Containment control of networked autonomous underwater vehicles with model uncertainty and ocean disturbances guided by multiple leaders

Information Sciences 316 (2015) 163–179

Contents lists available at ScienceDirect

Information Sciences journal homepage: www.elsevier.com/locate/ins

Containment control of networked autonomous underwater vehicles with model uncertainty and ocean disturbances guided by multiple leaders Zhouhua Peng a,b,⇑, Dan Wang a, Yang Shi c, Hao Wang a, Wei Wang a a

School of Marine Engineering, Dalian Maritime University, Dalian 116026, China School of Control Science and Engineering, Dalian University of Technology, Dalian 116026, China c Department of Mechanical Engineering, University of Victoria, Victoria, B.C. V8W 3P6, Canada b

a r t i c l e

i n f o

Article history: Received 13 July 2014 Received in revised form 4 April 2015 Accepted 11 April 2015 Available online 17 April 2015 Keywords: Containment Autonomous underwater vehicles Predictor Dynamic surface control Output feedback

a b s t r a c t This paper considers the containment control of networked autonomous underwater vehicles guided by multiple dynamic leaders over a directed network. Each vehicle is subject to model uncertainty and unknown time-varying ocean disturbances. A new predictor-based neural dynamic surface control design approach is presented to develop the adaptive containment controllers, under which the trajectories of vehicles converge to the convex hull spanned by those of the leaders. Specifically, iterative neural updating laws, based on prediction errors, are constructed, which enable the accurate identification of the unknown dynamics for each vehicle, not only in steady state but also in transient state. Furthermore, this result is extended to the output-feedback case where only the position-yaw information can be measured. A neural observer is developed to recover the unmeasured velocity information. Based on the observed velocities of neighboring vehicles, distributed output-feedback containment controllers are devised, under which the containment can be achieved regardless of model uncertainty, unknown ocean disturbances, and unmeasured velocity information. For both cases, Lyapunov–Krasovskii functionals are used to prove the uniform ultimate boundedness of the closed-loop error signals. Comparative studies are given to show the performance improvement of the proposed methods. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Coordinated control of autonomous marine vehicles, including autonomous underwater vehicles (AUVs) and autonomous surface vehicles (ASVs), has drawn great attention from control communities [17,30,7,8,25,27]. This is partially due to their broad applications for marine operations, such as cooperative search and rescue, coordinated exploration and exploitation, sensor networks, etc. In general, the objective of coordinated control is to seek collaborative policies such that the collective behaviors among vehicles can emerge [13,42,19,12,40,43,6,46,39]. Coordinated control of marine vehicles in the presence of single leader has been investigated in [17,30,7,8,25,27]. In [17], a nonlinear control law is derived to steer two marine vehicles along identical parallel paths, targeted on combined ASV/AUV control. In [30], a range-based formation control scheme is developed for joint ASV/AUV operations. Three vehicles, with one ⇑ Corresponding author at: School of Marine Engineering, Dalian Maritime University, Dalian 116026, China. Tel.: +86 0411 84728286. E-mail address: [email protected] (Z. Peng). http://dx.doi.org/10.1016/j.ins.2015.04.025 0020-0255/Ó 2015 Elsevier Inc. All rights reserved.

164

Z. Peng et al. / Information Sciences 316 (2015) 163–179

being AUV and the other two ASVs, are required to shape a triangular formation. In [7], network networks (NNs) and backstepping techniques are combined to develop the leader–follower formation controllers for underactuated AUVs. In [8], decentralized control laws are developed for synchronization of multiple AUVs both for state- and output-feedback cases. This solution requires that the reference trajectory of the leader is known to each vehicle. In [25], distributed cooperative tracking of ASVs with a time-varying dynamic leader is investigated. In [27], a dynamic surface control (DSC) design technique is employed to devise the adaptive node controllers for distributed coordinated tracking of AUVs. In the presence of multiple leaders, related works can be found in [2,3,21–23,28,20,44,45] on containment control of multi-agent systems, in which the control objective is to drive all followers to the convex hull spanned by multiple leaders. The vehicle dynamics considered in previous works correspond to first-order systems, second-order systems, high-order systems, and Lagrange systems [2,3,21–23,28,20,44,45]. A typical application that matches the containment problem in marine industry is the long-range seafloor exploration and monitoring. In this scenario, multiple AUVs are deployed at different depths, monitoring the sea environment and implementing the communication relays using the vertical acoustic communication channel. Then, the data collected by the AUVs can be transmitted to the ASVs which act as communication relays between AUVs and mother vessel. From a practical perspective, in contrast to other vehicles moving in the air or at sea, this still poses great challenges for AUVs working in underwater due to the following reasons. (i) The communications can be severely restricted due to the acoustic communication channel adopted. (ii) The complex vehicle dynamics should be taken into account in the controller design. The application of adaptive control to marine vehicles holds promise both for safety and robustness, especially in the presence of large model uncertainty and ocean disturbances. As a result, numerous adaptive control design methods have been proposed. In particular, adaptive backstepping [16] and DSC [31] design techniques have been widely applied [7,25,27,18,32,4,14,9,29,24]. From a practical standpoint, the state information of vehicles can be far different from the virtual control signals or filtered virtual control signals during transient (i.e., in the initial stage or learning phase), which may cause poor learning transient and even result in instability for system performance. For containment control of multiple AUVs, this situation is getting worse due to the large initial conditions in the motion control setup. Motivated by the above observations, this paper considers the containment control of multiple AUVs in the presence of multiple dynamic leaders. Each AUV is subject to model uncertainty and unknown ocean disturbances. It is assumed that the information of leaders is only available to a small fraction of followers. A new predictor-based neural dynamic surface control (PNDSC) design approach is proposed to develop the adaptive containment controllers. NNs with iterative updating laws, based on the prediction errors, are used to accurately identify the unknown vehicle dynamics. Further, this result is extended to the output-feedback case, and a local neural observer is developed to recover the unmeasured velocity information. Then, distributed output-feedback containment controllers are developed, under which the containment problem can be solved regardless of model uncertainty, unknown disturbances, and unmeasured velocity information. For both cases, the stability properties are established building on Lyapunov–Krasovskii functionals, and the containment errors converge to a residual set. Comparative studies are given to illustrate the efficacy and performance improvement of the proposed scheme. The contribution of this paper is threefold. (i) In contrast to the traditional neural DSC approach (NDSC) [38,41], a new PNDSC design methodology, by combining a predictor, NNs and a DSC technique, is proposed. Iterative updating laws using prediction errors are employed to identify the unknown dynamics of AUVs, which enables accurate learning not only in steady state but also in transient state. The design methodology is an enhanced version of NDSC approach proposed for single system in [38] and for multi-agent systems in [41]. The poor learning transient using NDSC approach due to large initial tracking errors can be avoided. (ii) For the first time, this paper studies the containment problem of multiple marine vehicles over a directed network both for state- and output-feedback cases. This is obviously different from the previous adaptive controllers developed for marine vehicles in [18,32,4,14,9,29,24] being lack of sharing information. (iii) A new distributed output-feedback control architecture is developed for containment control of AUVs in the presence of model uncertainty, unknown ocean disturbances, and unmeasured velocity information, where only one NN is used in the entire design. This is also different from previous works in [15,10,26] where two NNs are constructed: One is in the control loop, and the other in the estimation loop. Hence, the proposed neural output-feedback containment controllers can be much simplified. The rest of this paper is organized as follows. Section 2 introduces some preliminaries and gives problem formulation. Section 3 presents the containment controller design together with the stability analysis. Section 4 extends the above result to the output-feedback case. Section 5 provides an example to illustrate the theoretical results. Section 6 concludes this paper.

2. Preliminaries and problem formulation 2.1. Preliminaries Throughout the paper, Rn denotes the n-dimensional Euclidean Space, and jj  jj the Euclidean norm. For a given square matrix, kðÞ, kmin ðÞ, and kmax ðÞ denote the eigenvalue, the smallest eigenvalue, and the largest eigenvalue, respectively. rðÞ is the smallest singular value of a matrix. diagfK1 ; . . . ; KN g represents a block-diagonal matrix with matrixes Ki ; i ¼ 1; . . . ; N, on its diagonal. 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 2 V  Vg. The element ðni ; nj Þ 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

Z. Peng et al. / Information Sciences 316 (2015) 163–179

165

form ðnik ; nikþ1 Þ; 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 contains 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  2 RNN associated with the graph G is defined as aij ¼ 1, if ðnj ; ni Þ 2 E; and aij ¼ 0, otherwise. In addition, self connections are not allowed, i.e., aii ¼ 0. The Laplacian matrix L associated with the graph G is defined P as L ¼ D  A where D ¼ diagfd1 ; . . . ; dN g with di ¼ Nj¼1 aij ; i ¼ 1; . . . ; N. Definition 1 [1]. A set E # Rn is convex if

kx1 þ ð1  kÞx2 2 E;

ð1Þ

whenever x1 2 E; x2 2 E, and 0 6 k  1. The convex hull CoðXÞ 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 CoðXÞ ¼ ki xi jxi 2 X; ki > 0; ki ¼ 1 : i¼1

ð2Þ

i¼1

2.2. Problem formulation Consider a multi-vehicle system consisting of M followers, labeled as AUV 1 to M. Neglecting their dynamics in heave, roll, and pitch, and assuming that each AUV has independent depth control, the dynamics of AUVs in the horizontal plane can be described as follows [11]



g_ i ¼ Rðwi Þmi ; M i m_ i ¼ C i ðmi Þmi  Di ðmi Þmi  g i ðgi Þ þ si þ siw ðtÞ;

where

2

cos wi 6 Rðwi Þ ¼ 4 sin wi 0

 sin wi cos wi 0

0

ð3Þ

3

7 0 5;

ð4Þ

1

gi ¼ ½xi ; yi ; wi T is a vector of the earth-fixed position-yaw information; mi ¼ ½ui ; v i ; ri T is a vector of body-fixed surge, sway and yaw velocities; M i ¼ MTi 2 R33 ; C i ðmi Þ 2 R33 ; Di ðmi Þ 2 R33 denote the inertia matrix, coriolis/centripetal matrix, and damping matrix, respectively; g i ðgi Þ 2 R3 is the restoring forces and moments; si ¼ ½siu ; siv ; sir T 2 R3 denotes the control force; siw ¼ ½siwu ; siwv ; siwr T 2 R3 is the disturbance vector. In the following design, C i ðmi Þ; Di ðmi Þ; g i ðgi Þ; siw ðtÞ are unknown, which means that the AUV dynamics are heterogeneous. Consider N  M ðN > 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 /i where i ¼ M þ 1; . . . ; N. The followers 1 to M have at least one neighbor, and the leaders 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. Since the leaders have no neighbors, L can be represented by

L¼



L1

L2

0ðNMÞM

0ðNMÞðNMÞ



;

ð5Þ

where L1 2 RMM and L2 2 RMðNMÞ . Assumption 1. For each follower, there exists at least one leader that has a directed path to that follower. Lemma 1. [20,41]. Under Assumption 1, all the eigenvalues of L1 have positive real parts, each entry of L1 1 L2 has a sum equal to 1. The control objective is to design a distributed adaptive control law si for each AUV with dynamics (3) such that the output trajectory of each AUV nearly converge to the convex hull spanned by the leaders, i.e.,

limkgi  hðtÞk < d;

ð6Þ

t!1

where i ¼ 1; . . . ; M; hðtÞ 2 Cof/Mþ1 ðtÞ; . . . ; /N ðtÞg, and d is a positive constant. T

3 Define /ðtÞ ¼ ½/TMþ1 ðtÞ; . . . ; /TN ðtÞ and /d ðtÞ ¼ ½/d1 ðtÞ; . . . ; /dM ðtÞ ¼ ðL1 1 L2  I 3 Þ/ðtÞ where /di ðtÞ 2 R . By Lemma 1 and Definition 1, we obtain the output trajectory of each AUV nearly converge to the convex hull spanned by the leaders provided that

limkgi  /di ðtÞk < d ;

t!1

where d is a positive constant.

ð7Þ

166

Z. Peng et al. / Information Sciences 316 (2015) 163–179

3. Containment control via state-feedback In this section, we consider the state-feedback case by imposing that both the position-yaw and velocity information are measurable. A PNDSC design approach, by combining a predictor, NNs, and a DSC technique, is proposed to develop the distributed adaptive containment controllers, under which the containment errors converge to a residual set. 3.1. Controller design Step 1. At first, we define two dynamic surface errors as follows

zi1 ¼ RT ðwi Þ

( M X

aij ðgi  gj Þ þ

j¼1

N X

) aij ðgi  /j Þ ;

j¼Mþ1

zi2 ¼ mi  mid ; i ¼ 1; . . . ; M; where aij is defined in Section 2;

mid 2 R3 is a filtered control signal to be specified later.

Remark 1. Compared with the traditional tracking error introduced for single vehicle [32,4,14,9,29]

zi1 ¼ RT ðwi Þðgi  /r Þ;

ð8Þ

where /r is a reference trajectory, here, graph-induced distributed tracking errors zi1 decomposed in the body-fixed reference frame are introduced to solve the containment problem of AUVs. Taking the time derivative of zi1 along the first equation of (3) yields

z_ i1 ¼ r i Szi1 þ di mi 

M N X X aij RT ðwi ÞRðwj Þmj  aij RT ðwi Þ/_ j ; j¼1

ð9Þ

j¼Mþ1

where di is defined in Section 2; S is defined as

2

3 0 1 0 6 7 S ¼ 4 1 0 0 5: 0 0 0

ð10Þ

To stabilize zi1 , a virtual control law ai1 based on the relative position-yaw information and the velocity information of neighboring AUVs is proposed as follows

1 ai1 ¼ di

(

) M N X X T T _ ki1 zi1 þ aij R ðwi ÞRðwj Þmj þ aij R ðwi Þ/j ; j¼1

ð11Þ

j¼Mþ1

where ki1 ¼ diagfki11 ; ki12 ; ki13 g with ki11 2 R; ki12 2 R; ki13 2 R being positive constants. In what follows, let ai1 pass through a first-order bank filter to obtain the filtered signal

mid as

ci1 m_ id ¼ ai1  mid ; mid ð0Þ ¼ ai1 ð0Þ;

ð12Þ

where ci1 2 R is a time constant. Step 2. Taking the time derivative of zi2 along the second equation of (3) produces

