TY - JOUR
AU - Shekher, Vineet
AB - Abstract This paper is concerned with the problem of stability and consensus of non-linear multi-agent system by utilizing the sampled-data control. The innovative part of this paper is that the nonlinearity of this class of nonlinear systems is considered to satisfy a quasi one-sided Lipschitz condition. Communication among agents are assumed to be a switching directed graph. The principle target of this paper is to design a sampled data controller such that for all permissible uncertainties, the resulting closed-loop system is stable in the sense of mean square. For this reason, through the development of an appropriate Lyapunov–Krasovskii functional with dual integral terms and usage of Kronecker product properties alongside the matrix inequality techniques, a new set of stability and consensus conditions for the prescribed system is obtained in the form of a linear matrix inequality, which can be easily solved by the well-known effective numerical programming. Finally numerical examples are given to show the validity of the proposed hypothetical results. 1. Introduction Multi-agent systems have delighted in expanding prominence by both researchers and practitioners, because they are effectively applied in circulated sensor systems, versatile mechanical autonomy, reconnaissance frameworks, market simulation, monitoring, system diagnosis and remedial actions (see Xiao et al., 2012; Wang et al., 2009; Chen & Wang, 2005; Ji et al., 2015). Generally, due to its necessity, consensus of multi-agent system is an innovative topic for the researchers (see Song et al., 2010; Ren & Beard, 2005; Qian et al., 2018). The ultimate target of the consensus problem is the multi-agent system converges to a common state. It can be done by choosing an applicable control protocol depending on the communication existing between an agent and its neighborhood agent. Practically, the nonlinear part may play an important role in the stability of the nonlinear system. For the past few decades, Lipschitzian nonlinear system attracted many researchers due to its importance and based on three classes we assumed nonlinear function conditions: Lipschitz condition (Zhu & Han, 2002; Zemouche et al., 2008), one-sided Lipschitz condition (Hu, 2008; Boutayeb et al., 2012; Zhang et al., 2012) and quasi-one-sided Lipschitz condition (Fu et al., 2012; Song & Hen, 2015; Zhu & Hu, 2009). A nonlinear system satisfies the Lipschitz condition neither globally nor locally. Hu (2005) introduced the one-sided Lipschitz nonlinear system to make use of the needful communication of the nonlinear part fully, which also leads to a vital part in the stability analysis of nonlinear differential equations. Latterly, quasi one-sided Lipschitz nonlinear system was designed by Hu to show more involvement in the useful transformation of the nonlinear part. Quasi one-sided Lipschitz condition is an extended condition of both Lipschitz and one-sided Lipschitz condition and when comparing to both of these it is much more less conservative. In the multi-agent system, switching behavior present in the topologies, it should be noted that the Laplacian matrix changes when the topology switches. So, the multi-agent system with switching topologies consists only continuous or discrete time subsystems i.e., for example, the multi-agent system is controlled either by a physically implemented regulator or by a digitally implemented one with a switching rule between them synchronously. Multi-agent system with switched topologies exists vastly in real life due to the limitation in communication capacities, time delay and the switching of communication topologies. Nowadays many researchers studied the consensus of multi-agent system with switching topologies (Zheng & Wang, 2016; Lin et al., 2017; Lin & Zheng, 2016). In Cheng et al. (2020b) the author addressed the quantized nonstationary filtering problem for network-based Markov switching repeated scalar nonlinear systems and in Cheng et al. (2020c) the author investigated the nonstationary control for a class of nonlinear Markovian switching systems with the Tagaki-Sugeno (T-S) fuzzy model. The finite-time static output feedback control of Markovian switching systems, with quantization effects are studied in Cheng et al. (2020a). In the multi-agent system, information are communicated between the interconnected agents. At the time of large network, it is normally eased to make the channel block for continuous information transmission, the unwanted information exchanges are reduced and it improves the efficiency of the system. The control protocols are mainly focusing on continuous time control. To face this situation, sampled data control is more feasible when comparing to the continuous-time control, which means control signals are not changed in the sampling interval but these are changed in the sampling instants. Based on the sampling interval, sampling may be periodic or aperiodic. In many practical applications, the information transformed between agents occur at discrete sampling instants due to the application of digital