{T_1}}-{\displaystyle \frac{\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}}\right)\omega \left(t\right)\hfill \\ {\dot{e}}_4\left(t\right)=e_5\left(t\right)\hfill \\ {\dot{e}}_5\left(t\right)=-T_5{e}_4\left(t\right)-T_4{e}_5\left(t\right)+T_6\alpha \left(t\right)\hfill \end{array}\right.\end{array}$$

(7)

Define

$$\begin{array}{c}\begin{array}{c}e\left(t\right)={\left[\begin{array}{cccc}{e_1}^T\left(t\right)& {e_2}^T\left(t\right)& \cdots & {e_5}^T\left(t\right)\end{array}\right]}^T,d\left(t\right)=-T_2{e}_3^3\left(t\right)+\omega \left(t\right).\hfill \end{array}\end{array}$$

We have

$$\begin{array}{c}\left\{\begin{array}{c}\dot{e}\left(t\right)=\left(A+\Delta A\right)e\left(t\right)+B\alpha \left(t\right)+\left(D+\Delta D\right)d\left(t\right)\hfill \\ y\left(t\right)=Ce\left(t\right),\hfill \end{array}\right.\end{array}$$

(8)

where

$$\begin{array}{c}\begin{array}{c}A=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& -{\displaystyle \frac{1}{T_1}}& {\displaystyle \frac{T_3}{T_1}}& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& -T_5& -T_4\end{array}\right],\Delta A=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& a_{33}& a_{34}& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right],\hfill \\ B={\left[\begin{array}{ccccc}0& 0& 0& 0& T_6\end{array}\right]}^T,C=\left[\begin{array}{ccccc}1& -1& 0& 0& 0\end{array}\right],\hfill \\ D={\left[\begin{array}{ccccc}0& 0& {\displaystyle \frac{T_2}{T_1}}& 0& 0\end{array}\right]}^T,\Delta D={\left[\begin{array}{ccccc}0& 0& d_3& 0& 0\end{array}\right]}^T,\hfill \\ a_{33}={\displaystyle \frac{\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}},a_{34}={\displaystyle \frac{T_1\Delta T_3-T_3\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}},d_3=-{\displaystyle \frac{\Delta T_1}{T_1\left(T_1+\Delta T_1\right)}}.\hfill \end{array}\end{array}$$

Remark 1.

In this paper, a nonlinear USV system subject to uncertainties and a second-order underdamped rudder system are modeled as an integrated system. While most previous CCS studies primarily focused on modeling the USVs, in practical scenarios, the control input inevitably acts on the rudder, which subsequently influences the heading of the unmanned vessel. Therefore, the integrated modeling of the rudder and the unmanned vessel is essential.

This paper resolves the uncertainty issues in the CCS system and the coupling between the ESO and the controller by selecting appropriate control parameters. To accomplish this, relevant assumptions are introduced, and previous lemmas are revisited to support the theoretical development.

Assumption 1.

For controller gain perturbation, $\Delta K_x$ can be assumed to be of the following type:

$$\begin{array}{c}\Delta K_x=M_1{F}_1\left(t\right)E_1,\end{array}$$

(9)

where $\Delta K_x$ is additive uncertainty. $M_1$ and $E_1$ are known matrices with appropriate dimensions, $F_1\left(t\right)$ is an unknown function that fulfills the following condition:

$$\begin{array}{c}{F_1}^T\left(t\right)F_1\left(t\right)\le I.\end{array}$$

Assumption 2.

For system internal parameter uncertainties, $\Delta A$ and $\Delta D$ satisfy

$$\begin{array}{c}\left[\begin{array}{cc}\Delta A& \Delta D\end{array}\right]=M_2{F}_2\left(t\right)\left[\begin{array}{cc}{E}_2& E_3\end{array}\right],\end{array}$$

(10)

where $M_2$, $E_2$, and $E_3$ are known matrices with appropriate dimensions, $F_2\left(t\right)$ is an unknown function that fulfills the following condition:

$$\begin{array}{c}{F_2}^T\left(t\right)F_2\left(t\right)\le I.\end{array}$$

Lemma 1

([23]). Let M be a symmetric matrix of the following form:

$$\begin{array}{c}M=\left[\begin{array}{cc}{m}_1& m_2\\ m_2^T& m_3\end{array}\right],\end{array}$$

(11)

and the following statements are equivalent:

(1) 

$M<0$,

(2) 

$m_3<0$ and $m_1-m_2{m}_3^{-1}{m}_2^T<0$,

(3) 

$m_1<0$ and $m_3-m_2^T{m}_1^{-1}{m}_2<0$.

Lemma 2

([24]). For scalars $f_1$ and $f_2$ satisfying $f_1<f_2$, a vector function $\xi \left(s\right):\left[\begin{array}{cc}{f}_1& f_2\end{array}\right]\to {\mathbb{R}}^n$, and any matrix $Z>0$, the following inequality holds:

$$\begin{array}{c}{\left[{\int }_{f_1}^{f_2}\xi \left(s\right)\right]}^{T}Z\left[{\int }_{f_1}^{f_2}\xi \left(s\right)\right]⩽\left[f_2-f_1\right]{\int }_{f_1}^{f_2}{\xi }^T\left(s\right)Z\xi \left(s\right)ds.\end{array}$$

(12)

Lemma 3

([25]). For matrices satisfying $\Lambda ={\Lambda }^T$, $\mathbf{X}$ and $\mathbf{Y}$ with appropriate dimensions, $\Lambda +\mathbf{XGY}+{\mathbf{Y}}^T{\mathbf{G}}^T{\mathbf{X}}^T<0$ with $\mathbf{G}$ satisfying ${\mathbf{G}}^T\mathbf{G}\le I$, if and only if there is a $\sigma >0$ such that

$$\begin{array}{c}\Lambda +\sigma {\mathbf{Y}}^T\mathbf{Y}+{\sigma }^{-1}\mathbf{X}{\mathbf{X}}^T<0.\end{array}$$

(13)

3. Main Results

In this section, an integrated design of a non-fragile course-keeping controller, an extended state observer, and a disturbance compesator is given to address the challenges posed by system uncertainties, controller implementation issues, and the coupling between the controller and the observer.

