Stability of General Linear Dynamic Multi-Agent Systems under Switching Topologies with Positive Real Eigenvalues

Shengo Een Li*, Zhito Wng Yng Zheng, Dinge Yng Keyou You

a State Key Lab of Automotive Safety and Energy, School of Vehicle and Mobility, Tsinghua University, Beijing 100084, China

b Department of Engineering Science, Balliol College, University of Oxford, Oxford OX1 3PJ, UK

c Department of Automation, Tsinghua University, Beijing 100084, China

Keywords:

Stability Multi-agent system Switching topologies Common Lyapunov function

A B S T R A C T

The time-varying network topology can significantly affect the stability of multi-agent systems. This paper examines the stability of leader-follower multi-agent systems with general linear dynamics and switching network topologies, which have applications in the platooning of connected vehicles. The switching interaction topology is modeled as a class of directed graphs in order to describe the information exchange between multi-agent systems, where the eigenvalues of every associated matrix are required to be positive real. The Hurwitz criterion and the Riccati inequality are used to design a distributed control law and estimate the convergence speed of the closed-loop system.A sufficient condition is provided for the stability of multi-agent systems under switching topologies. A common Lyapunov function is formulated to prove closed-loop stability for the directed network with switching topologies.The result is applied to a typical cyber-physical system—that is, a connected vehicle platoon—which illustrates the effectiveness of the proposed method.

1. Introduction

In recent years, the coordination control of multi-agent-based cyber-physical systems has attracted considerable research attention due to theoretical breakthrough and wide-ranging engineering applications. Research topics in coordination control include consensus control[1],rendezvous control[2],flocking control,and formation control [3]. In addition, coordination control has a broad range of applications due to its efficiency and reliability, such as vehicle platooning, the formation of multiple unmanned aerial vehicles (UAVs), collaborative assembly systems [4], and sensor networks [5,6].

One central topic is the design of a distributed control law to stabilize a multi-agent system or reach a certain consensus,where each agent only uses local information from its neighbors for feedback[7].Graph Laplacians play an important role in describing the interaction topologies and analyzing the stability of multi-agent systems [8,9]. The theoretical framework for proving the stability with graph Laplacians was introduced in the seminal work by Olfati-Saber et al.[10,11],where each agent of the multi-agent system is a single integrator. By extending this framework into double-integrator dynamics,Ren and colleagues[12,13]presented sufficient and necessary conditions for the stability of multi-agent systems from a graph-theoretic perspective,where the transformation of the Jordan normal form was applied to analyze the closedloop matrices. For high-order dynamics, Ni and Cheng [14]designed a stability algorithm based on the Riccati and Lyapunov inequality. Zheng et al. [15] proved the stability under interconnected topologies whose matrix has positive real eigenvalues using matrix decomposition and the Hurwitz criterion. Hong et al. [16]proposed a rigorous proof for the stability with an extension of LaSalle’s invariance principle.Beyond the abovementioned control law, Zheng et al. [17]also designed a distributed model predictive controller for multi-agent nonlinear systems and formulated a Lyapunov function to prove the asymptotic stability of a connected vehicle platoon.Wu et al.[18]presented a distributed sliding mode controller for multi-agent systems with positive definite topologies and exploited the asymptotic stability in the Lyapunov sense.Barooah et al. [19] introduced a mistuning-based control method to improve the stability margin of vehicular platoons.Ploeg et al.[20]developed an H-infinity control law to achieve the string stability of multi-agent systems.

The variation of interaction topologies is quite common due to link failures/creations in networks or obstruction between interactional agents. The stability of multi-agent systems under switching topologies has also attracted considerable research attention. For example, Tanner et al. [21] proposed a control law in combination with the attractive and alignment forces,which could stabilize the flocking system under dynamic topology. Olfati-Saber et al. [10] introduced a common Lyapunov function that could ensure the stability of single-integrator linear systems based on matrix theory and algebraic graph theory. Ren[12] considered a multi-agent system with double-integrator dynamics and showed that a set of connected, undirected, or directed topologies could stabilize the switching system by proving that the Lyapunov function is locally Lipschitz continuous. Ni and Cheng [14] expanded this study into a high-order integrator dynamic system and discussed the problem under the jointly connected undirected graph using Cauchy’s convergence criteria.Theoretically, the stability analysis of directed graphs is more challenging than the case of an undirected graph [10]. The methods for undirected topologies cannot naturally be applied to problems with directed topologies due to the lack of a positive definite property in directed topologies. In addition, it is more challenging to find a common Lyapunov function for switching directed topologies. Some pioneering studies have focused on the stability analysis of multi-agent systems with special switching directed topologies. For example, Qin et al. [22] analyzed a Lyapunov function of switching directed topologies systems and proved that system stability can be achieved under balanced directed graphs.Dong et al. [23] explored an explicit expression of the timevarying formation reference function and showed that the stability can be maintained if the dwell time is greater than a positive threshold.

