Prescribed Performance-Based Formation Control for Multiple Autonomous Underwater Helicopters with Complex Dynamic Characteristics

Abstract

This research addresses the challenge of formation control among multiple homogeneous autonomous underwater helicopters (AUHs) in the presence of external disturbances and complex dynamic characteristics. The study introduces a novel approach by integrating both disturbance and state observers within the control law framework to manage external disturbances and the immeasurability of velocity, respectively. Concurrently, localized radial basis function neural networks (RBFNNs) of identical configurations are incorporated into the formation control law to assimilate model uncertainties. Building upon this integration, an experience-based formation control strategy is developed, leveraging accumulated knowledge to diminish computational demands while maintaining stipulated performance criteria. Furthermore, the incorporation of a finite-time prescribed performance control (FTPPC) technique enhances the learning process’s efficiency by expediting convergence. Numerical simulations are presented to validate the efficacy of the proposed methodology.

1. Introduction

Underwater exploration plays a vital role in advancing our understanding of marine ecosystems, resource management, and climate regulation [1]. Over the past few decades, humans have sought to explore the vast “universe under oceans” to uncover and utilize hidden resources for the benefit of mankind. However, despite these efforts, only 5% of the world’s oceans have been explored to date, leaving an immense frontier of unknowns [2]. However, conducting underwater exploration poses significant challenges, including harsh environmental conditions, limited visibility, and high-pressure zones, which can impede data collection and system functionality. The development of robust and efficient underwater vehicles, such as autonomous underwater vehicles (AUVs), gliders, and remotely operated systems, has been instrumental in overcoming these challenges [3,4].

Among these technologies, AUVs have attracted significant attention, particularly in formation control, which is critical for cooperative and efficient underwater missions. Researchers have applied various methods to achieve cooperative formation of AUVs, including the leader–follower, virtual structure, and behavior-based methods [5,6,7,8]. Compared to current AUVs, an autonomous underwater helicopter (AUH), as shown in Figure 1, provides more flexible steering, hovering, and landing due to its special shape and propulsion direction, which is more suitable for underwater formation operations [9,10,11].

The complexity of the underwater environment, including its unmeasurable speed, model dynamics, and disturbances, poses significant challenges in designing control law for AUVs. Much work has been carried out to address these issues. An AUV velocity estimation method derived from the Nussbaum state observer was investigated to tackle the problem of unmeasurable velocity [12]. Li et al. [13] presented a finite-time sliding-mode disturbance observer to estimate accurately unknown disturbances in finite time. For disturbances, Gao et al. [14] and Li et al. [15] combined external disturbances with model uncertainties as lumped disturbances and introduced a disturbances observer to estimate them in a finite time. In [16,17], the extended state observer was designed to estimate velocity and disturbances simultaneously. The neural network method has been developed to deal with model uncertainties due to its excellent generalization ability. RBFNNs were used to approximate nonlinear uncertainties in [18,19]. An estimation method derived from the neural-based robustness controller was used to approximate the lumped uncertainty composed of uncertain dynamics, disturbances, and approximate error [20]. Building upon coordinate transformation techniques and a hybrid linear–nonlinear differentiator to estimate the derivative of surge displacement, Zhang et al. [21] introduced an output-feedback adaptive backstepping control strategy tailored for AUVs operating without direct velocity measurements.

The NN methods mentioned above are based on online adaptive neural networks, which means that the learning ability of NN is ignored. Therefore, adaptive methods need to be reused even if AUVs are dealing with similar model uncertainties, which increases the computational burden. Wang et al. [22] proposed the deterministic learning theory to address this problem, which fully utilized the spatially localized learning capabilities of the localized RBFNN. Based on this, a distributed learning strategy composed of a distributed observer and learning controller was proposed in [23]. The cooperative learning controller was presented in [24], which can learn the model uncertainties cooperatively and use knowledge to tackle the uncertainties. In [25], reinforcement learning (RL) is employed to handle partially observable systems by compensating for the lack of state measurements. This algorithm uses a finite history of input–output data to approximate the system dynamics and optimize control actions. In [26], the RL algorithm is utilized to achieve optimal control at a low level for individual agents within the multi-agent formation. It transforms the time-varying system into an equivalent autonomous system, ensuring stability and optimality of the control strategy.

It is important to note that AUHs starts to learn only after getting into the steady state. Therefore, the error convergence rate will affect the learning speed, which poses demands on the convergence rate. The prescribed performance control limits the transient convergence process of tracking errors by performing the error transformation via performance function. In [27,28,29,30], prescribed performance control was proposed to achieve faster convergence, which is realized by using the exponential decay performance function. The traditional method was improved by introducing a finite-time performance function to ensure the trace error gets into the compact set in a specified time [31,32,33].

This study focuses on the formation control problem for homogeneous AUHs with external disturbance and model uncertainty. A cooperative formation event-triggered controller is established based on prescribed performance. Multiple observers estimate the unmeasurable high-order states and external disturbances separately. Finally, the dynamic uncertainties are learned cooperatively, and the learned knowledge is used to establish an empirically based controller.

To contextualize the advancements and challenges in formation control techniques, we conducted a comprehensive review of recent related works.

Table 1. Comparison of related works on formation control techniques.

Study	Dynamic Model Consideration	Robustness to Disturbances	Learning Mechanism	Performance Metrics
[12,18,22]	Unknown dynamics	Moderate	None	UUB errors
[13]	Feedback linearization	High	None	Finite-time convergence
[14,15]	Unknown dynamics, external disturbances	High	None	Fixed-time convergence
[16]	Thruster faults, unknown disturbances	Moderate	Reinforcement learning	UUB errors
[17,27,29]	Unknown disturbances	Moderate	None	UUB errors
[19]	Thruster faults, unknown disturbances	Low	None	Specified convergence time
[20]	Unknown dynamics, input saturation	Low	None	UUB errors
[23]	Uncertainties	High	Deterministic learning	UUB errors
[24]	Unknown disturbances, uncertainties	High	Cooperative learning	UUB errors
[26]	Dynamic models	Moderate	Reinforcement learning	UUB errors
[30]	Full-state constraints, disturbances	Moderate	None	Specified convergence time
Proposed	Unknown dynamics, uncertainties, external disturbances	High	Experience-based learning	Specified convergence time

provides a detailed comparison of these studies, focusing on key aspects such as the dynamic models considered, robustness to disturbances, the integration of learning mechanisms, and performance metrics. This comparison highlights the strengths and limitations of existing approaches, serving as a foundation to underscore the unique contributions of the proposed method in addressing current gaps in the field.

