Formation control of multiple underwater vehicles subject to communication faults and uncertainties

Applied Ocean Research 82 (2019) 109–116

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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Formation control of multiple underwater vehicles subject to communication faults and uncertainties

T

Tingting Yang, Shuanghe Yu , Yan Yan ⁎

College of Marine Electrical Engineering, Dalian Maritime University, Dalian 116026, China

ARTICLE INFO

ABSTRACT

Keywords: AUVs Port-controlled Hamiltonian systems Fault tolerant internal model Lur's system

This paper proposes a novel approach to analyze and design the formation keeping control protocols for multiple underwater vehicles in the presence of communication faults and possible uncertainties. First, we formulate the considered vehicle model as the Port-controlled Hamiltonian form, and introduce the spring-damping system based formation control. Next, the dynamics of multiple underwater vehicles under uncertain relative information is reformulated as a network of Lur’e systems. Moreover, the agents under unknown disturbances generated by an external system are considered, where the internal model is applied to tackle the uncertainties, which still can be regulated as the Lur’e systems. In each case, the formation control is derived from solving LMI problems. Finally, a numerical example is introduced to illustrate the effectiveness of the proposed theoretical approach.

1. Introduction Formation control, a typical behavior in various aspects of system, has received considerable attention due to its wide applications in spacecraft formation flying, deep sea inspections and underwater vehicles. Among many of the typical systems, the autonomous underwater vehicles (AUVs) share information with the neighbors in a distributed fashion to obtain the goal in the complex ocean environment. Considerable efforts have been made on the formation control in previous research, see [1–7]. In various kinds of formation objectives, the agents exchange the information with each other in a distributed way [8–10]. For instance, in [4] some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems are considered. The second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics are considered in [11]. The consensus of leader-following nonlinear multi-agent systems with stochastic sampling are considered in [12]. Most of the aforementioned works are based on the case without fault effects and uncertainties. However, in the complex ocean environment, constraints on the inputs, states, or relative states of agents are inevitable in the multiple AUV systems due to the physical limitations of AUVs or the uncertain communication channel. Considering the relative constraints on the multi-agent systems, [13] proposes a projected consensus control algorithm for solving the optimization problem, and is extended in [14] by considering the

⁎

communication delay. In [15], the communication channel constraints of discarded consensus are analyzed. However, the dynamics is limited to the single integrator system and the result is only local. In other works, [16] and [17] reformulate the constrained consensus problem to a Lur’e system, meanwhile, extend the dynamics to the general linear multi-agent systems. In addition, the global consensus is achieved by Lyapunov theorem and solving LMI problems. Although some effective fault tolerant schemes can be found in the existing literature, only a few papers have addressed the fault problems in multiple AUV systems. Therefore, it is significant to investigate and analyze this problem. In [1], the consensus control for multiple AUVs under imperfect information caused by communication faults is considered, and the uncertain fault is bounded and linear to the relative constraints. The assumption is limited. Moreover, the disturbances may also exist due to the complex ocean environment. On the other hand, the multiple underwater vehicles formation control under relative state constraints caused by the actuator fault has been studied in [1], the sliding mode principle is utilized in the control algorithm. Nevertheless, the chatting phenomenon can exist. Moreover, in the complex environment, the disturbance is a practical issue that needed to be considered. In this paper, we will formulate the autonomous underwater vehicle as the port-controlled Hamiltonian form, which can formulate a mathematical model that high lights the dissipative properties of the system. In [18–20], the model of AUV has been formulated as the Portcontrolled Hamiltonian (PCH) form, which is also so-called energy-

Corresponding author. E-mail address: [email protected] (S. Yu).

https://doi.org/10.1016/j.apor.2018.10.024 Received 24 July 2018; Received in revised form 27 September 2018; Accepted 31 October 2018 0141-1187/ © 2018 Elsevier Ltd. All rights reserved.

Applied Ocean Research 82 (2019) 109–116

T. Yang et al.

shaping models [21–25]. Based on the passivity theory, the formation control or the consensus control algorithm is developed by adding the virtual spring. Following the ideas of achieving the formation, we propose a novel approach to analyze and design the formation control for multiple underwater vehicles under the relative state constraints and uncertainties based on the Port-Hamiltonian system. First, by the given Port-controlled Hamiltonian multiple underwater vehicles system with the case of uncertain relative information, we reformulated the system as a network of Lur’e systems, and this method can be extended to a large scale of physical system. Compared with the previous work [1], the uncertain constraints satisfy the sector-bounded condition and are nonlinear to the relative information. Then, the agents under unknown disturbances generated by an external system are considered, where the internal model is applied to tackle the uncertainties, which still can be regulated as the Lur’e systems. In each case, the formation control is derived from solving LMI problems. The outline of the paper is as follows: in Section 2, we recall the model of autonomous underwater vehicles, Port-Controlled Hamitonian sytems, communication constraints or uncertainties, and the objectives of this paper. In Section 3, we introduce the main results of this paper. The numerical example is introduced in Section 4. Finally, Section 5 concludes the paper. Notations: Throughout this paper, 1n denotes the n × 1 column vector whose elements are all ones. In and 0n denote the identity matrix and null matrices, respectively. The symbol ⊗ denotes the Kronecker product and * is the symmetric entries of matrices. For a scalar function n H (x ) of a vector x , its gradient is defined as H H q H (x ) = [ x , …, x ]T . Furthermore, given a function f (x ) , we define

