_{∞}leader-following consensus of multi-agent systems with channel fading under switching topologies: a semi-Markov kernel approach

This paper focuses on the leader-following consensus problem of discrete-time multi-agent systems subject to channel fading under switching topologies. First, a topology switching-based channel fading model is established to describe the information fading of the communication channel among agents, which also considers the channel fading from leader to follower and from follower to follower. It is more general than models in the existing literature that only consider follower-to-follower fading. For discrete multi-agent systems, the existing literature usually adopts time series or Markov process to characterize topology switching while ignoring the more general semi-Markov process. Based on the advantages and properties of semi-Markov processes, discrete semi-Markov jump processes are adopted to model network topology switching. Then, the semi-Markov kernel approach for handling discrete semi-Markov jumping systems is exploited and some novel sufficient conditions to ensure the leader-following mean square consensus of closed-loop systems are derived. Furthermore, the distributed consensus protocol is proposed by means of the stochastic Lyapunov stability theory so that the underlying systems can achieve ℋ_{∞} consensus performance index. In addition, the proposed method is extended to the scenario where the semi-Markov kernel of semi-Markov switching topologies is not completely accessible. Finally, a simulation example is given to verify the results proposed in this paper. Compared with the existing literature, the method in this paper is more effective and general.

_{∞}leader-following consensus

With the rapid development of computer technology and networks, distributed cooperative control has drawn increasing attention due to its application in various fields, especially computer science and automation control. A fundamental and important research topic of distributed system control is the consensus problem of multi-agent systems. It has attracted considerable interest among many researchers in different fields in the past few decades, due to its significant applications in civilians and militaries, such as unmanned air vehicles ^{[1-3]}, autonomous underwater vehicles ^{[4]}, multiple surface vessels ^{[5]}, robot formation ^{[6, 7]}. The consensus problem essentially refers to a team of agents reaching the same state by designing proper and available distributed control algorithms that only utilize local information exchange with neighbors. Over the past decade, there have been a wealth of interesting and instructive achievements focusing on consensus problem of multi-agent systems, including leaderless consensus ^{[8-13]} and leader-following consensus ^{[14-21]}. The leaderless output consensus problem of multi-agent systems composed of agents with different orders was studied by transforming the original system through feedback linearization. Static feedback and dynamic feedback controllers are designed to solve the consensus problem and sinusoidal synchronization problem under uniformly jointly strongly connected topologies ^{[8]}. Under cyber-attacks, literature ^{[9]} proposed a fully distributed adaptive control protocol to solve the leaderless consensus problem of uncertain high-order nonlinear systems. The work ^{[11]} discussed the event-triggered coordination problem for general linear multi-agent systems based on a Lyapunov equation method. Leader-following consensus means that the states of all follower agents are expected to approach the state of the leader agent. In many practical situations, leader-following consensus can accomplish more complex tasks by enhancing inter-agent communication. Compared with leaderless consensus, leader-following consensus can be beneficial in reducing control costs and save energy. The key to the leader-follower consensus problem is how to design a distributed control protocol to synchronize the states of all follower agents and the leader agent. The work ^{[14]} proposed a novel distributed observer-type consensus controller for high-order stochastic strict feedback multi-agent systems based only on relative output measurements of neighbors. The 1-moment exponential leader-following consensus of the underlying system is ensured by adopting appropriate state transformation. In ^{[15]}, the sampled-data leader-following consensus problem for a family of general linear multi-agent systems was addressed, and the distributed asynchronous sampled-data state feedback control law was designed. The event-based secure leader-following consensus control problem of multi-agent systems with multiple cyber attacks, which contain reply attacks and DoS attacks simultaneously, was studied in ^{[16]}. The fixed-time leader-following group consensus of multi-agent systems composed of first-order integrators was realized under a directed graph^{[17]}. By designing a nonlinear distributed controller, the follower agents of every group can reach an agreement with its corresponding leader within a specified convergence time. In ^{[19]}, the author considered the problem of resilient practical cooperative output regulation of heterogeneous linear multi-agent systems, in which the dynamics of exosystem are unknown and switched under DOS attacks. A new cooperative output regulation scheme consisting of distributed controller, distributed resilient observer, auxiliary observer and data-driven learning algorithm was proposed to ensure the global uniform boundedness of the regulated output. More results can be seen in ^{[18, 20, 21]} and references cited therein.