Hybrid observer design: In contrast to the observer proposed in [14,15], which estimates the lumped uncertainties composed of model uncertainty and disturbance, and the extended state observer presented in [17], which estimates the higher-order states and disturbances simultaneously, the NN-based Luenberger observer and disturbance observer are used to estimate the high-order states and disturbance, respectively, which achieves higher estimation accuracy.

Experience-based control: Compared to the online adaptive neural network used in [18,19], this study maximizes the learning ability of local RBFNN, which enables AUH to learn dynamic uncertainties when moving along the periodic reference trajectory. Based on this, an experience-based control law is designed using the gained knowledge, which means the similar dynamic uncertainties is dealt with by experience rather than by reusing the adaptive method, thus reducing the calculation burden.

Finite-time prescribed performance control: Compared to the current results [23,24,26,34], the finite-time prescribed performance control (FTPPC) is proposed to guarantee that the tracking error converges to a compact set in a finite time, which accelerates the learning process.

The rest of this work is organized as follows. Section 2 contains the preliminaries and control objectives. Section 3 investigates the cooperative formation controller. Section 4 establishes an experience-based cooperative formation controller. Section 5 shows the effectiveness of the strategy through simulation experiments. Finally, Section 6 presents the conclusions.

2. Preliminaries and Problem Formulation

2.1. AUHs Dynamics

This study examines the control problem for a set of $N$ AUHs. The dynamic model of the AUH is based on references [19,35], and its specific expressions are as follows:

$$\begin{array}{l}{\dot{\eta }}_i=J({\eta }_i){\nu }_i\\ M{\dot{\nu }}_i+C({\nu }_i){\nu }_i+D({\nu }_i){\nu }_i+\Delta ({\eta }_i,{\nu }_i)+g={\tau }_{d,i}(t)+{\tau }_i\end{array}$$

(1)

where the $i\in 𝒩=\left[1,\dots ,N\right]$ signifies the i-th AUH. $M$ denotes the inertia matrix. ${\eta }_i={\left[x_i,y_i,z_i,{\phi }_i,{\theta }_i,{\psi }_i\right]}^{\mathrm{T}}$ denotes the position and declination of the AUH in the earth-fixed frame. ${\nu }_i={\left[u_i,{\upsilon }_i,w_i,p_i,q_i,r_i\right]}^{\mathrm{T}}$ represents the linear and angular velocities of the AUH in the body-fixed frame. $C({\nu }_i)$ represents the uncertain Coriolis force and the centripetal force matrix. $D({\nu }_i)$ denotes the uncertain hydrodynamic damping matrix. $\Delta ({\eta }_i,{\nu }_i)$ represent unmodeled dynamics. $g$ is the force and matrix of gravity and buoyancy. ${\tau }_{d,i}(t)$ and ${\tau }_i$ represent external disturbances and control inputs, respectively.

The rotation matrix $J({\eta }_i)$ between the earth-fixed frame and body-fixed frame can be defined as:

$$J\left({\eta }_i\right)=\left[\begin{array}{cc}{J}_a& 0_{3\times 3}\\ 0_{3\times 3}& J_b\end{array}\right]$$

(2)

with

$$J_a=\left[\begin{array}{ccc}\cos \psi \cos \theta & \cos \psi \sin \theta \sin \varphi -\sin \psi \cos \varphi & \sin \psi \sin \varphi +\cos \psi \sin \theta \cos \varphi \\ \sin \psi \cos \theta & \sin \theta \sin \psi \sin \varphi +\cos \psi \cos \varphi & \sin \psi \sin \theta \cos \varphi -\cos \psi \sin \theta \\ -\sin \theta & \cos \theta \sin \varphi & \cos \theta \cos \varphi \end{array}\right]$$

and

$$J_b=\left[\begin{array}{ccc}1& \sin \varphi \tan \theta & \cos \varphi \tan \theta \\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi /\cos \theta & \cos \varphi /\cos \theta \end{array}\right]$$

According to Fossen’s handbook [35], the representation singularity at $\theta \ne \pm \pi /2$ in the expression for $J_b$ implies that the inverse matrix $J^{-1}({\eta }_i)$ does not exist at this value.

The dynamic model can be reformatted as:

$$\begin{array}{l}{\dot{x}}_{1i}=x_{2i}\\ {\dot{x}}_{2i}=J_1({\eta }_i){\tau }_i+J_1({\eta }_i){\tau }_{d,i}(t)+F_i\end{array}$$

(3)

where ${\left[x_{1i},x_{2i}\right]}^{\mathrm{T}}={\left[{\eta }_i,{\dot{\eta }}_i\right]}^{\mathrm{T}}$, $J_1({\eta }_i)=J({\eta }_i)M^{-1}$, and to simplify the notation, $J({\eta }_i)$, $J_1({\eta }_i)$ is denoted as $J$ and $J_1$, respectively, and $F_i=-J{\dot{J}}^{-1}{\dot{\eta }}_i-J{M}^{-1}C({\nu }_i)J^{-1}{\dot{\eta }}_i-J{M}^{-1}D({\nu }_i)J^{-1}{\dot{\eta }}_i-J{M}^{-1}\Delta ({\eta }_i,{\nu }_i)-J{M}^{-1}g$ represent the total set of dynamic uncertainties.

2.2. Formation Structure and Graph Theory

The ideal AUH formation consists of a virtual leader and $N$ followers; the virtual leader follows the reference path ${\eta }_d$ and the i-th follower travels along the reference path ${\eta }_{r,i}$.

$${\eta }_{r,i}={\eta }_d+{\eta }_i^{*}$$

(4)

where ${\eta }_i^{*}$ is the relative placement of the i-th follower and the virtual leader that establish the formation structure and can be specified by the designer.

The communication topology is dictated by an undirected graph $𝒢=\left(𝒱,\epsilon \right)$, where $𝒱=\left\{{\vartheta }_1,\dots ,{\vartheta }_N\right\}$ represents the node set and $\epsilon \subseteq \left\{\left.\left({\vartheta }_i,{\vartheta }_k\right)\right|{\vartheta }_i,{\vartheta }_k\in 𝒱,{\vartheta }_i\ne {\vartheta }_k\right\}$ represents the edge set. The correlation adjacency matrix $𝒜=[a_{ik}]$ is determined as $a_{ik}=1$ if the i-th AUH is capable of receiving information from the k-th AUH; otherwise, $a_{ik}=0$. Consider ${𝒱}_i=\left\{\left.{\vartheta }_k\right|\left({\vartheta }_i,{\vartheta }_k\right)\in \epsilon \right\}$ to be the neighbor set of ${\vartheta }_i$, with $k\in {𝒩}_i$ if ${\vartheta }_k\in {𝒱}_i$. The Laplacian matrix $ℒ$ is defined by $ℒ==\left[l_{ik}\right]\in {ℝ}^{n\times n}$ where $l_{ii}={\displaystyle \sum _{k\in {𝒩}_i}{a}_{ik}}$, $l_{ik}=-a_{ik}$.

2.3. RBFNN

Neural networks are frequently used to estimate unknown or complex functions. This study employed RBFNN to approximate the dynamic uncertainties $F_i(x_i)$.

$$F_i(x_i)={W_i^{*}}^{\mathrm{T}}S\left(x_i\right)+{\epsilon }_i$$

(5)

where $x_i={[x_{1i}^{\mathrm{T}},{\widehat{x}}_{2i}^{\mathrm{T}}]}^{\mathrm{T}}\in {ℝ}^{12}$ stands for the i-th RBFNN input variables and $S\left(x_i\right)={\left[S_1^{\mathrm{T}}\left(x_i\right),\dots ,S_6^{\mathrm{T}}\left(x_i\right)\right]}^{\mathrm{T}}$ represents the Gaussian basis function

$$S_j\left(x_i\right)=\exp \left(-{\displaystyle \frac{{\left(x_i-{\mu }_j\right)}^T\left(x_i-{\mu }_j\right)}{{\mathrm{\sigma }}^2}}\right)$$

(6)

where ${\mu }_j={\left[{\mu }_{j,1},\dots ,{\mu }_{j,q}\right]}^{\mathrm{T}}\in {ℝ}^q$ indicates the center vector of the j-th node, $\mathrm{\sigma }$ denotes the basis width, ${W_i^{*}}^{\mathrm{T}}=\mathrm{blockdiag}\left[{W_{i,1}^{*}}^{\mathrm{T}},\dots ,{W_{i,6}^{*}}^{\mathrm{T}}\right]\in {ℝ}^{6\times 6q}$ is the ideal weight coefficient matrix, and $W_{ij}^{*}\in {ℝ}^q$, ${\epsilon }_i\in {\left[{\epsilon }_{i,1},\dots ,{\epsilon }_{i,6}\right]}^{\mathrm{T}}$ represents the inherent error.

The control law design is based on the following assumptions.

Assumption 1. 

All followers can access the information of virtual leader, and the communication among the followers is described by a connected undirected graph, which means there exists an undirected path between every pair of nodes.

Assumption 2. 

The disturbances ${\tau }_{d,i}(t)$ and their derivative ${\dot{\tau }}_{d,i}(t)$  are bounded.

Assumption 3. 

The tracking path ${\eta }_d$ and its derivative is bounded, periodic, or semi-periodic.

The control objectives are listed as follows: the AUHs are driven to maintain the preset formation configuration while tracking the reference trajectory, and the specified performance can be achieved within the limits of the performance function. The experience-based control law is designed to reduce the computing burden by using the experience obtained from learning to approximate the dynamic uncertainty.

3. NN-Based AUH Formation Control Mechanism

3.1. FTPPC

Definition 1 

[19]. A smooth function $\beta \left(t\right)$ can be defined as a finite-time performance function (FTPF) if it satisfies  $\beta \left(t\right)={\beta }_{tf}$  for  $t\ge t_f$,  where  ${\beta }_{tf}$  is a small constant and  $t_f$  is the specified time.

According to Definition 1, an FTPF candidate is proposed:

$$\dot{\beta }(t)=\left\{\begin{array}{ll}-k_1\left[{(\beta (t)-{\beta }_{\infty })}^{2-k_2}+{(\beta (t)-{\beta }_{\infty })}^{k_2}\right],& t\le t_f\\ 0& ,\hspace{0.33em}t>t_f\end{array}\right.$$

(7)

where $k_1={\left(\left(1-k_2\right)t_f\right)}^{-1}{\tan }^{-1}\left({\left({\beta }_0-{\beta }_{\infty }\right)}^{1-k_2}\right)$, $0<k_2<1$, and ${\beta }_0$ and ${\beta }_{\infty }$ are design parameters to constrain the overshoot and stable state error, respectively. Next, we will prove that (7) is a FTPF.

Proof of Definition 1. 

Consider the following Lyapunov function:

$$V_{\beta }={\displaystyle \frac{1}{2}}{e}_{\beta }^2$$

(8)

where $e_{\beta }=\beta (t)-{\beta }_{\infty }$. Combined with (7), we obtain:

$$\begin{array}{cc}{\dot{V}}_{\beta }& =-k_1{e}_{\beta }\left[{\left(\beta \right(t)-{\beta }_{\infty })}^{2-k_2}+{\left(\beta \right(t)-{\beta }_{\infty })}^{k_2}\right]\\ & =-k_1\left({e_{\beta }}^{2\times {\displaystyle \frac{3-k_2}{2}}}+{e_{\beta }}^{2\times {\displaystyle \frac{1+k_2}{2}}}\right)\hfill \\ & =-k_1\left({\mu }_1{V}_{\beta }^{1-k_2}+{\mu }_2\right)V_{\beta }^{{\displaystyle \frac{1+k_2}{2}}}\hfill \end{array}$$

(9)

where ${\mu }_1=2^{{\displaystyle \frac{3-k_2}{2}}}$ and ${\mu }_2=2^{{\displaystyle \frac{1+k_2}{2}}}$. ${\displaystyle \frac{1+k_2}{2}}<1$ since $0<k_2<1$. Moreover, when $e_{\beta }\ne 0$, ${V_{\beta }}^{1-k_2}>0$. Thus, $e_{\beta }$ may converge to 0 in finite time.

