Non-fragile sampled data control for stabilization of non-linear multi-agent system with additive time varying delays, Markovian jump and uncertain parameters

doi:10.1016/j.nahs.2019.100830

Nonlinear Analysis: Hybrid Systems

Volume 36, May 2020, 100830

https://doi.org/10.1016/j.nahs.2019.100830 Get rights and content

Abstract

This paper establishes a novel non-fragile sampled data control framework for non linear multi-agent systems with additive time varying delays and Markovian jump parameters. The Laplacian matrix represents the interconnection between the agents which are denoted by an undirected graph. Relevant Lyapunov Krasovskii functional (LKF) is constructed which contain major information about the additive time-varying delays. The major goal of this paper is to model a non-fragile sampled-data control scheme which guarantees the stabilization for the proposed system. Apart from that, the Jensen’s and some improved integral inequalities are used for deriving the derivatives of LKFs with single, double and triple integral terms and the adequate conditions are expressed in terms of linear matrix inequalities. At last two numerical examples are given to verify the theoretical results.

Introduction

Over the past few years the concept of stability and consensus for the multi-agent system is an effective research tool for the authors due to its ubiquitous spread in the real world like multi-robot systems, aircraft, satellites, sensor networks, formation control, swarms, flocking, etc. [1], [2], [3], [4], [5], [6], [7]. Multi-agent system is formed with the help of many interacting intelligent agents which are interconnected with one another for communication. Consensus means that all the agent can achieve the goal or best solution at the time. The idea for consensus originated from statistical consensus theory by DeGroot (1974) [8]. Consensus will achieve after that the closed loop system is stable. So the stability of multi-agent system is important to analyze the consensus of multi-agent system which are deeply discussed in [9], [10], [11]. The procedure for analyzing the consensus problem is regularly depends upon the connection between the dynamics of each agents and network communication. Generally, the three main elements, i.e., a system explaining a dynamics of the agents; a graph explaining the interaction or communication between the each agents which may be directed or undirected or switched or fixed; a control input (protocol) explaining the interaction between the agents with one another based on a graph (communication topology) are very important to frame the multi-agent system. The stability and consensus of linear or nonlinear multi-agent system will be achieved by utilizing the concept of matrix theory, algebraic graph theory, control theory and properties of Kronecker product. These are all hugely discussed by many authors in their research works for example [12], [13].

Practically, delay is inevitable one in the case of communication among the agents in multi-agent system. In fact, time delay is pervasive in multi-agent systems due to the slow process of interactions between agents. Control protocol without taking the effect of time delays may cause the instability of MASs. Due to its necessity, an numerous interesting results on time delay i.e., input time delay, fixed time delay, multiple time delay, time varying delay have been obtained by using different protocols in [14], [15], [16], [17], [18]. For the most part, in pragmatic circumstances, signals transmitted from one point to another may experience a couple of fragments of systems, which can instigate progressive delays with various properties because of the variable system transmission conditions. By virtue of the network transmission conditions, the two delays are usually time-varying with distinct properties. When compare to single delay, the system with additive delay have better applications. Recently, lots of literature results are established in additive time delay see for example [19], [20].

It is well known that there are lots of control methods are usable by the literatures to solve the uncontrolled systems, such as event triggered control [21], impulsive control [22], [23], sampled-data control [24], $H_{\infty}$ control [25] and so on. In current years, compare to many control methods, the importance of the sampled-data control scheme has been increasing as the digital hardware and communication technologies are rapidly developing. Commonly sampled-data control system describes a control system in which continuous-time plant is controlled with a digital device. In sampled-data control systems, control signals are in constants during sampling intervals and are allowed to change only at sampling instants. Because of this reason, the control signals in sampled-data control systems have stepwise form and these discontinuous signals cannot be applied directly to stabilize control systems. Consensus of multi-agent system via sampled data control with leader or leaderless are illustrated in [26], [27]. Besides, non-fragile control has the capability of adjusting the gain fluctuations in the controller design and is also used to maintain the system stability in the presence of small perturbations in the controller design. Due to its good transient response, fast responsibility and high insensitivity to parameter variations and uncertainties, few impressive works have been expressed on non-fragile control problem in MASs [28].

