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Distributed Consensus Tracking for Multiple Uncertain Nonlinear Strict-Feedback Systems Under a Directed Graph Sung Jin Yoo
Abstract— In this brief, we study the distributed consensus tracking control problem for multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. It is assumed that the leader’s output is time-varying and has been accessed by only a small fraction of followers in a group. The distributed dynamic surface design approach is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader. The function approximation technique using neural networks is employed to compensate unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem, it is shown that the consensus errors are cooperatively semiglobally uniformly ultimately bounded and converge to an adjustable neighborhood of the origin. Index Terms— Consensus, function approximation technique, networked nonlinear systems, unmatched uncertainties.
I. I NTRODUCTION The limitation of communication flow among multi-agent systems in a group has caused active research of the distributed consensus problem in the lack of shared information. Especially, most of these studies only allow for the consensus of multi-agent systems with single-integrator or double-integrator dynamics (see [1]–[6] and references therein for more detail). Even consensus results on the time delays and switching topology were limited to single-integrator or double-integrator systems [7]–[10]. To overcome the limitation, the consensus problems in the communication graph topology have recently extended into nonlinear multi-agent systems [11]–[20] and networked Lagrangian systems, including mechanical dynamics [21]–[25]. However, since practical applications have more complicated dynamics with nonlinearities unmatched in a control input, the previous approaches cannot be applied to the distributed consensus problem of multi-agent systems with unmatched nonlinearities. On the other hand, strict-feedback or lower triangular systems have been widely regarded as a control target due to the description of various physical systems [26]. The systems require the recursive and systematic control design procedure because the control input is not matched with system nonlinearities. The adaptive backstepping design was proposed to satisfy this requirement (see [26], [27] and references therein). Adaptive neural network backstepping controllers Manuscript received May 29, 2012; revised December 25, 2012; accepted December 30, 2012. Date of publication January 18, 2013; date of current version February 13, 2013. This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology under Grant 2012001440. The author is with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2013.2238554
were developed for a class of strict-feedback nonlinear systems [28], [29]. Although the backstepping has become one of the most popular design methods for the strict-feedback systems, it has the “explosion of complexity” problem, which is caused by the repeated differentiations of virtual controllers in each backstepping design step. To overcome this problem, a dynamic surface design technique was proposed for strictfeedback systems [30]. Its main idea is introducing a first-order filtering of the synthesized virtual control law at each step of the backstepping design procedure. The design method was extended and applied to several uncertain nonlinear systems (see [31], [32] and references therein). Despite these efforts, to the best of our knowledge, there are no other any research results for the distributed consensus problem of strict-feedback systems in the presence of networked communications. Motivated by these observations, in this brief, a distributed dynamic surface design approach is proposed for consensus tracking of multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. It is assumed that only a small fraction of followers in the group have access to the information about one time-varying leader. The local consensus controller for each follower is designed using only the neighbors’ information to achieve the consensus tracking between the followers and the leader where the graph-based error surfaces are introduced for the distributed dynamic surface design. The function approximation technique using neural networks is employed to adaptively compensate unknown nonlinearities included in each communication block. The contributions of this brief are two-fold: 1) compared with the existing literature on the consensus problem, the proposed approach can deal with multi-agent systems consisting of several different agents with unmatched nonlinearities, which are unknown on the total communication graph topology, and 2) since the dynamic surface design approach is firstly applied to the distributed consensus tracking problem, the simple local controllers can be designed although the order of the followers and the complexity of communication links increase (see Remark 8 for more detail). The weights for neural networks and the bound of residual approximation error terms are estimated using adaptive laws derived from the Lyapunov stability theorem. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach. II. P RELIMINARIES A. Graph Theory Let G (V, E) be a directed graph with the set of nodes or vertices V {1, . . . , M} and the set of edges or arcs E ⊆ V × V. An edge ( j, i ) ∈ E means that agent i can obtain information from agent j , but not vice versa where j and i are the parent node and child node, respectively. The set of neighbors of a node i is Ni = { j |( j, i ) ∈ E}, which is the set of nodes with edges incoming to node i . A directed path from node i 1 to node i k is a sequence of edges of the form (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k−1 , i k ) in a directed graph. A directed tree is a directed graph where every node has exactly one parent except
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for the root and the root has directed paths to every other node. A directed graph has a directed spanning tree if there exists at least one agent that has directed paths to all other agents. The adjacent matrix A = [ai j ] ∈ R M×M related with G is defined as ai j > 0 if ( j, i ) ∈ E, and ai j = 0 otherwise. Self-edges are not allowed, i.e., aii = 0. The (nonsymmetric) Laplacian matrix L is defined asL = D − A ∈ R M×M where M D = diag[d1 , . . . , d M ]; di = j =1, j =i ai j is the diagonal element of the degree matrix D.
B. Problem Statement Suppose that there exist M followers, labeled as agents 1 to M, and a leader, labeled as an agent 0 under a directed communication graph topology. The dynamic models of M followers in strict-feedback form are considered as: x˙i,k = x i,k+1 + f i,k (x¯i,k ) x˙i,ni = u i + fi,ni (x i ) yi = x i,1
(1)
where i = 1, . . . , M, k = 1, . . . , n i − 1, x¯ i,k = , . . . , x ] ; x p with l = 1, . . . , k, x [x i,1 i,l ∈ R i = i,k pn [x i,1 , . . . , x i,ni ] ∈ R i , and u i ∈ R p are the state vector and the control input of the i th follower, respectively, and yi ∈ R p is the output of the i th follower. f i,k (x¯i,k ) ∈ R p are C 1 nonlinear function vectors. It is assumed that the leader’s motion is independent of the motions of followers. The communication topology for the M + 1 agents is described by a directed graph G (V, E) with V {0, 1, 2, . . . , M}. To represent the communications among ¯ E) ¯ with V¯ followers, we define a subgraph as G¯ (V, {1, 2, . . . , M}. The adjacency matrix A¯ of the subgraph G¯ is ¯ ai j = 0 otherwise, A¯ = [ai j ] ∈ R M×M ; ai j > 0 if ( j, i ) ∈ E, and aii = 0. Then, the Laplacian matrix L is defined as 0 01×M (2) L= −b L¯ + B where b = [b1 , . . . , b M ] , with bi > 0 if the leader 0 ∈ Ni and bi = 0 otherwise, denotes the communication weight from ¯ A¯ the leader to followers, B = diag[b1 , . . . , b M ], and L¯ = D− with D¯ = diag[d1 , . . . , d M ] is the Laplacian matrix of the subgraph denoting the communication among followers. Remark 1: If the directed graph G has a spanning tree, ¯ ¯ rank(L) = M [4]. Then, rank(L+B) = M from (L+B)1 M = b where 1 M is an M-vector of all ones. Therefore, L¯ + B is invertible. Assumption 1: The leader output signal r (t) ∈ R p and its derivative r˙ (t) ∈ R p are bounded and available for the i th followers satisfying 0 ∈ Ni , i = 1, . . . , M. Assumption 2: The states x i,1 and x i,2 of the i th follower are only known and available for the j th followers satisfying i ∈ N j , i = 1, . . . , M, j = 1, . . . , M, and i = j . Assumption 3: The C 1 nonlinear functions f i,k are unknown on a directed graph G. The following definition extends the standard one of semiglobal uniform ultimate boundedness [33] to the distributed consensus control problem.
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Definition 2.1: The distributed consensus tracking errors for nonlinear followers (1) under the communication graph are said to be cooperatively semiglobally uniformly ultimately bounded (CSUUB) if there exist adjustable constants c1 > 0, c2 > 0, and the bounds β1 > 0, β2 > 0, independent of t0 , and for every α1 ∈ (0, c1 ) and α2 ∈ (0, c2 ), there is a time T ≥ 0, independent of t0 , such that yi (t0 ) − r (t0 ) ≤ α1 ⇒ yi (t) − r (t) ≤ β1 and yi (t0 )− y j (t0 ) ≤ α2 ⇒ yi (t)− y j (t) ≤ β2 for all t ≥ t0 + T where i = 1, . . . , M, j = 1, . . . , M, i = j , and k = 1, . . . , n. The objective of this brief is to design neural-networksbased distributed consensus control laws u i for M followers (1) with unknown nonlinearities so that under the directed graph, the follower outputs yi synchronize to the dynamic leader output r while all signals in the total closed-loop systems are bounded. Remark 2: The strict-feedback system (1) can describe many state-space models of nonlinear systems, i.e., various physical systems, such as flight systems, biochemical process, jet engine, robotic systems, and so on [27]. Therefore, the systems (1) under a graph topology can represent multiagent systems consisting of several practical applications with different dynamics. Remark 3: 1) Notice that the followers (1) can have various forms. That is, a group of the followers with the different nonlinear functions and order of the dynamics can be considered in this brief. 2) Assumption 1 means that the leader’s information is available for only a subset of a group of M followers and even the followers only require r and r˙ of the leader although the order of the followers increases. 3) Compared with the previous consensus works [1]–[25], this brief considers the consensus problem of a group of agents consisting of nonlinear followers with nonlinearities unmatched in the control input. Besides, the nonlinearities are unknown under the total communication topology, as stated in Assumption 3.
III. M AIN R ESULTS In this section, we focus on the distributed consensus tracking problem of multiple strict-feedback systems with unknown nonlinearities on the directed graph G (V, E). The neuralnetworks-based distributed dynamic surface design approach is presented for the controller design where the graph-based error surfaces are used and the following function approximation technique is employed to compensate unknown nonlinearities derived from the controller design procedure.