Mi z_ i2 ¼ si  f i ðni ; tÞ  Mi m_ id ; where

f i ðni ; tÞ ¼ C i ðmi Þmi þ Di ðmi Þmi þ g i ðgi Þ  siw ðtÞ;

ð13Þ

T

and ni ¼ ½1; gTi ; mTi  . If the perfect knowledge on f i ðni ; tÞ is available, a desired control law

si

¼ ki2 zi2 þ f i ðni ; tÞ þ Mi m_ id ;

si is proposed as follows ð14Þ

where ki2 ¼ diagfki21 ; ki22 ; ki23 g with ki21 2 R; ki22 2 R; ki23 2 R being positive constants. However, the accurate knowledge of f i ðni ; tÞ may not be available in practice, and thus additional scheme should be developed. To move on, the following assumption is required. Assumption 2. The nonlinear function f i ðni ; tÞ can be parameterized by

f i ðni ; tÞ ¼ W Ti ðtÞui ðni Þ; 8ni 2 X;

ð15Þ

Z. Peng et al. / Information Sciences 316 (2015) 163–179

167

where W i ðtÞ is an unknown time-varying matrix satisfying kW i ðtÞkF 6 W i with W i 2 R being a positive constant;

ui ðni Þ : X ! Rs is a known vector function defined by ui ðni Þ ¼ ½bi ; ui1 ðni Þ; ui2 ðni Þ; . . . ; uiðs1Þ ðni ÞT satisfying kui k 6 ui with ui being a positive constant; X is a compact set. Remark 2. Assumption 2 includes broader classes of uncertainties than the commonly used approximation f i ðni Þ ¼ W Ti ui ðni Þ þ eðni Þ, where W i is an constant ideal weight matrix and eðni Þ is the approximation error, due to the fact the time dependent variations are permitted in the unknown ideal weight matrix. Note that if the NNs are replaced by fuzzy logic systems [35,5,36,34,37,33], similar results can be derived without any difficulty. Then, a practical control law is constructed as follows

c T u ðni Þ; si ¼ ki2 zi2 þ Mi m_ id þ W i i

ð16Þ

c i is an estimate of W i . where W c i , consider a predictor as To design the update law for W

c T u ðni Þ þ si  ðki2 þ 1 Þm ^_ i ¼  W ~ ^ Mi m i i ; mi ð0Þ ¼ mid ð0Þ; i i ~i ¼ m ^i  mi and where m

1i ¼ diagf1i11 ; 1i12 ; 1i13 g with c i ðtÞ is designed as The update law for W

ð17Þ

1i11 2 R; 1i12 2 R; 1i13 2 R being positive constants.

c i ðtÞ ¼ ji W c i ðt  t d Þ þ Ci u ðni Þm ~Ti ; W i

ð18Þ

where t d 2 R is a positive updating interval. ji 2 Rss and Ci 2 R satisfy 0 < jTi ji < j I with 0 < j < 1, and Ci > 0, respectively. From (11) (16) and (17), the error dynamics of the closed-loop network can be written as

8 ~i þ qi1 Þ; z_ ¼ ki1 zi1  r i Szi1 þ di ð^zi2  m > < i1 T _ f ~ ~ M i mi ¼ ðki2 þ 1i Þmi  W i ui ðni Þ; > : _ ~; M ^z ¼ k ^z  1 m i i2

i2 i2

ð19Þ

i i

^i  mid . where qi1 ¼ mid  ai1 and ^zi2 ¼ m Taking the time derivative of qi1 gives

q_ i1 ¼ 

qi1

ci1

 a_ i1 :

ð20Þ

By integration of (20), it follows that c t

qi1 ðtÞ ¼ e

i1

qi1 ð0Þ 

Z

t

e

s t c i1

a_ i1 ðsÞds;

ð21Þ

0

from which we can compute an upper bound for qi1 as

kqi1 ðtÞk 6 e

c t

i1

kqi1 ð0Þk þ ci1 ai1 ;

ð22Þ

where ka_ i1 k1 6 ai1 with ai1 being a positive constant. Since the energy to drive the AUV is limited and the derivatives of the reference trajectory are bounded, the boundedness of a_ i1 is naturally satisfied for marine vehicles. Then, there exists a positive constant qi1 such that kqi1 ðtÞk 6 qi1 . ~i , Remark 3. The main difference between the above design and the NDSC method [38,41] is that the prediction errors m c instead of the dynamic surface errors zi2 , are used to update the NN parameters W i . By substituting the control law (16) into the Eq. (13), we derive that

f T u ðni Þ: Mi z_ i2 ¼ ki2 zi2  W i i

ð23Þ

~i from (19), we can observe that m ~i converges faster than zi2 by choosThus, compared with the prediction error dynamics of m ~i is almost the same as zi2 . This freedom to select the ing a positive parameter 1i . In fact, when setting 1i ¼ 0, the response of m parameter 1i aids in improving the transient behavior of the NN-based DSC controllers, which will be demonstrated in the simulation part.

168

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Remark 4. Note that the predictor design (17) is independent of the controller design, since there is no need to care about what kind of control input si is applied. This provides additional avenues to identify the unknown dynamics for AUVs, and the advantage is twofold. First, the identification process will not be affected by the control objective, since large error con^i ð0Þ ¼ mid ð0Þ, the undesirable tranditions in the initial stage may deteriorate the learning process. Second, by initializing m sient learning errors due to the initialization errors can be avoided. To the best of our knowledge, it is the first attempt to address the transient behavior of NN-based DSC design. Remark 5. The iterative updating laws (18) are advantageous over the integrator-type adaptive laws, due to the fact that they are easier to implement in digital processors since algebraic equations are used for updating the NN weights. In implementations, td can be chosen as the sampling-time interval in a sampled-data control system, or as an integer multiple of the sampling-time intervals. Remark 6. In [23], the altitude containment control problem of rigid bodies is considered. Note that the altitudes of the rigid bodies are transformed into Lagrange form; while in our paper, the rigid body dynamics of AUV is expressed in body-fixed reference frame other than Lagrange form. Hence, the controller design method given in [23] cannot be applied in our case.In [44], containment control of multi-agent systems with measurement noises is discussed where the disturbances exist in the communication channel; while in this paper, the disturbances exist in the AUV dynamics, and the proposed method is able to compensate for the disturbances actively. 3.2. Stability analysis In this subsection, the stability of the closed-loop network will be analyzed. To begin with, we state the first result of this paper. Theorem 1. Consider the closed-loop network consisting of the AUV dynamics (3), the control law (16), the adaptive law (18), the first-order filter (12), together with the predictor (17) under Assumptions 1,2. Then, the error signals in the closed-loop network are uniformly ultimately bounded (UUB), and the containment errors between the follower AUVs and the leaders converge to a residual set. Proof 1. Define the following Lyapunov–Krasovskii functional

( ) Z t N 1X T T T T f f ^ ^ ~ ~ V1 ¼ z zi1 þ zi2 M i zi2 þ mi M i mi þ qi trð W i ðsÞ W i ðsÞÞds ; 2 i¼1 i1 ttd

ð24Þ

where qi 2 R is a positive constant. Its time derivative along (19) can be expressed by

