Denoting the set of real numbers as R , we define R m × n as the set of m × n real matrices, and R n as the set of n × 1 real vectors. The identity matrix is symbolized as I . The vector with all elements being 1 in an n -dimensional space is represented as 1 n . The sets S + n and S − n + stand for real symmetric n × n and positive definite matrices, respectively. A block diagonal matrix with matrices X 1 , X 2 , … , X p on its main diagonal is denoted by diag { X 1 , X 2 , … , X p } . A ⊗ B signifies the Kronecker product of matrices A and B . For a matrix A , A ⃗ is the vectorization of A by stacking its columns on top of each other. For a series of column vectors x 1 , … , x n , col { x 1 , … , x n } represents a column vector formed by stacking them together. Given two integers k 1 and k 2 with k 1 < k 2 , I [ k 1 , k 2 ] = { k 1 , k 1 + 1 , … , k 2 } . For a vector x ∈ R n , its norm is defined as | x | ≔ ( x T x ) 1 / 2 . For a square matrix A , λ i ( A ) denotes its i -th eigenvalue, while λ min ( A ) and λ max ( A ) represent its minimum and maximum eigenvalues, respectively.

A directed graph G = ( V , E ) comprises nodes in the set V = { 1,2 , … , N } and edges in E ⊆ V × V . An edge from node i to node j is represented as ( i , j ) , with i as the parent node and j as the child node. Node i is also termed a neighbor of node j . N i is considered as the subset of V consisting of the neighbors of node i . A sequence of edges in G , ( i 1 , i 2 ) , ( i 2 , i 3 ) , … , ( i k , i k + 1 ) , is called a path from node i 1 to node i k + 1 . Node i k + 1 is reachable from node i 1 . A directed tree is a graph where each node, except for a root node, has exactly one parent. The root node is reachable from all other nodes. A directed graph G contains a directed spanning tree if at least one node can reach all other nodes. The weighted adjacency matrix of G is a non-negative matrix A = [ a i j ] ∈ R N × N , where a i i = 0 and a i j > 0   ⇒   ( j , i ) ∈ E . The Laplacian of G is denoted as L = [ l i j ] ∈ R N × N , where l i i = ∑ j = 1 N a i j and l i j = − a i j if i ≠ j . It is established that L has at least one eigenvalue at the origin, and all nonzero eigenvalues of L have positive real parts. From Ren and Beard (2005), L has one zero eigenvalue and remaining eigenvalues with positive real parts if and only if G has a directed spanning tree.

The RBFNN Networks can be described as f n n ( Z ) = ∑ i = 1 N w i s i ( Z ) = W T S ( Z ) , where Z ∈ Ω Z ⊆ R q and W = w 1 , … , w N T ∈ R N as input and weight vectors respectively (Park and Sandberg, 1991). N indicates the number of NN nodes, S ( Z ) = [ s 1 ( ‖ Z − μ i ‖ ) , … , s N ( ‖ Z − μ i ‖ ) ] T with s i ( ⋅ ) is a radial basis function, and μ i ( i = 1 , … , N ) is distinct points in the state space. The Gaussian function s i ( ‖ Z − μ i ‖ ) = exp − ( Z − μ i ) T ( Z − μ i ) η i 2 is generally used for radial basis function, where μ i = [ μ i 1 , μ i 2 , … , μ i N ] T is the center and η i is the width of the receptive field. The Gaussian function categorized by localized radial basis function s in the sense that s i ( ‖ Z − μ i ‖ ) → 0 as ‖ Z ‖ → ∞ . Moreover, for any bounded trajectory Z ( t ) within the compact set Ω Z , f ( Z ) can be approximated using a limited number of neurons located in a local region along the trajectory f ( Z ) = W ζ * S ζ ( Z ) + ϵ ζ . ζ denotes the indices of active RBFNN nodes where | s j i ( Z ) | > ι , based on the state Z ( t ) . ϵ ζ is the approximation error, with ϵ ζ = O ( ϵ ) = O ( ϵ * ) , S ζ ( Z ) = [ s j 1 ( Z ) , … , s j ζ ( Z ) ] T ∈ R N ζ , W ζ * = [ w j 1 * , … , w j ζ * ] T ∈ R N ζ , N ζ < N n , and the integers j i = j 1 , … , j ζ are defined by | s j i ( Z p ) | > ι ( ι > 0 is a small positive constant) for some Z p ∈ Z ( t ) . This holds if ‖ Z ( t ) − ξ j i ‖ < ϵ for t > 0 . The following lemma regarding the persistent excitation (PE) condition for RBFNN is recalled from Wang and Hill (2018).

