Research on Practical Path Tracking Control of Autonomous Underwater Vehicle Based on Constructive Dynamic Gain Controller

Abstract

In this study, the stochastic nonlinear system-based trajectory tracking control problem of an autonomous underwater vehicle (AUV) is studied. We investigate the time-varying gain adaptive control method to find possible approaches to reduce the excessive computational burden. Enhanced adaptive algorithms are devised by considering the dynamic characteristics of AUV motion. By transforming the original controller design problems into parameter selection problems and subsequently solving them using the functional time-varying observer technical theorem, we can achieve optimal control performance. The control system is shown to constrain system state error due to stochastic disturbances within arbitrarily small domains. A coordinate transformation is proposed for all system states to meet boundedness conditions. We show that the closed-loop stability is confirmed, the system is asymptotically probabilistically stable, and contraction limits given in the stability analysis may be used to certify the convergence of the AUV trajectory errors. A large number of simulation studies using an underwater vehicle model have proved the effectiveness and robustness of the proposed approach. A real-time, time-varying gain constructive control strategy is further developed for the hardware-in-the-loop simulation; the effectiveness of the controller design is confirmed by introducing the controller into the AUV actuator model.

1. Introduction

Scientific research and underwater exploration both frequently use autonomous underwater vehicles (AUVs). In AUV research, the control system of underwater thrusters can be said to be its neural center and the most complex technology, so the control of underwater thrusters is one of the most difficult issues in the control industry. Due to the crucial role played by AUVs in deep-sea environmental resource exploration in recent decades, tracking control, state maintenance, and the path planning of AUVs have received attention from researchers, causing a pressing need to develop new control theories in the theoretical and engineering fields of AUV output feedback control, making them fit for intricate systems impacted by a variety of elements, including randomness, delays in time and nonlinearity. In addition, stochastic nonlinear systems no longer work with a control framework developed for deterministic systems [1,2,3,4,5,6,7,8,9,10,11]. As a result, it is very difficult to use the classical control theory and other associated mathematical techniques to address the output feedback control problem for stochastic nonlinear systems. In engineering practice, there is a high probability that the input and output of a controlled system are random signals. Therefore, stochastic nonlinear systems that combine nonlinear and stochastic characteristics are a field of study that the community of control theory is very interested in.

The application of trajectory tracking control is essential for autonomous underwater robots. However, due to the complex system, uncertain working environments, and the highly coupled nonlinear characteristics of underwater robots [1,2], the controller design of AUVs becomes very challenging. In the past decades, the design of trajectory tracking controllers [3] for underwater vehicles has been a problem worthy of attention. Traditional PID, LQR, and Kalman filtering [4] can control the trajectory of an underwater robot. In addition, nonlinear PID controllers also have good performance in control. However, when the target trajectory being tracked is nonlinear, the curved trajectory makes nonlinear PID no longer applicable in vehicle driving. Feedback linearization [5] presents a powerful tool for dealing with nonlinear characteristics. However, applying feedback linearization to the AUV requires a system model with highly accurate hydraulic dynamic coefficients [6]. In this context, the adaptive [7,8,9] Lyapunov method has become the mainstream method for AUV trajectory control. In article [10], a dynamic controller is used for the first time in the backstepping control technique. AUV output backstep control can be found in article [11,12]; another popular AUV trajectory tracking control method is sliding mode control. It is well-known that this method has an excellent and robust effect when the parameters are uncertain, but the sliding effect of the sliding mode control tends to appear discrete. The sliding mode control is frequently used in conjunction with other control strategies, such as robust control, adaptive control, and PID, to lessen this problem [13,14,15].

Nonetheless, the above control methods have, as a general deficiency, the inability to deal with the stochastic characteristics and uncertain disturbances of the system. For AUVs, these factors are prevalent [16,17], and the actuator must manage them in the working environment. Adaptive control is featured in dealing with randomness and uncertainties [18], proposing a robust control structure in the face of this wide range of control problems. In addition, adaptive control can solve complex nonlinear problems for the control of AUV dynamic systems. So far, we do not have a very effective control method for the trajectory tracking control problems. In article [19], a self-correcting adaptive method is proposed, to solve problems including dynamic parameters, where the model structure often changes. Trajectory tracking control algorithms for AUV stochastic uncertain nonlinear systems have been proposed. In paper [20], the concept of the stochastic system model is creatively introduced. In article [21,22], a comprehensive trajectory tracking control and path planning problem is researched. A unified optimization framework [23] is developed for the problem of combined motion control for AUVs.

Although an adaptive control based on a key idea in high-order sliding modes provides a good algorithm for nonlinear systems with random and uncertain disturbances [13] when solving, this method heavily burdens the computer. In theoretical research, the computation time is often ignored, but in practice, the computational difficulty increases exponentially with the increase in stochastic and uncertain interference problems. Due to the short sampling period, many strategies such as iterative methods, precomputation, and numerical continuation [24] have been proposed to decrease the computational complexity and reduce the computation time. Ref. [25] explores the motion characteristics of underwater robots, combining a dynamic-static system model with a high-gain observer to develop an effective output feedback adaptive control algorithm. However, the closed-loop system’s stability evidence is omitted. Dynamic gain control algorithms may fix the underwater robots’ trajectory tracking and control issues. The stability analysis of the closed-loop system is made more difficult by the implicit coupling of the system state and the control signal. It is urgent to find a better stability analysis method. Refs. [26,27,28,29] mainly focus on the study of control strategies in finite time using multi-parameter uncertainty mathematical models, and the robust control part has an important reference value for our paper.

