1. IntroductionCoordination of multiple autonomous underwater vehicles (multi-AUVs) has attracted much attention as it has broad applications in both civilian and military underwater tasks.[1–3] When single AUV is used in complex missions, it has to be self-sufficient and thus requires sufficient power, sensors, and other devices. However, a single AUV is sometimes unable to complete the required tasks safely and stably by itself in severe ocean environments.[4] In contrast, multiple AUVs can be adapted to offsetting the weaknesses of a single AUV. The multi-AUVs can take more sensors and provide higher efficiency and applicability than single AUV. This means that the multi-AUVs can receive and process the information obtained from the sensors, and deal with the communication problems caused by the underwater sensors. In a practical AUV system, the data package sent by sonar consists of state information and flag bits, such as starting flag bit, checking flag bit and ending flag bit. If the receiver neither obtains nor mismatches flag bits, the data packet is ineffective. So the large data packet means more time delays and more data-packet dropouts. The method of dividing the whole data packet into several smaller ones can reduce the influence.
In the AUV research, stability is a common target in both single AUV and multi-AUVs. In the literature, numerous methods have been proposed to solve this problem, such as nonlinear control theory,[5] sliding mode control,[6,7] fuzzy control,[8] and backstepping control.[9,10] However, methods used in single-AUV scenarios cannot completely resolve control problem of multi-AUVs, as each member has to complete the mission while coordinate with the others based on common information. Further, the communication topology used for sharing such information among the members usually contains time delays or data packet dropouts. In addition, underwater communication, positioning, and navigation all depend heavily on acoustics, which limits the communication bandwidth, communication distance, and communication volume. The dynamic behaviours of AUVs are nonlinear and coupled,[11] and the stable control of multi-AUVs is even more difficult when practical and complex dynamics and communication constraints are involved. Qi,[12] Cui et al.,[13] and Yang and Gu[14] all proposed the methods in which various nonlinear approaches used in single AUV with good performance are extended to the control of underactuated AUVs. Their proposed methods navigate all the members in a formation to track a reference path or maintain the formation by communicating with a leader or a common target. In their methods, only the position of the leader is required to ensure that his motion follows the AUVs. However, in all the proposed methods, the motion information of the following AUVs is ignored. Consequently, the proposed methods lack coordination and autonomous ability.
Various methods used in multi-agents systems have recently been adopted and applied to coordinate control or group stability.[15,16] With the aid of these methods, autonomous ability and cooperation have been strengthened. Li and Wang[17] applied the consensus method to multi-AUVs with using a finite-time position consensus algorithm. Their algorithm enables the AUVs to avoid obstacles in the path and change configuration to avoid collisions while maintaining connectivity. However, communication constraints exist when the AUVs share information about underwater. The stability of multi-agent systems with delays can be analyzed and discussed by Lyapunov–Razumikhin and Lyapunov–Krasovskii theories.[18–20] Shang[21] and Xie and Cheng[22] proposed new methods in which the Lyapunov theory is used to solve the consensus problem under Markovian switching topology with time delays. Switching topologies are significant as they could simulate the communication constraints in ocean environments better.
Motivated by the considerations above, we focus on consensus of multi-AUVs under limited communication by using double independent Markovian switching topologies. The contributions of this paper consist of three parts. Firstly, the nonlinear mathematical model of AUV is transformed into a linear dynamic model via state feedback linearization with minimal inaccuracy. This process is accomplished by the coordinate transformation instead of the method in which the nonlinear parameter presented in Ref. [23] is ignored. Secondly, just as done in Ref. [24], the communication topology is divided into two parts, i.e., position and velocity, which are respectively used to deliver position and velocity information. This method reduces the capacity of per data packet. Subsequently, the same time-varying delays in both topologies are used to verify the feasibility of two independent position and velocity topologies. Thirdly, as the same delays in double topologies are difficult to realize in real ocean environment, two different time-varying delays are analyzed and discussed in this paper. This means that the multi-AUVs operate in the case of different time-varying delays with double independent Markovian switching topologies. Furthermore, in both delay cases, the sufficient conditions for uniformity are ultimately obtained via Lyapunov–Krasovskii theory.
