Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design

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Research Article

Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design Zhouhua Peng a,b,n, Dan Wang a, Wei Wang a, Lu Liu a a b

School of Marine Engineering, Dalian Maritime University, Dalian 116026, PR China School of Control Science and Engineering, Dalian University of Technology, Dalian 610031, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 5 May 2015 Received in revised form 14 September 2015 Accepted 28 September 2015 This paper was recommended for publication by Dr. Jeff Pieper

This paper investigates the containment control problem of networked autonomous underwater vehicles in the presence of model uncertainty and unknown ocean disturbances. A predictor-based neural dynamic surface control design method is presented to develop the distributed adaptive containment controllers, under which the trajectories of follower vehicles nearly converge to the dynamic convex hull spanned by multiple reference trajectories over a directed network. Prediction errors, rather than tracking errors, are used to update the neural adaptation laws, which are independent of the tracking error dynamics, resulting in two time-scales to govern the entire system. The stability property of the closed-loop network is established via Lyapunov analysis, and transient property is quantified in terms of L2 norms of the derivatives of neural weights, which are shown to be smaller than the classical neural dynamic surface control approach. Comparative studies are given to show the substantial improvements of the proposed new method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Autonomous underwater vehicles Containment Dynamic surface control (DSC) Predictor Neural networks

1. Introduction In recent years, coordinated control of autonomous surface vehicles (ASVs) and autonomous underwater vehicles (AUVs) has drawn great attention from control communities, due to its wide applications in marine industry, such as cooperative search and rescue, collaborative explorations, and sensor networks [1–3]. Coordinated control enables multiple vehicles working together to achieve a collective objective, offering enhanced reliability, performance, and effectiveness over a singe one [4–8]. Coordinated control of multiple AUVs was reported in [9–14]. In [9], a nonlinear path following controller was derived for two underwater vehicles along identical parallel paths, while guaranteeing that the lateral distance between them remains constant. In [10], a coordinated path following scheme was developed for networked AUVs in the presence of communication losses and time delays. The coordination between AUVs was achieved by exchanging the path variables assigned for them. In [11], a synchronized path following scheme for homogeneous AUVs was presented, and a decentralized speed adaptation mechanism was introduced to assure that the reference velocity for each vehicle converges to a constant value. In [12], a leader–follower formation control scheme was presented for multiple AUVs. By constructing a virtual vehicle n

Corresponding author. E-mail address: [email protected] (Z. Peng).

that converges to a reference trajectory of the follower, robust adaptive formation control laws were designed based on backstepping and function approximation techniques. In [13], decentralized control laws were developed for multiple fully actuated AUVs both for state- and output-feedback cases. In [14], distributed coordinated tracking of multiple AUVs with unknown dynamics was discussed, and the reference trajectory does not require to be known to all vehicles. In all aforementioned studies, a key feature is that only one leader exists in the motion control setup. In the presence of multiple leaders, the objective is to drive the followers to converge to the convex hull spanned by the leaders, which is called as the containment problem. The vehicle dynamics considered in the previous works correspond to first-order systems, second-order systems, high-order systems, and Lagrange systems [15–22]. A typical application that matches the containment in marine industry is the automatic seafloor exploration and monitoring. In this formation control scenario, a group of AUVs are guided by another group of ASVs, which are equipped with sensors to detect the hazardous obstacles and play as the communication relays between the AUVs and mother vessel. The ASVs can shape a safe area for the AUVs to follow, ensuring that the follower AUVs are contained within the moving safety area formed by the leader ASVs. Meanwhile, the data collected by the AUVs can be transmitted to the ASVs on the sea surface, which further send them to the mother vessel. Obviously, such coordinated control scheme has not been fully explored for networked marine vehicles.

http://dx.doi.org/10.1016/j.isatra.2015.09.018 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Peng Z, et al. Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.018i

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Adaptation and robustness is critical for high-performance control of AUVs. However, this is inhibited by the fact the AUV dynamics are intrinsic nonlinear and uncertain. It contains model parameters such as hydrodynamics damping, and external disturbances such as ocean currents, which are very hard to model accurately. A number of results have been reported that rely on their exact model knowledge, which is obtained from empirical studies. Obviously, the stability of the resulting controllers are difficult to guarantee. In order to preserve the stability and robustness in the presence of model uncertainty and ocean disturbances, adaptive control schemes have been widely suggested [24–28]. In [24], a direct adaptive control law was developed for way-point tracking of underactuated AUVs in the presence of ocean currents. In [25], neural networks (NNs) combined with a variable structure method were employed to derive a robust adaptive tracking control law. In [26], a dynamic recurrent fuzzy NN was used to estimate the dynamic uncertainties for AUV, and the improved tracking performance can be achieved due to their memory features. In [27], an adaptive fuzzy sliding mode control law was developed for kinematic control of AUV. In [28], a nonlinear control law was derived for a fully actuated AUV based on a robust integral of the sign of the error (RISE) structure with a feedforward NN term. In these studies, although the stability results are available, the design and applications of adaptive/neural controllers are overly challenging due to the following facts. First, oscillations will exacerbate with the increment of adaptation gains, which may exceed the available bandwidths of actuators. As such, it leads to constraints on the speed of adaptation [29,31]. Second, any nonzero tracking errors during the transient phase can result in control signals of large magnitude, which are unacceptable by actuators. This situation gets worse due to the nonzero initial conditions in the formation control setup. These challenges may limit their applications. This paper presents a new design method, named predictor-based neural dynamic surface control (PNDSC) design approach, to develop the containment controllers for multiple AUVs in the presence of model uncertainty and unknown ocean disturbances. In order to achieve the desired performance, adaptation is indeed necessary because AUVs contain plenty of model uncertainty and ocean disturbances [24–28]. For this problem, although the classical neural dynamic surface control (NDSC) approach can be applied [23,32,33]; it may not be efficient due to the fact the speed of adaptation is limited. To overcome the limitations of the NDSC approach, by incorporating a predictor design into the classical NDSC approach, a new PNDSC architecture is proposed that allows for using the prediction errors, instead of tracking errors, to update the neural weights. The key is that the prediction error dynamics can be made to converge faster than the tracking error dynamics by choosing a particular parameter, and results in two timescales to govern the closed-loop system dynamics. The stability property of the proposed scheme is established based on Lyapunov theory, and the transient property is quantified in terms of L2 norms of the derivatives of neural weights. Comparative studies are given to illustrate the substantial improvement of the proposed scheme. The contribution of this paper is three-fold:


In contrast to the NDSC approach [32,33], a new type of PNDSC




architecture for multi-input multi-output (MIMO) system is proposed. The prediction errors, rather than tracking errors, are used to update the neural adaptive laws, which are independent of the tracking error dynamics. It is rigorously proved that the transient property of the proposed PNDSC architecture performs better than the NDSC approach, with smaller L2 norms of the derivatives of neural weights. In contrast to the contributions in [9–14] and [24–28] where the motion controllers are developed for AUVs in the presence of a single leader, this paper addresses the containment control problem of multiple AUVs in the presence of multiple reference trajectories over a directed graph. This is done by introducing a




distributed diffeomorphic coordinate transformation, which has not been reported for AUV control. Distributed adaptive containment controllers for AUVs are developed based on the new PNDSC approach, with the guaranteed stability and transient performance. Besides, the model uncertainty and external disturbances can be reconstructed using the sampled input and output data.

The rest of this paper is organized as follows. Section 2 introduces some preliminaries and gives problem formulation. Section 3 presents the PNDSC method for an MIMO system. Section 4 gives the containment controller design, together with stability and transient analysis. Section 5 provides an example for illustrations. Section 6 concludes this paper. Notations: Rn denotes the n-dimensional Euclidean Space. J  J denotes the Euclidean norm. λðÞ, λmin ðÞ, and λmax ðÞ denote the eigenvalue, the smallest eigenvalue, and the largest eigenvalue of a square matrix ðÞ, respectively. σ ðÞ denotes the smallest singular value of a given matrix. diagfa1 ; …; aN g is a block-diagonal matrix with matrixes ai ; i ¼ 1; …; N, on its diagonal. Given p Z1 and v A Rn , the Lp R1 norm and truncated Lp norm is defined by J v J Lp ¼ ð 0 J vðsÞ J p dsÞ1=p R t and J v J Lp ;t  ¼ ð 0 J vðsÞ J p dsÞ1=p with t  4 0, respectively.

2. Preliminaries and problem formulation 2.1. Preliminaries A graph G ¼ fV; Eg consists of a node set V ¼ fn1 ; …; nN g and an edge set E ¼ fðni ; nj Þ A V  Vg with element ðni ; nj Þ that describes the communication from node i to node j. A path from node ni1 to node nil is a sequence of ordered edges of the form nik ; nik þ 1 , k ¼ 1; …; l  1. A directed path in the graph is an ordered sequence of nodes such that any two consecutive nodes in the sequence are an edge of the graph. A digraph has a spanning tree if there is a node called as the root, such that there is a directed path from the root to every other node in the graph. The adjacency matrix A ¼ ½aij  A RNN associated with the directed graph G is defined as aij ¼ 1, if ðnj ; ni Þ A E; and aij ¼ 0, otherwise. Self-connections are not allowed, i.e., aii ¼ 0. The Laplacian matrix L associated with the graph G is defined as L ¼ D  A where D ¼ diagfd1 ; …; dN g with PN di ¼ j ¼ 1 aij ; i ¼ 1; …; N. Definition 1 (Brualdi and Ryser [34]). The set E D Rn is convex if

λx1 þ ð1  λÞx2 A E;

ð1Þ

whenever x1 A E; x2 A E, and 0 r λ r 1. The convex hull Co(X) for a set of points X ¼ fx1 ; …; xn g is the minimal convex set containing all points in X and is defined as ( ) n n X X λi xi j xi A X; λi 4 0; λi ¼ 1 : ð2Þ CoðXÞ ¼ i¼1

i¼1

Lemma 1 (Cui et al. [12]). For bounded initial conditions, if there exists C1 continuous and positive definite Lyapunov function VðξÞ satisfying κ 1 ð J ξ J Þ r VðξÞ r κ 2 ð J ξ J Þ, such that V_ ðξÞ r μVðξÞ þ ϵ, where κ 1 ; κ 2 : Rn -R are class K functions and ϵ is a positive constant, then the solution ξ ¼ 0 is uniformly ultimately bounded. 2.2. Problem formulation Consider a network of multi-vehicle system consisting of M followers, labeled as AUV 1 to M. In the horizontal plane, the

Please cite this article as: Peng Z, et al. Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.018i

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dynamics of AUVs can be described as [1,13] ( η_ i ¼ Rðψ i Þνi ; M i ν_ i ¼  C i ðνi Þνi  Di ðνi Þνi  g i ðηi Þ  τiw ðtÞ þ τi ;

converges to the convex hull spanned by the leaders provided that 

ð3Þ

where 2

cos ψ i

6 Rðψ i Þ ¼ 4 sin ψ i 0

3

 sin ψ i

0

cos ψ i

7 0 5;

0

1

ð4Þ

ηi ¼ ½xi ; yi ; ψ i T A R3 with ðxi ; yi Þ A R2 being the north-east position and ψ i A ðπ ; π  being the yaw angle; νi ¼ ½ui ; vi ; r i T A R3 are the surge, sway velocities and yaw rate; M i ¼ M Ti A R33 ; C i ðνi Þ A R33 , and Di ðνi Þ A R33 are the inertia matrix, coriolis/centripetal matrix, and damping matrix, respectively; g i ðηi Þ A R3 denotes the restoring forces and moments; τi ¼ ½τiu ; τiv ; τir T A R3 is the control input; τiw ¼ ½τiwu ; τiwv ; τiwr T A R3 represents the disturbance vector. In the following design, we assume that the AUVs have well defined depth control. In practice, the parameters M i , C i ðνi Þ, Di ðνi Þ, g i ðηi Þ, and τiw ðtÞ are very hard to identify accurately, and are treated to be totally unknown here. To facilitate the controller design, rewrite the AUV dynamics as M i ν_ i ¼  f i ðηi ; νi ; tÞ þ τi ;

3

ð5Þ

ν ν ν ν η τ  where f i ðηi ; νi ; tÞ ¼ 1   M Þ τ . M is a nominal inertial matrix which can be M i ðM 1 i i i i accurately obtained from trials. Besides, M i is realistically assumed to be symmetric and positive definite. In the controller design, we show how the inertial matrix variations can also be included in the NN approximation by using the above transformations. Consider N  M ðN 4 MÞ virtual leaders, labeled as M þ 1 to N. The motion of leaders are assumed to be independent of that of followers, and their trajectories are denoted by ϕi where i ¼ M þ 1; …; N. The follower 1 to M have at least one neighbor, and the leader M þ 1 to N have no neighbors. Let the communication topologies among the followers and leaders be described by the graph G, and L be its Laplacian matrix. Due to the fact that the leaders have no neighbors, L can be represented by " # L2 L1 ; ð6Þ L¼ 0ðNMÞM 0ðNMÞðNMÞ M i M 1 i ðC i ð i Þ i þ Di ð i Þ i þ g i ð i Þ þ iw ðtÞÞ

lim J ηi ðt Þ  ϕdi ðt Þ J r δ ;

ð8Þ

t→∞



where δ is a positive constant. In the following design, we first present a new PNDSC method for an MIMO system. Then, we employ the PNDSC method to develop the distributed adaptive containment controllers for AUVs in Section 4.

