Automatica 46 (2010) 2014–2021

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Distributed adaptive control for synchronization of unknown nonlinearnetworked systems

Abhijit Das ∗, Frank L. LewisAutomation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX, USA

a r t i c l e i n f o

Article history:Received 2 December 2009Received in revised form2 July 2010Accepted 19 July 2010Available online 31 August 2010

Keywords:Nonlinear multiagent systemsDistributed adaptive controlSynchronizationConsensusNetworked systems

a b s t r a c t

This paper is concerned with synchronization of distributed node dynamics to a prescribed targetor control node dynamics. A design method is presented for adaptive synchronization controllers fordistributed systems having non-identical unknown nonlinear dynamics, and for a target dynamicsto be tracked that is also nonlinear and unknown. The development is for strongly connecteddigraph communication structures. A Lyapunov technique is presented for designing a robust adaptivesynchronization control protocol. The proper selection of the Lyapunov function is the key to ensuringthat the resulting control laws thus found are implementable in a distributed fashion. Lyapunov functionsare defined in terms of a local neighborhood tracking synchronization error and the Frobenius norm. Theresulting protocol consists of a linear protocol and a nonlinear control term with adaptive update law ateach node. Singular value analysis is used. It is shown that the singular values of certain key matrices areintimately related to structural properties of the graph.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Coordination and consensus of distributed groups of agentsis inspired by naturally occurring phenomena such as flockingin birds, swarming in insects, circadian rhythms in nature,synchronization and phase transitions in physical and chemicalsystems, and the laws of thermodynamics (Hui & Haddad, 2008).Early work has been done in the control systems community byFax and Murray (2004), Jadbabaie, Lin, and Morse (2003), Olfati-Saber and Murray (2004), Ren and Beard (2005), and Tsitsiklis(1984), which by now arewell known. Consensus has been studiedfor systems on communication graphs with fixed or varyingtopologies and communication delays. The average consensusproblem has garnered much interest. Synchronization to time-varying trajectories has been studied based on physical or naturalsystems by Chopra and Spong (2009), Kuramoto (1975), Strogatz(2000), and Vicsek et al. (1995). Synchronization of nonlinearpassive dynamical systems has been studied by Chopra and Spong(2006). Consensus using nonlinear protocols has been consideredthere and in Hui and Haddad (2008).

Convergence of consensus to a virtual leader or header node hasbeen studied in Jadbabaie et al. (2003) and Jiang and Baras (2009).

This work is supported by AFOSR grant FA9550-09-1-0278, NSF grant ECCS-0801330, and ARO grant W91NF-05-1-0314. This paper was not presented atany conference. This paper was recommended for publication in revised form byAssociate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic.∗ Corresponding author. Tel.: +1 817 521 0584; fax: +1 817 272 5989.

E-mail addresses: abhijit.das@ieee.org (A. Das), lewis@uta.edu (F.L. Lewis).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.08.008

Dynamic consensus for tracking of time-varying signals has beenpresented in Spanos, Olfati-Saber, andMurray (2005). Recently, thepinning control has been introduced for synchronization control ofcoupled complex dynamical systems (Li, Wang, & Chen, 2004; Li,Duan, & Chen, 2009; Lu & Chen, 2005; Wang & Chen, 2002; Yu,Chen, & Lü, 2009). Pinning control is a powerful technique thatallows controlled synchronization of interconnected dynamicalsystems by adding a control or leader node that is connected(pinned) into a small percentage of nodes in the network. Thesepinned or controlled nodes view the control node simply asanother neighbor, and consider the control node’s state value incomputing their local protocols. Analysis shows that all nodesconverge to the state of the control node, which may be time-varying. Analysis has been done using Lyapunov techniques byassuming either a Jacobian linearization of the nonlinear nodedynamics or a Lipschitz condition. A related idea is soft control(Han, Li, & Guo, 2006) where a shill node moves through thenetwork, and is perceived by existing nodes simply as anotherneighbor for purposes of computation of their own averagingprotocols. Proper placement and motion of the shill agent resultsin consensus to the state of the shill. The idea of pinning control forundirected graph topology using V -stability is discussed in Xiangand Chen (2007).

Consensus and collective motion of distributed agents havebeen analyzed using the theory of graphs and/orMarkov processes.Recent publications allow analysis using traditional control theorynotions including matrix analysis, Lyapunov theory, etc., upon theintroduction of certain key definitions including irreducibility, M-matrices, Frobenius form, special Lyapunov forms, etc. Notable are

A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021 2015

the books by Qu (2009), and Wu (2007). Such techniques allowone to bring in the machinery of matrix analysis (Bernstein, 2005).Instrumental in this analysis are the techniques employed in Khoo,Xie, and Man (2009).

Distributed multiagent systems with unknown nonlineardynamics and disturbances were studied in Hou, Cheng, and Tan(2009) where distributed adaptive controllers were designed toachieve robust consensus. That treatment assumed undirectedgraphs and solved the consensus problem, that is, the nodes reacha steady-state consensus that depends on the initial conditions.Expressions for the consensus value were not given. A method foradaptive tuning of coupling gainswas presented in Yu et al. (2009).

The study of control protocols on digraphs is significantly moreinvolved than their study on undirected graphs, where the graphLaplacian can be taken as a Lyapunov function. Pinning control fordirected graph topologies was studied by Lu, Li, and Rong (2010),and Wu (2007). There, the dynamics of all nodes were assumed tobe the same.

In this paper we present a Lyapunov technique for design ofprotocols for robust synchronization to tracking of a leader orcontrol node. Design techniques are developed for general stronglyconnected digraphs. The leader has unknown nonlinear dynamics,and the nodes have unknown, non-identical, nonlinear dynamicsand disturbances. Suitable control protocols are derived using aLyapunov functionwhich is carefully crafted to depend on a speciallocal synchronization errorwhich can be computed in a distributedfashion. The control laws thus derived are distributed in natureand can be implemented locally by each node. They consist ofa linear protocol plus a nonlinear adaptive learning term. Theseprotocols are robust to uncertain disturbances and dynamics, andto modeling errors in a sense to be made precise. It is shownthat the synchronization error converges to a residual set. Singularvalue analysis is used. It is shown that the singular values of certainkey matrices are intimately related to structural properties of thegraph. Simulation results are given to show the effectiveness of theproposed method.

2. Synchronization control formulation

Consider a graph G = (V , E) with a nonempty finite set of Nnodes V = v1, . . . , vN and a set of edges or arcs E ⊆ V × V . Weassume the graph is simple, e.g. no repeated edges and (vi, vi) ∈

E, ∀i no self loops. General directed graphs are considered. Denotethe connectivity matrix as A = [aij] with aij > 0 if (vj, vi) ∈ E andaij = 0 otherwise. Note aii = 0. The set of neighbors of a node viis Ni = vj : (vj, vi) ∈ E, i.e. the set of nodes with arcs incomingto vi. Define the in-degree matrix as a diagonal matrix D = [di]with di =

∑j∈Ni

aij the weighted in-degree of node i (i.e. ith rowsum of A). Define the graph Laplacian matrix as L = D − A, whichhas all row sums equal to zero. Define doi =

∑j aji, the (weighted)

out-degree of node i, that is the ith column sum of A.We assume the communication digraph is strongly connected,

i.e. there is a directed path from vi to vj for all distinct nodesvi, vj ∈ V . Then A and L are irreducible (Qu, 2009). That is theyare not cogredient to a lower triangular matrix, i.e., there is nopermutation matrix U such that

L = U[∗ 0∗ ∗

]UT . (1)

The results of this paper can easily be extended to graphs havinga spanning tree (i.e. not necessarily strongly connected) using theFrobenius form in (1).

2.1. Synchronization control problem

Consider node dynamics defined for the ith node as

xi = fi(xi) + ui + wi(t) (2)

where xi(t) ∈ R is the state of node i, ui(t) ∈ R is the controlinput, and wi(t) ∈ R is a disturbance acting upon each node.Note that each nodemay have its own distinct dynamics. Standardassumptions for existence of unique solutions are made, e.g. fi(xi)either continuously differentiable or Lipschitz. The overall graphdynamics is

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

where the overall (global) state vector is x =x1 x2 · · · xN

T∈ RN , global nodedynamics vector is f (x) = [f1 (x1) f2 (x2) · · ·

fN (xN)]T ∈ RN , input u =u1 u2 · · · uN

T∈ RN , and w =

w1 w2 · · · wNT

∈ RN .

Definition 1. The local neighborhood synchronization error for nodei is defined as (Khoo et al., 2009; Li et al., 2004)

ei =

−j∈Ni

aijxj − xi

+ bi (x0 − xi) (4)

with pinning gains bi ≥ 0, and bi > 0 for at least one i. Then,bi = 0 if and only if there exists an arc from the control node tothe ith node in G. We refer to the nodes i for which bi = 0 as thepinned or controlled nodes.

Note that (4) represents the information that is available to anynode i for control purposes.

The state of the leader or control node is x0(t) which satisfiesthe (generally nonautonomous) dynamics

x0 = f (x0, t). (5)

A special case is the standard constant consensus value with x0 =

0. The drift term could represent, e.g., motion dynamics of thecontrol node. That is, we assume that the control node can havea time-varying state.

