Distributed Interval Observers with Switching Topology Design for Cyber-Physical Systems

Abstract

In this paper, the distributed interval estimation problem for networked Cyber-Physical systems suffering from both disturbances and noise is investigated. In the distributed interval observers, there are some connected interval observers built for the corresponding subsystems. Then, due to the communication burden in Cyber-Physical systems, we consider the case where the communication among distributed interval observers is switching topology. A novel approach that combines $L_{\infty }$ methodology with reachable set analysis is proposed to design distributed interval observers. Finally, the performance of the proposed distributed interval observers with switching topology is verified through a simulation example.

1. Introduction

Cyber-Physical Systems (CPSs) are the combinations of physical procedures, high-efficiency computation, communication, and effective control defined by [1]. Architecturally, from [2], a typical CPS can be divided into three layers, which are composed of the sensing layer, the network layer, and the control layer. The development of distributed sensing and networking technologies such as [3,4] has enabled omnipresent sensing and computing capabilities. This has led to the implementation of CPSs in large-scale networks. CPSs are widely used in industrial informatics [5] manufacturing [6], healthcare [7], electrical grids [8], and so on. State estimation and observer design are crucial research areas in CPSs. Ref. [9] used a sliding mode observer and integrated the event-triggered mechanism to estimate the state of CPSs from sensor measurements. Ref. [10] introduced a security estimator combined with a Kalman filter to improve the practical performance of state estimation for CPSs. Ref. [11] accomplished state estimation and resilient control of CPSs using finite time observer techniques and switching schemes. It should be noted that the state estimation for CPSs with bounded disturbance and noise has not been investigated sufficiently.

On the other hand, disturbances and noise always exist in real systems, and the interval observer serves as a powerful estimator of upper and lower bounds for uncertain systems with disturbances and noise. In [12], the concept, as well as the framework of interval observer, were presented. Using the monotone system theory, Refs. [13,14] proposed the approach of coordinate transformation that serves as an efficient strategy to reduce the strict conditions for interval observer design. In recent years, the set-membership estimation method was applied effectively in interval observer design. A two-step interval observer design methodology that combines reachability analysis with robust observer was first presented in [15]. In [15], the $H_{\infty }$ method and reachable set analysis are combined to design interval observers that eliminate the effects of perturbations and noise on the system. At the same time, the $L_{\infty }$ (or $L_2$) method is also a powerful, robust property, and it is widely used in control and observation fields, such as [16,17,18]. It has been recently shown by [19,20,21] that the reachability analysis estimation method can not only enhance the accuracy of estimation but also increase the design freedom. Concurrently, in the context of distributed systems, several recent studies have been conducted on distributed interval observers, such as [22,23,24]. Ref. [22] designed distributed interval observers based on the monotone system approach for multiagent systems. Ref. [23] considered a distributed interval observers design problem for a class of linear time-invariant systems with uncertainty. At the same time, Ref. [24] improved distributed interval observers by using a set-membership estimation approach. However, the topology of interval observer in the aforementioned work [22,23,24] is fixed, and it is usually switching with respect to time in practice.

In light of the above discussion, for the problem of state estimation for uncertain CPSs, we apply distributed interval observers to such systems. Since the state estimation of CPSs with bounded disturbances and noise has not been investigated sufficiently, it is meaningful to study the state estimation problem for uncertain CPSs. Each interval observer has two types of observer gain: one is obtained by using the traditional observer design method, and the other one is determined by employing neighborhood information. Considering the practical communication problem of the network layer in CPSs, we suppose that the communication among distributed interval observers is described by switching topology. There are three main challenges: one is to design $L_{\infty }$ observers with optimal performance for networked CPSs, another is to construct a reachable set analysis framework for CPSs, which then gives upper and lower bounds on the state of the system, and the last is to solve the switching topology problem among the observers. The contributions of this paper are summarized in aspects below.

(1)

A distributed interval observer methodology for CPSs is proposed. Compared with the monotone system method, the estimation accuracy is greatly improved by using the two-step method. The $L_{\infty }$ technique is used to deal with the effects of uncertainty in observer design.

(2)

The switching topology with average dwell time (ADT) among distributed interval observers is taken into account and is more closely aligned with the actual system. It can also reduce the communication burden of CPSs.

This paper is structured as follows, with the rest of the paper starting with the graph theory, system model, and some basics presented in Section 2. In Section 3, the optimal robust observer is designed using the $L_{\infty }$ technique. In order to complete the interval estimation, a reachability analysis methodology is used to design the distributed interval observer. In Section 4, the paper simulates a networked CPS with four Unmanned Aerial Vehicles (UAV) models to illustrate the effectiveness of the distributed interval observer. Finally, Section 5 concludes the paper.