This paper hopes to provide a method to eliminate the influence of random and uncertain disturbances on AUV trajectory tracking and simultaneously ensure closed-loop system stability. Here, a reference augmentation technique is applied in modeling the AUV system so that the coupling between the systems is weakened [30]. Then, a dynamic gain method is studied to reduce the computational stress of stochastic uncertain nonlinear systems. The complex control problem has thereby been changed into a parameter selection and construction problem through dynamic gain. The computational complexity is significantly reduced, and the computation time is significantly reduced. The following are the article’s contributions:

• An established stochastic uncertain nonlinear dynamic adaptive control algorithm is offered to examine the dynamic properties of the AUV’s motion with the goal of controlling the problem of tracking the AUV’s trajectory.
• The computational complexity of the control method is dramatically reduced via a novel dynamic-static combination of high-gain observers that are provided, turning the controller design challenge into a problem of parameter calculation and selection.
• We investigate the fundamental properties of closed-loop systems. The control time and control error of numerical nonlinear model trajectory tracking is remarkably reduced, the control effect is optimized, and the sensitivity is improved. According to [7], the numerical model in our paper can match the actual AUV model in certain key feature parameters.
• Numerical simulations and additional experiments reveal that the proposed dynamic-static high-gain adaptive control algorithm has excellent and robust performance against stochastic and uncertain disturbances.

The remainder of this essay is structured as follows: The motion model for AUVs is discussed in the second part. The third section is the dynamic and static high-gain adaptive control algorithm. Section 4 presents simulation studies and additional experiments. The fifth section is the conclusion and prospects.

2. Description of Dynamic Modelling of Robot

The dynamics study of the AUV is completed in this part. Through the analysis of the power, resistance, and inertial force when moving underwater, we can obtain general dynamic equations. To model the complexity and uncertainty of the underwater environment [2,26,31], we have added a stochastic process to the model. This work is based on an established underwater robot model with underactuated control, as illustrated in Figure 1.

The following is an example of the 6-DOF dynamic model of the AUV:

$$M\dot{𝓋}+C(𝓋)𝓋+D(𝓋)𝓋+g\left(ϵ\right)=F$$

(1)

where the inertia mass matrix, which includes an added mass term to model the effect of the surrounding water on the acceleration of the AUV, is $M=M_A+M_{RB}$. $C(𝓋)$ is the underwater vehicles’ Coriolis force matrix. The underwater vehicle fluid damping matrix is represented by the symbol $D(𝓋)$. $g\left(ϵ\right)$ stands for the undersea vehicle’s operating gravity and the buoyancy-generated restoring force (moment) matrix. $F={\left[F_x,F_y,F_z,F_k,F_m,F_n\right]}^T$ stands for the resulting forces and moments. $\dot{v}={\left[\dot{u},\dot{𝓋},\dot{\omega },\dot{p},\dot{q},\dot{r}\right]}^T$ are the linear and angular accelerations of the body (moving) frame in the direction of pitch, roll, and heave; $v={\left[u,𝓋,\omega ,p,q,r\right]}^T$ stands for the angular and linear speed in relation to the body (moving) frame; $u$ stands for surge velocity; $𝓋$ stands for sway velocity; $\omega$ stands for heave velocity; $p$ stands for roll rate; $q$ stands for pitch rate; and $r$ stands for yaw rate. $ϵ={\left[x,y,z,\alpha ,\beta ,\gamma \right]}^T$ are the position and orientation in the inertial (earth-fixed) frame; $x$ stands for surge position, $y$ for sway position, $z$ for heave position, $\alpha$ for roll angle, $\beta$ for pitch angle, and $\gamma$ for yaw angle.

According to the above, the six-degree-of-freedom (6-DOF) model of the underwater robot has complex nonlinearity and state coupling. If we want to design a controller with six degrees of freedom, these factors will pose a great challenge to the controller design and physical characteristics. Here, we decompose the 6-DOF motion model into two kinematic models, separating linear velocity variables $v_1={\left[u,𝓋,\omega \right]}^T$ and angular velocity variables $v_2={\left[p,q,r\right]}^T$. The pose can similarly be divided into two kinematic models in the earth-fixed frame, underwater robot position ${ϵ}_1={\left[x,y,z\right]}^T$ and underwater robot orientation ${ϵ}_2={\left[\alpha ,\beta ,\gamma \right]}^T$, which can highly simplify the AUV model. In this study, we solely consider velocity variables $v_1={\left[u,v,\omega \right]}^T$ and position ${ϵ}_1={\left[x,y,z\right]}^T$ of the AUV underwater.

In terms of linear velocity, the fixed frame and body frame have the following relationship:

$${\dot{ϵ}}_1=J\left(ϵ\right)v_1$$

(2)

where J(ϵ) is the kinematic transformation matrix of the following structure:

$$J\left(ϵ\right)=\left(\begin{array}{ccc}c\gamma c\beta & -s\gamma c\alpha +c\gamma s\beta s\alpha & s\gamma s\alpha +c\gamma c\beta s\alpha \\ s\gamma c\beta & c\gamma c\alpha +s\gamma s\beta s\alpha & -c\gamma s\alpha +s\gamma s\beta c\alpha \\ -s\beta & c\beta s\alpha & c\beta c\alpha \end{array}\right)$$

where, accordingly, $s=\sin$, $c=\cos$ and ${ϵ}_2={\left[\alpha ,\beta ,\gamma \right]}^T$ is the angle between the direction of the surge, sway, and heave, and the earth frame. When $\beta =\pm {90}^{°}$, this transformation is undefined, and the quaternion method must be considered. In practice, most robots are made to work at pitch angles below $\pm {90}^{°}$. Thus, this restriction does not matter in this case. To better understand and analyze the motion state, we will study the AUV system based on the earth reference frame. To unify the signal states, using Equation (2) for coordinate transformation $\left({ϵ}_1,v_1\right)\stackrel{\mu }{\to }\left({ϵ}_1,{\dot{ϵ}}_1\right)$, we obtain:

$$\left(\begin{array}{c}{ϵ}_1\\ {\dot{ϵ}}_1\end{array}\right)=\left[\begin{array}{cc}I& 0\\ 0& J\left(ϵ\right)\end{array}\right]\left(\begin{array}{c}{ϵ}_1\\ v_1\end{array}\right)$$