The Synchronization control design problem confronted hereinis as follows: design control protocols for all the nodes in G tosynchronize to the state of the control node, i.e. one requiresxi(t) → x0(t), ∀i. It is assumed that the dynamics of the controlnode is unknown to any of the nodes in G. It is assumed furtherthat both the node nonlinearities fi(.) and the node disturbanceswi(t) are unknown. Thus, the synchronization protocols must berobust to unmodelled dynamics and unknown disturbances.

In fact, (5) is a command generator. Therefore we areconsidering a distributed command generator tracker problemwith unknown node dynamics and unknown command generatordynamics.

2.2. Synchronization error dynamics

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

e = − (L + B)x − 1x0

= − (L + B)

x − x0

(6)

where e =e1 e2 · · · eN

T∈ RN and x0 = 1x0 ∈ RN .

B ∈ RN×N is a diagonal matrix with diagonal entries bi and 1 isthe N-vector of ones.

Differentiating (6),

e = − (L + B)x − x0

= − (L + B)

f (x) − f (x0, t) + u + w(t)

(7)

where f (x0, t) = 1f (x0(t), t) ∈ RN .

2016 A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021

Remark 1. If the node states are vectors xi(t) ∈ Rn and x0(t) ∈ Rn,then x, e ∈ RnN and (6) becomes

e = − [(L + B) ⊗ In]f (x) − f (x0, t) + u + w(t)

(8)

with ⊗ the Kronecker product. To avoid obscuring the essentials,throughout the paperwe take thenode states as scalars, xi(t) ∈ R. Ifthey are vectors, all of the followingdevelopment is easilymodifiedby introducing the Kronecker product terms as appropriate. In fact,a simulation example is presented for the case xi(t) ∈ R2, namely2-D motion control for coupled inertial agents.

Remark 2. Note that

δ =x − x0

(9)

is the disagreement vector in Olfati-Saber and Murray (2004). Wedo not use this error herein because it is a global quantity thatcannot be computed locally at each node, in contrast to the localneighborhood error (4). As such, it is suitable for analysis but notfor distributed controls design using Lyapunov techniques.

Remark 3. We take the communication digraph as stronglyconnected. Therefore, if bi = 0 for at least one i then (L + B) is anirreducibly diagonally dominant M-matrix and hence nonsingular(Qu, 2009).

An M-matrix is a square matrix having its off-diagonal entriesnonpositive and all principal minors nonnegative. Based onRemark 3, the next result is therefore obvious from (6) and theCauchy Schwartz inequality (see also Khoo et al., 2009).

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

‖δ‖ ≤ ‖e‖ /σ(L + B) (10)

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

x(t) = x0(t). (11)

2.3. Synchronization control design

The control problem confronted in this paper is to design acontrol strategy so that local neighborhood error e(t) is boundedto a small residual set. Then according to Lemma 1 all nodessynchronize so that ‖xi(t) − x0(t)‖ is small ∀i. It is assumed thatthe dynamics f (x0, t) of the control node is unknown to any of thenodes in G. It is assumed further that both the node nonlinearitiesfi(.) and the node disturbances wi(t) are unknown. As such, thesynchronization protocolsmust be robust to unmodelled dynamicsand unknown disturbances.

To achieve this goal, define the input ui for node i as

ui = vi − fi(xi) (12)

where fi(xi) is an estimate of fi(xi) and vi(t) is an auxiliary controlsignal to be designed via Lyapunov techniques in Theorem 1. Thiscan be written in vector form for the overall network of N nodes as

u = v − f (x) (13)

where f (x) =f1 (x1) f2 (x2) · · · fN (xN)

T∈ RN and v =

v1 v2 · · · vNT

∈ RN . Then from (7) one gets

e = − (L + B)f (x) − f (x0, t) − f (x) + v + w

. (14)

Following the techniques in Lewis, Jagannathan, and Yesildirek(1999), assume that the unknown nonlinearities in (2) are locallysmooth and thus can be approximated on a compact set Ωi ∈ R by

fi (xi) = W Ti ϕi(xi) + εi (15)

with ϕi(xi) ∈ Rνi a suitable basis set of νi functions at each node iandWi ∈ Rνi a set of unknown coefficients. According to the neuralnetwork (NN) approximation literature (Hornik, Stinchombe, &White, 1989), a variety of basis sets can be selected, includingsigmoids, gaussians, etc. There ϕi(xi) ∈ Rνi is known as the NNactivation function vector and Wi ∈ Rνi as the NN weight matrix.The ideal approximating weights Wi ∈ Rνi in (15) are assumedunknown. The intention is to select only a small number νi of NNneurons at each node (see Simulations).