Fig.1. A depiction of the relationship between the discussed topologies. Positive real eigenvalues topology has the property of all the eigenvalues of matrix （L+P）being positive real. The followers in the forward-back topology can receive information from the same number of agents both forward and backward.It is clear that the forward-back type of topology is both a balanced graph and a positive real eigenvalues topology.

The rest of this paper is organized as follows: Section 2 introduces the algebraic graph theory. In Section 3, a class of positive real eigenvalues topologies is introduced and a linear controller designed with a common Lyapunov function and Riccati inequality is proposed.In Section 4,the stability and convergence speed of the closed-loop systems under switching topologies are proved. Section 5 illustrates the method through numerical simulation, and Section 6 concludes this paper.

2. Preliminaries and problem statement

This paper considers a multi-agent system that consists of one leader and N followers. The dynamics of each agent are homogeneous and linear. It is assumed that all the eigenvalues of the matrices （L+P） describing the interaction topologies are positive and real numbers.

2.1. Communication graph topology

To represent the information flow between the leader and followers, a pinning matrix P is defined as P =diag｛p1， p2， ...， pN｝,where pi=1 if the agent can obtain the information from the leader; otherwise, pi=0. Based on the pinning matrix P, a leaderreachable set could be defined as Pi= ｛0 ｝ if pi=1; otherwise,Pi=?. Then, an information-reachable set is defined as Ii=Ni∪Pito represent the nodes from which agent i can obtain information.

2.2. Agent dynamics

The dynamics of each agent is:

where xi（t ）∈Rndenotes the state vector, ui（t ）∈Rmis the control input, n and m are the dimension of state and control variable respectively,A ∈Rn×nand B ∈Rn×mare the system matrix and input matrix, respectively. The system is assumed to be stable by choosing an appropriate value of the pair （A， B）.

The leader has the following linear dynamic:

where x0∈Rnis the state of the leader.

2.3. Stability of multi-agent systems

The objective of multi-agent consensus control is to make the state of each following agent consistent with that of the leader.For every agent i ∈ ｛1， ...， N｝, a distributed controller ui（t ） is required to realize

For the simplicity of the subsequent stability analysis, a new tracking error is defined as follows:

The state space function of the tracking error is

3. Design of the controller

The interconnected topology of a multi-agent system varies with time due to some communication breakdown or obstacle between agents. In a switching topology problem, the information-reachable set of every agent varies with time. The notation （L+P）σ is used to describe the time-dependence of information flow,in which σ: ［0， ∞）→∑is a switching signal at time t,and ∑is the index set of a group of graphs containing all the topologies. Consider an infinite sequence of nonempty time intervals ［tk， tk+1）， k=0， 1， ...with t0=0， tk+1-tk≤Tcfor some constant Tc. It is assumed that σ is constant in each interval and the graph can be denoted as Gσ.In order to ensure stability under varying topologies,an appropriate controller and the graph set ｛G∑｝are designed in this section.

3.1. Linear control law

For each agent,the controller is distributed and can only use the information from its information-reachable set Ii. The following control law is used [24]:

where K ∈Rm×nis a linear feedback gain.Substituting Eq.(6)to Eq.(5),the closed-loop dynamics of agent i can be obtained as follows:

To describe the dynamic of the multi-agent system, the collective states of the system are defined as follows:

Recall the definition of Laplacian matrix L and pinning matrix P; the closed-loop dynamics of the leader-follower multi-agent system are

where INis the identity matrix and symbol ?is the Kronecker product. The overall closed-loop system matrix is defined as follows:

For a linear system, the stability is associated with the eigenvalues of the closed-loop system matrix. From Eq. (10), it can be seen that the eigenvalues of Acdepend on （L+P）. In other words, the interconnected topology influences the stability of the multi-agent system. In the following subsections, we will discuss a class of topologies that ensures that the eigenvalues of （L+P）are positive real numbers.

3.2. Interconnected topologies with positive real eigenvalues

The method proposed in this paper is suitable for a topology with positive real eigenvalues that lacks an exact uniform mathematic description. Therefore, a specific type of topology with a positive real property is particularly focused on in this paper.

