arXiv:1703.02685v1 [cs.SY] 8 Mar 2017 hn gat6417 n 1037 n ue rvnilEd funding. Provincial for Hubei T201302) (grant and partment 61501337) and 61471275 (grant China lzn ihdmninldt nrcn er 91] Ther [9–11]. years recent in data high-dimensional alyzing domain. con- frequency spatial the graph in of of processing development insights signal recent new the con- provide exploring One by to problem is sensus 8]. paper [7, this domain Nyquis of frequency tribution the the utilized in studies criterion Other stability time-domain the [4–6]. on model based state-space problem analyz consensus researches the quantities previous solved of certain agents and Most on of eventually. agreement network interest an whole of reach the to information and coordinate get neighbors, may only local can dis- its agent appropriate from each design that to [1–3]. protocols is decade consensus tributed last for the problem key in The studied individuals extensively been autonomous has of which behaviors fundamental collective a is in (MASs) problem systems multi-agent of Consensus Consensus Average cessing, effective- the methods. demonstrate Several our to of filter. ness given protocol are the examples on based numerical cases are both connected protocols for consensus is designed The other topology. the unknown bound, with eigenvalue graphs fixed a Laplacian estimated and cases, an by two modeled existing consider networks we uncertain by is paper, with one this deal In a to methods. be difficult time-domain to is u which out for protocols networks, turns consensus certain also effective design it to consensus, tool powerful average n the provides only of not relation insights This signals. graph an of systems r filtering multi-agent direct of the consensus average establish between we lation graph protocol, the consensus defining the By from filter perspective. processing signal fr graph consensus a multi-agent of problem the revisits paper This Crepnigato.Tak oteNtoa cec Foun Science National the to Thanks author. *Corresponding rp inlpoesn a rw ra neet o an- for interests great drawn has processing signal Graph Terms Index E EUT NMLIAETSSE OSNU:AGAHSIGNAL GRAPH A CONSENSUS: SYSTEM MULTI-AGENT ON RESULTS NEW — .INTRODUCTION 1. ut-gn ytm rp inlPro- Signal Graph System, Multi-agent ABSTRACT ua nvriyo cec n ehooy ua,China Wuhan, Technology, and Science of University Wuhan colo nomto cec n Engineering and Science Information of School RCSIGPERSPECTIVE PROCESSING { iige,chaili yijingwen, igWnY n iChai* Li and Yi Jing-Wen cto De- ucation ainof dation om ew ed n- e- d e t , } hc ossso naetset agent an of consists which Let Theory Graph Spectral 2.1. methods. proposed the demonstrat of effectiveness to given the is are bound examples error method Numerical consensus design presented. The also new gain. a protocol provide consensus we the de- of graph, maximum network the the and of connectivity gree the assuming except ma- Laplacian trix unknown completely with base MASs model For state-space methods. time-domain existing difficult the is by which analyze error, consensus well asymptotic as the gains protocol estimate consensus c distributed perspective design ma- processing to signal Laplacian help graph the estimated un- that with show on we MASs MASs trix, For in problems networks. challenging certain some method solve new a to presents also ology it insights, new provides only not graph. network the of matrix Laplacian o the property and MAS closed-loop the the from distributed derived a by protocol implemented consensus components be can frequency filter ma- graph higher designed Laplacian The other graph surrogating the while of (corre- eigenvalue trix) graph to zero the filter the of consensus to protocol component average sponding frequency appropriate low its an the reach designing only can keep by MAS time MASs. the finite of in that consensus shown and is signals It graph of filtering between transfor Fourier graph 16–19]. [10, frequencies as mation graph eigenvectors orthonormal as eigenvalues the and Laplacian inte graph which processing spectral the signal prets and es- classical algebraic other with merging concepts The graph by framework generalized [11–15]. and the matrix form tablishes adjacency normal the Jordan of the eigenbasis on based graph conc and on structure processing signal On discrete processing. the develops signal graph on frameworks different two are set @wust.edu.cn V × V ⊆ E iwn Asfo rp inlpoesn perspective processing signal graph from MASs Viewing connection explicit the reveal first will we paper, this In G ( = V , E , n daec matrix adjacency a and , A ) ea nietdgaho order of graph undirected an be .PRELIMINARIES 2. V = { v 1 , v A 2 , . . . , [ = a ij v N ] N } ℜ ∈ nedge an , ( N N ≥ × epts N an 2 as to r- d e e f ) - - . . An edge eij = (vi, vj ) ∈ E if and only if aij = aji 6= 0, The average consensus of multi-agent system (1) un- that means there exist communications between agent i and der protocol (2) is said to be achieved asymptotically if N agent j. Moreover, self-edges are not allowed, i.e., eii = 1 lim xi(k) = N xi(0), i = 1, 2,...,N. And the av- (vi, vi) ∈/ E and aii = 0. The Laplacian matrix of graph is k i=1 ∆ →∞ P defined by L = D − A, where D = diag{d1,...,dN } is the erage consensus is said to be reached at time T if xi(k) = N N , and di = j aij . The Laplacian matrix L 1 =1 N xi(0), i =1, 2,...,n, hold for all k ≥ T. is symmetric and all the eigenvalues are real. i=1 P P Lemma 1. [20] For an undirected graph G, the Laplacian From (1) and (2), the dynamics of the multi-agent system matrix L has at least one zero eigenvalue, and all the non-zero can be calculated that eigenvalues are positive, i.e., 0= λ1 ≤ λ2 ≤···≤ λN . Fur- x(k +1)=(I − εkL)x(k) thermore, zero is a simple eigenvalue of L with the associated T (4) = V diag{h(λ1, k), ··· ,h(λN , k)}V x(0). orthonormal eigenvector 1 ~1 , where ~1 =∆ [1,..., 1]T ∈ √N N N ℜN , if and only if the G is connected. Let the initial state x(0) and the current state x(k + 1) of the MAS be the collective signal values of the original graph sig- 2.2. Graph Signal Processing nal and the filtered one, respectively. Then, the MAS plays a role of the filter for the graph signal with x(0) and G, and the Consider a graph G = (V, E, A), its Laplacian matrix can be the transfer function of the filter is h(λ, k) defined in Defini- T written as L = V ΛV , where Λ = diag{λ1, λ2,...,λN }, tion 1. The following result gives the property of the protocol N N V = [~v1, ~v2,...,~vN ] ∈ℜ × is a . The graph filter achieving average consensus. signal x is the collection of the signal values on all the agents, Theorem 1. For the MAS (1) on a connected graph G, T N i.e., x = [x1, x2, ··· , xN ] ∈ ℜ . Similarly to classical of which the Laplacian matrix has eigenvalues as 0 = λ1 < signal processing, the Fourier transform in the graph signal λ2 ≤ ... ≤ λN , assume the consensus protocol is in the processing [10] is defined on the graph spectra as xˆ = V T x, form of (2). Then the MAS reaches average consensus at and the inverse is given by x = V xˆ, time T if and only if the corresponding protocol filter h(λ, T ) T N where xˆ = [ˆx1, xˆ2, ··· , xˆN ] ∈ℜ . defined by (3) satisfies h(0,T )=1 and h(λi,T )=0 for Let h(·) be the transfer function of a filter, and yˆ = i =2,...,N. T N [ˆy1, yˆ2,..., yˆN ] ∈ℜ be the filtered graph signal in spatial The proof of Theorem 1 is omitted due to space limita- frequency domain, then the graph spectral filtering can be tion. It is easy to verify that the consensus state is x(T ) = T defined as yˆi = h(λi)ˆxi. Taking the inverse graph Fourier ~v1~v1 x(0). Theorem 1 shows that the protocol filter can be transform, the filtered graph signal in time domain can be viewed as a low-pass filter with zero at high frequency com- T obtained as y = V diag{h(λ1), ··· ,h(λN )}V x, where ponents λ2,...,λN . It followsfrom Theorem1 that the MAS T N y = [y1,y2,...,yN ] ∈ℜ is the filted graph signal in time with N agents on a connected graph can definitely reach the domain. average consensus at time N − 1 by properly choosing the control gain εk. The following corollary shows that the con- 3. MULTI-AGENT CONSENSUS VIA GRAPH sensus time can be smaller than N − 1. SIGNAL PROCESSING Corollary 1. For the MAS (1) on a connected graph with p distinct nonzero eigenvalues (0= λ1 < λ2 < ··· < λp+1), Consider the dynamics of the MAS on a graph G = (V, E, A) take the control gains as with N agents described by 1 , k =0, 1,...,p − 1, λ +1− εk = p k (5) xi(k +1)= xi(k)+ ui(k), i =1, 2,...,N (1)  0, otherwise. where xi(k), ui(k) ∈ ℜ is the state and the control input, Then the consensus protocol (2) makes the MAS reach aver- respectively. A commonly used control protocol [1] is as fol- N 1 lows age consensus at time p, that is, x(p)= N xi(0). (2) i=1 ui(k)= εk aij (xj (k) − xi(k)), Remark. The fact that finite time consensusP can be jXNi ∈ achieved by choosing the control gains equal to the reciprocal where εk is the control gain at time k, and A = [aij ]N N is of nonzero Laplacian eigenvalues is not new. It has been × the of graph G. obtained by using different methods, for example, matrix Definition 1. For a control protocol within time T given factorization method [21], minimal polynomial method [22]. by (2), the corresponding protocol filter is defined as By defining the protocol filter h(λ, T ), Theorem 1 derives

