Distributed H_∞-consensus control of nonlinear multi-agent systems under weighted try-once-discard protocol

Abstract

In this paper, the $H_{\mathrm{\infty }}$-consensus control problem is investigated for a class of nonlinear multi-agent systems with time-varying parameters under Weighted Try-Once Discard (WTOD) protocol. In order to reduce the data collision and improve the transmission efficiency between one agent and its neighbouring agents, the WTOD communication protocol is adopted to regulate the transmission orders of the neighbouring agents. The data transmission between each agent and its neighbouring agents is implemented via a constrained communication channel where only one neighbouring agent is allowed to transmit data at each time instant. The purpose of this paper is to design an effective control scheme such that, under the WTOD communication protocol, the prespecified $H_{\mathrm{\infty }}$-consensus performance is achieved. Sufficient conditions are established for the existence of the desired controller in which the controller parameters are obtained by solving the linear matrix inequalities. Finally, a numerical simulation example is provided to illustrate the effectiveness of the theoretical results.

Keywords:

• Multi-agent systems
• $H_{\mathrm{\infty }}$-consensus control
• weighted try-once discard protocol

I. Introduction

Multi-agent systems (MASs) can be defined as a complex computing entity in which each agent has the capability of measuring and perceiving its surrounding environments and its neighbours independently. Moreover, the agents share their information through the network and therefore guiding their own actions by certain calculations so as to achieve the purpose of cooperation. During the past two decades, the MASs have received tremendous research attention which have been extensively applied to computer network, unmanned aerial vehicle formation, military and other domains due to their strong scalability, high reliability and easy maintenance (Li, Yang, & Wei, 2018; Ren, Moore, & Chen, 2006; Smith, Hanssmann, & Leonard, 2001; Wolfe, Chichka, & Speyer, 1996). In order to show the superiority of MASs, it is of great importance to explore the distributed coordination and cooperation of all the agents which has both theoretical and practical significance (Ding, Wang, Shen, & Wei, 2015; Ma, Wang, Han, & Liu, 2017; Olfati-Saber, Fax, & Murray, 2007).

In the research of MASs consensus control problem, the appearance of the interferences is inevitable because of the limitation of the operating environment. Accordingly, it is of necessity to take the external disturbance into consideration in the model establishment and control performance analysis. Up to now, the $H_{\mathrm{\infty }}$-consensus control problems for the MASs with external disturbance have been discussed widely, see Lin and Jia (2010), Wang, Ding, Dong, and Shu (2013), Xu, Wang, and Ho (2018) and the reference therein. Nevertheless, it is worth pointing out that most of the the existing results are concerned with the linear model. However, as a universal phenomenon, the nonlinearities always arise in practical engineering applications which would have great influence on the dynamic behaviour of the agents (Dunbar, 2007; Franco, Magni, Parisini, & Polycarpou, 2008; He, Chen, Han, & Qian, 2017; Hu, Wang, & Gao, 2018). For the sake of dealing with the nonlinearities, different kinds of methods have been proposed. For instance, the nonlinearities have been constrained by sector-bounded conditions (Hu, Guo, Hu, & Yang, 2015; Hu, Wang, Alsaadi, & Hayat, 2017) and Lipschitz conditions (Jenabzadeh & Safarinejadian, 2018; Li, Liu, Fu, & Xie, 2012; Zuo, Zhang, & Wang, 2014), respectively.

In a MAS, every individual needs to communicate with each other to exchange information in order to obtain global information. At present, the research on the consensus control is based on the assumption that the communication medium is ideal, i.e. each agent can obtain the accurate information of all its neighbouring agents timely and accurately. Note that in reality, due to the constraints of the communication resources such as the limited bandwidth of the network, an agent can not communicate with all the neighbouring agents simultaneously. To solve this problem, the communication protocol has been exploited to determine which neighbouring agent has the privilege to send information to the certain agent. There are many types of communication protocols commonly used in industry such as Round-Robin (RR) protocol (Bu, Dong, Han, Hou, & Li, 2019; Wan, Wang, Wu, & Liu, 2019), stochastic communication (SC) protocol (Wan, Wang, Han, & Wu, 2018) and weighted try-once-discard (WTOD) protocol (Shen, Wang, Shen, Alsaadi, & Alsaadi, 2020; Wang, Wang, Shen, & Li, 2019). It should be emphasized that the RR and SC protocols are independent on the measurement information, however, the WTOD protocol orchestrates the transmission orders in accordance with a designed quadratic selection rule which is dependent on the current measurement outputs. So far, the protocol-based consensus control problem has received some initial research attention (Song, Han, Fu, & Liu, 2019; Wan, Wang, Wu, & Liu, 2018; Wan et al., 2019; Xu, Lu, Shi, Li, & Xie, 2018; Zou, Wang, & Gao, 2016; Zou, Wang, Gao, & Alsaadi, 2017; Zou, Wang, & Gao, 2017), for example, the RR protocol and SC protocol have been respectively considered in Song et al. (2019) and Zou et al. (2017) in the control scheme design for MASs.

Based on the above discussions, in this paper, we aim to investigate the $H_{\mathrm{\infty }}$-consensus control problem for a class of nonlinear MASs with time-varying parameters under the WTOD communication protocol. The novelties of this paper are summarized as follows: (1) the model of the MASs under consideration is comprehensive which includes nonlinearities and time-varying parameters; (2) the WTOD protocol is employed during the communication among agents to reduce data transmission congestions and save precious energy; and (3) a consensus controller is designed for each agent to ensure the desirable $H_{\mathrm{\infty }}$-consensus control performance and the gain parameters are derived by solving a set of recursive linear matrix inequalities (RLMIs).

The remaining of this paper is organized as follows. Section II formulates the $H_{\mathrm{\infty }}$-consensus controller design issue for nonlinear MASs under WTOD communication protocol. Section III presents the main results, where the sufficient conditions ensuring the predefined $H_{\mathrm{\infty }}$ performance constraint are obtained for the MASs. Moreover, the gains of the controllers are computed in terms of the solutions to the RLMIs. In addition, Section IV gives a simulation example to illustrate the effectiveness of the designed control scheme. Finally, Section V outlines the conclusions of this paper.

Notation: The notations used in this paper is fairly standard except where otherwise stated. $A^{\mathrm{T}}$ represents the transpose of A. ${\mathbb{R}}^n$ represents the n dimensional Euclidean space and ${\mathbb{R}}^{m\times n}$ is the set of all $m\times n$ real matrices. The notation P>0 means that P is a real, positive definite matrix. The shorthand $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\cdots \}$ stands for a block-diagonal matrix. * always represents the symmetric block in a symmetric matrix. $\delta \left(y\right)$ is a binary function which equals to 1 for y = 0 and equals to 0 for $y\ne 0$. The operator ⊗ stands for the Kronecker product. ° denotes the Hadamard product operation. ${\mathbf{1}}_{\mathbf{n}\times \mathbf{n}}$ is an $n\times n$ matrix with all elements being 1. The N-dimensional identity matrix is denoted as $I_N$ or simply I, if no confusion is caused.

II. Problem formulation

Graph theory is widely used to model the communication topology of the network communication relationship of the agents in a MAS. Therefore, some necessary and required concepts regarding graph theory are introduced. In this paper, the communication network of a MAS compose of n agents is modelled by an undirected graph $\mathcal{D}$. Let $\mathcal{D}\mathcal{=}\mathcal{(}\mathcal{H}\mathcal{,}\mathcal{G}\mathcal{,}\mathcal{A}\mathcal{)}$ be a graph of order n where $\mathcal{H}\mathcal{=}\{1,2,\dots ,n\}$, $\mathcal{G}\mathcal{=}\mathcal{H}\times \mathcal{H}$, and $\mathcal{A}\mathcal{=}\left[a_{ij}\right]$ represent the set of agents, the set of edges and a weighted adjacency matrix with non-negative adjacency elements $a_{ij}\ge 0$, respectively. An edge $g_{ij}=(g_i,g_j)\in \mathcal{G}$ in an undirected graph $\mathcal{D}$ denotes that agent i and j can receive information from each other. The adjacency elements associated with the edges of a weighted graph are positive $a_{ij}>0$ if $g_{ij}\in \mathcal{G}$, otherwise $a_{ij}=0$ and it is assumed that self-edges $(i,i)$ are not permitted, i.e. $g_{i,i}\notin \mathcal{G}$. The set of neighbouring agents of agent i is represented by ${\mathcal{N}}_i\triangleq \{j\in \mathcal{H}\mathcal{:}(i,j)\in \mathcal{G}\}$ with $N_i$ being the number of neighbours of agent i.

Consider a discrete-time nonlinear MAS with n identical agents, the dynamics of the ith agent is depicted by the following discrete-time system over the finite time horizon $[0,\mathcal{N}-1]$: (1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) where $x_{i,k}\in {\mathbb{R}}^{n_x}$ is the agent state, $u_{i,k}\in {\mathbb{R}}^{n_u}$ is the control input, $y_{i,k}\in {\mathbb{R}}^{n_y}$ is the measurement output and $z_{i,k}\in {\mathbb{R}}^{n_z}$ is the controlled output. ${\omega }_{i,k}\in l_2\left(\right[0,\mathcal{N}-1];{\mathbb{R}}^{n_{\omega }})$ is the external disturbances. The time-varying matrices $A_k$, $B_k$, $C_k$, $D_k$, $E_k$, $F_k$ and $M_k$ are system parameters with proper dimensions, and $B_k$ is a full-column rank matrix. The nonlinear vector-valued function $f:{\mathbb{R}}^{n_x}\to {\mathbb{R}}^{n_x}$ contains all possible nonlinearities which is assumed to be continuous and satisfies the following sector bounded condition: (2) $\begin{array}{rl}& \left[f\right(x)-f(y)-{\mathrm{\Gamma }}_1(x-y\left){]}^{\mathrm{T}}\right[f\left(x\right)-f\left(y\right)-{\mathrm{\Gamma }}_2(x-y)]\le 0,\\ & \mathrm{\forall }x,y\in {\mathbb{R}}^{n_x}\end{array}$(2) where ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are two real matrices with ${\mathrm{\Gamma }}_1>{\mathrm{\Gamma }}_2$. Then, we know that (3) ${\left[\begin{array}{c}x\\ f\left(x\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\mathcal{T}}_1& {\mathcal{T}}_2\\ \ast & I\end{array}\right]\left[\begin{array}{c}x\\ f\left(x\right)\end{array}\right]\le 0$(3) where ${\mathcal{T}}_1=({\mathrm{\Gamma }}_1^{\mathrm{T}}{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_2^{\mathrm{T}}{\mathrm{\Gamma }}_1)/2$, ${\mathcal{T}}_2=-({\mathrm{\Gamma }}_1^{\mathrm{T}}+{\mathrm{\Gamma }}_2^{\mathrm{T}})/2$.

In order to alleviate the communication resource occupations, the WTOD communication protocol is introduced between agent i $(i=1,2,\dots ,n)$ and its neighbouring agents. The WTOD protocol can decide which neighbouring agent has the authority to communicate with agent i at a particular time point. Denoting $y_{j,k}$ and ${\overline{y}}_{j,k}$ as the original measurement outputs and the received measurement output of agent i from the neighbouring agent j, repectively, the updating rule of ${\overline{y}}_{j,k}$ under the WTOD protocol is expressed as follows: (4) ${\overline{y}}_{j,k}\triangleq \left\{\begin{array}{ll}{y}_{j,k},& j={\mu }_{i,k}\\ {\overline{y}}_{j,k-1},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$(4) where (5) ${\mu }_{i,k}\triangleq \mathrm{a}\mathrm{r}\mathrm{g}\underset{j\in {\mathcal{N}}_i}{max}({\overline{y}}_{j,k-1}-y_{i,k}{)}^{\mathrm{T}}{\overline{R}}_j({\overline{y}}_{j,k-1}-y_{i,k}),$(5) in which ${\overline{R}}_j\triangleq R{\mathrm{\Psi }}_j$ with ${\mathrm{\Psi }}_j\triangleq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{\delta \right(j-1),\delta (j-2),\dots ,\delta (j-N_i\left)\right\}$. R is a given positive-definite weighting matrix.

Under the scheduling of the WTOD protocol, the controller for each agent is constructed as follows: (6) $u_{i,k}=K_k{a}_{i,{\mu }_{i,k}}({\overline{y}}_{j,k}-y_{i,k})=K_k\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i(y_{j,k}-y_{i,k})$(6) where $K_k\in {\mathbb{R}}^{n_x\times n_y}$ is the consensus control gain matrix to be computed later and ${\gamma }_{j,k}^i=\delta (j-{\mu }_{i,k})(j\in {\mathcal{N}}_i)$.

Remark II.1

In this paper, we consider the WTOD protocol to save communication energy and prevent data collisions. Under the WTOD protocol, agent i can only receive the measurement output from one neighbouring agent at time instant k. How to select the neighbouring agent j is dependent on the absolute error between the measurement output of agent i and the last updated measurement output of all the neighbouring agents, i.e. the agent with larger absolute error has the chance to transmit data. That is to say, among all the neighbouring agents of agent i, only the jth ($j\in {\mathcal{N}}_i$) has the access right at each time instant which is denoted by ${\overline{y}}_{j,k}=y_{j,k}$. At the same time, the measurement outputs of the neighbouring agents except agent j keep the values of the last time instant, i.e. ${\overline{y}}_{l,k}={\overline{y}}_{l,k-1}$ ($l\in {\mathcal{N}}_i$, $l\ne j$).

Before proceeding further, the following lemmas are introduced for facilitating the subsequent derivations.

Lemma II.1

Horn & Johnson, 1994

For any matrices X, Y, A and B with appropriate dimensions, the properties of Kronecker product ⊗ are described as follows:

1. $(X+Y)\otimes A=X\otimes A+Y\otimes A,$

2. $(X\otimes Y)(A\otimes B)=\left(XA\right)\otimes \left(YB\right),$

3. $(X\otimes Y{)}^{\mathrm{T}}=X^{\mathrm{T}}\otimes Y^{\mathrm{T}}$.

Lemma II.2

Styan, 1973

For any matrices $U\in {\mathbb{R}}^{n\times n}$ and $V\in {\mathbb{R}}^{n\times n},$ the row-sums of $U\circ V$ are diagonal entries of $U{V}^{\mathrm{T}}$ $\sum _{b=1}^n(U\circ V{)}_{a,b}=(U{V}^{\mathrm{T}}{)}_{a,a}$ where $(U{)}_{a,b}$ represents the element in the ath row and bth column of the matrix U.

Defining $\begin{array}{rl}{u}_k& =[u_{1,k}^{\mathrm{T}}{u}_{2,k}^{\mathrm{T}}\cdots u_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},\\ y_k& =[y_{1,k}^{\mathrm{T}}{y}_{2,k}^{\mathrm{T}}\cdots y_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},\end{array}$ it is deduced from (6(6) $u_{i,k}=K_k{a}_{i,{\mu }_{i,k}}({\overline{y}}_{j,k}-y_{i,k})=K_k\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i(y_{j,k}-y_{i,k})$(6) ) that (7) $u_k=(I_n\otimes K_k)({\mathcal{A}}_{\mu ,k}\otimes I_{ny})y_k$(7) where $\begin{array}{rl}{\mathcal{A}}_{\mu ,k}& =\mathcal{A}\circ \mathrm{\Delta }-d_k,\mathrm{\Delta }=[{\gamma }_{j,k}^i{]}_{n\times n},\\ d_k& =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{d_{1,k},d_{2,k},\dots ,d_{n,k}\},\\ d_{i,k}& =\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i,i=1,2,\dots ,n.\end{array}$ Obviously, the matrix ${\mathcal{A}}_{\mu ,k}$ is determined by the sequence ${\mu }_{i,k},i\in \{1,2,\dots ,n\}$ which represents the selected neighbouring agent.

For simplification, the following symbols are introduced: $\begin{array}{rl}{x}_k& \triangleq [x_{1,k}^{\mathrm{T}}{x}_{2,k}^{\mathrm{T}}\cdots x_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},\\ z_k& \triangleq [z_{1,k}^{\mathrm{T}}{z}_{2,k}^{\mathrm{T}}\cdots z_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},\\ {\omega }_k& \triangleq [{\omega }_{1,k}^{\mathrm{T}}{\omega }_{2,k}^{\mathrm{T}}\cdots {\omega }_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},\\ f\left(x_k\right)& \triangleq \left[f^{\mathrm{T}}\right(x_{1,k}\left)f^{\mathrm{T}}\right(x_{2,k})\cdots f^{\mathrm{T}}(x_{n,k}){]}^{\mathrm{T}}.\end{array}$ Combining (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) with (7(7) $u_k=(I_n\otimes K_k)({\mathcal{A}}_{\mu ,k}\otimes I_{ny})y_k$(7) ), the closed-loop system is described as (8) $\begin{array}{rl}{x}_{k+1}& =(I_n\otimes A_k+{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k)x_k\\ & +({\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k+I_n\otimes E_k){\omega }_k+(I_n\otimes D_k)f\left(x_k\right)\\ y_k& =(I_n\otimes C_k)x_k+(I_n\otimes F_k){\omega }_k,\\ z_k& =(I_n\otimes M_k)x_k.\end{array}$(8) Letting ${\overline{x}}_k\triangleq [{\overline{x}}_{1,k}^{\mathrm{T}}{\overline{x}}_{2,k}^{\mathrm{T}}\cdots {\overline{x}}_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}},{\overline{z}}_k\triangleq [{\overline{z}}_{1,k}^{\mathrm{T}}{\overline{z}}_{2,k}^{\mathrm{T}}\cdots {\overline{z}}_{n,k}^{\mathrm{T}}{]}^{\mathrm{T}}$ with ${\overline{x}}_{i,k}=x_{i,k}-\left(1/n\right)\sum _{j=1}^n{x}_{j,k}$, ${\overline{z}}_{i,k}=z_{i,k}-\left(1/n\right)\sum _{j=1}^n{z}_{j,k}$, we obtain that ${\overline{x}}_k=(\mathcal{F}\otimes I_{n_x})x_k$ and ${\overline{z}}_k=(\mathcal{F}\otimes I_{n_z})z_k$ where $\mathcal{F}=I_n-1/n{\mathbf{1}}_{n\times n}$. In addition, it is deduced from Lemma II.2 that (9) $\begin{array}{rl}\sum _{j=1}^n({\mathcal{A}}_{\mu ,k}{)}_{i,j}& =\sum _{j=1}^n(\mathcal{A}\circ \mathrm{\Delta }{)}_{i,j}-d_{i,k}=(\mathcal{A}{\mathrm{\Delta }}^{\mathrm{T}}{)}_{i,i}\\ & -\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i=0.\end{array}$(9) According to the characteristics of the matrix $\mathcal{F}$, we have ${\mathcal{A}}_{\mu ,k}\mathcal{F}={\mathcal{A}}_{\mu ,k}$. Then, the closed-loop dynamics of ${\overline{x}}_k$ is represented in the following form: (10) $\begin{array}{rl}{\overline{x}}_{k+1}& =(I_n\otimes A_k+\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k){\overline{x}}_k\\ & +(\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k+\mathcal{F}\otimes E_k){\omega }_k\\ & +(\mathcal{F}\otimes D_k)f\left(x_k\right)\\ {\overline{z}}_k& =(I_n\otimes M_k){\overline{x}}_k.\end{array}$(10) Letting ${\xi }_k=[x_k^{\mathrm{T}}{\overline{x}}_k^{\mathrm{T}}{]}^{\mathrm{T}}$, we acquire the following augmented system that represents the dynamics of the considered MAS: (11) $\begin{array}{rl}{\xi }_{k+1}& ={\overline{A}}_k{\xi }_k+{\overline{D}}_{k}f\left({\xi }_k\right)+{\overline{E}}_k{\omega }_k,\\ {\overline{z}}_k& ={\overline{M}}_k{\xi }_k\end{array}$(11) where $\begin{array}{rl}{\overline{A}}_k& \triangleq \left[\begin{array}{cc}{I}_n\otimes A_k+{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k& 0\\ 0& I_n\otimes A_k\\ & +\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k\end{array}\right],\\ {\overline{D}}_k& \triangleq \left[\begin{array}{cc}{I}_n\otimes D_k& 0\\ \mathcal{F}\otimes D_k& 0\end{array}\right],\\ {\overline{E}}_k& \triangleq \left[\begin{array}{c}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k+I_n\otimes E_k\\ \mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k+\mathcal{F}\otimes E_k\end{array}\right],\\ {\overline{M}}_k& \triangleq \left[\begin{array}{cc}0& 0\\ 0& I_n\otimes M_k\end{array}\right],f\left({\xi }_k\right)\triangleq \left[f^{\mathrm{T}}\right(x_k\left)f^{\mathrm{T}}\right({\overline{x}}_k){]}^{\mathrm{T}}.\end{array}$ It follows from (3(3) ${\left[\begin{array}{c}x\\ f\left(x\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\mathcal{T}}_1& {\mathcal{T}}_2\\ \ast & I\end{array}\right]\left[\begin{array}{c}x\\ f\left(x\right)\end{array}\right]\le 0$(3) ) that (12) ${\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\overline{\mathcal{T}}}_1& {\overline{\mathcal{T}}}_2\\ \ast & I_2\end{array}\right]\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]\le 0,$(12) where $\begin{array}{rl}{\overline{\mathcal{T}}}_1& \triangleq \frac{{\overline{\mathrm{\Gamma }}}_1^{\mathrm{T}}{\overline{\mathrm{\Gamma }}}_2+{\overline{{\mathrm{\Gamma }}_2}}^{\mathrm{T}}\overline{{\mathrm{\Gamma }}_1}}{2},{\overline{\mathcal{T}}}_2\triangleq -\frac{{\overline{\mathrm{\Gamma }}}_1^{\mathrm{T}}+{\overline{\mathrm{\Gamma }}}_2^{\mathrm{T}}}{2},\\ {\overline{\mathrm{\Gamma }}}_1& \triangleq \left[\begin{array}{cc}{I}_n\otimes {\mathrm{\Gamma }}_1& 0\\ 0& I_n\otimes {\mathrm{\Gamma }}_1\end{array}\right],{\overline{\mathrm{\Gamma }}}_2\triangleq \left[\begin{array}{cc}{I}_n\otimes {\mathrm{\Gamma }}_2& 0\\ 0& I_n\otimes {\mathrm{\Gamma }}_2\end{array}\right].\end{array}$ Before proceeding further, we present the desirable performance index.

Definition II.1

For the given disturbance attenuation level $\text{β}>0$, a positive definite matrix $S=S^{\mathrm{T}}>0$ and the initial state ${\eta }_{i,0}$. The MAS (11(11) $\begin{array}{rl}{\xi }_{k+1}& ={\overline{A}}_k{\xi }_k+{\overline{D}}_{k}f\left({\xi }_k\right)+{\overline{E}}_k{\omega }_k,\\ {\overline{z}}_k& ={\overline{M}}_k{\xi }_k\end{array}$(11) ) with a prescribed connected topology is said to satisfy the $H_{\mathrm{\infty }}$-consensus control performance constraint over a finite time horizon $[0,\mathcal{N}-1]$ if the following inequality holds: (13) $\sum _{i=1}^n\parallel {\overline{z}}_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2-\sum _{i=1}^n{\text{β}}^2\parallel {\omega }_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2\le {\text{β}}^2\sum _{i=1}^n{\eta }_{i,0}^{\mathrm{T}}S{\eta }_{i,0}.$(13) The objective of this paper is to design $H_{\mathrm{\infty }}$-consensus controllers for the time-varying nonlinear MASs (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) under the WTOD communication protocol. In other words, we are concentrating on developing a controller described by (6(6) $u_{i,k}=K_k{a}_{i,{\mu }_{i,k}}({\overline{y}}_{j,k}-y_{i,k})=K_k\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i(y_{j,k}-y_{i,k})$(6) ) for each agent such that the above $H_{\mathrm{\infty }}$-consensus control performance is achieved over a finite time horizon $[0,\mathcal{N}-1]$.

III. Main results

In this section, by resorting to the Lyapunov stability theorem, the sufficient conditions are derived to ensure that the augmented systems (11(11) $\begin{array}{rl}{\xi }_{k+1}& ={\overline{A}}_k{\xi }_k+{\overline{D}}_{k}f\left({\xi }_k\right)+{\overline{E}}_k{\omega }_k,\\ {\overline{z}}_k& ={\overline{M}}_k{\xi }_k\end{array}$(11) ) achieves the desired $H_{\mathrm{\infty }}$ performance constraint (13(13) $\sum _{i=1}^n\parallel {\overline{z}}_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2-\sum _{i=1}^n{\text{β}}^2\parallel {\omega }_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2\le {\text{β}}^2\sum _{i=1}^n{\eta }_{i,0}^{\mathrm{T}}S{\eta }_{i,0}.$(13) ) under the given initial condition for ${\omega }_k\ne 0$. Moreover, the controller parameters are obtained in terms of the solutions to the certain RLMIs.

To proceed, the following three lemmas are useful for the subsequent developments.

Lemma III.1

(S-procedure)(Boyd, Ghaoui, Feron, & Balakrishnan, 1994)

Let ${\mathcal{V}}_0\left(\chi \right),{\mathcal{V}}_1\left(\chi \right),\dots ,{\mathcal{V}}_p\left(\chi \right)$ be quadratic functions of $\chi \in {\mathbb{R}}^n,$ ${\mathcal{V}}_i\left(\chi \right)={\chi }^{\mathrm{T}}{T}_i\chi ,$ $(i=0,1,\dots ,p)$ with $T_i^{\mathrm{T}}=T_i$. Then, ${\mathcal{V}}_1\left(\chi \right)\le 0,\dots ,{\mathcal{V}}_p\left(\chi \right)\le 0⟹{\mathcal{V}}_0\left(\chi \right)\le 0$ is true if there exist scalars ${\tau }_1,\dots ,{\tau }_p>0$ such that (14) $T_0-\sum _{i=1}^p{\tau }_i{T}_i\le 0$(14) holds.

Lemma III.2

Han, Wei, Ding, & Song, 2017

For a full-column rank matrix $B_k\in {\mathbb{R}}^{n_x\times n_u}$ $(n_x>n_u),$ there always exist two orthogonal matrices $Q_k\in {\mathbb{R}}^{n_x\times n_x}$ and $N_k\in {\mathbb{R}}^{n_u\times n_u}$ satisfying (15) $B_k=Q_k\left[\begin{array}{c}\sum _k\\ 0\end{array}\right]N_k=\left[Q_{1,k}{Q}_{2,k}\right]\left[\begin{array}{c}\sum _k\\ 0\end{array}\right]N_k$(15) where $Q_{1,k}\in {\mathbb{R}}^{n_x\times n_u},$ $Q_{2,k}\in {\mathbb{R}}^{n_x\times (n_x-n_u)},$ $\sum _k=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{{\phi }_1,\dots ,{\phi }_{n_u}\}$ and ${\phi }_j(j=1,2,\dots ,n_u)$ are non-zero singular values of matrix $B_k$. Therefore, if the structure of matrix $P_{k+1}$ has the following form: (16) $\begin{array}{rl}{P}_{k+1}& =\left[Q_{1,k}{Q}_{2,k}\right]\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{P_{1,k+1},P_{2,k+1}\}[Q_{1,k}{Q}_{2,k}{]}^{\mathrm{T}}\\ & =Q_{1,k}{P}_{1,k+1}{Q}_{1,k}^{\mathrm{T}}+Q_{2,k}{P}_{2,k+1}{Q}_{2,k}^{\mathrm{T}}\end{array}$(16) where $P_{1,k+1}\in {\mathbb{R}}^{n_u\times n_u}>0,$ $P_{2,k+1}\in {\mathbb{R}}^{(n_x-n_u)\times (n_x-n_u)}>0,$ then there exists a nonsingular matrix ${\tilde{P}}_{k+1}\in {\mathbb{R}}^{n_u\times n_u}$ such that $B_k{\tilde{P}}_{k+1}=P_{k+1}{B}_k$. Moreover, ${\tilde{P}}_{k+1}=N_k^{\mathrm{T}}\sum _k^{-1}{P}_{1,k+1}\sum _k{N}_k$.

Theorem III.1

Let the disturbance attenuation level $\text{β}>0$ and the controller gain matrices $\{K_k{\}}_{0\le k\le \mathcal{N}-1}$ in (6(6) $u_{i,k}=K_k{a}_{i,{\mu }_{i,k}}({\overline{y}}_{j,k}-y_{i,k})=K_k\sum _{j\in {\mathcal{N}}_i}{a}_{ij}{\gamma }_{j,k}^i(y_{j,k}-y_{i,k})$(6) ) be known. For the given positive definite matrix S and the initial condition ${\overline{P}}_0\le {\text{β}}^{2}S,$ the nonlinear MAS (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) under WTOD communication protocol (5(5) ${\mu }_{i,k}\triangleq \mathrm{a}\mathrm{r}\mathrm{g}\underset{j\in {\mathcal{N}}_i}{max}({\overline{y}}_{j,k-1}-y_{i,k}{)}^{\mathrm{T}}{\overline{R}}_j({\overline{y}}_{j,k-1}-y_{i,k}),$(5) ) satisfy the $H_{\mathrm{\infty }}$-consensus control performance (13(13) $\sum _{i=1}^n\parallel {\overline{z}}_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2-\sum _{i=1}^n{\text{β}}^2\parallel {\omega }_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2\le {\text{β}}^2\sum _{i=1}^n{\eta }_{i,0}^{\mathrm{T}}S{\eta }_{i,0}.$(13) ) for all ${\omega }_k\ne 0$ if there exist a sequence of symmetric positive definite matrices $\{{\overline{P}}_{k+1}{\}}_{0\le k\le \mathcal{N}-1}$ and a family of positive scalars $\{{\tau }_k{\}}_{0\le k\le \mathcal{N}-1}$ satisfying the following recursive matrix inequality: (17) ${\mathrm{\Omega }}_k=\left[\begin{array}{ccc}{\mathrm{\Omega }}_{11,k}& \ast & \ast \\ {\mathrm{\Omega }}_{21,k}& {\mathrm{\Omega }}_{22,k}& \ast \\ {\mathrm{\Omega }}_{31,k}& {\mathrm{\Omega }}_{32,k}& {\mathrm{\Omega }}_{33,k}\end{array}\right]<0$(17) where $\begin{array}{rl}{\mathrm{\Omega }}_{11,k}& \triangleq {\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k+{\overline{M}}_k^{\mathrm{T}}{\overline{M}}_k-{\overline{P}}_k-{\tau }_k{\overline{\mathcal{T}}}_1,\\ {\mathrm{\Omega }}_{21,k}& \triangleq {\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k-{\tau }_k{\overline{\mathcal{T}}}_2^{\mathrm{T}},{\mathrm{\Omega }}_{31,k}\triangleq {\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k,\\ {\mathrm{\Omega }}_{22,k}& \triangleq {\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{D}}_k-{\tau }_{k}I,{\mathrm{\Omega }}_{32,k}\triangleq {\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{D}}_k,\\ {\mathrm{\Omega }}_{33,k}& \triangleq {\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{E}}_k-{\text{β}}^{2}I.\end{array}$

Proof.

Defining the following function for system (11(11) $\begin{array}{rl}{\xi }_{k+1}& ={\overline{A}}_k{\xi }_k+{\overline{D}}_{k}f\left({\xi }_k\right)+{\overline{E}}_k{\omega }_k,\\ {\overline{z}}_k& ={\overline{M}}_k{\xi }_k\end{array}$(11) ) (18) $J_k={\xi }_{k+1}^{\mathrm{T}}{\overline{P}}_{k+1}{\xi }_{k+1}-{\xi }_k^{\mathrm{T}}{\overline{P}}_k{\xi }_k,$(18) then one has (19) $\begin{array}{rl}{J}_k& ={\xi }_k^{\mathrm{T}}{\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k{\xi }_k+{\xi }_k^{\mathrm{T}}{\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{D}}_{k}f\left({\xi }_k\right)+{\xi }_k^{\mathrm{T}}{\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{E}}_k{\omega }_k\\ & +f^{\mathrm{T}}\left({\xi }_k\right){\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k{\xi }_k\\ & +f^{\mathrm{T}}\left({\xi }_k\right){\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{D}}_{k}f\left({\xi }_k\right)+f^{\mathrm{T}}\left({\xi }_k\right){\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{E}}_k{\omega }_k\\ & +{\omega }_k^{\mathrm{T}}{\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k{\xi }_k\\ & +{\omega }_k^{\mathrm{T}}{\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{D}}_{k}f\left({\xi }_k\right)+{\omega }_k^{\mathrm{T}}{\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{E}}_k{\omega }_k-{\xi }_k^{\mathrm{T}}{\overline{P}}_k{\xi }_k\\ & ={\eta }_k^{\mathrm{T}}{\overline{\mathrm{\Omega }}}_k{\eta }_k\end{array}$(19) where $\begin{array}{rl}{\overline{\mathrm{\Omega }}}_k& \triangleq \left[\begin{array}{ccc}{\overline{\mathrm{\Omega }}}_{11,k}& \ast & \ast \\ {\overline{\mathrm{\Omega }}}_{21,k}& {\overline{\mathrm{\Omega }}}_{22,k}& \ast \\ {\mathrm{\Omega }}_{31,k}& {\mathrm{\Omega }}_{32,k}& {\overline{\mathrm{\Omega }}}_{33,k}\end{array}\right],\\ {\eta }_k& \triangleq \left[{\xi }_k^{\mathrm{T}}{f}^{\mathrm{T}}\right({\xi }_k){\omega }_k^{\mathrm{T}}{]}^{\mathrm{T}},{\overline{\mathrm{\Omega }}}_{11,k}\triangleq {\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k-{\overline{P}}_k,\\ {\overline{\mathrm{\Omega }}}_{21,k}& \triangleq {\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k,{\overline{\mathrm{\Omega }}}_{22,k}\triangleq {\overline{D}}_k^{\mathrm{T}}{\overline{P}}_{k+1}\overline{D},\\ {\overline{\mathrm{\Omega }}}_{33,k}& \triangleq {\overline{E}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{E}}_k.\end{array}$ Considering the nonlinear constraint (12(12) ${\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\overline{\mathcal{T}}}_1& {\overline{\mathcal{T}}}_2\\ \ast & I_2\end{array}\right]\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]\le 0,$(12) ) and Lemma III.1, we obtain (20) $J_k\le {\eta }_k^{\mathrm{T}}{\overline{\mathrm{\Omega }}}_k{\eta }_k-{\tau }_k{\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\overline{\mathcal{T}}}_1& {\overline{\mathcal{T}}}_2\\ \ast & I_2\end{array}\right]\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]={\eta }_k^{\mathrm{T}}{\tilde{\mathrm{\Omega }}}_k{\eta }_k$(20) where $\begin{array}{rl}{\tilde{\mathrm{\Omega }}}_k& \triangleq \left[\begin{array}{ccc}{\tilde{\mathrm{\Omega }}}_{11,k}& \ast & \ast \\ {\mathrm{\Omega }}_{21,k}& {\mathrm{\Omega }}_{22,k}& \ast \\ {\mathrm{\Omega }}_{31,k}& {\mathrm{\Omega }}_{32,k}& {\overline{\mathrm{\Omega }}}_{33,k}\end{array}\right],\\ {\tilde{\mathrm{\Omega }}}_{11,k}& \triangleq {\overline{A}}_k^{\mathrm{T}}{\overline{P}}_{k+1}{\overline{A}}_k-{\overline{P}}_k-{\tau }_k{\overline{\mathcal{T}}}_1.\end{array}$ Adding the zero term ${\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k-{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k-{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k$ to (20(20) $J_k\le {\eta }_k^{\mathrm{T}}{\overline{\mathrm{\Omega }}}_k{\eta }_k-{\tau }_k{\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\overline{\mathcal{T}}}_1& {\overline{\mathcal{T}}}_2\\ \ast & I_2\end{array}\right]\left[\begin{array}{c}{\xi }_k\\ f\left({\xi }_k\right)\end{array}\right]={\eta }_k^{\mathrm{T}}{\tilde{\mathrm{\Omega }}}_k{\eta }_k$(20) ) yields (21) $\begin{array}{rl}{J}_k& \le {\eta }_k^{\mathrm{T}}{\tilde{\mathrm{\Omega }}}_k{\eta }_k+{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k-{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k-{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k\\ & ={\eta }_k^{\mathrm{T}}{\mathrm{\Omega }}_k{\eta }_k-{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k.\end{array}$(21) It follows from (17(17) ${\mathrm{\Omega }}_k=\left[\begin{array}{ccc}{\mathrm{\Omega }}_{11,k}& \ast & \ast \\ {\mathrm{\Omega }}_{21,k}& {\mathrm{\Omega }}_{22,k}& \ast \\ {\mathrm{\Omega }}_{31,k}& {\mathrm{\Omega }}_{32,k}& {\mathrm{\Omega }}_{33,k}\end{array}\right]<0$(17) ) that ${\eta }_k^{\mathrm{T}}{\mathrm{\Omega }}_k{\eta }_k<0$. As a result, one has (22) $J_k\le -{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k.$(22) Then, summing up both sides of (22(22) $J_k\le -{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k.$(22) ) with respect to k from 0 to $\mathcal{N}-1$, we derive (23) $\begin{array}{rl}\sum _{k=0}^{\mathcal{N}-1}{J}_k& ={\eta }_{\mathcal{N}}^{\mathrm{T}}{\overline{P}}_{\mathcal{N}}{\eta }_{\mathcal{N}}-{\eta }_0^{\mathrm{T}}{\overline{P}}_0{\eta }_0\\ & \le -\sum _{k=0}^{\mathcal{N}-1}{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k+\sum _{k=0}^{\mathcal{N}-1}{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k.\end{array}$(23) Taking the ${\eta }_{\mathcal{N}}^{\mathrm{T}}{\overline{P}}_{\mathcal{N}}{\eta }_{\mathcal{N}}>0$ and ${\overline{P}}_0\le {\text{β}}^{2}S$ into account, one finally has $\sum _{k=0}^{\mathcal{N}-1}{\overline{z}}_k^{\mathrm{T}}{\overline{z}}_k\le \sum _{k=0}^{\mathcal{N}-1}{\text{β}}^2{\omega }_k^{\mathrm{T}}{\omega }_k+{\text{β}}^2{\eta }_0^{\mathrm{T}}S{\eta }_0,$ which is clear to see that J<0 if ${\mathrm{\Omega }}_k<0$ and the $H_{\mathrm{\infty }}$ performance (13(13) $\sum _{i=1}^n\parallel {\overline{z}}_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2-\sum _{i=1}^n{\text{β}}^2\parallel {\omega }_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2\le {\text{β}}^2\sum _{i=1}^n{\eta }_{i,0}^{\mathrm{T}}S{\eta }_{i,0}.$(13) ) is therefore achieved.

The proof is now complete.

Remark III.1

In Theorem III.1, the $H_{\mathrm{\infty }}$-consensus control problem for the addressed nonlinear discrete-time MASs with time-varying parameters has been analyzed. A sufficient condition has been developed to ensure the $H_{\mathrm{\infty }}$ performance constraint with ${\omega }_k\ne 0$ under the WTOD protocol. In what follows, the controller parameters will be derived by resorting to some matrix inequalities.

Theorem III.2

Consider the nonlinear discrete-time MASs (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) with time-varying parameters under the WTOD communication protocol (5(5) ${\mu }_{i,k}\triangleq \mathrm{a}\mathrm{r}\mathrm{g}\underset{j\in {\mathcal{N}}_i}{max}({\overline{y}}_{j,k-1}-y_{i,k}{)}^{\mathrm{T}}{\overline{R}}_j({\overline{y}}_{j,k-1}-y_{i,k}),$(5) ). For the given disturbance attenuation level $\text{β}>0$ as well as the positive definite matrices $S_1>0$ and $S_2>0$ with $S=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{S_1,S_2\},$ the closed-loop system (11(11) $\begin{array}{rl}{\xi }_{k+1}& ={\overline{A}}_k{\xi }_k+{\overline{D}}_{k}f\left({\xi }_k\right)+{\overline{E}}_k{\omega }_k,\\ {\overline{z}}_k& ={\overline{M}}_k{\xi }_k\end{array}$(11) ) satisfies the $H_{\mathrm{\infty }}$ performance constraint (13(13) $\sum _{i=1}^n\parallel {\overline{z}}_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2-\sum _{i=1}^n{\text{β}}^2\parallel {\omega }_{i,k}{\parallel }_{[0,\mathcal{N}-1]}^2\le {\text{β}}^2\sum _{i=1}^n{\eta }_{i,0}^{\mathrm{T}}S{\eta }_{i,0}.$(13) ) if there exist a sequence of scalars $\{{\tau }_k{\}}_{0\le k\le \mathcal{N}-1}>0,$ a set of matrices $\{{\tilde{K}}_k,{\}}_{0\le k\le \mathcal{N}-1}$ and a series of positive definite matrices $\{P_{1,k+1},P_{2,k+1}{\}}_{0\le k\le \mathcal{N}-1}$ satisfying the initial condition $P_{1,0}\le {\text{β}}^2{S}_1,$ $P_{2,0}\le {\text{β}}^2{S}_2$ and the following RLMI:

(24) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ A_{61,k}& 0& A_{63,k}& 0& A_{65,k}& -I_n\otimes P_{k+1}& \ast \\ 0& A_{72,k}& A_{73,k}& 0& A_{75,k}& 0& -I_n\otimes P_{k+1}\end{array}\right]<0$(24) where $\begin{array}{rl}{A}_{11,k}& \triangleq -P_k-{\tau }_k{I}_n\otimes {\mathcal{T}}_1,\\ A_{22,k}& \triangleq I_n\otimes M_k^{\mathrm{T}}{M}_k-P_k-{\tau }_k{I}_n\otimes {\mathcal{T}}_1,\\ A_{31,k}& \triangleq -{\tau }_k{I}_n\otimes {\mathcal{T}}_2^{\mathrm{T}},\\ A_{42,k}& \triangleq -{\tau }_k{I}_n\otimes {\mathcal{T}}_2^{\mathrm{T}},A_{61,k}\triangleq I_n\otimes P_{k+1}{A}_k\\ & +{\mathcal{A}}_{\mu ,k}\otimes B_k{\tilde{K}}_k{C}_k,A_{63,k}\triangleq I_n\otimes P_{k+1}{D}_k,\\ A_{65,k}& \triangleq I_n\otimes P_{k+1}{E}_k+{\mathcal{A}}_{\mu ,k}\otimes B_k{\tilde{K}}_k{F}_k,\\ A_{72,k}& \triangleq I_n\otimes P_{k+1}{A}_k+\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{\tilde{K}}_k{C}_k,\\ A_{73,k}& \triangleq \mathcal{F}\otimes P_{k+1}{D}_k,A_{75,k}\triangleq \mathcal{F}\otimes P_{k+1}{E}_k\\ & +\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{\tilde{K}}_k{F}_k,\\ P_{k+1}& \triangleq Q_{1,k}{P}_{1,k+1}{Q}_{1,k}^{\mathrm{T}}+Q_{2,k}{P}_{2,k+1}{Q}_{2,k}^{\mathrm{T}}.\end{array}$ Furthermore, if the above inequality is feasible, the controller gain matrix $K_k$ is computed by (25) $K_k=N_k\sum _k{P}_{1,k+1}^{-1}\sum _k^{-1}{N}_k^{\mathrm{T}}{\tilde{K}}_k.$(25) The matrices $N_k,$ $Q_{1,k},$ $Q_{2,k}$ have been defined in Lemma III.2.

Proof.

By using Schur Complement Lemma, the inequality (17(17) ${\mathrm{\Omega }}_k=\left[\begin{array}{ccc}{\mathrm{\Omega }}_{11,k}& \ast & \ast \\ {\mathrm{\Omega }}_{21,k}& {\mathrm{\Omega }}_{22,k}& \ast \\ {\mathrm{\Omega }}_{31,k}& {\mathrm{\Omega }}_{32,k}& {\mathrm{\Omega }}_{33,k}\end{array}\right]<0$(17) ) is transformed into the following inequality:

(26) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ {\overline{A}}_{61,k}& 0& {\overline{A}}_{63,k}& 0& {\overline{A}}_{65,k}& -I_n\otimes P_{k+1}^{-1}& \ast \\ 0& {\overline{A}}_{72,k}& {\overline{A}}_{73,k}& 0& {\overline{A}}_{75,k}& 0& -I_n\otimes P_{k+1}^{-1}\end{array}\right]<0$(26) where $\begin{array}{rl}{\overline{A}}_{61,k}& \triangleq I_n\otimes A_k+{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k,{\overline{A}}_{63,k}\triangleq I_n\otimes D_k,\\ {\overline{A}}_{65,k}& \triangleq I_n\otimes E_k+{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k,\\ {\overline{A}}_{72,k}& \triangleq I_n\otimes A_k+\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{C}_k,{\overline{A}}_{73,k}\triangleq \mathcal{F}\otimes D_k,\\ {\overline{A}}_{75,k}& \triangleq \mathcal{F}\otimes E_k+\mathcal{F}{\mathcal{A}}_{\mu ,k}\otimes B_k{K}_k{F}_k,\\ {\overline{P}}_{k+1}& \triangleq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{I_n\otimes P_{k+1},I_n\otimes P_{k+1}\}.\end{array}$ Next, in order to convert the inequality (26(26) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ {\overline{A}}_{61,k}& 0& {\overline{A}}_{63,k}& 0& {\overline{A}}_{65,k}& -I_n\otimes P_{k+1}^{-1}& \ast \\ 0& {\overline{A}}_{72,k}& {\overline{A}}_{73,k}& 0& {\overline{A}}_{75,k}& 0& -I_n\otimes P_{k+1}^{-1}\end{array}\right]<0$(26) ) into a linear matrix inequality, perform the congruent transformation to (26(26) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ {\overline{A}}_{61,k}& 0& {\overline{A}}_{63,k}& 0& {\overline{A}}_{65,k}& -I_n\otimes P_{k+1}^{-1}& \ast \\ 0& {\overline{A}}_{72,k}& {\overline{A}}_{73,k}& 0& {\overline{A}}_{75,k}& 0& -I_n\otimes P_{k+1}^{-1}\end{array}\right]<0$(26) ) by $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{I,I,I,I,I_n\otimes P_{k+1},I,I\}$. Then, from Lemma III.2, one has $B_k{\tilde{P}}_{k+1}=P_{k+1}{B}_k$, $P_{k+1}=Q_{1,k}{P}_{1,k+1}{Q}_{1,k}^{\mathrm{T}}+Q_{2,k}{P}_{2,k+1}{Q}_{2,k}^{\mathrm{T}}$ and ${\tilde{P}}_{k+1}=N_k^{\mathrm{T}}\sum _k^{-1}{P}_{1,k+1}\sum _k{N}_k$. Denoting ${\tilde{K}}_k={\tilde{P}}_{k+1}{K}_k$, we obtain (24(24) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ A_{61,k}& 0& A_{63,k}& 0& A_{65,k}& -I_n\otimes P_{k+1}& \ast \\ 0& A_{72,k}& A_{73,k}& 0& A_{75,k}& 0& -I_n\otimes P_{k+1}\end{array}\right]<0$(24) ) and (25(25) $K_k=N_k\sum _k{P}_{1,k+1}^{-1}\sum _k^{-1}{N}_k^{\mathrm{T}}{\tilde{K}}_k.$(25) ), which ends the proof.

According to Theorem III.2, the $H_{\mathrm{\infty }}$-consensus controller design (CCD) algorithm is summarized as follows:

Table

Algorithm CCD:
Step 1.	Set the disturbance attenuation level β, the positive definite matrix S and the initial condition ${\eta }_0$, the value for matrix $P_{1,0}$ and $P_{2,0}$ according to the initial condition constraints $P_{1,0}\le {\text{β}}^2{S}_1$, $P_{2,0}\le {\text{β}}^2{S}_2$, and assign k = 0.
Step 2.	Obtain the values of the matrices $P_{1,k+1}$, $P_{2,k+1}$ and the indirect variable ${\tilde{K}}_k$ by solving the RLMI (24(24) $\left[\begin{array}{ccccccc}{A}_{11,k}& \ast & \ast & \ast & \ast & \ast & \ast \\ 0& A_{22,k}& \ast & \ast & \ast & \ast & \ast \\ A_{31,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast & \ast \\ 0& A_{42,k}& 0& -{\tau }_{k}I& \ast & \ast & \ast \\ 0& 0& 0& 0& -{\text{β}}^{2}I& \ast & \ast \\ A_{61,k}& 0& A_{63,k}& 0& A_{65,k}& -I_n\otimes P_{k+1}& \ast \\ 0& A_{72,k}& A_{73,k}& 0& A_{75,k}& 0& -I_n\otimes P_{k+1}\end{array}\right]<0$(24) ) at time instant k.
Step 3.	Calculate the controller gain matrix $K_k$ by computing (25(25) $K_k=N_k\sum _k{P}_{1,k+1}^{-1}\sum _k^{-1}{N}_k^{\mathrm{T}}{\tilde{K}}_k.$(25) ).
Step 4.	If $k\le \mathcal{N}-1$, set k = k + 1 and go to step 2, else go to step 5.
Step 5.	Stop.

Remark III.2

Up to now, we have investigated the $H_{\mathrm{\infty }}$-consensus control problem for a class of nonlinear MASs with time-varying parameters under WTOD communication protocol over the finite-horizon. A new consensus controller has been designed via utilizing the scheduled measurement outputs of the neighbouring agents. The sufficient conditions have been established such that the augmented system is stable with satisfactory $H_{\mathrm{\infty }}$ performance constraint. Compared with the existing literature, the peculiarities of the main results obtained in this paper are shown as follows: (1) the consensus control scheme designed in this paper is applicable to handle the control problem of the nonlinear MASs with time-varying parameters under the WTOD communication protocol; (2) for saving communication resources, the protocol-based controller is put forward, where each agent can only receive one neighbouring agent's information at each time instant; and (3) the sufficient conditions has been acquired for the underlying system to achieve the desired $H_{\mathrm{\infty }}$-consensus control performance constraint over a finite-time horizon.

IV. An illustrative example

In this part, a numerical example is presented to illustrate the validity of the proposed $H_{\mathrm{\infty }}$-consensus controller design algorithm for nonlinear MASs with time-varying parameters under WTOD communication protocol simultaneously.

For a given finite-time horizon $[0,20]$, we consider a MAS with 3 agents whose communication topology is described by an undirected graph $\mathcal{D}=(\mathcal{H},\mathcal{G},\mathcal{A})$ with the set of agents $\mathcal{H}=\{1,2,3\}$ and the corresponding adjacency matrix $\mathcal{A}$ is given as follows: $\mathcal{A}=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right].$ The system parameters of (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) are set as follows: $\begin{array}{rl}{A}_k& =\left[\begin{array}{cc}-\sin \left(0.3k\right)& -0.5\\ -0.1& -0.4\end{array}\right],B_k=\left[\begin{array}{c}3\\ 0.6\sin \left(3k\right)\end{array}\right],\\ C_k& =\left[\begin{array}{cc}0.3\cos \left(k\right)& 0.22\\ \cos \left(k\right)& 0.2\end{array}\right],M_k=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\\ D_k& =\left[\begin{array}{cc}0.06\cos \left(3k\right)& 0.03\\ 0.01& 0.02\end{array}\right],\\ E_k& =\left[\begin{array}{cc}0.2& 0\\ 0& 0.1\sin \left(2k\right)\end{array}\right],F_k=\left[\begin{array}{cc}-0.01& 0.03\\ -0.01& 0.02\end{array}\right].\end{array}$ It is assumed that the nonlinear function $f\left(x_k\right)=\left[\mathrm{s}\mathrm{i}\mathrm{n}\right(x_{1,k}\left)\mathrm{s}\mathrm{i}\mathrm{n}\right(x_{2,k}){]}^{\mathrm{T}}$, and ${\mathrm{\Gamma }}_1=\left[0.90.9\right]$, ${\mathrm{\Gamma }}_2=\left[0.30.3\right]$. Moreover, we set the positive definite matrices as $S_1=1$, $S_2=1$, the weighting matrix $R=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{1,1\}$. The initial values of the states are selected as $x_{1,0}=[-22{]}^{\mathrm{T}}$, $x_{2,0}=[1-1{]}^{\mathrm{T}}$, $x_{3,0}=[-22{]}^{\mathrm{T}}$ where $x_{i,k}=[x_{i,k}^1{x}_{i,k}^2{]}^{\mathrm{T}}$. The external disturbance is chosen as ${\omega }_k=0.1\sin \left(k\right)$ and the $H_{\mathrm{\infty }}$ consensus control performance index is set as $\text{β}=0.8$.

In the simulation results, Figures 1–2 plot the actual states of the underlying agents. Figures 3–4 plot the state consensus errors of the underlying agents, where $z_{i,k}=[z_{i,k}^1,z_{i,k}^2]$, $i=1,2,3.$ Figures 5–6 presents the measurement outputs of the underlying agents, where $y_{i,k}=[y_{i,k}^1,y_{i,k}^2]$, $i=1,2,3.$ From the simulation results, it is obvious to see that the MAS (1(1) $\begin{array}{rl}{x}_{i,k+1}& =A_k{x}_{i,k}+B_k{u}_{i,k}+D_{k}f\left(x_{i,k}\right)+E_k{\omega }_{i,k},\\ y_{i,k}& =C_k{x}_{i,k}+F_k{\omega }_{i,k},\\ z_{i,k}& =M_k{x}_{i,k}\end{array}$(1) ) has good control performance under the proposed control scheme in this paper.

Figure 1. The first state of three agents $x_{1,k}^1$, $x_{2,k}^1$ and $x_{3,k}^1$.

Figure 2. The second state of three agents $x_{1,k}^2$, $x_{2,k}^2$ and $x_{3,k}^2$.

Figure 3. The first state consensus error of three agents $z_{1,k}^1$, $z_{2,k}^1$ and $z_{3,k}^1$.

Figure 4. The second state consensus error of three agents $z_{1,k}^2$, $z_{2,k}^2$ and $z_{3,k}^2$.

Figure 5. The first measurement output of three agents $y_{1,k}^1$, $y_{2,k}^1$ and $y_{3,k}^1$.

Figure 6. The second measurement output of three agents $y_{1,k}^2$, $y_{2,k}^2$ and $y_{3,k}^2$.

V. Conclusion

This paper has investigated the $H_{\mathrm{\infty }}$-consensus control problem for a class of time-varying nonlinear MASs under WTOD communication protocol over a finite-time horizon. The nonlinearities considered in this paper satisfy the sector-bounded conditions. The WTOD communication protocol has been adopted to schedule the information transmission orders of the neighbouring agents at each time instant for the purpose of mitigating the data collisions. Sufficient conditions have been derived for the designed consensus controller to meet the prescribed $H_{\mathrm{\infty }}$ performance constraints. The explicit expression of the controller gain matrix has been characterized in terms of the solution to a certain set of RLMI. Finally, the effectiveness of the proposed controller design scheme has been demonstrated by an illustrative example.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grants 61873058, the Natural Science Foundation of Heilongjiang Province of China under Grant ZD2019F001, the Open Fund of The Key Laboratory for Metallurgical Equipment and Control of Ministry of Education in Wuhan University of Science and Technology under Grants 2018A01 and MECOF2019B01, the Northeast Petroleum University Guided Innovation Foundation of China under Grant 2018YDL-01.