Improved Long Short-Term Memory-Based Fixed-Time Fault-Tolerant Control for Unmanned Marine Vehicles with Signal Quantization

Abstract

This paper presents a fixed-time fault-tolerant control strategy based on an improved long short-term memory network for dynamic positioning of unmanned marine vehicles subject to signal quantization, disturbances, and input saturation. Firstly, an improved long short-term memory network optimized by an adaptive mixed-gradient algorithm is developed to accurately estimate external disturbances. Secondly, a fixed-time extended state observer is designed to rapidly predict thruster faults. Subsequently, within a fixed-time control framework, a novel terminal sliding-mode surface incorporating signal quantization parameters is constructed. In addition, a dynamic uniform quantization strategy with tunable sensitivity is introduced to effectively alleviate the performance degradation induced by quantization errors. Based on this, a fixed-time fault-tolerant controller is constructed. Finally, simulation results and comparative experiments are provided to demonstrate the effectiveness of the proposed control scheme.

1. Introduction

In recent years, the motion control of unmanned marine vehicles (UMVs) has garnered increasing attention owing to their wide-ranging applications in scientific exploration, marine resource surveying, and defense operations [1,2]. To satisfy the demanding requirements of maritime missions for fast, precise, and robust performance, various advanced control strategies have been developed, including dynamic positioning (DP) control [3], tracking control [4,5], and path-following control [6]. Due to the highly nonlinear and strongly coupled characteristics of ships, DP control has consistently remained a challenging problem.

The finite-time control approach has recently gained popularity due to its high precision and rapid convergence. For UMVs, a resilient finite-time control methodology was proposed in [7]. Additionally, a nonsingular terminal sliding-mode (TSM) controller augmented with an extended state observer was developed to achieve finite-time convergence [8]. Furthermore, an adaptive parameter adjustment scheme leveraging both system tracking discrepancies and adaptive estimation errors was introduced, leading to the design of an adaptive finite-time controller based on second-order terminal sliding-mode control (TSMC) [9,10]. However, a significant limitation of finite-time control methods is their inherent dependence on the initial state, resulting in settling times that vary with different initial conditions. To address these limitations, Polyakov first proposed the fixed-time stability theory in [11]. Unlike finite-time stability, fixed-time stability guarantees that the upper bound of convergence time is independent of initial conditions, relying solely on design parameters. In [12], a novel fixed-time sliding-mode control strategy incorporating a time-varying threshold and additional parameters was proposed to address the trajectory tracking problem for autonomous surface vehicles. Further, an adaptive practical fixed-time sliding-mode controller was developed in [13] to handle model uncertainties and external disturbances affecting autonomous surface vehicles. In [14], a fast fixed-time sliding-mode trajectory tracking control strategy is proposed for autonomous surface vehicles, and faster convergence is achieved through a dual-phase state acceleration mechanism with a gain scheduling function. In [15], a global fixed-time stability result was derived by combining the power and hyperbolic cosine functions, and a fixed-time sliding-mode controller was developed accordingly. Moreover, when sensor measurements and control commands of UMVs are transmitted over bandwidth-limited network channels, signal quantization inevitably introduces quantization errors, degrading the control accuracy and robustness [16,17]. However, most existing fixed-time control methods based on terminal sliding mode under quantization conditions can only confine system states within a bounded region, failing to achieve exact convergence to the origin. Therefore, how to design a fixed-time control scheme with a dynamic parameter adjustment strategy is the first motivation of this study.

On the other hand, UMVs are routinely exposed to environmental disturbances, uncertainties in hydrodynamic coefficients, modeling errors, and parameter variations, further complicating accurate dynamic modeling [18,19]. To address this problem, recurrent neural networks have emerged as an effective solution due to their capability for approximating unknown parameters [20,21]. However, traditional recurrent neural networks structures suffer from short-term memory limitations and are prone to losing critical historical information, thus restricting their effectiveness in capturing highly fluctuating temporal features [22]. In estimating system uncertainties, long short-term memory networks have shown considerable advantages owing to their superior learning capabilities. In [23], a long short-term memory estimator integrating a selective update strategy was proposed to capture rapidly fluctuating temporal information for accurate sideslip angle estimation, based on which a learning line-of-sight guidance law for path following was developed. In [24], a long short-term memory model incorporating an enhanced forget-gate mechanism was developed, demonstrating superior performance in wind power forecasting tasks. Moreover, a read-first long short-term memory architecture introduced in [25] employs augmented read gates to enhance interactions among control gates, improve feature extraction during the encoding phase, and strengthen long-term memory capabilities. However, traditional long short-term memory networks exhibit certain limitations in capturing temporal dependencies within the gradient flow, which motivates the current study.

With the growing demand for high performance, safety, and reliability in UMV control systems, it is essential to develop effective fault-tolerant control (FTC) strategies, as actuator failures may lead to severe performance degradation or even mission failure. The authors of [26] propose an FTC strategy that employs an adaptive mechanism to estimate the upper bounds of time-varying stuck faults and external disturbances, without presuming prior knowledge of the lower bounds of thruster faults. The authors of [27] develop an integral sliding-mode, output-feedback FTC scheme for UMVs subject to unknown dynamics, thruster faults, and external disturbances; the dynamic-positioning objective is ensured via a high-gain compensator using the available outputs. To our knowledge, designing an effective fixed-time FTC scheme for UMVs under quantization errors and ocean disturbances remains a formidable challenge.

Based on the above analysis, this paper proposes a fixed-time FTC strategy utilizing an improved long short-term memory network to address the DP problem of UMVs under signal quantization, disturbances, and input saturation constraints. The main contributions are as follows:

(1) Unlike the long short-term memory networks in [23,24], this paper proposes a long short-term memory model integrated with adaptive mixed gradient (AMG) optimization algorithm, which enables more efficient capture of temporal features in gradient dynamics and substantially enhances the network’s capacity for modeling marine disturbances.

(2) Compared with [16,17], an improved dynamic quantization adjustment strategy is proposed by presetting the initial quantization threshold, which expands the upper bound of the quantization parameter range and enables more effective compensation for quantization errors.

(3) This paper proposes a fixed-time fault-tolerant controller that ensures the DP error converges to a small neighborhood of the equilibrium point within a finite time, independent of the initial state.

The rest of this article is organized as follows. The problem description and the quantizer model are provided in Section 2. A long short-term memory network structure combined with a long short-term memory optimization algorithm is developed in Section 3. The controller design and stability analysis are presented in Section 4. Simulation studies are demonstrated in Section 5. Finally, Section 6 concludes this article.

Notation: We write $\mathbb{R}$ for the set of real numbers, ${\mathbb{R}}^n$ for the space of n-dimensional real column vectors and ${\mathbb{R}}^{m\times n}$ for the space of $m\times n$ real matrices. For a matrix A, $\parallel A\parallel$ denotes the induced Euclidean norm, and $A^{\top }$ denotes the transpose of A. Let diag $(x_1,\dots ,x_n)$ be the diagonal matrix with diagonal entries $x_1,\dots ,x_n$. For $x={[x_1,\dots ,x_n]}^{\top }\in {\mathbb{R}}^n$, define ${sig}^r\left(x\right):=\left[\right|x_1{|}^{r}sgn\left(x_1\right),\dots ,|x_n{{|}^{r}sgn\left(x_n\right)]}^{\top }$, where $sgn(·)$ is the sign function. $round(·)$ denotes the rounding (integer-valued) function, $min(·,·)$ denotes the minimum of its arguments.

2. Problem Statement

2.1. UMV Modeling

In this section, a thruster-actuated three-degree-of-freedom UMV is considered, with the earth-fixed and body-fixed reference frames illustrated in Figure 1. As the focus is DP, the analysis is restricted to planar motion. Under low–speed operation, light to moderate sea states, and small attitude deviations, coupling with heave, roll, and pitch is negligible and therefore omitted. The kinematic and dynamic equations describing the motion of the UMV in the surge, sway, and yaw directions are adopted from the model presented in [3] and are given as follows:

$$\begin{array}{cc}\hfill M\dot{\vartheta }\left(t\right)+D\vartheta \left(t\right)+Q\xi \left(t\right)& =B{\tau }^F\left(t\right)+d\left(t\right),\hfill \\ \hfill \dot{\xi }\left(t\right)& =W\left(\psi \right(t\left)\right)\vartheta \left(t\right)\hfill \end{array}$$

(1)

where $\xi \left(t\right)={\left[\begin{array}{ccc}x\left(t\right)& y\left(t\right)& \psi \left(t\right)\end{array}\right]}^T\in {\mathbb{R}}^3$, with $x\left(t\right)$ and $y\left(t\right)$ denoting the position coordinates and $\psi \left(t\right)$ the yaw angle; $\vartheta \left(t\right)={\left[\begin{array}{ccc}u\left(t\right)& v\left(t\right)& r\left(t\right)\end{array}\right]}^T\in {\mathbb{R}}^3$, and $u\left(t\right)$, $v\left(t\right)$, $r\left(t\right)$ denoting the surge velocity, sway velocity, and yaw velocity, respectively. The vector ${\tau }^F\left(t\right)\in {\mathbb{R}}^6$ denotes the control input after thruster fault, while $d\left(t\right)\in {\mathbb{R}}^3$ captures unknown disturbances from wind, waves and currents. $B\in {\mathbb{R}}^{3\times 6}$ denotes the thruster configuration matrix. $M\in {\mathbb{R}}^{3\times 3}$ denotes the three-DOF inertia matrix satisfying $M=M^T>0$, $D\in {\mathbb{R}}^{3\times 3}$ denotes the hydrodynamic damping matrix, and $Q\in {\mathbb{R}}^{3\times 3}$ denotes the mooring-force matrices. Moreover, the rotation matrix $W\left(\psi \right(t\left)\right)$ is given by

$$\begin{array}{c}\hfill W\left(\psi \left(t\right)\right)=\left[\begin{array}{ccc}cos\left(\psi \right(t\left)\right)& -sin\left(\psi \right(t\left)\right)& 0\\ sin\left(\psi \right(t\left)\right)& cos\left(\psi \right(t\left)\right)& 0\\ 0& 0& 1\end{array}\right].\end{array}$$

(2)

The fault model follows the formulation presented in [26] and is detailed as follows:

$$\begin{array}{c}\hfill {\tau }^F\left(t\right)=\lambda \tau \left(t\right)+\mu {\tau }_s\left(t\right),\end{array}$$

(3)

where $\lambda$ is a diagonal semi-positive definite matrix representing the effectiveness of each thruster. It is defined as $\lambda \in \left\{{\lambda }^j\mid {\lambda }^j=\mathrm{diag}({\lambda }_1^j,{\lambda }_2^j,\dots ,{\lambda }_m^j),{\lambda }_i^j\in [{\underline{\lambda }}_i^j,{\overline{\lambda }}_i^j]\right\}$, where $0\le {\underline{\lambda }}_i^j\le {\overline{\lambda }}_i^j\le 1$. Fault factor $\mu \in \left\{{\mu }^j\mid {\mu }^j=\mathrm{diag}({\mu }_1^j,{\mu }_2^j,\dots ,{\mu }_m^j),{\mu }_i^j=0or{\mu }_i^j=1\right\}$. Here, the index $i=1,2,\dots ,m$ denotes the i-th thruster, and $j=1,2,\dots ,n$ denotes the j-th fault mode. ${\tau }_s\left(t\right)$ denotes the fault parameter of the thruster under conditions of time-varying jamming and full-deflection faults, satisfying $\parallel {\tau }_s\left(t\right)\parallel \le {\overline{\tau }}_s$, where ${\overline{\tau }}_s>0$ is the unknown constant upper bound on ${\tau }_s\left(t\right)$. Then, system (1) can be rewritten as

$$M\dot{\vartheta }\left(t\right)+D\vartheta \left(t\right)+Q\xi \left(t\right)=B(\lambda \tau \left(t\right)+\mu {\tau }_s\left(t\right))+d\left(t\right).$$

(4)

Due to the uncertainties in hydrodynamic coefficients and modeling errors, the UMV model exhibits dynamic uncertainties [28]. Accordingly, the matrices M, D, and Q are represented as the sum of their nominal components and associated uncertainty terms, given by

$$M=M_1+\overline{M},D=D_1+\overline{D},Q=Q_1+\overline{Q},$$

(5)

where $\overline{M},\overline{D},\overline{Q}$ represent standard term, and $M_1,D_1,Q_1$ denote uncertain terms, and the system (4) can be rewritten as

$$\begin{array}{cc}\hfill \overline{M}\dot{\vartheta }\left(t\right)+\overline{D}\vartheta \left(t\right)+\overline{Q}\xi \left(t\right)=& B(\lambda \tau \left(t\right)+\mu {\tau }_s\left(t\right))+\mathrm{\Delta }\left(t\right),\hfill \end{array}$$

(6)

where the lumped model uncertainties are denoted by $\mathrm{\Delta }\left(t\right)$, expressed as $\mathrm{\Delta }\left(t\right)=-M_1\dot{\vartheta }\left(t\right)-D_1\vartheta \left(t\right)-Q_1\xi \left(t\right)+d\left(t\right)$. Define $F=-{\overline{M}}^{-1}\overline{D},N=-{\overline{M}}^{-1}\overline{Q}$, ${\mathrm{\Delta }}_1\left(t\right)={\overline{M}}^{-1}\mathrm{\Delta }\left(t\right)$ and $B_0={\overline{M}}^{-1}B$. Then, the system (6) transforms into

$$\begin{array}{cc}\hfill \dot{\vartheta }\left(t\right)=& F\vartheta \left(t\right)+N\xi \left(t\right)+B_0(\lambda \tau \left(t\right)+\mu {\tau }_s\left(t\right))+{\mathrm{\Delta }}_1\left(t\right).\hfill \end{array}$$

(7)

Due to physical constraints such as motor drive capacity, propeller efficiency, and power supply limits, the actual thrust generated by the thrusters is inherently bounded, which may readily cause input saturation. Hence, the control input is imposed with the following symmetric saturation constraints.

$$-{\tau }_H\le \tau \left(t\right)\le {\tau }_H,$$

(8)

where ${\tau }_H$ is the limit of the control input, such that

$$\tau \left(t\right)=\left\{\begin{array}{cc}{\tau }_H,\hfill & \mathrm{if}\tau \ge {\tau }_H\hfill \\ {\tau }_0,\hfill & \mathrm{if}-{\tau }_H\le \tau <{\tau }_H\hfill \\ -{\tau }_H,\hfill & \mathrm{if}\tau <-{\tau }_H.\hfill \end{array}\right.$$

(9)

Thus, the control input $\tau \left(t\right)$ is specified by $\tau \left(t\right)={\tau }_0\left(t\right)+\mathrm{\Delta }\tau \left(t\right),$ where ${\tau }_0\left(t\right)$ is nominal part and $\mathrm{\Delta }\tau \left(t\right)$ is given as

$$\mathrm{\Delta }\tau \left(t\right)=\left\{\begin{array}{cc}{\tau }_H-{\tau }_0,\hfill & \mathrm{if}{\tau }_0\ge {\tau }_H\hfill \\ 0,\hfill & \mathrm{if}-{\tau }_H\le {\tau }_0<{\tau }_H\hfill \\ -{\tau }_H-{\tau }_0,\hfill & \mathrm{if}{\tau }_0<-{\tau }_H.\hfill \end{array}\right.$$

(10)

2.2. Quantizer Model

In this subsection, we consider quantization effects appearing in the sensor controller and controller thruster links. Since the DP vessel is connected to the remote control station via a communication network, bandwidth constraints unavoidably produce quantization errors in transmitted signals. Following the quantizer defined in [3], let $\varphi \in {\mathbb{R}}^p$ denote the variable subject to quantization. The static quantizer is defined as a mapping $q:{\mathbb{R}}^n\to \{q_1,\dots ,q_n\}$, where $\{q_1,\dots ,q_n\}\subset {\mathbb{R}}^n$ are the quantization points. The quantizer $q(·)$ exhibits the following characteristics:

$$\begin{array}{cc}\hfill & \parallel q\left(\varphi \right)-\varphi \parallel \le \iota ,\mathrm{if}\left|\varphi \right|\le H_q,\hfill \\ \hfill & \parallel q\left(\varphi \right)-\varphi \parallel >\iota ,\mathrm{if}\left|\varphi \right|>H_q,\hfill \end{array}$$

(11)

where $\iota$ denotes the quantization parameter, defined as $\iota =\sqrt{{\displaystyle \frac{p}{2}}}$, with p being the dimension of $\varphi$. The parameter $H_q$ represents the range of the quantizer $q(·)$. Accordingly, the dynamic quantizer $q_D(·)$ is defined as follows:

$$q_D\left(\varphi \right):=bq\left({\displaystyle \frac{\varphi }{b}}\right):=bround\left({\displaystyle \frac{\varphi }{b}}\right),b>0,$$

(12)

where b is a parameter of the quantizer, and the quantization error is defined as $E_b=q\left(\varphi \right)-\varphi$. Then, one has

$$\begin{array}{cc}\hfill & |E_{b_1}|=|q_D\left(x\left(t\right)\right)-x\left(t\right)|\le e_1{b}_1,\hfill \\ \hfill & |E_{b_2}|=|q\left(\tau \left(t\right)\right)-\tau \left(t\right)|\le e_2{b}_2,\hfill \end{array}$$

(13)

where $q_D\left(x\left(t\right)\right)$ denotes the dynamic quantizer and $q\left(\tau \right(t\left)\right)$ represents the static quantizer. The terms $e_1$ and $e_2$ represent quantization error parameters, where $b_1$ is a time-varying parameter and $b_2$ is a constant.

