Leader-Following Multiple Unmanned Underwater Vehicles ConsensusControl under theFixedandSwitchingTopologieswith Unmeasurable Disturbances

We propose a consensus control strategy for multiple unmanned underwater vehicles (multi-UUVs) with unmeasurable disturbances under the fixed and switching topologies. .e current methods presented in the literature mainly solve tracking consensus problems with the disturbances under the time-invariant and time-varying topologies, respectively. In the paper, considering the complex nonlinear and couple model of the UUV, the technique of the feedback linearization is employed to transform the nonlinear UUV model into a second-order integral UUV model. For unmeasurable disturbances consisting of unknown model uncertainties and external disturbance for each UUV, we design the distributed extended state observer (DESO) to estimate the disturbances using the UUV position information relative to its neighbours. Moreover, leader-following multiUUVs consensus control algorithm that enables all following UUVs to track the leader UUV state information based on the estimation state information from the DESO is proposed for two types of topologies, the fixed and switching topologies. Finally, simulation results are shown to demonstrate the effectiveness of the algorithm proposed in the paper.


Introduction
An unmanned underwater vehicle (UUV) is a small underwater vehicle and is capable of propelling itself beneath the water as well as on the surface of water. e UUV is also known as an autonomous underwater vehicle (AUV). e UUV and AUV are autonomous, which means they perform the assigned mission without any human intercession [1]. Recently, the cooperative control of the multi-UUVs has been becoming a hot topic in various applications, including ocean exploration, submarine rescue, minesweeping, and other fields [2][3][4]. Compared to the individual UUV which has the limited processing power and operational capability [5] to complete a single task, the multi-UUVs as a whole can perform the complex task, thus having more superiority in the harsh ocean environment, such as enhancing reliability and reducing cost. Designing the cooperative control law for the multi-UUVs is a current challenge, which utilizes the UUV location information relative to its neighbours, such that the multi-UUVs agree on the specific quantities of the interest, namely, reaching the consensus control. e development process of the consensus algorithm is reviewed in [6][7][8], and the algorithm has recently been reached considerably in the context of the cooperative control for multiagents systems (MASs) with first-order, second-order, or even high-order dynamics. In [9], the adaptive tracking control method was investigated for a second-order system. For designing the tracking control, a novel distributed estimation scheme was proposed to estimate the relative velocity information which was used to design the tracking control. In order to guarantee consensus tracking being reached, linearly parameterized models were first applied to present unknown disturbances and then designed the decentralized adaptive laws for them. As we know, the proposed to obtain consensus conditions. Some novel leader-following consensus control methods [20][21][22] were proposed to solve the problem of time-delays for the multi-UUVs in our past research work. e communication topology discussed above is time-invarying communication topology, but the topology is often changing, since the communication link may be unreliable underwater for the multi-UUVs.
As an urgent desire has come up for the high-precision control in the MASs, disturbance attenuation has become increasingly important and much attention has been paid to MASs with disturbance. Event-triggered control for consensus problem in MASs with quantized relative state measurements and external disturbance were investigated in [23]. Two type protocols for the quantized control and the event-triggered control were designed to investigate the bounded consensus of general linear MASs with quantized relative state measurements and nonbounded disturbance in the paper. For tackling the problem of disturbance, a high gain method was proposed to recompense for the effects of the external disturbances. In [24], a robust consensus for fractional MASs with external disturbances was investigated. Analyzing the disturbance rejection properties for both linear and nonlinear systems was through the combination of the tools of Mittag-Leffler stability theory, the inequality techniques and Laplace transform. In view of the consensus algorithm for high-order MASs system, the work [25] investigated the a bipartite consensus problem for a high-order MASs with unknown disturbances and cooperative-competitive interactions, in which linearly parameterized approaches were firstly applied to present the time-varying unknown disturbances, and then distributed adaptive laws were designed for the unknown parameters in the disturbances. e consensus performance of the multi-UUVs is also affected by the unmeasurable disturbances, such as uncertainties, modelling errors, and environmental disturbances. erefore, solving the multi-UUVs consensus problem with unmeasurable disturbances has been becoming important. Some control strategies have been investigated to compensate for the disturbances. e robust compensation algorithm was proposed to attenuate the disturbance for the multi-AUVs in [26,27], in which the graph theory and the robust compensation theory were used to reach the multi-AUVs consensus with disturbances. In [28,29], the disturbance observer which estimated the disturbances to compensate the disturbances was proposed for the multi-AUVs, where the disturbance in each AUV is bounded. However, the disturbance observer is based on both the position state information and the velocity state information from the neighbours to estimate the disturbance. erefore, it is inappropriate for the multi-UUVs to transmit a large amount of information timely and accurately under unstable communication conditions in the ocean. e research of nonlinear control theory has been playing an increasingly important role for MASs [9,25,[30][31][32] and has also become necessary in the design of the consensus control for the multi-UUVs, including the back stepping-based [33], adaptive-based [34,35], slide mode-2 Complexity based [36], and neutral networks-based [37][38][39] consensus control.
In this paper, a leader-following multi-UUVs consensus algorithm with unmeasurable disturbances under the fixed and switching topologies is discussed. Firstly, we transform the nonlinear UUV model into a second-order linear dynamic model by taking the method of the feedback linearization. e coordinate transformation presents the basic idea with feedback linearization instead of the conventional control techniques proposed by Shojaei and Arefi [40]. Secondly, the leader-following multi-UUVs consensus algorithm with unmeasurable disturbances under the fixed topology and the switching topology is proposed, and the stability of the multi-UUVs system is then analyzed. e main contribution of this paper is summarized as follows: (1) e leader-following multi-UUVs consensus with disturbances under switching topologies is mainly investigated in this paper. Achieving the multi-UUVs consensus with fixed topology is relatively simple but reaching the leader-following multi-UUVs consensus with changing topology as time is becoming challenging. Meanwhile, the case of the switching topology is more practical underwater due to unreliable and limited communication conditions. Unlike [41], which considered MASs under the fixed topologies in the presence of disturbances, it is not appropriate for multi-UUVs performing tasks in the complex ocean environment. (2) e DESO based on the relative position state for its neighbours is designed to estimate the lumped disturbance consisting of unknown model uncertainties and external disturbances in a multi-UUVs system under switching topology. Compared to the disturbance observer based on both the position information state and the velocity information state to estimate the lumped disturbance in [29], the DESO proposed in the literature estimates lumped disturbance only through the position state information. is can reduce the communication burden between the multi-UUVs in the unreliable underwater communication condition. e remaining part of this paper is organized as follows: Section 2 contains some necessary preliminaries and problem formulation. Section 3 discusses the leader-following consensus algorithm with the time-invarying communication topology. Section 4 extends the result investigated in Section 3 to the case of the time-varying communication topology. Section 5 illustrates the simulations to verify the result proposed in the paper. Section 6 gives a brief conclusion.

Notation.
e following notions are utilized throughout this literature:R n×m and R n represent the set of n × m real matrices and n × 1 real vectors, respectively. I n , 0 n , and 0 n×m denote n × n identity matrix, n × n zero matrix, and n × m zero matrix. 1 n shows n × 1 column vector of all ones. ‖ · ‖ denotes the Euclidean norm. For a symmetric matrix F, F > 0( ≥ 0, < 0 or ≤ 0) denotes that F is a positive definite matrix (positive semidefinite matrix, negative definite matrix, or negative semidefinite matrix). λ i (A), i � 1, 2, . . . , n shows the i-th eigenvalue of a matrix A. λ min (A) and λ max (A) mean the minimum and maximum nonzero eigenvalue of the matrix A, respectively. diag(a 1 , . . . , a n ) and diag(A 1 , . . . , A n ) represent a diagonal matrix and a block diagonal matrix. For the multi-UUVs, i shows the index of the UUV in the paper, i.e., i � 1, . . . , N. ⊗ means the Kronecker product referring to [42]. For any matrix W, X, Y Z, the following arguments associated with the Kronecker product is used in the sequel:

Graph eory.
For the multi-UUVs system in the paper, one leader UUV labeled as the node 0 and following UUVs labeled as node from 1 to N. We use a graph to represent the communication topology among the multi-UUVs. A graph G � (V, ε) is denoted as the communication topology for the following UUVs, where V � 1, . . . , N { } describes a set of nodes meaning N UUVs and ε � (i, j): i, j ∈ V ⊂ V × V shows a set of edges. For the undirected graph, the edge (i, j) means that the i-th UUV and the j-th UUV can receive information from each other. e adjacency matrix e subpart of the graph G describes the subgraph which is connected in the graph G. For a set s, |s| denotes the number of the element in the set s. A graph G includes the graph G, the node UUV 0, and edges between the leader and the followers. Define the matrix Q � diag(q 1 , . . . , q N ) where q i > 0 means that the leader is a neighbour of UUV i and q i � 0 otherwise. Define the structure matrix F � L + Q. For more detailed explanation, refer to the literature [42].