The remainder of this paper is organized as follows. In Section 2, we introduce preliminary graph theory notions and transform the single AUV nonlinear mathematical model into a double-integrator dynamic model via state feedback linearization. In Section 3, we present two main results respectively for the cases of same and different time delays in double independent Markovian switching topologies based on the simplified mathematical model. In Section 4 we give the simulation results that validate the theoretical results. Finally, in Section 5, we draw some conclusions from the study in this paper.
2. Problem formulation2.1. Graph theoryLet
represent a directed graph, where
is a set of nodes (or AUVs), and
is a set of edges. The adjacency matrix
is defined by
if
, which indicates that node (AUV) j can receive information from node (AUV) i, and
otherwise. The neighbor set of node (AUV) i is denoted as
. Define the in-degree matrix as a diagonal matrix
, where
and
are the in-degree and out-degree of the node (or AUV)
and
. The G is said to be balanced if
for all
.[25] Define the Laplacian matrix as
, which has all row sums equal to zero.
A directed tree is a graph in which each node has exactly one parent, except for one node, called the root, which has no parent and a directed path to each node. A directed spanning tree of a graph is a directed tree formed by graph edges that connect all the nodes of the graph. A graph has a directed spanning tree if there exists at least one node with a directed path to all the other nodes.[26] According to the lemma in Ref. [27], L has exactly one zero eigenvalue if and only if the directed graph associated with G has a spanning tree. It is obvious that
is the smallest non-zero eigenvalue.
In this paper, the communication topology is divided into two parts: position topology and velocity topology. Two topologies are respectively used to send position and velocity information. In what follows,
and
represent the position and velocity topologies, respectively. Similarly,
, respectively represent the Laplacian matrices of
and
and the neighbor sets of the i-th AUV. For a given positive integer N, the finite topology graphs can be represented as
and
associated with
and
signifies
or
. Let
denote the union of position graphs and
represent the union of velocity graphs. Then,
where
and
denote the unions of nodes and edges. Similarly,
. The
and
denote the unions of Laplacian matrices with
and
, respectively. The
and
represent the Laplacian matrices of position and velocity topologies at time t, and
and
. Then, the union disoriented Laplacian matrices are designed as
and
.
2.2. The AUV model and state feedback linearizationThe dynamic model of AUV can typically be described by a 6-DOF model according to the body-fixed and earth-fixed coordinate systems as indicated in the following equations. As rolling has little influence on translational motion, the roll speed is ignored in this paper.[28] Thus, the nonlinear and coupled model can be presented as follows:
where
describes the states of the position/Euler angles, and
describes the states of velocities for the
i-th AUV.
Mi,
, and
denote inertia, Coriolis, and damping, respectively, while
is the magnituge of a vector of generalized gravitational and buoyancy forces.
Ti is the control inputs.
is the Jacobian matrix from body-fixed frame to earth-fixed frame.
In this paper, an AUV is assumed to be symmetrical in plane and height and have a torpedo-like shape that can provide good hydrodynamic performance. This shape also means that some parameters can be ignored or simplified. It also means that
is equal to 0. Noting that both Mi and
contain the body-rigid part and hydrodynamic part for added mass, and are expressed as
where
is the rigid-body mass matrix,
is the rigid-body Coriolis and centripetal matrix.
is the inertia matrix of added mass terms, and
is the hydrodynamic Coriolis and Centripetal matrix of added mass terms. The hydrodynamic parameters
,
,
,
,
are presented in Refs. [
28] and [
29].
Consider that
where
represents the forces and rudder angles,
can be described as
is the hydrodynamic coefficients caused by rudders.
According to Eqs. (1) and (2), we can derive the following equation,
Then we take mathematical model of single AUV for example. The standard nonlinearization function can be described as
where
,
,
, and
.
Furthermore, for the first five elements of
, the Lie derivative of
can be defined as
and the second Lie derivative of
is
Then, we can design two new vectors
and
as follows:
where
,
,
.
The new control input, which is used in the linearization system, can be defined as
where
is the input of the nonlinear system in Eq. (
2).