Under the presence of both quantization and input saturation, system (7) is reformulated as follows:

$$\dot{\vartheta }\left(t\right)=F\vartheta \left(t\right)+N\xi \left(t\right)+B_0\left({\tau }_q\left(t\right)+\mathrm{\Delta }{\tau }_q\left(t\right)\right)+{\omega }_d\left(t\right)+{\mathrm{\Delta }}_1\left(t\right),$$

(14)

where ${\tau }_q=q\left(\tau \left(t\right)\right)$ denotes the quantized control input, and ${\omega }_d\left(t\right)=B_0\mu {\tau }_s\left(t\right)+\mathrm{\Delta }{B}_0\left({\tau }_q\left(t\right)+\mathrm{\Delta }{\tau }_q\left(t\right)\right)+E\left(t\right)$ represents a nonlinear term. Here, $\mathrm{\Delta }{B}_0=B_0(\lambda -I)$ and $E\left(t\right)=-B_0\lambda E_{\tau }$, where $E_{\tau }$ is the quantization error associated with the control input. Define the error coordinates $x_1=\xi \left(t\right)-{\xi }_d$, $x_2=R\vartheta \left(t\right)$, and $x_e=x_2-x_d$, where ${\xi }_d$ and $x_d$ denote the desired position and velocity, respectively. Then, one obtains

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)=& x_2\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)=& W\left(\psi \right)S\vartheta \left(t\right)+W\left(\psi \right)(F\vartheta \left(t\right)+N\xi \left(t\right)+B_0({\tau }_q\left(t\right)+\mathrm{\Delta }{\tau }_q\left(t\right))+{\omega }_d\left(t\right)+{\mathrm{\Delta }}_1\left(t\right)),\hfill \\ \hfill =& F_1\vartheta \left(t\right)+N_1\xi \left(t\right)+B_1({\tau }_q\left(t\right)+\mathrm{\Delta }{\tau }_q\left(t\right))+{\omega }_R\left(t\right)+h_t,\hfill \end{array}\hfill \end{array}\right.$$

(15)

where $F_1=W\left(\psi \right)(S+F)$, $N_1=W\left(\psi \right)N$, $B_1=W\left(\psi \right)B_0$, ${\omega }_R\left(t\right)=W\left(\psi \right){\omega }_d\left(t\right)$, $h_t=W\left(\psi \right){\mathrm{\Delta }}_1$, and $S=\left[\begin{array}{ccc}0& -r\left(t\right)& 0\\ r\left(t\right)& 0& 0\\ 0& 0& 0\end{array}\right]$ is a skew-symmetric matrix.

Remark 1.

$q_D\left(x\left(t\right)\right)$ denotes the dynamic quantizer and $q\left(\tau \left(t\right)\right)$ denotes the static quantizer. The parameters $e_1$ and $e_2$ characterize the corresponding quantization errors, with $b_1\left(t\right)$ a time-varying coefficient and $b_2$ a constant.

To guarantee the efficacy of the designed fixed-time FTC strategy, the following Lemma and Assumptions are presented.

Assumption 1.

The existence of unknown positive constants ${\overline{\tau }}_s$ and $\overline{d}$ is implied by the fact that both the thruster stuck fault ${\tau }_s\left(t\right)$ and external disturbance $d\left(t\right)$ are piece-wise continuous bounded functions.

$$\parallel {\tau }_s\left(t\right)\parallel \le {\overline{\tau }}_s,\parallel d\left(t\right)\parallel \le \overline{d}.$$

Assumption 2.

The prediction error between the long short-term memory outputs and the actual values is denoted by ϵ, which is assumed to be bounded as $ϵ\le {ϵ}_l$, where ${ϵ}_l$ is a constant.

Remark 2.

Assumption 1 is necessary to compensate for the adverse effects caused by time-varying stuck faults and external disturbances. Since the external disturbance $d\left(t\right)$ and the stuck-fault signal are both constrained by physical limits and vary slowly over time, it is reasonable to model them as piecewise-continuous bounded signals [26]. Under bounded inputs, a long short-term memory with bounded activation functions and bounded weights produces bounded outputs. Therefore, Assumption 2 that the estimation error admits a constant upper bound is reasonable.

Lemma 1

([29]). Suppose there exists a positive definite and continuous Lyapunov function $V\to \mathbb{U}$ such that $\dot{V}\left(t\right)\le -\beta V^h\left(t\right)-\phi V^g\left(t\right)+\overline{\mathrm{\Delta }}$, where $\beta ,\phi >0$, $0<h<1$, $g>1$ and $0<\overline{\mathrm{\Delta }}<\infty$. Then, the system state will converge to a neighborhood ${\mathrm{\Omega }}_1$ of the equilibrium point. Moreover, the convergence time t satisfies

$$t\le t_{\max }={\displaystyle \frac{1}{\overline{a}\beta (1-h)}}+{\displaystyle \frac{1}{\overline{a}\phi (g-1)}},$$

where $\overline{a}$ represents a scalar and fulfills $0<\overline{a}<1$, and the neighborhood Ω can be expressed as $\mathrm{\Omega }=\{x|\parallel x\parallel \le min\{{\phi }^{-1/g}{({\displaystyle \frac{\overline{\mathrm{\Delta }}}{1-\overline{a}}})}^{1/g},{\beta }^{-1/h}{({\displaystyle \frac{\overline{\mathrm{\Delta }}}{1-\overline{a}}})}^{1/h}\}\}$, where $x\left(t\right)$ is system state.

2.3. Control Objective

The objective of this study is to design a fixed-time fault-tolerant controller based on long short-term memory learning for DP vessels subject to signal quantization, external disturbances, and input saturation constraints. The proposed control scheme ensures accurate DP within a fixed time, while effectively compensating for various uncertainties and enhancing the robustness and reliability of the system. The system architecture is illustrated in Figure 2.

3. Improved LSTM Network

The long short-term memory network, a variant of recurrent neural networks, effectively addresses the challenges of capturing long-term dependencies inherent in sequential data. Central to long short-term memory is its unique gating mechanism, comprising three primary gates: the input gate, forget gate, and output gate. These gates collaboratively regulate the flow and retention of information within the network, dynamically updating the internal cell states. Through this sophisticated gating architecture, long short-term memory significantly mitigates common issues in traditional recurrent neural networks, such as gradient vanishing and exploding, thereby achieving enhanced performance in processing and predicting temporal sequences compared to conventional models. Therefore, to effectively predict the lumped disturbances in system (15), a long short-term memory network is employed in this paper. The network structure is shown in Figure 3 and can be expressed as follows.

$$\begin{array}{cc}\hfill i_t& =\sigma \left(W_{ii}{x}_t+b_{ii}+W_{hi}{\widehat{h}}_{(t-1)}+b_{hi}\right)\hfill \\ \hfill f_t& =\sigma \left(W_{if}{x}_t+b_{if}+W_{hf}{\widehat{h}}_{(t-1)}+b_{hf}\right)\hfill \\ \hfill {\tilde{x}}_t& =tanh\left(W_{ig}{x}_t+b_{ig}+W_{hg}{\widehat{h}}_{(t-1)}+b_{hg}\right)\hfill \\ \hfill O_t& =\sigma \left(W_{io}{x}_t+b_{io}+W_{ho}{\widehat{h}}_{(t-1)}+b_{ho}\right)\hfill \\ \hfill x_t& =f_t\ast x_{t-1}+i_t\ast g_t\hfill \\ \hfill {\widehat{h}}_t& =O_t\ast tanh\left(x_t\right),\hfill \end{array}$$

(16)

where $b_{ii},b_{hi},b_{if},b_{hf},b_{ig},b_{hg},b_{io}$, and $b_{ho}$ represent bias vectors, $W_{ii},W_{hi},W_{if},W_{hf},W_{ig}$, $W_{hg},W_{io}$ and $W_{ho}$ denote weight matrices, and ${\tilde{x}}_t$ is an intermediate variable. The central innovation of long short-term memory lies in its memory cell $x_t$, which selectively accumulates and updates state information using three gating mechanisms: the input gate $i_t$, forget gate $f_t$, and output gate $O_t$.

To precisely capture temporal dependencies within the gradient flow, this paper introduces an AMG optimization algorithm, whose detailed implementation is presented in Algorithm 1.

Algorithm 1 AMG Optimization Algorithm with Temporal Dependencies

Input: Learning rate ${\eta }_r$; decay rates ${\pi }_1,{\pi }_2\in [0,1)$; memory factor $\varrho \in [0.6,1)$; stochastic objective function $g\left(x\right)$ is defined to describe the performance index of the system under stochastic disturbances; initial values $m_0=0$, $v_0=0$, $t=0$ Output: Optimized parameters $x_t$   1: Initialize $t\leftarrow 0$, $m_0\leftarrow 0$, $v_0\leftarrow 0$   2: while not converged do   3:       $t\leftarrow t+1$   4:       Compute gradient: ${\iota }_t\leftarrow \nabla g\left(x_t\right)$   5:       Update learning rate: ${\eta }_t\leftarrow {\eta }_r/\sqrt{t}$   6:       Update first moment: $m_t\leftarrow {\pi }_1{m}_{t-1}+(1-{\pi }_1){\iota }_t$   7:       Temporal average: ${\widehat{m}}_t\leftarrow \varrho {\widehat{m}}_{t-1}+(1-\varrho )m_t$   8:       Update second moment: $v_t\leftarrow {\pi }_2{v}_{t-1}+(1-{\pi }_2){\iota }_t^2$   9:       Temporal average: ${\widehat{v}}_t\leftarrow \varrho {\widehat{v}}_{t-1}+(1-\varrho )v_t$    10:        Update parameters: $x_{t+1}\leftarrow {\mathrm{\Pi }}_X(x_t-{\eta }_t{\displaystyle \frac{{\widehat{m}}_t}{\sqrt{{\widehat{v}}_t}}})$    11:  end while

Subsequently, a fixed-time extended state observer is constructed to estimate the thruster faults $w_R$, whose dynamics are expressed as follows:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\dot{\widehat{x}}}_{q_1}\left(t\right)=& {\widehat{x}}_{q_2}\left(t\right)+{\beta }_1{\mathrm{sig}}^{n_1}\left(e\left(t\right)\right)+k_1{\mathrm{sig}}^{L_1}\left(e\left(t\right)\right)\hfill \\ \hfill {\dot{\widehat{x}}}_{q_2}\left(t\right)=& {\widehat{\omega }}_R\left(t\right)+F_1\vartheta \left(t\right)+N_1\xi \left(t\right)+B_1({\tau }_q\left(t\right)+\hfill \\ & \mathrm{\Delta }{\tau }_q\left(t\right))+h_t+{\beta }_2{\mathrm{sig}}^{n_2}\left(e\left(t\right)\right)+k_2{\mathrm{sig}}^{L_2}\left(e\left(t\right)\right)\hfill \\ \hfill {\dot{\widehat{\omega }}}_R\left(t\right)=& {\beta }_3{\mathrm{sig}}^{n_3}\left(e\left(t\right)\right)+k_3{\mathrm{sig}}^{L_3}\left(e\left(t\right)\right),\hfill \end{array}\hfill \end{array}\right.$$

(17)

where $e\left(t\right)=x_{q_1}\left(t\right)-{\widehat{x}}_{q_1}\left(t\right)$ and ${\beta }_i>0$, $n_i=i{n}_i-i+1$, $L_i=L_i+(i-1),(i=1,2,3)$, which satisfies $n_1\in (2/3,1)$, and $L_1=1/n_1$. ${\widehat{\omega }}_R$ denotes the estimate of the thruster fault $w_R$. Subsequently, the error dynamics will be derived as follows: $x_d\left(t\right)$

$$\left\{\begin{array}{c}\dot{e}\left(t\right)=e_1\left(t\right)-{\beta }_1{\mathrm{sig}}^{n_1}\left(e\left(t\right)\right)-k_1{\mathrm{sig}}^{L_1}\left(e\left(t\right)\right)\hfill \\ {\dot{e}}_1\left(t\right)=e_2\left(t\right)-{\beta }_2{\mathrm{sig}}^{n_2}\left(e\left(t\right)\right)-k_2{\mathrm{sig}}^{L_2}\left(e\left(t\right)\right)\hfill \\ {\dot{e}}_2\left(t\right)={\widehat{\omega }}_D-{\beta }_3{\mathrm{sig}}^{n_3}\left(e\left(t\right)\right)-k_3{\mathrm{sig}}^{L_3}\left(e\left(t\right)\right),\hfill \end{array}\right.$$

where $e_1\left(t\right)=x_{q_2}\left(t\right)-{\widehat{x}}_{q_2}\left(t\right)$, $e_2\left(t\right)={\omega }_R-{\widehat{\omega }}_R$; according to the Theorem 2 in [1], the errors $e\left(t\right),e_1\left(t\right)$ and $e_2\left(t\right)$ will converge to zero in fixed-time $t_d$. The convergence time $t_d$ is

$$t_d\le {\displaystyle \frac{1}{c_3(1-d_m)}}+{\displaystyle \frac{2}{c_1(d_n-1)}}+{\displaystyle \frac{1}{b{c}_2(1-d_m)}},$$

(18)

where $0<d_m<1$, $d_n>1$, $0<b<1$, $c_1$, $c_2$ and $c_3$ are positive constants.

Remark 3.

The ${\omega }_R\left(t\right)$ exhibits continuous differentiability over time, satisfying the condition $\parallel {\dot{\omega }}_R\left(t\right)\parallel \le F_d$, where $F_d$ denotes the upper bound of ${\dot{\omega }}_R\left(t\right)$.

4. Main Results

This section first presents the design of a fixed-time quantized TSM surface. On this basis, we develop a fixed-time TSM controller with anti-saturation capability.

4.1. Fixed-Time Quantized TSM Surface Design

The fixed-time TSM surface is designed as

$$S\left(x\right)=K{x}_1\left(t\right)+{\mathrm{sig}}^r\left(x_2\left(t\right)\right),$$

(19)

where $K=\mathrm{diag}(K_1,K_2,K_3)$ and K is defined as follows

$$K=\left(z\right|x_1{\left(t\right)|}^{p-(1/r)}+l|x_1\left(t\right){{|}^{o-(1/r)})}^r,$$

(20)

where $o={\displaystyle \frac{1+\theta }{2}}+{\displaystyle \frac{\theta -1}{2}}\mathrm{sign}\left(\right|x_1\left(t\right)|-1),z>0,l>0,r>1,\theta >1,1/r<p<1$. When $S\left(x\right)=0$, solving (19), yields the following fixed-time upper bound on the settling time $t_s$: $t_s\le {\displaystyle \frac{1}{l(1-p)}}ln(1+{\displaystyle \frac{z}{l}})+{\displaystyle \frac{1}{l(\theta -1)}}$.

Lemma 2.

The fixed-time quantized TSM surface is designed in the following form:

$$S_q\left(x\right)=K{x}_{q_1}\left(t\right)+{\mathrm{sig}}^r\left(x_{q_2}\left(t\right)\right),$$

(21)

when the system state $x\left(t\right)$ lies within the neighborhood

$$\mathrm{\Omega }=\left\{x\left(t\right):{\varsigma }_1{\rho }_1(\left(z+l\parallel l_{t_3}\parallel )∥l_{t_1}∥+z^r\right)+2{\varsigma }_2{\rho }_2∥l_{t_2}∥\right\},$$

(22)

it can be guaranteed that $\mathrm{sign}\left(S\left(x\right)\right)=\mathrm{sign}\left(S_q\left(x\right)\right)$, where $l_{t_i}(i=1,2,3)$ are Taylor expansion residual terms, and $x_{q_1},x_{q_2}$ denote the quantized system states.

Proof. 

By utilizing Equations (19) and (21), we can obtain

$$\begin{array}{cc}\hfill S_q\left(x\right)-S\left(x\right)=& K{x}_{q_1}\left(t\right)+{\mathrm{sig}}^r\left(x_{q_2}\left(t\right)\right)-K{x}_1\left(t\right)-{\mathrm{sig}}^r\left(x_2\left(t\right)\right).\hfill \end{array}$$

(23)

Case 1: When $|x_1\left(t\right)|>1$, the fixed-time TSM surface takes the following form:

$$\begin{array}{cc}\hfill & S\left(x\right)=\left(z\right|x_1{\left(t\right)|}^p+l|x_1\left(t\right){{|}^{\theta })}^r+{\mathrm{sig}}^r\left(x_2\left(t\right)\right),\hfill \\ \hfill & S_q\left(x\right)=\left(z\right|x_{q_1\left(t\right)}{|}^p+l|x_{q_1\left(t\right)}{{|}^{\theta })}^r+{\mathrm{sig}}^r\left(x_{q_2}\left(t\right)\right).\hfill \end{array}$$

(24)

Based on the terminal sliding-mode surface before quantization (19) and its quantized counterpart (21), the following result can be obtained

$$S\left(x\right)-S_q\left(x\right)<{\left(z\left|x_1\left(t\right)\right|+l{\left|x_1\left(t\right)\right|}^{\theta }\right)}^r-{\left(z\left|x_{q_1}\left(t\right)\right|+l{\left|x_{q_1}\left(t\right)\right|}^{\theta }\right)}^r+{\left|x_2\left(t\right)\right|}^r-{\left|x_{q_2}\left(t\right)\right|}^r.$$

(25)