Five Degrees of Freedom UUV Models.
e roll motion which is the rotation about the longitudinal axis has less influence on the UUV performance because of the torpedolike shape of the UUV in [43], so we consider five motion components of the UUV consisting of the surge, sway, heave, pitch, and yaw in this paper (see Table 1). e nonlinear kinematics and dynamic model of an UUV based on the body-fixed and earth-fixed coordinate system in Figure 1 is where η � [x, y, z, θ, ψ] T ∈ R 5 is the state information of the position/Euler angles, and υ � [u, v, w, q, r] T ∈ R 5 is the Complexity 3 state information of velocities for an UUV. J(η), M RB , C RB (υ), and τ are the Jacobian matrix, the inertia matrix, Coriolis and centripetal forces, and a vector of generalized forces, respectively. e detailed explanation about terms in (1) is described in the following parts referring to [35]. e representation of the Jacobian matrix J(η) is where J 1 (η) � cos ψ cos θ − sin ψ cos ψ sin θ sin ψ cos θ cos ψ sin ψ sin θ − sin θ 0 cos θ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and J 2 (η) � diag(1, 1/cos θ).
In the paper, it is assumed that the coordinate system locates in the center of gravity so that [x g , y g , z g ] � [0, 0, 0] and the center of buoyancy is [0, 0, z B ]. Commonly assume that the UUV has homogeneous distribution and xz-plane symmetry such that the products of inertia is I xy � I yx � I xz � I zx � I yz � I zy � 0. Under the assumptions mentioned above, the inertia matrix M RB and Coriolis and centripetal forces C RB (υ) in (1) can be represented as follows: where m is the mass of the UUV.
For τ in (1), it consists of hydrostatics τ B , hydrodynamics τ H , control input τ R , such as rudders, propellers, and bow thrusters, and disturbances τ d , then τ is written as It is common that the neutral UUV would satisfy where W and B mean the weight of the UUV and buoyancy force, respectively. According to the condition of W � B, the center of gravity and buoyancy mentioned above, hydrostatics τ B can be shown as follows: Hydrodynamic forces and moment τ H in (2) mainly includes added mass τ add and viscous damping τ vis : where the coefficient of the added mass τ add is shown as follows: and the viscous damping τ vis relating to skin friction, wave drift damping, vortex shedding, and lift/drag in [43]. e viscous damping τ vis can be given as follows: e control inputs τ R in (2) caused by rudders, propellers, and bow thrusters gives e lumped disturbances τ d in (2) can be shown as follows: According to the terms mentioned above, expanding the dynamic model in (1) yields Complexity where L is the length of the UUV. e mathematical models of a single UUV can be further shown in the compact form according to equations (1)- (16): 6 Complexity comprising the left-hand-side of the kinematics model in (1), and f 2 ′ (ξ) consisting of the remaining terms in the left-hand-side of equations (12)- (16) . en (17) can be rewritten as where is paper mainly investigates the cooperative control method for the UUV based on the earth-fixed coordinate system, so the position (x, y, z), pitch angle θ, and yaw angle ψ are defined as the output h(ξ):

Feedback Linearization. Feedback linearization has been
becoming an important technique to nonlinear control design since it can tackle some practical control problems. e idea of the technique is to transform the nonlinear model into the linear one, and then some linear control design method can be applied.
Considering the following multiple input multiple output (MIMO) nonlinear system where x ∈ R m , u ∈ R p , and y ∈ R q are the state variables, the input, and output of the system, respectively. f(x), g(x), and h(x) are system function with an appropriate dimension.
e nonlinear system (23) can be transformed into the linearization system if the following conditions can be satisfied: (1) e affine nonlinear system requires input and output numbers are the same, p � q (2) e affine nonlinear system exists the relative degree ρ 1 , ρ 2 , . . . , ρ p (3) e sum of relative degree ρ i is the same as the dimension of the nonlinear system, ρ � m Proof. e affine nonlinear system exists the relative degree ρ 1 , ρ 2 , . . . , ρ p , so a new set of state variables can be given and the last state equation is where where According to Definition 1, the matrix Υ(ϕ) is nonsingular, so the feedback input of the system (23) can be shown as follows: Consequently, after the state and the input of the system transformed, we can obtain the following linear equations: □

Feedback Linearization for the UUV Model.
e nonlinear model for an UUV in 5 DOF and the feedback linearization technique are introduced above. In this subsection, the nonlinear UUV model is transformed into a double-integrator dynamic model by the feedback linearization technique.
e system (20) and the system output (22) can be written in the form As mentioned above, the vector f(ξ) is known as follows: From terms 6 to 10 in (33), they are denoted by the symbol f 6 (ξ)-f 10 (ξ), since they are less affected on the feedback linearization method.
e system output function is According to (17), represented as follows: In the first column of g(ξ), In the second column of g(ξ), 8 Complexity In the third column of g(ξ), In the fourth column of g(ξ), In the fifth column of g(ξ), According to the matrix g(ξ), we can conclude that for any i and j, 1 ≤ i ≤ 5, 1 ≤ j ≤ 5, so that en the Lie derivative of h(ξ) with respect to f(ξ) is shown as follows: Complexity From (33)- (41), it follows that Υ(ξ) can be calculated as Υ(ξ) � g 6,1 cos ψ cos θ − g 7,2 sin ψ g 8,3 cos ψ sin θ g 8,4 cos ψ sin θ − g 7,5 sin ψ g 6,1 sin ψ cos θ g 7,2 cos ψ g 8,3 sin ψ sin θ g 8,4 sin ψ sin θ g 7,5 cos ψ − g 6,1 sin θ 0 g 8,3 cos θ g 8,4 cos θ 0 0 0 g 9,3 g 9,4 0 0 g 10,2 cos θ 0 0 g 10,5 cos θ It can see that Υ(ξ) is nonsingular according to g ij (ξ) when u ≠ 0, so the nonlinear model of UUV exists the relative degree: It can be seen that ρ 1 + ρ 2 + ρ 3 + ρ 4 + ρ 5 � 10 � m, so the technique of the state feedback linearization is applicable to the nonlinear model of the UUV according to Lemma 1, then the two vectors are defined as follows: Define the following control input of the UUV which would be applied in the linearization system: + w cos ψ sin θf 5 (ξ) + sin ψ cos θf 6 (ξ) + cos ψf 7 (ξ) + sin ψ sin θf 8 (ξ) + D 2 , 5 ] T are the disturbances. Since the input of the UUV can be shown as u � Φ(ξ) + Υ(ξ)u, we can further obtain: Consequently, the standard second-order integral form of the mathematical model of the UUV can be denoted as follows: where x ∈ R 5 , v ∈ R 5 , and d ∈ R 5 present the position information, the velocity information, and the disturbances of the UUV, respectively. To summarize, the second-order linear model of the UUV (48) can be accomplished by the state feedback linearization method.
e coupled system can be transformed into a decoupled system, which contributes to the design of the cooperative controller quickly and conveniently. However, as the physical meanings of the partial state and the input of the UUV model can vary in the feedback linearization method, the new output of the controller by state feedback linearization method would not directly be applied in the original UUV model. In order to solve the problem, we apply the method of linearization transformation. After using the transformation, the output of the cooperative controller can be applied to the original UUV model.

e Extended State Observer.
e extended state observer (ESO) method to reject the uncertainties in the system was proposed by Han [45]. By using a trackingdifferentiator with the observer form, the ESO for a class of uncertain systems of the form x, . . . , x (n− 1) , t) + ω(t) was given in the paper. It was shown, by choosing a proper nonlinear function and related parameters of the observer, the ESO can track the states of a class of uncertain plants.

Problem Formulation
Consider the multi-UUVs system with unmeasurable disturbances, including N followers and a leader. e i-th UUV dynamics of the followers is shown as follows: where x i (t) ∈ R 5 , v i (t) ∈ R 5 , and d i (t) ∈ R 5 are the position information state, the velocity information state, and unmeasurable disturbances, respectively. e dynamics (49) can be expressed in the following compact form: where B � 0 5 I 5 T , and C � I 5 0 5 . Note that the i-th UUV can only exchange the position information state with its local neighbours in the paper. e dynamics of the leader UUV is where x 0 ∈ R 5 and v 0 ∈ R 5 are the position state and velocity state of the leader. Write the compact form obtained in (21) to the case of the dynamics as follows: where ξ 0 � x T 0 (t) v T 0 (t) T ∈ R 10 , A � 0 5 I 5 0 5 0 5 , and C � I 5 0 5 .
Note that the leader is independent of the following UUVs and sends the information state to the part of the followers.
To design the consensus algorithm and analyze the stability for the system (50) and (52), the following definition, lemma, and assumption are necessary Remark 1. e lumped disturbances which have a constant value in the steady state consist of unknown uncertainties and external disturbances. For more details about the lumped disturbances, please refer to [46]. In the relative work [29], we can find out the method on addressing the lumped disturbances in a multi-AUVs system, in which the disturbance observer estimated the lumped disturbances to compensate for the disturbances for multi-AUVs.

Leader-Following Multi-UUVs Consensus with Unmeasurable Disturbances under the Fixed Topology.
In the subsection, the leader-following multi-UUVs consensus is discussed using the algorithm proposed in the paper, in Complexity which each UUV suffered from unmeasurable disturbances in the fixed communication network. e controller design is heavily inspired by the work [41]; however, the detailed analysis and derivation for the stability in the proof of the eorem 1 are from a different perspective in the subsection, which is beneficial to have preliminary knowledge about the next subsection, namely, the case with switching topologies.
In the subsection, the DESO which uses the relative local neighbour UUVs position information state to estimate the velocity information and disturbances is firstly shown.
en the leader-followers consensus algorithm which uses the estimation information from local neighbors DESO is exhibited. Finally, it is presented that the consensus control described in the subsection can be accomplished by Lyapunov function method Riccati inequality.
According to the control method of the standard ESO [45] adding an extended variable d i (t) ∈ R 5 to the system (49), the extended system for each following UUV can be presented as follows: where s i (t) ∈ R 5 . e compact extended system for (53) can be written as and v i (t) are the position and velocity state estimation for each UUV, respectively, d i (t) means the disturbances estimation and with a ij and q i are shown in the subsection 2.2 Graph theory, the observer gain E ∈ R 15×5 is discussed later, and y i (t) � y i (t) − y i (t). e consensus algorithm in the case of fixed topology is shown as where K ∈ R 5×10 is feedback control gain to be explained later. Note that the algorithm (57) is determined by the state estimation information instead of the information themselves and adopts the active disturbance rejection control scheme which would remove the disturbances effectively.
To analyze the stability of the closed-loop multi-UUVs system in the fixed topology, the following assumption is proposed as follows: e graph G among following UUVs is undirected, and the graph G is connected, in which the leader UUV has a directed path to the any following UUVs. e states and disturbances estimation error are defined as follows: where T . Combining (54), (55), and (58), the estimation error equation can be given as follows: en the tracking error for the leader UUV and the following UUVs is defined as follows: Combining (50), (52), and (57), the tracking error equation is given as follows: Lemma 3 (see [42]). e matrix F is symmetric positive defined under Assumption 2.
Finally, we show the main result of the leader-following consensus algorithm under unmeasurable disturbances with the fixed topology.

Complexity
Proof. As the structure matrix F is symmetric positive definite under Assumption 2 and Lemma 3, there exists orthogonal matrix U ∈ R N×N such that where λ i > 0, i � 1, . . . , N is the eigenvalue of the matrix F. Setting ε(t) � (U T ⊗ I 15 )ε(t), e(t) � (U T ⊗ I 10 )e(t), and S(t) � (U T ⊗ I 5 )S(t), (59) and (61) can be written as follows: (66) It thus follows that the leader-following multi-UUVs consensus can be reached asymptotically under designing the feedback control gain K and observer gain E such that the error systems (65) and (66) can asymptotically converge to zero, as t ⟶ ∞ by taking two steps.

