I. INTRODUCTION

Recently,the study of cooperative control of multi-agent systems has been received much attention from many fields. As an important issue of cooperative control,the agents should be able to reach a common trajectory based on a consensus protocol under the shared information in the presence of information exchange and dynamically changing interaction topologies^{[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]}. It is widely applied to flocks^{[1, 2]},vehicle formations^[3] and attitude alignment^[4]. The central problem is to design a distributed algorithm by which all the agents reach an agreement on their states or outputs,using local information exchange under the communication topologies. The consensus problem of multi-agent systems with leaderless was widely investigated in ^{[5, 6]}. Pioneering contributions to various distributed strategies that achieved consensus were described by ^{[1, 3, 5]}. They dealt with consensus problems for networks of dynamic agents under directed networks with fixed and switching topologies and undirected networks with communication time-delays and fixed topology. Diverse applications of consensus problems were discussed in ^[6] such as fast consensus in small-world networks,and Markov processes load balancing in networked dynamic systems.

In leader-following problems,a subsystem named leader moves along a predefined trajectory,while other subsystems named followers have the same trajectories as the leader^{[13, 14, 15, 16, 17, 18, 19]}. Reference ^[15] introduced the optimal tracking problem based on the adaptive dynamic programming method. In many applications,there might exist leaders in the multi-agent networks. However,some variables of an active leader cannot be measured,^[16] proposed an "observer" by inserting an integrator into the loop for each agent to estimate the leader$'$s velocity. Reference ^[17] considered the leader-following formation problem of multi-robot systems with switching interconnection topologies using dynamic output feedback with the help of a canonical internal model. Kinematics equations of the robots are given in ^[19] and it uses adaptive control law to solve the robust leader-following formation problem.

A further concept termed output regulation has been introduced to characterize the output consensus of leader-following multi-agent systems. Output regulation theory is to control one or more than one plant so that their outputs can track a reference signal (and/or reject a disturbance) produced by an exosystem. Well-known results about output regulation of linear systems and nonlinear systems have been posed and solved by ^{[20, 21, 22]}. In ^[23],the tracked signal or rejected disturbance is the same for all the nodes but only partial nodes can access the state information of exosystem and a synchronous manifold method was employed in it. But the majority of existing algorithms use the internal model principle method^{[24, 25, 26, 27]}. Based on it,a distributed control approach for robust output regulation problem was considered in ^[24]. A general result was given in ^[27] for linear uncertain multi-agent systems using distributed dynamic state feedback control law and distributed dynamic output feedback control law. Since not all the agents can access the exosystem information,^{[28, 29]} gave a dynamic compensator for the un-access part to solve the output regulation problem for heterogeneous agents.

There have been a lot of results about output regulation problem of continuous-time linear or nonlinear multi-agent systems,but few papers dealt with discrete-time multi-agent systems. In view of these issues,this technical note presents a framework on output regulation of discrete-time multi-agent systems. Every subsystem is identical and uncertain. Each agent also has different disturbance. Therefore,we propose an internal model with $n$ adjustable parameters. Then a distributed consensus protocol is given which not only makes the subsystem stable regardless of external disturbance but also regulates the errors to converge to zero.

The outline of this paper is as follows: Section I introduces graph theory and mathematical preliminaries. Then the network models are established. Section II presents the consensus protocols and solves the robust output regulation problem of discrete-time multi-agent systems. An example is given in Section III. Finally,concluding remarks are drawn in Section IV.

Some standard notations of matrix are used throughout this paper. ${\bf R}$ and ${\bf C}$ denote the sets of real numbers and complex numbers. The set of positive real numbers is denoted by ${\bf R}^+$. ${\bf R}^{m\times n}$ is a set of matrices with $m$ rows and $n$ columns. For matrix $A=[a_{ij}]\in {\bf R}^{n\times n},$ $A>0$ means $A$ is a positive definite matrix and the vector $vec (A)$ is defined by $vec(A)=(a_{11},\cdots,a_{n1},\cdots,a_{1n},\cdots,a_{nn})^{\rm T}.$ Given $\lambda \in {\bf C}$,${\rm Re}(\lambda)$,${\rm Im}(\lambda)$ and $\left| \lambda\right|$ are the real part,the imaginary part and the modulus of $\lambda$,respectively. $1_n\in {\bf R}^n$ is the vector with all entries being 1.

II. PRELIMINARIES AND NETWORK MODELS A. Basic Knowledge

In this section,some basic concepts and results are introduced. For more details,please refer to ^{[30, 31]}.

1)Graph theory: A weighted digraph ${\rm{g}} = (v,\varepsilon ,A)$ of order $n$ is composed of a vertex set $\varepsilon = \{ {v_1},{v_2},\cdots ,{v_n}\} $,an edge set $\mathcal{E}=\{e_{ij}=$ $(v_i,v_j)\}\in \mathcal{V}\times\mathcal{V}$ and a weighted adjacency matrix $A = [{a_{ij}}]$ with nonnegative adjacent elements $a_{ij}$. Moreover,$a_{ij}>0$ if $(v_i,v_j)\in \mathcal{E}$,$a_{ij}=0$ if $(v_i,v_j)\not\in \mathcal{E}$ and $a_{ii}=0$ for all $i$ $=$ $1$,$\cdots$,$n$. $(v_i,v_j)\in \mathcal{E}$ if and only if $i$th agent can receive information from $j$th agent directly. $\mathcal{N}_i=\{v_j\in \mathcal{V}: (v_i,v_j)$ $\in$ $\mathcal{E}\}$ is defined as the set of neighbors of node $v_i$. In association with $\mathcal{G}$,the in-degree and out-degree of node $v_i$ are defined as follows:

$$deg_{\rm in}(v_i)=\sum_{j=1}^n a_{ji},~~~deg_{\rm out}(v_i)=\sum_{j=1}^n a_{ij}.$$

The degree matrix $\mathcal{D}={\rm diag}\{d_1,d_2,\cdots,d_n\} $ is a diagonal matrix,whose diagonal elements are given as $d_i=deg_{\rm out}(v_i)$. The Laplacian matrix $\mathcal{L}=\mathcal{D}-\mathcal{A}=[l_{ij}]$ is defined by $l_{ij}=-a_{ij}$ and $l_{ii}=\sum_{j=1}^n a_{ij}$. A directed path is a sequence of edges in a directed graph of the form $(v_1,v_2)$,$(v_2,v_3)$,$\cdots$,where $v_i\in \mathcal{V}$. A directed graph contains a directed spanning tree if there exists at least one agent which is called root node that has a directed path to every other agents.

2)Matrix theory:Lemma 1^[30]. Let $A\in {\bf R}^{n\times n}$ and $\lambda_1,\lambda_2,\cdots,\lambda_n$ be the eigenvalues of $A$. Then for $c\in {\bf C}$,$\lambda_1+c$,$\lambda_2+c$,$\cdots$,$\lambda_n+c$ are the eigenvalues of $A+c I_n$.

Lemma 2. For any real symmetric matrix $Q>0$,there exists real symmetric matrix $\check{Q}>0$ such that $Q=\check{Q}\check{Q}.$

Proof. From the expression of $Q$,there is a real orthogonal matrix $U$ such that

\begin{align*} Q=U\Lambda_QU^{\rm T}=U\sqrt{\Lambda_Q}U^{\rm T}U\sqrt{\Lambda_Q}U^{\rm T}, \end{align*}

with $\lambda_{qi}>0$,$i=1,2,\cdots,N$ being the eigenvalues of $Q$ and $\Lambda_Q={\rm diag}\{\lambda_{q1},\cdots,\lambda_{qN}\},~\sqrt{\Lambda_Q}={\rm diag}\{\sqrt{\lambda_{q1}},\cdots$,$\sqrt{\lambda_{qN}}\}.$ Let $\check{Q}=U\sqrt{\Lambda_Q}U^{\rm T}$; then the proof is completed.

Lemma 3^[30](Geršgorin). Let $A=[a_{ij}]\in {\bf R}^{n\times n},$ and

\begin{align*} R_i^{'}(A)\equiv \sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|, \ \ 1\leq i\leq n, \end{align*}

and denote the deleted absolute row sums of $A$. Then all the eigenvalues of $A$ are located in the union of $n$ discs

\begin{align*} \bigcup_{i=1}^N\{z\in {\bf C}: \left|z-a_{ii} \right|\leq R_i^{'}(A) \}\equiv G(A). \end{align*} B. Network Models

Let the digraph $\bar{\mathcal{G}}=(\bar{\mathcal{V}},\bar{\mathcal{E}},\bar{\mathcal{A}})$ of order $N+1$ consist of a vertex set $\bar{\mathcal{V}}=(v_0,v_1,\cdots,v_N)$ and $v_0$ is defined as the exosystem. $a_{i0}>0 $ if agent $i$ can receive information from the exosystem and else $a_{i0}=0$. $\bar{\mathcal{L}}$ is defined as the Laplacian matrix of subgraph $\bar{\mathcal{G}}$ except node $v_0$. The adjacency matrix of digraph $\bar{\mathcal{G}}$ is obtained as

$\bar{\mathcal{A}}=\left( \begin{array}{cc} 0 & \textbf{0} \\ \mathcal{A}_0\textbf{1}_N & \bar{\mathcal{A}}_s \\ \end{array} \right),$

(1)

in which $\mathcal{A}_0={\rm diag}\{a_{10},a_{20},\cdots,a_{N0}\}$ and $\bar{\mathcal{A}}_s $ is the adjacency matrix of subgraph $\bar{\mathcal{G}}_s$ with vertex set $\bar{\mathcal{V}_s}=(v_1,v_2$,$\cdots$,$v_N)$. Then the Laplacian matrix $\bar{\mathcal{L}}$ of digraph $\bar{\mathcal{G}}$ is

$\begin{align}\label{H} \bar{\mathcal{L}}=\left( \begin{array}{cc} 0 & \textbf{0} \\ -\mathcal{A}_0\textbf{1}_N & \mathcal{A}_0+\bar{\mathcal{L}}_s \\ \end{array} \right), \end{align}$

(2)

in which $\bar{\mathcal{L}}_s=\bar{\mathcal{D}}_s-\bar{\mathcal{A}}_s$ is the Laplacian matrix of subgraph $\bar{\mathcal{G}}_s$. $\bar{\mathcal{D}}_s={\rm diag}\{\bar{d}_1,\bar{d}_2,\cdots,\bar{d}_N\}$ and $\bar{\mathcal{A}}_s$ are the degree matrix and adjacency matrix of subgraph $\bar{\mathcal{G}}_s$,respectively.

Lemma 4^[32]. All the eigenvalues of $\mathcal{H}=\mathcal{A}_0+\bar{\mathcal{L}}_s$ defined in (2) have positive real parts if and only if the digraph $\bar{\mathcal{G}}$ contains a spanning tree and node $v_0$ as its root.

A group of $N$ discrete-time linear agents with uncertain dynamics are described by

$\left\{ {\begin{array}{*{20}{l}} {x_{i}^{\dagger } = \bar A{x_i} + \bar B{u_i} + {{\bar E}_i}\omega ,}&{{x_i}(0) = {x_{i0}},}\\ {{y_i} = \bar C{x_i},}&{i = 1,2,\cdots ,N,} \end{array}} \right.$

(3)

where $x_i\in {\bf R}^n$ is the state,$y_i\in {\bf R}^p$ is the output and $x(k+1)$ is defined as $x_{i}^{\dagger }$,$u_i\in {\bf R}^q$ denotes the consensus protocol to be designed which depends on agent $i$ and its neighbors,$\bar{E}_i\omega$ denotes the external disturbance of the $i$th agent. The matrices $\bar{A}$,$\bar{B}$,$\bar{C}$ and $\bar{E}_i$ are uncertain and they could be written as follows:

$\begin{align} &\bar{A}=A+\Delta A,~~\bar{B}=B+\Delta B,\nonumber\\ &\bar{C}=C+\Delta C,~~\bar{E}_i=E_i+\Delta E_i, \end{align}$

(4)

where $\Delta A$,$\Delta B$,$\Delta C$ and $\Delta E_i$ are the perturbations of nominal matrices $A$,$B$,$C$ and $E_i$,respectively. It is convenient to identify the system uncertainties with a vector

\begin{align*} \Delta=\left( \begin{array}{c} vec(\Delta A,\Delta B,\Delta E_1,\cdots,\Delta E_N) \\ vec(\Delta C) \\ \end{array} \right) \in {\bf R}^{n(m+n+p+Nq)}. \end{align*}

System (3) with $\Delta=0$ is called a nominal system.

In addition,the reference inputs and the disturbances can be lumped together as the exosystem:

$\begin{align}\label{omega} \left\{ \begin{array}{ll} \omega^\dagger=S \omega,& {\omega(0)=\omega_0},\\ y_{r}=F\omega,& \hbox{} \end{array} \right. \end{align}$

(5)

with $\omega^\dagger \in {\bf R}^s$ denoting $\omega(k+1)$ and $y_{r}\in {\bf R}^p$ the reference signal. Thus,the consensus output regulation objectives may be prescribed in terms of the tracking errors $e_i\in {\bf R}^p$ as the following form:

$\begin{align}\label{error} e_i=y_i-y_r=\bar Cx_i-F \omega,\quad i=1,2,\cdots,N. \end{align}$

(6)

III. CONSENSUS PROTOCOL AND MAIN RESULTS

Assume that each agent can communicate with its neighbor agents and exosystem by sharing their output states. The output-coupling variable relationship between agent $i$ and its neighbors $j\in \mathcal{N}_i$ is defined as follows:

$\begin{align} \delta_i=\sum\limits_{j\in \mathcal{N}_i}a_{ij}(y_i-y_j)+a_{i0}(y_i-y_{r}),~~i=1,2,\cdots,N. \end{align}$

(7)

The internal model technique which will be employed to synthesize protocols has important advantages that the parameters of the internal model are not sensitive to the controlled systems^[33]. It is also necessary to solve the output regulation for uncertain multi-agent systems. Thus,the internal model servocompensator $z_i\in {\bf R}^{p(s_m)}$ under the network topology is built as follows:

$\begin{align}\label{z} z_i^\dagger=G_1z_i+G_2\phi_i\delta_i,~~i=1,2,\cdots,N, \end{align}$

(8)

with $\phi_i\in {\bf R}^+$,$i=1,2,\cdots,N. $ The pair of matrices $(G_1$,$G_2)$ is said to incorporate a $p$-copy internal model of matrix $S$ with

$$\begin{align*} G_1=&\ {\rm block} ~{\rm diag} \{\beta_1,\beta_2,\cdots,\beta_p\},\\ G_2=&\ {\rm block} ~{\rm diag} \{\sigma_1,\sigma_2,\cdots, \sigma_p\}, \end{align*}$$

for all $i=1,\cdots,p$,and $\beta_i$ is a constant square matrix. $\sigma_i$ is a constant column vector such that $(\beta_i,\sigma_i)$ is controllable and the minimal polynomial of $S$ divides the characteristic polynomial of $\beta_i$.

To solve the regulator problem,we give the following distributed consensus protocols:

$\begin{align}\label{protocol} u_i=K_1\phi_i\left(\sum\limits_{j\in \mathcal{N}_i}a_{ij}(x_i-x_j)+a_{i0}x_i\right)+K_2z_i, \end{align}$

(9)

with $i=1,2,\cdots,N$,$K_1\in {\bf R}^{q\times n},$ and $K_2\in {\bf R}^{q\times p(s_m)}$ being the gain matrices to be designed later.

Let $x=(x_1^{\rm T},x_2^{\rm T},\cdots,x_N^{\rm T})^{\rm T},$ $z=(z_1^{\rm T},z_2^{\rm T},\cdots,z_N^{\rm T})^{\rm T}$ and $\tilde{\omega}$ $=$ $\textbf{1}_N \otimes \omega$. Then using protocol (9),the uncertain discrete-time linear dynamics (3) and the internal model servocompensator (8) can be written as

$\begin{align} x^\dagger=&\ (I_N\otimes \bar{A}+\Phi \mathcal{H}\otimes \bar{B}K_1)x+(I_N\otimes \bar{B}K_2)z+\bar{E}\tilde{\omega},\nonumber\\ z^\dagger=&\ (\Phi \mathcal{H}\otimes G_2\bar{C})x+(I_N\otimes G_1)z-(\Phi \mathcal{H}\otimes G_2F)\tilde{\omega}, \end{align}$

(10)

where

$$\begin{align*} &\Phi={\rm diag}\{\phi_1,\phi_2,\cdots,\phi_N\},\\ &\bar{E}={\rm block} ~{\rm diag}\{\bar{E}_1,\bar{E}_2,\cdots,\bar{E}_N\},\\ &(\Phi \mathcal{A}_0\otimes G_2F)\tilde{\omega}=(\Phi \mathcal{H}\otimes G_2F)\tilde{\omega}. \end{align*}$$

Also,let $\zeta=(x^{\rm T},z^{\rm T})^{\rm T}$. Then the closed loop system can be written in a compact form as

$\begin{align}\label{zonghe} \zeta^\dagger=\bar{A}_c\zeta+\bar{E}_c\tilde{\omega}, \end{align}$

(11)

where

$$\begin{align*} &\bar{A}_c=\left( \begin{array}{cc} I_N\otimes \bar{A}+\Phi \mathcal{H}\otimes \bar{B}K_1 & I_N\otimes \bar{B}K_2 \\ \Phi \mathcal{H}\otimes G_2\bar{C} & I_N\otimes G_1 \\ \end{array} \right), \nonumber\\ &\bar{E}_c=\left( \begin{array}{c} \bar{E} \\ -\Phi \mathcal{H}\otimes G_2F \\ \end{array} \right). \end{align*}$$

Problem 1. The robust cooperative output regulation of discrete-time multi-agent systems is solved if there is the consensus protocol (9) such that

1) The nominal closed-loop system matrix $A_{c0}=\left( \begin{array}{cc} I_N\otimes A+\Phi \mathcal{H}\otimes BK_1 & I_N\otimes BK_2 \\ \Phi \mathcal{H}\otimes G_2C & I_N\otimes G_1 \\ \end{array} \right)$ is Schur.

2) There exists an open neighborhood $W$ of $\Delta=0$. For the closed-loop system (11),the local tracking error $e_i(k)\to 0$ as $k\to \infty,$ for any initial conditions $x_{i0}\in {\bf R}^n$,$\omega_0\in {\bf R}^s.$

The following basic assumptions and lemmas are necessary to solve the output regulation problem:

Assumption 1. The pair $(A,B)$ is stabilizable.

Assumption 2. $S$ has no eigenvalues in the interior of the unit circle in the $z$-plane.

Remark 1. If all the eigenvalues of $S$ lay in the interior of unit circle in the $z$-plane,the trajectories of $\omega$ will decay exponentially to zero and not affect the output regulation. Without loss of generality,we assume that all the eigenvalues lay on or out of the unit circle.

Assumption 3. For all $\lambda\in \sigma(G_1)$,

$$\begin{align*} {\rm rank}\left( \begin{array}{cc} A-\lambda I_n & B \\ C & 0 \\ \end{array} \right)=n+p. \end{align*}$$

Assumption 4. Digraph $\bar{\mathcal{G}}$ contains a spanning tree and node $v_0$ as its root.

Lemma 5^[33]. If Assumptions 1$\sim$3 hold,the pair $(G_1$,$G_2)$ incorporates a $p$-copy internal model of matrix $S$. Let

$$\begin{align*}\mathfrak{A}=\left( \begin{array}{cc} A & 0 \\ G_2C & G_1 \\ \end{array} \right),\quad \mathfrak{B}=\left( \begin{array}{c} B \\ 0 \\ \end{array} \right), \end{align*}$$

then pair $(\mathfrak{A},\mathfrak{B})$ is stabilizable.

Lemma 6^[33]. Under Assumption 2,assume $(G_1,G_2)$ incorporates a $p$-copy internal model of $S$. If the matrix equation

$$\begin{align*} \Pi_2S=G_1\Pi_2+G_2\Omega \end{align*}$$

has a solution $\Pi_2$,then $\Omega=0.$

Lemma 7. If $(\mathfrak{A},\mathfrak{B})$ is stabilizable and the gain matrix $K$ is defined as $K=-(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}$,then

$$\begin{align*} \rho=\frac{1}{\lambda_{\max}[\check{Q}^{-1}\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}\check{Q}^{-1}]}, \end{align*}$$

with $P=P^{\rm T}>0$,$Q=Q^{\rm T}>0$ and $Q=\check{Q}\check{Q} $ for symmetric matrix $\check{Q}>0$. Matrix $\mathfrak{A}+s\mathfrak{B}K$,$s\in {\bf C}$ is stable if and only if $s$ lies in the stability region

$\begin{align} \Psi=\{s\in {\bf C}: \left| s-1\right|^2<\rho \}. \end{align} $

(12)

Proof. Since $(\mathfrak{A},\mathfrak{B})$ is stabilizable,for any $Q=Q^{\rm T}>0$,the following discrete-time algebraic Riccati equation:

$\begin{align} \mathfrak{A}^{\rm T}P\mathfrak{A}-P-\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}+Q=0 \end{align}$

(13)

has a unique solution $P=P^{\rm T}>0.$

Under Lyapunov stability theorem,for $P>0$,$\mathfrak{A}+s\mathfrak{B}K$ is stable for some $K$ if and only if

$$\begin{align*} (\mathfrak{A}+s\mathfrak{B}K)^*P(\mathfrak{A}+s\mathfrak{B}K)-P<0 \end{align*}$$

Necessity. Since the gain matrix $K$ is defined as $K=-(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}$,it follows that

$\begin{align}\label{LQR} &(\mathfrak{A}+s\mathfrak{B}K)^*P(\mathfrak{A}+s\mathfrak{B}K)-P= \nonumber\\ &\qquad \mathfrak{A}^{\rm T}P\mathfrak{A}+s^*K^{\rm T}\mathfrak{B}^{\rm T}P\mathfrak{A}+s\mathfrak{A}^{\rm T}P\mathfrak{B}K+\nonumber\\ &\qquad ss^*K^{\rm T}\mathfrak{B}^{\rm T}P\mathfrak{B}K-P=\nonumber\\ &\qquad \mathfrak{A}^{\rm T}P\mathfrak{A}+(-s-s^*+ss^*)\mathfrak{A}^ {\rm T}P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}-\nonumber\\ &\qquad P=\mathfrak{A}^{\rm T}P\mathfrak{A}-\mathfrak{A}^{\rm T} P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}-P+\nonumber\\ &\qquad (1-s-s^*+ss^*)\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^ {{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}=\nonumber\\ &\qquad-Q+\left| s-1\right|^2\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}<0. \end{align}$

(14)

From Lemma 2,there exists symmetric matrix $\check{Q}$ that $Q=\check{Q}\check{Q}$. (14) is rewritten as

$$\begin{align*} -I_n+\left| s-1\right|^2\check{Q}^{-1}\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}\check{Q}^{-1}<0, \end{align*}$$

therefore,the stability region is given as

$\begin{align} \Psi=\{s\in {\bf C}: \left| s-1\right|^2<\rho \}, \end{align}$

(15)

with $\rho = \frac{1}{{{\lambda _{\max }}[\check{Q}^{-1}\mathfrak{A}^{\rm T} P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T} P\mathfrak{A}\check{Q}^{-1}]}}$.

Sufficiency. When $s$ lies in the region $\Psi$,and if $\lambda_{ri}$,$i= 1$,2,$\cdots,n$ are the eigenvalues of $\check{Q}^{-1}\mathfrak{A}^{\rm T} P\mathfrak{B}(\mathfrak{B}^{{\rm T}}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}$ $\times$ $P\mathfrak{A}\check{Q}^{-1}$,there has $\frac{1}{\left| s-1\right|^2}>\frac{1}{\rho}\geq$ $ \lambda_{ri}$,i.e.,

$ \begin{align}\label{eig} \lambda_{ri}\left|s-1\right|^2-1<0^{\rm T}. \end{align}$

(16)

Under Lemma 1,$\lambda_{ri}\left|s-1\right|^2-1$ are the eigenvalues of

$$\begin{align*}-I_n+\left|s-1\right|^2\check{Q}^{-1}\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}\check{Q}^{-1} \end{align*}$$

and (16) implies that

$\begin{align}\label{cc1} -I_n+\left|s-1\right|^2\check{Q}^{-1}\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}\check{Q}^{-1}<0. \end{align}$

(17)

Pre-and post-multiplying (17) by $\check{Q}$,yields

$$\begin{align*} &-Q+\left|s-1\right|^2\mathfrak{A}^{\rm T}P\mathfrak{B}(\mathfrak{B}^{T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A}\\\ &\qquad (\mathfrak{A}+s\mathfrak{B}K)^*P(\mathfrak{A}+s\mathfrak{B}K)-P<0. \end{align*}$$

Then $\mathfrak{A}+s\mathfrak{B}K$ is stable.

Theorem 1. Under Assumptions 1$\sim$4,if $\phi_i=\frac{1}{a_{i0}+\bar{d}_i},$ $ K$ $=$ $\left( \begin{array}{cc} K_1 & K_2 \\ \end{array} \right)=-(\mathfrak{B}^{\rm T}P\mathfrak{B})^{-1}\mathfrak{B}^{\rm T}P\mathfrak{A} $ and $\sigma(\Phi \mathcal{H})\subset\Psi$,then the consensus robust output regulation problem can be solved by the distributed consensus protocol (9).

Proof. Let $T_1=\left( \begin{array}{cc} I_{Nn} & 0 \\ 0 & (\Phi\mathcal{H})\otimes I_{p(s_m)} \\ \end{array} \right).$ The nominal closed-loop system matrix $A_{c0}$ can be transformed into

$\begin{align} \Lambda_1=&\ T_1^{-1}A_{c0}T_1=\nonumber\\ &\left( \begin{array}{cc} I_N\otimes A+\Phi \mathcal{H}\otimes BK_1 & \Phi\mathcal{H}\otimes BK_2 \\ I_N\otimes G_2C & I_N\otimes G_1 \\ \end{array} \right)=\nonumber\\ &\ I_N\otimes\mathfrak{A}+\Phi \mathcal{H}\otimes\mathfrak{B}K, \end{align}$

(18)

with $K=\left( \begin{array}{cc} K_1 & K_2 \\ \end{array} \right). $ Denote the eigenvalues of $\Phi H$ by $\lambda_i$,$i=1,\cdots,N$. There exists a unitary matrix $U$ such that $U^{-1}\Phi \mathcal{H}U=\mathcal{T}=[t_{ij}]$ is upper triangular,with diagonal entries $t_{ii}=\lambda_i$,$i=1,\cdots,N$. Let $T_2$ $=$ $\left( \begin{array}{cc} U\otimes I_n & 0 \\ 0 & U\otimes I_{p(s_m)} \\ \end{array} \right). $ Then

$\begin{align} \Lambda_2=&\ T^{-1}_2\Lambda_1T_2=\nonumber\\ &\left( \begin{array}{cc} I_N\otimes A+\mathcal{T}\otimes BK_1 & \mathcal{T}\otimes BK_2 \\ I_N\otimes G_2C & I_N\otimes G_1 \\ \end{array} \right)=\nonumber\\ &\ I_N\otimes \mathfrak{A}+\mathcal{T}\otimes \mathfrak{B}K \end{align}$

(19)

Because the elements of the transformed system matrix $\Lambda_2$ are either block diagonal or block upper-triangular,$\Lambda_2$ is Schur if and only if $\mathfrak{A}+\lambda_i\mathfrak{B}K$,$i=1,\cdots,N$ is Schur.

By the Ger\v{s}gorin disc theorem,all the eigenvalues $\lambda_i$,$i=$ $1$,$\cdots,N$ of $\Phi H$ are located in the union of $N$ discs

$\begin{align}\label{disc} \bigcup_{i=1}^N\left\{z\in {\bf C}: \left|z-\phi_i(\bar{d}_i+a_{i0}) \right|\leq\phi_i\bar{d}_i\right\}, \end{align}$

(20)

If $\phi_i=\frac{1}{a_{i0}+\bar{d}_i},$ then (20) is rewritten as

$\begin{align}\label{disc b} \bigcup_{i=1}^N\left\{z\in {\bf C}: \left|z-1 \right|\leq \frac{\bar{d}_i}{a_{i0}+\bar{d}_i}\right\}, \end{align}$

(21)

which represents that the eigenvalues of $\Phi H$ span in the neighbouring of 1. By Lemmas 5 and 7,the stability region of $\mathfrak{A}+s\mathfrak{B}K$ is

$\begin{align} \Psi=\left\{s\in {\bf C}: \left| s-1\right|^2<\rho \right\}, \end{align}$

(22)

If all the eigenvalues $\lambda_i$,$i=1,\cdots,N$ are located in $\Psi$,i.e.,$\sigma(\Phi H)\subset \Psi$,then $\mathfrak{A}+\lambda_i\mathfrak{B}K$,$i=1,\cdots,N$ is Schur. Therefore,the nominal closed-loop system matrix $A_{c0}$ is Schur.

For each $\Delta\in W,$ where $W$ is an open neighborhood of $\Delta=0$ such that $\bar{A}_{c}$ is Schur and under Assumption 2,$\lambda_i(S)+\lambda_j(\bar{A}_{c})\neq0$,$i=1,\cdots,s$,$j= 1,\cdots$,$N(n$ $+$ $p(s_m)).$ Then there exists a unique solution $\Pi$ $\in$ ${\bf R}^{N(n+p(s_m))\times N(n+p(s_m))}$ of Sylvester equation

$\begin{align}\label{sylvester} \Pi S=\bar{A}_c\Pi+\bar{E}_c(1_N\otimes I_s). \end{align}$

(23)

Let $\Pi=\left( \begin{array}{c} \Pi_1 \\ \Pi_2 \\ \end{array} \right),$ with $\Pi_1\in {\bf R}^{Nn\times Nn}$,$\Pi_2\in {\bf R}^{Np(s_m)\times Np(s_m)}.$ Equation (23) has the following form:

$\begin{align} \Pi_1 S=&\ (I_N\otimes \bar{A}+\Phi H\otimes \bar{B}K_1)\Pi_1+\nonumber\\ &\ (I_N\otimes BK_2)\Pi_2+\bar{E}(1_N\otimes I_s),\nonumber\\ \Pi_2 S=&\ (\Phi H\otimes G_2\bar{C})\Pi_1+(I_N\otimes G_1)\Pi_2-\nonumber\\ &\ (\Phi H\otimes G_2F)(1_N\otimes I_s)=\nonumber\\ &\ (I_N\otimes G_2)(\Phi H\otimes I_q)((I_N\otimes \bar{C})\Pi_1-1_N\otimes F)+\nonumber\\ &\ (I_N\otimes G_1)\Pi_2. \end{align}$

(24)

Since $(I_N\otimes G_1,I_N\otimes G_2)$ incorporates a $pN$-copy internal model of $S$,under Lemma 6 and by the invertibility of $\Phi H\otimes I_q$,one gets

$\begin{align}\label{eq00} (I_N\otimes \bar{C})\Pi_1-1_N\otimes F=0. \end{align}$

(25)

Leting $\hat{\zeta}=\zeta-\Pi\omega$ and calculating the difference of $\hat{\zeta}$ yields

$\begin{align} \hat{\zeta}^\dagger&=\bar{A}_c\zeta+\bar{E}_c(1_N\otimes I_s)\omega-\Pi S\omega=\bar{A}_c\hat{\zeta} \end{align}$

(26)

Let $e=(e_1^{\rm T},e_2^{\rm T},\cdots,e_N^{\rm T})^{\rm T}.$ The tracking errors (6) have the following form:

$$\begin{align*} e=& \left(I_N\otimes \bar C\right)x-(1_N\otimes F)\omega=\\ &\left( \begin{array}{cc} I_N\otimes \bar C & 0 \\ \end{array} \right)\zeta-(1_N\otimes F)\omega=\\ &\left( \begin{array}{cc} I_N\otimes \bar C & 0 \\ \end{array} \right)(\hat{\zeta}+\Pi\omega)-(1_N\otimes F)\omega=\\ &\left( \begin{array}{cc} I_N\otimes \bar C & 0 \\ \end{array} \right)\hat{\zeta}+((I_N\otimes \bar{C})\Pi_1-1_N\otimes F)\omega. \end{align*}$$

From (25),one gets

$$ e=\left( \begin{array}{cc} I_N\otimes \bar C & 0 \\ \end{array} \right)\hat{\zeta}. $$

Since $\bar{A}_c$ is Schur,$\hat{\zeta}\rightarrow 0$ $(t\rightarrow\infty).$ Thus,we get

$$\begin{align*} \displaystyle\lim_{k\rightarrow\infty}e=\displaystyle\lim_{k\rightarrow\infty}\left( \begin{array}{cc} I_N\otimes \bar C & 0 \\ \end{array} \right)\hat{\zeta}=0. \end{align*}$$

That is,the consensus robust output regulation problem is solved.

Remark 2. For the system dynamics (3),if $\Delta=0,$ robust output regulation problem for uncertain discrete-time multi-agent systems becomes output regulation problem for certain systems.

Remark 3. The robust output regulation for linear multi-agent systems has been deeply studied in ^[27] and a common gain matrix $K$ could be found to ensure $N$ nominal closed-loop systems stability. We can determine the gain matrix only based on continuous-time algebraic Riccati equation for any system matrix and any information graph. But for discrete-time multi-agent systems,it is hard to find a gain $K$ to make the closed-loop system matrix in the form of $\mathfrak{A}+\mathcal{H}\otimes \mathfrak{B}K$ stable without considering the relationship between system matrix $\mathfrak{A}$ and Laplacian matrix $\mathcal{H}$ of the information graph. Theorem 3.2 in ^[34] gave the relationship between them that can make the closed-loop system matrix stable. In our paper,parameters $\phi_i$ are used to regulate all the eigenvalues of $\mathcal{H}$ to lie in the neighbor of 1,which can reduce the conservatism.

IV. AN EXAMPLE

In this section,an example is given to validate the effectiveness of the theoretical result. Consider a network of discrete-time multi-agent systems described by (3) with

$$%\begin{eqnarray*} \begin{align*}&x_{i}=(x_{i1},x_{i2})^{\rm T},~~i=1,\cdots,7,\\[2mm] &A=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right),~~B=\left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right),~~C=\left( \begin{array}{cc} 1 & 0 \\ \end{array} \right),\\ & \Delta A=\left( \begin{array}{cc} 0 & 0.1 \\ 0.1 & 0 \\ \end{array} \right),\Delta B=\left( \begin{array}{c} 0 \\ 0.1 \\ \end{array} \right),\Delta C=\left( \begin{array}{cc} 0 & 0.2 \\ \end{array} \right),\\[3mm] &E_i=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & i \\ \end{array} \right),~~i=1,2,3,\\ &E_i=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right),~~i=4,5,6,7\\ &\Delta E_i=\left( \begin{array}{ccc} 0 & 0 & 0.1i \\ 0 & 0 & 0 \\ \end{array} \right),~~i=1,\cdots,7.\end{align*}$$

The exosystem is in the form of (5) with $\omega=$ $(\omega_1$,$\omega_2$,$\omega_3)^{\rm T}$ and

$$\begin{align*}S=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 &-1 \\ 0 & 1 & 0 \\ \end{array} \right),~~F=\left( \begin{array}{ccc} 1 & 0 & 0 \\ \end{array} \right). \end{align*}$$

By the exosystem matrix $S$,we could find a pair of matrices

$$\begin{align*}G_1=\left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 &-1 \\ 0 & 0 & 0 \\ \end{array} \right),~~G_2=\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ \end{array} \right), \end{align*}$$

which incorporate a $1$-copy internal model of $S$. It is easy to check that Assumptions 1$\sim$3 hold.

The system topology is described by the digraph showed in Fig. 1 with $\mathcal{\bar{V}}=(v_0,v_1,\cdots,v_7)$ and the adjacent weighted values are also showed in Fig. 1. Obviously,digraph $\mathcal{\bar{G}}$ contains a spanning tree and node 0 as its root. The parametric matrix $\Phi={\rm block}~{\rm diag}\{1,1,1,1/4,1/2,1,1/4\}.$ By some simple calculations,the stability region is shown as

$$\begin{align*} \Psi=\{s\in {\bf C}: \left| s-1\right|^2<0.2981 \}. \end{align*}$$

The control gain $K_1=\left( \begin{array}{cc}-0.4514 & 0 \\ \end{array} \right),$ and $K_2=\left( \begin{array}{ccc} 0 &-0.5486 & 0 \\ \end{array} \right)$ which can make $\bar{A}_c$ stable in an open neighborhood of $\Delta=0.$ Thus,we obtain the simulation results as shown in Fig. 2. The regulated errors converge to the origin asymptotically.

V. CONCLUSIONS

This paper has investigated the design of distributed consensus protocol for the consensus robust output regulation problem for discrete-time multi-agent systems. A directed graph has been used to describe the information exchange among agents. We assume that the digraph should contain a spanning tree so that all the nodes could communicate with their neighbors and there are no isolated nodes. The internal model method is also used to solve the robust regulation problem. We have also provided an example to illustrate the result.