It is well known that there is a large amount of data transmission in the control process of multi-agents. Data packets or signals between agents are usually transmitted through wireless communication networks. However, some special physical phenomena (such as reflection, refraction, diffraction) may occur during the transmission of a signal or data packet through a communication link or channel, which will result in the loss of signal energy and lead to the signal distorted. This type of phenomenon is often referred to the channel fading. Typically, fading effects are closely related to multipath propagation and shadows from obstacles. In practical applications, the factors that cause channel fading mainly include time, geographic location, and radar frequency. As a result, the phenomenon of channel fading may result in the degradation of signal quality due to the inability to receive accurate transmission information, thereby deteriorating the desired performance of the system. This also shows that it is meaningful to consider channel fading effects for the distributed control of multi-agent systems. In view of this, some results on channel fading have been published, such as channel fading of single systems ^{[22-26]}, channel fading for multi-agent systems ^{[27-29]}. The reference ^{[22]} designed a nonparallel distribution compensation interval type-2 fuzzy controller to address dynamic event-triggered control problems for interval type-2 fuzzy systems subject to fading channel, where the fading phenomenon is characterized by a time-varying random process. The literature ^{[23]} focused on the finite-horizon ^{[25]}, the event-triggered asynchronous guaranteed that cost control problem for Markov jump neural networks subject to fading channels could be addressed, where a novel rice fading model was established to consider the effects of signal reflections and shadows in wireless networks. The consensus tracking problem of second-order multi-agent systems with channel fading was investigated using the sliding mode control method, and the feasible distributed sliding mode controller was designed by introducing the statistical information of channel fading to the measure functions of the consensus errors ^{[27]}. It should be pointed out that most of the literatures mentioned above on channel fading in multi-agent systems only consider the fading effect among the follower agents and ignore the fading effect between the leader agent and the follower agents. As stated earlier, the leader plays a crucial role in the leader-following consensus problem. To improve the applicability of the controller and the ability to deal with the problem, it is reasonable and necessary to consider both the fading effect of leader-to-follower and follower-to-follower agents at the same time in the channel fading problem of multi-agent systems. This is one of the motivations of this paper.

On the other hand, the communication topology of multi-agent systems may change in practice due to various factors, such as sudden changes in the environment, communication range limitations, link failures, packet loss, malicious cyber attacks, etc. Given this, many researchers assume that the topology among agents is time-varying or Markov switching. Some good consensus results for multi-agent systems under time-varying topology and Markov switching topology have been reported in the past decade ^{[30-34]}. For example, the work ^{[33]} investigated the coupled group consensus problem for general linear time-invariant multi-agent systems under continuous-time homogeneous Markov switching topology. The designed linear consensus protocol can achieve coupled group consensus of the considered system under some algebraic and topological conditions. It is worth noting that since the transition probability in Markov jump process is constant and there is no memory characteristic, there are still some limitations in using Markov jump process to model topology switching among agents. Recently, a class of more general semi-Markov jump processes with a non-exponential distribution of sojourn-time (the time interval between two consecutive jumps) and time-varying transition probabilities has attracted interest of many scholars and has been used to characterize the topological switching among agents ^{[35-38]}. For example, the leader-following consensus of a multi-agent system under a sampled-data-based event-triggered transmission scheme was realized ^{[35]}, where a semi-Markov jump process was employed to model the switching of the network topologies. The containment control problem concerning semi-Markov jump multi-agent systems with semi-Markov switching topologies was studied by designing static and dynamic containment controllers ^{[37]}. Under a semi-Markov switching topology with partially unknown transition rates, the ^{[38]}. However, most of the above literature on semi-Markov switching topology is considered for the continuous system case. As a matter of fact, in the discrete-time case, the semi-Markov jump process can exert a stronger modeling ability and have a larger application range. The reason is that the probability density function of sojourn-time in the discrete semi-Markov jump process can be of different types in different modes, or of the same type but with different parameters. In order to make the modeling of switching topology more realistic, it is very necessary and valuable to employ discrete-time semi-Markov switching topologies. Naturally, how to deal with the consensus problem of multi-agent systems in this situation is a key point. Recently, the discrete semi-Markov jump process was adopted to model general linear systems and a semi-Markov kernel method was proposed to address its stability and stabilization problems ^{[39, 40]}. This also provides an idea for solving the consensus problem of multi-agent systems under discrete semi-Markov switching topology. To the best of our knowledge, to date, there have been few results on the leader-following consensus problem for discrete-time multi-agent systems subject to channel fading under semi-Markov switching topologies. Therefore, how to design a suitable distributed control protocol and how to establish leader-following consensus criteria for multi-agent systems with channel fading under discrete semi-Markov switching topologies are the key issues. This inspires us to carry out this work.