V_ 1 6

N n X

f T ðtÞu ðni Þ ~i þ qi1 Þ  ^zTi2 ki2 ^zi2  ^zTi2 1i m ~i  m ~Ti ð1i þ ki2 Þm ~i  m ~Ti W zTi1 ki1 zi1 þ di zTi1 ð^zi2  m i i

i¼1

h io f T ðtÞ W f i ðtÞ þ l W f T ðtÞ W f i ðtÞ  W f T ðt  t d Þ W f i ðt  td Þ ; þ qi tr .i W i i i i where

ð25Þ

li ¼ 1 þ .i . Define ^ ¼W c i ðtÞ  ji W c i ðt  td Þ; P ¼ W i ðtÞ  ji W i ðt  t d Þ; P

ð26Þ

and it follows that

f i ðtÞ ¼ ji W f i ðt  t d Þ þ P ^  P: W

ð27Þ

Substituting (27) into (25) produces

V_ 1 6

N n X

f T ðt  td ÞjT u ðni Þ ~i þ qi1 Þ  ^zTi2 ki2 ^zi2  ^zTi2 1i m ~i  m ~Ti ð1i þ ki2 Þm ~i  m ~Ti W zTi1 ki1 zi1 þ di zTi1 ð^zi2  m i i i

i¼1

n ^ T u ðni Þ þ m f T ðtÞ W f i ðtÞ þ l ½P ^ TP ^ þ PT P þ W f T ðt  t d ÞjT ji W f i ðt  t d Þ ~Ti P ~Ti PT ui ðni Þ þ qi tr .i W m i i i i i oo ^ T P þ 2P ^TW f i ðt  t d Þ  2PT ji W f i ðt  t d Þ  W f T ðt  td Þ W f i ðt  t d Þ :  2P i Using Young’s inequality, we have

f i ðt  td Þ 6 tr tr½2li PT ji W



l2i

”i0



f T ðt  t d ÞjT ji W f i ðt  t d Þ; P2 þ tr½”i0 W i i

ð28Þ

169

Z. Peng et al. / Information Sciences 316 (2015) 163–179

where ”i0 > 0 and kPkF 6 P with P ¼ W  ð1 þ kji kÞ. Letting qi li ¼ C1i , the inequality (28) can be put into

V_ 1 6

N n X f T ðtÞ W f i ðtÞ ~i þ qi1 Þ  ^zTi2 ki2^zi2  ^zTi2 1i m ~i  m ~Ti ð1i þ ki2 Þm ~i  qi .i tr½ W zTi1 ki1 zi1 þ di zTi1 ð^zi2  m i i¼1

f T ðt  td Þ½Is  ðl þ ”i0 ÞjT ji  W f i ðt  t d Þ þ q ðl þ  qi tr½ W i i i i i

l2i ”i0

 ÞP2 :

ð29Þ

2 2 2 kmax ð1i Þ”i1 T ^ 1 1 ^ T ~ 1 1 ~ 2 ^T ~ Further, noting that jzTi1 qi1 j 6 12 kzi1 k2 þ 12 q2 k^zi2 k2 þ i1 , jzi1 zi2 j 6 2 kzi1 k þ 2 kzi2 k , jzi1 mi j 6 2 kzi1 k þ 2 kmi k , zi2 1i mi 6 2 kmax ð1i Þ k~i k2 2”i1

m

with ”i1 > 0, it follows that

     N   X 3di T di kmax ð1i Þ”i1 T k ð1 Þ T ^zi2^zi2  kmin ð1i þ ki2 Þ  max i m ~i ~i m zi1 zi1  kmin ðki2 Þ    kmin ðki1 Þ  2”i1 2 2 2 i¼1 o f T ðtÞ W f i ðtÞ  q trf W f T ðt  t d Þ½Is  ðl þ ”i0 ÞjT ji  W f i ðt  t d Þg þ 1 ;  qi .i tr½ W i i i i i

V_ 1 6

where

1 ¼

PN ndi i¼1

2

o l2i 2 . q2 i1 þ qi ðli þ ” ÞP i0

Define and let

8 hi11 ¼ kmin ðki1 Þ  3d2 i > 0; > > > > > di kmax ð1i Þ”i1 > > 0; > 2 < hi12 ¼ kmin ðki2 Þ  2 

ð1i Þ hi13 ¼ kmin ðki2 þ 1i Þ  kmax > 0; 2”i1 > > > > > hi14 ¼ qi .i > 0; > > : hi15 ¼ qi kmin ½Is  ðli þ ”i0 ÞjTi ji  > 0;

ð30Þ

and then (30) can be expressed as

V_ 1 6 

M n o X f i ðtÞk2 þ hi15 k W f i ðt  t d Þk2 þ 1 ; ~Ti m ~i þ hi14 k W hi11 zTi1 zi1 þ hi12 ^zTi2 ^zi2 þ hi13 m F F

ð31Þ

i¼1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i ðtÞk > 1 = ~i k > 1 = Observing that either kzi1 k > 1 = hi11 , or k^zi2 k > 1 = hi12 , or km hi13 , or k W hi14 , or F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i ðt  t d Þk > 1 = _ < 0. It follows that the error signals of zi1 ; ^zi2 ; m f i ðtÞ; W f i ðt  t d Þ are UUB. Noting the fact ~ kW , renders ; h V W i15 i F ~i , it follows that zi2 is bounded. that zi2 ¼ ^zi2  m n qffiffiffiffiffiffio n qffiffiffiffiffiffio n qffiffiffiffiffiffio 1 1 _ ~i k 6 h1 \ \ pi : k^zi2 k 6 hi12 \ pi : km Note that VðtÞ < 0 outside the compact set Sa ¼ pi : kzi1 k 6 hi11 i13 n qffiffiffiffiffiffio n qffiffiffiffiffiffio 1 1 f i ðt  td Þk 6 f i ðtÞk 6 c c ^ ~ \ p , where p z pi : k W : k ¼ ½z ; ; m ; ; ðt  t Þ. Hence, VðtÞ cannot grow outside W W W i1 i2 i i i d i i F F   hi15 hi14 the set Sa due to the fact that VðtÞ is non-decreasing. It follow that VðtÞ is upper bounded by

VðtÞ 6 maxVðtÞ; t 0:

ð32Þ

pi 2Sa

The definition of the Lyapunov function VðtÞ yields

( ) Z t N 1X T T T T f f ~i M i m ~i þ qi VðtÞ ¼ z zi1 þ ^zi2 M i ^zi2 þ m trð W i ðsÞ W i ðsÞdsÞ ; 2 i¼1 i1 ttd ( ) Z t N 1X f T ðsÞ W f i ðsÞdsÞ ; ~i k2 þ qi 6 kzi1 k2 þ kmax ðM i Þk^zi2 k2 þ kmax ðMi Þkm trð W i 2 i¼1 tt d   N 1X 1 kmax ðMi Þ1 kmax ðMi Þ1 qi td 1 6 : þ þ þ hi12 hi13 hi14 2 i¼1 hi11 By selecting parameters such that

o PN n 1 kmax ðM i Þ1 d 1 i Þ1 < 2-1 with þ kmaxhðM þ qhi ti14 i¼1 h  i11 þ  hi12 i13

-1 > 0, it follows that VðtÞ 6 -1 is an

invariant set, i.e., if Vð0Þ 6 -1 ; then VðtÞ 6 -1 for all t > 0. T

Defining z1 ¼ ½zT11 ; . . . ; zTN1  , and noting that 2

kz1 k 6 VðtÞ; 2

ð33Þ

it follows that z1 is ultimately bounded by

kz1 k 6 where #1 ¼

pffiffiffiffiffiffiffiffiffiffi #1 1 :

PN

1 i¼1 f hi11

ð34Þ ðM i Þ ðM i Þ þ kmax þ kmax þ qh i td g.   h h i12

i13

i14

170

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Observing that

z1 ¼ R½ðL1  I3 Þg þ ðL2  I3 Þ/

ð35Þ

with R ¼ diagðRðw1 Þ; . . . ; RðwN ÞÞ, it follows that

R1 z1 ¼ ðL1  I3 Þe;

ð36Þ

where e ¼ ½eT1 ; . . . ; eTN  and ei ¼ gi  /di ðtÞ. Then, the containment error e is ultimately bounded by

kek 6

pffiffiffiffiffiffiffiffiffiffi #1 1 ; kmin ðL1 Þ

ð37Þ

implying (6). The proof is complete. Remark 7. The containment error can be reduced by appropriately choosing the control parameters.

