J Control Theory Appl 2008 6 (4) 357–364 DOI 10.1007/s11768-008-7044-8
Control of group of mobile autonomous agents via local strategies Lixin GAO 1 , Daizhan CHENG 2 , Yiguang HONG 2 (1.Institute of Operations Research and Control Sciences, Wenzhou University, Wenzhou Zhejiang 325027, China; 2.Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, China)
Abstract: This paper considers the formation control problem of multi-agent systems in a distributed fashion. Two cases of the information propagating topologies among multiple agents, characterized by graphics model, are considered. One is fixed topology. The other is switching topology which represents the limited and less reliable information exchange. The local formation control strategies established in this paper are based on a simple modification of the existing consensus control strategies. Moreover, some existing convergence conditions are shown to be a special case of our model even in the continuous-time consensus case. Therefore, the results of this paper extend the existing results about the consensus problem. Keywords: Formation control; Distributed control; Multi-agent coordination; Mobile autonomous agent
1
Introduction
Over the past few years, the problem of coordinating the motion of multiple autonomous agents has attracted significant attention. This is partly due to broad applications of multi-agent systems in many areas including cooperative control of unmanned aircraft, flocking of birds, schooling for underwater vehicles, distributed sensor networks, attitude alignment for a cluster of satellites, and congestion control in communication networks [1∼4]. Although the application backgrounds are different, the fundamental principles in the coordination of multiple spacecraft, robots, and even animals are very similar, that is, coordinating multiple agents to achieve a goal of the whole system by local information. The investigated feedback scheme is inspired by the aggregates of individuals in nature. Flocks of birds and schools of fish achieve coordinated motions of large numbers of individuals without a central controlling mechanism [5, 6]. Several researchers in the area of statistical physics and theory of complexity have addressed flocking and schooling behaviors in systems of self-propelled particles [7]. A computer graphics model to simulate flock behavior has been presented in [3]. In real applications, the interacting topology between agents may change dynamically. For example, in the case of interaction via communications, the communication links between vehicles may be unreliable due to disturbances and/or subject to communication range limit. In the case of interaction via sensing, the agents that can be sensed by a certain agent may change over time. In a pioneer work [8], each agent’s heading was updated using a simple local rule, and it was shown that for a large class of switching signals and for any initial set of headings that the headings of all the agents will converge to the same steady-state value. The approach in [8] is based on bidirectional information exchange, modeled by undirected graph. [9] extended the work of Jadbabaie et al. [8]
to the case of directed graphs and explored the minimum requirements to reach global consensus. [9] used weighted factor in an update scheme which provides more flexibility to take into account the relative confidence and reliability of information from different agents. [10] solved the average-consensus problem with directed graphs, which requires the graph to be strongly connected and balanced. The problem considered in [10] is essentially the same as in [8,9] for continuous-time consensus scheme. Other related works are [11∼14]. These problems, termed as consensus problems, have a long history in the field of computer science. The notion of a communication graph can be found in [2], where the Laplacian of a formation graph was used to determine a Nyquist-type criterion for stability analysis of multi-vehicle formations in a closed-loop form with linear controllers. It turns out that the information flow among the vehicles plays an important role in the stability of the closed-loop formation dynamics [2, 10]. The formation stability is proved by describing the connectivity properties of the graph that represents agent interconnections in a vector fashion instead of a scalar one (as in [2]), which may resolve the troubles caused by redundant distance information [2]. In [15], the stability of asynchronous swarms with a fixed communication topology was studied, where the stability was used to characterize the cohesiveness of a swarm. Coordinated control of multi-vehicle systems involves many research aspects. One of the most important and fundamental topics is formation control. In real applications, the individual vehicles should also be capable of collectively accomplishing tasks using only locally sensed information and little or no direct communication, which is called local control or decentralized control. [16] uses the method of Lyapunov function control to coordinate a collection of robots in such a way that they maintain a preassigned formation. Formation stabilization of a group of agents with linear dynamics (in the form of double-integrators) was
Received 9 March 2007; revised 30 January 2008. This work was supported by the National Natural Science Foundation of China (No.60674071).
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studied using structural potential functions in [1]. An alternative approach is to use artificial potential functions and virtual leaders as in [1, 17]. [18] investigates a number of formation problems of geometric patterns of robots in the plane. The problem of geometric patterns is closely related to certain agreement problems. [19] uses cyclic pursuit strategies for groups of mobile autonomous agents to achieve a formation, in which the sensor graph is fixed and strongly connected. The purpose of this paper is to develop some decentralized control strategies to achieve a global design goal: the overall system of multi-agents converging to a preassigned formation. The local formation control strategies are designed based on modifying consensus control strategies proposed in [8, 9, 19]. Thus, all the results established in this paper can also be applied to the consensus problems. For instance, the convergence conditions of the continuous-time consensus problem [8, 9, 19] are shown to be special cases of our results. The control strategies are presented as distributed state feedback laws for each agent. To obtain the main results of this paper, we also introduce some new concepts of graphs. For simplicity we assume that each mobile agent is a dynamical system moving on the plane. All results of this paper can be directly generalized to three or even higher dimensions dynamics. This paper is organized as follows. In Section 2 we introduce the notions and formulation of the problem. Local control strategies for formation stabilization in fixed topology with balanced graph are obtained in Section 3, and those for switching topology are established in Section 4.
2
Preliminaries and problem formulation
The group we wish to coordinate consists of n mobile agents and each mobile agent is a dynamical system moving on a common plane. For simplicity we let each agent be described by (1) z˙i = ui , T where zi = (xi , yi ) represents the position of the agent and the control input ui = (uxi , uyi )T represents its velocity. Let the relative position vector between agents i and j be denoted by zij = zi − zj . Stability analysis of the group of agents is based on several results of algebraic graph theory. In this section, we first introduce some basic concepts and notations in graph theory that will be used throughout this paper. More details are available in [20]. A weighted digraph of order n is denoted by G = {V, , A}, where V = {v1 , v2 , · · · , vn } is the set of vertices, ⊂ V × V is the set of edges of the digraph and each edge is denoted by (vi , vj ) ∈ , i, j ∈ {1, 2, · · · , n}, and a weighted adjacency matrix A = [aij ] has nonnegative adjacency elements aij . Throughout this paper, we assume that all the graphs have no edges from a node to itself. A weighted graph is call undirected (or bi-directional) if ∀ (vi , vj ) ∈ ⇒ (vj , vi ) ∈ and aij = aji . Otherwise, the graph is called a directed graph. The adjacency elements associated with the edges of the graph are positive, i.e. (vi , vj ) ∈ ⇔ aij > 0, and assume aii = 0. The weighted matrix is usually used to represent possibly relative reliabilities of different information exchange links
between agents. For a given non-negative matrix A = [aij ]n×n , the directed graph associated with matrix A is denoted by G(A) with a set of vertices V = {vi , i = 1, 2, · · · , n} such that there is a directed edge in G(A) from vi to vj if and only if aij = 0, i = j [21]. The controller of an agent i requires state information from a subset of the agent’s flockmates, called neighbor set. Neighboring relations may reflect physical proximity between two agents, or the existence of a communication channel. A neighboring relation induces a control interconnection between the two neighbors. If (vi , vj ) ∈ , then vj is said to be a neighbor of vi , which means that the information flow is from agent j to agent i. We denote the set of all neighbor vertices of vertex vi by Ni = {j|(vi , vj ) ∈ }. The number of neighbors of each vertex is its out-degree. A path from vertex vi to vertex vj is a sequence of distinct vertices starting from vi and ending at vj such that a consecutive pair of vertices make an edge of the digraph. If there is a path from one node vi to another node vj , then vj is said to be reachable from vi . If a node vi is reachable from every other node of the digraph, then it is said to be globally reachable. A directed graph G is called weakly connected if there exists a node which is globally reachable. A directed tree is a directed graph. The node vj is called the parent of node vi if (vi , vj ) is an edge of the directed tree. Only one node of the directed tree, which is called the root of tree, has no parent, and all other nodes of the directed tree have exactly one parent. A spanning tree of a directed graph is a directed tree formed by graph edges that connect all the vertices of the graph. It is not difficult to see the fact that a directed graph G is weakly connected if and only if it has a spanning tree. A digraph is said to be strongly connected if any two distinct nodes of the graph are connected by a path, which means that each node of the graph is globally reachable. A weakly connected undirected graph must be strongly connected, so it is simply termed as a connected graph. The degree matrix of G is a diagonal matrix denoted by Δ = diag{Δ1 , Δ2 , · · · , Δn }, n where Δi = aij . The weighted graph Laplacian matrix j=1
associated with G is defined as L = L(G) = Δ − A. It is well known that the Laplacian matrix captures many interesting properties of the graphs. The following lemma relates Laplacian matrix with its directed graph, as well as supplying an algebraic characterization. The result of Lemma 1 can be found in [9], which was proved by using the spanning tree. Lemma 1 [9] The weighted graph Laplacian matrix L associated with graph G has at least one zero eigenvalue and all of the non-zero eigenvalues are located on the open right half plane. Furthermore, L has exactly one zero eigenvalue if and only if the directed graph G is weakly connected. For a weakly connected digraph, the n-dimensional eigenvector associated with the single zero eigenvalue is the vector of ones, l. For undirected weight graph G, L is symmetric, positive semi-definite and all eigenvalues of L are
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real and nonnegative. For a connected undirected graph G, the following well-known property holds [20]: min
x=0,lT x=0
xT L(G)x = λ2 (L(G)), xT x
(2)
where λ2 (L(G)) is the second small eigenvalue of positive semi-definite matrix L(G), l = (1, 1, · · · , 1)T . To characterize the class of digraphs, [10] introduced the concepts of balanced graph and mirror graph which are denoted as follows. Definition 1 A weighted graph G = (V, , A) is said to be balanced if n n aij = aji , ∀i. (3) j=1
j=1
