Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control 2011; 21:1509–1524Published online 26 August 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1647

Cooperative adaptive control for synchronization of second-ordersystems with unknown nonlinearities

Abhijit Das∗,† and Frank L. Lewis

Automation and Robotics Research Institute, The University of Texas at Arlington,

7300 Jack Newell Blvd., S. Fort Worth, TX 76118, U.S.A.

SUMMARY

This paper studies synchronization to a desired trajectory for multi-agent systems with second-orderintegrator dynamics and unknown nonlinearities and disturbances. The agents can have different dynamicsand the treatment is for directed graphs with fixed communication topologies. The command generatoror leader node dynamics is also nonlinear and unknown. Cooperative tracking adaptive controllers aredesigned based on each node maintaining a neural network parametric approximator and suitably tuningit to guarantee stability and performance. A Lyapunov-based proof shows the ultimate boundedness of thetracking error. A simulation example with nodes having second-order Lagrangian dynamics verifies theperformance of the cooperative tracking adaptive controller. Copyright � 2010 John Wiley & Sons, Ltd.

Received 30 April 2010; Revised 15 July 2010; Accepted 15 July 2010

KEY WORDS: cooperative control; multi-agent system; nonlinear adaptive control; Lyapunov; neuralnetwork

1. INTRODUCTION

Synchronization behavior among agents is found in flocking of birds, schooling of fish, andother natural systems. Synchronization among coupled oscillators was studied by [1]. Much workhas extended consensus and synchronization techniques to man-made systems such as UAV toperform various tasks including surveillance, moving in formation, etc. We refer to consensus andsynchronization in terms of control of manmade dynamical systems. Early work on cooperativedecision and control for distributed systems includes [2]. The reader is referred to the book andsurvey papers [3–6]. Consensus has been studied for systems on communication graphs with fixedor varying topologies and communication delays. See [7–10], which proposed basic synchronizingprotocols for various communication topologies.

Early work on consensus studied leaderless consensus or the cooperative regulator problem,where the consensus value reached depends on the initial conditions of the node states and cannotbe controlled. On the other hand, the cooperative tracker problem seeks consensus or synchro-nization to the state of a control or leader node. Convergence of consensus to a virtual leader orheader node was studied in [10, 11]. Dynamic consensus for tracking of time-varying signals waspresented in [12]. The pinning control has been introduced for synchronization tracking control ofcoupled complex dynamical systems [13–16]. Pinning control allows controlled synchronizationof interconnected dynamical systems by adding a control or leader node that is connected (pinned)

∗Correspondence to: Abhijit Das, Automation and Robotics Research Institute, The University of Texas at Arlington,7300 Jack Newell Blvd., S. Fort Worth, TX 76118, U.S.A.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

1510 A. DAS AND F. L. LEWIS

into a small percentage of nodes in the network. Analysis has been done using Lyapunov andother techniques by assuming either a Jacobian linearization of the nonlinear node dynamics, or aLipschitz condition or contraction analysis. The agents are homogeneous in that they all have thesame nonlinear dynamics.

The study of second-order and higher order consensus is required to implement synchronizationin most real-world applications such as formation control and coordination among UAVs, whereboth position and velocity must be controlled. Note that Lagrangian motion dynamics and roboticsystems can be written in the form of second-order systems. Moreover, second-order integratorconsensus design (as opposed to first-order integrator node dynamics) involves more details aboutthe interaction between the system dynamics/control design problem and the graph structure asreflected in the Laplacian matrix. As such, second-order consensus is interesting because thereone must confront more directly the interface between control systems and communication graphstructure. See the book [3] in Section 3 (Chapters 4 and 5). Khoo et al. [17] studied the case ofhigher order consensus for linear chained integrator systems. The detailed analysis is performedfor third-order systems but it extends to the higher order cases. Zhu et al. [18] study the generalsecond-order consensus problem for double integrator systems. Khoo et al. [19] studied second-order consensus in finite time using sliding mode error and a Lyapunov analysis. References[20–22] discuss consensus with time delays for second-order integrator (Type II) systems. Seoet al. [23] proposed second-order consensus using the output feedback for agents with identicallinear dynamics.

Few papers study second-order consensus for unknown nonlinear systems. Few papers studyconsensus for heterogeneous agents with different unknown nonlinear dynamics. Leaderless oruncontrolled synchronization with nonlinear non-identical passive systems is reported in [24],which provided a Lyapunov proof valid for balanced graph structures. In contrast, this paperconcerns controlled consensus or the multi-agent tracker problem on general directed graphs.

Neural networks (NN) have a universal approximation property [25] and learning capabilitiesthat make them ideal for cooperative tracking control of multi-agent systems with non-identicalunknown nonlinear dynamics. NN have been used since 1990s to extend abilities of adaptivecontrollers to handle larger classes of unknown nonlinear dynamical systems. Novel NN weighttuning algorithms have been developed to make NN suitable for the online control of dynamicalsystems with real-time learning along the system trajectories. Rigorous proofs of convergence,performance, and stability have been offered. The reader is referred to [26–32] for early worksand the extensive literature since then is well known and hence not covered here. Neural adaptivecontrol has not been fully explored for control of multi-agent systems.

Distributed multi-agent systems with unknown nonlinear non-identical dynamics and distur-bances were studied in [33], where distributed neural adaptive controllers were designed to achieverobust consensus. This treatment assumed undirected graphs and solved the leaderless or uncon-trolled consensus problem, that is, the nodes reach a steady-state consensus that depends on theinitial conditions and cannot be controlled. Expressions for the consensus value were not given.Higher order consensus was proposed using a complex backstepping approach.

Many control system graph structures are non-symmetric in that communication links are unidi-rectional, with information flowing only in one way between subsystems. Moreover, in mostproblems it is important to be able to specify the desired synchronization trajectory. This corre-sponds to a multi-agent tracker problem. An example is the directed tree structure of formationcontrol, where all agents sense (either directly or indirectly through intermediate neighbors) thestate of the control node, but the control node sets the prescribed course and speed. Therefore,Das and Lewis [34] studied the cooperative tracker problem for agents on general digraphs havingnon-identical unknown dynamics and disturbances. First-order integrator dynamics was studied. Adistributed adaptive control technique was given that used pinning control to achieve synchroniza-tion to a desired command trajectory. Performance and stability were shown using a Lyapunovapproach.

In this paper we extend [34] to consensus for agents having second-order dynamics inthe Brunovsky form. We confront the second-order synchronization tracking problem for

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 2011; 21:1509–1524DOI: 10.1002/rnc

COOPERATIVE ADAPTIVE CONTROL FOR SYNCHRONIZATION 1511

heterogeneous nodes with non-identical unknown nonlinear dynamics with unknown disturbances.Herein, ‘synchronization control’ means the objective of enforcing all node trajectories to follow(in a ‘close-enough’ sense to be made precise) the trajectory of a leader or control node. Thecommunication structures considered are general directed graphs with fixed topologies. Analysisof digraphs is significantly more involved than in undirected graphs. The dynamics of the leader orcommand node is also assumed nonlinear and unknown. A distributed adaptive control approach istaken, where cooperative adaptive controllers are designed at each node. A parametric NN structureis introduced at each node to estimate the unknown dynamics. The choices of control protocol aswell as neural net tuning laws are the key factors in stabilizing the networked multi-agent systems.A Lyapunov analysis shows how to tune the NNs cooperatively, and guarantees the stability andthe performance of the networked systems. The error bounds obtained from the Lyapunov proofare dependent on control design and NN tuning parameters, which can be chosen to suitablymanage the tracking error and estimation error. The simulation results for networked agents withLagrangian dynamics are provided to show the effectiveness of the proposed method.

In Section 2 we formulated the synchronization tracking control problem for second-ordersystems with non-identical unknown nonlinear dynamics. In Section 3 a Lyapunov techniqueis used to design cooperative adaptive controllers based on NN approximation methods. Theperformance and stability guarantees are given for the networked systems. Section 4 presents thesimulation results.

2. SYNCHRONIZATION CONTROL FORMULATION

Consider a graph G = (V, E) with a non-empty finite set of N nodes V ={v1, . . . ,vN } and a setof edges or arcs E ⊆V ×V . We assume that the graph is simple, e.g. no repeated edges and(vi ,vi ) /∈ E, ∀i no self-loops. General directed graphs are considered. Denote the adjacency orconnectivity matrix as A= [aij] with aij>0 if (v j ,vi )∈ E and aij =0 otherwise. Note aii =0. Theset of neighbors of a node vi is Ni ={v j : (v j ,vi )∈ E}, i.e. the set of nodes with arcs incoming tovi . Define the in-degree matrix as a diagonal matrix D =diag{di } with di =

∑j∈Ni

aij the weightedin-degree of node i (i.e. i th row sum of A). Define the graph Laplacian matrix as L = D− A,which has all row sums equal to zero. Define do

i =∑ j aji, the (weighted) out-degree of node i ,that is the i th column sum of A.

