Distributed leader-follower consensus of nonlinear multi-agent systems with unconsensusable switching topologies and its application to flexible-joint manipulators

ABSTRACT This paper investigates the distributed consensus problem of nonlinear multi-agent systems with switching topologies under a leader-follower framework. In particular, the unconsensusable switching topology is considered for modelling communication failures, which is more practical in the applications. By adopting the novel mode-dependent average dwell time, sufficient consensus conditions are established and the corresponding topology-dependent consensus gains are designed in term of linear matrix inequalities. In the end, an illustrative example with application to multiple flexible-joint manipulators is provided to verify the effectiveness of our proposed design method.


Introduction
During the past decade, researches on multi-agent systems have gained great interest from both academic and engineering areas, and have obtained successful achievements. This is due to their distinct advantages compared with a traditional single system, which includes high robustness, high reliability, and high efficiency with relatively low cost (Oh, Park, & Ahn, 2015;Panait & Luke, 2005;Van der Hoek & Wooldridge, 2008;Yuan, Wang, Zhang, & Dong, 2018). With these advanced features, there are many successful applications with multi-agent systems, such as cooperation of intelligent robots (Egerstedt & Hu, 2001), formation of unmanned aerial vehicles (Anderson, Fidan, Yu, & Walle, 2008) and autonomous underwater vehicles (Cui, Ge, How, & Choo, 2010), synchronization of sensor networks (Rault, Bouabdallah, & Challal, 2014), and so on. In particular, the consensus is a fundamental yet significant research front of multiagent systems, which means that by local information communications, certain agreement can be reached by the group of agents (Guo, Ding, & Han, 2014;Li, Wen, Duan, & Ren, 2015). Generally speaking, there are two types of consensus architecture: the leaderless consensus and the leader-follower consensus (Ding & Guo, 2015;Qiu, Xie, & Hong, 2016). For the leader-follower consensus problem, since it has a very practical background in industrial applications, many effective consensus protocols are of multi-agent systems based on the MDADT approach (Wang & Yang, 2016;Zheng, Zhang, & Zheng, 2016). However, it should be pointed out that sometimes the topology may be unconsensusable which would result in divergence behaviours of multi-agents. Although this problem is similar to the switched systems with unstable subsystems, it is more difficult for multi-agent systems by graph theory. So far, to the best of the authors' knowledge, there are still few results for the consensus problem with unconsensusable switching topologies by applying the merit of switched systems, which motivates us for this study.
Following the above issues, in this paper, the leaderfollower consensus of nonlinear multi-agent systems with unconsensusable switching topologies is studied with the novel MDADT approach. Compared with the existing literature, the main contributions of our paper can be summarized as follows: (1) this paper makes further attempts to solve the consensus problem of nonlinear multi-agent systems with unconsensusable switching topologies by MDADT. The nonlinear dynamics are more applicable in modelling the practical systems. (2) The switching topologies consisted of both consensusable and unconsensusable topologies are concerned, which is more general than the common assumption on all consensusable topologies, and would bring more design flexibility and robustness for the multi-agents systems.
The rest of this paper is arranged as follows. In Section 2, some preliminaries are introduced on graph theory and the leader-follower consensus problem is formulated. Section 3 gives the main theoretical findings with MDADT in detail and Section 4 provides the consensus simulation results with flexible-joint manipulators to show the effectiveness of the developed theoretical results. In the end, Section 5 draws the conclusions and our future work.
Notation: R n and R m×n stand for the n-dimensional Euclidean space and the space of m × n real matrices, respectively. A − B 0 (A − B ≺ 0) means that A−B is positive definite (negative definite). A ⊗ B denotes the Kronecker product. sym{A} denotes A + A T and diag{· · · } represents the block-diagonal matrix. Finally, all matrices are compatible with algebraic operations.

Graph theory
The directed graph G = {V, E, A}, I = {1, . . . , N} is adopted to describe the information topology of the agents. A = [a ij ] ∈ R N×N represents the weighted adjacency matrix with (1) V(G) = {v 1 , . . . , v N } and E denote the sets of nodes and edges, respectively. The L = (l ij ) N×N denotes the Laplacian matrix with If G has a directed spanning tree, then 0 is the eigenvalue of L. More details of the algebraic graph theory can be found in Yu, Chen, and Cao (2010). In this paper, it is assumed that there are N followers whose communication topology is G and one leader. As a consequence, the adjacency matrix between the leader and the followers is defined by (3) Without loss of generality, there is at least one follower connected with the leader.
Furthermore, since the communication among the agents may fail in practical applications, the switching topologies are studied. Let {G p , B p : p ∈ S = 1, 2, . . . , n} denote all possible switching graphs, where σ (t) represents the index set for all graphs. The switching signal σ (t) : [0, ∞) → p and its value is the index of graph at time t. The switching sequence is t 1 , t 2 , . . . , where [t h , t h+1 ), h = 1, 2, . . . is uniformly bounded nonoverlapping.