sensors and finite bandwidth communication channels. Flexibility, robustness and low cost are the benefits of the sampled data control, these are well examined by lots of researchers (Ding & Zheng, 2016; Wang et al., 2017; Lee et al., 2019; Lee et al., 2018; Ding & Guo, 2015). By above mentioned deliberations, the main aim of this paper is to solve the consensus problem of quasi one-sided nonlinear multi-agent system by using the sampled data control with the switching directed topologies. The main contribution of this paper is summarized as follows: |$\maltese $| Both stability and consensus problem of non-linear multi-agent system were taken into account. |$\maltese $| The communication between the agents is very important for its dynamics and it can be in the switched directed form. Two switches are considered. |$\maltese $| The main novelty of this paper is that the nonlinear term satisfies the quasi one-sided Lipschitz condition. |$Notations$|⁠: The notations used throughout the paper are quite standard. |$ \mathbb{R}^n$| denotes |$n$| dimensional Euclidean space and |$\mathbb{R}^{n\times m}$| is the set of |$n\times m $| matrices. In symmetric block matrices, * is used as ellipsis for terms induced by symmetry. The superscript |$T$| is the transpose operator. |$\textbf{A}^{-1}$| is the inverse of |$\mathcal{A}$|⁠. |$\textbf{I}$| is the unit matrix, notation |$\textbf{X}>0$| denotes a real symmetric positive definite matrix. Notation |$||\cdot ||$| and |$\otimes $| denote the Euclidean norm and the Kronecker product, respectively. 2. Model description and preliminaries Consider a multi-agent system composed of |$\textbf{N}$| agents with non-linear dynamics, it is described by $$\begin{align}& \dot{y}_i(\textrm{t}_{\!\!c})=\textbf{A}y_i(\textrm{t}_{\!\!c})+ \textbf{B}y_i(\textrm{t}_{\!\!c}-\delta)+\textbf{C}u_i(\textrm{t}_{\!\!c})+f(y_i(\textrm{t}_{\!\!c}),y_i(\textrm{t}_{\!\!c}-\delta)), \ \ i=1,2,...,\textbf{N}, \end{align}$$(2.1) where |$y_i(\textrm{t}_{\!\!c}) \in \mathbb{R}^n$| is the state vector, |$\delta $| is the positive constant time delay, |$u_i(\textrm{t}_{\!\!c}) \in \mathbb{R}^m$| is the control protocol, |$f(y_i(\textrm{t}_{\!\!c}), y_i(\textrm{t}_{\!\!c}-\delta )) \in \mathbb{R}^n$| denotes the non-linear dynamics of the |$ith$| agent with respect to |$y, \ y_{\textrm{t}_{\!\!c}-\delta }$|⁠, |$\textbf{A}, \ \textbf{B}, \ \textbf{C}$| are the constant matrices representing the linear components of the agents. Let |$\mathcal{G}_\sigma = (\mathcal{V}, \mathcal{E}_\sigma , \mathcal{A}_\sigma )$| be a switching directed graph, where |$\mathcal{V}=\{v_1,v_2,...,v_{\textbf{N}}\}$| denotes the set of agents, |$\mathcal{E}\subseteq \mathcal{V} \times \mathcal{V}$| is the set of edges and |$\mathcal{A}_\sigma =[a_{ij}^\sigma ]_{\textbf{N}\times \textbf{N}}$| represents the adjacency matrix. The elements in the |$\mathcal{A}_\sigma $| are defined by |$a_{ij}^{\sigma }> 0$| if |$(v_i, v_j) \in \mathcal{E}_\sigma $| and |$a_{ij}^{\sigma }>0$| otherwise. A switching signal |$\sigma $| is defined by |$\sigma :[0, \infty ) \rightarrow \{1,2,..,\mathcal{S}\},$| where |$\mathcal{S}$| is the number of possible topologies. The neighbor set of |$v_i$| can be represented by |$\mathcal{N}_i=\{j: (v_i, v_j) \in \mathcal{E} \}.$| The degree matrix of the graph is |$\mathcal{D}=diag\{d_1,d_2,...,d_N\}$| with |$d_i=\sum _{j \in \mathcal{N}_i}a_{ij}.$||$\mathcal{L}^{\ \sigma }= [l_{ij}^{\sigma }]_{\textbf{N}\times \textbf{N}}$| be the Laplacian matrix, which can be calculated by |$\mathcal{L}^{\ \sigma} = \mathcal{D}-\mathcal{A}_\sigma $| and guarantees |$\sum _{j=1}^{\mathcal{N}} l_{ij}^{\sigma }=0.$| Presently it is the right time to structure an consensus protocol for the prescribed multi-agent system. Sampled data control is framed for the multi-agent system (1) under the switching topology scheme, $$\begin{align}& u_i(\textrm{t}_{\!\!c})=-c\textbf{K}\sum_{j=1}^{\textbf{N}}a_{ij}^{\sigma} (y_i(\textrm{t}_{\!\!c k})-y_j(\textrm{t}_{\!\!c k})), \ \ \ i=1,2,..., \textbf{N}, \end{align}$$(2.2) where |$\textbf{K}$| is the control co-efficient that to be determined, |$c$| is the coupling strength and |$\textrm{t}_{\!\!c k}$| is the updating instant time of the zero-order hold. That the sampling intervals |$\textrm{t}_{\!\!c k+1}-\textrm{t}_{\!\!c k}=\eta_k \leqslant \hat{\eta}$| are bounded for any integer |$k \geqslant 0.$| Let us define |$\eta(\textrm{t}_{\!\!c})=\textrm{t}_{\!\!c}-\textrm{t}_{\!\!c k}.$| Therefore we have, $$\begin{align}& u_i(\textrm{t}_{\!\!c})=-c\textbf{K}\sum_{j=1}^{\textbf{N}}a_{ij}^{\sigma}(y_i(\textrm{t}_{\!\!c}-\eta (\textrm{t}_{\!\!c})) -y_j(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))), \ \ \textrm{t}_{\!\!c k} \leqslant \textrm{t}_{\!\!c} \leqslant \textrm{t}_{\!\!c k+1}, \end{align}$$(2.3) Substitute (2.3) in (2.1), we get $$\begin{align} \dot{y}_i(\textrm{t}_{\!\!c})&=\textbf{A}y_i(\textrm{t}_{\!\!c})+\textbf{B} y_i(\textrm{t}_{\!\!c}-\delta)+c\textbf{C}\textbf{K}\sum_{j=1}^{\textbf{N}}a_{ij}^{\sigma}(y_i(\textrm{t}_{\!\!c} -\eta(\textrm{t}_{\!\!c}))-y_j(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})))\nonumber\\ &\quad +f(y_i(\textrm{t}_{\!\!c}),y_i(\textrm{t}_{\!\!c}-\delta)). \end{align}$$(2.4) Then the system can be rewritten as, $$\begin{align} \dot{y}(\textrm{t}_{\!\!c})&=(I_N \otimes \textbf{A})y(\textrm{t}_{\!\!c})+ (I_N \otimes \textbf{B}) y(\textrm{t}_{\!\!c}-\delta)+c(\textbf{L}^{\sigma} \otimes \textbf{C}\textbf{K})(y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))\nonumber \\ &\quad+\textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta)), \end{align}$$(2.5) where, |$y(\textrm{t}_{\!\!c})=[y_1^T(\textrm{t}_{\!\!c}), \ y_2^T(\textrm{t}_{\!\!c}),..., y_{\textbf{N}}^T(\textrm{t}_{\!\!c})],$||$\textbf{F}(y(\textrm{t}_{\!\!c}), y(\textrm{t}_{\!\!c}-\delta ))=[f(y_1(\textrm{t}_{\!\!c}), y_1(\textrm{t}_{\!\!c}-\delta )),..., f(y_{\textbf{N}}(\textrm{t}_{\!\!c}), y_{\textbf{N}}(\textrm{t}_{\!\!c}-\delta ))].$| Before moving forward, we first introduce the following Definition, Assumption and Lemmas for further derivation to achieve a stability and consensus of multi-agent system. Definition 2.1 Zhu & Hu (2009) For any |$x, \ y \in \mathbb{R}^n,$| $$\begin{align} <\mathbb{F},x> \leqslant \left[ \begin{array}{c} x \\ y \\ \end{array} \right]^T \varOmega \left[ \begin{array}{c} x \\ y \\ \end{array} \right], \end{align}$$(2.6) where |$\mathbb{F}=\textbf{P}f(x,y), \ \textbf{P}$| is some symmetric positive definite matrix to be determined later. (2.6) is called the quasi-one sided Lipschitz condition and the symmetric matrix $$\varOmega =\left [ \begin{array}{cc} \textbf{D} & 0 \\ 0 & \textbf{E} \\ \end{array} \right ],$$ which can be indefinite to |$x$| and |$y,$| where |$\textbf{D}, \ \textbf{E}$| are any real symmetric matrices. Remark 2.1 To replace Lipschitz condition, one-sided or quasi-one-sided Lipschitz condition has been widely applied in the observer design, robust control and so on. Generally, the one-sided Lipschitz constant is much smaller than the classical Lipschitz constant, that is, one-sided Lipschitz condition is weaker than Lipschitz condition. Then the concept of one-sided Lipschitz condition is extended to the quasi one-sided Lipschitz condition by Hu. Quasi one-sided Lipschitz condition is weaker than one-sided Lipschitz condition, which is the main difference between Lipschitz and one-sided, quasi-one-sided Lipschitz condition. Quasi one-sided Lipschitz is less conservative when compared to one-sided Lipschitz and Lipschitz condition. Definition 2.2 Li et al. (2015) The consensus of system (2.1) is said to be achieved asymptotically in the sense of mean square if, for each agent |$i \in \{1,2,...,\textbf{N}\}$|⁠, there is a local state feedback |$u_i$| of |$ \{y_j: j \in \mathcal{N}_i\}$| such that the closed-loop system (6) satisfies $$\begin{align*} lim_{t \rightarrow\infty} \mathbb{E}\{||y_i(t)-y_j(t)||^2\}=0. \end{align*}$$Assumption: Zhu & Hu (2009)|$f(x,y)$| of non-linear time delay system (1) satisfies the quasi-one-sided Lipschitz condition (7) and also the following bounded conditions: $$\begin{align} <f(x,y), f(x,y)>\leqslant a^2 x^T x +b^2 y^T y \ \ \forall \ x, \ y \in \mathbb{R}^n. \end{align}$$(2.7) Lemma 2.1 For any constant symmetric matrix |$\textbf{S} \in \mathbb{R}^{n \times n},$| real scalars |$\alpha , \ \beta $| satisfying |$\alpha < \beta $| and vector-valued function |$y \in [\alpha , \beta ] \rightarrow \mathbb{R}^n, $| the following integral inequality holds: $$\begin{align*} -\int_{\alpha}^{\beta} \dot{y}^T(s)\textbf{S}\dot{y}(s) \textrm{d}s \leqslant - \frac{1}{\beta-\alpha} \chi_1^T \textbf{S} \chi_1 - \frac{3}{\beta-\alpha} \chi_2^T \textbf{S} \chi_2- \frac{5}{\beta-\alpha} \chi_3^T \textbf{S} \chi_3, \end{align*}$$ where $$\begin{align*} \chi_1&=y(\beta)-y(\alpha),\\ \chi_2&=y(\alpha)+y(\beta)-\frac{2}{\beta-\alpha}\int_{\alpha}^{\beta}y(s) \textrm{d}s,\\ \chi_3&=\chi_1+\frac{6}{\beta-\alpha}\int_{\alpha}^{\beta}y(s)ds-\frac{12}{(\beta-\alpha)^2}\int_{\alpha}^{\beta} \int_{\theta}^{\beta}y(s) \textrm{d}s \textrm{d}\theta. \end{align*}$$ Lemma 2.2 (Boyd et al., 1994) Assume that |$\varLambda , \ M_i, \ E_i $| are real matrices with appropriate dimensions and |$F_i^T(t) F_i(t) \leqslant I.$| Then, the inequality |$\varLambda +M_i F_i(t)E_i+[M_i F_i(t)E_i]^T <0$| holds if and only if there exists a scalar |$\epsilon>0$| and satisfying |$\varLambda + \epsilon ^{-1} M_iM_i^T+\epsilon E_i^T E_i <0.$| Lemma 2.3 (Boyd et al., 1994) Let |$\varLambda _1, \ \varLambda _2, \ and \ \varLambda _3$| be given constant matrices with |$\varLambda _1= \varLambda _1^T, \ \varLambda _2=\varLambda _2^T \> 0.$| Then |$ \varLambda _1+ \varLambda _3^T \varLambda _2^{-1} \varLambda _3 < 0$| if and only if $$\begin{align*} \left[ \begin{array} {cc} \varLambda_1 & \varLambda_3^T\\[0.2cm] \varLambda_3 & -\varLambda_2 \end{array} \right]< 0. \end{align*}$$ 3. Main results In this section, we first derive the abundant conditions for the consensus of non-linear multi-agent system. Then the control gain matrix is designed for the closed loop system (2.5). Theorem 3.1 For given positive scalars |$\delta , \ \hat{\eta}, \ c,$| the multi-agent system satisfies quasi one-sided Lipschitz condition becomes stable, if there exists a positive definite matrices |$\textbf{P},\ \textbf{Q}_u, (u=1,2,3,4)$| and any matrix |$\textbf{R}>0,$| such that the following inequality hold: $$\begin{align} \boldsymbol{\varPhi}=\left[ \begin{array}{ccccccccc} \boldsymbol{\varPhi}_{11} & \boldsymbol{\varPhi}_{12} & \boldsymbol{\varPhi}_{13} & 0 & \boldsymbol{\varPhi}_{15} & \boldsymbol{\varPhi}_{16} & 0 & (I_{\textbf{N}} \otimes\textbf{A})^T (I_{\textbf{N}} \otimes \textbf{Q}_3) & (I_{\textbf{N}} \otimes\textbf{A})^T (I_{\textbf{N}} \otimes \textbf{Q}_4)\\ * & \boldsymbol{\varPhi}_{22} & 0 & 0 & \boldsymbol{\varPhi}_{25} & \boldsymbol{\varPhi}_{26} & 0 & (I_{\textbf{N}} \otimes\textbf{B})^T (I_{\textbf{N}} \otimes \textbf{Q}_3) & (I_{\textbf{N}} \otimes\textbf{B})^T (I_{\textbf{N}} \otimes \textbf{Q}_4) \\ * & * & \boldsymbol{\varPhi}_{33} & \boldsymbol{\varPhi}_{34} & 0 & 0 & 0 & c(\textbf{L}^{\sigma} \otimes \textbf{C}\textbf{K})^T & c(\textbf{L}^{\sigma} \otimes \textbf{C}\textbf{K})^T \\ * & * & * & \boldsymbol{\varPhi}_{44} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & \boldsymbol{\varPhi}_{55} & \boldsymbol{\varPhi}_{56} & 0 & 0 & 0 \\ * & * & * & * & * & \boldsymbol{\varPhi}_{66} & 0 & 0 & 0 \\ * & * & * & * & * & * & \boldsymbol{\varPhi}_{77} & (I_{\textbf{N}} \otimes \textbf{Q}_3) & (I_{\textbf{N}} \otimes \textbf{Q}_4) \\ * & * & * & * & * & * & * & \frac{(I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta} & 0 \\ * & * & * & * & * & * & * & * & \frac{(I_{\textbf{N}} \otimes \textbf{Q}_4)}{\hat{\eta}^2} \\ \end{array} \right] <0, \end{align}$$(3.1) where, |$\boldsymbol{\varPhi }_{11}=(I_{\textbf{N}} \otimes \textbf{P})(I_{\textbf{N}} \otimes \textbf{A})+(I_{\textbf{N}} \otimes \textbf{A})^T(I_{\textbf{N}} \otimes \textbf{P})^T+2(I_{\textbf{N}} \otimes \textbf{D})+(I_{\textbf{N}} \otimes \textbf{Q}_1)+(I_{\textbf{N}} \otimes \textbf{Q}_2)-9(I_{\textbf{N}} \otimes \textbf{Q}_3)-(I_{\textbf{N}} \otimes \textbf{Q}_4)+\textbf{R} a^2, \ \boldsymbol{\varPhi }_{12}=(I_{\textbf{N}} \otimes \textbf{P})(I_{\textbf{N}} \otimes \textbf{B})+3(I_{\textbf{N}} \otimes \textbf{Q}_3), \ \boldsymbol{\varPhi }_{13}=(I_{\textbf{N}} \otimes \textbf{Q}_4)+(I_{\textbf{N}} \otimes \textbf{P})c(\textbf{L}^{\sigma } \otimes \textbf{C}\textbf{K}), \ \boldsymbol{\varPhi }_{15}=-\frac{24 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta }, \ \boldsymbol{\varPhi }_{16}=\frac{60 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta }, \ \boldsymbol{\varPhi }_{22}=(I_{\textbf{N}} \otimes \textbf{E})-(I_{\textbf{N}} \otimes \textbf{Q}_1)+ \textbf{R} b^2-9(I_{\textbf{N}} \otimes \textbf{Q}_3), \ \boldsymbol{\varPhi }_{25}=\frac{36(I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta }, \ \boldsymbol{\varPhi }_{26}=-\frac{60 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta }, \ \boldsymbol{\varPhi }_{33}=-2(I_{\textbf{N}} \otimes \textbf{Q}_4),\ \boldsymbol{\varPhi }_{34}=(I_{\textbf{N}} \otimes \textbf{Q}_4), \ \boldsymbol{\varPhi }_{44}=-(I_{\textbf{N}} \otimes \textbf{Q}_4)-(I_{\textbf{N}} \otimes \textbf{Q}_2), \ \boldsymbol{\varPhi }_{55}=-\frac{192 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta ^2}, \ \boldsymbol{\varPhi }_{56}=\frac{360 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta ^2}, \ \boldsymbol{\varPhi }_{66}=-\frac{720 (I_{\textbf{N}} \otimes \textbf{Q}_3)}{\delta ^2}, \ \boldsymbol{\varPhi }_{77}=\hat{\eta}^2\textbf{Q}_4-\textbf{R}.$| Proof. For the multi-agent system, we consider the following Lyapunov–Krasovskii functional candidate as follows: $$\begin{align} V(y(\textrm{t}_{\!\!c}))= \sum_{z=1}^5 V_z(y(\textrm{t}_{\!\!c})), \end{align}$$(3.2) where, $$\begin{align*} V_1(y(\textrm{t}_{\!\!c}))&=y^T(\textrm{t}_{\!\!c}) (I_{\textbf{N}} \otimes \textbf{P}) y(\textrm{t}_{\!\!c}),\\ V_2(y(\textrm{t}_{\!\!c}))&=\int_{\textrm{t}_{\!\!c}-\delta}^{\textrm{t}_{\!\!c}} y^T(s) (I_{\textbf{N}} \otimes \textbf{Q}_1) y(s) \textrm{d}s,\\ V_3(y(\textrm{t}_{\!\!c}))&=\int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}} y^T(s) (I_{\textbf{N}} \otimes \textbf{Q}_2) y(s) \textrm{d}s,\\ V_4(y(\textrm{t}_{\!\!c}))&=\delta \int_{-\delta}^{0} \int_{\textrm{t}_{\!\!c}+\theta}^{\textrm{t}_{\!\!c}} \dot{y}^T(s) (I_{\textbf{N}} \otimes \textbf{Q}_3) \dot{y}^T(s) \textrm{d}s \textrm{d}\theta,\\ V_5(y(\textrm{t}_{\!\!c}))&=\hat{\eta} \int_{-\hat{\eta}}^{0} \int_{\textrm{t}_{\!\!c}+\theta}^{\textrm{t}_{\!\!c}} \dot{y}^T(s) (I_{\textbf{N}} \otimes \textbf{Q}_4) \dot{y}^T(s) \textrm{d}s \textrm{d}\theta. \end{align*}$$ Then taking the derivative of |$V(y(\textrm{t}_{\!\!c}))$| along the trajectory of the closed loop system (2.5), we get the succeeding equations, $$\begin{align} \dot{V}_1(y(\textrm{t}_{\!\!c}))&= 2y^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{P})\dot{y}(\textrm{t}_{\!\!c}),\nonumber\\ &= 2y^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{P})(I_N \otimes \textbf{A})y(\textrm{t}_{\!\!c})+ (I_N \otimes \textbf{B} y(\textrm{t}_{\!\!c}-\delta)+c(\textbf{L}^{\sigma} \otimes \textbf{C}\textbf{K})(y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))\nonumber\\ &\quad+\textbf{F}(y(\textrm{t}_{\!\!c}), y(\textrm{t}_{\!\!c}-\delta)) .