Lemma 1

Consider any continuous recurrent trajectory ¹ Z ( t ) : [ 0 , ∞ ) → R q . Z ( t ) remains in a bounded compact set Ω Z ⊂ R q . Then for an RBFNN W T S ( Z ) with centers placed on a regular lattice (large enough to cover the compact set Ω Z ), the regressor subvector S ζ ( Z ) consisting of RBFNN with centers located in a small neighborhood of Z ( t ) is persistently exciting.

A multi-agent system comprising N AUVs with heterogeneous nonlinear uncertain dynamics is considered. The dynamics of each AUV can be expressed as Fossen (1999): η ̇ i = J i η i ν i , M i ν ̇ i + C i ν i ν i + D i ν i ν i + g i η i + Δ i χ i = τ i . (1)

In this study, the subscript i ∈ I [ 1 , N ] identifies each AUV within the multi-agent system. For every i ∈ I [ 1 , N ] , the vector η i = [ x i , y i , ψ i ] T ∈ R 3 represents the i -th AUV’s position coordinates and heading in the global coordinate frame, while ν i = [ u i , v i , r i ] T ∈ R 3 denotes its linear velocities and angular rate of heading relative to a body-fixed frame. The positive definite inertial matrix M i = M i T ∈ S 3 + , Coriolis and centripetal matrix C i ( ν i ) ∈ R 3 × 3 , and damping matrix D i ( ν i ) ∈ R 3 × 3 characterize the AUV’s dynamic response to motion. The vector g i ( η i ) ∈ R 3 × 1 accounts for the restoring forces and moments due to gravity and buoyancy. The term Δ i ( χ i ) ∈ R 3 × 1 , with χ i ≔ col { η i , ν i } , describes the vector of generalized deterministic unmodeled uncertain dynamics for each AUV.

The vector τ i ∈ R 3 represents the control inputs for each AUV. The associated rotation matrix J i ( η i ) is given by: J i η i = cos ψ i sin ψ i 0 − sin ψ i cos ψ i 0 0 0 1 ,

Unlike previous work Yuan et al. (2017), which assumed known values for the AUV’s inertia and rotation matrices, this study considers all matrix coefficients, including C i ( ν i ) , D i ( ν i ) , g i ( η i ) , and Δ i ( χ i ) , as well as the inertia matrix, as completely unknown. The adaptive estimation process inherently addresses the effects of external forces and disturbances on system dynamics. This eliminates the need for explicit parameter estimation of these forces, as disturbances like water flow, varying currents, or depth variations are directly incorporated into the control input through adaptive estimation. This makes the controller universally applicable to any AUV, regardless of its design, weight, or environmental conditions by addressing both internal and external dynamic variation at the same time.

Internally, it handles unknown parameters such as mass and damping coefficients, as well as unmodeled nonlinear interactions and couplings. Externally, it accounts for unpredictable disturbances, including fluctuating water currents, depth-dependent pressures, and changes in hydrodynamic forces.

By avoiding reliance on predefined models, the proposed approach is robust and adaptable to diverse mission scenarios and unexpected environmental changes, ensuring reliable performance even in highly uncertain conditions.

In the context of leader-following formation tracking control, the following virtual leader dynamics generates the tracking reference signals: χ ̇ 0 = A 0 χ 0 , (2) with “0” marking the leader node, the leader state χ 0 ≔ col { η 0 , ν 0 } with η 0 ∈ R 3 and ν 0 ∈ R 3 , A 0 ∈ R 6 × 6 is a constant matrix available only to the leader’s neighboring AUV agents.

Considering the system dynamics of multiple AUVs (Equation 1) along with the leader dynamics (Equation 2), we establish a non-negative matrix A = [ a i j ] , where i , j ∈ I [ 0 , N ] such that for each i ∈ I [ 1 , N ] , a i 0 > 0 if and only if agent i has access to the reference signals η 0 and ν 0 . All remaining elements of A are arbitrary non-negative values, such that a i i = 0 for all i . Correspondingly, we establish G = ( V , E ) as a directed graph derived from A , where V = { 0,1 , … , N } designates node 0 as the leader, and the remaining nodes correspond to the N AUV agents. We proceed under the following assumptions:

Assumption 1

All the eigenvalues of A 0 in the leader’s dynamics (Equation 2) are located on the imaginary axis.