4. Containment control via output-feedback In the preceding section, the velocity information are assumed to be measurable for the feedback design. However, the velocity information may not be available for saving the sensor costs or due to the sensor faults. Therefore the purpose of this section is to present an output-feedback design to solve the containment problem of AUVs in the presence of model uncertainty, unknown ocean disturbances, and unmeasured velocity information. For simplicity, we continue to use the notations defined for the state-feedback case. If they need to be changed, we redefine them explicitly. 4.1. Predictor design based on output information Since only the position-yaw information can be obtained and the AUV kinetic dynamics are totally unknown, the predictors to be designed has two objectives. One is to reconstruct the unmeasurable velocity information; and the other is to identify the unknown dynamics for each AUV. In order to facilitate the predictor design, rewrite the AUV dynamics as follows

(

g_ i ¼ Rðwi Þmi ; m_ i ¼ M1 i ½si  f i ðni ; tÞ:

ð38Þ

^i be the estimates of gi and ^ i and m Let g

(

mi , respectively. Then, a predictor is constructed as

g^_ i ¼ K i1 g~ i þ Rðwi Þm^i ; cT ^ m^_ i ¼ K i2 RT ðwi Þg~ i þ M1 i ½si  W i ðtÞuðni Þ;

ð39Þ

^Ti T ; g ~i ¼ g ^ Ti ; m ^ i  gi ; K i1 2 R33 and K i2 2 R33 are positive constant matrices, which are designed to commute where ^ ni ¼ ½1; g with the rotation Rðwi Þ. ~i ¼ m ^i  mi , and the resulting error dynamics in terms of g ~i can be described as ~ i and m Let m

(

g~_ i ¼ K i1 g~ i1 þ Rðwi Þm~i ; fT ^ m~_ i ¼ K i2 RT ðwi Þg~ i1 þ M1 i ½ W i ðtÞuðni Þ þ ei0 ;

ð40Þ

where ei0 ¼ W Ti ðtÞ½uðni Þ  uð^ ni Þ satisfying kei0 k 6 ei0 with reshaped into

ei0 being a positive constant. The error dynamics (40) can be

e_ i ¼ Ai X e i þ Bi ½ W f T ðtÞuð^ni Þ þ ei0 ; g e i; ~ i ¼ C i0 X X i where T e i ¼ ½g ~Ti  ; Ai ¼ ~ Ti ; m X



K i1 T

K i2 R ðwi Þ

Rðwi Þ 0



; Bi ¼

ð41Þ 

0 M 1 i



; C i0 ¼ ½I3 ; 0:

ð42Þ

Since Ai contains wi which is time-varying, it can not be directly used to analyze the stability of the predictor. In order to eliminate its dependence on the state wi , define a block-diagonal non-singular transformation matrix

T i ¼ diagðRT ðwi Þ; I3 Þ;

ð43Þ

using which we obtain

(

e i þ Bi ½ W f T ðtÞuð^ni Þ þ ei0 ; e_ i ¼ T T Ai0 T i X X i i g~ i ¼ C i0 Xe i ;

ð44Þ

Z. Peng et al. / Information Sciences 316 (2015) 163–179

171

where

 Ai0 ¼

K i1

I3

K i2

0

 :

Ai0 can be made Hurwitz by choosing parameters K i1 and K i2 . e i and letting ST ¼ diagfST ; 0g, we have Using the mapping Z i ¼ T i X

f T ðtÞuð^ni Þ þ ei0 : Z_ i ¼ ðAi0 þ r i ST ÞZ i þ Bi ½ W i

ð45Þ

The above mapping produces a constant matrix Ai0 ; however, we have to deal with the time-varying signal ri additionally. Physically, ri describes the yaw rate of AUV, and this quantity is bounded provided that appropriate control is applied. Therefore, if a set of simultaneous Lyapunov inequalities are satisfied at the minimum and maximum of r i , the stability of the predictor can be established. Let jr i j 6 ri with r i 2 R being a positive constant. Then, consider the following simultaneous Lyapunov inequalities

8 T T T >  T > < Ai0 Pi þ Pi Ai0 þ Q i þ P i BB Pi þ ‘i Ei Ei þ r i ðST Pi þ Pi ST Þ < 0; ATi0 Pi þ Pi Ai0 þ Q i þ P i BBT Pi þ ‘i Ei ETi  r i ðSTT Pi þ Pi ST Þ < 0; > > : Ei ¼ C Ti  Pi Bi ;

ð46Þ

where Q i is positive definite; ‘i 2 R is a positive constant. The following lemma establishes the stability property of the predictor. c i be given by Lemma 2. Consider the system defined by the plant (3) and the predictor (39). Let the update law for W

c i ðtÞ ¼ ji W c i ðt  t d Þ  Ci uð^ni Þg ~ Ti Rðwi Þ: W

ð47Þ

f i ðtÞ are UUB. ~i and W ~i ; m Then, for bounded initial conditions, the error signals of g Proof 2. Define the following Lyapunov–Krasovskii functional

( ) Z t 1 T T f f V i0 ¼ Z i P i Z i þ qi trð W i ðsÞ W i ðsÞdsÞ : 2 tt d

ð48Þ

Taking the time derivative of V i0 along (45) gives

1 f T uð^ni Þ V_ i0 ¼ Z Ti ½P i Ai0 þ ATi0 Pi þ ri ðPi ST þ STT Pi ÞZ i þ Z Ti P i Bi ei0  Z Ti Ei W i 2

f T ðtÞ W f i ðtÞ þ l W f T ðtÞ W f i ðtÞ  W f T ðt  td Þ W f i ðt  t d Þ ; þ qi tr .i W i i i i

ð49Þ

Using

8 < :

eT ^ T ^ e f T uð^ni Þ 6 ‘i Z T Ei ET Z i þ W i uðni Þu ðni Þ W i ; Z Ti Ei W i i 2 i 2‘i

ð50Þ

Z Ti Pi Bi ei0 6 12 Z i Pi Bi BTi Pi Z i þ 12 e2 i0 ;

where ‘i is defined in (46), it follows that

u2 f T f 1 1 V_ i0 6 Z Ti ½Pi Ai0 þ ATi0 Pi þ r i ðPi ST þ STT Pi Þ þ Pi Bi BTi Pi þ ‘i Ei ETi þ Q i Z i  Z Ti Q i Z i  ðqi .i  i Þtr½ W i ðtÞ W i ðtÞ 2 2 2‘i f T ðt  td Þ½Is  ðl þ ”i0 ÞjT ji  W f i ðt  td Þg þ 2  qi trf W i i i where

2 ¼

PM

i¼1

n

l2

ð51Þ

o

e þ qi ðli þ ”i0i ÞP2 .

1 2 2 i0

Note that Z Ti ½Pi Ai0 þ ATi0 P i þ ri ðP i ST þ STT Pi Þ þ P i Bi BTi P i þ ‘i Ei ETi þ Q i Z i is a convex function of r i . Thus, the inequality (46) is equivalent to Z Ti ½Pi Ai0 þ ATi0 P i þ P i Bi BTi P i þ r i ðP i ST þ STT Pi Þ þ ‘i Ei ETi þ Q i Z i < 0, which leads to

f i ðtÞk2  h23 k W f i ðt  td Þk2 þ 2 ; V_ i0 6  h21 kZ i k2  h22 k W F F

ð52Þ

8 hi21 ¼ 12 kmin ðQ i Þ > 0; > > < u2 hi22 ¼ qi .i  2ji > 0; i > > : hi23 ¼ qi kmin ½Is  ðli þ ”i0 ÞjTi ji  > 0;

ð53Þ

where

172

Z. Peng et al. / Information Sciences 316 (2015) 163–179

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i ðtÞk > 2 = f i ðt  td Þk > 2 = Note that either kZ i k > 2 = hi21 , or k W hi22 , or k W hi23 renders V_ i0 < 0. It follows that the F F f i ðtÞ; W f i ðt  td Þ are UUB. ~i ; W ~i; m errors signals of g Remark 8. The proposed predictor does not require the strictly positive real (SPR) assumption on system fAi0 ; Bi ; C i g. 4.2. Controller design ^i , we have Step 1. Taking the time derivative of zi1 and using the predicted state of m

^i  z_ i1 ¼ r i Szi1 þ di m

M N M X X X ^j  ~i þ ~j : aij RT ðwi ÞRðwj Þm aij RT ðwi Þ/_ j  di m aij RT ðwi ÞRðwj Þm j¼1

j¼Mþ1

ð54Þ

j¼1

A virtual control law ai2 based on the relative position-yaw and predicted velocity information of neighboring AUVs is proposed as follows

ai2 ¼

1 di

( ki1 zi1 þ

) M N X X ^j þ aij RT ðwi ÞRðwj Þm aij RT ðwi Þ/_ j : j¼1

ð55Þ

j¼Mþ1

Let ai2 pass through a first-order bank filter to obtain the filtered control signal

mid as follows

ci2 m_ id þ mid ¼ ai2 ; ai2 ð0Þ ¼ mid ð0Þ;

ð56Þ

where ci2 2 R is a time constant. Step 2. In light of (40), the dynamics of the second surface tracking error ^zi2 can be written as

c T ðtÞuð^ni Þ  Mi m_ id : ~ i þ si þ W Mi ^z_ i2 ¼ M i K i2 RT ðwi Þg i Then, a practical controller

si

ð57Þ

si is proposed as follows

c T ðtÞuð^ni Þ þ M i m_ id þ M i K i2 RT ðw Þg ~i: ¼ ki2^zi2  W i i

ð58Þ

The resulting error dynamics for zi1 and ^zi2 can be described by

8 M X > > < z_ i1 ¼ ki1 zi1  r i Szi1 þ di ð^zi2 þ qi2 Þ  di m ~i þ ~j ; aij RT ðwi ÞRðwj Þm > > :

j¼1

ð59Þ

Mi ^z_ i2 ¼ ki2 ^zi2 ;

where qi2 is redefined as qi2 ¼ mid  ai2 . Taking the time derivative of qi2 gives

q_ i2 ¼ 

qi2

ci2

 a_ i2 ;

from which we have c t

qi2 ðtÞ ¼ e

i2

qi2 ð0Þ 

Z

t

s t c

e

i2

a_ i2 ðsÞds:

ð60Þ

0

Then, there exists a positive constant qi2 such that kqi2 ðtÞk 6 qi2 . 4.3. Stability analysis It is the position to state the second result of this paper. Theorem 2. Consider the closed-loop system consisting of the AUV dynamics (3), the control law (58), the adaptive law (47), the first-order filter (12), together with the predictor (39) under Assumptions 1,2. Then, the error signals in the closed-loop system are UUB, and the containment errors between the follower AUVs and the leaders converge to a residual set. Proof 3. Define the following Lyapunov function

V2 ¼

N  T 1X z zi1 þ ^zTi2 Mi ^zi2 þ V i0 : 2 i¼1 i1

ð61Þ

Z. Peng et al. / Information Sciences 316 (2015) 163–179

173

Its time derivative along (59) satisfies N X

(

N X 1 ~j  kmin ðQ i ÞZ Ti Z i aij zTi1 RT ðwi ÞRðwj Þm 2 i¼1 j¼1    2 u f T ðtÞ W f i ðtÞ  q trf W f T ðt  t d Þ½Is  ðl þ ”i0 ÞjT ji  W f i ðt  t d Þg þ 2 :  qi .i  i tr½ W i i i i i 2‘i

V_ 2 6

~i þ kmin ðki1 ÞzTi1 zi1 þ di zTi1 ð^zi2 þ qi2 Þ  kmin ðki2 Þ^zTi2^zi2  di zTi1 m

ð62Þ

~i 6 12 zTi1 zi1 þ 12 m ~Ti m ~i ; zTi1 RT ðwi ÞRðwj Þm ~j 6 12 zTi1 zi1 þ 12 m ~Tj m ~j , we have Using the inequalities zTi1 m

    N  X di T 1 1 ^zi2 ^zi2  kmin ðQ i Þg ~Ti m ~i ~ Ti g ~i  ½kmin ðki1 Þ  2di zTi1 zi1  kmin ðk2 Þ  kmin ðQ i Þ  di m 2 2 2 i¼1    u2 f T ðtÞ W f i ðtÞ  q trf W f T ðt  t d Þ½Is  ðl þ ”i0 ÞjT ji  W f i ðt  t d Þg þ 3 :  qi .i  i tr½ W i i i i i 2‘i

V_ 2 6

where

3 ¼

PN ndi i¼1

2

1 2 q2 i2 þ 2 ei0 þ qi



2



ð63Þ

o

li þ ”li0i P2 . h

Define and let

8 hi21 > > > > > > hi22 > > > > < hi23 hi24 > > > > > > hi25 > > > > : hi26

¼ kmin ðki1 Þ  2di > 0; ¼ kmin ðki2 Þ  d2i > 0; ¼ kmin ðQ i Þ > 0; ¼ kmin ðQ i Þ  di > 0; ¼ qi .i 

u2 i 2‘i

ð64Þ

> 0;

