Consensus Analysis in Multi-Agent Systems Subject to Delays and Switching Topology∗

Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of the 12th IFAC Workshop on Time Delay Systems June 28-30, 2015. Ann Arbor, USA Proceedings of 12th IFAC Workshop on Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems June 28-30, 2015. Ann Arbor, MI, USA Available online at www.sciencedirect.com June 28-30, 2015. Ann Arbor, MI, USA June 28-30, 2015. Ann Arbor, MI, USA

ScienceDirect IFAC-PapersOnLine 48-12 (2015) 147–152

Consensus Consensus Consensus Subject to Subject Subject to to

Analysis in Analysis in Analysis in Delays and Delays and Delays and

Multi-Agent Systems Multi-Agent Systems ⋆⋆ Multi-Agent Systems Switching Topology Switching Topology Switching Topology ⋆

Carlos R. P. dos Santos Junior ∗∗ Fernando O. Souza ∗∗ ∗ Fernando ∗∗ Carlos R. P. P. dos Santos O. Souza ∗∗ ∗ ∗ Fernando Carlos Junior O. HeitorJunior J. Savino Carlos R. R. P. dos dos Santos Santos Junior Fernando O. Souza Souza ∗∗ ∗ ∗ Heitor J. Savino Heitor Heitor J. J. Savino Savino ∗ ∗ Graduate Program in Electrical Engineering - Universidade Federal ∗ ∗ Graduate Program in Electrical Engineering - Universidade Federal ∗ Graduate Program in Electrical Engineering --Brazil Universidade de Minas Gerais Belo Horizonte, MG, (e-mails:Federal Graduate Program in Electrical Engineering Universidade [email protected]; Minas Gerais Belo Horizonte, Horizonte, MG, Brazil (e-mails:Federal de Minas Gerais --- Belo MG, Brazil (e-mails: [email protected]). de Minas Gerais Belo Horizonte, MG, Brazil (e-mails: [email protected]; [email protected]). ∗∗ [email protected]; Department of Electronic Engineering [email protected]). - Universidade Federal de [email protected]; [email protected]). ∗∗ ∗∗ Department of Electronic Engineering - Universidade Federal de ∗∗ Department of Engineering -- Universidade Federal Minas Gerais - Belo Horizonte, Brazil (e-mail: Department of Electronic Electronic EngineeringMG, Universidade Federal de de Minas Gerais -- Belo Horizonte, MG, Brazil (e-mail: Minas Gerais Belo Horizonte, MG, Brazil (e-mail: [email protected]). Minas Gerais [email protected]). - Belo Horizonte, MG, Brazil (e-mail: [email protected]). [email protected]). Abstract: This paper presents a consensus analysis method for multi-agent systems subject to Abstract: paper presents a consensus analysis method for multi-agent systems subject to Abstract: This paper analysis method for systems subject nonuniform This and non-differentiable time-varying delays and switching topology. It is considered Abstract: This paper presents presents a a consensus consensus analysis method for multi-agent multi-agent systems subject to to nonuniform and non-differentiable time-varying delays and switching topology. It is considered nonuniform and non-differentiable time-varying delays and switching topology. It is considered that the topology changes according to a continuous time Markov chain and the existence of nonuniform and non-differentiable time-varying delays and switching topology. It is considered that the topology changes according to aa continuous time Markov chain and the existence of that the topology changes according to continuous time Markov chain and the existence of nonuniform time-varying delays in the agents control laws. The main result formulated as linear that the topology changes according to a continuous time Markov chain and the existence of nonuniform time-varying delays the laws. main linear nonuniform time-varying delaysbyin inshowing the agents agents control laws. The The main result result formulated assystem linear matrix inequalities is obtained thatcontrol the consensus analysis in theformulated multi-agentas nonuniform time-varying delays in the agents control laws. The main result formulated as linear matrix is obtained showing that the multi-agent system matrix inequalities is by showing that the consensus analysis in the system can be inequalities performed by analyzingby the stability of the an consensus associated analysis Markov in jump linear system. The matrix inequalities is obtained obtained bythe showing that the consensus analysis in the multi-agent multi-agent system can be performed by analyzing stability of an associated Markov jump linear system. The can be performed by analyzing the stability of an associated Markov jump linear system. The results are illustrated by numerical examples. can be performed by analyzing the stability of an associated Markov jump linear system. The results are by numerical examples. results are illustrated illustrated by examples. results illustrated by numerical numerical © 2015, are IFAC (International Federation examples. of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: multi-agent system, consensus analysis, time-varying delay, switching topology, Keywords: multi-agent system, consensus time-varying Keywords: multi-agent system,linear consensus analysis, time-varying delay, delay, switching switching topology, topology, Markov jump linear systems, matrixanalysis, inequalities. Keywords: multi-agent system, consensus analysis, time-varying delay, switching topology, Markov jump linear systems, linear matrix inequalities. Markov jump linear systems, linear matrix inequalities. Markov jump linear systems, linear matrix inequalities. 