Lemma 1[15]: Let λi， i=1， 2， ...， N, be the eigenvalues of（L+P）, then all the eigenvalues are positive real numbers; that is, λi＞0， i=1， 2， ...， N, if there exists a directed spanning tree whose root is the leader and one of the following conditions holds:

(1) The interconnected topology of the following agents is the forward type; that is, Ni= ｛i-hu， ...， i-hl｝∩ ｛1， ...， N｝, where huand hlare the upper and lower bound of forward communication range respectively.

(2) The interconnected topology of the following agents is the forward-backward type; that is, Ni= ｛i-h， ...， i+h｝∩｛1， ...， N｝/｛i｝, where h is the communication range.

(3) The communication topology of the following agents is the undirected type; that is, j ∈Ni??i ∈Nj.

Remark 1:For single-integrator or double-integrator dynamics,it is proved that switching directed topologies with a directed spanning tree is sufficient to stabilize the system; for example,see Refs. [10,12].

Remark 2:In Ref. [14], stability under the switching of jointly connected undirected topologies is discussed. Our paper considered directed topologies; disconnected conditions are not considered, and will be studied in further work.

3.3. Design of the coefficient matrix

Since the pair （A， B）is stabilizable,there exists a solution P ＞0 for the following Riccati inequality:

where δ is a positive number, which can be designed to influence the convergence of the system [25], and I is the identity matrix.The feedback matrix K can be constructed as follows:

where α is the scaling factor that satisfies the following:

Lemma 2:is the well-known Gershgorin Disk Criterion.

Theorem 1:For the topology described inLemma 1, （L+P） is transformed to a Jordan diagonal canonical form J. Then He（J ） is a positive definite matrix.

Proof:For the topology defined as (2) and (3) inLemma 1,matrix （L+P）is real symmetric.It is obvious that He（J ）is positive definite,since J is a diagonal matrix.For the topology defined as(1)inLemma 1,the eigenvalues of （L+P）are larger than or equal to 1. J can be written as follows:

For each block of He（J ）, it has the following form:

Remark 4: Theorem 1shows that the minimum eigenvalue of He（J ）can influence the stability margin of the multi-agent system.It can be seen from Table 1 that the stability margin of the PF and BD topologies will get worse as the size N of followers increases,while the stability margin of the PLF,TPF,TPLF,and BDL topologies is independent of size N.The information from the leader is important for the stability margin of the system,and a suitable selection of topology,such as PLF and BDL,can improve the stability margin of the system. The result of the undirected topologies BD and BDL is the same as shown in Ref. [27]. A strict theoretical analysis will be conducted in future.

4. Stability under switching topologies

It is obvious that for a finite switching system, stability can be realized if the final topology can stabilize the system with the control law proposed in Section 3.Under infinite switching conditions and under a class of topologies, the system will be stabilized with the control law shown in Eq.(6).The speed of convergence can also be ensured.

Lemma 4[28]:Given a family fσ,σ ∈Σ of functions from Rnto Rn, where Σ is some index set. This can represent a family of systems x˙=fσ（x ）,σ ∈Σ. If all systems in the family share a common Lyapunov function,then the switching system x˙=fσ （x ）is globally uniform asymptotically stable.

This theorem will be used to prove our main theoretical result. Before the proof, some lemmas in matrix theory will be introduced.

Lemma 5:Consider a positive definite real matrix M, and a positive real number ξ ＜min｛λ（ M）｝, where λ（ M） denotes the eigenvalues of M. The matrix M-ξI is still positive definite.

Proof:If λiis an eigenvalue of M, there exists an eigenvector xisatisfying Mxi=λixi. Then, we have （M-ξI）xi= （λi-ξ）xi. Since 0 ＜ξ ＜min｛λ（M ）｝, all the eigenvalues of （M-ξI ） are positive. It is obvious that （M-ξI） remains symmetric. Therefore, M-ξI is a positive definite matrix.

The main result of this paper is stated as follows.

?Number of followerPFPLFTPFTPLFBDBDL 5 0.26792220.16202 6 0.19812220.11622 7 0.15222220.08742 8 0.12062220.06812 9 0.09792220.05462 100.08102220.04472

Proof:Following the control law in Eq.(12)and Inequality(13),the following inequality can be obtained:

The closed-loop dynamics of the multi-agent system are

For a positive real topology, （L+P）σ is transformed to a Jordan diagonal canonical form. The closed-loop dynamic matrix can also be transformed to a diagonal block matrix:

Substituting Inequality (13) into Eq. (19), we have

The matrix

is still symmetric. He （Jσ） is a positive definite matrix, according to

Theorem 1.