T 1 the consensus result from a graph signal processing perspec- − tive. In the next section, we will show that this method is a h(λ, T )= (1 − εkλ). (3) powerful tool to solve the consensus of MASs on uncertain kY=0 networks, which is difficult (sometimes unable) to deal with 4 by existing methods. 2

x(k) 0 −2 −4 4. CONSENSUS ON UNCERTAIN NETWORKS 0 1 2 3 4 5 k

4 In the previous section, we assume that the graph structure 2 of MASs is completely known. The exact average consensus x(k) 0 −2 in finite-time can be reached by designing a appropriate pro- −4 0 2 4 6 8 10 tocol filter. This section will discuss the average consensus k problem in two non-ideal cases: estimated Laplacian matrix 4 2

and unknown network topology. Denote the consensus error x(k) 0 N N −2 1 2 −4 as e(t) = (xi(t) − N xi(0)) , the consensus perfor- 0 5 10 15 i=1 i=1 k mance willP be analyzed in bothP cases.

4.1. Consensus for MASs on Graphs with Estimated Fig. 1. (a) The trajectory of agent state x(k) for Ts = 5. Laplacian Matrix (b) The trajectory of agent state x(k) for Ts = 10. (c) The trajectory of agent state x(k) for Ts = 15. Consider the MAS on a connected graph G˜ with Laplacian matrix L˜. Denote the eigenvalues of L˜ as 0 = λ˜1 < λ˜2 ≤ T 50 ... ≤ λ˜N . Let L0 = V Λ0V be the estimated Laplacian matrix of L˜, where Λ0 = diag{λ1, λ2,...,λN }. For δ¯ > 0, e(k) denote 0 δ¯2 N 2 λ + δ¯ 0 1 2 3 4 5 ϕ = − (1 − 2 )2. (6) k 2 k 50 λ2 Y =1 λk+2

Theorem 2. For the MAS (1) on the graph G˜ under the e(k) 0 consensus protocol (2), take the control gain as 0 2 4 6 8 10 k 50 k =0, 1,...,N − 2, ε = 1 , (7) k+j(N 1) λN−k − j =0, 1,..., ∞, e(k)

0 0 5 10 15 where λ2,...,λN are eigenvalues of the estimated Laplacian k matrix L0. Then the MAS reaches average consensus asymp- totically if λ˜i − λi ≤ δ¯ and ϕ < 1, where ϕ is defined in Fig. 2. (a) The consensus square error e(k) for Ts = 5. (b) (6). Moreover, at time Tj = j(N − 1), the consensus error j 2 The consensus square error e(k) for Ts = 10. (c) The con- satisfies e(Tj ) ≤ ϕ kx(0)k . sensus square error e(k) for Ts = 15. Proof Outline: The corresponding protocol filter at each N 2 control period can be written as h(λ, j(N − 1)) = − (1 − k=0 Let the simulation time be Ts = 5, 10, 15, respectively, then λ j Q ) . Then the consensus error at time Tj = j(N − 1) the evolution of agent states and the consensus error are λN−k N 2 shown in Fig. 1 and Fig. 2. It can be seen that the MAS T can be calculated by e(Tj) = h(λi,Tj )~vi~vi x(0) ≤ (1) under the designed protocol (2) reaches average consen- i=2 sus asymptotically, and the consensus errors at each Ts are j 2 P ϕ kx(0)k . It is easy to see that lim e(Tj )=0 since ϕ< 1. j e(5) = 0.9675, e(10) = 0.1376, and e(15) = 0.0017. Thus, the MASs (1) can reach the→∞ average consensus asymp- totically by the protocol (2). 4.2. Consensus for MASs with Unknown Network Topol- Example 1: Consider the MAS on a connected graph G˜ ogy with 6 agents, the Laplacian matrix is given as L˜ = L0 + T T Consider a connected graph G with unknown network topol- δV¯ diag{0, IN 1}V , where L0 = V diag{0, 1, 1, 3, 3, 4}V is the estimated− Laplacian matrix corresponding to an un- ogy, the methods proposed in previous sections have no weighted cycle network, and δ¯ = 0.5 is the estimation error application to solve the average consensus problem in this bound. From Theorem 2, the control gain in one control pe- case. Assume the maximum degree of the graph is given, 1 1 1 i.e., max {di} = d¯. Then, divide [0, 2d¯] into T0 uniform riod can be designed as εk = 4 , 3 , 3 , 1, 1 for k =0, 1, 2, 3, 4. i=1,...,N 2d¯ intervals, and the length of each interval is . A general State of Unweighted Cycle T0 result for average consensus can be derived as follows. 6 4 Theorem 3. For the MAS (1) on an unknown connected 2 graph G with maximum degree d¯, take the control gains as x(k) 0

−2 0 5 10 15 20 25 30 T0 k =0, 1,...,T0 − 1, k εk+jT0 = , (8) State of Unweighted Path 2(T0 k)d¯ − j =1,... ∞. 6

4

2 Then the consensus protocol (2) makes the MAS reach aver- x(k) age consensus asymptotically. Furthermore, assume the lower 0 ¯ bound of the algebraic connectivity satisfies λ ≥ 2d , then 2 αT0 −2 0 5 10 15 20 25 30 the consensus error at time Tj = jT0 satisfies e(jT0) ≤ k ϕ2j kx(0)k2, where ϕ = max{1 − 1 ln(T + 1), 1 } < 1. α 0 2T0 Proof Outline: From the design of the control gain, the Fig. 3. (a) The agent state trajectory of the unweighted cycle corresponding protocol filter at the end of each control pe- G1. (b) The agent state trajectory of the unweighted path G2. T0 1 − λT0 j riod can be derived as h(λ,jT0) = (1 − ) . 2(T0 k)d¯ k=0 − Consensus Error of Unweighted Cycle Q 60 It can be calculated that |h(λ, T0)| ≤ ψ, where ψ = T0 1 − max { (1 − λ )} < 1. And the consensus error 40 (T0 k)φ λ (0,2d¯] k=0 − ∈ Q e1(k) N 2 20 T can be calculated as e(jT0)= h(λi,jT0)~vi~vi x(0) ≤ i 0 =2 0 5 10 15 20 25 30 2j 2 P ψ kx(0)k . It is easy to see that lim e(jT0)=0 since k j Consensus Error of Unweighted Path ψ< 1. Then the MAS reaches the average→∞ consensus asymp- 60 totically under the protocol (2). 40

Assume the lower bound of the algebraic connectivity sat- e2(k) ¯ 20 isfies λ ≥ 2d , a more accurate upper bound of the protocol 2 αT0 1 0 filter can be derived as |h(λ, T0)| ≤ max{1 − α ln(T0 + 0 5 10 15 20 25 30 1), 1 }. Then the consensus error at time T = jT satisfies k 2T0 j 0 2j 2 e(Tj) ≤ ϕ kx(0)k . Fig. 4. (a) The consensus error e1(T ) of the unweighted cycle It follows from Theorem 3 that the algebraic connectiv- G1. (b) The consensus error e2(T ) of the unweighted path G2. ity of the graph plays an important role in reaching consen- sus, and the higher algebraic connectivity corresponding toa better consensus performance and a lower consensus error at 5. CONCLUSION each control period.

Example 2: Consider a MAS with 6 agents on two dif- This paper has established the explicit connection between fil- ferent graphs, one is an unweighted cycle G1, the other is an tering of graph signals and consensus of MASs. It has been unweighted path G2. The maximumdegrees of the two graphs shown that the MAS can reach its average consensus in fi- ¯ ¯ are the same, i.e., d1 = d2 = 2. Divide [0, 4] into 5 uniform nite time by designing an appropriate protocol filter, which ¯ intervals, then T = 5 and 2d = 0.8. From Theorem 3, the 0 T0 can be implemented by a distributed consensus protocol. By 1 control gain in one period can be derived as εk = 4 0.8k , using the concept of the protocol filter, we provide new meth- − k = 0,..., 4. For the MAS (1) on two graphs G1 and G2 re- ods to solve the average consensus problem in cases of esti- spectively, the protocol (2) with the designed periodic control mated Laplacian matrix and unknown network topology. The gain can solve the average consensus asymptotically as shown asymptotic consensus error has been analyzed in both cases. in Fig 3. Moreover, it is easy to verify that the algebraic con- While the protocol filter is defined for MASs on undirected nectivity of the cycle is higher than that of the path, thus the graphs, it can be easily extended to direct graphs. We only consensus performance of the graph G1 is much better than consider MASs consisting of agents with first-order dynam- the graph G2, and the consensus errors of the two graphs are ics, it is interesting to extend our methods to MASs with high- shown in Fig 4. order dynamics. 6. REFERENCES [12] J. Mei and J. M. F. Moura, “Signal processing on graphs: Estimating the structure of a graph,” in Proceed- [1] R. Olfati-Saber and R. M. Murray, “Consensus problems ings of 2015 IEEE International Conference on Acous- in networks of agents with switching topology and time- tics, Speech and Signal Processing (ICASSP), pp. 5495- delays,” IEEE Transactions on Automatic Control, vol. 5499, 2015. 49, no. 9, pp. 1520-1533, 2004. [13] A. Sandryhaila and J. M. F. Moura, “Discrete Sig- [2] K. You, Z. Li and L. Xie, “Consensus condition for lin- nal Processing on Graphs: Frequency Analysis,” IEEE ear multi-agent systems over randomly switching topolo- Transactions on Signal Processing, vol. 62, no. 12, pp. gies,” Automatica, vol. 49, no. 10, pp. 3125-3132, 2013. 3042-3054, 2014.

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