Recent advances in communication and computing have made the control and coordination of dynamic network agents to become an area of multidisciplinary research at the intersection of the theory of control systems, communication and linear algebra. The advances of the research in multi-agent systems are strongly supported by their critical applications in different areas as for example in consensus problem of communication networks, or formation control of mobile robots. Mainly, the consensus problem has been studied from the point of view of stability. Nevertheless, recently some researchers have started to analyze the controllability problems. The study of controllability is motivated by the fact that the architecture of communication network in engineering multi-agent systems is usually adjustable. Therefore, it is meaningful to analyze how to improve the controllability of a multi-agent system. In this work we analyze the stability and controllability of multiagent systems consisting of k + 1 agents with dynamics x¿i = Aixi + Biui, i = 0, 1, . . . , kPostprint (published version

García Planas, María Isabel

Tarragona, Sonia

English

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Int. J. Complex Systems in Sciencevol.6(1) (2016), pp. 9–15On stability and controllability of multi-agent linearsystemsM. I. Garc´ıa-Planas1,† and S. Tarragona21 Departament de Matema`tiques, Universitat Polite`cnica de Catalunya2 Departmento de Matema´ticas, Universidad de Leo´nAbstract. Recent advances in communication and computing have made thecontrol and coordination of dynamic network agents to become an area of multi-disciplinary research at the intersection of the theory of control systems, commu-nication and linear algebra. The advances of the research in multi-agent systemsare strongly supported by their critical applications in different areas as for ex-ample in consensus problem of communication networks, or formation control ofmobile robots. Mainly, the consensus problem has been studied from the point ofview of stability. Nevertheless, recently some researchers have started to analyzethe controllability problems. The study of controllability is motivated by the factthat the architecture of communication network in engineering multi-agent sys-tems is usually adjustable. Therefore, it is meaningful to analyze how to improvethe controllability of a multi-agent system. In this work we analyze the stabilityand controllability of multiagent systems consisting of k+1 agents with dynamicsx˙i = Aixi + Biui, i = 0, 1, . . . , k.Keywords: Stability, Controllability, ConsensusMSC 2000: 93D15, 93B05† Corresponding author: maria.isabel.garcia@upc.eduReceived: July 27th, 2016Published: November 30th, 20161. IntroductionIt is well known the great interest created in many research communities aboutthe study of control multi-agents system, as well as the increasing interest indistributed control and coordination of networks consisting of multiple au-tonomous (potentially mobile) agents (see for example [3], [7], [9], [11]). It isdue to the fact that the multi-agents appear in different areas as for example10 On stability and controllability of multi-agent linear systemsin consensus problem of communication networks [8], or formation control ofmobile robots [1].Jinhuan Wang, Daizhan Cheng and Xiaoming Hu in [9] study the con-sensus problem in the case of multiagent systems in which all agents have anidentical linear dynamics and it is a stable linear system. In [3], this result isgeneralized to the case where the dynamic of each agent is controllable.Wei Ni and Daizhan Cheng in [6] analyze the case where A0 = A1 = . . . =Ak and B1 = . . . = Bk; this particular case has practical scenarios as the flightof groups of birds. It is obvious that in this case, the mechanic of the firstsystem is independent of the others, then consensus under a fixed topologycan be easily obtained and it follows from the motion of the first equation.This consensus problem is known as leader-following consensus problem [6],[4].In this paper multiagent systems consisting of k agents of the same orderwith dynamicsx˙i = Aixi + Biui 1 ≤ i ≤ k (1)is considered.2. PreliminariesThe topology of the system is defined by means of the well-known topologyinduced by a graph in the following manner: we consider a graph G = (V, E)of order k with the set of vertices V = {1, . . . , k} 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 childnode and j is in the neighbor of i, specifically we define the neighbor of i andwe denote it by Ni to the set Ni = {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 anytwo vertices, otherwise is called disconnected.Associated to the graph we consider a matrix G = (gij) called (un-weighted) adjacency matrix defined as follows gii = 0, gij = 1 if (i, j) ∈ E , andgij = 0 otherwise.In a more general case we can consider that a weighted adjacency matrixis G = (gij) with gii = 0, gij > 0 if (i, j) ∈ E , and gij = 0 otherwise.The Laplacian matrix of the graph isL = (lij) =|Ni| if i = j−1 if j ∈ Ni0 otherwise(2)M. I. Garc´ıa-Planas et al 11Taking into account that the graph is undirected the matrix L is symmet-ric, consequently there exists an orthogonal matrix P such that PLP t = D.Moreover 0 is an eigenvalue of L and (1, . . . , 1)t is the associated eigenvector,and in the case where the graph is also connected the eigenvalue 0 is simple.(For more details about graph theory see [10]).It is helpful to remember that given A = (aij) ∈ Mn×m(C) and B =(bij) ∈Mp×q(C) the Kronecker product of the matrices is defined asA⊗B = (aijB) ∈Mnp×mq(C).See [5] for more information and properties.Finally, recall that the dynamical system x˙ = Ax + Bu is said to becontrollable if for every initial condition x(0) and every vector x1 ∈ Rn, thereexist a finite time t1 and control u(t) ∈ Rm, t ∈ [0, t1], such that x(t1) = x1.This definition requires only that any initial state x(0) can be steered toany final state x1 at some time t1. However, the trajectory of the dynamicalsystem between 0 and t1 is not specified. Furthermore, there is no constraintsposed on the control vector u(t) and the state vector x(t).It is well-known that the controllability can be ensured computing therank of the matrixC = ( B AB A2B . . . An−1B ) . (3)called controllability matrix, according to the following proposition.Proposition 1 The dynamical system x˙ = Ax+Bu is controllable if and onlyif rank C = n.As it has been said, controllability of the dynamical system x˙ = Ax + Buimplies that each initial state can be steered to 0 on a finite time-interval. Ifone only requires this condition to happen asymptotically for t→∞, one hasthe definition of stabilizable systems. That is to say:The system x˙ = Ax + Bu is called stabilizable if for each initial statex0 ∈ Rn there exists a (piece-wise continuous) control input u : [0,∞) −→ Rmsuch that the state-response with x(0) = x0 verifieslimt→∞x(t) = 0.(Stability for singular dynamical linear systems was studied by M.I. Garc´ıa-Planas in [2]).12 On stability and controllability of multi-agent linear systems3. Consensus problemRoughly speaking, we can define the consensus as a collection of processes suchthat each process starts with an initial value, where each one is supposed tooutput the same value and there is a validity condition that relates outputs toinputs. More specifically, the consensus problem is a canonical problem thatappears in the coordination of multi-agent systems.Given initial values (scalar or vector) of agents, the objective is to establishconditions under which and through local interactions and computations, theagents asymptotically agree upon a common value; that is to say: to reach aconsensus.Let us consider now, a multi-agent where the dynamic of each agent isgiven by the following dynamical system as (1) where matrices Ai and Bi arenot necessarily equal.Communication topology among agents are defined in § 2.The consensus of the system (1) is achieved using local information if thereexists a state feedbackui = Ki∑j∈Ni(xi − xj), 1 ≤ i ≤ ksuch thatlimt→∞ ‖xi − xj‖ = 0, 1 ≤ i, j ≤ k.For simplicity we define zi =∑j∈Ni(xi − xj), 1 ≤ i ≤ k.The closed-loop system obtained under this feedback is as followsX˙ = AX + BKZwhereX = ( x1 . . . xk )t , X˙ = ( x˙1 . . . x˙k )tA = diag (A1, . . . , Ak), B = diagB1, . . . , Bk)K = diag (K1, . . . ,Kk), Z =( ∑j∈N1 x1 − xj . . . ∑j∈Nk xk − xj )t .FromZ = (L ⊗ In)Xthe following proposition is deduced.Proposition 2 ([3])) The closed-loop system can be deduced in terms of ma-trices A, B and K in the following manner.X˙ = (A+ BK(L ⊗ In))X (4)M. I. Garc´ıa-Planas et al 13The interest is in Ki such that the consensus is achieved.Proposition 3 ([3]) Let us consider the system (1) with a connected adjacenttopology. If the system (1) is stable, the consensus problem has a solution.Corollary 3 Let us consider the system (1) with a connected adjacent topol-ogy. If the matrices Ai defining the dynamic are Hurwitz stable, then theconsensus is achieved.Remark 1 The system 4 can be written asX˙ = AX + BU with U = K(L ⊗ In)X .So, we have the following proposition.Proposition 4 A necessary (but not sufficient) condition for consensus to bereached is that the systemX˙ = AX + BUis stabilizable.Corollary 4A necessary condition for consensus to be reached is that the systemsx˙i = Aixi + Biui, ∀i = 1, . . . , kare stabilizable.Remark 2 The feedback K obtained from the feedbacks stabilizing the sys-tems x˙i = Aixi + Biui does not necessarily stabilize the system X˙ = (A +BK(L ⊗ In))X .Example.Let us consider the following two one-dimensional systemsx1 = u1x2 = x2 + u2The communication topology is defined by the graph V = {1, 2}, E = {(1, 2)} ⊂V × V.Taking K =(1 −16 −6)we have (A + BK(L ⊗ In)) =(1 −16 −5)witheigenvalues −0.2679, and −3.7321, then the system is stable.Taking k1 = −1 and k2 = −2, clearly these feedbacks stabilize the systems,but taking K =(k1k2)=( −1−2)we have (A + BK(L ⊗ In)) =( −1 12 −1)with eigenvalues 0.4142, and −2.4142, then the system is notstable.14 On stability and controllability of multi-agent linear systems4. ConclusionsIn this work, it is showed that the stability and controllability of multiagentsystems consisting of k + 1 agents with dynamics x˙i = Aixi + Biui, i =0, 1, . . . , k permit to obtain the consensus of the multisystem.References[1] A. Fax, R. Murray, Information flow and cooperative control of vehi-cle formations, IEEE Trans. Automat. Control. 49, (9), pp. 1453–1464,(2004).[2] M.I. Garc´ıa-Planas, Sensivity and stability of singular systems underproportional and derivative feedback, Wseas Transactions on Mathemat-ics, 8, (11), pp. 635–644, (2009).[3] M.I. Garc´ıa-Planas, Obtaining Consensus of Multi-agent Linear Dy-namic Systems, Advances in Applied and Pure Mathematics, pp. 91–95,(2014).[4] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobileautonomous agents using nearest neighbor rules, IEEE Transaction onAutomatic Control. 48, (6), pp. 943–948, (2007).[5] P. Lancaster, M. Tismenetsky, The Thoery of Matrices. AcademicPress. San Diego, (1985).[6] W. Ni, D. Cheng, Leader-following consensus of multi-agent systemsunder fixed switching topologies, Systems and Control Letters. 59, pp.209–217, (2010).[7] R.O. Saber, R.M. Murray, Consensus Protocols for Networks of Dy-namic Agents, Report.[8] R.O. Saber, R.M. Murray, Consensus problems in networks of agentswith switching topology and time-delays, IEEE Trans. Automat. Control.49, (9), pp. 1520–1533, (2004).[9] J. Wang, D. Cheng, X. Hu, Consensus of multi-agent linear dynamicssystems, Asian Journal of Control 10, (2), pp. 144–155, (2008).[10] D. West Introduction to Graph Theory Prentice Hall (3rd Edition),(2007).M. I. Garc´ıa-Planas et al 15[11] G. Xie, L. Wang, Consensus control for a class of networks of dynamicagents: switching topology, Proc. 2006 Amer. Contro. Conf., pp. 1382–1387, (2007).

On stability and controllability of multi-agent linear systems

Recent advances in communication and computing have made the control and coordination of dynamic network agents to become an area of multidisciplinary research at the intersection of the theory of control systems, communication and linear algebra. The advances of the research in multi-agent systems are strongly supported by their critical applications in different areas as for example in consensus problem of communication networks, or formation control of mobile robots. Mainly, the consensus problem has been studied from the point of view of stability. Nevertheless, recently some researchers have started to analyze the controllability problems. The study of controllability is motivated by the fact that the architecture of communication network in engineering multi-agent systems is usually adjustable. Therefore, it is meaningful to analyze how to improve the controllability of a multi-agent system. In this work we analyze the stability and controllability of multiagent systems consisting of k + 1 agents with dynamics x¿i = Aixi + Biui, i = 0, 1, . . . , 

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On stability and controllability of multi-agent linear systems

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