Let $x_{\beta }=V_{\beta }^{{\displaystyle \frac{1-k_2}{2}}}$; then, (9) can be rewritten as follows:

$$\begin{array}{c}{V}_{\beta }^{-{\displaystyle \frac{1+k_2}{2}}}{\displaystyle \frac{\mathrm{d}{V}_{\beta }}{\mathrm{d}t}}=-k_1\left({\mu }_1{V}_{\beta }^{1-k_2}+{\mu }_2\right)\\ {\displaystyle \frac{2}{1-k_2}}{\displaystyle \frac{\mathrm{d}{V}_{\beta }^{\frac{1-k_2}{2}}}{\mathrm{d}t}}=-k_1\left({\mu }_1{V}_{\beta }^{1-k_2}+{\mu }_2\right)\\ {\displaystyle \frac{2}{1-k_2}}{\displaystyle \frac{\mathrm{d}{x}_{\beta }}{\mathrm{d}t}}=-k_1\left({\mu }_1{x}_{\beta }^2+{\mu }_2\right)\\ {\displaystyle \frac{1}{{\mu }_1{x}_{\beta }^2+{\mu }_2}}\mathrm{d}{x}_{\beta }=-{\displaystyle \frac{\left(1-k_2\right)k_1}{2}}\mathrm{d}t\end{array}$$

(10)

By integrating both sides of (10), we obtain:

$${\left({\mu }_1{\mu }_2\right)}^{-{\displaystyle \frac{1}{2}}}{\tan }^{-1}\left({\left({\mu }_1/{\mu }_2\right)}^{{\displaystyle \frac{1}{2}}}{x}_{\beta }\left(t\right)\right)={\left({\mu }_1{\mu }_2\right)}^{-{\displaystyle \frac{1}{2}}}{\tan }^{-1}\left({\left({\mu }_1/{\mu }_2\right)}^{{\displaystyle \frac{1}{2}}}{x}_{\beta }\left(0\right)\right)-{\displaystyle \frac{\left(1-k_2\right)k_1}{2}}t$$

(11)

Substituting ${\mu }_1{\mu }_2=4$ into (11) yields:

$${\tan }^{-1}\left({\left({\mu }_1/{\mu }_2\right)}^{{\displaystyle \frac{1}{2}}}{x}_{\beta }\left(t\right)\right)={\tan }^{-1}\left({\left({\mu }_1/{\mu }_2\right)}^{{\displaystyle \frac{1}{2}}}{x}_{\beta }\left(0\right)\right)-\left(1-k_2\right)k_{1}t$$

(12)

Therefore, there exists $t_f$ satisfying $x_{\beta }(t_f)=0$.

$$\begin{array}{c}{t}_f={\displaystyle \frac{2{\left({\mu }_1{\mu }_2\right)}^{-\frac{1}{2}}}{\left(1-k_2\right)k_1}}{\tan }^{-1}\left({\left({\mu }_1/{\mu }_2\right)}^{{\displaystyle \frac{1}{2}}}{x}_{\beta }\left(0\right)\right)\\ ={\displaystyle \frac{1}{\left(1-k_2\right)k_1}}{\tan }^{-1}\left({\left({\beta }_0-{\beta }_{\infty }\right)}^{1-k_2}\right)\end{array}$$

(13)

Then, we know that $\beta (t)$ tends to ${\beta }_{\infty }$ in a finite time $t_f$.

The tracking error transformations are described in the following formula:

$$e_{1,ij}={\beta }_{ij}(t)T_{ij}(z_{1,ij})\hspace{1em}\hspace{1em}\hspace{1em}j=1,\dots ,6$$

(14)

where ${\beta }_{ij}(t)$ is the FTPF, whose update law is provided in (7), $z_{1,ij}$ are the transformed errors, and $T_{ij}(z_{1,ij})$ refers to the error transformation function which can be provided by:

$$T_{ij}(z_{1,ij})={\displaystyle \frac{\exp \left(z_{1,ij}\right)-\exp \left(-z_{1,ij}\right)}{\exp \left(z_{1,ij}\right)+\exp \left(-z_{1,ij}\right)}}$$

(15)

Substituting (15) into (14) yields:

$$z_{1,ij}={\displaystyle \frac{1}{2}}\ln \left(1+{\displaystyle \frac{e_{1,ij}}{{\beta }_{ij}(t)}}\right)-{\displaystyle \frac{1}{2}}\ln \left(1-{\displaystyle \frac{e_{1,ij}}{{\beta }_{ij}(t)}}\right)$$

(16)

□

3.2. State Observer Design

We introduce state observer to estimate the high-order states $x_{2i}$. The RBFNN, disturbance observer, and Luenberger observer are combined to estimate the state variables $x_{2i}$ due to the uncertainties and disturbances in the dynamic model. The state observer based on RBFNN and disturbance observer are designed as follows:

$$\begin{array}{l}{\dot{\widehat{x}}}_{1i}={\widehat{x}}_{2i}+L_{1i}(y_i-{\widehat{y}}_i)\\ {\dot{\widehat{x}}}_{2i}=J_1{\tau }_i+J_1{\widehat{\tau }}_{d,i}+{\widehat{F}}_i+L_{2i}(y_i-{\widehat{y}}_i)\\ {\widehat{y}}_i={\widehat{x}}_{1i}\end{array}$$

(17)

where ${\widehat{x}}_{1i}$, ${\widehat{x}}_{2i}$, and ${\widehat{y}}_i$ represent the estimation of the state variables and output variables, respectively. ${\widehat{\tau }}_{d,i}$ denotes the estimation of disturbances, ${\widehat{F}}_i$ stands for the approximation of complex uncertainties using RBFNN, and $L_{1i}$ and $L_{2i}$ represent the gain matrix to be designed. The state estimation error can be calculated as follows:

$$\begin{array}{l}{\dot{\tilde{x}}}_{1i}=-L_{1i}{\tilde{x}}_{1i}+{\tilde{x}}_{2i}\\ {\dot{\tilde{x}}}_{2i}=-L_{2i}{\tilde{x}}_{1i}+J_1{\tilde{\tau }}_{d,i}+{\tilde{F}}_i\end{array}$$

(18)

where ${\tilde{x}}_{1i}=x_{1i}-{\widehat{x}}_{1i}$ and ${\tilde{x}}_{2i}=x_{2i}-{\widehat{x}}_{2i}$ stand for state estimation errors, ${\tilde{\tau }}_{d,i}$ stands for disturbances estimation error, and ${\tilde{F}}_i$ denotes the approximation error. If ${\tilde{x}}_i={\left[{\tilde{x}}_{1i},{\tilde{x}}_{2i}\right]}^{\mathrm{T}}$, then (18) is expressed in matrix form.

$${\dot{\tilde{x}}}_i=A{\tilde{x}}_i+B{\tilde{\overline{F}}}_i$$

(19)

where $A=\left(\begin{array}{cc}-L_{1i}& I\\ -L_{2i}& 0\end{array}\right)$, $B=\left(\begin{array}{c}0\\ I\end{array}\right)$, ${\tilde{\overline{F}}}_i=J_1{\tilde{\tau }}_{d,i}+{\tilde{F}}_i$.

Remark 1. 

Although the Luenberger observer is traditionally applied to linear systems, this study extends its application by integrating it with an RBFNN to address the nonlinearities in the AUH model. Specifically, the RBFNN approximates the nonlinear components of the system dynamics, while the Luenberger observer estimates the linearized states. This hybrid design leverages the Luenberger observer’s efficiency in linear dynamics estimation and the RBFNN’s capability to handle nonlinearities.