□
Step 1. According to (65), there exists a symmetrical positive-define matrix P 1 and we have (67) It follows from (62) that (69) which means the matrix (I N ⊗ A) + (Λ ⊗ EC) is a Hurwitz matrix based on the Lyapunov stability theorem. Under Assumption 1, lim t⟶∞ S(t) � 0 and (I N ⊗ A) + (Λ ⊗ EC) being a Hurwitz matrix, it is concluded from Lemma 2 that estimation error system (65) satisfy lim t⟶∞ ε(t) � 0, which implies lim t⟶∞ ε(t) � 0. It can be observed the DESO can estimate unmeasurable disturbances.
Step 2. Take the method like step 1 to show the multi-UUVs consensus tracking error can asymptotically converge to zero, as t ⟶ ∞.
According to (66), there exists a symmetrical positivedefine matrix P 2 and we have Defining K � − ςB T P 2 with ς ≥ (1/λ min (Λ)) and substituting K into (70) yields It follows from (63) that is implies that the state of the following UUVs can follow the state of the leader UUV. e above analysis shows that lim t⟶∞ ε(t) � 0 and lim t⟶∞ e(t) � 0 under Assumption 1 and Lemma 2, which means estimation error and the tracking error can converge to zero as t ⟶ ∞ and the multi-UUVs system can achieve consensus under the consensus algorithm (57). e proof is completed.

Leader-Following Multi-UUVs Consensus with Unmeasurable Disturbances under the Switching Topology.
is section extends the result in the previous section to the case multi-UUVs system under switching topologies with disturbances. From the previous section and this section, the difference and comparison between the case of fixed and switching topologies can be shown clearly. For the extended system (54), the DESO in the condition of changing topology based on the relative position information state is designed as follows: ). e consensus algorithm under changing topology is designed as follows: (74)

Complexity 13
Note that the consensus algorithm in the section is different from the case in the last section. In the section, each UUV needs information from its neighbours and the communication topology is changing dynamically because of unreliable communication link underwater. N i (t), a ij (t), and q i (t) represent different communication topologies among UUVs over time.
Similar to the last section, the estimation error equation and the tracking error equation can be written as follows: In the section, the graph shows that the communication topology may be dynamically changing over time. Use G ρ(t) : ρ(t) ⟶ ℓ to denote a graph set of all possible graphs for the leader UUV and following UUVs, where ℓ is the index set for all graphs and ρ: [0, ∞) ⟶ ℓ is changing signal at time t. G ρ(t) : ρ(t) ⟶ ℓ shows a graph set of all possible graphs among the following UUVs.
Definition 3 (see [48]). e union of a collection of graphs is a graph whose node and edge sets are the unions of the node and edge sets of the graphs in the collection. It is said that the union of the collection of the graphs is jointly connected when the union graph is connected.
For the time-varying communication topology, as treated in [49], consider an infinite sequence of a contiguous, nonempty, and bounded time interval [t k , t k+1 ), k � 0, 1, . . ., starting with t 0 � 0 with t k+1 − t k ≤ T, T > 0. Suppose each interval [t k , t k+1 ) can be divided into z k nonoverlapping Example 1. Consider one leader UUV is labeled as 0 and four following UUVs are labeled as 1 to 4. e graph G f consists of two subpart graphs G 1 f and G 2 f seen in Figure 2. e structure matrices F 1 f and F 2 f can be represented as follows: 14 Complexity We define, . . , N, the node corresponding to nonzero eigenvalue of structure maxtix .
According to the definition above, we have N(f) � 1 3 4 in example 1.
For each graph f ∈ ℓ, λ min (F f ) means the minimum eigenvalue of the structure matrix F f described in Notation, and we further define c min � min λ min (F f ), f ∈ ℓ in this section.
We now show the main result of this section.

Theorem 2. Suppose that Assumption 1 is satisfied and the switching graphs G ρ(t) associated with changing communication topology satisfies Assumption 3. For systems (49) and (51) under control law (74), the leader-following multi-UUVs
Proof. It takes two steps to illustrate that error systems (75) and (76) can asymptotically converge to zero, as t ⟶ ∞.

