# Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances

^{*}

## Abstract

**:**

## 1. Introduction

- The introduction of Halanay inequality into the area of discrete multiagent system control with various delays. Comparing with Lyapunov dependent methods, the advantage of Halanay inequality lies in the easy extension of results from undirected graphs to directed graphs. At the same time, unlike the stochastic matrix theory, no row sum condition is necessary to guarantee the consensus by Halanay inequality. Even though the method proposed in this paper is preliminary and not capable of solving all kinds of consensus problems in multiagent systems, it can still be viewed as a supplement of traditional tools for situations where they can hardly be applied.
- Extension of the discrete Halanay inequality into a more general one. With the extension, the obtained result is more suitable for complex problems with disturbances. It should also be noticed that the extension we have made is a fundamental one and the potential applications are not limited in multiagent systems.

## 2. Problem Formulation

_{i}(k) ∈ ℝ indicates the state of agent i, k is the k

_{th}update instant, T(k) is the sampling period between instant k and k + 1 with assumption that $\underset{\xaf}{T}\le T(k)\le \overline{T},{u}_{i}(k)\in \mathbb{R}$ denotes the control input of agent i. In this paper, it is assumed that all agents are identical, that is, ∀i ∈ {1, 2,⋯, n}, p

_{i}= p and q

_{i}= q. In addition, the states of all agents are scalars to avoid possible complexity in expression and all results obtained in this paper can be easily extended into multiple states agents with kronecker product.

**Remark 1.**From Equation (1), it is assumed that the sampling period of the whole system is time varying. Under this assumption, the obtained results can be applied to more general situations. Usually, for feedback systems based on sensors, the sampling period is determined by configurations and always fixed. However, for the situation where centralized feedback system is deployed, such as the vision-based position feedback of indoor multi-UAVs system, the sampling period maybe time-varying from time to time due to the computing time of the central server.

_{ij}= 1. Adjacency matrix of the graph can be denoted as A = (a

_{ij})

_{n}

_{×}

_{n}and the neighbor set of agent i is N

_{i}= {j ∈ υ : (i, j) ∈ ε}. A path in a graph is denoted as a sequence of ordered edges (i, i

_{1}), (i

_{1}, i

_{2}),⋯, (i

_{m}, j) and if there is a path to i from any vertex in the graph, i is called a globally reachable vertex. Besides, diagonal matrix is defined as D = diag{d

_{1}, d

_{2}, ⋯, d

_{n}} with ${d}_{i}={\sum}_{j\in {N}_{i}}{a}_{ij}$. In the rest of this paper, it is assumed that the graph is connected and undirected.

**Definition 1.**The consensus of multiagent system consists of Equation (1) is reached if for any initial states of agents, there exists

**Definition 2.**The bounded consensus of multiagent system consists of Equation (1) is called achieved if there exist a constant C > 0, that

_{i}(k) depending only on local information to reach consensus or bounded consensus for the multiagent system with consideration of various delays and disturbances. To solve the consensus problems proposed as above, the following control protocol is constructed at first

_{i}and β

_{i}are positive control parameters to be designed, τ(k) is bounded sending delay satisfying 1 ≤ τ(k) ≤ r. In addition, with receiving delays μ(k) concerned in this paper, it should be noticed that shared information from neighbored agents will not be effective until agent i receives it. As a result, the control input of agent i should be revised as

**Remark 2.**From the formulation process of the controller (6), the time delays between agents are formulated as combination of sending delay and receiving delay in this paper. Through dividing delays into two different parts, a more clear description about sources of delays is obtained. It is assumed that τ(k) ≥ 1, indicating that agents cannot instantly send the feedback data out to neighbors due to the limitation from system resources or structures. In addition, since the discrete systems are sampled-data dependent, τ(k) is an integer number. On the other hand, the receiving delay depicts the information delay between agents within one sampling period. From practical point of view, the kind of delay always results from instruments or transmissions. Combining two kinds of delays, the total information delay between two connected agents can be calculated as$\mu (k)+{\sum}_{j=k-\tau (k)}^{k}T(j)$. In previous researches, conditions about the total delay are usually discussed. With this kind of construction, we can analyze the influence of delays on the system with better clearness through considering the sending delay and receiving delay separately.

## 3. Main Results

_{1}(k), x

_{2}(k),⋯, x

_{n}(k)]

^{T}, P = pI

_{n}and Q = qI

_{n}, the dynamics of multiagent system can be expressed in compact form as

_{α}(k) is a diagonal matrix with i

_{th}diagonal item as α

_{i}(T(k)−μ(k)), and T

_{β}(k) is a weighted adjacency matrix as

_{i}= α and β

_{i}= β. Considering about the consensus error defined as (3), the state error vector of multiagent system can be presented as

_{i}(k) = x

_{i}(k) − x

_{n}(k). According to the Equation (8), the dynamics of error vector can be derived. Since the following relation exists

_{n}

_{−1}is the first n − 1 rows and the first n − 1 columns of matrix D and A

_{n}

_{−1}has similar definitions. ${A}_{n-1}^{n}$ is formed by the first n − 1 items of n

_{t}h rows of matrix A. M

_{C}∈ R

^{1×(}

^{n}

^{−1)}can also be calculated, the detailed structure of M

_{C}is omitted here. Now, the discrete dynamics of state error vector E(k) can be obtained as follows

**Lemma 1.**Let 0 < a(n) ≤ M

_{a}< 1, 0 < b(n) ≤ 1, and let {y

_{n}}

_{n}

_{≥−}

_{r}be a sequence of real numbers satisfying the Halanay-type inequality

_{n}= y

_{n}

_{+1}− y

_{n}If there exist positive constants Γ ∈ Z

^{+}, and λ

_{l}∈ (0, 1) such that

**Theorem 1.**For multiagent system consist of agent with dynamics as Equation (1), with the control protocol designed as Equation (6) and assumptions made about the sending and receiving delays, the consensus of the system is guaranteed if following conditions are satisfied by proper choice of control parameters α and β.

- sup λ
_{α}(k) < 1 - ${\lambda}_{\beta}(k)<\frac{1}{{\lambda}_{C}}$
- ${\lambda}_{\beta}(k){\lambda}_{C}\le \phantom{\rule{0.2em}{0ex}}{d}_{0}{\lambda}_{\alpha}^{r}(k)-{\lambda}_{\alpha}^{r+1}(k)$

_{0}< 1, λ

_{α}(k) = |p − qα(T(k) − μ(k))|, λ

_{β}(k) = |p − qβ(T(k) − μ(k))| and λ

_{C}= ║C║ is constant value for fixed communication graph.

**Proof**. According to Equation (14), the error dynamics of multiagent system can be expressed as follows.

_{α}, λ

_{β}and λ

_{C}

^{2}(k), the consensus of the multiagent system is reached. Theorem 1 is proved.

**Remark**3. During the discussion of Theorem 1, it can be concluded that one advantage of implementing the Halanay inequality in proving stability of the multiagent system is the avoidance of dependence on symmetry of communication graph. As a matter of fact, even though the communication graph is assumed to be connected and undirected, conditions (i–iii) in Theorem 1 are not dependent on this assumption.

**Remark**4. The existence of α and β satisfying conditions (i) and (ii) is actually decided by the dynamics of agents, receiving delay and communication graph together. Since the situations can vary from case to case, it is difficult to obtain general expressions for conditions of α and β. However, with particular dynamics of agents and communication graph, the values of α and β can be easily chosen according to definitions of λ

_{α}and λ

_{β}. Here, a typical situation where p > 0 and q > 0 is presented for a better illustration. According to conditions (i) and (ii), we have

_{i}(k) and δ

_{j}(k − τ(k)) denote bounded state disturbances of agent i and j respectively. Due to the uncertainty of disturbance, δ

_{j}(k − τ(k)) can also be presented as δ

_{j}(k). Based on the protocol, the dynamics of the multiagent system can be presented as

_{δ}= T

_{α}(k) − T

_{β}(k) and Ξ(k) = [δ

_{1}(k), δ

_{2}(k),⋯, δ

_{n}(k)]

^{T}. The error dynamics of the system should be revised as

_{1}(k) − δ

_{n}(k),⋯, δ

_{n}

_{−1}(k) − δ

_{n}(k)]

^{T}.

**Lemma 2.**Let 0 < a(n) ≤ M

_{a}< 1, 0 < b(n) ≤ 1, c ≥ 0 and let {z

_{n}}

_{n}

_{≥−}

_{r}be a sequence of real numbers satisfying the inequality

^{+}and 0 < λ

_{l}< 1 such that

**Proof**. Define an auxiliary system as below

_{1}≥ 0 such that z

_{s}= y

_{s}, s ∊ {n

_{1}, n

_{1}− 1, …, n

_{1}− r}, then

_{n}

_{1}= y

_{n}

_{1}, we have

_{0}and ω as

_{0}and ω have the following relations

_{n}≤ y

_{n}for n > n

_{1},

**Theorem 2.**For multiagent system consist of agents with dynamics as Equation (1), with the control protocol presented as Equation (23) and conditions assumed in Theorem 1, the bounded consensus of the system is guaranteed.

**Proof**. According to Equation (25), similar as the proof in Theorem 1, we have

_{δ}= λ

_{α}+λ

_{β}. It is obviously that above equation can satisfy the condition of Lemma2 with Γ = 1. Since the environmental disturbances concerned are bounded, the value of ω = sup

_{k}

_{→∞}{λ

_{δ}λ

_{C}║Λ(k)║} is also bounded. As a consequence, the boundedness of the multiagent system is guaranteed. And the bound of consensus error can be calculated as

## 4. Simulations

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Yan, Z.; Wu, D.; Liu, Y.
Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances. *Entropy* **2014**, *16*, 6524-6538.
https://doi.org/10.3390/e16126524

**AMA Style**

Yan Z, Wu D, Liu Y.
Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances. *Entropy*. 2014; 16(12):6524-6538.
https://doi.org/10.3390/e16126524

**Chicago/Turabian Style**

Yan, Zheping, Di Wu, and Yibo Liu.
2014. "Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances" *Entropy* 16, no. 12: 6524-6538.
https://doi.org/10.3390/e16126524