According to the Lemma 5, the inequality can be derived as follows:

The following inequality can be derived according toLemma 5:

Remark 6:In practice, the switching topologies may be unknown, which makes the selection of α nontrivial. A larger α is helpful to stabilize the switching system in this situation. In fact,Inequality (13) is only a sufficient condition for the system stability, which ensures the stability in theory. In our simulation, an α inconsistent with this inequality can also stabilize the system.

5. Simulation results

The vehicle platoon is a typical multi-agent system, which has attracted increasing attention because of its benefit in traffic[24]. The (L+P） matrices of typical topologies that describe the information flow among the vehicles in a platoon have positive real eigenvalues [15]. We conducted simulations of a homogeneous platoon with six identical vehicles (one leader and five followers)in order to validate the effectiveness. For platoon control, a thirdorder state space model is derived for each vehicle [17]:

Fig.2. Switching topologies and are all positive real eigenvalue topologies. and are the forward type and is the forward-back type. In the simulations, the topology switches among these three topologies.

The eigenvalues of He（J ） for the three topologies are listed in Table 2.All the eigenvalues are positive real and,considering their minimum value,the scaling factor α can be chosen to be 10.Three scenarios have been simulated,with two stable scenarios of different response coefficients δ and one unstable scenario.The controller parameters in Scenarios 1 and 2 are designed as in Theorem 2.However,the parameters in Scenario 3 do not satisfy the stability condition in Ref.[15].All the parameters are listed in Table 3.

Fig.4 shows the state error of the vehicle platoon under the switching topologies.The simulation result shows that the control law designed according to Eq.(12)and Inequality(13)can stabilize the vehicle platoon. Compared with Fig.5, it demonstrates that a larger δ tends to make the system converge to the stable state more quickly. Fig.6 illustrates the performance of a controller whose parameters are chosen as the unstable region criterion in Ref.[15], which can show the effectiveness of our controller design method.It should be noted thatTheorem 2is only a sufficient condition for the system stability, which means that the selection of controller parameters—that is, if α does not meet the condition of Inequality (13)—may also stabilize the switching system.

6. Conclusions

This paper examines the stability of multi-agent systems under a class of switching topologies,where all the eigenvalues of(L+P)matrices are positive real numbers. Graph theory is used to describe the interconnected topology. The Hurwitz criterion andRiccati inequality are applied to design the control law in order to stabilize the multi-agent system and adjust the convergence speed of the system. By using the common Lyapunov function theorem,the stability of switching topology systems is proved. We have shown that stability can be achieved if the （L+P）matrices’eigenvalues of all the topologies are positive real numbers and present a sufficient condition for the switching system.The exponential stability and convergence speed can be influenced by the response coefficients δ in our controller.

?Switching topologyEigenvalue of He J（ ）～G10.27, 1.00, 2.00, 3.00, and 3.73～G20.59, 1.00, 2.00, 3.00, and 3.41～G30.16, 1.38, 3.43, 5.66, and 7.37

?ParametersScenario 1Scenario 2Scenario 3 K 10.071.6010.00 24.008.642.10 8.003.204.00 α 10.0010.00—δ 0.500.20—

Fig.3. Switching signal. The dwell time is set as 2 s.

Fig.4. Stability performance under switching topologies with δ = 0.5. (a), (b), and (c) show the tracking error of the position, velocity, and acceleration, respectively. The switching system achieved stability in 15 s.

Fig.5. Stability performance under switching topologies with δ = 0.2. (a), (b), and (c) show the tracking error of the position, velocity, and acceleration, respectively.Compared with the controller in Scenario 1, this controller tends to have a longer convergence time of about 25 s.

Fig.6. Stability performance under switching topologies with an unstable controller. (a), (b), and (c) show the tracking error of the position, velocity, and acceleration,respectively. The parameters are designed from the unstable region presented in Ref. [15]. This illustrates the effectiveness of our controller design method.

Acknowledgements

This work is supported by International Science and Technology Cooperation Program of China (2019YFE0100200) and Beijing Natural Science Foundation(JQ18010).It is also partially supported by Tsinghua University-Didi Joint Research Center for Future Mobility.

Compliance with ethics guidelines

Shengbo Eben Li, Zhitao Wang, Yang Zheng, Diange Yang, and Keyou You declare that they have no conflict of interest or financial conflicts to disclose.