A. Nonlinear Function Approximation Suppose that i,k (·) is an unknown smooth function vector from Rq to R p . Then, the function approximator i,k (·) can approximate any unknown nonlinear function i,k (·), to a sufficient degree of accuracy over the compact set Kνi,k as
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follows: i,k (νi,k ) =
Step 1: The derivative of si,1 along (1), (7), and (7) is s˙i,1 = (di + bi ) si,2 + z i,2 + v i,2 + i,1 (νi,1 )
∗
i,k (νi,k |Wi,k ) + εi,k (νi,k )
∗
i,k ) + [ = i,k (νi,k |W i,k (νi,k |Wi,k )
i,k )] + εi,k (νi,k ) − i,k (νi,k |W
(3)
where νi,k ∈ Kνi,k ⊂ Rq and i,k (·) are an input and an output of the function approximator, respectively, εi,k (νi,k )
i,k = diag[W
i,k,l ]; denotes a reconstruction error vector, W l = 1, . . . , p is an estimate weighting matrix of the optimal ∗ = diag[W ∗ ] defined as W ∗ = weighting matrix Wi,k i,k,l i,k
i,k )]. i,k (νi,k |W arg min W
i,k [ supνi,k ∈Kν i,k (νi,k ) − i,k Assumption 4: Assume that the optimal weighting matrix is ∗ ≤ W ¯ i,k where W¯ i,k is a positive constant bounded as Wi,k F and · F denotes the Frobenius norm. Note that the bounded value W¯ i,k is not required to implement the controller proposed in this brief. This value is used only for the stability analysis of the proposed control ∗ ) system. Taking the Taylor series expansion of i,k (νi,k |Wi,k
i,k for the training of all weights of the function around W approximator, we can obtain [34] ∗ ∗
i,k ) = W i,k
)− i,k (νi,k |W i,k + Hi,k (Wi,k , Wi,k ) i,k (νi,k |Wi,k
i,k (t) = W ∗ − W
i,k (t), i,k = [∂ where W i,k,1 / i,k
∂ Wi,k,1 , . . . , ∂ i,k, p /∂ Wi,k, p ] , and Hi,k (·) is the high-order term. Substituting it into (3) gives [28]
i,k ) + W i,k i,k (νi,k ) = i,k (νi,k |W i,k + αi,k αi,k ≤ α¯ i,k
(4) (5)
where αi,k = Hi,k (·)+εi,k (νi,k ). α¯ i,k > 0 is an unknown value used for the stability analysis of the proposed control system and α¯ i,k is estimated online by an adaptive estimate denoted by αˆ i,k . Remark 4: Notice that the approximator can be used with function approximators such as radial basis function neural networks (RBFNNs), wavelet neural networks, and fuzzy systems, without any difficulty as stated in [35]. B. Distributed Consensus Controller Design The design procedure on the i th follower contains n i steps. From the distributed dynamic surface design, define the graphbased error surfaces si,k and the boundary layer errors z i,k for the i th follower as si,1 =
M
ai j (yi − y j ) + bi (yi − r )
ai j x j,2 − bi r˙ ,
(6)
si,k = x i,k − v¯i,k and (7)
where i = 1, . . . , M and k = 2, . . . , n i , v i,k and v¯i,k are the virtual control and the filtered virtual control, respectively. Remark 5: Compared with the previous dynamic surface design [30]–[32], the graph-based error surfaces si,1 = M a j =1 i j (yi − y j ) + bi (yi − r ) are considered to treat the distributed consensus tracking problem so that the followers’ outputs yi synchronize to the leader output r .
(8)
j =1
where i,1 (νi,1 ) = f i,1 −(1/(di +bi )) M j =1 ai j f j,1 with νi,1 = [x i,1 , x j,1 ] , j ∈ Ni . To stabilize (8), the distributed first virtual control law v i,2 for the i th follower is designed as 1 v i,2 = ai j x j,2 + bi r˙ − i,1 si,1 + di + bi j ∈Ni si,1
− i,1 (νi,1 |Wi,1 ) − αˆ i,1 tanh (9) i,1 where i,1 and i,1 are positive constants, tanh(si,1 /i,1 ) = [ tanh(si,1,1 /i,1 ), . . . , tanh(si,1, p /i,1 )] , and the approxima i,1 ) approximates the unknown nonlinear functor i,1 (νi,1 |W
i,1 and αˆ i,1 are estimates of W ∗ and tion vector i,1 (νi,1 ). W i,1 α¯ i,1 , respectively, and are tuned by the following adaptation laws: ˙
i,1,l (10) W i,1,l = λi,1,l (di + bi ) i,1,l si,1,l − λi,1,l σi,1 W s i,1 α˙ˆ i,1 = ηi,1 (di + bi )si,1 tanh − ηi,1 ςi,1 αˆ i,1 (11) i,1 where λi,1,l > 0 and ηi,1 > 0 are tuning gains, σi,1 and ςi,1 are positive constants for σ -modification [36], and si,1,l and
i,1,l are the lth element of the error i,1,l = ∂ i,1,l (νi,1 )/∂ W vector si,1 and i,1 , respectively. Choose the Lyapunov candidate function Vi,1 as 1 1 2 −1 s si,1 + tr(Wi,1 λi,1 Wi,1 ) + (12) α˜ Vi,1 = 2 i,1 ηi,1 i,1 where λi,1 = diag[λi,1,l ], l = 1, . . . , p, and αi,1 = α¯ i,1 − αˆ i,1 . Differentiating (12) along (4), (5), (8), and (9) yield si,1 + (di + bi )si,1 (si,2 + z i,2 ) V˙i,1 ≤ − i,1 si,1 si,1 1 ˙ + α˜ i,1 (di + bi )si,1 tanh αˆ i,1 − i,1 ηi,1 p 1 ˙ i,1,l W W (di + bi ) i,1,l si,1,l − + i,1,l λi,1,l l=1
+ (di + bi )0.2785 pi,1 α¯ i,1
j =1
z i,k = v¯i,k − v i,k
−
M
(13)
tanh(s / ) ≤ 0.2785 p . Applying where 0 ≤ si,1 −si,1 i,1 i,1 i,1 the adaptation laws (10) and (11) into (13), we get si,1 + (di + bi )si,1 (si,2 + z i,2 ) + ςi,1 α˜ i,1 αˆ i,1 V˙i,1 ≤ − i,1 si,1 i,1 W
i,1 ) + (di + bi )0.2785 pi,1 α¯ i,1 . +σi,1 tr(W (14)
Then, in order to obtain the filtered virtual control vector v¯i,2 , we pass v i,2 through the low-pass first-order filter with a small time constant τi,2 > 0 τi,2 v˙¯i,2 + v¯i,2 = v i,2 , v¯i,2 (0) = v i,2 (0).
(15)
Step k (k = 2, . . . , n i − 1): After differentiating si,k along (1), (7), and (7), we get s˙i,k = si,k+1 + z i,k+1 + v i,k+1 +
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i,k (νi,k ) − v˙¯i,k where i,k (νi,k ) = f i,k with νi,k = x¯ i,k . The distributed kth virtual control law v i,k for the i th follower is proposed as follows:
i,k ) v i,k+1 = − i,k si,k − i,k (νi,k |W v i,k − v¯i,k − αˆ i,k tanh(si,k /i,k ) + τi,k
(16)
where i,k and i,k are positive design parameters, tanh(si,k /i,k ) = [tanh(si,k,1 /i,k ), . . . , tanh(si,k, p /i,k )] , and
i,k ) approximates the unknown the approximator i,k (νi,k |W
i,k and αˆ i,k are estimates of nonlinear function i,k (νi,k ). W ∗ and α Wi,k ¯ i,k , respectively, and are tuned by
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Consider the Lyapunov candidate function Vi,ni as 1 1 2 −1 i,n Vi,ni = W λ ) + α ˜ si,ni si,ni + tr(W (25) i,n i i i,n i 2 ηi,ni i,ni where λi,ni = diag[λi,ni ,l ], l = 1, . . . , p, and αi,ni = α¯ i,ni − αˆ i,ni . Differentiating (25) and substituting s˙i,ni , (22)–(24) into it yield V˙i,ni ≤ − i,ni si,n s + ςi,ni α˜ i,ni αˆ i,ni i i,n i i,n + σi,ni tr(W Wi,ni ) + 0.2785 pi,ni α¯ i,ni . (26) i
Remark 6: The proposed adaptive consensus tracking scheme is summarized as the distributed virtual and actual (17) control [(9), (16), (22)] with adaptive laws [(10), (11), (17), (18) (18), (23), (24)] and the first-order filters [(15), (21)]. Remark 7: Compared with the control methods for strictwhere λi,k,l > 0 and ηi,k > 0 are tuning gains, σi,k and ςi,k are feedback systems [26]–[32] and the consensus or synchroniza i,k,l tion methods for multi-agent systems with nonlinear dynamics positive constants, and si,k,l and i,k,l = ∂ i,k,l (νi,k )/∂ W are the lth element of the error vector si,k and i,k , respec- [11]–[20], the proposed adaptive consensus approach using tively. neural networks has the following advantages. Consider the Lyapunov candidate function Vi,k as 1) In [26]–[32], the controllers were designed for single strict-feedback systems, without considering the com1 1 −1 2 i,k λi,k Wi,k ) + α˜ i,k Vi,k = (19) si,k si,k + tr(W munication among agents. However, this brief considers 2 ηi,k the multiple strict-feedback systems (1) under the direct αi,k = α¯ i,k − αˆ i,k . where λi,k = diag[λi,k,l ], l = 1, . . . , p, and graph topology, i.e., the lack of shared information in Proceeding similarly, differentiate (19) and substitute s˙i,k , communication links. That is, all of the followers cannot (16), (17), and (18) into it. Then, we easily obtain directly receive the information of the leader during the motion and can only use the information of their V˙i,k ≤ − i,k si,k si,k + si,k (si,k+1 + z i,k+1 ) + ςi,k α˜ i,k αˆ i,k neighbors, as reported in Assumptions 1 and 2. i,k + σi,k tr(W (20) Wi,k ) + 0.2785 pi,k α¯ i,k . 