Output-based adaptive distributed observer for general linear leader systems over periodic switching digraphs

Abstract

In this paper, we present a sufficient condition for the exponential stability of a class of linear switched systems. As an application of this stability result, we establish an output-based adaptive distributed observer for a general linear leader system over a periodic jointly connected switching communication network, which extends the applicability of the output-based adaptive distributed observer from a marginally stable linear leader system to any linear leader system and from an undirected switching graph to a directed switching graph. This output-based adaptive distributed observer will be applied to solve the leader-following consensus problem for multiple double-integrator systems.

1 Introduction

The distributed observer approach is an effective tool for dealing with a variety of cooperative control problems of multi-agent systems, such as the leader-following consensus problem [1–3], the containment control problem [4], the cooperative output regulation problem [5–8], the leader-following output synchronization problem [9–11], and the rendezvous problem [12, 13].

A distributed observer for the leader system is a distributed dynamic compensator that asymptotically estimates the state of the leader system over a communication network. The distributed observer for linear leader systems was first established over a connected static network in [7] and then over a jointly connected switching network in [8]. However, the distributed observers in [7, 8] have two drawbacks. First, they assume all followers know the system matrix of the leader system. Second, they assume the state of the leader system is available. The first drawback was overcome in [14] by proposing the so-called adaptive distributed observer, which can estimate both the state and the system matrix of a neutrally stable leader system over a jointly connected switching network. The adaptive distributed observer in [14] was strengthened to handle a general linear leader system provided that the switching signal is periodic in [15]. The second drawback was addressed in [16] by proposing the so-called output-based distributed observer which only makes use of the output of the leader system. By combining the ideas of the adaptive distributed observer in [14] and the output-based distributed observer in [16], reference [17] further developed the output-based adaptive distributed observer such that it can estimate the state of a leader system over a jointly connected switching network based on the output of the leader system only and without requiring all followers to know the system matrix of the leader system. Nevertheless, the existing output-based adaptive distributed observer over a jointly connected switching network still requires that the system matrix of the leader system be marginally stable [17]. This requirement is restrictive as it cannot even handle the frequently encountered double-integrator systems. For this reason, in this paper, we will explore the possibility of establishing an output-based adaptive distributed observer for a general linear leader system over a jointly connected switching network. Indeed, by establishing a key stability result of a class of linear periodic switched systems, we manage to ascertain the existence of an output-based adaptive distributed observer for a general linear leader system over a periodic jointly connected switching network.

The rest of this paper is organized as follows. We summarize some useful stability results for periodic linear switched systems and establish a key stability result of a class of linear switched systems in Sect. 2. In Sect. 3, we apply the stability result to develop an output-based adaptive distributed observer for a general linear leader system. In Sect. 4, we apply the output-based adaptive distributed observer to a leader-following consensus problem for multiple double-integrator systems. Section 5 closes this paper with some remarks.

Notation

$\mathbb{R}$ denotes the set of real numbers. $\Re (\cdot )$ denotes the real part of a complex number. $\|x\|$ denotes the Euclidean norm of vector x and $\|A\|$ denotes the induced norm of matrix A by the Euclidean norm. For a square matrix A, $\bar{\lambda}_{\min}(A)$ denotes an eigenvalue of A with the smallest real part and $\bar{\lambda}_{\max}(A)$ denotes an eigenvalue of A with the greatest real part. $\boldsymbol{1}_{N}$ denotes the N-dimensional column vector with all elements being 1. For $A_{i}\in \mathbb{R}^{p_{i}\times m}$, $i=1,\ldots,n$, $\operatorname{col}(A_{1},\ldots,A_{n})= \begin{bmatrix} A_{1}^{T} & \cdots & A_{n}^{T} \end{bmatrix} ^{T}$. ⊗ denotes the Kronecker product of matrices. block $\operatorname{diag}(A_{1},\ldots , A_{n})$ denotes a block diagonal matrix whose diagonal block elements are $A_{1},\ldots ,A_{n}$.

2 Preliminaries

Let $\sigma (t) : [0,\infty ) \to \mathcal{P} = \{1, 2, \ldots , \rho \}$ be a piecewise constant switching signal with dwell time $\tau _{0}>0$. Consider a switching digraph¹$\bar{\mathcal{G}}_{\sigma (t)}=(\bar{\mathcal{V}}, \bar{\mathcal{E}}_{ \sigma (t)})$ where $\bar{\mathcal{V}}=\{0, 1, \ldots , N\}$ and $\bar{\mathcal{E}}_{\sigma (t)}\subseteq \bar{\mathcal{V}}\times \bar{\mathcal{V}}$ for $t\ge 0$. Assume that the switching digraph $\bar{\mathcal{G}}_{\sigma (t)}$ satisfies the following assumptions.

Assumption 1

There exists a subsequence $\{i_{k}| k=0,1, \ldots \}$ of $\{i| i=0,1,\ldots \}$ with $t_{i_{k+1}}-t_{i_{k}}<\nu$ for some $\nu >0$, such that the union graph $\bigcup_{j=i_{k}}^{i_{k+1}-1}\bar{\mathcal{G}}_{\sigma (t_{j})}$ has the property that there is a path from node 0 to every other node.

Assumption 2

The switching signal $\sigma (t)$ is periodic with period T.

Remark 2.1

Assumption 1 is called the jointly connected condition which allows the switching digraph to be disconnected at any time instant. As noted in [15], under Assumption 2, we can assume without loss of generality that the switching signal takes the form

$$\begin{aligned} \sigma (t)= \textstyle\begin{cases} 1,&\text{if } sT\le t< ( s+\omega _{1}) T, \\ 2,&\text{if } (s+\omega _{1}) T\le t< ( s+\sum_{p=1}^{2}\omega _{p}) T, \\ &\vdots \\ \rho , &\text{if } (s+\sum_{p=1}^{\rho -1}\omega _{p})T\le t< (s+1) T, \end{cases}\displaystyle \end{aligned}$$

(1)

where $s=0, 1, \ldots$ , and $\omega _{p}$, $p=1,\ldots ,\rho$, are positive constants satisfying $\sum_{p=1}^{\rho }\omega _{p}=1$. Under Assumptions 1 and 2, without loss of generality, we may assume that, for any $k=0,1,\ldots$ , the union graph $\bigcup_{j=i_{k}}^{i_{k+1}-1}\bar{\mathcal{G}}_{\sigma (t_{j})}= \bigcup_{p=1}^{\rho }\bar{\mathcal{G}}_{p}$. A switching digraph $\bar{\mathcal{G}}_{\sigma (t)}$ satisfying Assumptions 1 and 2 is said to be periodic jointly connected.

Remark 2.2

Let $H_{\sigma (t)} \in \mathbb{R}^{N\times N}$ be obtained from the Laplacian matrix $\bar{\mathcal{L}}_{\sigma (t)}$ of $\bar{\mathcal{G}}_{\sigma (t)}$ by removing the first row and the first column of $\bar{\mathcal{L}}_{\sigma (t)}$. Under Assumption 1, by Corollary 4 of [8], the origin of the linear switched system

$$\begin{aligned} \dot{\phi} = -\mu (H_{\sigma (t)} \otimes I_{q} ) \phi, \end{aligned}$$

(2)

where μ is an arbitrary positive real number and $\phi \in \mathbb{R}^{Nq}$, is exponentially stable. Moreover, under Assumptions 1 and 2, consider the switching signal $\sigma (t)$ defined in (1). From the proof of Lemma 2.2 of [15], all the eigenvalues of the matrix $\sum_{p=1}^{\rho} \omega _{p} H_{p}$ have positive real parts.

Let

$$\begin{aligned} \bar{\mu}_{0}= \frac{\Re (\bar{\lambda}_{\max}(G))}{\Re (\bar{\lambda}_{\min} (\sum_{p=1}^{\rho} \omega _{p} H_{p} ) )}. \end{aligned}$$

(3)

The following lemma is rephrased from Lemma 2.2 of [15].

Lemma 2.1

Consider the linear switched system as follows:

$$\begin{aligned} \dot{\vartheta} & = \bigl(I_{N}\otimes G -\mu (H_{\sigma (t)} \otimes I_{n}) \bigr) \vartheta, \end{aligned}$$

(4)

where $\sigma (t)$ is defined in (1); $\vartheta \in \mathbb{R}^{Nn}$ is the state; $G\in \mathbb{R}^{n\times n}$ is a constant matrix; and $\mu \in \mathbb{R}$ is a positive real number. Under Assumptions 1and 2, there exists a positive constant $\bar{T}_{0}$ such that, for any $\mu >\bar{\mu}_{0}$ and any $0< T<\bar{T}_{0}$, the origin of system (4) is exponentially stable.

In what follows, we establish a stability result of the following class of linear switched systems:

$$\begin{aligned} \begin{aligned} \dot{x} & = \bigl(I_{N} \otimes S - \mu \bigl(H_{\sigma (t)} \otimes (L C) \bigr) \bigr)x, \end{aligned} \end{aligned}$$

(5)

where N is a positive integer; μ is a positive real number; $x\in \mathbb{R}^{Nq}$ is the state; $S\in \mathbb{R}^{q \times q}$ and $C\in \mathbb{R}^{p \times q}$ are constant matrices; and $L \in \mathbb{R}^{q\times p}$ is a gain matrix to be designed.

Assumption 3

The pair $(C, S)$ is detectable.

Remark 2.3

Under Assumption 3 and the assumptions that S is marginally stable and the matrices $H_{p}\in \mathbb{R}^{N \times N}$, $p\in \mathcal{P}$ with $\mathcal{P}$ being the switching index set, are positive semi-definite, the asymptotic stability of system (5) was established in [16] and the exponential stability of system (5) was established in [17].

Remark 2.4

Under Assumption 3, the following algebraic Riccati equation:

$$\begin{aligned} SP+PS^{T}-PC^{T}CP+I_{q}=0 \end{aligned}$$

(6)

admits a unique positive definite solution $P\in \mathbb{R}^{q \times q}$ [18].

Let us first establish the exponential stability of system (5) with S being a general square matrix and $\sigma (t)$ being a periodic signal.

Lemma 2.2

Under Assumptions 1, 2, and 3, let $\lambda _{1},\ldots ,\lambda _{N}$ be the N eigenvalues of the matrix $\sum_{p=1}^{\rho} \omega _{p} H_{p}$, let $\bar{\mu}_{1} = \frac{1}{2}\delta ^{-1}$ where $\delta =\min_{i=1,\ldots ,N}\{\Re (\lambda _{i})\}$. Then, there exists a positive constant $\bar{T}_{1}$ such that, for any $\mu \ge \bar{\mu}_{1}$ and any $0< T<\bar{T}_{1}$, the origin of system (5) with $L=PC^{T}$ is exponentially stable.

Proof

Let

$$\begin{aligned} \begin{aligned} M_{p} & = I_{N} \otimes S-\mu \bigl(H_{p}\otimes \bigl(PC^{T}C\bigr) \bigr), \quad p=1, \ldots ,\rho, \\ M_{c} & = \sum_{p=1}^{\rho} \omega _{p} M_{p}. \end{aligned} \end{aligned}$$

(7)

Under Assumptions 1 and 2, by Remark 2.2, the N eigenvalues $\lambda _{1},\ldots ,\lambda _{N}$ of the matrix $\sum_{p=1}^{\rho} \omega _{p} H_{p}$ have positive real parts. Let $Q\in \mathbb{R}^{N\times N}$ be a nonsingular matrix such that $J=Q (\sum_{p=1}^{\rho} \omega _{p} H_{p} )Q^{-1}$ where $J\in \mathbb{R}^{N\times N}$ is the Jordan form of $\sum_{p=1}^{\rho} \omega _{p} H_{p}$. Since $\sum_{p=1}^{\rho }\omega _{p}=1$, $M_{c}$ can be rewritten as

$$\begin{aligned} M_{c} ={}& \bigl(Q^{-1}\otimes I_{q}\bigr) \bigl(I_{N} \otimes S-\mu \bigl(J\otimes \bigl(PC^{T}C \bigr)\bigr) \bigr)(Q\otimes I_{q}). \end{aligned}$$

(8)

The block triangular structure of J implies that the eigenvalues of $M_{c}$ coincide with the eigenvalues of $S-\mu \lambda _{i} PC^{T}C$, $i=1,\ldots ,N$. By Lemma 2.12 of [17], for any $\mu \ge \frac{1}{2}\delta ^{-1}$ with $\delta =\min_{i=1,\ldots ,N}\{\Re (\lambda _{i})\}$, the matrices $S-\mu \lambda _{i} PC^{T}C$, $i=1,\ldots ,N$, are Hurwitz. Hence, for any $\mu \ge \frac{1}{2}\delta ^{-1}$, $M_{c}$ is Hurwitz. Invoking Lemma 3.22 of [19] with $\mu \ge \bar{\mu}_{1}= \frac{1}{2}\delta ^{-1}$ completes the proof. □

Remark 2.5

Since Lemma 2.2 imposes no restriction on the eigenvalues of the matrix S and does not require $H_{p}$, $p=1,\ldots ,\rho$, to be symmetric, it has partially extended the results in [16, 17] from a marginally stable matrix S to any matrix S and from an undirected graph to a directed graph provided that the switching signal ${\sigma (t)}$ satisfies Assumption 2 with the period T being smaller than some threshold.

3 Output-based adaptive distributed observer

In this section, we further apply Lemma 2.2 to establish the output-based adaptive distributed observer over a jointly connected graph for a general linear leader system of the following form:

$$\begin{aligned} \dot{v}=Sv, \qquad y=Cv, \end{aligned}$$

(9)

where $v\in \mathbb{R}^{q}$, $y\in \mathbb{R}^{p}$, and $S\in \mathbb{R}^{q\times q}$ and $C\in \mathbb{R}^{p\times q}$ are constant matrices.

As in [17], for $i=1,\ldots ,N$, consider a distributed dynamic compensator as follows:

$$\begin{aligned} &\dot{S}_{i} =\mu _{1} \sum _{j=0}^{N} a_{i j}(t) (S_{j}-S_{i} ), \end{aligned}$$

(10a)

$$\begin{aligned} &\dot{C}_{i} =\mu _{2} \sum _{j=0}^{N} a_{i j}(t) (C_{j}-C_{i} ), \end{aligned}$$

(10b)

$$\begin{aligned} &\dot{L}_{i} =\mu _{3} \sum _{j=0}^{N} a_{i j}(t) (L_{j}-L_{i} ) , \end{aligned}$$

(10c)

$$\begin{aligned} &\dot{\xi}_{i} =S_{i} \xi _{i}+\mu _{4} L_{i} \sum_{j=0}^{N} a_{i j}(t) (C_{j} \xi _{j}-C_{i} \xi _{i} ), \end{aligned}$$

(10d)

where $S_{0}=S$, $C_{0}=C$, $L_{0}=L\in \mathbb{R}^{q\times p}$ is a gain matrix to be designed, and $\xi _{0}=v$; $\mu _{1}$ to $\mu _{4}$ are positive real numbers to be specified; and, for $i=1,\ldots ,N$, $S_{i} \in \mathbb{R}^{q \times q}$, $C_{i} \in \mathbb{R}^{p \times q}$, $L_{i} \in \mathbb{R}^{q \times p}$, $\xi _{i} \in \mathbb{R}^{q}$, and $a_{ij}(t)$ is the element of the weighted adjacency matrix.

Given the leader system (9) and a group of N dynamic compensators (10a)–(10d), we can define a switching digraph $\bar{\mathcal{G}}_{\sigma (t)}=(\bar{\mathcal{V}}, \bar{\mathcal{E}}_{ \sigma (t)})$ where $\bar{\mathcal{V}}=\{0, 1, \ldots , N\}$ and $\bar{\mathcal{E}}_{\sigma (t)}\subseteq \bar{\mathcal{V}}\times \bar{\mathcal{V}}$ for $t\ge 0$. In the set $\bar{\mathcal{V}}$, node 0 is associated with the leader system (9) and node i, $i=1, \ldots , N$, is associated with the ith dynamic compensator of (10a)–(10d). For $i=1, \ldots , N$, $j=0, 1, \ldots , N$, $i \neq j$, $(j, i)\in \bar{\mathcal{E}}_{\sigma (t)}$ if and only if the ith dynamic compensator of (10a)–(10d) can make use of the information of agent j at time t. The weighted adjacency matrix of the digraph $\bar{\mathcal{G}}_{\sigma (t)}$ is a nonnegative matrix $\bar{\mathcal{A}}_{\sigma (t)}=[a_{ij}(t)]_{i,j=0}^{N}\in \mathbb{R}^{(N+1) \times (N+1)}$, where, for $t\ge 0$, $a_{ii}(t)=0$, and, for $i\neq j$, $a_{ij}(t)>0 \Leftrightarrow (j, i)\in \bar{\mathcal{E}}_{\sigma (t)}$. Let $H_{\sigma (t)}=[h_{ij}(t)]_{i,j=1}^{N}\in \mathbb{R}^{N\times N}$, where $h_{ii}(t)=\sum_{j=0}^{N}a_{ij}(t)$ and $h_{ij}(t)=-a_{ij}(t)$ if $i\neq j$, then $H_{\sigma (t)}$ is the matrix obtained from the Laplacian matrix $\bar{\mathcal{L}}_{\sigma (t)}$ of $\bar{\mathcal{G}}_{\sigma (t)}$ by removing the first row and the first column of $\bar{\mathcal{L}}_{\sigma (t)}$. Let $\mathcal{G}_{\sigma (t)}=(\mathcal{V}, \mathcal{E}_{\sigma (t)})$ with $\mathcal{V}=\{1, \ldots , N\}$ and $\mathcal{E}_{\sigma (t)}\subseteq \mathcal{V}\times \mathcal{V}$ be the subgraph obtained from the switching digraph $\bar{\mathcal{G}}_{\sigma (t)}$ by removing all the edges between node 0 and the nodes in $\mathcal{V}$.

Assumption 4

The subgraph $\mathcal{G}_{\sigma (t)}$ is undirected for all $t\ge 0$.

Assumption 5

The matrix S is marginally stable, that is, there exists a positive definite matrix $P_{0}$ satisfying $P_{0}S^{T}+SP_{0}\le 0$.

The dynamic compensator (10a)–(10d) is a so-called output-based adaptive distributed observer for the leader system (9) due to the following lemma rephrased from Theorem 4.7 of [17].

Lemma 3.1

Consider systems (9) and (10a)–(10d). Under Assumptions 1, 3, 4, and 5, there exists a constant matrix $L\in \mathbb{R}^{q\times p}$ such that, for any $\mu _{1}, \mu _{2}, \mu _{3}, \mu _{4}>0$ and any initial conditions $v(0)$, $S_{i}(0)$, $C_{i}(0)$, $L_{i}(0)$, and $\xi _{i}(0)$, $i=1,\ldots ,N$, the solutions of systems (9) and (10a)–(10d) exist for all $t\ge 0$ and satisfy

$$\begin{aligned}& \lim_{t\to \infty}\bigl(S_{i}(t)-S\bigr) =0 , \end{aligned}$$

(11a)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(C_{i}(t)-C\bigr) =0 , \end{aligned}$$

(11b)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(L_{i}(t)-L\bigr) =0 , \end{aligned}$$

(11c)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(\xi _{i}(t)-v(t)\bigr) =0, \quad i=1,\ldots ,N \end{aligned}$$

(11d)

all exponentially.

However, Lemma 3.1 is quite restrictive since it only applies to a marginally stable leader system and requires the graph to be undirected. Thus, it is desirable to establish an output-based adaptive distributed observer for the leader system (9) without these restrictive assumptions, provided that the switching signal ${\sigma (t)}$ satisfies Assumption 2 with the period T being smaller than some threshold.

Theorem 3.1

Consider systems (9) and (10a)–(10d). Under Assumptions 1, 2, and 3, let $\lambda > \Re (\bar{\lambda}_{\max}(S))$ be some positive real number and let

$$\begin{aligned} \bar{\mu}_{2} & = \frac{\lambda}{\Re (\bar{\lambda}_{\min} (\sum_{p=1}^{\rho} \omega _{p} H_{p} ) )}, \end{aligned}$$

(12)

$$\begin{aligned} \bar{\mu}_{3} & = \frac{1}{2}\times \frac{1}{\Re (\bar{\lambda}_{\min} (\sum_{p=1}^{\rho} \omega _{p} H_{p} ) )}. \end{aligned}$$

(13)

Then there exists a positive constant $\bar{T}_{2}$, such that, if $\mu _{1},\mu _{2}, \mu _{3}>\bar{\mu}_{2}$, $\mu _{4}\ge \bar{\mu}_{3}$, and $0< T<\bar{T}_{2}$, then, for any initial conditions $v(0)$, $S_{i}(0)$, $C_{i}(0)$, $L_{i}(0)$, and $\xi _{i}(0)$, $i=1,\ldots ,N$, the solutions of systems (9) and (10a)–(10d) with $L=PC^{T}$ exist for all $t\ge 0$ and satisfy

$$\begin{aligned}& \lim_{t\to \infty}\bigl(S_{i}(t)-S\bigr) =0, \end{aligned}$$

(14a)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(C_{i}(t)-C\bigr) =0, \end{aligned}$$

(14b)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(L_{i}(t)-L\bigr) =0, \end{aligned}$$

(14c)

$$\begin{aligned}& \lim_{t\to \infty}\bigl(\xi _{i}(t)-v(t)\bigr) =0, \quad i=1,\ldots ,N \end{aligned}$$

(14d)

all exponentially.

Proof

Let $\tilde{S}_{i}= S_{i}-S$, $\tilde{C}_{i}= C_{i}-C$, $\tilde{L}_{i}= L_{i}-L$, and $\tilde{v}_{i}= \xi _{i}-v$, $i=1,\ldots ,N$. Also, let $\tilde{S}= \operatorname{col}(\tilde{S}_{1},\ldots ,\tilde{S}_{N})$, $\tilde{C}= \operatorname{col}(\tilde{C}_{1},\ldots ,\tilde{C}_{N})$, $\tilde{L}= \operatorname{col}(\tilde{L}_{1},\ldots ,\tilde{L}_{N})$, $\tilde{v}= \operatorname{col}(\tilde{v}_{1},\ldots ,\tilde{v}_{N})$, $\tilde{S}_{d}= \text{block diag}(\tilde{S}_{1}, \ldots ,\tilde{S}_{N})$, $\tilde{C}_{d}= \text{block diag}(\tilde{C}_{1},\ldots ,\tilde{C}_{N})$, $\tilde{L}_{d}= \text{block diag}(\tilde{L}_{1}, \ldots ,\tilde{L}_{N})$, and $L_{d}= \text{block diag}(L_{1},\ldots ,L_{N})$. Then, from (9) and (10a)–(10d), as shown in Sect. 4.4.3 of [17], S̃, C̃, L̃, and ṽ are governed by

$$\begin{aligned}& \dot{\tilde{S}} =-\mu _{1} (H_{\sigma (t)}\otimes I_{q} ) \tilde{S} , \end{aligned}$$

(15a)

$$\begin{aligned}& \dot{\tilde{C}} =-\mu _{2} (H_{\sigma (t)}\otimes I_{p} ) \tilde{C}, \end{aligned}$$

(15b)

$$\begin{aligned}& \dot{\tilde{L}} =-\mu _{3} (H_{\sigma (t)}\otimes I_{q} ) \tilde{L} , \end{aligned}$$

(15c)

$$\begin{aligned}& \dot{\tilde{v}} = \bigl(A_{\sigma (t)}+ M_{d}(t) \bigr) \tilde{v}+F(t) , \end{aligned}$$

(15d)

where

$$\begin{array}{c}{A}_{\sigma (t)}=I_N\otimes S-{\mu }_4(H_{\sigma (t)}\otimes (LC)),\\ \begin{array}{rl}{M}_d(t)=& {\tilde{S}}_d-{\mu }_4{\tilde{L}}_d(H_{\sigma (t)}\otimes C)\\ & -{\mu }_4{L}_d(H_{\sigma (t)}\otimes I_p){\tilde{C}}_d,\end{array}\\ F(t)=({\tilde{S}}_d-{\mu }_4{L}_d(H_{\sigma (t)}\otimes I_p){\tilde{C}}_d)({\mathbf{1}}_N\otimes v).\end{array}$$

(16)

For $i=1,\ldots ,q$, let $\tilde{s}_{i}\in \mathbb{R}^{Nq}$ and $\tilde{c}_{i}\in \mathbb{R}^{Np}$ be the ith column of S̃ and C̃, respectively. For $j=1,\ldots ,p$, let $\tilde{l}_{j}\in \mathbb{R}^{Nq}$ be the jth column of L̃. Then, (15a), (15b), and (15c) can be written as

$$\begin{aligned}& \dot{\tilde{s}}_{i} = -\mu _{1}(H_{\sigma (t)} \otimes I_{q}) \tilde{s}_{i}, \end{aligned}$$

(17a)

$$\begin{aligned}& \dot{\tilde{c}}_{i} = -\mu _{2}(H_{\sigma (t)} \otimes I_{p}) \tilde{c}_{i}, \quad i=1,\ldots ,q, \end{aligned}$$

(17b)

$$\begin{aligned}& \dot{\tilde{l}}_{j} = -\mu _{3}(H_{\sigma (t)} \otimes I_{q}) \tilde{l}_{j}, \quad j=1,\ldots ,p. \end{aligned}$$

(17c)

Under Assumption 1, by Remark 2.2, the origin of system (17a)–(17c) is exponentially stable for any $\mu _{1}, \mu _{2}, \mu _{3}>0$. Thus, we have $\lim_{t\to \infty}\tilde{S}_{d}(t)=0$, $\lim_{t\to \infty}\tilde{C}_{d}(t)=0$, and $\lim_{t\to \infty}\tilde{L}_{d}(t)=0$ all exponentially, which implies (14a), (14b), and (14c). Since (14c) implies that $L_{d}(t)$ is bounded over $[0, \infty )$, we have $\lim_{t\to \infty}M_{d}(t)=0$ exponentially.

In what follows, we show (14d). For this purpose, we first show that there exists a positive constant $\bar{T}_{3}$, such that, for any $\mu _{1},\mu _{2}, \mu _{3}>\bar{\mu}_{2}$ and any $0< T<\bar{T}_{3}$, $\lim_{t\to \infty} F(t)=0$ exponentially, where $F(t)$ is defined in (16). Let $\bar{S}= I_{q}\otimes \lambda$, $\hat{S}= I_{p}\otimes \lambda$, $\zeta _{i}= (I_{N}\otimes e^{\bar{S}t})\tilde{s}_{i}$, and $\gamma _{i}= (I_{N}\otimes e^{\hat{S}t})\tilde{c}_{i}$, $i=1,\ldots ,q$. Then, $\zeta _{i}$ is governed by

$$\begin{aligned} \dot{\zeta}_{i} & = \bigl(I_{N} \otimes \bigl(\bar{S}e^{\bar{S}t}\bigr) \bigr) \tilde{s}_{i}+ \bigl(I_{N}\otimes e^{\bar{S}t}\bigr)\dot{\tilde{s}}_{i} \\ & =(I_{N}\otimes \bar{S}) \bigl(I_{N}\otimes e^{\bar{S}t}\bigr)\tilde{s}_{i}- \mu _{1} \bigl(I_{N}\otimes e^{\bar{S}t}\bigr) (H_{\sigma (t)} \otimes I_{q}) \tilde{s}_{i} \\ & =(I_{N}\otimes \bar{S}) \bigl(I_{N}\otimes e^{\bar{S}t}\bigr)\tilde{s}_{i}- \mu _{1}(H_{\sigma (t)} \otimes I_{q}) \bigl(I_{N}\otimes e^{\bar{S}t} \bigr) \tilde{s}_{i} \\ & = \bigl(I_{N}\otimes \bar{S}-\mu _{1}(H_{\sigma (t)} \otimes I_{q}) \bigr)\zeta _{i}, \quad i=1,\ldots ,q. \end{aligned}$$

(18)

Similarly, $\gamma _{i}$ is governed by

$$\begin{aligned} \dot{\gamma}_{i} & = \bigl(I_{N} \otimes \hat{S}-\mu _{2}(H_{\sigma (t)} \otimes I_{p}) \bigr)\gamma _{i}, \quad i=1,\ldots ,q. \end{aligned}$$

(19)

For all $i=1,\ldots ,q$, system (18) (respectively, system (19)) is in the form of (4) with $G=\bar{S}$, $\mu =\mu _{1}$, and $\vartheta =\zeta _{i}$ (respectively, $G=\hat{S}$, $\mu =\mu _{2}$, and $\vartheta =\gamma _{i}$). Since $\Re (\bar{\lambda}_{\max}(\bar{S}))=\Re (\bar{\lambda}_{\max}( \hat{S}))=\lambda$, under Assumptions 1 and 2, by Lemma 2.1, there exists a positive constant $\bar{T}_{3}$, such that, for any $\mu _{1}, \mu _{2}>\bar{\mu}_{2}$ and any $0< T<\bar{T}_{3}$, the origins of systems (18) and (19) are exponentially stable. Thus, for $i=1,\ldots ,N$, we have

$$\begin{aligned} &\lim_{t\to \infty} \bigl\Vert e^{\bar{S}t} \tilde{S}_{i}(t) \bigr\Vert = \lim_{t \to \infty} e^{\lambda t} \bigl\Vert \tilde{S}_{i}(t) \bigr\Vert = 0, \end{aligned}$$

(20)

$$\begin{aligned} &\lim_{t\to \infty} \bigl\Vert e^{\hat{S}t} \tilde{C}_{i}(t) \bigr\Vert = \lim_{t \to \infty} e^{\lambda t} \bigl\Vert \tilde{C}_{i}(t) \bigr\Vert = 0 \end{aligned}$$

(21)

exponentially. Since $\| e^{St} \| \le \beta e^{\lambda t}$ for some positive real number β, from (20) and (21), for $i=1,\ldots ,N$, we have

$$\begin{aligned} \lim_{t\to \infty} \bigl\Vert \tilde{S}_{i}(t)v(t) \bigr\Vert & = \lim_{t\to \infty} \bigl\Vert \tilde{S}_{i}(t) e^{S t} v(0) \bigr\Vert \\ & \leq \lim_{t\to \infty} \beta e^{\lambda t} \bigl\Vert \tilde{S}_{i}(t) \bigr\Vert \bigl\Vert v(0) \bigr\Vert = 0, \end{aligned}$$

(22)

$$\begin{aligned} \lim_{t\to \infty} \bigl\Vert \tilde{C}_{i}(t)v(t) \bigr\Vert & = \lim_{t\to \infty} \bigl\Vert \tilde{C}_{i}(t) e^{S t} v(0) \bigr\Vert \\ & \leq \lim_{t\to \infty} \beta e^{\lambda t} \bigl\Vert \tilde{C}_{i}(t) \bigr\Vert \bigl\Vert v(0) \bigr\Vert = 0 \end{aligned}$$

(23)

exponentially. Thus, it holds that

$$\underset{t\to \mathrm{\infty }}{lim}{\tilde{S}}_d(t)({\mathbf{1}}_N\otimes v(t))=0,$$

(24)

$$\underset{t\to \mathrm{\infty }}{lim}{\tilde{C}}_d(t)({\mathbf{1}}_N\otimes v(t))=0$$

(25)

exponentially. Since $L_{d}(t)$ is bounded over $[0, \infty )$, from (16), (24), and (25), we have $\lim_{t\to \infty} F(t)=0$ exponentially.

Now, we are ready to show (14d). For this purpose, consider the following system:

$$\begin{aligned} \dot{\tilde{v}} & = \bigl(I_{N}\otimes S- \mu _{4} \bigl(H_{\sigma (t)} \otimes (LC ) \bigr) \bigr)\tilde{v} \end{aligned}$$

(26)

which is obtained from system (15d) by setting $M_{d}(t)$ and $F(t)$ to zero. Under Assumptions 1, 2, and 3, by Lemma 2.2, there exists a positive constant $\bar{T}_{4}$, such that, for any $\mu _{4}\ge \bar{\mu}_{3}$ and any $0< T<\bar{T}_{4}$, the origin of system (26) is exponentially stable. Let $\bar{T}_{2}= \min \{\bar{T}_{3}, \bar{T}_{4}\}$. Applying Lemma 1 of [20] to system (15d) completes the proof. □

Remark 3.1

The exponential stability of the linear switched system (26) with a general square matrix S is crucial in establishing the output-based adaptive distributed observer (10a)–(10d). This stability result of (26) cannot be obtained by any existing approach. Theorem 3.1 has made the output-based adaptive distributed observer (10a)–(10d) applicable to any linear leader system over a directed switching graph $\bar{\mathcal{G}}_{\sigma (t)}$ that satisfies Assumption 2 with the period T being smaller than a threshold.

4 Example

In this section, we apply the output-based adaptive distributed observer (10a)–(10d) to solve the leader-following consensus and disturbance rejection problem of four double-integrator systems as follows:

$$\begin{aligned} \begin{gathered} \dot{x}_{i1} = x_{i2}, \\ \dot{x}_{i2} = u_{i}+d_{i}, \\ e_{i} = \begin{bmatrix} x_{i1} \\ x_{i2} \end{bmatrix} - \begin{bmatrix} v_{1} \\ v_{2} \end{bmatrix}, \\ y_{mi} = x_{i1}, \quad i=1,2,3,4 ,\end{gathered} \end{aligned}$$

(27)

where, for $i=1,2,3,4$, $x_{i1}, x_{i2}\in \mathbb{R}$; $u_{i}\in \mathbb{R}$ is the control input to be designed; $d_{i}=i\times v_{3}\in \mathbb{R}$ is the external disturbance; $e_{i}\in \mathbb{R}^{2}$ is the tracking error; $y_{mi}\in \mathbb{R}$ is the measurement output; and $v=\operatorname{col}(v_{1},v_{2},v_{3})\in \mathbb{R}^{3}$ is the exogenous signal generated by the following leader system:

$$\begin{aligned} \begin{gathered} \dot{v}=Sv= \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}v, \\ y = Cv= \begin{bmatrix} 1 & 0 & 1 \end{bmatrix}v. \end{gathered} \end{aligned}$$

(28)

Hence, the four double-integrator systems are subject to exponentially growing disturbances. Since the pair $(C, S)$ is observable, Assumption 3 is satisfied.

We can interpret the above problem as a cooperative output regulation problem (see Definition 1 of [8]) by rewriting the four double-integrator systems in the following standard form:

$$\begin{aligned} &\dot{x}_{i}= A_{i}x_{i}+B_{i}u_{i}+E_{i}v, \\ &y_{mi}= C_{mi}x_{i}+D_{mi}u_{i}+F_{mi}v, \\ &e_{i}= C_{ri}x_{i}+D_{ri}u_{i}+F_{ri}v, \quad i=1,2,3,4 , \end{aligned}$$

(29)

where, for $i=1,2,3,4$, $x_{i}=\operatorname{col}(x_{i1}, x_{i2})$,

$$\begin{aligned} \begin{gathered} A_{i} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix},\qquad B_{i}= \begin{bmatrix} 0 \\ 1 \end{bmatrix},\qquad E_{i}= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & i \end{bmatrix}, \\ C_{ri} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad D_{ri}= \begin{bmatrix} 0 \\ 0 \end{bmatrix},\\ F_{ri}= \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix}, \\ C_{mi} = \begin{bmatrix} 1 & 0 \end{bmatrix}, \qquad D_{mi}=0, \qquad F_{mi}= \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}. \end{gathered} \end{aligned}$$

Then it can be verified that the pair $(A_{i}, B_{i})$ is controllable; the pair $(C_{mi}, A_{i})$ is observable; and the solution of the regulator equation

$$\begin{aligned} \begin{aligned} &X_{i}S =A_{i}X_{i}+B_{i}U_{i}+E_{i}, \\ &0=C_{ri}X_{i}+D_{ri}U_{i}+F_{ri}, \end{aligned} \quad i=1,2,3,4 \end{aligned}$$

(30)

is given by $X_i=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$ and $U_{i}= [ -1 \ 0 \ - i ]$.

Consider the following periodic switching signal:

$$\begin{aligned} \sigma (t)= \textstyle\begin{cases} 1,&\text{if }sT\le t< ( s+\frac{1}{4} )T, \\ 2,&\text{if } ( s+\frac{1}{4} )T\le t< ( s+ \frac{1}{2} )T, \\ 3,&\text{if } ( s+\frac{1}{2} )T\le t< ( s+1 )T, \end{cases}\displaystyle \end{aligned}$$

(31)

where $T=1$ and $s=0, 1, \ldots$ . Thus, $\sigma (t)$ is in the form of (1) with $\omega _{1}=\frac{1}{4}$, $\omega _{2}=\frac{1}{4}$, and $\omega _{3}=\frac{1}{2}$. The three digraphs $\bar{\mathcal{G}}_{i}$, $i=1, 2, 3$, associated with $\sigma (t)$ are shown in Fig. 1. Although the switching digraph $\bar{\mathcal{G}}_{\sigma (t)}$ is disconnected at every time instant $t\ge 0$, it can be seen that Assumptions 1 and 2 are satisfied. We let $a_{ij}(t)=1$ whenever $(j,i)\in \bar{\mathcal{E}}_{\sigma (t)}$.

Figure 1

Communication graphs: (a) $\bar{\mathcal{G}}_{1}$, (b) $\bar{\mathcal{G}}_{2}$, and (c) $\bar{\mathcal{G}}_{3}$

The three matrices associated with the three digraphs $\bar{\mathcal{G}}_{i}$, $i=1,2,3$, are given by

$$H_{1} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ,\qquad H_{2}= \begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ,$$

and

$$H_{3} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ -1 & 0 & 0 & 1 \end{bmatrix} . $$

The four eigenvalues of the matrix $\sum_{p=1}^{3} \omega _{p} H_{p}$ are $\lambda _{1} = 0.5$, $\lambda _{2} = 0.5$, $\lambda _{3} = 0.25$, and $\lambda _{4} = 0.25$, thus $\Re (\bar{\lambda}_{\min} (\sum_{p=1}^{3} \omega _{p} H_{p} ) )=\min_{i=1,2,3,4}\{\Re (\lambda _{i})\}=0.25$. From (13), $\bar{\mu}_{3}=2$. From (28), $\Re (\bar{\lambda}_{\max}(S))=1$. Then we let $\lambda = 1.2$ which satisfies $\lambda > \Re (\bar{\lambda}_{\max}(S))$. From (12), we have $\bar{\mu}_{2}=4.8$.

By Theorem 3.1, we can design an output-based adaptive distributed observer of the form (10a)–(10d) for (28) with $\mu _{1}=10$, $\mu _{2}=10$, $\mu _{3}=10$, $\mu _{4}=10$, and $L=\operatorname{col}(-0.2, 1.4, 4.2)$. As shown in [8], the cooperative output regulation problem of the multi-agent system composed of (28) and (29) can be solved by the following dynamic measurement output feedback control law:

$$\begin{aligned}& u_{i} = K_{1i}\eta _{i} +K_{2i}\xi _{i}, \end{aligned}$$

(32a)

$$\begin{aligned}& \begin{aligned} \dot{\eta}_{i} =& A_{i} \eta _{i} + B_{i} u_{i} + E_{i} \xi _{i} \\ & {} +L_{mi}(C_{mi}\eta _{i}+D_{mi}u_{i}+F_{mi} \xi _{i}-y_{mi}), \end{aligned} \end{aligned}$$

(32b)

where, for $i=1,2,3,4$, $\xi _{i}\in \mathbb{R}^{3}$ is generated by (10a)–(10d), $L_{mi}=\operatorname{col}(-22,-120)$, $K_{1i} = [ -12 \ -7 ]$, and $K_{2i} = U_{i}-K_{1i}X_{i}= [ 11 \ 7 \ -i ]$. In particular, for $i=1,2,3,4$, $K_{1i}$ and $L_{mi}$ are such that $A_{i}+B_{i}K_{1i}$ and $A_{i}+L_{mi}C_{mi}$ are Hurwitz.

Our simulation is conducted with $v(0)=\operatorname{col}(1,1,1)$ and other initial conditions randomly generated within the range $[-1.5, 1.5]$. The estimation errors of the distributed observer are shown in Figs. 2 to 5. The tracking errors of the followers are shown in Fig. 6. As expected, satisfactory performance is observed.

Figure 2

Estimation errors of matrix S

Figure 3

Estimation errors of matrix C

Figure 4

Estimation errors of matrix L

Figure 5

Estimation errors of state v

Figure 6

Tracking errors $e_{i}=\operatorname{col}(e_{i1}, e_{i2})$

5 Conclusion

In this paper, having established an exponential stability property for a class of linear switched systems, we have further developed an output-based adaptive distributed observer for a general linear leader system over a periodic jointly connected switching communication network, which extends the applicability of the output-based adaptive distributed observer from a marginally stable linear leader system to a general linear leader system and from an undirected switching graph to a directed switching graph.