Consensus Control of Leader-Following Multi-Agent Systems in Directed Topology With Heterogeneous Disturbances

Qinglai Wei, Senior Member, IEEE, Xin Wang, Xiangnan Zhong, Member, IEEE, and Naiqi Wu, Fellow, IEEE

Abstract—This paper investigates the consensus problem for linear multi-agent systems with the heterogeneous disturbances generated by the Brown motion. Its main contribution is that a control scheme is designed to achieve the dynamic consensus for the multi-agent systems in directed topology interfered by stochastic noise. In traditional ways, the coupling weights depending on the communication structure are static. A new distributed controller is designed based on Riccati inequalities,while updating the coupling weights associated with the gain matrix by state errors between adjacent agents. By introducing time-varying coupling weights into this novel control law, the state errors between leader and followers asymptotically converge to the minimum value utilizing the local interaction. Through the Lyapunov directed method and Itô formula, the stability of the closed-loop system with the proposed control law is analyzed.Two simulation results conducted by the new and traditional schemes are presented to demonstrate the effectiveness and advantage of the developed control method.

I. INTRODUCTION

CONSENSUS problem is one of important issues in coordination control of multi-agent (MA) systems, which aims to design appropriate protocols and algorithms such that all agents in a system can asymptotically converge to an agreement by communicating with their neighbours [1], [2]. In resent years, this research field has been widely studied owing to its potential applications on multi-vehicle cooperative control, fault-tolerant control and consensus approach for distributed energy storage systems [3], [4]. Vicsek et al. [5]pioneer this research topic by proposing a simple discrete-time model of particles that are all moving in a plane at the same speed with different headings. Since then, coordination control for multi-agent systems has been an active research filed over the past two decades [6]–[8]. The consensus problem for single-integrator agents is addressed by Olfati-Saber and Murray [9]. Distributed consensus control and its convergence are studied for agents in networks with a fixed or switching topology [10]. In [11], robust consensus tracking is addressed for multi-agent systems under a fixed topology in the presence of communication disturbances and delays. In[12], effective local controllers together with neighbour-based state estimation rules are designed to solve the consensus tracking problem for multi-agent systems with a leader. In[13]–[15], the authors pay attention to the multi-agent systems with time-delay. Especially in [15], in order to optimize the inventory control in a distributed system with communication delay, a new control policy is introduced to generate smaller holding costs while maintaining the same service level as the classic scheme. Event-triggered consensus control is utilized in [16], [17] to reduce the computation and communication burden. With widespread applications of machine learning,various learning methods are introduced to settle the optimal control problems [18]–[22]. For instance, in [23], [24], the optimal consensus control problem of multi-agent systems is solved by deliberately designed reinforcement learning algorithms. Meanwhile, adaptive control is another significant development in the control field, which has greatly improved the control performance for diverse kinds of problems[25]–[27].

As multi-agent systems are normally distinguished with large scale, complex structures and multi-point communication, they are naturally modeled by graphs in which vertices are used to describe agents and edges are introduced to represent the topological relationship between agents. With the employment of graphs, the descriptions of systems become simpler and visualized. In general, there are two key elements in consensus problems of MA systems, i.e.,the dynamics of agents and the communications among them.However, the entire communication graph presents the global information. In other words, these consensus protocols cannot be computed and implemented by each agent in a fully distributed fashion. Different static and dynamic consensus protocols are designed in [28], requiring the knowledge of the communication graph to be known by each agent to determine the bound of the coupling weights. On the other hand,disturbance is unavoidable in practical applications. The consensus control for multi-agent systems with external disturbances is designed in [29], [30], where the disturbances are deterministic and defined to be known in advance or observable. But in most situations, disturbances are stochastic and unknown to the system. Although an approach of consensus control for multi-agent systems with stochastic disturbances is proposed in [31], it only considers the undirected situation. In general cases, directed graphs are much more widely applied to describe practical multi-agent systems such as distributed energy storage and inventory systems. How to achieve a dynamic consensus of multi-agent systems over general directed graphs is more tricky due to the fact that the Laplacian matrix of a directed graph is asymmetric. A distributed PI controller is proposed to tackle the static consensus of heterogeneous multi-agent systems with directed graphs in [32]. However, since the agent’s dynamics are heterogeneous, a time-varying state consensus can no longer be achieved by the designed protocol.