2) In [11]–[20], the consensus or synchronization methods for multi-agent systems with nonlinear dynamics were Let v i,k+1 pass through the kth low-pass first-order filter to presented. These methods require some information on obtain the kth filtered virtual control vector v¯i,k+1 , namely nonlinear functions (i.e, bounding conditions such as ˙ QUAD). Besides, the considered nonlinear functions are τi,k+1 v¯i,k+1 + v¯i,k+1 = v i,k+1 , v¯i,k+1 (0) = v i,k+1 (0) (21) matched in the control input. That is, the control input where τi,k+1 is a small positive time constant. can influence nonlinear functions in each state-space Step n i : Using (1) and (7), the derivative of si,ni can equation. Therefore, these methods [11]–[20] limit their be represented by s˙i,ni = u i + i,ni (νi,ni ) − v˙¯i,ni where applications to nonlinear systems with matched nonlini,ni (νi,ni ) = f i,ni with νi,ni = x i . We propose a distributed earities. However, the proposed approach in this brief actual control input on the i th follower as can deal with the follower systems with C 1 nonlinearities unmatched in the control input. Also, the informa ) u i = − i,ni si,ni − (ν |W i,ni i,ni i,ni tion on all nonlinearities under the total directed commuv i,ni − v¯i,ni si,ni nication graph topology is not required to implement the + (22) −αˆ i,ni tanh i,ni τi,ni local controllers of the followers since the approximators using neural networks are designed to estimate the where i,ni are positive design parameters, i,ni and nonlinear functions, including neighbors’ nonlinearities tanh si,ni /i,ni = [tanh(si,ni ,1 /i,ni ), . . . , tanh(si,ni , p /i,ni )] ,
(i.e., the approximator term i,1 in (9) estimates the and the approximator i,ni (νi,ni |Wi,ni ) approximates the
nonlinear functions including neighbors’ nonlinearities unknown nonlinear function i,ni (νi,ni ). Wi,ni and αˆ i,ni are M ∗ (ν ) = f − (1/(d + b )) a f i,1 i,1 i,1 i i estimates of Wi,ni and α¯ i,ni , respectively, and are tuned by j =1 i j j,1 ). Remark 8: Notice that the proposed consensus control ˙
i,ni ,l
W (23) scheme has the following advantage in the sense of the i,n i ,l = λi,n i ,l i,n i ,l si,n i ,l − λi,n i ,l σi,n i W α˙ˆ i,ni = ηi,ni si,ni tanh(si,ni /i,ni ) − ηi,ni ςi,ni αˆ i,ni (24) dynamic surface design. The first virtual controller (9) include the neighbors’ state information and the function approxiwhere λi,ni ,l > 0 and ηi,ni > 0 are tuning gains, σi,ni mator term estimating the neighbors’ nonlinearities. If the and ςi,ni are positive constants, and si,ni ,l and i,ni ,l = backstepping approach is used for the follower systems under
i,ni ,l are the lth element of the error vector the network communication topology, the actual control law ∂ i,ni ,l (νi,ni )/∂ W si,ni and i,ni , respectively. requires the repeated differentiations of the virtual controllers, ˙
i,k,l W i,k,l = λi,k,l i,k,l si,k,l − λi,k,l σi,k W s i,k − ηi,k ςi,k αˆ i,k α˙ˆ i,k = ηi,k si,k tanh i,k
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including (9). Therefore, the complexity of the actual control law drastically grows due to the increase of the repeated differentiations of the virtual controllers, including neural networks and the neighbors’ states as the complexity of the communication links and the order of the followers increase. This problem becomes more serious in the case of the group consisting of a large number of followers. However, the proposed approach based on the dynamic surface design does not require the repeated derivative terms of states of neighbors communicated in the directed graph and the neural network approximators because of the use of the filters (15) and (21) at the design steps. Accordingly, the simpler controller can be designed compared with the backstepping design. Remark 9: In [24], the connection weights between outputs and hidden nodes of RBFNNs are only trained by the adaptation laws. That is, the centers and widths of basis functions are randomly chosen in a big compact set. This method can degrade the approximation performance of the RBFNNs because it is difficult to initially design the optimal centers and widths of the basis functions for estimating uncertain nonlinear functions. Therefore, in our control system, the tuning laws for all weights, including centers and widths of the basis functions of approximators, are derived from the stability analysis. C. Stability Analysis Differentiating (7) yields z i,2
i,1 , αˆ i,1 , s j,3 , z j,3 , R + i,2 si,1 , si,2 , z i,2 , W z˙ i,2 = − τi,2 (27) z i,k+1 z˙ i,k+1 = − + i,k+1 si,1 , . . . , si,k+1 , z i,2 , . . . , z i,k+1 , τi,k+1
Wi,1 , . . . , Wi,k+1 , αˆ i,1 , . . . , αˆ i,k+1 , R (28) where i = 1, . . . , M, k = 2, . . . , n i − 1, j ∈ Ni , R = [r , r˙ , r¨ ] , i,2 (·) = (1/(di + bi ))( i,1 s˙i,1 − ˙ ˙ˆ i,1 tanh(si,1 /i,1 ) +
j ∈Ni ai j x˙ j,2 − bi r¨ ) + i,1 (νi,1 | Wi,1 ) + α 2 αˆ i,1 (1 − tanh (si,1 /i,1 ))(˙si,1 /i,1 ) and i,k+1 (·) =
i,k ) + α˙ˆ i,k tanh(si,k /i,k ) + αˆ i,k (1 − ˙ i,k (νi,k |W
i,k s˙i,k + tanh2 (si,k /i,k ))(˙si,k /i,k ) + z˙ i,k /τi,k are continuous functions. For the stability analysis, choose the total Lyapunov candidate function V as ni n i −1 M 1 Vi,k + z i,k+1 z i,k+1 . (29) V = 2 i=1
k=1
k=1
Based on the proposed design method, the main result of this brief is presented in the following theorem. Theorem 3.1: Consider the multiple uncertain nonlinear strict-feedback systems (1) controlled by the distributed consensus controller (22) and assume that the leader has directed paths to all followers 1 to M. Under Assumptions 1–4, for any initial conditions satisfying V (0) ≤ μ, the consensus tracking errors between the leader and the followers in the overall closed-loop system are CSUUB and can be made arbitrarily small.
Proof: Differentiating (29) and substituting (14), (20), (26)–(28) into it, we get ni M ˙ − i,k si,k si,k +σi,k tr(Wi,k Wi,k )+ςi,k α˜ i,k αˆ i,k V = i=1
k=1
+(di + bi )si,1 (si,2 + z i,2 ) +
n i −1
+(di + bi )0.2785 pi,1 α¯ i,1 + +
n i −1 k=1
k=2 ni
si,k (si,k+1 + z i,k+1 )
0.2785 pi,k α¯ i,k
k=2
−
z i,k+1 z i,k+1
τi,k+1
+
z i,k+1 i,k+1
.
W Using Young’s inequality1 and the inequalities tr(W i,k i,k ) = 1 1 2 2 i,k F W¯ i,k −W i,k = − W i,k − (W i,k F − W¯ i,k )2 + W F F 2 2 1 ¯2 1 ¯2 2 = −1α 2 2 ˜ i,k |α¯ i,k − α˜ i,k 2 Wi,k ≤ 2 ( Wi,k − Wi,k F ) and |α 2 ˜ i,k − 1 1 2 1 2 2 2 ˜ i,k | − α¯ i,k ) + 2 α¯ i,k ≤ 2 (α¯ i,k − α˜ i,k ), we have 2 (|α V˙ ≤
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where ζ > constant and M 0niis a positive 2 2 ) ¯ [ ((σ /2) W + (ςi,k /2)α¯ i,k C = i,k i=1 k=1 ni,k i ¯ i,k + + (d + bi )0.2785 pi,1 α¯ i,1 + k=2 0.2785 pi,k α ni −1i (ζ /2)]. k=1 i k s ( h=1 (sl,h Consider the sets i,k = { l=1 l,h + k−1 −1 2 tr(W λ Wl,h ) + (1/ηl,h )α˜ l,h ) + h=1 z l,h+1 z l,h+1 ) + l,h l,h r + r˙ r˙ + r¨ r¨ ≤ R } s s ≤ 2μ} and = {r m,k+1 0 m∈Ni m,k+1 where i = 1, . . . , M, k = 1, . . . , n i , R0 > 0 is a constant. i,k and are compact in Rdim(i,k ) and R3 p , respectively. Then, since i,k × is also compact in Rdim(i,k )+3 p , there exist positive constants Q i,k+1 such that i,k+1 ≤ Q i,k+1 on i,k × , where dim(i,k ) denotes the dimension of the set i,k . ∗ , Choosing the parameters i,1 = (di + bi ) + i,1 i,k = 1 + ∗ ∗ ((di +bi )/2)+ i,k , i,ni = 1/2+ i,ni , 1/τi,2 = ((di +bi )/2)+ ∗ , and 1/τ 2 ∗ (Q 2i,2 /2ζ ) + τi,2 i,k+1 = 1/2 + (Q i,k+1 /2ζ ) + τi,k+1 ∗ ∗ with k = 2, . . . , n i − 1, positive constants i,k , τi,k+1 , ζ , we finally get V˙ ≤ −γ V + C (30) ∗ , 2τ ∗ , σ λ where γ = min [2 i,k i,h i,k i,k,l , ςi,k ηi,k ] > 0 with i = 1, . . . , M, k = 1, . . . , n i , h = 2, . . . , n i , and l = 1, . . . , p. The inequality (30) implies V˙ < 0 on V = μ when γ > C/μ. Therefore, V ≤ μ is an invariant set, i.e., if V (0) ≤ μ, then V (t) ≤ μ for all t ≥ 0. Accordingly, the consensus tracking errors in the overall closed-loop system are CSUUB. In 1 ab ≤ a p / p + bq /q where a,b ≥ 0 and p, q > 0 such that 1/ p + 1/q = 1.
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Fig. 3. Outputs of function approximators. (a) ˆ 1,1 and ˆ 2,1 . (b) ˆ 3,1 and ˆ 4,1 . (c) ˆ 1,2 and ˆ 2,2 . (d) ˆ 3,2 and ˆ 4,2 .
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(b) Fig. 2. Consensus results. (a) Output trajectories of the followers with one leader in the 2-D space and (b) ei,o .
e−γ t V (0)
addition, from (30), we know that V (t) ≤ + (C/γ )[1 − e−γ t ]. Using 1/2s1 2 ≤ V (t) with s1 = , . . . , s ] , we get s 2 ≤ 2e −γ t V (0) + (2C/γ )[1 − [s1,1 1 M,1 −γ e t ]. Therefore, as time increases, all error surfaces s1 exponentially converge to the compact set = {s1 |s1 ≤ √ 2C/γ }. The compact set can be kept arbitrarily small by increasing γ . Then, from Remark 1 and s1 = ((L¯ + B) ⊗ ] , I is an identity I p )(y − (1 M ⊗r )) where y = [y1 , . . . , y M p matrix of order p, and ⊗ stands for the Kronecker product, the consensus tracking errors in the overall closed-loop system can be reduced. Remark 10: The design parameters i,k , λi,k , ηi,k , σi,k , ςi,k , i,k are only a sufficient condition and provide a guideline for the designers. From the proof of Theorem 3.1, some suggestions are given for the choice of some key design parameters as follows: (1) increasing i,k , λi,k , ηi,k helps to increase γ , subsequently reduces the bound 2C/γ of error, and (2) decreasing σi,k , ςi,k , i,k helps to decrease C, and reduces 2C/γ .
In this section, we present simulation results to validate the proposed theoretical result. We consider a group of one leader and four followers with the following nonlinear strict-feedback dynamics in the 2-D space (i.e., p = 2) described by x˙i,1 = x i,2 + f i,1 (x i,1 ) x˙i,2 = u i + fi,2 (x i ) yi = x i,1
(31)
where i = 1, . . . , 4, x i,k = [x i,k1 , x i,k2 ] ; k = 1, 2, x i = , x ] , y = [y , y ] , f (x ) = [ sin(x [x i,1 i i,1 i,2 i,1 i,1 i,11 )x i,12 , i,2 cos(x i,12 )] , and f i,2 (x i ) = e−0.4xi,11 x i,21 [ cos(x i,21 ), x i,22 ] . The directed network topology for the simulation is shown in Fig. 1. We choose ai j = 1 on j ∈ Ni , ai j = 0 otherwise, b1 = b2 = b3 = 0, and b4 = 1. The leader output r (t) is r (t) = [r1 (t), r2 (t)] = [4 sin(t/20), 2 sin(0.8t) + (t/5)] and the initial states of the four followers are set to x 1 (0) = [ − 1, 0, 1, −1] , x 2 (0) = [ − 1, −2, 1, −3] , x 3 (0) = [0, −2, 2, −2] , and x 4 (0) = [ − 1, −1, 2, 3] . To display the consensus tracking errors between the leader and the followers, we use ei,o = x i,1 (t) − r (t) where i = 1, . . . , 4. For the simulation, RBFNN approximators are used to estimate unknown nonlinear functions i,k where i = 1, . . . , 4 and k = 2. The design parameters for distributed consensus controllers are chosen as i,1 = 15, i,2 = 10, τi,2 = 0.005, λi,1,1 = 0.2, λ1,1,2 = λ2,1,2 = λ4,1,2 = 0.1, λ3,1,2 = 0.05, λi,2,l = 0.001, ηi,k = 0.02, i,k = 0.1, and σi,k = ςi,k = 0.001 with i = 1, . . . , 4, k = 1, 2, and l = 1, 2. Fig. 2(a) shows the outputs of the four followers and one leader in the 2-
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D space where the square boxes denote the initial points of the followers and the circles are the output snapshots of the followers at t = 5 s, t = 10 s, and t = 15 s. The consensus tracking errors ei,o drop quickly in less than a few seconds, as shown in Fig. 2(b). The outputs of function approximators are plotted in Fig. 3. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology, although the followers have the unknown nonlinearities unmatched in the control input. V. C ONCLUSION We proposed a distributed adaptive consensus control approach for multiple strict-feedback systems with unknown nonlinearities under the directed communication graph. Each local distributed consensus controller was designed by applying the dynamic surface design to the distributed consensus tracking problem. The graph-based error surfaces for the distributed dynamic surface design are used to guarantee the consensus tracking between the followers and one leader. The function approximation technique using neural networks was employed to compensate the unknown nonlinearities unmatched in the control input of followers. Finally, the effectiveness of the proposed approach was proved through a computer simulation. ACKNOWLEDGMENT The author would like to thank Associate Editor and the anonymous reviewers for their helpful comments and constructive suggestions with regard to this brief. R EFERENCES [1] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. [2] Y. Hong, L. Gao, D. Cheng, and J. Hu, “Lyapunov-based approach to multi-agent systems with switching jointly-connected interconnection,” IEEE Trans. Automat. Control, vol. 52, no. 5, pp. 943–948, May 2007. [3] W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control. New York: Springer-Verlag, 2008. [4] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Trans. Autom. Control, vol. 53, no. 6, pp. 1053–1059, Jul. 2008. [5] H. Su, X. Wang, and G. Chen, “Rendezvous of multiple mobile agents with preserved network connectivity,” Syst. Control Lett., vol. 59, no. 5, pp. 313–322, 2010. [6] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Int. J. Robust Nonlinear Control, vol. 17, nos. 10–11, pp. 941–959, 2007. [7] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. [8] P. Lin, Y. Jia, and L. Li, “Distributed robust H1 consensus control in directed networks of agents with time-delay,” Syst. Control Lett., vol. 57, no. 8, pp. 643–653, 2008. [9] Y. Zhnag and Y. Tian, “Consentability and protocol design of multiagent systems with stochastic switching topology,” Automatica, vol. 45, no. 5, pp. 1195–1201, 2009. [10] Y. Tian and C. Liu, “Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations,” Automatica, vol. 45, no. 5, pp. 1347–1353, 2009. [11] W. Yu, G. Chen, M. Cao, and J. Kurths, “Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 881–891, Jun. 2010.
[12] H. Su, G. Chen, X. Wang, and Z. Lin, “Adaptive second-order consensus of networked mobile agents with nonlinear dynamics,” Automatica, vol. 47, no. 2, pp. 368–375, 2011. [13] X. Liu, T. Chen, and W. Lu, “Consensus problem in directed networks of multi-agents via nonlinear protocols,” Phys. Lett. A, vol. 373, no. 35, pp. 3122–3127, 2009. [14] X. Liu and T. Chen, “Cluster synchronization in directed networks via intermittent pinning control,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1009–1020, Jul. 2011. [15] P. De Lellis, M. di Bernardo, and F. Garofalo, “Novel decentralized adaptive strategies for the synchronization of complex networks,” Automatica, vol. 45, no. 5, pp. 1312–1318, 2009. [16] P. De Lellis, M. di Bernardo, F. Garofalo, and M. Porfiri, “Evolution of complex networks via edge snapping,” IEEE Trans. Circuits Syst. I, Regular Paper, vol. 57, no. 8, pp. 2132–2143, Aug. 2010. [17] P. DeLellis, M. di Bernardo, T. E. Gorochowski, and G. Russo, “Synchronization and control of complex networks via contraction, adaptation and evolution,” IEEE Circuits Syst. Mag., vol. 10, no. 3, pp. 64–82, Aug. 2010. [18] X.-Z. Jin, G.-H. Yang, and W.-W. Che, “Adaptive pinning control of deteriorated nonlinear coupling networks with circuit realization,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 9, pp. 1345–1355, Sep. 2012. [19] J. Liang, Z. Wang, X. Liu, and P. Louvieris, “Robust synchronization for 2-D discrete-time coupled dynamical networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 6, pp. 942–953, Jun. 2012. [20] J. Lu, J. Kurths, J. Cao, N. Mahdavi, and C. Huang, “Synchronization control for nonlinear stochastic dynamical networks: Pinning impulsive strategy,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 285–292, Feb. 2012. [21] G. Chen and F. L. Lewis, “Distributed adaptive tracking control for synchronization of unknown networked Lagrangian systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 3, pp. 805–816, Jun. 2011. [22] W. Ren, “Distributed leaderless consensus algorithms for networked Euler-Lagrange systems,” Int. J. Control, vol. 82, no. 11, pp. 2137–2149, 2009. [23] J. Mei, W. Ren, and G. Ma, “Distributed containment control for Lagrangian networks with parameteric uncertaities under a directed graph,” Automatica, vol. 48, no. 4, pp. 653–659, 2012. [24] Z. G. Hou, L. Cheng, and M. Tan, “Decentralized robust adaptive control for multiagent system consensus problem using neural networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 3, pp. 636–647, Jun. 2009. [25] S. Chung and J. J. E. Slotine, “Cooperative robot control and concurrent synchronization of Lagrangian systems,” IEEE Trans. Robot., vol. 25, no. 3, pp. 686–700, Jun. 2009. [26] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Autom. Control, vol. 36, no. 11, pp. 1241–1253, Nov. 1991. [27] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley Interscience, 1995. [28] M. M. Polycarpou and M. J. Mears, “Stable adaptive tracking of uncertain systems using nonlinearly parameterized on-line approximators,” Int. J. Control, vol. 70, no. 3, pp. 363–384, 1998. [29] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Automat. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [30] D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. 45, no. 10, pp. 1893–1899, Oct. 2000. [31] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005. [32] S. J. Yoo, J. B. Park, and Y. H. Choi, “Adaptive dynamic surface control for stabilization of parametric strict-feedback nonlinear systems with unknown time delays,” IEEE Trans. Autom. Control, vol. 52, no. 12, pp. 2360–2365, Dec. 2007. [33] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996. [34] F. J. Lin, T. S. Lee, and C. H. Lin, “Robust H∞ controller design with recurrent neural network for linear synchronous motor drive,” IEEE Trans. Ind. Electron., vol. 50, no. 3, pp. 456–470, Jun. 2003. [35] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2001. [36] P. A. Ioannou and P. V. Kokotovic, Adaptive Systems with Reduced Models. New York: Springer-Verlag, 1983.