f (x ) x

1

2.1. Port-controlled Hamiltonian system An input-state-output Port-Controlled Hamiltonian system has the following form:

x= [I (x )

i

+ Di ( i )

i

=

(3)

uT y.

where the function H(x) is assumed to be the Lyapunov function of the system. Next, the AUV is designed to be Port-Hamiltonian form by introducing the coordination transformation. 2.2. Port-Hamiltonian dynamics of AUV Define the state transformation below, i = qi , Mi i = pi .

(4)

Then system (1) is equivalent to the following PCH form as

n

i

[ H (x )]T R (x ) H (x ) + uT y

H (x ) =

pi = qi

Ri (pi ) Ji (qi )

JiT (qi ) 0

pi

Hi (pi , qi )

qi

Hi (pi , qi )

+

Gi i , 0

(5)

with the whole damping matrix

Ri (pi ) = Ci ((Mi) 1pi ) + Di ((Mi ) 1pi )

0,

and the function Hi (pi , qi ) = pi 2(M ) 1 + Vi (qi ) with the potential eni ergy Vi(qi). The input matrix is defined as Gi(xi) = In to be the identity matrix, that is to say that we consider the case of full-actuated system, e.g. m = n. As usual, the formation of multi-agent Port-Hamiltonian system is solved by introducing a virtual spring-damping system. This idea builds upon the theory of Port-Hamiltonian systems on graphs ([23]) and has been applied before to fully actuated systems in [24] and satellites in [25]. The M edges represent the virtual couplings, meaning that the AUVs exchanges information interconnected by the virtual spring and damper. That is to say the main idea behind the method is that the edges of the graph represent the virtual couplings of the nodes corresponding to the agents. 1 2

We consider a system of N AUVs with i = 1, …, N. The nonlinear maneuvering model of each AUV can be described below [Fossen, 2002]:

+ Ci ( i )

(2) m

where x is the state vector, u ∈ R and y ∈ R are control input vector and output vector. The matrix I (x ) is skew-symmetric, R (x ) is symmetric and positive semi-define, G (x ) is the input matrix and H (x ) is the system's Hamiltonian function. All these matrices depend smoothly on the state x. The system is strictly passive from the input u to the output y if

2. Preliminaries

i

+ G (x ) u ,

m

the q × n Jacobian matrix. Denote the Euclidean norm

= Ji ( i ) i,

H (x ) x

H (x ) G T (x ) x ,

y=

n×n ∥x ∥ 2 : = xTx, and the weighted norm x 2 = xT x with . The communication topology is represented by an undirected graph g = {Δ, B, A} of the N agents, where = {v1, v2, …, vN } is a finite nonempty set of nodes. Let us denote B as the incidence matrix associated to graph g = {Δ, E, A}. The columns of B are indexed by the edge set, and the i-th row entry takes the value 1 and −1, respectively, as the initial node and the terminal node of the corresponding edge, and zero otherwise. The n×n weighted adjacency matrix A = [aij] is defined such that aij is positive if (vi, vj ) B , while aij = 0 otherwise.

Mi

R (x )]

gi ( i ) + i, (1)

where ηi = [ni, ei, oi, ϕi, θi, ψi]T is the standard position vector in inertial coordinate system, i = [ui , i , wi, pi , qi , ri]T is the standard velocity vector in body coordinate system ni, ei, oi are respectively the position in north, east and down, ui , i , wi are respectively the velocity in surge, sway and heave. Moreover, the variables ϕi, θi, ψi and pi, qi, ri are the angles and rates in roll,pitch and yaw, respectively. The matrix Ci(νi) represents rigid-body Coriolis-centripetal matrix and Di(νi) is the Ci ( i ) = CiT ( i ) damping matrix, satisfying that and Di ( i ) + DiT ( i ) 0 , respectively. gi(ηi) is the matrix of restoring forces. Ji(ηi) denotes the kinematic transformation matrix from the body-fixed reference frame to the inertial frame. They are assumed to be known constant matrices of compatible dimensions. In this section, the AUV model is described as Port-Controlled Hamiltonian form, and the method of formation control will be benefit from this structure by interconnecting with the virtual spring-damping system, which is a general way to deal with the multi-agent PortHamiltonian systems. Firstly, the structure of Port-Controlled Hamiltonian system is introduced below.