Inspired by the discussions above, we focus on the dynamic consensus of linear multi-agent systems in a directed topology with heterogenous stochastic disturbance. To settle this kind of consensus problem, we employ the Riccati inequality method [33] to design an adaptive tracking controller for each agent. Compared with the traditional static controller, the coupling weights of the adaptive controller can be updated by state errors between adjoining agents. Then, through the Lyapunov directed method, the stability of the closed-loop system with the designed control law is analyzed. An example that includes contrast test is given to show the effectiveness and advantage of the method. The results indicate that consensus can be reached for leader-following multi-agent systems under a directed topology with stochastic external disturbances by the proposed scheme. The contribution of the paper is mainly reflected in two aspects. First, comparing with most relative researches over directed graphs that settle the static consensus of multi-agent systems, such as [32], [34],this paper pursues the protocol that achieves time varying consensus between each agent and the leader. Second, a control policy that addresses the consensus problem of multiagent systems with a directed topology, which contains stochastic disturbances, is proposed for the first time.Although all agents have the same system matrix, the heterogeneous property of the multi-agent system appears in its disturbance fractions that are made up of different functions and multiplied by Gauss white noise.

The remaining sections of this paper are organized as follows. Section II introduces the preliminaries of graphs and multi-agent systems. Section III describes the leaderfollowing consensus problem, presents the designed consensus control law of a multi-agent system under a directed topology, and identifies its stability. Section IV performs simulations to validate the effectiveness of the proposed control scheme and demonstrate its advantage by contrast test that compares the tracking performance with that of the classic method. Finally, Section V draws conclusions.

II. PRELIMINARIES AND PROBLEM STATEMENT

A. Graph Theory

From this definition, we can know that the sum of the elements in a row element in the Laplacian matrix L is zero.Note that an undirected graph can be seen as a special case of a directed one where aij=aji, i ≠j , i,j ∈{1,2,...,N}. For any undirected graph G, its Laplacian matrix L is symmetric,meanwhile the Laplacian matrix of a directed graph is asymmetric.

As shown in Fig.1 , the structures that contain directed topology and followers with globally reachable leader are considered in this paper. For multi-agent systems with one leader and N followers, we define that 0 represents the leader and 1,2,...,N denote the N followers, respectively. A diagonal matrix is defined as D= diag[di],i ∈{1,2,...,N}. To simplify the calculation, we define that di=1 if there is a connection between the leader and the ith follower; otherwise,di=0. Then, Gris defined on the vertices 0,1,...,N.

Fig.1. The directed topology of multi-agent systems.

In this paper, RN×Ndenotes the set of n×n real matrices, INrepresents an N-dimensional identity matrix. For a symmetric matrix P, the matrix inequality P>0 means that P is positive definite. A⊗B is the Kronecker product of matrix A and B.

There are some matrices defined for simplification,W1= diag(Wi)∈RN×NandW2=[Wji]∈RN×Ni,j∈{1,2,...,N}.H=L+D ∈RN×N, M1= diag(Mi)∈RN×N, M2=[Mji]∈RN×N,

Before moving on, the following lemma referring to [35] is introduced.

Lemma 1: For matrices A,B,C and D with appropriate dimensions, we have

1)(A⊗B)T=AT⊗BT;

2)A⊗(B+C)=A⊗B+A⊗C;

3)(A⊗B)(C ⊗D)=AC ⊗BD;

4)(A⊗B)−1=A−1⊗B−1, for any given invertible matrices A and B.

B. Consensus Control

Consensus control is a typical and basic problem in the research field of multi-agent systems. The so-called consensus refers to the system’s behavior that all the individual agents in the system eventually reach the same state through interaction.For the study of consensus control, people often start from a simple first-order system [36], then extend to the second-order and even more general cases with a higher-order [37], [38].

Throughout this paper, the first-order differential dynamics model of a multi-agent system is investigated. Since the consensus of a multi-agent system is achieved when the states of the followers satisfy x1=x2=···=xN=x0, the simple consensus controller can be given as follows:

where uiis the control action of the ith agent and decided by the information interaction of each agent adjacent to the ith.

