#### Abstract

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.

#### 1. Introduction

Synchronization phenomena were observed in different fields, such as synchronization of oscillators, swarms of birds, schools of fish, and robot rendezvous [1–5]. Synchronization is one of the fundamental aspects of self-organization in networks of systems. Many interesting results have been obtained. The analysis and control of the synchronized states thus becomes an interesting task. Recently, more and more researchers are putting their attention on consensus control, and it is well accepted as one of the most important and fundamental issues in the fields of coordination control for multiagent systems, which aims to develop distributed control policies that enable a group of agents to reach an agreement on some quantities [6].

The pioneering work [6] gave the theoretical explanation for the consensus behavior of the well-known Vicsek’s model. Till now, a great number of interesting and useful results have been established for the consensus problems with different agent’s dynamics, including first-order systems [6–8], second-order systems [9], general linear system [10–12], descriptor system [13], discrete-time systems [14], time-delay systems [15], fractional-order systems [16], and nonlinear system [17]. In many practical systems, some state variables cannot be obtained directly due to the technical constraints or economic cost. In this case, state observers may be adopted to estimate those unmeasurable state variables. The observer-based consensus protocols were investigated extensively [9–14].

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 [18]. Signed graphs can be applied in scenarios of social networks, predator-prey dynamics, and biological systems [19–21]. Based on conventional consensus theory, the bipartite consensus problem based on signed graph was introduced by the authors in [18]. The bipartite consensus problem of first-order multiagent systems under directed signed graphs was discussed in [22]. A state feedback bipartite consensus law was proposed for linear time-invariant (LTI) single-input systems in [23]. The equivalence of bipartite consensus and conventional consensus under state feedback and output control laws were discussed in [24]. A unified framework for bipartite output synchronization of heterogeneous linear multiagent systems was proposed in [25]. In [26], the authors discussed the effects of measurement noises on bipartite consensus problem with first-order multiagent systems under undirected signed graphs. An adaptive bipartite consensus was proposed for high-order multiagent systems with unknown disturbances in [27]. An adaptive protocol was proposed in [28] to solve the bipartite output consensus problem over signed graphs. The sign consensus problem and bipartite sign consensus problem under signed graphs have been discussed in [29]. The bipartite containment tracking problem for leader-following networks associated with signed digraphs was addressed in [30]. Most previous investigations of the interaction topology have focused on edge-weighted graphs, and node weights have been largely neglected. In [31], an algorithmic technique was proposed to deal with the graph covering problem in node-weighted graphs.

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 [32], some sufficient conditions were proposed to guarantee the cluster synchronization. An intermittent control law was proposed to solve cluster synchronization problem in [33]. Cluster synchronization problem of linear multiagent systems was investigated via a pinning control strategy under directed interaction topology in [34]. An adaptive pinning control strategy for cluster synchronization problem was proposed in [35]. The couple-group consensus for the discrete-time heterogeneous MASs with input and communication time delays was investigated in [36].

Motivated by the above studies, especially by [24], this paper considers the distributed weighted consensus problem for multiagent systems with general continuous-time linear dynamics. The main contributions of this paper are listed as follows: (1) We first propose weighted consensus problem, which generalizes the concept of conventional consensus to weighted consensus case. The conventional, bipartite, and cluster consensus problems can be unified in our given weighted consensus frame. (2) To solve weighted consensus problem, the edge- and node-weighted digraph is adopted to model the interaction topology among agents, by which the distributed protocol to solve the weighted consensus problem is provided. (3) The equivalence between weighted consensus and conventional consensus is established, by which most conventional consensus protocols can be generalized to solve the weighted consensus problem directly. Since the conventional, bipartite, and cluster consensus problems can be unified in our given weighted consensus frame, the established results for the weighted consensus problem can be applied to solve the conventional, bipartite, and cluster consensus problems directly. For output feedback case, we propose a unified frame to construct the state-observer-based weighted consensus protocols, which extends the existed results of [10–12] even in conventional consensus case.

The rest of the paper is organized as follows. In Section 2, the considered weighted consensus problem is formulated with the help of graph theory. Then, the equivalence between weighted consensus and conventional consensus is discussed in Section 3. The distributed weighted state feedback consensus protocols and output feedback consensus protocols are proposed in Section 4. The observer’s gain matrix construct approaches and the bipartite consensus case are discussed in Section 5. Following that, Section 6 provides a numerical example to illustrate our established results, and finally, the concluding remarks are given in Section 7.

The notations of this paper are standard. Let be the set of real matrices. The real part of complex number *s* is denoted by . . represents the identity matrix with dimension *n*. and represent transpose and conjugate transpose of matrix *A*, respectively. represents the rank of matrix *A*. represents a diagonal matrix with diagonal elements . Matrix *A* is called Hurwitz matrix if every eigenvalue of *A* has strictly negative real part. represents *i*-th eigenvalue of *A*. denotes the Kronecker product.

#### 2. Preliminaries and Problem Formulation

##### 2.1. Interaction Topology

To formulate our considered problem, the interaction topology of the multiagent systems is modeled by a simple edge- and node-weighted digraph . Let be an edge- and node-weighted digraph with node set , edge set , a vector with () being the weight associated to node *i*, and a nonnegative weighted adjacency matrix . if and only if , and otherwise. We do not consider the graph with self-loops, that is, all . When the node-weight is neglected, the edge- and node-weighted digraph degenerates to the normal digraph. The neighbor set of node is denoted by . The degree matrix of graph is . Then, the Laplacian matrix of is defined as , which has the following property:

Lemma 1 (see [8]). *Zero is an eigenvalue of L with 1 as a right eigenvector, and all nonzero eigenvalues have positive real parts. Furthermore, zero is a simple eigenvalue of L if and only if has a directed spanning tree.*

##### 2.2. Weighted Consensus

The considered multiagent system is composed of *N* agents, whose dynamics is assumed to be modeled by a linear time-invariant system (LTI).where , , and are the state, measured output, and control input of agent *i*, respectively. *A*, *B*, and *C* are constant system matrices with compatible dimensions. Throughout this paper, we always assume that is stabilizable and is detectable. Without loss of generality, we assume that matrix *C* is full row rank, that is, .

*Definition 1 (weighted consensus). *Multiagent system (1) achieves weighted consensus with weights if there exists some nontrivial trajectory such that , .

From Definition 1, it is easy to see that multiagent system (1) achieves weighted consensus with weights if the states of all agents satisfy thatWe say the protocol can solve the weighted consensus problem, if the closed-loop feedback system achieves weighted consensus. Our main objective is to construct the distributed control law to solve the weighted consensus problem.

*Definition 2 (conventional consensus [6]). *Multiagent system (1) achieves conventional consensus if there exists some nontrivial trajectory such that , .

Obviously, while all weights are taken by , weighted consensus problem reduces to the well-known conventional consensus problem. Till now, the conventional consensus problems have been widely investigated by many researchers, and numerous interesting and useful results have been obtained.

*Definition 3 (bipartite consensus [18]). *Multiagent system (1) achieves bipartite consensus if there exists some nontrivial trajectory such that , and , where and .

The signed graph was introduced by [18] to probe the bipartite consensus problem. In contrast to the normal weighted digraph, the weight associated with edge can be negative. Let the signature matrices set as . A signed directed graph is structurally balanced if and only if , such that is a nonnegative matrix. To achieve bipartite consensus, is required to be structurally balanced [18]. Based on the above analysis, we have the following lemma.

Lemma 2. *If a signed graph is structurally balanced, then such that can be modeled by a simple edge- and node-weighted digraph with and nonnegative weighted matrix .*

It is easy to see that while the weights are taken by , and , , the weighted consensus problem degenerates to the bipartite consensus problem. Thus, the bipartite consensus can be unified in the frame of weighted consensus. By our proposed approach, the bipartite consensus problems of [18, 22–24] can be translated into the equivalent conventional consensus problems to be addressed. By our established results, some results of [18, 22–24] can be obtained directly.

Let be a partition of the set . If , , and for , , the partition is called as cluster partition. For , let be the subscript of the subset in which the number *i* is, i.e., .

*Definition 4 (cluster consensus [32]). *Multiagent system (1) achieves cluster consensus with partition if the agent’s states satisfy , and , .

Let be weight set associated with cluster partition . When the multiagent system achieves the weighted consensus, the multiagent system must achieve cluster consensus. Thus, our proposed approaches for weighted consensus problem can be used to solve the cluster consensus problem directly.

#### 3. Equivalence between Weighted Consensus and Conventional Consensus

For multiagent systems, we will pay more attention to the distributed control protocols. To solve the conventional consensus problem, the authors in [7] proposed a general form of the distributed state feedback for first-order multiagent systems. Similarly, for weighted consensus problem, the state feedback protocol for agent *i* is said to be distributed if it has the form

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.

##### 3.1. Equivalence via State Feedback Law

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

*Problem 1 (state feedback weighted consensus). *Consider a multiagent system (1) under a directed interaction topology with node weights , whose interaction topology contains a directed spanning tree. The closed-loop multiagent system can achieve weighted consensus via a distributed state feedback protocols with the formwhere is the given vector function.

*Problem 2 (state feedback conventional consensus). *Consider the following multiagent systemunder a directed interaction topology , whose interaction topology contains a directed spanning tree. The closed-loop multiagent system can achieve conventional consensus via a distributed state feedback protocol with the formThe equivalence between Problem 1 and Problem 2 is given in the following theorem.

Theorem 1. *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 (7) can solve Problem 2, then it means that the distributed state feedback protocol (5) can also solve Problem 1, and vice versa.*

*Proof. *For Problem 1, it is not too difficult to rewrite the dynamics for closed-loop system aswhere , , and . On the contrary, for Problem 2, the dynamics for closed-loop system can be rewritten aswhere .

For simplicity, we use the same notation to define a state transformation . It follows the from (8) thatwhich is exactly the closed-loop dynamics (9) for Problem 2. Thus, it is easy to see that if the protocol (7) can solve Problem 2, then the protocol (5) can also solve Problem 1. Since Problem 2 is a special case of Problem 1, the converse part is self-evident.

##### 3.2. Equivalence via Dynamical Output Feedback Law

Since the state of agent *i* is not available directly, only the output can be used in the control protocols. To achieve the control objective, state observers are adopted to estimate the unmeasured states.

*Problem 3 (dynamical output feedback weighted consensus). *Consider a multiagent system (1) under a directed interaction topology with node weights , whose interaction topology contains a directed spanning tree. The closed-loop multiagent system can achieve weighted consensus via a distributed observer-based protocols with the formwhere and are the state estimation and output estimation for agent *i*, respectively.

*Problem 4 (dynamical output feedback conventional consensus). *Consider a multiagent system (6) under a directed interaction topology , whose interaction topology contains a directed spanning tree. The closed-loop multiagent system can achieve conventional consensus via a distributed state feedback protocol with the formwhere and are the state estimation and output estimation for agent *i*, respectively.

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

Theorem 2. *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 (12) can solve Problem 4, the it means that the distributed state feedback protocol (11) can also solve Problem 3, and vice versa.*

#### 4. Distributed Weighted Consensus Protocol

##### 4.1. State Feedback Protocol

In this subsection, we investigate weighted consensus problem via state feedback control law. The state feedback protocol for agent *i* is taken aswhere *c* is the positive coupling strength and is the feedback gain matrix.

Theorem 3. *Consider a multiagent system (1) under a directed interaction topology with node weights , whose interaction topology contains a directed spanning tree. Takewith P being the unique positive definite solution of the Riccati equationwhere Q and R are given positive definite matrices with appropriate dimensions, and the coupling strength c is selected to satisfy*

*Then, the multiagent system achieves weighted consensus via the proposed protocol (13).*

*Proof. *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 [11].

*Remark 1. *While all nodes’ weights , , are taken same constant the protocol (13) degenerates to the well-known conventional state consensus protocol , which has been discussed in many studies. Here, the feedback gain matrix *K* and the convergence condition for *c* are same as those in [11]. Unfortunately, the eigenvalue of the Laplacian matrix involved in (16) depends on the entire communication graph and belongs to the global information, which implies that the proposed protocols might not be implemented in a fully distributed fashion. To overcome this limitation, the adaptive parameter approach can be adopted to solve this problem (see [37–39]). By following the line of the equivalence result presented in Section 3, the adaptive parameter laws of [37–39] can be modified to solve the weighted consensus problem.

##### 4.2. Output Feedback Protocol via Local Observer

The weighted consensus proposed for agent *i* is proposed with the formwhere is the observer’s state and is the estimated state of .

Theorem 4. *Consider a multiagent system (1) under a directed interaction topology with node weights , whose interaction topology contains a directed spanning tree. Gain matrixes , , , , and are chosen to satisfy the following condition:**The coupling strength and feedback gain matrix K are chosen as (16) and (14), respectively. Then, the multiagent system can achieve weighted consensus via the distributed protocols (15).*

*Proof. *Take variable transformation , , , and . According to the equivalence between Problem 3 and Problem 4, the result of Theorem 4 can be obtained if the following protocolcan solve the conventional consensus problem for the multiagent system (6) with same coupling strength *c* and gain matrices *K*, , , , , and .

Let and . From (6), (18), and (19), we haveFrom (17) and (18), we haveFrom (1), (19), and (21), we haveDenote *i*-th column of Laplacian matrix *L* by . Let . Then, by (20) and (22), the equivalent dynamics of the closed-loop system can be expressed in a compact form as follows:Since , there exists a Schur orthogonal matrix with the form such thatBy using variable substitution to system (23), we havefrom which a subsystem can be obtained with the formwhere with being its first *n* components.

According to Lemma 1, all and (). It follows the from (16) that . From (14) and (15), we havewhich implies that () are stable. Noticing that we know that is stable. On the contrary, all are stable. Thus, system (26) is also stable, which implies . Moreover, we havewhich means . Thus, the multiagent system (6) achieves consensus via protocol (19).

According to the equivalence between Problem 3 and Problem 4, the multiagent system (1) can achieve weighted consensus via the distributed protocols (17). Now, the proof is completed.

##### 4.3. Output Feedback Protocol via Cooperative Observer

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

Then, the weighted consensus proposed for agent *i* is proposed with the formwhere is the observer’s state and is the estimated output.

Theorem 5. *Consider a multiagent system (1) under a directed interaction topology with node weights , whose interaction topology contains a directed spanning tree. Gain matrixes , , , , and are chosen to satisfy condition (18). The feedback gain matrix K is chosen as (14), and the coupling strength is selected to satisfy (16). Then, the multiagent system can achieve weighted consensus via the distributed protocols (30).*

*Proof. *Similarly, take variable transformation , , , and . According to the equivalence between Problem 3 and Problem 4, the result of Theorem 4 can be obtained if the following protocolcan solve the conventional consensus problem for the multiagent system (6) with the same coupling strength *c* and gain matrices *K*, , , , , and .

Let and . From (6), (18), and (31), we haveFrom (30) and (18), we haveFrom (1), (31), and (33), we haveThen, from (20) and (22), the equivalent dynamics of the closed-loop system can be expressed as The rest proof is omitted because it is very similar to that of Theorem 4.

*Remark 2. *Based on the proof process of Theorem 4, we know that the following dynamical estimatoris the state observer to estimate the state . Similarly, according to the proof process of Theorem 5, the dynamical estimator with the form is the state observer to estimate the relative weighted state information of its neighbors . Most existed references adopted the observers with identical gain matrices for all agents. According to Theorems 4 and 5, different agents can choose observers with different gain matrices which only satisfy condition (18). Thus, all observers can be designed parallelly. The solvability of condition (18) is discussed in the next section.

#### 5. More Discussions

##### 5.1. Gain Matrix Construct Approach for Observer

In the sequel, we present some special solutions of condition (18). Correspondingly, the consensus protocols can be obtained from (17). For brevity, we adopt identical observers for all agents here. Certainly, the agents can equip observer with different forms.

*Special Case 1. *Take , , , and . Then, . Since is observable, it is easy to choose *G* such that *F* is stable. Then, a kind of distributed full-order observer-based consensus protocols are obtained as

*Special Case 2. *Select a Hurwitz matrix with a set of desired eigenvalues that contains no eigenvalues in common with those of A. Solve Sylvester equation to get a nonsingular matrix *T*. If *T* is singular, select another *G* to solve Sylvester equation until *T* is nonsingular. Take , , and . Then, another kind of distributed full-order observer-based consensus protocols are obtained as *Special Case 3*. Without lost of generalization, *C* is assumed to have full row rank *q*. Select a Hurwitz matrix with a set of desired eigenvalues that contains no eigenvalues in common with those of *A*. Select randomly such that is controllable. Solve Sylvester equation to get matrix *T* until is nonsingular, otherwise select another G. Let , where and . Take , , and . Then, a kind of distributed reduce-order observer-based consensus protocols are obtained as

*Special Case 4. **C* is assumed to have full row rank *q*. Select such that is nonsingular. Let , where and . Thus, we know that . While is detectable, we can prove that is also detectable. Then, choose such that is stable. Take and . Then, we have , from which *G* is chosen as such that is satisfied. Obviously, take and , which satisfies . Then, another kind of distributed reduce-order observer-based consensus protocols are obtained as

*Remark 3. *Based on the above analysis, the observer’s gain matrices involved in (38)–(41) satisfy condition (18). According to Theorem 4, while interaction topology contains a directed spanning tree, and , the weighted consensus problem can be solved via all kinds of protocols (38)–(41), respectively. Certainly, based on the protocol (30), we can also provide special kinds of protocols as (38)–(41). Some conventional consensus protocols discussed by [10–12] can be unified in our proposed protocol (17). By using our proposed design approach, the protocol containing a neighbor-based controller together with a neighbor-based observer proposed in [11] can also be generalized to solve the weighted consensus problem easily.

##### 5.2. Bipartite Consensus under Signed Graph

In [18,22–24], the bipartite consensus problem under signed graphs was addressed. When the signed graph is structurally balanced, there exists such that is a nonnegative matrix, by which an edge- and node-weighted digraph with and can be constructed. When interaction topology modeled by a signed graph, all () are unknown. It follows from the nonnegative matrix that . Then, we know that . Thus, the protocol (13) can be expressed as follows:

This protocol was proposed in [24] to solve the bipartite consensus problem under structurally balanced signed graphs.

Furthermore, by using a similar approach to (17) and (30), the following two protocols are obtained to solve the bipartite consensus problem under signed graphs

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

Corollary 1. *Consider a multiagent system (1) under the signed directed graph . Suppose that is structurally balanced and contains a directed spanning tree. Gain matrixes , , , , and are chosen to satisfy condition (18). The coupling strength and feedback gain matrix K are chosen as (16) and (14), respectively. Then, the two distributed protocols (43) and (44) can solve the bipartite consensus problem.*

#### 6. Numerical Example

Consider a multiagent system with six agents labeled by , whose dynamics is modeled by the third-order integrator systemwhere

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

By taking a positive definite matrix , solve the Riccati equation (15) to get the unique positive definite solution . Then, .

The interaction topology is directed and contains a directed spanning tree, whose Laplacian matrix is given by

By simple calculation, take , which satisfies . When all node weight are taken same, it is easy to see that all adopted in the protocols (13), (17), and (30) can be removed. Then, the protocols (13), (17), and (30) degenerate to the conventional consensus protocols, which have been discussed in many studies [10–12].

Here, we only discuss the case that the node weights are different. The weighted vector is chosen as , by which six agents are divided into three groups , , and . According to Definition 1, when the system achieves weighted consensus, each group converges to the same value, the consensus value of group is double the consensus value of group , and the consensus values of group and group are opposite numbers. By using the state feedback protocol (13) to solve the weighted consensus problem, the trajectories of , , and () are depicted in Figures 1–3, respectively, which shows that the multiagent system can achieve weighted consensus.

When the output feedback can be used in the protocols, we can use condition (18) to compute gain matrices involved in the protocols. By using the design approach proposed in subsection 5.1, four different observers including two full-order observers and two reduced-order observers are adopted for the six agents, whose gain matrices are taken as , , , , , , , , , , , , , , , , , , , .

The weighted consensus protocols via local observers (17) are used to solve the weighted consensus problem. The trajectories of , , and () are depicted in Figures 4–6, respectively, which shows that the consensus values are compatible with the weighted vector . Although four different observers are adopted, the multiagent system also achieves weighted consensus.

The weighted consensus protocols via cooperative observers (30) are used to solve the weighted consensus problem. The trajectories of , , and () are depicted in Figures 7–9, respectively, which also show that the multiagent system achieves weighted consensus.

#### 7. Conclusion

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.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

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.