2.3. Formation control using virtual coupling Design z˜k = z k fined as

z˜k= k=

k, Hk z˜k

(z˜k ) + Dkz

zkr . Then the dynamics of a virtual spring is de-

k,

k = 1, …, M .

(6)

n is the relative position error and the Hamiltonian funcwhere z˜k 1 tion Hk = 2 z k zkr 2 k with the nominal spring length zkr and the spring constant gain Δk. Moreover, the term Dkz is the dissipation matrix. The compact form of M virtual spring-damping systems gives

z˜=

,

=

H z˜

with

(z˜) + D z , z˜ =

T T [˜z1T , …, z˜M ] ,

(7)

=[

T 1 , …,

T T M] ,

D z = diag{D1z , …, DNz } . The total Hamiltonian H =

110

=[ M k =1

T T T 1 , …, M ]

Hk =

1 2

M k=1

and z˜T

k z˜

Applied Ocean Research 82 (2019) 109–116

T. Yang et al.

with Δk as the identity matrix for simplification. Then the corresponding interconnection constraint to achieve formation is given by ([23], [24], [25])

=

(In

= (In

bandwidth between AUVs, the exchange information obtained by each AUV could suffer certain uncertainties or constraints. Assume that constraints could be described as follows, for any j ∈ [1, M], yj, k = j, k (˜zj, k ), k = 1, …, n , where yj, k is the kth component of the signal exchanged through the edge j: j, k : is a continuous function that satisfies the following sector-bounded condition:

B) B )T

H p

(8)

(

This method will be utilized to deal with the path following formation control in the next section.

Define function τi,k = fi,k(ϱi,k), ∀k = 1, …, m, where ϱi,k is the aggregated signal that the kth input of agent i received and m is the dimension of control input. fi,k is a continuous function satisfying the following sector-bounded conditon: i, k )

k ,1 i, k )(fik ( i, k )

k ,2 i, k )

0,

with

x j (t )

zir = 0,

i , j = 1, …, N .

t t

lim qi (t )

qj (t )

lim qi (t )

qir (t ) = 0,

lim pi (t )

pir (t ) = lim pie (t ) = 0

+ + +

zir = 0,

t

the

Before solving this problem, the following lemma below is necessary to be introduced. First of all, the matrix L¯ = BT B .

(10)

Lemma 1. The following statement holds if g is connected. L¯ has exactly N − 1 non-zero eigenvalues, which are equal to positive eigenvalues of L while all other eigenvalues of L¯ if existing are zero. Proof. Given (λ, υ) as the eigen-pair of L with λ ≠ 0. Clearly, ¯ T , hence, one knows that (λ, BTυ) is also BT = BT = BT L = LB an eigen-pair of L¯ . Due to the fact that the Laplacian matrix L has one zero eigenvalue and N − 1 non-zero positive ones. Then L¯ has the same N−1 non-zero eigenvalues as L. Moreover, trace(L) = trace(BBT ) = trace(BT B ) = trace(L¯ ) , so if there exist all other eigenvalues of L¯ , they must be zero. With Eq. (13), the relative information z˜ obtained by each AUV can z , hence the actual control input derived from (13) is be described as (˜) given as

i, j = 1, …, N .

+

0

3.2. Controller design

where N and M are the numbers of AUVs and the edges, respectively. In addition, zir is the desired relative constant distance. It means to make the relative position zik between AUV i and j converge to a desired position zir if an edge k exists between nodes i and j. t

(13)

power preserving interconnecting matrix k (Im B) , and the gain k w is positive value, which k (Im B )T 0 dramatically influences the convergence rate and stability. Next, we will establish sufficient conditions for the existence of a dynamic protocol (13) that achieves the formation objective (11) with the constraint conditions (12).

(9)

Definition 1. The AUV multi agent systems with dynamics represented by (1) and the information exchanged among agents represented by g are said to achieve a formation if +

(12)

k w (Im B ) ( ), = k w (Im B ) ye ,

2.5. Objectives

lim x i (t )

0,

k=

where δk,1 and δk,2 are known constants with δk,1 < δk,2. Consequently, in presence of constraints and uncertainties described above, each agent tries to approach the formation defined as follows.

t

k ,2 z˜j, k )

where k,1 and k,2 are known constants with σk,1 < σk,2. Then the path-following formation control law τk considering the uncertainties (12) can be obtained according to Eq. (8) by interconnecting system (7) and (8) as follows,

2.4. Constraints/uncertainties

(fik (

z j, k ) k ,1 z˜j, k )( j, k (˜

z j, k ) j, k (˜

(11)

where pie(t) = pi(t) − pir(t) is the error between the momentum and its desired value. 3. Main results

k

=

k w (Im

B) ( ) =

k w (Im

B ) (˜) z

(14)

Rv ,

corresponds to be utilized to add the damping term with where Rv = k w (Im B) D z ( ).

Rv

The purpose of this section is to design control law to achieve the desired path-following formation under relative information uncertainties. To analysis this problem, first, the error Port-Hamiltonian model about the desired trajectory is introduced, preparing for analyzing the formation control law by interconnecting with a virtual spring-damping system. Then with the existence of relative information constraints or uncertainties, the formation control law is achieved by solving specific limitations of LMI.

pe z˜

=

(R + R v ) IN k w (Im BT )

k w (Im 0

B)

V (x˜) pe V (x˜) z˜

+

B1

z 1 (˜) 0

, (15)

B1 = k w (Im B), z = z˜ where and 1 (˜) 1 1 V (x˜) = 2 peT M 1pe + 2 z˜T z˜ . The partial derivative

3.1. Formation control design under relative information uncertainties

(˜) z . Moreover, V (x˜) T ˜T ]T . = [ p , z e x˜

x˜= Ax˜ + B1 , = [

As we know the relative information z = (Im ⊗ B)(q − qr) + zr and B )(J (q) J (qr ) r ) with qr and pr are rethe derivative z = (Im spectively the desired position and moment, satisfying qr = J (q) M 1pr = J (q) r to maintain the unactuated position states with J(q) = J(qr). Since the existence of qr, the path-following formation control is designed and it is necessary to consider the error dynamics of q qr . Then we design the reference velocity controller based on the generalized canonical coordinate transformations. Due to the practical limitation of the communication range and

T ˜) 1 (z

T

0] .

where the matrix A = the new control input. □

(16)

(R + Rv ) IN k w (Im BT )

k w (Im 0

B)

and τ denotes

Remark 1. This process can be extended to solve the interconnection uncertainties for the generalized Port-Hamiltonian system, which maybe our further study. Given two Port-Hamiltonian systems Ξ1 and Ξ2 below,

111

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T. Yang et al.

x1= [J1 (x )

1:

y1 = g1T

y2 = g2T

1

(17) H2x x2

R2 (x )]

+ g2 u2,

J1 (x )

(18)

R1 (x )

g1

=

1(

g2 y1

J2 (x )

y2 )

H1x x1

0

0

R2 (x )

+

H2x x2

1 2 1 1

1( 1

2

+

1( 1

2)

+

2)

1( 1

+

2)

)

< 0,

IM

(26)

IM

((

1 2

1( 1

+

2)

)

IN

T T

)

, (27)

IM ,

IM +

(

1 4

(

1 2 k (R 4 w

2 1( 1

+

+ Rv + In) 2)

)

2R 1 2

)

1

BT B

IM < 0,

(28)

Let U be the orthogonal matrix that diagonalizes L¯ , then ¯ = U T LU with Γ is a diagonal matrix whose terms are the eigenvalues of L¯ . Multiply UT ⊗ IN and U ⊗ IN respectively to left and right side of equation above, then inequality of (28) is equivalent to N ×N

IM +

1

+

(

1 4

(

1 2 k (R 4 w

2 1( 1

+

+ Rv + In)

)

1 2 2) R2

)

1

IM < 0,

(29)

Denote λi(i = 1, …, N) is the eigenvalue of matrix L¯ . By some simple mathematical manipulations, subsequently, we obtain

(20) (21)

is symmetric,

1

1 2

1

1

+

<0

R2

+ In > 0,

*

1

(19)

1 2

*

(

which is equivalent to

,

.

1 k 2 (R ) 1 4 i w 1

R2

P3=

Lemma 2. Consider the system (5), the controller (14) renders the system (5) to achieve the formation task under relative constraints and uncertainties, if there exist a positive definite matrix Ψ1 and kω that solve the LMI below

+

*

1 T B 2 1

P2=

then the problem is transformed as the dynamic model (16).

1

=

P1= diag{R1, R2},

H2x , x2

With the unit negative feedback interconnection uncertainties, the closed-loop compact form can be described as

x1 = x2

1 B 2 1

R1 0

+ g1 u1,

H1x , x1

x2= [J2 (x )

2:

H1x x1

R1 (x )]

1

+

1 k 2 (R ) 1 4 i w 1

1 2

1( 1

+

2)

<0

IN + IN × n and R2 =− (Ψ1Π1Π2) ⊗ IM + ϵIM×n, where R1 = (R + Ψ1 are symmetric positive definite matrices, the matrices 1 = diag{ k ,1}k = 1, … , n and 2 = diag{ k ,2}k = 1, … , n . λi(i = 1, …, N) are the eigenvalue of matrix L¯ with L¯ = BT B .

achieving the objective of (11). Hence, the proof is completed. □

1 1 Proof. Define the Lyapunov function V (x˜) = 2 peT M 1pe + 2 z˜T z˜ . Taking the derivative of V (x˜) gives

Remark 2. The interconnection structure between system (5) and system (6) is represented as

1 2

Rv )

V (x˜)=

=

V (x˜)T pe

((R + Rv )

V (x˜)T pe

(k w (Im

V (x˜)T pe

((R + Rv )

IN )

V (˜) x pe

+

IN )

V (˜) x pe

+

BT )

k

w

V (˜) x pe

V (x˜)T B1 1 (˜), z pe

Accordingly, with the condition (12) and M

Finally,

V (x˜)T B1 1 (˜) z pe

B )) z˜ + z˜T k w (Im

z 1 (˜)

[

1 (z j, k )

k ,1 z j, k ][ 1 (z j, k )

k,2 z j, k ]

0,

with

k,1

=1

k ,1

and M j=1

V (x˜) + V (x˜)

k,2

(23)

(

1 (z j )

T 1z j )

1 ( 1 (z j )

(24)

pi = qi

where Ψ1 is a symmetric positive definite matrix, the matrices and Denote 1 = diag{ k ,1}k = 1, … , n 2 = diag{ k ,2}k = 1, … , n . x˜1 = [peT , z˜T , 1T (z )]T . Alternatively using the S-procedure theorem, the formation control problem with relative uncertainties can be transformed as solving the inequality x˜1T x˜1 0 with the matrix Ω given as

R1 = k w (Im

k w (Im BT )

*

R2 *

1 B 2 1

B)

(

1 2

1( 1

+ 1

2)

)

IM

(30)

lim V (˜)( x t ) = 0 gives +

0 k (Im

k (Im 0

B )T

t

lim pe (t ) = 0 and +

ye

B)

t

.

lim z˜ (t ) = 0 , +

(31)

t 0

[ yeT (s )

T

(s )]

Due to the complex ocean environments, the system (5) may suffer from external disturbances di, described as

2 zj )

< 0,

R2

3.3. Disturbance interconnection uncertainties

k ,2 .

=1

2)

k (s ) ds = 0 , which means the w (s ) structure is power preserving, moreover, the interconnection is a Dirac structure, due to the fact yeT k + T w = 0 . The next step is to state the controller to overcome the disturbances generated from external environment.

n

j=1 k=1

=

t

+

Then the power

(22)

(˜) z , we obtain

= z˜

1( 1

Ri (pi ) Ji (qi )

JiT (qi ) 0

pi

Hi (pi , qi )

qi

Hi (pi , qi )

Gi i + di , 0

+

(32)

Immediately, system (15) is rewritten as

pe z˜

=

(R + R v ) IN k w (Im BT )

k w (Im 0

B)

V pe V z˜

+

0

, (33)

with ϖ = B1ϕ1(z) + τd + d. Assume the external unknown disturbance d is generalized by an exosystem as follows

IM , (25)

i= Ti H i , di= ¯ i1 H i ,

IN + IN ×n and R2 =− (Ψ1Π1Π2) ⊗ IM + ϵIM×n. with R1 = (R + Rv ) Subsequently, removing the skew-symmetric terms results in solving a novel inequality below

112

(34)

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T. Yang et al.

where di is the periodical disturbance of i-th AUV, and the Hamiltonian 1 H i = 2 | i |2 , TiT + Ti 0 . For being written by the PCH system form, we

R1

can make JTi = 2 (Ti TiT ) and RTi = 2 (Ti + TiT ) . Assume for the simple calculation that all the AUVs are influenced by the same disturbance, meaning that T = Ti for i = 1, …, N, and ¯ 1 = ¯ i1 for i = 1, …, N. Then a controller based on the internal model principle is written, 1

ˆ = (T dˆ= ( ¯ 1

1

IN ) H ˆ + ( ¯ T1

H ˆ ) + ( ¯ T1

IN )( H

dˆ , then the compact

diag R1 , R2, (37)

¯

k ,4 j, k )

0,

1 T 3

(38)

1=

[

T ¯ 2 ( )

0] ,

(39)

(R + R v ) IN k w (Im BT ) T 1

k w (Im 0

IN

B)

0

0

kw (Im

¯ =

( 1T )

kw (Im

B)

BT )

R2

0

IN

0

R3

* *

* *

* *

1 B1 2

( 12

1( 1 +

)

2)

IM

1

1

(

3 1 4

1( 1

+

2)

2

1 1 2

)

IN

IN ,

) IM

1

0

0

IN

2

(

1 (T 2

+ TT)

+

2)

2

3

In

4

2

1 1

In

2

1 k ( 1 4 w

2)( 1

)

+

2)

)

IN

< 0,

IN

2)( 1 1

i

} < 0,

2)

<0 (43)

4

(44)

V T (x˜) ¯ V (˜) x A p pe e V T (x˜) ((R pe

+

V T (x˜) B1 1 (˜) z pe

+ Rv )

+

+

V (x˜) pe

V T (x˜) B2 2 ( ¯ ) pe

+

V T (x˜) B1 1 (z˜) pe

B )) z˜ + z˜T k w (Im

V T (x˜) B2 2 ( ¯ ) pe V T (x˜) ((R pe

=

IN )

+

V T (x˜) pe

+ Rv )

IN )

¯ + ¯T V (x˜) pe

+

BT )

V (x˜) pe

T V (x˜) pe V T (x˜) B1 1 (z ) pe

V T (x˜) B2 2 ( ¯ ), pe

(45)

4)

IN

0

0

2

IN

1 4

2( 3

,

T 3

4)

0 1 2( 3 + 2

0

+

IN × n

V T (x˜) (k w (Im pe

1 B2 2

IN

2( 3

1 T 1 1 p M 1pe + z˜T z˜ + ¯ T ¯ , 2 e 2 2

(40)

IN

1

+

=

Next, we establish the conditions for the existence of the dynamic protocol (39) for overcoming the disturbances and interconnection uncertainties of the formation control of AUVs. R1

4

* *

1( 1

V (x¯)=

,

T

3

1 2

Taking the derivative of V (x˜) gives

IN

1

0

Proof. Define the Lyapunov function

(d˜) with the matrix B2 = Ξ1 ⊗ IN. τ and τ1 are where B2 2 ( ¯ ) = d˜ the control inputs of this Lur’e system. The matrix A¯ is described as A¯ =

(

V (x¯) =

T

2)

where λ′ is the eigenvalues of L. The matrices R1 = (R + Rv) IN IN × n , R2 = ( 1 1 2) IM IM × n , R3 = (T 2 3 4) IN IN × n , the matrices Δi(i = 1, …, 5) are denoted in the proof below.

T

0] ,

)

+

0

3 1 4

1 k ( 1 4 w

¯ ¯ + B1 + B2 1, x¯ = Ax T z 1 (˜)

+ TT)

5

IN ) ( ¯ ) = d˜ (d˜) , k,3 with ( 1 and k,4 are known T constants with σk,3 < σk,4. Define the new variable x¯ = [peT , z˜T , ¯ ]T . Then we obtain a novel Lur’e system as

= [

1( 1

IN )

IM 1 (T 2

* *

{

(d˜)

k ,3 j, k )( j, k ( j, k )

1 2

1 ( 1 2

B)

Theorem 1. Consider the system (33), the controllers (14) and (37) render the system (33) to achieve the formation control with relative constraints and disturbance interconnection feedback uncertainties, if there exist matrices Ψ1, Ψ2, variables ϵ′ and k w such that the LMIs described below are feasible,

(36)

H ˆ ),

˜

(

1 k (I 2 w m

(42)

IN ) u1,

V (x ) pe

¯ j, k ( j, k )

0

(

0

IN

be a continuous function that satisfies the following Let j, k : sector-bounded condition:

(

0

* *

Interconnecting system (5) with system (36), the controller τd to overcome disturbance d gives

u1=

R2

2

ˆ and d˜ = d

where H ˆ = | ˆ |2 . Define ¯ = model is obtained, 1 2

d=

0

(35)

IN )( H

0

IN ) u1,

IN ) H ˆ ,

¯ = (T d˜= ( ¯ 1

¯1 =

0

(41)

<0

1

(

1 ( 3 16

+

4)

2 T 1

(

+

4)

2

1 k ( 1 4 w

1

+ 2 (T + TT )

+

2)

BT

2 3

4

In

)

1

)

1 k ( 1 4 w

IN

(

1 4

1( 1

+

2)

2

+

2) 1 1 2

B In

)

, IN

<0

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T. Yang et al. M

n

¯

[

¯

2 ( j, k )

¯

¯

k ,3 j, k ][ 2 ( j, k )

k ,4 j, k ]

0

U T LU = with Γ′ is a diagonal matrix whose terms are the eigenIN and U′ ⊗ IN respectively to left and right values of L. Multiply U T side of equation above, then inequality of (29) is equivalent to

