Dynamic H∞ Consensus of Higher-Order Nonlinear Multi-Agent Systems with General Directed Topology

This paper investigates the leaderless H∞ consensus problem of multi-agent systems with higher-order Lipschitz dynamics and external disturbance. The topology is assumed to be directed. Distributed controllers using only relative outputs information of the adjacent agents are constructed. In order to deal with the non-symmetric property of the associated Laplacian matrix of directed topology, some properties of Laplacian matrix are introduced. By introducing an appropriate state transformation, the H∞ consensus problem is converted to a low-dimensional system stability problem which is solved using Lyapunov stability analysis method. The effectiveness of the proposed control design is demonstrated through simulation examples.


I. INTRODUCTION
The consensus of multi-agent systems have received compelling attention due to its wide-spread applications in satellites formation [1], unmanned systems [2], and distributed reconfigurable sensor networks [3,4]. Since that agents are coupled together through a communication network, the consensus achievement is also affected by the system dynamics and network topology [5]. Leaderless consensus is one of the important research directions. Under various topology assumptions, many research results have been achieved including both integrator types [6][7][8][9] and general dynamics [10,11].
Considering the fact that some system are usually described with nonlinear dynamics, researchers have studied the consensus problem of higher-order nonlinear multi-agent systems. In [12], the leaderless global H∞ consensus of Lipschitz multi-agent systems was studied with strongly connected graphs. Then the results were extended to the case that the topology containing a directed spanning tree [13]. In [14], the leader-following consensus was studied with switching directed topologies using multiple Lyapunov functions method. It was proved that the consensus could be achieved if each topology contains a spanning tree and the dwell time was larger than a positive threshold. In [15], the leaderless consensus problem with general directed fixed and switching topologies was invested. Several topology depending Lyapunov functions were constructed. The leaderless consensus of higher-order Lipschitz multi-agent systems was studied in [16] with jointly connected topologies, where common Lyapunov function method and Cauchy convergence criterion were used. Consensus controllers using relative state feedback were introduced for one-sided Lipschitz nonlinear multi-agents systems with directed strongly connected topology [17] and undirected topology [18], separately. In [19], leader-following exponential consensus was investigated using sampled-data information. When the union graph having a directed spanning tree rooted at the leader, [20] considered the performance consensus tracking problem of singular Lipschitz multi-agent systems. Both the leaderless and leader-following guaranteed-performance consensus problems were investigated in [21] with strongly connected and balanced topologies.
It is worth mentioning that the results in [12,[14][15][16][17][18][19][20][21] are obtained with the assumption that the relative state information of neighboring agents is available. However, in some real applications, only output information is available and there may be external disturbances. Inspired by this, output based consensus controllers are constructed with different topology conditions. Under fixed directed topology， [22] investigated the leader-following consensus using the output information. [23] addressed the leader-following system states cannot be obtained directly. In this regard, it is more practical than the results in which the states feedback based consensus controller.
Third, this paper considers the H∞ leaderless consensus problem with external disturbances. It is worth noting that from the perspective of analytical difficulty, the leaderless consensus problem is more difficult than the leaderfollowing consensus problem, because that there is no specified leader in the leaderless consensus problem to construct the consensus error system with directed topology.
The rest of this work is organized as follows. Section 2 introduces some useful definitions and notations. In section 3, output based consensus controllers are introduced. In section 4, two simulation examples are given. Section 5 is the conclusion.

II. PRELIMINARIES
In this paper,  denotes the Kronecker product. For   , The following conclusion will be used in the following sections.
Lemma 1 [28]: Assume that D and S are real matrices with compatible dimensions. For any given , n xy  , and matrices 0 P  , the following inequality holds x DSy x DPD x y S P Sy   .

A. Problem Formulation
Let there is a group of N identical agents. The dynamics of the i-th agent are specified as   12 1, 2,..., ,

B. Main Results
Here, by using only relative output information of neighboring agents, the following dynamic consensus controllers are introduced: We can see that the row sum of U is zero and U is a special Laplacian matrix of a completed graph.
Before moving forward, the following conclusions obtained in our earlier works are introduced to obtain the results.
Lemma 2 [13,29]: For a matrix R whose row sum is zero and a full row rank matrix one can find a matrix Lemma 3 [13,29]: For a matrix R with the eigenvalues having positive real parts, there exist a matrix 0 Q  and a scalar 0 In light of the property of matrix U in (3)    This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
where   The derivation of 1 V along the trajectory of system (6) is According to the Lipschitz condition and Lemma 1, we where 0  is a positive constant.
Then (11) can be rewritten as  The derivation of 2 V along the trajectory of system (6) is According to Lemma 2 and Lemma 3, we can obtain the following conclusion Similarly, using the Lipschitz condition, the following conclusion can be obtained  T  T  T  T   TT   T  T  T   T  T  T   T  T  T  n   T  T  T   TT   T  T Using (14) and (19) (7) and (8) Using (24) and (25)   By the Schur complement lemma, inequalities (7) and (8) imply that 10 Considering (30), (31), (32), and (33) at the same time, we can conclude that inequalities (28) and (29) Since that   00 V  , the following inequality is satisfied That means that system (1) achieves the H  consensus. The proof is completed.
Remark 1: Leaderless consensus problems of Lipschitz multi-agent systems using output information were studied in [24,26,27]. However, the results were constrained with some special topologies, such as undirected or strongly connected topology. It is worth noting that the topology used here can include undirected or strongly connected topology as special cases. Comparing with [24,26,27], the topology constraints are relaxed to a more general situation. In practical applications, the communication between agents is usually not bidirectional due to many factors such as obstructions and communication capabilities. Thus, the results obtained in this paper are less conservative.

Remark 2:
In [14,22,23], the leader-following output based consensus controllers for Lipschitz multi-agent systems are constructed under directed topologies. There is a specific leader in the system, which provides a great convenience for constructing the consensus error system. In contrast with the results in [14,22,23], leaderless consensus problem is investigated here. By using the properties of the Laplacian matrix obtained in our earlier works [13,29], the leaderless H  consensus problem is very succinctly transformed into a H  control problem of a low-dimensional system. It is worth pointing out that, in the homogeneous multi-agent systems, leaderless consensus problem can include leader-following consensus problem as a special case. Thus, the conclusions obtained here can also solve the leader-following consensus problem. A simulation example will be provided to verify the applicability of this conclusion.

Remark 3:
It should be pointed out that although this article assumes that the directed topology graph is fixed, based on the processing method of this article, the H  consensus problem can also be dealt with using multiply Lyapunov function method with directed switching topologies.
Remark 4: The traditional method of analyzing the H  consensus problem is to use the diagonalization method to decompose the system into the H  control problem of N-1 subsystems corresponding to the N-1 non-zero eigenvalues of the Laplaican matrix. However, when the topology is directional and the system has non-linear terms at the same time, the traditional analysis method is very difficult. Here, a brief analysis method is used. The leaderless H  consensus problem is very succinctly transformed into a lowdimensional system H  control problem which is solved using Lyapunov function method. The solution of the problem also benefits from the properties of the Laplacian matrix given in Lemma 2.
Remark 5: It should be pointed out that the global information of the topology structure is needed in the proposed approach. Specifically, the structural information of the topology is required to calculate the parameter  using the smallest nonzero eigenvalue of the graph Laplacian matrix in Lemma 3. The research results have certain limitations. However, this shortcoming can be overcome if we can know the structure of the topology in advance.

Remark 6:
It can be seen that the performance index  affects the feasibility of the inequality (7). Actually, if inequality (7) holds, the following inequality is a necessary condition.  (7) and (8) can be easily checked from standard LMI routines. Note that a possible solution 1 P of (7) can be calculated first with a positive constant 0  . Then, the possible solution 2 P of (8) with the fixed 1 P can be obtained. If it is assumed that the system is linear and there is no external interference, the following corollary can be obtained.

IV SIMULATION
In this section, two simulation examples are provided to verify the applicability of consensus controller proposed above with two different topology conditions. Consider a group of one-link manipulators with revolute joints actuated by a DC, as shown in Figure.

Example 2:
In this example, the leader-following consensus problem without external disturbance is investigated with the topology shown in Figure. 13. It is shown that the topology contains a directed spanning tree with agent 1 being the leader.  can track the states of the leader. That means that the system can also achieve leader-following consensus under the proposed output-feedback based consensus controller.

V CONCLUSIONS
This paper has investigated the leaderless H∞ consensus of higher-order Lipschitz nonlinear multi-agent systems with external disturbance under general directed topology. The only constraint on the topology structure is having a directed spanning tree, which is a quite general condition. By using relative outputs information, dynamic consensus controllers have been proposed. Sufficient conditions have been obtained, under which the Lipschitz multi-agent systems can achieve leaderless H∞ consensus with a guaranteed H∞ performance for a group of nonlinear agents subject to external disturbances. The key technology used here is the properties of Laplacian matrix in our earlier work.
However, the controller constructed in this article requires continuous communication. This brings a lot of communication burden. An interesting topic is to consider the H∞ consensus problem with discontinuous communication.