Notation: For a matrix of real symmetry $E\in R^{N\times N},E\succ 0$ demonstrates that E is positive definite, while $E\prec 0$ demonstrates that E is negative and $He\left(E\right)=E+E^T$. The maximal (minimum) eigenvalue of the matrix Q is denoted by ${\lambda }_{max}\left(Q\right)\left({\lambda }_{min}\left(Q\right)\right)$. The norm $L_2$ of the vector v is represented by ${∥v∥}_2$. In other words, ${∥v∥}_2=\sqrt{v^{T}v}$. Similarly, the norm $L_{\infty }$ of the vector v is denoted by ${∥v∥}_{\infty }$. The symbol * means the term that can arise due to symmetry in the symmetric matrices.

2. Preliminaries

2.1. Graph Theory

For a digraph G which has N vertices, $\mathcal{A}=\left[a_{ij}\right]\in R^{N\times N}$ is called the adjacency matrix. The weight associated with the edge $(i,j)\in S$ that connects node i to node j is called $a_{ij}$, and $\mathcal{A}$ is provided by $a_{ij}=0$. The length path from vertex i to vertex j is made up of $t+1$ distinct vertices with consecutive vertices adjacent to each other. If there is a path connecting any two vertices of the graph $\mathcal{G}$, the graph $\mathcal{G}$ is considered connected. $\mathcal{L}=\mathcal{D}-\mathcal{A}$ is defined as Laplacian matrix of a graph $\mathcal{G}$. $\mathcal{D}$ is referred to as a degree matrix of $\mathcal{G}$. The Laplacian matrix $\mathcal{L}$ of a connected graph has a single zero eigenvalue, and $1_N$ is the associated eigenvector. Furthermore, $0={\lambda }_1\left(\mathcal{G}\right)\le \cdots \le {\lambda }_N\left(\mathcal{G}\right)$ if $\mathcal{G}$ is connected, where ${\lambda }_i\left(\mathcal{G}\right)$$(i=1,2,\cdots ,N)$ is the eigenvalue of a Laplacian matrix $\mathcal{L}$.

Lemma 1

([25]). For a strongly connected graph $\mathcal{G}$, denote $t_i(i=1,2,\cdots ,N)$ as the left eigenvector with a 0 eigenvalue, and $T=diag\{t_1,t_2,\cdots \cdots ,t_N\}$, then, we can obtain $T\mathcal{L}+{\mathcal{L}}^{T}T\ge 0$.

Lemma 2

([26]). Suppose that graph $\mathcal{G}$ is a strongly connected and balanced graph, and the algebraic connectivity of $\mathcal{G}$ is defined by $a\left(\mathcal{L}\right)=\underset{t^{T}x=0,x\ne 0}{min}\frac{x^T(T\mathcal{L}+{\mathcal{L}}^{T}T)x}{2x^{T}Tx}$ where $T=diag\{t_1,t_2,\cdots \cdots ,t_N\}$. Then, we can obtain $a\left(\mathcal{L}\right)={\lambda }_{min}\left(\frac{He\left(\mathcal{L}\right)}{2}\right)$.

2.2. System Model

For a networked CPS, consider a network with N subsystems. The following is the ith subsystem with disturbances and noise:

$$\begin{array}{c}\begin{array}{cc}\hfill x_i(k+1)& =A{x}_i\left(k\right)+B{u}_i\left(k\right)+p_i\left(k\right),\hfill \\ \hfill y_i\left(k\right)& =C{x}_i\left(k\right)+q_i\left(k\right),\hfill \end{array}\end{array}$$

(1)

where $x_i\left(k\right)\in R^n$ is the state, $u_i\left(k\right)\in R^n$ is the control input, $y_i\left(k\right)\in R^m$ is the output, $p_i\left(k\right)\in R^n$ is the disturbance, $q_i\left(k\right)\in R^m$ is the noise. $A\in R^{n\times n}$, $B\in R^{n\times n}$ and $C\in R^{m\times n}$ are constant matrices.

