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Adaptive flocking control of nonlinear multi-agent systems with directed switching topologies and saturation constraints
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Journal of the Franklin Institute 350 (2013) 1545–1561 www.elsevier.com/locate/jfranklin
Adaptive flocking control of nonlinear multi-agent systems with directed switching topologies and saturation constraints Hajar Atrianfar, Mohammad Haerin Advanced Control Systems Lab, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran Received 12 September 2012; received in revised form 27 February 2013; accepted 6 March 2013 Available online 22 March 2013
Abstract In this paper, we propose and analyze flocking algorithms in a network of second-order agents with bounded control inputs and nonlinear intrinsic dynamics. We consider a general switching network topology, for velocity information exchange, rather than undirected or fixed directed network topology with a directed spanning tree. The proposed adaptive controller architecture applies a leader-following strategy in which the pinning scheme is defined based on the interaction topology. Finally, some examples are presented to illustrate the theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Flocking is a coordinated motion of a large group of agents with local interactions. This issue has a historical background in physics, biology and computer science [1,2] and has attracted significant attention in the control and system community because of its broad applications in formation control of mobile robots, mobile sensor networks, and cooperative coordination of unmanned aerial vehicles [3,4]. The first studies of flocking in swarm robotics were inspired by Reynolds [5], which obtained a realistic computer animation of a flock of birds through three simple behaviors: separation, cohesion and alignment.
n
Corresponding author. Fax: þ98 21 66023261. E-mail address: [email protected] (M. Haeri).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.03.002
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Motivated by the principles of Reynolds model [5], variety of the algorithms has been proposed for flocking [6–8]. Olfati-Saber [6] provided a theoretical framework for analysis of flocking algorithms. In [7], a non-smooth framework has been developed to analyze the stabilization of flocking algorithms in networks with discontinuous control laws. Leader-following flocking is a special case of flocking where there is a virtual leader who specifies the objective for the whole group. In [8], a connectivity-preserving flocking algorithm with bounded potential function is proposed to solve the flocking problem of multi-agent systems with a dynamical virtual leader. In practice, it is also possible that only a portion of agents have access to the states of the virtual leader. This strategy, which is known as pinning control, may reduce the number of controlled agents [9–12]. In many flocking algorithms designed to address the tracking problem, including those mentioned above, it has been commonly assumed that the virtual leader is single. A new algorithm was developed in [13] to study the problem of controlling a group of mobile autonomous agents to track multiple virtual leaders with varying velocities. Since nonlinear intrinsic dynamics is inevitable in real systems, it is necessary and beneficial to study second-order flocking algorithms in a network of nonlinear dynamical systems. To apply adaptive strategies is common in most existing works on the synchronization problems, e.g. [12,14–16]. The other common applications of adaptive techniques are in the networks that vary in time according to different environmental conditions [17,18]. Most existing references that consider nonlinear dynamics intrinsically [12,14,19,20] only investigated second-order consensus of multi-agent systems and the flocking problem is not addressed. Also in these references, the network topology is assumed to be either undirected or fixed weakly connected, which poses an obvious limitation. Recently, an adaptive Leader-following flocking algorithm was proposed for multi-agent systems in which all agents and the virtual leader share the same intrinsic nonlinear dynamics [21]. Their idea only works for connected initial networks and is based on the connectivity preserving of the network during the system evolution. Another problem which should be addressed for coordination strategies to be successful is the actuator saturation. This issue is inevitable in real control systems due to existing physical constraints. All real-world applications of feedback control involve actuators with amplitude and rate limitations. In particular, any physical electromechanical device can provide only a limited force, torque, flow capacity or linear/angular rate. In [22], consensus algorithms with bounded control input for double-integrator dynamics under undirected interaction graph were proposed. The authors in [23] extended the results in [22] and proposed consensus algorithms for double-integrator dynamics without velocity measurements and in the presence of input saturation constraints. In [24], the result in [22] is extended for both single-integrator and double-integrator in the cases of undirected and directed interaction graphs. In [25], it is proved that, when there are saturation constraints, a general consensus protocol widely used in the literatures [26,27] for single-integrator dynamics remains valid when the topology of the connections is fixed and contains a spanning tree. More recently the problem has been impliedly and partly solved in [28] for undirected topology. Zheng et al. [29] discussed the consensus problem of heterogeneous multi-agent system with the saturated consensus protocol. The synchronization problems of networked passive systems with strongly connected communication topologies and actuator saturation are discussed in [30]. As far as we know, there has not been a formal article to systematically address the saturated flocking protocols in the presence of adaptive strategies. On the other hand, the connectivity-preserving second-order consensus algorithm in [12, 14–16,21] was only concerned with agents governed by nonlinear intrinsic dynamics. In this paper, however, we consider an adaptive saturated flocking protocol for a class of multi-agent
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systems where the intrinsic dynamics of the agents are nonlinear and the control input should be bounded. In many situations, velocity information of neighboring agents is not available in all directions. In fact, this problem is faced when agents are not equipped with all-directional velocity sensors, to save cost, space and weight, or velocity is not precisely measured. Moreover, sensed information flow which plays a central role in flocking is typically not bidirectional. Also, unidirectional communication is of considerable importance in practical applications and can easily be incorporated for example, via broadcasting. Therefore, we allow for non-bidirectional communication patterns too. Finally, another aspect of our work is time-dependent communication patterns which become important when we take into consideration some issues such as link failure and link creation and nearest neighbor coupling. The contributions of this paper are threefold. First, we propose an adaptive leader-following flocking algorithm, in which the time-varying pinning scheme depends on the pattern of velocity information exchange among agents. Second, we assume a general switching network topology rather than undirected connected or fixed directed network topology with a directed spanning tree, thereby extending some of the results reported in [12,14–18,21]. Finally, the third contribution concerns with considering both actuator saturation and nonlinear intrinsic dynamics. Other studies that consider this issue such as [22–27] typically do not work with adaptive laws. An outline of this paper is shown as follows. In Section 2, we present some concepts in graph theory and introduce the model of communication for both position and velocity information. The adaptive leader-following flocking problem of the multi-agent system is discussed with the saturated protocol and switching velocity communication graph in Section 3. Furthermore, an extension to the obtained results is given for the case which the condition of all agents' awareness of the virtual leader position is relaxed. Simulation examples are presented to illustrate the analytical results in Section 4. Conclusions are drawn in Section 5. 2. Graph preliminaries To model the communication topology among agents in coordination problems, graph theory is helpful. A time-varying weighted directed graph GðtÞ ¼ ðV,ℰðtÞ,AðtÞÞ consists of a node set V ¼ 1,2,…,N, an edge set ℰðtÞ⊂V V, and a weighted adjacency matrix A(t). An edge (i, j) in a weighted directed graph denotes that agent i can get information from agent j, but not necessarily vice versa. In contrast, the pairs of nodes in a weighted undirected graph are unordered, where an edge (i, j) denotes that agents i and j can exchange information mutually with the same weights. An edge ði,jÞϵℰðtÞ is said to be bidirectional whenever ðj,iÞϵℰðtÞ too and corresponding weights are equal. Otherwise, it is an unidirectional edge. The set of all neighbors of node i is denoted by N i ðtÞ ¼ j : ði,jÞϵℰðtÞ which is the subset of V. In many situations, it is difficult for some agents to measure the relative velocity difference between themselves and neighboring agents in all directions and only relative position data is available. Thus we consider the interaction graph for velocity information exchange as a directed weighted graph. As an example, these topologies are feasible in networks with agents that could only sense the velocity of agents in their conic sensing neighborhoods. Since the sensing mechanism of relative position between agents is simpler, we assume that all agents are equipped with position spherical sensors with identical radii. This causes the interaction graph for position information exchange to be an undirected one. A (directed) path in a graph G(t) is a sequence i1 , …,ik of vertices such that ðij ,ijþ1 ÞϵℰðtÞ for j¼ 1,..,k−1. If there is a path of edges between any two nodes of an undirected graph G(t), then G(t) is connected,