To compensate for unknown nonlinearities, each node willmaintain a neural network locally to keep track of the currentestimates for the nonlinearities. The idea is to use the informationof the states from the neighbors of node i to evaluate theperformance of the current control protocol alongwith the currentestimates of the nonlinear functions. Therefore, select the localnode’s approximation fi(xi) as

fi (xi) = W Ti ϕi(xi) (16)

where Wi ∈ Rνi is a current estimate of the NN weights for nodei, and νi is the number of NN neurons maintained at each node i.It will be shown in Theorem 1 how to select the estimates of theparameters Wi ∈ Rνi using the local neighborhood synchronizationerrors (4).

The global node nonlinearity f (x) for G is now written as

f (x) = W Tϕ(x) + ε (17)

where ϕ (x) =ϕT1 (x1) ϕT

2 (x2) · · · ϕTN (xN)

T,W T

=

diagW Ti , ε =

ε1 ε2 · · · εN

T∈ RN . The estimate f (x) is

f (x) = W Tϕ(x) (18)

with W T= diagW T

i . Now, the error dynamics (14) takes the form

e = − (L + B)f (x) + v + w(t) − f (x0, t)

(19)

where the parameter estimation error is Wi = Wi − Wi and thefunction estimation error is

f (x) = f (x) − f (x) = W Tϕ(x) + ε (20)

with W T= W T

− W T . Therefore, one obtains finally the errordynamics

e = − (L + B)W Tϕ(x) + v + ε + w(t) − f (x0, t)

(21)

with v(t) an auxiliary control yet to be designed.

3. Lyapunov design for networked systems: distributed NNtuning protocols

Now we show how to select the auxiliary control v(t) and NNweight tuning laws such as to guarantee that all nodes synchronizeto the desired control node signal, i.e., xi(t) → x0(t), ∀i. It isassumed that the dynamics f (x0, t) of the control node (whichcould represent itsmotion) are unknown to any of the nodes inG. Itis assumed further that the node disturbances wi(t) are unknown.The Lyapunov analysis technique approach of Lewis et al. (1999)and Lewis, Yesildirek, and Liu (1996) is used, though there arecomplications arising from the fact that v(t) and the NN weighttuning laws must be implemented as distributed protocols. Thisentails a careful selection of the Lyapunov function.

The singular values of a matrixM are denoted σi(M)with σ (M)the maximum singular value and σ(M) the minimum singularvalue. The Frobenius norm is ‖M‖F =

trMTM with tr · the

trace. The Frobenius inner product of two matrices is ⟨M1,M2⟩F =trMT

1M2.The following Fact gives two standard results used in neural

adaptive control (Lewis et al., 1999).

A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021 2017

Fact 1. Let the nonlinearities f (x) in (17) be smooth on a compactset Ω ∈ RN . Then, the NN estimation error ε(x) is bounded by‖ε‖ ≤ εM on Ω , with εM a fixed bound (Hornik et al., 1989; Lewiset al., 1999) Weierstrass higher-order approximation theorem.Select the activation functions ϕ(x) as a complete independentbasis (e.g. polynomials). Then the NN estimation error ε(x)converges uniformly to zero on Ω as νi → ∞, i = 1,N . Thatis ∀ξ > 0 there exist νi, i = 1,N such that νi > νi, ∀i impliessupx∈Ω ‖ε(x)‖ < ξ (Stone, 1948).

The following standard assumptions are required. Although thebounds mentioned are assumed to exist, they are not used in thedesign 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 used to estimate these bounds includingRovithakis (2000).)

Assumption 1. (a) The unknown disturbance wi is bounded forall i. Thus the overall disturbance vector w is also bounded by‖w‖ ≤ wM with wM a fixed bound.

(b) The unknown consensus variable dynamics f (x0, t) is boundedso that

f (x0, t) ≤ FM , ∀t .(c) The target trajectory is in a bounded region, e.g. ‖x0(t)‖ <

X0, ∀t , with X0 a constant bound.(d) Unknown ideal NN weight matrix W is bounded by ‖W‖F ≤

WM .(e) NN activation functionsϕi are bounded∀i, so that one canwrite

for the overall network that ‖ϕ‖ ≤ φM .

The next definitions extend standard notions (Khalil, 1996; Lewiset al., 1999) to synchronization for distributed systems.

Definition 2. The global neighborhood error e(t) ∈ RN is uniformlyultimately bounded (UUB) if there exists a compact set Ω ⊂ RN sothat ∀e(t0) ∈ Ω there exists a bound B and a time tf (B, e(t0)), bothindependent of t0 ≥ 0, such that ‖e(t)‖ ≤ B ∀t ≥ t0 + tf .

Definition 3. The control node trajectory x0(t) given by (5) iscooperative UUB with respect to solutions of node dynamics (2)if there exists a compact set Ω ⊂ R so that ∀ (xi(t0) − x0(t0)) ∈

Ω , there exist a bound B and a time tf (B, (x(t0) − x0(t0))), bothindependent of t0 ≥ 0, such that ‖xi(t) − x0(t)‖ ≤ B ∀i, ∀t ≥

t0 + tf .