3.1. Extended State Observer

Define the extended state variable as

$$\begin{array}{c}{e}_6\left(t\right)=d\left(t\right).\end{array}$$

The extended system can be described as follows:

$$\begin{array}{c}\left\{\begin{array}{c}\dot{\beta }\left(t\right)=(\overline{A}+\Delta \overline{A})\beta \left(t\right)+\overline{B}\alpha \left(t\right)+Gh\left(t\right)\hfill \\ \overline{z}\left(t\right)=\overline{C}\beta \left(t\right),\hfill \end{array}\right.\end{array}$$

(14)

where

$$\begin{array}{c}\begin{array}{c}\beta \left(t\right)=\left[\begin{array}{c}e\left(t\right)\\ e_6\left(t\right)\end{array}\right],h\left(t\right)={\displaystyle \frac{dd\left(t\right)}{dt}},\overline{A}={\left[\begin{array}{cc}{A}_{5\times 5}& D_{5\times 1}\\ 0_{1\times 5}& 0_{1\times 1}\end{array}\right]}_{6\times 6},\overline{B}={\left[\begin{array}{c}{B}_{5\times 1}\\ 0_{1\times 1}\end{array}\right]}_{6\times 1},\hfill \\ E={\left[\begin{array}{c}{0}_{5\times 1}\\ 1_{1\times 1}\end{array}\right]}_{6\times 1},\overline{C}={\left[\begin{array}{cc}C& 0_{1\times 1}\end{array}\right]}_{1\times 6},\Delta \overline{A}={\left[\begin{array}{cc}\Delta A_{5\times 5}& 0_{5\times 1}\\ 0_{1\times 5}& 0_{1\times 1}\end{array}\right]}_{6\times 6},\hfill \\ \alpha \left(t\right)=\overline{\alpha }\left(t\right)+\Delta \alpha \left(t\right),\overline{\alpha }\left(t\right)=K_{x}e\left(t-\tau \left(t\right)\right)+K_d\widehat{d}\left(t\right),\Delta \alpha \left(t\right)=\Delta K_{x}e\left(t-\tau \left(t\right)\right).\hfill \end{array}\end{array}$$

For system (14), establish the following extended state observer:

$$\begin{array}{c}\left\{\begin{array}{c}\dot{\widehat{\beta }}\left(t\right)=\overline{A}\widehat{\beta }\left(t\right)+\overline{B}\overline{\alpha }\left(t\right)+L\left(\overline{y}\left(t\right)-\widehat{y}\left(t\right)\right)\hfill \\ \widehat{\overline{z}}\left(t\right)=\overline{C}\widehat{\beta }\left(t\right),\hfill \end{array}\right.\end{array}$$

(15)

where $\widehat{\beta }\left(t\right)={\left[\begin{array}{cc}\widehat{e}{\left(t\right)}^T& {\widehat{e}}_6{\left(t\right)}^T\end{array}\right]}^T$, and $\widehat{e}\left(t\right)$, and ${\widehat{e}}_6\left(t\right)$ are estimates of the system (14) species $e\left(t\right)$ and $e_6\left(t\right)$. The matrix L is the observer gain to be designed. The state and disturbance errors are defined as follows:

$$\begin{array}{c}\left\{\begin{array}{c}{e}_x\left(t\right)=\widehat{e}\left(t\right)-e\left(t\right)\hfill \\ e_d\left(t\right)=\widehat{d}\left(t\right)-d\left(t\right).\hfill \end{array}\right.\end{array}$$

(16)

Combining Equations (14)–(16) gives the error dynamic as follows:

$$\begin{array}{c}\left\{\begin{array}{c}\dot{\tilde{e}}\left(t\right)=\left(\overline{A}-L\overline{C}\right)\tilde{e}\left(t\right)-\overline{B}\Delta \alpha \left(t\right)-Gh\left(t\right)-\Delta \overline{A}\overline{e}\left(t\right)\hfill \\ \tilde{z}\left(t\right)=\overline{C}\tilde{e}\left(t\right),\hfill \end{array}\right.\end{array}$$

(17)

where

$$\begin{array}{c}\tilde{e}\left(t\right)=\left[\begin{array}{c}{e}_x\left(t\right)\\ e_d\left(t\right)\end{array}\right],\tilde{z}\left(t\right)=\widehat{z}\left(t\right)-\overline{z}\left(t\right).\end{array}$$

Remark 2.

System (17) shows that the controller and observer are coupled and the separation principle does not hold true. Consequently, the controller and observer designs for CCS systems under uncertainty are reformulated as an integrated design problem and addressed through the LMI process.

3.2. Integrated Design of Controller, Observer and Compensator

Substitute the composite control input $\alpha \left(t\right)$ into system (8) and define the observation error of the ESO in relation to the disturbance as follows:

$$\begin{array}{c}{e}_{dd}\left(t\right)=-B{K}_d\widehat{d}\left(t\right)+(D+\Delta D)d\left(t\right).\end{array}$$

(18)

By incorporating $e_{dd}\left(t\right)$ into system (8), the closed-loop system can be expressed as follows:

$$\begin{array}{c}\left\{\begin{array}{cc}\hfill \dot{e}\left(t\right)& =(A+\Delta A)e\left(t\right)-B(K_x+\Delta K_x)e(t-\tau \left(t\right))+e_{dd}\left(t\right)\hfill \\ \hfill y\left(t\right)& =Ce\left(t\right)\hfill \end{array}\right.\end{array}$$

(19)

Theorem 1.

