In this paper, the concept of consensus is generalized to weighted consensus, by which the conventional consensus, the bipartite consensus, and the cluster consensus problems can be unified in the proposed weighted consensus frame. The dynamics of agents are modeled by the general linear time-invariant systems. The interaction topology is modeled by edge- and node-weighted directed graphs. Under both state feedback and output feedback control strategies, the weighted consensus problems are transformed into the equivalent conventional consensus problems. Then, some distributed state feedback and output feedback protocols are proposed to solve the weighted consensus problems. For output feedback case, a unified frame to construct the state-observer-based weighted consensus protocols is proposed, and different design approaches are discussed. As special cases, some related results for bipartite consensus and cluster consensus can be obtained directly. Finally, a simple example is given to illustrate the effectiveness of our proposed approaches.

Synchronization phenomena were observed in different fields, such as synchronization of oscillators, swarms of birds, schools of fish, and robot rendezvous [

The pioneering work [

The conventional consensus is achieved through collaboration, which normally focuses on cooperative systems. Since only collaborative interactions are considered in the systems, the edge weights of the interaction topology among agents are assumed to be nonnegative. When both collaborative and antagonistic interactions coexist within a group of agents, the interaction topology can be more suitably modeled by signed graphs, in which a positive edge means collaboration and a negative edge represents an antagonistic interaction [

Cluster synchronization requires that the system splits into several clusters and synchronization occurs in each cluster, which can find many applications in biological, social, and technological networks. In [

Motivated by the above studies, especially by [

The rest of the paper is organized as follows. In Section

The notations of this paper are standard. Let

To formulate our considered problem, the interaction topology of the multiagent systems is modeled by a simple edge- and node-weighted digraph

Zero is an eigenvalue of

The considered multiagent system is composed of

Multiagent system (

From Definition 1, it is easy to see that multiagent system (

We say the protocol

Multiagent system (

Obviously, while all weights are taken by

Multiagent system (

The signed graph

If a signed graph

It is easy to see that while the weights are taken by

Let

Multiagent system (

Let

For multiagent systems, we will pay more attention to the distributed control protocols. To solve the conventional consensus problem, the authors in [

While the state information cannot be available directly, the distributed output feedback protocol is considered, which has the form

In this section, while the state feedback and output feedback control protocols have a particular form, we will prove that the weighted consensus problem is equivalent to the conventional consensus problem. Furthermore, a systematic design approach can be provided to construct weighted consensus protocols from the well-investigated conventional consensus problems.

We first introduce the state feedback problems related with weighted consensus problem and conventional consensus problem.

Consider a multiagent system (

Consider the following multiagent system

The equivalence between Problem 1 and Problem 2 is given in the following theorem.

The state feedback weighted consensus problem (Problem 1) is equivalent to the state feedback conventional consensus problem (Problem 2), that is, if the distributed state feedback protocol (

For Problem 1, it is not too difficult to rewrite the dynamics for closed-loop system as

For simplicity, we use the same notation to define a state transformation

Since the state of agent

Consider a multiagent system (

Consider a multiagent system (

Similarly, Problem 3 is equivalent to Problem 4. The proof is omitted because it is very similar to that of Theorem 1.

The dynamical output feedback weighted consensus problem (Problem 3) is equivalent to the dynamical output feedback conventional consensus problem (Problem 4), that is, if the distributed state feedback protocol (

In this subsection, we investigate weighted consensus problem via state feedback control law. The state feedback protocol for agent

Consider a multiagent system (

Then, the multiagent system achieves weighted consensus via the proposed protocol (

According to the equivalence between Problem 1 and Problem 2, this result can be obtained directly from the related result for the conventional consensus in [

While all nodes’ weights

The weighted consensus proposed for agent

Consider a multiagent system (

The coupling strength and feedback gain matrix

Take variable transformation

Let

From (

From (

Denote

Since

By using variable substitution

According to Lemma 1, all

According to the equivalence between Problem 3 and Problem 4, the multiagent system (

Suppose that each agent can only obtain the relative input and output measurements with its neighbors.

Then, the weighted consensus proposed for agent

Consider a multiagent system (

Similarly, take variable transformation

Let

From (

From (

Then, from (

The rest proof is omitted because it is very similar to that of Theorem 4.

Based on the proof process of Theorem 4, we know that the following dynamical estimator

In the sequel, we present some special solutions of condition (

Take

Select a Hurwitz matrix

Based on the above analysis, the observer’s gain matrices involved in (

In [

This protocol was proposed in [

Furthermore, by using a similar approach to (

According to Theorems 4 and 5, the following result can be obtained directly.

Consider a multiagent system (

Consider a multiagent system with six agents labeled by

The kinematic model of several real physical and mechanical systems can be transformed into the integrators system.

By taking a positive definite matrix

The interaction topology

By simple calculation, take

Here, we only discuss the case that the node weights are different. The weighted vector is chosen as

Trajectories of

Trajectories of

Trajectories of

When the output feedback can be used in the protocols, we can use condition (

The weighted consensus protocols via local observers (

Trajectories of

Trajectories of

Trajectories of

The weighted consensus protocols via cooperative observers (

Trajectories of

Trajectories of

Trajectories of

In this paper, the concept of consensus was generalized to weighted case. The conventional consensus, bipartite consensus, and cluster consensus problems can be unified in our proposed weighted problem frame. To solve our proposed problem, the edge- and node-weighted digraph is used to describe the interaction topology, which is different to the conventional consensus and bipartite consensus cases. The equivalence between the conventional consensus problems and the weighted consensus problems provides an approach to construct the weighted consensus protocols. For linear multiagent systems, both state feedback and state-observer-based protocols were provided to solve the weighted consensus problem. By our proposed approaches, the involved feedback gain matrix and the observer’s gain matrices are decoupled to the interaction topology. The algebraic Riccati equation are used to design the feedback gain matrix. The unified form of full-order and reduced-order observers was given in our proposed protocols, and an algebraical condition was established. Furthermore, all agents can adopt different forms of state observers, which means that the state observers can be designed parallelly. Since the bipartite consensus and cluster consensus problems can be viewed as the special cases, our proposed method can be applied to solve the bipartite consensus and cluster consensus problems directly. More generalized cases such as uncertain dynamics, time delays, and adaptive control will be probed in our future work.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant nos. LY17F030003 and LY15F030009 and the National Natural Science Foundation of China under Grant no. 61501331.