According to the quantization error in (23) and the factorization formula $A^n-B^n=(A-B){\sum }_{i=0}^{n-1}{A}^i{B}^{n-1-i}$, it follows that

$$\begin{array}{cc}\hfill S\left(x\right)-S_q\left(x\right)\le & \left(z\right(|x_1\left(t\right)|-|x_{q_1}\left(t\right)\left|\right)+l\left(\right|x_1\left(t\right)|-|x_{q_1}\left(t\right)\left|\right)\sum _{i=0}^{\theta -1}|x_1{\left(t\right)|}^i|x_{q_1}\left(t\right){|}^{\theta -1-i})\hfill \\ & +\sum _{i=0}^{r-1}\left(z\right|x_1\left(t\right)|+l|x_1{\left(t\right)\left|\right)}^i\left(z\right|x_{q_1\left(t\right)}|+l|x_{q_1}\left(t\right){\left|\right)}^{r-1-i}\hfill \\ \hfill & +\left(\right|x_2\left(t\right)|-|x_{q_2}\left(t\right)\left|\right)\sum _{i=0}^{r-1}|x_2{\left(t\right)|}^i{\left|x_{q_2}\left(t\right)\right|}^{r-1-i}\hfill \\ \hfill \le & {\varsigma }_1{\rho }_1(z+l\parallel l_{t_3}\parallel )\parallel l_{t1}\parallel +{\varsigma }_2{\rho }_2\parallel l_{t_2}\parallel .\hfill \end{array}$$

(26)

where $l_{t_1}={\sum }_{i=0}^{r-1}{\left(z|x_1\left(t\right)|+l|x_1\left(t\right)|\right)}^i{\left(z|x_{q_1}\left(t\right)|+l|x_{q_1}\left(t\right)|\right)}^{r-1-i}$, $l_{t_2}={\sum }_{i=0}^{r-1}|x_2{\left(t\right)|}^i{\left|x_{q_2}\left(t\right)\right|}^{r-1-i}$, and $l_{t_3}={\sum }_{i=0}^{\theta -1}|x_1{\left(t\right)|}^i{\left|x_{q_1}\left(t\right)\right|}^{\theta -1-i}$ represent the corresponding remainder terms.

Case 2: When $|x_1\left(t\right)|\le 1$, the fixed-time TSM surface takes the following form

$$\begin{array}{cc}\hfill & S\left(x\right)=\left(z\right|x_1{|}^p+l|x_1{\left|\right)}^r+{\mathrm{sig}}^r\left(x_2\right)\hfill \\ \hfill & S_q\left(x\right)=\left(z\right|x_{q_1}{|}^p+l|x_{q_1}{\left|\right)}^r+{\mathrm{sig}}^r\left(x_{q_2}\right),\hfill \end{array}$$

(27)

we have

$$\begin{array}{cc}\hfill S\left(x\right)-S_q\left(x\right)=& \left(z\right|x_1{\left(t\right)|}^p+l|x_1{\left(t\right)\left|\right)}^r+{\mathrm{sig}}^r\left(x_2\left(t\right)\right)-\left(z\right|x_{q_1}{\left(t\right)|}^p+l|x_{q_1}\left(t\right){\left|\right)}^r-{\mathrm{sig}}^r\left(x_{q_2}\left(t\right)\right)\hfill \\ \hfill <& \left(z\right|x_1{\left(t\right)|}^p{)}^r-\left(z\right|x_{q_1}{\left(t\right)|}^p{)}^r+|x_2{\left(t\right)|}^r-{\left|x_{q_2}\left(t\right)\right|}^r\hfill \\ \hfill <& z^r|x_1\left(t\right)|-z^r|x_{q_1}\left(t\right)|+{\varsigma }_2{\rho }_2\parallel l_{t_2}\parallel \hfill \\ \hfill <& z^r{\varsigma }_1{\rho }_1+{\varsigma }_2{\rho }_2\parallel l_{t_2}\parallel .\hfill \end{array}$$

(28)

In conclusion, it follows that

$$\begin{array}{cc}\hfill S\left(x\right)-S_q\left(x\right)& \le {\varsigma }_1{\rho }_1\left(\right(z+l\parallel l_{t_3}\parallel )\parallel l_{t_1}\parallel +z^r)+2{\varsigma }_2{\rho }_2\parallel l_{t_2}\parallel .\hfill \end{array}$$

(29)

The proof is completed.    □

Remark 4.

Compared with the conventional finite-time integral terminal sliding-mode surface, the terminal sliding surface designed in this section introduces a higher-order integral term $S\left(x\right)=K{x}_1\left(t\right)+{\mathrm{sig}}^r\left(x_2\left(t\right)\right)$, which addresses the difficulty of accurately obtaining the system’s initial state. Consequently, its convergence time depends only on the design parameters, thereby accelerating the system’s convergence.

Remark 5.

This study examines the relationship between the quantized fixed-time TSM surface and its non-quantized counterpart. By utilizing Equation (29), it is further shown that the system state converges to a specified neighborhood, thereby providing the basis for the stability analysis in the subsequent section.

4.2. Controller Design and Stability Analysis

To solve the DP problem of UMVs under signal quantization, disturbances, and input saturation, we propose the following long short-term memory-based fixed-time fault-tolerant anti-saturation controller

$${\tau }_n\left(t\right)=-B_1^T{\left(B_1{B}_1^T\right)}^{-1}({\tau }_{q_0}\left(t\right)+{\tau }_{q_1}\left(t\right)),$$

(30)

where

$$\begin{array}{cc}\hfill {\tau }_{q_0}\left(t\right)=& F_1\vartheta \left(t\right)+N_1\xi \left(t\right)+{\widehat{\omega }}_R\left(t\right)+F\mathrm{sig}\left(S\left(x\right)\right)+\widehat{h}\left(t\right)\hfill \\ & +{\displaystyle \frac{1}{r}}\left|x_2\left(t\right)\right|{\mathrm{sig}}^{2-r}(\overline{K}\left(x_1\left(t\right)\right)x_2\left(t\right)+K{x}_2\left(t\right)),\hfill \end{array}$$

and the switching control component ${\tau }_{q_1}\left(t\right)$ is designed as

$$\begin{array}{cc}\hfill {\tau }_{q_1}\left(t\right)=& {\displaystyle \frac{1}{r}}{f}_{ϵ}{\left|x_2\left(t\right)\right|}^{1-r}({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)+\kappa +K_{s}S\left(x\right)),\hfill \end{array}$$

where $K_s=\mathrm{diag}[K_{s_1},K_{s_2},K_{s_3},\dots ,K_{s_n}]$ is a diagonal positive definite matrix, with each diagonal element satisfying $K_{s_i}>0$ for $i=1,2,\dots ,n$. $\widehat{h}\left(t\right)$ denotes the long short-term memory estimate of the lumped disturbance $\mathrm{\Delta }\left(t\right)$. The constants ${\sigma }_1$ and ${\sigma }_2$ are positive, and the exponents satisfy $0<m_1<1$, $m_2>1$. The anti-windup compensator $\kappa$ is then defined as follows:

$$\dot{\kappa }=-K_{\zeta }\kappa -\left({\sigma }_3{\mathrm{sig}}^{m_1}\left(\kappa \right)+{\sigma }_4{\mathrm{sig}}^{m_2}\left(\kappa \right)\right)+S\left(x\right)+\mathrm{\Delta }\tau ,$$

(31)

where $K_{\zeta }$ is a diagonal positive definite matrix, ${\sigma }_3$ and ${\sigma }_4$ are positive constants, and the function $f_{ϵ}$ is defined as follows:

$$f_{ϵ}=\left\{\begin{array}{cc}sin\left({\displaystyle \frac{\pi |x_2{\left(t\right)|}^{r-1}}{2}}\right),\hfill & \mathrm{if}|x_2{\left(t\right)|}^{r-1}\le {ϵ}_2\hfill \\ 1,\hfill & \mathrm{if}|x_2{\left(t\right)|}^{r-1}>{ϵ}_2\hfill \end{array}\right.$$

(32)

where $ϵ>0$, we can get $f_{ϵ}\in \left[0,1\right]$, and as $x_2$ approaches infinity, $f_{ϵ}$ tends to zero, thus mitigating the issue of singularity.

Theorem 1.

Consider the UMV DP system (1) under Assumptions 1 and 2. Apply the fixed-time extended state observer (17), the fixed-time sliding-mode fault-tolerant controller (30), and the adaptive law (31). Then the state $x\left(t\right)$ converges, within a user-assigned fixed time t, to the compact set

$${\mathrm{\Omega }}_1=\{x\left(t\right):\parallel x\left(t\right)\parallel \le min({\varpi }_2^{-{\displaystyle \frac{2}{1+m_1}}}{\left(\frac{{\overline{\mathrm{\Delta }}}_1}{1-\overline{a}}\right)}^{{\displaystyle \frac{2}{1+m_1}}},{\varpi }_1^{-2}{\left(\frac{{\overline{\mathrm{\Delta }}}_1}{1-\overline{a}}\right)}^2\left)\right\},$$

where

$$\begin{array}{cc}\hfill x\left(t\right)& =\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right],{\overline{\mathrm{\Delta }}}_1={\rho }_1{\left|{\mathrm{\Omega }}_1\right|}^{1+m_1}+{\rho }_1{\kappa }_1\left|{\mathrm{\Omega }}_1\right|,\hfill \\ \hfill {\varpi }_1& =\sqrt{2}{\rho }_1\parallel {\kappa }_1\parallel ,{\varpi }_2=\sqrt{2^{1+m_1}{\rho }_1},\hfill \end{array}$$

and the total settling time t is given by

$$t=t_1+t_2+t_d+t_s,$$

with

$$\begin{array}{cccc}\hfill t_1& \le {\displaystyle \frac{2}{\overline{a}{\psi }_{l1}(1-m_1)}}+{\displaystyle \frac{2}{\overline{a}{\psi }_{l2}(m_2-1)}},\hfill & \hfill t_2& \le {\displaystyle \frac{1}{\overline{a}zp}}+{\displaystyle \frac{1}{\overline{a}lo}},\hfill \\ \hfill t_d& \le {\displaystyle \frac{1}{{\overline{a}}_1(1-d_m)}}+{\displaystyle \frac{2}{{\overline{a}}_2(d_n-1)}}+{\displaystyle \frac{1}{b{\overline{a}}_3(1-d_m)}},\hfill & \hfill t_s& \le {\displaystyle \frac{1}{l(1-p)}}ln\left(1+\frac{z}{l}\right)+{\displaystyle \frac{1}{l(\theta -1)}}.\hfill \end{array}$$

Here, $t_1$ and $t_2$ denote the fixed times required for the sliding variable to evolve from $S\left(x\right)\ne 0$ to $S\left(x\right)=0$, $t_d$ is the fixed-time convergence bound of the extended state observer, and $t_s$ is the time needed for the state error to contract to the equilibrium neighbourhood ${\mathrm{\Omega }}_1$.

Proof. 

Choose the Lyapunov functional candidate as

$$V\left(t\right)={\displaystyle \frac{1}{2}}{S}^T\left(x\right)S\left(x\right)+{\displaystyle \frac{1}{2}}{\kappa }^T\kappa ,$$

(33)

Taking the time derivative of $V\left(t\right)$ along the trajectory of the closed-loop system (15), we obtain

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)=& S^T\left(x\right)(\overline{K}\left(x_1\left(t\right)\right)x_2\left(t\right)+K{x}_2\left(t\right)+r{\left|x_2\left(t\right)\right|}^{r-1}(F_1\vartheta \left(t\right)+N_1\xi \left(t\right)\hfill \\ \hfill & +B_1({\tau }_q\left(t\right)+\mathrm{\Delta }{\tau }_q\left(t\right))+{\omega }_R\left(t\right)+h_t\left)\right)+{\kappa }^T\dot{\kappa }\hfill \\ \hfill \le & S^T\left(x\right)rX\left({\omega }_R\left(t\right)+{ϵ}_l-{\widehat{\omega }}_R\left(t\right)-F\mathrm{sig}\left(S\left(x\right)\right)+B_1\mathrm{\Delta }{\tau }_q\left(t\right)\right)\hfill \\ \hfill & -S^T\left(x\right)f_{ϵ}\left({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)+\kappa +K_{s}S\left(x\right)\right)+{\kappa }^T\dot{\kappa }.\hfill \end{array}$$

(34)

where $X=\mathrm{diag}\left\{\right|x_{2_1}{|}^{r-1},|x_{2_2}{|}^{r-1},|x_{2_3}{|}^{r-1}\}$. Based on Assumption 2 and Remark 3, the above inequality can be simplified as follows:

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & S^T\left(x\right)rX\left({ϵ}_l+B_1\mathrm{\Delta }{\tau }_q\left(t\right)\right)-S^T\left(x\right)f_{ϵ}({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)\hfill \\ \hfill & +{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)+\kappa +K_{s}S\left(x\right))+{\kappa }^T\dot{\kappa }.\hfill \end{array}$$

According to Young’s inequality, it can be obtained that $S^T\left(x\right)rX{B}_1\mathrm{\Delta }{\tau }_q\left(t\right)\le {\displaystyle \frac{r}{2}}{S}^T\left(x\right)X{B}_1{B}_1^T{X}^{T}S\left(x\right)+{\displaystyle \frac{r}{2}}\mathrm{\Delta }{\tau }_q^T\left(t\right)\mathrm{\Delta }{\tau }_q\left(t\right)$, and by a similar approach, it can be further simplified as

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -S^T\left(x\right)f_{ϵ}\left({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)\right)-{\kappa }^T\left({\sigma }_3{\mathrm{sig}}^{m_1}\left(\kappa \right)+{\sigma }_4{\mathrm{sig}}^{m_2}\left(\kappa \right)\right)\hfill \\ \hfill & +\left({\displaystyle \frac{1+r}{2}}\right)\mathrm{\Delta }{\tau }_q^T\left(t\right)\mathrm{\Delta }{\tau }_q\left(t\right)+{\displaystyle \frac{r}{2}}{ϵ}_l^T{ϵ}_l-{\kappa }^T\left(K_{\kappa }-{\displaystyle \frac{1}{2}}{I}_3\right)\kappa \hfill \\ \hfill & +\left({\displaystyle \frac{r}{2}}{\parallel X{B}_1\parallel }^2-{\displaystyle \frac{1}{2}}\right)\parallel f_{ϵ}{\parallel }^2-\parallel f_{ϵ}{K}_s\parallel +{\displaystyle \frac{1}{2}}{I}_3+{\displaystyle \frac{r}{2}}{\parallel X\parallel }^2{\parallel S\left(x\right)\parallel }^2.\hfill \end{array}$$

(35)

When the system state satisfies $\parallel x_2{\left(t\right)\parallel }^{r-1}>{ϵ}_2$, we obtain

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -S^T\left(x\right)\left({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)\right)-{\kappa }^T\left({\sigma }_3{\mathrm{sig}}^{m_1}\left(\kappa \right)+{\sigma }_4{\mathrm{sig}}^{m_2}\left(\kappa \right)\right)\hfill \\ \hfill & +\left({\displaystyle \frac{1+r}{2}}\right)\mathrm{\Delta }{\tau }_q^T\left(t\right)\mathrm{\Delta }{\tau }_q\left(t\right)+{\displaystyle \frac{r}{2}}{ϵ}_l^T{ϵ}_l-{\kappa }^T\left(K_{\kappa }-{\displaystyle \frac{1}{2}}{I}_3\right)\kappa \hfill \\ \hfill & +\left({\displaystyle \frac{r}{2}}\parallel X{B}_1{\parallel }^2-\parallel K_s\parallel +{\displaystyle \frac{r}{2}}{\parallel X\parallel }^2\right){\parallel S\left(x\right)\parallel }^2.\hfill \end{array}$$

By choosing parameter values to ensure $\left({\displaystyle \frac{r}{2}}\parallel X{B}_1{\parallel }^2-\parallel K_s\parallel +{\displaystyle \frac{r}{2}}{\parallel X\parallel }^2\right)>0$ and $\left(K_{\kappa }-{\displaystyle \frac{1}{2}}{I}_3\right)>0$, the above inequality simplifies to

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -S^T\left(x\right)\left({\sigma }_1{\mathrm{sig}}^{m_1}\left(S\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S\left(x\right)\right)\right)\hfill \\ \hfill & -{\kappa }^T\left({\sigma }_3{\mathrm{sig}}^{m_1}\left(\kappa \right)+{\sigma }_4{\mathrm{sig}}^{m_2}\left(\kappa \right)\right)\hfill \\ \hfill & +\left({\displaystyle \frac{1+r}{2}}\right)\mathrm{\Delta }{\tau }_q^T\left(t\right)\mathrm{\Delta }{\tau }_q\left(t\right)+{\displaystyle \frac{r}{2}}{ϵ}_l^T{ϵ}_l.\hfill \end{array}$$

(36)

According to $\mathrm{sign}\left(S\left(x\right)\right)=\mathrm{sign}\left(S_q\left(x\right)\right)$, and based on the banded area (26) and adaptive law (31), we obtain