3. PNDSC for MIMO system

(

Consider a second-order MIMO uncertain nonlinear system as X_ 1 ¼ X 2 ; X_ 2 ¼ f ðX 1 ; X 2 ; tÞ þ U;

ð9Þ

where X 1 A Rn and X 2 A Rn are system states; U A Rn is the control input; f ðX 1 ; X 2 ; tÞ A Rn represents the unknown dynamics that contains model uncertainty and external disturbances. The control objective is to stabilize X1 in the presence of unknown dynamics f ðX 1 ; X 2 ; tÞ. Before presenting the PNDSC approach, we recall the approximation property of NN. The following standard assumption will be used in this paper.Note that if the NN is replace by Fuzzy system [35,36]. The same results can be derived. Assumption 3. A nonlinear continuous function f ðξÞ can be represented by f ðξÞ ¼ W T φðξÞ þ εðξÞ;

8 ξ A Ω;

where W is an unknown matrix satisfying J W J F r W with W  A R being a positive constant; φðξÞ : Ω-Rs þ 1 is a known vector function of the form φðξÞ ¼ ½b; φ1 ðξÞ; φ2 ðξÞ; …; φs ðξÞT with Ω being a compact set; εðξÞ is the approximation error satisfying J εðξÞ J r ε with ε being a positive constant. Note that the function f ðX 1 ; X 2 ; tÞ in (9) contains time-varying terms, which means that it cannot be directly approximated by NN. Here, we use the sampled input and output data to reconstruct f ðX 1 ; X 2 ; tÞ as [37]. Lemma 3. Given ε 4 0, there exists a set of bounded weights W, such that the continuous function f ðX 1 ; X 2 ; tÞ can be approximated over a compact set Ω by an NN as

where L1 A RMM and L2 A RMðNMÞ .

f ðX 1 ; X 2 ; tÞ ¼ W T φðξÞ þ εðξÞ;

Assumption 1. For each follower, there exists at least one leader that has a directed path to that follower.

using the input vector

Lemma 2 (Cao et al. [15], Li et al. [22]). Under Assumption 1, the eigenvalues of L1 have positive real parts, and each entry of L1 1 L2 has a sum equal to 1.

with t d 4 0, and

_ ðtÞ, ϕ € ðtÞ, i ¼ M þ1; …; N, Assumption 2. The trajectories of ϕi ðtÞ, ϕ i i are bounded. The control objective is to design a distributed adaptive control law τi for each AUV with the dynamics (3) such that under a directed graph, the output trajectory of each AUV nearly converges to the convex hull spanned by the leaders; i.e., lim J ηi ðt Þ  hðt Þ J r δ;

t→∞

ð7Þ

where i ¼ 1; …; M; hðtÞ A CofϕM þ 1 ðtÞ; …; ϕN ðtÞg, and δ is a positive constant. T T Let ϕðtÞ ¼ ½ϕM þ 1 ðtÞ; …; ϕN ðtÞT and ϕd ðtÞ ¼ ½ϕd1 ðtÞ; …; ϕdM ðtÞ ¼ 1 ðL1 L2  I 3 ÞϕðtÞ where ϕdi ðtÞ A R3 . By Lemma 2 and Definition 1, we obtain the output trajectory of each AUV which nearly

ð10Þ 

ξ ¼ ½1; X T2 ðtÞ; X T2 ðt  t d Þ; U T ðtÞT ; J εðξ Þ J r ε ;

ð11Þ

ð12Þ

ð13Þ

provided there exists a suitable basis of activation function φðÞ on the compact set Ω. Proof. Note that the derivative of X2 based on past values is X 2 ðt Þ  X 2 ðt  t d Þ : X_ 2 ¼ lim td t d →0

ð14Þ

where t d 4 0. This means that for any ε 4 0, there exists a δðεÞ, such that 8 t d o δðεÞ:

J

J

X 2 ðt Þ  X 2 ðt  t d Þ X_ 2 ðt Þ  o ε: td

ð15Þ

Since X_ 2 ðtÞ can be approximated to arbitrary accuracy using sufficiently dense data of X 2 ðtÞ and X 2 ðt  t d Þ, it follows from f ðX 1 ; X 2 ; tÞ ¼ X_ 2  U that f ðX 1 ; X 2 ; tÞ can be reconstructed by an NN, operating on the current value of X 2 ðtÞ, U(t), and one past value

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4

X 2 ðt  t d Þ. This dependence is well known as the finite difference theory.□

3.2. Stability analysis The following theorem states the stability property of the closed-loop system (25):

3.1. Controller design Step 1: To stabilize the first dynamical equation of (9), a virtual control signal α is designed as

α ¼  k1 Z 1 ;

ð16Þ

where Z 1 ¼ X 1 and k1 ¼ diagfk11 ; …; k1n g with k1i A R ; i ¼ 1; …; n, being positive constants. Let α pass through a first-order filter to obtain an estimate of α as follows:

Theorem 1. Consider the closed-loop system defined by the plant (9), the control law (20), the adaptive law (23), the first-order filter (17), together with the predictor (22) under Assumption 3. Then, there exist control parameters k1 ; k2 ; κ ; Γ ; kW , and γ such that the error ~ , are signals in the closed-loop system, i.e., Z 1 ; Z 2 ; X~ 2 ; Z^ 2 ; q, and W uniformly ultimately bounded (UUB). Proof. Construct the following Lyapunov function: n o T T 1 ~ TW ~ Þ : V ¼ 1 Z T Z 1 þ X~ X~ 2 þ Z^ Z^ 2 þ qT q þ Γ trðW

ð17Þ

where γ A R is a positive constant. Step 2: Define the second tracking error Z2 as follows: Z 2 ¼ X 2  X 2d :

Taking the time derivative of (26) and using (25) and (23) yield

T

Z^ 2 k2 Z^ 2 

ð18Þ

Z_ 2 ¼ f ðX 1 ; X 2 ; tÞ þU  X_ 2d ;

ð19Þ

To stabilize (19), the control law for U is proposed as ^ T φðξÞ; U ¼  k2 Z 2 þ X_ 2d  W

where k2 ¼ diagfk21 ; …; k2n g with k2i A R ; i ¼ 1; …; n, being positive constants. ^ can be Following the NDSC approach, the adaptive law for W designed as

Remark 1. Even though the stability of the controller using the NDSC approach can be established as in [32,33]; the transient performance of the NDSC approach can be very poor due to the following reasons. First, oscillations will increase with the adaptation gain, which may exceed the available bandwidths of actuators. Second, nonzero tracking error of Z2 during the transient phase may lead to control signals of large magnitude, which cannot be implemented by actuators. To address the above problems, consider a state predictor as ð22Þ

where X~ 2 ¼ X^ 2  X 2 , and κ ¼ diagfκ 1 ; … ., κ n g with κ i A R; i ¼ 1; …; n, being positive constants. ^ is designed as Then, the update law for W ð23Þ

Let q ¼ X 2d  α, and its time derivative with (18) is given by q

q_ ¼  þBðÞ;

ð24Þ

γ

2 where BðÞ ¼ k1 Z 1 þk1 X~ 2  k1 Z^ 2  k1 q. ^ Define Z 2 ¼ X^ 2  X 2d , then the resulting closed-loop system in the new coordinates Z 1 ; X~ 2 ; Z^ 2 , and q can be described as

Z_ 1 ¼  k1 Z 1  X~ 2 þ Z^ 2 þq; _ ~ T φðξÞ þ ε; X~ 2 ¼  ðk2 þ κ ÞX~ 2  W q_ ¼ 

q

γ

þ BðÞ:

T

þ qT BðÞ  Z^ 2 κ X~ 2

_ Z^ 2 ¼  k2 Z^ 2  κ X~ 2 ;

ð27Þ

T Using Yong's inequality, one has j X~ 2 j r 12 J X~ 2 J 2 þ 12 2 , j Z T1 qj r T ^ 2 2 2 1 1 1 1 ^ j Z 1 Z 2 j r 2 J Z 1 J þ 2 J Z 2 J 2 , j Z T1 X~ 2 j r 12 J Z 1 J 2 þ 2 J Z1 J þ 2 J q J , 2 1 ~ T q BðÞ r ð2ϱ3  min ðk1 ÞÞ J q J 2 þ 32ϱð J Z 1 J 2 þ J Z^ 2 J 2 þ J X~ 2 J 2 Þ 2JX2 J , T 2 with ∋ 40 and ¼ maxf max ðk1 Þ; max ðk1 Þg, Z^ 2 X~ 2 r λmax2 ðκ Þ J Z^ 2 J 2 T λmax ðκÞ ~ 2 kW kW 2 ~ ~ ^ þ 2 J X 2 J , kW trðW W Þ r  2 ‖W ‖F þ 2 W 2 ,

ε

λ

λ

λ

ε

κ

which lead to 





3 3ϱ T 1 λmax ðκ Þ λmin ðk1 Þ   Z 1 Z 1  λmin ðk2 Þ   2 2 2 2   3 ϱ ^ T ^ 1 ϱ 1 T q q Z 2Z 2    þ λ ðk Þ  2 γ min 1 2 3 2  kW ~ 2 λmax ðκ Þ ‖F  λmin ðκ þ k2 Þ  1   ‖W 2 2 3 ϱ ~ T ~   X 2X 2 þϵ ; 2

V_ r 

where kW A R and Γ A R are positive constants.

^ : ^_ ¼ Γ ½φðξÞX~ T  kW W W 2

γ

T ~ TW ^ Þ þ X~ T ε: X~ 2 ðκ þ k2 ÞX~ 2  kW trðW 2

ð21Þ

_ ^ T φðξÞ þ U  ðk2 þ κ ÞX~ 2 ; X^ 2 ¼ W

qT q

ϱ

ð20Þ

^ ; ^_ ¼ Γ ½φðξÞZ T  kW W W 2

ð26Þ

2

2

V_ ¼  Z T1 k1 Z 1 þ Z T1 Z^ 2  Z T1 X~ 2 þ Z T1 q

The time derivative of Z2 along the second dynamical equation of (9) is given by

(

1

2

γ X_ 2d ¼ α  X 2d ;

where ϵ ¼ 12ε2 þ 12kW W 2 . Let 8 μ ¼ λmin ðk1 Þ  32  ∋ϱ > 2 4 0; > > 1 > λmax ðκÞ > 1 > μ ¼ λmin ðk2 Þ  2  2  ∋ϱ > 2 4 0; > < 2 ϱ μ3 ¼ 1γ þ λmin ðk1 Þ  2∋  12 40; > > > > μ4 ¼ λmin ðk2 þ κ Þ  1  λmax2 ðκÞ  ∋ϱ > 2 4 0; > > > : μ5 ¼ k2W 4 0;

ð28Þ

ð29Þ

and then (28) can be expressed in a compact form: V_ r  μ V þ ϵ ;

ð30Þ

where μ ¼ minf2μ1 ; 2μ2 ; 2μ3 ; 2μ4 ; 2μ5 Γ g. The inequality (30) implies V_ o 0 as V 4 ϵ =μ . Therefore, the error signals in the ~ , are UUB by Lemma 1. closed-loop system, i.e., Z 1 , X~ 2 ; Z^ 2 ; q, and W As J Z 2 J r J X~ 2 J þ J Z^ 2 J ; then Z 2 is UUB. Solving the inequality (30) produces 

Vr

ϵ     1  eμ t þ Vð0Þeμ t : μ

ð31Þ

Noting that 1 2

J Z^ 1 J 2 r V;

1 2

J Z 1 J 2 rV ;

1 2

J X~ 2 J 2 r V;

ð32Þ

it follows that Z 1 ; Z^ 2 , and X 2 are bounded by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J Z 1 J r 2ϵ =μ ; J Z^ 1 J r 2ϵ =μ ; J X 2 J r 2ϵ =μ ; as t-1. Also, observe that

ð25Þ

J Z 2 J ¼ J  X~ 2 þ Z^ 2 J r J X~ 2 J þ J Z^ 2 J ;

ð33Þ

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~ J have the transient Then, we can derive that J X~ 2 J and J W bounds: sffiffiffiffiffiffiffiffi ~ ð0Þ J F 2ϵ1 JW ð42Þ J X~ 2 J r þ J X~ 2 ð0Þ J þ pffiffiffiffi ;

from which we can obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J Z 2 J r 2 2ϵ =μ ; as t-1. The proof is complete.□

μT

and 3.3. Transient analysis In the preceding subsection, we have derived the stability property of the proposed PNDSC method; however, no evidence shows that the transient performance is improved. In this subsection, we quantify the transient performance of the proposed PNDSC architecture by deriving the truncated L2 norm of the derivatives of NN weights, which correspond to their frequency characteristics. In general, a larger L2 norm of the derivative of a signal in a specified time duration implies more oscillations con^_ tained in that signal. Then, we demonstrate that the L2 norm of W of the proposed PNDSC method is smaller than that of the NDSC approach, yielding the improved performance. Recalling (25), the prediction error dynamics of X~ 2 can be written as _ ~ T φðξÞ þ ε: X~ 2 ¼  ðk2 þ κ ÞX~ 2  W

ð34Þ

The following theorem states the second result of this paper: Theorem 2. Consider the prediction error dynamics (34) together with the adaptive law (23); then, the truncated L2 norms of X~ 2 and ^_ satisfy W ~ ð0Þ J F 1 JW J X~ 2 J L2 ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J X~ 2 ð0Þ J þ pffiffiffiffi Γ 2λmin ðκ þ k2 Þ  1 qffiffiffiffiffiffiffiffiffiffiffiffi þ 2ε1 t  ;

ð35Þ

ð36Þ

Proof. Consider the following Lyapunov function candidate: ð37Þ

Recalling the inequality (39), it follows that  λmin ðκ þ k2 Þ  12 J X~ 2 J 2 r  V_ ðtÞ þ ϵ1 ;



ð43Þ

ð44Þ



by integration of which over t A ½0; t  gives J X~ 2 J 2L2 ;t r

ϵ1 t  Vð0Þ þ : λmin ðκ þ k2 Þ  1=2 λmin ðκ þ k2 Þ  1=2

By applying

ð45Þ

pffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi a þ b r a þ b with a Z0 and b Z 0, one has

~ ð0Þ J F pffiffiffiffiffiffiffiffiffiffi J X~ 2 ð0Þ J J W Vð0Þ r pffiffiffi þ pffiffiffiffiffiffiffi ; 2 2Γ

ð46Þ

which leads to (35). ^_ J . To this Next, we explicitly derive an upper bound for J W _ ^ end, recalling the expression of W in (23), we have ^_ J F r Γ J φðξÞ J J X~ 2 J þ Γ kW J W ^ JF: JW

ð47Þ

ð48Þ

which results in

Remark 2. The state of the predictor (22) can be easily initialized ^_ such that X^ 2 ð0Þ ¼ X 2 ð0Þ; then, the truncated L2 norms of X~ 2 and W are ! ~ ð0Þ J F qffiffiffiffiffiffiffiffiffiffiffiffi 1 JW   ~  pffiffiffiffi þ 2ε1 t ; J X 2 J L2 ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð51Þ Γ 2λmin ðκ þ k2 Þ  1 and

The time derivative of V along (34) and (23) is given by ~ κ þ k2 ÞX~ 2  kW trðW

~ ð0Þ J F : þ JW

It follows from (35) and (43) that one finally has (36).□

where t  is a specified constant.

V_ ¼ 

μT

! þ J X~ 2 ð0Þ J

~  þW  Þ2 t  ; ^_ J 2  r 2Γ 2 φ2 J X~ 2 J 2  þ 2Γ 2 k2 ðW ð49Þ JW L2 ;t L2 ;t W pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ~ ð0Þ J F . In what follows, ~ ¼ Γ ð 2ϵ =μ þ J X~ 2 ð0Þ J Þ þ J W where W T 1 we can obtain pffiffiffi pffiffiffi pffiffiffiffi ~  þ W Þ t: ^_ J L ;t r 2Γφ J X~ 2 J L ;t þ 2Γ kW ðW JW ð50Þ 2 2

μT

1 T 1 ~ Þ: ~ TW V ¼ X~ 2 X~ 2 þ trðW 2 2Γ

sffiffiffiffiffiffiffiffi 2ϵ1

Using the bounds for φðξÞ and W, it follows that

pffiffiffi ~ ð0Þ J 2Γφ JW ^_ J L ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi J X~ 2 ð0Þ J þ pffiffiffiffi F JW 2 Γ 2λmin ðκ þ k2 Þ  1 sffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffi 2Γϵ1 þ 2ε1 t  þ 2Γ kW þ J X~ 2 ð0Þ J Γ pffiffiffiffi ~ ð0Þ J F þW  þ JW t;

pffiffiffiffi ~ JF r Γ JW

Γ

~  þ W  Þ; ^_ J F r Γφ J X~ 2 J þ Γ kW ðW JW

and

T X~ 2 ð

5

T

^ Þ þ X~ T W 2

ε;

which can be further put into   1 ~T ~ kW ~ 2 X 2X 2  ‖W ‖F þ ϵ1 ; V_ r  λmin ðκ þ k2 Þ  2 2

ð38Þ

ð39Þ

! pffiffiffi ~ ð0Þ J F qffiffiffiffiffiffiffiffiffiffiffiffi JW 2Γφ  t ^_ J L ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffi þ 2 ε JW 2 1 Γ 2λmin ðκ þ k2 Þ  1 sffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffiffi 2Γϵ1 ~ ð0Þ J F þW  t : þ JW þ 2Γ kW

μT

ð52Þ

where ϵ1 ¼ 12ε2 þ k2W W 2 . Letting μT ¼ minf2λmin ðκ þ k2 Þ  1; Γ kW g and noting that μT 4 0 under the condition (29), it follows that (39) can be written as

Remark 3. From (51) and (52), it can be observed that by increasing λmin ðκ Þ, one can further decrease the L2 norms of X~ 2 and ^_ ; i.e., reduce the oscillations in neural adaptive control signals. W

V_ r  μT V þ ϵ1 ;

Remark 4. In summary, the PNDSC approach takes the following properties:

ð40Þ

by integration of which produces Vr

 ϵ1  1  eμT t þ Vð0ÞeμT t : μT

ð41Þ


By introducing the first-order filters, it avoids tedious differential operations in virtual control signals, and thus reduces the

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6







complexity of controller, which is easier to implement in practice. This advantage will be sharper as the system order increases. Using sampled input and output data, the uncertainties in the system dynamics can be reconstructed by NNs. It is shown that the error signals of the closed-loop system are bounded regardless of both time-varying unknown dynamics and external disturbances. By incorporating a predictor into the NDSC design, it results in two time-scales to govern the entire system, i.e., a fast prediction error dynamics and a relatively slow tracking error dynamics. By allowing the two time-scales to be separate, the transient performance can be controlled without compromising the learning rate of adaptation.

To compare, we next practice the transient analysis of the NDSC approach. The error dynamics of Z~ 2 using the NDSC approach can be expressed as _ ~ T φðξÞ þ ε: Z~ 2 ¼  k2 Z~ 2  W

ð53Þ

The following corollary holds for the NDSC approach: Corollary 1. Consider the tracking error dynamics (53) together with ^_ the adaptive law (21); then, the truncated L2 norms of Z 2 and W satisfy ! ~ ð0Þ J F qffiffiffiffiffiffiffiffiffiffiffiffi 1 JW J Z 2 J L2 ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J Z 2 ð0Þ J þ pffiffiffiffi þ 2ε1 t  ; ð54Þ Γ 2λmin ðk2 Þ  1 and

! pffiffiffi ~ ð0Þ J F qffiffiffiffiffiffiffiffiffiffiffiffi 2Γφ JW _   ^ J W J L2 ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J Z 2 ð0Þ J þ pffiffiffiffi þ 2ε1 t Γ 2λmin ðk2 Þ  1 sffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffi pffiffiffi pffiffiffiffi 2Γϵ1  ~ t : þ J Z 2 ð0Þ J Γ þ J W ð0Þ J F þW þ 2Γ kW

μT

ð55Þ

Proof. The proof can be completed by replacing X~ 2 with Z 2 in the proof of Theorem 2, and is omitted here.□ Remark 5. Based on the above derivations and proofs, the comparisons between the NDSC and PNDSC are stated as follows, from which we can see the pros and cons of the PNDSC architecture.


Compared to the classical NDSC approach [32,33], the proposed







new PNDSC design methodology provides two avenues (extra ^_ in the presence of the same freedom) to deduce the L2 norm of W adaptation gain Γ . One is done by choosing a positive matrix κ ; and the other is practiced by initializing X^ 2 ð0Þ ¼ X 2 ð0Þ such that the undesired learning transient due to initial errors can be completely avoided. It is obvious that the above two advantages cannot be provided by using the classical NDSC approach. For both NDSC and PNDSC approaches, oscillations will exacerbate as the adaptation gains increase, implying that too large adaptation gains are not permitted. However, the constraints on the speed of adaptation law for a specified transient performance can be relaxed by using PNDSC approach; i.e., a larger adaptive gain is allowed than the NDSC approach. As far as the complexity is concerned, both the NDSC and PNDSC can avoid the tedious differential operations for virtual control signals, and thus reduce the complexity of controller. However, the PNDSC method is slightly more complex than the NDSC approach due to the introduction of the predictor in the control process. While such disadvantage may not be remarkable by using recently developed digital processors.