It is valid to mention that the model of multiplicative uncertainty is rapidly used in systems because of its many practical systems such as production line, environmental risk assessments and economic systems [29]. On the other point of view, stability and performance of numerous dynamical systems are greatly impacted by the sudden changes occurs in the systems model and its parameters. Due to the occurrence of this changes, dynamical system model will show some difficulties in their performance like any oscillation, unexpected environment changes, interconnection failures, slow and poor performance and also may lead to the divergence in eminent cases. To face these situations, Markovian jump systems is used and it gives a effectual results for the system model which are affected by random switching behavior. The most significant investigation on Markovian jump system is to assume that the data on change probabilities is altogether known. All things considered, in numerous real systems, the transition probability of Markov jumping may not be quantifiable precisely, or possibly just few part of the transition probabilities is accessible. Thus, it is fundamental to concentrate such general Markovian jump systems having somewhat known transition probabilities. Because of this induction, lots of impressive results are exists for Markovian jumping parameters in multi-agent systems due to its very immense applications in various fields [30], [31], [32], [33], [34].

Inspired by the above given deliberation, the non-fragile sampled data control with additive time varying delay and markovian jumping parameters for the stabilization and consensus of nonlinear multi-agent system is investigated in this paper. However the main goal of this paper consists in bid to give a new approach for the study of MASs with additive time varying delay.

The main contribution of this paper is listed below:

$•$ The combination of non-fragile and sampled data control are taken into account to achieve the consensus for non-linear multi-agent system.

$•$ The uncertainties term is considered in the control input and also linear form of matrix coefficients. Along with this additive time varying delay and Markovian jump parameters are considered.

$•$ Less conservative results are obtained by taking the suitable Lyapunov Krasovskii functional with single, double and triple integral terms which are solved by utilizing the integral techniques.

$•$ Finally, numerical examples are given to show the potency of the proposed theoretical results.

$N o t a t i o n s$ : The notations used in this paper are standard. $R^{n}$ denotes $n$ dimensional Euclidean space and $R^{n \times m}$ is the set of all $n \times m$ real matrices. The script $*$ denotes the symmetric term in the matrix. The transpose and inverse for the matrix $A$ is denoted by $A^{T}$ and $A^{- 1}$ , respectively. $I$ is the identity matrix with appropriate dimension, $X > 0$ is symmetric positive definite matrix, $A \otimes B$ denotes the Kronecker product of matrices $A$ and $B$ .

Algebraic graph theory, is an essential structure to study the problems of multi-agent system. It is very useful tool to denote the communications among the agents through directed or undirected graph. Let us consider an undirected graph $G = (V, E, A)$ with order $N$ , where $V = {v_{1}, v_{2}, \dots, v_{N}}$ denotes the set of all agents, $E \subseteq V \times V$ represents the edges set, and $A = {[a_{i j}]}_{N \times N}$ is the adjacency matrix with $a_{i j} > 0$ if $(v_{i}, v_{j}) \in E$ and $a_{i j} = 0$ otherwise. The neighbor set of $v_{i}$ can be represented as $N_{i} = {j : (v_{i}, v_{j}) \in E}$ . Let $D = d i a g {d e g (1), d e g (2), \dots, d e g (i)}$ be the degree matrix of the undirected graph $G$ with entries $d e g (i) = \sum_{j \in N_{i}} a_{i j}$ . Then the Laplacian matrix of $G$ can be expressed as $L = D - A$ .

Section snippets

Problem description and preliminaries

We consider nonlinear multi-agent system with Markov jump parameter consists of $N$ agents, whose dynamics of each agent is represented as follows: ${\dot{γ}}_{i} (t) = A_{1} (σ (t)) γ_{i} (t) + A_{2} (σ (t)) γ_{i} (t - ρ_{1} (t) - ρ_{2} (t)) + B (σ (t)) u_{i} (t) + δ (γ_{i} (t), t), i = 1, \dots, N,$ where $γ_{i} (t) = {[γ_{i 1} (t), γ_{i 2} (t), \dots, γ_{i n} (t)]}^{T} \in R^{n}$ denotes the state vector of the agent $i$ , $n$ is the number of states of agent, $u_{i} (t) = {[u_{i 1} (t), u_{i 2} (t), \dots, u_{i n} (t)]}^{T} \in R^{n}$ is the control input acting on agent $i$ , $δ (γ_{i} (t), t) = {[δ_{1} (γ_{i 1} (t), t), δ_{2} (γ_{i 2} (t), t), \dots, δ_{n} (γ_{i n} (t), t)]}^{T}$ is a nonlinear vector

Main results

The stabilization and consensus criteria for the multi-agent system without uncertain parameters and with uncertain parameters are presented in this section. The following notations are used in this paper for the sake of simplicity. $Π_{1} = c o l {γ (t) γ (t - {\bar{ρ}}_{1}) \frac{1}{{\bar{ρ}}_{1}} \int_{t - {\bar{ρ}}_{1}}^{t} γ (s) d s}, Π_{2} = c o l {γ (t) γ (t - {\bar{ρ}}_{2}) \frac{1}{{\bar{ρ}}_{2}} \int_{t - {\bar{ρ}}_{2}}^{t} γ (s) d s},$ $Π_{3} = c o l {γ (t) γ (t - ρ_{a}) \frac{1}{ρ_{a}} \int_{t - ρ_{a}}^{t} γ (s) d s}, Π_{4} = c o l {γ (t - {\bar{ρ}}_{1}) γ (t - ρ_{a}) \frac{1}{\bar{ρ_{2}}} \int_{t - ρ_{a}}^{t - \bar{ρ_{1}}} γ (s) d s},$ $Π_{5} = c o l {γ (t - {\bar{ρ}}_{2}) γ (t - ρ_{a}) \frac{1}{{\bar{ρ}}_{1}} \int_{t - ρ_{a}}^{t - {\bar{ρ}}_{2}} γ (s) d s}, Π_{6} = [γ (t) - \frac{1}{{\bar{ρ}}_{1}} \int_{t - {\bar{ρ}}_{1}}^{t} γ (s) d s],$ $Π_{7} = [γ (t) + \frac{2}{{\bar{ρ}}_{1}} \int_{t - {\bar{ρ}}_{1}}^{t} γ (s) d s - \frac{6}{{\bar{ρ}}_{1}^{2}} \int_{t - {\bar{ρ}}_{1}}^{t} \int$

Numerical examples

To check the correctness of the theoretical results presented in the previous section, numerical examples are given in this section.

Example 4.1

We consider a nonlinear multi-agent system without uncertain parameters (6) with four agents and two jump modes. The coefficient matrices are given as

Mode 1: $A_{11} = [\begin{matrix} - 1 & 0 \\ 0 & - 1.8 \end{matrix}], A_{21} = [\begin{matrix} 1.6 & - 2.4 \\ 1.9 & - 3.8 \end{matrix}], B_{1} = [\begin{matrix} 3.5 & - 1.2 \\ - 0.5 & 2.8 \end{matrix}],$

Mode 2: $A_{12} = [\begin{matrix} - 2.5 & 0 \\ 0 & - 1.4 \end{matrix}], A_{22} = [\begin{matrix} - 1.0 & - 1.4 \\ - 0.9 & - 1.8 \end{matrix}], B_{2} = [\begin{matrix} - 3.8 & 1.9 \\ 0.7 & - 2.9 \end{matrix}] .$ As mentioned in Section 2 the interconnection between the agents are taken as an undirected graph

Conclusion

This paper analysis the problem of stability and consensus of nonlinear multi-agent system with additive time-varying delay, uncertain and Markovian jumping parameters. Non-fragile sampled data control has been composed for the proposed system. Communication between the agents are transmitted without any specific leader subject to undirected graph. Consensus condition is reached after the stabilization of the multi-agent system. The stabilization conditions are derived by picking the augmented

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^☆: No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.nahs.2019.100830.

¹: The work of the authors was supports by SERB, India - MATRICS project No. MTR/2018/000931. This work was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1A6A1A03013567), (NRF-2018R1A2A2A14023632) and by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (Grant number: 20194030202300).

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Non-fragile sampled data control for stabilization of non-linear multi-agent system with additive time varying delays, Markovian jump and uncertain parameters☆

Abstract

Introduction

Section snippets

Problem description and preliminaries

Main results

Numerical examples

Conclusion

Adv. Biophys.

Automatica

J. Franklin Inst. B

IFAC

Neurocomputing

Neurocomputing

Neurocomputing

J. Franklin Inst.

Neurocomputing

Automatica

Nonlinear Anal. Hybrid Syst.

J. Difference Equ.

Appl. Math. Lett.

Partial stability approach to consensus problem of linear multi-agent systems

Acta Autom. Sin.

Information consensus in multivehicle cooperative control

IEEE Control Syst. Mag.

Coordination of groups of mobile autonomous agents using nearest neighbor rules

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Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs

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Distributed parameter estimation in sensor networks: nonlinear observation models and imperfect communication

IEEE Trans. Inform. Theory

Software agents for peer-to-peer video streaming

IET Softw.