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION In recent years, studies involving multi-agent systems have In studies multi-agent systems have In recent recent years, years, studies involving involving multi-agent systemsfields. have attracted the interest of researchers from several In recent years, studies involving multi-agent systems have attracted the interest of researchers from several fields. attracted the interest of researchers from several fields. Certainly, this comprehensiveness is due to advances in attracted the interest of researchers fromtoseveral fields. Certainly, this comprehensiveness is due advances in Certainly, this comprehensiveness is due to advances in communication systems and its diversity of applications, Certainly, this comprehensiveness is due to applications, advances in communication systems and its diversity communication systems and of its unmanned diversity of ofaerial applications, such as coordinated control vehicles communication systems and its diversity of applications, such coordinated control of such as as coordinated control of unmanned unmanned aerial vehicles (Ren et al., 2007), flight formation (Giuliettiaerial et al.,vehicles 2000), such as coordinated control of unmanned aerial (Ren et al., 2007), flight formation (Giulietti et al., al.,vehicles 2000), (Ren et al., 2007), flight formation (Giulietti et 2000), cooperative control of autonomous agents (Pimenta et al., (Ren et al., 2007), flight formation (Giulietti et al., 2000), cooperative control of autonomous agents (Pimenta et al., al., cooperative control of autonomous agents (Pimenta et 2013), and alignment of satellites (Beard et al., 2001). An cooperative control of of autonomous agentset (Pimenta et al., 2013), and alignment satellites (Beard al., 2001). An 2013), and and alignment of satellites satellites (Beard et et al., al., 2001). can An overview of alignment recent applications of multi-agent systems 2013), of (Beard 2001). An overview overview of recent applications of multi-agent multi-agent systems systems can can be found of in recent Cao etapplications al. (2013). of overview of recent applications of multi-agent systems can be be found found in in Cao Cao et et al. al. (2013). (2013). be Cao et al. (2013). Onefound of theinmajor concerns related to multi-agent systems One the concerns related to One ofone theofmajor major concerns related law to multi-agent multi-agent systems is theof determining a control that allows systems a group One of the major concerns related to multi-agent systems is the one of determining aa control law that allows aaa group is the one of determining control law that allows group of agents to achieve an agreement on the value of given is the one of determining a control law that allows a of agents to achieve an agreement on the value of aa group given of agents to achieve an agreement on the value of given magnitude, e.g. velocity and position. This is known as of agents to e.g. achieve an agreement on the value of a given magnitude, magnitude,problem. e.g. velocity velocity and and position. position. This This is is known known as as consensus magnitude, e.g. velocity and position. This is known as consensus problem. consensus problem. consensus Vicsek et problem. al. (1995) introduced the dynamic interaction Vicsek al. the interaction Vicsek et etneighbors al. (1995) (1995)forintroduced introduced the dynamic dynamic interaction between the ordination of a particle system Vicsek et al. (1995) introduced the dynamic interaction between neighbors for the ordination of aa particle system between neighbors for the ordination of particle system moving in a plane, showing that a distributed control between neighbors for the ordination of a particle system moving in aa plane, showing that aaa distributed control moving in plane, showing that distributed control law is able to govern all agents into common direction. moving in to a plane, showing that aa distributed control law govern all into law is is able able et to al. govern all agents agents into a common common direction. Jadbabaie (2003) were the first to modeldirection. consenlaw is able to govern all agents into a common direction. Jadbabaie et al. were first model consenJadbabaie et using al. (2003) (2003) wereofthe the first to tograph modeltheory consensus problems concepts algebraic to Jadbabaie et al. (2003) were the first to model consensus problems using concepts of algebraic graph theory to sus problems using concepts of algebraic graph theory to represent the relations between neighbors. In Olfati-Saber sus problems using concepts of algebraic graph theory to represent the relations between neighbors. In Olfati-Saber represent the relations between neighbors. In Olfati-Saber (2006), the control law that dictates the interaction berepresent the relations between neighbors. In Olfati-Saber (2006), the law dictates interaction be(2006),agents the control control law that that dictates the the interaction between was named as consensus protocol and many (2006), the control law that dictates the interaction between agents was named as consensus protocol and many tween agents was named as consensus protocol and many particularities of the consensus problemprotocol were treated, such tween agents was named as consensus and many particularities of consensus problem such particularities of the the consensusnetworks, problem were were treated, such as directed and undirected fixedtreated, or variable particularities of the consensus problem were treated, such as directed and undirected networks, fixed or variable as directed and undirected networks, fixed or variable topologies, the time delay effects, etc. as directed andtime undirected networks, fixed or variable topologies, topologies, the the time time delay delay effects, effects, etc. etc. topologies, the delay effects, etc. ⋆ This research was partially supported by the Brazilian funding ⋆ This research research was partially supported the Brazilian funding ⋆ This agencies FAPEMIG, CNPq. by was partially supported by ⋆ This research was CAPES, partially and supported by the the Brazilian Brazilian funding funding agencies FAPEMIG, CAPES, and CNPq. agencies FAPEMIG, CAPES, and CNPq. agencies FAPEMIG, CAPES, and CNPq.

In practical applications there always exist time delays in In practical applications there always exist time delays in In practical applications there always exist in the agents interactions. is due to information processIn practical applicationsThis there always exist time time delays delays in the agents interactions. This is due to information processthe agents interactions. This is due to information processing, physical limitations in communication channels, timethe agents interactions. This is due to information processing, limitations in channels, timeing, physical physical limitationsetc. in communication communication channels, many timeresponse of actuators, Based on this situation, ing, physical limitations in communication channels, timeresponse of actuators, etc. Based on this situation, many response of actuators, etc. Based on this situation, many works have dealt with etc. the Based consensus problem subject to response of actuators, on this situation, many works have dealt the consensus subject to works have dealt with the consensus problem subject to time delays. Somewith considered constantproblem delays for all the works have dealt with the consensus problem subject to time delays. Some considered constant delays for all the time delays. Some considered constant delays for all the agents’ interactions (Lin et al., 2008), others considered time delays. Some considered constant delays for all the agents’ interactions (Lin et et al., (Zhang 2008), others others considered agents’ interactions (Lin 2008), considered them constant and nonuniform et al., 2011), and agents’ interactions (Lin et al., al., (Zhang 2008), others considered them constant and nonuniform et al., 2011), and them constant and nonuniform (Zhang et al., 2011), and yet there were other studies that considered nonuniform them constant and nonuniform (Zhang et al., 2011), and yet there were other studies that considered nonuniform yet there were other studies that considered nonuniform time-varying delays (Savino et al., 2014). yet there were other studies that considered nonuniform time-varying delays (Savino et 2014). time-varying delays (Savino et al., al., time-varying delays (Savino al., 2014). 2014). There are some works in theetliterature considering multiThere are some works in the literature considering multiThere are some works in the literature considering multiagent systems subject to switching topology. In real enThere are somesubject works in the literature considering multiagent systems to switching topology. In real enagent systems subject to switching topology. In real vironments these switches are due to temporary commuagent systems subject to switching topology. In commureal enenvironments these switches are due to temporary vironments these switches are due to temporary communication losses or switches agents failure, changescommuin the vironments these are dueand/or to temporary nication losses or agents failure, and/or changes in the nication losses agents failure, changes in the arrangement of or agents. A description of this problem nication losses or agents failure, and/or and/or changes in can the arrangement of agents. A description of this problem can arrangement of agents. A description of this problem can be seen in Ren et al. (2007), whereas the switching topolarrangement of agents. A description of this problem can be seen in Ren et al. (2007), whereas the switching topolbe seen in et whereas the switching topology is not stochastic. Currently, there works in the be seen in Ren Ren et al. al. (2007), (2007), whereas theare switching topology is not stochastic. Currently, there are works in the ogy is not stochastic. Currently, there are works in the literature treating the Currently, switching there topology as a Markov ogy is not stochastic. are works in the literature treating the switching topology as a Markov literature treating the switching topology as a Markov process. In Zhao et al. (2011), analysis conditions, for literature treating the switching topology as a Markov process. In Zhao et al. (2011), analysis conditions, for process. In Zhao et al. (2011), analysis conditions, for mean square consensus for second order multi-agent sysprocess. In Zhao et al.for (2011), analysis conditions, sysfor mean square consensus second order multi-agent meanwith square consensus for second second orderare multi-agent system Markov switching topology, formulatedsysas mean square consensus for order multi-agent tem Markov switching topology, are formulated as tem with Markov switching topology, are formulated as linearwith matrix inequalities (LMIs). It is also considered tem with Markov switching topology, are formulated as linear matrix inequalities (LMIs). It is also considered linear matrix inequalities (LMIs). It is also considered the cases where the transition probability rates are not linear matrix inequalities (LMIs). It is also considered the cases where the transition not the cases where the transition probability rates are not precisely system probability is subject torates time are delays. the cases known where and the the transition probability rates are not precisely known and the system is subject to time delays. precisely known known and and the the system system is is subject subject to to time time delays. delays. precisely This paper extends previous results of consensus analysis This paper extends previous results of consensus analysis This paper previous results of for multi-agent systems, considering and This paper extends extends previous results non-differentiable of consensus consensus analysis analysis for multi-agent systems, considering non-differentiable and for multi-agent multi-agent systems, considering considering non-differentiable and nonuniform time-varying delays, to non-differentiable the case of switching for systems, and nonuniform time-varying delays, to the case of switching nonuniform time-varying delays, to the case of switching topologies described as a delays, continuous time Markov chain. nonuniform time-varying to the case of switching topologies described as aa based continuous time Markov chain. topologies described as continuous time Markov The analysis proposed on the transformation of topologies described as is a based continuous time Markov chain. chain. The analysis proposed is on the transformation of The analysis proposed is based on the transformation of the multi-agent system into a Markov jump linear system The analysis proposed is based on thejump transformation of the multi-agent system into a Markov linear system the multi-agent system into a Markov jump linear system (MJLS), so thatsystem we analyze stability of thelinear transformed the multi-agent into a Markov jump system (MJLS), The so that that we analyze analyze stability of the the transformed (MJLS), so we stability of system. proposed method is sufficient and based (MJLS), so that we analyze stability of the transformed transformed system. The proposed method is sufficient and based system. The proposed method is sufficient system. The proposed method is sufficient and and based based

