Adaptive Guaranteed-Performance Consensus Control for Multiagent Systems With an Adjustable Convergence Speed

Adaptive guaranteed-performance consensus control problems for multi-agent systems are investigated, where the adjustable convergence speed is discussed. This paper firstly proposes a novel adaptive guaranteed-performance consensus protocol, where the communication weights can be adaptively regulated. By the state space decomposition method and the stability theory, sufficient conditions for guaranteed-performance consensus are obtained, as well as the guaranteed-performance cost. Moreover, since the convergence speed is usually adjusted by changing the algebraic connectivity in existing works, which increases the communication burden and the load of the controller, and the system topology is always given in practical applications, the lower bound of the convergence coefficient for multi-agent systems with the adaptive guaranteed-performance consensus protocol is deduced, which is linearly adjustable approximately by changing the adaptive control gain. Finally, simulation examples are introduced to demonstrate theoretical results.


Introduction
In recent years, by the incentive effects of spacious applications, such as synchronization [1,2] , formation control [3][4][5] , cluster flight [6][7][8] and other fields [9][10][11] , there is considerable attention in distributed cooperative control of multi-agent systems. As a significant topic in cooperative control, consensus, which means that all agents in a multiagent system achieve an agreement on some factors by designing reasonable consensus control protocols, arouses extensive interest from investigators, and some meaningful relevant works have been developed in [12][13][14][15], where the consensus performance is not taken into account.
In some practical complex multi-agent systems, except for the requirements of achieving consensus, the consensus performance also need to be taken into consideration. There is a representative example in [16], and it is shown that when a particular task is carried out by multiple mobile vehicles, the distance performance is likely to be a critical factor due to restricted resource. Generally speaking, when consensus should be achieved in multi-agent systems and under the condition that certain cost functions included in constraints are identified as minimum or maximum, one can model these correlative issues as optimal or suboptimal consensus. It should be pointed out that researchers always use guaranteedperformance consensualization to realize optimal consensus control. In order to deal with the optimal consensus problem for first-order multi-agent systems with both continuous-time and aerospace are constrained by limited energy. However, this method is not available in [12][13][14][15]. Secondly, the communication weights are regulated by both state errors and the adaptive control gain, while the communication weights cannot be regulated in [12][13][14][15]. Thirdly, in view of the given system topology and the huge communication burden, by changing the adaptive control gain rather than the algebraic connectivity related to the topology network, the convergence speed can be adjusted in this paper, but the adjustable convergence speed is not mentioned in [12][13][14][15].
The remainder of the paper is organized as follows. Section 2 states several important conclusions of graph theory and the problem description in which the adaptive guaranteedperformance consensus protocol is proposed. Section 3 shows sufficient conditions which can ensure the guaranteed-performance consensus of multi-agent systems and the convergence coefficient associated with the convergence speed.

Preliminaries and problem description
This section mainly introduces some basic concepts of graph theory and presents the problem description.

A. Algebraic graph theory
. The adjacency element ij w corresponds to the interaction strength between node i v and node j v , where 1  . It is assumed that there is no self edge for each node; that is, According to the definition of connected undirected graphs, one eigenvalue of L is zero, and the remaining eigenvalues are positive.

B. Problem description
Consider a complex multi-agent system with M nodes with each node modeled as x t  R is the state of the k th node to be coordinated and ( ) k u t  R is the control input of the k th node to be designed according to the information of its neighboring nodes. The system topology among nodes is modeled as a connected undirected graph.
The following adaptive guaranteed-performance control protocol is proposed: where  is the adaptive control gain,  represents the performance coefficient with 0   , k N denotes the neighbor set of node k , kj  stands for the connection state of the edge between node j and node k , which equals to one if node j is connected to node k and identically equals to zero otherwise, and ( ) kj w t and ( ) kj w t  are the adaptively regulated weight of the edge from node j to node k and its time derivative, respectively. It can be found that if    , ( ) kj w t is non-decreasing and time-varying and its variation is closely associated with the state errors between two agents. Furthermore, the growth rate of ( ) kj w t is faster when the state errors are greater, which means that the greater effect is given to adjust the state errors between them. Especially, ( ) kj w t is not changed when the states of two agents are equal. Without loss of generality, it is assumed that the initial value of weight (0) kj w is 1 and the upper bound of ( ) kj w t is kjm w . In consideration of the undirected topology G , one can easily see that nodes in the topology correspond to agents of multi-agent systems. In this case, let , then a compact form of system (1) with control protocol (2) is produced as follows where ( ) w t L stands for the Laplacian matrix of G . The definition of the adaptive guaranteed-performance consensus for multi-agent system (1) is described as follows.
Definition1. Combined with control protocol (2), multi-agent system (1) is said to achieve adaptive guaranteed-performance consensus if there exists  such that lim ( ( ) J  is known as the guaranteed-performance cost. Remark 1. The aim of this paper is to obtain a suitable adaptive control gain and the guaranteed-performance cost, so that multi-agent system (1) can achieve adaptive guaranteed-performance consensus. Moreover, there are two main features in adaptive guaranteed-performance control protocol (2). Firstly, the guaranteed-performance function r J associated with the performance coefficient and state errors is proposed. In this case, the consensus performance of multi-agent systems is guaranteed, which is more meaningful compared with [12][13][14][15]. Secondly, the adaptive control gain is designed, and it can effectively regulate the time-varying communication weights, which plays an important role in the consensus analysis for multi-agent systems. However, this factor is not referred in [12][13][14][15].