¼ qi kmin ½Is  ðli þ ”i0 ÞjTi ji  > 0;

and we have

V_ 2 6 

N n o X f T ðtÞ W f i ðtÞÞ þ hi26 trð W f T ðt  td Þ W f i ðt  td ÞÞ þ 3 : ~Ti m ~i þ hi25 trð W ~ Ti g ~ i þ hi24 m hi21 zTi1 zi1 þ hi22 ^zTi2 ^zi2 þ hi23 g i i i¼1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i ðtÞk > 3 = ~i k > 3 = ~ i k > 3 = Note that either kzi1 k > 3 = hi21 , or kzi2 k > 3 = hi22 , or kg hi23 , or km hi24 , or k W hi25 , or F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i ðt  t d Þk > 3 = f i ðtÞ; W f i ðt  t d Þ are UUB. ~i ; W ~i; m kW hi26 , renders V_ 2 < 0. It follows that the error signals of zi1 ; zi2 ; g F Following the same steps in proving Theorem 1, we can derive that z1 and e are ultimately bounded by

pffiffiffiffiffiffiffiffiffiffi #2 3 ;

ð65Þ

pffiffiffiffiffiffiffiffiffiffi #2  3 ; kmin ðL1 Þ

ð66Þ

kz1 k 6 and

kek 6 where #2 ¼

PN n i¼1

1  hi21

o ðM i Þ ðPi Þ ðP i Þ i td . The proof is complete. þ kmax þ kmax þ kmax þ qhi25    hi22 hi23 hi24

Remark 9. A distinguished feature of the proposed output-feedback scheme is that only one NN is used entire design, which is in contrast to the previous works [15,10,26] where two NNs, with one being in control loop, and the other in estimation loop, are used. The main advantage is that the AUV dynamics identification process is separated from the controller design, which is capable of producing an accurate learning.

5. An example To illustrate the control performance of the proposed method, a networked system consisting of five follower AUVs and three leaders is considered. The model parameters for each AUV are given in Table 1. In addition, some time-varying ocean disturbances are introduced. They are modeled as the first-order Gauss-Markov processes s_ iwu þ ciu siwu ¼ w1 , s_ iwv þ civ siwv ¼ w2 ; s_ iwr þ cir siwr ¼ w3 , where ciu > 0; civ > 0; cir > 0 and w1 ; w2 ; w3 are white noises. 5.1. State feedback The information exchange topology among the five AUVs and three leaders is given by Fig. 1. The NN activation x . The NN learning rates CiW can be chosen as fast as desired, and the transient learning function is chosen as 1e 1þex profile can be regulated by properly choosing 1i . Accordingly, the control parameters are selected as

174

Z. Peng et al. / Information Sciences 316 (2015) 163–179 Table 1 Model parameters. Parameters

Value

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

200 250 80 250v 200u 70 þ 100juj 100 þ 200jv j 50 þ 100jrj

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

Fig. 1. Communication topology.

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

ki1 ¼ diagf5; 5; 5g; ki2 ¼ diagf2000; 2500; 800g; 1i ¼ diagf2000; 2500; 800g; CiW ¼ 10000, kW ¼ 0:01; ci1 ¼ 0:02. For comparisons, the same adaptive parameters are taken for the direct adaptive laws. Simulation results are provided in Figs. 2–6. Fig. 2 shows the geometric formation of the five AUVs projected in 2D plane. It can be seen that the five AUVs converge to the convex hull spanned by the three leaders. Fig. 3 plots the norms of the surface tracking errors zi1 . It reveals that the error signals converge to a small neighborhood of origin. The unknown AUV c T u ðni Þ dynamics including model uncertainty and ocean disturbances and the outputs of NNs uiad ¼ ½uu ; uv ; ur T ¼ W iad

iad

iad

i

i

using NDSC and PNDSC approaches, corresponding to the first AUV, are depicted in Figs. 4 and 5, respectively. It can be seen that the PNDSC approach performs better than NDSC approach in transient learning, with smaller approximation errors. Fig. 6 demonstrates the boundedness of control signals. 5.2. Output feedback In this section, the containment controllers provided in Theorem 2 are applied to the control of the network. The parameters for the predictor are selected as

2

21:9

6 K i1 ¼ 4 0 0

0 21:8 0

0

3

2

7 6 0 5; K i2 ¼ 4 21:8

118:1

0

0

0

115:5

0

0

0

116:0

3 7 5;

such that kðAi0 Þ ¼ f10; 10; 10; 12; 12; 12g. The parameters for NNs are taken as CiW ¼ 10000; kW ¼ 0:01.

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Fig. 3. Norms of surface tracking errors zi1 .

Fig. 4. Uncertainty and disturbances (dotted line), outputs of NNs (solid line) using NDSC approach.

Fig. 5. Uncertainty and disturbances (dotted line), outputs of NNs (solid line) using PNDSC approach.

175

176

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Fig. 6. Control inputs for AUVs.

Fig. 7. Formation pattern projected in 2D plane.

Fig. 8. Norms of surface tracking errors zi1 .

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Fig. 9. Comparisons of observed states and velocities for AUV 1.

Fig. 10. Uncertainty and disturbances (dotted line), outputs of NNs (solid line) using PNDSC approach.

Fig. 11. Control inputs for AUVs.

177

178

Z. Peng et al. / Information Sciences 316 (2015) 163–179

Fig.

7

shows

the

geometric

formation

pattern

of

the

five

AUVs

with

the

time-varying

trajectory

g0 ¼ ½0:1t; 6sinð0:1tÞ; atan2ð0:6cosð0:1tÞ; 0:1ÞT . Fig. 8 plots the norms of the surface tracking errors zi1 that converge to a small neighborhood of origin. Fig. 9 shows that unknown velocity information can be recovered by the proposed observer. Fig. 10 demonstrates that the unknown dynamics can be learned by NN not only in steady state but also in transient state. Fig. 11 shows the control signals of the five AUVs. 6. Conclusions This paper addressed the containment control of multiple AUVs in the presence of model uncertainty and time-varying ocean disturbances both for state- and output-feedback cases. A PNDSC approach was proposed to develop the adaptive state- and output-feedback containment controllers, under which containment can be reached if the leaders have directed paths to each follower. For both cases, Lyapunov–Krasovskii functionals are employed to demonstrate that all signals in the closed-loop network are UUB, and containment errors converge to a residual set. An illustrative example is given to show the performance improvement of the proposed control scheme, compared with traditional NDSC approach. The AUV dynamics considered in this paper is fully-actuated, and hence, future works include an extension to the underactuated case. Acknowledgement This work was in part supported by the National Nature Science Foundation of China under Grants 51209026, 61273137, 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, 3132014321. References [1] R.A. Brualdi, H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, 1991. [2] Y.C. Cao, W. Ren, M. Egerstedt, Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks, Automatica 48 (8) (2012) 1586–1597. [3] Y.C. Cao, D. Stuart, W. Ren, Z.Y. Meng, Distributed containment control for multiple autonomous vehicles with double-integrator dynamics: algorithms and experiments, IEEE Trans. Control Syst. Technol. 19 (4) (2011) 929–938. [4] M. Chen, S.S. Ge, B.V.E. How, Y.S. Choo, Robust adaptive position mooring control for marine vessels, IEEE Trans. Control Syst. Technol. 21 (2) (2013) 395–409. [5] W.S. Chen, L.C. Jiao, R.H. Li, J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Trans. Fuzzy Syst. 18 (4) (2010) 674–685. [6] W.S. Chen, X.B. Li, L.C. Jiao, Quantized consensus of second-order continuous-time multi-agent systems with a directed topology via sampled data, Automatica 49 (7) (2013) 2236–2242. [7] R.X. Cui, S.S. Ge, B.V.E. How, Y.S. Choo, Leader-follower formation control of underactuated autonomous underwater vehicles, Ocean Eng. 37 (17–18) (2010) 1491–1502. [8] R. X Cui, W.S. Yan, D.M. Xu, Synchronization of multiple autonomous underwater vehicles without velocity measurements, Sci. China: Inform. Sci. 55 (7) (2012) 1693–1703. [9] S.L. Dai, C. Wang, F. Luo, Identification and learning control of ocean surface ship using neural networks, IEEE Trans. Indust. Inform. 8 (4) (2012) 801– 810. [10] T. Dierks, S. Jagannathan, Output feedback control of a quadrotor UAV using neural networks, IEEE Trans. Neural Netw. 21 (1) (2009) 50–66. [11] T. Fossen, Marine Control System. Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles, Marine Cyernetics, Trondheim, Norway, 2002. [12] Y.G. Hong, X.L. Wang, Z.P. Jiang, Distributed output regulation of leader-follower multi-agent systems, Int. J. Robust Nonlinear Control 23 (1) (2013) 48–66. [13] Z.G. Hou, L. Cheng, M. Tan, Decentralized robust adaptive control for the multiagent system consensus problem using neural networks, IEEE Trans. Syst. Man Cybernet. Part B 39 (3) (2009) 636–647. [14] B.V.E. How, S.S. Ge, Y.S. Choo, Dynamic load positioning for subsea installation via adaptive neural control, IEEE J. Oceanic Eng. 35 (2) (2013) 366–375. [15] Y.H. Kim, F.L. Lewis, Neural network output feedback control of robot manipulators, IEEE Trans. Robot. Autom. 15 (2) (1999) 301–309. [16] M. Krstic´, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. [17] L. Lapierre, D.M. Soetanto, A.M. Pascoal, Coordinated motion control of marine robots, in: Proceedings of the 6th IFAC Conference on Manoeuvering and Control of Marine Craft, Girona, Spain, 2003. [18] J.H. Li, P.M. Lee, B.H. Jun, Y.K. Lim, Point-to-point navigation of underactuated ships, Automatica 44 (12) (2008) 3201–3205. [19] Z.K. Li, X.D. Liu, W. Ren, L.H. Xie, Distributed tracking control for linear multi-agent systems with a leader of bounded unknown input, IEEE Trans. Autom. Control 58 (2) (2012) 518–523. [20] Z.K. Li, W. Ren, X.D. Liu, M.Y. Fu, Distributed containment control of multi-agent systems with general linear dynamics in the presence of multiple leaders, Int. J. Robust Nonlinear Control 23 (5) (2013) 534–547. [21] J. Li, W. Ren, S. Xu, Distributed containment control with multiple dynamic leaders for double-integrator dynamic using only position measurements, IEEE Trans. Autom. Control 57 (6) (2012) 1553–1559. [22] J. Mei, W. Ren, G.F. Ma, Distributed containment control for Lagrangian networks with parameteric uncertaities under a directed graph, Automatica 48 (4) (2012) 653–659. [23] Z.Y. Meng, W. Ren, Z. You, Distributed finite-time attitude containment control for multiple rigid bodies, Automatica 46 (12) (2010) 2092–2099. [24] Z.H. Peng, D. Wang, Z.Y. Chen, X.J. Hu, W.Y. Lan, Adaptive dynamic surface control for formations of autonomous surface vehicles with uncertain dynamics, IEEE Trans. Control Syst. Technol. 21 (2) (2013) 513–520. [25] Z.H. Peng, D. Wang, T.S. Li, Z.L. Wu, Leaderless and leader-follower cooperative control of multiple marine surface vehicles with unknown dynamics, Nonlinear Dynam. 74 (1–2) (2013) 95–106. [26] Z.H. Peng, D. Wang, Hugh H.T. Liu, G. Sun, H. Wang, Distributed robust state and output feedback controller designs for rendezvous of networked autonomous surface vehicles using neural networks, Neurocomputing 115 (4) (2013) 130–141. [27] Z.H. Peng, D. Wang, H. Wang, W. Wang, A predictor-based neural DSC design approach to distributed coordinated control of multiple autonomous underwater vehicles, in: Proceedings of Chinese Control Conference, 2014, pp. 1214–1219.

Z. Peng et al. / Information Sciences 316 (2015) 163–179

179

[28] G.D. Shi, Y.G. Hong, K.H. Johansson, Connectivity and set tracking of multi-agent systems guided by multiple moving leaders, IEEE Trans. Autom. Control 57 (3) (2012) 663–676. [29] R. Skjetne, T.I. Fossen, P.V. Kokotovic, Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory, Automatica 41 (2) (2005) 289–298. [30] J.M. Soares, A.P. Aguiar, A.M. Pascoal, M. Gallieri, Triangular formation control using range measurements: an application to marine robotic vehicles, IFAC Workshop on Navigation, Guidance and Control of Underwater Vehicles, Portugal, 2012. [31] D. Swaroop, J. Hedrick, P. Yip, J. Gerdes, Dynamic surface control for a class of nonlinear systems, IEEE Trans. Autom. Control 45 (10) (2000) 893–1899. [32] K.P. Tee, S.S. Ge, Control of fully actuated ocean surface vessels using a class of feedforward approximators, IEEE Trans. Control Syst. Technol. 14 (4) (2006) 750–756. [33] S.C. Tong, B. Huo, Y.M. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 1–15. [34] S.C. Tong, Y.M. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Trans. Fuzzy Syst. 20 (1) (2012) 168–180. [35] S.C. Tong, Y.M. Li, G. Feng, T.S. Li, Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems, IEEE Trans. Syst. Man Cybernet. Part B 41 (4) (2011) 1124–1135. [36] S.C. Tong, Y.M. Li, P. Shi, Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems, IEEE Trans. Fuzzy Syst. 20 (4) (2012) 771–785. [37] S.C. Tong, T. Wang, Y.M. Li, B. Chen, A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control, IEEE Trans. Fuzzy Syst. 21 (2) (2013) 314–327. [38] D. Wang, J. Huang, Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Neural Netw. 6 (1) (2005) 195–202. [39] G.H. Wen, Z.S. Duan, G.R. Chen, W.W. Yu, Consensus tracking of multi-agent systems with lipschitz-type node dynamics and switching topologies, IEEE Trans. Circ. Syst. 61 (4) (2014) 499–511. [40] G.H. Wen, G.Q. Hu, W.W. Yu, G.R. Chen, Distributed H1 consensus of higher order multiagent systems with switching topologies, IEEE Trans. Circ. Syst. 61 (5) (2014) 359–363. [41] S.J. Yoo, Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form, Automatica 49 (7) (2013) 2145– 2153. [42] H.W. Zhang, F.L. Lewis, A. Das, Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback, IEEE Trans. Autom. Control 56 (8) (2011) 1948–1952. [43] Y.S. Zheng, L. Wang, Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements, Syst. Control Lett. 61 (8) (2012) 871–878. [44] Y.S. Zheng, T. Li, L. Wang, Containment control of multi-agent systems with measurement noises, arXiv:1405.4681, 2014. [45] Y.S. Zheng, L. Wang, Containment control of heterogeneous multi-agent systems, Int. J. Control 87 (1) (2014) 1–8. [46] Y. Zheng, Y. Zhu, L. Wang, Consensus of heterogeneous multi-agent systems, IET Control Theory Appl. 5 (16) (2011) 1881–1888.