Motivated by the above discussion, in this paper, the ^{[27, 41]}, a more general channel fading model based on discrete semi-Markov switching topologies is established to characterize the possible effects of inter-agent signal transmission. The influences of channel fading between leader and follower, follower and follower agents are simultaneously considered to explore the influence of channel fading on system consensus, rather than considering only channel fading among follower agents in literature ^{[27, 41]}. (Ⅱ) As mentioned in the previous paragraph, discrete semi-Markov processes are more powerful in modeling ability and application range than Markov and continuous-time semi-Markov processes. For this reason, different from the Markov switching and continuous-time semi-Markov switching topologies adopted in ^{[32-38]}, this paper employs a discrete semi-Markov process to describe the network topology switching among agents and switching for channel fading. A set of novel sufficient conditions to guarantee that leader-following mean square consensus of multi-agent systems under semi-Markov switching topologies is derived via a semi-Markov kernel approach. (Ⅲ) The distributed consensus controller design scheme based on fading relative states is proposed to solve the

Notations: Denote that

In this paper, we employed an undirected graph

Consider a linear discrete-time multi-agent system consist of one leader and

where

In the real world, the communication network topology among agents is more likely to be time-varying. In this paper, a switching signal

To describe semi-Markov chain more formally, the following concepts are introduced. (Ⅰ) The stochastic process

^{[39]} The stochastic process

where

Denote the matrix

where

^{[35, 37, 38]} studied the leader-following consensus and containment control problems for multi-agent systems with semi-Markov switching topologies, respectively. A continuous-time semi-Markov jump process is employed to describe the switching of the topology. Accordingly, the probability density function of the sojourn-time can only be of a fixed probability distribution type for the different modes. This limits its practical application. In this paper, a discrete semi-Markov chain is introduced to characterize topology switching among agents. The introduced probability density function of sojourn time depends on both the current mode and the next mode, so that different parameters of the same distribution or different types of probability distributions can coexist. Hence, the probability density function introduced in this paper is more applicable than that in the literature ^{[35, 37, 38]}.

^{[39]} The stochastic process

Based on the above introduction to the semi-Markov chain, the switching topology among agents in this paper can be denoted as

Every topological graph

In practice, the communication process between an agent and its neighbors is often affected by channel noise and fading. Motivated by the channel fading model in ^{[28, 29]}, in this paper we assume that each agent obtains relative state information from its neighbors through fading channels. Correspondingly, the channel fading model among agents can be expressed as

where

where

Multi-agent systems with channel fading under semi-Markov switching topologies

^{[27-29]}, the channel fading model introduced in this paper not only considers the fading influence from leader to follower and follower to follower, but also introduces the effect of channel interference on signal transmission and the influence of topology switching on the statistical characteristics of fading coefficients. This means that the models presented in this paper are more general than those in the previous literature. When ^{[27]}. If the values of

Define the consensus error as

with

where

Before the subsequent analysis, some definition and assumptions are introduced.

^{[39]} The dynamic system (6) is said to be

for given the upper bound of sojourn-time

According to the above analysis and discussion, it can be found that the leader-following consensus of system (1) under the semi-Markov switching topology

with

Consequently, the objective of this paper is to design the distributed consensus controller such that the following two conditions are satisfied:

(Ⅰ) when

(Ⅱ) the inequality

In the subsection, the leader-following consensus and

hold for any

where

On the other hand, the stability of the system during mode switching needs to be considered. It can be derived along the trajectory of systems (6) and (8) that

with

Together with the condition (12), there exists a positive constant

Then, it follows that

Summing both sides of the above equation from 0 to

Let

Next, the

Denote

where

Further, we can obtain

under the zero initial conditions. Thus, the

In Theorem 1, the leader-following consensus and

In this case, the following corollary gives the leader-following mean square consensus analysis of the system (18) under no-channel interference fading model.

hold for

The proof of Corollary 1 is similar to that of Theorem 1, so it is omitted.

Although Theorem 1 addresses a family of leader-following consensus conditions, these conditions cannot be directly utilized to solve controller gains

hold for any

then the system (6) is leader-following consensus in mean square sense and possess a

According to the literature ^{[40]}, we can obtain that for the positive definite matrix

can ensure

for any

Supposing that the channel inferences

for any

In this subsection, it is assumed that the information of the semi-Markov kernel ^{[40]}, the index set

where

In this paper, only the case of

Then, the following Theorem proposes the leader-following mean-square consensus conditions for systems (6) and (8) under incompletely accessible semi-Markov kernel of switching topologies.

hold for any

then systems (6) and (8) is leader-following mean square consensus under incompletely accessible semi-Markov kernel of switching topologies and possess a

Similar to inequality (13), we have

for

with

Then,

By Schur complement lemma, the above inequality can be further transformed into

where

Applying congruence transformation

with

^{[28, 29]}, the above literature only considers leaderless multi-agent systems. They ignore the fading effects from leader to follower agents, and the edge Laplacian method introduced cannot be used to tackle the models considered in this paper. Therefore, it is interesting and meaningful to investigate the non-identical channel fading problem within the framework of the fading model proposed in this paper. No better method has been proposed to solve the problem of non-identical channel fading under model (3). This also encourages us to continue to study this issue in future work.

and

for fully known semi-Markov kernel and incompletely available semi-Markov kernel.