Thus, the mathematical model of the i-th AUV can be modified into a standard double integrator dynamic model by state feedback linearization as follows:
where
,
,
.
2.3. Basic lemmas and notationsIn this section, various lemmas that play important roles in the subsequent analysis are introduced.
3. Main resultIn this work, the multi-AUVs operate in the form of a stochastically switching topology. The switching process is assumed to be driven by a finite ergodic Markov chain with a topology set
and transition probability matrix
. Note that
is a transition rate matrix whose row summation is zero and all off-diagonal elements are nonnegative.[33]
Given a probability space
, where U is the sample space, F is the algebra of events, and P is the probability measure defined on F, then
where
,
γij is the transition rate from
i to
j if
, otherwise,
,
denotes an infinitesimal of a higher order than
, which means
.
Next, the stability conditions for multi-AUVs based on the simplified model defined by Eq. (9) under protocol (14) are discussed:
where
and
are the common consensus gain matrices for position and velocity topologies, respectively, and
and
are the
entries of the adjacency matrices
and
.
Thus,
is defined. Combination of protocols (14) and (9) can be changed into
where
, and
,
.
For
, the error vector is defined as
, where
is the average state vector. Thus, equation (15) can be rewritten as
where
and
are defined to filter position and velocity states in
.
Let
, then it will follow that
Note that
is stabilizable if there is a feedback gain matrix KAB such that
is of Hurwitz.[34] Based on the matrix Riccati equation,[35] there exists
that satisfies
Thus, the gain matrices
and
are designed as
where
,
, and
and
are positive constants.
3.1. Consensus of multi-AUVs with the same delays in double Markovian switching topologiesIn this section, the time delays in switching topologies are assumed to be the same. Thus, the control protocol
is described as
In what follows, V, Vi, and ξ are, respectively, short for
,
, and
.
Let
represent the expectation of the Lyapunov function,
and for the
i-th topology at time
t,
.
Note that
represents the Lyapunov function for the i-th topology and
represents the Lyapunov function for the i-th part in Eq. (28). Thus, the Lyapunov function can be changed into
and the derivatives of the Lyapunov function can be rewrite as
The derivatives of these parts are
According to Lemma 2, we can obtain that
Then the integrations of
and
can be transformed based on the characteristics of derivatives. Moreover, equation (31) can be changed into the following equation:
where
and
are the abbreviations of
and
respectively.
By Lemma 3, equation (33) can be rewritten as
Note that
and
,
,
. For the union of switching topologies,
can be summarized as follows:
where
,
, and
.
Then
can be rewrite as
where
, and
ϕ and
are defined in Eqs. (
24) and (
25).
With
,
and
where
,
,
, and
.
According to Lemma 1, and when condition (27) in Theorem 1 holds, it can be derived that
Thus, it is easy to obtain
for
. Moreover, there exists a constant
, satisfying
This means that system (9) is asymptotically stable in the case of the same time-varying delays.
3.2. Consensus of multi-AUVs with different delays in double Markovian switching topologiesIn this subsection, the time delays in double switching topologies are assumed to be different. The following assumption, which is similar to Assumption 2, is needed.
Just as indicated in Subsection 3.1, ξ,
, and
are abbreviations of
,
, and
respectively; the definitions of
, Vi, Vi,
, and
are also the same as those in the above section.
The derivatives of three parts are
The elements of
can be transformed in accordance with Lemma 2,
For convenient calculation,
is determined to represent
in the following processing stage. Then,
and
By Lemma 3,
can be rewritten as
Substituting
and
into Eq. (52) and noting that
and
,
where
,
,
, and
.
Substituting
, which represent ξ,
,
,
, and
respectively, into Eq. (55),
where
,
are determined from Eqs. (
44) and (
45).
With the new vector
, it is easy to obtain
and
where
,
,
,
,
,
,
, and
.
According to Lemma 1 and condition (47) in Theorem 2, it is easy to obtain
Thus, we can obtain
for
. Moreover, there exists a constant
satisfying
This means that system (9) is asymptotically stable with two independent topologies in the case of two different time-varying delays.