Nonlinear multi-agent system
Consider a nonlinear multi-agent system under a leaderfollower framework. The followers are described by the following dynamics: The dynamics of leader are given as follows: where x 0 (t) ∈ R n denotes the state of the leader.
Consequently, define e i (t) := x i (t) − x 0 (t) and it can be obtained thaṫ Moreover, the following consensus protocol is designed for the followers: where K σ (t) denotes the topology-dependent consensus gain to be determined. Then, the closed-loop dynamics of the multi-agent system can be obtained bẏ which can be further rewritten as follows: As a result, the leader-follower consensus can be achieved if it holds that lim t→∞ e(t) = 0, which implies that Before proceeding, the following definitions are given. Without loss of generality, suppose that there are r consensusable topologies with σ (t) = 1, 2, . . . , r and n−r consensusable topologies with r + 1, r + 2, . . . , n. All the possible communication topology can be divided into the consensusable set denoted as CT and the unconsensusable set denoted as UCT , respectively.
To this end, some well-known concepts of switched systems are introduced for later analysis and synthesis.
holds for τ a > 0 with an integer N 0 ≥ 0, then τ a is called ADT and N 0 is called the chattering bound.
Remark 2.2: Compared with ADT, it can be found that the MDADT switching can reduce the conservativeness by considering the mode-dependent features for each subsystems.

Main results
In this section, sufficient consensus conditions are derived for the multi-agent systems with unconsensusable switching topologies.
Theorem 3.1: Consider multi-agent system (4) with switching topologies, for giving constants α i , μ i > 1, i ∈ S and the topology-dependent consensus gains K i , if there exist matrices P i 0, ∀(i, j) ∈ S × S, i = j, such that and P i μ i P j .
Then, the leader-follower consensus can be achieved by any switching signal with MDADT where Proof: Choose the following mode-dependent multiple Lyapunov functions:

Moreover, one has l 2 e T (t)e(t) − F T (e(t))F(e(t)) ≥ 0.
Then, it can be derived thaṫ

ξ(t) := [e T (t), F T (e(t))] T and
Consequently, for ∀t > 0 and ∀t ∈ [t k , t k+1 ), it can be obtained by (7) and (8) that where T i (t, 0) denotes the running time of the ith mode. By assuming γ + T + − γ − T − < −γ * t, it can be verified that when the MDADT conditions can be satisfied, then V σ (t) (e(t)) converges to zero as t → ∞. Therefore, one has lim t→∞ e(t) → 0 as t → ∞, which completes the proof.
Remark 3.1: It is worth mentioning that the common Lyapunov functions may be difficult to find in most cases, such that the mode-dependent multiple Lyapunov functions are chosen for the consensus problem. This can considerably reduce the conservatism.

Remark 3.2:
It can be observed that the established conditions are applicable for the slow switching communication topologies, which is more practical than the fast switching in multi-agent systems.
Based on the results of Theorem 1, the following theorem is given to calculate the desired consensus gains in terms of matrix transformation.

Theorem 3.2: Consider multi-agent system (4) with switching topologies, for giving constants
Then, the leader-follower consensus can be achieved by any switching signal with MDADT T i (t, 0), In addition, the desired consensus gains can be obtained by Proof: LetP i := P −1 i , K iPi = W i and perform congruent transformation with I ⊗P i andP i to (7) and (8), respectively. Then, the rest of the proof can follow directly by Theorem 1.
Based on the above results, the corresponding distributed consensus algorithm can be given as follows.

Illustrative example
In this section, an application example of flexible-joint manipulators is provided to demonstrate the effectiveness of our proposed method.
In the simulation, the nonlinear dynamics of flexiblejoint manipulators (see Figure 1) can be given by the form of (4) as Ma and Qiao (2017): is the angular rotation of the motor, x i2 (t) is the angular velocity of the motor, x i3 (t) is the angular position of the link and x i4 (t) is the angular velocity of the link. It is assumed that there are four manipulators and their switching communication topologies are illustrated in Figure 2.
It can be verified that G 1 , G 2 are consensusable and G 3 is unconsensusable.
For giving parameters α 1 = −2, α 2 = −1, α 3 = 2 and μ 1 = 1.5, μ 2 = 2, μ 3 = 3 and τ a3 = 2, the feasible solutions of the conditions in Theorem 2 can be obtained     Figures 3-10, respectively. It can be seen that all the followers can well track the state trajectory of the leader, which supports our theoretical results. Moreover, it should be pointed out that since the developed consensus conditions are with the l, f (x i (t)) will affect the solvability of the conditions. In detail, with same other       parameters, the feasible solution of Theorem 3.2 may not be obtained while l increases. This implies that in the practical applications, the value of l should be carefully addressed in the consensus problem.

Conclusions
In this paper, the distributed consensus problem for nonlinear multi-agent systems with switching topologies is concerned in a leader-follower architecture. Based on the results of model transformation, the MDADT method combined with the multiple Lyapunov method is applied to deal with both slow switching consensusable and unconsensusable topologies. Sufficient consensus criteria are further established in the form of LMIs such that the followers can track the leader. Finally, an application example of consensus for multiple flexible-joint manipulators is presented to illustrate our obtained results. Our future work will be extending the obtained results to the cases with finite-time or fixed-time requirements, which can further reduce the conservatism of the consensus conditions.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by the National Natural Science Foundation of China (grant numbers 61703038, 61627808 and 9164820), and the National Key Research and Development Program of China (grant number SQ2017YFB130092). This work is also supported by the Strategic Priority Research Program of the CAS (grant number XDB02080003).