A similarity transformation in a linear system is an analog of the coordinate transformation, which is a global diffeomorphism. The dynamic underwater vehicle model in a fixed reference frame for the earth looks like this:

$$M_{{ϵ}_1}{\ddot{ϵ}}_1+C_{{ϵ}_1}{\dot{ϵ}}_1+D_{{ϵ}_1}{\dot{ϵ}}_1+g_{{ϵ}_1}=F_{{ϵ}_1}$$

(3)

where

$$\begin{array}{l}{M}_{{ϵ}_1}=J{(ϵ)}^{-T}MJ{(ϵ)}^{-1}\\ C_{{ϵ}_1}=J{(ϵ)}^{-T}\left(C\left(\nu \right)-MJ{(ϵ)}^{-1}\dot{J}\left(ϵ\right)\right)J{(ϵ)}^{-1}\\ D_{{ϵ}_1}=J{(ϵ)}^{-T}D\left(\nu \right)J{(ϵ)}^{-1}\\ g_{{ϵ}_1}=J{(ϵ)}^{-T}g\left(ϵ\right)\\ F_{{ϵ}_1}=J{(ϵ)}^{-T}F\end{array}$$

(4)

We need some assumptions as follows.

Assumption 1.

In this paper, there is only consideration for the velocity between the self-position and the fixed frame, that the velocity is $\dot{l}={\left({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2\right)}^{\frac{1}{2}}$. Suppose the origin state ${ϵ}_1\left(0\right)=\left[0 0 0\right]$ and ${\dot{ϵ}}_1\left(0\right)=\left[0 0 0\right]$.

Assumption 2.

In this paper, the position ${ϵ}_1=\left[x y z\right]$ and the angle ${ϵ}_2=\left[\alpha \beta \gamma \right]$ are known from the sensors.

For a class of stochastic uncertain nonlinear systems, we concentrate on the problem of trajectory tracking control in this study. For the underwater robot’s trajectory tracking control issue with nonlinear dynamics, to better design the control algorithm, we convert the dynamic model of the system into a broad numerical model as follows.

$$\begin{array}{c}\mathrm{d}{l}_1=l_{2}dt+{\phi }_1\left(t,l\left(t\right),u\right)d\omega \\ \mathrm{d}{l}_2=udt+{\phi }_2\left(t,l\left(t\right),u\right)d\omega \\ y=l_1-y_r \end{array}$$

(5)

In stochastic nonlinear systems, $l={\left(l_1,l_2\right)}^T\in R^2$, $u\in \mathrm{R}$, $y\in \mathrm{R}$ represent the system’s states, input, and output, respectively; ${\mathrm{y}}_r$ is the target trajectory to be tracked. Here, state $l_2$ and $y_r$ are unmeasurable. The stochastic process of the system is introduced in this paper: An $m$-dimensional standard Wiener process called $\omega$ defined on the entire probability space $\left(\Omega , \Gamma , P\right)$, where $p$ is the probability measure, $\mathsf{\Gamma }$ is the filter, and $\mathsf{\Omega }$ is the sample space. The nonlinear term ${\phi }_i:{ℝ}^{+}\times ℝ\times {ℝ}^2,i=1,2$ is satisfied continuous polynomial growth conditions and local Lipschitz.

In this paper, we assume that system (5) satisfies the following assumptions and implements output feedback tracking control.

Assumption 3.

There is a known integer $q\ge 1$ and a positive constant $c_0\ge 0$ such that the following disparity exists

$$\left|{\phi }_i\left(t,l,u\right)\right|\le c_0\left(1+{\left|l_1\right|}^q\right)\left(\left|l_1\right|+\left|l_2\right|+\cdots +\left|l_i\right|\right)+c_0, i=1,2$$

(6)

This assumption shows that an output polynomial growth rate governs this system (5).

Assumption 4.

The target trajectory $max\left(\left|y_r\left(t\right)\left|, \right|{\dot{y}}_r\left(t\right)\right|\right)\le c_1$, $c_1$ is a known constant. According to the description of the above system, the purpose of this part is to develop a time-variable gain constructive control algorithm: for any given tolerance $\delta >0$. All of the states in the closed-loop system must be clearly defined and globally bounded on $\left[0,+\infty \right)$ in finite time $T_{\delta }>0$, for instance, $\left|y\left(t\right)\right|=\left|l_1\left(t\right)-y_r\left(t\right)\right|\le \delta , \forall t\ge T_{\delta }$.

Remark 1.

It should be pointed out that the output feedback tracking control problems studied in previous pieces of literature [32,33] are mainly aimed at nonlinear systems or nonlinear systems with parameter uncertainty and unknown control direction. The system (5) under study in this research is a stochastic uncertain nonlinear system with the introduction of random elements. At the same time, system (5) is observable by Assumption 3, which depends on the unmeasurable state and nonlinear growth conditions satisfying a polynomial growth condition based on Assumption 4. As can be observed, the upper bound value of the derivative as well as the upper bound value of the reference trajectory $y_r$ are provided; therefore, it is not essential to provide a specific description, function, or additional details for the reference trajectory $y_r$.

Based on Assumptions 3 and 4 and system (5), for $l_1=y+y_r, i=1, 2$, we have:

$$\begin{array}{c}\left|{\phi }_i\left(t,l,u\right)\right|\le c_0\left(1+{\left|y+y_r\right|}^q\right)\left(\left|l_1\right|+\left|l_2\right|+\cdots +\left|l_i\right|\right)+c_0\\ \le c\left(1+{\left|y\right|}^q\right)\left(\left|l_1\right|+\left|l_2\right|+\cdots +\left|l_i\right|\right)+c\end{array}$$

(7)

where $c=c_{0}max\left\{1+2^{q-1}{c}_1^q, 2^{q-1}\right\}$ is a known constant.

3. Dynamic and Static High-Gain Adaptive Control Algorithm

The complexity of the stochastic uncertain nonlinear underwater robot system will be studied in two parts using the dynamic gain observer technique that we suggest in this section, which will provide a dynamic adaptive tracking controller. The first step is the dynamic gain observer and controller design; the second step is stability analysis and implementation.

3.1. Dynamic Gain Observer and Controller Design

We introduce the following state transformation:

$$\begin{array}{l}{z}_1=y=l_1-y_r\hfill \\ z_2=l_2 \hfill \end{array}$$

(8)

Then, combining (5) and (8), the updated system can be obtained as:

$$\begin{array}{l}{\dot{z}}_1=z_2+f_1\left(t,z,u\right)\dot{\omega }\hfill \\ {\dot{z}}_2=u+f_2\left(t,z,u\right)\dot{\omega }\hfill \\ y=z_1 \hfill \end{array}$$

(9)

where

$$\begin{array}{l}{f}_1\left(t,z,u\right)={\phi }_1\left(t,z_1+y_r,z_2,u\right)-\frac{d{y}_r}{d\omega }\hfill \\ f_2\left(t,z,u\right)={\phi }_2\left(t,z_1+y_r,z_2,u\right) \hfill \end{array}$$

(10)

Since the states of the stochastic system are unmeasurable except for ${\mathrm{l}}_1$, which is a measurable state, the time-varying observers are first constructed:

$$\begin{array}{c}{\dot{\widehat{z}}}_1={\mathrm{z}}_2+K^1{h}_1\left(z_1-{\widehat{z}}_1\right),\\ {\dot{\widehat{z}}}_2=u+K^2{h}_2\left(z_1-{\widehat{z}}_1\right), \end{array}$$

(11)

We choose constant design parameters $h_i, i=1, 2$, satisfying the Hurwitz condition for the polynomial $s^2+h_{1}s+h_2$. The dynamic gain parameter $K=AB\left(t\right)$ is made up of two components: a constant ($A>1$) and a time-varying function $B\left(t\right)$ that is updated by

$$\begin{array}{l}\dot{B}\left(t\right)=-{\mu }_1{B}^2+{\mu }_2{\left(1+{\left|y\right|}^q\right)}^{2}B\\ B\left(0\right)=1\end{array}$$

(12)

Theorem 1.

Assuming that all closed-loop system states are bounded on $\left[0, T\right)$, we can deduce that $T$ must be infinite. Otherwise, the closed-loop system’s continuity will be violated since at least one state will escape at time $T$. The type of stochastic uncertain nonlinear system (5), has an output of the form $y=l_1-y_r$, and its nonlinear term satisfies Assumptions 3 and 4. If we choose the suitable parameters$A, {\mu }_1$ ${\mu }_2$ and use observer (11), the design of an output feedback controller can be as follows:

$$u=-\left(K^2{a}_1{\widehat{z}}_1+K{a}_2{\widehat{z}}_2\right)$$

(13)

where the parameter $a={\left[a_1, a_2\right]}^T$ is a constant vector satisfying Hurwitz’s condition $h_i=a_{3-i}$. Next, we will demonstrate that when $A$ is large enough, practical global tracking can be accomplished. Furthermore, it is possible to track paths effectively. There is a finite period $T_{\delta }$ such that $\left|y\left(t\right)\right|\le \delta , \forall t\ge T_{\delta }$ for any $\delta >0$.

Remark 2.

Through Equation (8), we can turn complex systems into simple ones. According to the unmeasurable state of the system, we introduce a dynamic gain observer (11). Unlike previous conditions, the observer gain in this paper is an observer combined with dynamic $B\left(t\right)$ and static $A$ gain parameters. The observed effect is close to the actual value of the system, which provides a good condition for the controller to control the error variable. The typical tracking control methods, including the Model Predictive Control (MPC) method, are no longer able to resolve the issues in this section due to the lack of information about the system and the tracking signal, as well as the instability of the system following the introduction of random elements. For these challenging problems, this article has used stochastic nonlinear systems satisfying Assumptions 3 and 4 to construct an output feedback dynamic adaptive tracking controller ground using a combination of static and dynamic gain observers to achieve control of the original system (5) that includes the tracking control target.

3.2. Stability Analysis and Implementation

The key findings of the paper are given and rigorously shown in this subsection. We require the next scaling transformation in the closed-loop system for the stability study.

$e_i=z_i-{\widehat{z}}_i, \left(i=1, 2\right)$ is the definition of the state estimate error. Then the dynamics of $e_i$ satisfy:

$$\begin{array}{l}{\dot{e}}_1=e_2-K^1{h}_1{e}_1+f_1\left(t,z,u\right)\dot{\omega }\hfill \\ {\dot{e}}_2=-K^2{h}_2{e}_1+f_2\left(t,z,u\right)\dot{\omega } \hfill \end{array}$$

(14)

To make the controller’s design easier, we need to transform the estimated state ${\widehat{z}}_i$ and the error state $e_i$ as follows:

$$\begin{array}{c}{\epsilon }_i=\frac{e_i}{K^{b+i-1}}\left(i=1,2\right)\\ {\tau }_i=\frac{{\widehat{z}}_i}{K^{b+i-1}}\left(i=1,2\right)\end{array}$$

(15)

where $0<b<\frac{1}{4q}$ is a known constant, $\epsilon ={\left({\epsilon }_1, {\epsilon }_2\right)}^T, \tau ={\left({\tau }_1, {\tau }_2\right)}^T, h={\left(h_1, h_2\right)}^T$. According to (15), the following conversions can be made for stochastic nonlinear systems (9) and (11):

$$\begin{array}{l}\dot{\epsilon }=KH\epsilon -\frac{\dot{K}}{K}{C}_b\epsilon +G\left(a,K\right)\dot{\omega }\hfill \\ \dot{\tau }=K{H}_b\tau -Kh{\epsilon }_1+\frac{\dot{h}}{h}{C}_b\tau \hfill \end{array}$$

(16)

Meantime, $G\left(z,K\right)={\left[\frac{f_1}{K^b},\frac{f_2}{K^{b+1}}\right]}^T,H=\left(\begin{array}{cc}-h_1& 1\\ -h_2& 0\end{array}\right),H_b=\left(\begin{array}{cc}0& 1\\ -h_1& -h_2\end{array}\right),C_b=\left(\begin{array}{cc}b& 0\\ 0& b+1\end{array}\right)$.

Then, we discuss the gain $B$ and the boundedness of the states $\epsilon$ and $\tau$. Above all, we must choose parameters $r_1,r_2, r_3, r_4$, so that the relationship between the positive definite matrices $P,Q$ and the matrix $H, H_b, C_b$ satisfies:

$$\begin{array}{cc}{H}^{T}P+PAH\le -I_2& r_1{I}_2\le C_{b}P+P{C}_b\le r_2{I}_2\\ H_b^{T}Q+Q{H}_b^T\le -2I_2& r_3{I}_2\le C_{b}Q+Q{C}_b\le r_4{I}_2\end{array}$$

(17)

Let $V\left(\epsilon \left(t\right), \tau \left(t\right)\right)={\epsilon }^T\left(t\right)P \epsilon \left(t\right)+{\tau }^{T}Q\tau \left(t\right)$ be a Lyapunov function candidate. The time derivative of $V$ on $\left[0, T\right)$ then meets the following inequality along the path of (16):

$$LV=+2{\epsilon }^{T}GP-{\left|\epsilon \right|}^{2}K-2{\epsilon }^T\frac{\dot{K}}{K}\left(C_{b}P+P{C}_b\right)\epsilon +2K{\tau }^T{\epsilon }_{1}h-2K{\left|\tau \right|}^2-Tr\left(G^T\left(P+Q\right)G\right)-\frac{\dot{K}}{K}{\tau }^T\left(C_{b}P+P{C}_b\right)\tau$$

(18)

From Remark 2, we can know $h>1$. And by Assumption 4, (8), (12), (15), (17) and the fact $h>1$ we have:

$$\begin{array}{c}-\frac{\dot{K}}{K}{\epsilon }^T\left(C_{b}P+P{C}_b\right)\epsilon \le r_2{\mu }_{1}B{\left|\epsilon \right|}^2-r_1{\mu }_2{\left(1+{\left|y\right|}^q\right)}^2{\left|\epsilon \right|}^2\\ -\frac{\dot{K}}{K}{\tau }^T\left(C_{b}Q+Q{C}_b\right)\tau \le r_4{\mu }_{1}B{\left|\tau \right|}^2-r_3{\mu }_2{\left(1+{\left|y\right|}^q\right)}^2{\left|\tau \right|}^2\end{array}$$

(19)

$$\left|G\right|={\left({\left|G_1\right|}^2+{\left|G_2\right|}^2\right)}^{\frac{1}{2}}={\left({\left(\frac{f_1}{K^b}\right)}^2+{\left(\frac{f_2}{K^{b+1}}\right)}^2\right)}^{\frac{1}{2}}$$

(20)

where

$$\begin{array}{c}\left|G_1\right|\le \left[1+\left(\left|\epsilon \right|+\left|\tau \right|+c_1\right)\left(1+{\left|y\right|}^q\right)\right]c+c_1 \\ \left|G_2\right|\le \left[1+\left(c_1+\sqrt{2}\left(\left|\epsilon \right|+\left|\tau \right|\right)\right)\left(1+{\left|y\right|}^q\right)\right]c+c_1\end{array}$$

(21)

Split and enlarge the terms in Formula (18), and we can get:

$$\begin{array}{c}2{G\epsilon }^{T}P\le 2‖\epsilon ‖‖P‖\left(c\left(1+{\left|y\right|}^q\right)\left(\left(\sqrt{2}\left(\left|\epsilon \right|+\left|\tau \right|\right)\right)+c_1\right)+c_1+c\right)\\ \le ‖P‖\left[c\left(1+c_1\right){\left({\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)\left(1+{\left|y\right|}^q\right)}^2+2K{\epsilon }_{1}h{\tau }^T\left(c{c}_1+c+c_1\right)+\left(5c+c_1\right){\left|\epsilon \right|}^2\right]\\ \le K\left({{\left|\tau \right|}^2+{\left|\epsilon \right|}^2\left|h\right|}^2\right)\end{array}$$

(22)

Therefore, combining Equations (14)–(22), the system (14)’s Ito differential equation is attainable as:

$$\begin{array}{c}\begin{array}{c}LV\le -K{\left|\epsilon \right|}^2+c‖P‖\left(1+c_1\right){\left(1+{\left|y\right|}^q\right)}^2\left({\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)+‖P‖\left(5c+c_1\right){\left|\epsilon \right|}^2+‖P‖\left(c{c}_1+c+c_1\right)\\ +{\alpha }_{1}B\left({r_2\left|\epsilon \right|}^2+r_4{\left|\tau \right|}^2\right)-{\alpha }_2{\left(1+{\left|y\right|}^q\right)}^2\left({r_1\left|\epsilon \right|}^2+r_3{\left|\tau \right|}^2\right)-2K\left({\left|\tau \right|}^2+{\tau }^T{\epsilon }_{1}h\right)+Tr\left(G^T\left(P+Q\right)G\right)\\ \le -K{\left|\epsilon \right|}^2+c‖P‖\left(1+c_1\right){\left(1+{\left|y\right|}^q\right)}^2\left({\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)+‖P‖\left(5c+c_1\right){\left|\epsilon \right|}^2+‖P‖\left(c{c}_1+c+c_1\right)\end{array}\\ +r_2{\alpha }_{1}B{\left|\epsilon \right|}^2-{\alpha }_2{\left(1+{\left|y\right|}^q\right)}^2\left({r_1\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)-2K{\left|\tau \right|}^2+r_4{\alpha }_{1}B{\left|\tau \right|}^2+K\left({\left|\tau \right|}^2+{\left|h\right|}^2{\left|\epsilon \right|}^2\right)\\ -\left(‖P‖+‖G‖\right){\left(c\left(1+{\left|y\right|}^q\right)\left(\sqrt{2}\left(\left|\epsilon \right|+\left|\tau \right|\right)+c_1\right)+c+c_1\right)}^2\end{array}$$

(23)

where the last term of Formula (23) can be enlarged as:

$$\begin{array}{c}\begin{array}{c}\left(‖G‖+‖P‖\right){\left(c\left(1+{\left|y\right|}^q\right)\left(\sqrt{2}\left(\left|\epsilon \right|+\left|\tau \right|\right)+c_1\right)+c+c_1\right)}^2\\ \le 2\left(‖P‖+‖G‖\right)\left(c^2{\left(1+{\left|y\right|}^q\right)}^2{\left(\sqrt{2}\left(\left|\epsilon \right|+\left|\tau \right|\right)+c_1\right)}^2+{\left(c+c_1\right)}^2\right)\\ \le 2\left(‖P‖+‖G‖\right)\left(2c^2{\left(1+{\left|y\right|}^q\right)}^2\left(2{\left(\left|\epsilon \right|+\left|\tau \right|\right)}^2+{c_1}^2\right)+{\left(c+c_1\right)}^2\right)\end{array}\\ \le 2\left(‖P‖+‖G‖\right)\left(2c^2{\left(1+{\left|y\right|}^q\right)}^2\left(4\left({\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)+{c_1}^2\right)+{\left(c+c_1\right)}^2\right)\\ =\left(‖P‖+‖G‖\right)\left(16c^2{\left(1+{\left|y\right|}^q\right)}^2\left({\left|\epsilon \right|}^2+{\left|\tau \right|}^2\right)+4c^2{\left(1+{\left|y\right|}^q\right)}^2{c}_1^2+2{\left(c+c_1\right)}^2\right)\end{array}$$

(24)

After completion, we receive:

$$\begin{array}{c}LV\le -\left(K(1-{\left|h\right|}^2)-\left(5c+c_1\right)‖P‖-r_{2}B{\mu }_2\right){\left|\epsilon \right|}^2-{\left({\left|y\right|}^q+1\right)}^2\left(r_1{\mu }_2-‖P‖c\left(c_1+1\right)-16\left(‖Q‖+‖P‖\right)\right){\left|\epsilon \right|}^2\\ +\left(r_4{\mu }_{1}B-K\right){\left|\tau \right|}^2-{\left({\left|y\right|}^q+1\right)}^2\left(r_3{\mu }_2-c‖P‖\left(1+c_1\right)-16c^2\left(‖Q‖+‖P‖\right)\right){\left|\epsilon \right|}^2\\ +‖P‖\left(c{c}_1+c+c_1\right)+\left(4c^2{\left(1+{\left|y\right|}^q\right)}^2{c}_1^2+2{\left(c+c_1\right)}^2\right)\left(‖P‖+‖Q‖\right)\end{array}$$

(25)

According to Formula (25), selection ${\mu }_2$ satisfies:

$$\begin{array}{l}{\mu }_2\ge \frac{c\left(1+c_1\right)‖P‖+16c^2\left(‖Q‖+‖P‖\right)}{r_1}\hfill \\ {\mu }_2\ge \frac{c\left(1+c_1\right)‖P‖+16c^2\left(‖Q‖+‖P‖\right)}{r_3}\hfill \\ {\mu }_2\ge {\mu }_1>0\hfill \end{array}$$

(26)

According to $B>1$, choose $A$ to satisfy:

$$\begin{array}{l}A\ge \frac{‖P‖\left(5c+c_1\right)+r_2{\mu }_1}{1-{\left|h\right|}^2}\hfill \\ A\ge r_4{\mu }_1\hfill \\ A\ge 1\hfill \end{array}$$