□
Step 1. e first step is to show that the estimation error can converge to zero, as t ⟶ ∞. We firstly investigate the case where the graph k , i � 0, . . . , z k − 1 is not changing, which implies the structure matrix F f is symmetric.
As the structure matrix F f is symmetric positive definite under Assumption 2 and Lemma 3, there exists orthogonal matrix U ∈ R N×N such that where λ i > 0, i � 1, . . . , N is the eigenvalue of the matrix F f . Setting ε(t) � (U T ⊗ I 15 )ε(t), e(t) � (U T ⊗ I 10 )e(t), and S(t) � (U T ⊗ I 5 )S(t), (75) and (76) can be written as follows: and According to (83), there exists a symmetrical positive-define matrix P 1 . Define the Lyapunov candidate function V(t) � ε T (t)(I N ⊗ P 1 )ε(t) which is continuously differentiable except the switching instant, and taking derivative of the Lyapunov candidate function with respect to time along the trajectory of estimation error dynamics, Defining E � − ΓP − 1 1 C T with Γ ≥ (1/c min ) and substituting E into (85), then it follows from (80) that _ V(t) ≤ 0 which means lim t⟶∞ V(t) exists.
Considering the infinite sequence of V(t), t ∈ t k+1 t k , k � 0, 1, . . ., according to Cauchy's convergence criteria, for any β > 0, there exists the positive integer N β , and for any k with k > N β , |V(t k+1 ) − V(t k )| < β, k � 0, 1, . . .. We can rewrite the inequality as follows: which implies where μ i , i � 1, . . . , N is a positive integer. Note that ε(t) is bounded according to _ It can be concluded that the DESO can effectively estimate the unmeasurable disturbances under the switching topology for multi-UUVs.
Step 2. e next step is to present that the position state and velocity state of the following UUVs can track that of the leader UUV with unmeasurable disturbances under switching topology. e method in the subsection is similar to the method in Step 1. We firstly discuss the graph G ρ(t) , ρ(t) � f, t � t i k t i+1 k , i � 0, . . . , z k − 1 is not changing, which implies the structure matrix F f is symmetric. According to (84), there exists the symmetrical positive-define matrix P 2 . Consider the Lyapunov candidate function V(t) � e T (t)(I N ⊗ P 2 )e(t) which is continuously differentiable except the switching instant. Under the previous result lim t⟶∞ ε(t) � 0 in step 1, taking derivative of the Lyapunov candidate function with respect to time along the tracking error dynamics, Defining K � − ΓB T P 2 with Γ ≥ (1/c min ) and substituting K into (91), then it follows from (81) that _ V(t) ≤ 0 which means lim t⟶∞ V(t) exists.
Considering an infinite sequence of V(t), t ∈ t k+1 t k , k � 0, 1, . . ., according to Cauchy's convergence criteria, for any η > 0, there exists the positive integer N η , and for any k with k > N η , For any k > N η , we have which implies where υ i , i � 1, . . . , N is a positive integer.
Note that e(t) is bounded according to _ V(t) ≤ 0; therefore,      It is clear from the discussion in the subsection that the estimation and tracking errors can asymptotically converge to zero as t ⟶ ∞ under the consensus algorithm (74).