The next key constructive result is needed. An M-matrix is asquare matrix having nonpositive off-diagonal elements and allprincipal minors nonnegative.

Lemma 2 (Qu (2009)). Let L be irreducible and B have at least onediagonal entry bi > 0. Then (L + B) is a nonsingular M-matrix. Define

q =q1 q2 · · · qN

T= (L + B)−1 1 (22)

P = diag pi ≡ diag 1/qi . (23)

Then P > 0 and the matrix Q defined as

Q = P (L + B) + (L + B)T P (24)

is positive definite.

The main result of this paper is given by the following result,which shows how to design the control protocols (12) and tunethe NN weights such that the local neighborhood cooperativeerrors for all nodes are UUB, which implies consensus variablex0(t) is cooperative UUB, thereby showing synchronization andcooperative stability for the whole network G.

Consider the following distributed adaptive control protocol:Select the auxiliary control signals in (12) as vi(t) = cei(t)with the

neighborhood synchronization errors ei(t) defined in (4) so thatthe local node control protocols are given by

ui = cei − fi(xi)

= c−j∈Ni

aijxj − xi

+ cbi (x0 − xi) − W T

i ϕi(xi) (25)

or

u = ce − W Tϕ(x) (26)

with control gains c > 0. Let local node NN tuning laws be givenby

˙W i = −FiϕieTi pi(di + bi) − κFiWi (27)

with Fi = ΠiIνi , Iνi the νi × νi identity matrix, Πi > 0 and κ > 0scalar tuning gains, and pi > 0 defined in (23).

The following main theorem details the design and perfor-mance of this protocol.

Theorem 1 (Distributed Adaptive Control Protocol for Synchroniza-tion). Consider the networked systems given by (2), (3) under Assump-tion 1. Let the communication digraph be strongly connected. Adoptthe distributed adaptive control protocol given by (25)–(27). SelectNN tuning gain κ =

12 cσ (Q ) and the control gain c so that

cσ(Q ) >12φM σ (P) σ (A) (28)

with P > 0, Q > 0 the matrices in Lemma 2 and A the graphadjacency matrix.

Then there exist numbers of neurons νi, i = 1,N such that forνi > νi, ∀i the overall local cooperative error vector e(t) and the NNweight estimation errors W are UUB, with practical bounds given by(46) and (47) respectively. Therefore the control node trajectory x0(t)is cooperative UUB and all nodes synchronize to x0(t). Moreover, thebounds on local consensus errors (4) can be made small by increasingthe control gains c.

Proof. Part a. We claim that for a fixed εM > 0, there existnumbers of neurons νi, i = 1,N such that for νi > νi, ∀i theNN approximation error is bounded by ‖ε‖ ≤ εM . The claim isproven in Part b of the proof. Consider now the Lyapunov functioncandidate

V =12eTPe +

12tr

W T F−1W

(29)

with e(t) the vector of local neighborhood cooperative errors (4),0 < P = PT

∈ RN×N the diagonal matrix defined in (23), and F−1 ablock diagonal matrix defined in terms of F = diagFi. Then,

V = eTPe + trW T F−1 ˙W

(30)

and from (21)

V = −eTP (L + B)W Tϕ(x) + ce + ε + w − f (x0, t)

+ tr

W T F−1 ˙W

(31)

V = −ceTP (L + B) e − eTP (L + B)ε + w − f (x0, t)

− eTP (L + B) W Tϕ(x) + tr

W T F−1 ˙W

(32)

V = −ceTP (L + B) e − eTP (L + B)ε + w − f (x0, t)

+ tr

W T

F−1 ˙W − ϕeTP (L + B)

(33)

V = −ceTP (L + B) e − eTP (L + B)ε + w − f (x0, t)

+ tr

W T

F−1 ˙W − ϕeTP (D + B − A)

. (34)

2018 A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021

Since L is irreducible and B has at least one diagonal entry bi >

0, then (L + B) is a nonsingular M-matrix. Defining therefore Qaccording to (24) one has

V = −12ceTQe − eTP (L + B)

ε + w − f (x0, t)

+ tr

W T

F−1 ˙W − ϕeTP (D + B)

+ tr

W TϕeTPA

. (35)

Adopt now theNNweight tuning law (27) or ˙W i = FiϕieTi pi(di+

bi) + κFiWi. Since P and (D + B) are diagonal and ˙W has the formin (18), one has

V = −12ceTQe − eTP (L + B)

ε + w(t) − f (x0, t)

+ κtr

W T (W − W )

+ tr

W TϕeTPA

. (36)

Therefore, for fixed εM > 0,

V ≤ −12cσ(Q ) ‖e‖2

+ ‖e‖ σ (P) σ (L + B) (εM + wM + FM)