Copyright © 2015, IFAC 2015 147 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 147 Copyright © IFAC 2015 147 Peer review under responsibility of International Federation of Automatic Copyright © IFAC 2015 147Control. 10.1016/j.ifacol.2015.09.368

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on LMIs. Simulation examples are given to show the performance of the proposed method. 2. BACKGROUND 2.1 Algebraic Graph Theory A directed graph is a representation of a set of nodes or vertices, and a set of edges, where the edges connecting the nodes are unidirectional and indicate the information flow. The graph is said to be connected when there exists a path from each node to all the other nodes. Mathematically, a graph is described by G = (V, E), with V representing the set of n nodes labeled as v1 , v2 , . . . , vn , and E the set of edges, denoted by eij = (vi , vj ), where the first element vi is said to be the parent node (tail) and the other vj to be the child node (head). In matrix form, we can write the adjacency matrix A = [aij ], where  = 0, if i = j or ∄ eji aij = 1, iff ∃ eji which elements are defined by aij = 0 if i = j, and aij = 1 if and only if there is an edge connecting the node vj to vi .

Moreover, it is defined the degree matrix D = [dii ] which is a diagonal matrix related to the nadjacency matrix. Its elements are defined by dii = j=1 aij . Then, the Laplacian matrix is defined as L = D − A, n i.e. L = [lij ] with lii = j=1 aij and lij = −aij , for i �= j.

An important characteristic of the Laplacian matrix is that the sum of its rows is null, which is given as follows: L1 = 0, (1) where 1 is an unitary column vector with proper dimension. 3. PROBLEM FORMULATION

Moreover, we assume that θt is given by a continuous time Markov chain with discrete states given by the set S = {1, 2, ..., s}, and the probability transition matrix Ψ = [ψpq ] (p and q represent the states of the Markov chain) defined by:  p �= q, π ∆ + o(∆) ψpq = P{θt+∆ = q|θt = p} = pq 1 + πpp ∆ + o(∆) p = q,

where P represents the probability of {·}, ∆ > 0, lim∆→0 o(∆) transition ma∆ = 0, and πpq the rates in the  trix Π = [πpq ], where πpq ≥ 0 and πpp = − sq=1,q�=p πpq .

For a compact representation of the multi-agent system, we consider that k¯ is the number of delays τij (t, θt ) in the topology with the highest number of different delays. Thus, we consider that the delays of the system are given ¯ by τk (t, θt ), with k = 1, 2, . . . , k. Therefore, the multi-agent system, where the agents’ dynamics are given by (2) with the consensus protocol in (3), is represented by: ˙ X(t) =−

¯ k 

Lk (θt )X(t − τk (t, θt )),

(4)

k=1

where X(·) = [x1 (·), x2 (·), . . . , xn (·)]T and Lk (θt ) is the Laplacian matrix of the subgraph connections associated only with the delay τk (t, θt ) on the topology θt . Note that k¯ k=1 Lk (θt ) = L(θt ) with L(θt ) the Laplacian matrix of all graph connections on the topology θt . Before progressing further, in order to formalize the problem of consensus analysis with stochastic switching topology, we assume the following definition: Definition 1. (Zhao et al. (2011)) Under stochastic switching topology, the multi-agent system (4) achieves consensus if, for any initial condition (X(0) ∈ Rn and θ0 ∈ S), |xi (t) − xj (t)| → 0 as t → ∞, ∀ i �= j.

3.2 Transformed Multi-Agent System

3.1 Consensus Problem A multi-agent system can be represented by a graph where the nodes represent agents, and the edges represent communication channels. Thus, consider a multi-agent system composed by n agents arranged in a directed network and the dynamics of each agent given by (2) x˙ i (t) = ui (θt , t − τij (t, θt )), i = 1, 2, ..., n, where xi and ui ∈ R represent the state and the input control of the ith agent, respectively, θt represents the topology of the multi-agent system at the instant of time t and is not subject to time delays, and τij (t, θt ) the delay between the agents i and j on the topology θt . The consensus protocol is given by: n    ui (θt , t) = − aij (θt ) xi (t) − xj (t) . (3) j=1

In this paper, we represent the delays by τij (t, θt ) = τ + µij (t, θt ), where τ is a constant value and µij (t, θt ) are time-varying perturbations that satisfy |µij (t, θt )| ≤ µ ¯<τ ∀ (i, j, θt ), such that τij (t, θt ) ∈ [τ − µ ¯, τ + µ ¯ ]. 148

In this section, we present how to construct a transformed multi-agent system, such that its stability dictate if the multi-agent system in (4) achieves consensus. The proposed transformed system is obtained based on the disagreement of the state variables of the multi-agent system. Then, it is defined the new variable: zi (t) = x1 (t) − xi+1 (t), such that for the consensus be achieved, it is necessary that lim zi (t) = 0 with i = 1, 2, . . . , n − 1. Thus, as in t→∞