In the following, it is supposed that the communication topology of the subsystem is strongly connected. The global dynamic system of (1) can be given:

$$\begin{array}{c}\begin{array}{cc}\hfill x(k+1)& =\overline{A}x\left(k\right)+\overline{B}u\left(k\right)+p\left(k\right),\hfill \\ \hfill y\left(k\right)& =\overline{C}x\left(k\right)+q\left(k\right),\hfill \end{array}\end{array}$$

(2)

where $x={[x_1^T,\cdots ,x_N^T]}^T$, $u={[u_1^T,\cdots ,u_N^T]}^T$, $y={[y_1^T,\cdots ,y_N^T]}^T$, $q={[q_1^T,\cdots ,q_N^T]}^T$, $p={[p_1^T,\cdots ,p_N^T]}^T$, $\overline{A}=diag\left\{\underset{N}{\underbrace{A,\cdots ,A}}\right\}$, $\overline{B}=diag\left\{\underset{N}{\underbrace{B,\cdots ,B}}\right\}$ and $\overline{C}=diag\left\{\underset{N}{\underbrace{C,\cdots ,C}}\right\}$.

Since the information can be received by a single subsystem from its neighborhood, we consider the case of switching topology of the observer system and then present $\varphi \left(k\right)$, a stepwise constant function that takes values from the finite collection $S=\left\{\begin{array}{cccc}1,& 2,& \cdots ,& N\end{array}\right\}$. The observer dynamics of ith subsystem are:

$$\begin{array}{c}\begin{array}{cc}\hfill {\widehat{x}}_i(k+1)& =A{\widehat{x}}_i\left(k\right)+B{u}_i\left(k\right)+L_i(y_i\left(k\right)-C{\widehat{x}}_i\left(k\right))+{\kappa }_{\varphi \left(k\right)}{M}_i\sum _{j=1}^N{a}_{ij}^{\varphi \left(k\right)}({\widehat{x}}_j\left(k\right)-{\widehat{x}}_i\left(k\right)),\hfill \end{array}\end{array}$$

(3)

where ${\kappa }_{\varphi \left(k\right)}$ is the coupled gain that needs to be designed and $M_i$ and $L_i$ are observer gains of the ith subsystem, and $a_{ij}^{\varphi \left(k\right)}$ represents the connectivity weight from subsystem i to subsystem j at moment k.

By subtracting (1) from (3), we obtain the error dynamics of a single subsystem:

$$\begin{array}{c}\begin{array}{cc}\hfill e_i\left(k\right)& =x_i\left(k\right)-{\widehat{x}}_i\left(k\right),\hfill \\ \hfill e_i(k+1)& =(A-L_{i}C)e_i\left(k\right)-{\kappa }_{\varphi \left(k\right)}{M}_i\sum _{j=1}^N{a}_{ij}^{\varphi \left(k\right)}({\widehat{x}}_j\left(k\right)-{\widehat{x}}_i\left(k\right))+D_i{d}_i\left(k\right),\hfill \end{array}\end{array}$$

(4)

with $D_i=\left[\begin{array}{cc}I& -L_i\end{array}\right]$ and $d_i\left(k\right)=\left[\begin{array}{c}{p}_i\left(k\right)\\ q_i\left(k\right)\end{array}\right]$.

Then we can obtain the dynamic system of the global observer

$$\begin{array}{c}\begin{array}{cc}\hfill \widehat{x}(k+1)& =\overline{A}\widehat{x}\left(k\right)+\overline{B}u\left(k\right)+{\overline{L}}_{\varphi \left(k\right)}(y\left(k\right)-\overline{C}\widehat{x}\left(k\right))\hfill \\ & +{\kappa }_{\varphi \left(k\right)}{\overline{M}}_{\varphi \left(k\right)}({\mathcal{L}}_{\varphi \left(k\right)}\otimes I_N)(x\left(k\right)-\widehat{x}\left(k\right)),\hfill \end{array}\end{array}$$

(5)

with $\overline{L}=diag\left\{{\underbrace{L_1,\cdots ,L_N}}_N\right\}$ and $\overline{M}=diag\left\{{\underbrace{M_1,\cdots ,M_N}}_N\right\}$.

The global error system $e\left(k\right)={[e_1^T\left(k\right),\cdots ,e_N^T\left(k\right)]}^T$ is as follows

$$\begin{array}{c}\begin{array}{cc}\hfill e(k+1)& =(\overline{A}-{\overline{L}}_{\varphi \left(k\right)}\overline{C}-{\kappa }_{\varphi \left(k\right)}{\overline{M}}_{\varphi \left(k\right)}({\mathcal{L}}_{\varphi \left(k\right)}\otimes I_N))e\left(k\right)+\overline{D}d\left(k\right),\hfill \end{array}\end{array}$$

(6)

and the compact form of (6) can be writen as

$$\begin{array}{cc}e(k+1)& ={\Gamma }_{\varphi \left(k\right)}e\left(k\right)+\overline{D}d\left(k\right),\end{array}$$

(7)

where $\overline{D}=diag\left\{\underset{N}{\underbrace{D_1,\cdots ,D_N}}\right\}$ and ${\Gamma }_{\varphi \left(k\right)}=\overline{A}-{\overline{L}}_{\varphi \left(k\right)}\overline{C}-{\kappa }_{\varphi \left(k\right)}{\overline{M}}_{\varphi \left(k\right)}({\mathcal{L}}_{\varphi \left(k\right)}\otimes I_N).$

Definition 1

([27]). If the following condition holds,

$$\begin{array}{c}{∥e\left(k\right)∥}_2\le \varkappa \sqrt{{∥d∥}_{\infty }^2+V\left(0\right){\omega }^k},\end{array}$$

(8)

where $\varkappa >0$ and $0<\omega <1$, $V\left(0\right)=e^T\left(0\right)P_{\varphi \left(k\right)}e\left(0\right)$ and $P_{\varphi \left(k\right)}\succ 0$. Then the observer (5) is a $L_{\infty }$ robust observer for system (2).

Definition 2

([28]). Let $N_{\psi }(k_1,k_2)$ be the switching times of $\psi \left(k\right)$ across the range $[k_1,k_2)$. If

$$\begin{array}{c}\begin{array}{c}\hfill N_{\psi }(k_1,k_2)\le N_0+\frac{k_2-k_1}{\tilde{\tau }},\end{array}\end{array}$$

(9)

for given $\tilde{\tau }>0$ and $N_0\ge 0$, $\tilde{\tau }$ is the average dwell time (ADT) of the switching signal $\psi \left(k\right)$. In this paper, we let $N_0=0$.

Definition 3

([29]). The definition of an α-dimensional zonotope is as follow

$$\begin{array}{c}\begin{array}{c}\hfill \mathsf{\Omega }=\nu \oplus H{Y}^{\alpha }=\nu +Hz,z\in Y^{\alpha },\end{array}\end{array}$$

(10)

where $\nu \in R^{\iota }$ represents a given vector, $H\in R^{\iota \times \alpha }$ represents a given matrix, $Y^{\alpha }$ is a unitary box made up of α unitary intervals and $Y^{\alpha }$ is a unitary interval. In the sequel, the zonotope Ω is represented as $\langle \nu ,H\rangle$ for the sake of simplicity.

Lemma 3

([30]). The following equation is satisfied for a zonotope defined in (10):

$$\langle {\nu }_1,H_1\rangle \oplus \langle {\nu }_2,H_2\rangle =\langle {\nu }_1+{\nu }_2,[H_1,H_2]\rangle ,$$

$$W\odot \langle \nu ,H\rangle =\langle W\nu ,WH\rangle ,$$

$$\langle \nu ,H\rangle \subseteq \langle \nu ,\overline{H}\rangle ,$$

where $H_1$ and $H_2$ represent the shape matrices of each of the zonotopes, and ${\nu }_1$ and ${\nu }_2$ are their centers. $\overline{H}\in R^{N\times N}$ means a diagonal matrix with ${\overline{H}}_{M,M}={\sum }_{i=1}^{\alpha }\left|H_{M,i}\right|,M=1,...,\rho$.

Remark 1.

$\overline{H}$ can be expressed in the form shown below:

$$\overline{H}=\left[\begin{array}{ccc}\sum _{i=1}^{\alpha }\left|H_{1,i}\right|& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \sum _{i=1}^{\alpha }\left|H_{\rho ,i}\right|\end{array}\right].$$

(11)

If $\alpha >\rho$, then $\langle \nu ,H\rangle \subseteq \langle \nu ,\overline{H}\rangle$ is applied to reduce the order of high order zonotopes.

Remark 2.

For zonotopes ${\mathsf{\Omega }}_M\subset R^N,M=1,...,\rho$, the Minkowski sum of them is

$${\displaystyle \underset{M=1}{\overset{\rho }{⨁}}{\mathsf{\Omega }}_M={\mathsf{\Omega }}_1\oplus {\mathsf{\Omega }}_2\oplus \cdots \oplus {\mathsf{\Omega }}_{\rho }.}$$

(12)

Definition 4

([19]). For an α-order zonotope, there is an interval hull Ω that could contain Ω in its entirety:

$$\mathsf{\Omega }\subset Box\left(\mathsf{\Omega }\right)=[a,b],$$

(13)

with $a={[a_1,\cdots ,a_{\alpha }]}^T$, $b={[b_1,\cdots ,b_{\alpha }]}^T$ and $Box(·)$ stands for the interval hull. For any zonotope, the interval hull is the smallest interval vector.

Lemma 4

([29]). If $\mathsf{\Omega }=\langle \nu ,H\rangle$, the components of its interval hull are

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill a_i& {\displaystyle ={\nu }_i-\sum _{j=0}^{\alpha }\left|H_{ij}\right|,i=1,\cdots ,\iota ,}\hfill \\ \hfill b_i& {\displaystyle ={\nu }_i+\sum _{j=0}^{\alpha }\left|H_{ij}\right|,i=1,\cdots ,\iota .}\hfill \end{array}\end{array}\right.$$

(14)

Lemma 5

([29]). Given zonotopes ${\mathsf{\Omega }}_i$, $i=1,2,\cdots ,m$

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle Box\left(\underset{m=1}{\overset{\rho }{⨁}}{\mathsf{\Omega }}_m\right)=\underset{m=1}{\overset{\rho }{⨁}}\left(Box\left({\mathsf{\Omega }}_m\right)\right).}\end{array}\end{array}$$

(15)

Lemma 6

([31]). For the given symmetric matrix $\left[\begin{array}{cc}\mathbb{A}& \mathbb{B}\\ {\mathbb{B}}^T& \mathbb{C}\end{array}\right]$, the following inequalities are equivalent:

$\left(1\right)\left[\begin{array}{cc}\mathbb{A}& \mathbb{B}\\ {\mathbb{B}}^T& \mathbb{C}\end{array}\right]\prec 0,$

$\left(2\right)\mathbb{C}\prec 0;\mathbb{A}-\mathbb{B}{\mathbb{A}}^{-1}{\mathbb{B}}^T\prec 0,$

$\left(3\right)\mathbb{A}\prec 0;\mathbb{C}-{\mathbb{B}}^T{\mathbb{C}}^{-1}\mathbb{B}\prec 0.$

Assumption 1.

The initial state of the ith subsystem satisfies the following condition

$${\underline{x}}_i\left(0\right)\le x_i\left(0\right)\le {\overline{x}}_i\left(0\right).$$

(16)

Assumption 2.

The disturbances and output noise in system (6) are bounded, which are:

$${∥d\left(k\right)∥}_2\le {∥d∥}_{\infty },$$

(17)

where ${∥d∥}_{\infty }$ is a constant.

Assumption 3.

The initial state, disturbance and noise, and initial error are represented by $x\left(0\right)$, $d\left(0\right)$ and $e\left(0\right)$, which can be wrapped as in the equations below:

$$\begin{array}{c}\begin{array}{cc}\hfill x\left(0\right)\in & \langle {\nu }_0,H_0\rangle =\mathcal{X}\left(0\right),\hfill \\ \hfill d\left(0\right)\in & \langle 0,E_d\rangle =\mathcal{D}\left(0\right),\hfill \\ \hfill e\left(0\right)\in & \langle 0,H_0\rangle =\mathcal{E}\left(0\right),\hfill \end{array}\end{array}$$

(18)

where $H_0$ is given matrix and $E_d=diag\left\{d_i\right\}$.

3. Main Results

In this part, we provide sufficient conditions for the observer with $L_{\infty }$ property for CPS. Then, a reachability analysis methodology is used to design the distributed interval observer.

Theorem 1.

Let η and θ be two given constants with $0<\eta <1$ and $\theta >1$. If there exists a constant χ and matrices $P_n\succ 0\in R^{N\times N}$, $P_m\succ 0\in R^{N\times N}$ such that

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill & min{\chi }^2,\hfill \\ \hfill & subjectto:\hfill \\ \hfill & \left[\begin{array}{ccc}-\eta P_n& \ast & \ast \\ 0& -{\chi }^2{I}_N& \ast \\ P_n\overline{A}-W_n\overline{C}-{\kappa }_n{Q}_n({\mathcal{L}}_n\otimes I_N)& P_n\overline{D}& -P_n\end{array}\right]\prec 0,\hfill \\ \hfill & P_n\prec \theta P_m,\hfill \\ \hfill & \frac{1}{\tilde{\tau }}+\frac{ln\eta }{ln\theta }<0,\hfill \\ & {\kappa }_n>\frac{1}{a\left({\mathcal{L}}_n\right)},\hfill \end{array}\end{array}\right.$$

(19)

where $W_n=P_n{\overline{L}}_n,Q_n=P_n{\overline{M}}_n$, $n\ne m,\forall n,m\in S$ and $\tilde{\tau }$ satisfying ADT. Then (5) is a robust $L_{\infty }$ observer for system (3).

Proof. 

Please see the Appendix A.    □

Remark 3.

In Theorem 1, the parameter ${\kappa }_n$ depends on $a\left({\mathcal{L}}_n\right)$. The algebraic connectivity $a\left({\mathcal{L}}_n\right)$ of a graph tends to increase with graph stability. As the stability of the graph increases, there will be a greater range of options for ${\kappa }_n$.

Remark 4.

In practice, the ADT $\tilde{\tau }$ and the disturbance attenuation level ${\chi }^2$ stand for the performance of the observer. Owing to the fact that the ADT ${\displaystyle \frac{1}{\tilde{\tau }}+\frac{ln\eta }{ln\theta }<0}$ depends on η and θ, it is necessary to select suitable values for η and θ to minimize the ADT $\tilde{\tau }$.

After completing the design of the optimal $L_{\infty }$ observer, we need to construct the interval observer by designing the interval hull that can completely wrap the system disturbances, noise, and errors. The real-time error from (5) can be wrapped by the zonotope as

$$\begin{array}{c}\begin{array}{cc}\hfill e\left(k\right)\in & \langle {\nu }_0,H\left(k\right)\rangle =\mathcal{E}\left(k\right).\hfill \end{array}\end{array}$$

(20)

Then, we add the reachability analysis methodology to the distributed $L_{\infty }$ observer designed in Theorem 1 and then give the following interval observer:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill \overline{x}& =\widehat{x}+\overline{e}\hfill \\ \hfill \underline{x}& =\widehat{x}+\underline{e}\hfill \end{array}\end{array}\right.$$

(21)

where $\overline{e}$ and $\underline{e}$ are the upper and lower bounds of $e\left(k\right)$.

Theorem 2.

An interval estimate of the system state is provided by (22), if Assumption 3 holds, then the error $e\left(k\right)$ has the following upper and lower bounds:

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle \left[\begin{array}{cc}\underline{e}\left(k\right),& \overline{e}\left(k\right)\end{array}\right]=Box\left(\left(\prod _{n=0}^{k-1}{\Gamma }_n\right)\mathcal{E}\left(0\right)\right)\oplus \underset{n=0}{\overset{k-1}{⨁}}Box\left(\left(\prod _{m=1}^{k-1}{\Gamma }_m\right){\Gamma }_n^{-1}D\mathcal{D}\right),}\end{array}\end{array}$$

(22)

where $Box\left(\mathcal{E}\left(0\right)\right)=\left[\begin{array}{cc}\underline{e}\left(0\right),& \overline{e}\left(0\right)\end{array}\right]$.

Proof. 

Based on Assumption 3, we can obtain $\mathcal{E}\left(1\right),$

$$\begin{array}{c}\begin{array}{c}\hfill \mathcal{E}\left(1\right)={\Gamma }_{\varphi \left(0\right)}\mathcal{E}\left(0\right)\oplus D\mathcal{D}\end{array}\end{array}.$$

(23)

When $k=1$, then we can obtain

$$\begin{array}{c}\begin{array}{cc}\hfill \mathcal{E}\left(2\right)& ={\Gamma }_{\varphi \left(1\right)}\mathcal{E}\left(1\right)\oplus D\mathcal{D}\hfill \\ & ={\Gamma }_{\varphi \left(1\right)}{\Gamma }_{\varphi \left(0\right)}\mathcal{E}\left(0\right)\oplus {\Gamma }_{\varphi \left(1\right)}D\mathcal{D}\oplus D\mathcal{D}.\hfill \end{array}\end{array}$$

(24)

Iterating the above process yields

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle \mathcal{E}(k+1)=\left(\prod _{n=0}^{k-1}{\Gamma }_n\right)\mathcal{E}\left(0\right)\oplus \underset{n=0}{\overset{k-1}{⨁}}\left(\prod _{m=1}^{k-1}{\Gamma }_m\right){\Gamma }_n^{-1}D\mathcal{D}.}\end{array}\end{array}$$

(25)

Then the interval hull below describes the set $\mathcal{E}\left(k\right)$

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle \mathcal{E}\left(k\right)=\left[\begin{array}{cc}\underline{e}\left(k\right),& \overline{e}\left(k\right)\end{array}\right]=Box\left(\left(\prod _{n=0}^{k-1}{\Gamma }_n\right)\mathcal{E}\left(0\right)\right)\oplus \underset{n=0}{\overset{k-1}{⨁}}Box\left(\left(\prod _{m=1}^{k-1}{\Gamma }_m\right){\Gamma }_n^{-1}D\mathcal{D}\right).}\end{array}\end{array}$$

(26)

   □

The proof of Theorem 2 is now completed.

Based on Theorems 1 and 2, the design of a distributed interval observer with switching topology can be implemented by Algorithm 1.

Remark 5.

According to Theorem 1, the robust $L_{\infty }$ observer is designed to reduce the effect of outside disturbance and output noise. It is obvious that a bounded interval hull exists based on the result of Theorem 1, which can include errors and disturbances. Then, using the given interval hull $\mathcal{E}\left(0\right)$ as a starting point, Theorem 2 gives the interval hull $\mathcal{E}\left(k\right)$. We propose a reachable set analysis technique by combining Theorem 1 with Theorem 2.

Algorithm 1: Algorithm for designing the distributed interval observer with switching topology.

(1) Model CPSs with given disturbances and noise. (2) Design distributed observers for subsystems. (3) Select the appropriate ${\kappa }_n$ according to the switching topology. (4) Solve the LMI problem in Theorem 1 using the information of the bounds of disturbances and noise. (5) Calculate the gains of the observers by ${\overline{M}}_n=P_n^{-1}{Q}_n,{\overline{L}}_n=P_n^{-1}{W}_n$. (6) Determine the ADT by given η and θ. (7) Obtain the zonotopes of the initial value according to (18) (8) Transform the zonotope at the moment $k=n$ into an interval hull starting from $n=0$. (9) Iterate the interval hull in step (6). (10) Obtain the interval hull at the moment $k=n+1$. (11) The interval observer is obtained through (21).

Remark 6.

If the modeling uncertainty is taken into consideration, the new model of this paper is as follows:

$$\begin{array}{c}\begin{array}{cc}\hfill x_i(k+1)& =(A+\Delta A\left(k\right))x_i\left(k\right)+(B+\Delta B\left(k\right))u_i\left(k\right)+p_i\left(k\right),\hfill \\ \hfill y_i\left(k\right)& =(C+\Delta C\left(k\right))x_i\left(k\right)+q_i\left(k\right),\hfill \end{array}\end{array}$$

where $\Delta A\left(k\right),\Delta B\left(k\right)$ and $\Delta C\left(k\right)$ represent the uncertainty of the system, respectively. Then, we may use the norm-bounded condition on $\Delta A\left(k\right),\Delta B\left(k\right)$ and $\Delta C\left(k\right)$ to design the $L_{\infty }$ interval observer. However, it is not an easy task to construct the interval hull of the corresponding error system since the time-varying terms are contained. In the near future, we will deal with systems with model uncertainty with the method proposed in this paper.

Remark 7.

In Theorems 1 and 2, limited by the current knowledge of authors, this paper only gives sufficient conditions for observer design. In the future, we will study the necessary conditions for the design of interval observers, and in conjunction with this paper, we will give the necessary and sufficient conditions for the design of interval observers.

4. Simulation

Among CPSs, UAVs have recently achieved widespread application. In this section, we simulate through a networked CPSs with four UAVs.

Referring to [11], the dynamical system of each UAV is as follows

$$\begin{gathered} \begin{array}{cc}\begin{array}{c}\hfill \left[\begin{array}{c}{\dot{\alpha }}_i\left(t\right)\\ {\dot{\beta }}_i\left(t\right)\end{array}\right]\end{array}& =A\left[\begin{array}{c}{\alpha }_i\left(t\right)\\ {\beta }_i\left(t\right)\end{array}\right]\end{array}+B{u}_i\left(k\right)+p_i\left(k\right), \\ {y}_i\left(k\right)=C\left[\begin{array}{c}{\alpha }_i\left(t\right)\\ {\beta }_i\left(t\right)\end{array}\right]+q_i\left(k\right), \end{gathered}$$

(27)

where the pitch rate and angle of attack of each UAV are indicated by ${\beta }_i\left(t\right)$ and ${\alpha }_i\left(t\right)$. The schematic of each UAV is shown in Figure 1, and the detailed derivation of the dynamics model is omitted here.

Table 1. The output noise and the external disturbance.

Subsystem	Output Noise	External Disturbance
1	$0.1cos\left(k\right)$	$0.2+0.1cos\left(0.5k\right)$
2	$0.1sin\left(k\right)$	$0.1cos\left(0.1k\right)$
3	$0.01cos\left(k\right)$	$cos\left(0.2\pi k\right)+0.3sin\left(0.2\pi k\right)$
4	$0.05sin\left(k\right)$	$cos\left(0.3\pi k\right)+0.1sin\left(0.3\pi k\right)$

displays the output noise and the external disturbance borrowed from [11], and the partial matrix values are as follows:

$$A=\left[\begin{array}{cc}0.8825& 0.0987\\ -0.8458& 0.9122\end{array}\right],B=\left[\begin{array}{cc}-0.0194& -0.0036\\ -1.9290& -0.3803\end{array}\right],C=\left[\begin{array}{cc}1& 0.2\end{array}\right].$$

For illustrative purposes, Figure 2 shows a switching communication topology ${\mathcal{G}}_1$,${\mathcal{G}}_2$ and ${\mathcal{G}}_3$ with four UAVs. Figure 3 displays the change in the switching signal $\varphi \left(k\right)$. Then, from Figure 2, we can give the corresponding matrix ${\mathcal{L}}_n$:

$${\mathcal{L}}_1=\left[\begin{array}{cccc}1& -1& 0& 0\\ -1& 3& -1& -1\\ 0& -1& 2& -1\\ 0& -1& -1& 2\end{array}\right],{\mathcal{L}}_2=\left[\begin{array}{cccc}1& -1& 0& 0\\ 0& 2& -1& -1\\ 0& 0& 1& -1\\ -1& -1& 0& 2\end{array}\right],{\mathcal{L}}_3=\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 0& -1\\ 0& 0& 1& -1\\ -1& -1& -1& 3\end{array}\right].$$

From Lemmas 1 and 2, we can obtain $a\left({\mathcal{L}}_1\right)=a\left({\mathcal{L}}_2\right)=a\left({\mathcal{L}}_3\right)=1$, then we have $\kappa >1$. In this simulation, ${\kappa }_1=1.8,{\kappa }_2=2.3,{\kappa }_3=3.2$ are chosen. We can determine $L_n$ and $M_n$ by solving (19), the observer gains for 1th subsystem are listed below

$$L_1=\left[\begin{array}{cc}47.8095& -46.2386\end{array}\right],L_2=\left[\begin{array}{cc}-9.5528& 47.7273\end{array}\right],L_3=\left[\begin{array}{cc}17.8244& -11.9289\end{array}\right].$$

$$M_1=\left[\begin{array}{cc}-41.7480& -36.5506\\ -1.8874& -6.9585\end{array}\right],M_2=\left[\begin{array}{cc}10.3302& 8.4984\\ -79.5140& -71.1991\end{array}\right],M_3=\left[\begin{array}{cc}-11.7589& -7.5478\\ -6.1219& -9.0834\end{array}\right].$$

The peak estimation error of Angle of Attack and Pitch Rate of 1th subsystem are $0.4184$ and $0.7668$, respectively. The disturbance attenuation level ${\chi }^2$ and ADT $\tilde{\tau }$ are as follows

$${\chi }^2=0.4984,\tilde{\tau }\le 0.1763.$$

Below are the results of the numerical simulation. The states of the original systems and the upper and lower observers are depicted in Figure 4 and Figure 5. $x_{ij}$ are the original states of the subsystems. $u_{ij}$ and $v_{ij}$ reflect the bounds produced via the interval hull technique used in this paper and the monotone system method in [32], where i denotes the ith subsystem and j denotes the jth state. Figure 6 shows the error system for a single subsystem, $e{u}_{ij}$, and $e{v}_{ij}$ reflect the observation error through the method used in this paper and the monotone system method in [32].

It is evident that the states of CPS are completely surrounded by those of the upper and lower observers. From Figure 4, Figure 5 and Figure 6, it can be seen that the distributed interval observer designed in this paper has higher estimation accuracy compared to the traditional monotone system approach in [32]. We design the optimal robust observer using the $L_{\infty }$ technique, which reduces the design requirements of the observer, unlike the $H_{\infty }$ technique applied in [24]. Different from [23] and most of the work, we consider for the first time the case of switching topology among distributed interval observers. Thus, the proposed distributed interval observers design method for CPSs is effective and feasible.

Remark 8.

For networked CPSs with perturbations and noise, we propose a class of distributed interval observer design methods with switching topology that combine the design of $L_{\infty }$ observers with interval hulls. This class of methods eliminates the need to consider the error system to be Schur and, therefore, eliminates the need to use coordinate transformation methods, significantly reducing the conservatism of the estimation. It can be seen from the observer form (3), sufficient conditions (19), and the proof of Theorem 1. In addition, the estimation accuracy is better than that in [32], as it can also be seen from (22).

5. Conclusions

In this paper, a distributed interval estimation method for uncertain CPSs is investigated. Due to the communication burden of networked CPSs, we consider the case of switching topology among distributed interval observers. To improve the accuracy of the estimation, a reachability analysis is introduced in conjunction with the $L_{\infty }$ technique. Finally, the validity of the main results of this paper is verified by one example. In the future, we may focus our research on the distributed interval estimation of the attacked CPSs.