+ κWM

WF− κ

W2

F+

WF‖e‖ φM σ (P) σ (A) (37)

V ≤ −

‖e‖W

F

T 1

2cσ(Q ) −

12φM σ (P) σ (A)

−12φM σ (P) σ (A) κ

×

‖e‖W

F

+

BM σ (P) σ (L + B) κWM

‖e‖W

F

with BM ≡ εM + wM + FM . Write this as

V ≤ −zTRz + rT z. (38)

Then V ≤ 0 if R is positive definite and

‖z‖ >‖r‖σ(R)

. (39)

According to (29) one has

12σ(P) ‖e‖2

+1

2Πmax

W2

F

≤ V ≤12σ (P) ‖e‖2

+1

2Πmin

W2

F(40)

12

‖e‖

WF

σ(P) 0

01

Πmax

‖e‖W

F

≤ V

≤12

‖e‖

WF

σ (P) 0

01

Πmin

‖e‖W

F

(41)

with Πmin, Πmax the minimum and maximum values of Πi. Definevariables to write 1

2 zT Sz ≤ V ≤

12 z

T Sz. Then

12σ(S) ‖z‖2

≤ V ≤12σ (S) ‖z‖2 . (42)

Therefore

V >12

σ (S) ‖r‖2

σ 2(R)(43)

implies (39).

One can write the minimum singular values of R as

σ (R) =

12 cσ (Q ) + κ

2

−

12 cσ (Q ) − κ

+

14ϕ

2M σ 2 (P) σ 2 (A)

2. (44)

To obtain a cleaner form for σ(R), select κ =12 cσ (Q ) Then

σ (R) =cσ (Q ) −

12ϕM σ (P) σ (A)

2(45)

which is positive under condition (28). Therefore, z(t) is UUB(Khalil, 1996).

In view of the fact that, for any vector z, one has ‖z‖1 ≥ ‖z‖2 ≥

· · · ≥ ‖z‖∞, sufficient conditions for (39) are

‖e‖ >BM σ (P) σ (L + B) + κWM

σ (R)(46)

orW >BM σ (P) σ (L + B) + κWM

σ (R). (47)

Now Lemma 1 shows that the consensus errors δ(t) areUUB. Then x0(t) is cooperative UUB. Note that increasing gain cdecreases the bound in (46).Part b. According to (38) V ≤ −σ(R) ‖z‖2

+‖r‖ ‖z‖ and accordingto (42)

V ≤ −αV + β√V (48)

with α ≡ 2σ(R)/σ (S), β ≡√2 ‖r‖ /

σ(S). Thence

V (t) ≤

V (0)e−αt/2

+β

α(1 − e−αt/2) ≤

V (0) +

β

α.

Using (42) one has

‖e(t)‖ ≤ ‖z(t)‖ ≤

σ (S)σ (S)

‖e(0)‖2

+

W (0)2

F+

σ (S)σ (S)

‖r‖σ(R)

.

Then (6) shows that

‖x(t)‖ ≤1

σ(L + B)‖e(t)‖ +

√N ‖x0(t)‖

‖x(t)‖ ≤1

σ(L + B)

σ (S)σ (S)

‖e(0)‖2

+

W (0)2

F

+σ (S)σ (S)

‖r‖σ(R)

+√NX0 ≡ r0 (49)

where ‖r‖ ≤ BM σ (P) σ (L + B) + κWM . Therefore, the state iscontained for all times t ≥ 0 in a compact setΩ0 = x(t)| ‖x(t)‖ ≤

r0. According to the Weierstrass approximation theorem, givenany NN approximation error bound εM there exist numbers ofneurons νi, i = 1,N such that νi > νi, ∀i implies supx∈Ω ‖ε(x)‖ <εM .

This completes the proof.

Discussion: If either (46) or (47) holds, the Lyapunov derivativeis negative and V decreases. Therefore, these provide practicalbounds for the neighborhood synchronization error and the NNweight estimation error.

Part a of the proof contains the new material relevant tocooperative control of distributed systems. Part b of the proof is

A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021 2019

standard in the neural adaptive control literature, see e.g. Ge andWang (2004) and Lewis et al. (1999). Note that the set definedby (49) depends on the initial errors and the graph structuralproperties. Therefore, proper accommodation of large initial errorsrequires a larger number of neurons in the NN. The proof alsoreveals that, for a given number of neurons, the admissible initialcondition set is bounded (Lewis et al., 1999).

It is important to select the Lyapunov function candidateV in (29) in terms of locally available variables, e.g. the localneighborhood synchronization error e(t) in (4) and (6). This meansthat any local control signals vi(t) and NN tuning laws developedin the proof are distributed and hence implementable at each node.The use of the Frobenius norm in the Lyapunov function is alsoinstrumental, since it gives rise to Frobenius inner products in theproof that only depend on trace terms, where only the diagonalterms are important. In fact, the Frobenius norm is ideally suitedfor the design of distributed protocols. Eq. (23) shows that P in (29)is chosen diagonal. This is important in allowing selection of theNNtuning law (27) that leads to expression (36).