3.3. NN-Based AUH Formation Controller

The tracking errors are defined as:

$$e_{1,i}={\eta }_i-{\eta }_{r,i}=x_{1i}-{\eta }_{r,i}$$

(20)

Combined with (16), ${\dot{z}}_{1,i}$ is calculated as:

$$\begin{array}{ll}{\dot{z}}_{1,i}=& {\mathrm{{\rm Y}}}_i{\dot{e}}_{1,i}-{\mathrm{\Theta }}_i{e}_{1,i}\\ & ={\mathrm{{\rm Y}}}_i\left(x_{2i}-{\dot{\eta }}_{r,i}\right)-{\mathrm{\Theta }}_i{e}_{1,i}\end{array}$$

(21)

where ${\mathrm{{\rm Y}}}_i=\mathrm{diag}\left[{\chi }_{i1},\dots ,{\chi }_{i6}\right]$, ${\mathrm{\Theta }}_i=\mathrm{diag}\left[{\theta }_{i1},\dots ,{\theta }_{i6}\right]$, ${\chi }_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{{\beta }_{ij}(t)+e_{1,ij}}}\right.$, $\left.+{\displaystyle \frac{1}{{\beta }_{ij}(t)-e_{1,ij}}}\right)$, and ${\theta }_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\dot{\beta }}_{ij}(t)}{{\beta }_{ij}(t)\left({\beta }_{ij}(t)+e_{1,ij}\right)}}+{\displaystyle \frac{{\dot{\beta }}_{ij}(t)}{{\beta }_{ij}(t)\left({\beta }_{ij}(t)-e_{1,ij}\right)}}\right)$.

Step 1: The Lyapunov function are selected as $V_1={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^N{\tilde{x}}_i^{\mathrm{T}}{\tilde{x}}_i}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^N{z_{1,i}}^{\mathrm{T}}{z}_{1,i}}$. According to (19) and (21), the derivative of $V_1$ can be calculated as:

$${\dot{V}}_1={\displaystyle \sum _{i=1}^N\left\{{{\tilde{x}}_i}^{\mathrm{T}}A{\tilde{x}}_i+{{\tilde{x}}_i}^{\mathrm{T}}B{\tilde{\overline{F}}}_i+{z_{1,i}}^{\mathrm{T}}\left[{\mathrm{{\rm Y}}}_i\left({\widehat{x}}_{2i}+{\tilde{x}}_{2i}-{\dot{\eta }}_{r,i}\right)-{\mathrm{\Theta }}_i{e}_{1,i}\right]\right\}}$$

(22)

Consider ${\widehat{x}}_{2i}$ as the virtual control variable, and $z_{2,i}$ is designed as:

$$z_{2,i}={\widehat{x}}_{2i}-{\alpha }_i$$

(23)

where the virtual control law ${\alpha }_i$ is designed as:

$${\alpha }_i=-{\mathrm{{\rm Y}}}_i^{-1}{K}_{1,i}{z}_{1,i}+{\dot{\eta }}_{r,i}+{\mathrm{{\rm Y}}}_i^{-1}{\mathrm{\Theta }}_i{e}_{1,i}$$

(24)

where $K_{1,i}=\mathrm{diag}[k_{1,i1},\dots ,k_{1,i6}] > 0$ is the gain matrix, and combining (22), (23), and (24), we obtain:

$${\dot{V}}_1={\displaystyle \sum _{i=1}^N\left({{\tilde{x}}_i}^{\mathrm{T}}A{\tilde{x}}_i+{{\tilde{x}}_i}^{\mathrm{T}}B{\tilde{\overline{F}}}_i-{z_{1,i}}^{\mathrm{T}}{K}_{1,i}{z}_{1,i}+{z_{1,i}}^{\mathrm{T}}{\mathrm{{\rm Y}}}_i{z}_{2,i}+{z_{1,i}}^{\mathrm{T}}{\mathrm{{\rm Y}}}_i{\tilde{x}}_{2,i}\right)}$$

(25)

Next, a nonlinear observer is proposed to estimate the ${\dot{\alpha }}_i$, which avoids the direct computation of ${\dot{\alpha }}_i$.

$$\begin{array}{l}{\dot{\wp }}_i={\Im }_i-{\varsigma }_1{\mathrm{sig}}^{1-1/{\varsigma }_3}({\wp }_i-{\alpha }_i)\\ {\dot{\Im }}_i=-{\varsigma }_2{\mathrm{sig}}^{1-2/{\varsigma }_3}({\wp }_i-{\alpha }_i)\end{array}$$

(26)

where ${\wp }_i$ and ${\Im }_i$ are the estimates of ${\alpha }_i$ and ${\dot{\alpha }}_i$, respectively. The function sig is defined by ${\mathrm{sig}}^a(m)=\mathrm{sign}(m){\left|m\right|}^a$. ${\varsigma }_1>0$, ${\varsigma }_2>0$, and ${\varsigma }_3>2$ are the parameters.

Considering the estimate error ${\tilde{\wp }}_i={\wp }_i-{\alpha }_i$ and ${\tilde{\Im }}_i={\Im }_i-{\dot{\alpha }}_i$, the error dynamic can be calculated as:

$$\begin{array}{l}{\dot{\tilde{\wp }}}_i={\tilde{\Im }}_i-{\varsigma }_1{\mathrm{sig}}^{1-1/{\varsigma }_3}({\tilde{\wp }}_i)\\ {\dot{\tilde{\Im }}}_i=-{\varsigma }_2{\mathrm{sig}}^{1-2/{\varsigma }_3}({\tilde{\wp }}_i)-{\ddot{\alpha }}_i\end{array}$$

(27)

According to [36], the error dynamic (27) is stable, which means the ${\tilde{\Im }}_i$ will converge to a compact set and satisfy $‖{\tilde{\Im }}_i‖\le {\overline{\Im }}_i$ with ${\overline{\Im }}_i$ being a small constant.

Remark 2. 

Note that direct computation of ${\dot{\alpha }}_i$ requires the use of higher order states $x_{2i}$, which is unmeasurable and unavailable. And the ${\dot{\alpha }}_i$ obtained by the differentiator directly is often not smooth. To address this problem, the nonlinear observer is used to estimate the signal ${\dot{\alpha }}_i$, which avoids both the direct computation of ${\dot{\alpha }}_i$ and the use of differentiator and is more practical for control law design.

Step 2: The derivative of $z_{2,i}$ can be computed as:

$$\begin{array}{ll}{\dot{z}}_{2,i}& =J_1{\tau }_i+J_1{\tau }_{d,i}(t)+F_i-{\dot{\tilde{x}}}_{2i}-{\dot{\alpha }}_i\\ & =J_1{\tau }_i+d_i+{W_i^{*}}^{\mathrm{T}}S\left(x_i\right)-{\dot{\tilde{x}}}_{2i}-{\dot{\alpha }}_i\end{array}$$

(28)

where $d_i=J_1{\tau }_{d,i}+{\epsilon }_i$ represents the sum of disturbances and RBFNN inherent errors, which can be estimated by the disturbance observer.

$$\begin{array}{l}{\dot{\xi }}_i=-l_i{\xi }_i-l_i\left[J_1{\tau }_i-{\Im }_i+{{\widehat{W}}_i}^{\mathrm{T}}S\left(x_i\right)+p_i(z_{2,i})\right]\\ {\widehat{d}}_i={\xi }_i+p_i(z_{2,i})\end{array}$$

(29)

where $l_i>0$ is the design parameter, $p_i(z_{2,i})=l_i{\displaystyle \int {\dot{z}}_{2,i}}dt$, and ${\xi }_i$ is the state variable.

The estimation error dynamic is computed as follows:

$$\begin{array}{ll}{\dot{\tilde{d}}}_i& ={\dot{d}}_i-{\dot{\widehat{d}}}_i\\ & =l_i{\xi }_i+l_i\left[J_1{\tau }_i-{\Im }_i+{{\widehat{W}}_i}^{\mathrm{T}}S\left(x_i\right)+p_i(z_{2,i})\right]-l_i{\dot{z}}_{2,i}+{\dot{d}}_i\\ & =-l_i{\tilde{d}}_i-l_i{{\tilde{W}}_i}^{\mathrm{T}}S\left(x_i\right)+l_i{\dot{\tilde{x}}}_{2i}-l_i{\tilde{\Im }}_i+{\dot{d}}_i\end{array}$$

(30)

where ${\tilde{d}}_i=d_i-{\widehat{d}}_i$, and ${\tilde{W}}_i=W_i^{\ast }-{\widehat{W}}_i$.

Based on the multi-agent consensus and graph theory, and taking full advantage of the information exchange between neighboring AUHs, the weight coefficient update law of cooperative neural network is devised as below:

$${\dot{\widehat{W}}}_{ij}={\mathrm{\Gamma }}_{1,ij}\left[S_j\left(x_i\right)z_{2,ij}+{\mathrm{\sigma }}_{ij}{\widehat{W}}_{ij}\right]-{\gamma }_{2,ij}{\displaystyle \sum _{k=1}^N{a}_{ik}\left({\widehat{W}}_{ij}-{\widehat{W}}_{kj}\right)}$$

(31)

where ${\mathrm{\Gamma }}_{1,ij}$, ${\mathrm{\sigma }}_{ij}$, and ${\gamma }_{2,ij}$ are positive constants, ${\mathrm{\Gamma }}_{1,ij}\left[S_j\left(x_i\right)z_{2,ij}+{\mathrm{\sigma }}_{ij}{\widehat{W}}_{ij}\right]$ represents the adaptive term for monomeric AUH, and ${\gamma }_{2,ij}{\displaystyle \sum _{k=1}^N{a}_{ik}\left({\widehat{W}}_{ij}-{\widehat{W}}_{kj}\right)}$ stands for the AUHs’ consensus cooperative adaptive term.

Remark 3. 

Although this work focuses on formation control, the consensus problem is introduced as a critical step to ensure consistent information sharing among the AUHs. Specifically, the connection and interaction facilitated through the cooperative neural network ensure rapid convergence of the weight coefficients, thereby enabling a robust and efficient formation control strategy. This perspective aligns with the broader understanding of consensus as a means to achieve coordinated behavior in multi-agent systems.

The NN-based cooperative formation controller is designed using a state observer (17), disturbance observer (29), and updated law (31).

$${\tau }_i={J_1}^{-1}\left[-K_{2,i}{z}_{2,i}-{\mathrm{{\rm Y}}}_i{z}_{1,i}+{\Im }_i-{\widehat{W}}_i^{\mathrm{T}}S\left(x_i\right)-{\widehat{d}}_i\right]$$

(32)

Theorem 1. 

If we consider the AUHs’ formation system (3), the state observer (17), the disturbance observer (29), the cooperative RBFNN adaptive update law (31), and the NN-based cooperative formation controller (32), then we have (i). All of the state variables of the system are uniformly ultimately bounded (UUB). (ii) The system achieves the specified performance, with trace errors converging in finite time within the limits of the performance function.

The relevant proof procedure for Theorem 1 can be seen in Proof A1 in Appendix A.

4. AUH Formation Control Using Experience

We propose the learning mechanism that enables the NN-based formation controller (32) to learn dynamic uncertainties $F_i(x_i)$ and store the acquired knowledge during the adaptive process. Furthermore, the gained experience will be utilized to establish an experience-based cooperative formation controller which deals with similar dynamic uncertainties $F_i(x_i)$ using experience instead of using the adaptive method repeatedly.

4.1. Learn from Formation Track Control

In this section, our goal is to establish a cooperative formation controller using the localized RBFNN which can learn dynamic uncertainties over a local range along the periodic reference trajectory.

Lemma 1 

([22]). In localized RBF networks, each individual basis function has a limited impact on the network output. In other words, localized RBFNN only has localized representation during deterministic learning. Therefore, for any bounded orbit $x_i(t)$, the model uncertainty $F_i(x_i)$ can be approximated using neurons located in a local region along the trajectory.

$$F_i(x_i)={W_{\zeta }}^{*\mathrm{T}}{S}_{\zeta }(x_i)+{\epsilon }_{\zeta ,i}$$

(33)

where ${\epsilon }_{\zeta ,i}$ is the inherent error, $S_{\zeta }(x_i)\in {ℝ}^{q_{\zeta }}$ is the regress subvector with the neural node center placed close to the trajectory, i.e., $‖{\mu }_j-x_i‖<d_N$, where ${\mu }_j$ is the center vector defined in Section 2.

Lemma 2 

([22]). Consider arbitrarily continuous periodic orbit $x_i(t):\left[0,\infty \right)\to {ℝ}^d$ within a bounded set ${\mathrm{\Omega }}_{x_i}\in {ℝ}^d$. Then, for RBFNN $W^{\mathrm{T}}S(x_i)$ with node centers located on a large enough regular lattice ${\mathrm{\Omega }}_{\zeta }$ that satisfies ${\mathrm{\Omega }}_{x_i}\in {\mathrm{\Omega }}_{\zeta }$, the regress subvector $S_{\zeta }(x_i)$ defined in Lemma 1 satisfies the persistently exciting (PE) condition.

Remark 4. 

It is important to note that the satisfaction of the PE condition is one of the bases of deterministic learning. In other words, deterministic learning occurs when the system tracking the periodic orbit and regress subvector satisfies the PE condition simultaneously.

Theorem 2. 

Consider the AUH system (3) with Assumptions 1~3. For each AUH, if there exists a compact set large enough that $x_i\in {\mathrm{\Omega }}_{x_i}$ holds at any time, then the following conclusion stands if ${[x_{1,i}^{\mathrm{T}}(0),x_{2,i}^{\mathrm{T}}(0)]}^{\mathrm{T}}\in {\mathrm{\Omega }}_{x_i(0)}$ and ${\widehat{W}}_{ij}(0)=0$.

(1) The partial PE condition of the subvector is satisfied.

(2) Along the periodic reference trace ${\phi }_{\zeta ,i}(x_i(t)){|}_{t\ge T_i}$, the weight of the regress subvector will converge to the optimal value $W_{\zeta ,ij}^{*}$.

(3) The local precise approximation of dynamic uncertainties  $F_i(x_i)$  can be realized by  ${{\widehat{W}}_i}^{\mathrm{T}}S\left(x_i\right)$  and  ${{\overline{W}}_i}^{\mathrm{T}}S\left(x_i\right)$  simultaneously.

$$\begin{array}{ll}{F}_i(x_i)& ={\widehat{W}}_i^{\mathrm{T}}S\left(x_i\right)+{\tilde{W}}_i^{\mathrm{T}}S\left(x_i\right)+{\epsilon }_i\\ & ={\overline{W}}_i^{\mathrm{T}}S\left(x_i\right)+{\overline{\epsilon }}_i\end{array}$$

(34)

where ${\overline{W}}_i^{\mathrm{T}}=\mathrm{blockdiag}\left[{\overline{W}}_{i,1}^{\mathrm{T}},\dots ,{\overline{W}}_{i,6}^{\mathrm{T}}\right]\in {ℝ}^{6\times 6q}$

$${\overline{W}}_{ij}=\mathrm{mean}{\widehat{W}}_{ij}(t),\hspace{1em}t\in \left[t_{a,i},t_{b,i}\right]$$

(35)

where $\left[t_{a,i},t_{b,i}\right]$ is a time interval after the system (3) converges to the transient range, and ${\overline{\epsilon }}_i={\left[{\overline{\epsilon }}_{i,1},\dots ,{\overline{\epsilon }}_{i,6}\right]}^{\mathrm{T}}$ is close to ${\epsilon }_i$.

Remark 5. 

The time interval $\left[t_{a,i},t_{b,i}\right]$ should be at least one orbit period after the system converges, which means that enough dynamic information is expressed, and the RBFNN is sufficiently trained.

The relevant proof procedure for Theorem 2 can be seen in Proof A2 in Appendix A.

4.2. Experience-Based AUH Formation Controller

According to Theorem 2, the experience-based RBFNN can locally accurately approximate the dynamic uncertainties along the periodic reference trajectory. The RBFNN weight obtained by the adaptive method is learned by the controller and stored in the ${{\overline{W}}_i}^{\mathrm{T}}$.

By using experience, the experience-based disturbance observer can be specified as follows:

$$\begin{array}{l}{\dot{\xi }}_i=-l_i{\xi }_i-l_i\left[J_1{\tau }_i-{\Im }_i+{\overline{W}}_i^{\mathrm{T}}S\left(x_i\right)+p_i(z_{2,i})\right]\\ {\widehat{d}}_i={\xi }_i+p_i(z_{2,i})\end{array}$$

(36)

The experience-based cooperative formation controller can be designed as:

$${\tau }_i={J_1}^{-1}\left[-K_{2,i}{z}_{2,i}-{\mathrm{{\rm Y}}}_i{z}_{1,i}+{\Im }_i-{{\overline{W}}_i}^{\mathrm{T}}S\left(x_i\right)-{\widehat{d}}_i\right]$$

(37)

Theorem 3. 

If we consider the AUH formation system (3) with Assumptions 1~3, the state observer (17), the empirically based disturbance observer (36), the empirically based cooperative formation controller (37), then we have (i). All states in the system are UUB. (ii) The system achieves the specified performance, with trace errors converging in finite time within the limits of the performance function.

The relevant proof procedure for Theorem 3 can be seen in Proof A3 in Appendix A.

5. Simulation Results

To validate the proposed method, a group of four homogeneous AUHs was simulated in a lozenge formation tracking a predefined reference trajectory. The AUHs were modeled with identical dynamic parameters, operating in an environment with external disturbances and communication constraints. This setup represents practical applications such as collaborative underwater inspections, environmental monitoring, and resource exploration. The AUH’s model dynamics is shown in [14].

Table 2. The initial information on AUH formation.

Terms	Values
The initial states	$\begin{array}{l}{\eta }_i\left(0\right)={[20,\hspace{0.33em}-30,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0]}^{\mathrm{T}}\\ v_i\left(0\right)={[0,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0]}^{\mathrm{T}}\end{array}$
The reference trajectory of the virtual leader ${\eta }_d$	$\begin{array}{ll}{\eta }_d=& [30\sin \left(0.1t\right)+20\cos \left(0.1t\right),\\ & 20\sin \left(0.1t\right)-30\cos \left(0.1t\right),\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0,\hspace{0.33em}0.1t{]}^{\mathrm{T}}\end{array}$
The external disturbance ${\tau }_{d,i}$	$\begin{array}{c}{\tau }_{d,i}=[20+5\cos \left(0.2t+{\displaystyle \frac{\pi }{4}}\right),\hspace{0.33em}15+10\sin \left(0.2t\right),0,\\ 5\cos \left(0.2t+{\displaystyle \frac{\pi }{4}}\right),\hspace{0.33em}0,\hspace{0.33em}10+5\sin \left(0.5t\right){]}^{\mathrm{T}}\end{array}$