\end{align}$$(3.3) By using (2.6) in Definition 2.1, the term |$2y^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{P})\textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta ))$| becomes $$\begin{align} 2y^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{P})\textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta)) \leqslant 2\left[ \begin{array}{c} y(\textrm{t}_{\!\!c}) \\ y(\textrm{t}_{\!\!c}-\delta) \\ \end{array} \right]^T \left[ \begin{array}{cc} (I_{\textbf{N}} \otimes\textbf{D}) & 0 \\ 0 & (I_{\textbf{N}} \otimes\textbf{E}) \\ \end{array} \right] \left[ \begin{array}{c} y(\textrm{t}_{\!\!c}) \\ y(\textrm{t}_{\!\!c}-\delta) \\ \end{array} \right] \end{align}$$(3.4) $$\begin{align} \dot{V}_2(y(\textrm{t}_{\!\!c}))&= y^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{Q}_1)y(\textrm{t}_{\!\!c})-y^T(\textrm{t}_{\!\!c}-\delta) (I_{\textbf{N}} \otimes \textbf{Q}_1) y(\textrm{t}_{\!\!c}-\delta), \end{align}$$(3.5) $$\begin{align} \dot{V}_3(y(\textrm{t}_{\!\!c}))&= y^T(\textrm{t}_{\!\!c})[(I_{\textbf{N}} \otimes \textbf{Q}_2)]y(\textrm{t}_{\!\!c})-y^T(\textrm{t}_{\!\!c}-\hat{\eta}) (I_{\textbf{N}} \otimes \textbf{Q}_2) y(\textrm{t}_{\!\!c}-\hat{\eta}), \end{align}$$(3.6) $$\begin{align} \dot{V}_4(y(\textrm{t}_{\!\!c}))&=\delta^2\dot{y}^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{Q}_3)\dot{y}(\textrm{t}_{\!\!c})-\delta \int_{\textrm{t}_{\!\!c}-\delta}^t\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_3)\dot{y}(s) \textrm{d}s, \end{align}$$(3.7) $$\begin{align} \dot{V}_5(y(\textrm{t}_{\!\!c}))&=\hat{\eta}^2\dot{y}^T(\textrm{t}_{\!\!c})(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(\textrm{t}_{\!\!c})-\hat{\eta} \int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s. \end{align}$$(3.8) By applying Lemma 2.1 for the integral terms in (3.7) we get, $$\begin{align} -\delta \int_{\textrm{t}_{\!\!c}-\delta}^{\textrm{t}_{\!\!c}}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_3)\dot{y}(s) \textrm{d}s &\leqslant -\boldsymbol{\varUpsilon}_1^T(I_{\textbf{N}} \otimes \textbf{Q}_3) \boldsymbol{\varUpsilon}_1-3\boldsymbol{\varUpsilon}_2^T(I_{\textbf{N}} \otimes \textbf{Q}_3) \boldsymbol{\varUpsilon}_2-5\boldsymbol{\varUpsilon}_3^T(I_{\textbf{N}} \otimes \textbf{Q}_3)\boldsymbol{\varUpsilon}_3, \end{align}$$(3.9) where, $$\begin{align*} \boldsymbol{\varUpsilon}_1&=y(\textrm{t}_{\!\!c})-y(\textrm{t}_{\!\!c}-\delta),\\ \boldsymbol{\varUpsilon}_2&=y(\textrm{t}_{\!\!c}-\delta)+y(\textrm{t}_{\!\!c})-\frac{2}{\delta}\int_{\textrm{t}_{\!\!c}-\delta}^{\textrm{t}_{\!\!c}} y(s) \textrm{d}s,\\ \boldsymbol{\varUpsilon}_3&=\boldsymbol{\varUpsilon}_1+\frac{6}{\delta}\int_{\textrm{t}_{\!\!c}-\delta}^{\textrm{t}_{\!\!c}} y(s) \textrm{d}s -\frac{12}{\delta}\int_{\textrm{t}_{\!\!c}-\delta}^{\textrm{t}_{\!\!c}} \int_{\theta}^{\textrm{t}_{\!\!c}} y(s) \textrm{d}s \textrm{d}\theta, \end{align*}$$ An integral term in (3.8) can be written as $$\begin{align*} &-\hat{\eta} \int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s \leqslant -\hat{\eta} \int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s\nonumber\\&-\hat{\eta} \int_{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}^{\textrm{t}_{\!\!c}}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s. \end{align*}$$ By applying Jensen’s inequality and simplifying the above terms we obtain, $$\begin{align} &-\hat{\eta} \int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s \leqslant -\bigg(\int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}\dot{y}(s) \textrm{d}s\bigg)^T(I_{\textbf{N}} \otimes \textbf{Q}_4)\bigg(\int_{\textrm{t}_{\!\!c}-\hat{\eta}}^{\textrm{t}_{\!\!c} -\eta(\textrm{t}_{\!\!c})}\dot{y}(s) \textrm{d}s\bigg), \nonumber \\ & \leqslant -[y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))-y(\textrm{t}_{\!\!c}-\hat{\eta})]^T (I_{\textbf{N}} \otimes \textbf{Q}_4) [y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))-y(\textrm{t}_{\!\!c}-\hat{\eta})], \end{align}$$(3.10) $$\begin{align} &-\hat{\eta} \int_{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}^{\textrm{t}_{\!\!c}}\dot{y}^T(s)(I_{\textbf{N}} \otimes \textbf{Q}_4)\dot{y}(s) \textrm{d}s \leqslant -\bigg(\int_{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}^{\textrm{t}_{\!\!c}} \dot{y}(s) \textrm{d}s\bigg)^T(I_{\textbf{N}} \otimes \textbf{Q}_4)\bigg(\int_{\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})}^{\textrm{t}_{\!\!c}} \dot{y}(s) \textrm{d}s\bigg), \nonumber \\ & \leqslant -[y(\textrm{t}_{\!\!c})-y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))]^T (I_{\textbf{N}} \otimes \textbf{Q}_4) [y(\textrm{t}_{\!\!c})-y(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c}))]. \end{align}$$(3.11) A non-linear time delay function |$\textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta ))$| satisfies the bounded condition. For positive diagonal matrix |$\textbf{R},$| therefore from Assumption we get, $$\begin{align} \textbf{R}[a^2 y^T(\textrm{t}_{\!\!c})y(\textrm{t}_{\!\!c})+ b^2 y^T(\textrm{t}_{\!\!c}-\delta) y(\textrm{t}_{\!\!c}-\delta)-\textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta))] \leqslant 0. \end{align}$$(3.12) By combining the equations from (3.3)—(3.12) we obtain, $$\begin{align} \dot{V}(y(\textrm{t}_{\!\!c})) \leqslant \xi^T(\textrm{t}_{\!\!c}) \boldsymbol{\varPhi} \xi(\textrm{t}_{\!\!c}) <0, \end{align}$$(3.13) where, |$ \xi ^T(\textrm{t}_{\!\!c})= [ y^T(\textrm{t}_{\!\!c}) \ \ \ y^T(\textrm{t}_{\!\!c}-\delta ) \ \ \ y^T(\textrm{t}_{\!\!c}-\eta(\textrm{t}_{\!\!c})) \ \ \ y^T(\textrm{t}_{\!\!c}-\hat{\eta}) \ \ \ \int _{\textrm{t}_{\!\!c}-\delta }^{\textrm{t}_{\!\!c}} y^T(s)ds \ \ \ \int _{\textrm{t}_{\!\!c}-\delta }^{\textrm{t}_{\!\!c}} \int _{\theta }^{\textrm{t}_{\!\!c}} y^T(s)ds d\theta \\ \textbf{F}(y(\textrm{t}_{\!\!c}),y(\textrm{t}_{\!\!c}-\delta ))].$| From (3.13) we get that the prescribed MASs is stable. This completes the proof. Followed by the proof of Theorem 3.1, now design the controller gain matrix to reach the consensus for closed loop system. Then the sampled data controller design is given in the following Theorem. Theorem 3.2 For given positive scalars |$\delta , \ \hat{\eta }, \ \rho _1, \ \rho _2, ;$| the multi-agent system satisfies quasi one-sided Lipschitz condition achieves consensus, if there exists a positive definite matrices |$\textbf{X},\ \hat{\textbf{Q}}_u, (u=1,2,3,4)$| and any matrix |$\hat{\textbf{R}}>0,$| such that the following inequality hold: $$\begin{align} \hat{\boldsymbol{\varPhi}}=\left[ \begin{array}{ccccccccc} \hat{\boldsymbol{\varPhi}}_{11} & \hat{\boldsymbol{\varPhi}}_{12} & \hat{\boldsymbol{\varPhi}}_{13} & 0 & \hat{\boldsymbol{\varPhi}}_{15} & \hat{\boldsymbol{\varPhi}}_{16} & 0 & (I_{\textbf{N}} \otimes\textbf{A})^T (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3) & (I_{\textbf{N}} \otimes\textbf{A})^T (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4)\\ * & \hat{\boldsymbol{\varPhi}}_{22} & 0 & 0 & \hat{\boldsymbol{\varPhi}}_{25} & \hat{\boldsymbol{\varPhi}}_{26} & 0 & (I_{\textbf{N}} \otimes\textbf{B})^T (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3) & (I_{\textbf{N}} \otimes\textbf{B})^T (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4) \\ * & * & \hat{\boldsymbol{\varPhi}}_{33} & \hat{\boldsymbol{\varPhi}}_{34} & 0 & 0 & 0 & \rho_1c(\textbf{L}^{\sigma} \otimes \textbf{C} \textbf{Y})^T & \rho_2c(\textbf{L}^{\sigma} \otimes \textbf{C} \textbf{Y})^T \\ * & * & * & \hat{\boldsymbol{\varPhi}}_{44} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & \hat{\boldsymbol{\varPhi}}_{55} & \hat{\boldsymbol{\varPhi}}_{56} & 0 & 0 & 0 \\ * & * & * & * & * & \hat{\boldsymbol{\varPhi}}_{66} & 0 & 0 & 0 \\ * & * & * & * & * & * & \hat{\boldsymbol{\varPhi}}_{77} & (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3) & (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4) \\ * & * & * & * & * & * & * & \frac{(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta^2} & 0 \\ * & * & * & * & * & * & * & * & \frac{(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4)}{\hat{\eta}^2} \\ \end{array} \right] <0, \end{align}$$(3.14) where, |$\hat{\boldsymbol{\varPhi }}_{11}=(I_{\textbf{N}} \otimes \textbf{X})(I_{\textbf{N}} \otimes \textbf{A})+(I_{\textbf{N}} \otimes \textbf{A})^T(I_{\textbf{N}} \otimes \textbf{X})^T+2(I_{\textbf{N}} \otimes \hat{\textbf{D}})+(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_1)+(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_2)-9(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)-(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4)+\hat{\textbf{R}} a^2, \ \hat{\boldsymbol{\varPhi }}_{12}=(I_{\textbf{N}} \otimes \textbf{X})(I_{\textbf{N}} \otimes \textbf{B})+3(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3), \ \hat{\boldsymbol{\varPhi }}_{13}=(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4)+c(\textbf{L}^{\sigma } \otimes \textbf{C}\textbf{Y}), \ \hat{\boldsymbol{\varPhi }}_{15}=-\frac{24 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta }, \ \hat{\boldsymbol{\varPhi }}_{16}=\frac{60 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta }, \ \hat{\boldsymbol{\varPhi }}_{22}=(I_{\textbf{N}} \otimes \hat{\textbf{E}})-(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_1)+ \hat{\textbf{R}} b^2-9(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3), \ \hat{\boldsymbol{\varPhi }}_{25}=\frac{36(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta }, \ \hat{\boldsymbol{\varPhi }}_{26}=-\frac{60 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta }, \ \hat{\boldsymbol{\varPhi }}_{33}=-2(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4),\ \hat{\boldsymbol{\varPhi }}_{34}=(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4), \ \hat{\boldsymbol{\varPhi }}_{44}=-(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_4)-(I_{\textbf{N}} \otimes \hat{\textbf{Q}}_2), \ \hat{\boldsymbol{\varPhi }}_{55}=-\frac{192 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta ^2}, \ \hat{\boldsymbol{\varPhi }}_{56}=\frac{360 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta ^2}, \hat{\boldsymbol{\varPhi }}_{66}=-\frac{720 (I_{\textbf{N}} \otimes \hat{\textbf{Q}}_3)}{\delta ^2}, \hat{\boldsymbol{\varPhi }}_{77}=\hat{\eta}^2\hat{\textbf{Q}_4}-\hat{\textbf{R}}.$| Moreover the control gain matrix is |$\textbf{K}=\textbf{Y}\textbf{X}^{-1}.$| Proof. To prove this Theorem, we assume the following conditions, |$\hat{\textbf{Q}}_m=\textbf{X}\textbf{Q}_m\textbf{X} \ (m=1,2,3,4), \textbf{Q}_3 < \rho _1 \textbf{P}, \ \textbf{Q}_4 < \rho _2 \textbf{P}$| and |$\textbf{Y}=\textbf{K}\textbf{X},$| where |$\rho _1>0, \ \rho _2>0.$| Then pre and post multiplying both sides of (3.1) by |$diag\{(I \otimes \textbf{X}),...,(I \otimes \textbf{X})\}$| and its transpose, respectively, we get |$\hat{\boldsymbol{\varPhi }}$| in (3.13), which implies that the system is stable and reach consensus in the sense of Definition 2.2. This completes the proof. Remark 3.1 It is highly pointed out that, in this paper, our proposed criterion is different from the existing results in Fu et al. (2012), Song & Hen (2015) and Zhu & Hu (2009). In these models quasi one-sided Lipschitz condition is used in MASs for the stability and consensus performance. Mostly, in the stability and consensus problem of MASs, LMIs technique is rarely used but our main results are based on LMIs technique, which is very useful in control theory. Remark 3.2 In Ding & Zheng (2016) and Wang et al. (2017), the authors discussed the consensus of MASs without the switching topologies but in these models switching topology is considered. It is worth mentioning that the main results in this paper are attained based on the common Lyapunov Krasovskii functional. For conservatism reduction purpose, in the future, the current results can be extended to fuzzy. 4. Numerical examples This part gives an illustrative example to prove the above mentioned results. In this example, the information communicated between the agents are denoted by two distinct switching directed graphs. Consider a multi-agent system (2.1) with $$\begin{align*} &\textbf{A}= \left[ \begin{array}{cc} -0.3 & 0.2 \\ 0.2 & -0.3 \\ \end{array} \right], \ \textbf{B}= \left[ \begin{array}{cc} 0.5 & 0\\ 0 & 0.5 \\ \end{array} \right], \ \textbf{C}= \left[ \begin{array}{cc} 0.11 & 0.3\\ 0.5 & 0\\ \end{array} \right]. \end{align*}$$ Let |$\delta =0.5, \ \hat{\eta }=0.8, \ c=1.1, \ a=4, \ b=2.5, \ \rho _1=0.4, \ \rho _2=1$| and the Laplacian matrices are $$\textbf{L}^{1}=\left [ \begin{array}{cccc} 2 & -1 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & -1 & 1 \\ \end{array} \right ], \ \textbf{L}^{2}=\left [ \begin{array}{cccc} 2 & -1 & -1 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 2 & 0 \\ -1 & -1 & 0 & 2 \\ \end{array} \right ].$$ Solving the LMIs in Theorem 3.2 by using the MATLAB numerical software we get the control gain matrix as $$\mathcal{K}=\left [ \begin{array}{cc} -0.1876 & 0.1108\\ 0.0219 & -0.0133\\ \end{array} \right ].$$ 5. Conclusion In this paper, we investigated the consensus of nonlinear multi-agent system with sampled data control under the switched directed communication among the agents. Quasi one-sided Lipschitz conditions is used for the non-linear functions. To be precise, by employing the suitable Lyapunov–Krosovskii functional, matrix theory and algebraic graph theory, some novel consensus conditions are achieved. Based on these conditions, controller gain matrix is obtained. At last, the proposed theoretical concept will be validated by the numerical examples. Hence the proposed technique can be extended to other models such as fuzzy multi-agent system, switched multi-agent system and stochastic systems. This will occur in the near future. Funding Science Engineering Research Board MATRIX Project (MTR/2018/000931). References Boutayeb , M. , Benallouch , M. & Zasadzinski , M. ( 2012 ) Observer design for one-sided Lipschitz discrete-time systems . Systems Control Lett. , 61 , 879 – 886 . Google Scholar OpenURL Placeholder Text WorldCat Boyd , S. , Ghoui , L. E., Feron , E. & Balakrishnan , V. ( 1994 ) Linear Matrix Inequalities in System and Control Theory . Philadelphia, PA : SIAM . Chen , Y. Q. & Wang , Z. ( 2005 ) Formation control: a review and a new consideration . IEEE/RSJ Intern. Conf. Intell. Robots Syst. , 3181 – 3186 . Google Scholar OpenURL Placeholder Text WorldCat Cheng , J. , Park , J. H., Cao , J. & Qi , W. ( 2020a ) A hidden mode observation approach to finite-time SOFC of Markovian switching systems with quantization . Nonlinear Dynam. , 100 , 509 – 521 . Google Scholar Crossref Search ADS WorldCat Cheng , J. , Park , J. H., Zhao , X., Karimi , H. R. & Cao , J. ( 2020b ) Quantized nonstationary filtering of network-based Markov switching RSNSs: a multiple hierarchical structure strategy . IEEE Trans. Automat. Control . 