(47)

j=1 k=1

with k,3 = 1 k ,3 and k,4 = 1 k,4 . Then our purpose is to seek the matrix simultaneously satisfying V (x˜) + V (x˜) < 0 where ϵ′ > 0. Similarly, using the S-procedure leads to M

V (x˜) + V (x˜)

¯

(

¯ T 3 j)

3 ( j)

¯

¯

2 ( 4 ( j)

4 j)

1 2 k 4 w i 1 ( 3 16 2

0

(48)

j =1

(

with a diagonal matrix Ψ2 = diag{ψ2k}, k = 1, …, n. Alternatively, the consensus problem with interconnection uncertainties can be transx¯1T ¯ x¯1 0 formed as satisfying the inequality with T T T T ¯ T T ¯ ¯ x¯1 = [pe , z˜ , , 1 (z ), 2 ( )] . The matrix is obtained as (41). Similarly, we get the matrix without skew-symmetric terms, then we obtain the matrix ¯ 1 (42). Define the variables P¯1, P¯2, P¯3 as follows,

(

P¯1= diag{R1 , R2 ,

1 k (I 2 w m

(

P¯2=

1 (T 2

1 2

1( 1

TT)

+

2

2)

)

IM ,

2( 3

+

IN }

with

(

4)

(49)

1 4

* *

1( 3

+

4)

IN < 0,

0

4

*

(50)

2

where 1=

1 2 k 4 w

2=

(

3=

1 k ( 1 4 w

4=

1 4

(

1

1

2( 3

1 4

(

BBT + +

4

+

1( 1

)2

2

1

1 (T 2

+

1 T 1

2

(R + Rv )

TT)

+

2

3

In In

4

)

)

IN IN

B

2)

+

1 4

)2

1 1

In

2

)

IN

(51)

Again, define new variables as 1

P¯1=

(

T 3

1 ( 3 4

P¯2=

(

P¯3=

1 4

3 1 4

1( 1

+

2( 3

+

2

)2

T 1

4)

+

4

1 1

IN 0

)2

1 (T 2

+

In

2

)

IN

T

+ TT)

2

3

In

4

)

4

In)

1

+

2)

2

2( 3

IN

+

1

1 16 i

1 1

2

4

)2

k w2 ( In

)

In 1 (T 2

+

1 1

+

2)

+ TT) 2

<0

(55)

1 k ( 1 4 w

1 4

2( 3

+ 5

+

2)( 1

1

= 4 k w2 4)

2

+

+

2)

i

1

1

1 (T 2

i

1 4 1 TT)

+

+

2)( 1 1

2

1 T 1

2

2)

<0 (56)

4

(R + R v ) 3

4

In

)

In 1

T 2

Similarly, using the Schur theorem, we have P¯1 P¯2 P¯ 3 P¯ < 0 and P¯1 < 0 , which is equivalent to (46). Considering the second term of Eq. (46) gives,

(

(

1

1 ( 3 16

2

3

+

4)

2 T 1

(

1

1)

In)

4

1 4

2( 3

+

4)

2

1

+ 2 (T + T T )

1 2 k ( 1 + 2)2 16 w (BBT ) 4 1 <

IN )

(53)

0

Substituting Δ1 and Δ4 into (29) gives 1 2 k 4 w

(

1

(

BBT +

1 ( 3 16 2

1 4

1

3

1( 1

Let U

2 T 4) 1

+

In)

4

+

( (

2

)2

N ×N

1

1 4

1

1 4

2

2( 3

1) 1 1

1 T 1

IN 2

+

(R + Rv ) 4)

2

+

1 (T 2

In

IN

+ TT)

1 2 k ( 1 + 2) 2 16 w 1 In (BBT ) <

)

)

0

+

4)

2 T 1

.

(52) 1

1 ( 3 16

In this section, we present the results obtained from the previous sections with simulations to demonstrate the effectiveness of the control protocols designed in this paper. The autonomous underwater vehicles are modeled as [1]. For simulation purpose, we choose the incidence matrix B = [−1, 0, 0;1, − 1, 0;0, 1, − 1 ;0, 0, 1]. and the initial generalized momentum p = 0 and generalized positions q = [1, 2, 3, 5, 6, 8, 4, 2, 3, 4, 5, 6]T. The distances of x axis and y axis between the agents is designed as zd = [5, 0, − 5, 0, 5, 0, 0, 0, 0]T, vr = [0.1, 0.1, 0.1, 0.1, 0.2, 0.2, 0.2, 0.2, 0, 0, 0, 0]T , respectively. In the first case of overcoming relative information uncertainties, we randomly generate those relative constraints in the interval [0.1, 0.8]. The matrix Ψ1 is obtained as diag{338.1102, 338.1187, 338.1036} and the control gain k w = 11.9577 . The relative distances of formation control between the agents are plotted in Figs. 1 and 2. It shows that all the agents eventually achieve the desired formation. It's necessary to explain here that the agents finally achieve the square form. The state ν is described in details in Fig. 3, respectively. It shows velocity of forward speed u, υ and r, which asymptotically converge to the desired velocity. At 50 second, the AUVs are affected by external disturbances. the AUVs are influenced by the periodical disturbances. The matrices in

