Output Consensus Controller Design for Nonlinear Relative Degree One Multi-Agent Systems with Delays*

Output Consensus Controller Design for Nonlinear Relative Degree One Multi-Agent Systems with Delays ? Ulrich M¨ unz ∗ Antonis Papachristodoulou ∗∗ Frank Allg¨ ower ∗ ∗

Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany {muenz,allgower}@ist.uni-stuttgart.de ∗∗ Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, U.K. [email protected] Abstract: Simple design conditions are presented for decentralized output feedback controllers that achieve output consensus between nonlinear, relative degree one Multi-Agent Systems (MAS) with stable zero dynamics. It is shown that consensus is achieved even if the exchange of information between the agents suffers from time-varying communication delays and switching communication topologies. In this way, the assumptions both on the network topology and the delays are minimal. The proofs are based on an invariance principle for LyapunovRazumikhin functions, which has been recently used to prove consensus between single integrator c IFAC 2009 MAS.Copyright ° Keywords: Multi-agent systems, relative degree one, nonlinear consensus, delays. 1. INTRODUCTION

results for single integrator MAS in M¨ unz et al. (2009, 2008), Papachristodoulou and Jadbabaie (2006).

Multi-Agent-Systems (MAS) are a general description of large-scale systems consisting of small subunits, called agents. A typical task for the MAS is to achieve consensus, e.g. all agents eventually move with the same velocity. The major difficulty in MAS design is to guarantee an overall group behavior with decentralized controllers using only local information from neighboring agents. Examples for consensus in MAS appear both in nature, e.g. schools of fish or synchronizing pacemaker cells (Strogatz, 2003), and in engineering problems, e.g. vehicle formation control, distributed computing, or internet congestion control (OlfatiSaber et al., 2007, Jadbabaie et al., 2003). Recent reviews on consensus and cooperation are given in Olfati-Saber et al. (2007) and Ren and Beard (2008).

In contrast to the paper at hand, most publications on MAS consider only linear subsystems and ideal communication channels without delay, cf. Wieland et al. (2008), Olfati-Saber et al. (2007). Nonlinear consensus problems without delay have been previously studied in Lin et al. (2007), Arcak (2007). Single integrator MAS with delays have been analyzed for example in M¨ unz et al. (2009, 2008), Bliman and Ferrari-Trecate (2008), Papachristodoulou and Jadbabaie (2006). Extensions to higher order nonlinear dynamics with relative degree one have been presented in Chopra and Spong (2006, 2008), Chopra et al. (2008). However, these publications only provide linear consensus controllers via fixed graphs that are either undirected or balanced. The results presented here provide nonlinear consensus controllers on fixed and switching directed graphs.

In this paper, we consider heterogeneous agents modeled as nonlinear, single-input single-output systems of relative degree one and stable zero dynamics. For these agents, we provide a decentralized nonlinear controller that achieves output consensus over a communication network with time-varying communication delays and minimal assumptions on the network connectivity. The only requirement for fixed network topologies is that the underlying graph contains a spanning tree. For networks with switching topology, just the union graph of all subgraphs that persist over time has to contain a spanning tree. The methodology we use is based on an invariance principle for LyapunovRazumikhin functions. The proofs are based on recent ? U. M¨ unz’s work was financially supported by The MathWorks Foundation and by the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems” of the German Research Foundation.

The paper is structured as follows: Some background information on time-delay systems and algebraic graph theory is given in Section 2. The problem is formulated in Section 3. In Section 4, we present conditions for consensus on fixed topologies. In Section 5, these results are extended to switching topologies. The paper is concluded in Section 6. 2. PRELIMINARIES 2.1 Stability of Functional Differential Equations This subsection gives a brief summary of stability results for functional differential equations. For more information, see Hale and Lunel (1993) and Haddock and Terj´eki (1983).

Let Rn denote the n-dimensional Euclidean space with the standard norm | · |. Let C([a, b], Rn ) denote the Banach space of continuous functions mapping the interval [a, b] ⊂ R into Rn with the topology of uniform convergence. Given T > 0, we define C = C([−T , 0], Rn ). The norm on C is defined as kϕk = sup−T ≤s≤0 |ϕ(s)|. Let ρ ≥ 0 and x ∈ C([−T , ρ], Rn ), then for any t ∈ [0, ρ], define a segment xt ∈ C by xt (s) = x(t + s), s ∈ [−T , 0]. Let Ω be a subset of C, f : Ω → Rn a given function, and ‘˙’ represent the right-hand Dini derivative, then we call x(t) ˙ = f (xt )

(1)

an autonomous Retarded Functional Differential Equation (RFDE) on Ω. The solution of (1) with initial condition ϕ ∈ Ω is denoted xt (ϕ). Note that xt (ϕ)(s) = x(ϕ)(t + s) for s ∈ [−T , 0]. An element φ ∈ C is called a steady-state or equilibrium of (1) if xt (φ) = φ for all t ≥ 0. Without loss of generality, we assume that φ = 0 is an equilibrium of (1). The stability of (1) around such a steady-state is defined in a way similar to the stability of nonlinear Ordinary Differential Equations (ODE) using an ²-δ-argument, see Hale and Lunel (1993). Let D ⊆ Rn . A Lyapunov-Razumikhin function V = V (x) is a continuous function V : D → R. The upper right-hand Dini derivative of V with respect to (1) is defined by 1 V˙ (ϕ) = lim sup (V (ϕ(0) + hf (ϕ)) − V (ϕ(0))). h→0+ h With this definition, we have the following LyapunovRazumikhin theorem: Theorem 1. Suppose f : Ω → Rn maps bounded subsets of Ω into bounded sets of Rn and consider (1). Suppose v, w : R+ → R+ are continuous, non-decreasing functions, v(s) positive for s > 0, v(0) = 0. If there is a LyapunovRazumikhin function V : D → R such that: V (x) ≥ v(|x|) ˙ V (ϕ(0)) ≤ −w(ϕ(0))

for x ∈ D, and if V (ϕ(0)) = max V (ϕ(s)), −T ≤s≤0

then the equilibrium x = 0 of (1) is stable. Note that the function V in Razumikhin’s theorem need not be non-increasing along the system trajectories, but may indeed increase within a delay interval. The proof of Razumikhin’s theorem is based on the fact that V (ϕ) = max V (ϕ(s)) −T ≤s≤0

(2)

is a Lyapunov-Krasovskii functional that is non-increasing along the system trajectories. This is an important fact that we will be using in our proofs. In this paper, we have to prove the attractivity of a subspace of Rn . Therefore, we will make repeated use of an invariance principle for RFDEs. For this, we use ω-limit sets ω(ϕ) of solutions of RFDEs (1) with initial condition ϕ, see Hale and Lunel (1993) for a formal definition. Remember that, if x(ϕ) is a solution of (1) that is defined and bounded on [−T , ∞), then ω(ϕ) is nonempty, compact, connected, and invariant, and xt (ϕ) → ω(ϕ) as t → ∞. For a given set Ω ⊂ C, define

EV = {ϕ ∈ Ω :

¾

max V (xt (ϕ)(s)) = max V (ϕ(s)), ∀t ≥ 0

s∈[−T ,0]

s∈[−T ,0]

(3)

MV : Largest set in EV that is invariant wrt. (1). (4) Here, EV is the set of functions ϕ ∈ Ω which can serve as initial conditions for (1) such that xt (ϕ) satisfies max V (xt (ϕ)(s)) = max V (ϕ(s)) s∈[−T ,0]

s∈[−T ,0]

for all t ≥ 0. Note that the above condition guarantees that V defined in (2) satisfies V˙ (ϕ) = 0. In particular, for a Lyapunov-Razumikhin function V and for any ϕ ∈ EV , we have V˙ (xt (ϕ)) = 0 for any t > 0 such that maxs∈[−T ,0] V (xt (ϕ)(s)) = V (xt (ϕ)(0)). We then have the following theorem: Theorem 2. Suppose there exists a closed set Ω that is positively invariant with respect to (1) and a LyapunovRazumikhin function V : Ω → R such that V˙ (ϕ) ≤ 0 ∀ϕ ∈ Ω s.t. V (ϕ(0)) = max V (ϕ(s)). (5) s∈[−T ,0]

Then, for any ϕ ∈ Ω such that x(ϕ) is defined and bounded on [−T , ∞), ω(ϕ) ⊆ MV ⊆ EV , and we have xt (ϕ) → MV as t → ∞. Theorem 2 will be used extensively in our work. It proves the attractivity of invariant subsets MV of Ω for the solutions of RFDE (1). 2.2 Algebraic Graph Theory The topology of the communication network linking the agents is represented by a graph. A graph G = (V, E) consists of a set of vertices (nodes) V = {vi }, i ∈ I = {1, . . . , N }, which represent the agents, and a set of edges (links) E ⊆ V × V, which represent the communication channels between the agents. If vi , vj ∈ V and eij = (vi , vj ) ∈ E, then there is an edge (a directed arrow) from node vi to node vj , i.e. agent j can receive data from agent i. In this paper, we assume that the graph G is directed, i.e. eij ∈ E does not necessarily imply that eji ∈ E. We also assume that the network topology does not contain self-loops, i.e. eii ∈ / E. The graph adjacency matrix A = [aij ], A ∈ RN ×N , is such that aij = 1 if eij ∈ E and aij = 0 if eij ∈ / E. If eji ∈ E, then vj is a parent of vi . A directed path from vi to vj is a sequence of edges out of E that takes the following form (vi , vi1 ), (vi1 , vi2 ), . . . , (vip , vj ). A directed tree is a directed graph where every vertex has exactly one parent except for one node, the so-called root vR . Clearly, there is a directed path from vR to all other nodes of the directed tree. ˜ E) ˜ of G is a graph with V˜ ⊆ V and E˜ ⊆ E. A subgraph (V, ˜ of G that is a directed If there exists a subgraph (V, E) tree, then we say that G contains a directed spanning tree. Hence, a graph G contains a directed spanning tree if and only if it contains at least one root, i.e. one node with a directed path to all other vertices. We denote the set of all roots of G as IR and IR = I \ IR . In the following, we also say spanning tree when referring to a directed spanning tree. The union graph of a set of P

³ S ´ graphs {(V, Ep )}, p ∈ P = {1, . . . , P }, is V, p∈P Ep . More details on algebraic graph theory can be found for example in Godsil and Royle (2000). 3. OUTPUT FEEDBACK CONSENSUS CONTROL We consider N heterogeneous single-input single-output agents with nonlinear dynamics, relative degree one, and stable zero dynamics of the form x˙ i (t) = fi (xi (t), ui (t)) (6) yi (t) = hi (xi (t)), with xi ∈ Rn , ui , yi ∈ R and smooth functions fi : Rn × R → Rn and hi : Rn → R, as well as smooth initial conditions. The aim of this work is twofold: (i) define suitable conditions for MAS (6) and (ii) design a scalable decentralized output feedback controller, in order to achieve output consensus asymptotically, i.e. limt→∞ yi (t) = y ∗ , ∀i, for some consensus point y ∗ ∈ R. A major difficulty is that the agents exchange their outputs on a network with time-varying communication delays τji , where τji : R+ → [0, T ], T > 0, describes the continuous, time-varying communication delay from agent j to agent i. Clearly, all τji must be bounded, i.e. T is finite, because otherwise it is straightforward to construct delays such that consensus can never be achieved. In order to solve the consensus problem, we propose the following decentralized controller ui (t) = −

N X

aji kji (yi (t) − yj (t − τji (t))) ,

(7)

j=1

where kji : R → R are static nonlinear control functions with kji (0) = 0 and aji are the elements of the adjacency matrix A of the underlying communication graph. For the RFDE (6), (7), we define the consensus state φy∗ ∈ C([−T , 0], RN ) such that all elements φy∗ ,i of φy∗ satisfy φy∗ ,i (η) = y ∗ , for all η ∈ [−T , 0]. The consensus set Θ of the MAS (6), (7) is then defined as [ Θ= {φy∗ }, (8) y ∗ ∈R

i.e. we say output consensus is reached asymptotically if limt→∞ yt = φy∗ for some φy∗ ∈ Θ where yt = [y1t , . . . , yN t ]T . First, we are interested in conditions such that all elements of Θ are steady states of (6), (7). These conditions imply that output consensus can be achieved at any consensus point y ∗ ∈ R. In particular, this excludes the trivial consensus problem where all agents converge to the origin. Note that ui = 0 for all yt ∈ Θ, cf. (7). Hence, the derivative of the outputs yi of (6) satisfies ∂hi fi (xi , 0), (9) y˙ i = ∂xi for all yt ∈ Θ. We conclude that all elements of Θ are i steady states of (6), (7) if ∂h ∂xi fi (xi , 0) = 0, for all i. Therefore, we assume the following on the agent dynamics: Assumption 3. The agent dynamics (6) have stable zero ∂hi i dynamics and satisfy ui ∂h ∂xi fi (xi , ui ) > 0 and ∂xi fi (xi , 0) = 0, ∀xi , ui 6= 0, i ∈ I.

i The passivity assumption ui ∂h ∂xi fi (xi , ui ) > 0 implies that the relative degree of (6) is well defined and always one. If i ui ∂h ∂xi fi (xi , ui ) < 0 for all xi , all following results still hold with negated ui .

In the next sections, we will show that the following design condition on the control functions kji guarantees output consensus: Design Condition 4. The continuous functions kji : R → R are locally passive, i.e. kji (0) = 0 and ηkji (η) > 0, for − + − + all η ∈ [−γji , γji ] \ {0}, where γji , γji > 0. This MAS model (6) with nonlinear consensus controller (7) includes previous results in the literature as special cases, e.g. passive agents (Chopra and Spong, 2006), Lagrangian agents (Chopra et al., 2008), and nonlinear relative degree one agents (Chopra and Spong, 2008) with linear controllers and communication delays. Yet, these publications have more restrictions on the delays and the network topology. 4. CONSENSUS OVER FIXED TOPOLOGIES We want to prove that the consensus set Θ is asymptotically attracting for appropriate initial conditions ϕ ∈ CD = C([−T , 0], D), where an underestimate of the region of attraction is given by n γo (10) D = [xT1 , . . . , xTN ]T ∈ RnN : |hi (xi )| ≤ 2 − + − + with γ = mini,j∈I {γij , γij }, where γij , γij are the bounds of the locally passive functions kij , cf. Design Condition 4. Note that D = RnN if kji are globally passive and in particular linear. For CD , we have the following result: Lemma 5. If Assumption 3 and Design Condition 4 hold, then CD = C([−T , 0], D) is a positively invariant set of (6), (7). Proof. Consider the Lyapunov-Razumikhin function candidate 1 2 1 (t), V (x(t)) = max h2i (xi (t)) = yM 2 i∈I 2 2 where M denotes the index that satisfies yM (t) = maxi∈I yi2 (t). If there are several possible indices, we choose that one which the maximal modulo of the derivative |y˙ M (t)|. If there are still several possible indices, we choose any one of them but fix the index M as long as it satisfies the maximum conditions. Note that index M depends on time t. The upper right-hand Dini derivative of V along solutions of (6), (7) is ∂hM V˙ (xt ) = yM ∂xM   N X × fM xM , − ajM kjM (yM (t) − yj (t − τjM (t))) . j=1

(11) The condition on |y˙ M (t)| is necessary in order to guarantee that (11) is indeed the upper right-hand derivative. Now, consider the behavior of V˙ if V (x(t)) = maxη∈[0,T ] V (x(t − η)), i.e. |yM (t)| = maxη∈[0,T ] maxj∈I |yj (t − η)|. Under this condition and Design Condition 4, we

conclude that kjM (yM (t) − yj (t − τjM (t))) ≥ 0 if yM (t) > 0 and kjM (yM (t) − yj (t − τjM (t))) ≤ 0 if yM (t) < 0 for all j and all xt ∈ CD . From Assumption 3, we M have ∂h ∂xM fM (xM , uM ) > 0 if uM > 0, i.e. yM (t) < 0, ∂hM and ∂xM fM (xM , uM ) < 0 if uM < 0, i.e. yM (t) > 0. Hence, V˙ ≤ 0 if V (x(t)) = maxη∈[0,T ] V (x(t − η)) and consequently xt (ϕ) ∈ CD for all t ≥ 0 if ϕ ∈ CD . ¤ Having established an underestimate of the region of attraction, we can now prove that the consensus set Θ is actually asymptotically attracting for any initial condition ϕ ∈ CD . Theorem 6. Given a MAS consisting of N agents with dynamics (6) that satisfy Assumption 3, with controllers (7) that satisfy Design Condition 4, and with initial condition ϕ ∈ CD , as well as an underlying network topology of a directed graph with a spanning tree, then this MAS achieves output consensus asymptotically. Proof. Following Theorem 2, we first need a closed set Ω that is positively invariant with respect to (6), (7). In general, CD is not closed. However, there is always a closed subset of CD that is positively invariant because of Lemma 5 and the stable zero dynamics of (6). Consider the Lyapunov-Razumikhin function candidates V1 = max hi (xi (t)) = yI (t), i∈I

V2 = − min hi (xi (t)) = −yJ (t). i∈I

Similar to the former proof, I and J denote those indices that satisfy yI (t) = maxi∈I yi (t) and yJ (t) = mini∈I yi (t), respectively. If there are several such indices, we choose those with the maximal derivative y˙ I (t) and minimal derivative y˙ J (t), respectively. If there are still several possible indices, we choose any one of them but fix the indices I and J as long as they satisfy the extremum conditions. Using this notation, the right-hand Dini derivatives of V1 and V2 along solutions of (6), (7) are ! Ã N X ¢ ¡ ∂hI V˙ 1 (xt ) =

∂xI

fI

à ∂hJ V˙ 2 (xt ) = − fJ ∂xJ

ajI kjI yI (t) − yj (t − τjI (t))

xI , −

j=1 N X

¡

ajJ kjJ yJ (t) − yj (t − τjJ (t))

xJ , −

,

! ¢

.

j=1

Following Theorem 2, we are interested in the behavior of V˙ k , k = 1, 2, whenever Vk (x(t)) = maxη∈[0,T ] Vk (x(t − η)), i.e. yI (t) = maxη∈[0,T ] maxj∈I yj (t − η) and yJ (t) = minη∈[0,T ] minj∈I yj (t − η), respectively. A similar argument as in the proof of Lemma 5 shows that V˙ k ≤ 0, k = 1, 2. Hence, condition (5) in Theorem 2 is fulfilled. Next, we have to find the sets EVk and MVk , k = 1, 2, according to (3) and (4). For every ϕ ∈ EVk , there is an yk∗ ∈ R such that maxη∈[0,T ] Vk (x(ϕ)(t−η)) = yk∗ for all t ≥ 0. Moreover, any ϕ ∈ EVk satisfies V˙ k (xt (ϕ)) = 0 for any t ≥ 0 whenever Vk (x(ϕ)(t)) = maxη∈[0,T ] Vk (x(ϕ)(t − η)), see Haddock and Terj´eki (1983). For V1 , this transforms into V˙ 1 = 0 if yI (t) = maxη∈[0,T ] maxj∈I yj (t − η) = y1∗ . Furthermore, V˙ 1 = 0 if uI (t) = 0, i.e. yI (t) = yj (t − τjI (t)) for all j with ejI ∈ E. Hence, all parents vj of vI must fulfill yj (t − τjI (t)) = y1∗ . Since fi , hi , kji , τji , and the initial conditions are continuous, all yi are differentiable.

As y1∗ is the maximum of all states and of all times t, this requires that y˙ j (t−τjI (t)) = 0. Thus, all the parents of the parents of vI also satisfy yκ (t − τjI (t) − τκj (t)) = y1∗ for all ejI , eκj ∈ E. This can be continued up to any root of the underlying graph of the MAS. There exists at least one root because the graph contains a spanning tree. Thus, we have ) ( ( max hi (xt,i (ϕ)(η)) = y1∗ , ∀ i ∈ IR [ η∈[−T ,0] EV1 =

ϕ ∈ CD :

∗ ∈R y1

max

η∈[−T ,0]

hi (xt,i (ϕ)(η)) ≤ y1∗ , ∀ i ∈ IR ∀t ≥ 0} .

(12)

With similar arguments for EV2 , we get ) ( ( min hi (xt,i (ϕ)(η)) = y2∗ , ∀ i ∈ IR [ η∈[−T ,0] EV2 =

ϕ ∈ CD :

∗ ∈R y2

min

η∈[−T ,0]

hi (xt,i (ϕ)(η)) ≥ y2∗ , ∀ i ∈ IR ∀t ≥ 0} ,

(13)

where y2∗ = minη∈[0,T ] minj∈I yj (t − η) for all t ≥ 0. Since both Lyapunov functions V1 and V2 satisfy the conditions of Theorem 2, we conclude that EV1 ∩ EV2 is asymptotically attracting to all solutions xt (ϕ) of (6), (7) with ϕ ∈ CD . Note moreover that all solutions xt (ϕ) of (6), (7) with initial condition ϕ ∈ EV1 ∩ EV2 satisfy maxη∈[−T ,0] hi (xt,i (η)) = y1∗ and minη∈[−T ,0] hi (xt,i (η)) = y2∗ for all t and all i ∈ IR . Hence, there are two possibilities: (i) xt (ϕ) generates a constant output, i.e. hi (xt,i (η)) = hi (ϕi (η)) = x∗1 = x∗2 for all t ≥ 0, i ∈ I, and η ∈ [−T , 0] or (ii) xt (ϕ) generate persistently oscillating outputs. The first case (i) implies that the output consensus is reached asymptotically. It remains to show why (6), (7) cannot generate persistently oscillating outputs for initial conditions ϕ ∈ EV1 ∩ EV2 . Consider i ∈ IR and t0 > 0 such that yi (t0 ) = y1∗ . Remember that yj (t0 + η) ∈ [y2∗ , y1∗ ] for all η ∈ [−T , ∞] and for all j ∈ I. For a persistently oscillating solution, we would need that yi reaches y2∗ in finite time, i.e. there exists an η ∈ [0, T ] such that yi (t0 + η) = y2∗ . Yet, we will show that yi approaches y2∗ at most asymptotically, but not in finite time.  Consider  N X ∂hi  fi xi , − aji kji (yi (t) − yj (t − τji (t))) y˙ i (t) = ∂xi j=1 (14) and note that yi & y2∗ requires yj (t − τji (t)) = y2∗ for some j and t > t0 because y˙ i (t) < 0 requires ui < 0 and this in turn requires yi (t) > yj (t − τji (t)) ≥ y2∗ for some j. As yi approaches y2∗ , it enters the set [y2∗ , y2∗ + ε], where ε > 0 is defined such that all kji (η) are monotonically i increasing for η ∈ [0, ε] and ∂h ∂xi fi (xi , η) is monotonically PN decreasing for all xi and for η ∈ [− j=1 kji (ε), 0]. For yi (t) ∈ [y2∗ , y2∗ + ε], a lower bound y˜i (t) for yi (t) results from   N X ∂hi  y˜˙ i (t) = fi x ˜i , − aji kji (˜ yi (t) − y2∗ ) ≤ y˙ i (t). ∂xi j=1 Note that y˜i cannot converge in finite time to y2∗ because of uniqueness of solutions and because y2∗ is a steady state of the above differential equation. Hence, yi (t) cannot converge in finite time to y2∗ . Therefore, (6), (7) cannot generate persistently oscillating outputs. ¤

5. CONSENSUS WITH SWITCHING TOPOLOGY We now turn to MAS with dynamic topologies. We assume a finite set of directed graphs {Gp } with p ∈ P = {1, . . . , P }. At any time t, one of the graphs Gp represents the topology of the communication network between the agents. The switching signal σ : [0, ∞) → P determines the index of the active graph at time t. σ is piecewise constant from the right and non-chattering, i.e. there is a dwell-time h > 0 between any two switching instants tl+1 − tl ≥ h for all l = 1, 2, . . .. We assume that the topology does not stop switching because otherwise this problem could be solved as in Section 4 considering only the last active graph. We denote all switching times when graph Gp becomes active tpν , tpν+1 > tpν , i.e. σ(t) = p for t ∈ [tpν , tpν +1 ) with ν = 1, 2, . . .. The set P∞ ⊆ P is such that every graph Gp , p ∈ P∞ , is infinitely often active, i.e. there are infinitely many switching times tpν . Finally, we define the set³of graphs that´ persist over time as the union S graph G∞ = V, p∈P∞ Ep . The controllers with switching topology are N X (σ) ui (t) = − aji kji (yi (t) − yj (t − τji (t))) ,

(15)

j=1 (p)

where A(p) = [aij ] is the adjacency matrix of graph Gp . The initial condition is x0 = ϕ. The consensus set of the MAS (6), (15) is Θ as defined in (8). Theorem 7. Given a MAS consisting of N agents with dynamics (6) that satisfy Assumption 3, with controllers (15) that satisfy Design Condition 4, and with initial condition ϕ ∈ CD , as well as an underlying switched network topology of directed graphs, such that the union graph G∞ has a spanning tree, then this MAS achieves output consensus asymptotically. Proof. The proof is based on a common Lyapunov function argument to allow for the arbitrary switching and on Theorem 2 to prove attractivity of the consensus set Θ. Note first that there exists always a closed subset of CD that contains the initial condition ϕ and is positively invariant with respect to an arbitrarily switching system (6), (15) because it is invariant with respect to every subsystem. Next, we briefly outline the remainder of the proof. In Part (i), we define two common Lyapunov Razumikhin function candidates V1 and V2 and prove that they satisfy condition (5). In order to determine the sets EVk and MVk , k = 1, 2, we adopt an invariance principle from Hespanha et al. (2005). This requires the definition of two functionals V k , k = 1, 2, which are based on V1 and V2 , cf. (2). Since the derivatives of V k cannot be determined easily, we just distinguish between the two important cases V˙ k = 0 and V˙ k < 0 through simple rules in Part (ii) of the proof. Using these conditions, we apply the result from Hespanha et al. (2005) and determine the sets EVk and MVk in Part (iii). Part (i): We consider the functions V1 and V2 from the proof of Theorem 6 as common Lyapunov-Razumikhin function candidates. The indices I and J are defined as in Theorem 6 to be the maximal and minimal states over all agents. The right-hand Dini derivatives of V1 and V2 along solutions of (6), (15) are

V1

V1

V 1(t)

V 1(t)

t−T+

t − ηI

t − T + = t − ηI

t

(a)

t

(b)

Fig. 1. Exemplary Lyapunov function V1 and exemplary Lyapunov functional V 1 , see text for details. ! Ã N X ¢ ¡ ∂hI (σ) (σ) V˙ 1

(xt ) =

∂xI

fI

à (σ) V˙ 2 (xt )

∂hJ fJ =− ∂xJ

ajI kjI yI (t) − yj (t − τjI (t))

xI , −

,

j=1 N X

xJ , −

(σ) ajJ

¡

¢

!

kjJ yJ (t) − yj (t − τjJ (t))

j=1

(p) Following the proof of Theorem 6, we know that V˙ k ≤ 0, k = 1, 2, for all p ∈ P, whenever Vk (x(t)) = maxη∈[0,T ] Vk (x(t − η)). We conclude that condition (5) is satisfied and that the functionals V k (xt ) = maxη∈[0,T ] Vk (x(t − η)), k = 1, 2, are nonincreasing.

Part (ii): It would be quite difficult to calculate the righthand Dini derivatives of V k along solutions of (6), (15). Instead, we determine simple rules to distinguish the two main cases V˙ k = 0 and V˙ k < 0. Therefore, we use the following notation: Let the values Iη and Jη indicate the indices at each time t that satisfy yIη (t−η) = maxi∈I yi (t− η) and yJη (t − η) = mini∈I yi (t − η), respectively. If there are several possible indices, we chose any one of them. Let ηI , ηJ ∈ [0, T ] be such that yIηI (t − ηI ) = max max yi (t − η), (16) η∈[0,T ] i∈I

yJηJ (t − ηJ ) = min min yi (t − η), η∈[0,T ] i∈I

(17)

i.e. V 1 (xt ) = yIηI (t − ηI ) and V 2 (xt ) = −yJηJ (t − ηJ ). Clearly, ηI and ηJ are changing with time and there might be several values ηI , ηJ ∈ [0, T ] that satisfy (16) and (17), respectively. Now, we can state the following about the derivatives of V k , k = 1, 2, along solutions of (6), (15): (p) • V˙ k (xt ) ≤ 0 for all p ∈ P. (p) • V˙ 1 (xt ) = 0 if and only if there exists an ηI ∈ [0, T ) that satisfies (16), see Figure 1(a). (p) • V˙ (x ) < 0 if and only if η = T satisfies (16) and 1

t

I

2

t

J

there does not exist an ηI∗ ∈ [0, T ) that satisfies (16), see Figure 1(b). (p) • V˙ 2 (xt ) = 0 if and only if there exists an ηJ ∈ [0, T ) that satisfies (17). (p) • V˙ (x ) < 0 if and only if η = T satisfies (17) and there does not exist an ηJ∗ ∈ [0, T ) that satisfies (17).

Part (iii): With these conditions, we can now turn to an invariance principle for switched RFDEs. Recall the definition of the switching times tpν given above, such that σ(t) = p for t ∈ [tpν , tpν +1 ) with ν = 1, 2, . . .. Since two

.

switching times tl are separated by a dwell time h, we have νp∗ Z t P X pν +1 X (p) V k (x(tl )) − V k (x(0)) = V˙ k (xt )dt, (18) p=1 ν=1

with

νp∗

tp ν

such that tpν ∗ +1 ≤ tl and tpν ∗ +1 > tl . From our p

p

(p) former arguments, we know that V˙ k (xt ) ≤ 0. Moreover, we know that the left hand side of (18) converges to a finite value for tl → ∞ because V k is nonincreasing and bounded from below. Following the proof of Theorem 7 in (p) Hespanha et al. (2005), we conclude that V˙ (x ) = 0 as k

REFERENCES

t (p∗ )

t → ∞ for all p ∈ P∞ and k = 1, 2. Clearly, V˙ k (xt ) = 0 is not necessary for those p∗ ∈ P \P∞ because these graphs are only active a finite number of times, i.e. νp∗∗ does not go to infinity for tl → ∞. Now, we have to determine the sets ¾ ½ (p) EVk = ϕ ∈ CD : V˙ k (xt (ϕ)) = 0 ∀ p ∈ P∞ , t ≥ 0 , k = 1, 2, in order to conclude that EVk is asymptotically attracting. We first consider EV1 . For every ϕ ∈ EV1 , there exists an y1∗ ∈ R such that V 1 (xt (ϕ)) = y1∗ , i.e. y1∗ = maxη∈[0,T ] maxi∈I yi (ϕ)(t − η), for all t ≥ 0. Following (p) our former arguments, we know that V˙ (x ) = 0 if and 1

These conditions hold for arbitrary delay sizes and variations and with minimal assumptions on the network topology. We only require that the underlying graph contains a spanning tree for the fixed topology case; and in the case of switching graphs, we require that the union graph of the set of graphs that persist over time contains a spanning tree.

t

only if there exists an ηI ∈ [0, T ) that satisfies (16). Hence, we know that the right-hand Dini derivatives of (p) the Razumikhin candidate V1 satisfy V˙ 1 (x(t − ηI )) = 0 (see Figure 1(a)) and this implies that yIηI (t − ηI ) − yj (t − ηI − τjIηI (t − ηI )) = 0, (19) for all j with ejIηI ∈ Ep . Since yIηI (t − ηI ) = y1∗ and since y1∗ is the maximum of all states at all times, we conclude that y˙ j (t − ηI − τjIηI (t − ηI )) = 0 for all eIjηI ∈ Ep . The same arguments hold for all p ∈ P∞ . Note that y1∗ depends on ϕ but not on p. Following the proof of Theorem 6, we conclude that, if the union graph G∞ has at least one root vi , i ∈ IR , then EV1 is given by (12). We know that G has at least one root because it contains a spanning tree. We use similar arguments for EV2 and obtain (13). As in Theorem 6, we conclude that Θ is asymptotically attracting for the outputs. ¤ This result shows that consensus is reached among heterogeneous, nonlinear MAS with relative degree one even if they exchange information over communication networks with an arbitrarily switching network topology. The assumption that G∞ has a spanning tree resembles the connected-over-time assumption in previous works, e.g. Jadbabaie et al. (2003). We have shown in M¨ unz et al. (2009) that the new condition is not more restrictive than the former conditions. Note that both Theorem 6 and 7 apply also for MAS with a leader following the same arguments as in M¨ unz et al. (2009). 6. CONCLUSIONS Theorem 6 and 7 provide consensus conditions for heterogeneous, nonlinear multi-agent systems with relative degree one and time-varying communication delays.

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