Any undirected weighted graph is balanced. Furthermore, a weighted graph is balanced if and only if lT L = 0 [10]. Definition 2 Let G = (V, , A) be a weighted digraph. Let ˜ be the set of reverse edges of G obtained by reversing the order of all the pairs in . The mirror of G denoted ˆ with the by an undirected graph in the form Gˆ = (V, ˆ, A) same set of nodes of G, the set of edges ˆ = ∪ ˜, and the symmetric adjacent matrix Aˆ = [ˆ aij ] with elements aij + aji a ˆij = . 2 For a balanced graph G, the Laplacian matrix for mirror graph Gˆ is T ˆ = (L(G) + L (G)) . L(G) 2 The Laplacian function ΦG (x) := xT L(G)x satisfies ΦG (x) = ΦGˆ(x). We will review some properties of matrices which are used in the sequel. If the dimensions of the matrices involved are such that all the operations below are defined, then [21, 22] a) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD). b) e(A⊗I) = eA ⊗ I. c) If AB = BA, then eA+B = eA eB . d) Let matrices A, B be real symmetry, and λ1 (A) λ2 (A) · · · λn (A), λ1 (B) λ2 (B) · · · λn (B), λ1 (A − B) λ2 (A − B) · · · λn (A − B). Then λi (B) + λ1 (A − B) λi (A) λi (B) + λn (A − B). The distributed property of the controllers of all agents is achieved by imposing a specific structure on a weighted graph associated with the formation of the agents. In other words, the interactions of the agents are designed in such a way that the system of the multiple agents converges to a unique and desired formation. The formation of geometric patterns considered in this paper are characterized by any feasible vector set {dij ∈ R2 , i, j = 1, 2, · · · , n}. By a perfect formation we mean that all position vectors of pairs of agents are fixed over time. It is obvious that dij satisfies dji = −dij and dij may not be independent. For the fixed formation of geometric patterns, the vector set
{dij , i, j = 1, 2, · · · , n} is fixed and there exists a vector set {ri , i = 1, 2, · · · , n} such that dij = ri − rj , n for ∀i, j, and ri = 0. From this reason, the vector set i=1
{dij , i, j = 1, 2, · · · , n} is said to be feasible if there exists a vector set {ri , i = 1, 2, · · · , n} such that dij = ri −rj , for n ∀i, j, and ri = 0. It is known that there are only n − 1 i=1
independent vectors in the vector set {dij } and the solution vector set {ri , i = 1, 2, · · · , n} is unique. In this paper, we always assume that the vector set {dij , i, j = 1, 2, · · · , n} is feasible. It is obvious that a feasible vector set can guarantee a unique geometric formation. Therefore, the overall system will converge to a desired formation, if for a given vector set {dij ∈ R2 } we have lim zij (t) = dij . t→∞
To get the desired formation, we propose the following local control strategies aij (zi (t) − zj (t) − dij ), i = 1, 2, · · · , n, ui (t) = − j∈Ni
(4) where Ni (t) is the set of agent i’s neighbors. If all dij = 0, then local control strategies degenerates to aij (zi (t) − zj (t)), i = 1, 2, · · · , n. (5) ui (t) = − j∈Ni
The set of agents is said to reach consensus asymptotically if lim (zi (t) − zj (t)) = 0 for any zi (0), i = 1, 2, · · · , n. t→∞ We say that a set of local control strategies (5) solves the consensus problem if there exists an asymptotically stable equilibrium x∗ satisfying x∗i = χ(z(0)) for the closed-loop feedback system. A special case is χ(z(0)) = Ave(z(0)) = n 1 zi (0), which is commonly called average-consensus n i=1 problem [10, 14].
3 Fixed topology with balanced graph Using control strategies (4) and denoting r := col(r1 , r2 , · · · , rn ), the overall system becomes z(t) ˙ = −(L ⊗ I2 )(z(t) − r).
(6)
Lemma 2 Assume G is a digraph with constant weighted Laplacian L. G is weakly connected if and only if (7) lim exp(−Lt) = lwlT , t→∞
where wl is an n-dimensional non-zero constant vector satisfying wlT L = 0 and lT wl = 1. Proof (Sufficiency) We prove it by contradiction. From Jordan decomposition of L = SJS −1 , we have exp(−Lt) = S exp(−Jt)S −1 . Suppose G is not weakly connected, which implies, by Lemma 1, that the algebraic multiplicity of eigenvalue λ = 0 of L is m > 1. Therefore, the rank of lim exp(−Jt) is at least m > 1. But, (7) imt→∞
plies that the rank of lim exp(−Jt) is 1, which leads to a t→∞ contradiction. (Necessity) Using the results of Lemma 1, the fact that graph G is weakly connected means the algebraic multiplicity of eigenvalue λ = 0 of L is 1. As t → ∞, all blocks in
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the diagonal of exp(−Jt) associated with non-zero eigenvalues vanish due to the property that non-zero eigenvalues of L are local in the open right half plane. Therefore, as t → ∞, exp(−Jt) converges to matrix Q = [qij ] with a single non-zero element q11 = 1. Thus, we have lim exp(−Lt) = SQS −1 . The first column of S is wr t→∞
satisfying Lwr = 0, and the first row of S −1 is wlT satisfying wlT L = 0. Without loss of generality, we set wr = l. Then, (7) and lT wl = 1 can be obtained by straightforward calculation. Theorem 1 With a time-invariant topology and a constant weighted matrix, using the local control strategies (4), the system converges to the desired formation asymptotically for any initial positions of the agents if and only if the associated graph is weakly connected. Furthermore, the centroid of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary if and only if the graph is balanced. Proof From (6), we can see that z(t) = exp(−(L ⊗ I2 )t)(z(0) − r) + r = (exp(−Lt) ⊗ I2 )(z(0) − r) + r. (8) Therefore, the system converges to the desired formation asymptotically for any initial positions of the agents if and only if lim exp(−Lt) = lcT .
