Resilient Time-Varying Output Formation Tracking of Linear Multi-Agent Systems Against Unbounded FDI Sensor Attacks and Unreliable Digraphs

Notation: let $\mathbb{R}$, $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times m}$ be the sets of the real numbers, real $n$-dimensional vectors and real $n\times m$ matrices, respectively. Let $\mathbb{R}_{>0}$ be the set of all positive real numbers and $\mathbb{N}$ denote the set of all positive natural numbers. Let 0 (1) be the vector with all zeros (ones) with proper dimensions, respectively. Denote col$(x_{1},...,x_{n})$ and diag$\{a_{1},...,a_{n}\}$ as a column vector with all entries $x_{i}$ and a diagonal matrix with all entries $a_{i}$, $i=1,\cdots,n$, respectively. $\otimes$ and $\left\|\cdot\right\|$ are the Kronecker product and Euclidean norm, respectively. Given a real matrix $M=M^{T}$, let $M>0$ be positive definite. Let $\lambda_{\min}(M)$, $\lambda_{\max}(M)$ be its minimum and maximum eigenvalues, respectively. Besides, $\sigma_{\max}(M)$ represents the maximum singular value of a matrix $M$.

Graph Theory: Let $\mathcal{G}$ $=$ $\left\{\mathcal{V},\mathcal{E}\right\}$ be a graph and $\mathcal{V}$ $\in$ $\left\{1,...,N\right\}$ be the set of vertices. The set of edges is denoted as $\mathcal{E}$ $\subseteq$ $\mathcal{V\times V}$. $\mathcal{N}_{i}$ $=$ $\left\{j\in\mathcal{V\mid}(j,i)\in\mathcal{E}\right\}$ is the neighborhood set of vertex $i$. For a directed graph $\mathcal{G}$, $(i,j)\in\mathcal{E}$ means that the information of node $i$ is accessible to node $j$, but not conversely. The matrix $\mathcal{A}=\left[a_{ij}\right]$ denotes the adjacency matrix of $\mathcal{G}$, where $a_{ij}>0$ if $(j,i)\in\mathcal{E}$, else $a_{ij}=0$. The matrix $\mathcal{L}=[l_{ij}]$ is called the Laplacian matrix of $\mathcal{G}$, where $l_{ii}=\sum^{N}_{j=1}a_{ij}$ and $l_{ij}=-a_{ij}$, $i\neq j$. The digraph $\mathcal{G}$ is said to contain a spanning tree if there exists a node from which there are directed paths to all other nodes. For a general directed graph, $\mathcal{L}$ is not necessarily symmetric. Let $\mathcal{\bar{G}}=(\mathcal{\bar{V}},\mathcal{\bar{E}})$ be a directed graph of a leader-follower network, where $\bar{\mathcal{E}}\subseteq\bar{\mathcal{V}}\times\bar{\mathcal{V}}$, $\mathcal{\bar{V}}=\{0,\cdots,N\}$, and the node $0$ is associated with the leader. Then, $\bar{\mathcal{N}}_{i}=\{j\neq i,(j,i)\in\mathcal{\bar{E}}\}$ is the neighbor set of node $i$. Clearly, $\mathcal{G}$ is a subgraph of $\mathcal{\bar{G}}$, where $\mathcal{E}$ is obtained from $\mathcal{\bar{E}}$ by removing all the edges between the node $0$ and the nodes in $\bar{\mathcal{V}}$. Define the Laplacian matrix of $\mathcal{\bar{G}}$ as $\mathcal{\bar{L}}=[0,\mathbf{0}^{T};-\mathcal{B}\mathbf{1},H]$, where $\mathcal{B}$ is a diagonal matrix with its $i$-th diagonal element being $a_{i0}$, ($a_{i0}>0$, if $(0,i)\in\mathcal{\bar{E}}$, and $a_{i0}=0$, otherwise), and $H=\mathcal{L}+\mathcal{B}$ is an information exchange matrix.

Consider a multi-agent system consisting of $N$ followers governed by the following heterogeneous linear dynamics:

where $x_{i}\in\mathbb{R}^{n_{i}}$ denotes the state of agent $i$, $u_{i}\in\mathbb{R}^{m_{i}}$ denotes the control input to agent $i$, $y_{i}\in\mathbb{R}^{p}$ is its output, and $A_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$, $B_{i}\in\mathbb{R}^{n_{i}\times m_{i}}$, $C_{i}\in\mathbb{R}^{p\times n_{i}}$ are constant system matrices.

The dynamics of the leader are described by

where $x_{0}\in\mathbb{R}^{r}$ is the leader’s state, $y_{0}\in\mathbb{R}^{p}$ is its measurable output, and $A_{0}\in\mathbb{R}^{r\times r},C_{0}\in\mathbb{R}^{p\times r}$ are constant matrices.

For this leader-follower system, suppose that the pair $(A_{i},B_{i})$ is stabilizable and $(A_{0},C_{0})$ is detectable. Moreover, the following linear matrix equation has a solution $(X_{i},U_{i})$ for each agent $i$

To specify the desired time-varying output formation tracking, a time-varying vector $h(t)=\text{col}(h_{1}(t),\cdots,h_{N}(t))$ is introduced, where each $h_{i}(t)$ is generated by

and the pair $(A_{hi},C_{hi})$ satisfies

where $(X_{hi},U_{hi})$ is the solution of (5) for a formation shape (4).

In a large-scale cyber-physical system, the wireless communication may not be always reliable due to the physical uncertainties such as failures, quantization errors, and packet losses in a digital communication. Hence, we study an unreliable network where all agents’ communication links are time-varying and switching. Let $\sigma(t):[0,\infty)\rightarrow\Xi=\{1,2,\cdots,\delta\}$ represent a piecewise constant switching signal used to describe the switching among topologies $\bar{\mathcal{G}}_{\sigma(t)}$ and $\delta\in\mathbb{N}$ indicates its cardinality. Suppose that there exists a sequence $t_{0}=0<t_{1}<t_{2}<\cdots$ with $t_{k+1}-t_{k}\geq\tau>0$ for a dwell time $\tau$ and $k\in\mathcal{\mathbb{N}}$ so that during $[t_{k},t_{k+1})$, $\sigma(t)=i$ for $i\in\Xi$ and this graph $\bar{\mathcal{G}}_{i}$ is time-invariant [12].

Suppose that $\Xi$ is divided into two subsets $\Xi_{c}$ and $\Xi_{b}$, i.e., $\Xi=\Xi_{c}\cup\Xi_{b}$, where $\Xi_{c}=\{1,2,\cdots,\bar{\delta}\}$ is used to index the set of graphs $\bar{\mathcal{G}}_{i}$ that contain a directed spanning tree with the leader being the root, while $\Xi_{b}=\{\bar{\delta}+1,\cdots,\delta\}$ indexes the set of graphs $\bar{\mathcal{G}}_{i}$ that are allowed to be disconnected. Then, we denote $T_{t_{\varepsilon}}^{c}(t)$ and $T_{t_{\varepsilon}}^{b}(t)$ as the total activation time when $\sigma(\varepsilon)\in\Xi_{c}$ and $\sigma(\varepsilon)\in\Xi_{b}$, respectively, during $\varepsilon\in[{t_{\varepsilon}},t)$, $\forall t_{\varepsilon}\geq t_{0}$ [12].

In this part, we describe the model of unknown and unbounded FDI attacks on the sensors of agents as shown in Fig. 1.

This work aims to achieve the resilient time-varying output formation tracking of linear multi-agent systems under FDI attacks and unreliable digraphs. We will develop a distributed estimator-based controller with corrupted individual output information and neighbor-based group output information. The control of agent $i$ is supposed to have the structure as depicted in Fig. 1. In the next section, we will specify the design procedure.

Denote an estimated state named as $\zeta_{i}$ to estimate the leader’s state $x_{0}(t)$. Then, a consensus tracking error is defined as

In light of (8), a reliable distributed leader estimator using only output information is designed with a constant gain matrix $K_{0}$,

Denote the tracking error $\tilde{\xi}_{i}(t)=\zeta_{i}(t)-x_{0}(t)$ and its collective form is given by $\tilde{\xi}(t)=\text{col}(\tilde{\xi}_{1}(t),\cdots,\tilde{\xi}_{N}(t))$. Then, combing (8) and (9) gives rise to the following closed-loop error system

Next, the following lemma is provided to establish a symmetric and positive definite matrix for directed graphs.

For notational convenience, denote

where $\Theta_{i}>0$, $i\in\Xi_{c}$ are defined in Lemma 2, and further, let

where $\Phi>0$ denotes the average of $\Theta_{i}$, $\forall i\in\Xi_{c}$, and moreover, it is not difficult to derive $\Phi<\mu\Theta_{i}$, $\forall i\in\Xi_{c}$.

In this subsection, we propose a novel resilient mechanism to achieve exponential output formation tracking in the presence of FDI sensor attacks in (6). Under sensor attacks, the output $y_{i}(t)$ is corrupted and only the corrupted $y^{c}_{i}(t)=y_{i}(t)+\phi^{a}_{i}y^{a}_{i}(t)$ can be measured. To deal with these FDI sensor attacks, a novel resilient distributed output feedback controller is developed as

where $K_{1i},K_{2i},K_{3i},L_{i}$ are controller and observer gain matrices to be determined later, $\hat{x}_{i}(t)$ represents the observed state of the observer, $\zeta_{i}(t)$ is the estimate of the leader’s state designed in (8) and (9), $h_{i}(t)$ is the time-varying formation vector defined in (4), and $\tilde{y}_{i}(t)$ denotes the measurable error term described by

where $\hat{y}_{i}(t)=C_{i}\hat{x}_{i}(t)$ is an estimation of the uncorrupted output measurement $y_{i}(t)$, and $\hat{y}^{a}_{i}(t)$ is an estimation of unknown sensor attacks $\phi^{a}_{i}y^{a}_{i}(t)$, which is updated through the following design:

where $M_{i}$ is a gain matrix with appropriate dimensions and $f^{a}_{i}(t)$ denotes a compensation signal to be determined later.

Next, we denote two estimated errors:

Then, based on (28)-(30), we can obtain that

To achieve the main objective of this paper, we need to analyze the global exponential convergence of $\tilde{x}_{i}(t)$ and $\tilde{y}^{a}_{i}(t)$. Denote $\varrho_{i}=\text{col}(\tilde{x}_{i}(t),\tilde{y}^{a}_{i}(t))\in\mathbb{R}^{n_{i}+p}$ as an augmented error variable. Further, it follows from (32)-(33) that we can derive the following closed-loop error system described by

Before presenting the convergence of $\varrho_{i}$, the following assumption is made to facilitate stability analysis.

Next, we present the global exponential convergence of $\varrho_{i}=\text{col}(\tilde{x}_{i}(t),\tilde{y}^{a}_{i}(t))$ in the closed-loop error system (34).