Γ can be selected as large as possible. κ determines pffiffiffiffi the damping of the NN learning, and a proper choice is κ ¼ 2 Γ  k2 . Remark 7. Note that the predictor-based design has been presented in [30,31]. In [30], the predominant concern is to extract information about the true parameters from the prediction errors. Besides, the authors aim to extract the parameter information from both prediction errors and tracking errors, resulting in the so-called composite adaptive control. The underlying mechanism of using prediction errors to improve transient performance was not explored. In [31], the usage of predictor aims to decouple the estimation loop from control loop, and the ability to arbitrary increase the adaptation gain is achieved by inserting a lowpass filter into the control channel. In the design, the time-scale of the prediction error dynamics is the same as the tracking error dynamics. While in our design, the prediction error dynamics is made faster than the tracking error dynamics, and a particular parameter is provided to regulate the transient performance. The control performance of the proposed the PNDSC method is illustrated by the following example: Example 1. Consider an uncertain nonlinear system described by ( x_ 1 ¼ x2 ; ð56Þ 2 x_ 2 ¼ x22 þ x1 x2 þ sin ðtÞ cos 2 ð0:5tÞ þ u;

For illustrations, we compare the PNDSC approach with the NDSC approach proposed in [32]. The control law by using NDSC approach is given by 8 α ¼  k1 Z 1 ; > > > > > ^ T φðξÞ; < U ¼  k2 Z 2 þ X_ 2d  W ð57Þ ^ ; ^_ ¼ Γ ½φðξÞZ T  kW W > W > 2 > > > : γ X_ ¼ α  X : 2d 2d The parameters for the two controllers are chosen as k1 ¼ 2; k2 ¼ 10 000; kw ¼ 0:0001; κ ¼ 198; x1 ð0Þ ¼  1; x2 ð0Þ ¼ 2; γ ¼ 0:1; Γ ¼ 0:1, and x^ 2 ð0Þ ¼ 0:1. The activation functions for NNs are chosen as 1 1 þ eax with a ¼ 2. Simulation results are provided in Fig. 1. Fig. 1(a) shows the output responses of the system using the NDSC and PNDSC approaches. Fig. 1(b) demonstrates the learning profile of NN for both methods. It can be seen that an accurate and smooth learning can be achieved using PNDSC approach. Fig. 1(c) depicts the control signals of the NDSC and PNDSC methods. It can be observed that the PNDSC performs better than the NDSC approach, with less oscillations and smaller magnitude in the control signal.

4. Containment controller design via PNDSC In this section, the proposed PNDSC method is employed to develop the adaptive containment controller, under which the trajectories of the follower AUVs nearly converge to the convex hull spanned by the multiple leaders. 4.1. Controller design Step 1: We start with defining a distributed diffeomorphic coordinate transformation as follows: 8 9 M N
Remark 6. In practical implementations, k1 and k2 are designed based on the desired output responses as well as the conditions (29). As far as the NN parameters are concerned, the adaptive gain

j ¼ Mþ1

which is expressed in the body-fixed frame, and aij is defined in Section 2.

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Taking the time derivative of zi1 along (3) yields z_ i1 ¼  r i Szi1 þ di νi 

M X

aij RT ðψ i ÞRðψ j Þνj

j¼1 N X



_ ; aij RT ðψ i Þϕ j

ð59Þ

j ¼ Mþ1

where S is defined as 2

0

6 S¼41 0

1

0

3

0

7 0 5:

0

0

ð60Þ

To stabilize zi1 , a virtual control law αi based on the information of neighboring AUVs is proposed as follows: 8 9 M N = X X 1< T T _ ; αi ¼ ki1 zi1 þ aij R ðψ i ÞRðψ j Þνj þ aij R ðψ i Þϕ j ; di : j¼1

j ¼ M þ1

where ki1 ¼ diagfki11 ; ki12 ; ki13 g with ki11 A R; ki12 A R, and ki13 A R being positive constants. Note that if the backstepping is used to design the dynamic controller, ( it will be very complex since the expansion of α_ i gives P M P T 1 α_ i ¼ di ki1 ðri Szi1 þ di νi  1 aij RT ðψ i ÞRðψ j Þνj  N j ¼ M þ 1 aij R ð j¼

P

T T 1 _ _ ψ i Þϕ_ j Þ þ M j ¼ 1 aij ðR ðψ i ÞRðψ j Þνj þ R ðψ i ÞRðψ j Þνj þ R ðψ i ÞRðψ j Þ½M j ð PN T _ þ C j ðνj Þνj  Dj ðνj Þνj  g j ðηj Þ τjw ðtÞ þ τj ÞÞ þ j ¼ M þ 1 aij ðR_ ðψ i Þϕ j T

 € Þ . To avoid this complexity, let α pass through a firstRT ðψ i Þϕ i j

order filter bank to obtain the filtered signal νid as

γ i ν_ id ¼ αi  νid ;

ð61Þ

where γ i A R is a time constant. Step 2: A second tracking error is defined as follows: zi2 ¼ νi  νid ;

i ¼ 1; …M;

ð62Þ

whose time derivative along (5) is given by M i z_ i2 ¼ τi  f i ðηi ; νi ; tÞ  M i ν_ id ; Let f i ðηi ; νi ; tÞ be approximated by an NN using sampled input and output data as follows: f i ðηi ; νi ; tÞ ¼  M i ν_ i þ τi ¼ W Ti φi ðξi Þ þ εi ðξi Þ; where ξi ¼ ½1; νTi ðtÞ; νTi ðt  t d Þ; τTi ðtÞT A R10 ; W i is the NN weights; φi ðξi Þ is the activation function; εðξÞ is the NN approximation error; Then, a practical control law is constructed as follows: T

^ φ ðξ Þ; τi ¼  ki2 zi2 þ Mi ν_ id þ W i i i

ð63Þ

^ i is an estimate of W i ; ki2 ¼ diagfki21 ; ki22 ; ki23 g with where W ki21 A R; ki22 A R, and ki23 A R being positive constants. ^ i , consider a state predictor as To design an update law for W ^ φ ðξ Þ þ τi  ðκ i þ ki2 Þν~ i ; M i ν^_ i ¼  W i i i T

ð64Þ

where ν~ i ¼ ν^ i  νi and κ i ¼ diagfκ i11 ; κ i12 ; κ i13 g with κ i11 A R; κ i12 A ^ i is R, and κ i13 A R being positive constants. The update law for W devised as follows: Fig. 1. Performance comparisons of the NDSC and PNDSC approaches (a) Output responses (b) Learning profile of NNs Uad¼ WTφ(ξ) (c) Control inputs.

^ i ; ^_ i ¼ Γ iW ½φ ðξ Þν~ T  kW W W i i i

ð65Þ

where Γ iW A R and kW A R are positive constants.

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As a consequence, the resulting closed-loop network can be described as 8 z_ ¼  ki1 zi1  r i Szi1 þdi ðν~ i þ z^ i2 þ qi Þ; > > < i1 ~ T φ ðξ Þ þ εi ; ð66Þ M i ν~_ i ¼  ðκ i þki2 Þν~ i  W i i i > > :  ^_ M z ¼  k z^  κ ν~ ; i

i2

i2 i2

i

i

where qi ¼ νid  αi and z^ i2 ¼ ν^ i  νid . Taking the time derivative of qi produces q_ i ¼ 

qi

γi

 α_ i :

ð67Þ

o N n P 2 1 2 1 2 1 where ϵ ¼ . Let 2qi þ 2εi þ 2kW W i i¼1 8 3di > > 4 0; > μi1 ¼ λmin ðki1 Þ  > > 2 > > > > di λmax ðκ i Þ > > 4 0; < μi2 ¼ λmin ðki2 Þ   2 2 d þ 1 λmax ðκ i Þ > > >  4 0; μi3 ¼ λmin ðki2 þ κ i Þ  i > > 2 2 > > > > k > W > 4 0; : μi4 ¼ 2

ð73Þ

and then (72) can be expressed in a compact form: V_ r  μV þ ϵ;

Solving Eq. (67) gives Z t eðtτÞ=γ i α_ i ðτÞ dτ; qi ðtÞ ¼ et=γ i qi ð0Þ 

ð74Þ

μ ¼ mini ¼ 1;…;N f2μi1 ; 2μi2 =λ μ λ μ Γ i g. The inequality (74) implies V_ o 0 when μ 4 ϵ=μ. Therefore, the ~ i , are error signals in the closed-loop network, i.e., zi1 ; z^ i2 ; ν~ i , and W UUB. Noting that J zi2 J r J ν~ i J þ J z^ i2 J , it follows that zi2 is UUB. In addition, solving the inequality (74) gives  ϵ V r 1  eμt þ Vð0Þeμt : ð75Þ   max ðM i Þ; 2 i3 = max ðM i Þ; 2 i4

where ð68Þ

0

from which we can compute an upper bound for qi as J qi ðtÞ J r et=γ i J qi ð0Þ J þ γ i αid ;

ð69Þ

where α is the upper bound for α_ i . Note that the bound for α_ i exists as long as the inputs are bounded and the reference signals _ and ϕ € are bounded. Since the energy to drive the AUVs is of ϕ j j limited, the boundedness of α_ i is naturally satisfied for practical marine applications. Hence, there exists a positive constant qi such that J qi ðtÞ J rqi .  id

μ

Note that J z1 J 2 =2 r V with z1 ¼ ½zT11 ; …; zTN1 T , and then the surface error J z1 J is bounded by pffiffiffiffiffiffiffiffiffiffiffi z1 J r 2ϵ=μ: ð76Þ By appropriately increasing μ, the bound can be reduced. Also, note that z1 ¼ R½ðL1  I 3 Þη þ ðL2  I 3 Þϕ where η ¼ ½ηT1 ; …; ηTN T

4.2. Stability analysis

and R ¼ diagðRT ðψ 1 Þ; …; RT ðψ M ÞÞ. It follows that there exists a positive constant δ such that

Theorem 3. Consider the closed-loop network defined by the AUV dynamics (3), the control law (63), the adaptive law (65), the firstorder filter (61), together with the predictor (64) under Assumptions 1–3. Then, there exist control parameters ki1 ; ki2 ; κ i ; Γ iW ; kW , and γ i , such that all error signals in the closed-loop network, i.e., zi1 ; zi2 ; ~ i , are UUB. z^ i2 ; ν~ i , and W Proof. Consider the following Lyapunov function candidate: V¼

M n o 1X T 1 ~ TW ~ iÞ : zTi1 zi1 þ z^ i2 M i z^ i2 þ ν~ Ti M i ν~ i þ Γ iW trðW i 2i¼1

ð70Þ

V_ r

n

zTi1 ki1 zi1 þ di zTi1 ðν~ i þ z^ i2 þ qi Þ  z^ i2 ki2 z^ i2  z^ i2 κ i ν~ i  ν~ Ti ðκ i T

T

i¼1

o ^ i Þ þ ν~ T εi : ~ TW þ ki2 Þν~ i  kW trðW i i

ð71Þ

T 1 Using the following inequalities j ν~ Ti εi j r 12 J ν~ i J 2 þ 12ε2 i ; j zi1 qi j r 2 J

zi1 J 2 þ 12 J qi J 2 ; j zTi1 z^ i2 j r 12 J zi1 J 2 þ 12 J z^ i2 J 2 ; j zTi1 ν~ i j r 12 J zi1 J 2 þ 12 J ν~ i T J ; z^ i2 i ~ i r λmax2ðκ i Þ J z^ i2 J 2 þ λmax2ðκi Þ J ~ i J 2 ;  ~ TW ~ i ‖2 þ kW W 2 , we can ^ i Þ r  kW ‖W kW trðW i i F 2 2 2

V_ r

κν

ν

which is directly implying (8) by Lemma 2. The proof is complete.□ 4.3. Transient analysis Recalling (66), the prediction error dynamics of ν~ i can be written as ~ T φ ðξ Þ þ εi : M i ν~_ i ¼  ðκ i þ ki2 Þν~ i  W i i i

ð78Þ

Theorem 4. Consider the prediction error dynamics (78) together the ^_ i satisfy adaptive law (65); then, the truncated L2 norms of ν~ i and W qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 J ν~ i J L2 ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λmax ðM i Þ J ν~ i ð0Þ J 2λmin ðκ i þ ki2 Þ  1 ! ~ i ð0Þ J F pffiffiffiffiffiffiffiffiffiffiffiffi JW  þ pffiffiffiffiffi þ 2ϵ1 t ; ð79Þ

Γi

and pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Γ i φi ^_ i J L ;t r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi JW λmax ðM i Þ J ν~ i ð0Þ J 2 2λmin ðκ i þ ki2 Þ  1

sffiffiffiffiffiffiffiffiffiffiffiffiffi ! ~ i ð0Þ J F pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi JW 2Γ i ϵ1  þ pffiffiffiffiffi þ 2ϵ1 t þ 2Γ i kW

derive that

Γi

 M  X  3d  λmin ðki1 Þ  i zTi1 zi1  λmin ðki2 Þ 2 i¼1

 d λmax ðκ i Þ ^ T ^ kW ~ 2  z i2 z i2  ‖W i ‖F  λmin ðκ i  i 2 2 2  d þ 1 λmax ðκ i Þ T  ν~ i ν~ i g þ ϵ; þ ki2 Þ  i 2 2

ð77Þ

The following theorem states the fourth result of this paper:

Taking the time derivative of V along (66) gives M X

J ηðtÞ þ ðL1 1 L2  I 3 ÞϕðtÞ J r δ

κ iT

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi ~ i ð0Þ J F þ W  t : þ λmax ðM i Þ J ν~ i ð0Þ J Γ i þ J W i ð80Þ Proof. Consider the Lyapunov function candidate ð72Þ

1 1 ~ i Þ; ~ TW V ¼ ν~ Ti M i ν~ i þ trðW i 2 2Γ i

ð81Þ

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L1

F1

F2

L2

F3

9

Table 1 Model parameters.

L3

F4

F5

Fig. 2. Communication topology.

Parameters

Value

m11 m22 m33 c13 ¼  c31 c23 ¼  c32 d11 d22 d33

200 250 80  250v 200u 70 þ 100j uj 100 þ 200j vj 50 þ 100j rj

kg kg kg kg/s kg/s kg/s kg/s kg/s

7 NDSC PNDSC

6

4

1

||z (t)||

5

3

2

1

Fig. 3. Formation pattern in projected 2D plane. 0 0

whose time derivative with (65) and (78) is given by ð82Þ

which can be further put into     1 T k ~ i ‖2 þ ϵ1 ; ν~ i ν~ i  W ‖W V_ r  λmin ðκ i þki2 Þ  F 2 2 and

ð83Þ

then

(83) ð84Þ

whose integration is  ϵ1  Vr 1  eκiT t þV ð0Þeκ iT t :

ð85Þ

κ iT

~ i J have the transient Similarly, we can derive that J ν~ i J and J W bounds as sffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ i ð0Þ J F 1 2ϵ1 JW  ~ ffi p ffiffiffiffiffi J ν~ i J r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ðM Þ J ν ð0Þ J þ þ ; ð86Þ max i i κ iT λmin ðMi Þ Γi

~ i JF r JW

pffiffiffiffiffi

Γi

sffiffiffiffiffiffiffiffi 2ϵ1

κ iT

! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ i ð0Þ J F JW  þ λmax ðM i Þ J ν~ i ð0Þ J þ pffiffiffiffiffi :

Γi

Recalling the inequality (83), one has   λmin ðκ i þ ki2 Þ  12 J ν~ i J 2 r  V_ ðtÞ þ ϵ1 ;

ð87Þ

Vð0Þ ϵ1 t  þ : λmin ðκ i þ ki2 Þ  1=2 λmin ðκ i þ ki2 Þ  1=2

40

50

60

70

80

Fig. 4. The tracking error norms of z1 .

By square of (89) and using pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ð0Þ J F λmax ðMi Þ J ν~ i ð0Þ J J W pffiffiffi þ piffiffiffiffiffiffiffiffi ; 2 2Γ i

ð90Þ

it follows that (79) is satisfied. ^_ i J . From (65), one has Next, we derive the upper bound for J W ^ i JF: ^_ i J F r Γ i J φ ðξ Þ J J ν~ i J þ Γ i kW J W JW i i

ð91Þ

Using the bounds for φi ðξi Þ and W i , and employing (86) and (87), we obtain ~  þ W  Þ; ^_ i J F r Γ i φ J ν~ i J þ Γ i kW ðW JW i i i

ð92Þ

which results in ~  þ W  Þ2 t  ; ^_ i J 2  r 2Γ 2 φ2 J ν~ i J 2  þ 2Γ 2 k2 ðW ð93Þ JW i i i W i L2 ;t L2 ;t i pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi    ~ ~ where W i ¼ Γ i ð 2ε1 =κ iT þ λmax ðM i Þ J ν~ i ð0Þ J þ J W i ð0Þ J F = Γ i Þ. By square of (93), one has pffiffiffi pffiffiffi pffiffiffiffi ^_ i J L ;t  r 2Γ i φ J ν~ i J L ;t þ 2Γ i kW ðW ~  þ W Þ t; JW ð94Þ i i i 2 2 Combining (79) with (94), one finally has (80). The proof is complete.□

ð88Þ

by integration of which over t A ½0; t   produces

J ν~ i J 2L2 ;t  r

30

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V_ r  κ iT V þ ϵ1 ;

and

20

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~ TW ^ i Þ þ ν~ T εi : V_ ¼  ν~ Ti ðκ i þ ki2 Þν~ i  kW trðW i i

2 kW where ϵ1 ¼ 12ε2 i þ 2 Wi . Let κ iT ¼ minf2λmin ðκ i þ ki2 Þ  1; Γ i kW  1g becomes

10

ð89Þ

Remark 8. In [2,3,12] and [25–28], the neural adaptive controllers are developed for marine vehicles in order to preserve stability and robustness in the presence of model uncertainty and ocean disturbances. However, they may not be able to achieve the desired performance in the presence of large model uncertainty and ocean disturbances. In this paper, a new PNDSC approach is used to

Please cite this article as: Peng Z, et al. Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.018i

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develop the adaptive containment controllers, and the transient performance can be improved dramatically by regulating an additional parameter. This means that less control efforts are needed and the proposed scheme results in energy-efficient controllers. Remark 9. By using the DSC technique [32,33,38,39], the repeated differentiations of velocity information of neighbors are

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not required. If the backstepping design approach [40] is employed to develop the containment controllers for the follower AUVs, the control inputs of the neighbors will appear in constructing the actual control laws for the follower AUVs. This may result in the “circular design” problem. However, the DSC technique can overcome this problem due to the use of the filters

Please cite this article as: Peng Z, et al. Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.018i

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11

that the PNDSC performs better with less control efforts and less oscillations, which results in energy-efficient controllers.

2500 NDSC PNDSC

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6. Conclusions In this paper, we investigated the containment control of multiple AUVs in the presence of model uncertainty and time-varying ocean disturbances over a directed network. A new PNDSC design approach was proposed to develop the adaptive containment controllers. Both stability and transient performance have been established via rigorous analysis. It was shown that the PNDSC performs better than the classical NDSC approach, with smaller L2 norms of the derivatives of neural weights. The simulation results verified the control performance of the proposed new control architecture.

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Fig. 7. Comparisons of control inputs using the NDSC and PNDSC approaches.

between the kinematic controller and the kinetic controller design. Remark 10. Note that the size of the compact set Ω is not actually used in the controller design. It is commonly assumed to be as large as desired. In this regard, the obtained result is semi-global. To obtain a global result, the readers are referred to [41,42]. Remark 11. By using the sliding mode technique as in [43,44], it is able to improve the design of adaptive laws such that finite-time estimation of unknown dynamics is achieved.

5. An example A networked system that consists of five follower AUVs and three reference trajectories is considered. Let the information exchange topology among the five AUVs and three reference trajectories be given by Fig. 2. The model parameters for each AUV are found in Table 1 [13]. Additionally, some time-varying ocean disturbances are introduced into the plant, and they are modeled as the first-order Gauss–Markov processes as τ_ iwu þ γ iu τiwu ¼ w1 , τ_ iwv þ γ iv τiwv ¼ w2 , τ_ iwr þ γ ir τiwr ¼ w3 , where γ iu 4 0; γ iv 4 0; γ ir 4 0 and w1 ; w2 ; w3 are white noises. The control parameters are chosen as ki1 ¼ diagf5; 5; 5g, ki2 ¼ diagf1000; 1250; 160g, κ i ¼ diagf2000; 2500; 800g, Γ iW ¼ 10 000, kW ¼ 0:0001, γ i ¼ 0:02. To illustrate the control performance of the proposed method, the PNDSC scheme is compared with the direct NDSC approach. The control parameters for Γ iW and kW are chosen as the same as the PNDSC approach. The simulation results are shown in Figs. 3–7 which demonstrate the geometric formation of the five AUVs projected in 2D plane, and the triangle denotes the snapshots of the three leaders. Fig. 4 shows the norms of the tracking errors z1 using the NDSC and PNDSC approach, respectively. It implies that the containment errors converge to a small neighborhood of origin for both approaches. The unknown AUV dynamics including model uncertainty and ocean disturbances, and the output of NN using the NDSC and PNDSC approach, respectively, are depicted in Figs. 5 and 6, where τiad ¼ u v r ½τuiad ; τviad ; τriad T and f i ðÞ ¼ ½f i ðÞ; f i ðÞ; f i ðÞT . It demonstrates that, using the PNDSC approach, the unknown dynamics for the AUV in each direction can be accurately identified by NN not only in steady time, but also in transient time, which cannot be achieved using the classical NDSC approach. Fig. 7 depicts the norm of the control input of τ ¼ ½τT1 ; …; τT5 T using the NDSC and PNDSC approach. It reveals

This work was in part supported by the National Nature Science Foundation of China under Grants 51209026, 61273137, and 51579023, 51579022, and in part by the China Postdoctoral Science Foundation under Grant 2015M570247, and in part by the Scientific Research Fund of Liaoning Provincial Education Department under Grant L2013202, and in part by the Fundamental Research Funds for the Central Universities under Grants 3132015021 and 3132014321.

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