We consider directed communication graphs with fixed topologies and assume the digraph isstrongly connected, i.e. there is a directed path from vi to v j for all distinct nodes vi ,v j ∈V . ThenA and L are irreducible [26, 35]. That is they are not cogradient to a lower triangular matrix, i.e.there is no permutation matrix U such that

L =U

[∗ 0

∗ ∗

]U T. (1)

The results of this paper can easily be extended to graphs having a spanning tree (i.e. not necessarilystrongly connected) using the Frobenius form in (1).

2.1. Cooperative tracking problem for synchronization of multi-agent systems

Consider second-order node dynamics defined for i th node in Brunovsky form as

x1i = x2

i ,

x2i = fi (xi )+ui +wi ,

(2)

where xi = [x1i x2

i ]T ∈ R2, ui (t)∈ R is the control input and wi (t)∈ R a disturbance acting upon eachnode. Note that each node may have its own distinct nonlinear dynamics. Standard assumptions for



existence of unique solutions are made, e.g. fi (xi ) either continuously differentiable or Lipschitz.The overall graph dynamics is

x1 = x2,

x2 = f (x)+u+w,(3)

where the overall (global) state vector is x2 = [x21 x2

2 · · · x2N ]T ∈ RN , x1 = [x1

1 x12 · · · x1

N ]T ∈RN , x = (x1, x2)T, f (x)= [ f1(x1) f2(x2) · · · fN (xN )]T ∈ RN , input u = [u1 u2 · · · uN ]T ∈ RN ,and w= [w1 w2 · · · wN ]T ∈ RN .

If the states xki are not scalars, this analysis carries over with a mere addition of the standard

Kronecker product term [34].

Definition 1The local neighborhood tracking synchronization errors (position and velocity) for node i aredefined as [19]

e1i = ∑

j∈Ni

aij(x1j −x1

i )+bi (x10 −x1

i ) (4)

and

e2i = ∑

j∈Ni

aij(x2j −x2

i )+bi (x20 −x2

i ) (5)

with pinning gains bi�0, and bi>0 for at least one i . Then, bi �=0 if and only if there exist an arcfrom the control node to the i th node in G. We refer to the nodes i for which bi �=0 as the pinnedor controlled nodes.

Note that (4) and (5) represent the information that is available for any node i for controlpurposes.

The state x0 = [x10 x2

0 ]T ∈ R2 of the leader or control node satisfies the (generally non-autonomous) dynamics in the Brunovsky form

x10 = x2

0 ,

x20 = f0(x0, t).

(6)

This can be regarded as a command or a reference generator. A special case is the standard constantconsensus value with x1

0 =0 and x20 absent. Here, we assume that the control node can have a

time-varying state.The Synchronization tracking control problem confronted herein is as follows: Design control

protocols for all the nodes in G to synchronize to the state of the control node, i.e. one requiresxk

i (t)→ xk0 (t), k =1,2,∀i . It is assumed that the dynamics of the control node is unknown to

any of the nodes in G. It is assumed further that both the node nonlinearities fi (.) and the nodedisturbances wi (t) are unknown. Thus, the synchronization protocols must be robust to unmodelleddynamics and unknown disturbances.

2.2. Synchronization tracking error

Define the consensus disagreement error vector

�= [�1 �2]T = [x1 −1x10 x2 −1x2

0 ]T. (7)

From (4), the global error vector for network G is given by

e1 =−(L + B)(x1−1x10 )=−(L + B)�1 (8)

and

e2 =−(L + B)(x2−1x20 )=−(L + B)�2, (9)



where, B =diag{bi } is the diagonal matrix of pinning gains, and ek = [ek1 ek

2 · · · ekN ]T ∈ RN , k =

1,2,∀i and 1∈ RN the vector of 1’s.

Lemma 1Let the graph is strongly connected and B �=0. Then

‖�k‖�‖ek‖/�(L + B), k =1,2 (10)

with �(L + B) the minimum singular value of (L + B), and e=0 if and only if the nodes synchronize,that is

xki (t)= xk

0 (t), k =1,2 ∀i. (11)

2.3. Synchronization control design and error dynamics

Differentiating (8) and (9),

e1 =−(L + B)(x2−1x20 ) (12)

and

e2 =−(L + B)(x2−1 f0(x0, t)). (13)

Note that e1 =e2. Define sliding mode error for node i

ri =e2i +�i e

1i (14)

or as a whole

r =e2 +�e1, (15)

where �=diag(�i )>0. The next result follows directly.

