Adaptive self-triggered control-based cooperative output regulation of heterogeneous multi-agent systems under sensor and actuator attack

Abstract

The cooperative output regulation problem for heterogeneous multi-agent systems under sensor and actuator attacks with an adaptive self-triggering strategy is considered in this article. The attack considered is the false data injection attack. To resolve the system, a dynamic internal model is introduced for each agent, and a distributed adaptive self-triggering control scheme is proposed. The results show that intermittent communication is possible for all agents using the proposed control mechanism and no agent shows Zeno behavior. The adaptive controller is used to guarantee that the regulated output for the heterogeneous multi-agent systemss is achieved with cooperative uniform ultimate boundedness under both actuator and sensor attacks. Given the whole state of information is unavailable, the adaptive dynamic system is designed based on receivable compromised or uncompromised output information. The effectiveness of the given control scheme is illustrated by two examples.

Introduction

Over the last few decades, research on the cooperative output regulation (COR) of multi-agent systems (MASs) has undergone extensive development.^1–5 Various tasks, such as underwater vehicles, spacecraft formation flying, satellite attitude control, and manipulator manipulation, require collaborative control of MASs. The primary objective of COR is to provide a control scheme that facilitates disturbance rejection and trajectory tracking. In recent years, several notable advancements have been made in this field.^6–10 Nevertheless, the majority of these studies have assumed the proper functioning of both the actuators’ security and the complete communication topology of each agent. In reality, cyber-physical attacks on sensors and actuators of MASs pose potential threats.

Consider the scenario where a microprocessor is embedded within an MAS to handle data acquisition and communication control of the drive. If the sensor network is attacked, the sensor data may be destroyed, and false actuator inputs or misleading data can significantly impact system performance, hindering the achievement of system-level tasks. Therefore, it is crucial to address attacks on sensors and actuators of MASs in cyber-physical systems.^11–16 Asymptotic stability can be guaranteed in networked MASs when external disturbances are uniformly ultimately bounded. Numerous studies have been carried out regarding systems suffering from external attacks. Specifically, a novel distributed adaptive control structure was developed ¹⁵ to ensure asymptotic stabilization of closed-loop dynamic systems. Dogan et al. ¹⁷ proposed a framework for control synthesis, which provides stability guarantees for linear time-invariant MASs with uncertainties in the system. Xu and Haddad ¹⁸ developed a novel control architecture that addresses the problem of networked MASs affected by random external disturbances in the event of damaged actuators and sensors. Moreover, the framework also addressed actuator attacks, and the proposed adaptive scheme guarantees uniform ultimate boundedness (UUB) of the state tracking error. Recently, Hu et al. ¹⁶ investigated the estimation problems under fake data injection attacks, while Gusrialdi et al. ¹⁹ proposed a resilient control method for cooperative systems that were attacked from outside. Mustafa and Modares ²⁰ conducted a rigorous analysis of the adverse effects of attacks on MASs and proposed a mitigation method against sensor and actuator attacks. In addition, optimal denial-of-service (DoS) attacks scheduled with energy constraints were analyzed^21,22; however, the studies focused on how to damage the performance of the system, and no consensus analysis was given. To address this issue, Lu and Yang ²³ presented a distributed control design for MAS, which showed the possibility of achieving consensus under DoS attacks. Jin et al. ²⁴ employed an adaptive control scheme that guaranteed the UUB of the closed-loop system to address sensor and actuator attacks among cyber-physical systems. Meng et al. ²⁵ proposed a novel adaptive controller with leader-following consensus through cooperative UUB for a heterogeneous MAS affected by sensor and actuator attacks. As a result, it is interesting to study the COR in the case of sensor networks under attack.

The existing results of attacks on the sensors and actuators of MAS in cyber-physical systems mostly consider continuous communication, which often leads to significant resource consumption. However, many application scenarios involve agents with limited computing and energy resources. To minimize energy consumption, event-triggered control, an intermittent communication mechanism, has gained widespread attention in this context.^26–37 Recently, researchers proposed distributed event-triggered control strategies for solving consensus problems among both heterogeneous and homogeneous linear MASs.^33,34 In addition, Tan et al. ³⁶ proposed a distributed event-triggered impulsive control method in which the actuator only executes the control input at event-triggering impulsive instants. However, limited results are addressing the COR problem of MAS by adaptive self-triggering mechanisms under sensor and actuator attacks.

From the review of the existing literature, this article faces two main challenges. First, most of the studies above focus on homogeneous MASs under network attacks. However, this article examines heterogeneous MASs under sensor and actuator attacks, which presents a challenge since agents do not necessarily have the same dynamics. In addition, intermittent communication between devices and actuators is necessary due to limited power resources. Based on the study of Cheng and Ugrinovskii, ²⁹ event-triggering mechanisms can be effective in both engineering and theory to overcome this challenge. As a result, an adaptive self-triggered scheme has been designed.

Inspired by the aforementioned works, the main contributions are summarized as follows: (1) compared with the work by Meng et al., ²⁵ where continuous communication among each heterogeneous agent is required, to avoid the limitations of continuous communication, this article presents a novel adaptive self-triggered scheme for heterogeneous MASs under attacks; thus, the communication is intermittent, the continuous communication can be avoided, and the communication loads of the whole systems can be reduced. Furthermore, Zeno behavior is excluded. (2) A novel adaptive observer is designed to obtain the system state information of each agent. This problem is usually encountered in practical applications but rarely considered in existing researches.^17,18 Besides, the state information of each heterogeneous agent can be accurately estimated by the observer even under sensor and actuator attacks, as well as with external disturbances in cyber-physical systems. (3) The points including heterogeneity, antagonistic interactions, external disturbances, attacks, observer-based control, and adaptive self-triggered scheme are given in the meantime. The existing results consider a part of points but not all points.

Notations

${\mathbf{R}}^m$ denotes the $m\times 1$ dimension Euclidean spaces. N represents all non-negative integer sets, ||·|| represents the Euclidean norm. ⊗ denotes the Kronecker product. Matrix $P>0$ means P is positive definite. $I_n$ means $n\times n$ identity matrix. $\mathrm{diag}\{d_1,d_2,...,d_n\}$ stands for a diagonal matrix with scalar $d_i,i=1,...,n$ . $\lambda (M)$ denotes the eigenvalue of matrix M.

Problem formulation

Graph theory

Suppose $\mathcal{G}$ is the undirected graph consisting of $(\tilde{V},\tilde{E},\tilde{A})$ , where $\tilde{V}=\{{\tilde{v}}_1,\dots ,{\tilde{v}}_N\}$ denotes the set of nodes, and $\tilde{E}\subseteq \tilde{V}\times \tilde{V}$ denotes the set of edges. The adjacency matrix is $\tilde{A}=[{\tilde{a}}_{\mathrm{ij}}{]}_{N\times N}$ , if $({\tilde{v}}_j,{\tilde{v}}_i)\in \tilde{E}$ , ${\tilde{a}}_{\mathrm{ij}}>0$ else, ${\tilde{a}}_{\mathrm{ij}}=0$ . In the graph, there is no self-loop, that is, ${\tilde{a}}_{\mathrm{ii}}=0$ . The Laplacian matrix $\mathcal{L}=[{\ell }_{\mathrm{ij}}]\in {\mathbf{R}}^{N\times N}$ is denoted as ${\ell }_{\mathrm{ii}}=\sum _{j\ne i}{\tilde{a}}_{\mathrm{ij}},{\ell }_{\mathrm{ij}}=-{\tilde{a}}_{\mathrm{ij}}$ . For MASs with a leader, the leader is considered as node 0. The communication graph can be described by $\overline{\mathcal{G}}=\mathcal{G}\cup \{0\}$ . Let $\mathcal{B}=\mathrm{diag}\{{\tilde{a}}_{10},{\tilde{a}}_{20},...,{\tilde{a}}_{N0}\}$ be the leader adjacency matrix. $a_{i0}>0$ only if agent i is able to accept information from the leader, otherwise, ${\tilde{a}}_{i0}=0$ .

Problem statement

Initially, we give a linear MAS with N heterogeneous agents. For each agent i, its dynamics can be determined as follows

$$\{\begin{array}{c}{\stackrel{·}{x}}_i(t)=A_i{x}_i(t)+B_i{u}_i(t)+E_{i}v(t)\hfill \\ \multicolumn{1}{c}{y_i(t)=C_i{x}_i(t),i=1,...,N}\end{array}$$

(1)

where $x_i(t)\in {\mathbf{R}}^{n_i}$ , $y_i(t)\in {\mathbf{R}}^p$ , and $u_i(t)\in {\mathbf{R}}^{m_i}$ are state, output, and input vectors, respectively. $A_i$ , $B_i$ , $C_i$ , and $E_i$ are constant matrices that have appropriate dimensions. The state $v(t)$ is assumed with

$$\{\begin{array}{c}\stackrel{·}{v}(t)=A_{0}v(t)\hfill \\ \multicolumn{1}{c}{y_0(t)=C_{0}v(t)}\end{array}$$

(2)

where $v(t)\in {\mathbf{R}}^r$ is the tracked exogenous signal or the disturbance signal to be rejected, and $y_0(t)\in {\mathbf{R}}^p$ is the measurable output. $A_0$ and $C_0$ are constant real matrices. The attacks on actuators are as follows

$$u_i^c(t)=u_i(t)+u_i^a(t)$$

(3)

where $u_i(t)\in {\mathbf{R}}^{m_i}$ is the uncompromised input, $u_i^a(t)\in {\mathbf{R}}^{m_i}$ is the unknown disturbance to the actuator of agent i, and $u_i^c(t)\in {\mathbf{R}}^{m_i}$ is the compromised input available to agent i in equation (1). The sensor attack is given below

$$y_i^c(t)=y_i(t)+y_i^a(t)$$

(4)

where $y_i^c(t)$ denotes the compromised output applied to the feedback, $y_i(t)$ represents the uncompromised measurable output, and $y_i^a(t)$ represents the unknown and captured attack on the sensor.

Remark 1. It can be seen that faults on sensors and actuators are modeled in the same way as in equations (3) and (4). The fact is that as in the work by Meng et al., ²⁵ faults and attacks have qualitatively different characteristics. Faults have random and unintentional consequences, while attacks mislead or cripple the entire network by intentional injection. Faults with random characteristics are easier to detect and mitigate. This is a simplified description of the motivation of the model.

Definition 1. The output $y_0(t)$ of the leader for heterogeneous MASs is cooperatively uniformly ultimately bounded; if there is a compact set $\mathcal{C}$ , such that $y_i(t_0)-y_0(t_0)\in \mathcal{C}$ , for any $i\in \{1,...,N\}$ , then there exists a bound $\delta$ , and a time instance $t_b(\delta ,y_i(t_0)-y_0(t_0))$ can be found, both of which are independent of $t_0$ , such that $||y_i(t)-y_0(t)||\le \delta$ , for $i=1,\dots ,N,\forall t\ge t_0+t_b$ .

In the case of cyber attacks, the COR performance may be seriously affected. The comprised dynamics can be described as follows

$$\{\begin{array}{c}{\stackrel{·}{x}}_i(t)=A_i{x}_i(t)+B_i(u_i(t)+u_i^a(t))+E_{i}v(t)\hfill \\ \multicolumn{1}{c}{y_i^c(t)=C_i{x}_i(t)+y_i^a(t)}\end{array}$$

(5)

Before we proceed, consider the following standard assumptions and lemmas.

Assumption 1. Attacks of both sensor and actuator $y_i^a(t)$ and $u_i^a(t)$ and their derivatives ${\stackrel{·}{y}}_i^a(t)$ and ${\stackrel{·}{u}}_i^a(t)$ have unknown bounds.

Assumption 2. Graph $\overline{\mathcal{G}}$ contains a directed spanning tree, and 0 is regarded as the root node.

Assumption 3. $A_0$ has no eigenvalues with positive real parts.

Assumption 4. The matrix pairs $(A_i,B_i),(A_0,C_0)$ are controllable and detectable, respectively.

Assumption 5. The matrix $(A_i,C_i{A}_i)$ is presumed detectable.

Remark 2. Assumptions 2–5 are standard and necessary conditions for solving OR problems for MASs with linear dynamics.^4–9 The matrix pair $(A_i,B_i)$ , $i\in N$ is stabilizable. The condition in Assumption 5 is replaceable, that is, $(A_i,C_i)$ is controllable, and for $i\in N$ , the condition guarantees that Assumption 5 is satisfied.

Lemma 1 (Young’s inequality). Suppose that there exist two positive real numbers $\overline{p}$ and $\overline{q}$ that satisfy $(1/\overline{p})+(1/\overline{q})=1$ , for two non-negative real numbers $\overline{a}$ and $\overline{b}$ , the inequality $\overline{a}\overline{b}\le ({\overline{a}}^{\overline{p}}/\overline{p})+({\overline{b}}^{\overline{q}}/\overline{q})$ holds.

Lemma 2. Suppose that $\overline{W}$ , $\overline{E}$ , and $\overline{Q}$ are constant matrices that have appropriate dimensions and $\overline{Q}>0$ . Then, for any vector, $\overline{x},\overline{y}\in R^n$ , $2{\overline{x}}^T\overline{W}\overline{E}\overline{y}\le {\overline{x}}^T\overline{W}\overline{Q}{\overline{W}}^T\overline{x}+{\overline{y}}^T{\overline{E}}^T{\overline{Q}}^{-1}\overline{E}\overline{y}$ .

Main results

In this section, a novel adaptive resilient control protocol is introduced for the considered heterogeneous MASs under the sensor and actuator attacks with a self-triggering mechanism. The COR problem for heterogeneous MASs will be achieved with cooperative UUB.

Control law design

The adaptive resilient control protocol is designed as follows

$$u_i(t)=K_{\mathrm{xi}}{\hat{x}}_i(t)+K_{\xi i}{\xi }_i(t)-{\hat{u}}_i^a(t)$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\xi }}_i(t)=A_0{\xi }_i(t)+K{C}_0\sum _{j=1}^N{a}_{\mathrm{ij}}({\xi }_i(t_k^i)-{\xi }_j(t_{k\prime }^j))}\\ \multicolumn{1}{c}{+d_{i0}K{C}_0({\xi }_i(t_k^i)-v(t))}\end{array}$$

(7)

$$\stackrel{·}{{\hat{x}}_i}(t)=A_i{\hat{x}}_i(t)+B_i{u}_i(t)+F_i{\tilde{y}}_i(t)+E_i{\xi }_i(t)+B_i{\hat{u}}_i^a(t)$$

(8)

where $k\prime =\underset{l\in N}{\arg max}\{t_l^j|t_l^j\le t\}$ , $t_k^i$ is the $k\mathrm{th}$ triggering time instant, for $k\in N$ , and $\underset{k\to \infty }{lim}{t}_k^i=\infty$ . The gain matrices K, $K_{\mathrm{xi}}$ , $K_{\xi i}$ , $F_i,i=1,...,N$ will be determined later. ${\xi }_i(t)\in {\mathbf{R}}^r$ denotes internal reference state. ${\hat{x}}_i(t)\in {\mathbf{R}}^{n_i}$ is the observer for the $i\mathrm{th}$ agent state $x_i(t)$ . The actuator $u_i^a(t)$ attack is estimated by ${\hat{u}}_i^a(t)$ . Define ${\tilde{y}}_i(t)=y_i^c(t)-C_i{\hat{x}}_i(t)-{\hat{y}}_i^a(t)$ , where ${\hat{y}}_i^a(t)$ represents the estimation of the sensor attack $y_i^a(t)$ . Moreover, the dynamics of ${\hat{y}}_i^a(t)$ and ${\hat{u}}_i^a(t)$ are described as follows

$$\stackrel{·}{{\hat{y}}_i^a}(t)=G_i{\tilde{y}}_i(t)$$

(9)

$$\stackrel{·}{{\hat{u}}_i^a}(t)=-{\alpha }_i{\hat{u}}_i^a(t)+H_i{\tilde{y}}_i(t)$$

(10)

where ${\alpha }_i>0$ , $H_i$ and $G_i$ are gains. The output $y_i^c(t)$ can be used to estimate unknown parameters in MASs. Considering ${\tilde{y}}_i(t)=y_i^c(t)-C_i{\hat{x}}_i(t)-{\hat{y}}_i^a(t)$ is accessible, the controller presented in equations (6)–(10) is based on the local measurable output.

Remark 3. Assumptions 1–2 are required. It is worth mentioning that the state $x_i(t)$ of agent i cannot be obtained directly, so the output $y_i^c(t)$ is used to estimate the unknown parameters in MASs (equation (5)). As mentioned above, the information of ${\tilde{y}}_i(t)=y_i^c(t)-C_i{\hat{x}}_i(t)-{\hat{y}}_i^a(t)$ is accessible, and the controller proposed in equations (6)–(10) is dependent on the local output information that is measurable.

Lemma 3. According to Assumptions 2–5, the matrix $\tilde{M}=L+D$ is in fact an M matrix, and a matrix $\tilde{Q}=\mathrm{diag}({\tilde{q}}_1,{\tilde{q}}_2,...,{\tilde{q}}_N)$ exists with ${\tilde{q}}_i>0,i=1,2,...,N$ , and $(({\tilde{q}}_1,{\tilde{q}}_2,...,{\tilde{q}}_N{)}^T={\tilde{M}}^{-T}{1}_N$ , such that $\tilde{Q}\tilde{M}+{\tilde{M}}^T\tilde{Q}>0$

Theorem 1. Assumptions 1–5 are considered hold, the gain matrix K is defined as $K=-P^{-1}{C}_0^T$ , such that $I_N\otimes A_0+M\otimes K{C}_0$ is Hurwitz, where M is defined based on Lemma 3. Let ${\delta }_0$ be a positive number, and the matrix $P>0$ that satisfies

$$A_{0}P+{\mathrm{PA}}_0^T-2C_0^T{C}_0+{\delta }_{0}P<0$$

(11)

Then, an adaptive self-triggering mechanism is introduced as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\zeta }_i(e_i(t),{\mathrm{\Xi }}_i(t_k^i))=(1+{\theta }_i(t)){e_i}^T(t)\Phi e_i(t)}\\ \multicolumn{1}{c}{-{{\mathrm{\Xi }}_i}^T(t_k^i)\Phi {\mathrm{\Xi }}_i(t_k^i)}\end{array}$$

(12)

where $e_i(t)={\xi }_i(t_k^i)-{\xi }_i(t)$ is the measurement error. ${\mathrm{\Xi }}_i(t_k^i)=\sum _{j=1}^N{a}_{(\mathrm{ij})}({\xi }_i(t_k^i)-{\xi }_i(t_{k\prime }^j))+d_{i0}({\xi }_i(t_k^i)-v(t))$ , $\Phi =P^{-1}{C}_0^T{C}_0{P}^{-1}$ is a symmetric and positive definite matrix. ${\theta }_i(t)>0$ is the adaptive gain that satisfies the following dynamics

$${\stackrel{·}{\theta }}_i(t)=e_i^T(t)\Phi e_i(t)$$

(13)

The next triggering time instant $t_{k+1}^i$ is predetermined by information from the previous triggering time, and more specifically, the next triggering time instant $t_{k+1}^i$ can be determined by $t_{k+1}^i=\underset{t>t_k^i}{inf}\{t\ge t_k^i|||e_i(t)||={\overline{\gamma }}_i{\mathrm{\Xi }}_i(t_k^i)\}$ , where ${\overline{\gamma }}_i=1/\sqrt{1+{\theta }_i(t_k^i)}$ . The adaptive self-triggering condition is enforced, when ${\zeta }_i(e_i(t),{\mathrm{\Xi }}_i(t_k^i))\le 0$ . Then, the COR problem of system (7) is achieved with cooperative UUB, and under self-triggering condition (12), there are no agents that exhibit Zeno behavior.

Proof. Define ${\tilde{\xi }}_i(t)={\xi }_i(t)-v(t)$ , the measurement error can be rewritten as $e_i(t)={\tilde{\xi }}_i(t_k^i)-{\tilde{\xi }}_i(t)$ . Let $\mathrm{\Xi }(t_k^i)=({\mathrm{\Xi }}_1^T(t_k^1),\dots ,{\mathrm{\Xi }}_N^T(t_k^N){)}^T$ , $\tilde{\xi }(t)=({\tilde{\xi }}_1^T(t),\dots ,{\tilde{\xi }}_N^T(t){)}^T$ , $e(t)=(e_1^T(t),\dots ,e_N^T(t){)}^T$ . Rewrite ${\mathrm{\Xi }}_i(t_k^i)$ in a compact form as $\mathrm{\Xi }(t_k^i)=(M\otimes I_r)\tilde{\xi }(t_k^i)=(M\otimes I_r)\tilde{\xi }(t)+(M\otimes I_r)e(t)$ . Next, the dynamics of equation (7) can be reformulated as follows

$$\stackrel{·}{\stackrel{~}{\xi }}(t)=(I_N\otimes A_0+M\otimes K{C}_0)\tilde{\xi }(t)+(M\otimes K{C}_0)e(t)$$

(14)

For closed-loop system (14), we consider a Lyapunov candidate as follows

$$V_{1i}(t)=\sum _{i=1}^N{{\tilde{\xi }}_i}^T(t)P^{-1}{\tilde{\xi }}_i(t)+\sum _{i=1}^N\frac{{({\theta }_i(t)-{\omega }_0)}^2}{4}$$

(15)

where ${\omega }_0$ is a properly chosen constant. Derivative of $V_{1i}(t)$ along trajectory (equation (14)) is as follows

$$\begin{array}{c}{\stackrel{·}{V}}_{1i}(t)=[{\tilde{\xi }}^T(t){(I_N\otimes A_0+M\otimes K{C}_0)}^T\hfill \\ +e^T(t){(M\otimes K{C}_0)}^T]·(I_N\otimes P^{-1})\tilde{\xi }(t)\hfill \\ +{\tilde{\xi }}^T(t)·(I_N\otimes P^{-1})·[(I_N\otimes A_0+M\otimes K{C}_0)\hfill \\ ·\tilde{\xi }(t)+(M\otimes K{C}_0)e(t)]\hfill \\ +\frac{1}{2}\sum _{i=1}^N({\theta }_i(t)-{\omega }_0){\stackrel{·}{\theta }}_i(t)\hfill \\ ={\tilde{\xi }}^T(t)(I_N\otimes ({A_0}^T{P}^{-1}+P^{-1}{A}_0))\tilde{\xi }(t)\hfill \\ -2{\tilde{\xi }}^T(t)(M\otimes \Phi )\tilde{\xi }(t)\hfill \\ -2{\tilde{\xi }}^T(t)(M\otimes \Phi )e(t)\hfill \\ +\frac{1}{2}\sum _{i=1}^N({\theta }_i(t)-{\omega }_0){\stackrel{·}{\theta }}_i(t)\hfill \end{array}$$

(16)

Consider the matrix M is symmetric and positive definite, then, according to Lemmas 1 and 2, we can obtain the following inequality

$$\begin{array}{c}\multicolumn{1}{c}{-2{\tilde{\xi }}^T(t)(M\otimes \Phi )e(t)\le {\tilde{\xi }}^T(t)(M\otimes \Phi )\tilde{\xi }(t)}\\ \multicolumn{1}{c}{+e^T(t)(M\otimes \Phi )e(t)}\end{array}$$

(17)

By substituting equation (17) into equation (16), one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{1i}(t)\le {\tilde{\xi }}^T(t)(I_N\otimes ({A_0}^T{P}^{-1}+P^{-1}{A}_0))\tilde{\xi }(t)}\\ \multicolumn{1}{c}{-2{\tilde{\xi }}^T(t)(M\otimes \Phi )\tilde{\xi }(t)+{\tilde{\xi }}^T(t)(M\otimes \Phi )\tilde{\xi }(t)}\\ \multicolumn{1}{c}{+e^T(t)(M\otimes \Phi )e(t)+\frac{1}{2}{e}^T(t)(\theta (t)\otimes \Phi )e(t)}\\ \multicolumn{1}{c}{-\frac{1}{2}{e}^T(t)({\omega }_0{I}_N\otimes \Phi )e(t)}\end{array}$$

(18)

Based on Lemmas 1 and 2, the following inequality is given as

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Xi }(t_k^i{)}^T(I_N\otimes \Phi )\mathrm{\Xi }(t_k^i)\le 2e^T(t)(M^{T}M\otimes \Phi )e(t)}\\ \multicolumn{1}{c}{+2{\tilde{\xi }}^T(t)(M^{T}M\otimes \Phi )\tilde{\xi }(t)}\end{array}$$

(19)

Next, consider the self-triggering condition $(1+{\theta }_i(t)){e_i}^T(t)\Phi e_i(t)\le {{\mathrm{\Xi }}_i}^T(t_k^i)\Phi {\mathrm{\Xi }}_i(t_k^i)$ , and substituting equation (19) into equation (18), one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{1i}(t)\le {\tilde{\xi }}^T(t)(I_N\otimes ({A_0}^T{P}^{-1}+P^{-1}{A}_0))\tilde{\xi }(t)}\\ \multicolumn{1}{c}{-{\tilde{\xi }}^T(t)(M\otimes \Phi )\tilde{\xi }(t)+e^T(t)(M\otimes \Phi )e(t)}\\ \multicolumn{1}{c}{+{\tilde{\xi }}^T(t)(M^{T}M\otimes \Phi )\tilde{\xi }(t)+e^T(t)(M^{T}M\otimes \Phi )e(t)}\\ \multicolumn{1}{c}{-\frac{1}{2}{e}^T(t)(I_N\otimes \Phi )e(t)-\frac{1}{2}{e}^T(t)({\omega }_0{I}_N\otimes \Phi )e(t)}\end{array}$$