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -S_q^T\left(x\right)\left({\sigma }_1{\mathrm{sig}}^{m_1}\left(S_q\left(x\right)\right)+{\sigma }_2{\mathrm{sig}}^{m_2}\left(S_q\left(x\right)\right)\right)\hfill \\ & -{\kappa }^T\left({\sigma }_3{\mathrm{sig}}^{m_1}\left(\kappa \right)+{\sigma }_4{\mathrm{sig}}^{m_2}\left(\kappa \right)\right)+{\mathrm{\Omega }}_3\hfill \\ & -\mathrm{\Omega }\left({\sigma }_1{\left|S_q\left(x\right)\right|}^{m_1+1}+{\sigma }_2{\left|S_q\left(x\right)\right|}^{m_2+1}\right)\hfill \\ \hfill \le & -(1+\mathrm{\Omega })\left({\sigma }_1{\left|S_q\left(x\right)\right|}^{m_1+1}+{\sigma }_2{\left|S_q\left(x\right)\right|}^{m_2+1}\right)\hfill \\ & -\left({\sigma }_3{\left|\kappa \right|}^{m_1+1}+{\sigma }_4{\left|\kappa \right|}^{m_2+1}\right)+{\mathrm{\Omega }}_3\hfill \\ \hfill \le & -(1+\mathrm{\Omega }){\left({\sigma }_1{\left|S\left(x\right)\right|}^{1+m_1}+{\sigma }_2{\left|S\left(x\right)\right|}^{1+m_2}\right)}^{\frac{1}{2}}\hfill \\ & -{\left({\sigma }_3{\left|\kappa \right|}^{1+m_1}+{\sigma }_4{\left|\kappa \right|}^{1+m_2}\right)}^{\frac{1}{2}}+{\overline{\mathrm{\Omega }}}_3\hfill \\ \hfill \le & -{\psi }_{l1}V{\left(t\right)}^{{\displaystyle \frac{1+m_1}{2}}}-{\psi }_{l2}V{\left(t\right)}^{{\displaystyle \frac{1+m_2}{2}}}+{\overline{\mathrm{\Omega }}}_3.\hfill \end{array}$$

(37)

where

$$\begin{array}{cc}\hfill {\psi }_{l1}& =2^{(1+m_1)/2}min\left((1+\mathrm{\Omega }){\sigma }_1,{\sigma }_3\right)>0,\hfill \\ \hfill {\psi }_{l2}& =2^{(1+m_2)/2}min\left((1+\mathrm{\Omega }){\sigma }_2,{\sigma }_4\right)>0,\hfill \\ \hfill {\overline{\mathrm{\Omega }}}_3& =\left({\displaystyle \frac{1+r}{2}}\right)\mathrm{\Delta }{\tau }^T\left(t\right)\mathrm{\Delta }\tau \left(t\right)+{\displaystyle \frac{r}{2}}{ϵ}_l^T{ϵ}_l>0.\hfill \end{array}$$

The system state can converge to the nearby neighborhood ${ϵ}_2$ a within a fixed time $t_1$, and the convergence time $t_1$ can be expressed as

$$t_1\le {\displaystyle \frac{2}{\overline{a}{\psi }_{l1}(1-m_1)}}+{\displaystyle \frac{2}{\overline{a}{\psi }_{l2}(m_2-1)}},$$

(38)

According to Lemma 1, the neighborhood can be described as

$${\mathrm{\Omega }}_1=\left\{x\left(t\right):\parallel x\left(t\right)\parallel \le min\left\{{\psi }_{l2}^{-2/(1+m_2)}{\left({\displaystyle \frac{{\overline{\mathrm{\Omega }}}_3}{1-\overline{a}}}\right)}^{2/(1+m_2)},{\psi }_{l1}^{-2/(1+m_1)}{\left({\displaystyle \frac{{\overline{\mathrm{\Omega }}}_3}{1-\overline{a}}}\right)}^{2/(1+m_1)}\right\}\right\}.$$

(39)

When the system state satisfies $\parallel x_2{\left(t\right)\parallel }^{r-1}\le {ϵ}_2$, it follows that $0<f_{ϵ}<1$, and we obtain

$$\parallel S\left(x\right)\parallel \le min\left\{{\psi }_{l2}^{-2/(1+m_2)}{\left({\displaystyle \frac{{\overline{\mathrm{\Omega }}}_3}{1-\overline{a}}}\right)}^{2/(1+m_2)},{\psi }_{l1}^{-2/(1+m_1)}{\left({\displaystyle \frac{{\overline{\mathrm{\Omega }}}_3}{1-\overline{a}}}\right)}^{2/(1+m_1)}\right\}.$$

(40)

Let $x_1\left(t\right)=-\mathrm{sig}\left(z{\mathrm{sig}}^p\left(x_1\left(t\right)\right)+l{\mathrm{sig}}^o\left(x_1\left(t\right)\right)\right)+S^{1/r}\left(x\right)$. Define the Lyapunov function as

$$V_{x_1}\left(t\right)=\sum _{i=1}^3\left|x_{1i}\right|=\sum _{i=1}^3\mathrm{sign}\left(x_{1i}\right){\dot{x}}_{1i}\left(t\right).$$

(41)

Taking the time derivative of $V_{x_1}\left(t\right)$ yields

$$\begin{array}{cc}\hfill {\dot{V}}_{x_1}\left(t\right)=& -\sum _{i=1}^3\left|z{\mathrm{sig}}^p\left(x_{1i}\left(t\right)\right)+l{\mathrm{sig}}^o\left(x_{1i}\left(t\right)\right)+\mathrm{sign}\left(x_{1i}\right)S^{1/r}\right|\hfill \\ \hfill \le & -\left(z{V}_{x_1}^p\left(t\right)+l{V}_{x_1}^o\left(t\right)\right)+\left|S^{1/r}\right|.\hfill \end{array}$$

(42)

The system state will converge to a neighborhood near the equilibrium point within a fixed time $t_2$, and the settling time $t_2$ can be expressed as

$$t_2\le {\displaystyle \frac{1}{\overline{a}zp}}+{\displaystyle \frac{1}{\overline{a}lo}}.$$

(43)

According to Lemma 1, the system state will converge to the neighborhood of the equilibrium point denoted by ${\mathrm{\Omega }}_1$. According to the following algorithm, when $b_1\left(t\right)\to 0$, we have $x\left(t\right)\to 0$. Eventually, the system state will converge to the equilibrium point in a fixed time t, which can be expressed as

$$t=t_1+t_2+t_d+t_s.$$

(44)

The proof is completed.    □

Next, we propose a dynamic adjustment strategy for the quantization parameter to ensure that $x\left(t\right)\to 0$ whenever $b_1\left(t\right)\to 0$.

5. Simulation Results

In this section, simulations are conducted on a floating production vessel to demonstrate the effectiveness of the proposed fixed-time FTC strategy. Following [28], the model matrices are adopted. The specific numerical values of M, D, Q, B, and the disturbance $d\left(t\right)$ used in system (1) are summarized in

Table 1. Model matrices for the reference floating production vessel (as in [28]).

Matrix	Numerical Value	Matrix	Numerical Value
M	$\left[\begin{array}{ccc}1.0852& 0& 0\\ 0& 2.0575& -0.4087\\ 0& -0.4087& 0.2153\end{array}\right]$	D	$\left[\begin{array}{ccc}0.0865& 0& 0\\ 0& 0.0762& 0.0151\\ 0& 0.0151& 0.0310\end{array}\right]$
Q	$\left[\begin{array}{ccc}0.0398& 0& 0\\ 0& 0.0266& 0\\ 0& 0& 0\end{array}\right]$	$d\left(t\right)$	$\left[\begin{array}{c}sin\left(0.01t\right)\\ -0.09sin\left(0.01t\right)\\ 0.9\end{array}\right]$
B	$\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 1& 1\\ 0.0472& -0.0472& -0.4108& -0.3858& 0.4554& 0.3373\end{array}\right]$

.

The model uncertainties account for $10\%$ and $d\left(t\right)$ is considered as $d_1\left(t\right)=sin\left(0.01t\right)$; $d_2\left(t\right))=-0.09sin\left(0.01t\right)$; $d_3\left(t\right)=0.9$. The data of the UMV were collected offline using an ocean disturbance dataset, which was employed to train the long short-term memory neural network in an offline manner. The long short-term memory model consists of five layers, with a seven-dimensional input comprising six input features and one target output. The objective function used for training is the minimization of the mean squared error of the prediction. To update the weights, the adaptive mixed gradient optimization algorithm optimizer is utilized, where the initial learning rate is 0.01 and the momentum factor is 0.7. The total uncertainty is estimated using the proposed long short-term memory network integrated with the AMG optimization algorithm. Figure 4 illustrates the predicted curve of the total uncertainty obtained by the long short-term memory. The minimal error between the predicted and actual values effectively demonstrates the accuracy of the proposed estimation approach. The following fault scenario is considered: the port main thruster experiences a 10% reduction in effectiveness, and the starboard main propeller is stuck at $0.1$, while the remaining thrusters operate normally. The corresponding design parameters are specified as follows: ${\tau }_H=0.1$, $r=m_2=h=1.2, m_1=0.8, p=0.85, z=l=0.5$, $\sigma ={\sigma }_1={\sigma }_2={\sigma }_3=0.2$, $K_{\zeta }=\mathrm{diag}[1,1,1]$, $K_S=\mathrm{diag}[1.5,1.5,1.5]$, $k\left(0\right)=0.5$, ${\gamma }^{\ast }=0.1$. All numerical simulations in this paper were carried out in the MATLAB R2024a environment. The gain adjustment follows a performance-driven procedure grounded in Lyapunov analysis. Starting from the prescribed performance pair $(T,\epsilon )$, the performance indices are fixed. We then derive a feasible region for the controller gains from the Lyapunov inequality/theorem and compute (or update) the gains by solving the associated LMI with the constraint $\dot{V}<0$. Finally, a scaling refinement is applied under actuator-saturation and anti-chattering requirements to ensure both a certified fixed-time convergence upper bound and practical implementability. To visualize the DP performance, the Figure 5 shows the UMV trajectory on the 3–D $xyz$ plane. The initial position is approximately $(0.1,0.1,0)\mathrm{m}$. Under the proposed controller, the vessel undergoes a brief adjustment and rotation, rapidly converges toward the origin, and performs small refinements in its vicinity. The terminal positioning error is about $\left(\right|x|,|y|,|z\left|\right)\approx (7.5\times {10}^{-5},5.5\times {10}^{-4},0)\mathrm{m}$. These results indicate that the proposed method achieves smooth, stable dynamic positioning with high-precision convergence in the plane.