The result of this Lyapunov design is the local node NN tuninglaw (27) which depends on the local cooperative error ei(t). Bycontrast, standard NN tuning laws (Lewis et al., 1999) depend on aglobal tracking error, equivalent to the disagreement error vectorδ =

x − x0

which is not available at each node. Note that (27) is

a distributed version of the sigma-mod tuning law of Ioannou. Theparameters pi in (27) are computed as in Lemma 2, which requiresglobal information of the graph. As such, they are not known locallyat the nodes. However, NN tuning parameters Fi > 0 are arbitrary,so that piFi > 0 can be arbitrary.

Each node maintains a local NN to provide estimates of itsnonlinear dynamics. The local nature of the NNs, as reflected inthe distributed tuning protocols (27), means that the number ofneurons νi at each node can be selected fairly small (see theSimulations). It is not necessary to select a large centralized NNto approximate the full global vector of nonlinearities f (x) in (14).The total number of neurons in the network is ν =

∑Ni=1 νi.

The unknown dynamics of the control node f (x0, t) in (5)(which could be, e.g., motion) are treated as a disturbance tobe rejected. The proof shows that even though these dynamicsare unknown, synchronization of all nodes to the generally time-varying control node state x0(t) is guaranteed, within a small error.This is verified in the simulations.

Note that the node control gains c are similar to the pinning gainparameters defined in Wang and Chen (2002). According to (46)and (45) increasing these gains results in a smaller synchronizationerror.

Remark 4. If the node states are vectors xi ∈ Rn and x0 ∈ Rn, thecontrol protocols (25) and the error bounds given by (46) or (47)will remain unaltered. The NN weight tuning protocols will thenbe given by

˙W i = −FiϕieTi (pi(di + bi) ⊗ In) − κFiWi. (50)

Remark 5. It is easy to extend the theorem to the case where thedigraph only contains a spanning tree using the Frobenius form(Qu, 2009).

4. Relation of error bounds to graph structural properties

According to the bounds (46) and (47) both the synchronizationerror and the NN weight estimation error increase with themaximum singular values σ (A), σ (L + B), and σ (P). It is desiredto obtain bounds on these singular values in terms of graphproperties. Recall that di =

∑j∈Ni

aij is the (weighted) in-degree

1

2

3

4

5

L

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

of node i, that is, the ith row sum of adjacency matrix A, and doi =∑j aji, the (weighted) out-degree of node i, that is the ith column

sum of A. It is direct to obtain upper bounds on these singularvalues in terms of graph properties. Recalling several results fromBernstein (2005) one can easily show the following.

Lemma 3 (Bounds on Maximum Singular Values of A and (L + B)).

σ (A) ≤

N−i=1

di ≡ vol(G), σ (A) ≤

max

i(di) × max

i(doi )

σ (L + B) ≤

N−i−1

(bi + di + doi )

σ (L + B) ≤

max

i(bi + di + doi ) × max

i(bi + 2di).

These results show that the residual synchronization error isbounded above in terms of the graph complexity expressed interms of volumes and maximum degree sums.

5. Simulation results

This section will show the effectiveness of the distributedadaptive protocol of Theorem 1. It is shown that the protocoleffectively enforces synchronization, using only a few NN nodesat each node, for unknown nonlinear control node dynamics,unknown nonlinear node dynamics, and unknown disturbances ateach node.

For this set of simulations, consider the five-node stronglyconnected digraph structure in Fig. 1 with a leader node connectedto node 3. The edge weights and the pinning gain in (4) were takenequal to 1.Case 1. Nonlinear node dynamics and disturbances

For the graph structure shown above consider the followingnode dynamics

x1 = x31 + u1 + d1, x2 = x22 + u2 + d2x3 = x43 + u3 + d3, x4 = x4 + u4 + d4x5 = x55 + u5 + d5

(51)

which has nonlinearities and disturbances at each node, allassumed unknown. The disturbance at ith node is wi = 0.1 ∗

randn (1) ∗ cos(t). (We use MATLAB symbology.)Consider now the control protocol of Theorem 1. Take the

desired consensus value of x0 = 2, i.e. x0 = f (x0, t) = 0. Thefollowing parameters are used in the simulation. Control Gainsc = 300, number of neurons at each node: νi = 3, κ = 0.8,Fi = 1500. Fig. 2 plots the disagreement vector δ =

x − x0

and

the NN estimation errors fi(xi) − fi(xi). The steady-state consensuserror is approximately zero. Table 1 shows that at steady-state, theNNs closely estimate the nonlinearities. (That is, the nonlinearities

2020 A. Das, F.L. Lewis / Automatica 46 (2010) 2014–2021

Fig. 2. Consensus errors andNNweight estimation errorswith distributed adaptivecontrol.

Table 1Steady-state values.

f (x) Estimated f (x)

8 7.98354 3.8701

16 15.80632 1.8051

32 31.5649

in (51) evaluated at the consensus value of x0 = 2. Recall there aresmall random disturbances wi(t) present.)Case 2. Synchronization of second-order dynamics.

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

q1i = q2i + u1i

q2i = J−1i

u2i − Br

i q2i − Migli sin(q1i) (52)

where qi =q1i , q2i

T∈ R2 is the state vector, Ji is the total inertia

of the link and the motor, Bri is overall damping coefficient, Mi is

total mass, g is gravitational acceleration and li is the distance fromthe joint axis to the link center ofmass for the node. Ji, Br

i ,Mi, g andli are considered unknown andmay be different for each node. Thisis similar to the inertial agent dynamics Jiqi +Br

i qi +Migli sin(qi) =

ui, however, here we take an input into each state component, soit is not the same as second-order consensus. It is easy to treatsecond-order consensus using the method of this paper and it willbe done in another paper.

The desired target node dynamics is taken as the inertial system

m0q0 + d0q0 + koq0 = u0 (53)

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

u0 = − [K1 (q0 − sin(βt)) + K2 (q0 − β cos(βt))]

+ d0q0 + k0q0 + β2m0 sin(βt) (54)

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

The cooperative adaptive control law of Theorem 1 wassimulated, including the Kronecker product with I2 as in Remark 4.Fig. 3 verifies the fact that every node dynamics synchronizes tothe target node dynamics. Fig. 4 is the phase plane plot for all thenodes along with the target node. It can be seen that the phaseplane trajectories of all the nodes, started with different initialconditions, synchronize to the target node phase plane trajectory,finally forming a Lissajous pattern.

Fig. 3. State synchronization errors δi and NN estimation errors.

Fig. 4. Synchronized motion phase plane plot.

Acknowledgements

The ideas in this paper were primarily inspired by the work ofQu (2009), Wang and Chen (2002), and Khoo et al. (2009). We areindebted to their insight. Also important was the excellent book ofBernstein (2005).

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Abhijit Das was born in Kharagpur, India, in 1980. Hereceived his B.E. degree from Bengal Engineering College(Deemed University), Shibpur, in 2003, M.S. degree fromIndian Institute of Technology, Kharagpur in 2006 andPh.D. degree from The University of Texas at Arlington in2010, all in Electrical Engineering. From 2003 to 2006 hewas involved with several projects with Defense Researchand Development Organization (DRDO), India. In 2007,he joined Automation and Robotics Research Institute asa Research Assistant. His Ph.D. dissertation won DeanDissertation Fellowship award in 2010. He is the author

of 1 book, 3 book chapters and several journal and conference articles. He is lifemember of Systems Society of India, studentmember of AIAA, IEEE, SIAM.His profileis also appeared in Marquis Who’s Who in America.

Frank L. Lewis, Fellow IEEE, Fellow IFAC, Fellow U.K. Insti-tute of Measurement & Control, PE Texas, U.K. CharteredEngineer, is Distinguished Scholar Professor andMoncrief-O’Donnell Chair at University of Texas at Arlington’s Au-tomation & Robotics Research Institute. He obtained theBachelor’s Degree in Physics/EE and the MSEE at Rice Uni-versity, the MS in Aeronautical Engineering from Univ. W.Florida, and the Ph.D. at Ga. Tech. He works in feedbackcontrol, intelligent systems, distributed control systems,and sensor networks. He is author of 6 US patents, 216journal papers, 330 conference papers, 14 books, 44 chap-

ters, and 11 journal special issues. He received the Fulbright Research Award, NSFResearch Initiation Grant, ASEE Terman Award, Int. Neural Network Soc. GaborAward 2009, U.K. Inst Measurement & Control Honeywell Field Engineering Medal2009. Received Outstanding Service Award fromDallas IEEE Section, selected as En-gineer of the year by Ft. Worth IEEE Section. Listed in Ft. Worth Business Press Top200 Leaders in Manufacturing. He served on the NAE Committee on Space Stationin 1995. He is an elected Guest Consulting Professor at South China University ofTechnology and Shanghai Jiao Tong University. Founding Member of the Board ofGovernors of the Mediterranean Control Association. Helped win the IEEE ControlSystems Society Best Chapter Award (as Founding Chairman of DFW Chapter), theNational Sigma Xi Award for Outstanding Chapter (as President of UTA Chapter),and the US SBA Tibbets Award in 1996 (as Director of ARRI’s SBIR Program).