For the given positive scalars $\overline{\tau }$, $\underline{\tau }$, μ, the course-keeping controller $K_x$ can be designed as $K_x=Y{\widehat{N}}^{-1}$, the disturbance compensator gain is chosen as

$$\begin{array}{c}{K}_d={\left[C{(A-BY{\widehat{N}}^{-1})}^{-1}B\right]}^{-1}C{(A-BY{\widehat{N}}^{-1})}^{-1}D,\end{array}$$

(20)

and the ESO gain can be designed as $L=N_2^{-1}{Y}_2$, where $\widehat{N}$ and $N_2$ are inverse matrices. Suppose there is a positive definite matrix $S>0$, $Q>0$, $W>0$, $V>0$, $\left[\begin{array}{cc}T& U\\ \ast & T\end{array}\right]>0$ and a positive scalar ${\gamma }_2>0$ such that inequalities (21) holds:

$$\begin{array}{c}\Psi =\left[\begin{array}{ccccc}{\overline{\Psi }}_2& \sigma {\Gamma }_1& {\Lambda }_1^T& \sigma {\Gamma }_2& {\Lambda }_2^T\\ \ast & -\sigma I& 0& 0& 0\\ \ast & \ast & -\sigma I& 0& 0\\ \ast & \ast & \ast & -\sigma I& 0\\ \ast & \ast & \ast & \ast & -\sigma I\end{array}\right]<0,\end{array}$$

(21)

where

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}{\overline{\Psi }}_2=\left[\begin{array}{ccc}{\overline{\Psi }}_{21}& {\sigma }_1{\Gamma }_3& {{\Lambda }_5}^T\\ \ast & -{\sigma }_{1}I& 0\\ \ast & \ast & -{\sigma }_{1}I\end{array}\right],{\overline{\Psi }}_{21}=\left[\begin{array}{ccc}{\overline{\Psi }}_{211}& {\Lambda }_3& {\Lambda }_4\\ \ast & -I& 0\\ \ast & \ast & -I\end{array}\right],\hfill \\ {\overline{\Psi }}_{211}=\left[\begin{array}{ccccccccc}{Z}_1& Z_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & -2N_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & \ast & {\widehat{Z}}_3& {\widehat{Z}}_4& {{\widehat{Z}}^{\prime }}_6& 0& \widehat{W}& 0& \widehat{N}\\ \ast & \ast & \ast & {\widehat{Z}}_5& {{\widehat{Z}}^{\prime }}_7& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & {{\widehat{Z}}^{\prime }}_8& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_9& {\widehat{Z}}_{10}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_{11}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right],\hfill \\ Z_1=N_2\overline{A}+{\overline{A}}^T{N}_2^T-Y_2\overline{C}-{\overline{C}}^T{Y}_2^T,Z_2=P-N_2-{\overline{C}}^T{Y}_2^T+{\overline{A}}^T{N}_2^T,\hfill \\ {\widehat{Z}}_3=He\left(\tilde{A}\widehat{N}\right)+Q_N-W_N,{\widehat{Z}}_4=\widehat{N}{\tilde{A}}^T+S_N-\widehat{N},{{\widehat{Z}}^{\prime }}_6=\widehat{N}{\tilde{A}}^T+BY,\hfill \\ {\widehat{Z}}_5={\overline{\tau }}^2{W}_N+{\left(\overline{\tau }-\underline{\tau }\right)}^2{V}_N-2\widehat{N},{{\widehat{Z}}^{\prime }}_7=BY-\widehat{N},{{\widehat{Z}}^{\prime }}_8=He\left(BY\right)-\left(1-\mu \right)\widehat{Q},\hfill \\ {\widehat{Z}}_9=-V_N+He(T-U),{\widehat{Z}}_{10}=V_N-He(T-U),{\widehat{Z}}_{11}=-W_N-V_N+He(T-U),\hfill \\ {\Lambda }_3={\left[\begin{array}{ccc}\overline{C}& 0& \begin{array}{cc}\cdots & 0\end{array}\end{array}\right]}^T,{\Lambda }_4={\left[\begin{array}{ccc}\begin{array}{cc}0& \begin{array}{cc}0& C\end{array}\end{array}& 0& \begin{array}{cc}\cdots & 0\end{array}\end{array}\right]}^T,X_2=\sigma N_2,\hfill \\ {\Gamma }_2={\left[\begin{array}{cccc}{N}_2\overline{B}{M}_1& N_2\overline{B}{M}_1& 0& \begin{array}{cc}\cdots & 0\end{array}\end{array}\right]}^T,\hfill \\ {\Gamma }_1={\left[00{\left(B{M}_1\right)}^T{\left(B{M}_1\right)}^T{\left(B{M}_1\right)}^{T}0\cdots 0\right]}^T,\hfill \\ {\Lambda }_1=\begin{array}{cccccc}\left[0\right.& 0& 0& 0& E_1{\widehat{N}}^T& 0\end{array}\begin{array}{ccccc}0& 0& 0& 0& \left.0\right]\end{array},\hfill \\ {\Lambda }_2=\begin{array}{cccccc}\left[0\right.& 0& 0& 0& E_1& 0\end{array}\begin{array}{ccccc}0& 0& 0& 0& \left.0\right]\end{array}.\hfill \end{array}\hfill \end{array}\end{array}$$

Then the system (19) is stable with an $H_{\infty }$ disturbance attenuate level γ.

Proof. 

The Lyapunov functional is designed as follows:

$$\begin{array}{c}\begin{array}{c}V\left(t\right)={\tilde{e}}^T\left(t\right)P\tilde{e}\left(t\right)+e^T\left(t\right)Se\left(t\right)+{\int }_{t-\tau \left(t\right)}^t{e}^T\left(u\right)Qe\left(u\right)du+\overline{\tau }{\int }_{-\overline{\tau }}^0{\int }_{t+\theta }^t{\dot{e}}^T\left(u\right)W\dot{e}\left(u\right)dud\theta \hfill \\ +\left(\overline{\tau }-\underline{\tau }\right){\int }_{-\overline{\tau }}^{-\underline{\tau }}{\int }_{t+v}^t{\dot{e}}^T\left(u\right)V\dot{e}\left(u\right)dudv\hfill \end{array}\end{array}$$

(22)

The result of deriving Equation (22) is

$$\begin{array}{c}\begin{array}{c}\dot{V}\left(t\right)={\dot{\tilde{e}}}^T\left(t\right)P\tilde{e}\left(t\right)+{\tilde{e}}^T\left(t\right)P\dot{\tilde{e}}\left(t\right)+{\dot{e}}^T\left(t\right)Se\left(t\right)+e^T\left(t\right)S\dot{e}\left(t\right)+e^T\left(t\right)Qe\left(t\right)\hfill \\ -\left(1-\dot{\tau }\left(t\right)\right)e^T\left(t-\tau \left(t\right)\right)Qe\left(t-\tau \left(t\right)\right)+{\overline{\tau }}^2{\dot{e}}^T\left(t\right)W\dot{e}\left(t\right)+{\left(\overline{\tau }-\underline{\tau }\right)}^2{\dot{e}}^T\left(t\right)V\dot{e}\left(t\right)\hfill \\ -{\int }_{t-\underline{\tau }}^t{\dot{e}}^T\left(u\right)W\dot{e}\left(u\right)du-\left(\overline{\tau }-\underline{\tau }\right){\int }_{t-\overline{\tau }}^{t-\underline{\tau }}{\dot{e}}^T\left(u\right)V\dot{e}\left(u\right)du.\hfill \end{array}\end{array}$$

(23)

Combining this with Lemma 2 yields

$$\begin{array}{c}\begin{array}{c}\dot{V}\left(t\right)\le {\dot{\tilde{e}}}^T\left(t\right)P\tilde{e}\left(t\right)+{\tilde{e}}^T\left(t\right)P\dot{\tilde{e}}\left(t\right)+{\dot{e}}^T\left(t\right)Se\left(t\right)+e^T\left(t\right)S\dot{e}\left(t\right)+e^T\left(t\right)Qe\left(t\right)\hfill \\ -\left(1-\mu \right)e^T(t-\tau \left(t\right))Qe(t-\tau \left(t\right))+{\overline{\tau }}^2{\dot{e}}^T\left(t\right)W\dot{e}\left(t\right)+{\left(\overline{\tau }-\underline{\tau }\right)}^2{\dot{e}}^T\left(t\right)V\dot{e}\left(t\right)\hfill \\ -e^T(t-\overline{\tau })We(t-\overline{\tau })+e^T(t-\overline{\tau })We\left(t\right)+e^T\left(t\right)We(t-\overline{\tau })-e^T\left(t\right)We\left(t\right)\hfill \\ -e^T(t-\overline{\tau })Ve(t-\overline{\tau })+e^T(t-\overline{\tau })Ve(t-\underline{\tau })+e^T(t-\underline{\tau })Ve(t-\overline{\tau })-e^T(t-\underline{\tau })Ve(t-\underline{\tau }).\hfill \end{array}\end{array}$$

(24)

By employing the free weight matrix technique, the following equation holds for any given matrix N and $N_2$:

$$\begin{array}{c}\begin{array}{c}\left[e^T\left(t\right)N+{\dot{e}}^T\left(t\right)N+e^T(t-\tau \left(t\right)N\right]O_1+{O_1}^T{\left[e^T\left(t\right)N+{\dot{e}}^T\left(t\right)N+e^T(t-\tau \left(t\right)N\right]}^T=0,\hfill \\ \left[{\tilde{e}}^T\left(t\right)N_2+{\dot{\tilde{e}}}^T\left(t\right)N_2\right]O_2+O_2^T{\left[{\tilde{e}}^T\left(t\right)N_2+{\dot{\tilde{e}}}^T\left(t\right)N_2\right]}^T=0,\hfill \end{array}\end{array}$$

(25)

where

$$\begin{array}{c}\begin{array}{c}{O}_1=Ae\left(t\right)+B{K}_{x}e\left(t-\tau \left(t\right)\right)+e_{dd}\left(t\right)-\dot{e}\left(t\right),\hfill \\ O_2=\left(\overline{A}-L\overline{C}\right)\tilde{e}\left(t\right)-\overline{B}\Delta u\left(t\right)-Gh\left(t\right)-\dot{\tilde{e}}\left(t\right).\hfill \end{array}\end{array}$$

Define

$$\begin{array}{c}m\left(t\right)=\left[\begin{array}{c}\tilde{z}\left(t\right)\\ y\left(t\right)\end{array}\right],\widehat{\omega }\left(t\right)=\left[\begin{array}{c}h\left(t\right)\\ e_{dd}\left(t\right)\end{array}\right].\end{array}$$

A control function with an $H_{\infty }$ performance index $\gamma$ is proposed, along with the sufficient conditions required for its validity:

$$\begin{array}{c}J=\dot{V}\left(t\right)+m^T\left(t\right)m\left(t\right)-{\gamma }^2{\widehat{\omega }}^T\left(t\right)\widehat{\omega }\left(t\right)\le {\eta }^T\left(t\right)\Psi \eta \left(t\right),\end{array}$$

(26)

where

$$\begin{array}{c}\begin{array}{c}\eta \left(t\right)=\begin{array}{cccc}\left[{\tilde{e}}^T\left(t\right)\right.& {\dot{\tilde{e}}}^T\left(t\right)& {\tilde{e}}^T\left(t\right)& {\dot{\tilde{e}}}^T\left(t\right)\end{array}\hfill \\ \begin{array}{ccc}{\tilde{e}}^T(t-\tau \left(t\right))& {\tilde{e}}^T(t-\underline{\tau })& {\tilde{e}}^T(t-\overline{\tau })\end{array}{\begin{array}{cc}{h}^T\left(t\right)& \left.{e^T}_{dd}\left(t\right)\right]\end{array}}^T\hfill \end{array}\end{array}$$

Combining Equation (24) to Equation (25) yields

$$\begin{array}{c}{\Psi }_1<0,\end{array}$$

(27)

where

$$\begin{array}{c}\begin{array}{c}{\Psi }_1=\left[\begin{array}{ccccccccc}{Z}_1& Z_2& 0& 0& Z_{12}& 0& 0& -N_{2}G& 0\\ \ast & -2N_2& 0& 0& Z_{12}& 0& 0& -N_{2}G& 0\\ \ast & \ast & Z_3& Z_4& Z_6& 0& W& 0& N\\ \ast & \ast & \ast & Z_5& Z_7& 0& 0& 0& N\\ \ast & \ast & \ast & \ast & Z_8& 0& 0& 0& N\\ \ast & \ast & \ast & \ast & \ast & Z_9& Z_{10}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & Z_{11}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right],\hfill \\ Z_1=N_2\left(\overline{A}-L\overline{C}\right)+{\left(\overline{A}-L\overline{C}\right)}^T{N}_2^T+{\overline{C}}^T\overline{C},Z_2=P+{\left(\overline{A}-L\overline{C}\right)}^T{N}_2^T-N_2,\hfill \\ Z_3=N\tilde{A}+{\tilde{A}}^{T}N+Q-W+C^{T}C,Z_4=S-N+{\tilde{A}}^{T}N,Z_5={\overline{\tau }}^{2}W+{\left(\overline{\tau }-\underline{\tau }\right)}^{2}V-2N,\hfill \\ Z_6=NB(K_x+\Delta K_x)+{\tilde{A}}^{T}N,Z_7=NB(K_x+\Delta K_x)-N,\hfill \\ Z_8=NB(K_x+\Delta K_x)+{(K_x+\Delta K_x)}^T{B}^{T}N-\left(1-\mu \right)Q,Z_9=-V+2T-U-U^T,\hfill \\ Z_{10}=V-2T+U+U^T,Z_{11}=-W-V+2T-U-U^T,Z_{12}=N_2\overline{B}\Delta K_x.\hfill \end{array}\end{array}$$

Apply Lemma 1 and perform the decoupling operation using left and right multiplication of $diag\left[\begin{array}{ccccccccc}I& I& \widehat{N}& \widehat{N}& \widehat{N}& I& I& I& I\end{array}\right]$, $diag\left[\begin{array}{ccccccccc}I& I& {\widehat{N}}^T& {\widehat{N}}^T& {\widehat{N}}^T& I& I& I& I\end{array}\right]$, respectively. Let $\widehat{N}=N^{-1}$, $\widehat{N}S{\widehat{N}}^T=S_N$, $\widehat{N}Q{\widehat{N}}^T=Q_N$, $\widehat{N}W{\widehat{N}}^T=W_N$, $\widehat{N}V{\widehat{N}}^T=V_N$, $Y=K_x{\widehat{N}}^T$, $\Delta Y=\Delta K_x{\widehat{N}}^T$, $Y_2=N_{2}L$. Then

$$\begin{array}{c}{\tilde{\Psi }}_1=\left[\begin{array}{ccc}{\tilde{\Psi }}_{11}& {\Lambda }_3& {\Lambda }_4\\ \ast & -I& 0\\ \ast & \ast & -I\end{array}\right]<0.\end{array}$$

(28)

where

$$\begin{array}{c}\begin{array}{c}{\tilde{\Psi }}_{11}=\left[\begin{array}{ccccccccc}{\widehat{Z}}_1& {\widehat{Z}}_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & -2N_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & \ast & {\widehat{Z}}_3& {\widehat{Z}}_4& {\widehat{Z}}_6& 0& \widehat{W}& 0& \widehat{N}\\ \ast & \ast & \ast & {\widehat{Z}}_5& {\widehat{Z}}_7& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & {\widehat{Z}}_8& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_9& {\widehat{Z}}_{10}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_{11}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right],\hfill \\ {\Lambda }_3={\left[\begin{array}{ccc}\overline{C}& 0& \begin{array}{cc}\cdots & 0\end{array}\end{array}\right]}^T,{\Lambda }_4={\left[\begin{array}{ccc}\begin{array}{cc}0& \begin{array}{cc}0& C\end{array}\end{array}& 0& \begin{array}{cc}\cdots & 0\end{array}\end{array}\right]}^T,\hfill \\ {\widehat{Z}}_1=N_2\overline{A}+{\overline{A}}^T{N}_2^T-Y_2\overline{C}-{\overline{C}}^T{Y}_2^T,{\widehat{Z}}_2=P+{\overline{A}}^T{N}_2^T-{\overline{C}}^T{Y}_2^T-N_2,\hfill \\ {\widehat{Z}}_3=\tilde{A}\widehat{N}+\widehat{N}{\tilde{A}}^T+Q_N-W_N,{\widehat{Z}}_4=\widehat{S}-\widehat{N}+\widehat{N}{\tilde{A}}^T,{\widehat{Z}}_5={\overline{\tau }}^2{W}_N+{\left(\overline{\tau }-\underline{\tau }\right)}^2{V}_N-2\widehat{N},\hfill \\ {\widehat{Z}}_6=B\left(Y+\Delta Y\right)+\widehat{N}{\tilde{A}}^T,{\widehat{Z}}_7=B\left(Y+\Delta Y\right)-\widehat{N},{\widehat{Z}}_9=-V_N+He(T-U),\hfill \\ {\widehat{Z}}_8=B\left(Y+\Delta Y\right)+{\left(Y+\Delta Y\right)}^T{B}^T-\left(1-\mu \right)\widehat{Q},{\widehat{Z}}_{10}=V_N-He(T-U),\hfill \\ {\widehat{Z}}_{11}=-W_N-V_N+He(T-U),Z_{12}=N_2\overline{B}\Delta K_x,\tilde{A}=A+\Delta A.\hfill \end{array}\end{array}$$

In system (8), the USVs are subject to external disturbance (e.g., wind, waves, currents), nonlinear characteristics, and parameter uncertainties $\Delta D$. Effective compensation of these problems is achieved by designing an ESO. In addition, the system suffers from the controller gain perturbation $\Delta K_x$, and the internal parameter uncertainty $\Delta A$ challenges the system. Next, each of these uncertainties will be analyzed and solved in detail.

Initially, the focus is on addressing the internal parameter uncertainty $\Delta A$. By separating the $\Delta A$ uncertainty from ${\tilde{\Psi }}_1$ in the system, the matrix $\Delta {\Psi }_1$ is constructed. Using Lemma 1 and Lemma 3, the resulting matrix ${\tilde{\Psi }}_2$ can then be derived.

$$\begin{array}{c}\begin{array}{c}{\tilde{\Psi }}_1={\overline{\Psi }}_1+\Delta {\Psi }_1={\overline{\Psi }}_1+{\Gamma }_3{F}_2\left(t\right){\Lambda }_5+{{\Lambda }_5}^T{F}_2^T\left(t\right){{\Gamma }_3}^T<0\hfill \\ \iff {\tilde{\Psi }}_2={\overline{\Psi }}_1+{\sigma }_1{\Gamma }_3{\Gamma }_3^T+{\sigma }_1^{-1}{\Lambda }_5^T{\Lambda }_5<0\hfill \end{array}\end{array}$$

(29)

$$\begin{array}{c}{\tilde{\Psi }}_2=\left[\begin{array}{ccc}{\overline{\Psi }}_1& {\sigma }_1{\Gamma }_3& {{\Lambda }_5}^T\\ \ast & -{\sigma }_{1}I& 0\\ \ast & \ast & -{\sigma }_{1}I\end{array}\right]<0\end{array}$$

(30)

where

$$\begin{array}{c}\begin{array}{c}{\overline{\Psi }}_1=\left[\begin{array}{ccc}{\overline{\Psi }}_{11}& {\Lambda }_3& {\Lambda }_4\\ \ast & -I& 0\\ \ast & \ast & -I\end{array}\right]<0,\Delta {\Psi }_1=\left[\begin{array}{ccc}\Delta {\Psi }_{11}& 0& 0\\ \ast & 0& 0\\ \ast & \ast & 0\end{array}\right]<0,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\begin{array}{c}{\overline{\Psi }}_{11}=\left[\begin{array}{ccccccccc}{\widehat{Z}}_1& {\widehat{Z}}_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & -2N_2& 0& 0& 0& 0& 0& -N_{2}G& 0\\ \ast & \ast & {\overline{Z}}_3& {\overline{Z}}_4& {\overline{Z}}_6& 0& \widehat{W}& 0& \widehat{N}\\ \ast & \ast & \ast & {\widehat{Z}}_5& {\widehat{Z}}_7& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & {\widehat{Z}}_8& 0& 0& 0& \widehat{N}\\ \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_9& {\widehat{Z}}_{10}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\widehat{Z}}_{11}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right],\hfill \\ \Delta {\Psi }_{11}=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \Delta Z_3& \Delta Z_4& \Delta Z_6& 0& 0& 0& 0\\ \ast & \ast & \ast & 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & 0\end{array}\right],\hfill \\ \begin{array}{c}{\overline{Z}}_3=A\widehat{N}+\widehat{N}{A}^T+\widehat{Q}-\widehat{W},\Delta Z_3=\Delta A\widehat{N}+\widehat{N}\Delta A^T,{\overline{Z}}_4=\widehat{S}-\widehat{N}+\widehat{N}{A}^T,\hfill \\ \Delta Z_4=\widehat{N}\Delta A^T,{\overline{Z}}_6=B\left(Y+\Delta Y\right)+\widehat{N}{A}^T,\Delta Z_6=\widehat{N}\Delta A^T,\hfill \\ {\Lambda }_5=\left[\begin{array}{cccccccc}0& 0& E_2{\widehat{N}}^T& E_2{\widehat{N}}^T& E_2{\widehat{N}}^T& 0& \cdots & 0\end{array}\right],\hfill \\ {\Gamma }_3={\left[\begin{array}{cccccccc}0& 0& 0& 0& {\left(E_1{\widehat{N}}^T\right)}^T& 0& \cdots & 0\end{array}\right]}^T.\hfill \end{array}\hfill \end{array}\end{array}$$

The controller gain uptake is treated by considering it as additive uptake, i.e., the $\Delta K_x$. Assume $\Delta K_x=M_1{F}_1\left(t\right)E_1$, ${F_1}^T\left(t\right)F_1\left(t\right)\le I$. ${\tilde{\Psi }}_2$ will contain the $\Delta K_x$ part extracted as a whole. Using Lemma 3, $\Psi$ can be derived:

$$\begin{array}{c}\begin{array}{c}{\tilde{\Psi }}_2={\overline{\Psi }}_2+\Delta {\Psi }_2={\overline{\Psi }}_2+{\Gamma }_1{F}_1\left(t\right){\Lambda }_1+{{\Lambda }_1}^T{F}_1^T\left(t\right){{\Gamma }_1}^T\hfill \\ +{\Gamma }_2{F}_1\left(t\right){\Lambda }_2+{{\Lambda }_2}^T{F}_1^T\left(t\right){{\Gamma }_2}^T<0\hfill \\ \iff \Psi ={\overline{\Psi }}_2+\sigma {\Gamma }_1{\Gamma }_1^T+{\sigma }^{-1}{\Lambda }_1^T{\Lambda }_1+\sigma {\Gamma }_2{\Gamma }_2^T+{\sigma }^{-1}{\Lambda }_2^T{\Lambda }_2<0,\hfill \end{array}\end{array}$$

(31)

where

$$\begin{array}{c}\begin{array}{c}\Delta {\Psi }_2=\left[\begin{array}{ccccc}\Delta {\Psi }_{21}& 0& 0& 0& 0\\ \ast & 0& 0& 0& 0\\ \ast & \ast & 0& 0& 0\\ \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0\end{array}\right],\hfill \\ \Delta {\Psi }_{21}=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & 0& 0& B\Delta Y& 0& 0& 0& 0\\ \ast & \ast & \ast & 0& B\Delta Y& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & 2B\Delta Y& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & 0\end{array}\right].\hfill \end{array}\end{array}$$

Apply Lemma 1 to $\Psi$, we can obtain inequalities (21).

If $\Psi <0$, then $\dot{V}\left(t\right)+m^T\left(t\right)m\left(t\right)-{\gamma }^2{\widehat{\omega }}^T\left(t\right)\widehat{\omega }\left(t\right)<0$, and J also holds. Given that $V\left(0\right)=0$, integrating Equation (26) yields the following result as $t\to \infty$:

$$\begin{array}{c}{\int }_0^{\infty }{m}^T\left(t\right)m\left(t\right)dt<{\int }_0^{\infty }{\gamma }^2{\widehat{\omega }}^T\left(t\right)\widehat{\omega }\left(t\right)dt.\end{array}$$

(32)

☐

The performance metric is then found, and the observer-controller gain can be obtained from $K_x=Y{\widehat{N}}^{-1}$ and $L=N_2^{-1}{Y}_2$. Due to the integrated design of the controller and observer gains, the resulting Sylvester equation becomes high-dimensional and computationally expensive to solve directly. To alleviate this issue, a multiplicative splitting iteration method, based on Hermitian and skew-Hermitian decompositions, can be employed to facilitate more efficient and robust computation [26]. The disturbance compensation gain $K_d$ is designed in Equation (20). Based on the estimation error (16), system (8) can be rewritten as

$$\begin{array}{c}\left\{\begin{array}{c}\dot{e}\left(t\right)=Ae\left(t\right)-B{K}_{x}e\left(t-\tau \left(t\right)\right)-B{K}_d{e}_d\left(t\right)\hfill \\ +\left((D-B{K}_d)d\left(t\right)\right)\hfill \\ y\left(t\right)=Ce\left(t\right).\hfill \end{array}\right.\end{array}$$

(33)

The transfer function from $d\left(t\right)$ to $y\left(t\right)$ can be described as

$$\begin{array}{c}{G}_{yd}\left(s\right)=C{(sI-A+e^{-\tau s}B{K}_x)}^{-1}(D-B{K}_d).\end{array}$$

(34)

Based on the final value theorem, the following condition must be satisfied to eliminate the effect of disturbances on the output channel:

$$\begin{array}{c}C{(A-B{K}_x)}^{-1}(D-B{K}_d)=0.\end{array}$$

(35)

This shows that disturbances are eliminated from the steady state output channel by the disturbance compensation gain.

4. Simulation Example

In the first section, the CCS and rudder system of the USV were thoroughly validated using real experimental data. Comparative experiments were conducted to further assess the reliability of the experimental data. In the second section, the effectiveness of the proposed control strategy was demonstrated through simulation experiments.

4.1. Model Analysis

Based on real-time environmental data provided by the Hangzhou Meteorological Bureau, the wave height during the testing period remained below 0.1 m, and the wind speed was consistently maintained under 1 mile per hour (approximately 0.45 m/s). These stable environmental conditions ensured minimal external interference with the heading stabilization control experiments conducted on the Qiantang River. Due to the unavoidable influence of noise and other factors in the detected data, data preprocessing to remove noise and outliers is essential [27]. The sensor data used in this study have been pre-processed to ensure the accuracy and reliability of subsequent analysis. During the navigation experiment, the control rudder signal $\alpha$ is input, while the corresponding rudder response signal ${\delta }_a\left(t\right)$ and heading angle signal ${\psi }_a\left(t\right)$ are measured using sensors. The parameters of systems (4) and (6) were derived through system identification methods as follows: $T_1=1.6283$, $T_2=0.028$, $T_3=0.0846$, $T_4=0.19$, $T_5=0.3466$, and $T_6=0.28$.

In Figure 2, the estimated model heading angle ${\psi }_m\left(t\right)$ and model rudder angle ${\delta }_m\left(t\right)$ obtained from the identification model are compared with the actual measured heading angle ${\psi }_a\left(t\right)$ and rudder angle ${\delta }_a\left(t\right)$, respectively. From the comparison results shown in Figure 3, it is evident that the model obtained through system identification demonstrates high effectiveness and accurately captures the dynamic characteristics of the actual system.