65 , 4816 – 4823 . Google Scholar Crossref Search ADS WorldCat Cheng , J. , Shan , Y., Cao , J. & Park , J. H. ( 2020c ) Nonstationary control for T-S fuzzy Markovian switching systems with variable quantization density . IEEE Trans. Fuzzy Syst. doi: 10.1109/TFUZZ.2020.2974440. Google Scholar OpenURL Placeholder Text WorldCat Ding , L. & Guo , G. ( 2015 ) Sampled-data leader-following consensus for nonlinear multi-agent systems with Markovian switching topologies and communication delay . J. Franklin Inst. B , 352 , 369 – 383 . Google Scholar Crossref Search ADS WorldCat Ding , L. & Zheng , W. X. ( 2016 ) Consensus tracking in heterogeneous nonlinear multi-agent networks with asynchronous sampled-data communication . Systems Control Lett. , 96 , 151 – 157 . Google Scholar Crossref Search ADS WorldCat Fu , F. , Hou , M. & Duan , G. ( 2012 ) Stabilization of quasi-one-sided Lipschitz nonlinear systems . IMA J. Math. Control Inform. , 29 , 1 – 16 . Google Scholar OpenURL Placeholder Text WorldCat Hu , G. ( 2005 ) Observers for one-sided Lipschitz non-linear systems . IMA J. Math. Control Inform. , 23 , 395 – 401 . Google Scholar Crossref Search ADS WorldCat Hu , G. ( 2008 ) A note on observer for one-sided Lipschitz nonlinear systems . IMA J. Math. Control Inform. , 25 , 297 – 303 . Google Scholar Crossref Search ADS WorldCat Ji , Z. , Lin , H. & Yu , H. ( 2015 ) Protocols design and uncontrollable topologies construction for multi-agent networks . IEEE Trans. Automat. Control , 60 , 781 – 786 . Google Scholar Crossref Search ADS WorldCat Lee , C. H. , Lee , S. H., Park , M. J. & Kwon , O. M. ( 2018 ) Stability and stabilization criteria for sampled-data control system via augmented Lyapunov–Krasovskii functionals . Int. J. Control, Automat. Syst. , 16 , 2290 – 2302 . Google Scholar Crossref Search ADS WorldCat Lee , S. H. , Park , M. J., Kwon , O. M. & Sakthivel , R. ( 2019 ) A sampled-data control problem of neural-network-based systems using an improved free-matrix-based inequality . J. Franklin Inst. B , 365 , 8344 – 8365 . Google Scholar OpenURL Placeholder Text WorldCat Li , Z. , Wen , G., Duan , Z. & Ren , W. ( 2015 ) Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs . IEEE Trans. Automat. Control. , 60 , 1152 – 1157 . Google Scholar Crossref Search ADS WorldCat Lin , X. & Zheng , Y. ( 2016 ) Finite-time consensus of switched multiagent systems . IEEE Trans. Syst. Man Cybern. Syst. , 47 , 1535 – 1545 . Google Scholar Crossref Search ADS WorldCat Lin , X. , Zheng , Y. & Wang , L. ( 2017 ) Consensus of switched multiagent systems with random networks . Internat. J. Control , 90 , 1113 – 1122 . Google Scholar Crossref Search ADS WorldCat Qian , W. , Gao , Y. & Yang , Y. ( 2018 ) Global consensus of multiagent systems with internal delays and communication delays . IEEE Trans. Syst. Man. Cybern. Syst. , 1 , 1 – 10 . Google Scholar OpenURL Placeholder Text WorldCat Ren , W. & Beard , R. W. ( 2005 ) Consensus seeking in multiagent systems under dynamically changing interaction topologies . IEEE Trans. Automat. Control , 50 , 655 – 661 . Google Scholar Crossref Search ADS WorldCat Song , J. & Hen , S. ( 2015 ) Finite-time |${\mathrm{H}}\_{\infty } $| control for quasi-one-sided Lipschitz nonlinear systems . Neurocomputing , 149 , 1433 – 1439 . Google Scholar Crossref Search ADS WorldCat Song , Q. , Cao , J. & Yu , W. ( 2010 ) Second-order leader-following consensus of nonlinear multi-agent systems via pinning control . Systems Control Lett. , 553 – 562 . Google Scholar OpenURL Placeholder Text WorldCat Wang , L. , Jiang , F., Xie , G. & Ji , Z. ( 2009 ) Controllability of multi-agent systems based on agreement protocols . Sci. China Ser. F Inf. Sci. , 52 , 2074 – 2088 . Google Scholar Crossref Search ADS WorldCat Wang , Y. , Zhou , H., Wang , Z., Liu , Z. & Hu , W. ( 2017 ) Sampled containment control for multi-agent systems with nonlinear dynamics . Neurocomputing , 219 , 242 – 247 . Google Scholar Crossref Search ADS WorldCat Xiao , F. , Wang , L. & Chen , T. ( 2012 ) Connectivity preservation for multiagent rendezvous with link failure . Automatica J. IFAC , 48 , 25 – 35 . Google Scholar Crossref Search ADS WorldCat Zemouche , A. , Boutayeb , M. & Bara , G. L. ( 2008 ) Observers for a class of Lipschitz systems with extension to |${\mathrm{H}}\_{\infty } $| performance analysis . Systems Control Lett. , 57 , 18 – 27 . Google Scholar Crossref Search ADS WorldCat Zhang , W. , Su , H., Wang , H. & Han , Z. ( 2012 ) Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations . Commun. Nonlinear Sci. Numer. Simul. , 17 , 4968 – 4977 . Google Scholar Crossref Search ADS WorldCat Zheng , Y. & Wang , L. ( 2016 ) Consensus of switched multiagent systems . IEEE Trans. Circuits Syst. II Express Briefs , 63 , 314 – 318 . Google Scholar Crossref Search ADS WorldCat Zhu , F. & Han , Z. ( 2002 ) A note on observers for Lipschitz nonlinear systems . IEEE Trans. Automat. Control , 47 , 1751 – 1754 . Google Scholar OpenURL Placeholder Text WorldCat Zhu , Q. & Hu , G. ( 2009 ) Stability analysis for uncertain nonlinear time-delay systems with quasi-one-sided Lipschitz condition . Acta Automat. Sinica , 35 . Google Scholar OpenURL Placeholder Text WorldCat © The Author(s) 2019. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Stability analysis of quasi one-sided Lipschitz non-linear multi-agent system via sampled data control subject to directed switching topology
JO - IMA Journal of Mathematical Control and Information
DO - 10.1093/imamci/dnab005
DA - 2021-05-10
UR - https://www.deepdyve.com/lp/oxford-university-press/stability-analysis-of-quasi-one-sided-lipschitz-non-linear-multi-agent-xBfx0rLb49
SP - 1
EP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -