The Cooperative Output Regulation Problem of Discrete-Time Linear Multi-Agent Systems by the Adaptive Distributed Observer

TL;DR

This paper introduces an adaptive distributed observer for discrete-time linear multi-agent systems, enabling followers to estimate leader signals and system matrices, and proposes control laws to solve the cooperative output regulation problem.

Contribution

It develops a novel adaptive distributed observer that estimates both leader signals and system matrices, facilitating a new solution to the cooperative output regulation problem.

Findings

Successful estimation of leader signals and system matrices by followers.

Effective discrete adaptive algorithms for regulator equations.

Achievement of cooperative output regulation using proposed control laws.

Abstract

In this paper, we first present an adaptive distributed observer for a discrete-time leader system. This adaptive distributed observer will provide, to each follower, not only the estimation of the leader's signal, but also the estimation of the leader's system matrix. Then, based on the estimation of the matrix S, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower, and obtain an estimated feedforward control gain. Finally, we solve the cooperative output regulation problem for discrete-time linear multi-agent systems by both state feedback and output feedback adaptive distributed control laws utilizing the adaptive distributed observer.

Equations

$$\dot{v}=Sv$$

$$v(t+1)=Sv(t).$$

$$\hbox{vec}(A)=\left[\begin{array}[]{c}A_{1}\\ \vdots\\ A_{n}\\ \end{array}\right]$$

$${M^{q}_{n}}(X)=\left[\begin{array}[]{ccc}X_{1}\cdots X_{q}\\ \end{array}\right]$$

$$x_i(t+1)$$

$$e_i(t)$$

$$y_{mi}(t)$$

$$\begin{split}u_{i}(t)=&k_{i}(z_{i}(t))\\ z_{i}(t+1)=&g_{i}\big{(}z_{i}(t),y_{mi}(t),z_{j}(t),y_{mj}(t),j\in\bar{\mathcal{N}}_{i}\big{)}\\ i&=1,\cdots,N\end{split}$$

$$\underset{t\to \infty }{\lim }{e}_i(t)=0,i=1,\dots ,N.$$

$$X_{i}S$$

$$0$$

$$\displaystyle\mbox{\rm rank}\left[\begin{array}[]{cc}A_{i}-\lambda I_{n_{i}}&B_{i}\\ C_{i}&D_{i}\end{array}\right]=n_{i}+m_{i},~{}~{}\forall~{}~{}\lambda\in\sigma(S).$$

$${\eta }_i(t+1)=S{\eta }_i(t)+\mu S\sum _{j=0}^N{a}_{ij}({\eta }_j(t)-{\eta }_i(t)),i=1,\dots ,N$$

$$\begin{split}\tilde{\eta}(t+1)&=\big{(}(I_{N}\otimes S)-\mu(H\otimes S)\big{)}\tilde{\eta}(t)\\ \end{split}$$

$$\underset{t\to \infty }{\lim }({\eta }_i(t)-v(t))=0.$$

$$\displaystyle\max_{l=1,\cdots,N}\bigg{\{}\frac{a_{l}-\sqrt{\Delta_{l}}}{a_{l}^{2}+b_{l}^{2}}\bigg{\}}<\mu<\min_{l=1,\cdots,N}\bigg{\{}\frac{a_{l}+\sqrt{\Delta_{l}}}{a_{l}^{2}+b_{l}^{2}}\bigg{\}}$$

$$\displaystyle 0<\mu<\min_{l=1,\cdots,N}\bigg{\{}\frac{2a_{l}}{a_{l}^{2}+b_{l}^{2}}\bigg{\}}.$$

$$S_i(t+1)$$

$${\eta }_i(t+1)$$

$$x(t+1)=Fx(t)+F_1(t)x(t)+F_2(t)$$

$$x(t+1)=Fx(t)+F_1(t)x(t).$$

$$V(t+1)-V(t)$$

$$\underset{t\to \infty }{\mathrm{limsup}}||x(t)||\le \gamma (\underset{t\to \infty }{\mathrm{limsup}}||F_1(t)x(t)+F_2(t)||).$$

$$\underset{t\to \infty }{\lim }{\tilde{S}}_i(t)=0$$

$$\underset{t\to \infty }{\lim }{\tilde{\eta }}_i(t)=0$$

$$\tilde{S}(t+1)=(I_{Nq}-{\mu }_1(H\otimes I_q))\tilde{S}(t)$$

$${\tilde{\eta }}_i(t+1)$$

$$=S{\tilde{\eta }}_i+{\tilde{S}}_i{\eta }_i+{\mu }_2{S}_i\sum _{j=0}^N{a}_{ij}({\tilde{\eta }}_j-{\tilde{\eta }}_i)$$

$$=S{\tilde{\eta }}_i+{\tilde{S}}_{i}v+{\tilde{S}}_i{\tilde{\eta }}_i+{\mu }_2{S}_i\sum _{j=0}^N{a}_{ij}({\tilde{\eta }}_j-{\tilde{\eta }}_i)$$

$$=S{\tilde{\eta }}_i+{\mu }_{2}S\sum _{j=0}^N{a}_{ij}({\tilde{\eta }}_j-{\tilde{\eta }}_i)$$

$$+{\tilde{S}}_{i}v+{\tilde{S}}_i{\tilde{\eta }}_i+{\mu }_2{\tilde{S}}_i\sum _{j=0}^N{a}_{ij}({\tilde{\eta }}_j-{\tilde{\eta }}_i).$$

$$\tilde{\eta }(t+1)=((I_N\otimes S)-{\mu }_2(H\otimes S))\tilde{\eta }+{\tilde{S}}_d(1_N\otimes v)$$

$$\displaystyle+\left(\tilde{S}_{d}-\mu_{2}\left[\begin{array}[]{c}H_{1}\otimes\tilde{S}_{1}\\ \vdots\\ H_{N}\otimes\tilde{S}_{N}\end{array}\right]\right)\tilde{\eta}$$

$$\underset{t\to \infty }{\lim }{\tilde{\eta }}_i(t)=0$$

$$Ax=b$$

Full text

I Introduction

The cooperative output regulation problem for continuous-time linear multi-agent systems using distributed observer approach was first studied in [16]. This problem aims to design a distributed control law for a multi-agent system to achieve asymptotic tracking of a class of reference inputs and rejection of a class of disturbances. Both reference inputs and disturbances are generated by a leader system. The problem is an extension of the classical output regulation problem [4, 6, 7] from a single plant to a multi-agent system. On the other hand, the problem can also be viewed as a generalization of the leader-following consensus problem [5, 9, 12, 14] because its objectives include not only asymptotic tracking but also disturbance rejection.

The core of the approach in [16] is the design of a distributed observer for the leader system of the following form,

[TABLE]

where $v\in{\mathbb{R}}^{q}$ is the state of the leader system representing the reference input to be tracked and/or the external disturbance to be rejected and $S\in{\mathbb{R}}^{q\times q}$ is a constant matrix. The distributed observer is capable of producing the estimation of the leader's signal to each follower so that a distributed control law can be synthesized to solve the problem. However, a drawback associated with the distributed observer approach is that each follower needs to know the information of the matrix $S$. To remove this assumption, recently, an adaptive distributed observer for the leader system (1) was proposed in [1].

The adaptive distributed observer not only estimates the leader's signal but also estimates the leader's dynamics. Thus, it does not require every follower know the matrix $S$. In this paper, we will first propose a discrete counterpart of the adaptive distributed observer in [1] for a discrete leader system of the following form

[TABLE]

Then, we will further develop an adaptive scheme to solve the cooperative output regulation problem for discrete linear multi-agent systems utilizing the discrete adaptive distributed observer. Technically, we offer three specific contributions. Firstly, we establish a stability result for a class of time-varying discrete-time systems that lends itself to the existence conditions of the adaptive distributed observer of the leader system (2). Secondly, based on the estimation of the matrix $S$, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower of the discrete linear multi-agent system, and obtain an estimated feedforward control gain. Finally, we design both state feedback and output feedback adaptive distributed control laws utilizing the discrete adaptive distributed observer to solve the cooperative output regulation problem for the discrete-time linear multi-agent system.

Notation. $\otimes$ denotes the Kronecker product of matrices. $||x||$ denotes the Euclidean norm of a vector $x\in{\mathbb{R}}^{n}$. $\mathbb{Z^{+}}$ denotes the set of nonnegative integers. Let $x:\mathbb{Z^{+}}\rightarrow{\mathbb{R}}^{n}$. Then we often denote $x(t)$, $t\in\mathbb{Z^{+}}$ by a shorthand notation $x$ when no confusion will occur. $\sigma(A)$ denotes the spectrum of $A$ and $\rho(A)=\max_{\lambda\in\sigma(A)}\{\frac{|\lambda|^{2}}{Re(\lambda)}\}$. $\mathbf{1}_{N}$ denotes an $N\times 1$ column vector whose elements are all $1$. For $X_{i}\in{\mathbb{R}}^{n_{i}\times p}$, $i=1,\dots,m$, col$(X_{1},\dots,X_{m})=[X_{1}^{T},\dots,X_{m}^{T}]^{T}$. For any matrix $A\in{\mathbb{R}}^{m\times n}$,

[TABLE]

where $A_{i}$ is the $i$th column of $A$. For any column vector $X\in{\mathbb{R}}^{nq}$ for some positive integers $n$ and $q$,

[TABLE]

where, for $i=1,\dots,q$, $X_{i}\in{\mathbb{R}}^{n}$, and are such that $X=\mbox{col}(X_{1},\dots,X_{q})$.

II Problem Formulation

Consider the following discrete-time linear multi-agent system:

[TABLE]

where $x_{i}\in{\mathbb{R}}^{n_{i}}$, $u_{i}\in{\mathbb{R}}^{m_{i}}$, $e_{i}\in{\mathbb{R}}^{m_{i}}$, $y_{mi}\in{\mathbb{R}}^{p_{mi}}$ are the state, control input, regulated output and measurement output of the $i$th subsystem, respectively.

Like in [16], we treat the system composed of (2) and (5) as a multi-agent system of $(N+1)$ agents with (2) as the leader and the $N$ subsystems of (5) as $N$ followers, and define a graph111See [17] or [8] for a summary of digraph. $\bar{\mathcal{G}}=(\bar{\mathcal{V}},\bar{\mathcal{E}})$ with $\bar{\mathcal{V}}=\{0,1,\dots,N\}$ and $\bar{\mathcal{E}}\subseteq\bar{\mathcal{V}}\times\bar{\mathcal{V}}$. Here the node [math] is associated with the leader system (2) and the node $i$, $i=1,\dots,N$, is associated with the $i$th subsystem of the follower system (5). For $i=0,1,\dots,N$, $j=1,\dots,N$, $(i,j)\in\bar{\mathcal{E}}$ if and only if agent $j$ can use the state or output of agent $i$ for control. Let $\bar{\mathcal{N}}_{i}$ denote the neighbor set of the node $i$ of $\bar{\mathcal{G}}$.

We will consider the output feedback control law of the following form:

[TABLE]

where $y_{m0}=v$, $z_{i}\in{\mathbb{R}}^{l_{i}}$ for some integer $l_{i}$, $k_{i}$ and $g_{i}$ are linear functions of their arguments whose specific form will be given in Section IV. It can be seen that, for each $i=1,\cdots,N$, $j=0,1,\cdots,N$, $u_{i}$ of (6) depends on $y_{mj}$ only if the agent $j$ is a neighbor of the agent $i$. Thus, the control law (6) is a distributed control law. The control law (6) contains the state feedback control law as a special case when $y_{mi}=x_{i}$, $i=1,\cdots,N$.

We now define the adaptive cooperative output regulation problem for (2) and (5) as follows.

Problem 1

Given systems (2), (5) and a graph $\bar{\mathcal{G}}$, design a distributed control law of the form (6) such that the trajectory of the closed-loop system starting from any initial state exists for all $t\geq 0$ and satisfies

• •

when $v$ is bounded, the trajectory of the closed-loop system is bounded for all $t\geq 0$;

• •

the regulated output satisfies

[TABLE]

We need the following assumptions for the solvability of Problem 1.

Assumption 1

All the eigenvalues of $S$ have modulus smaller than or equal to $1$.

Assumption 2

For $i=1,\dots,N$, $(A_{i},B_{i})$ are stabilizable.

Assumption 3

For $i=1,\dots,N$, $(C_{mi},A_{i})$ are detectable.

Assumption 4

For $i=1,\dots,N$, the following linear matrix equations

[TABLE]

have unique solution pairs $(X_{i},U_{i})$.

Assumption 5

The graph $\bar{\mathcal{G}}$ contains a spanning tree with the node [math] as the root.

Remark 1

A large class of signals such as the step function, ramp function, and sinusoidal function satisfies Assumption 1.

In the classical linear output regulation problem, equations (8) are called the regulator equations whose solvability imposes a necessary condition for the solvability of the output regulation problem. By Theorem 1.9 of [10], for any matrices $E_{i}$ and $F_{i}$, the regulator equations (8) are solvable if and only if

[TABLE]

Assumption 5 is a standard assumption in the literature of the cooperative control of multi-agent systems subject to static networks.

III Adaptive Distributed Observer

The distributed observer for a discrete leader system of the form (2) was proposed in [17] and takes the following form:

[TABLE]

where $\eta_{0}=v$, $\eta_{i}\in{\mathbb{R}}^{q}$, $i=1,\dots,N$, $\mu>0$, and $\bar{\mathcal{A}}=[a_{ij}]_{i,j=0}^{N}$ denote the weighted adjacency matrix of $\bar{\mathcal{G}}$. Let $\eta(t)=\mbox{col}(\eta_{1}(t),\cdots,\eta_{N}(t))$, $\hat{v}(t)=\mathbf{1}_{N}\otimes v(t)$, and $\tilde{\eta}(t)=\eta(t)-\hat{v}(t)$. Then (12) can be put in the following form:

[TABLE]

where $H=[h_{ij}]\in{\mathbb{R}}^{N\times N}$ with $h_{ii}=\sum_{j=0}^{N}a_{ij}$ and $h_{ij}=-a_{ij}$, for any $i\neq j$. By Lemma 1 of [9], under Assumption 5, all the eigenvalues of $H$ have positive real parts.

If there exists some $\mu$ such that the matrix \big{(}(I_{N}\otimes S)-\mu(H\otimes S)\big{)} is Schur, then, for any $v(0)$, and $\eta_{i}(0)$, $i=1,\dots,N$, we have

[TABLE]

Thus, we call the system (12) a distributed observer of the leader system if and only if the system (13) is asymptotically stable. A detailed discussion on the stability of the system (13) is summarized in Lemma 3.1 of [17]. In particular, denote the eigenvalues of $S$ by $\{\lambda_{1},\cdots,\lambda_{q}\}$ where $0\leq|\lambda_{1}|\leq\cdots\leq|\lambda_{q}|$, and the eigenvalues of $H$ by $\{a_{l}\pm jb_{l}\}$, where $b_{l}=0$ when $1\leq l\leq N_{1}$ with $0\leq N_{1}\leq N$ and $b_{l}\neq 0$ when $(N_{1}+1)\leq l\leq N_{2}$ where $N_{1}+2(N_{2}-N_{1})=N$. Then, under Assumptions 1 and 5, the matrix $((I_{N}\otimes S-\mu(H\otimes S))$ is Schur for all $\mu$ satisfying

[TABLE]

where $\Delta_{l}=\frac{(a_{l}^{2}+b_{l}^{2})}{|\lambda_{l}|^{2}}-b^{2}_{l}$. If $|\lambda_{l}|=1$ for all $l=1,\cdots,q$, then $\Delta_{l}=a_{l}^{2}$ and (15) reduces to

[TABLE]

However, in (12), the matrix $S$ is used by every follower, which may not be realistic in some applications. Here, we will further propose the following so-called adaptive distributed observer candidate:

[TABLE]

where $S_{0}=S$, $S_{i}\in{\mathbb{R}}^{q\times q}$, $i=1,\dots,N$, $\mu_{1},\mu_{2}>0$. If there exist some $\mu_{1},\mu_{2}>0$ such that, for $i=1,\dots,N$, for any initial condition, the solution to (17) satisfies $\lim_{t\rightarrow\infty}(S_{i}(t)-S)=0$, $\lim_{t\rightarrow\infty}({\eta}_{i}(t)-v(t))=0$, then (17) is called the adaptive distributed observer for the leader system. It can be seen from (17) that only those followers who are the children of the leader know the matrix $S$. Thus, the adaptive distributed observer is more realistic than the distributed observer.

To find the conditions under which (17) is an adaptive distributed observer for the leader system, we first establish the following lemma.

Lemma 1

Consider the following system

[TABLE]

where $x\in{\mathbb{R}}^{n}$, $F\in{\mathbb{R}}^{n\times n}$ is Schur, $F_{1}(t)$ and $F_{2}(t)$ are well defined for all $t\in\mathbb{Z^{+}}$. If $F_{1}(t),F_{2}(t)\rightarrow 0$ (exponentially) as $t\rightarrow\infty$, then, for any $x(0)\in{\mathbb{R}}^{n}$, $x(t)\rightarrow 0$ (exponentially) as $t\rightarrow\infty$.

Proof: If $F_{2}(t)=0$, then (18) reduces to

[TABLE]

Since $F$ is Schur, for any symmetric and positive definite matrix $N\in{\mathbb{R}}^{n\times n}$, there exist a symmetric positive definite matrix $M\in{\mathbb{R}}^{n\times n}$ such that $F^{T}MF-M=-N$. Let $V(t)=x^{T}(t)Mx(t)$. Then, along the trajectory of (19)

[TABLE]

Since $F_{1}(t)\rightarrow 0$ as $t\rightarrow\infty$, there exist $T\in\mathbb{Z^{+}}$ and $\lambda>0$ such that $x^{T}(t)(N-2F^{T}MF_{1}(t)-F^{T}_{1}(t)MF_{1}(t))x(t)>\lambda||x(t)||^{2}$ for all $t\geq T$. Thus, the system (19) is exponentially stable. Therefore, for any initial condition, the solution to (18) is bounded for all $t\in\mathbb{Z^{+}}$. Since $F$ is Schur, system (18) is input-to-state stable with $F_{1}(t)x+F_{2}(t)$ as input. Thus, by Lemma 3.8 of [13], the system (18) has ${\cal K}$ asymptotic gain property, i.e., there exists a class $\mathcal{K}$ function $\gamma$ such that, for any $x(0)\in{\mathbb{R}}^{n}$, the solution $x(t)$ of (18) satisfies

[TABLE]

Since $x(t)$ is bounded, if $F_{1}(t),F_{2}(t)$ tend to zero (exponentially), so does $x(t)$. $\Box$

We now establish the main result of this section.

Lemma 2

*Given the systems (2) and (17), let $\tilde{S}_{i}=S_{i}-S$, $\tilde{\eta}_{i}={\eta}_{i}-v$. Then, for any $S_{i}(0)$ and $\eta_{i}(0)$, we have

(i) Under Assumption 5, for any $\mu_{1}$ satisfying $0<\mu_{1}<\frac{2}{\rho(H)}$, for $i=1,\dots,N$,*

[TABLE]

*exponentially, and

(ii) Under Assumptions 1 and 5, let $\mu_{1}$ satisfy $0<\mu_{1}<\frac{2}{\rho(H)}$ and let $\mu_{2}$ be such that the matrix $(I_{N}\otimes S)-\mu_{2}(H\otimes S)$ is Schur. Then, for $i=1,\dots,N$, for any $\tilde{\eta}_{i}(0)$,*

[TABLE]

exponentially.

Proof: Part (i). Let $\tilde{S}=\mbox{col}({\tilde{S}}_{1},\dots,{\tilde{S}}_{N})$. Then (17a) can be put in the following form

[TABLE]

Under Assumption 5, by Lemma 1 of [9], all the eigenvalues of $H$ have positive real parts. Thus, for any $\mu_{1}$ satisfying $0<\mu_{1}<\frac{2}{\rho(H)}$, the matrix $((I_{Nq}-\mu_{1}(H\otimes I_{q}))$ is Schur. Therefore, $\lim_{t\rightarrow\infty}\tilde{S}(t)=0$ exponentially, that is, for $i=1,\dots,N$, $\lim_{t\rightarrow\infty}\tilde{S}_{i}(t)=0$ exponentially.

Part (ii). By (17), we have

[TABLE]

Let $\tilde{\eta}=\mbox{col}(\tilde{\eta}_{1},\dots,\tilde{\eta}_{N})$ and $\tilde{S}_{d}=\hbox{block diag}\{\tilde{S}_{1},\dots,\tilde{S}_{N}\}$. Then, (23) can be rewritten in the following compact form

[TABLE]

where, for $i=1,\cdots,N$, $H_{i}$ is the $i{th}$ row of $H$. By assumption, the matrix $((I_{N}\otimes S)-\mu_{2}(H\otimes S))$ is Schur. By Part (i) of this Lemma, $\lim_{t\rightarrow\infty}\tilde{S}(t)=0$ exponentially, Thus, under Assumption 1, $\tilde{S}_{d}(t)(\mathbf{1}_{N}\otimes v(t))$ will decay to zero exponentially, too. It follows from Lemma 1 that, for any $\tilde{\eta}_{i}(0)$,

[TABLE]

exponentially and the proof is completed. $\Box$

IV Main Result

When the matrix $S$ is known by every follower, a control law utilizing the solution to the regulator equations has been designed in [17] for solving our problem. Since, in this paper, we assume those followers which are not the children of the leader do not know $S$, we cannot directly use the solution to the regulator equations to design our control law. We will propose to adaptively calculate the solution to the regulator equations based on the estimation $S_{i}$ of $S$. For this purpose, we need to establish the following lemma.

Lemma 3

Consider the following linear algebraic equation:

[TABLE]

where $x\in{\mathbb{R}}^{n}$, $A\in{\mathbb{R}}^{n\times n}$ is nonsingular, and $b\in{\mathbb{R}}^{n}$. Let $B(t)\in{\mathbb{R}}^{n\times n}$ be well defined for all $t\in\mathbb{Z^{+}}$ such that $\tilde{A}(t)\triangleq B(t)-A\rightarrow 0$ exponentially as $t\rightarrow\infty$. Then, for any $x(0)\in{\mathbb{R}}^{n}$ and $0<\varepsilon<\frac{2}{\rho(A^{T}A)}$, the solution $x(t)$ to the following system

[TABLE]

is such that

[TABLE]

exponentially.

Proof:

[TABLE]

where

[TABLE]

Let $\bar{x}={x}-{x}^{*}$ where $x^{*}=A^{-1}b$. Then, from (29), we have

[TABLE]

Since $\lim_{t\rightarrow\infty}\tilde{A}(t)=0$ exponentially, both $F(t)$ and $d(t)$ will decay to zero exponentially. Also, since our choice of $\varepsilon$ is such that $(I_{n}-\varepsilon A^{T}A)$ is Schur, by Lemma 1, for any $x(0)\in{\mathbb{R}}^{n}$,

[TABLE]

exponentially. Therefore, the proof is completed. $\Box$

Remark 2

Using a differential equation to solve a linear algebraic equation of the form (26) was studied in [3] for the case where $A$ is known, and was studied in [1] for the case where $A$ is unknown. Lemma 3 here further shows how to solve (26) with $A$ unknown by the difference equation (27).

Using the information of $S_{i}$, an adaptive algorithm was proposed in [1] to calculate the solution to the regulator equations. This algorithm is governed by a set of nonlinear differential equations. Here we will develop a discrete counterpart of the adaptive algorithm in [1] to calculate the solution to the regulator equations by a set of difference equations. For this purpose, like in [1], for $i=1,\dots,N$, let

[TABLE]

[TABLE]

and

[TABLE]

where $S_{i}(t)$ is generated by (17a). It is noted that the dimensions of $x_{i}$, $Q_{i}$ and $G_{i}(t)$ are $q(n_{i}+m_{i})\times 1$, $q(n_{i}+m_{i})\times q(n_{i}+m_{i})$ and $q(n_{i}+m_{i})\times q(n_{i}+m_{i})$, respectively.

We have the following result.

Lemma 4

Under Assumption 4, for $i=1,\dots,N$, for any initial condition $\zeta_{i}(0)$, each of the following equations

[TABLE]

where $0<\mu_{3i}<\frac{2}{\rho(Q_{i}^{T}Q_{i})}$, $i=1,\cdots,N$, has a bounded solution for all $t\in\mathbb{Z^{+}}$. Moreover, let $\Xi_{i}(t)=M_{(n_{i}+m_{i})}^{q}(\zeta_{i}(t))$. Then,

[TABLE]

exponentially.

Proof: The regulator equations (8) can be put in the following form

[TABLE]

By Theorem 1.9 of [10], (38) can be transformed into the following form

[TABLE]

Moreover, Assumption 4 holds if and only if $Q_{i}$ is nonsingular for $i=1,\dots,N$. Thus, for $i=1,\dots,N$, the linear algebraic equations (39) has a unique solution $Q_{i}^{-1}b_{i}$. By (20), $\lim_{t\rightarrow\infty}(G_{i}(t)-Q_{i})=0$ exponentially. Therefore, by Lemma 3, for any ${\zeta}_{i}(0)$, the solution $\zeta_{i}(t)$ to (36) is such that

[TABLE]

exponentially. (40) implies (37) since

[TABLE]

$\Box$

Remark 3

It is noted that, for $i=1,\cdots,N$, $\mu_{3i}$ in (36) depends on the matrix $S$. Since $\mu_{3i}$ can be calculated off-line, the algorithm (36) itself does not need to know $S$ once $\mu_{3i}$ has been predetermined.

We now ready to show how to solve Problem 1 by state feedback control. Partition $\Xi_{i}(t)$ as $\Xi_{i}(t)=[X_{i}(t)^{T},U_{i}(t)^{T}]^{T}$, where $X_{i}(t)\in{\mathbb{R}}^{n_{i}\times q}$ and $U_{i}(t)\in{\mathbb{R}}^{m_{i}\times q}$. Since $(A_{i},B_{i})$ is stabilizable, let $K_{xi}$ be such that $\tilde{A}_{i}\triangleq A_{i}+B_{i}K_{xi}$ is Schur and $K_{\eta i}(t)$ be given as

[TABLE]

For $i=1,\dots,N$, we design the following state feedback controller

[TABLE]

We have the following result.

Theorem 1

Under Assumptions 1, 2, 4 and 5, let $\mu_{1}$ satisfy $0<\mu_{1}<\frac{2}{\rho(H)}$, $\mu_{2}$ be such that the matrix $(I_{N}\otimes S)-\mu_{2}(H\otimes S)$ is Schur, and $0<\mu_{3i}<\frac{2}{\rho(Q_{i}^{T}Q_{i})}$, $i=1,\cdots,N$. Then, Problem 1 is solvable by the control law composed of (17), (36) and (43).

Proof: Let $\tilde{x}_{i}(t)=x_{i}(t)-X_{i}v(t)$, $\tilde{u}_{i}(t)=u_{i}(t)-U_{i}v(t)$, $K_{\eta i}=U_{i}-K_{xi}X_{i}$, and $\tilde{K}_{\eta i}(t)={K}_{\eta i}(t)-K_{\eta i}$. By making use of the solution to the regulator equations (8), we obtain, for $i=1,\dots,N$,

[TABLE]

and

[TABLE]

Further, we have

[TABLE]

Substituting (46) to (44) gives

[TABLE]

where $f_{i}(t)=B_{i}K_{\eta i}(t)\tilde{\eta}_{i}+B_{i}\tilde{K}_{\eta i}(t)v$. By Lemma 2, $\tilde{\eta}_{i}(t)$ decays to zero exponentially. Since $\tilde{K}_{\eta i}(t)=(U_{i}(t)-U_{i})-K_{xi}(X_{i}(t)-X_{i})$, by Lemma 4, $\lim_{t\rightarrow\infty}\tilde{K}_{\eta i}(t)=0$ exponentially. Thus, $\tilde{K}_{\eta i}(t)v$ decays to zero exponentially, and hence, $\lim_{t\rightarrow\infty}f_{i}(t)=0$ exponentially. Moreover, since $(A_{i}+B_{i}K_{xi})$ is Schur, by Lemma 1, for any initial condition $\tilde{x}_{i}(0)$, $\lim_{t\rightarrow\infty}\tilde{x}_{i}(t)=0$ exponentially. Thus, $\lim_{t\rightarrow\infty}\tilde{u}_{i}(t)=0$ by (46), and hence $\lim_{t\rightarrow\infty}{e}_{i}(t)=0$ by (45). $\Box$

Next, we will further consider solving Problem 1 by measurement output feedback control. Let $K_{xi}$ and $K_{\eta i}(t)$ be defined as in the state feedback control law (43). Since $(C_{mi},A_{i})$ is detectable, there exists $L_{i}$ such that $A_{i}+L_{i}C_{mi}$ is Schur. For $i=1,\dots,N$, we design the following measurement output feedback controller

[TABLE]

We have the following result.

Theorem 2

Under Assumptions 1-5, let $\mu_{1}$, $\mu_{2}$, and $\mu_{3i}$ be the same as those in Theorem 1. Then, Problem 1 is solvable by the control law composed of (17) , (36) and (48).

Proof: Let $\hat{x}_{i}=\xi_{i}-x_{i}$. Then we have

[TABLE]

Since $\lim_{t\rightarrow\infty}\tilde{\eta}_{i}(t)=0$ exponentially and $(A_{i}+L_{i}C_{mi})$ is Schur, by Lemma 1, $\lim_{t\rightarrow\infty}\hat{x}_{i}(t)=0$ exponentially. Note that in this case, (44) and (45) still hold. Next, similar to (46), a simple calculation gives

[TABLE]

The rest of the proof follows from the proof of Theorem 1 by noticing that $K_{xi}\hat{x}_{i}$ will also decay to zero exponentially. $\Box$

V An Example

In this section, we consider the cooperative output regulation problem of four agents. The leader system is given by (2) with

[TABLE]

Clearly, Assumption 1 is satisfied.

The four followers are given by (5) with

[TABLE]

Thus, Assumption 2 is also satisfied. Let us take $K_{xi}=[0.2,0]$ such that, for $i=1,2,3,4$, the eigenvalues of $(A_{i}+B_{i}K_{xi})$ are $\{0.447,-0.447\}$.

It can be verified that, for $i=1,2,3,4$, the solution to the regulator equations are as follows:

[TABLE]

Thus, Assumption 4 is verified

The communication graph is shown in Fig. 1. Thus, Assumption 5 is satisfied with

[TABLE]

By Theorems 1 and 2, the cooperative output regulation problem for this system is solvable by both state and output feedback control laws. Since $\sigma(H)=\{2.420\pm j0.606,1,0.161\}$, we can take $\mu_{1}$ such that $0<\mu_{1}<0.778$. Since $\sigma(S)=0.707\pm j0.707$, by (16), for all $\mu_{2}$ satisfying $0<\mu_{2}<1.414$, (13) is stable. To obtain $\mu_{3i}$, note that, for $i=1,2,3,4$,

[TABLE]

Thus, we can take $\mu_{3i}$ such that $0<\mu_{3i}<0.615$. Thus, we can design a state feedback control law with $\mu_{1}=0.3$, $\mu_{2}=0.4$, and, for $1,2,3,4$, and $\mu_{3i}=0.1$. It can be verified that Assumption 3 is also satisfied. Thus, it is also possible to obtain an output feedback control law.

With the initial condition $v(0)=\mbox{col}(0,2)$, the solution to the leader system is $v(t)=\mbox{col}(2\sin\frac{\pi}{4}t,2\cos\frac{\pi}{4}t)$. Figs. 2 and 3 show the estimation errors of the first and second components of the leader's signal by the adaptive distributed observer, respectively. Fig. 4 further shows the tracking errors of the four followers under state feedback control law. Satisfactory tracking performance is observed.

VI Conclusion

In this paper, we have studied the adaptive cooperative output regulation problem for discrete-time linear multi-agent systems utilizing an adaptive distributed observer. Compared with the existing distributed observer based approach, the approach of this paper does not require that the system matrix of the leader system be known by each follower.

One of the common challenges for the control of multi-agent systems is the the delay and the communication delay [9, 15, 17]. A natural extension of the current paper is to further consider same problem for systems with such delays.