Similar to Corollary 2, the following corollary gives the mean square consensus controller design for system (18) with incompletely accessible semi-Markov kernel.

for any

The proof of Corollary 3 can be obtained in a similar way to Theorem 3, which is omitted here.

^{[6, 11, 16]}. The advantages and disadvantages of these methods cannot be directly compared. Similarly, event-triggered control and sampled-data control methods can also be applied to the problems considered in this paper. Naturally, event-triggered control and sampled-data control can also be studied in a distributed framework.

In this section, a numerical example is provided to demonstrate the validity of the proposed results. Consider a multi-agent system consisting of four followers and one leader with the following parameter matrices

In this paper, the information exchange between agents is represented by an undirected switching topology network. The topology graphs are shown in

The communication topology graph.

A discrete semi-Markov jump process with semi-Markov kernel is employed to describe the switching of topologies. The transition probability matrix of EMC and the probability density function of sojourn time are provided by

Let the upper bound of the sojourn-time for each topology mode be

First, we assume that there is no channel fading phenomenon between the leader and the follower, that is,

By solving simplified consensus conditions and simulating, we can obtain the state response curves of the system consensus error in this case as the solid line in

The consensus error responses under reduction conditions of Theorem 2 and additional leader-to-follower fading.

Next, consider the simultaneous existence of channel fading between leader and follower and between follower and follower. Choosing

In addition, the

The state responses of

The state responses of

In

The norm-squared responses of

A possible mode jumping diagram of communication topologies.

Then, the relationship between the statistical properties of channel fading and the

1 | 2.3705 | |

2.4327 | ||

2.7232 | ||

2 | $$\sigma_{2}=0.15$$, $$\sigma_{02}=0.25$$, $$\sigma_{3}=0.2$$, $$\sigma_{03}=0.15$$ | 2.3705 |

$$\sigma_{2}=0.1$$, $$\sigma_{02}=0.2$$, $$\sigma_{3}=0.15$$, $$\sigma_{03}=0.1$$ | 2.2259 | |

$$\sigma_{2}=0.05$$, $$\sigma_{02}=0.1$$, $$\sigma_{3}=0.1$$, $$\sigma_{03}=0.05$$ | 2.1583 | |

3 | $$\sigma_{1}=0.15$$, $$\sigma_{2}=0.25$$, $$\sigma_{3}=0.3$$, $$\sigma_{1}=0.2$$, $$\sigma_{2}=0.35$$, $$\sigma_{3}=0.25$$ | 2.7265 |

$$\sigma_{1}=0.2$$, $$\sigma_{2}=0.3$$, $$\sigma_{3}=0.35$$, $$\sigma_{1}=0.25$$, $$\sigma_{2}=0.4$$, $$\sigma_{3}=0.3$$ | 3.1614 | |

$$\sigma_{1}=0.25$$, $$\sigma_{2}=0.35$$, $$\sigma_{3}=0.4$$, $$\sigma_{1}=0.3$$, $$\sigma_{2}=0.45$$, $$\sigma_{3}=0.35$$ | 3.9035 |

According to ^{[27]}, Theorem 3.2 of^{[41]}, and Simplified Theorem 2 in the absence of leader-to-follower fading additionally consider the channel fading of leader-to-follower. The control method proposed in Theorem 2 in this paper can still ensure the consensus performance of the system considering the channel fading between the leader-to-follower and follower-to-follower agents at the same time. This further proves that the model proposed in this paper is more general and the results are more effective than existing ones.

Comparative simulations with literature ^{[27, 41]} for consensus performance

Theorem 3.2 in ^{[27]} |
Consensus | Inconsistent |

Theorem 3.2 in ^{[41]} |
Consensus | Inconsistent |

Simplified Theorem 2 without leader-to-follower fading | Consensus | Consensus performance deterioration |

Theorem 2 | Consensus | Consensus |

In this paper, the

Made substantial contributions to the research, idea generation, algorithm design, simulation, wrote and edited the original draft: Yang H

Performed critical review, commentary and revision, as well as provided administrative, technical, and material support: Zhang H, Wang Z, Zhou X

Not applicable.

This work is supported by National Natural Science Foundation of China(61922063), Shanghai Hong Kong Macao Taiwan Science and Technology Cooperation Project (21550760900), Shanghai Shuguang Project (18SG18), Shanghai Natural Science Foundation(19ZR1461400), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100) and Fundamental Research Funds for the Central Universities. (Corresponding author: Zhang H).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2022.

Zhang D, Deng C, Feng G. Resilient cooperative output regulation for nonlinear multi-agent systems under DoS sttacks.

10.1109/TAC.2022.3184388