As reported in

Table 2. Performance of optimizers.

Optimizers	AMG	Adam	EKF	RMS
RMSE	0.168	0.212	0.174	0.241
MAE	0.124	0.301	0.156	0.241
MAPE	8.43%	10.6%	7.93%	8.47%
SMAPE	9.44%	12.9%	10.4%	15.6%

, the AMG algorithm delivers the best accuracy: relative to Adaptive Moment (Adam), root-mean-square error (RMSE) and mean absolute error (MAE) drop by 4.5% and 17%, and symmetric mean absolute percentage error (SMAPE) reaches a minimum of 3.46%.

5.1. Hardware-in-the-Loop Validation of the Proposed Control Scheme

To further validate the feasibility of the proposed control method, a hardware-in-the-loop (HiL) simulation platform is established for testing. As shown in Figure 6, the HiL platform is composed of a Raspberry Pi 4B controller that runs Ubuntu 20.04 and ROS Noetic, and a model computer that runs Windows 11 and MATLAB 2024a. Due to the hardware limitations of the Raspberry Pi and cross-system data transfer requirements, it is necessary to increase the solution step size and modify multiple design parameters. The other settings are the same as described above.

The pose data of the UMV are recorded via data acquisition modules in Simulink and exported to MATLAB for plotting, the specific results are shown in Figure 7. Although the results are inferior to software simulation, the control objectives are achieved, which demonstrates the feasibility for future engineering applications.

Remark 6.

The proposed control method is implemented on a Raspberry Pi 4B development board for hardware-in-the-loop simulation testing. The results demonstrate that the method exhibits a certain degree of real-time capability. Future work will explore model compression techniques (e.g., quantization, pruning) and efficient architectures to enable real-time operation under hardware constraints.

Remark 7.

The optimizer’s hyperparameters—learning rate ($\eta ={10}^{-3}$), momentum ($\beta =0.7$), and decay factor ($\gamma =0.95$)—were selected via grid search and validated across simulation scenarios. Performance remained stable within moderate ranges (e.g., $\eta \in [{10}^{-4},{10}^{-2}]$), whereas extreme values led to divergence or oscillation.

5.2. Comparative Analysis of Fixed-Time TSMC and Finite-Time TSMC

The simulation results in Figure 8 show that the proposed approach achieves noticeably faster convergence of both positions and velocities than the finite-time TSMC and fixed-time TSMC baselines. In particular, relative to the finite-time controller in [7], the settling time is substantially shortened. This acceleration translates into enhanced closed-loop stability and a quicker transient response properties that are essential for real-time applications demanding high precision and rapid regulation. Furthermore, the corresponding control efforts in Figure 9 and Figure 10 remain uniformly bounded, with smaller peaks and quicker decay than the finite-time TSMC baseline.

5.3. Comparative Analysis of Dynamic Adjustment of Quantization Parameters

To assess the proposed scheme for dynamic adjustment of quantization parameters in Algorithm 2, we compare it against the uniform static-quantization controller reported in [17]. As shown in Figure 11, the dynamic-tuning strategy drives the state to equilibrium noticeably faster. Continuous adjustment of the quantization parameters suppresses the error throughout the transient, whereas the static design attains stability more slowly and with larger overshoot. These results confirm the superiority of the dynamic adjustment mechanism.

Algorithm 2 Adjustment Strategy of Quantization Parameter $b_1\left(t\right)$

1: Input: Select $b_1\left(t_0\right)=\gamma b_1\left(t_1\right)$, the parameter $\gamma$ is chosen as ${\gamma }^{\ast }/{\mathrm{\Omega }}_1<\gamma <1$, ${\mathrm{\Omega }}_1=\left\{x\left(t\right)\right|\parallel x\left(t\right)\parallel \le min\{{\varpi }_2^{-2/1+m_1}{\left({\displaystyle \frac{{\overline{\mathrm{\Delta }}}_1}{1-\overline{a}}}\right)}^{2/1+m_1},{\varpi }_1^{-2}{\left({\displaystyle \frac{{\overline{\mathrm{\Delta }}}_1}{1-\overline{a}}}\right)}^2\}\}$, where ${\gamma }^{\ast }$ is a positive constant and ${\mathrm{\Omega }}_1$ is defined in Theorem 1.    2: Output: $b_1\left(t_i\right)$    3: $i=0$;    4: $j=1$;    5: while $j=1$ do    6: ${\displaystyle q\left(\frac{x\left(t\right)}{b_1\left(t_i\right)}\right)=bround\left(\frac{x\left(t\right)}{k^i{b}_1\left(t_0\right)}\right)}$    7:   if $\parallel q\left({\displaystyle \frac{x\left(t\right)}{b_1\left(t_i\right)}}\right){\parallel }_2>{\mathrm{\Omega }}_1$, then    8:     $b_1\left(t_{i+1}\right)=k^i{b}_1\left(t_i\right)$; $j=0$;    9:   else 10:       $i=i+1$; 11:      $b_1\left(t_{i+1}\right)=k^i{b}_1\left(t_i\right)$; 12:   end if 13: end while

5.4. Comparative Analysis of the Impact of Input Saturation

To further evaluate the performance of the proposed anti-saturation compensation controller, we simulated the controller with and without input-saturation compensation. As depicted in Figure 12, the saturation-aware scheme drives the position, yaw, and velocity states to the origin more rapidly and with reduced overshoot. After a short transient, all states remain bounded and converge to zero, whereas the saturation-neglecting controller develops increasing oscillations and ultimately destabilizes the system. Hence, explicitly incorporating thruster limits markedly improves closed-loop robustness.

5.5. Comparative Analysis of the Impact of Thruster Faults

In this subsection, we compare a baseline without fault accommodation to the proposed FTC. We consider the following fault scenario: for $t\le 7 \mathrm{s}$, all thrusters operate normally; for $t>7 \mathrm{s}$, the port main thruster and aft tunnel thruster I are stuck at $0.001$ and $0.01$, respectively; the starboard main thruster experiences a $30\%$ loss of effectiveness; the remaining thrusters operate nominally. As shown in Figure 13 and Figure 14, without fault handling the surge/sway/yaw velocities and the position/attitude states exhibit pronounced oscillations and noticeable steady-state offsets. In contrast, under the proposed FTC all states and control inputs remain bounded and converge rapidly to the desired equilibrium despite thruster faults, demonstrating closed-loop stability and effective fault mitigation.

5.6. Comparative Analysis of the Impact of Long Short-Term Memory Network and Adaptive Law

To provide a more comprehensive evaluation of the proposed controller, we compared its performance with that of the controller based on the adaptive law presented in [7,9]. The simulation results are shown in Figure 15. As shown in Figure 15, due to the superior capability of neural network in approximating nonlinear functions, the state curve demonstrates enhanced stability and robustness when subjected to time-varying disturbances. However, the adaptive parameters vary with these time-varying disturbances, which can lead to degraded control performance and cause the system state to diverge.

Remark 8.

In the comparative simulations, the same disturbances, input constraints, initial conditions, simulation time, and solution step size are set, although different control methods are used for the comparison, which may have resulted in different design parameters for the controllers. However, all parameters are adjusted as much as possible according to the same standards (such as control accuracy, control input size, and response speed). Therefore, a fair comparison may be guaranteed.

6. Conclusions

This paper proposed a fixed-time FTC scheme for DP of UMVs in the presence of signal quantization, external disturbances, and thruster saturation. As a key technical contribution, we designed a long short-term memory network enhanced with the adaptive mixed gradient optimizer to accurately learn lumped disturbances in real time. Next, a fixed-time extended state observer compensates for thruster effectiveness loss within a pre-assigned settling time that is independent of the initial state. To counteract quantization effects, we construct a terminal sliding-mode manifold that explicitly incorporates quantization parameters and employ an adaptive uniform quantizer with online-tunable sensitivity. By integrating the disturbance learner, fault estimates, and quantization mechanism, the resulting fixed-time FTC law guarantees that all closed-loop states converge to a small neighborhood of the origin within a fixed time, regardless of initial conditions. In our future work, the prescribed-time FTC of UMVs will be considered as a meaningful research topic. However, the research is conducted under ideal communication conditions without considering communication delays and data packet loss. Therefore, SMC methods that account for practical communication issues will be investigated in future research.