P¯1 < 0 3

(

1 4

(R + R v )

4. Simulation results

Using Schur theorem leads to,

1

1( 1

1 k ( 1 4 w

IN }

2

+

2 T 4) 1

1

5

0 1 2

1

)

1 T 1

+

for i = 2, …, N, since that U TLU = diag{0, 2 , …, N } . The case λ1 = 0 is included in the condition of Δ5 < 0 given below. Due to the fact that the graph is undirected, then the eigenvalues are all real. Then it can be written as

IN )

IM

0 P¯3= diag{

In

4

1 ( 1 2

B)

+

3

1 4

3

1

1 4

2

1

Fig. 1. Profiles of states zi, i = 1, 3, 5.

(54)

be the orthogonal matrix that diagonalizes L = BBT, then 114

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T. Yang et al.

Fig. 5. Profiles of states z(t).

Fig. 2. Profiles of states zi, i = 2, 4, 6, 7, 8, 9.

Fig. 6. Profiles of states u(t). Fig. 3. Profiles of states ν(t).

Fig. 7. Profiles of states v (t).

Fig. 4. Profiles of states d, dˆ and d˜ .

0.6 0.6 0 0 2 and ¯ = 10* 0.6 0.6 0 0 . 0.1 0.6 0.6 0 0 In this simulation, we randomly generate those disturbances interconnection uncertainties in the interval [0.3, 0.8]. It shows that the internal model controller is effective to estimate the periodic disturbances, although the existance of fault error disturbance d˜ , see Fig. 4, and are rejected perfectly, which illustrates the effectiveness of the designed control laws. Moreover, the desired relative disturbances z and the velocity can be achieved in Figs. 5–8. The matrix Ψ1 is

(35) are given as T = I2

0.1 2

517.3137 0.0376 0 0 0.0376 517.3173 0.0385 0 0 0.0385 517.3161 0.0369 0 0.0000 0.0369 517.3098

Fig. 8. Profiles of states r(t).

The matrix Ψ2 is diag(521.8388, 521.8388, 521.8385, 521.8385), k w is 17.6483. It proves that the formation keeping control for multiple underwater vehicles in the presence of communication faults and possible uncertainties is achieved.

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5. Conclusions

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We have addressed a novel approach to analyze and design the formation keeping control protocols for multiple underwater vehicles. The multi-agent underwater vehicles under uncertain relative information and unknown disturbances can be regulated as the Lur’e systems. In each case, the formation robust control are derived from solving LMI problem. In the future, fault detection for multiple underactuated AUV systems will be studied. Moreover, the effect of parameter uncertainties are interesting topics for further investigation. Acknowledgements This work was supported by the National Natural Science Foundation of China (61503080), Natural Science Foundation of Liaoning Province (20180520036) and Fundamental Research Funds for the Central Universities of China (3132017131). References [1] S. Chen, D.W. Ho, Consensus control for multiple AUVs under imperfect information caused by communication faults, Inf. Sci. 370 (2016) 565–577. [2] Y. Yan, S. Yu, Sliding mode tracking control of autonomous underwater vehicles with the effect of quantization, Ocean Eng. 151 (2018) 322–328. [3] E. Vos, J.M. Scherpen, A.J. van der Schaft, A. Postma, Formation control of wheeled robots in the port-Hamiltonian framework, IFAC Proc. 47 (3) (2014) 6662–6667. [4] Y. Wang, S. Yu, An improved dynamic quantization scheme for uncertain linear networked control systems, Automatica 92 (6) (2018) 244–248. [5] J. Wei, A.J. van der Schaft, Load balancing of dynamical distribution networks with flow constraints and unknown in/outflows, Syst. Control Lett. 62 (2013) 1001–1008. [6] S. Knorn, A. Donaire, J.C. Agero, et al., Passivity-based control for multi-vehicle systems subject to string constraints, Automatica 50 (12) (2014) 3224–3230. [7] M. Andreasson, D.V. Dimarogonas, H. Sandberg, et al., Distributed control of networked dynamical systems: static feedback, integral action and consensus, IEEE Trans. Autom. Control 59 (7) (2014) 1750–1764. [8] Q. Shen, P. Shi, Y. Shi, Distributed adaptive fuzzy control for nonlinear multiagent systems via sliding mode observers, IEEE Trans. Cybern. 46 (12) (2016) 3086–3097.

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