(20)

On account of Lemma 3, the matrix M is symmetric. Thus, an orthogonal matrix U exists and enables $M=U^T\Lambda U$ , $\Lambda =\mathrm{diag}\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N\}$ . Let $\hat{\xi }(t)=(U\otimes I_r)\tilde{\xi }(t)$ and $\hat{e}(t)=(U\otimes I_r)e(t)$ , then one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{1i}(t)\le \sum _{i=1}^N{{\hat{\xi }}_i}^T(t)(2\Vert \Phi \Vert +{{\lambda }_N}^2\Vert \Phi \Vert -{\lambda }_1\Vert \Phi \Vert }\\ \multicolumn{1}{c}{-{\delta }_0\Vert P^{-1}\Vert ){\hat{\xi }}_i(t)+\sum _{i=1}^N{{\hat{e}}_i}^T(t)({\lambda }_N\Vert \Phi \Vert +{{\lambda }_N}^2\Vert \Phi \Vert }\\ \multicolumn{1}{c}{-\frac{\Vert \Phi \Vert ({\omega }_0+1)}{2}){\hat{e}}_i(t)}\end{array}$$

(21)

where ${\omega }_0$ and ${\delta }_0$ are chosen sufficiently large such that ${\omega }_0>2({\lambda }_N+{{\lambda }_N}^2)-1$ and ${\delta }_0>||\Phi ||(2+{{\lambda }_N}^2-{\lambda }_1)/||P^{-1}||$ . One has

$${\stackrel{·}{V}}_{1i}(t)\le \sum _{i=1}^N{{\hat{\xi }}_i}^T(t)(2\Vert \Phi \Vert +{{\lambda }_N}^2\Vert \Phi \Vert -{\lambda }_2\Vert \Phi \Vert -{\delta }_0\Vert P^{-1}\Vert ){\hat{\xi }}_i(t)$$

In light of LaSalle’s invariance principle, $\hat{\xi }(t)$ globally converges to the largest invariance set contained in $\{\hat{\xi }(t)\in {\mathbf{R}}^r|{\stackrel{·}{V}}_{1i}(\hat{\xi }(t))=0\}$ , note that $(U\otimes I_r)\tilde{\xi }(t)=\hat{\xi }(t)$ , $(U\otimes I_r)$ is nonsingular, so one has $\underset{t\to \infty }{lim}\tilde{\xi }(t)=0$ , when $t\to \infty$ , for any initial condition. Therefore, one has $\underset{t\to \infty }{lim}||\tilde{\xi }(t)||=0$ . Then, it can be proved that under self-triggering condition (12), no agent will show Zeno behavior.

The upper right-hand derivative $D^{+}{e}_i(t)$ satisfies the following differential inequality

$$\begin{array}{c}\multicolumn{1}{c}{D^{+}\Vert e_i(t)\Vert =\frac{{e_i}^T(t){\stackrel{·}{e}}_i(t)}{\Vert e_i(t)\Vert }\le \Vert {\stackrel{·}{\stackrel{~}{\xi }}}_i(t)\Vert }\\ \multicolumn{1}{c}{\le \Vert A_0\Vert \Vert e_i(t)\Vert +{\hat{\alpha }}_i}\end{array}$$

(22)

where ${\hat{\alpha }}_i=||A_0||||{\tilde{\xi }}_i(t_k^i)||+||K{C}_0||||{\mathrm{\Xi }}_i(t_k^i)||$ is bounded, and according to the definition of $\mathrm{\Xi }(t_k^i)$ , there exists a positive constant $\overline{p}$ such that $||{\tilde{\xi }}_i(t_k^i)||\le \overline{p}||{\mathrm{\Xi }}_i(t_k^i)||$ . Remind the fact that $||e_i(t_k^i)||=0$ . An upper bound of $||e_i(t)||$ for $t\in (t_k^i,t_{k+1}^i)$ is given by $||e_i(t)||\le ({\hat{\alpha }}_i/||A_0||)[e^{||A_0||(t-t_k^i)}-1]$ . Under self-triggering condition (12), there exists

$$\frac{{\hat{\alpha }}_i}{\Vert A_0\Vert }[e^{\Vert A_0\Vert (t_{k+1}^i-t_k^i)}-1]\ge \frac{\Vert {\mathrm{\Xi }}_i(t_k^i)\Vert }{\sqrt{1+{\theta }_i(t_k^i)}}$$

(23)

We consider the following two cases to indicate that a minimum triggering time ${\overline{\tau }}_i=t_{k+1}^i-t_k^i$ is strictly positive and can be obtained as follows

$${\overline{\tau }}_i=\frac{1}{\Vert A_0\Vert }·\ln [\frac{\Vert A_0\Vert }{{\hat{\alpha }}_i}·\frac{\Vert {\mathrm{\Xi }}_i(t_k^i)\Vert }{\sqrt{1+{\theta }_i(t_k^i)}}+1]$$

(24)

• Case 1: $||{\mathrm{\Xi }}_i(t_k^i)||\ne 0$ , one has ${\overline{\tau }}_i>0$ .

• Case 2: $\underset{k\to \infty }{lim}||{\mathrm{\Xi }}_i(t_k^i)||=0$ . Simple transposition of ${\hat{\alpha }}_i$ leads to

$$\underset{k\to \infty }{lim}\frac{\Vert \mathrm{\Xi }(t_k^i)\Vert }{{\hat{\alpha }}_i}=\frac{1}{\overline{p}\Vert A_0\Vert +\Vert K{C}_0\Vert }>0$$

(25)

Considering equation (25) and self-triggering condition (12), one has

$$\underset{k\to \infty }{lim}(t_{k+1}^i-t_k^i)\ge$$

$$\frac{1}{\Vert A_0\Vert }·\ln (\frac{\Vert A_0\Vert }{\sqrt{1+{\theta }_i(t_k^i)}(\overline{p}\Vert A_0\Vert +\Vert K{C}_0\Vert )}+1)>0$$

(26)

As for Zeno behavior, it means that there are an infinite number of triggering instants in a finite time.^27,37 Based on the self-triggering algorithm, the minimum triggering time ${\overline{\tau }}_i$ can be calculated as $t_{k+1}^i-t_k^i>0$ for each agent i when $||{\mathrm{\Xi }}_i(t_k^i)||\ne 0$ . For the case of $\underset{k\to \infty }{lim}||{\mathrm{\Xi }}_i(t_k^i)||=0$ . It can be seen that the minimum triggering time $\underset{k\to \infty }{lim}(t_{k+1}^i-t_k^i)$ is also strictly positive; thus, no agent will show Zeno behavior. The proof is complete.

Then, the self-triggering algorithm will be presented. The upper right derivative $D^{+}{e}_i(t)$ satisfies the following differential inequality

$$D^{+}\Vert e_i(t)\Vert \le \Vert A_0\Vert \Vert e_i(t)\Vert +{\hat{\alpha }}_i(t)\le w_i(t)$$

(27)

Note that $w_i(t)$ remains to be $w_i(t_k^i)$ and ${\theta }_i(t)$ remains to be ${\theta }_i(t_k^i)$ , until new ${\xi }_i(t_k^i)$ is received from any of its neighbors. Adaptive parameter ${\theta }_i(t)$ can avoid introducing the global information of the minimum non-zero eigenvalue of the communication topology Laplacian matrix and realize distributed control.

Inspired by Hu et al., the following condition is considered as

$$\underset{t_k^i}{\overset{t_{k+1}^i}{\int }}{w}_i(t)\mathrm{dt}={\overline{\gamma }}_i{\mathrm{\Xi }}_i(t_k^i)$$

(28)

We propose an iterative algorithm for agent i to compute the next triggering time instant $t_{k+1}^i$ . Determine that the update time instants of $w_i(t)$ be represented as $\{{\tilde{t}}_0^i,{\tilde{t}}_1^i,\dots ,{\tilde{t}}_m^i\}$ , where $m\ge 1$ , ${\tilde{t}}_0^i=t_k^i$ , and ${\tilde{t}}_m^i=t_{k+1}^i$ . Let $s_k^i({\tilde{t}}_k^i)={\overline{\gamma }}_i{\mathrm{\Xi }}_i(t_k^i)$ . Based on the literature, ³⁷ the iterative algorithm is introduced as follows

$$\underset{{\tilde{t}}_l^i}{\overset{{\tilde{t}}_{l+1}^i}{\int }}{w}_i(t)\mathrm{dt}=s_k^i\left({\tilde{t}}_l^i\right)$$

(29)

Rewrite the above iterative algorithm in the following form

$$w_i({\tilde{t}}_l^i)({\tilde{t}}_{l+1}^i-{\tilde{t}}_l^i)=s_k^i\left({\tilde{t}}_l^i\right)$$

(30)

where the initial condition can be set as $l=0$ , ${\tilde{t}}_0^i=t_k^i$ , $s_k^i({\tilde{t}}_0^i)=||{\mathrm{\Xi }}_i(t_k^i)||/\sqrt{1+{\theta }_i(t_k^i)}$ .

Remark 4. On account of the self-triggering algorithm, when the instant time $t_k^i$ is reached, the agent i samples the information from its neighboring agents to obtain ${\xi }_i(t_k^i)$ and sends the value to its neighbors. The next time instant $t_{k+1}^i$ of agent i can be calculated according to its own information ${\xi }_i(t_k^i)$ based on the current triggering time $t_k^i$ , or the most recently received neighboring information, ${\xi }_i(t_{k\prime }^j)$ , $j\in N$ .

Consider the defined equations ${\tilde{x}}_i(t)=x_i(t)-{\hat{x}}_i(t)$ , ${\tilde{y}}_i^a(t)=y_i^a(t)-{\hat{y}}_i^a(t)$ , ${\tilde{u}}_i^a(t)=u_i^a(t)-{\hat{u}}_i^a(t)$ . Since ${\tilde{y}}_i(t)=y_i^c(t)-C_i{\hat{x}}_i(t)-{\hat{y}}_i^a(t)=C_i{\tilde{x}}_i(t)+{\tilde{y}}_i^a(t)$ , one has

$$\stackrel{·}{{\stackrel{~}{x}}_i}\left(t\right)=(A_i-F_i{C}_i){\tilde{x}}_i\left(t\right)-F_i{\tilde{y}}_i^a\left(t\right)-E_i{\tilde{\xi }}_i\left(t\right)+B_i{\tilde{u}}_i^a\left(t\right)$$

(31)

$$\stackrel{·}{{\stackrel{~}{y}}_i^a}\left(t\right)=-G_i{C}_i{\tilde{x}}_i\left(t\right)-G_i{\tilde{y}}_i^a\left(t\right)+{\stackrel{·}{y}}_i^a\left(t\right)$$

(32)

Let ${\eta }_i(t)=({{\tilde{x}}_i}^T(t),{\tilde{y}}_i^a(t{)}^T{)}^T$ , the dynamics of ${\eta }_i(t)$ can be denoted as follows

$$\stackrel{·}{{\eta }_i}\left(t\right)=A_{\mathrm{Fi}}{\eta }_i\left(t\right)+B_{i1}{\tilde{u}}_i^a\left(t\right)-E_{i1}{\tilde{\xi }}_i\left(t\right)+B_{i2}{\stackrel{·}{y}}_i^a\left(t\right)$$

(33)

where $A_{\mathrm{Fi}}=\left[\begin{array}{cc}{A}_i-F_i{C}_i\hfill & -F_i\hfill \\ \multicolumn{1}{c}{-G_i{C}_i}& -G_i\hfill \end{array}\right]$ , $B_{i1}=\left[\begin{array}{c}{B}_i\hfill \\ \multicolumn{1}{c}{0_{p\times m_i}}\end{array}\right]$ , $E_{i1}=\left[\begin{array}{c}{E}_i\hfill \\ \multicolumn{1}{c}{0_{p\times r}}\end{array}\right]$ , $B_{i2}=\left[\begin{array}{c}{0}_{n_i\times p}\hfill \\ \multicolumn{1}{c}{I_p}\end{array}\right].$

Theorem 2. Consider the COR problem of MAS under sensor and actuator attacks. Assuming that Assumptions 1–5 hold, the resilient adaptive control scheme is given in equations (6)–(10), and the self-triggering condition is satisfied. Then, $({\eta }_i(t),{\tilde{u}}_i^a(t))$ of the corresponding closed-loop system achieves UUB.

Proof. Considering $A_{\mathrm{Fi}}=\left[\begin{array}{cc}{A}_i\hfill & 0_{n_i\times p}\hfill \\ \multicolumn{1}{c}{0_{p\times n_i}}& 0_{p\times p}\hfill \end{array}\right]+\left[\begin{array}{c}-F_i\hfill \\ \multicolumn{1}{c}{-G_i}\end{array}\right]\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]$ , $(\left[\begin{array}{cc}{A}_i\hfill & 0_{n_i\times p}\hfill \\ \multicolumn{1}{c}{0_{p\times n_i}}& 0_{p\times p}\hfill \end{array}\right],\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right])$ can be observed if and only if $\mathrm{rank}\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \\ \multicolumn{1}{c}{C_i{A}_i}& 0_{p\times p}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill \\ \multicolumn{1}{c}{C_i{A_i}^{n_i+p-1}}& 0_{p\times p}\hfill \end{array}\right]=n_i+p$ .

The formula is equivalent to

$$\mathrm{rank}{\left[\begin{array}{cc}{\left(C_i{A}_i\right)}^T,\dots ,\hfill & {\left(C_i{A}_i^{{n_i}^{n_i+p-1}}\right)}^T\hfill \end{array}\right]}^T=n_i$$

(34)

Since $A_i\in {\mathbf{R}}^{n_i\times n_i}$ , $p\ge 1$ . According to Cayley–Hamilton theorem, one has

$$\mathrm{rank}{\left[\begin{array}{cccc}{\left(C_i{A}_i\right)}^T,\hfill & {\left(C_i{A^2}_i\right)}^T,\hfill & \dots ,\hfill & {\left(C_i{A}_i^{n_i}\right)}^T\hfill \end{array}\right]}^T=n_i$$

(35)

Suppose Assumptions 1–5 hold, the appropriate controller gains $G_i$ , $F_i$ can be properly chosen, and there exists a positive definite matrix $Q_i\in R^{(n_i+p)\times (n_i+p)}$ , such that $P_i{A}_{Fi}+{A_{Fi}}^T{P}_i<-Q_i$ and $-{\lambda }_{min}(Q_i)+||P_i||<0$ .

Choose a Lyapunov function as

$$V_{2i}(t)={\eta }_i^T(t)P_i{\eta }_i(t)+{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\tilde{u}}_i^a\left(t\right)$$

(36)

Deriving the above equation yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{2i}(t)=2{\eta }_i^T(t)P_i{\stackrel{·}{\eta }}_i(t)+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T\stackrel{·}{{\stackrel{~}{u}}_i^a}\left(t\right)}\\ \multicolumn{1}{c}{={\eta }_i^T(t)(P_i{A}_{\mathrm{Fi}}+A_{\mathrm{Fi}}^T{P}_i){\eta }_i(t)+2{\eta }_i^T(t)P_i{B}_{i1}{\tilde{u}}_i^a}\\ \multicolumn{1}{c}{-2{\eta }_i^T(t)P_i{E}_{i1}{\tilde{\xi }}_i\left(t\right)+2{\eta }_i^T(t)P_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)}\\ \multicolumn{1}{c}{+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T({\stackrel{·}{u}}_i^a\left(t\right)-\stackrel{·}{{\hat{u}}_i^a}\left(t\right))}\\ \multicolumn{1}{c}{\le -{\eta }_i^T(t)Q_i{\eta }_i(t)+2{\eta }_i^T(t)P_i{B}_{i1}{\tilde{u}}_i^a\left(t\right)}\\ \multicolumn{1}{c}{+2{\eta }_i^T(t)P_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)-2{\eta }_i^T(t)P_i{E}_{i1}{\tilde{\xi }}_i\left(t\right)}\\ \multicolumn{1}{c}{+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T({\stackrel{·}{u}}_i^a\left(t\right)+{\alpha }_i{\hat{u}}_i^a\left(t\right))}\\ \multicolumn{1}{c}{+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{H}_i{\tilde{y}}_i\left(t\right)}\end{array}$$

(37)

Using Lemma 2, one obtains

$$\begin{array}{c}\multicolumn{1}{c}{-2{\eta }_i^T(t)P_i{E}_{i1}{\tilde{\xi }}_i\left(t\right)\le \frac{1}{4}{\eta }_i^T(t)P_i{\eta }_i(t)}\\ \multicolumn{1}{c}{+4{\tilde{\xi }}_i^T\left(t\right)E_{\mathrm{Fi}}^T{P}_i{E}_{i1}{\tilde{\xi }}_i\left(t\right)}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{2{\eta }_i^T(t)P_i{B}_{i1}{\tilde{u}}_i^a\left(t\right)\le \frac{1}{2}{\eta }_i^T(t)P_i{\eta }_i(t)}\\ \multicolumn{1}{c}{+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{B}_{i1}^T{P}_i{B}_{i1}{\tilde{u}}_i^a\left(t\right)}\end{array}$$

$$\begin{array}{c}2{\eta }_i^T(t)P_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)\le \frac{1}{2}{\eta }_i^T(t)P_i{\eta }_i(t)\hfill \\ +2{\left({\stackrel{·}{y}}_i^a\left(t\right)\right)}^T{B}_{i2}^T{P}_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)\hfill \end{array}$$

$$\begin{array}{c}2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\stackrel{·}{u}}_i^a\left(t\right)\le \frac{{\alpha }_i}{2}{\left({\tilde{u}}_i^a\left(t\right)\right)}^T\left({\tilde{u}}_i^a\left(t\right)\right)\hfill \\ +\frac{2}{{\alpha }_i}{\left({\stackrel{·}{u}}_i^a\left(t\right)\right)}^T{\stackrel{·}{u}}_i^a\left(t\right)\hfill \end{array}$$

$$\begin{array}{c}2{\alpha }_i{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\hat{u}}_i^a\left(t\right)=2{\alpha }_i{\left(u_i^a\left(t\right)\right)}^T{u}_i^a\left(t\right)-2{\alpha }_i{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\tilde{u}}_i^a\hfill \\ \le 2{\alpha }_i{\left(u_i^a\left(t\right)\right)}^T{u}_i^a\left(t\right)\hfill \\ -\frac{3}{2}{\alpha }_i{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\tilde{u}}_i^a\left(t\right)\hfill \end{array}$$

and

$$\begin{array}{c}-2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{H}_i{\tilde{y}}_i\left(t\right)=-2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{H}_i\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]{\eta }_i(t)\hfill \\ \le \frac{1}{4}{{\eta }_i}^T(t)P_i{\eta }_i\left(t\right)\hfill \\ +4{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{H}_i\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]{P_i}^{-1}{\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]}^T{H_i}^T{\tilde{u}}_i^a\left(t\right)\hfill \end{array}$$

Combining the above inequalities, we get

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{2i}(t)\le -{\eta }_i^T(t)Q_i{\eta }_i(t)+{\eta }_i^T(t)P_i{\eta }_i(t)}\\ \multicolumn{1}{c}{-{\alpha }_i{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{\tilde{u}}_i^a\left(t\right)+2{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{B}_{i1}^T{P}_i{B}_{i1}{\tilde{u}}_i^a\left(t\right)}\\ \multicolumn{1}{c}{+2{\left({\stackrel{·}{y}}_i^a\left(t\right)\right)}^T{B}_{i2}^T{P}_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)+4{{\tilde{\xi }}_i}^T\left(t\right)E_{i1}^T{P}_i{E}_{i1}{\tilde{\xi }}_i}\\ \multicolumn{1}{c}{+\frac{2}{{\alpha }_i}{\left({\stackrel{·}{u}}_i^a\left(t\right)\right)}^T{\stackrel{·}{u}}_i^a\left(t\right)+2{\alpha }_i{\left(u_i^a\left(t\right)\right)}^T{u}_i^a\left(t\right)}\\ \multicolumn{1}{c}{+4{\left({\tilde{u}}_i^a\left(t\right)\right)}^T{H}_i\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]P_i^{-1}{\left[\begin{array}{cc}{C}_i\hfill & I_p\hfill \end{array}\right]}^T{H}_i^T{\tilde{u}}_i^a}\end{array}$$

(38)

Next, ${\alpha }_i$ is chosen to satisfy ${\alpha }_i={\tilde{\alpha }}_{i1}+2\left|\right|{B_{i1}}^T{P}_i{B}_{i1}\left|\right|+4\left|\right|H_i[\begin{array}{cc}{C}_i& I_p\end{array}]{P_i}^{-1}{[\begin{array}{cc}{C}_i& I_p\end{array}]}^T{H_i}^T\left|\right|$ , where ${\tilde{\alpha }}_{i1}$ is a positive constant. Choosing a positive definite matrix $Q_i$ , such that $-{\lambda }_{min}(Q_i)+||P_i||=-{\overline{a}}_{i1}<0$ , ${\overline{a}}_{i1}$ is a positive constant, then one yields

$${\stackrel{·}{V}}_{2i}(t)\le -{\vartheta }_i{V}_{2i}(t)+{\overline{c}}_i$$

(39)

where ${\vartheta }_i=min\{{\overline{a}}_{i1}/||P_i||,{\tilde{\alpha }}_{i1}\}$ , ${\overline{c}}_i$ is a bound of

$$\begin{array}{c}\multicolumn{1}{c}{2{\left({\stackrel{·}{y}}_i^a\left(t\right)\right)}^T{B_{i2}}^T{P}_i{B}_{i2}{\stackrel{·}{y}}_i^a\left(t\right)+4{{\tilde{\xi }}_i}^T\left(t\right){E_{i1}}^T{P}_i{E}_{i1}{\tilde{\xi }}_i\left(t\right)}\\ \multicolumn{1}{c}{+\frac{2}{{\alpha }_i}{\left({\stackrel{·}{u}}_i^a\left(t\right)\right)}^T{\stackrel{·}{u}}_i^a\left(t\right)+2{\alpha }_i{\left(u_i^a\left(t\right)\right)}^T{u}_i^a\left(t\right)}\end{array}$$

In light of the comparison lemma, ²⁵ and it follows from equation (39) that

$$V_{2i}(t)\le (V_{2i}(0)-\frac{{\overline{c}}_i}{{\vartheta }_i})e^{-{\vartheta }_{i}t}+\frac{{\overline{c}}_i}{{\vartheta }_i}$$

(40)

Due to $e^{-{\vartheta }_{i}t}$ asymptotically converges to 0, when $t\to \infty$ , so there exists a positive constant $T_i$ , such that for any $t\ge T_i$ , $e^{-{\vartheta }_{i}t}<{\hat{c}}_i/{\vartheta }_i|V_{2i}(0)-({\overline{c}}_i/{\vartheta }_i)|$ , ${\hat{c}}_i$ is a positive constant, $V_{2i}(0)-({\overline{c}}_i/{\vartheta }_i)$ is non-zero. For any $t\ge T_i$ , $V_{2i}(t)\le c_i/{\vartheta }_i$ , $c_i={\overline{c}}_i+{\hat{c}}_i>0$ . Note that $||{\eta }_i(t)|{|}^2$ is less than ${\lambda }_{min}^{-1}(P_i)V_{2i}(t)$ and $||{\tilde{u}}_i^a\left(t\right)|{|}^2\le V_{2i}(t)$ , thus $({\eta }_i(t),{\tilde{u}}_i^a(t))$ is UUB.

Remark 5. ${\hat{x}}_i(t)$ , ${\hat{y}}_i^a(t)$ , and ${\hat{u}}_i^a(t)$ are the estimated values of $x_i^a(t)$ , $y_i^a(t)$ , and $u_i^a(t)$ and each may have little error threshold, which lays the primary foundation for guaranteeing UUB.

Theorem 3. Suppose Assumptions 1–5 hold, self-triggering condition (12) is satisfied. Let $K_{\xi i}=Y_i-K_{\mathrm{xi}}{X}_i$ , and select the matrix $K_{\mathrm{xi}}$ to ensure that $A_i+B_i{K}_{\mathrm{xi}}$ is Hurwitz. Let ${\epsilon }_i(t)={\hat{x}}_i(t)-X_i{\xi }_i(t)$ . Then, under the proposed control protocol and the triggering mechanism, if the following regulated equations exist solutions $X_i$ and $Y_i$ , the trajectory of ${\epsilon }_i(t)$ is achieved with UUB

$$X_i{A}_0=A_i{X}_i+B_i{Y}_i+E_i$$

(41)

$$C_0=C_i{X}_i$$

(42)

Proof. As analyzed in Theorem 1, for any initial condition, $\underset{t\to \infty }{lim}||\tilde{\xi }(t)||=0$ . In addition, it can be derived that there exists a positive constant $\nu$ and a time instant $T^{*}$ , when $t\ge T^{*}$ , one has $||\tilde{\xi }(t)|{|}^2<\nu$ , where $\tilde{\xi }(t)=({\tilde{\xi }}_1^T(t),{\tilde{\xi }}_2^T(t),\dots ,{\tilde{\xi }}_N^T(t){)}^T$ . Similarly, there exists a constant $\overline{\nu }>0$ , according to the triggering condition, one has $||e(t)|{|}^2<\overline{\nu }$ . Define $\epsilon (t)=({\epsilon }_1^T(t),{\epsilon }_2^T(t),\dots ,{\epsilon }_N^T(t){)}^T$ , $\tilde{y}(t)=({\tilde{y}}_1^T(t),{\tilde{y}}_2^T(t),\dots ,{\tilde{y}}_N^T(t){)}^T$ , $X=\mathrm{diag}\{X_1,\dots ,X_N\}$ , $F=\mathrm{diag}\{F_1,\dots ,F_N\}$ , $A=\mathrm{diag}\{A_1,\dots ,A_N\}$ , $B=\mathrm{diag}\{B_1,\dots ,B_N\}$ , $K_x=\mathrm{diag}\{K_{x1},\dots ,K_{\mathrm{xN}}\}$ .

The derivative of ${\epsilon }_i(t)$ the following compact form can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\epsilon }\left(t\right)=(A+B{K}_x)\epsilon \left(t\right)+F\tilde{y}\left(t\right)}\\ \multicolumn{1}{c}{-X(M\otimes K{C}_0)\tilde{\xi }\left(t\right)-X(M\otimes K{C}_0)e\left(t\right)}\end{array}$$

(43)

Choose the Lyapunov function as

$$V_{\epsilon }\left(t\right)={\epsilon }^T\left(t\right)P_x\epsilon \left(t\right)$$

(44)

where $P_x=\mathrm{diag}\{P_{x1},\dots ,P_{\mathrm{xN}}\}$ . $P_{\mathrm{xi}}>0$ that satisfies

$$P_{\mathrm{xi}}(A_i+B_i{K}_{\mathrm{xi}})+{(A_i+B_i{K}_{\mathrm{xi}})}^T{P}_{\mathrm{xi}}<-2P_{\mathrm{xi}}$$

(45)

for $i=1,2,...,N$ . The time derivative of $V_{\epsilon }(t)$ is as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\epsilon }\left(t\right)=2{\epsilon }^T\left(t\right)P_x\stackrel{·}{\epsilon }\left(t\right)}\\ \multicolumn{1}{c}{\le -2{\epsilon }^T\left(t\right)P_x\epsilon \left(t\right)+2{\epsilon }^T\left(t\right)P_{x}F\tilde{y}\left(t\right)}\\ \multicolumn{1}{c}{-2{\epsilon }^T\left(t\right)P_{x}X(M\otimes K{C}_0)\tilde{\xi }\left(t\right)}\\ \multicolumn{1}{c}{-2{\epsilon }^T\left(t\right)P_{x}X(M\otimes K{C}_0)e\left(t\right)}\end{array}$$

(46)

According to Lemma 2, one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\epsilon }\left(t\right)\le -\frac{1}{2}{\epsilon }^T\left(t\right)P_x\epsilon \left(t\right)+2{\tilde{y}}^T\left(t\right)F^T{P}_{x}F\tilde{y}\left(t\right)}\\ \multicolumn{1}{c}{+2{\tilde{\xi }}^T\left(t\right){(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\tilde{\xi }\left(t\right)}\\ \multicolumn{1}{c}{+2e^T\left(t\right){(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)e\left(t\right)}\end{array}$$

(47)

As analyzed previously, the following inequalities are obtained

$$\begin{array}{c}{\tilde{y}}^T\left(t\right)F^T{P}_{x}F\tilde{y}\left(t\right)\hfill \\ \le 2\Vert F^T{P}_{x}F\Vert ({\Vert C\Vert }^2{\Vert \tilde{x}\left(t\right)\Vert }^2+{\Vert {\tilde{y}}^a\left(t\right)\Vert }^2)\hfill \end{array}$$

(48)

$$\begin{array}{c}{\tilde{\xi }}^T\left(t\right){(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\tilde{\xi }\left(t\right)\hfill \\ \le \Vert {(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\Vert \nu \hfill \end{array}$$

(49)

$$\begin{array}{c}{e}^T\left(t\right){(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)e\left(t\right)\hfill \\ \le \Vert {(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\Vert \overline{\nu }\hfill \end{array}$$

(50)

Then, the ultimate equation is obtained as follows

$${\stackrel{·}{V}}_{\epsilon }\left(t\right)\le -\frac{1}{2}{V}_{\epsilon }\left(t\right)+\tilde{b}$$

(51)

where $\tilde{b}$ is bounded, for $t\ge max\{T_1,T_2,\dots ,T_N,T^{*}\}$ , $\tilde{b}=4N\Vert F^T{P}_{x}F\Vert ({\Vert C\Vert }^2+1)\underset{i=1,\dots ,N}{max}\left\{\frac{c_i}{{\lambda }_{min}\left(P_i\right){\vartheta }_i}\right\}+$ $2(\Vert {(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\Vert )\nu +2\Vert {(M\otimes K{C}_0)}^T{X}^T{P}_{x}X(M\otimes K{C}_0)\Vert \overline{\nu }$ .

Similar to the proof of Theorem 2, there exists a positive number $\overline{b}$ , such that $||\epsilon (t)|{|}^2<{\lambda }_{min}^{-1}(P_x)\overline{b}$ , for any $t\ge T^{*}$ , $\epsilon (t)$ is uniformly ultimately bounded.

Theorem 4. Suppose Assumptions 1–5 hold, under the distributed control protocol and the triggering mechanism, the heterogeneous MASs (1) uniformly ultimately bounded.

Proof. According to the proposed equation

$$\begin{array}{c}\Vert y_i\left(t\right)-y_0\left(t\right){\Vert }^2=\Vert C_i{x}_i\left(t\right)-C_i{\hat{x}}_i\left(t\right)+C_i{\hat{x}}_i\left(t\right)\hfill \\ -C_i{X}_i{\xi }_i\left(t\right)+C_i{X}_i{\xi }_i\left(t\right)-C_{0}v\left(t\right){\Vert }^2\hfill \\ \le 3\Vert C_i{\Vert }^2\Vert {\tilde{x}}_i\left(t\right){\Vert }^2+3\Vert C_i{\Vert }^2\Vert {\epsilon }_i\left(t\right){\Vert }^2\hfill \\ +3\Vert C_0{\Vert }^2\Vert {\tilde{\xi }}_i\left(t\right){\Vert }^2\hfill \end{array}$$

(52)

Therefore, for any $t\ge T^{*}$ , $y_i(t)-y_0(t)$ is uniformly ultimately bounded. Moreover, the regulated output for the heterogeneous MASs is achieved with cooperative UUB.

Remark 6. Based on Theorem 4 and Meng et al., ²⁵ the output regulation with cooperative UUB is independent of the upper number of the compromised agent, as long as the communication topology satisfies Assumption 2.

Remark 7. From the proof presented throughout this section, the designed adaptive controller still applies if there is only a constant sensor attack, once the following terms $u_i^a(t)$ , ${\stackrel{·}{u}}_i^a(t)$ , and ${\stackrel{·}{y}}_i^a(t)$ are removed; thus, COR can still be achieved, that is, the error $y_i(t)-y_0(t)$ is close to zero when $t\to \infty ,i=1,2,...,N$ . Furthermore, the algorithm could also be extended to the situation for MASs under actuator dynamics.

Illustrative examples

In this section, two examples, including a numerical example and an application to a spacecraft formation flying problem, are given to validate the effectiveness of the proposed self-triggered schemes.

Example 1

A graph composed of seven heterogeneous agents is considered, as shown in Figure 1. The dynamic characteristics of each agent can be expressed as follows

$$\begin{array}{c}{A}_i=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & a_i\hfill \\ \multicolumn{1}{c}{0}& -b_i\hfill & c_i\hfill \end{array}\right],B_i=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{d_i}\end{array}\right],C_i=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{E_i=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\end{array}$$

The parameters $\{a_i,b_i,c_i,d_i\}$ , i = 1, 5 are set as $\{2,5,1,1\}$ , $\{3,2,1,1\}$ , respectively

$$\begin{array}{c}{A}_i=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& -a_i\hfill \end{array}\right],B_i=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{b_i}\end{array}\right],C_i=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{E_i=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\end{array}$$

Assign $\{a_i,b_i\}$ , i = 2, 6, and 7 to $\{2,1\}$ , $\{2.5,1.5\}$ , and $\{3,2\}$ , respectively

$$\begin{array}{c}{A}_i=\left[\begin{array}{cc}0\hfill & a_i\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],B_i=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{b_i}\end{array}\right],C_i=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{E_i=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\end{array}$$

The parameters $\{a_i,b_i\}$ , i = 3, 4 are chosen as $\{3,2\}$ , $\{4,1\}$ , respectively.

Figure 1.

The communication topology among heterogeneous MASs.

Furthermore, according to the design procedure, $P=[1.6030,0;0,1.6030]$ is solved by linear matrix inequality (LMI). The given parameters are set as follows: $A_0=[0,1;-1,0]$ , $C_0=[1,0;0,1]$ . $u_i^a(t)=0.1·\cos (t)$ , $y_i^a(t)=0.1·\sin (t)$ . The initial conditions can also be selected randomly.

The other consensus gains, $K_{1i}$ , $K_{2i}$ , and $F_i$ , can be calculated as follows

$$\begin{array}{c}\multicolumn{1}{c}{K_{11}=[-5,-5,-5],K_{21}=[-6.5,7.5],F_1=[-15,-10,0{]}^T,}\\ \multicolumn{1}{c}{K_{12}=[-6,-3],K_{22}=[6,0],F_2=[-3,-12{]}^T,}\\ \multicolumn{1}{c}{K_{13}=[-1.5,-5],K_{23}=[1.5,0],F_3=[-5,-1.5{]}^T,}\\ \multicolumn{1}{c}{K_{14}=[-1,-2.5],K_{24}=[1,0],F_4=[-5,-2{]}^T,}\\ \multicolumn{1}{c}{K_{15}=[-5,-5,-5],K_{25}=[-3,5.333],F_5=[-5,-10,0{]}^T,}\\ \multicolumn{1}{c}{K_{16}=[-4,-\frac{10}{3}],K_{26}=[4,0],F_6=[-2.5,-12.25{]}^T,}\\ \multicolumn{1}{c}{K_{17}=[-3,-1],K_{27}=[3,0],F_7=[-2,-12{]}^T.}\end{array}$$

The variable $v(t)$ is described as follows

$$\stackrel{·}{v}(t)=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill \end{array}\right]v(t)$$

The adaptive coupling gains are defined as ${\theta }_i$ , for $i=1,...,7$ , and the regulated outputs are defined as ${\tilde{e}}_i=y_i-y_0$ , for $i=1,...,7$ . The simulation results are displayed as follows. Figure 2 shows the adaptive gains ${\theta }_i$ , which converges to positive constant values. By utilizing the proposed adaptive self-triggered control strategies, the measurement error $||e_3(t)||$ and the threshold of agent 3 are shown in Figure 3, which converges to zero. Triggering instants for each agent are indicated in Figure 4, which indicates that the Zeno behavior can be excluded via the given self-triggering mechanism; furthermore, the model has intermittent communication. The regulated outputs are depicted in Figure 5; it can be concluded that the regulated outputs are regularly bounded, which indicates that cooperative UUB can be achieved.

Figure 2.

The adaptive coupling gains ${\theta }_i$ .

Figure 3.

The evolution of $||e_3(t)||$ under proposed self-triggered strategy.

Figure 4.

The triggering instants of each agent.

Figure 5.

The evolution of regulated output ${\tilde{e}}_i$ via self-triggered control.

Example 2: application to a spacecraft formation flying problem in the low Earth orbit

In this example, the proposed adaptive self-triggering control scheme approach is illustrated to be equally applicable to solve the formation flight problem of a spacecraft in low Earth orbit. A related work is referred by Hu et al., ³⁷ which addresses the formation consensus problem for spacecraft in low Earth orbit under event-triggered and self-triggered algorithms.

Consider a virtual reference spacecraft moving in a circular orbit of radius $R_0$ . Concerning the virtual spacecraft, the relative dynamics of the other spacecraft can be linearized in a coordinate system, with the origin at the mass center of the virtual spacecraft, the x-axis running along the velocity vector, the y-axis being aligned with the position vector, and the z-axis along the right-handed coordinate system. The linearized dynamics equation of the $i\mathrm{th}$ satellite relative to virtual satellites can be described by Hill equation, as follows

$$\begin{array}{c}{\stackrel{··}{\tilde{x}}}_{\mathrm{si}}-2{\omega }_0{\stackrel{·}{\stackrel{~}{y}}}_{\mathrm{si}}=u_{x_i}\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{\tilde{y}}}_{\mathrm{si}}+2{\omega }_0{\stackrel{·}{\stackrel{~}{x}}}_{\mathrm{si}}-3{\omega }_0^2{y}_{\mathrm{si}}=u_{y_i}}\\ \multicolumn{1}{c}{{\stackrel{··}{\tilde{z}}}_{\mathrm{si}}+{\omega }_0^2{\tilde{z}}_{\mathrm{si}}=u_{z_i}}\end{array}$$

(53)

where ${\tilde{x}}_s$ , ${\tilde{y}}_s$ , and ${\tilde{z}}_s$ indicate the position vectors in rotational coordinates; $u_x$ , $u_y$ , and $u_z$ denote the control inputs; ${\omega }_0$ represents the angular rate of the virtual satellite. A very small distance between the $i\mathrm{th}$ satellite and the virtual satellite compared to the orbital radius $R_0$ is a general assumption in Hill equation.

Let $r_i=[{\tilde{x}}_{\mathrm{si}}^T,{\tilde{y}}_{\mathrm{si}}^T,{\tilde{z}}_{\mathrm{si}}^T{]}^T$ and $u_i=[u_{\mathrm{xi}}^T,{\tilde{u}}_{\mathrm{yi}}^T,{\tilde{u}}_{\mathrm{zi}}^T{]}^T$ denote the position vector and the control vector, respectively. Satellites can be described to achieve formation flight if the velocity vectors tend to converge to the same constant value and the position vectors remain at a specified level of separation, that is, for all $i,j\in N$ , $r_j-h_j\to r_i-h_i$ , ${\stackrel{·}{r}}_i\to {\stackrel{·}{r}}_j$ , when $t\to \infty$ , where $h_i=[{\tilde{h}}_{\mathrm{xi}}^T,{\tilde{h}}_{\mathrm{yi}}^T,{\tilde{h}}_{\mathrm{zi}}^T{]}^T$ , $h_j-h_i$ denotes the constant distance value.

Denote $x_i=\mathrm{col}({\overline{r}}_i,{\stackrel{·}{r}}_i)$ , where ${\overline{r}}_i=r_i-h_i$ . Then, equation (53) can be rewritten as follows

$${\stackrel{·}{x}}_i(t)=\left[\begin{array}{cc}0\hfill & I_3\hfill \\ \multicolumn{1}{c}{A_1}& A_2\hfill \end{array}\right]x_i(t)+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{I_3}\end{array}\right]{\overline{u}}_i$$

where $A_1=\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -3{\omega }_0^2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{\omega }_0^2\hfill \end{array}\right]$ , $A_2=\left[\begin{array}{ccc}0\hfill & 2{\omega }_0\hfill & 0\hfill \\ \multicolumn{1}{c}{-2{\omega }_0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]$ , ${\overline{u}}_i=u_i+A_1{h}_i$ .

To illustrate the effectiveness of the method proposed in this article to solve a practical engineering problem, a scenario is considered where the formation flight model has five satellites, and the communication topology is shown in Figure 6, similar to Figure 1.

Figure 6.

The communication topology among five satellites.

Define the satellite position error as ${\overline{r}}_i-{\overline{r}}_j=\mathrm{col}({\overline{x}}_{\mathrm{si}},{\overline{y}}_{\mathrm{si}},{\overline{z}}_{\mathrm{si}})$ , where ${\overline{x}}_{\mathrm{si}}$ , ${\overline{y}}_{\mathrm{si}}$ , and ${\overline{z}}_{\mathrm{si}}$ indicate the satellite position error agents i and j. The consensus gains $K_{1i}$ and $K_{2i}$ are calculated as $K_{1i}=[-5,-5,-5,0,-1,0]$ and $K_{2i}=[3.9980,0.0100;4.9980,0.0100;4.9980,0.0100]$ . The attacks can be set as $u_i^a(t)=0.13·\cos (t)$ , $y_i^a(t)=0.14·\sin (t)$ . The constant is set as ${\omega }_0=0.001$ . The satellite position error and velocity responses on the y-axis through the self-triggered algorithm are given in Figures 7 and 8, respectively. These results show that the velocity vectors of the satellites converge to the same value and the position vector errors remain at the prescribed interval level; therefore, the proposed control scheme is applied to describe the satellites to achieve formation flight.

Figure 7.

The position outputs via the self-triggered control scheme on the y-axis.

Figure 8.

The velocity outputs via the self-triggered control scheme on the y-axis.

Conclusion

This article has addressed the COR problem for heterogeneous linear MASs with fixed graphs under sensor and actuator attacks through an adaptive self-triggered control approach. A dynamic internal model has been introduced for each agent, and an adaptive self-triggered control approach has been given. The results have shown that the regulated output for the heterogeneous MASs could be achieved with cooperative UUB and no agent would show Zeno behavior. In addition, the proposed method would be applied to engineering examples as well. Future research topics could be oriented to systems with switching topologies or time-varying delays.