t→∞
Then, the first part of this theorem follows directly from Lemma 2. For the second part of the theorem, the centroid zc (t) of the points {z1 (t), z2 (t), · · · , zn (t)} satisfies n 1 z˙c (t) = z˙i (t) n i=1 1 = − (lT ⊗ I2 )(L ⊗ I2 )(z(t) − r) n 1 (9) = − ((lT L) ⊗ I2 )(z(t) − r). n Therefore, the centroid is stationary if and only if lT L = 0, which implies that the graph is balanced. Corollary 1 With a time-invariant topology and constant weighted matrix, the local control strategies (5) solve the consensus problem if and only if the neighbors’ graph is weakly connected. Furthermore, the local control strategies (5) solve the average-consensus problem if and only if the neighbors’ graph is weakly connected and balanced. For a time-invariant weakly connected and balanced topology and constant weighted matrix, system (6) converges to the desired formation asymptotically for any initial positions of the agents and the centroid zc of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. Therefore, the trajectory looks like zi (t) = zc + ri + δi (t). By choosing 1 Lyapunov function V (δ) = δ T δ, the following theorem 2 can be obtained directly. The proof is omitted because it is similar to the proof of Theorem 2 which appears later. Theorem 2 Consider a time-invariant topology G that is weakly connected and balanced. Using local control strategies (4), the system globally exponentially converges to the desired formation with at least an exponent speed ˆ which means the error is less than κ = λ2 (L(G)),
M e−kt for certain M > 0, and the centroid of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. Remark 1 A known result is that [19] the cyclic pursuit control strategy for a group of mobile agents z˙i = zi+1 − zi − di(i+1) , i = 1, 2, · · · , n − 1, (10) z˙n = z1 − zn − dn1 guarantees that the system globally asymptotically converges to a desired formation with at least an exponent speed κ = λ2 (L), where ⎡ ⎤ 2 −1 · · · · · · −1 ⎢ .. .. ⎥ ⎢ −1 2 −1 . .⎥ ⎢ ⎥ 1⎢ . . . . .. ⎥ ⎢ ⎥ . . . . L= ⎢ . . . . .⎥ , 2⎢ ⎥ .. .. ⎢ ⎥ ⎣ −1 . . 2 −1 ⎦ −1 −1 · · · −1 2 and the centroid of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. It is obvious that this result is a particular case of Theorem 2.
4
Local control with switching topology
In this section, we consider local control strategies with switching topology. Note that the interaction topology G may be dynamically changing over time due to unreliable transmission or limited communication, sensing range, etc. That is, the system’s information topology may be timevarying. We also allow the weighted matrix to be dynamically varying to represent possible time-varying relative confidence of each agent’s information variable or relative reliabilities of different information exchange links between agents. Let Γ1 = {Gp , p ∈ P} denote the class of all possible weakly connected and balanced weighted graphs of order n, and all the arbitrary constant weighted factors aij satisfy aij amin > 0. Let Γ2 denote the class of all possible connected and non-weighted undirected graphs of order ˆ n. For any G ∈ Γ1 , the weighted Laplacian matrix L(G) ˆ ¯ of its mirror graph G is positive semi-definite. Let G be a non-weighted mirror graph of G, denoted by G¯ ∈ Γ2 , with the property that all nonzero weighted factors are 1. ¯ of its mirror So the non-weighted Laplacian matrix L(G) graph Gˆ is also positive semi-definite. From the above de¯ ˆ − 1 amin L(G) finition, we can get that the matrix L(G) 2 ˆ is a positive semi-definite matrix, which implies λ2 (G) 1 ¯ > 0 by using the matrix property d). The set amin λ2 (G) 2 ¯ > 0. Γ2 is finite, which implies min (λ2 (G)) ¯ 2 G∈Γ
At time t, let the topology graph be Gp(t) and let the corresponding matrix be Lp(t) . Using local control strategies (4), the overall system can be expressed as z(t) ˙ = −(Lp(t) ⊗ I2 )(z(t) − r).
(11)
The signal p(t) switches among P as t progresses, which means the topology graph or weighting factor is varying. It is assumed that chattering does not occur, that is, that p(t)
L. GAO et al. / J Control Theory Appl 2008 6 (4) 357–364
switches a finite number of times in every finite time interval. Then the system (6) has a well-defined solution. Let Sdwell be the set of switching signals p(t) which are piecewise constant and any consecutive discontinuities are separated by no less than a positive constant tmin . Theorem 3 Let t1 , t2 , · · · be an infinite time sequences at which the topological graph or weighting factors switch, and let G(ti ) ∈ Γ1 be a switching topological graph at time ti . Assume the switching signal p(t) ∈ Sdwell . Then, the local control strategies (4) guarantee that the system globally asymptotically converges to the desired formation with at least an exponential speed of κ∗ =
1 amin min (λ2 (G)) G∈Γ2 2
(12)
and the centroid of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. Proof Since all G(ti ) ∈ Γ1 , the centroid zc of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. Therefore, the trajectory can be expressed as zi (t) = zc + ri + δi (t). Let δ(t) = col(δ1 (t), δ2 (t), · · · , δn (t)) be the error vector which satisfies lT δ(t) = 0. By choosing Lyapunov function 1 V (δ) = δ T δ. For any t ∈ [tk , tk+1 ), we have 2 V˙ = −δ T (L(G(tk )) ⊗ I2 )δ ˆ k )) ⊗ I2 )δ = −δ T (L(G(t ˆ k ))δ T δ −λ2 (L(G(t 1 ¯ k ))δ T δ − amin (L(G(t 2 ∗ (13) −2κ V (δ), which implies V (t) V (δ(0)) exp(−2κ∗ t),
(14)
then we can obatin
δ(t) δ(0) exp(−κ∗ t).
(15)
Thus, δ(t) globally asymptotically converges to 0 with at least an exponential speed of κ∗ . Then, we have lim (zi (t) − zj (t) − dij ) t→∞
= lim {zi (t) − zj (t) − ri + rj } t→∞
= lim {δi (t) − δj (t)} = 0. t→∞
(16)
Remark 2 From the result of Theorem 3, we know that the system will converge more quickly as large weighted factors are chosen. The result of Theorem 3 can be extended to more general cases in which all topological graphs G(ti ) at time ti are balanced, and there exists an infinite sequence of bounded, non-overlapping time intervals [tik , tik +lk ) such that at least once in each time interval the topological graph is balanced and weakly connected. Then, using local control strategies (4), the system globally asymptotically converges to the desired formation with at tmin , where T is an upleast an exponential speed of κ∗ T per bound of |tik − tik +lk |, and the centroid of the points {z1 (t), z2 (t), · · · , zn (t)} is stationary. If r = 0 and the dimension is one, the problem involved in Theorem 3 is termed as an agreement problem, which
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is essentially the same problem as that the headings of all the agents will converge to the same steady-state value for any initial set of headings [8∼10]. We also have generalized the finite set of weighted factors in [10] to arbitrary constant weighted factor cases. Each agent regulates its position based on the position of its neighbors, so the control law (4) is a distributed cooperative control. When some communication channels are broken, the control law guarantees asymptotic stabilization of the agents’ formation only if the neighbors’ graph made by the rest communication channels is still connected. A simulation to achieve a formation is shown in Fig. 1. The initial locations of the five agents are randomly produced and the feasible formation vector set {dij , i, j = 1, 2, · · · , n} is also randomly produced. The topological graph and weighting factor satisfy the condition given in Theorem 3. From Fig. 1, it is shown that the local control strategies (4) guarantee that the system globally asymptotically converges to the desired formation and the centroid of the formation is stationary.
Fig. 1 Achieving a formation for five agents.
A more general case is when the topological graph is a directed graph. To obtain the main result of this paper, we now introduce some new concepts about the graph. Definition 3 For an ordered set of {Gp1 , Gp2 , · · · , Gpm }, where every Gpi is a simple digraph with vertices given by {v1 , v2 , · · · , vn } and edge set given by pi , a joint path of length l from vj0 to vjl is an ordered set of distinct nodes {vj0 , vj1 , · · · , vjl } such that (vji−1 , vji ) ∈ pki , i = 1, 2, · · · , l, pki ∈ {p1 , p2 , · · · , pm } and ki ki+1 . If there is a joint path from one node vi to another node vj , then vj is said to be jointly reachable from vi . If a node vj is jointly reachable from every other node in the digraph, it is said to be jointly globally reachable. Note that if such an ordered set contains at least one weakly connected graph, then the ordered set must have a jointly globally reachable node. Given a nonnegative matrix A, if all its row sums are 1 (i.e. Al = l), A is said to be a (row) stochastic matrix [21]. For any Laplacian matrix L, −L can be written as the sum of a nonnegative M and −ηL I, where ηL is the maximum absolute value of diagonal entries of L. From matrix property (c) we know that e−LΔt = e−ηL ΔtI eM Δt = e−ηL Δt eM Δt is nonnegative matrix for any Δt > 0. It can also be verified that e−LΔt l = 0. Thus, matrix e−LΔt is a stochastic matrix for any Δt > 0. We are ready to present our main result on local struc-
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tural stabilization of formations for multiple agents in the directed graph case using a time update scheme. To do this, we need the following lemmas. Lemma 3 [19] Let {P1 , P2 , · · · } be a finite or infinite set of row stochastic matrices satisfying 0 λ(Pi ) β < 1, where λ(Pi ) is defined by λ(Pi ) = 1 − min min{Pi (i1 , j), Pi (i2 , j)}. i1 ,i2 j
Then, for each infinite sequence, Pk1 , Pk2 , · · · Pki , · · · , ki ∈ {1, 2, · · · }, there exists a vector c such that lim Pki Pki−1 · · · Pk1 = lcT .
Assume the switching signal p(t) ∈ Sdwell and there exists an infinite sequence of bounded, non-overlapping time intervals [tik , tik +lk ), k = 1, 2, · · · , starting at ti1 = t0 and satisfying tik+1 = tik +lk , with the property that each interval [tik , tik +lk ) is bounded and the ordered set of neighbors’ graphs across each such interval has a jointly globally reachable node. Then the local control strategies (4) guarantee that the system globally asymptotically converges to the desired formation. Proof Set y(t) = z(t) − r. Thus, the dynamic system can be represented as y(t) ˙ = −(Lp(t) ⊗ I2 )y(t).
i→∞
Lemma 4 For any given nonnegative n × n matrices Apk , the (i, j) element of A satisfies Apk (i, j) amin > 0, for all Apk (i, j) = 0 and i = j, and the ordered set {Gp1 , Gp2 , · · · , Gpm }, where graph Gpk associated with weighted matrix Aij , has a jointly globally reachable node. Then there exist ε(amin ) > 0 and a j such that matrix P = [pij ]n×n := eApm eApm−1 · · · eAp1 satisfies pij ε, ∀i. Proof Note that eApk = {I + Apk +
A2pk Anpk + ··· + + · · ·}. 2! n!
(17)
Assume the node j is the jointly globally reachable node of the ordered set {Gp1 , Gp2 , · · · , Gpm }. For any vi , i = j, there exists a joint path from vi to vj with the path length l satisfying l n. Assume the joint path is an ordered set of distinct nodes {vi , vj1 , · · · , vjl−1 , vj } such that (vjq−1 , vjq ) ∈ pkq , q = 1, 2, · · · , l, pkq ∈ {p1 , p2 , · · · , pm } and kq kq+1 , which means the (i, j) element of matrix Apk1 Apk2 · · · Apkl is greater than 0 and at least has a non-zero term Apk1 (i, j1 )Apk2 (j1 , j2 ) · · · Apkl (jl−1 , j) by noting that all the matrices are nonnegative. Assume kl1 := k1 = k2 = · · · = kl2 −1 > kl2 = · · · = kl3 −1 > · · · > klq = · · · = kl . Then, the matrix Apk1 Apk2 · · · Apkl can be expressed as
Alp2k−l1 Alp3k−l2 l l 1
2
l−l +1
· · · Apklqq
. From (17),
1 (l2 − l1 )!(l3 − l2 )! · · · (l − lq + 1)! × Apk1 (i, j1 )Apk2 (j1 , j2 ) · · · Apkl (jl−1 , j) al ak min min { min }. k=0,1,··· ,n l! k! Let Γ3 = {Gp , p ∈ P} denote the class of all possible graphs of order n, and the constant weighted factor aij ∈ [amin , amax ] and amin > 0. The set Γ3 is an infinite set. If there are actually only finitely many switches, the final time tm , we can artificially define tm+j = tm + jb, j = 1, 2, · · · , where b is a finite positive value. Let t1 , t2 , · · · be an infinite time sequence at which the topological graph or weighting factors switch, the topological graph Gp(t) and the weighted Laplacian matrix Lp(t) are time-invariant in [ti , ti+1 ). Now we present the result about the jointly connected graph with arbitrary positive weighted numbers. Theorem 4 Let t1 , t2 , · · · be an infinite time sequence at which the sensor graph or weighting factors switch, and let G(ti ) ∈ Γ3 be a switching topological graph at time ti . pij
(18)
The solution of systems (18) is y(t) = e−(Lti ⊗I2 )(t−ti ) e−(Lti−1 ⊗I2 )(ti −ti−1 ) · · · e−(Lt0 ⊗I2 )(t1 −t0 ) y(0) = [(e−Lti (t−ti ) e−Lti−1 (ti −ti−1 ) · · · e−Lt0 (t1 −t0 ) ) ⊗ I2 ]y(0).
(19)
The fact that p(t) ∈ Sdwell and [tik , tik +lk ) is bounded implies that there exists tmin such that ti+1 − ti tmin for ∀i. For [tik , tik +lk ), define ϕk as −L
(t
−t
)
ϕk = e tik +lk −1 ik +lk ik +lk −1 · · · −L (t −t ) −L (t −t ) e tik +1 ik +2 ik +1 e tik ik +1 ik . (20) −L
(t
−t
)
All matrices e tik +q ik +q+1 ik +q , q = 0, 1, · · · , lk − 1, are stochastic, so the matrix ϕk is a stochastic matrix. Let η = namax and Aq := (−Ltik +q−1 +ηI)(tik +q −tik +q−1 ), q = 1, 2, · · · , lk . (21) From the matrix property (c), the matrix ϕk can be expressed as ϕk = eAlk eAlk −1 · · · eA1 e−η(tik +lk −tik ) .