Assumption 2

The directed graph G contains a directed spanning tree with the node 0 as its root.

Assumption 1 is crucial for ensuring that the leader dynamics produce stable, meaningful reference trajectories for formation control. It ensures that all states of the leader, represented by χ 0 = col { η 0 , ν 0 } , remain within the bounds of a compact set Ω 0 ⊂ R 6 for all t ≥ 0 . The trajectory of the system, starting from χ 0 ( 0 ) and denoted by ϕ 0 ( χ 0 ( 0 ) ) , generates periodic signal. This periodicity is essential for maintaining the Persistent Excitation (PE) condition, which is pivotal for achieving parameter convergence in Distributed Adaptive (DA) control systems. Modifications to the eigenvalue constraints on A 0 mentioned in Assumption 1 may be considered when focusing primarily on formation tracking control performance, as discussed later.

Additionally, Assumption 2 reveals key insights into the structure of the Laplacian matrix L of the network graph G . Let Ψ be an N × N non-negative diagonal matrix where each i -th diagonal element is a i 0 for i ∈ I [ 1 , N ] . The Laplacian L is formulated as: L = ∑ j = 1 N a 0 j − a 01 , … , a 0 N − Ψ 1 N H , where a 0 j > 0 if ( j , 0 ) ∈ E and a 0 j = 0 otherwise. This results in H 1 N = Ψ 1 N since L 1 N + 1 = 0 . As cited in Su and Huang (2011), all nonzero eigenvalues of H , if present, exhibit positive real parts, confirming H as nonsingular under Assumption 2.

Problem 1

In the context of a multi-AUV system (Equation 1) integrated with virtual leader dynamics (Equation 2) and operating within a directed network topology G , the aim is to develop a distributed NN learning control protocol that leverages only local information. The specific goals are twofold:

1) Formation Control: Each of the N AUV agents will adhere to a predetermined formation pattern relative to the leader, maintaining a specified distance from the leader’s position η 0 .

Remark 1

The leader dynamics described in Equation 2 are designed as a neutrally stable LTI system. This design choice facilitates the generation of sinusoidal reference trajectories at various frequencies which is essential for effective formation tracking control. This approach to leader dynamics is prevalent in the literature on multiagent leader-following distributed control systems like Yuan (2017) and Jandaghi et al. (2024).

Remark 2

It is important to emphasize that the formulation assumes formation control is required only within the horizontal plane, suitable for AUVs operating at a constant depth, and that the vertical dynamics of the 6 degrees of freedom (DOF) AUV system, as detailed in Prestero (2001), are entirely decoupled from the horizontal dynamics.

As shown in Figure 1, a two-layer hierarchical design approach is proposed to address the aforementioned challenges. The first layer, the Cooperative Estimator, enables information exchange among neighboring agents. The second layer, known as the Decentralized Deterministic Learning (DDL) controller, processes only local data from each individual AUV. The development and formulation of the first layer are discussed in detail in Section 3.1, while the DDL control strategy, along with its corresponding controller design and analysis, is provided in Section 3.2.

In the context of leader-following formation control, not all AUV agents may have direct access to the leader’s information, including tracking reference signals ( χ 0 ) and the system matrix ( A 0 ) . This necessitates collaborative interactions among the AUV agents to estimate the leader’s information effectively. Drawing on principles from multiagent consensus and graph theories Ren and Beard (2008), we propose to develop a distributed adaptive observer for the AUV systems as: χ ̂ ̇ i t = A ^ i t χ ̂ i t + β i 1 ∑ j = 1 N a i j χ ̂ j t − χ ̂ i t , ∀ i ∈ I 1 , N . (3)

The observer states for each i -th AUV, denoted by χ ^ i = [ η ^ i , ν ̂ i ] T ∈ R 6 , aim to estimate the leader’s state, χ 0 = [ η 0 , ν 0 ] T ∈ R 6 . As t → ∞ , these estimates are expected to converge, such that η ̂ i approaches η 0 and ν ̂ i approaches ν 0 , representing the leader’s position and velocity, respectively. Equation 3 accounts for the communication graph by including the adjacency matrix information through the term a i j . Note that A ^ i ( t ) ∈ R 6 × 6 represents an estimate computed by agent i of the leader’s matrix dynamics A 0 ( t ) ∈ R 6 × 6 which is also not available to the agents. Therefore, each agent estimates such matrix using a cooperative adaptation law: A ^ ̇ i t = β i 2 ∑ j = 1 N a i j A ^ j t − A ^ i t , ∀ i ∈ I 1 , N , (4) which borrowed from Ren and Beard (2008) as well. The constants β i 1 and β i 2 are all positive numbers and are subject to design.

Remark 3

Each AUV agent in the group is equipped with an observer configured as specified in Equations 3, 4, comprising two state variables, χ i and A i . For each i ∈ I [ 1 , N ] , χ ̂ i estimates the virtual leader’s state χ 0 , while A ^ i estimates the leader’s system matrix A 0 . The real-time data necessary for operating the i -th observer includes: (1) the estimated state χ ̂ i and matrix A i ^ , obtained from the i -th AUV itself, and (2) the estimated states χ ̂ j and matrices A j ^ for all j ∈ N i , obtains from the j -th AUV’s neighbors. Note that in Equations 3, 4, if j ∉ N i , then a i j = 0 , indicating that the i -th observer does not utilize information from the j -th AUV agent. This configuration ensures that the proposed distributed observer can be implemented in each local AUV agent using only locally estimated data from the agent itself and its immediate neighbors, without the need for global information such as the size of the AUV group or the network interconnection topology.

To verify the convergence properties, we need to compute the error dynamics. Now we define the estimation error for the state and the system matrix for agent i as χ ̃ i = χ ̂ i − χ 0 and A ̃ i = A ^ i − A 0 , and then we derive the error dynamics: χ ̃ ̇ i ⁢ t = A ^ i ⁢ t ⁢ χ ̂ i ⁢ t − A 0 ⁢ χ 0 ⁢ t + β i 1 ∑ j = 1 N a i j ⁢ χ ̂ j ⁢ t − χ ̂ i ⁢ t = A ^ i ⁢ t ⁢ χ ̂ i ⁢ t − A 0 ⁢ χ ̂ i ⁢ t + A 0 ⁢ χ ̂ i ⁢ t − A 0 ⁢ χ 0 ⁢ t + β i 1 ∑ j = 1 N a i j ⁢ χ ̂ j ⁢ t − χ 0 ⁢ t + χ 0 ⁢ t − χ ̂ i ⁢ t = A 0 ⁢ χ ̃ i ⁢ t + A ̃ i ⁢ t ⁢ χ ̃ i ⁢ t + A ̃ i ⁢ t ⁢ χ 0 ⁢ t + β i 1 ∑ j = 1 N a i j ⁢ χ ̃ j ⁢ t − χ ̃ i ⁢ t A ̃ ̇ i ⁢ t = β i 2 ∑ j = 1 N a i j ⁢ A ̃ j ⁢ t − A ̃ i ⁢ t , ∀ i ∈ I ⁡ 1 , N .

Define the collective error states and adaptation matrices: χ ̃ = col { χ ̃ 1 , … , χ ̃ N } for the state errors, A ̃ = col { A ̃ 1 , … , A ̃ N } for the adaptive parameter errors, A ̃ b = diag { A ̃ 1 , … , A ̃ N } representing the block diagonal of adaptive parameters, B β 1 = diag { β 11 , … , β N 1 } and B β 2 = diag { β 12 , … , β N 2 } for the diagonal matrices of design constants. With these definitions, the network-wide error dynamics can be expressed as: χ ̃ ̇ ⁢ t = I N ⊗ A 0 − B β 1 ⁢ H ⊗ I 6 ⁢ χ ̃ ⁢ t + A ̃ b ⁢ t ⊗ I 6 ⁢ χ ̃ ⁢ t + A ̃ b ⁢ t ⁢ 1 N ⊗ χ 0 ⁢ t , A ̃ ̇ ⁢ t = − B β 2 ⁢ H ⊗ I 6 ⁢ A ̃ ⁢ t . (5)

Theorem 1

Consider the error system Equation 5. Under Assumptions 1, 2, and given that β 1 , β 2 > 0 , it follows that for all i ∈ I [ 1 , N ] and for any initial conditions χ 0 ( 0 ) , χ i ( 0 ) , A i ( 0 ) , the error dynamics of the adaptive parameters and the states will converge to zero exponentially. Specifically, lim t → ∞ A ̃ i ( t ) = 0 and lim t → ∞ χ ̃ i ( t ) = 0 .

This convergence is facilitated by the independent adaptation of each agent’s parameters within their respective error dynamics, represented by the block diagonal structure of A ̃ b and control gains B β 1 and B β 2 . These matrices ensure that each agent’s parameter updates are governed by local interactions and error feedback, consistent with the decentralized control framework.

Proof: We begin by examining the estimation error dynamics for A ̃ as presented in Equation 5. This can be rewritten in the vector form: A ̃ ⃗ ̇ 0 t = − β 2 I 6 ⊗ H ⊗ I 6 A ̃ ⃗ 0 t . (6)

Under Assumption 2, all eigenvalues of H possess positive real parts according to Su and Huang (2011). Consequently, for any positive β 2 > 0 , the matrix − β 2 ( I 6 ⊗ ( H ⊗ I 6 ) ) is guaranteed to be Hurwitz, Which implies the exponential stability of system (Equation 6). Hence, it follows that lim t → ∞ A ̃ ⃗ 0 ( t ) = 0 exponentially, leading to lim t → ∞ A ̃ i 0 ( t ) = 0 exponentially for all i ∈ I [ 1 , N ] . Now, we analyze the error dynamics for χ 0 in Equation 5. Based on the previous discussions, We have lim t → ∞ A ̃ b ( t ) = 0 exponentially, and the term A ̃ b ( t ) ( 1 N ⊗ χ 0 ( t ) ) will similarly decay to zero exponentially. Based on Cai et al. (2015), if the system defined by χ ̃ ̇ 0 t = I N ⊗ A 0 − β 1 H ⊗ I 6 χ ̃ 0 t . (7) is exponentially stable, then lim t → ∞ χ 0 ( t ) = 0 exponentially. With Assumption 1, knowing that all eigenvalues of A 0 have zero real parts, and since H as nonsingular with all eigenvalues in the right-half plane, system (Equation 7) is exponentially stable for any positive β 1 > 0 . Consequently, this ensures that lim t → ∞ χ 0 ( t ) = 0 , i.e., lim t → ∞ χ i 0 ( t ) = 0 exponentially for all i ∈ I [ 1 , N ] .

Now, each individual agent can accurately estimate both the state and the system matrix of the leader through cooperative observer estimation Equations 3, 4. This information will be utilized in the DDL controller design for each agent’s second layer, which will be discussed in the following subsection.

To fulfill the overall formation learning control objectives, in this section, we develop the DDL control law for the multi-AUV system defined in Equation 1. We use d i * to denote the desired distance between the position of the i -th AUV agent η i and the virtual leader’s position η 0 . Then, the formation control problem is framed as a position tracking control task, where each local AUV agent’s position η i is required to track the reference signal η d , i ≔ η 0 + d i * . Besides, due to the inaccessibility of the leader’s state information χ 0 for all AUV agents, the tracking reference signal η ̂ d , i ≔ η ̂ 0 , i + d i * is employed instead of the reference signal η d , i . As established in Theorem 1, η ̂ d , i is autonomously generated by each local agent and will exponentially converge to η d , i . This ensures that the DDL controller is feasible and the formation control objectives are achievable for all i ∈ I [ 1 , N ] using η ̂ d , i .

To design the DDL control law that addresses the formation tracking control and the precise learning of the AUVs’ complete nonlinear uncertain dynamics at the same time, we will integrate renowned backstepping adaptive control design method outlined in Krstic et al. (1995) along with techniques from Wang and Hill (2018) and Yuan et al. (2017) for deterministic learning using RBFNN. Specifically, for the i -th AUV agent described in system (Equation 1), we define the position tracking error as z 1 , i = η i − η ̂ d , i for all i ∈ I [ 1 , N ] . Considering J i ( η i ) J i T ( η i ) = I for all i ∈ I [ 1 , N ] , we proceed to: z ̇ 1 , i = J i η i ν i − η ̂ ̇ i , ∀ i ∈ I 1 , N . (8)

To frame the problem in a more tractable way, we assume ν i as a virtual control input and α i as a desired virtual control input in our control strategy design, and by implementing them in the above system we have: z 2 , i = ν i − α i , α i = J i T η i − K 1 , i z 1 , i + η ̂ ̇ i , ∀ i ∈ I 1 , N . (9)

A positive definite gain matrix K 1 , i ∈ S 3 + is used for tuning the performance. Substituting ν i = z 2 , i + α i into Equation 8 yields: z ̇ 1 , i = J i η i z 2 , i − K 1 , i z 1 , i , ∀ i ∈ I 1 , N .