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Distributed Consensus Tracking for Multiple Uncertain Nonlinear Strict-Feedback Systems Under a Directed Graph Sung Jin Yoo
Abstract— In this brief, we study the distributed consensus tracking control problem for multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. It is assumed that the leader’s output is time-varying and has been accessed by only a small fraction of followers in a group. The distributed dynamic surface design approach is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader. The function approximation technique using neural networks is employed to compensate unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem, it is shown that the consensus errors are cooperatively semiglobally uniformly ultimately bounded and converge to an adjustable neighborhood of the origin. Index Terms— Consensus, function approximation technique, networked nonlinear systems, unmatched uncertainties.
I. I NTRODUCTION The limitation of communication flow among multi-agent systems in a group has caused active research of the distributed consensus problem in the lack of shared information. Especially, most of these studies only allow for the consensus of multi-agent systems with single-integrator or double-integrator dynamics (see [1]–[6] and references therein for more detail). Even consensus results on the time delays and switching topology were limited to single-integrator or double-integrator systems [7]–[10]. To overcome the limitation, the consensus problems in the communication graph topology have recently extended into nonlinear multi-agent systems [11]–[20] and networked Lagrangian systems, including mechanical dynamics [21]–[25]. However, since practical applications have more complicated dynamics with nonlinearities unmatched in a control input, the previous approaches cannot be applied to the distributed consensus problem of multi-agent systems with unmatched nonlinearities. On the other hand, strict-feedback or lower triangular systems have been widely regarded as a control target due to the description of various physical systems [26]. The systems require the recursive and systematic control design procedure because the control input is not matched with system nonlinearities. The adaptive backstepping design was proposed to satisfy this requirement (see [26], [27] and references therein). Adaptive neural network backstepping controllers Manuscript received May 29, 2012; revised December 25, 2012; accepted December 30, 2012. Date of publication January 18, 2013; date of current version February 13, 2013. This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology under Grant 2012001440. The author is with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2013.2238554
were developed for a class of strict-feedback nonlinear systems [28], [29]. Although the backstepping has become one of the most popular design methods for the strict-feedback systems, it has the “explosion of complexity” problem, which is caused by the repeated differentiations of virtual controllers in each backstepping design step. To overcome this problem, a dynamic surface design technique was proposed for strictfeedback systems [30]. Its main idea is introducing a first-order filtering of the synthesized virtual control law at each step of the backstepping design procedure. The design method was extended and applied to several uncertain nonlinear systems (see [31], [32] and references therein). Despite these efforts, to the best of our knowledge, there are no other any research results for the distributed consensus problem of strict-feedback systems in the presence of networked communications. Motivated by these observations, in this brief, a distributed dynamic surface design approach is proposed for consensus tracking of multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. It is assumed that only a small fraction of followers in the group have access to the information about one time-varying leader. The local consensus controller for each follower is designed using only the neighbors’ information to achieve the consensus tracking between the followers and the leader where the graph-based error surfaces are introduced for the distributed dynamic surface design. The function approximation technique using neural networks is employed to adaptively compensate unknown nonlinearities included in each communication block. The contributions of this brief are two-fold: 1) compared with the existing literature on the consensus problem, the proposed approach can deal with multi-agent systems consisting of several different agents with unmatched nonlinearities, which are unknown on the total communication graph topology, and 2) since the dynamic surface design approach is firstly applied to the distributed consensus tracking problem, the simple local controllers can be designed although the order of the followers and the complexity of communication links increase (see Remark 8 for more detail). The weights for neural networks and the bound of residual approximation error terms are estimated using adaptive laws derived from the Lyapunov stability theorem. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach. II. P RELIMINARIES A. Graph Theory Let G (V, E) be a directed graph with the set of nodes or vertices V {1, . . . , M} and the set of edges or arcs E ⊆ V × V. An edge ( j, i ) ∈ E means that agent i can obtain information from agent j , but not vice versa where j and i are the parent node and child node, respectively. The set of neighbors of a node i is Ni = { j |( j, i ) ∈ E}, which is the set of nodes with edges incoming to node i . A directed path from node i 1 to node i k is a sequence of edges of the form (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k−1 , i k ) in a directed graph. A directed tree is a directed graph where every node has exactly one parent except
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for the root and the root has directed paths to every other node. A directed graph has a directed spanning tree if there exists at least one agent that has directed paths to all other agents. The adjacent matrix A = [ai j ] ∈ R M×M related with G is defined as ai j > 0 if ( j, i ) ∈ E, and ai j = 0 otherwise. Self-edges are not allowed, i.e., aii = 0. The (nonsymmetric) Laplacian matrix L is defined asL = D − A ∈ R M×M where M D = diag[d1 , . . . , d M ]; di = j =1, j =i ai j is the diagonal element of the degree matrix D.
B. Problem Statement Suppose that there exist M followers, labeled as agents 1 to M, and a leader, labeled as an agent 0 under a directed communication graph topology. The dynamic models of M followers in strict-feedback form are considered as: x˙i,k = x i,k+1 + f i,k (x¯i,k ) x˙i,ni = u i + fi,ni (x i ) yi = x i,1
(1)
where i = 1, . . . , M, k = 1, . . . , n i − 1, x¯ i,k = , . . . , x ] ; x p with l = 1, . . . , k, x [x i,1 i,l ∈ R i = i,k pn [x i,1 , . . . , x i,ni ] ∈ R i , and u i ∈ R p are the state vector and the control input of the i th follower, respectively, and yi ∈ R p is the output of the i th follower. f i,k (x¯i,k ) ∈ R p are C 1 nonlinear function vectors. It is assumed that the leader’s motion is independent of the motions of followers. The communication topology for the M + 1 agents is described by a directed graph G (V, E) with V {0, 1, 2, . . . , M}. To represent the communications among ¯ E) ¯ with V¯ followers, we define a subgraph as G¯ (V, {1, 2, . . . , M}. The adjacency matrix A¯ of the subgraph G¯ is ¯ ai j = 0 otherwise, A¯ = [ai j ] ∈ R M×M ; ai j > 0 if ( j, i ) ∈ E, and aii = 0. Then, the Laplacian matrix L is defined as 0 01×M (2) L= −b L¯ + B where b = [b1 , . . . , b M ] , with bi > 0 if the leader 0 ∈ Ni and bi = 0 otherwise, denotes the communication weight from ¯ A¯ the leader to followers, B = diag[b1 , . . . , b M ], and L¯ = D− with D¯ = diag[d1 , . . . , d M ] is the Laplacian matrix of the subgraph denoting the communication among followers. Remark 1: If the directed graph G has a spanning tree, ¯ ¯ rank(L) = M [4]. Then, rank(L+B) = M from (L+B)1 M = b where 1 M is an M-vector of all ones. Therefore, L¯ + B is invertible. Assumption 1: The leader output signal r (t) ∈ R p and its derivative r˙ (t) ∈ R p are bounded and available for the i th followers satisfying 0 ∈ Ni , i = 1, . . . , M. Assumption 2: The states x i,1 and x i,2 of the i th follower are only known and available for the j th followers satisfying i ∈ N j , i = 1, . . . , M, j = 1, . . . , M, and i = j . Assumption 3: The C 1 nonlinear functions f i,k are unknown on a directed graph G. The following definition extends the standard one of semiglobal uniform ultimate boundedness [33] to the distributed consensus control problem.
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Definition 2.1: The distributed consensus tracking errors for nonlinear followers (1) under the communication graph are said to be cooperatively semiglobally uniformly ultimately bounded (CSUUB) if there exist adjustable constants c1 > 0, c2 > 0, and the bounds β1 > 0, β2 > 0, independent of t0 , and for every α1 ∈ (0, c1 ) and α2 ∈ (0, c2 ), there is a time T ≥ 0, independent of t0 , such that yi (t0 ) − r (t0 ) ≤ α1 ⇒ yi (t) − r (t) ≤ β1 and yi (t0 )− y j (t0 ) ≤ α2 ⇒ yi (t)− y j (t) ≤ β2 for all t ≥ t0 + T where i = 1, . . . , M, j = 1, . . . , M, i = j , and k = 1, . . . , n. The objective of this brief is to design neural-networksbased distributed consensus control laws u i for M followers (1) with unknown nonlinearities so that under the directed graph, the follower outputs yi synchronize to the dynamic leader output r while all signals in the total closed-loop systems are bounded. Remark 2: The strict-feedback system (1) can describe many state-space models of nonlinear systems, i.e., various physical systems, such as flight systems, biochemical process, jet engine, robotic systems, and so on [27]. Therefore, the systems (1) under a graph topology can represent multiagent systems consisting of several practical applications with different dynamics. Remark 3: 1) Notice that the followers (1) can have various forms. That is, a group of the followers with the different nonlinear functions and order of the dynamics can be considered in this brief. 2) Assumption 1 means that the leader’s information is available for only a subset of a group of M followers and even the followers only require r and r˙ of the leader although the order of the followers increases. 3) Compared with the previous consensus works [1]–[25], this brief considers the consensus problem of a group of agents consisting of nonlinear followers with nonlinearities unmatched in the control input. Besides, the nonlinearities are unknown under the total communication topology, as stated in Assumption 3.
III. M AIN R ESULTS In this section, we focus on the distributed consensus tracking problem of multiple strict-feedback systems with unknown nonlinearities on the directed graph G (V, E). The neuralnetworks-based distributed dynamic surface design approach is presented for the controller design where the graph-based error surfaces are used and the following function approximation technique is employed to compensate unknown nonlinearities derived from the controller design procedure.
A. Nonlinear Function Approximation Suppose that i,k (·) is an unknown smooth function vector from Rq to R p . Then, the function approximator i,k (·) can approximate any unknown nonlinear function i,k (·), to a sufficient degree of accuracy over the compact set Kνi,k as
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follows: i,k (νi,k ) =
Step 1: The derivative of si,1 along (1), (7), and (7) is s˙i,1 = (di + bi ) si,2 + z i,2 + v i,2 + i,1 (νi,1 )
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i,k )] + εi,k (νi,k ) − i,k (νi,k |W
(3)
where νi,k ∈ Kνi,k ⊂ Rq and i,k (·) are an input and an output of the function approximator, respectively, εi,k (νi,k )
i,k = diag[W
i,k,l ]; denotes a reconstruction error vector, W l = 1, . . . , p is an estimate weighting matrix of the optimal ∗ = diag[W ∗ ] defined as W ∗ = weighting matrix Wi,k i,k,l i,k
i,k )]. i,k (νi,k |W arg min W
i,k [ supνi,k ∈Kν i,k (νi,k ) − i,k Assumption 4: Assume that the optimal weighting matrix is ∗ ≤ W ¯ i,k where W¯ i,k is a positive constant bounded as Wi,k F and · F denotes the Frobenius norm. Note that the bounded value W¯ i,k is not required to implement the controller proposed in this brief. This value is used only for the stability analysis of the proposed control ∗ ) system. Taking the Taylor series expansion of i,k (νi,k |Wi,k
i,k for the training of all weights of the function around W approximator, we can obtain [34] ∗ ∗
i,k ) = W i,k
)− i,k (νi,k |W i,k + Hi,k (Wi,k , Wi,k ) i,k (νi,k |Wi,k
i,k (t) = W ∗ − W
i,k (t), i,k = [∂ where W i,k,1 / i,k
∂ Wi,k,1 , . . . , ∂ i,k, p /∂ Wi,k, p ] , and Hi,k (·) is the high-order term. Substituting it into (3) gives [28]
i,k ) + W i,k i,k (νi,k ) = i,k (νi,k |W i,k + αi,k αi,k ≤ α¯ i,k
(4) (5)
where αi,k = Hi,k (·)+εi,k (νi,k ). α¯ i,k > 0 is an unknown value used for the stability analysis of the proposed control system and α¯ i,k is estimated online by an adaptive estimate denoted by αˆ i,k . Remark 4: Notice that the approximator can be used with function approximators such as radial basis function neural networks (RBFNNs), wavelet neural networks, and fuzzy systems, without any difficulty as stated in [35]. B. Distributed Consensus Controller Design The design procedure on the i th follower contains n i steps. From the distributed dynamic surface design, define the graphbased error surfaces si,k and the boundary layer errors z i,k for the i th follower as si,1 =
M
ai j (yi − y j ) + bi (yi − r )
ai j x j,2 − bi r˙ ,
(6)
si,k = x i,k − v¯i,k and (7)
where i = 1, . . . , M and k = 2, . . . , n i , v i,k and v¯i,k are the virtual control and the filtered virtual control, respectively. Remark 5: Compared with the previous dynamic surface design [30]–[32], the graph-based error surfaces si,1 = M a j =1 i j (yi − y j ) + bi (yi − r ) are considered to treat the distributed consensus tracking problem so that the followers’ outputs yi synchronize to the leader output r .
(8)
j =1
where i,1 (νi,1 ) = f i,1 −(1/(di +bi )) M j =1 ai j f j,1 with νi,1 = [x i,1 , x j,1 ] , j ∈ Ni . To stabilize (8), the distributed first virtual control law v i,2 for the i th follower is designed as 1 v i,2 = ai j x j,2 + bi r˙ − i,1 si,1 + di + bi j ∈Ni si,1
− i,1 (νi,1 |Wi,1 ) − αˆ i,1 tanh (9) i,1 where i,1 and i,1 are positive constants, tanh(si,1 /i,1 ) = [ tanh(si,1,1 /i,1 ), . . . , tanh(si,1, p /i,1 )] , and the approxima i,1 ) approximates the unknown nonlinear functor i,1 (νi,1 |W
i,1 and αˆ i,1 are estimates of W ∗ and tion vector i,1 (νi,1 ). W i,1 α¯ i,1 , respectively, and are tuned by the following adaptation laws: ˙
i,1,l (10) W i,1,l = λi,1,l (di + bi ) i,1,l si,1,l − λi,1,l σi,1 W s i,1 α˙ˆ i,1 = ηi,1 (di + bi )si,1 tanh − ηi,1 ςi,1 αˆ i,1 (11) i,1 where λi,1,l > 0 and ηi,1 > 0 are tuning gains, σi,1 and ςi,1 are positive constants for σ -modification [36], and si,1,l and
i,1,l are the lth element of the error i,1,l = ∂ i,1,l (νi,1 )/∂ W vector si,1 and i,1 , respectively. Choose the Lyapunov candidate function Vi,1 as 1 1 2 −1 s si,1 + tr(Wi,1 λi,1 Wi,1 ) + (12) α˜ Vi,1 = 2 i,1 ηi,1 i,1 where λi,1 = diag[λi,1,l ], l = 1, . . . , p, and αi,1 = α¯ i,1 − αˆ i,1 . Differentiating (12) along (4), (5), (8), and (9) yield si,1 + (di + bi )si,1 (si,2 + z i,2 ) V˙i,1 ≤ − i,1 si,1 si,1 1 ˙ + α˜ i,1 (di + bi )si,1 tanh αˆ i,1 − i,1 ηi,1 p 1 ˙ i,1,l W W (di + bi ) i,1,l si,1,l − + i,1,l λi,1,l l=1
+ (di + bi )0.2785 pi,1 α¯ i,1
j =1
z i,k = v¯i,k − v i,k
−
M
(13)
tanh(s / ) ≤ 0.2785 p . Applying where 0 ≤ si,1 −si,1 i,1 i,1 i,1 the adaptation laws (10) and (11) into (13), we get si,1 + (di + bi )si,1 (si,2 + z i,2 ) + ςi,1 α˜ i,1 αˆ i,1 V˙i,1 ≤ − i,1 si,1 i,1 W
i,1 ) + (di + bi )0.2785 pi,1 α¯ i,1 . +σi,1 tr(W (14)
Then, in order to obtain the filtered virtual control vector v¯i,2 , we pass v i,2 through the low-pass first-order filter with a small time constant τi,2 > 0 τi,2 v˙¯i,2 + v¯i,2 = v i,2 , v¯i,2 (0) = v i,2 (0).
(15)
Step k (k = 2, . . . , n i − 1): After differentiating si,k along (1), (7), and (7), we get s˙i,k = si,k+1 + z i,k+1 + v i,k+1 +
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i,k (νi,k ) − v˙¯i,k where i,k (νi,k ) = f i,k with νi,k = x¯ i,k . The distributed kth virtual control law v i,k for the i th follower is proposed as follows:
i,k ) v i,k+1 = − i,k si,k − i,k (νi,k |W v i,k − v¯i,k − αˆ i,k tanh(si,k /i,k ) + τi,k
(16)
where i,k and i,k are positive design parameters, tanh(si,k /i,k ) = [tanh(si,k,1 /i,k ), . . . , tanh(si,k, p /i,k )] , and
i,k ) approximates the unknown the approximator i,k (νi,k |W
i,k and αˆ i,k are estimates of nonlinear function i,k (νi,k ). W ∗ and α Wi,k ¯ i,k , respectively, and are tuned by
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Consider the Lyapunov candidate function Vi,ni as 1 1 2 −1 i,n Vi,ni = W λ ) + α ˜ si,ni si,ni + tr(W (25) i,n i i i,n i 2 ηi,ni i,ni where λi,ni = diag[λi,ni ,l ], l = 1, . . . , p, and αi,ni = α¯ i,ni − αˆ i,ni . Differentiating (25) and substituting s˙i,ni , (22)–(24) into it yield V˙i,ni ≤ − i,ni si,n s + ςi,ni α˜ i,ni αˆ i,ni i i,n i i,n + σi,ni tr(W Wi,ni ) + 0.2785 pi,ni α¯ i,ni . (26) i
Remark 6: The proposed adaptive consensus tracking scheme is summarized as the distributed virtual and actual (17) control [(9), (16), (22)] with adaptive laws [(10), (11), (17), (18) (18), (23), (24)] and the first-order filters [(15), (21)]. Remark 7: Compared with the control methods for strictwhere λi,k,l > 0 and ηi,k > 0 are tuning gains, σi,k and ςi,k are feedback systems [26]–[32] and the consensus or synchroniza i,k,l tion methods for multi-agent systems with nonlinear dynamics positive constants, and si,k,l and i,k,l = ∂ i,k,l (νi,k )/∂ W are the lth element of the error vector si,k and i,k , respec- [11]–[20], the proposed adaptive consensus approach using tively. neural networks has the following advantages. Consider the Lyapunov candidate function Vi,k as 1) In [26]–[32], the controllers were designed for single strict-feedback systems, without considering the com1 1 −1 2 i,k λi,k Wi,k ) + α˜ i,k Vi,k = (19) si,k si,k + tr(W munication among agents. However, this brief considers 2 ηi,k the multiple strict-feedback systems (1) under the direct αi,k = α¯ i,k − αˆ i,k . where λi,k = diag[λi,k,l ], l = 1, . . . , p, and graph topology, i.e., the lack of shared information in Proceeding similarly, differentiate (19) and substitute s˙i,k , communication links. That is, all of the followers cannot (16), (17), and (18) into it. Then, we easily obtain directly receive the information of the leader during the motion and can only use the information of their V˙i,k ≤ − i,k si,k si,k + si,k (si,k+1 + z i,k+1 ) + ςi,k α˜ i,k αˆ i,k neighbors, as reported in Assumptions 1 and 2. i,k + σi,k tr(W (20) Wi,k ) + 0.2785 pi,k α¯ i,k . 2) In [11]–[20], the consensus or synchronization methods for multi-agent systems with nonlinear dynamics were Let v i,k+1 pass through the kth low-pass first-order filter to presented. These methods require some information on obtain the kth filtered virtual control vector v¯i,k+1 , namely nonlinear functions (i.e, bounding conditions such as ˙ QUAD). Besides, the considered nonlinear functions are τi,k+1 v¯i,k+1 + v¯i,k+1 = v i,k+1 , v¯i,k+1 (0) = v i,k+1 (0) (21) matched in the control input. That is, the control input where τi,k+1 is a small positive time constant. can influence nonlinear functions in each state-space Step n i : Using (1) and (7), the derivative of si,ni can equation. Therefore, these methods [11]–[20] limit their be represented by s˙i,ni = u i + i,ni (νi,ni ) − v˙¯i,ni where applications to nonlinear systems with matched nonlini,ni (νi,ni ) = f i,ni with νi,ni = x i . We propose a distributed earities. However, the proposed approach in this brief actual control input on the i th follower as can deal with the follower systems with C 1 nonlinearities unmatched in the control input. Also, the informa ) u i = − i,ni si,ni − (ν |W i,ni i,ni i,ni tion on all nonlinearities under the total directed commuv i,ni − v¯i,ni si,ni nication graph topology is not required to implement the + (22) −αˆ i,ni tanh i,ni τi,ni local controllers of the followers since the approximators using neural networks are designed to estimate the where i,ni are positive design parameters, i,ni and nonlinear functions, including neighbors’ nonlinearities tanh si,ni /i,ni = [tanh(si,ni ,1 /i,ni ), . . . , tanh(si,ni , p /i,ni )] ,
(i.e., the approximator term i,1 in (9) estimates the and the approximator i,ni (νi,ni |Wi,ni ) approximates the
nonlinear functions including neighbors’ nonlinearities unknown nonlinear function i,ni (νi,ni ). Wi,ni and αˆ i,ni are M ∗ (ν ) = f − (1/(d + b )) a f i,1 i,1 i,1 i i estimates of Wi,ni and α¯ i,ni , respectively, and are tuned by j =1 i j j,1 ). Remark 8: Notice that the proposed consensus control ˙
i,ni ,l
W (23) scheme has the following advantage in the sense of the i,n i ,l = λi,n i ,l i,n i ,l si,n i ,l − λi,n i ,l σi,n i W α˙ˆ i,ni = ηi,ni si,ni tanh(si,ni /i,ni ) − ηi,ni ςi,ni αˆ i,ni (24) dynamic surface design. The first virtual controller (9) include the neighbors’ state information and the function approxiwhere λi,ni ,l > 0 and ηi,ni > 0 are tuning gains, σi,ni mator term estimating the neighbors’ nonlinearities. If the and ςi,ni are positive constants, and si,ni ,l and i,ni ,l = backstepping approach is used for the follower systems under
i,ni ,l are the lth element of the error vector the network communication topology, the actual control law ∂ i,ni ,l (νi,ni )/∂ W si,ni and i,ni , respectively. requires the repeated differentiations of the virtual controllers, ˙
i,k,l W i,k,l = λi,k,l i,k,l si,k,l − λi,k,l σi,k W s i,k − ηi,k ςi,k αˆ i,k α˙ˆ i,k = ηi,k si,k tanh i,k
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including (9). Therefore, the complexity of the actual control law drastically grows due to the increase of the repeated differentiations of the virtual controllers, including neural networks and the neighbors’ states as the complexity of the communication links and the order of the followers increase. This problem becomes more serious in the case of the group consisting of a large number of followers. However, the proposed approach based on the dynamic surface design does not require the repeated derivative terms of states of neighbors communicated in the directed graph and the neural network approximators because of the use of the filters (15) and (21) at the design steps. Accordingly, the simpler controller can be designed compared with the backstepping design. Remark 9: In [24], the connection weights between outputs and hidden nodes of RBFNNs are only trained by the adaptation laws. That is, the centers and widths of basis functions are randomly chosen in a big compact set. This method can degrade the approximation performance of the RBFNNs because it is difficult to initially design the optimal centers and widths of the basis functions for estimating uncertain nonlinear functions. Therefore, in our control system, the tuning laws for all weights, including centers and widths of the basis functions of approximators, are derived from the stability analysis. C. Stability Analysis Differentiating (7) yields z i,2
i,1 , αˆ i,1 , s j,3 , z j,3 , R + i,2 si,1 , si,2 , z i,2 , W z˙ i,2 = − τi,2 (27) z i,k+1 z˙ i,k+1 = − + i,k+1 si,1 , . . . , si,k+1 , z i,2 , . . . , z i,k+1 , τi,k+1
Wi,1 , . . . , Wi,k+1 , αˆ i,1 , . . . , αˆ i,k+1 , R (28) where i = 1, . . . , M, k = 2, . . . , n i − 1, j ∈ Ni , R = [r , r˙ , r¨ ] , i,2 (·) = (1/(di + bi ))( i,1 s˙i,1 − ˙ ˙ˆ i,1 tanh(si,1 /i,1 ) +
j ∈Ni ai j x˙ j,2 − bi r¨ ) + i,1 (νi,1 | Wi,1 ) + α 2 αˆ i,1 (1 − tanh (si,1 /i,1 ))(˙si,1 /i,1 ) and i,k+1 (·) =
i,k ) + α˙ˆ i,k tanh(si,k /i,k ) + αˆ i,k (1 − ˙ i,k (νi,k |W
i,k s˙i,k + tanh2 (si,k /i,k ))(˙si,k /i,k ) + z˙ i,k /τi,k are continuous functions. For the stability analysis, choose the total Lyapunov candidate function V as ni n i −1 M 1 Vi,k + z i,k+1 z i,k+1 . (29) V = 2 i=1
k=1
k=1
Based on the proposed design method, the main result of this brief is presented in the following theorem. Theorem 3.1: Consider the multiple uncertain nonlinear strict-feedback systems (1) controlled by the distributed consensus controller (22) and assume that the leader has directed paths to all followers 1 to M. Under Assumptions 1–4, for any initial conditions satisfying V (0) ≤ μ, the consensus tracking errors between the leader and the followers in the overall closed-loop system are CSUUB and can be made arbitrarily small.
Proof: Differentiating (29) and substituting (14), (20), (26)–(28) into it, we get ni M ˙ − i,k si,k si,k +σi,k tr(Wi,k Wi,k )+ςi,k α˜ i,k αˆ i,k V = i=1
k=1
+(di + bi )si,1 (si,2 + z i,2 ) +
n i −1
+(di + bi )0.2785 pi,1 α¯ i,1 + +
n i −1 k=1
k=2 ni
si,k (si,k+1 + z i,k+1 )
0.2785 pi,k α¯ i,k
k=2
−
z i,k+1 z i,k+1
τi,k+1
+
z i,k+1 i,k+1
.
W Using Young’s inequality1 and the inequalities tr(W i,k i,k ) = 1 1 2 2 i,k F W¯ i,k −W i,k = − W i,k − (W i,k F − W¯ i,k )2 + W F F 2 2 1 ¯2 1 ¯2 2 = −1α 2 2 ˜ i,k |α¯ i,k − α˜ i,k 2 Wi,k ≤ 2 ( Wi,k − Wi,k F ) and |α 2 ˜ i,k − 1 1 2 1 2 2 2 ˜ i,k | − α¯ i,k ) + 2 α¯ i,k ≤ 2 (α¯ i,k − α˜ i,k ), we have 2 (|α V˙ ≤
ni M i=1
k=1
− i,k si,k 2 −
σi,k 2 ςi,k 2 Wi,k F − α˜ 2 2 i,k
1 1 +(di + bi )(si,1 2 + si,2 2 + z i,2 2 ) 2 2 n i −1 1 1 + (si,k 2 + si,k+1 2 + z i,k+1 2 ) 2 2 k=2 n i −1 z i,k+1 2 i,k+1 2 z i,k+1 2 + + − +C τi,k+1 2ζ k=1
where ζ > constant and M 0niis a positive 2 2 ) ¯ [ ((σ /2) W + (ςi,k /2)α¯ i,k C = i,k i=1 k=1 ni,k i ¯ i,k + + (d + bi )0.2785 pi,1 α¯ i,1 + k=2 0.2785 pi,k α ni −1i (ζ /2)]. k=1 i k s ( h=1 (sl,h Consider the sets i,k = { l=1 l,h + k−1 −1 2 tr(W λ Wl,h ) + (1/ηl,h )α˜ l,h ) + h=1 z l,h+1 z l,h+1 ) + l,h l,h r + r˙ r˙ + r¨ r¨ ≤ R } s s ≤ 2μ} and = {r m,k+1 0 m∈Ni m,k+1 where i = 1, . . . , M, k = 1, . . . , n i , R0 > 0 is a constant. i,k and are compact in Rdim(i,k ) and R3 p , respectively. Then, since i,k × is also compact in Rdim(i,k )+3 p , there exist positive constants Q i,k+1 such that i,k+1 ≤ Q i,k+1 on i,k × , where dim(i,k ) denotes the dimension of the set i,k . ∗ , Choosing the parameters i,1 = (di + bi ) + i,1 i,k = 1 + ∗ ∗ ((di +bi )/2)+ i,k , i,ni = 1/2+ i,ni , 1/τi,2 = ((di +bi )/2)+ ∗ , and 1/τ 2 ∗ (Q 2i,2 /2ζ ) + τi,2 i,k+1 = 1/2 + (Q i,k+1 /2ζ ) + τi,k+1 ∗ ∗ with k = 2, . . . , n i − 1, positive constants i,k , τi,k+1 , ζ , we finally get V˙ ≤ −γ V + C (30) ∗ , 2τ ∗ , σ λ where γ = min [2 i,k i,h i,k i,k,l , ςi,k ηi,k ] > 0 with i = 1, . . . , M, k = 1, . . . , n i , h = 2, . . . , n i , and l = 1, . . . , p. The inequality (30) implies V˙ < 0 on V = μ when γ > C/μ. Therefore, V ≤ μ is an invariant set, i.e., if V (0) ≤ μ, then V (t) ≤ μ for all t ≥ 0. Accordingly, the consensus tracking errors in the overall closed-loop system are CSUUB. In 1 ab ≤ a p / p + bq /q where a,b ≥ 0 and p, q > 0 such that 1/ p + 1/q = 1.
5
0 −5
0
5
20
5
2,2
−0.5
0
0.5
1
y
1.5
2
2.5
3
3.5
i,1
(a) 2.5 2
40 20 0 −20 −400
2 0
−2
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10 15 Time(sec)
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10 15 Time(sec) (b)
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4 2 0
−2
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10 15 Time(sec)
10 15 Time(sec)
20 0 −20 −40 −60 0
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−400
−2 −3 −1
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−20
t=5
−1
10 15 Time(sec) (a)
4
3,2
Estimates of Γ1,2
0
Estimates of Γ
yi,2
t=15
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10 15 Time(sec)
4,1
−5 0
Estimates of Γ 3,1
0
Estimates of Γ
t=10
4
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Leader Followers
5
10
Estimates of Γ4,2
6
Estimates of Γ2,1
Fig. 1. Network topology for a group of one leader (L) and four followers F1 to F4.
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5 0 −5 −10 −15 0
(c)
5
10 15 Time(sec)
20
5
10 15 Time(sec) (d)
20
Fig. 3. Outputs of function approximators. (a) ˆ 1,1 and ˆ 2,1 . (b) ˆ 3,1 and ˆ 4,1 . (c) ˆ 1,2 and ˆ 2,2 . (d) ˆ 3,2 and ˆ 4,2 .
ei,o
1.5 1
IV. S IMULATION R ESULTS 0.5 0
0
5
10 Time(sec)
15
20
(b) Fig. 2. Consensus results. (a) Output trajectories of the followers with one leader in the 2-D space and (b) ei,o .
e−γ t V (0)
addition, from (30), we know that V (t) ≤ + (C/γ )[1 − e−γ t ]. Using 1/2s1 2 ≤ V (t) with s1 = , . . . , s ] , we get s 2 ≤ 2e −γ t V (0) + (2C/γ )[1 − [s1,1 1 M,1 −γ e t ]. Therefore, as time increases, all error surfaces s1 exponentially converge to the compact set = {s1 |s1 ≤ √ 2C/γ }. The compact set can be kept arbitrarily small by increasing γ . Then, from Remark 1 and s1 = ((L¯ + B) ⊗ ] , I is an identity I p )(y − (1 M ⊗r )) where y = [y1 , . . . , y M p matrix of order p, and ⊗ stands for the Kronecker product, the consensus tracking errors in the overall closed-loop system can be reduced. Remark 10: The design parameters i,k , λi,k , ηi,k , σi,k , ςi,k , i,k are only a sufficient condition and provide a guideline for the designers. From the proof of Theorem 3.1, some suggestions are given for the choice of some key design parameters as follows: (1) increasing i,k , λi,k , ηi,k helps to increase γ , subsequently reduces the bound 2C/γ of error, and (2) decreasing σi,k , ςi,k , i,k helps to decrease C, and reduces 2C/γ .
In this section, we present simulation results to validate the proposed theoretical result. We consider a group of one leader and four followers with the following nonlinear strict-feedback dynamics in the 2-D space (i.e., p = 2) described by x˙i,1 = x i,2 + f i,1 (x i,1 ) x˙i,2 = u i + fi,2 (x i ) yi = x i,1
(31)
where i = 1, . . . , 4, x i,k = [x i,k1 , x i,k2 ] ; k = 1, 2, x i = , x ] , y = [y , y ] , f (x ) = [ sin(x [x i,1 i i,1 i,2 i,1 i,1 i,11 )x i,12 , i,2 cos(x i,12 )] , and f i,2 (x i ) = e−0.4xi,11 x i,21 [ cos(x i,21 ), x i,22 ] . The directed network topology for the simulation is shown in Fig. 1. We choose ai j = 1 on j ∈ Ni , ai j = 0 otherwise, b1 = b2 = b3 = 0, and b4 = 1. The leader output r (t) is r (t) = [r1 (t), r2 (t)] = [4 sin(t/20), 2 sin(0.8t) + (t/5)] and the initial states of the four followers are set to x 1 (0) = [ − 1, 0, 1, −1] , x 2 (0) = [ − 1, −2, 1, −3] , x 3 (0) = [0, −2, 2, −2] , and x 4 (0) = [ − 1, −1, 2, 3] . To display the consensus tracking errors between the leader and the followers, we use ei,o = x i,1 (t) − r (t) where i = 1, . . . , 4. For the simulation, RBFNN approximators are used to estimate unknown nonlinear functions i,k where i = 1, . . . , 4 and k = 2. The design parameters for distributed consensus controllers are chosen as i,1 = 15, i,2 = 10, τi,2 = 0.005, λi,1,1 = 0.2, λ1,1,2 = λ2,1,2 = λ4,1,2 = 0.1, λ3,1,2 = 0.05, λi,2,l = 0.001, ηi,k = 0.02, i,k = 0.1, and σi,k = ςi,k = 0.001 with i = 1, . . . , 4, k = 1, 2, and l = 1, 2. Fig. 2(a) shows the outputs of the four followers and one leader in the 2-
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 4, APRIL 2013
D space where the square boxes denote the initial points of the followers and the circles are the output snapshots of the followers at t = 5 s, t = 10 s, and t = 15 s. The consensus tracking errors ei,o drop quickly in less than a few seconds, as shown in Fig. 2(b). The outputs of function approximators are plotted in Fig. 3. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology, although the followers have the unknown nonlinearities unmatched in the control input. V. C ONCLUSION We proposed a distributed adaptive consensus control approach for multiple strict-feedback systems with unknown nonlinearities under the directed communication graph. Each local distributed consensus controller was designed by applying the dynamic surface design to the distributed consensus tracking problem. The graph-based error surfaces for the distributed dynamic surface design are used to guarantee the consensus tracking between the followers and one leader. The function approximation technique using neural networks was employed to compensate the unknown nonlinearities unmatched in the control input of followers. Finally, the effectiveness of the proposed approach was proved through a computer simulation. ACKNOWLEDGMENT The author would like to thank Associate Editor and the anonymous reviewers for their helpful comments and constructive suggestions with regard to this brief. R EFERENCES [1] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. [2] Y. Hong, L. Gao, D. Cheng, and J. Hu, “Lyapunov-based approach to multi-agent systems with switching jointly-connected interconnection,” IEEE Trans. Automat. Control, vol. 52, no. 5, pp. 943–948, May 2007. [3] W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control. New York: Springer-Verlag, 2008. [4] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Trans. Autom. Control, vol. 53, no. 6, pp. 1053–1059, Jul. 2008. [5] H. Su, X. Wang, and G. Chen, “Rendezvous of multiple mobile agents with preserved network connectivity,” Syst. Control Lett., vol. 59, no. 5, pp. 313–322, 2010. [6] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Int. J. Robust Nonlinear Control, vol. 17, nos. 10–11, pp. 941–959, 2007. [7] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. [8] P. Lin, Y. Jia, and L. Li, “Distributed robust H1 consensus control in directed networks of agents with time-delay,” Syst. Control Lett., vol. 57, no. 8, pp. 643–653, 2008. [9] Y. Zhnag and Y. Tian, “Consentability and protocol design of multiagent systems with stochastic switching topology,” Automatica, vol. 45, no. 5, pp. 1195–1201, 2009. [10] Y. Tian and C. Liu, “Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations,” Automatica, vol. 45, no. 5, pp. 1347–1353, 2009. [11] W. Yu, G. Chen, M. Cao, and J. Kurths, “Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 881–891, Jun. 2010.
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