(22)
All matrices Aq , q = 1, 2, · · · , lk are nonnegative matrices and satisfy Aq (i, j) amin tmin for all Aq (i, j) = 0 and i = j. Because of the bound of intervals [tik , tik +lk ), there exists a T such that |tik − tik +lk | T for k = 1, 2, · · · . Using the result of Lemma 4, there exists a constant μ = e(amin tmin ) > 0 and j0 such that the matrix ϕk satisfies ηT ϕk (i, j0 ) μ, ∀i. Then we get λ(ϕk ) = 1 − min min(ϕk (i1 , j1 ), ϕk (i2 , j1 )) i1 ,i2 j1
1 − μ < 1, (23) which implies, by Lemma 3, limk→∞ ϕk ϕk−1 · · · ϕ1 = lc. Thus, we have lim y(tik ) = [(ϕk ϕk−1 · · · ϕ1 ) ⊗ I2 ]y(0) = l ⊗ a,
k→∞
where a = (cT ⊗ I2 )y(0). The result lim y(t) = l ⊗ a can t→∞
be shown using the fact that for all t ∈ [tik , tik+1 ) it satisfies that [19] max yi1 (t) − yi2 (t) max yj1 (tik ) − yj2 (tik ) . i1 ,i2
j1 ,j2
Thus we obtained that each yi (t) asymptotically converges to the same point a, where a depends only on y(0) and switching signals p(t). It follows that lim (zi (t) − zj (t) − dij ) t→∞
L. GAO et al. / J Control Theory Appl 2008 6 (4) 357–364
= lim {zi (t) − zj (t) − ri + rj } t→∞
= lim {yi (t) − yj (t)} = 0. t→∞
(24)
Definition 4 By the union of a collection of simple graphs, {Gp1 , Gp2 , · · · , Gpm }, each with vertex set V, we mean that a simple graph G with vertex set V and edge set equaling the union of the edge sets of all of the graphs in the collection. We say that such a collection is jointly weakly connected if the union of its members is a weakly connected graph. For undirected graphs the jointly weakly connected union must be jointly connected. The concept of jointly connected union can be found in [8], based on which some interesting results are established [8, 9]. It is obvious that the union of graphs is jointly weakly connected if the ordered set of graphs has a jointly globally reachable node. On the other hand, a collection can be jointly weakly connected but the the associated ordered set has no jointly globally reachable node. Let Γ denote the set of all weakly connected graph. Since the set Γ is a finite set, let n ¯ be the number of elements in Γ . For characterizing the relationship between jointly weakly connected and jointly globally reachable, we will prove the following claim: For non-overlapping time intervals [tik , tik +lk ), k = 1, 2, · · · , starting at ti1 = t0 and satisfying tik+1 = tik +lk , with the property that each interval [tik , tik +lk ) is bounded and the union of neighbors’ graphs across each such interval [tik , tik +lk ) is jointly weakly connected, the ordered set of neighbors’ graphs across each interval [ti(m−1)(n¯n)+1 , tim(n¯n)+1 ), m = 1, 2, 3, · · · , must be bounded and have at least a jointly globally reachable node. For the contiguous, nonempty, bounded, time-intervals n, the union of neighbors’ [tik , tik+1 ), k = 1, 2, · · · , n¯ graphs of time-intervals [tik , tik+1 ) is jointly weakly connected. Therefore, there must exist at least n such intervals [tikj , tikj +1 ), j = 1, 2, · · · , n, kj < kj+1 , such that all unions of neighbors’ graphs is the same, which is assumed as G ∈ Γ . Since G is weakly connected, there exists a globally reachable node j. For any node vj0 , j0 = j, there must exist a path of length l, l n, from vj0 to vj , which is an ordered set of distinct nodes {vj0 , vj1 , · · · , vjl−1 , vj }. In time-interval [ti1 , tin¯n+1 ), there must exist a joint path from vj0 to vj by noting that at least a neighbors’ graph of [tikn−i+1 , tikn−i+1 +1 ) contains the edge (vji−1 , vji ). Thus, the ordered set of neigh) has at least a bors’ graphs across interval [ti1 , tinn+1 ¯ jointly globally reachable node j. Similarly, we can prove that [ti(m−1)(n¯n)+1 , tim(n¯n)+1 ) has at least a jointly globally reachable node. It is obvious that [ti(m−1)(n¯n)+1 , tim(n¯n+1) ), m = 1, 2, · · · , is bounded. So the above claim is proved. By using the above claim, the following theorem can be easily obtained from Theorem 4. Theorem 5 Let t1 , t2 , · · · be an infinite time sequence at which the sensor graphs or weighting factors switch, and let G(ti ) ∈ Γ3 be a switching topological graph at time ti . Assume the switching signal p(t) ∈ Sdwell and there exists an infinite sequence of bounded, non-overlapping time intervals [tik , tik +lk ), k = 1, 2, · · · , starting at ti1 = t0 and satisfying tik+1 = tik +lk , with the property that each interval [tik , tik +lk ) is bounded and the union of neighbors’ graphs across each such interval [tik , tik +lk ) is jointly
363
weakly connected. Then the local control strategies (4) guarantee that the system globally asymptotically converges to the desired formation. Remark 3 For the continuous model used in [8], the switching times of the interaction graph is assumed to be separated by τd (sample time) time units, where τd is a constant dwell time. The continuous update scheme used in [9] allows the switching times to be within an infinite set of positive numbers generated by any finite set of positive numbers. However, we only assume the dwell times is no less than some positive constant in this paper, which guarantees that chattering does not occur, and is better suited to simulating the random switching of interaction graphs. Unlike the update schemes in [9], we do not constrain the weighting factors in a finite set of positive numbers as long as they are positive and bounded. As a result, the continuous update scheme used in [8, 9] can be considered as a special case of Theorem 5. Remark 4 Consider a piecewise continuous positive weighing factor aij (t) ∈ [amin , amax ] and amin > 0, and it has only discontinuities of the first type (i.e. jumps). The control strategies (4) have a general form as follows aij (t)(zi (t) − zj (t) − dij ). (25) ui (t) = − j∈Ni
Let t1 , t2 , · · · be an infinite time sequence at which the topology graph switch or the discontinuous point of weighting factors, and all other conditions of Theorem 5 are satisfied. Using similar proof of Theorem 5, it is not too difficult to prove that the control strategies (25) guarantee that the system globally asymptotically converges to the desired formation. [19] shows that consensus of information (the heading of each agent in their context) can be achieved if there exists a positive T such that sensor graphs at least one within each time interval of length T are strongly connected graphs. This paper demonstrates that the same result can be achieved as long as the union of the graphs is weakly connected, which is a milder requirement than being strongly connected and is therefore more suitable for practical applications. We also allow the relative weighting factors to be positive and time-varying, which provides more flexibility to take into account the relative confidence and relative reliability for information flow between different agents [9]. Thus, the convergence conditions and local control strategies given in [19] are also shown to be a special case of our local control strategies with weighting factors and weakly jointly connected graphs.
Fig. 2 Achieving a formation for five agents.
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Fig. 2 shows a simulation for five agents. The initial locations of the five agents are randomly produced; we set all dij = 0 for simplicity. The topological graph and weighting factor satisfy the condition given in Theorem 5. From Fig. 2, it is shown that the local control strategies (4) guarantee that the system globally asymptotically converges to a point.
5
Conclusions
The problem that a group of mobile agents converges to a common location is an instance of agreement problems. Besides being of interest in its own right, if convergence to a point is achievable, then other formations are achievable by a simple modification of the control law. For the fixed topological graph, we proved that a group of agents converges to a desired formation if and only if the sensor graph is weakly connected. So this paper extended the local consensus control strategies to local formation control strategies. We have revealed the close connection between graph theory and formations control, and introduced a local control law under dynamically changing interaction topology with weighting factors for a group of mobile agents which guarantees that the system globally asymptotically converges to the desired formation. For simplicity, we assume each mobile agent is a dynamical system moving on the plane. All results of this paper can be easily generalized to three-dimensional or even higher dimensional cases. References [1] R. Olfati-Saber, R. M. Murray. Distibuted cooperative control of multiple vehicle formations using structural potential functions[C] //Proceedings of the 15th IFAC World Congress, Barcelona, Spain, June 2002. [2] A. Fax, R. M. Murray. Information flow and cooperative control of vehicle formations[J]. IEEE Transactions on Automatic Control, 2004, 49(9): 1453 – 1464. [3] C. W. Reynolds. Flocks, herds, and schools: a distributed behavioral model, Computer Graphics[C]// ACM SIGGRAPH ’87 Conference Proceedings, 1997, 21(4): 25 – 34. [4] D. M. Stipanovic, G. Inalhan, R. Teo, C. J. Tomlin. Decentralized overlapping control of a formation of unmanned aerial vehicles[J]. Automatica, 2004, 40(8): 1285 – 1296 [5] Y. Liu, K. M. Passino. Stable social foraging swarms in a noisy environment[J]. IEEE Transactions on Automatic Control, 2004, 49(1): 30 – 44. [6] A. Okubo. Dynamical aspects of animal grouping: swarms, schools, flocks and herds[J]. Advances in Biophysics, 1986, 22(1): 1 – 94. [7] T. Vicsek, A. Czirok, E. B. Jacob, I. Cohen, O. Schochet. Novel type of phase transitions in a system of self-driven particles[J]. Physical Review Letters, 1995, 75(2): 1226 – 1229. [8] A. Jadbabaie, J. Lin, A. S. Morse. Coordination of groups of mobile agents using nearest neighbor rules[J]. IEEE Transactions on Automatic Control, 2003, 48(6): 988 – 1001. [9] W. Ren, R. W. Beard. Consensus seeking in multi-agent systems using dynamically changing interaction topologies[J]. IEEE Transactions on Automatic Control, 2005, 50(5): 655 – 661. [10] R. Olfati-Saber, R. M. Murray. Consensus Problems in networks of agents with switching topology and time-delays[J]. IEEE Transactions on Automatic Control, 2004, 49(9): 1520 – 1533. [11] A. V. Savkin. Coordinated collective motion of groups of autonomous mobile robot: analysis of Vicsek’s model[J]. IEEE Transactions on Automatic Control, 49(6): 981 – 983. [12] L. Gao, D. Cheng. Comment on Coordination of groups of mobile autonomous agents using nearest neighbor rules[J]. IEEE
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[email protected]. Daizhan CHENG graduated from Tsinghua University in 1970, received M.S. from Graduate School, Chinese Academy of Sciences in 1981, Ph.D. from Washington University, St. Louis, in 1985. Since 1990, he is a professor with Institute of Systems Science, AMSS, CAS. His research interests include nonlinear control systems, switched systems, Hamiltonian systems and numerical realization for control design. He is currently Chairman of Technical Committee on Control Theory, Chinese Association of Automation, Chairman of IEEE CSS Beijing Chapter, and IEEE Fellow. Yiguang HONG received his B.S. and M.S. from Department of Mechanics, Peking University, and his Ph.D. from Chinese Academy of Sciences in China. He is currently a professor in Institute of Systems Science, Chinese Academy of Sciences. He is a recipient of “Guan Zhaozhi” Award in Chinese Control Conference, Young Author Prize of the International Federation of Automatic Control (IFAC) World Congress, the US National Research Council Research Associateship Award, Young Scientist Prize of Chinese Academy of Science, Outstanding Youth Award of NSF of China, and Youth Scientific Award of China. He is also a vice chairman of IEEE CSS Beijing Chapter. His research interests include nonlinear dynamics and control, robot control, multi-agent systems, and reliability and performance analysis of computer and communication systems.
