Adaptive Tracking Control of Second-Order Multiagent Systems with

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 2756046, 11 pages http://dx.doi.org/10.1155/2016/2756046

Research Article Adaptive Tracking Control of Second-Order Multiagent Systems with Jointly Connected Topologies Lei Chen,1 Kaiyu Qin,1 and Jiangping Hu2 1

School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

2

Correspondence should be addressed to Lei Chen; [email protected] Received 7 April 2016; Revised 23 June 2016; Accepted 3 July 2016 Academic Editor: Xuejun Xie Copyright © 2016 Lei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper considers a consensus problem of leader-following multiagent system with unknown dynamics and jointly connected topologies. The multiagent system includes a self-active leader with an unknown acceleration and a group of autonomous followers with unknown time-varying disturbances; the network topology associated with the multiagent system is time varying and not strongly connected during each time interval. By using linearly parameterized models to describe the unknown dynamics of the leader and all followers, we propose a decentralized adaptive tracking control protocol by using only the relative position measurements and analyze the stability of the tracking error and convergence of the adaptive parameter estimators with the help of Lyapunov theory. Finally, some simulation results are presented to demonstrate the proposed adaptive tracking control.

1. Introduction As one kind of the major research content of distributed coordination control for multiagent systems, leader-following problem had attracted a host of researchers. For example, Ren proposed and analyzed consensus tracking algorithms in [1] and solved the leader-following problem that only a few agents can obtain a time-varying consensus reference state. Hong et al. investigated the leader-following problem, using an “observer” to solve how to track the leader with unknown velocity in [2]. Hu and Hong considered a leader-following consensus problem of a group of autonomous agents with time-varying coupling delays in [3] and investigated two different cases of coupling topologies. Peng and Yang studied the problem of multiple time-varying delays for secondorder multiagent systems in [4]. Song et al. achieved leaderfollowing consensus in a network of agents with nonlinear second-order dynamics in [5] by presenting a pinning control algorithm. Meanwhile, estimation strategies with partial measurements and adaptive control schemes about unknown disturbances had captured some individuals’ attention. Hong et al. designed the distributed observers for the second-order

agents in [6], such that the velocity of the active leader cannot be measured. Hu et al. solved an event-triggered tracking problem in [7] by using an observer-based consensus tracking control, which is designed on the basis of a novel distributed velocity estimation technique. Zhang and Yang proposed two bounded control laws, which are independent of velocity information in [8], to deal with the finite-time consensus tracking problem. Bauso et al. considered stationary consensus protocols for networks of dynamic agents in [9] in which the neighbors’ states are affected by unknown but bounded disturbances. Hu and Zheng just used the relative position measurements to design a dynamic output-feedback tracking control together with decentralized adaptive laws in [10]. Li et al. designed a distributed adaptive consensus protocol in [11] based on the agent dynamics and the relative states of neighboring agents and achieved leader-follower consensus for any communication graph which contains a directed spanning tree with the leader as the root node. Bai et al. considered the situation where the reference velocity information is available only to a leader in [12] and then developed an adaptive design to recover the desired formation. The work about dynamically changing topologies, such as jointly connected topology, also appeared in some research.

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Mathematical Problems in Engineering

In [13] Hong et al. adopted a neighbor-based rule to realize local control strategies for these autonomous agents and made all the agents converge to a common value by using a Lyapunov-based approach. In [14], Lin and Jia investigated consensus problems in networks of continuous-time agents with time delays and jointly connected topologies. In this paper, we consider a leader-following problem about second-order multiagent system, which has unknown time-varying disturbances and the system is partial measurement. Different from some existing research, we consider the network of the system is jointly connected and prove a lemma to solve the the jointly connected topologies problem about leader-following system. This method can apply to some other jointly connected problems. Moreover, the leader’s velocity and acceleration in the multiagent system are unknown; we propose a state variable to estimate the relative velocity and design a control law to guarantee the agents to follow the leader by using relative position measurement only. In addition, we propose the decentralized adaptive laws for the unknown disturbances, and with the help of a prudently chosen common Lyapunov function under a persistent excitation condition, we prove both the tracking errors and disturbances parameter estimate errors can converge to zero. The subsequent sections are organized as follows: In Section 2, we introduce some preliminaries and present the leader-following multiagent model. In Section 3, we propose a dynamic output-feedback tracking control with two decentralized adaptive laws for each follower. Then we analyze the consensus of the system and obtain the main results in Section 4. In Section 5, we give the numerical simulation results. Finally, some conclusions are drawn in Section 6.

G with vertex set V = V ∪ {V0 }, which contains graph G of 𝑛 followers and the leader with directed edges, if any, from some vertices of G to the leader vertex. Use 𝐵 to describe the leader adjacency matrix and 𝐵 = diag (𝑏1 , 𝑏2 , . . . , 𝑏𝑛 ), where 𝑏𝑖 > 0 if the leader is a neighbor of agent 𝑖 and 𝑏𝑖 = 0 otherwise. When there is at least one directed edge from vertices of the graph G to the leader vertex V0 , the graph G is said to be connected. Consider an infinite sequence of nonempty, bounded, and continuous-time intervals [𝑡𝑟 , 𝑡𝑟+1 ), 𝑟 = 0, 1, . . ., with 𝑡0 = 0, 𝑡𝑟+1 − 𝑡𝑟 = 𝑇 for some constant 𝑇 > 0. Suppose that, in each interval [𝑡𝑟 , 𝑡𝑟+1 ), there is a sequence of nonoverlapping subintervals

2. Problem Statement

Lemma 1 (Godsil and Royle [15]). If the graph 𝐺 is connected, its Laplacian 𝐿 satisfies the following: (1) zero is a simple eigenvalue of 𝐿, and 1𝑛 is the corresponding eigenvector; (2) the remaining 𝑛 − 1 eigenvalues are all positive and real.

2.1. Algebraic Graph Theory. Firstly we introduce the graph theory; we use it to describe the communication between agents in a multiagent system. Consider a tracking problem for a multiagent system about 𝑛 followers and 𝑜𝑛𝑒 leader. The interconnection topology of 𝑛 followers can be conveniently described by a undirected graph G = (V, 𝐸, 𝐴) of order 𝑛, where V = {V1 , V2 , . . . , V𝑛 } is the set of 𝑛 nodes, 𝐸 ⊆ V × V is the set of edges, and 𝐴 = [𝑎𝑖𝑗 ] is a weighted adjacency matrix. The node indexes belong to a finite index set I = {1, 2, . . . , 𝑛}. An edge of G is denoted by 𝑒𝑖𝑗 = (V𝑖 , V𝑗 ). The adjacency elements associated with the edges are positive. The adjacency matrix is defined as 𝑎𝑖𝑖 = 0 and 𝑎𝑖𝑗 = 𝑎𝑗𝑖 ≥ 0. When node V𝑖 has edge to V𝑗 , 𝑎𝑖𝑗 > 0, the vertex 𝑗 is called a neighbor of vertex 𝑖; it means that agent 𝑗 is communicating to agent 𝑖, denoted by 𝑁𝑖 . Then 𝑁𝑖 (𝑡) = {𝑗 ∈ V : (𝑖, 𝑗) ∈ 𝐸, 𝑗 ≠ 𝑖}. The out-degree matrix of G is 𝐷 = diag (𝑑1 , . . . , 𝑑𝑛 ) ∈ 𝑅𝑛×𝑛 , where 𝑑𝑖 = ∑𝑗∈𝑁𝑖 (𝑡) 𝑎𝑖𝑗 are the diagonal elements for 𝑖 = {1, 2, . . . , 𝑛}. The Laplacian of the undigraph G is defined as 𝐿 = 𝐷 − 𝐴. The leader (labeled 0) is represented by vertex V0 , and the connection between the followers and the leader is directed. In the context of this paper, there are only parts of the followers having edges to the leader. Then, we have a simple graph

[𝑡𝑟0 , 𝑡𝑟1 ) ⋅ ⋅ ⋅ [𝑡𝑟𝑗 , 𝑡𝑟𝑗+1 ) ⋅ ⋅ ⋅ [𝑡𝑟𝑚 −1 , 𝑡𝑟𝑚 ) , 𝑟

𝑟

(1)

(𝑡𝑟 = 𝑡𝑟0 , 𝑡𝑟+1 = 𝑡𝑟𝑚 ) 𝑟

satisfying 𝑡𝑟1 − 𝑡𝑟0 = ⋅ ⋅ ⋅ = 𝑡𝑟𝑚 − 𝑡𝑟𝑚 −1 = 𝑡Δ , 0 ≤ 𝑗 < 𝑚𝑟 , for 𝑟 𝑟 some integer 𝑚𝑟 ≥ 0 and given constant 𝑡Δ > 0. It is clear that there are 𝑚∗ = 𝑇/𝑡Δ subintervals in each interval. During each of the subintervals, the interconnection topology described by G𝑚 (𝑚 = 1, . . . , 𝑚∗ ) is stable and changing at each time 𝑡𝑟𝑗+1 . The graph G𝑚 (𝑚 = 1, . . . , 𝑚∗ ) has the same node set V, and the union of the collection is defined as G1−𝑚∗ , whose node set is V and edge set equals the union of the edge sets of all of the graphs in the collection. However, the graph G𝑚 (𝑚 = 1, . . . , 𝑚∗ ) may be not strongly connected, but its union graph is connected; then we say the network is jointly connected.