Sun and Wang (2009), we define: z(t) = U X(t), (5) (6) X(t) = x1 (t)1 + W z(t),   T 0 where U = [ 1 −In−1 ] and W = , with 0 a null −In−1 column vector of proper dimension, I an identity matrix, and z(t) = [z1 (t), . . . , zn−1 (t)]. Therefore to obtain the transformed system, we take the time-derivative of (5) and combine it with the Equations (1), (4), and (6), as follows

IFAC TDS 2015 June 28-30, 2015. Ann Arbor, MI, USA Carlos R. P. dos Santos Junior et al. / IFAC-PapersOnLine 48-12 (2015) 147–152

˙ z(t) ˙ = U X(t) = −U

¯ k 

lim E

t→∞

Lk (θt )X(t − τk (t, θt ))

k=1 ¯ k

= −U



Lk (θt )[x1 (t − τk (t, θt ))1+W z(t − τk (t, θt ))]

k=1 ¯ k

=−



U Lk (θt )W z(t − τk (t, θt ))].

Then, the proposed transformed system is given by: ¯ k 

¯ k (θt )z(t − τk (t, θt )), L

(7)

k=1

¯ k (θt ) = U Lk (θt )W ∈ R(n−1)×(n−1) . where L Now, based on the transformed system (7) we state the main result of this section. Proposition 2. The system (7) reaches the origin if and only if system (4) achieves consensus. Proof. It is known that the system (7) reaches the origin when limt→∞ z(t) = 0 and that the multi-agent system (4) achieves consensus when limt→∞ X(t) = β(t)1, with β(t) ∈ R. Initially, consider (5) and assume that the multi-agent system (4) achieves consensus, thus: lim z(t) = lim U X(t) = U lim X(t) t→∞

t→∞

t→∞

= β(t)U 1 = 0. Therefore, we have that if the multi-agent system (4) achieves consensus, then (7) reaches the origin. Now, consider (6) and assume that the system (7) reaches the origin, thus: lim X(t) = lim [x1 (t)1 + W z(t)] t→∞

t→∞

= lim x1 (t)1 + W lim z(t) t→∞

t

0

 z T (ξ)z(ξ)dξ < ∞,

where E represents the mathematical expectancy. Definition 5. (Fei et al. (2009)) Consider a stochastic process defined by {θt , t ∈ [0, +∞)}. Then, the infinitesimal generator L applied in a function f (θt ) is given by:   f (θt+∆ ) − f (θt ) . Lf (θt ) = lim E ∆→0 ∆ Hereinafter, for a more compact notation of the stochastic variables, the argument θt is replaced by the subscript index ℓ, if no confusion occurs. For example, Y (θt ) is replaced by Yℓ and τk (t, θt ) is replaced by τkℓ (t).

k=1

z(t) ˙ =−



149

t→∞

= lim x1 (t)1 = β(t)1. t→∞

Then, if the system (7) reaches the origin, the multi-agent system (4) achieves consensus. This completes the proof.  The above proposition essentially establishes that the task of consensus analysis of the multi-agent system in (4), can be performed by studying the stability of the reduceddimension transformed system in (7). 4. PRELIMINARIES The following are typical results found in the literature. Lemma 3. (Gu (2000)) For any constant matrix M = M T > 0 and scalar τ > 0, the following inequality holds:  t  t  t 1 xT (ξ)M x(ξ)dξ ≥ xT (ξ)dξ M x(ξ)dξ. τ t−τ t−τ t−τ Definition 4. (Fei et al. (2009)) A MJLS will be stochastically stable if, for any initial condition (z(0) ∈ Rn and θ0 ∈ S), the following inequality is satisfied. 149

5. CONSENSUS ANALYSIS Now we are in position to enounce our main result. Theorem 6. Consider the multi-agent system in (4) with ¯, τ + µ ¯] for all k and θt . Let τ > 0 and τk (t, θt ) ∈ [τ − µ 0 ≤ µ ¯ < τ be given. Then, the multi-agent system (4) achieves consensus for all τk (t, θt ) ∈ [τ − µ ¯, τ + µ ¯], if there exist (n − 1) × (n − 1) matrices Pℓ = PℓT , Qℓ , R1 = R1T , ¯ R2 , R3 = R3T , Sℓ = SℓT , and Zk = ZkT , for k = 1, 2, ..., k, where the constant k¯ is the number of delays in the topology with the maximum number of different delays, and ℓ = 1, 2, ..., s, where s is the number of topologies, such that the following LMIs hold ∀ℓ = 1, 2, ..., s:   Pℓ Qℓ 1 > 0, (8) ∗ Sℓ  τ  R1 R2T ¯ R= > 0, (9) R2 R3 and LMI (10) in the top of next page. Proof. Due to Proposition 2, the stochastic stability of system (7) implies consensus for the system (4) given that, when system (7) reaches the origin, the condition imposed by Definition 4 is satisfied. Furthermore, by the definition of z(t) as in (6), we have that system (4) achieves consensus by Definition 1. Thus, the proof of this theorem relies on the fact that if the conditions presented here are satisfied, the system (7) will be stochastically stable by Definition 4. First, we show that if the proposed LMIs hold, then the inequalities V (z(t), ℓ) > 0 and LV (z(t), ℓ) < 0 are satisfied, where L is the infinitesimal generator operator, and V (z(t), ℓ) is the following Lyapunov–Krasovskii stochastic functional: V (zt , ℓ) = V1 (z(t), ℓ) + V2 (zt , ℓ) (11) + V3 (zt ) + V4 (zt , ℓ) + V5 (zt ), ¯, t], where zt corresponds to z(σ) for σ ∈ [t − τ − µ T V1 (z(t), ℓ) = z (t)Pℓ z(t),  t T V2 (zt , ℓ) = 2z (t)Qℓ z(ξ)dξ, t−τ  0  t ¯ z (ξ)dξdζ, z¯T (ξ)R¯ V3 (zt ) = −τ t+ζ  0 z T (t + ξ)Sℓ z(t + ξ)dξ, V4 (zt , ℓ) = V5 (zt ) =



−τ µ ¯  t

−¯ µ

t−τ +ζ

T

z˙ (ξ)

¯ k 

k=1

Zk z(ξ)dξdζ, ˙

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

 φ¯11  ℓ     ∗      ∗  φ¯ℓ =    ∗    ∗    ∗    ∗ ∗ with φ¯11 ℓ =

s �

¯ k �

¯ kℓ ¯ 2ℓ . . . µ ¯ 1ℓ µ ¯ kℓ − 1 R2 + ¯Fℓ L ¯Fℓ L ¯Fℓ L πℓm Qm µ L ¯   τ  m=1 k=1  ¯ k  � ¯ kℓ ¯ 1ℓ µ ¯ 2ℓ . . . µ ¯¯  −Gℓ Qℓ µ ¯ Gℓ L ¯Gℓ L ¯Gℓ L L kℓ   k=1  1 1   R2 0 0 ... 0 − R3 − Sℓ  τ τ s  < 0, � 1 1  0 0 ... 0 πℓm Sm ∗ − R1 +   τ τ m=1   ∗ ∗ −¯ µZ1 0 ... 0   . .. ..  . ∗ ∗ ∗   ..  . ∗ ∗ 0 ∗ ∗ ∗ ∗ ... ∗ −¯ µZk¯

1 φ¯12 ℓ −Qℓ + R3 − Fℓ τ φ¯22 ∗ ∗ ∗ ∗ ∗ ∗

(10)

¯ k

πℓm Pm + Qℓ +

QTℓ

m=1

� 1 T ¯22 = τ R3 + 2¯ + τ R1 − R3 + Sℓ , φ¯12 µ Zk − Gℓ − GTℓ . ℓ = Pℓ + τ R2 − Fℓ and φ τ k=1

� ¯ = R ¯T = given z¯T (ξ) = z T (ξ) z˙ T (ξ) , Pℓ = PℓT , Qℓ , R � � R1 R2T , Sℓ = SℓT , and Zk = ZkT . R2 R3 �

Next, we present the condition for V (zt , ℓ) > 0 to be satisfied. Applying Lemma 3 in (11), we have: � � � 0 � t Pℓ Qℓ T ¯ z (ξ)dξdζ 1 η+ V (zt , ℓ) ≥ η z¯T (ξ)R¯ ∗ Sℓ −τ t+ζ τ ¯ � µ¯ � t k � T + z˙ (ξ) Zk z(ξ)dξdζ, ˙ −¯ µ

t−τ +ζ

with Λ = [z T (t)Fℓ + z˙ T (t)Gℓ ]. Then, applying the inequality 2aT b ≤ aT Xa + bT X −1 b, where a and b are vectors and X is a matrix defined positive, in (13) we have ¯ � −τ k � ¯ kℓ )Z −1 (ΛL ¯ kℓ )T dξ v(t) ≤ (ΛL k k=1 −τkℓ (t) ¯ � −τ k �

with η T = z T (t)

�0

−τ

k=1

k=1

z T (t + ξ)dξ .

Now, we prove the LMI condition to guarantee that LV (zt , ℓ) < 0. Initially, consider the following null term:   ¯ k � ¯ kℓ z(t − τkℓ (t)) 0 = 2[z T (t)Fℓ + z˙ T (t)Gℓ ] −z˙ − L k=1

� −τ z(t ˙ + ξ)dξ 

−τkℓ (t)

k=1



˙ − = 2[z (t)Fℓ + z˙ (t)Gℓ ]−z(t)

¯ k �

k=1



¯ kℓ )¯ ¯ kℓ )T (ΛL µZk−1 (ΛL

+

�

t−τ +¯ µ

z˙ T (ξ)

t−τ −¯ µ

¯ k �

Zk z(ξ)dξ. ˙

k=1

Moreover, invoking the operator of the infinitesimal generator in (11), we have LV (zt , ℓ) = LV1 (z(t), ℓ) + LV2 (zt , ℓ) + LV3 (z(t)) +LV4 (zt , ℓ) + LV5 (z(t)), �

T ˙ (t) LV1 = z˙ T (t)Pℓ z(t)+z T (t)Pℓ z(t)+z

LV2 = 2z˙ T (t)Qℓ

�

150

s �

�

πℓm Pm z(t),

m=1

t

z(ξ)dξ

t−τ

+ 2z T (t)Qℓ z(t) − 2z T (t)Qℓ z(t − τ ) �� � s t � πℓm Qm z(ξ)dξ, + 2z T (t) �

t−τ

m=1 t T

¯ z (ξ)dξ, z¯ (ξ)R¯

t−τ T

−τkℓ (t)

(14)

where

¯ z (t) − LV3 = z¯T (t)τ R¯

¯ kℓ z(t − τ ) + v(t), L

where Fℓ and Gℓ have appropriated dimensions, and ¯ � −τ k � ¯ v(t) = 2Λ z(t ˙ + ξ)dξ, (13) Lkℓ k=1

¯ k �

−τkℓ (t)

k=1

Note that, if LMIs (8) and (9) hold, then the first and the second terms of the right side of the inequality (12) are positive. Also, if the LMI (10) holds, then the terms −Zk (∀k) in the principal diagonal are defined negative, which implies that the third term on the right side of inequality (12) is positive. Thus, a sufficient condition to check if V (zt , ℓ) > 0, is to verify if LMIs (8), (9), and (10) are satisfied.

T

≤

�

= 2[z T (t)Fℓ + z˙ T (t)Gℓ ]  � ¯ � k � ¯ kℓ z(t − τ ) − × −z− ˙ L

z˙ T (t + ξ)Zk z(t ˙ + ξ)dξ

+

(12)

�

T



s �

LV4 = z T (t)Sℓ z(t) − z (t − τ )Sℓ z(t − τ ) � � s � 0 � T z (t + ξ) πℓm Sm z(t + ξ)dξ, + −τ

m=1

IFAC TDS 2015 June 28-30, 2015. Ann Arbor, MI, USA Carlos R. P. dos Santos Junior et al. / IFAC-PapersOnLine 48-12 (2015) 147–152

LV5 = z˙ T (t)2¯ µ

¯ k �

Zk z(t)− ˙

k=1

�

t−τ +¯ µ

z˙ T (ξ) t−τ −¯ µ

¯ k �

τ1,1 (t) Zk z(ξ)dξ. ˙

1

LV (zt , ℓ) ≤ ΥT Φℓ Υ +

¯ kℓ )¯ ¯ kℓ )T , (ΛL µZk−1 (ΛL

(15)

2

Then, the negativeness of the right side of the inequality in (15), is a sufficient condition to guarantee that LV (zt , ℓ) < 0. Now, by Schur’s complement we rewrite (15) as LV (zt , ℓ) ≤ ΥT φ¯ℓ Υ. Thus, if LMI (10) holds, then LV (zt , ℓ) < 0. Finally, it is shown that if LMIs (8), (9), and (10) hold, then the system in (7) is stochastically stable according to the Definition 4. Assume that the LMIs in the Theorem are satisfied then LV (zt , ℓ) < −αz T (t)z(t) is true for some α > 0 sufficiently small. Then applying the Dynkin’s Formula, it yields �� t � LV (z(ξ), ℓξ )dξ E [V (zt , ℓ)] = V (z(0), ℓ0 ) + E 0

and through some manipulations:

t T

�

z (ξ)z(ξ)dξ , E [V (zt , ℓ)] − V (z(0), ℓ0 ) ≤ −αE 0 �� t � αE z T (ξ)z(ξ)dξ ≤ V (z(0), ℓ0 ) − E [V (zt , ℓ)] , 0

≤ V (z(0), ℓ0 ),

E

��

0

t

�

z T (ξ)z(ξ)dξ ≤

1 τ2,1 (t)

2

V (z(0), ℓ0 ) , α

it reveals that � � �� t � V (z(0), ℓ0 ) lim E , z T (ξ)z(ξ)dξ ≤ lim t→∞ t→∞ α 0 V (z(0), ℓ0 ) ≤ < ∞. α Therefore, if the LMIs proposed by Theorem 6 hold, the system (7) is stochastically stable according to the Definition 4. Consequently, from Proposition 2, the system (4) achieves consensus according to the Definition 1.  6. NUMERICAL EXAMPLE Consider a multi-agent system which has two possible topologies as depicted by the graphs Ga and Gb in Fig. 1. In this example, it is worth noting that when the system is on the topology Ga , the agents 1 and 3 do not receive information from the other agents, therefore if the system never leaves this topology it will not reach consensus, unless the agents 1 and 3 are in consensus occasionally. On the other hand, when the system is on the topology Gb , the agent 4 is disconnected from the others, therefore, 151

τ2,2 (t)

τ3,1 (t) 3

4

k=1

�0 T z (t + ξ)dξ] where ΥT = [z T (t) z˙ T (t) z T (t − τ ) −τ and Φℓ is the top-left 4×4 sub-block matrix of φ¯ℓ in (10), with entries correspond to rows 1 through 4 and columns 1 through 4 of φ¯ℓ .

��

τ1,2 (t)

k=1

¯ and z¯, adding the null term, and applying Expanding R Lemma 3 in (14), we have: ¯ k �

151

3

(Ga )

4 (Gb )

Fig. 1. Graphs Ga and Gb representing the possible topologies of the multi-agent system. if the system never leaves this topology it will not reach consensus, unless the agent 4 is in consensus occasionally. Although the system does not reach consensus on each individual topology, when it switches between the topologies Ga and Gb it can achieve consensus according to the transition rates. This fact is due the absence of communication link in one of the topologies be supplied by the communication link from the other one. In the following it is illustrated one such case. Initially, to represent the multi-agent system as in (4) we have that k¯ = 3, since the topology Ga has the highest number of delays, i.e. subject to three delays. Then, it yields the following Lkℓ :       0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 1 0 −1 −1 1 0 0 , L = , L = L1,1 =  0 0 0 0 2,1 0 0 0 0  3,1 0 0 0 0 0 0 −1 1 00 0 0 0 0 00 (16) and       0 0 0 0 0 00 0 1 −1 0 0  0 0 0 0 0 0 0 0 0 0 0 0 L1,2 =  ,L = , ,L = 0 0 0 0 2,2 −1 0 1 0 3,2 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 (17) where the sub-index ℓ = 1 and ℓ = 2 represent the topologies Ga and Gb , respectively. Moreover, assume that the switching topology dynamic can be described by a continuous time Markov chain, where the matrix of transition rates is given by � � −1 1 Π= . (18) 1 −1

Besides, consider that the system is subject to nondifferentiable and non-uniform time-varying delays τkℓ (t) ∈ [τ − µ ¯, τ + µ ¯], k = 1, 2, 3 and ℓ = 1, 2. Therefore, in order to evaluate the efficiency of the proposed method, we determine the largest value of µ ¯, for given values of τ , such that the LMIs in the Theorem 6 hold. The obtained results are listed in the Table 1. Table 1. Largest perturbation allowed µ ¯ for different values of τ by Theorem 6. τ µ ¯

0.15 0.150

0.20 0.200

0.25 0.227

0.30 0.200

0.35 0.183

In order to illustrate the applicability of the main result showed in this paper, we run a numerical simulation considering the multi-agent system (4) with data presented in this section and choosing τ = 0.4 and µ ¯ = 0.1, such that

IFAC TDS 2015 152 Carlos R. P. dos Santos Junior et al. / IFAC-PapersOnLine 48-12 (2015) 147–152 June 28-30, 2015. Ann Arbor, MI, USA

the time delays can vary in the interval τkℓ (t) ∈ [0.30, 0.50], for all k and ℓ. Note that by Theorem 6, see Table 1, the multi-agent system achieves consensus in this interval. The upper graphic of Figure 6 presents the state trajectories of the agents and the lower graphic the states of the switching topology, where 1 and 2 correspond to the topologies Ga and Gb , respectively. Figure 3 illustrates the behavior of the delays τ1,1 (t) and τ1,2 (t), and shows that even belonging the same interval of domain they are different.

7. CONCLUSION This paper presented LMI sufficient conditions for consensus analysis in multi-agent systems subject to nondifferentiable and non-uniform time-varying delays and switching topology according to a continuous time Markov chain. We illustrated the performance of the proposed method by performing numerical simulation in an example. In future results we will also consider uncertainties in the transition rates of the Markov chain. REFERENCES

States

15 10 5 0 0

2

4

6 Time

8

10

12

2

4

6 Time

8

10

12

Topology

3

2 1

0 0

Fig. 2. Upper graphic: state trajectories of the multi-agent system in (4) which has the topologies in Fig. 1 that yield Lk (θt ) for θt = 1 and θt = 2 given by Lk,1 in (16) and Lk,2 in (17), respectively, and subject to time-varying delays τkℓ (t) ∈ [τ − µ ¯, τ + µ ¯ ], k = 1, 2, 3, ℓ = 1, 2, with τ = 0.4 and µ ¯ = 0.1 (see Fig. 6). Moreover the switches between the graphs Ga and Gb is governed by a continuous time Markov chain with transition rates matrix (18). Lower graphic: switching topology for one simulation, where 1 indicates that the multi-agent system assumes the behavior of graph Ga and 2 the behavior of graph Gb .

0,5

τ1,1 (t)

0,45 0,4

0,35 0

2

4

6 Time

8

10

12

0

2

4

6 Time

8

10

12

0,5

τ1,2 (t)

0,45 0,4

0,35

Fig. 3. Illustration of behavior of delays τ1,1 (t) and τ1,2 (t). The illustration of the others delays are omitted since they present similar behavior.

152

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