Simulation Result
Some numerical simulations are to verify the effectiveness of the theoretical results proposed above in the section. e multi-UUVs system has five vehicles consisting of one UUV leader indexed by 0 and four UUV followers indexed by 1, 2, 3, 4 under switching communication topology (Figure 3) with unmeasurable disturbances (98) using the consensus algorithm (74). We consider the nonlinear UUV model (1) and the double-integrator dynamic model (49) and (51), noting that the leader is independent of the following UUVs and sends the state information to the part of the followers, and the followers only exchange the position information state with their the local neighbours in the paper. e parameters of the nonlinear UUV model refer to [50]. ese five UUVs are randomly located in the threedimensional space of DESO is used to estimate the velocity information state and disturbances timely for each following UUV. e following disturbances are considered to be imposed on each UUV system. For simplicity, we assume that the possible communication topology for these five UUVs are the set G 1 , G 2 , G 3 , G 4 , G 5 , G 6 shown in Figure 3, and switches sequentially circularly from the set, namely,   and the dwell time of each graph is for 1/3 sec.. G 1 ∪ G 2 ∪ G 3 and G 4 ∪ G 5 ∪ G 6 are connected; then we can choose t 0 k � k, t 1 k � k + (1/3), t 2 k � k + (2/3), t 3 k � k + 1 with k � 0, 1, . . .. According to Figure 3, we have the structure matrix F i , i � 1, . . . , 6 associated with G i , i � 1, . . . , 6 and further obtain c min � 0.1981. By solving (80) and (81), the solutions P 1 and P 2 are shown as follows: (100) Figure 4 shows the position and attitude states response curves of the leader-following multi-UUVs in presence of unmeasurable disturbances under the switching topology. It can be observed from the position figures x, y, and z that the following UUVs absolutely track the trajectory of the leader UUV. It is seen that the position and attitude states vary at 15 sec., since the changing lumped disturbances are added to each UUV system at 15 sec. e attitude figure θ indicates that the Euler angles θ of the following UUVs cannot completely converge to that of the leader UUV in presence of disturbances under the switching topologies, since the disturbances imposed on a multi-UUV system include the bounded function sin((1/2)t), which leads to estimation error converge to zero as time goes to infinity. However, the errors between the leader UUV and following UUVs are very small and within the allowable range, and the lumped disturbances can be attenuated according to the conclusion in the previous section, which means the consensus algorithm (74) can compensate for the disturbances in the UUV system. Figure 5 presents the velocity states of the leader-following multi-UUVs system under the switching topology. As shown in the figure, the velocity state of each UUV can follow that of leader UUV about 15 sec and then undergo changes because of imposing varying disturbances from 15 sec on the UUV system; however, the velocity state of each UUV finally converges that of the leader UUV.
From Figures 4 and 5, the leader-following multi-UUV system can reach consensus using the consensus algorithm (74) with unmeasurable disturbances under the switching topology.
e response curves of real disturbance and disturbance estimation for each UUV are shown in Figures 6-9. e disturbance d i , i � 1, 2, 3, 4 acts on the UUV system over time. It can be observed from Figures 6-9 that DESO can estimate the changing lumped disturbances timely and the method proposed in the paper can obtain better disturbance rejection performance.
To illustrate the effectiveness of the algorithm proposed in the paper, the following consensus algorithm without considering the disturbance rejection for the multi-UUVs in [34] is employed for comparison. e feedback controller is shown as To have a fair comparison, the switching topologies have the same structures shown in Figure 3. e gain K is taken the same as the previous derivation approach for K, where K � − ςB T P, ς ≥ (1/λ min (F)) and P is the symmetrical positive-define matrix solution to the following Riccati inequality: ⊗ I 5×5 . Figures 10 and 11 show the position and attitude states and the velocity states response curves under the algorithm (101) when there are disturbances (100) imposed on each UUV system. As shown in Figures 10 and 11, the leaderfollowing consensus cannot be achieved for the multi-UUVs according to Definition 2. Especially, comparing with the 22 Complexity algorithm (74), the algorithm (101) cannot reach consensus control for multi-UUVs with disturbances under the switching topology before 15 sec. Note the proposed method (101) obtains poor control performance with big chattering in Figure 11. It is observed that the methods fail to reject the disturbances for multi-UUVs effectively.  Figure 11: Velocity states of leader-following multi-UUVs with algorithm.

Complexity
As compared with the algorithm (101), the proposed method in the paper has exhibited the superiority of rejecting the disturbances for multi-UUVs system under the switching topology.

Conclusion
e paper tackles the problems of tracking consensus problems with the disturbances under the fixed and switching topologies for the multi-UUVs system. A complex nonlinear and couple model of the UUV is transformed into a second-order integral UUV model by the method of the feedback linearization method. For the unmeasurable disturbance consisting of unknown model uncertainties and external disturbance for each UUV, we design DESO to estimate the disturbances utilizing the UUV position information relative to its neighbour UUVs. Moreover, leaderfollowing multi-UUVs consensus control algorithm that enables all following UUVs to track the leader UUV state information based on the estimation state information from the DESO is proposed for two types of topologies, the fixed and switching topologies. Finally, simulation results are shown to demonstrate the multi-UUVs system in presence of unmeasurable disturbances under switching topologies can accomplish consensus control asymptotically.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.