Now we derive the first derivatives of the virtual control input and the desired control input as follows: z ̇ 2 , i = ν ̇ i − α ̇ i = M i − 1 − C i ν i ν i − D i ν i ν i − g i η i − Δ i χ i + τ i − α ̇ i , α ̇ i = J ̇ i T ⁢ η i ⁢ − K 1 , i ⁢ z 1 , i + η ̂ ̇ i + J i T ⁢ η i ⁢ K 1 , i ⁢ η ̂ ̇ i − K 1 , i ⁢ J i ⁢ η i ⁢ ν i + η ̂ ̈ i , ∀ i ∈ I ⁡ 1 , N . (10)

As previously discussed, unlike earlier research that only identified the matrix coefficients C i ( ν i ) , D i ( ν i ) , g i ( η i ) , and Δ i ( χ i ) as unknown system nonlinearities while assuming the mass matrix M i to be known, this work advances significantly by also considering M i as unknown. Consequently, all system dynamic parameters are treated as completely unknown, making the controller fully independent of the robot’s configuration such as its dimensions, mass, or any appendages and the uncertain environmental conditions it encounters, like depth, water flow, and viscosity. This independence is critical as it ensures that the controller does not rely on predefined assumptions about the dynamics, aligning with the main goal of this research. To address these challenges, we define a unique nonlinear function F i ( Z i ) that encapsulates all nonlinear uncertainties as follows: F i Z i = M i α ̇ i + C i ν i ν i + D i ν i ν i + g i η i + Δ i χ i , (11) where F i ( Z i ) = [ f 1 , i ( Z i ) , f 2 , i ( Z i ) , f 3 , i ( Z i ) ] T and Z i = col { η i , ν i } ∈ Ω Z i ⊂ R 6 , with Ω Z i being a bounded compact set. We then employ the following RBFNN to approximate the model dynamics in (Equation 11) expressed by nonlinear functions F i ( Z i ) with f k , i for all i ∈ I [ 1 , N ] and k ∈ I [ 1,3 ] as follows: f k , i Z i = W k , i * T S k i Z i + ϵ k , i Z i , (12) where W k , i * is the ideal constant NN weights, and ϵ k , i ( Z i ) is the approximation error ϵ k , i * > 0 for all i ∈ I [ 1 , N ] and k ∈ I [ 1,3 ] , which satisfies | ϵ k , i ( Z i ) | ≤ ϵ k , i * . This error can be made arbitrarily small given a sufficient number of neurons in the network. A self-adaptation law is designed to estimate the unknown W k , i * online. We aim to estimate W k , i * with W ^ k , i by constructing the DDL feedback control law as follows: τ i = − J i T η i z 1 , i − K 2 , i z 2 , i + W ^ i T S i F Z i . (13) K 2 , i ∈ S 3 + is a feedback gain matrix that can be tuned to achieve the desired performance. To approximate the unknown nonlinear function vector F i ( Z i ) in (Equation 11) along the trajectory Z i within the compact set Ω Z i , we use: W ^ i T S i F Z i = W ^ 1 , i T S 1 , i Z i W ^ 2 , i T S 2 , i Z i W ^ 3 , i T S 3 , i Z i .

Then, from Equations 1, 13 we have: M i ⁢ ν ̇ i + C i ⁢ ν i ⁢ ν i + D i ⁢ ν i ⁢ ν i + g i ⁢ η i + Δ i ⁢ χ i = τ i = − J i T ⁢ η i ⁢ z 1 , i − K 2 , i ⁢ z 2 , i + W ^ k , i T ⁢ S k , i ⁢ Z i .

By subtracting W k , i * T S k , i ( Z i ) + ϵ k , i ( Z i ) from both sides and considering Equations 9, 11, we define W ̃ k , i ≔ W ^ k , i − W k , i * , leading to: z ̇ 2 , i = M i − 1 − J i T η i z 1 , i − K 2 , i z 2 , i + W ^ k , i T S k , i Z i − ϵ k , i Z i .

