Secure state estimation and attack reconstruction for cyber-physical system with intermediate and robust observer

Abstract

An intermediate and robust observer is presented in this article to robustly estimate secure state and reconstruct actuator attack for continuous-time/discrete-time cyber-physical system. First, we propose a general form of cyber-physical system under actuator attack and external disturbance, and then some assumptions are stated in order to facilitate further research. Second, a novel intermediate and robust observer is designed, based on which an augmented error equation is deduced, necessary and sufficient conditions for continuous-time/discrete-time cyber-physical system to obtain gain matrices are derived separately. Third, strict linear matrix inequality conditions are deduced by transforming the original problem into convex optimization problem. Finally, we conduct two simulation experiments with continuous-time and discrete-time unmanned ground vehicle model to show the effectiveness of the proposed secure state estimation and attack reconstruction approach.

Introduction

As the core technology of Industry 4.0, cyber-physical system (CPS) is becoming a global technology industry to support and lead a new round of technological evolution, and have a significant impact on economics and social lives. Up to now, CPS has been widely used in many domains; typical application scenarios include manufacturing, transportation, healthcare, urban construction, and so on.

CPS requires tight integration and cooperation of computing, communication, and control technologies ¹ (Figure 1). Due to such integration, CPS has greatly improved system efficiency; meanwhile make the system more vulnerable to attacks. ² In addition to traditional cyber-attacks, CPS also suffers from attacks that break down physical plants (such as sensors and actuators), owing to the various physical plants involve in CPS (Figure 2). Break down of these plants usually result in great damages, for example, the Stuxnet worm attack incident happened in 2010 that targeting the centrifuges of Iran’s Bushehr nuclear power plant, and the Maroochy-Shire water breach happened in 2000 that targeting sewage pumping stations. Against such attacks, traditional information security techniques ³ (such as access control, data encryption, network isolation, authentication, and so on) seem insufficient, as they do not make full use of the knowledge about whether the measurements are compatible with the corresponding physical process or control mechanism, and they cannot deal with insider attacks and attacks targeting the physical dynamics. ⁴

Figure 1.

Cyber-physical system.

Figure 2.

Attacks that break down physical nodes.

Summary of related work

For the above reasons, the research of CPS secure state estimation and attack reconstruction from the perspective of control theory has received plenty of attentions in last several years. ⁵ For instance, in Fawzi et al., ⁶ secure state estimation and control for continuous-time CPS under adversarial attacks is investigated, while attack reconstruction is not considered, and it assumes that the set of attacked nodes is fixed. In Chang et al., ⁷ Kalman filter is used for CPS secure estimation, and the limitation about fixed attacked nodes is released. In Lu and Yang, ⁸ switched Luenberger observer is adopted for secure state estimation with CPS case, but only disturbance and sensor attack are considered, and does not take attack reconstruction into account. There are also some bibliographies focus on attack reconstruction. In Song et al., ⁹ two adaptive sliding mode observers with on line parameter estimation are designed to solve the problem of attack detection and reconstruction, though the estimation of sensor attack and actuator attack are executed separately, and external disturbance has not been considered. In Corradini and Cristofaro, ¹⁰ state and sensor attack observer is designed using sliding mode, and external disturbance is taken into account in order to achieve robust estimation performance, while discrete-time CPS case is not considered. In Shi et al., ¹¹ a stochastic modeling framework is introduced for joint state and attack estimation in CPS; nevertheless, actuator attack is absent, and discrete-time CPS case is not considered yet. In Wan et al., ¹² distributed observers are designed to reconstruct the node states of multi-agent system with Lipschitz-type nonlinear dynamics, and the reconstructed states are used for feedback control, but only sufficient conditions are derived, and the discrete-time case is not considered. There are also some other results and developments in this domain, see in Hu et al., ¹³ Corradini and Cristofaro, ¹⁴ Lu and Yang, ¹⁵ Xie and Yang, ¹⁶ and the references therein.

Main contributions

In this article, we focus on how to estimate secure state and reconstruct actuator attack simultaneously for continuous-time/discrete-time CPS. While most existing results only consider continuous-time CPS case, and there are few results about discrete-time case. To achieve the aforementioned goal, an intermediate and robust observer is proposed. The main contributions of this article are as follows. (1) A new observer is designed, based on which the reconstruction of secure state and actuator attack are executed simultaneously with $H_{\infty }$ performance. By introducing error parameters, the original model of CPS is transformed into an augmented error dynamics form, then bounded real lemma is used to obtain the observer gain matrices. (2) By introducing the intermediate variable proposed in Zhu and colleagues,^17,18 the designed observer is more general than traditional Luenberger observer; therefore, we can get better results. In addition, discrete-time CPS as well as continuous-time case is studied in our work.

Organization

The rest of this article is organized as follows. In section “Methodology,” a common CPS model is described with continuous-time/discrete-time form, and some reasonable assumptions are stated. Next, an intermediate and robust observer is designed, then an augmented form is deduced, based on which two theorems and their corollaries are presented to obtain the observer gain matrices. The illustrated examples are presented in section “Results and discussion” to prove the availability of our proposed scheme, and conclusion is given in section “Conclusion.”

Notation

For a matrix P, $P^T$ denotes its transpose. $\mathrm{rank}(P)$ is the rank of P. $\Vert x\Vert$ represents the Euclidean norm of vector x. $\mathrm{O}(\alpha ,r)$ presents a circle with center $(\alpha ,0)$ and radius r.

Methodology

Preliminaries and problem formulation

Consider the following common form of continuous-time/discrete-time CPS model

$$\delta \left[x\left(t\right)\right]=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+\mathrm{Dd}\left(t\right)+f_a\left(t\right)$$ (1)

$$y\left(t\right)=\mathrm{Cx}\left(t\right)$$ (2)

where operator $\delta [·]$ represents $\stackrel{·}{x}(t)$ for continuous-time CPS or $x(t+1)$ for discrete-time case. $x(t)\in {\mathbb{R}}^n$ is the state vector, $u(t)\in {\mathbb{R}}^m$ is the input vector, $y(t)\in {\mathbb{R}}^p$ is the output vector, $f_a(t)\in {\mathbb{R}}^n$ denotes actuator attack vector, and $d(t)\in {\mathbb{R}}^{n_d}$ is the external disturbance vector. $A\in {\mathbb{R}}^{n\times n}$ , $B\in {\mathbb{R}}^{n\times m}$ , $C\in {\mathbb{R}}^{p\times n}$ , and $D\in {\mathbb{R}}^{n\times n_d}$ are all known real matrices. Meanwhile, it is supposed that $(A,B)$ is stabilizable and $(A,C)$ is observable.

Remark 1

The CPS model (equations (1) and (2)) describes a general system that suffers from actuator attack and external disturbance, which is very common in actual scenarios. Our goal is to estimate the happened attack and system state simultaneously for continuous-time/discrete-time CPS, which is a challenging work and has not been completely solved yet in the past literature. Future study will focus on joint reconstruction of actuator attack and sensor attack, and security control for CPS under attacks, on the basis of reconstruction approach proposed in this article.

Some assumptions are put forward as follows.

Assumption 1

For every complex number s with non-negative real part

$$\mathrm{rank}\left[\begin{array}{cc}A-s{I}_n\hfill & I_n\hfill \\ \multicolumn{1}{c}{C}& 0\hfill \end{array}\right]=2n$$

Assumption 2

The external disturbance $d(t)$ satisfies $\Vert d(t)\Vert \le {\alpha }_1$ for continuous-time/discrete-time CPS, where ${\alpha }_1\ge 0$ .

Assumption 3

The actuator attack $f_a(t)$ satisfies $\Vert {\stackrel{·}{f}}_a(t)\Vert \le {\alpha }_2$ for continuous-time CPS, and $\Vert \Delta f_a(t)\Vert \le {\alpha }_3$ for discrete-time case, where the scalars ${\alpha }_2\ge 0$ and ${\alpha }_3\ge 0$ .

Remark 2

Assumption 1 signifies the invariant zeros of the system $(A,C,I_n)$ lie in the open left-hand complex plane. In the physical sense, Assumption 1 signifies that all the actuator attack components can affect the outputs, which implies that actuator attack is detectable. Assumption 2 implies that external disturbance is norm bounded, which is reasonable in practical systems.

Remark 3

Assumption 3 only requires the first derivative of actuator attack to be norm bounded, while the attack itself could be unbounded, this is more general than the previous approaches presented in Song et al. ⁹ and Corradini and Cristofaro.^10,14

Observer design

Here, an intermediate variable^17,18 is defined as

$$\xi \left(t\right)=f_a\left(t\right)-\omega x\left(t\right)$$ (3)

where $\omega >0$ is a scalar to be chosen. Combining equations (1) and (3), for continuous-time case, we yield

$$\stackrel{·}{\xi }\left(t\right)={\stackrel{·}{f}}_a\left(t\right)-\omega [\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+\xi \left(t\right)+\omega x\left(t\right)+\mathrm{Dd}\left(t\right)]$$ (4)

for discrete-time case, we yield

$$\begin{array}{c}\multicolumn{1}{c}{\xi (t+1)=\xi \left(t\right)+\Delta f_a\left(t\right)}\\ -\omega [\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+\xi \left(t\right)+\omega x\left(t\right)+\mathrm{Dd}\left(t\right)-x\left(t\right)]\hfill \end{array}$$ (5)

where $\Delta f_a(t)=f_a(t+1)-f_a(t)$ .

Then, based on equations (1)–(5), the following observer is constructed as

$$\delta \left[\hat{x}\left(t\right)\right]=A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+{\hat{f}}_a\left(t\right)+L(y\left(t\right)-\hat{y}\left(t\right))$$ (6)

for continuous-time case, the differential equation of the observed intermediate variable is

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\hat{\xi }}\left(t\right)=-\omega [A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+\hat{\xi }\left(t\right)+\omega \hat{x}\left(t\right)]}\\ +F(y\left(t\right)-\hat{y}\left(t\right))\hfill \end{array}$$ (7)

for discrete-time case, the difference equation of the observed intermediate variable is

$$\begin{array}{c}\multicolumn{1}{c}{\hat{\xi }(t+1)=-\omega [A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+\hat{\xi }\left(t\right)+\omega \hat{x}\left(t\right)-\hat{x}\left(t\right)]}\\ +F(y\left(t\right)-\hat{y}\left(t\right))+\hat{\xi }\left(t\right)\hfill \end{array}$$ (8)

for both continuous-time and discrete-time cases, the output equation under observer condition is

$$\hat{y}\left(t\right)=C\hat{x}\left(t\right)$$ (9)

the actuator attack reconstruction equation under observer condition is

$${\hat{f}}_a\left(t\right)=\hat{\xi }\left(t\right)+\omega \hat{x}\left(t\right)$$ (10)

where $\hat{x}(t)$ , $\hat{\xi }(t)$ , $\hat{y}(t)$ , and ${\hat{f}}_a(t)$ are the estimated values of $x(t)$ , $\xi (t)$ , $y(t)$ , and $f_a(t)$ , respectively. $L\in {\mathbb{R}}^{n\times p}$ and $F\in {\mathbb{R}}^{n\times p}$ are observer gain matrices need to be calculated.

According to equations (6)–(10), it is obvious that the design of intermediate and robust observer is summed up to seek out adequate L, F to realize robust state estimation and attack reconstruction, and by adjusting the scalar $\omega$ , we can obtain different results.

Denote the error variables

$$\{\begin{array}{c}{e}_x\left(t\right)=x\left(t\right)-\hat{x}\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)=\xi \left(t\right)-\hat{\xi }\left(t\right)}\\ \multicolumn{1}{c}{e_{\mathrm{fa}}\left(t\right)=f_a\left(t\right)-{\hat{f}}_a\left(t\right)=[\xi \left(t\right)+\omega x\left(t\right)]-[\hat{\xi }\left(t\right)+\omega \hat{x}\left(t\right)]}\\ \multicolumn{1}{c}{=e_{\xi }\left(t\right)+\omega e_x\left(t\right)}\end{array}$$

Then, the error dynamics for continuous-time case are given by

$$\{\begin{array}{c}{\stackrel{·}{e}}_x\left(t\right)=(A+\omega I_n)e_x\left(t\right)-\mathrm{LC}{e}_x\left(t\right)+e_{\xi }\left(t\right)+\mathrm{Dd}\left(t\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\xi }\left(t\right)=(-\omega A-{\omega }^2{I}_n)e_x\left(t\right)-\mathrm{FC}{e}_x\left(t\right)-\omega e_{\xi }\left(t\right)-\omega \mathrm{Dd}\left(t\right)+{\stackrel{·}{f}}_a\left(t\right)}\end{array}$$

and for discrete-time case, the error dynamics are given by

$$\{\begin{array}{c}{e}_x(t+1)=(A+\omega I_n)e_x\left(t\right)-\mathrm{LC}{e}_x\left(t\right)+e_{\xi }\left(t\right)+\mathrm{Dd}\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }(t+1)=(-\omega A-{\omega }^2{I}_n+\omega I_n)e_x\left(t\right)-\mathrm{FC}{e}_x\left(t\right)+(I_n-\omega I_n)e_{\xi }\left(t\right)-\omega \mathrm{Dd}\left(t\right)+\Delta f_a\left(t\right)}\end{array}$$

After analyzing the above error dynamics, the following augmented system is obtained in order to calculate gain matrices L, F, for continuous-time case

$$\begin{array}{c}\delta \left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]=\left[\begin{array}{cc}A+\omega I_n\hfill & I_n\hfill \\ \multicolumn{1}{c}{-\omega A-{\omega }^2{I}_n}& -\omega I_n\hfill \end{array}\right]\left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]-\left[\begin{array}{c}L\hfill \\ \multicolumn{1}{c}{F}\end{array}\right]\hfill \\ \left[\begin{array}{cc}C\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]+\left[\begin{array}{cc}D\hfill & 0\hfill \\ \multicolumn{1}{c}{-\omega D}& I_n\hfill \end{array}\right]\left[\begin{array}{c}d\left(t\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{f}}_a\left(t\right)}\end{array}\right]\hfill \end{array}$$ (11a)

and for discrete-time case

$$\begin{array}{c}\multicolumn{1}{c}{\delta \left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]=\left[\begin{array}{cc}A+\omega I_n\hfill & I_n\hfill \\ \multicolumn{1}{c}{-\omega A-{\omega }^2{I}_n+\omega I_n}& I_n-\omega I_n\hfill \end{array}\right]\left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]}\\ -\left[\begin{array}{c}L\hfill \\ \multicolumn{1}{c}{F}\end{array}\right]\left[\begin{array}{cc}C\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{e}_x\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\xi }\left(t\right)}\end{array}\right]+\left[\begin{array}{cc}D\hfill & 0\hfill \\ \multicolumn{1}{c}{-\omega D}& I_n\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}d\left(t\right)\hfill \\ \multicolumn{1}{c}{\Delta f_a\left(t\right)}\end{array}\right]\hfill \end{array}$$ (11b)

Let $\overline{e}(t)=\left[\begin{array}{c}{e}_x(t)\hfill \\ \multicolumn{1}{c}{e_{\xi }(t)}\end{array}\right]$ , $\overline{A}=\left[\begin{array}{cc}A+\omega I_n\hfill & I_n\hfill \\ \multicolumn{1}{c}{-\omega A-{\omega }^2{I}_n}& -\omega I_n\hfill \end{array}\right]$ for continuous-time cases or $\overline{A}=\left[\begin{array}{cc}A+\omega I_n\hfill & I_n\hfill \\ \multicolumn{1}{c}{-\omega A-{\omega }^2{I}_n+\omega I_n}& I_n-\omega I_n\hfill \end{array}\right]$ for discrete-time cases, $\overline{L}=\left[\begin{array}{c}L\hfill \\ \multicolumn{1}{c}{F}\end{array}\right]$ , $\overline{C}=\left[\begin{array}{cc}C\hfill & 0\hfill \end{array}\right]$ , $\overline{D}=\left[\begin{array}{cc}D\hfill & 0\hfill \\ \multicolumn{1}{c}{-\omega D}& I_n\hfill \end{array}\right]$ , $\overline{d}(t)=\left[\begin{array}{c}d(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{f}}_a(t)}\end{array}\right]$ for continuous-time cases or $\overline{d}(t)=\left[\begin{array}{c}d(t)\hfill \\ \multicolumn{1}{c}{\Delta f_a(t)}\end{array}\right]$ for discrete-time cases.

Then, equations (11a) and (11b) can be simplified as follows

$$\delta \left[\overline{e}\left(t\right)\right]=(\overline{A}-\overline{L}\overline{C})\overline{e}\left(t\right)+\overline{D}\overline{d}\left(t\right)$$ (12)

Remark 4

From equation (12), we can see that $\overline{A}$ , $\overline{C}$ , and $\overline{D}$ are already known, so the design of intermediate and robust observer is further summed up to seek out gain matrix $\overline{L}$ to realize robust state estimation and attack reconstruction.

Remark 5

We can also find that the system dynamics is finally deduced as error dynamics form as shown in equation (12), then the existing result presented in Gahinet and Apkarian ¹⁹ may be used, which helps turning the $H_{\infty }$ suboptimal constraints into a matrix inequality. Besides, the error term $\overline{d}(t)$ is formed from $d(t)$ and ${\stackrel{·}{f}}_a(t)$ for continuous-time case and $d(t)$ and $\Delta f_a(t)$ for discrete-time case, based on Assumption 2 and Assumption 3, $\overline{d}(t)$ is norm bounded as well.

Gain matrices design

The following theorems give the solutions to obtain gain matrices L and F for continuous-time /discrete-time CPS.

Theorem 1 (for continuous-time CPS)

Provided that Assumptions 1–3 hold, $H_{\infty }$ index $\gamma$ and circle $\mathrm{O}(\alpha ,r)$ are preseted. Error equation (12) for continuous-time case satisfies $\Vert e_{\mathrm{fa}}(t)\Vert <\gamma \Vert \overline{d}(t)\Vert$ , and the eigenvalues of $(\overline{A}-\overline{L}\overline{C})$ belong to $\mathrm{O}(\alpha ,r)$ if and only if there exist two symmetric positive definite matrices ${\overline{P}}_1$ , ${\overline{P}}_2\in {\mathbb{R}}^{2n\times 2n}$ and matrix $\overline{L}\in {\mathbb{R}}^{2n\times p}$ such that the inequations listed below are satisfied

$$\left[\begin{array}{ccc}{\overline{P}}_1(\overline{A}-\overline{L}\overline{C})+{(\overline{A}-\overline{L}\overline{C})}^T{\overline{P}}_1\hfill & {\overline{P}}_1\overline{D}\hfill & \overline{I}\hfill \\ \multicolumn{1}{c}{*}& -\gamma I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\gamma I\hfill \end{array}\right]<0$$ (13)

$$\left[\begin{array}{cc}-{\overline{P}}_2\hfill & {\overline{P}}_2(\overline{A}-\overline{L}\overline{C})-\alpha {\overline{P}}_2\hfill \\ \multicolumn{1}{c}{*}& -r^2{\overline{P}}_2\hfill \end{array}\right]<0$$ (14)

Proof

By directly using Lemma 4.1 in Gahinet and Apkarian, ¹⁹ which is called Bounded Real Lemma, equation (13) is obtained.

For equation (12), let $\overline{A}-\overline{L}\overline{C}=A$ , ${\overline{P}}_2=P$ , using Theorem 1 in Garcia and Bernussou, ²⁰ and pre- and post-multiply the result by $\left[\begin{array}{cc}P\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& I\hfill \end{array}\right]$ , equation (14) is obtained.

Theorem 2 (for discrete-time CPS)

Provided that Assumptions 1–3 hold, $H_{\infty }$ index $\gamma$ and circle $\mathrm{O}(\alpha ,r)$ are preseted. Error equation (12) for discrete-time case satisfies $\Vert e_{\mathrm{fa}}(t)\Vert <\gamma \Vert \overline{d}(t)\Vert$ , and the eigenvalues of $(\overline{A}-\overline{L}\overline{C})$ belong to $\mathrm{O}(\alpha ,r)$ if and only if there exist two symmetric positive definite matrices ${\overline{P}}_1$ , ${\overline{P}}_2\in {\mathbb{R}}^{2n\times 2n}$ and matrix $\overline{L}\in {\mathbb{R}}^{2n\times p}$ such that the inequations listed below are satisfied

$$\left[\begin{array}{cccc}-{\overline{P}}_1\hfill & {\overline{P}}_1(\overline{A}-\overline{L}\overline{C})\hfill & {\overline{P}}_1\overline{D}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& -{\overline{P}}_1\hfill & 0\hfill & \overline{I}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\gamma I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -\gamma I\hfill \end{array}\right]<0$$ (15)

$$\left[\begin{array}{cc}-{\overline{P}}_2\hfill & {\overline{P}}_2(\overline{A}-\overline{L}\overline{C})-\alpha {\overline{P}}_2\hfill \\ \multicolumn{1}{c}{*}& -r^2{\overline{P}}_2\hfill \end{array}\right]<0$$ (16)

Proof

Similar to Theorem 1, the proof is omitted here.

Remark 6

The obtained conditions (equations (13)–(16)) are not strict linear matrix inequality (LMI) conditions due to the two different Lyapunov matrices ${\overline{P}}_1$ , ${\overline{P}}_2$ ; thus, we cannot use LMI tools to solve the inequalities, in order to convert them into convex optimization problems, we introduce matrices $\overline{P}={\overline{P}}_1={\overline{P}}_2$ , and $\overline{Y}$ such that $\overline{Y}=\overline{P}\overline{L}$ , then the following two corollaries are deduced.

Corollary 1 (for continuous-time CPS)

Provided that Assumptions 1–3 hold, $H_{\infty }$ index $\gamma$ and circle $\mathrm{O}(\alpha ,r)$ are preseted. If there exist a symmetric positive definite matrix $\overline{P}\in {\mathbb{R}}^{2n\times 2n}$ and $\overline{Y}\in {\mathbb{R}}^{2n\times p}$ such that the inequations listed below are satisfied

$$\left[\begin{array}{ccc}\overline{P}\overline{A}+{\overline{A}}^T\overline{P}-\overline{Y}\overline{C}-{\overline{C}}^T{\overline{Y}}^T\hfill & \overline{P}\overline{D}\hfill & \overline{I}\hfill \\ \multicolumn{1}{c}{*}& -\gamma I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\gamma I\hfill \end{array}\right]<0$$ (17)

$$\left[\begin{array}{cc}-\overline{P}\hfill & \overline{P}\overline{A}-\overline{Y}\overline{C}-\alpha \overline{P}\hfill \\ \multicolumn{1}{c}{*}& -r^2\overline{P}\hfill \end{array}\right]<0$$ (18)

then, error equation (12) for continuous-time case satisfies $\Vert e_{\mathrm{fa}}(t)\Vert <\gamma \Vert \overline{d}(t)\Vert$ , the eigenvalues of $(\overline{A}-\overline{L}\overline{C})$ belong to $\mathrm{O}(\alpha ,r)$ , and the observer gain matrix is calculated by $\overline{L}={\overline{P}}^{-1}\overline{Y}$ .

Corollary 2 (for discrete-time CPS)

Provided that Assumptions 1–3 hold, $H_{\infty }$ index $\gamma$ and circle $\mathrm{O}(\alpha ,r)$ are preseted. If there exist a symmetric positive definite matrix $\overline{P}\in {\mathbb{R}}^{2n\times 2n}$ and $\overline{Y}\in {\mathbb{R}}^{2n\times p}$ such that the inequations listed below are satisfied

$$\left[\begin{array}{cccc}-\overline{P}\hfill & \overline{P}\overline{A}-\overline{Y}\overline{C}\hfill & \overline{P}\overline{D}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& -\overline{P}\hfill & 0\hfill & \overline{I}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\gamma I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -\gamma I\hfill \end{array}\right]<0$$ (19)

$$\left[\begin{array}{cc}-\overline{P}\hfill & \overline{P}\overline{A}-\overline{Y}\overline{C}-\alpha \overline{P}\hfill \\ \multicolumn{1}{c}{*}& -r^2\overline{P}\hfill \end{array}\right]<0$$ (20)

then, error equation (12) for discrete-time case satisfies $\Vert e_{\mathrm{fa}}(t)\Vert <\Vert \gamma \overline{d}(t)\Vert$ , the eigenvalues of $(\overline{A}-\overline{L}\overline{C})$ belong to $\mathrm{O}(\alpha ,r)$ , and the observer gain matrix is calculated by $\overline{L}={\overline{P}}^{-1}\overline{Y}$ .

Remark 7

Based on Corollary 1 and Corollary 2, the minimization problem of $H_{\infty }$ performance level $\gamma$ can be formulated as follows:

• Minimize $H_{\infty }$ performance level $\gamma$ ;

• Subject to equations (17) and (18) for continuous-time case;

• Or equations (19) and (20) for discrete-time case.

Results and discussion

In this part, we consider an unmanned ground vehicle (UGV) system presented in Shoukry and Tabuada, ²¹ include continuous-time/discrete-time case, to prove the availability of our proposed scheme.

Example 1

Consider the continuous-time UGV model under actuator attack and external disturbance, which is described as follows

$$\begin{array}{c}\left[\begin{array}{c}\stackrel{·}{x}\left(t\right)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{v}\left(t\right)}\end{array}\right]=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& -\frac{b}{M}\hfill \end{array}\right]\left[\begin{array}{c}x\left(t\right)\hfill \\ \multicolumn{1}{c}{v\left(t\right)}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\frac{1}{M}}\end{array}\right]F+\left[\begin{array}{c}0.1\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right]d\left(t\right)+\left[\begin{array}{c}{f}_{a1}\left(t\right)\hfill \\ \multicolumn{1}{c}{f_{a2}\left(t\right)}\end{array}\right]\hfill \\ \left[\begin{array}{c}{y}_1\left(t\right)\hfill \\ \multicolumn{1}{c}{y_2\left(t\right)}\\ \multicolumn{1}{c}{y_3\left(t\right)}\\ \multicolumn{1}{c}{y_4\left(t\right)}\end{array}\right]=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \end{array}\right]\left[\begin{array}{c}x\left(t\right)\hfill \\ \multicolumn{1}{c}{v\left(t\right)}\end{array}\right]\hfill \end{array}$$

where $x(t)$ and $v(t)$ represent the UGV position and linear velocity, respectively. M is the mechanical mass, b is the translational friction coefficient, F is the UGV input force, $d(t)$ is the external disturbance, $y_i(t)$ ( $i\in \{1,2,3,4\}$ ) is the UGV output, and $f_{\mathrm{ai}}(t)$ ( $i\in \{1,2\}$ ) is the actuator attack, respectively. Here, the parameters M and b are specified as $M=0.8$ and $b=1$ .

It is obvious that the above given UGV model is the form of equations (1) and (2). The external disturbance is set as 0, the actuator attack is created as in Table 1, which meet Assumption 2 and Assumption 3.

Table 1.

The actuator attacks $f_a$ .

Time (t)	$[0,10\mathrm{s})$	$[10\mathrm{s},20\mathrm{s})$	$[20\mathrm{s},30\mathrm{s})$
$f_{a1}$	$0$	$10(1-e^{-(t-10)})$	$10-5(1-e^{-(t-20)})$
$f_{a2}$	0	$e^{-(t-10)}-1$	$e^{-(t-10)}-1$
Time (t)	$[30\mathrm{s},40\mathrm{s})$	$[40\mathrm{s},50\mathrm{s})$	$[50\mathrm{s},80\mathrm{s})$
$f_{a1}$	$5-10(1-e^{-(t-30)})$	$-5+7(1-e^{-(t-40)})$	$2-2(1-e^{-(t-50)})$
$f_{a2}$	$-1+7(1-e^{-(t-30)})$	$-1+7(1-e^{-(t-30)})$	$6-6(1-e^{-(t-50)})$

Solving the inequations in Corollary 1 with $\omega =0$ and $\mathrm{O}(-8,7.5)$ , we can obtain ${\gamma }_{min}=0.1938$ with

$$\begin{array}{c}\overline{P}=\left[\begin{array}{cccc}380.1944\hfill & -252.5026\hfill & -25.8403\hfill & 17.3956\hfill \\ \multicolumn{1}{c}{-252.5026}& 380.1944\hfill & 17.3956\hfill & -25.8403\hfill \\ \multicolumn{1}{c}{-25.8403}& 17.3956\hfill & 2.2900\hfill & -1.3955\hfill \\ \multicolumn{1}{c}{17.3956}& -25.8403\hfill & -1.3955\hfill & 2.2900\hfill \end{array}\right]\hfill \\ \overline{Y}=1.0e+03*\left[\begin{array}{cccc}3.6866\hfill & -3.3033\hfill & 0.3795\hfill & 0.3795\hfill \\ \multicolumn{1}{c}{-3.4346}& 3.5234\hfill & 0.0972\hfill & 0.0972\hfill \\ \multicolumn{1}{c}{-0.2671}& 0.0993\hfill & 0.0221\hfill & 0.0221\hfill \\ \multicolumn{1}{c}{0.2301}& -0.1342\hfill & -0.0195\hfill & -0.0195\hfill \end{array}\right]\hfill \end{array}$$

The observer gain matrix is

$$\overline{L}=\left[\begin{array}{cccc}4.1213\hfill & -17.4241\hfill & 8.2965\hfill & 8.2965\hfill \\ \multicolumn{1}{c}{-6.1435}& 15.1520\hfill & 2.1562\hfill & 2.1562\hfill \\ \multicolumn{1}{c}{-37.4553}& -189.6145\hfill & 92.4616\hfill & 92.4616\hfill \\ \multicolumn{1}{c}{-22.9624}& 129.1968\hfill & 9.1356\hfill & 9.1355\hfill \end{array}\right]$$

For given $K=[\begin{array}{cc}-40.3877\hfill & -96.7544\hfill \end{array}]$ , simulation results are displayed as follows. Figure 3 shows actuator attack components and their estimated values. Figure 4 shows the observation errors of $f_{a1}$ and $f_{a2}$ . Figure 5 shows the system state components and their estimated values. Figure 6 shows the observation errors of $x(t)$ and $v(t)$ .

Figure 3.

Actuator attack components and their estimated values.

Figure 4.

Observation errors of actuator attack components.

Figure 5.

System state components and their estimated values.

Figure 6.

Observation errors of system state components.

From Figures 3 and 4, we can see that the observation value of actuator attack can track the real attack simultaneously, and its observation error has a peak at the moment of attack occur and converges quickly. Figures 5 and 6 show that the observation value of UGV position and linear velocity can also track the real state well. Meanwhile, due to the observation gain matrix is calculated offline, the possible performance penalty introduced by proposed solution is negligible. It can be illustrated that the proposed solution has achieved desired estimation performance under continuous-time CPS case.

Example 2

Consider discrete-time CPS model (equations (1) and (2)) with

$$\begin{array}{c}A=\left[\begin{array}{cc}1.0000\hfill & 0.0485\hfill \\ \multicolumn{1}{c}{0}& 0.9394\hfill \end{array}\right],B=\left[\begin{array}{c}0.0015\hfill \\ \multicolumn{1}{c}{0.0606}\end{array}\right],\hfill \\ C=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \end{array}\right],D=\left[\begin{array}{c}0.1\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right]\hfill \end{array}$$

which is the discretization of the aforementioned Example 1 using zero-order holder with a sampling period $T=0.05\mathrm{s}$ . The external disturbance and actuator attack are set as Example 1.

Solving the inequations in Corollary 2 with $\omega =1$ and $\mathrm{O}(0,1)$ , we can obtain ${\gamma }_{min}=1.7147$ with the observer gain matrix

$$\overline{L}=\left[\begin{array}{cccc}-107.1456\hfill & -109.5526\hfill & 54.8005\hfill & 54.8005\hfill \\ \multicolumn{1}{c}{-23.8345}& -21.4395\hfill & 11.9172\hfill & 11.9172\hfill \\ \multicolumn{1}{c}{-16.5268}& -15.5823\hfill & 7.7669\hfill & 7.7669\hfill \\ \multicolumn{1}{c}{1.9171}& 0.9846\hfill & -0.9586\hfill & -0.9586\hfill \end{array}\right]$$

For given $K=[\begin{array}{cc}-34.7114\hfill & -3.8262\hfill \end{array}]$ , simulation results are displayed as follows. Figure 7 shows the observation errors of $f_{a1}$ and $f_{a2}$ . Figure 8 shows the observation errors of $x(t)$ and $v(t)$ . It can be seen that the proposed solution has also achieved desired estimation performance under discrete-time case.

Figure 7.

Observation errors of actuator attack components.

Figure 8.

Observation errors of system state components.

Conclusion

In this article, we study the problem of secure state estimation and actuator attack reconstruction simultaneously for continuous-time and discrete-time CPS. By introducing an intermediate variable, a novel intermediate observer is designed. By denoting the error variables about system state, intermediate variable, and actuator attack, an augmented error dynamics equation is obtained based on matrix augmentation technique. Then, a regional pole placement method and a Bounded Real Lemma are used to obtain robust and less conservatism results. The proposed scheme is verified effectively by the illustrative examples.

The present work only considered CPS that suffers from actuator attack and external disturbance, sensor attack is absent, so our future study will focus on secure state estimation and attack reconstruction that under both actuator attack and sensor attack.