Lemma 2The velocity error is bounded according to

‖e2‖�‖r‖+ �(�)‖e1‖. (16)

Now by differentiating r one obtains the error dynamics

r = −(L + B)(x2−1 f0(x0, t))−�(L + B)(x2−1x20 )

= −(L + B)( f (x)+u+w)+(L + B)1 f0(x0, t)+�e2. (17)

Following the techniques in [27, 36], assume that the unknown nonlinearities in (2) are smoothand thus can be approximated on a compact set S ∈ R by

fi (xi )=W Ti �i (xi )+εi (18)

with �i (xi )∈ R�i a suitable basis set of �i functions at each node i with �i number of neurons andWi ∈ R�i a set of unknown coefficients. According to the Weierstrass higher order approximationtheorem [37] a polynomial basis set suffices to approximate fi (xi ) as well as its derivatives, whenthey exist, and moreover, the approximation error εi →0 uniformly as �i →∞. According to theNN approximation literature [25], a variety of basis sets can be selected, including sigmoids,gaussians, etc. There �i (xi )∈ R�i is known as the NN activation function vector and Wi ∈ R�i as theNN weight matrix. Then it is shown that εi is bounded on a compact set. The ideal approximatingweights Wi ∈ R�i in (18) are assumed unknown. The intention is to select only a small number �iof NN neurons at each node (see Simulations).



Here, to avoid distractions from the main issues being introduced, we assume a linear-in-the-parameters NN, i.e. the basis set of activation functions is fixed and only the output weights aretuned. It is straightforward to use a two-layer NN whereby the first- and second-layer weights aretuned. Then, one has a nonlinear-in-the-parameters NN and the basis set is automatically selectedin the NN. Then, the below development can be easily modified as in [27].

To compensate for unknown nonlinearities, each node will maintain a NN locally to keep trackof the current estimates for the nonlinearities. The idea is to use the information of the states fromthe neighbors of node i to evaluate the performance of the current control protocol along withthe current estimates of the nonlinear functions. Therefore, select the local node’s approximationfi (xi ) as

fi (xi )= W Ti �i (xi ), (19)

where Wi ∈ R�i is the current estimate of the NN weights for node i , and �i is the number of NNneurons maintained at each node i . It will be shown in Theorem 1 how to select the estimates ofthe parameters Wi ∈ R�i using the local neighborhood synchronization errors (4), (5). The globalnode nonlinearity f (x) for graph G is now written as

f (x)=W T�(x)+ε=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

W T1

W T2

. . .

. . .

W TN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1(x1)

�2(x2)

...

...

�N (xN )

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸�(x)

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ε1

ε2

...

...

εN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

and estimate f (x) as

f (x)= W T�(x)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

W T1

W T2

. . .

. . .

W TN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1(x1)

�2(x2)

...

...

�N (xN )

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸�(x)

. (21)

Consider an input

ui =− fi (xi )+�i (x, t) (22)

or

u =− f (x)+�(x, t) (23)

with �i (x, t) an auxiliary input for the i th node yet to be specified. Then using (23) the errordynamics (17) becomes

r =−(L + B)( f (x)+�(x, t)+w)+(L + B)1 f0(x0, t)+�e2, (24)

where f (x)= f (x)− f (x)= W�(x) with W =diag(W1 −W1,W2 −W2, . . . ,WN −WN )T.



3. LYAPUNOV DESIGN FOR COOPERATIVE ADAPTIVE TRACKING CONTROL

It is now shown how to select the auxiliary control �(t) and NN weight tuning laws such as toguarantee that all nodes synchronize to the desired control node signal, i.e. xi (t)→ x0(t), ∀i . Itis assumed that the dynamics f (x0, t) of the control node (which could represent its motion) isunknown to any of the nodes in G. It is assumed further that the node nonlinearities fi (xi ) anddisturbances wi (t) are unknown. The Lyapunov analysis technique approach of [27, 28] is used,although there are some complications arising from the fact that �(x, t) and the NN weight tuninglaws must be implemented as distributed protocols. This entails a careful selection of the Lyapunovfunction.

The maximum and minimum singular values of a matrix M are denoted �(M) and �(M),respectively. The Frobenius norm is ‖M‖F =

√tr{MT M} with tr{·} the trace. The Frobenius inner

product of two matrices is 〈M1, M2〉F =√

tr{MT1 M2}.

The following fact gives two standard results used in neural adaptive control [27].

Fact 1Let the nonlinearities f (x) in (3) be smooth on a compact set �⊂ RN . Then:

(a) The NN estimation error ε(x) is bounded by ‖εt‖�εM on �, with εM a fixed bound [25, 27].(b) Weierstrass higher order approximation theorem. Select the activation functions �(x) as a

complete independent basis (e.g. polynomials). Then NN estimation error ε(x) convergesuniformly to zero on � as �i →∞, i =1, N . That is ∀�>0 there exist �i , i =1, N such that�i>�i , ∀i implies supx∈� ‖ε(x)‖<� [37].

The following standard assumptions are required. Although the bounds mentioned are assumedto exist, they are not used in the design and do not have to be known. They appear in the errorbounds in the proof of Theorem 1. (Though not required, if desired, standard methods can be usedto estimate these bounds including [27].)

Assumption 1

(a) The unknown disturbance wi is bounded for all i . Thus the overall disturbance vector w isalso bounded by ‖w‖�wM with wM a fixed bound.

(b) Unknown ideal NN weight matrix W is bounded by ‖W‖F�WM .(c) NN activation functions i are bounded ∀i , hence that one can write for the overall network

that ‖�‖�M .(d) The unknown consensus variable dynamics f0(x0, t) as well as the target output vector are

bounded so that ‖ f0(x0, t)‖�FM ,∀t respectively.(e) The target trajectory is bounded so that ‖x1

0 (t)‖<X10, ‖x2

0 (t)‖<X20, ∀t .

The following definition for robust practical stability or uniform ultimate boundedness is standardand the following definition extends it to multi-agent systems.

Definition 2Any vector time function y(t) is said to be uniformly ultimately bounded (UU B) [27] if there exista compact set �⊂ RN so that ∀y(t0)∈� there exist a bound Bm and a time t f (Bm, y(t0)) such that‖x(t)− y(t)‖�Bm ∀t�t0 + t f .

Definition 3The control node state x0(t) is said to be cooperative uniformly ultimately bounded (CUUB) ifthere exist a compact set �⊂ R so that ∀ (xi (t0)−x0(t0)∈� there exist a bound Bm and a timet f (Bm, x(t0), x0(t0)) such that ‖xi (t)−x0(t)‖�Bm ∀i, ∀t�t0 + t f .

The next key constructive result is needed. An M-matrix is a square matrix having non-positiveoff-diagonal elements and all principal minors non-negative [26, 35].



Lemma 3 (Qu [26])Let L be irreducible and B have at least one diagonal entry bi>0. Then (L + B) is a non-singularM-matrix. Define

q = [q1 q2 · · · qN ]T = (L + B)−11, (25)

P = diag{pi }≡diag{1/qi }. (26)

Then P>0 and the matrix Q defined as

Q = P(L + B)+(L + B)T P (27)

is positive definite.

It is important that the matrix P is diagonal, as will be seen in the upcoming proof. The mainresult of the paper is now presented.

Theorem 1 (Distributed adaptive control protocol for synchronization)Consider the networked systems given by (3) under Assumption 1. Let the communication digraphbe strongly connected. Select the auxiliary control signal �(x, t) in (23) so that the local nodecontrol protocols are given by

ui =cri − fi (xi )+ �i

di +bie2

i (28)

with �i =�>0 ∀i , control gains c>0, and ri (t) defined in (14). Then

u =cr −W T�(x)+�(D+ B)−1e2 (29)

with local node NN tuning laws given by

˙W i =−Fi�i rTi pi (di +bi )−Fi Wi (30)

with Fi =�i I�i, I�i

the �i ×�i identity matrix, �i>0 and >0 scalar tuning gains, and pi>0defined in Lemma 3. Define

�=√

�(D+ B)

�(P)�(A)(31)

and select the control gain c and NN tuning gain so that

(i) c= 2

�(Q)

(1√�

+�

)> 0

(ii)1

2�m �(P)�(A)� � �−1

(32)

with P>0, Q>0 defined in Lemma 3 and A the graph adjacency matrix.Then there exist number of neurons �i , i =1, N such that for �i>�i , ∀i the overall sliding mode

cooperative error vector r (t), the local cooperative error vectors e1(t),e2(t) and the NN weightestimation errors W are UUB, with practical bounds given by (53)–(55) respectively. Moreoverthe consensus variable x0(t)= [x1

0 , x20 ]T is cooperative UUB and all nodes synchronize such that

‖x1i (t)−x1

0 (t)‖→0, ‖x2i (t)−x2

0 (t)‖→0. Moreover, the bounds (53)–(55) can be made small bymanipulating NN and control gain parameters.