(27)

According to the above selection of parameters, the Formula (25) is capable of transformation:

$$\begin{array}{c}LV\le -K\left({\left|\tau \right|}^2+{\left|\epsilon \right|}^2\right)+‖P‖\left(c{c}_1+c_1+c\right)+\left(‖P‖+‖Q‖\right)\left(4c_1^2{c}^2{\left(1+{\left|y\right|}^q\right)}^2+2{\left(c+c_1\right)}^2\right)\\ \le -\frac{AV}{max\left({\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\right)}+\left(c{c}_1+c_1+c\right)‖P‖+\left(‖P‖+‖Q‖\right)\left(4c^2{c}_1^2{\left(1+{\left|y\right|}^q\right)}^2+2{\left(c+c_1\right)}^2\right)\end{array}$$

(28)

These are the defined relevant parameters:

$$\begin{array}{l}{E}_1=\frac{A}{max\left({\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\right)}\hfill \\ E_2=‖P‖\left(c{c}_1+c+c_1\right)+\left(4c^2{\left(1+{\left|y\right|}^q\right)}^2{c}_1^2+2{\left(c+c_1\right)}^2\right)\left(‖P‖+‖Q‖\right)\hfill \end{array}$$

(29)

The closed-loop system, as was already mentioned, has a singular solution on the maximal time period $[0,T)$, where $0<T\le +\infty$.

$$V\left(\epsilon \left(t\right),\tau \left(t\right)\right)\le V\left(\epsilon \left(0\right),\tau \left(0\right)\right)e^{-\frac{E_1}{A}t}+\frac{E_2}{E_1}, t\in [0,T)$$

(30)

which shows that $\tau$ are confined to $[0,T)$.

Then demonstrate the confinement of $B$ on $[0,\mathrm{T})$ because $\epsilon \mathrm{and} \tau$ are bounded on the interval in $[0,\mathrm{T})$, and ${\epsilon }_1+{\tau }_1=\frac{z_1}{K^b}=\frac{y}{K^b}$, we get $\left|y\right|\le {\left(AB\right)}^b{E}_3$. Then from this, (12) and $0<b<\frac{1}{4q}$, it is easy to get that:

$$\begin{array}{c}\dot{B}=-{\mu }_1{B}^2+{\mu }_2{\left(1+{\left|y\right|}^q\right)}^{2}B\\ \begin{array}{c}\le -{\mu }_1{B}^2+{\mu }_2{\left({\left({\left(AB\right)}^b{E}_3\right)}^q+1\right)}^{2}B\\ \le -{\mu }_1{B}^2+2{\mu }_2\left({\left({\left(AB\right)}^b{E}_3\right)}^{2q}+1\right)B\\ \le -{\mu }_1{B}^2+2{\mu }_{2}B\left({\left(AB\right)}^{\frac{1}{2}}{E}_3^{2q}+1\right)\end{array}\end{array}$$

(31)

It means that:

$$B\left(t\right)\le \frac{4{\mu }_2^2{\left(1+E_3^{2q}\sqrt{A}\right)}^2}{{\mu }_1^2}, \forall t\in [0,T)$$

(32)

And hence $B$ is bounded on $[0,T)$.

Proof. 

In the open neighborhood of the initial conditions of $\left(x,\widehat{x},A\right)$, it is simple to confirm that the ensuing closed-loop system is locally Lipschitz. Therefore, the closed-loop system has a singular solution on a narrow interval $[0,t)$. The greatest interval over which a singular solution occurs is $[0,T)$, where $0<T\le +\infty$. Equation (30) asserts that the closed-loop system states have been established on $[0,+\infty )$ when $T=+\infty$.

$$V\left(\epsilon \left(0\right),\tau \left(0\right)\right)e^{-\frac{E_1}{A}t}\le \frac{E_2}{E_1},\forall t>T$$

(33)

In the next step we can get:

$$V\left(\epsilon \left(t\right),\tau \left(t\right)\right)\le \frac{2E_2}{E_1}, \forall t>T$$

(34)

The presence of parameter $\mathrm{A}$ in $E_1$ should be highlighted at this point. From this, it is evident that:

$${\lambda }_{min}\left(P\right){\left|\epsilon \right|}^2+{\lambda }_{min}\left(Q\right){\left|\tau \right|}^2\le V\left(\epsilon \left(t\right),\tau \left(t\right)\right)$$

(35)

Next, we have:

$${\epsilon }_1^2\left(t\right)+{\tau }_1^2\left(t\right)\le \frac{E_4}{A}$$

(36)

Combining (34), (36) and $\left|y\left(t\right)\right|\le {\left(AB\right)}^b{E}_3$, we can get:

$$\left|y\left(t\right)\right|\le A^b{B}^b\left(\left|{\epsilon }_1\left(t\right)\right|,\left|{\tau }_1\left(t\right)\right|\right)\le \sqrt{2\left({\epsilon }_1^2\left(t\right)+{\tau }_1^2\left(t\right)\right)}\le \sqrt{\frac{2E_4}{A}}, \forall t>T$$

(37)

Then, we are aware that for all $\forall t\in [T,+\infty )$,

$$B\left(t\right)\le \frac{4{\mu }_2^2}{{\mu }_1^2}{\left(1+{\left(\frac{2E_4}{A}\right)}^q\sqrt{A}\right)}^2$$

(38)

Combining (37) and in turn:

$$y^2\left(t\right)={\left(A^b{B}^b\right)}^2{\left(\left|{\epsilon }_1\left(t\right)\right|,\left|{\tau }_1\left(t\right)\right|\right)}^2\le 2A^{2b}{B}^{2b}\left({\epsilon }_1^2\left(t\right)+{\tau }_1^2\left(t\right)\right)\le 2\frac{{16}^b{\mu }_2^{4b}{E}_4}{{\mu }_1^{4b}{A}^{1-2b}}{\left(1+\frac{2^q{E}_4^q}{A^{q-\frac{1}{2}}}\right)}^{2b}$$

(39)

From this and $1-2b>1-\frac{1}{2q}>0(since 0<b<\frac{1}{4q})$, when $\mathrm{t}>\mathrm{T}$, by choosing $\mathrm{A}$ big enough to make $\left|y\left(t\right)\right|$ arbitrarily small, so as to realize the actual path tracking. □

Remark 3.

By analyzing the dynamic gain parameter $B\left(t\right)$ of the dynamic observer, we can see that this method turns the control problem into a parameter selection problem through analysis, from which it can be deduced that the more prominent the parameter $B\left(t\right)$ selection area is, the better the control effect will be. A dynamic high-gain state observer is constructed. The real tracking controller for output feedback is created using the Ito stochastic calculus theory. The closed-loop stochastic nonlinear system’s state and high-gain parameters are guaranteed to be constrained by choosing the right design parameters, and the system tracking error can converge to zero in a limited area. Statistical systems are included in the investigation of output feedback tracking control for nonlinear systems. First-ever research is carried out on the actual tracking of output feedback for a group of stochastic nonlinear systems that satisfy the growth condition of the output polynomial function.

4. Model of an Underwater Robot Transportation System

In this piece of literature, we will simulate and verify the proposed stochastic uncertain dynamic gain adaptive control algorithm through specific numerical simulation. The dynamic model is converted into a specific mathematical model according to the dynamic characteristics of the underwater vehicle. For simplicity, consider the following stochastic nonlinear system:

$$\begin{array}{l}{\dot{l}}_1=l_2+sin{l}_1\dot{\omega } \hfill \\ {\dot{l}}_2=u+l_{2}l{n}^{\left(1+l_1^2\right)}\dot{\omega }\hfill \\ y=l_1-y_r \hfill \end{array}$$

(40)

which has no explicit relationship to the dynamics given in Section 2. In this section, we apply Theorem 1 and Assumptions 3 and 4 to specify the underwater robot path tracking control system described in Figure 1 with $c_0=0.21,I=2$, and $c_1=1$. So, by utilizing Remark 3, one can create an adaptive output feedback controller. The implementation of the relevant control laws starts at $u=-K^2{a}_1{\widehat{z}}_1+K{a}_2{\widehat{z}}_2$. As well-known in Section 2, the estimates of $y$ and $l_2$ are ${\widehat{z}}_1$and ${\widehat{z}}_2$, respectively. These variables are selected as $\mathrm{b}=0.1, h={\left[0.5,1\right]}^T, h={\left[1,0.5\right]}^T, {\mu }_1=7.2, {\mu }_2=9.8$ and $N=117$.

Initial stations $l_0={\left[1,-0.5\right]}^T$ and ${\widehat{z}}_0=0$ indicate that Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 display the simulation’s outcomes. These figures show $l,\widehat{z},B$ and $\dot{B}$ are all bounded, which must be confirmed to determine whether the route tracking controller is functional. In addition, the tracking error $\left|l_1-y_r\right|\le 0.001$ occurs after 0.04 s.

Through experimental results, the theoretical achievements of the control strategy in this paper have been demonstrated to successfully achieve hover stability control of AUVs. The control time is relatively short, which has played a very good control effect on robot movement under realistic working conditions, achieving safe and stable operation of the robot.

At present, the methods existing in the literature mainly focus on PID control. By comparing the curves of the two controllers in Figure 9 and Figure 10, our control strategy is significantly superior to PID control in terms of time required and energy consumption. According to the curves of the two controllers, the PID control strategy requires a much longer time than our latest control strategy. The energy consumed by PID control is much greater than that of our controller compared to PID control. Our control strategy is more stable, with fewer fluctuations, and can achieve hover control stability of the AUV in the shortest possible time.

5. Concluding and Future Prospects

In this study, we researched the application of this dynamic adaptive control to AUV trajectory tracking. The nonlinear characteristics of underwater robot motion were studied, and a novel dynamic-static gain observer strategy was proposed to reduce the computational burden. Numerical simulations and additional experiments validate the efficiency and robustness of the proposed strategy and highlight the advantages of dynamic-static high-gain adaptive control algorithms. It can decrease the controlled system’s reaction time and the control state’s error, and it also has a great inhibitory effect on stochastic and uncertain interference.

In applied engineering and research, we identified a family of nonlinear systems with piecewise linearization. A nonlinear system is broken down into several finite or infinite linear subsystems using piecewise linearization. This combination can approximate nonlinear systems. In the control process, piecewise linearization has the same characteristics as nonlinear and simplifies the control process and method. According to the known situation, this method has been applied in many practical projects and has achieved a perfect control effect. However, the current application scope of this method is still minimal and has special conditions for nonlinear systems. The following work will expand the application range of piecewise linearization of nonlinear systems, which will be the next hotspot of nonlinear control research.

In our current research, we have started to apply the simulation results obtained in this article to actual underwater robots. At present, we have the experimental pool and experimental robot, and we have started to implant our latest control algorithm in the electronic cabin. Meanwhile, in the conclusion section of the paper submitted after the latest revisions, we have added prospects for future practical applications in robotics research.