Adaptive Tracking Control of Second-Order Multiagent Systems with Jointly Connected Topologies

This paper considers a consensus problem of leader-following multiagent system with unknown dynamics and jointly connected topologies.Themultiagent 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.


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.
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.

Problem Statement
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  followers and  leader.The interconnection topology of  followers can be conveniently described by a undirected graph G = (V, , ) of order , where V = {V 1 , V 2 , . . ., V  } is the set of  nodes,  ⊆ V × V is the set of edges, and  = [  ] is a weighted adjacency matrix.The node indexes belong to a finite index set I = {1, 2, . . ., }.An edge of G is denoted by   = (V  , V  ).The adjacency elements associated with the edges are positive.The adjacency matrix is defined as   = 0 and   =   ≥ 0. When node V  has edge to V  ,   > 0, the vertex  is called a neighbor of vertex ; it means that agent  is communicating to agent , denoted by   .Then   () = { ∈ V : (, ) ∈ ,  ̸ = }.The out-degree matrix of G is  = diag ( 1 , . . .,   ) ∈  × , where   = ∑ ∈  ()   are the diagonal elements for  = {1, 2, . . ., }.The Laplacian of the undigraph G is defined as  =  − .
The leader (labeled 0) is represented by vertex V 0 , 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 G with vertex set V = V ∪ {V 0 }, which contains graph G of  followers and the leader with directed edges, if any, from some vertices of G to the leader vertex.Use  to describe the leader adjacency matrix and  = diag ( 1 ,  2 , . . .,   ), where   > 0 if the leader is a neighbor of agent  and   = 0 otherwise.When there is at least one directed edge from vertices of the graph G to the leader vertex V 0 , the graph G is said to be connected.
The graph G  ( = 1, . . .,  * ) has the same node set V, and the union of the collection is defined as G 1− * , 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 G  ( = 1, . . .,  * ) may be not strongly connected, but its union graph is connected; then we say the network is jointly connected.
Lemma 1 (Godsil and Royle [15]).If the graph  is connected, its Laplacian  satisfies the following: (1) where   , V  ,   ⊂ ,   , V  are the position and velocity vectors of the th agent, respectively, and   is the control input.(  , ) is the dynamics of agent , which is assumed to be an unknown time-varying disturbance.
The dynamics of the leader in the multiagent system is described by where  0 () 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., lim

Adaptive Control Design
Before giving the adaptive control law, we propose two variables to estimate  0 () and (  , ).
Secondly, we define two variables to describe the relative measurement of position and velocity.
Differentiating the two relative measurements   () and Thus, we take Then the system can be simplified as Furthermore,
is the Laplacian for the  followers; the leader adjacency matrix  is an  ×  diagonal matrix whose th diagonal element is   () at time  and is utilized to represent the connections between the followers and the leader.Now, we consider the control protocol.If V 0 ,   , and   are known, we can design the control protocol as but in our cases, V 0 ,   , and   are unknown; we define   () as the estimate of   () by agent .Then the control protocol is From ( 14),   () is unknown, so we design the parameter input of   () as Lemma 4. When ( 15) is satisfied, without consideration of the parameter error of () 0 and ()  ,   () is the estimate of   (); that is, lim →∞ (  () −   ()) = 0.
In our case,   () is unknown, so we define a variable η for each agent  and set and then (15) can be rewritten as From ( 19) and (20), we can use only related position measurement to estimate the relative velocity measurement, so the tracking control is And equality is 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 parameter variables ω0 () and ω () and let ω0 () = ω0 () Then lim →∞ ω0 () = ω0 () , Consider lim →∞   () = 0.That is to say, we can design the adaptive laws for ω0 () and ω () to get the value of ω0 () and ω ().
We design the adaptive laws for the two variables ω0 () and ω () as Equation ( 26) can be rewritten as

Consensus Analysis
From the above design and definition, we can rewrite the system as which is equal to That is, when lim →∞ () = 0, then lim →∞ () = 0 and lim →∞ V() = 0.

Mathematical Problems in Engineering
By using the definitions and properties of jointly connected topology, we have The matrix   is a permutation matrix,      =   , so we get Meanwhile, so According to the discussion above, in each subinterval, the control scheme of each connected component is Lemma 5. When graphs G  are jointly connected, the leader connects to one follower at least; then where     is the minimum eigenvalue of    ( = 1, . . .,   ).

𝐻 𝜖
is the maximal eigenvalue of    .
Proof.Consider a common Lyapunov function candidate Then the derivative of () along the trajectory of system (32) is given by From Lemma 5, Let From (49), we have Q  ,     are the maximal eigenvalue of Q  and    , respectively.

Mathematical Problems in Engineering
After calculation, we have  Q  < 0, when , , and  satisfy where
Theorem 9.If the PE condition (61) (which will be mention later) is satisfied and φ  ( = 0, 1, . . ., ) 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 ] . (60) The matrix Φ  () is persistently exciting (PE) (Marino and Tomei [17]); that is, there exist two positive real numbers,  0 and , such that Define a function And we have We know lim →∞ () = (∞); from (59) and ( 46 Thus where By the sign-preserving theorem of continuous functions, there exists a time interval [   ,    + Δ) with    ≥  3 , Δ > 0 such that So which contradicts (66).Therefore, conjecture (62) is not true and then the parameter convergence is guaranteed.Hence, the proof is complete.

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  = 3; then by using Theorem 8, we choose  = 3,  = 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 (V  − V 0 ) between the followers and the leader, and Figure 3 shows the trajectories of position errors (  −  0 ) 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 Ω 0 of each agent (ω() 0 − () 0 ), and Figure 5 shows the parameter estimation errors of Ω  of each agent (ω()  − ()  ).From the figures, we can ensure that the parameter estimation errors of Ω 0 and Ω can converge to zero; it means the decentralized adaptive laws (26) can estimate the unknown time-varying disturbance ((  , ) =     ) and the unknown acceleration of the leader ( 0 () =   ()  ).

Conclusion
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

zero is a simple eigenvalue of 𝐿, and 1 𝑛 is the corresponding eigenvector; (2) the remaining 𝑛 − 1 eigenvalues are all positive and real. Lemma 2 (Hong et al
[2]]).Denote  =  + , where L is the weighted Laplacian of graph G and  is the leader adjacency matrix as defined in Section 2. If graph G is connected, then the symmetric matrix  associated with G is positive definite.Moreover, matrices  1 , . . .,   * are associated with the graphs G 1 , G 2 , . . ., G  * , respectively;   is positive semidefinite because both   and   are positive semidefinite.