Lemma 2 (Hong et al. [2]). Denote 𝐻 = 𝐿 + 𝐵, where L is the weighted Laplacian of graph G and 𝐵 is the leader adjacency matrix as defined in Section 2. If graph G is connected, then the symmetric matrix 𝐻 associated with G is positive definite. Moreover, matrices 𝐻1 , . . . , 𝐻𝑚∗ are associated with the graphs G1 , G2 , . . . , G𝑚∗ , respectively; 𝐻𝑖 is positive semidefinite because both 𝐿 𝑖 and 𝐵𝑖 are positive semidefinite. Lemma 3 (Hong et al. [13] and Lin et al. [16]). Graph G is jointly connected and has 𝑙𝜎 ≥ 1 connected topology in each subinterval. The corresponding nodes sets of the connected components are denoted by 𝜀𝑟1𝑗 , 𝜀𝑟2𝑗 , . . . , 𝜀𝑟𝑙𝜎𝑗 , 𝑑𝜎𝜖 denotes the 𝑙

𝜎 𝑑𝜎𝜖 = 𝑛. From some knowledge number of nodes in 𝜀𝑟𝜖𝑗 , and ∑𝜖=1

of matrix theory, there exists a permutation matrix 𝐸𝜎 ∈ 𝑅𝑛×𝑛 , which satisfies 𝐸𝜎 𝑇 𝐻𝜎 𝐸𝜎 = diag [𝐻𝜎 1 , 𝐻𝜎 2 , . . . , 𝐻𝜎 𝑙𝜎 ] .

(2)

Mathematical Problems in Engineering

3

2.2. Problem Statement. In this paper, the dynamics of each agent is described by

𝜗𝑖 (𝑡) = ∑ 𝑎𝑖𝑗 (V𝑖 (𝑡) − V𝑗 (𝑡)) + 𝑏𝑖 (V𝑖 (𝑡) − V0 (𝑡)) ,

𝑥̇ 𝑖 (𝑡) = V𝑖 (𝑡) ,

𝑗∈𝑁𝑖 (𝑡)

V̇𝑖 (𝑡) = 𝑢𝑖 (𝑡) + 𝑓 (𝑥𝑖 , 𝑡) ,

(3) 𝑖 = 1, 2, . . . , 𝑛,

where 𝑥𝑖 , V𝑖 , 𝑢𝑖 ⊂ 𝑅, 𝑥𝑖 , V𝑖 are the position and velocity vectors of the 𝑖th agent, respectively, and 𝑢𝑖 is the control input. 𝑓(𝑥𝑖 , 𝑡) is the dynamics of agent 𝑖, which is assumed to be an unknown time-varying disturbance. The dynamics of the leader in the multiagent system is described by 𝑥̇ 0 (𝑡) = V0 (𝑡) , V̇0 (𝑡) = 𝑎0 (𝑡) ,

(4)

Before giving the adaptive control law, we propose two variables to estimate 𝑎0 (𝑡) and 𝑓(𝑥𝑖 , 𝑡). For leader-follower system, the acceleration 𝑎0 (𝑡) and the disturbances 𝑓(𝑥𝑖 , 𝑡) (𝑖 = 1, . . . , 𝑛) are unknown. By the techniques in classical adaptive control (Marino and Tomei [17]) and multiagent systems (Bai et al. [12], Hu and Zheng [10]), they can be parameterized, respectively, as follows:

𝑓 (𝑥𝑖 , 𝑡) = 𝜙𝑖 (𝑡) 𝜔𝑖 (𝑡) ,

̃ (𝑥 , 𝑡) = 𝜙 𝜔 𝑓 𝑖 𝑖 ̃ 𝑖 (𝑡) ,

for 𝑖 = 1, . . . , 𝑛.

(2) lim𝑡→∞ (𝑥 − 𝑥0 ) = 0 and lim𝑡→∞ (V − V0 ) = 0. Differentiating the two relative measurements 𝜃𝑖 (𝑡) and 𝜗𝑖 (𝑡) 𝜃̇ 𝑖 (𝑡) = 𝜗𝑖 (𝑡) , 𝜗̇ 𝑖 (𝑡) = ∑ 𝑎𝑖𝑗 (𝑢𝑖 (𝑡) − 𝑢𝑗 (𝑡) + 𝑓 (𝑥𝑖 , 𝑡) − 𝑓 (𝑥𝑗 , 𝑡)) (9) + 𝑏𝑖 (𝑢𝑖 (𝑡) + 𝑓 (𝑥𝑖 , 𝑡) − 𝑎0 (𝑡)) . Thus, we take 𝑥𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝑥0 (𝑡) ,

𝑥 = col {𝑥1 (𝑡) , 𝑥2 (𝑡) , . . . , 𝑥𝑛 (𝑡)} , V = col {V1 (𝑡) , V2 (𝑡) , . . . , V𝑛 (𝑡)} ,

(6)

(7)

(10)

𝜃 (𝑡) = col {𝜃1 (𝑡) , 𝜃2 (𝑡) , . . . , 𝜃𝑛 (𝑡)} , 𝜗 (𝑡) = col {𝜗1 (𝑡) , 𝜗2 (𝑡) , . . . , 𝜗𝑛 (𝑡)} . Then the system can be simplified as 𝜃 (𝑡) = 𝐻𝜎 𝑥 (𝑡) , 𝜗 (𝑡) = 𝐻𝜎 V (𝑡) .

for 𝑖 = 1, . . . , 𝑛. Secondly, we define two variables to describe the relative measurement of position and velocity. The relative position measurement is 𝜃𝑖 (𝑡) = ∑ 𝑎𝑖𝑗 (𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)) + 𝑏𝑖 (𝑥𝑖 (𝑡) − 𝑥0 (𝑡)) ,

(1) lim𝑡→∞ 𝜃 = 0 and lim𝑡→∞ 𝜗 = 0.

(5)

where 𝜙𝑜 (𝑡), 𝜙𝑖 (𝑡) ∈ 𝑅𝑚 are basis function vectors and 𝜔𝑜 , 𝜔𝑖 ∈ 𝑅𝑚 are unknown constant parameter vectors that will be estimated. Each follower 𝑖 estimates the parameter vectors 𝜔𝑜 , 𝜔𝑖 by ̃ 𝑖 (𝑡) ∈ 𝑅𝑚 and estimates 𝑎0 (𝑡), 𝑓(𝑥𝑖 , 𝑡) by 𝑎𝑖 (𝑡), and ̃ 𝑜𝑖 (𝑡), 𝜔 𝜔 ̃ , 𝑡), respectively. So we have 𝑓(𝑥 𝑖 ̃ 𝑜𝑖 (𝑡) , 𝑎𝑖 (𝑡) = 𝜙𝑜 (𝑡) 𝜔

for 𝑖 = 1, . . . , 𝑛. So from the above definition, the following statements are equivalent:

V𝑖 (𝑡) = V𝑖 (𝑡) − V0 (𝑡) ,

3. Adaptive Control Design

𝑎0 (𝑡) = 𝜙𝑜 (𝑡) 𝜔𝑜 (𝑡) ,

(8)

𝑗∈𝑁𝑖 (𝑡)

where 𝑎0 (𝑡) is an unknown acceleration of the leader. Our aim is to design a decentralized control scheme for each agent and study under what conditions the agents can follow the leader (i.e., lim𝑡→∞ (𝑥𝑖 (𝑡) − 𝑥0 (𝑡)) = 0, lim𝑡→∞ (V𝑖 (𝑡) − V0 (𝑡)) = 0).

𝑗∈𝑁𝑖 (𝑡)

The relative velocity measurement is

(11)

Furthermore, 𝜃̇ (𝑡) = 𝜗 (𝑡) , 𝜗̇ (𝑡) = 𝐻𝜎 [𝑢 (𝑡) − 𝑎0 (𝑡) 1𝑛 + 𝑓 (𝑥, 𝑡)] , 𝑓 (𝑥1 , 𝑡) 𝑓 (𝑥, 𝑡) = (

𝑓 (𝑥2 , 𝑡) .. . 𝑓 (𝑥𝑛 , 𝑡)

), (12)

𝑢1 𝑢2 𝑢 = ( . ). .. 𝑢𝑛 1𝑛 denote a column vector where all the elements are 1.

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Mathematical Problems in Engineering

𝜎: [0, ∞) → P{1, 2, . . . , 𝑚∗ } (𝑚∗ denotes the total number of all possible graphs) is a switching signal that determines the communication topology G. 𝐿 is the Laplacian for the 𝑛 followers; the leader adjacency matrix 𝐵 is an 𝑛 × 𝑛 diagonal matrix whose 𝑖th diagonal element is 𝑏𝑖 (𝑡) at time 𝑡 and is utilized to represent the connections between the followers and the leader. Now, we consider the control protocol. If V0 , 𝜔𝑜 , and 𝜔𝑖 are known, we can design the control protocol as 𝑢𝑖 (𝑡) = 𝜙𝑜 (𝑡) 𝜔𝑜 − 𝛾𝜃𝑖 (𝑡) − 𝑘𝜗𝑖 (𝑡) − 𝜙𝑖 (𝑡) 𝜔𝑖 ,

(13)

In our case, 𝜗𝑖 (𝑡) is unknown, so we define a variable 𝜂̂𝑖 for each agent 𝑖 and set 𝜂̂𝑖 = 𝜂𝑖 − 𝑙𝜃𝑖 , and then (15) can be rewritten as 𝜂̂̇ 𝑖 (𝑡) = − (𝛾 + 𝑙𝑘) [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

but in our cases, V0 , 𝜔𝑜 , and 𝜔𝑖 are unknown; we define 𝜂𝑖 (𝑡) as the estimate of 𝜗𝑖 (𝑡) by agent 𝑖. Then the control protocol is

𝜂̇ 𝑖 (𝑡) = −𝑙 (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡))

(20)

− 𝑘 [ ∑ 𝑎𝑖𝑗 (̂𝜂𝑖 (𝑡) − 𝜂̂𝑗 (𝑡)) + 𝑏𝑖 𝜂̂𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

̃ 𝑜𝑖 (𝑡) − 𝛾𝜃𝑖 (𝑡) − 𝑘𝜂𝑖 (𝑡) − 𝜙𝑖 (𝑡) 𝜔 ̃ 𝑖 (𝑡) . (14) 𝑢𝑖 (𝑡) = 𝜙𝑜 (𝑡) 𝜔 From (14), 𝜂𝑖 (𝑡) is unknown, so we design the parameter input of 𝜂𝑖 (𝑡) as

(19)

− 𝑙̂𝜂𝑖 (𝑡) − 𝑙2 𝜃𝑖 (𝑡) . From (19) and (20), we can use only related position measurement to estimate the relative velocity measurement, so the tracking control is 𝜂̂̇ 𝑖 (𝑡)

− 𝛾 [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

= − (𝛾 + 𝑙𝑘) [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

(15)

− 𝑘 [ ∑ 𝑎𝑖𝑗 (𝜂𝑖 (𝑡) − 𝜂𝑗 (𝑡)) + 𝑏𝑖 𝜂𝑖 (𝑡)] , [𝑗∈𝑁𝑖 (𝑡) ]

− 𝑘 [ ∑ 𝑎𝑖𝑗 (̂𝜂𝑖 (𝑡) − 𝜂̂𝑗 (𝑡)) + 𝑏𝑖 𝜂̂𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

𝑙 > 1. Lemma 4. When (15) is satisfied, without consideration of the parameter error of 𝜔(𝑡)0 and 𝜔(𝑡)𝑖 , 𝜂𝑖 (𝑡) is the estimate of 𝜗𝑖 (𝑡); that is, lim𝑡→∞ (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) = 0.

(21)

− 𝑙̂𝜂𝑖 (𝑡) − 𝑙2 𝜃𝑖 (𝑡) , 𝑢𝑖 (𝑡) ̃ 𝑖 (𝑡) = − (𝛾 + 𝑘𝑙) 𝜃𝑖 (𝑡) − 𝑘̂𝜂𝑖 (𝑡) − 𝜙𝑖 𝜔

Proof. From (14) and (15),

̃ 𝑜𝑖 (𝑡) . + 𝜙𝑜 (𝑡) 𝜔

𝜂̇ 𝑖 (𝑡)

And equality is

= −𝑙 (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) + ∑ 𝑎𝑖𝑗 (𝑢𝑖 (𝑡) − 𝑢𝑗 (𝑡) + 𝑓 (𝑥𝑖 , 𝑡) − 𝑓 (𝑥𝑗 , 𝑡))

𝜂̂̇ (𝑡) = − (𝑙2 𝐼𝑛 + 𝑙𝑘𝐻𝜎 + 𝛾𝐻𝜎 ) 𝜃 (𝑡)

(16)

𝑗∈𝑁𝑖 (𝑡)

− (𝑘𝐻𝜎 + 𝑙𝐼𝑛 ) 𝜂̂ (𝑡) ,

+ 𝑏𝑖 (𝑢𝑖 (𝑡) + 𝑓 (𝑥𝑖 , 𝑡) − 𝑎0 (𝑡)) .

̃ 𝑜 (𝑡) 𝑢 (𝑡) = − (𝛾 + 𝑘𝑙) 𝜃 (𝑡) − 𝑘̂𝜂 (𝑡) + Φ𝑜 (𝑡) Ω

̃ 𝑖 (𝑡) = 𝜔𝑖 (𝑡), from (9) ̃ 0𝑖 (𝑡) = 𝜔0 (𝑡) and 𝜔 When 𝜔 𝜂̇ 𝑖 (𝑡) = −𝑙 (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) + 𝜗̇ 𝑖 (𝑡) , 𝜂̇ 𝑖 (𝑡) − 𝜗̇ 𝑖 (𝑡) = −𝑙 (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) .

̃ (𝑡) , − ΦΩ (17)

That is, lim𝑡→∞ (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) = 0. This completes the proof.

where 𝜂̂ = col {̂𝜂1 , 𝜂̂2 , . . . , 𝜂̂𝑛 } , Φ𝑜 (𝑡) = diag {𝜙0 , . . . , 𝜙0 } ,

It is equal to 𝑑 (𝜂 (𝑡) − 𝜗𝑖 (𝑡)) = −𝑙 (𝜂𝑖 (𝑡) − 𝜗𝑖 (𝑡)) . 𝑑𝑡 𝑖

(22)

(18)

Φ (𝑡) = diag {𝜙1 , . . . , 𝜙𝑛 } , ̃ 0 = col {̃ ̃ 0𝑛 } , Ω 𝜔01 , . . . , 𝜔 ̃ 0 = col {̃ ̃ 𝑛} . Ω 𝜔1 , . . . , 𝜔

(23)

Mathematical Problems in Engineering

5 ̂ 01 (𝑡) , . . . , Ω ̂ 0𝑛 (𝑡)} , ̂ 0 (𝑡) = col {Ω Ω

Thus we have designed the control protocol only using the relative position measurement, and it is similar to the protocol, which Hu and Zheng designed in [10]. Now we design the adaptive laws. Firstly we define two ̂ 𝑖 (𝑡) and let ̂ 𝑖0 (𝑡) and 𝜔 parameter variables 𝜔

̂ 𝑛 (𝑡)} , ̂ (𝑡) = col {Ω ̂ 1 (𝑡) , . . . , Ω Ω

(28)

Φ0 (𝑡) = 𝐼𝑛 ⊗ 𝜙0 (𝑡) , Φ (𝑡) = diag {𝜙1 (𝑡) , . . . , 𝜙𝑛 (𝑡)} .

̃ 0𝑖 (𝑡) ̂ 0𝑖 (𝑡) = 𝜔 𝜔 + 2𝜙0 [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] , [𝑗∈𝑁𝑖 (𝑡) ] ̃ 𝑖 (𝑡) ̂ 𝑖 (𝑡) = 𝜔 𝜔

Set

(24)

̃ 0𝑛 (𝑡) − 𝜔0 (𝑡)} , Ω0 (𝑡) = col {̃ 𝜔01 (𝑡) − 𝜔0 (𝑡) , . . . , 𝜔 ̃ 𝑛 (𝑡) − 𝜔𝑛 (𝑡)} . Ω (𝑡) = col {̃ 𝜔1 (𝑡) − 𝜔1 (𝑡) , . . . , 𝜔

(29)

̇ ̃̇ 0 (𝑡) and Ω(𝑡) ̃̇ and = Ω(𝑡), Then we know Ω̇ 0 (𝑡) = Ω adaptive laws (27) can be transformed to

− 2𝜙𝑖 [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] . [𝑗∈𝑁𝑖 (𝑡) ]

Ω̇ 0 (𝑡) = −Φ0 (𝑡) 𝐻𝜎 𝜃 (𝑡) − 2Φ0 (𝑡) 𝐻𝜎 𝜗 (𝑡)

Then ̂ lim 𝜔 𝑡→∞ 0𝑖

+ Φ0 (𝑡) 𝐻𝜎 𝜂 (𝑡) ,

̃ 0𝑖 (𝑡) , (𝑡) = 𝜔

̂ 𝑖 (𝑡) = 𝜔 ̃ 𝑖 (𝑡) . lim 𝜔

(25)

𝑡→∞

Consider lim𝑡→∞ 𝜃𝑖 (𝑡) = 0. That is to say, we can design ̂ 𝑖 (𝑡) to get the value of 𝜔 ̃ 0𝑖 (𝑡) ̂ 0𝑖 (𝑡) and 𝜔 the adaptive laws for 𝜔 ̃ 𝑖 (𝑡). and 𝜔 ̂ 0𝑖 (𝑡) We design the adaptive laws for the two variables 𝜔 ̂ 𝑖 (𝑡) as and 𝜔 ̂̇ 0𝑖 (𝑡) = (𝑙𝜙0 − 𝜙0 + 2𝜙̇ 0 ) 𝜔

̂̇ 𝑖 (𝑡) = − (𝑙𝜙𝑖 − 𝜙𝑖 + 2𝜙̇ 𝑖 ) 𝜔

From the above design and definition, we can rewrite the system as

(31)

𝜂̇ (𝑡) = 𝑙𝜗 (𝑡) − 𝛾𝐻𝜎 𝜃 (𝑡) − (𝑙 + 𝑘𝐻𝜎 ) 𝜂 (𝑡) , (26)

which is equal to 𝛿̇ = 𝐹𝜎 𝛿 (𝑡) + Δ (𝑡) ,

(32)

𝜃 [𝜗] 𝛿 = [ ], [𝜂] 0 𝐼𝑛 [−𝛾𝐻 0 𝐹𝜎 = [ 𝜎

Equation (26) can be rewritten as

0 −𝑘𝐻𝜎

] ],

(33)

[−𝛾𝐻𝜎 𝑙𝐼𝑛 − (𝑙𝐼𝑛 + 𝑘𝐻𝜎 )]

̂̇ 0 (𝑡) = (𝑙Φ0 (𝑡) − Φ0 (𝑡) + 2Φ̇ 0 (𝑡)) 𝐻𝜎 𝜃 (𝑡) Ω

− Φ𝐻𝜎 𝜂̂ (𝑡) ,

4. Consensus Analysis

− 𝐻𝜎 ΦΩ (𝑡) ,

− 𝜙𝑖 [ ∑ 𝑎𝑖𝑗 (̂𝜂𝑖 (𝑡) − 𝜂̂𝑗 (𝑡)) + 𝑏𝑖 𝜂̂𝑖 (𝑡)] . [𝑗∈𝑁𝑖 (𝑡) ]

̂̇ (𝑡) = − (𝑙Φ (𝑡) − Φ (𝑡) + 2Φ̇ (𝑡)) 𝐻𝜎 𝜃 (𝑡) Ω

− Φ (𝑡) 𝐻𝜎 𝜂 (𝑡) .

𝜗̇ (𝑡) = −𝛾𝐻𝜎 𝜃 (𝑡) − 𝑘𝐻𝜎 𝜂 (𝑡) + 𝐻𝜎 Φ0 Ω0 (𝑡)

⋅ [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ]

+ Φ0 𝐻𝜎 𝜂̂ (𝑡) ,

(30)

𝜃̇ (𝑡) = 𝜗 (𝑡) ,

⋅ [ ∑ 𝑎𝑖𝑗 (𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡)) + 𝑏𝑖 𝜃𝑖 (𝑡)] [𝑗∈𝑁𝑖 (𝑡) ] + 𝜙0 [ ∑ 𝑎𝑖𝑗 (̂𝜂𝑖 (𝑡) − 𝜂̂𝑗 (𝑡)) + 𝑏𝑖 𝜂̂𝑖 (𝑡)] , [𝑗∈𝑁𝑖 (𝑡) ]

Ω̇ (𝑡) = Φ (𝑡) 𝐻𝜎 𝜃 (𝑡) + 2Φ (𝑡) 𝐻𝜎 𝜗 (𝑡)

(27)

0 ] [ ] Δ (𝑡) = [ [𝐻𝜎 Φ0 (𝑡) Ω0 (𝑡) − 𝐻𝜎 Φ (𝑡) Ω (𝑡)] . 0 ] [ That is, when lim𝑡→∞ 𝛿(𝑡) = 0, then lim𝑡→∞ 𝑥(𝑡) = 0 and lim𝑡→∞ V(𝑡) = 0.

6

Mathematical Problems in Engineering

By using the definitions and properties of jointly connected topology, we have 𝑇

According to the discussion above, in each subinterval, the control scheme of each connected component is

𝑇

𝑇

𝑥𝑇 (𝑡) 𝐸𝜎 = [𝑥1𝜎 (𝑡) , 𝑥2𝜎 (𝑡) , . . . , 𝑥𝑙𝜎𝜎 (𝑡)] , 𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

𝜖 𝛿̇ 𝜎 (𝑡) = 𝐹𝜎𝜖 𝛿𝜎𝜖 (𝑡) + Δ𝜖𝜎 (𝑡) ,

V𝑇 (𝑡) 𝐸𝜎 = [V1𝜎 (𝑡) , V2𝜎 (𝑡) , . . . , V𝑙𝜎𝜎 (𝑡)] ,

(34)

𝜃𝜎𝜖

[ ] 𝛿𝜎𝜖 = [𝜗𝜎𝜖 ] ,

𝜂𝑇 (𝑡) 𝐸𝜎 = [𝜂𝜎1 (𝑡) , 𝜂𝜎2 (𝑡) , . . . , 𝜂𝜎𝑙𝜎 (𝑡)] .

get

𝜖

[𝜂𝜎 ]

The matrix 𝐸𝜎 is a permutation matrix, 𝐸𝜎𝑇 𝐸𝜎 = 𝐼𝑛 , so we 𝐹𝜎𝜖

𝜃𝑇 (𝑡) 𝐸𝜎 = 𝑥𝑇 (𝑡) 𝐸𝜎 𝐸𝜎𝑇 𝐻𝑇 𝐸𝜎 𝑇

0 0 𝐼𝑑𝜎𝜖 ] [ 𝜖 ], −𝑘𝐻𝜎𝜖 =[ ] [−𝛾𝐻𝜎 0 𝜖 𝜖 𝜖 𝜖 −𝛾𝐻 𝑙𝐼 − (𝑙𝐼 + 𝑘𝐻 ) 𝑑𝜎 𝑑𝜎 𝜎 𝜎 ] [

(39)

𝑇

𝑇

= [𝜃𝜎1 (𝑡) , 𝜃𝜎2 (𝑡) , . . . , 𝜃𝜎𝑙𝜎 (𝑡)] ,

0 ] [ 𝜖 𝜖 𝜖 𝜖 𝜖 𝜖 ] Δ𝜖𝜎 (𝑡) = [ [𝐻𝜎 Φ0𝜎 (𝑡) Ω0𝜎 (𝑡) − 𝐻𝜎 Φ𝜎 (𝑡) Ω𝜎 (𝑡)] . 0 ] [

(35)

𝜗𝑇 (𝑡) 𝐸𝜎 = V𝑇 (𝑡) 𝐸𝜎 𝐸𝜎𝑇 𝐻𝑇 𝐸𝜎 𝑇

(38)

𝑇

𝑇

= [𝜗𝜎1 (𝑡) , 𝜗𝜎2 (𝑡) , . . . , 𝜗𝜎𝑙𝜎 (𝑡)] .

Lemma 5. When graphs G𝜎 are jointly connected, the leader connects to one follower at least; then

Meanwhile, 𝐸𝜎𝑇 𝐻𝜎 Φ0 (𝑡) Ω0 (𝑡) = 𝐸𝜎𝑇 𝐻𝜎 𝐸𝜎 𝐸𝜎𝑇 Φ0 (𝑡) Ω0 (𝑡) ,

𝜆𝐻𝜎𝜖 ≥ 0,

𝐸𝜎𝑇 𝐻𝜎 Φ (𝑡) Ω (𝑡) = 𝐸𝜎𝑇 𝐻𝜎 𝐸𝜎 𝐸𝜎𝑇 Φ (𝑡) Ω (𝑡) , 𝑙𝜎

𝐸𝜎𝑇 Φ0 (𝑡) Ω0 (𝑡) =

(36)

1 [Φ10𝜎 (𝑡) Ω0𝜎

𝐸𝜎𝑇 Φ (𝑡) Ω (𝑡)

=

𝑙𝜎 𝑇 𝑙 (𝑡) , . . . , Φ0𝜎𝜎 (𝑡) Ω0𝜎 1 (𝑡) Ω𝜎

[Φ1𝜎

(𝑡)] ,

(𝑡) , . . . , Φ𝑙𝜎𝜎

𝑙𝜎 (𝑡) Ω𝜎

where 𝜆𝐻𝜎𝜖 is the minimum eigenvalue of 𝐻𝜎𝜖 (𝜖 = 1, . . . , 𝑙𝜎 ).

𝑇

(𝑡)] ,

Proof. Assuming the leader is connected to one follower in the 𝜒th connected components, thus 𝐵𝜎𝜒 > 0 and 𝐵𝜎𝜖 = 0 (𝜖 ≠ 𝜒); then 𝐿 𝜎 𝜖 = 𝐻𝜎 𝜖 (𝜖 ≠ 𝜒), and

𝐸𝜎𝑇 𝐻𝜎 Φ0 (𝑡) Ω0 (𝑡) 1

𝑙

𝑙

𝐸𝜎𝑇 𝐻𝜎 Φ (𝑡) Ω (𝑡) =

𝐸𝜎 𝑇 𝐻𝜎 𝐸𝜎 = diag [𝐿 𝜎 1 , 𝐿 𝜎 2 , . . . , 𝐻𝜎 𝜒 , . . . , 𝐿 𝜎 𝑙𝜎 ] .

𝑇

𝑇

= [𝐻𝜎 1 Φ10𝜎 (𝑡) Ω0𝜎 (𝑡) , . . . , 𝐻𝜎 𝑙𝜎 Φ0𝜎𝜎 (𝑡) Ω0𝜎𝜎 (𝑡)] ,

[𝐻𝜎 Φ1𝜎

(40)

𝜖=1

𝑇

so

1

∑𝜆𝐻𝜎𝜖 > 0,

(41)

(37) According to Lemmas 1 and 2, 𝜆𝐻𝜎𝜒 > 0 and when 𝑖 ≠ 𝜒,

1 (𝑡) Ω𝜎

𝑙𝜎

(𝑡) , . . . , 𝐻𝜎 Φ𝑙𝜎𝜎

𝑙𝜎 (𝑡) Ω𝜎

𝑇

(𝑡)] ,

𝑙

𝜎 𝜆𝐻𝜎𝜖 > 0. 𝜆𝐻𝜎𝜖 = 0. So ∑𝜖=1 The proof is complete.

Lemma 6. Consider a function 𝑓(𝛿) = 𝛿𝑇 Υ𝛿, when Υ =

where

𝑏1 𝑎1 𝑎2 𝑏2 𝑎3 𝑎2 𝑎3 𝑏3

[ 𝑎1

𝜖 = 1, . . . , 𝑙𝜎 , 𝜖

𝜖

𝐿 𝜎 𝜖 ∈ 𝑅𝑑𝜎 ×𝑑𝜎 is the Laplacian of the corresponding connected graph, 𝜖

] ⊗ 𝐼𝑛 , and 𝑎1 , 𝑎2 , 𝑎3 , 𝑏1 , 𝑏2 , 𝑏3 are constants; then

𝑑𝜎𝜖 ×𝑑𝜎𝜖

is the leader adjacency matrix of each 𝐵𝜎 ∈ 𝑅 connected graph. Thus 𝐻𝜎𝜖 = 𝐿𝜖𝜎 + 𝐵𝜎𝜖 . Each block matrix 𝐻𝜎𝜖 describes the connection of the corresponding connected components.

𝑙𝜎

𝑇

𝛿𝑇 Υ𝛿 = ∑𝛿𝜎𝜖 Υ𝜎𝜖 𝛿𝜎𝜖 , 𝜖=1

𝑏1 𝑎1 𝑎2 [𝑎 𝑏 𝑎 ] 𝜖 Υ𝜎 = [ 1 2 3 ] ⊗ 𝐼𝑑𝜎𝜖 . [𝑎2 𝑎3 𝑏3 ]

(42)

Mathematical Problems in Engineering

7

Proof. From Lemma 3 and 𝐸𝜎𝑇 𝐸𝜎 = 𝐼𝑛 , thus

Then the derivative of 𝑉(𝑡) along the trajectory of system (32) is given by

𝐸𝜎𝑇 0

0 [ ] ] 𝑇 ] 𝛿 Υ𝛿 = 𝛿 [ 0 𝐸𝜎 0 ] [ [ 0 𝐸𝜎 0 ] Υ [ 0 0 𝐸𝜎 ] [ 0 0 𝐸𝜎𝑇 ] 𝑇[

𝑇

𝐸𝜎 0

0

𝑉̇ (𝑡) = 𝛿 (𝑡)𝑇 (𝐹𝜎𝑇 𝑃 + 𝑃𝐹𝜎 ) 𝛿 (𝑡) + 2𝛿 (𝑡)𝑇 𝑃Δ (𝑡) 𝑇 𝑇 + 2Ω0 (𝑡) Ω̇ 0 (𝑡) + 2Ω (𝑡) Ω̇ (𝑡)

𝐸𝜎𝑇

0 0 𝐸𝜎 0 0 ] [ 0 𝐸 0 ][ 𝑇 ] ×[ ][ 𝜎 [ 0 𝐸𝜎 0 ] 𝛿 [ 0 0 𝐸𝜎 ] [ 0 0 𝐸𝜎𝑇 ]

= 𝛿 (𝑡)𝑇 𝑄𝜎 𝛿 (𝑡) ,

= 𝜃𝑇 𝐸𝜎 𝐸𝜎𝑇 Υ𝐸𝜎 𝐸𝜎𝑇 𝜃 + 𝜗𝑇 𝐸𝜎 𝐸𝜎𝑇 Υ𝐸𝜎 𝐸𝜎𝑇 𝜗

−2𝛾𝐻𝜎 𝛾𝐼𝑛 − 𝛾𝐻𝜎 −𝑘𝐻𝜎 [𝛾𝐼 − 𝛾𝐻 2𝐼 ] =[ 𝑛 𝜎 𝑛 (1 − 𝑙) 2𝑙𝐼𝑛 − 𝑘𝐻𝜎 ] .

(43)

+ 𝜂𝑇 𝐸𝜎 𝐸𝜎𝑇 Υ𝐸𝜎 𝐸𝜎𝑇 𝜂 =

𝑇 𝑇 𝜃𝜎1 Υ𝜎1 𝜃𝜎1 𝑇

+ ⋅⋅⋅ +

𝑇 𝑇 𝜃𝜎𝑙𝜎 Υ𝜎𝑙𝜎 𝜃𝜎𝑙𝜎

𝑇

𝑇

+

[ −𝑘𝐻𝜎

𝑇 𝑇 𝜗𝜎1 Υ𝜎1 𝜗𝜎1 𝑇

𝑇

𝑙𝜎

𝑙𝜎

−2𝛾𝐻𝜎𝜖

Theorem 8. Consider the leader-follower system (32). The interconnection network of the system is jointly connected across each time interval [𝑡𝑟 , 𝑡𝑟+1 ) (𝑟 = 0, 1, . . .) and 𝜙𝑖 , (𝑖 = 0, 1, . . . , 𝑛) are uniformly bounded. When 𝑘 and 𝛾 satisfy (44), the consensus tracking is achieved:

𝛾𝐼𝑑𝜎𝜖 − 𝛾𝐻𝜎𝜖

(49)

−2𝛾𝐻𝜎𝜖

𝛾𝐼𝑑𝜎𝜖 − 𝛾𝐻𝜎𝜖

2

8𝑙 (𝜆𝐻𝜎𝜖 + 1)

−1

𝛾𝐻𝜎𝜖 − 𝛾𝐼𝑑𝜎𝜖

−𝑘𝐼𝑑𝜎𝜖 ] [ −1 ] [ 𝜖−1 𝜖−1 = [𝛾𝐻𝜎 − 𝛾𝐼𝑑𝜎𝜖 2 (1 − 𝑙) 𝐻𝜎 2𝑙𝐻𝜎𝜖 − 𝑘𝐼𝑑𝜎𝜖 ] ] [ 𝜖−1 𝜖−1 𝜖 𝜖 −𝑘𝐼 2𝑙𝐻 − 𝑘𝐼 −2𝑙𝐻 𝑑𝜎 𝑑𝜎 𝜎 𝜎 ] [

,

(51)

̂ 𝜖 ⊗ 𝐻𝜖 , ⊗ 𝐻𝜎𝜖 = 𝑄 𝜎 𝜎

2

𝑀 = 𝜆𝐻𝜎𝜖 [3𝑘2 − 4𝑙2 − 8𝑘𝑙 + 𝜆𝐻𝜎𝜖 (4𝑙2 − 𝑙 + 𝑘2 + 8𝑘𝑙) (45)

9 2 − 𝜆𝐻𝜎𝜖 𝑘2 ] , 4

−𝑘𝐻𝜎𝜖

[ 𝜖 𝜖] ] 𝑄𝜎𝜖 = [ [𝛾𝐼𝑑𝜎𝜖 − 𝛾𝐻𝜎 2𝐼𝑑𝜎𝜖 (1 − 𝑙) 2𝑙𝐼𝑑𝜎𝜖 − 𝑘𝐻𝜎 ] 𝜖 2𝑙𝐼𝑑𝜎𝜖 − 𝑘𝐻𝜎𝜖 −2𝑙𝐼𝑑𝜎𝜖 ] [ −𝑘𝐻𝜎

(44)

4

(50)

Let

−2𝛾𝐼𝑑𝜎𝜖

√ 16𝑙 (𝜆𝐻𝜖 − 1) (𝑘2 𝜆𝐻𝜖 (𝑙 − 1)) + 𝑀2 − 𝑀 𝜎 𝜎

−𝑘𝐻𝜎𝜖

[ 𝜖 𝜖] ] 𝑄𝜎𝜖 = [ [𝛾𝐼𝑑𝜎𝜖 − 𝛾𝐻𝜎 2𝐼𝑑𝜎𝜖 (1 − 𝑙) 2𝑙𝐼𝑑𝜎𝜖 − 𝑘𝐻𝜎 ] . 𝜖 2𝑙𝐼𝑑𝜎𝜖 − 𝑘𝐻𝜎𝜖 −2𝑙𝐼𝑑𝜎𝜖 ] [ −𝑘𝐻𝜎

, 2

𝛾>

𝑇

𝜖=1

Lemma 7 (Barbalat’s lemma, Popov [18]). If a function 𝑓(𝑡) is 𝑡 uniformly continuous and lim𝑡→0 ∫0 𝑓(𝑠)𝑑𝑠 exists and is finite, then lim𝑡→0 𝑓(𝑡) = 0.

𝜆𝐻𝜎𝜖

]

𝑉̇ (𝑡) = 𝛿 (𝑡)𝑇 𝑄𝜎 𝛿 (𝑡) = ∑𝛿𝜎𝜖 (𝑡) 𝑄𝜎𝜖 𝛿𝜎𝜖 (𝑡) ,

𝑇 ∑𝛿𝜎𝜖 Υ𝜎𝜖 𝛿𝜎𝜖 . 𝜖=1

4𝑙𝛾

−2𝑙𝐼𝑛

From Lemma 5,

𝑇

The proof is complete.

𝑘 < 2√

2𝑙𝐼𝑛 − 𝑘𝐻𝜎

+ ⋅⋅⋅

+ 𝜗𝜎𝑙𝜎 Υ𝜎𝑙𝜎 𝜗𝜎𝑙𝜎 + 𝜂𝜎1 Υ𝜎1 𝜂𝜎1 + ⋅ ⋅ ⋅ + 𝜂𝜎𝑙𝜎 Υ𝜎𝑙𝜎 𝜂𝜎𝑙𝜎 =

(48)

𝑄𝜎 = (𝐹𝜎𝑇 𝑃 + 𝑃𝐹𝜎 )

̂𝜖 𝑄 𝜎

−1

𝛾𝐻𝜎𝜖 − 𝛾𝐼𝑑𝜎𝜖

−𝑘𝐼𝑑𝜎𝜖 ] [ −1 ] [ 𝜖−1 𝜖−1 = [𝛾𝐻𝜎 − 𝛾𝐼𝑑𝜎𝜖 2 (1 − 𝑙) 𝐻𝜎 2𝑙𝐻𝜎𝜖 − 𝑘𝐼𝑑𝜎𝜖 ] . ] [ 𝜖−1 𝜖−1 𝜖 𝜖 −𝑘𝐼 2𝑙𝐻 − 𝑘𝐼 −2𝑙𝐻 𝑑𝜎 𝑑𝜎 𝜎 𝜎 ] [ −2𝛾𝐼𝑑𝜎𝜖

2

where 𝜆𝐻𝜎𝜖 is the maximal eigenvalue of 𝐻𝜎𝜖 .

From (49), we have

Proof. Consider a common Lyapunov function candidate 𝑇

𝑉 (𝑡) = 𝛿 (𝑡) 𝑃𝛿 (𝑡) +

𝑇 Ω0 (𝑡) Ω0

2𝛾𝐼𝑛 𝐼𝑛 0 [ 𝐼 2𝐼 −𝐼 ] 𝑃=[ 𝑛 𝑛 𝑛 ] > 0, [ 0

−𝐼𝑛 𝐼𝑛 ]

𝑇

𝑙𝜎

(𝑡) + Ω (𝑡) Ω (𝑡) ,

(46)

𝛾 > 1.

(47)

𝑙𝜎

󵄩󵄩 𝜖 󵄩󵄩2 𝜖𝑇 𝜖 𝑉̇ (𝑡) ≤ ∑𝜆𝑄 ̂𝜖 𝜆𝐻𝜎𝜖 𝛿𝜎 𝛿𝜎 ≤ ∑ 𝜆𝑄 ̂𝜖 𝜆𝐻𝜎𝜖 󵄩 󵄩𝛿𝜎 󵄩󵄩 . 𝜖=1

𝜎

𝜖=1

𝜎

𝜖

(52)

𝜖 ̂ 𝜆𝑄 ̂𝜖 , 𝜆𝐻𝜎𝜖 are the maximal eigenvalue of 𝑄𝜎 and 𝐻𝜎 , respec𝜎 tively.

8

Mathematical Problems in Engineering After calculation, we have 𝜆𝑄 ̂𝜖 < 0, when 𝑘, 𝛾, and 𝑙 satisfy 𝜎

4𝑙𝛾

𝑘 < 2√

𝜆𝐻𝜎𝜖

, 2

𝛾>

𝑀=

2 8𝑙 (𝜆𝐻𝜎𝜖 2 𝜆𝐻𝜎𝜖

(53)

4

√ 16𝑙 (𝜆𝐻𝜖 − 1) (𝑘2 𝜆𝐻𝜖 (𝑙 − 1)) + 𝑀2 − 𝑀 𝜎 𝜎 + 1)

Theorem 9. If the PE condition (61) (which will be mention later) is satisfied and 𝜙̇ 𝑖 (𝑖 = 0, 1, . . . , 𝑛) are uniformly bounded, by using adaptive law (26), the parameter estimation errors converge to zero. Before proving the theorem, we introduce the PE condition. We take Φ𝑟 (𝑡) = [

,

(54)

9 2 − 𝜆𝐻𝜎𝜖 𝑘2 ] , 4 2

where 𝜆𝐻𝜎𝜖 is the maximal eigenvalue of 𝐻𝜎𝜖 . We know 𝜆𝐻𝜎𝜖 ≥ 0, so from (52) and Lemma 4, we get 𝑙𝜎

𝑇

∫

(55)

∫

𝜖=1

∫

𝑡𝑠+1

𝑡𝑠

or ∫

𝑡𝑠+1

𝑡𝑠

𝑡𝑠+𝜏

𝑡𝑠

𝑡+𝑇0

𝑉̇ (𝑡) 𝑑𝑡 > −𝑐.

𝛿𝑇 (𝑡) 𝛿 (𝑡) 𝑑𝑡 < ∑ ∫

𝑡𝑠+1

𝜖=1 𝑡𝑠

(61) Φ𝜖𝑟𝜎

𝑇 (𝑡) Φ𝜖𝑟𝜎

(𝑡) 𝑑𝑠 ≥ 𝜌𝐼𝑛 , ∀𝜖 = 1, . . . , 𝑙𝜎 .

(62)

∀𝑡 ≥ 𝑇1 , ∀𝜖 = 1, . . . , 𝑙𝜎 .

𝑇

𝛿𝜎𝜖 (𝑡) 𝛿𝜎𝜖 (𝑡) 𝑑𝑡 (57)

𝑙𝜎

Φ𝑟 (𝑠) Φ𝑇𝑟 (𝑠) 𝑑𝑠 ≥ 𝜌𝐼𝑛 ,

󵄩󵄩 󵄩 󵄩󵄩Ω0 (𝑡)󵄩󵄩󵄩 > 𝑐, 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩Ω (𝑡)󵄩󵄩󵄩 > 𝑐, 󵄩 󵄩 󵄩󵄩 𝜖 󵄩󵄩 󵄩󵄩󵄩Ω0𝜎 (𝑡)󵄩󵄩󵄩 > 𝑐, 󵄩󵄩 𝜖 󵄩󵄩 󵄩󵄩Ω𝜎 (𝑡)󵄩󵄩 > 𝑐, 󵄩 󵄩

(56)

𝑙𝜎

(60)

Proof. We prove Theorem 9 by contradiction. Assume that, for any constant 𝑐 > 0, there exists 𝑇1 > 0; then

𝑉̇ (𝑡) 𝑑𝑡 < 𝑐

From (53) and Lemma 5, it follows that ∫

𝑡+𝑇0

𝑡

Therefore, lim𝑡→∞ 𝑉(𝑡) = 𝑉(∞) exists. Next, we will show that lim𝑡→∞ 𝛿(𝑡) = 0. Consider the infinite sequence {𝑉(𝑡𝑠 ) | 𝑠 = 0, 1, . . .}. From the Cauchy convergence criteria, we know, for any 𝑐 > 0, there exists a positive integer 𝑁𝑐 such that |𝑉(𝑡𝑠+1 ) − 𝑉(𝑡𝑠 )| < 𝑐, ∀𝑠 > 𝑁𝑐 . Then we have

],

The matrix Φ𝑟 (𝑡) is persistently exciting (PE) (Marino and Tomei [17]); that is, there exist two positive real numbers, 𝑇0 and 𝜌, such that

𝑡

𝑉̇ (𝑡) ≤ ∑𝛿𝜎𝜖 (𝑡) 𝑄𝜎𝜖 𝛿𝜎𝜖 (𝑡) ≤ 0.

Φ (𝑡)

Φ𝜖0 (𝑡) (𝑡) = [ 𝜖𝜎 ] . Φ𝜎 (𝑡)

Φ𝜖𝑟𝜎

[3𝑘2 − 4𝑙2 − 8𝑘𝑙 + 𝜆𝐻𝜎𝜖 (4𝑙2 − 𝑙 + 𝑘2 + 8𝑘𝑙)

Φ0 (𝑡)

𝑐 < −∑ , 𝜖 ̂ 𝜆𝐻𝜎𝜖 𝜖=1 𝜆𝑄

From Section 2.1, for the infinite sequence of time intervals [𝑡𝑟 , 𝑡𝑟+1 ), 𝑟 = 0, 1, . . ., has the subintervals [𝑡𝑟𝑗 , 𝑡𝑟𝑗+1 ), with identical length 𝑡Δ ; that is, 𝑡𝑟𝑗+1 = 𝑡𝑟𝑗 + 𝑡Δ . Define a function Ψ (𝑡) =

𝜎

1 𝑇 [Ω (𝑡 + 𝑡Δ ) Ω𝐸 (𝑡 + 𝑡Δ ) − Ω𝑇𝐸 (𝑡) Ω𝐸 (𝑡)] , 2 𝐸 𝑇

(63)

Ω𝐸 (𝑡) = [Ω0 (𝑡) , Ω (𝑡)] .

which implies that 𝑡𝑠+𝜏

lim ∫

𝑡→∞ 𝑡 𝑠

𝛿𝑇 (𝑡) 𝛿 (𝑡) = 0.

(58)

Moreover, from the Lyapunov function 𝑉(𝑡) and (52), 𝛿(𝑡), ̇ Ω(𝑡)0 and Ω(𝑡) are uniformly bounded and 𝛿(𝑡) is also bounded. By the assumption that 𝜙0 (𝑡) and 𝜙𝑖 (𝑡) are uniformly bounded, therefore 𝛿𝑇 (𝑡)𝛿(𝑡) is uniformly continuous. Invoking Barbalat’s lemma (Lemma 7), we get lim 𝛿𝑇 (𝑡) 𝛿 (𝑡) = 0,

𝑡→∞

(59)

which implies that lim𝑡→∞ 𝜃(𝑡) = 0 and lim𝑡→∞ 𝜗(𝑡) = 0; thus the consensus tracking is achieved.

And we have Ψ𝜎𝜖 (𝑡) 𝑙

=

𝑇 𝑇 1 𝜎 ∑ [Ω𝜖 (𝑡 + 𝑡Δ ) Ω𝜖𝐸𝜎 (𝑡 + 𝑡Δ ) − Ω𝜖𝐸𝜎 (𝑡) Ω𝜖𝐸𝜎 (𝑡)] , (64) 2 𝜖=1 𝐸𝜎

𝜖

𝑇

𝜖

Ω𝜖𝐸𝜎 (𝑡) = [Ω0𝜎 (𝑡) , Ω𝜎 (𝑡)] . We know lim𝑡→∞ 𝑉(𝑡) = 𝑉(∞); from (59) and (46) we have lim𝑡→∞ Ω𝑇𝐸 (𝑡)Ω𝐸 (𝑡) = 𝑉(∞); it shows that lim Ψ (𝑡) = 0.

𝑡→∞

(65)

Mathematical Problems in Engineering

9

From (63), for ∀𝛼 > 0, there exists 𝑇𝛼 > 0 such that 󵄩󵄩 󵄩 󵄩󵄩Ψ (𝑡) − Ψ (𝑡󸀠 )󵄩󵄩󵄩 < 𝛼, ∀𝑡, 𝑡󸀠 > 𝑇𝛼 . 󵄩 󵄩

2

⋅𝑐 ∫

𝑡𝑟𝑗

(66)

𝑙𝜎

𝑡𝑟𝑗 +𝑡Δ

𝑙𝜎

= ∑∫

𝑡𝑟𝑗 +𝑡Δ

𝑙𝜎

= ∑∫

𝑡𝑟𝑗 +𝑡Δ

𝜖=1 𝑡𝑟𝑗

𝜖=1

𝑇

Thus

𝑇

𝑇

{[𝛿𝜎𝜖 (𝑠) 𝐾𝜎𝜖 𝐻𝜎𝜖 Φ𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠) − Ω𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠)]

𝑙𝜎

𝑡𝑟𝑗 +𝑡Δ

𝜖=1 𝑡𝑟𝑗

𝑇

2

𝑇

𝑇

Ω𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠) 𝐻𝜎𝜖 Φ𝜖𝑟𝜎 (𝑠) Ω𝜖𝑟𝜎 (𝑠) 𝑑𝑠 = 𝜉1 − 𝜉2 ,

𝑙𝜎

1 Ψ̇ (𝑡) = 𝜉1 − 𝜉2 ≤ − 𝜌𝑐2 ∑𝜆2𝐻𝜎𝜖 , 2 𝜖=1

𝜖𝑇

𝜖

Ω𝜖𝑟𝜎 (𝑠) = col [−Ω0𝜎 (𝑡), Ω𝜎 (𝑡)], So

𝜉1 and 𝜉2 denote the first and the second integrals in the third equality of (67), respectively. 𝜖

We know 𝑉(𝑡) is bounded, so 𝛿𝜎𝜖 (𝑡) and Ω𝐸𝜎 are bounded. Then we can set two constants 𝑌𝛿 , 𝑌Ω > 0, which satisfy 𝑌𝛿 > ‖𝛿𝜎𝜖 ‖ and 𝑌Ω > ‖Ω𝜖𝐸𝜎 ‖ for all 𝜖 = 1, . . . , 𝑙𝜎 . 𝜖 Meanwhile, Φ𝜖𝑟𝜎 (𝑠) and Φ̇ 𝑟𝜎 (𝑠) are assumed to be bounded by 𝑌Φ and 𝑌Φ̇ , respectively, for all 𝜖 = 1, . . . , 𝑙𝜎 . Then we have 𝜉1 ≤ ∑𝑌𝜎𝜖 ∫

𝑡𝑟𝑗 +𝑡Δ

𝑡𝑟𝑗

𝜖=1

󵄩󵄩 𝜖 󵄩󵄩 󵄩󵄩𝛿𝜎 (𝑠)󵄩󵄩 𝑑𝑠,

(68)

where 𝑌𝜎𝜖 = √6𝜆𝐻𝜎𝜖 [√6𝜆𝐻𝜎𝜖 𝑌𝛿 𝑌Φ2 + 𝑌Ω 𝑌Φ̇ + ‖𝐸𝜎𝜖 ‖𝑌Ω 𝑌Φ ]. ‖𝐸𝜎𝜖 ‖ denotes the norm bound of 𝐸𝜎𝜖 , which depends on 𝛾, 𝑘, 𝑙, and 𝜆𝐻𝜎𝜖 . Since lim𝑡→∞ 𝛿(𝑡) = 0, we obtain that ∀𝑐 > 0; there exists 𝑇2 > 0, such that 𝑙

𝜎 3 𝜉1 ≤ 𝜌𝑐2 ∑𝜆2𝐻𝜎𝜖 , ∀𝑡𝑟𝑗 ≥ 𝑇2 . 2 𝜖=1

(69)

Now we consider 𝜉2 . From the assumption (62), we have ‖Ω𝑟 (𝑡)‖ > 𝑐 and ‖Ω𝜖𝑟𝜎 (𝑡)‖ > 𝑐. From the PE condition (61), we get 𝑙𝜎

𝜉2 ≥ 2∑𝜆2𝐻𝜎𝜖 ∫ 𝜖=1 𝑙𝜎

≥ 2∑𝜆2𝐻𝜎𝜖 𝜖=1

𝑡𝑟𝑗 +𝑡Δ

𝑡𝑟𝑗

𝑇

(72) ∀𝑡 ∈ [𝑡𝑟𝑗 , 𝑡𝑟𝑗 + Δ𝑇] .

𝑇 Φ𝜖𝑟𝜎 (𝑠) = col [Φ𝜖0𝜎 (𝑡), Φ𝜖𝜎 (𝑡)], 𝐾𝜎𝜖 = col [1, 2, −1] ⊗ 𝐼𝑑𝜎𝜖 ,

𝑙𝜎

(71)

where 𝑇3 = max [𝑇1 , 𝑇𝛼 , 𝑇2 ]. By the sign-preserving theorem of continuous functions, there exists a time interval [𝑡𝑟𝑗 , 𝑡𝑟𝑗 + Δ𝑇) with 𝑡𝑟𝑗 ≥ 𝑇3 , Δ𝑇 > 0 such that

𝑇

𝑙𝜎

1 Ψ̇ (𝑡𝑟𝑗 ) = 𝜉1 − 𝜉2 ≤ − 𝜌𝑐2 ∑𝜆2𝐻𝜎𝜖 , ∀𝑡𝑟𝑗 ≥ 𝑇3 , 2 𝜖=1

(67)

⋅ 𝐻𝜎𝜖 𝐾𝜎𝜖 − Ω𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠) 𝐻𝜎𝜖 𝐾𝜎𝜖 𝐸𝜎𝜖 } 𝛿𝜎𝜖 (𝑠) 𝑑𝑠 − 2∑ ∫

∀𝑡𝑟𝑗 ≥ 𝑇1 . (70)

𝑇 𝑑 [Ω𝜖 (𝑠) Φ𝜖𝑟𝜎 (𝑠) (𝐼𝑛 + 2𝐼𝑛 − 𝐼𝑛 ) 𝐻𝜎𝜖 𝛿𝜎𝜖 (𝑠)] 𝑑𝑠 𝑑𝑠 𝑟𝜎

𝜖=1 𝑡𝑟𝑗

≥ 2𝜌𝑐2 ∑𝜆2𝐻𝜎𝜖 ,

𝑇 𝑑 𝜖 [Ω𝜖 (𝑠) Ω̇ 𝐸𝜎 (𝑠)] 𝑑𝑠 𝑑𝑠 𝐸𝜎

𝜖=1 𝑡𝑟𝑗

𝑇

Ω𝜖𝑟𝜎 (𝑠) Ω𝜖𝑟𝜎 (𝑠) 𝜖 𝜖𝑇 Φ Φ (𝑠) (𝑠) 󵄩 󵄩 𝑑𝑠 󵄩󵄩 𝜖 󵄩󵄩 𝜖 𝑟𝜎 󵄩󵄩Ω𝑟 (𝑠)󵄩󵄩󵄩 𝑟𝜎 󵄩󵄩Ω𝑟 (𝑠)󵄩󵄩󵄩 󵄩 𝜎 󵄩 󵄩 𝜎 󵄩

𝑙𝜎

The time derivative of Ψ(𝑡) at time instant 𝑡𝑟𝑗 is given by Ψ̇ (𝑡𝑟𝑗 ) = ∑ ∫

𝑡𝑟𝑗 +𝑡Δ

𝑇

𝑇

Ω𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠) Φ𝜖𝑟𝜎 (𝑠) Ω𝜖𝑟𝜎 (𝑠) 𝑑𝑠

∫

𝑡𝑟𝑗 +Δ𝑇

𝑡𝑟𝑗

𝑙

𝜎 1 Ψ̇ (𝑡) 𝑑𝑡 = 𝜌𝑐2 Δ𝑇∑𝜆2𝐻𝜎𝜖 . 2 𝜖=1

𝑙

𝜎 𝜆2𝐻𝜎𝜖 From Lemma 5, ∑𝜖=1

2)𝜌𝑐

2

𝑙𝜎 Δ𝑇 ∑𝜖=1

𝜆2𝐻𝜎𝜖 ;

>

0, we set 𝛼

(73) =

(1/

that is to say,

Ψ (𝑡𝑟𝑗 ) − Ψ (𝑡𝑟𝑗 + Δ𝑇) > 𝛼,

(74)

which contradicts (66). Therefore, conjecture (62) is not true and then the parameter convergence is guaranteed. Hence, the proof is complete.

5. Simulation Results Some numerical simulations will be given in this section to illustrate the results of this paper. Figure 1 shows six different graphs each with six followers (labeled by 1−6) and one leader (labeled by 0). The communication topology switches every 0.5 s. We set 𝑙 = 3; then by using Theorem 8, we choose 𝑘 = 3, 𝛾 = 2, and using the control protocol (21) and adaptive law (26), we obtain the simulation results about tracking error as shown in Figures 2 and 3 and the performance of parameter estimation as shown in Figures 4 and 5, respectively. Figure 2 shows the trajectories of velocity errors (V𝑖 − V0 ) between the followers and the leader, and Figure 3 shows the trajectories of position errors (𝑥𝑖 − 𝑥0 ) between the followers and the leader. It is clear that all the line will converge to zero; it means that the tracking errors of velocity and position of each agent will become zero; that is to say all followers can follow the leader. Figure 4 shows the parameter estimation errors of Ω0 of each agent (̃ 𝜔(𝑡)0𝑖 − 𝜔(𝑡)0 ), and Figure 5 shows the parameter

10

Mathematical Problems in Engineering 0

0

0

1

2

3

1

2

3

1

2

3

4

5

6

4

5

6

4

5

6

G1

G2

0

G3 0

0

1

2

3

1

2

3

1

2

3

4

5

6

4

5

6

4

5

6

G4

G5

G6

20

20

15

15

10 Position tracking error

Velocity tracking error

Figure 1: Six undirect graphs.

5 0 −5 −10

5 0 −5 −10

−15 −20

10

0

5

10

Agent 1 Agent 2 Agent 3

15 Time

20

25

30

−15

0

5

10

15

20

25

30

Time Agent 4 Agent 5 Agent 6

Agent 1 Agent 2 Agent 3

Agent 4 Agent 5 Agent 6

Figure 2: The velocity error.

Figure 3: The position error.

estimation errors of Ω𝑖 of each agent (̃ 𝜔(𝑡)𝑖 − 𝜔(𝑡)𝑖 ). From the figures, we can ensure that the parameter estimation errors of Ω0 and Ω can converge to zero; it means the decentralized adaptive laws (26) can estimate the unknown time-varying disturbance (𝑓(𝑥𝑖 , 𝑡) = 𝜙𝑖 𝜔𝑖 ) and the unknown acceleration of the leader (𝑎0 (𝑡) = 𝜙𝑜 (𝑡)𝜔𝑜 ).

and employed decentralized adaptive laws to estimate the unknown disturbances. Afterwards, with the help of a common Lyapunov function method and a PE condition, we ensured the tracking stability and the parameter convergence. Finally, simulation results are given to demonstrate the performance of the proposed adaptive tracking control.

6. Conclusion

Competing Interests

In this paper, we study the adaptive tracking control designed for a second-order leader-following system in jointly connected topology. Moreover, the multiagent system contained unknown disturbance dynamics and the velocity of the leader that is unmeasurable by the followers. To solve such a consensus tracking problem, we proposed a dynamic output-feedback control protocol for tracking the leader

The authors declare that they have no competing interests.

Acknowledgments This work is supported by (1) the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20130185110023, (2) National Natural Science

Mathematical Problems in Engineering

11

0.2

The parameter estimation errors

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

0

5

10

15 Time

Agent 1 Agent 2 Agent 3

20

25

30

Agent 4 Agent 5 Agent 6

Figure 4: The parameter estimation errors of 𝜔0 . 0.8

The parameter estimation errors

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

5

Agent 1 Agent 2 Agent 3

10

15 Time

20

25

30

Agent 4 Agent 5 Agent 6

Figure 5: The parameter estimation errors of 𝜔𝑖 .

Foundation of China under Grants nos. 61104104, 61473061, and 71503206, and (3) the Program for New Century Excellent Talents under Grant no. NCET-13-0091.

References [1] W. Ren, “Multi-vehicle consensus with a time-varying reference state,” Systems & Control Letters, vol. 56, no. 7-8, pp. 474–483, 2007. [2] Y. Hong, J. Hu, and L. Gao, “Tracking control for multiagent consensus with an active leader and variable topology,” Automatica, vol. 42, no. 7, pp. 1177–1182, 2006.

[3] J. Hu and Y. Hong, “Leader-following coordination of multiagent systems with coupling time delays,” Physica A: Statistical Mechanics and its Applications, vol. 374, no. 2, pp. 853–863, 2007. [4] K. Peng and Y. Yang, “Leader-following consensus problem with a varying-velocity leader and time-varying delays,” Physica A: Statistical Mechanics and its Applications, vol. 388, no. 2-3, pp. 193–208, 2009. [5] Q. Song, J. Cao, and W. Yu, “Second-order leader-following consensus of nonlinear multi-agent systems via pinning control,” Systems & Control Letters, vol. 59, no. 9, pp. 553–562, 2010. [6] Y. Hong, G. Chen, and L. Bushnell, “Distributed observers design for leader-following control of multi-agent networks,” Automatica, vol. 44, no. 3, pp. 846–850, 2008. [7] J. Hu, J. Geng, and H. Zhu, “An observer-based consensus tracking control and application to event-triggered tracking,” Communications in Nonlinear Science and Numerical Simulation, vol. 20, no. 2, pp. 559–570, 2015. [8] Y. Zhang and Y. Yang, “Finite-time consensus of secondorder leader-following multi-agent systems without velocity measurements,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 377, no. 3-4, pp. 243–249, 2013. [9] D. Bauso, L. Giarr´e, and R. Pesenti, “Consensus for networks with unknown but bounded disturbances,” SIAM Journal on Control and Optimization, vol. 48, no. 3, pp. 1756–1770, 2009. [10] J. Hu and W. X. Zheng, “Adaptive tracking control of leaderfollower systems with unknown dynamics and partial measurements,” Automatica, vol. 50, no. 5, pp. 1416–1423, 2014. [11] Z. Li, G. Wen, Z. Duan, and W. Ren, “Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs,” IEEE Transactions on Automatic Control, vol. 60, no. 4, pp. 1152–1157, 2015. [12] H. Bai, M. Arcak, and J. T. Wen, “Adaptive design for reference velocity recovery in motion coordination,” Systems and Control Letters, vol. 57, no. 8, pp. 602–610, 2008. [13] Y. Hong, L. Gao, D. Cheng, and J. Hu, “Lyapunov-based approach to multiagent systems with switching jointly connected interconnection,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 943–948, 2007. [14] P. Lin and Y. Jia, “Consensus of a class of second-order multiagent systems with time-delay and jointly-connected topologies,” IEEE Transactions on Automatic Control, vol. 55, no. 3, pp. 778–784, 2010. [15] C. Godsil and G. Royle, Algebraic Graph Theory, vol. 207 of Graduate Texts in Mathematics, Springer, 2001. [16] P. Lin, K. Qin, H. Zhao, and M. Sun, “A new approach to average consensus problems with multiple time-delays and jointlyconnected topologies,” Journal of the Franklin Institute, vol. 349, no. 1, pp. 293–304, 2012. [17] R. Marino and P. Tomei, Nonlinear Control Design: Geometric, Adaptive and Robust, Prentice Hall, London, UK, 1995. [18] V.-M. Popov, Hyperstability of Control Systems, Springer, New York, NY, USA, 1973.

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