4.2. Example

By incorporating the identified parameters into (8), the dynamics of the heading error for the “Qingshan” unmanned surface vehicle can be derived as follows:

$$\begin{array}{c}\begin{array}{c}A=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& -0.61412& 0.052& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& -0.3446& -0.19\end{array}\right],\hfill \\ B={\left[\begin{array}{ccccc}0& 0& 0& 0& 0.28\end{array}\right]}^T,C=\left[\begin{array}{ccccc}1& -1& 0& 0& 0\end{array}\right],\hfill \\ D={\left[\begin{array}{ccccc}0& 0& 0.61412& 0& 0\end{array}\right]}^T.\hfill \end{array}\end{array}$$

The system parameter uncertainty is denoted as

$$\begin{array}{c}\begin{array}{c}\Delta \mathrm{A}=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 1\end{array}\right],\Delta D={\left[\begin{array}{ccccc}0& 0& 1& 0& 0\end{array}\right]}^T.\hfill \end{array}\end{array}$$

The environmental disturbance is defined as

$$\begin{array}{c}w\left(t\right)=\left\{\begin{array}{c}3sin(t+2),\hfill \\ 6sin\left(1.5\right(t+3\left)\right),\hfill \end{array}\right.\begin{array}{c}0\le t<65,\\ 65\le t<180,\end{array}\end{array}$$

which persists throughout the operation of the heading stabilization control system. The parameters associated with the time-varying delay are specified as $\overline{\tau }=0.2$, $\underline{\tau }=0$, $\mu =0.797$. Select the additive uptake of the controller as $\Delta K_x=\left[\begin{array}{ccccc}sin\left(t\right)& sin\left(t\right)& sin\left(t\right)& sin\left(t\right)& sin\left(t\right)\end{array}\right]$. $F_1\left(t\right)=sin\left(t\right)$ satisfy $F_1\left(t\right)F_1^T\left(t\right)\le I$. The augmentation system in Section 3.1 can be represented as follows:

$$\begin{array}{c}\begin{array}{c}\overline{A}=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& -0.61412& 0.052& 0& 0.6412\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& -0.3446& -0.19& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ \overline{B}={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0.28\end{array}\right]}^T,\overline{C}=\left[\begin{array}{cccccc}1& -1& 0& 0& 0& 0\end{array}\right],\hfill \\ G={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 1\end{array}\right]}^T.\hfill \end{array}\end{array}$$

Based on Theorem 1 obtained in Section 3.2, the following observer gain, controller gain, and disturbance compensation gain can be obtained by using the LMI solver toolbox in Matlab R2019b:

$$\begin{array}{c}\begin{array}{c}L=\left[\begin{array}{cccc}13.1534& 10.3539& -3.2044& -0.0864\end{array}\right.{\left.\begin{array}{cc}0.0439& -3.8723\end{array}\right]}^T,\hfill \\ K_x=\left[\begin{array}{cccc}-0.1419& -2.0468& 3.2221& 0.3155\end{array}\right.\left.\begin{array}{c}0.5154\end{array}\right],K_d=18.7101.\hfill \end{array}\end{array}$$

Figure 4 shows the heading angle output and the rudder angle output of the CCS, respectively. In addition, a backstepping method with disturbance observer compensation [28] is introduced as a comparative experiment in Figure 4 to further validate the effectiveness of the proposed controller design. As can be observed from Figure 4, the red line represents the command heading, the blue line represents the control effect of the proposed method, and the yellow line represents the control effect of the backstepping method with disturbance observer compensation. Compared with the backstepping method, the proposed method exhibits superior control performance, characterized by a smaller overshoot, faster convergence speed, and lower steady-state oscillation amplitude, thereby achieving more accurate tracking of the commanded heading. Although the backstepping method is eventually able to track the commanded heading, it suffers from significant oscillations and a slower response speed, resulting in relatively poor control performance. Moreover, under the conditions of controller parameter perturbations and system uncertainties, the control performance of the backstepping method is not ideal, showing difficulty in effectively suppressing disturbances and leading to degraded system stability and tracking accuracy. Figure 5 presents the observed disturbance and the observation error between the observed value and the actual disturbance of the observer output. In Figure 5, it can be seen that, when the CCS encounters mismatched and energy-unbounded disturbances, the observer gain obtained from Theorem 1 is reasonable. The disturbance observer can accurately estimate the actual disturbance within a short period of time, thereby quickly reducing the observation error to zero and compensating in a timely manner. Overall, it can be concluded that the controller gain and observer gain obtained by the previously proposed integrated controller-observer design method exhibit excellent control and observation performance, respectively.

5. Conclusions

This paper investigates the CCS problem of nonlinear USVs with parameter uncertainties under mismatched energy-unbounded disturbances. First, considering the unavoidable external environmental disturbances and the resulting model nonlinearities, the USV and rudder systems are modeled using a Norrbin nonlinear model with uncertainties and a second-order linear underdamped system, respectively, leading to the development of an integrated state-space model. Then, addressing the scenario where the separation principle does not hold in the control system design process, an integrated control method based on an ESO is proposed. This method incorporates the $H_{\infty }$ performance index to achieve target heading control, real-time state observation, compensation for mismatched disturbances, and effective suppression of observation errors. The feasibility of the proposed method is demonstrated through theoretical analysis. Finally, the proposed method is compared with a backstepping method incorporating a disturbance observer, and its effectiveness and robustness are validated through numerical simulations under typical external disturbance scenarios.

The handling of the CCS problem for USVs in this paper has certain limitations. Future research directions can be anticipated as follows: In practical CCS applications, the data acquired by sensors inevitably suffer from biases, noise, and other factors, which may result in inaccuracies. Therefore, it is essential to further investigate how to ensure the stable control of the CCS system in the presence of sensor faults or measurement inaccuracies, and to improve the robustness of the system in complex environments.