For updating W ^ k , i online, a robust self-adaptation law is constructed using the σ -modification technique Ioannou and Sun (1996) as follows: W ^ ̇ k , i = − Γ k , i S k , i Z i z 2 k , i + σ k i W ^ k , i . (14) where z 2 , i = [ z 21 , i , z 22 , i , z 23 , i ] T , Γ k , i = Γ k , i T > 0 , and σ k , i > 0 are free parameters to be designed for all i ∈ I [ 1 , N ] and k ∈ I [ 1,3 ] . Integrating Equations 9, 13, 14 yields the following closed-loop system: z ̇ 1 , i = − K 1 , i z 1 , i + J i η i z 2 , i , z ̇ 2 , i = M i − 1 − J i T η i z 1 , i − K 2 , i z 2 , i + W ^ k i T S k i Z i − ϵ k , i Z i , W ̃ ̇ k , i = − Γ k , i S k , i Z i z 2 , k , i + σ k , i W ^ k , i , (15) where, for all i ∈ I [ 1 , N ] and k ∈ I [ 1,3 ] , W ̃ i T S i ( Z i ) = [ W ̃ 1 , i T S 1 , i ( Z i ) , W ̃ 2 , i T S 2 , i ( Z i ) , W ̃ 3 , i T S 3 , i ( Z i ) ] T , and ϵ i ( Z i ) = [ ϵ 1 , i ( Z i ) , ϵ 2 , i ( Z i ) , ϵ 3 , i ( Z i ) ] T .

Remark 4

Unlike the first-layer DA observer design, the second-layer control law is fully decentralized for each local agent. It utilizes only the local agent’s information for feedback control, including χ i , χ ̂ i , and W k , i , without involving any information exchange among neighboring AUVs.

The following theorem summarizes the stability and tracking control performance results of the overall system: Theorem 2

Consider the local closed-loop system (Equation 15). For each i ∈ I [ 1 , N ] , if there exists a sufficiently large compact set Ω Z i such that Z i ∈ Ω Z i for all t ≥ 0 , then for any bounded initial conditions, we have: 1) All signals in the closed-loop system remain uniformly ultimately bounded (UUB). 2) The position tracking error η i − η d , i converges exponentially to a small neighborhood around zero in finite time T i > 0 by choosing the design parameters with sufficiently large λ ̲ ( K 1 , i ) > 0 and λ ̲ ( K 2 , i ) > 2 λ ̄ ( K 1 , i ) > 0 , and sufficiently small σ k , i > 0 for all i ∈ I [ 1 , N ] and k ∈ I [ 1,3 ] .

Proof: 1) Consider the following Lyapunov function candidate for the closed-loop system (Equation 15):

Evaluating the derivative of V i along the trajectory of Equation 15 for all i ∈ I [ 1 , N ] yields: V ̇ i = z 1 , i T − K 1 , i z 1 , i + J i η i z 2 , i + z 2 , i T − J i T η i z 1 , i − K 2 , i z 2 , i + W ̃ k , i T S k , i Z i − ϵ k , i Z i − ∑ k = 1 3 W ̃ k , i T S k , i Z i z 2 k , i + σ k , i W ^ k , i = − z 1 , i T K 1 , i z 1 , i − z 2 , i T K 2 , i z 2 , i − z 2 , i T ϵ k , i Z i − ∑ k = 1 3 σ k , i W ̃ k , i T W ^ k , i , ∀ i ∈ I 1 , N .

Choose K 2 , i = K 1 , i + K 22 , i such that K 1 , i , K 22 , i ∈ S 3 + . Using the completion of squares, we have: − σ k , i W ̃ k , i T W ^ k , i ≤ − σ k , i ‖ W ̃ k , i ‖ 2 2 + σ k , i ‖ W k , i * ‖ 2 2 , − z 2 , i T K 22 , i z 2 , i − z 2 , i T ϵ i Z i ≤ ϵ i T Z i ϵ i Z i 4 λ ̲ K 22 , i ≤ ‖ ϵ i * ‖ 2 4 λ ̲ K 22 , i , where ϵ i * = [ ϵ 1 , i * , ϵ 2 , i * , ϵ 3 , i * ] T . Then, we obtain: V ̇ i ≤ − z 1 , i T K 1 , i z 1 , i − z 2 , i T K 1 , i z 2 , i + ‖ ϵ i * ‖ 2 4 λ ̲ K 22 , i + ∑ k = 1 3 − σ k , i ‖ W ̃ k , i ‖ 2 2 + σ k , i ‖ W k , i * ‖ 2 2 .