presents the initial pattern of AUH formation. The desired formation shape is a lozenge, and the vertex of the lozenge represents the AUH’s barycenter. We choose the desired relative offsets as ${\eta }_1^{*}={\left[0,-5,0\right]}^{\mathrm{T}}$, ${\eta }_2^{*}={\left[-5,0,0\right]}^{\mathrm{T}}$, ${\eta }_3^{*}={\left[0,5,0\right]}^{\mathrm{T}}$, and ${\eta }_4^{*}={\left[5,0,0\right]}^{\mathrm{T}}$ to achieve a lozenge formation shape. The neighboring sets ${𝒱}_1=\left\{{\vartheta }_2,{\vartheta }_3\right\}$, ${𝒱}_2=\left\{{\vartheta }_1,{\vartheta }_4\right\}$, ${𝒱}_3=\left\{{\vartheta }_1,{\vartheta }_4\right\}$, and ${𝒱}_4=\left\{{\vartheta }_2,{\vartheta }_3\right\}$ describe the communication topology. The modeling uncertainties are provided by $\Delta \left({\eta }_i,v_i\right)={\left[{\Delta }_1,{\Delta }_2,{\Delta }_3,{\Delta }_4,{\Delta }_5,{\Delta }_6\right]}^{\mathrm{T}}$, where ${\Delta }_1=\cos \left(u_i^2\right)+1$, ${\Delta }_2=\sin \left(v_i^2\right)+1$, ${\Delta }_3=0.1w_i^2+1$, ${\Delta }_4=1-\cos \left(p_i^2\right)$, ${\Delta }_5=1-\sin \left(q_i^2\right)$, and ${\Delta }_6=0.2r_i^2$.

We construct the RBFNNs ${\widehat{W}}_{ij}^{\mathrm{T}}{S}_j\left(v_i\right),\hspace{0.33em}i=1-4\hspace{0.33em}\mathrm{and}\hspace{0.33em}j=1-6$ to approximate the modeling uncertainties using 200 nodes with the centers evenly spaced on [−4, 4] and the widths being 6.

The parameters in FTPPC are provided as ${\rho }_{i0}=5$, ${\rho }_{i\infty }=0.03$, $k_2=0.8$, and $t_f=20$. The parameters in state observer (17), disturbance observer (29), adaptive law (31), and formation control law (32) are given as $L_{1i}=\mathrm{diag}\left[15,15,15,15,15,15\right]$, $L_{2i}=\mathrm{diag}\left[50,50,50,50,50,50\right]$, $K_{1,\hspace{0.33em}i}=\mathrm{diag}\left[5,5,5,5,5,5\right]$, $K_{2,\hspace{0.33em}i}=\mathrm{diag}\left[4,4,4,4,4,4\right]$, ${\varsigma }_1=4$, ${\varsigma }_2=4$, and ${\varsigma }_3=2.5$. The training interval $[t_{a,i},t_{b,i}]$ is set to be $[20,20+20\pi ]$, where 20 is the converge time specified by FTPPC, and $20\pi$ is the orbit period.

Figure 2 illustrates the shape and trajectory of AUH formation. Figure 3 shows the tracking error using the proposed experience-based cooperative formation control law (56), which converges in a finite time within the limit of the performance function. Furthermore, the tracking error remains in a compact set when $t>20+20\pi$, which means the tracking performance is guaranteed when experience is used to deal with the dynamic uncertainty. Figure 4 illustrates the RBFNN approximation error for dynamic uncertainty, which converges to a compact set during the adaptive process. Moreover, it can be seen that the approximation error remains in an admissible set after experience is used at $t=(20+20\pi )$ s.

Several comparison experiments are provided to further demonstrate the superiority of the proposed algorithm. The experience-based control method in this paper is compared with the adaptive NN-based control method used in [18], which is conducted on a computer with Intel i5-11400 2.60 GHz and 16 GB RAM, and the solver step is set to be 0.01 s in MATLAB 2021a. The result in

Table 3. The comparison experiment results.

Methods	The Adaptive NN-Based Control Method	The Experience-Based Control Method
Simulation time setting (s)	200	200
Actual running time (s)	97.62	83.67
Tracking error in translation degree (m)	<0.03	<0.03
Tracking error in rotation degree (rad)	<0.03	<0.03

shows that the solver running time is shorter using experience-based control method under the same tracking accuracy, which means the computing burden is reduced. The NN-based state observer (NN-SO) is compared with the ESO proposed in [17], and the result is shown in Figure 5. We can see that the NN-SO has a better estimate accuracy and smaller oscillation magnitude compared to the ESO. Figure 6 provides the results of comparison between the disturbance observer (DO) used in this paper and the ESO proposed in [17]. It can be seen that the disturbance observer has better estimate accuracy compared with the ESO. A comparison between the FTPPC used in our work and the PPC in [18] is shown in Figure 7. We can observe that the FTPPC leads to a faster convergence speed, which is more suitable for deterministic learning control methods.

To highlight the advantages of the proposed formation control algorithm, we conducted a comparative simulation study against a recently published method [37]. This method also employs a prescribed performance control strategy and was chosen for its relevance and novelty in addressing formation control challenges.

To ensure a fair and rigorous comparison, the following conditions were maintained for both algorithms: (1) both controllers were initialized with identical formation configurations and tracking errors. (2) The parameters defining the prescribed performance functions were set equivalently. (3) Both simulations included the same model uncertainty and external disturbances.

The comparative results, presented in Figure 8, demonstrate the following advantages of the proposed algorithm: the proposed method achieved faster error decay during the transient phase compared to the reference algorithm, indicating quicker adaptation to the desired trajectory. The steady-state tracking error achieved by the proposed method was significantly smaller, reflecting higher accuracy and stability under disturbances and uncertainties. These findings confirm that the proposed control algorithm outperforms the reference method in both transient and steady-state performance.

6. Conclusions

This study presents a novel formation control framework for AUHs that integrates prescribed performance control, hybrid observers, and experience-based learning. The proposed method offers several advantages: the hybrid observer, which combines RBFNN with a Luenberger observer, effectively addresses the nonlinear dynamics of AUHs, resulting in higher estimation accuracy. Additionally, the incorporation of experience-based learning reduces the need for continuous adaptation, thereby enhancing computational efficiency. Furthermore, the adoption of finite-time prescribed performance control ensures faster convergence of tracking errors compared to conventional approaches, contributing to improved overall performance. Nevertheless, the proposed method has certain limitations. It relies on reliable communication among AUHs, necessitates sufficient training data to support the experience-based framework, and imposes moderate computational demands on onboard systems. Moreover, a key limitation of this study is the absence of experimental validation using real robots.

To address these limitations, future research will focus on developing decentralized communication frameworks to enhance robustness and designing training methods suitable for highly dynamic environments. In parallel, we are actively pursuing the miniaturization and manufacturing of additional AUH prototypes. These efforts aim to facilitate future experiments with real robots, which will validate the proposed control strategy under realistic operating conditions and further demonstrate its practical applicability.