Main results
In the following theoretical analysis, sufficient conditions for multi-agent system (1) to achieve adaptive guaranteed-performance consensus are obtained by designing the adaptive control gain  , then the guaranteed-performance cost r J  is determined. Moreover, the consensus convergence speed is discussed, and the lower bound of the convergence coefficient is presented to indicate the convergence speed. We prove that it is an effective way to linearly regulate the consensus convergence speed by changing  .

C. adaptive guaranteed-performance consensus analysis
In the first place, the state space decomposition approach is proposed to transform the dynamics of multi-agent system (1). Denoted by Then according to matrix theory, one can find that there exists an orthogonal matrix , then one can get that multi-agent system (3) can be rewritten as denote M -dimensional unit vectors with the i th element 1 and 0 elsewhere. Define Due to 2 ( ) , ( ) , one can obtain from (8) (11) and (12). ; in other words, subsystems (6) and (7) are presented respectively to indicate the consensus motion and relative state motion of multi-agent system (1).
In the following theorem, sufficient conditions for adaptive guaranteed-performance consensus are obtained, which means that distributed guaranteed-performance consensus design for multi-agent system (1) is derived.

Theorem 1.
Multi-agent (1) with control protocol (2) achieves adaptive guaranteedperformance consensus if 2    . In this case, the guaranteed-performance cost satisfies that Proof. To begin with, we prove lim μ( ) Consider a Lyapunov function candidate as follows.
Because of ( ) kjm kj w w t  , one can get that ( ) 0 V t  in spite of the value of  . Furthermore, taking the time derivative of ( ) V t along the trajectory of subsystem (7), one has ( ) ( Due to the assumption of the undirected connected graph G , it can be seen that ( ) Then since equation Then by (2) and (16),it can be seen that In addition, it can be also obtained that Substituting (17) and (18) into (15), it can be deduced that (19) Moreover, it should be pointed that ,then if    , the following inequality holds Accordingly, ( ) t μ converges to 0 ; that is, multi-agent system (1) under control protocol (2) can be adaptively consensualizable.

D. Convergence speed analysis
It should be noted that the Laplacian matrix L related to the undirected graph G is a semidefinite matrix. Then being the minimum nonzero eigenvalue of L , 2  is also said to be the algebraic connectivity satisfying According to [23], one can see that in case there is no gain matrix in the consensus control protocol, the convergence speed is associated with the minimum nonzero eigenvalue 2  . In other words, the larger the value of 2  is, the faster the convergence speed is.
For the sake of verifying the assumption that the proposed method can effectively regulate the convergence speed of multi-agent system (1), we give the definition of the convergence coefficient as follows.

Definition 2.
For multi-agent system (1), the convergence coefficient ( ) t  can be defined as Then the lower bound of ( ) t  is shown as min  which means the minimum convergence speed of multi-agent system (1).
As a matter of fact, equation (28) originates from Definition 2. Then with regard to multiagent system (1) under the standard control protocol in [23], where ( ) It can be seen that min ; that is, the lower bound of ( ) t  is directly associated with the algebraic connectivity. Since there exists the relationship between min  and the algebraic connectivity, it is rational and convenient to use min  to describe the convergence speed to some extent.
In the following, we determine the convergence coefficient of multi-agent system (1) under control protocol (2) and compare it with the convergence coefficient under the standard consensus protocol. In order to ensure the effectiveness of this comparison, the control gain  is also considered in the standard consensus protocol as a reference, which means that ( ) In this case, one can directly obtain that min,1 Then substituting (17) into (19), the convergence coefficient of multi-agent system (1) under control protocol (2) can be described as Therefore, the following theorem can be obtained.

Remark 3.
As the improvement of the convergence speed can save working time of multiagent systems in practice to some degree, it is significant to investigate how to improve the convergence speed when multi-agent systems achieve consensus. Thus, the adjustable convergence speed is deduced, and the lower bound of the convergence coefficient under control protocol (2) is derived in this paper, which means that one can regulate the convergence speed by changing the adaptive control gain or the algebraic connectivity. However, in [12][13][14][15], the convergence speed is not considered; that is, one cannot regulate the consensus convergence according to the requirements in different environment. that  can play a more important role in achieving the adjustable convergence speed. In other words, with the increase of the value of  , min,2  is increased obviously, and is less affected by the minimum nonzero eigenvalue of L . In this case, the convergence speed can be promoted more quickly. Furthermore, min,2  is linearly associated with  on account of fixed 2  . As a result, the convergence speed of multi-agent system (1) can be linearly regulated approximately. However, although many approaches have been used to adjust the convergence speed in [33][34][35][36][37][38], all of them depend on the minimum nonzero eigenvalue of L associated with the system topology.

Results and Discussion
In this section, a simulation example is given to demonstrate the theoretical results shown in the previous analysis.  1/ T c /s is much less; that is, the convergence speed is faster. Fig. 4 5 shows the relationship between the convergence time and 1  . One can obtain that the convergence time can be positively correlative approximately with 1  , provided that  is small enough.

Conclusions
A new adaptive guaranteed-performance consensus scheme for multi-agent systems with an adjustable convergence speed was proposed in this paper. The adaptive guaranteedperformance consensus protocol was presented by adjusting the communication weights among agents in the system topology. Sufficient conditions for adaptive guaranteedperformance consensus was obtained and the guaranteed-performance cost was deduced. Then in order to indicate the convergence speed of multi-agent systems, the convergence coefficient was defined, and it was proved that the convergence speed can be approximately linear adjustable by changing the adaptive control gain. Furthermore, another interesting work in the following years is to consider the external disturbance for the guaranteedperformance consensus of multi-agent systems.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.

Funding Statement
This work is supported by National Natural Science Foundation under grant #61867005.