Obtaining Consensus of Multi-agent Linear Dynamic Systems

Transcription

1 Obtaining Consensus of Multi-agent Linear Dynamic Systems M I GRCÍ-PLNS Universitat Politècnica de Catalunya Departament de Matemàtica plicada Mineria 1, arcelona SPIN mariaisabelgarcia@upcedu bstract: In this paper the consensus problem is considered for multi-agent systems, in which all agents have an identical linear dynamic mode that can be of any order generalization to the case all agents are of the same order but do not have the same linear dynamic ey Words: Multi-agent systems, consensus, control 1 Introduction It is well known the attraction created in many research communities, the study of multi-agents system, and there are an amount of literature as for example [3, 6, 8] It is due to the multi-agents appear in different areas as for example in consensus problem of communication networks [4], or formation control of mobile robots [] Jinhuan Wang, Daizhan Cheng and Xiaoming Hu in [6] study the consensus problem in the case of multiagent systems in which all agents have an identical linear dynamics and it is an stable linear system In this paper, we generalize to the case where the dynamic of the agents are controllable Finally a brief study of the case of all agents are not identical linear systems is introduced Preliminaries 1 lgebraic Graph theory We consider a graph G = (V, E) of order N with the set of vertices V = {1,, N} and the set of edges E = {(i, j) i, j V} V V Given an edge (i, j) i is called the parent node and j is called the child node and j is in the neighbor of i, concretely we define the neighbor of i and we denote it by N i to the set N i = {j V (i, j) E} The graph is called undirected if verifies that (i, j) E if and only if (j, i) E The graph is called connected if there exists a path between any two vertices, otherwise is called disconnected ssociated to the graph we consider a matrix G = (g ij called (unweighted) adjacency matrix defined as follows g ii = 0, g ij = 1 if (i, j) E, and g ij = 0 otherwise (In a more general case we can consider a weighted adjacency matrix is G = (g ij with g ii = 0, g ij > 0 if (i, j) E, and g ij = 0 otherwise) The Laplacian matrix of the graph is N i if i = j L = (l ij ) = 1 if j N i 0 otherwise Remark 1 i) If the graph is undirected then the matrix L is symmetric, then there exist an orthogonal matrix P such that P LP t = D ii) If the graph is undirected then 0 is an eigenvalue of L and (1,, 1) t is the associated eigenvector iii) If the graph is undirected and connected the eigenvalue 0 is simple For more details about graph theory see [7] ronecker product Remember that = (a ij ) M n m (IC) and = (b ij ) M p q (IC) the ronecker product (see [5] for more information) is defined as follows Definition Let = (a i j ) M n m(ic) and M p q (IC) be two matrices, the ronecker product of and, write, is the matrix = a 1 1 a1 a1 m a 1 a a m a n 1 an an m M np mq(ic) ISN:

2 ronecker product verifies the following properties 1) ( + ) C = ( C) + ( C) ) ( + C) = ( ) + ( C) 3) ( ) C = ( C) 4) ( ) t = t t 5) If Gl(n; IC) and Gl(p; IC)), then Gl(np; IC)) and ( ) 1 = 1 1 6) ( )(C D) = (C) (D) ssociated to the ronecker product, can be defined the vectorizing operator that transforms any matrix into a column vector, by placing the columns in the matrix one after another, Definition 3 Let X = (x i j ) M n m(ic) be a matrix, and we denote x i = ( i,, xn i )t for 1 i m the i-th column of the matrix X We define the vectorizing operator vec, as vec : M n m (IC) X Obviously, vec is an isomorphism M nm 1 (IC) x x m 3 Dynamic of multi-agent having identical dynamical mode Let us consider a group of k identical agents The dynamic of each agent is given by the following linear dynamical systems ẋ 1 = + u 1 (1) ẋ k = x k + u k x i IR n, u i IR m, 1 i k We consider the undirected graph G with i) Vertex set: V = {1,, k} ii) Edge set: E = {(i, j) i, j V } V V defining the communication topology among agents Definition 4 Consider the system 4 We say that the consensus is achieved using local information if there is a state feedback such that u i = (x i x j ), 1 i k lim t xi x j = 0, 1 i, j k For simplify we will write z i = (x i x j ), 1 i k The closed-loop system obtained under this feedback is as follows where = = X = X = X + Z, x k X = = Z = ẋ 1 ẋ k j N 1 x j j N k x k x j Following this notation we can conclude the following Proposition 5 ([6]) The closed-loop system can be described as X = ((I k ) + (I k )(L I n ))X Proof: Calling = = ISN:

3 and observing that = I k = I k Z = (L I n )X, we can conclude that closed-loop system is X = ((I k ) + (I k )(L I n ))X Taking into account that the graph is undirected, following remark 1, we have that there exists an orthogonal matrix P Gl(k; IR) such that P LP t = D = diag (λ 1,, λ k ), (λ 1 λ k ) Corollary 6 The closed-loop system can be described in terms of the matrices,, the feedback and the eigenvalues of L Proof: Following properties of ronecker product we have that (P I n )(I k )(P t I n ) = (I k ) (P I n )(I k )(P t I n ) = (I k ) (P I n )(L I n )(P t I n ) = (D I n ) and calling X = (P I n )X = P P x P x 3, we have X = ((I k ) + (I k )(D I n )) X Equivalently, X = + λ 1 + λk X () Remark 7 lim t P x i P x j = 0 if and only if lim t x i x j = 0 31 Consensus problem It would seem that if the graph is connected the consensus problem would be solvable of there is a such that the system is stabilized ut taking into account that λ 1 = 0 this system is only stabilized if ẋ 1 = is stable Suppose now, that the system (, ) is controllable, so there exist such that the close loop system ẋ = ( + )x = x is stable and we apply all results presented in 3 over the group of k identical agents, where the dynamic of each agent is given by the following linear dynamical systems ẋ 1 = + u 1 (3) ẋ k = x k + u k x i IR n, u i IR m, 1 i k Example 1 We consider 3 identical agents with the following dynamics of each agent ẋ 1 = + u 1 ẋ = x + u (4) ẋ 3 = x 3 + u 3 ( ) ( ) with = and = The communication topology is defined by the graph (V, E): V = {1,, 3} E = {(i, j) i, j V } = {(1, ), (1, 3)} V V and the adjacency matrix: G = The neighbors of the parent nodes are N 1 = {, 3}, N = {1}, N 3 = {1} The Laplacian matrix of the graph is L = with eigenvalues λ 1 = 0, λ = 1, λ 3 = 3 u i = ( (x i x j )) = z i (5) u 1 = (( x ) + ( x 3 )) = = ( x x 3 ) u = (x ) u 3 = (x 3 ) Taking into account that the system ẋ 1 = is not stable but (, ) is( a controllable ) system, we 0 1 consider = + = with appropriate a b values for a and b ISN:

4 Then, the close loop system of 4 with control 5 is ẋ 1 = + ( x x 3 ) = = ( + ) x x 3 ẋ = x + x ) = ( + )x ẋ 3 = x 3 + x 3 ) = ( + )x 3 Or in a (formal)-matrix form: X = X (6) The basis change matrix diagonalizing the matrix L is 1 0 P = and we obtain the following equivalent system X = X The eigenvalues of the system are: b± b +4a, (corresponding to the eigenvalues of the system ẋ 1 = ), b+d± b +bd+d +4a+4c, (corresponding to the eigenvalues of the system ẋ = ( + )x ), b+3d± b +6bd+9d +4a+1c, (corresponding to the eigenvalues of the system ẋ 3 = ( + 3)x 3 ) Then, there exist and (defined by a, b, c, d), which assign the eigenvalues as negative as possible In a more general case we have the following lemma Lemma 8 ([6]) Let (, ) be a controllable pair of matrices and we consider the set of k-linear systems ẋ i = x i + λ i u i, 1 i k with λ i > 0 Then, there exist a feedback which simultaneously assigns the eigenvalues of the systems as negative as possible More concretely, for any M > 0, there exist u i = x i for 1 i k such that Re σ( + λ i ) < M, 1 i k (σ(+λ i ) denotes de spectrum of +λ i for each 1 i k) s a corollary we can consider the consensus problem Corollary 9 We consider the system 4 with a connect adjacent topology If (, ) is a controllable pair then, the consensus is achieved by means the feedback defined in 4 and a feedback stabilizing (, ) Proof: Taking into account that the adjacent topology is connected we have that 0 = λ 1 < λ λ k and (1,, 1) t = 1 k is the eigenvector corresponding to the simple eigenvalue λ 1 = 0 On the other hand we can find stabilizing (, ) and then we can find stabilizing the associate system, then we find Z such that lim t Z = 0 Using Z = (L I n )X we have that lim t X = 1 k v for some vector v IR n and the consensus is obtained 4 Dynamic of multi-agent having no identical dynamical mode The dynamic of each agent is given by the following dynamical systems ẋ 1 = u 1 (7) ẋ k = k x k + k u k x i IR n, u i IR m, 1 i k The communication topology among agents is defined by means the undirected graph G with i) Vertex set: V = {1,, k} ii) Edge set: E = {(i, j) i, j V } V V an in a similar way as before, we have the following Definition 10 Consider the system 7 We say that the consensus is achieved using local information if there exists a state feedback u i = i (x i x j ), 1 i k such that lim t xi x j = 0, 1 i, j k For simplicity we define z i = (x i x j ), 1 i k The closed-loop system obtained under this feedback is as follows ISN:

5 where = = Calling X = 1 1 = and observing that X = X + Z x k k k k X = = Z = 1 k k ẋ 1 ẋ k 1 k j N 1 x j j N k x k x j k = References: [1] Z Li, Z Duan, G Chen, Consensus of Multiagent Systems and Synchronization of Complex Networks: unified Viewpoint, IEEE Trans on Circuits and Systems, 57, (1), pp 13-4, (010) [] Fax, R Murray, Information flow and cooperative control of vehicle formations, IEEE Trans utomat Control 49, (9), pp , (004) [3] RO Saber, RM Murray, Consensus Protocols for Networks of Dynamic gents, Report [4] RO Saber, RM Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans utomat Control 49, (9), pp , (004) [5] P Lancaster, M Tismenetsky, The Thoery of Matrices cademic Press San Diego (1985) [6] J Wang, D Cheng, X Hu, Consensus of multiagent linear dynamics systems, sian Journal of Control 10, (), pp , (008) [7] D West Introduction to Graph Theory Prentice Hall (3rd Edition), (007) [8] G Xie, L Wang, Consensus control for a class of networks of dynamic agents: switching topology, Proc 006 mer Contro Conf, pp , (007) Z = (L I n )X we deduce the following proposition Proposition 11 The closed-loop system can be deduced in terms of matrices, and in the following manner X = ( + (L I n ))X (8) We are interested in i such that the consensus is achieved Proposition 1 We consider the system 7 which a connected adjacent topology If the system 8 is stable the consensus problem has a solution 5 Conclusions In this paper the consensus problem is considered for multi-agent systems, in which all agents have an identical linear dynamic mode that can be of any order generalization to the case all agents are of the same order but do not have the same linear dynamic ISN: