Finite-Time Consensus Problems for Networks of Dynamic Agents

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1 Finite- Consensus Problems for Networks of Dynamic Agents arxiv:math/774v [math.ds] 5 Jan 7 Long Wang and Feng Xiao Abstract In this paper, finite-time state consensus problems for continuous-time multi-agent systems are discussed, and two distributive protocols, which ensure that the states of agents reach an agreement in a finite time, are presented. By employing the method of finite-time Lyapunov functions, we derive conditions that guarantee the two protocols to solve the finite-time consensus problems respectively. Moreover, one of the two protocols solves the finite-time weighted-average consensus problem and can be successively applied to the systems with switching topology. Upper bounds of convergence times are also established. Simulations are presented to show the effectiveness of our results. Index Terms Multi-agent systems, finite-time consensus problems, switching topology, consensus protocols, coordination control. I. INTRODUCTION The theory of consensus or agreement problems for multi-agent systems has emerged as a challenging new area of research in recent years. It is a basic and fundamental problem in decentralized control of networks of dynamic agents and has attracted great attention of researchers. This is partly due to its broad applications in cooperative control of unmanned This work was supported by NSFC (66745 and 6587), National 973 Program (CB3), and -5 project (A633). The authors are with the Intelligent Control Laboratory, Center for Systems and Control, Department of Industrial Engineering and Management, and Department of Mechanics and Space Technologies, College of Engineering, Peking University, Beijing 87, China ( longwang@pku.edu.cn; fengxiao@pku.edu.cn). February, 8

2 air vehicles, formation control of mobile robots, control of communication networks, design of sensor networks, flocking of social insects, swarm-based computing, etc. In [], Vicsek et al. proposed a simple but interesting discrete-time model of finite agents all moving in the plane. Each agent s motion is updated using a local rule based on its own state and the states of its neighbors. The Vicsek model can be viewed as a special case of a computer model mimicking animal aggregation proposed in [] for the computer animation industry. By using graph theory and nonnegative matrix theory, Jadbabaie et al. provided a theoretical explanation of the consensus property of the Vicsek model in [3], where each agent s set of neighbors changes with time as system evolves. The typical continuous-time model was proposed by Olfati-Saber and Murray in [4], where the concepts of solvability of consensus problems and consensus protocols were first introduced. The authors used a directed graph to model the communication topology among agents and studied three consensus problems, namely, directed networks with fixed topology, directed networks with switching topology, and undirected networks with communication time-delays and fixed topology. And it was assumed that the directed topology is balanced and strongly connected. In [5], Ren and Beard extended the results of [3], [4] and presented mathematically weaker conditions for state consensus under dynamically changing directed interaction topology. In the past few years, consensus problems of multi-agent systems have been developing fast and many research topics have been addressed, such as agreement over random networks [6], [7], asynchronous information consensus [8], dynamic consensus [9], networks with nonlinear consensus protocols [], consensus filters [], and networks with communication time-delays [4], [], [3], [4]. For details, see the survey [5] and references therein. By long-time observation of animal aggregations, such as schools of fish, flocks of birds, groups of bees, and swarms of social bacteria, it is believed that simple, local motion coordination rules at the individual level can result in remarkable and complex intelligent behavior at the group level. We call those local motion coordination rules distributive protocols. In the study of consensus problems, they are called consensus protocols. In the analysis of consensus problems, convergence rate is an important performance index of the proposed consensus protocol. In [4], a linear consensus protocol was given and it was shown that the second smallest eigenvalue of interaction graph Laplacian, called algebraic connectivity of graph, quantifies the convergence speed of the consensus algorithm. In [6], Kim and Mesbahi February, 8

3 3 considered the problem of finding the best vertex positional configuration so that the second smallest eigenvalue of the associated graph Laplacian is maximized, where the weight for an edge between two vertices was assumed to be a function of the distance between the two corresponding agents. In [7], Xiao and Boyd considered and solved the problem of the weight design by using semi-definite convex programming, so that algebraic connectivity is increased. If the communication topology is a small-world network, it was shown that large algebraic connectivity can be obtained [8]. Although by maximizing the second smallest eigenvalue of interaction graph Laplacian, we can get better convergence rate of the linear protocol proposed in [4], the state consensus can never occur in a finite time. In some practical situations, it is required that the consensus be reached in a finite time. Therefore, finite-time consensus is more appealing and there are a number of settings where finite-time convergence is desirable. The main contribution of this paper is to address finite-time consensus problems and discuss two effective distributed protocols that can solve consensus problems in finite times. Furthermore, the method used in this paper is of interest itself, which is partly motivated by the work of [9], in which continuous finite-time differential equations were introduced as fast accurate controllers for dynamical systems, and partly by the results of finite-time stability of homogeneous systems in []. The proposed protocols in this paper are continuous state feedbacks, but they do not satisfy the Lipschitz condition at the agreement states, which is the least requirement for finite-time consensus problems because Lipschitz continuity can only lead to asymptotical convergence. To prove that the two protocols effectively solve finite-time consensus problems, we adopt the theory of finite-time Lyapunov stability [9], []. Although for systems under general communication topology, it is difficult and even impossible to find the valid Lyapunov functions, we have succeeded in reducing the general case into several special cases, in which appropriate Lyapunov functions can be found. This paper shows that the convergence time is closely related to the underlying communication topology, especially, the algebraic connectivity for the undirected case. Large algebraic connectivity can greatly reduce the convergence time. We also compare the convergence rates of two systems under the same protocol but with different protocol parameters and draw the conclusion that one converges faster when agents states differ a lot while the other converges faster when agents states differs a little. This conclusion encompasses the case when one of the two systems adopts the linear consensus protocol presented in [4]. Therefore, in order February, 8

4 4 to get shorter convergence time, we can change those adjustable parameters in the proposed protocols according to agents states. By the same method, we also study the case where the topology is dynamically changing. This paper is organized as follows. Section II presents the preliminary results in algebraic graph theory. Section III sets the basic setting in which the problem is formulated. The main theoretical results are established in Section IV and the simulation results are given in Section V. Finally, concluding remarks are provided in Section VI. II. PRELIMINARIES In this section, we list some basic definitions and results in algebraic graph theory. More comprehensive discussions can be found in []. Directed graphs will be used to model the communication topologies among agents. A directed graph G of order n consists of a vertex set V(G) = {v i : i =,,..., n} and an edge set E(G) {(v i, v j ) : v i, v j V(G)}. If (v i, v j ) E(G), v i is called the parent vertex of v j and v j is called the child vertex of v i. The set of neighbors of vertex v i is denoted by N(G, v i ) = {v j : (v j, v i ) E(G), j i}. The associated index set is denoted by N(G,i) = {j : v j N(G,v i )}. A subgraph G s of directed graph G is a directed graph such that the vertex set V(G s ) V(G) and the edge set E(G s ) E(G). If V(G s ) = V(G), we call G s a spanning subgraph of G. For any v i, v j V(G s ), if (v i, v j ) E(G s ) if and only if (v i, v j ) E(G), G s is called an induced subgraph. In this case, G s is also said to be induced by V(G s ). A path in directed graph G is a finite sequence v i,...,v ij of vertices such that (v ik, v ik+ ) V(G) for k =,..., j. A directed tree is a directed graph, where every vertex, except one special vertex without any parent, has exactly one parent, and the special vertex, called the root vertex, can be connected to any other vertices through paths. A spanning tree of G is a directed tree that is a spanning subgraph of G. We say that a directed graph has or contains a spanning tree if a subset of the edges forms a spanning tree. Directed graph G is strongly connected if between every pair of distinct vertices v i, v j in G, there is a path that begins at v i and ends at v j (that is, from v i to v j ). A strongly connected component of a directed graph is an induced subgraph that is maximal, subject to being strongly connected. Since any subgraph consisting of one vertex is strongly connected, it follows that each vertex lies in a strongly connected component, and therefore the strongly connected components of a given directed graph partition its vertices. February, 8

5 5 Matrix A is called a nonnegative matrix, denoted by A, if all its entries are nonnegative, and is called a positive matrix, denoted by A >, if all its entries are positive. A weighted directed graph G(A) is a directed graph G plus a nonnegative weight matrix A = [a ij ] R n n such that (v i, v j ) E(G) a ji >. And a ji is called the weight of edge (v i, v j ). Moreover, if A T = A, then G(A) is also called an undirected graph. In this case, (v i, v j ) G(A) (v j, v i ) G(A), and G(A) having a spanning tree is equivalent to G(A) being strongly connected. Those strongly connected undirected graphs are usually called to be connected. In this paper, the induced subgraph of weighted directed graph G(A) is also a weighted directed graph which inherits its weights in G(A). Notations: Let = [,,...,] T with compatible dimensions, I n = {,,, n}, span() = {ξ R n : ξ = r, r R}, and ρ(a) denote the spectral radius of a square matrix A. For any p, p denotes the l p -norm on R n and ip is its induced norm on R n n. If ω = [ω, ω,...,ω n ] T R n, then diag(ω) is the diagonal matrix with the (i, i) entry being ω i. Lemma ([4], [5], []): Let L(A) = [l ij ] R n n denote the graph Laplacian of G(A), which is defined by Then n k=,k i l ij = a ik, j = i a ij, j i (i) is an eigenvalue of L(A) and is the associated eigenvector; (ii) If G(A) has a spanning tree, then eigenvalue is algebraically simple and all other eigenvalues are with positive real parts; (iii) If G(A) is strongly connected, then there exists a positive column vector ω R n such that ω T L(A) = ; If G(A) is undirected, namely, A T properties:. = A, and connected, then L(A) has the following (iv) ξ T L(A)ξ = n i,j= a ij(ξ j ξ i ) for any ξ = [ξ, ξ,...,ξ n ] T R n, and therefore L(A) is semi-positive definite, which implies that all eigenvalues of L(A) are nonnegative real numbers; (v) The second smallest eigenvalue of L(A), which is denoted by λ (L(A)) and called the algebraic connectivity of G(A), is larger than zero; February, 8

6 6 (vi) The algebraic connectivity of G(A) is equal to min ξ, T ξ= ξt L(A)ξ, and therefore, if ξ T ξ T ξ =, then Proof: ξ T L(A)ξ λ (L(A))ξ T ξ. Statement (i) follows directly from the definition of L(A). (ii) is a corollary of Lemma 3.3 or Lemma 3. in [5]. The last three statements appeared in [4] (see equations (5) (7) and Theorem in [4]). Here, we provide the proof of (iii) for completeness, which can be extended to more general case, see []. Let d = max i l ii, ε = max i a ii + and let d = d + ε. Then L(A) = di + ( L(A) + di), where I is the identity matrix with compatible dimensions. By direct observation, L(A) + di A (component-wise) is a nonnegative matrix with positive diagonal entries. Since G(A) is strongly connected, G( L(A) + di) is also strongly connected. By Lemma 4 (in the Appendix), L(A) + di is irreducible, equivalently, L(A) T + di is also irreducible. By Geršgorin Disk Theorem (in the Appendix), all the eigenvalues of L(A) are located in the following region: n {c C : c + l ii l ij = l ii } {c C : c + d d } j= j i The eigenvalues of L(A) + di are located in the right circle. Fig.. Eigenvalues of L(A) + di Therefore, the eigenvalue of L(A) + di are located in the region {c C : c ε February, 8

7 7 d }, see Fig.. Consequently, its spectral radius, ρ( L(A) + di), is not larger than d. From ( L(A) + di) = ( + d) = d, we have that d is an eigenvalue of L(A) + di, and thus ρ( L(A) + di) = d. By Perron-Frobenius Theorem (in the Appendix), there exists a positive column vector ω such that ( L(A) T +di)ω = dω. Therefore, ω T L(A) = ω T (di) dω T =. Corollary : Suppose G(A) is strongly connected, and let ω > such that ω T L(A) =. Then diag(ω)l(a) + L(A) T diag(ω) is the graph Laplacian of the undirected weighted graph G(diag(ω)A + A T diag(ω)). And therefore it is semi-positive definite, is its algebraically simple eigenvalue and is the associated eigenvector. Proof: Since the definition of graph Laplacian L(A) is independent of the diagonal entries of A, without loss of generality, we assume that the diagonal entries of A are all zeros (in the subsequent sections, A will be assumed to be with zero diagonal entries). Then by the definition of graph Laplacian, we have that L(A) = diag(a) A. () Then diag(ω)l(a) = diag(ω)(diag(a) A) = diag(diag(ω)a) diag(ω)a =L(diag(ω)A) (by ()). () Since ω T L(A) =, T diag(ω)l(a) =. Therefore, = T diag(ω)l(a) = T L(diag(ω)A) = T diag(diag(ω)a) T (diag(ω)a) = T A T diag(ω) T (diag(ω)a). Thus, diag(ω)a = A T diag(ω). (3) February, 8

8 8 Therefore, L(A) T diag(ω) =L(diag(ω)A) T (by ()) = diag(diag(ω)a) (diag(ω)a) T = diag(a T diag(ω)) A T diag(ω) (by (3)) =L(A T diag(ω)) (by ()). Moreover, by (), we can see that for any nonnegative matrices A and A, L(A )+L(A ) = L(A + A ). Therefore, diag(ω)l(a)+l(a) T diag(ω) = L(diag(ω)A) + L(A T diag(ω)) = L(diag(ω)A + A T diag(ω)). Corollary : Let b = [b, b,...,b n ] T, b, and let G(A) be undirected and connected. Then L(A) + diag(b) is positive definite. Proof: By Lemma (iv), L(A) + diag(b) is semi-positive definite. If there exists a vector ξ R n such that ξ T (L(A) + diag(b))ξ =, then by lemma, we have (i) ξ span(); (ii) ξ T diag(b)ξ =, which implies that some entries of ξ are zeros. Hence, ξ = and L(A) + diag(b) is positive definite. III. PROBLEM FORMULATION The distributed dynamic system studied in this paper consists of n autonomous agents, e.g. particles or robots, labeled through n. All these agents share a common state space R. The state of agent i is denoted by x i, i =,,..., n and let x = [x, x,...,x n ] T. Suppose that agent i, i I n, is with the following dynamics ẋ i (t) = u i (t), (4) where u i (t) is the protocol to be designed. In this multi-agent system, each agent can communicate with some other agents which are defined as its neighbors. Protocol u i is a state feedback, which is designed based on the state information received by agent i from its neighbors. We use the weighted directed graph G(A) to represent the communication topology, where A = [a ij ] is a given n n nonnegative matrix, with February, 8

9 9 diagonal entries all being zeros. Vertex v i represents agent i. Edge (v i, v j ) N(G(A)) represents an available information channel from agent i to agent j. The neighbors of agent i are those agents whose information can be received directly by agent i. Thus, the set of neighbors of agent i just corresponds to the set N(G(A), v i ). The leaders of this system are the agents, the vertices corresponding to which are the roots of some spanning trees of G(A), i.e., the vertices that can be connected to any other vertices through paths [3]. The agents that are not leaders are called followers. Given certain agents, the local communication topology of them is the subgraph of G(A), which is induced by the vertices corresponding to those agents. If A is time-dependent, then the underlying topology is said to be switching. In order to reflect the dependency on time, we use A(t) instead of A in this case. The switching topology will be discussed in detail in Section IV.D. Unless it is explicitly specified, A is assumed to be time-invariant. Given protocol u i, i I n, we say that u i or this multi-agent system solves a consensus or agreement problem if for any given initial states and any j, k I n, x j (t) x k (t), as t, and we say that it solves a finite-time consensus problem if it solves a consensus problem, and given any initial states, there exist a time t and a real number κ such that x j (t) = κ for t t and all j I n. If the final consensus state is a function of the initial states, namely, x j (t) χ(x()) for all j I n as t, where χ : R n R is a function, we say that it solves the χ-consensus problem [4]. Specially, if χ(x()) = average-consensus problem. P n x i() n, the system is said to solve the We are now in a position to present two consensus protocols, which will be shown to solve finite-time consensus problems: (i) For i I n, if N(G(A), v i ) = φ, then u i =, else u i = sign a ij (x j x i ) j N(G(A),i) j N(G(A),i) α i a ij (x j x i ) ; (5) (ii) For i I n, if N(G(A), v i ) = φ, then u i =, else u i = a ij sign(x j x i ) x j x i α ij, (6) j N(G(A),i) where < α i, α ij <, is the absolute value of real numbers, and sign( ) is the sign function February, 8

10 defined as sign(r) =, r >, r =, r <. For simplicity, sign(r) r α is denoted by sig (r) α for any α >. Remark: sig (r) α, α >, is a continuous function on R, which leads to the continuity of protocols (5) and (6). That will be discussed in Property. If we set α i =, i I n and α ij =, i, j I n in the above protocols (5) and (6) respectively, then they will become the typical linear consensus protocol studied in [4] and [5], and they solve a consensus problem asymptotically provided that G(A) has a spanning tree. If we set α i = and α ij = in (5) and (6), they will become discontinuous. The discontinuous case of protocol (5) was studied by Cortés in [4], where G(A) is assumed to be a connected undirected graph and A is also a matrix. In the next section, we will investigate the mathematical conditions that guarantee protocol (5) or protocol (6) to solve a finite-time consensus problem. A. Basic Properties and Lemmas Property : Protocols (5) and (6) are continuous with respect to state variables x, x,..., x n. Moreover, under either of those two protocols, there exists at least one solution of differential equations (4) on [, ) for any initial state x(). Furthermore, max i x i (t) is non-increasing and min i x i (t) is non-decreasing. Hence, x(t) is also non-increasing and x(t) x() for all t. Proof: Consider function f(r) = sig (r) α, where < α <. If r >, then f(r) = r α is continuous, and lim r + r α =. If r <, then f(r) = ( r) α is also continuous, and lim r ( r) α =. From α =, we have that f(r) is continuous on R, and therefore, protocol (5) and (6) are continuous with respect to state variables x, x,...,x n. Let E be the set {(t, ξ) : ξ < x() +, r < t < r, r > }. Then E is open. Consider the solution when the domain of u i is restricted in the open set E. By Peano s Existence Theorem (in the Appendix), there exists at least one solution of differential equations (4) on [, t + ] for some t + >. We extend the obtained solution over a maximal interval of existence (ω, ω + ). At time t, t [, ω + ), if x j (t) = max i x i (t), then ẋ j (t), and if x j (t) = min i x i (t), then ẋ j (t). Therefore max i x i (t) is non-increasing and min i x i (t) is non-decreasing. With the February, 8

11 same arguments, x(t) is also non-increasing for t < ω +. By Extension Theorem (in the appendix), (t, x(t)) tends to the boundary of E as t ω +, which only occurs when ω + = r. Because of the arbitrariness of r, there exists at least one solution of differential equation (4) on [, ) for any initial state x(). Remark: In fact, any solution of the system can be extended over [, ) because it is bounded by x(). Moreover, one notices that (5) and (6) are not Lipschitz at some points. As all solutions reach subspace span() in finite time, there is nonuniqueness of solutions in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations. The following property shows that the equilibrium point set of the considered differential equations ẋ i = u i, i I n, is the set of all consensus states. Property : With protocol (5) or (6), the equilibrium point set of the differential equations ẋ i = u i, i I n, is span(), provided that G(A) has a spanning tree. Proof: Since G(A) has a spanning tree, there exists at most one agent, whose neighbor set is empty. Let ξ = [ξ, ξ,...,ξ n ] T satisfy the equations u i x=ξ =, i I n. We prove ξ span() by contradiction. Assume that min i ξ i max i ξ i. Let M = {j : ξ j = min i ξ i } and let H = {j : ξ j = max i ξ i }. Then M H = φ. Suppose that the root of the spanning tree is vertex v j. Then j M or j H or both hold. Without loss of generality, assume that j M. For any k M, there exists a path v j = v i, v i,..., v is = v k from v j to v k. Let l be the number, such that i l M but i l M. Since ξ il > min i ξ i, i l N(G, i l ) and ξ il = min i ξ i, we have that u il >, which is a contradiction. Thus, min i ξ i = max i ξ i, namely, ξ span(). To prove the converse, suppose that ξ span(). Then all u i x=ξ, i I n, are all zeros. Therefore, ξ belongs to the equilibrium point set. In order to establish our main results, we need the following Lemmas. Lemma : Let ξ, ξ,..., ξ n and let < p. Then ( ) p ( ) p ξ i ξ p i n p ξ i. Proof: Obviously ξ i = ξ p i =. February, 8

12 Let U = {ζ = [ζ, ζ,...,ζ n ] T R n : n ζ i = and ζ }. If n n ξp i ( n ξ i) p = ( ξi n ξ i) p inf ζ U ζ p i, ξ i, ξ where the last inequality follows from that U. T ξ For any ζ U, n ζp i. Since n ζp i is continuous and U is a bounded closed set, inf n ζ U ζp i exists and is larger than. It can be calculated directly. Precisely, ζ p i = min{n p, } =. Thus, the left inequality holds. inf ζ U With the same arguments, n ξp i ( n ξ i) p sup ζ U Therefore, the right inequality also holds. ζ p i = max{n p, } = n p. Lemma 3 (cf. [], Theorem ): Suppose that function V (t) : [, ) [, ) is differentiable (the derivative of V (t) at is in fact its right derivative), such that dv (t) KV (t) α, where K > and < α <. Then V (t) will reach zero at finite time t V (t) = for all t t. Proof: Let f(t) satisfy differential equation df(t) = Kf(t) α. Given initial value f() = V () >, its unique solution is f(t) = ( K( α)t + f() α ) α, t < V () α K( α),, t V () α K( α) V () α K( α), and Since V () = f(), by Comparison Principle of differential equations (in the Appendix), V (t) f(t), t. Hence, V (t) will reach zero in time V () α dv (t). Since V (t) and, K( α) V (t) cannot leave zero once it reaches it. February, 8

13 3 IV. MAIN RESULTS A. Networks Under Protocol (5) In this subsection, protocol (5) is studied. The communication topology G(A) is supposed to have a spanning tree, which is the least requirement when the topology is time-invariant. Otherwise, if G(A) does not have a spanning tree, then there exist at least two groups of agents, between which there does not exist information exchange, and therefore it is impossible for the system to solve a consensus problem through distributive protocols. Furthermore, we hope that the state of the system reaches span () in a finite time. Now, we present our first main result. Theorem : If the communication topology G(A) has a spanning tree, then system (4) solves a finite-time consensus problem when protocol (5) is applied. Proof: This theorem is proved through the following three steps. Step : Suppose that the communication topology G(A) is strongly connected. By Lemma, there exists a vector ω = [ω, ω,..., ω n ] T R n such that ω > and ω T L(A) =. Let y i = n j= a ij(x j x i ), i I n, and let y = [y, y,...,y n ] T. Then y = L(A)x, and ẋ i = sig (y i ) α i. Since ω T y(t) = ω T L(A)x(t) = x(t) =, y(t) ω. Consider nonnegative function ω i V (t) = α i + y i αi+, which will be proved to be a valid Lyapunov function for the disagreement of agents states. If y i <, y i α i+ = ( y i ) α i+ and d( y i) α i + and dyαi+ i dy i right derivatives of y i αi+ at are all. Consequently, for any y i, y i >, y i α i+ = y α i+ i dy i = (α i + )y α i i d y i α i+ dy i = (α i + ) sig (y i ) α i. = (α i + )( y i ) α i = (α i + ) sig (y i ) α i. If = (α i + ) sig (y i ) α i. Furthermore, the left and February, 8

14 4 Thus, dv (t) = = ω i sig (y i ) α dy i i ( ) ω i sig (y i ) α i a ij (sig (y j ) α j sig (y i ) α i ). Let α = [α, α,...,α n ] T, let sig (y) α = [sig (y ) α, sig (y ) α,...,sig (y n ) αn ] T sig ( y T)α = (sig (y) α ) T. Then V (t) j= = sig ( y T) α diag(ω)l(a) sig (y) α. and let Let α = max i α i and let B = (diag(ω)l(a) + L(A)T diag(ω)). Suppose that V (t). Then y, and dv (t) = sig ( y T) α (diag(ω)l(a) + L(A) T diag(ω)) sig (y) α = sig ( y T) α B sig (y) α sig sig (y T ) α sig (y) α ( ) y T α sig (y) α V V (t) α (t) α (7) Since < α <, < α <. In order to apply Lemma 3, we need to find lower bounds of the first two qualities in the right side of equality (7). Let U = {ξ : nonzero terms of ξ, ξ,...,ξ n are not with the same sign } and let U = U {ξ : ξ T ξ = }. Obviously, U is a bounded closed set, namely, a compact set. From y ω and ω >, we have y U, and from that function sig (r) α conserves the sign of r, we have sig (y) α U. Let ξ U. Then ξ span() and therefore by Corollary, ξ T Bξ >. Because the function ξ T Bξ is continuous and U is compact, min ξ U ξ T Bξ, denoted by K, exists and is larger than zero. Hence sig ( y T) α B sig (y) α sig (y T ) α sig (y) α = The last inequality follows from that sig ( y T) α sig (y) α sig (yt ) α sig (y) αb sig (yt ) α sig (y) K. α sig(y) α sig(y T ) α sig(y) α U. February, 8