C. The Leader-Following Consensus Problem

In this paper, we consider a linear multi-agent system containing heterogeneous external disturbance with an active leader indexed by 0 and N agents denoted by 1,2,...,N. The dynamics of the ith agent is represented as

where xi∈Rnis the state of the ith agent, i ∈{1,2,...,N} and ui∈Rmis the control input of the ith agent which can use the information of its neighbours and itself only. A and B are constant real matrices. The fi(·,·):Rnis the noise intensity function vector and the v(t) is the Gauss white noise generated by the Brown motion w(t). Hence, (2) can be written as the following differential equation:

where w(t) is a one-dimensional matrix and E{w(t)} =0,E{[w(t)]2}=dt [39]. The dynamics of the leader denoted by 0 is represented as

where the state of the leader is x0. There is no control input for the leader and the dynamics of the leader is independent of others. f0(·,·):Rnis the external disturbance acting on the leader and w(t) is a standard Wiener process. There are some assumptions given for the multi-agent systems.

Assumption 1: The pair (A,B) is stabilizable.

Assumption 2: The disturbance function vector fi(·,·):Rn,i ∈{0,1,...,N} satisfies the Lipschitz condition, i.e., there is a constant matrix γ that satisfies the following inequality for all the u,v ∈Rn,i ∈{0,1,...,N}

Distributed control is adopted in this paper to drive the followers to track the leader with local information. Consider the existence of the stochastic disturbances, the leaderfollowing consensus can be achieved if the following equation is satisfied under any initial condition xi(0),i ∈{0,1,...,N}[40].

where E[·] is the mathematical expectation of a given random variable.

III. LEADER-FOLLOwING CONSENSUS CONTROL IN DIRECTED TOPOLOGY

In this section, a control policy for leader-following multiagent systems in a directed topology is developed such that the closed-loop system can be stabilized, and all the following agents are gradually consistent with the leader. According to the knowledge studied before, it is necessary to introduce the following assumption.

Assumption 3: The vertex 0 associated with the leader is globally reachable in Gr.

As we know, based on the relative state difference between neighbouring agents, the static control protocol for linear leader-following multi-agent systems can be designed as

where aijis the (i,j)th entry of the adjacency matrix ∆associated with graph Grand diis the ith diagonal entry of the leader adjacency matrix D. Let g>0 be the coupling weight among neighbouring agents and K ∈Rm×nbe a feedback gain matrix. qij>0 , i,j ∈{1,2,...,N} is a fixed weight of each edge or arc, which can be seen in Fig.2.

The coupling weight g in (7) is dependent on λ1, where λ1≤λ2≤···≤λNare the eigenvalues of H [41]. However, it is not easy to calculate λ1when the size of a multi-agent network is large. Moreover, the communication topology of the system is also needed. For these reasons, we attempt to design an adaptive controller without requiring explicit λ1.Meanwhile the coupling weights of the controller can be updated through the information from neighbouring agents.To achieve the two features above, the distributed adaptive controller is presented as

Fig.2. Directed topology of multi-agent systems with weights.

The parameter gijis a time-varying coupling weight between agents i and j , gidenotes the weight between agents i and 0. For a directed topology, which means aij≠ajiand qij ≠qji, the values of g˙ijand g˙jiare different. Besides, both ηijand ηiare positive constants.

Let εi=xi−x0. Then, by integrating (3) and (4), the state error εican be written as follows:

The positive definite matrix P is the solution of the following inequality

where τ>0 is a tunable parameter. The feedback gain matrix can be considered as

For the symmetry and tidiness, the constant gain matrix in(8) is represented as

From (13), it is obvious that Γ ≥0 such that the coupling weights gijand giare nondecreasing. In summary, the adaptive Controller (8) can solve the consensus problem of a leader-following system if the state error εi=xi−x0satisfies(6) under any initial condition xi(0),i ∈{0,1,...,N}.

Theorem 1: Consider the multi-agent systems given by (3)and (4). Under Assumptions 1–3, the state error ε can satisfy Equation (6) under any initial condition xi(0), i ∈{0,1,...,N}, by employing Controller (8).