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otherwise it is disconnected. We say that agent i is connected to agent j if there is a path from agent i to agent j. The weighted adjacency matrix of G(t) is denoted by AðtÞ ¼ ½aij ðtÞϵRNN , where aii(t) ¼ 0 and aij(t)40 if there is an edge between agent i and agent j. Its degree matrix ΔAðtÞ ¼ diagðΔ1 ðtÞ,Δ2 ðtÞ,…,ΔN ðtÞÞϵRNN is a diagonal matrix, where diagonal elements Δi(t) ¼ ∑j≠iaij(t). Then the Laplacian of a graph is defined as L(t) ¼ ΔA(t)−A(t) which is symmetric only for undirected ones (see e.g. [31]). In the sequel, the position and velocity interaction graphs are denoted by Gp ðtÞ ¼ ðV,ℰp ðtÞ,Ap ðtÞÞ and Gv ðtÞ ¼ ðV,ℰv ðtÞ,Av ðtÞÞ, respectively. Also, the neighboring sets for agent i in position and velocity interaction graphs are defined as N pi ðtÞ and N vi ðtÞ. Also, we mean by A⊕B, disjoint summation of two sets A and B. 3. Problem formulation and mathematical preliminaries Consider a network consisting of N41 identical agents with nonlinear dynamics described as ( q_ i ¼ pi , ð1Þ p_ i ¼ f ðpi Þ þ ui , where qi , pi ϵRm are the position and velocity of the ith node, f : Rm -Rm is a continuously differentiable vector function and ui is the control input of agent i. Our objective is to design control inputs ui so that all agents asymptotically move with the same velocity as the virtual leader. In this transition, it is desired that the velocities of all agents asymptotically approach each other. The other feature which distinguishes the proposed flocking protocol from the existing second-order consensus algorithms is that it enables the agents in the group to reach a conformation in which the inter-agent distances not necessarily come to zero. Also the maximum number of collisions in the network can be predefined. This geometry resembles the real-world flocks. In this regard, the initial condition of the agents is defined such that collision-free motion can be guaranteed. Also the distance between each agent and the virtual leader should not exceed a maximum value which can be calculated based on the initial conditions. Since the network consists of the agents with unknown nonlinear intrinsic dynamics and actuator saturation, a novel distributed adaptive flocking protocol with bounded control input is proposed. ui ¼ − ∑ apij ðt Þtanh ‖qij ‖s −d α jϵN pi ðtÞ
qij tanhðmij Þ av ðt Þtanhðpj −pi Þ þ ∑ pffiffiffiffiffiffiffiffiffiffi 1 þ ε‖qij ‖s jϵN vi ðtÞ N−1 cosh2 ðmij Þ ij
tanhðc2i Þ tanhð pi −pr Þ −c1 tanhðqi −qr Þ−hi pffiffiffiffiffiffiffiffiffiffi N−1 cosh2 ðc2i Þ
ð2Þ
where c140, qij ¼ qi−qj and (qr, pr) is the state vector of the virtual leader whose acceleration is the same as agents' nonlinear intrinsic dynamics. avij ðtÞ and apij ðtÞ are the adjacency elements of velocity and position information graph, respectively. tanh(.) is defined component-wise. ‖qij ‖s is a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi differentiable map everywhere even at qij ¼ 0 and is defined as ‖qij ‖s ¼ 1=εð 1 þ ε‖qij ‖2 −1Þ with a fixed parameter ε40. Introducing this type of norm is motivated by [6] and causes the smoothness
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of collective potential field. Also, we have dα ¼ ‖d‖s , d40. ( _ ij ¼ avij ðtÞðpi −pr ÞT tanhðpi −pr Þ if ði,jÞϵℰv b ðtÞ m _ ij ¼ 0 m if not c_ 2i ¼ hi ni ðpi −pr ÞT tanhðpi −pr ÞÞ
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ð3Þ ð4Þ
mij and c2i are adaptive parameters representing the velocity coupling strengths between agents and the weights of velocity navigational feedbacks, respectively. (3) implies that the adaptive protocol only updates the coupling weights associated with the bidirectional edges of the velocity information graph at each time instant t and the coupling weights associated with the unidirectional edges remain constant. Also, c2i is updated only for informed agents at each time instant t. The edge set of the velocity information graph consists of two sets of bidirectional edges (ℰv b ðtÞ) and unidirectional edges (ℰv u ðtÞ), i.e. ℰv ðtÞ ¼ ℰv b ðtÞ⊕ℰv u ðtÞ, which both are timevarying throughout the network evolution. When agent i is informed about the virtual leader's velocity at time instant t, then hi(t)¼ 1 and it is called a pinned agent or an informed agent; otherwise, hi(t)¼ 0. Here, the pinning strategy is applied on a fraction of agents at each time instant t such that each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ. Therefore, the minimum number of informed agents is time-varying and equals to the connected components of this undirected graph. As in [14,15], we say that a function f : Rm -Rm is QUAD if and only if, for x any yϵRm , we have: ðx − yÞT ½f ðx,tÞ− f ðy,tÞ− ðx − yÞT Δðx − yÞ≤ −ωðx − yÞT ðx − yÞ
ð5Þ
where Δ is an arbitrary diagonal matrix of order m and ω is a positive scalar. Note that the above inequality is a Lipschitz-type condition, which is satisfied by many chaotic systems. A large class of nonlinear systems is QUAD with Δ ¼ 0 (see [15,32]). For example, as stated in [33], adopting the algebraic manipulation makes it possible to verify that the Chua's circuit is QUAD with Δ ¼ 0. Respectively, up to now, adaptive strategies for the synchronization of the first-order nonlinear multi-agent systems which are QUAD with Δ ¼ 0 have been extensively studied ([15,17,18]), but there has not been a formal article to systematically address the flocking problem of nonlinear multi-agent systems which are QUAD with Δ ¼ 0, in presence of input saturation. Therefore, the following assumption about the nonlinear part f(.) is proposed and used throughout the paper. Assumption 1. A nonlinear function f : Rm -Rm is QUAD with Δ ¼ 0 and positive scalar ω≥2=27. 4. Main results Before stating the main results, the following lemmas are needed. Lemma 1. [34]. For any x and yϵRn and any symmetric positive definite matrix R, we have: 2xT y≤xT R−1 x þ yT Ry
ð6Þ
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Lemma 2. Suppose that Gv ðtÞ ¼ ðV,ℰv ðtÞ,Av ðtÞÞ, vi ϵRm , M ¼ ½mij ðtÞϵRNN and Av ¼ ½avij ðtÞϵ RNN . If M and Av are symmetric matrices for all t, then 1 2
tanhðmij Þ v tanhðm Þ a ðt Þðvi −vj ÞT tanhðvi −vj Þ ¼ ∑ cosh2 ðmij Þ avij ðt Þvi T ∑ tanhðvi −vj Þ cosh2 ðmij Þ ij ij i jϵN vi ði,jÞϵℰv ðtÞ
∑
ð7Þ
Proof. The proof is similar to that of [22] and is omitted.& ~ ~ ¼ ðV~ , ℰðtÞ,CðtÞ Lemma 3. Let GðtÞ ¼ ½cij ðtÞÞ is a time-varying undirected graph, where ~ ~ V ¼ 1,2,…,M is the node set, ℰðtÞis the edge set and C(t) is the associated adjacency matrix. ~ Also, di ðtÞ40, ∀i, ∀t. Consider the graph GðtÞ, which consists of graph GðtÞ, node 0 (leader) and bidirectional edges between the leader (node 0) and informed agents (of the leader's state). Then, it follows that: AðvÞ ¼
M 1 M M ∑ ∑ cij ðvi −vj ÞT tanhðvi −vj ÞÞ þ ∑ hi di vi T tanhðvi Þ 2i¼1j¼1 i¼1
ð8Þ
is positive definite if and only if the graph GðtÞ is connected at all t. ~ consists of mðtÞ ~ connected subgraphs for Proof. Without loss of generality, we assume that GðtÞ ~ ~ mðtÞ 1 ~ 1 ðtÞ,…, ℰ ~ mðtÞ ~ ~ any t. These subgraphs are labeled G ðtÞ,…, G ðtÞ with edge sets ℰ ðtÞ, respectively. Let π ¼ fijhi ¼ 1g: Since tan h(.) has the same sign as its variable component-wise, we know T that∑M 0, ∀iϵπ g. Furthermore, i ¼ 1 hi d i vi tanhðvi Þ≥0 and takes zero value only at fvjvi ¼ n T 1 M M ~ k ðtÞ, kϵ vjvi ¼ vj ,∀ði,jÞϵℰ 2 ∑i ¼ 1 ∑j ¼ 1 cij ðvi −vj Þ tanhðvi −vj Þ≥0 and takes zero value only at o ~ 1,…, mðtÞ : Therefore, A(v) is positive definite if and only if each connected subgraph contains at least one informed agent which means that the graph G should be connected.& ~ node 0 and edges Lemma 4. [35]. Consider the undirected graph G, which consists of graph G, between the leader (node 0) and informed agents. Then matrix LþZ is positive definite if and only ~ and Z ¼ diagðz1 ,z2 ,…, zN Þ if the graph G is connected, where L is the Laplacian of the graph G and zi40 if there is an edge between the leader and node i and zi ¼ 0, otherwise. Now, we can prove the following theorem for the distributed control of the flocking behavior of a group of N agents. Theorem 1. Consider a group of N mobile agents modeled by (1) with the nonlinear intrinsic dynamics, f(pi), satisfying Assumption 1 at each time instant t≥0. Suppose that the position information graph is a fixed undirected graph. Then under the feedback control strategy (2)–(4) with any arbitrary directed velocity information topology, the followings hold: (i) The control input of agent i is saturated by 1 2 2 ‖ui ‖∞ ≤ pffiffiffi ∑jϵN pi apij þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑jϵN vi ðtÞ avij ðt Þ þ c1 þ hi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 3 3ðN−1Þ 3 3ðN−1Þ
ð9Þ
(ii) The velocity of all agents asymptotically converges to the velocity of the virtual leader.
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(iii) The final position of all agents constitutes a configuration which is a local minimum of V1 ¼
N 1 N ‖x ‖ ∑ ∑ apij ∫dα ij σ tanhðu−d α Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ: 2 i¼1 jϵN p i¼1 i
(iv) If the initial energy, U0 (defined in the lines of proof) of the group is less than ðk þ 1Þ max apij ∫0dα tanhðuÞdu, then at most k distinct inter-agent collisions could occur for i;jϵf1;…;N g all t>0. (v) If the initial energy, U0, is less than c1 logðcoshðlÞÞ, then the relative distance between each agent and the virtual leader is less than l for all t>0. pffiffiffi Proof. Part i can be proved directly by the fact that 0≤qij =ð1 þ ε‖qij ‖σ Þ≤ð1= εÞ; ∀qij and tanhðaÞ 2 ≤ pffiffiffi cosh2 ðaÞ 3 3
tanhðaÞ≤1, ∀a:
and
Considering xi ¼ qi−qr and vi ¼ pi−pr as the position and velocity of agent i in the moving frame centered at the virtual leader, apij ðtÞ ¼ apij and N pi ðtÞ ¼ N pi for ∀t, the system equation would be as follows: 8 x_ i ¼ vi > > > > v_ i ¼ f ðvi þ pr Þ−f ðpr Þ−∑jϵN p apij tanhð‖xij ‖s −dα Þ xij > 1þε‖xij ‖s > i > > > tanhðmij Þ > v ffi > þ∑jϵN vi ðtÞ pffiffiffiffiffiffi a ðt Þtanhðvj −vi Þ−c1 tanhðxi Þ > N−1cosh2 ðmij Þ ij > > < tanhðc ffi 2i Þ tanhðvi Þ −hi pffiffiffiffiffiffi ð10Þ N−1cosh2 ðc2i Þ > > > > T b > _ ij ¼ avij ðtÞðvi −vj Þ tanhðvi −vj Þ if ði,jÞϵℰv ðtÞ m > > > > > _ ij ¼ 0 m if not > > > > : c_ 2i ¼ hi ni vi T tanhðvi Þ where xij ¼ xi−xj. Consider the energy function candidate as: Z ‖xij ‖s N 1 N p Uðx,v, m, c2 Þ ¼ tanhðu−dα Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ ∑ ∑ aij 2 i ¼ 1 jϵN p i¼1 dα i
þ
N ðtanhðc Þ−wÞ2 1 N 1 N N ðtanhðmij Þ−wÞ2 2i pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi þ ∑ ∑ ‖vi ‖2 þ ∑ ∑ 2i¼1 2i¼1j¼1 2 N−1 2 N−1 ni i¼1
ð11Þ
where log(.) and cosh(.) are component-wise functions. x ¼ ðx1 T ,x2 T ,…,xTN ÞT and v ¼ ðv1 T ,v2 T ,…,vTN ÞT are the collective position and velocity vectors of the new system. Also, c2 ¼ (c21,c22,…,c2N)T and m ¼ (m11,…,m1N,…,mN1,…,mNN)T are collective adaptive parameters of the system. Moreover, w is a bounded positive constant to be determined later.
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The time derivative of Uðx,v, m, c2 Þ along the trajectories of (10) gives: tanhðmij Þ _ avij ðt Þtanhðvj −vi Þ Uðx,v, m, c2 Þ ¼ ∑ vi T f ðvi þ pr Þ−f ðpr Þ þ ∑ pffiffiffiffiffiffiffiffiffiffi 2 v i¼1 jϵN i ðtÞ N−1cosh ðmij Þ tanhðc2i Þ tanhðv −hi pffiffiffiffiffiffiffiffiffiffi Þ i N−1cosh2 ðc2i Þ 1 1 þ pffiffiffiffiffiffiffiffiffiffi ∑ ðtanhðmij Þ−wÞ av ðt Þðvi −vj ÞT tanhðvi −vj Þ cosh2 ðmij Þ ij 2 N−1 ði,jÞϵℰv b ðtÞ N
N 1 1 þ pffiffiffiffiffiffiffiffiffiffi ∑ hi ðtanhðc2i Þ−wÞ vT ðtÞtanhðvi Þ: cosh2 ðc2i Þ i N−1 i ¼ 1
ð12Þ
Thus, Assumption 1, Lemmas 1 and 2 imply that: " # N 1 tanh2 ðmij Þ T T T v2 _ Uðx,v, m, c2 Þ≤− ∑ vi ωvi þ ∑ vi vi þ aij ðtÞtanhðvj −vi Þ tanhðvj −vi Þ 4 i¼1 ði,jÞϵℰv u ðtÞ 2 ðN−1Þcosh ðmij Þ −
w 1 ∑ pffiffiffiffiffiffiffiffiffiffi av ðt Þðvi −vj ÞT tanhðvi −vj Þ 2 ði,jÞϵℰv b ðtÞ N−1cosh2 ðmij Þ ij
1 vTi tanhðvi Þ −w∑ hi pffiffiffiffiffiffiffiffiffiffi N−1cosh2 ðc2i Þ i
ð13Þ
As said before, each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ for all t≥0. It then follows from Lemma 3 that 1 1 1 b ∑ av ðt Þ ðvi −vj ÞT tanhðvi −vj Þ þ ∑ hi vTi tanhðvi Þ 2 2 ði,jÞϵℰv ðtÞ cosh2 ðmij Þ ij cosh ðc Þ i 2i
ð14Þ
is positive definite. Therefore, by appropriate choice of constant w, we can get the right side of inequality (13) to be negative definite. pffiffiffi By using jtanhðaÞ=cosh2 ðaÞj≤2=ð3 3Þ and the fact that tanhðjajÞ≤jaj, ∀a, one can show: N w 2 T 1 v2
_ Uðx,v, m, c2 ÞÞ≤ ∑ −ω þ aij ðt Þ tanhðvj −vi ÞT tanhðvj −vi Þ vi vi þ ∑ u 27 2 2 i¼1 ði,jÞϵℰv ðtÞ 1 ∑ pffiffiffiffiffiffiffiffiffiffi avij ðt Þðvi −vj ÞT tanhðvi −vj Þ 2 ði,jÞϵℰv b ðtÞ N−1cosh ðmij Þ 1 −w∑ hi pffiffiffiffiffiffiffiffiffiffi tanhðvTi Þ tanhðvi Þ: ð15Þ N−1cosh2 ðc2i Þ i Noting that ω≥
2 27
and tanhðx þ yÞ ¼
tanhðxÞ þ tanhðyÞ , 1 þ tanhðxÞtanhðyÞ
one can see that: _ Uðx,v, m, c2 Þ≤tanhðvT ÞLs tanhðvÞ−wtanhðvT ÞLM tanhðvÞ−wtanhðvT ÞHtanhðvÞ,
ð16Þ
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where Ls ¼ ΔS−S and LM ¼ ΔM−M, S ¼ ½sij ðtÞ, M ¼ ½mij ðtÞ, 8 !2 > <1 v 2 2 v 1 ða ðtÞ þ aji ðtÞ Þ min ð1−tanhðvi Þtanhðvj ÞÞ ði,jÞϵΨ sij ðtÞ ¼ 2 ij ði,jÞϵℰuv > : 0 O:W: n o Ψ ¼ ði,jÞði,jÞϵℰv u ðt Þ or ðj,iÞϵℰv u ðt Þ
mij ðtÞ ¼
8 > < 1 pffiffiffiffiffiffiffi 1 > :
2
1553
ð17Þ
ð18Þ
!2
av ð t Þ N−1cosh2 ðmij Þ ij
0
1 max ð1−tanhðvi Þtanhðvj ÞÞ
ði,jÞϵℰv b ðtÞ
ði,jÞϵℰv b ðtÞ
ð19Þ
O:W:
1 1 1 p ffiffiffiffiffiffiffiffiffi ffi H¼ diag h1 ðtÞ ,…,hN ðtÞ : cosh2 ðc21 Þ cosh2 ðc2N Þ N−1
ð20Þ
Definition of mij ðtÞ and the fact that −1≤tanhðvi Þ≤1, ∀i, prove that the associated graph with LM has the same edge set as ℰv b ðtÞ. Then, by Lemma 4 and the fact that each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ, it is easy to see that λmin ðLM þ HÞ40. On the other hand, the edge weights avij ðtÞ are chosen from a finite interval [0, amax], where amax is the upper bound on the adjacency elements of velocity information graph. Therefore, there must be a negative scalar μ such that max λmax ðLs Þ≤μ: Then, by choosing a t bounded scalar w≥max λmax ðLS Þ=min λmin ðLM þ HÞ, one can easily show that: t t _ Uðx,v, m, c2 Þ≤0: ð21Þ Therefore, U is a non-increasing function for all t≥0 and the set Ω0 ¼ ðx,vÞ, Uðx,vÞ≤U 0 is an invariant set, where U0 ¼ U(0)o∞. Boundedness of U0 proves the boundedness of xi and vi. This in turn implies that Ω0 is a compact invariant set. Therefore, LaSalle's principle reveals that all solutions of the system starting in Ω0 converge to a largest invariant set in _ ¼0 . Π ¼ ðx,vÞϵΩ0 jU Then based on (5), U_ ¼ 0 implies that vi ¼ 0 for all i. Consequently, it follows that pi ¼ pr for all i. Hence the velocity of all agents gradually converges to the velocity of virtual leader. It thus follows from (9) that: xij ∑ apij tanhð‖xij ‖s −dα Þ þ c1 tanhðxi Þ ¼ 0 ð22Þ p 1 þ ε‖xij ‖s jϵN i
Solutions to above relation are equivalent to the local minima of the total potential function Z ‖xij ‖s N 1 N p V 1 ðxÞ ¼ ∑ ∑ aij tanhðu−d α Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ ð23Þ 2 i ¼ 1 jϵN p i¼1 dα i
The proof of part (ii) and (iii) is completed. Assume that there is more than k (at least kþ1) distinct collisions between agents at time t40. Then, (4) reveals that Z dα p UðtÞ≥ðk þ 1Þ max aij tanhðuÞdu: i,jϵf1,…,N g
0
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But, U is a non-increasing function. Thus, Z dα Uð0Þ≥UðtÞ≥ðk þ 1Þ max apij tanhðuÞdu, i,jϵf1,…,N g
ð24Þ
0
which is in contradiction with the assumption of theorem that Z dα p Uð0Þoðk þ 1Þ max aij tanhðuÞdu: i,jϵf1,…,N g
0
Therefore, no more than k distinct inter-agent collisions could occur in the network. This proves part (iv). Assume that the distance between agent i and the virtual leader is equal or more than l at time t40. Therefore, we have UðtÞ≥c1 logðcoshðlÞÞ which implies that Uð0Þ≥c1 logðcoshðlÞÞ which is in contradiction with the assumption of theorem that Uð0Þoc1 logðcoshðlÞÞ. Therefore, relative distance between each agent and the virtual leader is less than l. This proves Part (v).& Remark 1. In the proposed algorithm, adaptive strategy is used to guarantee the stability of flocking behavior in the presence of directed switching velocity interactions. Incorporating adaptive coupling gains in networks which vary in time according to different environmental conditions is common in the literature. This approach is introduced for undirected switching topologies in [17,18]. Assumption 2. The edge set of the velocity information graph consists of two sets of fixed bidirectional edges (ℰv b ) and time-varying unidirectional edges (ℰv u ðtÞ), i.e. ℰv ¼ ℰv b ⊕ℰv u ðtÞ. A typical example of above assumption is a network of agents in which some agents are equipped with non-identical conical velocity sensors and a few agents are equipped with spherical velocity sensors with very large radii. Another common example is a network in which some agents are equipped with non-identical conical velocity sensors and also a fixed wireless communication network is dedicated to gather and communicate mutually velocity data between agents. Also, fixed directed and undirected graphs are of those topologies which satisfy above assumption. Since, the pinning scheme is dependent on the edge set ℰv b , therefore, the time-independency of this set causes the pinning scheme not change as system evolves. Consequently, the parameters hi can be chosen fixed during the time. Based on this fact, we will show in the following that how the flocking behavior is achieved for a different class of networks in which the condition of all agents' awareness of the virtual leader position is relaxed. For this purpose, the new control input is defined as below: qij tanhðmij Þ ui ¼ −∑jϵN pi ðtÞ apij ðt Þtanh ‖qij ‖s −d α þ ∑jϵN vi ðtÞ pffiffiffiffiffiffiffiffiffiffi avij ðt Þtanhðpj −pi Þ 1 þ ε‖qij ‖s N−1cosh2 ðmij Þ tanhðc2i Þ tanhð p −hi c1 tanhðqi −qr Þ þ pffiffiffiffiffiffiffiffiffiffi −p Þ ð25Þ i r N−1cosh2 ðc2i Þ ( _ ij ¼ avij ðtÞðvi −vj ÞT tanhðvi −vj Þ if ði,jÞϵℰv b m ð26Þ _ ij ¼ 0 m if not c_ 2i ðtÞ ¼ hi ni ðpi ðtÞ−pr ðtÞÞT tanhðpi ðtÞ−pr ðtÞÞ:
ð27Þ
Next, we extend the result in Theorem 1 to the case that only a fraction of agents are informed of the position of the virtual leader.
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Theorem 2. Consider a group of N mobile agents modeled by (1) with the nonlinear intrinsic dynamics, f(pi), satisfying Assumption 1 at each time instant t≥0. Suppose that the position information graph is a fixed undirected graph in which each agent is connected to an informed agent. Then, under the feedback control strategy (25)–(27) with any arbitrary directed velocity information topology satisfying Assumption 2, the followings hold: (i) The control input of agent i is saturated by ! ∑jϵN pi apij 2∑jϵN vi ðtÞ avij ðtÞ 2 ‖ui ðtÞ‖∞ ≤ pffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ hi c1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ε 3 3ðN−1Þ 3 3ðN−1Þ (ii) The velocity of all agents asymptotically converges to the velocity of the virtual leader. (iii) The final position of all agents constitutes a configuration which is a local minimum of Z ‖xij ‖s N 1 N p V 2 ðxÞ ¼ ∑ ∑ aij tanhðu−d α Þdu þ c1 ∑ hi 1Tm logðcoshðxi ÞÞ: 2 i ¼ 1 jϵN p i¼1 dα i
(iv) If the initial energy, U0 (defined in the lines of proof) of the group is less than ðk þ 1Þ Rd max apij 0 α tanhðuÞdu, then at most k distinct inter-agent collisions could occur for all i,jϵf1,…,N g
t40. (v) If the initial energy, U0, is less than c1 logðcoshðlÞÞ, then the relative distance between each informed agent and the virtual leader is less than l for all t40. Proof. Proof of part (i) is obvious. As stated before, Assumption 2 implies that the pinning scheme is fixed during the system evolution. Therefore, we introduce the new energy function as: Uðx,v, m, c2 Þ ¼
1 N 1 N ∑ ‖vi ‖2 þ ∑ ∑ apij 2i¼1 2 i ¼ 1 jϵN p i
N
þ c1 ∑ hi 1Tm logðcoshðxi ÞÞ þ i¼1
Z
‖xij ‖s
tanhðu−dα Þdu
dα
N ðtanhðc Þ−wÞ2 1 N N ðtanhðmij Þ−wÞ2 2i pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi þ ∑ ∑ ∑ 2i¼1j¼1 2 N−1 2 N−1ni i¼1
ð28Þ
Then, by applying above energy function and mimicking a similar proof as for Theorem 1, parts (ii), (iii), (iv) and (v) are proved. Also, it should be notified that boundedness of xi is concluded from the boundedness of U0 ¼ U(0) in addition to the property that each agent is connected to an informed agent in the position information graph.& 5. Simulation results To validate our theoretical results in Theorem 1, we consider the flocking problem in a network of 10 agents. The system dynamics is assumed similar to (1) with the adaptive flocking protocol with bounded control input defined in (2)–(4). The nonlinear function f is described by: f ðpÞ ¼ ð−4px þ jpx þ 1j−jpx −1j þ 1, −py −1, 14:87ð−pz þ 0:5ÞÞ,
ð29Þ
where p ¼ (px, py, pz). Obviously, f satisfies Assumption 1 with ω ¼ 1. The parameter values applied in the simulation of this section are ε ¼ 0.1, a ¼ b¼ c1 ¼ 0.5, d ¼ 3 and ni ¼ 0.2 for all i. The sampling time is selected as T s ¼ 0:02 s . Also, we select the
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initial positions and initial velocities of all agents randomly from the cube [−25, 18] [−25, 18] [−25, 18] and the cubic [−2, 3] [−2, 3] [−2,3], respectively. The initial states of the virtual leader are qr ð0Þ ¼ ½3, 3, 3 and pr ð0Þ ¼ ½0:1, 0:1, 0:1. Moreover, c2i(0) ¼ 0 for all i and mij(0) ¼ 0.1 for all (i, j). Position neighboring topology in network applying control protocol (2)–(4) is a fixed unconnected graph illustrated in Fig. 1a and position neighboring topology in network applying control protocol (25)–(27) is a fixed connected graph illustrated in Fig. 1b. The adjacency elements of position interaction topologies are assumed to be one. Two patterns A1 and A2 are considered for velocity information exchange and the corresponding simulation results are given in Figs. 2 and 3, respectively. In pattern A1, we assume that all agents communicate their velocity information with agents in their conic sensing region defined as below: ( ! ) qyji π −1 ℰv ðt Þ ¼ ði,jÞ : ‖qi −qj ‖ ≤ r&− ≤ tan ≤π : ð30Þ 2 qxji where qlji ¼ qlj −qli , qli is the lth component of qi and the sensing radius of all agents is r¼ 4. The velocity neighboring rule in pattern A2 is depicted as: ( ) ! qyji π π −1 ℰv ðt Þ ¼ ði,jÞ : ‖qi −qj ‖ ≤ r&− ≤ tan : ð31Þ ≤ 2 2 qxji The weight of existing communication links between each pair of agents in both patterns equals to 1. Definition of conic sensing regions in (30) and (31) reveals that the communication pattern A1 includes both bidirectional and unidirectional links, while pattern A2 only consists of unidirectional links. In illustrated figures, triangular objects are used to indicate the position of agents. Different symbols show the type of agents (triangular for general type, star for virtual leader). Velocity of agents is displayed by arrow where its length represents magnitude of the velocity. The velocity neighboring relations among the agents are expressed by solid blue arrows. In Figs. 2 and 3, plot (a) illustrates the initial positions and velocities. Plot (b) demonstrates the group configuration at an intermediate snapshot. Plot (c) shows the final group configuration, from which one can realize that the group reaches consensus on velocity of the virtual leader and also forms a flock with the final configuration minimizing V1. Plot (d) shows that the velocity difference between the center of mass and the virtual leader converges to zero. Plots (e) and (f) show the adaptive velocity coupling strengths between agents and the adaptive weights of velocity navigational feedback, respectively. It can be seen that only the coupling weights associated with the bidirectional
Fig. 1. Position neighboring topology.
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Fig. 2. 3-D flocking of 10 agents under protocol (2)–(4) and velocity neighboring rule (30).
edges are updated at each time instant t. Plot (g) shows the number of informed agents as time evolves. It is observed that in pattern A1, the number of informed agents is much fewer than in pattern A2 in which all agents should be informed all the time. This advantage is achieved on account of the existing bidirectional links in pattern A1. Plot (h) shows the bounded control inputs for 10 agents. These observations are consistent with our theoretical predictions in Theorem 1.
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Fig. 3. 3-D flocking of 10 agents under protocol (2)–(4) and velocity neighboring rule (31).
Flocking of 10 agents applying control protocol (25)–(27) is depicted in Fig. 4. For simplicity, ℰv b ðtÞ ¼ ℰv b is considered connected and subsequently, one agent in the group, i.e. agent 1, is chosen to be informed. Moreover, c2i(0) ¼ 0 for all i and mij(0) ¼ 0.1 for all (i, j).
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Fig. 4. 3-D flocking of 10 agents under protocol (25)–(27).
In addition to the permanent neighboring links defined in ℰv b , we assume that all agents communicate their velocity information with a unidirectional pattern similar to what is defined in (31). All other parameters are chosen the same as those in the simulations shown in Figs. 2 and 3. Plots (a) and (b) demonstrate the initial and the final group configurations, respectively. It can be seen from plot (b) that the group reaches consensus on the velocity of the virtual leader and also forms a flock with the final configuration minimizing V2. It should be emphasized that this minimum is reached when the position of the informed agent and the virtual leader overlap. This fact can be easily observed from plot (b). Plot (c) shows that the velocity difference between the center of the mass and the virtual leader converges to zero. Plots (d) and (e) show the adaptive velocity coupling strengths between agents and the adaptive weights of velocity navigational feedback, respectively, which all converge to constants. It can be seen that only the coupling weights associated with the fixed bidirectional edges and the weight of velocity navigational feedback of agent 1 are updated at each time instant t. Plot (f) shows the bounded control inputs for 10 agents. These observations are consistent with our theoretical predictions in Theorem 2.
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6. Conclusion and future work We considered the leader-following flocking problem for second-order dynamics with nonlinear intrinsic dynamics and with input constraints, under switching directed velocity information flow. Some adaptive techniques have been proposed to calculate appropriate coupling strengths for achieving flocking behavior. The effective pinning scheme is designed based on the bidirectional communication links of the velocity information between agents. Furthermore, the adaptive flocking protocol has been enhanced for networks with fixed bidirectional structure, so that the condition of all agents' awareness of virtual leader position is relaxed too. Simulations also verify our theoretical results. Although this paper focuses on studying flocking over an undirected fixed position neighboring graph, a similar analysis may be extended to account for the case of a directed switching position neighboring graph. This will be one of our future research directions. Future investigations will also deal with the extension of our results for networks in which the nonlinear intrinsic dynamics of agents depends on both position and velocity.
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Journal of the Franklin Institute 350 (2013) 1545–1561 www.elsevier.com/locate/jfranklin
Adaptive flocking control of nonlinear multi-agent systems with directed switching topologies and saturation constraints Hajar Atrianfar, Mohammad Haerin Advanced Control Systems Lab, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran Received 12 September 2012; received in revised form 27 February 2013; accepted 6 March 2013 Available online 22 March 2013
Abstract In this paper, we propose and analyze flocking algorithms in a network of second-order agents with bounded control inputs and nonlinear intrinsic dynamics. We consider a general switching network topology, for velocity information exchange, rather than undirected or fixed directed network topology with a directed spanning tree. The proposed adaptive controller architecture applies a leader-following strategy in which the pinning scheme is defined based on the interaction topology. Finally, some examples are presented to illustrate the theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Flocking is a coordinated motion of a large group of agents with local interactions. This issue has a historical background in physics, biology and computer science [1,2] and has attracted significant attention in the control and system community because of its broad applications in formation control of mobile robots, mobile sensor networks, and cooperative coordination of unmanned aerial vehicles [3,4]. The first studies of flocking in swarm robotics were inspired by Reynolds [5], which obtained a realistic computer animation of a flock of birds through three simple behaviors: separation, cohesion and alignment.
n
Corresponding author. Fax: þ98 21 66023261. E-mail address: [email protected] (M. Haeri).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.03.002
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Motivated by the principles of Reynolds model [5], variety of the algorithms has been proposed for flocking [6–8]. Olfati-Saber [6] provided a theoretical framework for analysis of flocking algorithms. In [7], a non-smooth framework has been developed to analyze the stabilization of flocking algorithms in networks with discontinuous control laws. Leader-following flocking is a special case of flocking where there is a virtual leader who specifies the objective for the whole group. In [8], a connectivity-preserving flocking algorithm with bounded potential function is proposed to solve the flocking problem of multi-agent systems with a dynamical virtual leader. In practice, it is also possible that only a portion of agents have access to the states of the virtual leader. This strategy, which is known as pinning control, may reduce the number of controlled agents [9–12]. In many flocking algorithms designed to address the tracking problem, including those mentioned above, it has been commonly assumed that the virtual leader is single. A new algorithm was developed in [13] to study the problem of controlling a group of mobile autonomous agents to track multiple virtual leaders with varying velocities. Since nonlinear intrinsic dynamics is inevitable in real systems, it is necessary and beneficial to study second-order flocking algorithms in a network of nonlinear dynamical systems. To apply adaptive strategies is common in most existing works on the synchronization problems, e.g. [12,14–16]. The other common applications of adaptive techniques are in the networks that vary in time according to different environmental conditions [17,18]. Most existing references that consider nonlinear dynamics intrinsically [12,14,19,20] only investigated second-order consensus of multi-agent systems and the flocking problem is not addressed. Also in these references, the network topology is assumed to be either undirected or fixed weakly connected, which poses an obvious limitation. Recently, an adaptive Leader-following flocking algorithm was proposed for multi-agent systems in which all agents and the virtual leader share the same intrinsic nonlinear dynamics [21]. Their idea only works for connected initial networks and is based on the connectivity preserving of the network during the system evolution. Another problem which should be addressed for coordination strategies to be successful is the actuator saturation. This issue is inevitable in real control systems due to existing physical constraints. All real-world applications of feedback control involve actuators with amplitude and rate limitations. In particular, any physical electromechanical device can provide only a limited force, torque, flow capacity or linear/angular rate. In [22], consensus algorithms with bounded control input for double-integrator dynamics under undirected interaction graph were proposed. The authors in [23] extended the results in [22] and proposed consensus algorithms for double-integrator dynamics without velocity measurements and in the presence of input saturation constraints. In [24], the result in [22] is extended for both single-integrator and double-integrator in the cases of undirected and directed interaction graphs. In [25], it is proved that, when there are saturation constraints, a general consensus protocol widely used in the literatures [26,27] for single-integrator dynamics remains valid when the topology of the connections is fixed and contains a spanning tree. More recently the problem has been impliedly and partly solved in [28] for undirected topology. Zheng et al. [29] discussed the consensus problem of heterogeneous multi-agent system with the saturated consensus protocol. The synchronization problems of networked passive systems with strongly connected communication topologies and actuator saturation are discussed in [30]. As far as we know, there has not been a formal article to systematically address the saturated flocking protocols in the presence of adaptive strategies. On the other hand, the connectivity-preserving second-order consensus algorithm in [12, 14–16,21] was only concerned with agents governed by nonlinear intrinsic dynamics. In this paper, however, we consider an adaptive saturated flocking protocol for a class of multi-agent
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systems where the intrinsic dynamics of the agents are nonlinear and the control input should be bounded. In many situations, velocity information of neighboring agents is not available in all directions. In fact, this problem is faced when agents are not equipped with all-directional velocity sensors, to save cost, space and weight, or velocity is not precisely measured. Moreover, sensed information flow which plays a central role in flocking is typically not bidirectional. Also, unidirectional communication is of considerable importance in practical applications and can easily be incorporated for example, via broadcasting. Therefore, we allow for non-bidirectional communication patterns too. Finally, another aspect of our work is time-dependent communication patterns which become important when we take into consideration some issues such as link failure and link creation and nearest neighbor coupling. The contributions of this paper are threefold. First, we propose an adaptive leader-following flocking algorithm, in which the time-varying pinning scheme depends on the pattern of velocity information exchange among agents. Second, we assume a general switching network topology rather than undirected connected or fixed directed network topology with a directed spanning tree, thereby extending some of the results reported in [12,14–18,21]. Finally, the third contribution concerns with considering both actuator saturation and nonlinear intrinsic dynamics. Other studies that consider this issue such as [22–27] typically do not work with adaptive laws. An outline of this paper is shown as follows. In Section 2, we present some concepts in graph theory and introduce the model of communication for both position and velocity information. The adaptive leader-following flocking problem of the multi-agent system is discussed with the saturated protocol and switching velocity communication graph in Section 3. Furthermore, an extension to the obtained results is given for the case which the condition of all agents' awareness of the virtual leader position is relaxed. Simulation examples are presented to illustrate the analytical results in Section 4. Conclusions are drawn in Section 5. 2. Graph preliminaries To model the communication topology among agents in coordination problems, graph theory is helpful. A time-varying weighted directed graph GðtÞ ¼ ðV,ℰðtÞ,AðtÞÞ consists of a node set V ¼ 1,2,…,N, an edge set ℰðtÞ⊂V V, and a weighted adjacency matrix A(t). An edge (i, j) in a weighted directed graph denotes that agent i can get information from agent j, but not necessarily vice versa. In contrast, the pairs of nodes in a weighted undirected graph are unordered, where an edge (i, j) denotes that agents i and j can exchange information mutually with the same weights. An edge ði,jÞϵℰðtÞ is said to be bidirectional whenever ðj,iÞϵℰðtÞ too and corresponding weights are equal. Otherwise, it is an unidirectional edge. The set of all neighbors of node i is denoted by N i ðtÞ ¼ j : ði,jÞϵℰðtÞ which is the subset of V. In many situations, it is difficult for some agents to measure the relative velocity difference between themselves and neighboring agents in all directions and only relative position data is available. Thus we consider the interaction graph for velocity information exchange as a directed weighted graph. As an example, these topologies are feasible in networks with agents that could only sense the velocity of agents in their conic sensing neighborhoods. Since the sensing mechanism of relative position between agents is simpler, we assume that all agents are equipped with position spherical sensors with identical radii. This causes the interaction graph for position information exchange to be an undirected one. A (directed) path in a graph G(t) is a sequence i1 , …,ik of vertices such that ðij ,ijþ1 ÞϵℰðtÞ for j¼ 1,..,k−1. If there is a path of edges between any two nodes of an undirected graph G(t), then G(t) is connected,
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otherwise it is disconnected. We say that agent i is connected to agent j if there is a path from agent i to agent j. The weighted adjacency matrix of G(t) is denoted by AðtÞ ¼ ½aij ðtÞϵRNN , where aii(t) ¼ 0 and aij(t)40 if there is an edge between agent i and agent j. Its degree matrix ΔAðtÞ ¼ diagðΔ1 ðtÞ,Δ2 ðtÞ,…,ΔN ðtÞÞϵRNN is a diagonal matrix, where diagonal elements Δi(t) ¼ ∑j≠iaij(t). Then the Laplacian of a graph is defined as L(t) ¼ ΔA(t)−A(t) which is symmetric only for undirected ones (see e.g. [31]). In the sequel, the position and velocity interaction graphs are denoted by Gp ðtÞ ¼ ðV,ℰp ðtÞ,Ap ðtÞÞ and Gv ðtÞ ¼ ðV,ℰv ðtÞ,Av ðtÞÞ, respectively. Also, the neighboring sets for agent i in position and velocity interaction graphs are defined as N pi ðtÞ and N vi ðtÞ. Also, we mean by A⊕B, disjoint summation of two sets A and B. 3. Problem formulation and mathematical preliminaries Consider a network consisting of N41 identical agents with nonlinear dynamics described as ( q_ i ¼ pi , ð1Þ p_ i ¼ f ðpi Þ þ ui , where qi , pi ϵRm are the position and velocity of the ith node, f : Rm -Rm is a continuously differentiable vector function and ui is the control input of agent i. Our objective is to design control inputs ui so that all agents asymptotically move with the same velocity as the virtual leader. In this transition, it is desired that the velocities of all agents asymptotically approach each other. The other feature which distinguishes the proposed flocking protocol from the existing second-order consensus algorithms is that it enables the agents in the group to reach a conformation in which the inter-agent distances not necessarily come to zero. Also the maximum number of collisions in the network can be predefined. This geometry resembles the real-world flocks. In this regard, the initial condition of the agents is defined such that collision-free motion can be guaranteed. Also the distance between each agent and the virtual leader should not exceed a maximum value which can be calculated based on the initial conditions. Since the network consists of the agents with unknown nonlinear intrinsic dynamics and actuator saturation, a novel distributed adaptive flocking protocol with bounded control input is proposed. ui ¼ − ∑ apij ðt Þtanh ‖qij ‖s −d α jϵN pi ðtÞ
qij tanhðmij Þ av ðt Þtanhðpj −pi Þ þ ∑ pffiffiffiffiffiffiffiffiffiffi 1 þ ε‖qij ‖s jϵN vi ðtÞ N−1 cosh2 ðmij Þ ij
tanhðc2i Þ tanhð pi −pr Þ −c1 tanhðqi −qr Þ−hi pffiffiffiffiffiffiffiffiffiffi N−1 cosh2 ðc2i Þ
ð2Þ
where c140, qij ¼ qi−qj and (qr, pr) is the state vector of the virtual leader whose acceleration is the same as agents' nonlinear intrinsic dynamics. avij ðtÞ and apij ðtÞ are the adjacency elements of velocity and position information graph, respectively. tanh(.) is defined component-wise. ‖qij ‖s is a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi differentiable map everywhere even at qij ¼ 0 and is defined as ‖qij ‖s ¼ 1=εð 1 þ ε‖qij ‖2 −1Þ with a fixed parameter ε40. Introducing this type of norm is motivated by [6] and causes the smoothness
H. Atrianfar, M. Haeri / Journal of the Franklin Institute 350 (2013) 1545–1561
of collective potential field. Also, we have dα ¼ ‖d‖s , d40. ( _ ij ¼ avij ðtÞðpi −pr ÞT tanhðpi −pr Þ if ði,jÞϵℰv b ðtÞ m _ ij ¼ 0 m if not c_ 2i ¼ hi ni ðpi −pr ÞT tanhðpi −pr ÞÞ
1549
ð3Þ ð4Þ
mij and c2i are adaptive parameters representing the velocity coupling strengths between agents and the weights of velocity navigational feedbacks, respectively. (3) implies that the adaptive protocol only updates the coupling weights associated with the bidirectional edges of the velocity information graph at each time instant t and the coupling weights associated with the unidirectional edges remain constant. Also, c2i is updated only for informed agents at each time instant t. The edge set of the velocity information graph consists of two sets of bidirectional edges (ℰv b ðtÞ) and unidirectional edges (ℰv u ðtÞ), i.e. ℰv ðtÞ ¼ ℰv b ðtÞ⊕ℰv u ðtÞ, which both are timevarying throughout the network evolution. When agent i is informed about the virtual leader's velocity at time instant t, then hi(t)¼ 1 and it is called a pinned agent or an informed agent; otherwise, hi(t)¼ 0. Here, the pinning strategy is applied on a fraction of agents at each time instant t such that each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ. Therefore, the minimum number of informed agents is time-varying and equals to the connected components of this undirected graph. As in [14,15], we say that a function f : Rm -Rm is QUAD if and only if, for x any yϵRm , we have: ðx − yÞT ½f ðx,tÞ− f ðy,tÞ− ðx − yÞT Δðx − yÞ≤ −ωðx − yÞT ðx − yÞ
ð5Þ
where Δ is an arbitrary diagonal matrix of order m and ω is a positive scalar. Note that the above inequality is a Lipschitz-type condition, which is satisfied by many chaotic systems. A large class of nonlinear systems is QUAD with Δ ¼ 0 (see [15,32]). For example, as stated in [33], adopting the algebraic manipulation makes it possible to verify that the Chua's circuit is QUAD with Δ ¼ 0. Respectively, up to now, adaptive strategies for the synchronization of the first-order nonlinear multi-agent systems which are QUAD with Δ ¼ 0 have been extensively studied ([15,17,18]), but there has not been a formal article to systematically address the flocking problem of nonlinear multi-agent systems which are QUAD with Δ ¼ 0, in presence of input saturation. Therefore, the following assumption about the nonlinear part f(.) is proposed and used throughout the paper. Assumption 1. A nonlinear function f : Rm -Rm is QUAD with Δ ¼ 0 and positive scalar ω≥2=27. 4. Main results Before stating the main results, the following lemmas are needed. Lemma 1. [34]. For any x and yϵRn and any symmetric positive definite matrix R, we have: 2xT y≤xT R−1 x þ yT Ry
ð6Þ
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Lemma 2. Suppose that Gv ðtÞ ¼ ðV,ℰv ðtÞ,Av ðtÞÞ, vi ϵRm , M ¼ ½mij ðtÞϵRNN and Av ¼ ½avij ðtÞϵ RNN . If M and Av are symmetric matrices for all t, then 1 2
tanhðmij Þ v tanhðm Þ a ðt Þðvi −vj ÞT tanhðvi −vj Þ ¼ ∑ cosh2 ðmij Þ avij ðt Þvi T ∑ tanhðvi −vj Þ cosh2 ðmij Þ ij ij i jϵN vi ði,jÞϵℰv ðtÞ
∑
ð7Þ
Proof. The proof is similar to that of [22] and is omitted.& ~ ~ ¼ ðV~ , ℰðtÞ,CðtÞ Lemma 3. Let GðtÞ ¼ ½cij ðtÞÞ is a time-varying undirected graph, where ~ ~ V ¼ 1,2,…,M is the node set, ℰðtÞis the edge set and C(t) is the associated adjacency matrix. ~ Also, di ðtÞ40, ∀i, ∀t. Consider the graph GðtÞ, which consists of graph GðtÞ, node 0 (leader) and bidirectional edges between the leader (node 0) and informed agents (of the leader's state). Then, it follows that: AðvÞ ¼
M 1 M M ∑ ∑ cij ðvi −vj ÞT tanhðvi −vj ÞÞ þ ∑ hi di vi T tanhðvi Þ 2i¼1j¼1 i¼1
ð8Þ
is positive definite if and only if the graph GðtÞ is connected at all t. ~ consists of mðtÞ ~ connected subgraphs for Proof. Without loss of generality, we assume that GðtÞ ~ ~ mðtÞ 1 ~ 1 ðtÞ,…, ℰ ~ mðtÞ ~ ~ any t. These subgraphs are labeled G ðtÞ,…, G ðtÞ with edge sets ℰ ðtÞ, respectively. Let π ¼ fijhi ¼ 1g: Since tan h(.) has the same sign as its variable component-wise, we know T that∑M 0, ∀iϵπ g. Furthermore, i ¼ 1 hi d i vi tanhðvi Þ≥0 and takes zero value only at fvjvi ¼ n T 1 M M ~ k ðtÞ, kϵ vjvi ¼ vj ,∀ði,jÞϵℰ 2 ∑i ¼ 1 ∑j ¼ 1 cij ðvi −vj Þ tanhðvi −vj Þ≥0 and takes zero value only at o ~ 1,…, mðtÞ : Therefore, A(v) is positive definite if and only if each connected subgraph contains at least one informed agent which means that the graph G should be connected.& ~ node 0 and edges Lemma 4. [35]. Consider the undirected graph G, which consists of graph G, between the leader (node 0) and informed agents. Then matrix LþZ is positive definite if and only ~ and Z ¼ diagðz1 ,z2 ,…, zN Þ if the graph G is connected, where L is the Laplacian of the graph G and zi40 if there is an edge between the leader and node i and zi ¼ 0, otherwise. Now, we can prove the following theorem for the distributed control of the flocking behavior of a group of N agents. Theorem 1. Consider a group of N mobile agents modeled by (1) with the nonlinear intrinsic dynamics, f(pi), satisfying Assumption 1 at each time instant t≥0. Suppose that the position information graph is a fixed undirected graph. Then under the feedback control strategy (2)–(4) with any arbitrary directed velocity information topology, the followings hold: (i) The control input of agent i is saturated by 1 2 2 ‖ui ‖∞ ≤ pffiffiffi ∑jϵN pi apij þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑jϵN vi ðtÞ avij ðt Þ þ c1 þ hi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 3 3ðN−1Þ 3 3ðN−1Þ
ð9Þ
(ii) The velocity of all agents asymptotically converges to the velocity of the virtual leader.
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(iii) The final position of all agents constitutes a configuration which is a local minimum of V1 ¼
N 1 N ‖x ‖ ∑ ∑ apij ∫dα ij σ tanhðu−d α Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ: 2 i¼1 jϵN p i¼1 i
(iv) If the initial energy, U0 (defined in the lines of proof) of the group is less than ðk þ 1Þ max apij ∫0dα tanhðuÞdu, then at most k distinct inter-agent collisions could occur for i;jϵf1;…;N g all t>0. (v) If the initial energy, U0, is less than c1 logðcoshðlÞÞ, then the relative distance between each agent and the virtual leader is less than l for all t>0. pffiffiffi Proof. Part i can be proved directly by the fact that 0≤qij =ð1 þ ε‖qij ‖σ Þ≤ð1= εÞ; ∀qij and tanhðaÞ 2 ≤ pffiffiffi cosh2 ðaÞ 3 3
tanhðaÞ≤1, ∀a:
and
Considering xi ¼ qi−qr and vi ¼ pi−pr as the position and velocity of agent i in the moving frame centered at the virtual leader, apij ðtÞ ¼ apij and N pi ðtÞ ¼ N pi for ∀t, the system equation would be as follows: 8 x_ i ¼ vi > > > > v_ i ¼ f ðvi þ pr Þ−f ðpr Þ−∑jϵN p apij tanhð‖xij ‖s −dα Þ xij > 1þε‖xij ‖s > i > > > tanhðmij Þ > v ffi > þ∑jϵN vi ðtÞ pffiffiffiffiffiffi a ðt Þtanhðvj −vi Þ−c1 tanhðxi Þ > N−1cosh2 ðmij Þ ij > > < tanhðc ffi 2i Þ tanhðvi Þ −hi pffiffiffiffiffiffi ð10Þ N−1cosh2 ðc2i Þ > > > > T b > _ ij ¼ avij ðtÞðvi −vj Þ tanhðvi −vj Þ if ði,jÞϵℰv ðtÞ m > > > > > _ ij ¼ 0 m if not > > > > : c_ 2i ¼ hi ni vi T tanhðvi Þ where xij ¼ xi−xj. Consider the energy function candidate as: Z ‖xij ‖s N 1 N p Uðx,v, m, c2 Þ ¼ tanhðu−dα Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ ∑ ∑ aij 2 i ¼ 1 jϵN p i¼1 dα i
þ
N ðtanhðc Þ−wÞ2 1 N 1 N N ðtanhðmij Þ−wÞ2 2i pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi þ ∑ ∑ ‖vi ‖2 þ ∑ ∑ 2i¼1 2i¼1j¼1 2 N−1 2 N−1 ni i¼1
ð11Þ
where log(.) and cosh(.) are component-wise functions. x ¼ ðx1 T ,x2 T ,…,xTN ÞT and v ¼ ðv1 T ,v2 T ,…,vTN ÞT are the collective position and velocity vectors of the new system. Also, c2 ¼ (c21,c22,…,c2N)T and m ¼ (m11,…,m1N,…,mN1,…,mNN)T are collective adaptive parameters of the system. Moreover, w is a bounded positive constant to be determined later.
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The time derivative of Uðx,v, m, c2 Þ along the trajectories of (10) gives: tanhðmij Þ _ avij ðt Þtanhðvj −vi Þ Uðx,v, m, c2 Þ ¼ ∑ vi T f ðvi þ pr Þ−f ðpr Þ þ ∑ pffiffiffiffiffiffiffiffiffiffi 2 v i¼1 jϵN i ðtÞ N−1cosh ðmij Þ tanhðc2i Þ tanhðv −hi pffiffiffiffiffiffiffiffiffiffi Þ i N−1cosh2 ðc2i Þ 1 1 þ pffiffiffiffiffiffiffiffiffiffi ∑ ðtanhðmij Þ−wÞ av ðt Þðvi −vj ÞT tanhðvi −vj Þ cosh2 ðmij Þ ij 2 N−1 ði,jÞϵℰv b ðtÞ N
N 1 1 þ pffiffiffiffiffiffiffiffiffiffi ∑ hi ðtanhðc2i Þ−wÞ vT ðtÞtanhðvi Þ: cosh2 ðc2i Þ i N−1 i ¼ 1
ð12Þ
Thus, Assumption 1, Lemmas 1 and 2 imply that: " # N 1 tanh2 ðmij Þ T T T v2 _ Uðx,v, m, c2 Þ≤− ∑ vi ωvi þ ∑ vi vi þ aij ðtÞtanhðvj −vi Þ tanhðvj −vi Þ 4 i¼1 ði,jÞϵℰv u ðtÞ 2 ðN−1Þcosh ðmij Þ −
w 1 ∑ pffiffiffiffiffiffiffiffiffiffi av ðt Þðvi −vj ÞT tanhðvi −vj Þ 2 ði,jÞϵℰv b ðtÞ N−1cosh2 ðmij Þ ij
1 vTi tanhðvi Þ −w∑ hi pffiffiffiffiffiffiffiffiffiffi N−1cosh2 ðc2i Þ i
ð13Þ
As said before, each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ for all t≥0. It then follows from Lemma 3 that 1 1 1 b ∑ av ðt Þ ðvi −vj ÞT tanhðvi −vj Þ þ ∑ hi vTi tanhðvi Þ 2 2 ði,jÞϵℰv ðtÞ cosh2 ðmij Þ ij cosh ðc Þ i 2i
ð14Þ
is positive definite. Therefore, by appropriate choice of constant w, we can get the right side of inequality (13) to be negative definite. pffiffiffi By using jtanhðaÞ=cosh2 ðaÞj≤2=ð3 3Þ and the fact that tanhðjajÞ≤jaj, ∀a, one can show: N w 2 T 1 v2
_ Uðx,v, m, c2 ÞÞ≤ ∑ −ω þ aij ðt Þ tanhðvj −vi ÞT tanhðvj −vi Þ vi vi þ ∑ u 27 2 2 i¼1 ði,jÞϵℰv ðtÞ 1 ∑ pffiffiffiffiffiffiffiffiffiffi avij ðt Þðvi −vj ÞT tanhðvi −vj Þ 2 ði,jÞϵℰv b ðtÞ N−1cosh ðmij Þ 1 −w∑ hi pffiffiffiffiffiffiffiffiffiffi tanhðvTi Þ tanhðvi Þ: ð15Þ N−1cosh2 ðc2i Þ i Noting that ω≥
2 27
and tanhðx þ yÞ ¼
tanhðxÞ þ tanhðyÞ , 1 þ tanhðxÞtanhðyÞ
one can see that: _ Uðx,v, m, c2 Þ≤tanhðvT ÞLs tanhðvÞ−wtanhðvT ÞLM tanhðvÞ−wtanhðvT ÞHtanhðvÞ,
ð16Þ
H. Atrianfar, M. Haeri / Journal of the Franklin Institute 350 (2013) 1545–1561
where Ls ¼ ΔS−S and LM ¼ ΔM−M, S ¼ ½sij ðtÞ, M ¼ ½mij ðtÞ, 8 !2 > <1 v 2 2 v 1 ða ðtÞ þ aji ðtÞ Þ min ð1−tanhðvi Þtanhðvj ÞÞ ði,jÞϵΨ sij ðtÞ ¼ 2 ij ði,jÞϵℰuv > : 0 O:W: n o Ψ ¼ ði,jÞði,jÞϵℰv u ðt Þ or ðj,iÞϵℰv u ðt Þ
mij ðtÞ ¼
8 > < 1 pffiffiffiffiffiffiffi 1 > :
2
1553
ð17Þ
ð18Þ
!2
av ð t Þ N−1cosh2 ðmij Þ ij
0
1 max ð1−tanhðvi Þtanhðvj ÞÞ
ði,jÞϵℰv b ðtÞ
ði,jÞϵℰv b ðtÞ
ð19Þ
O:W:
1 1 1 p ffiffiffiffiffiffiffiffiffi ffi H¼ diag h1 ðtÞ ,…,hN ðtÞ : cosh2 ðc21 Þ cosh2 ðc2N Þ N−1
ð20Þ
Definition of mij ðtÞ and the fact that −1≤tanhðvi Þ≤1, ∀i, prove that the associated graph with LM has the same edge set as ℰv b ðtÞ. Then, by Lemma 4 and the fact that each agent is connected to an informed agent in the undirected graph defined with the node set V and the edge set ℰv b ðtÞ, it is easy to see that λmin ðLM þ HÞ40. On the other hand, the edge weights avij ðtÞ are chosen from a finite interval [0, amax], where amax is the upper bound on the adjacency elements of velocity information graph. Therefore, there must be a negative scalar μ such that max λmax ðLs Þ≤μ: Then, by choosing a t bounded scalar w≥max λmax ðLS Þ=min λmin ðLM þ HÞ, one can easily show that: t t _ Uðx,v, m, c2 Þ≤0: ð21Þ Therefore, U is a non-increasing function for all t≥0 and the set Ω0 ¼ ðx,vÞ, Uðx,vÞ≤U 0 is an invariant set, where U0 ¼ U(0)o∞. Boundedness of U0 proves the boundedness of xi and vi. This in turn implies that Ω0 is a compact invariant set. Therefore, LaSalle's principle reveals that all solutions of the system starting in Ω0 converge to a largest invariant set in _ ¼0 . Π ¼ ðx,vÞϵΩ0 jU Then based on (5), U_ ¼ 0 implies that vi ¼ 0 for all i. Consequently, it follows that pi ¼ pr for all i. Hence the velocity of all agents gradually converges to the velocity of virtual leader. It thus follows from (9) that: xij ∑ apij tanhð‖xij ‖s −dα Þ þ c1 tanhðxi Þ ¼ 0 ð22Þ p 1 þ ε‖xij ‖s jϵN i
Solutions to above relation are equivalent to the local minima of the total potential function Z ‖xij ‖s N 1 N p V 1 ðxÞ ¼ ∑ ∑ aij tanhðu−d α Þdu þ c1 ∑ 1Tm logðcoshðxi ÞÞ ð23Þ 2 i ¼ 1 jϵN p i¼1 dα i
The proof of part (ii) and (iii) is completed. Assume that there is more than k (at least kþ1) distinct collisions between agents at time t40. Then, (4) reveals that Z dα p UðtÞ≥ðk þ 1Þ max aij tanhðuÞdu: i,jϵf1,…,N g
0
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But, U is a non-increasing function. Thus, Z dα Uð0Þ≥UðtÞ≥ðk þ 1Þ max apij tanhðuÞdu, i,jϵf1,…,N g
ð24Þ
0
which is in contradiction with the assumption of theorem that Z dα p Uð0Þoðk þ 1Þ max aij tanhðuÞdu: i,jϵf1,…,N g
0
Therefore, no more than k distinct inter-agent collisions could occur in the network. This proves part (iv). Assume that the distance between agent i and the virtual leader is equal or more than l at time t40. Therefore, we have UðtÞ≥c1 logðcoshðlÞÞ which implies that Uð0Þ≥c1 logðcoshðlÞÞ which is in contradiction with the assumption of theorem that Uð0Þoc1 logðcoshðlÞÞ. Therefore, relative distance between each agent and the virtual leader is less than l. This proves Part (v).& Remark 1. In the proposed algorithm, adaptive strategy is used to guarantee the stability of flocking behavior in the presence of directed switching velocity interactions. Incorporating adaptive coupling gains in networks which vary in time according to different environmental conditions is common in the literature. This approach is introduced for undirected switching topologies in [17,18]. Assumption 2. The edge set of the velocity information graph consists of two sets of fixed bidirectional edges (ℰv b ) and time-varying unidirectional edges (ℰv u ðtÞ), i.e. ℰv ¼ ℰv b ⊕ℰv u ðtÞ. A typical example of above assumption is a network of agents in which some agents are equipped with non-identical conical velocity sensors and a few agents are equipped with spherical velocity sensors with very large radii. Another common example is a network in which some agents are equipped with non-identical conical velocity sensors and also a fixed wireless communication network is dedicated to gather and communicate mutually velocity data between agents. Also, fixed directed and undirected graphs are of those topologies which satisfy above assumption. Since, the pinning scheme is dependent on the edge set ℰv b , therefore, the time-independency of this set causes the pinning scheme not change as system evolves. Consequently, the parameters hi can be chosen fixed during the time. Based on this fact, we will show in the following that how the flocking behavior is achieved for a different class of networks in which the condition of all agents' awareness of the virtual leader position is relaxed. For this purpose, the new control input is defined as below: qij tanhðmij Þ ui ¼ −∑jϵN pi ðtÞ apij ðt Þtanh ‖qij ‖s −d α þ ∑jϵN vi ðtÞ pffiffiffiffiffiffiffiffiffiffi avij ðt Þtanhðpj −pi Þ 1 þ ε‖qij ‖s N−1cosh2 ðmij Þ tanhðc2i Þ tanhð p −hi c1 tanhðqi −qr Þ þ pffiffiffiffiffiffiffiffiffiffi −p Þ ð25Þ i r N−1cosh2 ðc2i Þ ( _ ij ¼ avij ðtÞðvi −vj ÞT tanhðvi −vj Þ if ði,jÞϵℰv b m ð26Þ _ ij ¼ 0 m if not c_ 2i ðtÞ ¼ hi ni ðpi ðtÞ−pr ðtÞÞT tanhðpi ðtÞ−pr ðtÞÞ:
ð27Þ
Next, we extend the result in Theorem 1 to the case that only a fraction of agents are informed of the position of the virtual leader.
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Theorem 2. Consider a group of N mobile agents modeled by (1) with the nonlinear intrinsic dynamics, f(pi), satisfying Assumption 1 at each time instant t≥0. Suppose that the position information graph is a fixed undirected graph in which each agent is connected to an informed agent. Then, under the feedback control strategy (25)–(27) with any arbitrary directed velocity information topology satisfying Assumption 2, the followings hold: (i) The control input of agent i is saturated by ! ∑jϵN pi apij 2∑jϵN vi ðtÞ avij ðtÞ 2 ‖ui ðtÞ‖∞ ≤ pffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ hi c1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ε 3 3ðN−1Þ 3 3ðN−1Þ (ii) The velocity of all agents asymptotically converges to the velocity of the virtual leader. (iii) The final position of all agents constitutes a configuration which is a local minimum of Z ‖xij ‖s N 1 N p V 2 ðxÞ ¼ ∑ ∑ aij tanhðu−d α Þdu þ c1 ∑ hi 1Tm logðcoshðxi ÞÞ: 2 i ¼ 1 jϵN p i¼1 dα i
(iv) If the initial energy, U0 (defined in the lines of proof) of the group is less than ðk þ 1Þ Rd max apij 0 α tanhðuÞdu, then at most k distinct inter-agent collisions could occur for all i,jϵf1,…,N g
t40. (v) If the initial energy, U0, is less than c1 logðcoshðlÞÞ, then the relative distance between each informed agent and the virtual leader is less than l for all t40. Proof. Proof of part (i) is obvious. As stated before, Assumption 2 implies that the pinning scheme is fixed during the system evolution. Therefore, we introduce the new energy function as: Uðx,v, m, c2 Þ ¼
1 N 1 N ∑ ‖vi ‖2 þ ∑ ∑ apij 2i¼1 2 i ¼ 1 jϵN p i
N
þ c1 ∑ hi 1Tm logðcoshðxi ÞÞ þ i¼1
Z
‖xij ‖s
tanhðu−dα Þdu
dα
N ðtanhðc Þ−wÞ2 1 N N ðtanhðmij Þ−wÞ2 2i pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi þ ∑ ∑ ∑ 2i¼1j¼1 2 N−1 2 N−1ni i¼1
ð28Þ
Then, by applying above energy function and mimicking a similar proof as for Theorem 1, parts (ii), (iii), (iv) and (v) are proved. Also, it should be notified that boundedness of xi is concluded from the boundedness of U0 ¼ U(0) in addition to the property that each agent is connected to an informed agent in the position information graph.& 5. Simulation results To validate our theoretical results in Theorem 1, we consider the flocking problem in a network of 10 agents. The system dynamics is assumed similar to (1) with the adaptive flocking protocol with bounded control input defined in (2)–(4). The nonlinear function f is described by: f ðpÞ ¼ ð−4px þ jpx þ 1j−jpx −1j þ 1, −py −1, 14:87ð−pz þ 0:5ÞÞ,
ð29Þ
where p ¼ (px, py, pz). Obviously, f satisfies Assumption 1 with ω ¼ 1. The parameter values applied in the simulation of this section are ε ¼ 0.1, a ¼ b¼ c1 ¼ 0.5, d ¼ 3 and ni ¼ 0.2 for all i. The sampling time is selected as T s ¼ 0:02 s . Also, we select the
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initial positions and initial velocities of all agents randomly from the cube [−25, 18] [−25, 18] [−25, 18] and the cubic [−2, 3] [−2, 3] [−2,3], respectively. The initial states of the virtual leader are qr ð0Þ ¼ ½3, 3, 3 and pr ð0Þ ¼ ½0:1, 0:1, 0:1. Moreover, c2i(0) ¼ 0 for all i and mij(0) ¼ 0.1 for all (i, j). Position neighboring topology in network applying control protocol (2)–(4) is a fixed unconnected graph illustrated in Fig. 1a and position neighboring topology in network applying control protocol (25)–(27) is a fixed connected graph illustrated in Fig. 1b. The adjacency elements of position interaction topologies are assumed to be one. Two patterns A1 and A2 are considered for velocity information exchange and the corresponding simulation results are given in Figs. 2 and 3, respectively. In pattern A1, we assume that all agents communicate their velocity information with agents in their conic sensing region defined as below: ( ! ) qyji π −1 ℰv ðt Þ ¼ ði,jÞ : ‖qi −qj ‖ ≤ r&− ≤ tan ≤π : ð30Þ 2 qxji where qlji ¼ qlj −qli , qli is the lth component of qi and the sensing radius of all agents is r¼ 4. The velocity neighboring rule in pattern A2 is depicted as: ( ) ! qyji π π −1 ℰv ðt Þ ¼ ði,jÞ : ‖qi −qj ‖ ≤ r&− ≤ tan : ð31Þ ≤ 2 2 qxji The weight of existing communication links between each pair of agents in both patterns equals to 1. Definition of conic sensing regions in (30) and (31) reveals that the communication pattern A1 includes both bidirectional and unidirectional links, while pattern A2 only consists of unidirectional links. In illustrated figures, triangular objects are used to indicate the position of agents. Different symbols show the type of agents (triangular for general type, star for virtual leader). Velocity of agents is displayed by arrow where its length represents magnitude of the velocity. The velocity neighboring relations among the agents are expressed by solid blue arrows. In Figs. 2 and 3, plot (a) illustrates the initial positions and velocities. Plot (b) demonstrates the group configuration at an intermediate snapshot. Plot (c) shows the final group configuration, from which one can realize that the group reaches consensus on velocity of the virtual leader and also forms a flock with the final configuration minimizing V1. Plot (d) shows that the velocity difference between the center of mass and the virtual leader converges to zero. Plots (e) and (f) show the adaptive velocity coupling strengths between agents and the adaptive weights of velocity navigational feedback, respectively. It can be seen that only the coupling weights associated with the bidirectional
Fig. 1. Position neighboring topology.
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Fig. 2. 3-D flocking of 10 agents under protocol (2)–(4) and velocity neighboring rule (30).
edges are updated at each time instant t. Plot (g) shows the number of informed agents as time evolves. It is observed that in pattern A1, the number of informed agents is much fewer than in pattern A2 in which all agents should be informed all the time. This advantage is achieved on account of the existing bidirectional links in pattern A1. Plot (h) shows the bounded control inputs for 10 agents. These observations are consistent with our theoretical predictions in Theorem 1.
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Fig. 3. 3-D flocking of 10 agents under protocol (2)–(4) and velocity neighboring rule (31).
Flocking of 10 agents applying control protocol (25)–(27) is depicted in Fig. 4. For simplicity, ℰv b ðtÞ ¼ ℰv b is considered connected and subsequently, one agent in the group, i.e. agent 1, is chosen to be informed. Moreover, c2i(0) ¼ 0 for all i and mij(0) ¼ 0.1 for all (i, j).
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Fig. 4. 3-D flocking of 10 agents under protocol (25)–(27).
In addition to the permanent neighboring links defined in ℰv b , we assume that all agents communicate their velocity information with a unidirectional pattern similar to what is defined in (31). All other parameters are chosen the same as those in the simulations shown in Figs. 2 and 3. Plots (a) and (b) demonstrate the initial and the final group configurations, respectively. It can be seen from plot (b) that the group reaches consensus on the velocity of the virtual leader and also forms a flock with the final configuration minimizing V2. It should be emphasized that this minimum is reached when the position of the informed agent and the virtual leader overlap. This fact can be easily observed from plot (b). Plot (c) shows that the velocity difference between the center of the mass and the virtual leader converges to zero. Plots (d) and (e) show the adaptive velocity coupling strengths between agents and the adaptive weights of velocity navigational feedback, respectively, which all converge to constants. It can be seen that only the coupling weights associated with the fixed bidirectional edges and the weight of velocity navigational feedback of agent 1 are updated at each time instant t. Plot (f) shows the bounded control inputs for 10 agents. These observations are consistent with our theoretical predictions in Theorem 2.
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6. Conclusion and future work We considered the leader-following flocking problem for second-order dynamics with nonlinear intrinsic dynamics and with input constraints, under switching directed velocity information flow. Some adaptive techniques have been proposed to calculate appropriate coupling strengths for achieving flocking behavior. The effective pinning scheme is designed based on the bidirectional communication links of the velocity information between agents. Furthermore, the adaptive flocking protocol has been enhanced for networks with fixed bidirectional structure, so that the condition of all agents' awareness of virtual leader position is relaxed too. Simulations also verify our theoretical results. Although this paper focuses on studying flocking over an undirected fixed position neighboring graph, a similar analysis may be extended to account for the case of a directed switching position neighboring graph. This will be one of our future research directions. Future investigations will also deal with the extension of our results for networks in which the nonlinear intrinsic dynamics of agents depends on both position and velocity.
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