15 5 Consider the second quality, sig ( y T)α sig (y) α V (t) α = ( n n y i α i ω i +α i y i +α i ) α = n ( ( n n i=n y i α i ) α ω i +α i ( ( n ( by Lemma ) y i (+α i) α y l α l ) ) α ω i +α i ) α ω i +α i y l (+α l) α y l α l (+α l) α, where l = arg max i y i (+α i) α. α Since < α l α <, α l +α l = α α l, which implies that α ( )(+α l ) l (+α l ) α. Therefore, function r α l (+α l ) α does not increase as r increases. Since < y l (t) y(t) = L(A)x(t) L(A) i x(t) L(A) i x(), y l α l (+α l ) α ( L(A) i x() ) α l (+α l ) α. Let Then K > and Thus by (7), K = min ( L(A) n ( ω i +α i ) α i x() ) α i (+α i ) +α i dv (t) sig ( y T)α sig (y) α α. V (t) α K. (8) K K V (t) α, t. By Lemma 3, the above differential inequality gives that V (t) reaches zero in finite time α ( )V () K K ( α ). If V (t) =, then y(t) =, which implies that u i =, i I n, and thus x(t) span() by Property. Therefore the system solves a finite-time consensus problem. Step : Next, suppose that there exists only one leader and the local communication topology among the followers are strongly connected. By protocol (5), the state of the leader is timeinvariant. Without loss of generality, suppose that agents,,..., m = n are the followers and agent n is the leader (because it is just a matter of relabeling the n agents). Consequently, a n, a n,...,a nn are all equal to zero. For notational simplicity, let b = a n, b = a n,...,b m = February, 8

16 6 a mn, let b = [b, b,...,b m ] T, let ᾱ = [α, α,...,α m ], and let Ā = [a ij ] i,j m R m m. Since agent n is the leader, b. Denote x i x n by x i, i I m. Then for any i I m, we have that ( m ) αi x i = ẋ i = sig a ij ( x j x i ) b i x i. Let y i = m j= a ij( x j x i ) b i x i, i I m, and let ȳ = [y, y,...,y m ] T. Then m ẏ i = a ij (sig (y j ) α j sig (y i ) α i ) b i sig (y i ) α i, i I m. j= Since the local communication topology among the followers is j= G(Ā) that is strongly connected, by Lemma, there exists ω = [ω, ω,...,ω m ] T R m such that ω > and ω T L(Ā) =. Define Lyapunov function V (t) = m ω i +α i y i +α i. Obviously, V (t) = if and only if ȳ =. Notice that diag( ω) diag( b) = diag([ω b, ω b,...,ω m b m ] T ). Then diag( ω)(l(ā) + diag( b)) + (L(Ā)T + diag( b)) diag( ω) is positive definite by Corollaries and, and thus L(Ā)+diag( b) is non-degenerate. And from ȳ = (L(Ā)+diag( b))[ x, x,..., x m ] T, we have that [ x, x,..., x m ] T = if and only if ȳ =. Therefore, V (t) = if and only x(t) span(). Differentiate it with respect to time. dv (t) Let B = that V (t). = sig ( ȳ T) ᾱ diag( ω)l( Ā) sig (ȳ)ᾱ sig ( ȳ T) ᾱ diag( ω) diag( b) sig (ȳ) ᾱ = sig ( ȳ T) ( ᾱ ( diag( ω)l( Ā) + L(Ā)T diag( ω) ) ) + diag( ω) diag( b) sig (ȳ)ᾱ ( diag( ω)l( Ā) + L(Ā)T diag( ω) ) +diag( ω) diag( b) and ᾱ = max i Im α i. Suppose dv (t) sig ( ȳ T)ᾱ B sig (ȳ) ᾱ sig ( ȳ T) ᾱ sig (ȳ) ᾱ V sig (ȳ T )ᾱ sig (ȳ)ᾱ V (t) ᾱ (t) ᾱ +ᾱ. +ᾱ By Corollaries and, B is real symmetric and positive definite. Let the smallest eigenvalue of B be λ ( B). Then λ ( B) >, and for any ξ R m, ξ T Bξ λ ( B)ξ T ξ. Therefore, sig ( ȳ T)ᾱ B sig (ȳ) ᾱ sig (ȳ T )ᾱ sig (ȳ)ᾱ λ ( B). With the same arguments as in the proof of inequality (8), there exists K 3 = ( m ) min ᾱ ω +ᾱ i +α i ᾱ ( L(A) i x() ) α i (+α i ) +ᾱ > i I m February, 8

17 7 such that Therefore, sig ( ȳ T)ᾱ sig (ȳ)ᾱ K V (t) ᾱ 3. +ᾱ dv (t) λ ( B)K 3 V (t) ᾱ +ᾱ. By Lemma 3, in this case, this system solves a finite-time consensus problem ' ' 7 (a) (b) (c) Suppose that (a) is the underlying communication topology, where the numbers in the circles are the indices of agents, and the numbers near the edges are the weights. The vertices in the dashed circle in (a) are the leaders, the local communication topology among which is strongly connected. After a period of time, the states of them will agree, and they can be viewed as one agent. The equivalent topology is (b), where represents the virtue agent. In (b), by the conclusion of the second step, agent 4 and agent 6 will also agree with the leaders separately after a time. And then the equivalent topology becomes (c). And finally, they will reach a consensus on states. Fig.. Demonstration of the third step of the proof of Theorem Step 3: Finally, suppose that the communication topology G(A) has a spanning tree. Consider another directed graph G, whose vertex set is the set of all strongly connected components of G(A), denoted by {u, u,...,u m }, m n. (u i, u j ) E(G) if and only if there exist v i V(u i ) and v j V(u j ) such that (v i, v j ) E(G(A)). Then G is uniquely determined by A and is a directed tree. Denote the root of G by u s, where its vertex set corresponds to the leader set. Consider the states of agents associated with E(u i ). Since G is a directed tree, February, 8

18 8 there exists a path u s, u s,..., u sl = u i from the root u s to u i. For the agents corresponding to the vertex set of u s, since the local communication topology u s among them is strongly connected, and the dynamics of them is not affected by others, by the conclusion of the first step, their states will reach a consensus in finite time t. After time t, the states of those agents are time-invariant. Therefore, under protocol (5), they can be viewed as one agent for the other agents. Denote this virtue agent by v. Define a new weighted directed graph G. The vertex set is { v } E(u s ). For any v j E(u s ), ( v, v j ) E(G ) if and only if there exists v k E(u s ) such that (v k, v j ) E(G(A)). And the weight of ( v, v j ) is the sum of the weights of all such edges as (v k, v j ). The induced weighted subgraph of G by the vertices of u s is also u s and inherits its weights in G(A). Notice that the dynamics of agents associated with E(u s ) under topology G(A) is the same as the dynamics under topology G. Hence, by the conclusion of the second step, there exists time t t, at which those agents will reach a consensus on states with the leaders. By induction, the states of agents corresponding to E(u j ) will be the same as the leaders final states in a finite time. The illustration of the proof is shown in Fig.. To conclude, the system solves a finite-time consensus problem. From the first step of the proof of Theorem, we can see that Corollary 3: Suppose that G(A) is strongly connected and there exists ω R n such that ω > and ω T L(A) =. If the following protocol u i = a ij (sig (x j ) α j sig (x i ) α i ), i I n, (9) j= where < α i, α j <, is taken and the initial state x() satisfies ω T x() =, then the states of agents will reach zero in finite time. Remark: Notice that ω T ẋ(t) = ω T L(A) sig (x) α =. Then ω T x(t) is time-invariant and ω T x(t). Thus the above system is the same as the system of y in the first step of the proof of Theorem. And it is easy to check that the above system satisfies the Lipschitz condition at all points in span() except the origin, and thus it is impossible for the states of agents reach consensus in finite time on other agreement points. February, 8

19 9 B. Networks Under Protocol (6) In this subsection, we present conditions under which protocol (6) solves a finite-time consensus problem. In fact when the topology is dynamically changing, protocol (6) is also applicable. This case will be discussed thoroughly in the subsequent subsection. Unlike the discussions on protocol (5), here will present several theorems, which are valid in different cases, and are of interest themselves. Theorem : Suppose that communication topology G(A) is undirected and connected. Then protocol (6) solves the finite-time average-consensus problem, if α ij = α ji for all i I n, j N(G(A), i). Proof: First, complete definition of α ij is provided. For any i I n, if j N(G(A), i), set α ij = r max k In,l N(G(A),k) α kl. Clearly in this case a ij =. Let Since a ij = a ji and α ij = α ji, for all i, j I n, we get that ẋ i (t) =. () κ = n x i (t). By (), κ is time-invariant. Let x i (t) = κ + δ i (t) and let δ(t) = [δ (t), δ (t),..., δ n (t)] T. Then δ i (t) = ẋ i (t). In [4], δ(t) is referred to as the group disagreement vector. Consequently, T δ(t) = n x i(t) nκ = n x i(t) n x i(t) =. We take Lyapunov function V 3 (δ(t)) = δi (t). The remainder of the proof is to show that V 3 (t) satisfies the conditions of Lemma 3 for some K and α and then V 3 (t) reaches zero in finite time, which yields that all agents states reach a consensus in finite time and the final state is κ, the average of their initial states. February, 8

20 Differentiate V 3 (t) with respect to t. dv 3 (t) = δ i (t) δ i (t) = (a ij δ i sig (δ j δ i ) α ij + a ji δ j sig (δ i δ j ) α ji ) () = i,j= a ij (δ i δ j ) sig (δ j δ i ) α ij i,j= = = a ij δ j δ i +α ij i,j= ( a i,j= ij ) ( +α (δj δ i ) )+α ij, where α = max ij α ij and equation () follows from that δ j δ i = x j x i. From < α <, < <. Suppose that V 3 (t). By Lemma, ( ) +α dv 3 (t) +α a ij ((δ i δ j ) ) +αij i,j= = n i,j= a ij ((δ i δ j ) ) +αij n i,j= a ij (δ i δ j ) The last equation follows from that n i,j= a ij n i,j= a ij (δ i δ j ) V 3 (t) V 3 (t) (δ i δ j ). In fact, if n i,j= a ij () (δ i δ j ) =, then by the connectivity of G(A), δ i = δ j for all i, j I n, namely, δ span(). Because T δ =, δ =, and thus V 3 (t) =, which contradicts our assumption. Therefore, n i,j= a ij (δ i δ j ). Next, we give the lower bounds of the first two qualities in the big brackets of equation (). By Property, max i x i min i x i is non-increasing. For any i, j I n, δ i (t) δ j (t) max k x k (t) min k x k (t) max k x k () min k x k (). Let K 4 = n i,j= a ij min i,j In a ij a ij (max x k () min k x k ()) ( +α ij k ), which is positive. Let (i, j ) = arg max i,j In (δ i δ j ). Since +α ij a ij, with the same February, 8

21 arguments as in the proof of inequality (8), n i,j= a ((δ i δ j ) ) +αij, ij n i,j= a ij (δ i δ j ) +α a i j ((δ i δ j ) ) ( n K 4. i,j= a ij +α i j ) (δ i δ j ) Now consider the second quality. Let B = [b ij ] R n n, where b ij = a and since δ, Therefore, i,j= a +α ij (δ i δ j ) = n i,j= a ij (δ i δ j ) V 3 (t) dv 3 (t) b ij (δ i δ j ) = δ T L(B)δ, i,j= = δt L(B)δ δt δ ij 4λ (L(B)) >. (4K 4λ (L(B))) V 3 (t). Thus, by Lemma 3, system (4) solves the finite-time average-consensus problem.. Then by Lemma The next theorem is on the case with a leader. Theorem 3: Suppose that the system consists of one leader and n followers and the local communication topology among the followers is undirected and connected. If α ij = α ji for all i, j such that agent i and agent j are neighbors of each other, then protocol (6) solves a finite-time consensus problem and the final common state is the state of the leader. Proof: First we define the undefined α ij as in the proof of Theorem and let m = n. Without loss of generality, suppose that agents,,..., m are the followers and agent n is the leader. From the provided conditions of this theorem, we have that a n = a n = = a nn =. Let Ā = [a ij ] i,j m R m m, let b = [b, b,...,b m ] T = [a n, a n,...,a mn ] T and let α = max ij α ij. Then ĀT = Ā and b. Rewrite protocol (6) m u i = a ij sig (x j x i ) α ij + b i sig (x n x i ) α in, i I m. j= Let δ i = x i x n, i I n. Then δ n and m δ i = ẋ i = a ij sig (δ j δ i ) α ij b i sig (δ i ) α in, i I m. j= February, 8

22 Also consider the Lyapunov function V 3 (t) = n δ i. ( dv 3 (t) m m ) = δ i a ij sig (δ j δ i ) α ij b i sig (δ i ) α in = = j= m a ij δ j δ i +α ij i,j= m ( a i,j= ( m a i,j= ij ij The last inequality is obtained by Lemma. For simplicity, let G (δ) = m G (δ) = m inequality (8), i,j= a ij (δ j δ i ) + m m b i δ i +α in ) +α δ j δ i (+αij) δ j δ i (+αij) + i,j= a ij b i m b m ( b i i δ j δ i (+αij) + m ) +α δ i (+α in) ) +α δ i (+α in). b i δ i (+α in) and let δi. With the same arguments as in the proof of G (δ) G (δ) K 4, where K 4 is defined in Theorem. +α Let B = [a ij ] i,j m and let b +α = [b +α, b,...,b m ] T. By Corollary, L( B)+diag( b) is positive definite. Its smallest eigenvalue is denoted by λ (L( B) + diag( b)). Then G (δ) V 3 (t) = [δ, δ,..., δ m ] T (L( B) + diag( b))[δ, δ,..., δ m ] [δ 4λ, δ,..., δ m ] T (L( [δ, δ,...,δ m ] B) + diag( b)). Let B be the same as in the proof of Theorem. We have B b B =. Therefore, L(B) = L( B) + diag( b) b. Its eigenvalues are all nonnegative numbers, and the second smallest eigenvalue of L(B), denoted by λ (L(B)), is λ (L( B) + diag( b)). Thus, dv 3 (t) (4K 4λ (L(B))) V 3 (t). February, 8

23 3 By Lemma 3, V 3 (t) will reach zero in finite time, which implies that all agents states reach the common value x n () in a finite time. In the above analysis, we only consider the case when G(A) is undirected or the local communication topology among the followers is undirected. To some extent, the above two results can be extended to the case when the communication topology belongs to the set of some special kinds of directed graphs. Definition ([5]): Communication topology G(A) is said to satisfy the detail balance condition in weights if there exist some scalars ω i >, i I n, such that ω i a ij = ω j a ji for all i, j I n. We can see that in the detail-balanced communication topology, the communication channels among agents are also bidirectional like in undirected graphs, but with different weights. Corollary 4: Protocol (6) solves a finite-time consensus problem if G(A) is strongly connected and detail-balanced, and α ij = α ji for all i I n, j N(G(A), i). Proof: Let ω = [ω, ω,...,ω n ] T such that ω i a ij = ω j a ji for all i, j I n and define the undefined α ij as in the proof of Theorem. It is easy to check that diag(ω)a is symmetric and ω i ẋ i (t) =. Let κ = P n ω i n ω ix i (t). Then κ is time-invariant. Let δ i (t) = x i (t) κ. Then ω T δ(t) =. We take Lyapunov function V 4 (t) = n ω iδ i (t). dv 4 (t) = ω i a ij δ j δ i +α ij. With the same argument as in the proof of Theorem, there exists K 5 > such that dv 4 (t) i,j= K 5 V 4 (t), where α = max ij α ij. The details are omitted. And thus this system solves a finite-time consensus problem, and the final state is average-consensus function. P n ω i n ω ix i (), which can be viewed as a weighted The following corollary is also obtainable but the proof is omitted. Corollary 5: Suppose that the conditions are the same as in Theorem 3, except that the local communication topology among the followers is strongly connected and detail-balanced. Then protocol (6) solves a finite-time consensus problem. February, 8

24 4 Based on the results of Corollary 4 and Corollary 5, we can still extend those results further, though, for protocol (6), we cannot obtain results as desirable as for protocol (5). Theorem 4: If G(A) has a spanning tree and each strongly connected component is detailbalanced, and α ij = α ji for all i, j such that agent i and agent j are neighbors of each other, then protocol (6) solves a finite-time consensus problem. The proof of the above theorem is the same as the third step of the proof of Theorem. C. Performance Analysis In this subsection, we investigate the relationship between the convergence time and the other factors, such as the underlying communication topology, parameters α i and α ij in the proposed protocols, and the initial states. The convergence time is defined as the time that the system spends to reach a consensus. For the studied nonlinear system, the precise time of reaching the consensus state is hard to given. However, we can analyze it by investigating its upper bound obtained by Lemma 3. We claim that V (t), V (t), V 3 (t) and V 4 (t) measure how much agents states differ from each other. V 3 (t) and V 4 (t) clearly measure the disagreement of agents current states with their common final state by their definitions. Now consider V (t). We take the same assumptions and notations as in Theorem. Let ω = {ξ R n : ω T ξ = }. Then ω is L(A)-invariant. Furthermore, for any ξ ω, L(A)ξ = ξ =. Therefore, ξ L(A) = L(A)ξ defines a vector norm on ω (see [6], Theorem 5.3.). Furthermore, for any x, let x = r+x, where r R and x ω. L(A)x = x T L(A) T L(A)x = x T L(A)T L(A)x = x L(A). And apparently, V (t) is a measure of the length of L(A)x. Especially, if all α i are equal to α, < α <, and ω =, V (x(t)) = +α ( L(A)x (+α)) +α. Therefore, V (t) can be seen as a measure of the length of vector x, which reflects the disagreement of x with subspace span(). In the same way, we can show that V (t) also measures the disagreement of agents states. Undoubtedly, larger initial value of V (t), V (t), V 3 (t) and V 4 (t) will result in longer convergence time of the considered system by Lemma 3. This fact also can be reflected in the estimation of K, K 3 and K 4. February, 8

25 5 It is well known that, if α i or α ij = for all i, j, then the system at most asymptotically solves a consensus problem, and thus the convergence time becomes infinitively long. In our model, since lim αi α = and lim αij =, by Lemma 3, the estimated upper bound of convergence time tends to infinity, if all α i or α ij tend to. In [4], it was proved that the larger the algebraic connectivity of the underlying communication topology is, the faster the system converges. To illustrate the dependency of the convergence time under protocol (5) or (6) on the algebraic connectivity of the communication topology, we assume that G(A) is undirected and connected, and all α i or α ij are all equal to α, < α <. For protocol (5), we consider Lyapunov function V 5 (t) = 4 n i,j= a ij(x j (t) x i (t)) instead of V (t). It can be shown that V 5 (t) = x span() and dv 5 (t) For protocol (6), K 4 can be taken to be, and where B = [a +α ij ]. dv 3 (t) (λ (L(A))) +α V5 (t) +α. (3) (4λ (L(B))) +α V3 (t) +α, (4) The above two differential inequalities show that larger algebraic connectivity of G(A) or G(B) can lead to shorter the convergence time (see Lemma 3). The following two theorems compare the convergence rates of protocol (5) or protocol (6) with different parameters α i or α ij under the same communication topology respectively. It is shown that smaller α i or α ij can get better convergence rate when agents states differ a little from each other, and larger α i or α ij can get better convergence rate when agents states differ a lot from each other. Therefore, in order to get better convergence rate, we can change the values of protocol parameters α i or α ij based on the states of agents as system evolves. Theorem 5: Suppose that G(A) is undirected and connected, protocol (5) is applied, and all α i are equal, denoted by α. Then the eigenvalues of L(A) are nonnegative numbers (by Lemma ) and let them be λ (L(A)), λ (L(A)),..., λ n (L(A)) in the increasing order. Since G(A) is connected, λ (L(A)) = and λ (L(A)) >. Given < α < α <, let ǫ = λ (L(A)) n α α α and let ǫ = ǫ, then λ n(l(a)). Consider Lyapunov function V 5 = 4 n i,j= a ij(x j (t) x i (t)). If V 5 (t) dv 5 (t) α=α dv 5(t), α=α February, 8

26 6 and if V 5 (t) ǫ, Proof: Since dv 5 (t) = dv 5 (t) a ij (x j x i ) j= by Lemma, ( ) n α a ij (x j x i ) and j= It is easy to see that +α dv 5(t) α=α +α = dv 5(t) α=α. ( ) a ij (x j x i ) j= +α ( ) a ij (x j x i ) j= ( a ij (x j x i )) = x T L(A) T L(A)x, j= V 5 (t) = xt L(A)x., (5) +α. (6) Let D = λ (L(A))... λ n (L(A)). Since L(A) is symmetric, there exists an orthogonal matrix T R n n such that L(A) = T T DT. And, Therefore, x T L(A) T L(A)x xt L(A)x = xt T T DTT T DTx x T T T DTx λ (L(A)) xt L(A) T L(A)x xt L(A)x = xt T T D Tx x T T T DTx. λ n (L(A)). (7) February, 8

27 7 If V 5 (t) ǫ, then dv 5 (t) α=α ( x T L(A) T L(A)x )+α ( by (6)) = ( x T L(A) T L(A)x ) +α ( x T L(A) T L(A)x ) α α ( x T L(A) T L(A)x ) +α (λ (L(A))V 5 (t)) α α ( by (7)) n α dv 5(t) ( x T L(A) T L(A)x ) +α (by V 5 (t) ǫ ) α=α ( by (6)). If V 5 (t) ǫ, by (7), ( a ij (x j x i )) λ n (L(A))V 5 (t). Therefore, for any i I n, By (5), j= ( n ( ) a ij (x j x i ) j= ) j= a ij(x j x i ). And because α < α, +α dv 5 (t) +α ( ) a ij (x j x i ) j= dv 5(t) α=α α=α. +α < +α.. Thus Theorem 6: Suppose that G(A) is undirected and connected, protocol (6) is applied, and all α ij are equal, denoted by α. Let B = [a +α ]. Then the eigenvalues of L(B) are nonnegative numbers ij (by Lemma ) and let the second smallest and the largest eigenvalues be λ (L(B)), λ n (L(B)) respectively (which are positive). Given < α < α <, let ǫ = n ( α ) α α (λ n (L(B α=α ))) +α α α (λ (L(B α=α ))) +α α α, and let ǫ = n ( α ) α α (λ n (L(B α=α ))) +α α α (λ (L(B α=α ))) +α α α. Consider Lyapunov function V 3 (δ(t)) = n δ i (t) (where δ(t) was defined in the proof of Theorem ). If V 3 (t) ǫ, then dv 3 (t) α=α dv 3(t), α=α February, 8

28 8 and if V 3 (t) ǫ, dv 3 (t) dv 3(t) α=α. α=α Proof: In the proof of Theorem, we get that dv 3 (t) = ( )+α a +α ij (δ j δ i ), and by Lemma, ( n α a +α ij i,j= i,j= a +α )+α (δ j δ i ) i,j= i,j= dv 3(t) ( a +α ij i,j= )+α (δ j δ i ). (8) Moreover, since n ij (δ j δ i ) = δ T L(B)δ, V 3 (t) = δt δ, and δ, by Lemma, 4λ (L(B))V 3 (t) a +α ij (δ j δ i ) 4λ n (L(B))V 3 (t). (9) By (8) and (9), we have that n α If V 3 (t) ǫ, then dv 3 (t) α=α (4λ n (L(B))V 3 (t)) +α dv 3(t) (4λ (L(B))V 3 (t)) +α. () (4λ (L(B α=α ))) +α V 3 (t) +α V 3 (t) α α ( by ()) (4λ (L(B α=α ))) +α n α α α (λ n (L(B α=α ))) +α (λ (L(B α=α ))) (+α ) V 3 (t) +α (by V 3 (t) ǫ ) = n α +α (λ n (L(B α=α ))) (+α ) V 3 (t) +α dv 3(t) (by ()). α=α The case when V 3 (t) ǫ can be proved in the same way. D. Networks With Switching Topology In practice, the information channel between any two agents may not be always available because of the restrictions of physical equipments or the interference in signals from external, such as exceeding the sensing range or existence of obstacles between agents. Therefore, it is more reasonable to assume that the communication topology is dynamically changing. February, 8

29 9 In order to describe the switching property of the topology, suppose that a ij is a piece-wise constant right continuous function of time, denoted by a ij (t), and takes value in a finite set, such that a ii (t) = for all i, t. And the changing topology is represented by G(A(t)). We only study the validity of protocol (6) when the underlying topology is switching. As for protocol (5), the finite-time convergence analysis is more challenging and we leave it as one future research topic. In fact, for protocol (6), the parameter α ij can also be changing to reflect the reliability of information channel (v j, v i ) as weight a ij or to get shorter convergence time as indicated in Theorem 6. By investigating the properties of the two protocols, we can see that the main difference between the two protocols is that the Lyapunov function V 3 (δ(t)) used in the proof of Theorem does not depend on the network topology. This property of V 3 (δ(t)) makes it a possible candidate as a common Lyapunov function for convergence analysis of the system with switching topology. Theorem 7: Suppose that G(A(t)) is undirected and connected all the time, all α ij are piecewise constant right continuous functions of time, denoted by α ij (t), and take values in a finite set such that α ij (t) = α ji (t) and < α ij < for all i, j, t. Then protocol (6) solves the finite-time average-agreement problem. Proof: The proof is similar to that of Theorem and we take the same notations as in the proof of Theorem, such as κ, δ(t) and B. Then κ is also time-invariant. Let α = max i,j,t α ij (t) and take the Lyapunov function V 3 (δ(t)). At any time t, we have an estimation of K 4 (t), which can be chosen to depend only on A(t) and the initial state x(). Since all possible A(t) and α ij (t) are finite, K 6 = min t K 4 (t)λ (L(B(t))) exists and is larger than. Then dv 3 (t) α K 6 V 3 (t). Therefore, V 3 (t) will reach zero in finite time t = α V 3 () α solves the finite-time average-agreement problem. ( α )K 6 and the switching system Remark: The conditions in Theorem 7 can be relaxed in several ways. For example, we can assume that the sum of time intervals, in which G(A(t)) is connected, is larger than t. In fact G(A(t)) is not necessarily undirected. This condition can be replaced by that G(A(t)) is always detail-balanced with a common ω > such that diag(ω) T A(t) is symmetric. In [3], [5] and [], it was shown that under certain conditions, if the union of graph G(A(t)) February, 8