Proof: Consider the following Lyapunov functional candidate of state error (10):

By (10) and Itô formula, the differential of V(t) can be calculated as

After that, substituting (16) and (17) into (15), according to(10), the differential of V(t) can be described as

Under Assumption 2, we get

where τ is the maximum eigenvalue of the matrix P. Since E{w(t)}=0, substituting (19) into (18), we obtain

Let αH −M1+M2=UTΛU, where U is an orthogonal matrix. Let δ=(U ⊗In)ε. Then, (21) becomes

where λi,i=1,...,n, are the eigenvalues of matrix αH −M1+M2. If αλi≥1,i ∈{1,2,...,N} holds, according to (11), (22)becomes

Thus, for any ε ≠0, dE{V(t)}<0. It implies that Equation(6) can hold by employing Controller (8) and the matrix Γ is a non-negative definite matrix such that gijand giwould not decrease. The constant α should be large enough in order to make αλi≥1, i ∈{1,2,...,N} hold. According to the above proof, the system described by (3), (4) and (8) is asymptotically stable in terms of mean square and all agents can follow the leader.■

IV. SIMULATIONS

In this paper, the directed topology of leader-following linear multi-agent systems is strongly connected, while the information transmitting among followers is affected by different weights.

Example 1: This example is used to illustrate that the control law proposed in this paper is effective to settle the consensus problem of leader-following multi-agent systems with stochastic disturbance. To demonstrate the advantage of this novel control method over the classic ones, a contrast test is conducted under the same situation.

Assume that a multi-agent system includes four followers which are denoted as 1, 2, 3, 4 and one leader which is labelled as 0. The dynamics of the ith agent satisfies (3) and (4) with

Remark 1: From Fig.2, we can see the 1st and 2nd agents are adjacent to the leader. Meanwhile the direction of information flowing is 1 →3 →4 →2 →1. Associated with different weights, Laplacian matrix of Grand the adjacency matrix of the leader can be obtained as

As it can be seen from the foregoing, (A,B) is stabilizable and the leader is globally reachable in graph Gr. It is obvious that the formation of the noise function can satisfy Assumption 2. By calculating the inequality, we can know the solutions of (11)–(13) as

and

Since V(t) is bounded in the sense of the mean square, the errors eialways tend to converge on origin and the coupling weights gijand gi, i,j ∈{1,2,3,4} are non-decreasing, which are shown in Figs. 5 and 6, respectively.

Fig.3. The trajectories of leader state and external disturbance.

Fig.4. The state error trajectories of the followers controlled by classic method.

Remark 2: Since the leader is autonomous and the stochastic noise is partly determined by the product of trigonometric functions and the state of each following agent, the system matrix A can affect the amplitude of the noise. If all eigenvalues of system matrix A have a negative real part, the leader is a stable system and the states of the leader can finally converge to zero. After that the noise is gradually equal to zero and the coupling weights gi,gijcould monotonically converge to a certain constant. In order to make the noise have an lasting effect on the system and effectively verify the tracking performance of the proposed control scheme, we choose the value of matrix A to make the leader an oscillatory system. Due to the existence of the noise, the state errors cannot be stabilized at zero and the coupling weights gi,gijgrow gradually.

Fig.5. The state error trajectories of the followers controlled by proposed method.

Fig.6. The trajectories of coupling weights in the example.

Remark 3: From the simulation results in Figs. 4 and 5, we can see that it is difficult to control the error of each state within ±0.2 through classic method in solving the consensus problem of a linear multi-agent system with stochastic disturbance. It takes nearly 4 seconds to make the errors of all states approximately converge to 0. Meanwhile, by employing the control scheme proposed in this paper, the error of each state will not exceed ±0.2 after simulation runs for two seconds — the time it takes to converge to zero for the first time. By comparing with the classic method, we can know that the advantage of the method designed in this paper is possessing faster convergence speed and higher tracking accuracy.

V. CONCLUSION

In this paper, an effective control policy is proposed for the first time to solve the consensus problem of a multi-agent system with directed topology, which contains stochastic disturbances. We develop an adaptive protocol based on Riccati inequalities with an adaptive law for adjusting coupling weights between neighbouring agents. Through the proofs of the theorem given in Section III, we demonstrate that the consensus control can be achieved by deploying the method proposed in this paper and the system stability is proved through the Lyapunov directed method. Simulation results show that the proposed control law can effectively make the system achieve the consensus. The contrast test demonstrates that the designed control scheme possesses a faster convergence speed and higher tracking performance compared with the classic method. The main limitation of the proposed method is that the convergence proof of the control scheme is not applicable to the consensus problem of fully heterogeneous multi-agent systems and it does not consider the communication delays. Future work will focus on dynamic consensus control protocols for heterogeneous multi-agent systems with communication delays and try to combine with inventory control and energy management system.