ProofPart A: We claim that for any fixed εM>0, there exist number of neurons �i , i =1, N such thatfor �i>�i , ∀i the NN approximation error is bounded by ‖ε‖�εM . The claim is proven in Part bof the proof. Consider now the Lyapunov function candidate

V = 12rT Pr+ 1

2 W T F−1W + 12 (e1)Te1 (33)



with P = PT>0 and F−1 = F−T>0. Then

V =rT Pr + tr{W T F−1 ˙W }+(e1)Te1 (34)

Using (24) and (29)

r =−(L + B)( f (x)+cr +w)+(L + B)1 f0(x0, t)+ A(D+ B)−1�e2. (35)

Therefore

V = −rT P(L + B)(W T�(x)+ε+w+cr)+rT P(L + B)1 f0(x0, t)

+rT P A(D+ B)−1�e2 + tr{W T F−1 ˙W }+(e1)Te2, (36)

V = −rT P(L + B)cr −rT P(L + B)W T�(x)−rT P(L + B)(ε+w)

+rT P(L + B)1 f0(x0, t)+rT P A(D+ B)−1�e2 + tr{W T F−1 ˙W }+(e1)T(r −�e1) (37)

V = −crT P(L + B)r −rT P(L + B){ε+w−1 f0(x0, t)}+rT P A(D+ B)−1�e2 + tr[W T(F−1 ˙W −(x)rT P(L + B))]+(e1)Tr −(e1)T�e1 (38)

V = −crT P(L + B)r −rT P(L + B){ε+w−1 f0(x0, t)}+rT P A(D+ B)−1�(r −�e1)

+tr[W T(F−1 ˙W −�(x)rT P(L + B))]+(e1)Tr −(e1)T�e1 (39)

V = −crT P(L + B)r −rT P(L + B){ε+w−1 f0(x0, t)}+ tr[W T(F−1 ˙W −�(x)rT P(D+ B))]

+tr[W T�(x)rT P A]+rT P A(D+B)−1�r−rT P A(D+B)−1�2e1+(e1)Tr−(e1)T�e1. (40)

As L is irreducible and B has at least one diagonal entry bi>0, then (L + B) is a non-singularM-matrix. Thus, according to Lemma 3, one can write

V = − 12 crT Qr −rT P(L + B){ε+w−1 f0(x0, t)}+ tr[W T(F−1 ˙W −�(x)rT P(D+ B))]

+tr[W T�(x)rT P A]+rT P A(D+B)−1�r−rT P A(D+B)−1�2e1+(e1)Tr−(e1)T�e1 (41)

Adopt the NN weight tuning law ˙W i = Fi�i rTi pi (di +bi )+Fi Wi . Taking norm on both sides in

(41) one has

V � −1

2c�(Q)‖r‖2 + �(P)�(L + B)BM‖r‖+WM‖W‖F −‖W‖2

F +M �(P)�(A)‖W‖F‖r‖

+ �(P)�(A)�(�)

�(D+ B)‖r‖2 +

(1+ �(P)�(A)�(�2)

�(D+ B)

)‖r‖‖e1‖−�(�)‖e1‖2, (42)

where BM = (εM +wM + FM ). Then

V �−[‖e1‖ ‖r‖ |W‖F ]

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�(�)1

2

(1+ �(P)�(A)�(�2)

�(D+ B)

)0

1

2

(1+ �(P)�(A)�(�2)

�(D+ B)

) (1

2c�(Q)− �(P)�(A)�(�)

�(D+ B)

)1

2M �(P)�(A)

01

2M �(P)�(A)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦

+[0 �(P)�(L + B)BM WM ]

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦ . (43)



Write this as

V �−zT H z+hTz. (44)

Clearly V �0 iff H�0 and

‖z‖> ‖h‖�(H )

. (45)

According to (33) this defines a level set of V (z), hence it is direct to show that V �0 for V largeenough such that (45) holds [38]. To show this, according to (33) one has

1

2�(P)‖e‖2 + 1

2�max‖W‖2

F + 1

2‖e1t‖2�V �1

2�(P)‖e‖2+ 1

2�min‖W‖2

F + 1

2‖e1‖2, (46)

1

2[‖e1‖ ‖r‖ ‖W‖F ]

⎡⎢⎢⎢⎣

�(P)

1

�max

1

⎤⎥⎥⎥⎦

︸︷︷︸S1

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦

�V �1

2[‖e1‖ ‖r‖ ‖W‖F ]

⎡⎢⎢⎢⎣

�(P)

1

�min

1

⎤⎥⎥⎥⎦

︸︷︷︸S2

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦ (47)

with �min, �max the minimum and maximum values of �i . Equation (47) is equivalent to

12 zTSz�V � 1

2 zT Sz, (48)

where S =�(S1) and S = �(S2). Then

12 �(S)‖z‖2�V � 1

2 �(S)‖z‖2. (49)

Therefore,

V >1

2

�(S)‖h‖2

�2(H )(50)

implies (45).For a symmetric positive-definite matrix, the singular values will be equal to its eigenvalues.

Define

�=�I, �=√

�(D+ B)

�(P)�(A), c= 2

�(Q)

(1√�

+�

)and �= 1

2�m �(P)�(A).

Then (43) can be written as

V �−[‖e1‖ ‖r‖ ‖W‖F ]

⎡⎢⎣

� 1 0

1 � �

0 �

⎤⎥⎦

︸︷︷︸H

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦+[0 �(P)�(L + B)BM WM ]

⎡⎢⎢⎣

‖e1‖‖r‖

‖W‖F

⎤⎥⎥⎦ . (51)



The transformed H matrix is symmetric and positive definite under assumption given by (32).Therefore from Gershgorin circle’s theorem

�(H )�−� (52)

with 0<��−1. Therefore z(t) is UUB [38].In view of the fact that, for any vector z, one has ‖z‖1�‖z‖2� · · ·�‖z‖∞, sufficient conditions

for (45) are:

‖r‖> BM �(P)�(L + B)+WM

− 12�m �(P)�(A)

(53)

or

‖e1‖> BM �(P)�(L + B)+WM

− 12�m �(P)�(A)

(54)

or

‖W‖> BM �(P)�(L + B)+WM

− 12�m �(P)�(A)

. (55)

Note that this shows UUB of r (t),e1(t), W (t). Therefore from Lemma 2, the boundedness of rand e1 implies bounded e2. Now Lemma 1 shows that the consensus error vector �(t) is UUB.Then x0(t) is cooperative UUB.

Part B: See [39].According to (44) V �−�(H )‖z‖2 +‖h‖‖z‖ and according to (49)

V �−�V + √

V (56)

with �≡2�(H )/�(S), ≡√2‖h‖/√�(S). Thence

√V (t)�

√V (0)e−�t/2 +

�(1−e−�t/2)�

√V (0)+

�. (57)

Using (49) one has

‖e1(t)‖�‖z(t)‖�√

�(S)

�(S)

√‖e(0)‖2 +‖W (0)‖2

F +‖e1(0)‖2 + �(S)

�(S)

‖h‖�(H )

≡� (58)

and

‖r (t)‖�‖z(t)‖�√

�(S)

�(S)

√‖e(0)‖2 +‖W (0)‖2

F +‖e1(0)‖2 + �(S)

�(S)

‖h‖�(H )

≡�. (59)

Then from (8)

‖x1(t)‖ � 1

�(L + B)‖e1(t)‖+

√N‖x1

0 (t)‖, (60)

‖x1(t)‖ � �

�(L + B)+

√N X1

0 ≡h10. (61)



Similarly from (59) and using (16) in Lemma 2

‖e2(t)‖ � �‖e1(t)‖+‖r (t)‖,� �(1+�). (62)

This implies

‖x2(t)‖� �(1+�)

�(L + B)+

√N X2

0 ≡h20, (63)

where X20 =‖x2

0 (t)‖ and ‖h‖�BM �(P)�(L + B)+WM . Therefore, the state is contained for alltimes t�0 in a compact set �0 ={x(t)|‖x1(t)‖<h1

0,‖x2(t)‖<h20}. According to the Weierstrass

approximation theorem (Fact 1), given any NN approximation error bound εM there exist numberof neurons �i , i =1, N such that �i>�i , ∀i implies supx∈� ‖ε(x)‖<εM . �

3.1. Discussion

If any one of (53), (54) or (55) holds, the Lyapunov derivative is negative and V decreases.Therefore, these provide practical bounds for the neighborhood synchronization error and the NNweight estimation error.

The elements pi of the positive definite matrix P =diag{pi } required in the NN tuning law(30) are computed as P−11= (L + B)−11 (see Lemma 3), which requires global knowledge of thegraph structure unavailable for individual nodes. However, due to the presence of the arbitrarydiagonal gain matrix Fi>0 in (30), one can choose pi Fi>0 arbitrary without loss of generality.

It is important to select the Lyapunov function candidate V in (33) as a function of locallyavailable variables, e.g. the local sliding mode error r (t) and cooperative neighborhood error e1(t)in (15) and (8), respectively. This means that any local control signals �i (x, t) and NN tuninglaws developed in the proof are distributed and hence implementable at each node. The use ofthe Frobenius norm in the Lyapunov function is also instrumental, since it gives rise to Frobeniusinner products in the proof that only depend on trace terms, where only the diagonal terms areimportant. In fact, the Frobenius norm is ideally suited for the design of distributed protocols.Finally, it is important that the matrix P of Lemma 3 is diagonal.

4. SIMULATION RESULT

For this set of simulations, consider the 5-node strongly connected digraph structure in Figure 1with a leader node connected to node 3. The edge weights and the pinning gain in (4) were takenequal to 1.

Consider the node dynamics for node i given by the second-order Lagrange form dynamics

q1i = q2i ,

q2i = J−1i [ui − Br

i q2i − Mi gli sin(q1i )],(64)

where qi = [q1i ,q2i ]T ∈ R2 is the state vector, Ji is the total inertia of the link and the motor, Br

i isthe overall damping coefficient, Mi is the total mass, g is the gravitational acceleration and li is thedistance from the joint axis to the link center of mass for node. Ji , Br

i , Mi and li are consideredunknown and may be different for each node.

The desired target node dynamics is taken as the inertial system

m0q0 +d0q0 +koq0 =u0 (65)

with known m0, d0, k0. Select the feedback linearization input

u0 =−[K1(q0 −sin( t))+K2(q0 − cos( t))]+d0q0 +k0q0 + 2m0 sin( t) (66)

for a constant >0. Then, the target motion q0(t) tracks the desired reference trajectory sin( t).



Figure 1. Five-node SC digraph with one leader node.

The cooperative adaptive control protocols of Theorem 1 were implemented at each node. As isstandard in neural and adaptive control systems, reasonable positive values were initially chosenfor all control and NN tuning parameters. Generally, the simulation results are not dependent onthe specific choice of parameters as long as they are selected as detailed in the Theorem andassumptions, e.g. positive. The number of NN hidden layer units can be fairly small with goodperformance resulting, normally in the range 5–10. As in most control systems, however, theperformance can be improved by trying a few simulation runs and adjusting the parameters toobtain good behavior. In online implementations, the parameters can be adjusted online to obtainbetter performance.

For NN activation function we use log-sigmoid of the form 1/(1+e−kt ) with positive slopeparameter k. Number of neurons used at each node is 3. Thus, �m ≈1 and �≈0.04. NN gainparameter we selected as =1.5. According to (32) the pinning gain should be selected large andit was taken as c=1000.

In the simulation plots we show the tracking performance of positions (q1i ) and velocities (q2i )for all i . At steady state all q1i and q2i are synchronized and follow the second-order single linkleader trajectory given by q0 and q0, respectively.

Figure 2 shows the tracking performance of the system. One can see from the figure thatpositions and velocities of all the five nodes are synchronized in two different final values whichare given by the final values of [q0, q0]T. More specifically q1i ’s are synchronized with q0 andq2i ’s are synchronized with q0 at steady state. The figure also shows the inputs ui for all agentswhile tracking.

Figure 3 describes the position and velocity consensus disagreement error vectors namely(q1i −q0,q2i − q0)T,∀i and also the NN estimation error in terms of [ fi (x)− fi (x)]T∀i . One caneasily see that all the errors are minimized almost to zero.

Figure 4 is the phase plane plot of all agents, i.e. the plot of q1i ∀i (along the x-axis) and q2i ∀i(along the y-axis). At steady state the Lissajous pattern formed by all five nodes is the targetnode’s phase plane trajectory.

5. CONCLUSION

This paper presents a method for synchronization control for multi-agent systems of order twowith unknown dynamics. It gives the design of distributed adaptive controllers for second-ordernonlinear systems communicating on general strongly connected digraph network structures. Theagent dynamics and command generator dynamics are considered unknown. Moreover the agent



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4State dynamics and input

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–300

–200

–100

0

100

200

Position

Velocity

Figure 2. Tracking performance (position and velocity) and control input.

0 0.2 0.4 0.6 0.8 1–1

0

1

2

3Error Plot

0 0.2 0.4 0.6 0.8 1–100

–50

0

50

100

150

Velocity Error

Position Error

Figure 3. Disagreement vector and NN estimation error vector.

dynamics need not be same. It is shown that with the use of pinning control based on the exchangeof cooperative neighborhood errors among the agents, one can guarantee synchronization of all therobots to the single command trajectory. A simple NN parametric approximator is introduced at eachnode to estimate the unknown dynamics and disturbances. The choices of control protocol as wellas neural net tuning laws are selected through a Lyapunov formulation to induce synchronization



–3.5 –3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5–3.5

–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

Sync. Motion

Figure 4. Phase plane plot for all the nodes.

within the networked multi-robot team. A Lyapunov-based proof shows the ultimate boundednessof the tracking error. The simulation results for Lagrangian agent dynamics are shown to illustratethe effectiveness of the proposed method.

ACKNOWLEDGEMENTS

This work is supported by AFOSR grant FA9550-09-1-0278, NSF grant ECCS-0801330, and ARO grantW91NF-05-1-0314.

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Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities