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# Covert Attack Detection and Secure Control for Cyber Physical Systems

Xianghua Wang, Ali Zolghadri, Changqing Wang

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Xianghua Wang, Ali Zolghadri, Changqing Wang. Covert Attack Detection and Secure Control for Cyber Physical Systems. 2023. [hal-04236000](https://hal.science/hal-04236000v1)

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Covert Attack Detection and Secure Control for Cyber Physical Systems

Xianghua Wang, Member, IEEE, Ali Zolghadri, Senior Member, IEEE, Changqing Wang

***Abstract***—The problem addressed in this paper is that of monitoring and secure control in cyber physical systems subject to cyber covert attacks. In this scenario, the attacker can manipulate sensor measurements and control actions while bypassing the classical monitoring schemes. Firstly, attack detection is investigated by constructing signal generators in both cyber and physical layers, whose output is injected into the transmission network to reveal the covert attacks. After the attack has been detected, the nominal output feedback controller is switched into a secure controller based on adaptive observer to ensure an output tracking performance level. The adaptive observer is used to suppress (and not to estimate) the attack signals, hence the control effect does not depend on the estimation precision. Moreover, there is no assumption on the boundedness of the attack signals. Finally, numerical simulations are provided to show the effectiveness of the proposed scheme.

***Index Terms***—Cyber physical systems, secure control, covert attacks, attack detection.

## I. INTRODUCTION

CYBER-Physical Systems (CPSs) are engineered systems with deep integration of computation, communication and networking, physical processes, and control systems. CPSs are set to change the shape of our daily lives as they concern many technological areas, including aerospace, automotive, energy, chemical industry, transportation, or health care. Monitoring and secure control of CPSs have spurred on substantial research activities during the last decade. Researches have been developed along several major lines, from both control-oriented and information/communication perspectives. Much effort has been devoted to cyber attack detection and secure control. See for example [1] and [2] for a recent survey.

Cyber attacks can be classified into two main categories: denial of service (DoS) attacks and deception attacks, which encompass replay attacks, covert attacks, zero dynamics attacks, and more [3]. A covert attacker possesses the capability to assess the impact of their actions in a manner that the dynamics of the controlled system under attack would appear unchanged from the nominal case [4]. Consequently, detecting covert attacks using conventional methods is highly challenging, and only a limited number of studies on this topic can be found in the literature. These methods can be categorized into two classes. The first class, known as moving

target-based methods, introduces elements such as a switched auxiliary system [5], linear time-varying external dynamics [6], or modulation matrices [7] to prevent the attacker from accurately identifying the plant dynamics. Nevertheless, these methods are effective only when operating under the assumption that the attacker remains unaware of the introduced matrices or dynamics. An alternative approach, presented in [8], involves a dual time-varying coding detection scheme. In this scheme, both the measured output signals and control input signals are encoded before transmission over the network and subsequently decoded upon reception by the next nodes. However, this approach necessitates a complex setup, requiring synchronous knowledge of the time-varying coding matrices in both the cyber and physical layers. It also assumes that the attacker cannot accurately distinguish between the time-varying coding matrices. The other class consists of distributed model-based methods [9], [10], which are primarily suitable for large-scale systems featuring interconnected plants. These methods rely on information from adjacent plants to detect attacks. Consequently, they are not applicable to single-plant systems, which are the focus of this paper.

In the realm of secure control, research on resilient control against covert attacks remains relatively sparse. Notably, in [9], a resilient control approach based on attack isolation was developed for large-scale systems facing covert attacks. This approach employs a two-stage fixed-time observer to enhance system resilience. In [10], the attack detection scheme was enhanced to estimate the actions of an attacker. Subsequently, an accommodation scheme was introduced to mitigate or neutralize the abnormal behavior exhibited by interconnected systems under covert attacks. It's worth noting that in [9], [10], the considered plant is assumed to be free from disturbances and noises. In [11], a method was introduced to counteract covert attacks by generating redundant control sequences, which were randomly selected on the actuator side to conceal the exploited control action. However, this approach needs a high computational cost. Fauser and Zhang [12] used frequency hopping spread spectrum to nullify the attack's influence on the system but required two attack detection filters to reveal the actual attack, resulting in both high computational cost and the absence of an attack signal estimate.

Given this overall picture, it seems that finding viable solutions to the problem of covert attacks detection and subsequent secure control remains an open problem. This paper addresses this issue by first presenting a procedure for covert attack detection, using signal generators placed in both the cyber and physical layers capable of generating and injecting signals into the transmission network. Subsequently, an output feedback

X. Wang is with the School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, China, XianghuaWang@bupt.edu.cn. The corresponding author.

A. Zolghadri is with the CNRS-IMS lab, University of Bordeaux, F-33400 Talence, France, ali.zolghadri@ims-bordeaux.fr

W. Wang is with the School of Automation, Northwestern Polytechnical University, Xi'an, China, wangcq@nwpu.edu.cn
secure control scheme, based on an adaptive observer, is proposed to enable the system to operate effectively even under attack conditions, ensuring a certain level of output tracking performance. The main contributions can be summarized as follows:

1) Regarding attack detection, we propose a method that revolves around the construction of signal generators in both the cyber and physical layers. These generators are tasked with generating and inserting external signals into the transmission network as soon as the CPS begins operation. These additional signals have no impact on the system's performance under nominal conditions (no attacks). However, they are designed to mislead potential adversaries and lead them to form an incorrect system model. As a result, the residual crosses a predefined threshold, revealing the presence of covert attacks. It's important to note that this approach differs from the moving target-based method [8], which relies on rapidly altering the system dynamics to prevent the adversary from accurately recognizing the system model.

2) Regarding subsequent secure control, we propose an original method designed to deal with covert attacks effectively. This approach is built upon an adaptive observer, which plays a pivotal role in ensuring output tracking performance. It's important to note that the primary function of the adaptive observer is to overcome and suppress the effects of attacks, rather than estimating state or attack signals. Consequently, the scheme is not reliant on observer performance, leading to a reduction in computational cost compared to [11], [12]. Furthermore, unlike [9], [10], our proposed scheme is applicable to systems subject to disturbances and noise.

The paper is organized as follows. Section II is devoted to system description under covert attacks. Section III presents our main results, including attack detection and secure control. A numerical example is provided in Section IV and some concluding remarks are given in Section V.

Notations: $I_r$ is an identity matrix of dimension $r$. $0$ is a matrix or vector with appropriate dimension and its all entries are $0$. For a vector $x \in \mathbb{R}^n$, $\|x\| = \sqrt{x^T x}$. For a matrix $W$, $He(W) = W + W^T$, $\star$ denotes the symmetric part of a matrix and Trace($W$) is the trace of $W$. $S_n$ is a diagonal matrix of dimension $n$ whose diagonal elements are all $s \neq 0$ and $s \neq -1$.

## II. PROBLEM FORMULATION AND PRELIMINARY

This section starts by presenting preliminary knowledge about the plant dynamics, the nominal controller, and the proposed signal generators. Subsequently, we conduct an analysis of system performance under normal conditions before introducing the attack scenario.

### A. System description

The cyber physical systems consist of two layers: physical layer and cyber layer, as shown in Fig. 1. The plant is in

<figure><img src="image_3.png" /><figcaption>Fig. 1. Block diagram for attack detection and secure control.</figcaption></figure>

the physical layer, which is considered to be continuous-time linear time invariant (LTI) described by

$$ \dot{x}_p = Ax_p + Bu_p + Ed, \quad y_p = Cx_p \quad (1) $$

where $x_p \in \mathbb{R}^n$ is the state vector, $u_p \in \mathbb{R}^m$ is the control input, $y_p \in \mathbb{R}^p$ is the measured output, $d \in \mathbb{R}^q$ is the unknown and bounded disturbance vector, $A, B, C$ and $E$ are matrices of appropriate dimensions.

For the plant (1), the following assumptions are made:

*Assumption 1:* It is assumed that the couple $(A, B)$ is controllable, and $(A, C)$ is observable.

*Assumption 2:* $\text{rank}(CB) = \text{rank}(B)$.

*Assumption 3:* The invariant zeros of the triple $(A, B, C)$ lie in $C_-$, $C_+$ denotes the left complex plane.

*Remark 1:* Assumption 2 (namely the system (1) satisfies the matching condition) and Assumption 3 (namely the system (1) is minimum-phase) are sufficient and necessary for sliding mode observer and unknown input observer [13].

The controller is in the cyber layer. As shown in Fig. 1, $y_c$ is the output signal received by the controller, $u_c$ is the control signal generated by the controller and defined by

$$ u_c = \begin{cases} u_{c,n}, & 0 \le t < T_d \\ u_{c,s}, & t \ge T_d \end{cases} \quad (2) $$

where $T_d$ is the time when the attack is detected and which will be given in Section III-A, and $u_{c,n}$ is the nominal controller whose expression is

$$ \hat{x}_p = A\hat{x}_p + Bu_c + L_n\tilde{y}_c, \quad \tilde{y}_c = y_c - C\hat{x}_p \quad (3) $$

$$ u_{c,n} = K_n \hat{x}_p + r \quad (4) $$

where $r$ is a reference command signal which acts on the closed-loop dynamics to generate $y_{ref}$

$$ \dot{x}_r = A_c x_r + Br, \quad y_{ref} = C x_r \quad (5) $$

where $A_c = A + BK_n$, $L_n$ and $K_n$ are design matrices given later in (17). $u_{c,s}$ is the secure controller and will be developed in Section III-B.
$\mathcal{E}_p$ and $\mathcal{E}_c$ are the proposed signal generators in the physical and cyber layers respectively, and are activated at the beginning of the system operation. The dynamics of signal generators $\mathcal{E}_c$ and $\mathcal{E}_p$ are given by

$$ 
\begin{aligned}
\dot{\xi}_c &= A_d \xi_c + B S_m r + L_n S_p y_c \\
\eta_c &= K_n \xi_c + S_m r
\end{aligned}
\qquad (6) $$

and

$$ 
\begin{aligned}
\dot{\xi}_p &= A_d \xi_p + B S_m r + L_n S_p y_p \\
\eta_p &= K_n \xi_p + S_m r
\end{aligned}
\qquad (7) $$

where $A_d = A + BK_n - L_n C$, $S_m$ and $S_p$ are design parameters and from Notations, the initial values are chosen as $\xi_c(0) = \xi_p(0) = \hat{x}_p(0)$. In the nominal case, $u_c = u_{c,n}$ and from (3)-(4) it follows

$$ u_{c,n} = [K_n(\lambda I - A_d)^{-1}B + I]r + K_n(\lambda I - A_d)^{-1}L_n y_c \quad (8) $$

where $\lambda$ is the Laplace operator. From (6), it follows

$$ \eta_c = s[K_n(\lambda I - A_d)^{-1}B + I]r + sK_n(\lambda I - A_d)^{-1}L_n y_c \quad (9) $$

By comparing (8) with (9), it is clear that

$$ \eta_c = su_{c,n} \qquad (10) $$

The objective is twofold:

* to detect the covert attack. In this work, the observer (3) is also used for attack detection which is a classical observer-based method and the detection mechanism is

- if $||\tilde{y}_c|| > \gamma_s||d||$ where $\tilde{y}_c$ is from (3), then an attack has occurred;

- otherwise, it is judged that there are no cyber attacks.

* to design $u_c$ in (2) such that

$$ \|y_p - y_{ref}\| \leq \gamma_s \|d\| \qquad (11) $$

where $\gamma_s$ is some desired output tracking performance level.

## B. Performance analysis for attack-free case

In the following, the closed-loop system performance in the nominal case (attack-free case) will be discussed.

Denote the signals propagated through the networks as $y_t$ and $u_t$, then from Fig. 1, it follows

$$ y_t = y_p, u_t = u_c + \eta_c \qquad (12) $$

When there are no cyber attacks, namely $y_a = 0$ and $u_a = 0$, the controller input $y_c$ and the plant input $u_p$ are respectively

$$ y_c = y_t, u_p = u_t - \eta_p \qquad (13) $$

Combine (12) with (13) to get $y_c = y_t = y_p$, which when substituted into (6) and (7), it follows $\eta_c = \eta_p$. Again from (12) and (13), we get $u_p = u_c + \eta_c - \eta_p = u_c = u_{c,n}$.

Denote $\tilde{x}_p = x_p - \hat{x}_p$, and subtract (3) from (1) (using $u_p = u_c$ and $y_c = y_p$) to get

$$ \dot{\tilde{x}}_p = A_o \tilde{x}_p + Ed, \tilde{y}_c = C \tilde{x}_p \qquad (14) $$

where $A_o = A - L_n C$.

Define $e_p = x_p - x_r$ and subtract (5) from (1) with the use of $u_p = u_c = u_{c,n}$ and (4) to get

$$ \dot{e}_p = A_c e_p - BK_n \tilde{x}_p + Ed, y_p - y_{ref} = C e_p \qquad (15) $$

By combining (14) with (15), we get

$$ 
\begin{bmatrix} \dot{e}_p \\ \dot{\tilde{x}}_p \end{bmatrix} = \underbrace{\begin{bmatrix} A_c & -BK_n \\ \mathbf{0} & A_o \end{bmatrix}}_{A_n} \begin{bmatrix} e_p \\ \tilde{x}_p \end{bmatrix} + \underbrace{\begin{bmatrix} E \\ E \end{bmatrix}}_{E_n} d \qquad (16) \\
\begin{bmatrix} y_p - y_{ref} \\ \tilde{y}_c \end{bmatrix} = \underbrace{\begin{bmatrix} C & \mathbf{0} \\ \mathbf{0} & C \end{bmatrix}}_{C_n} \begin{bmatrix} e_p \\ \tilde{x}_p \end{bmatrix}
$$

By virtue of Bounded Real Lemma [14], for a given $\gamma_s > 0$, if there exist matrices $K_n$, $L_n$, and a symmetric positive definite (s.p.d.) matrix $P_n = \text{diag}\{P_c, P_o\}$ with $P_c = P_c^T$, $P_o = P_o^T$ such that

$$ \begin{bmatrix} A_n^T P_n + P_n A_n + C_n^T C_n & P_n E_n \\ \star & -\gamma_s^2 \mathbf{I}_q \end{bmatrix} < \mathbf{0} \qquad (17) $$

then $\|y_p - y_{ref}\| \leq \gamma_s \|d\|$ (hence the acceptable performance (11) is ensured) and $\|\tilde{y}_c\| \leq \gamma_s \|d\|$ (hence there are no false alarms for the attack-free case).

Using the structures of $A_n$, $P_n$, $E_n$ and $C_n$, the inequality (17) becomes

$$ \begin{bmatrix} \Pi_1 & -P_c BK_n & P_c E \\ \star & \Pi_2 & P_o E \\ \star & \star & -\gamma_s^2 \mathbf{I}_q \end{bmatrix} < \mathbf{0} \qquad (18) $$

where $\Pi_1 = P_c A + A^T P_c + P_c BK_n + K_n^T B^T P_c + C^T C$, $\Pi_2 = P_o A + A^T P_o - P_o L_n C - C^T L_n^T P_o + C^T C$. Pre-multiplying and post-multiplying both sides of (18) with $\text{diag}\{P_c^{-1}, \mathbf{I}_n, \mathbf{I}_q\}$ and using Schur complement [15], (18) is equivalent to the following inequality

$$ \begin{bmatrix} \Theta_1 & \mathbf{0} & E & P_c^{-1}C^T \\ \star & \Theta_2 & P_o E & \mathbf{0} \\ \star & \star & -\gamma_s^2 \mathbf{I}_q & \mathbf{0} \\ \star & \star & \star & -\mathbf{I}_p \end{bmatrix} + \text{He} \left\{ \begin{bmatrix} -BK_n \\ \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{0} & \mathbf{I}_n & \mathbf{0} & \mathbf{0} \end{bmatrix} \right\} < \mathbf{0} \qquad (19) $$

where $\Theta_1 = AP_c^{-1} + BK_n P_c^{-1} + P_c^{-1}A^T + P_c^{-1}K_n^T B^T$, $\Theta_2 = P_o A + A^T P_o - P_o L_n C - C^T L_n^T P_o + C^T C$. Now, from the Young's inequality [16], it follows that

$$ \text{He} \left\{ \begin{bmatrix} -BK_n \\ \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{0} & \mathbf{I}_n & \mathbf{0} & \mathbf{0} \end{bmatrix} \right\} \leq \gamma_a \left[ \begin{aligned} & -P_c^{-1} K_n^T B^T \mathbf{0} \mathbf{0} \mathbf{0} \mathbf{0}^T P_c \begin{bmatrix} -P_c^{-1} K_n^T B^T \mathbf{0} \mathbf{0} \mathbf{0} \mathbf{0} \end{bmatrix} \\ & + \gamma_a^{-1} \begin{bmatrix} \mathbf{0} & \mathbf{I}_n & \mathbf{0} & \mathbf{0} \end{bmatrix}^T P_c \begin{bmatrix} \mathbf{0} & \mathbf{I}_n & \mathbf{0} & \mathbf{0} \end{bmatrix} \end{aligned} \right] \qquad (20) $$
where $0 < \gamma_a < 1$. Substitute (20) into (19) and use Schur complement [15] again to get

$$ \begin{bmatrix} \Theta_1 & 0 & E & P_c^{-1}C^T & -BK_nP_c^{-1} & 0 \\ \star & \Theta_2 & P_oE & 0 & 0 & I_n \\ \star & \star & -\gamma_s^2\mathbf{I}_q & 0 & 0 & 0 \\ \star & \star & \star & -I_p & 0 & 0 \\ \star & \star & \star & \star & -\gamma_a^{-1}P_c^{-1} & 0 \\ \star & \star & \star & \star & \star & -\gamma_a P_c^{-1} \end{bmatrix} < 0 \quad (21) $$

Define $\hat{P}_c = P_c^{-1}$, $\hat{K}_n = K_n P_c^{-1}$ and $\hat{L}_n = P_o L_n$, then (21) is linear with respect to $\hat{P}_c$, $\hat{K}_n$, $\hat{L}_n$, and $P_o$ and an LMI solver (e.g. LMI toolbox of MATLAB) can be used to solve it. Then $K_n = \hat{K}_n P_c$, and $L_n = P_o^{-1} \hat{L}_n$.

**Remark 2:** It can be seen from above that the introduction of signal generators $\mathcal{E}_p$ and $\mathcal{E}_c$ will neither affect the nominal operation nor trigger a false alarm.

**Remark 3:** From (18), it follows $P_c A_c + A_c^T P_c < 0$, $P_o A_o + A_o^T P_o < 0$ and

$$ \begin{bmatrix} P_o A_o + A_o^T P_o + C^T C & P_o E \\ E^T P_o & -\gamma_s^2 \mathbf{I}_p \end{bmatrix} < 0 \quad (22) $$

which implies $\|C(\lambda I_n - A_o)^{-1}E\| \le \gamma_s$.

### III. MAIN RESULTS

In this section, the main results including attack detection and secure control are given. In order to derive the main results, the following lemma is firstly introduced.

**Lemma 1:** [17] For a system in the form of (1), there exist a s.p.d. matrix $P$, and matrices $L, K$ such that

$$ (A - LC)^T P + P(A - LC) < 0, \quad B^T P = KC $$

if and only if Assumption 2 and Assumption 3 hold.

In the following, a detection scheme is firstly presented to reveal the covert attack. Then, a secure controller is designed such that (11) can be achieved.

#### A. Attack Detection

The covert attacks operate in two phases.

In **Phase 1**, the attacker seizes the network signals $u_t$ and $y_t$ and uses them to establish a model of the plant. In this phase, the attack signals $u_a$ and $y_a$ have not yet been injected into the network, and the nominal controller is adopted, namely $u_c = u_{c,n}$. From (10) and (12), it follows that $y_t = y_p$ and $u_t = u_c + \eta_c = \alpha u_c$ with $\alpha = s + 1$. As stated after (13), in this case, $u_p = u_c$, hence $u_p = \alpha^{-1} u_t$, and when substituted into (1) yields

$$ \dot{x}_p = Ax_p + \alpha^{-1} Bu_t + Ed, \quad y_t = Cx_p \quad (23) $$

As in [8], the covert agent is sophisticated, and has access to the transmitted signals $u_t$ and $y_t$, and perfect knowledge of the disturbance-free and noise-free plant model [18]. Therefore, from (23) the covert agent can use $u_t$ and $y_t$ to establish a model of the plant as (some system identification technique may be used to estimate them):

$$ \dot{x}_a = Ax_a + \alpha^{-1} Bu_a, \quad y_a = Cx_a \quad (24) $$

where $u_a \in \mathbb{R}^m$ is the covert agent's control input, $y_a \in \mathbb{R}^p$ is the output signal of the covert agent.

In **Phase 2**, the covert attack is carried out, and define $T_a$ as the time when this happens. The covert misappropriation is performed by combining two aspects: one is adding the covert controller signal $u_a$ into $u_t$, and the other is subtracting the covert agent output $y_a$ from $y_t$, hence $u_p$ (the actual control input signal of the plant) and $y_c$ (the actual output signal received by the controller) in (13) become respectively

$$ u_p = u_t + u_a - \eta_p; \quad (25) $$

and

$$ y_c = y_t - y_a. \quad (26) $$

From (1), (12) and (24), (25) and (26) can be rewritten as

$$ u_p = u_c + u_a + \eta_c - \eta_p, \quad y_c = Cx_p - Cx_a \quad (27) $$

In the following, we will show with signal generators $\mathcal{E}_p$ and $\mathcal{E}_c$, the attack will be uncovered.

Subtract (3) from (1) and use (27) to get

$$ \begin{aligned} \dot{x}_p &= A_o \tilde{x}_p + B u_a + Ed + L_n C x_a + B(\eta_c - \eta_p), \\ \tilde{y}_c &= C \tilde{x}_p - C x_a \end{aligned} \quad (28) $$

where $\eta_c - \eta_p = K_n(\xi_c - \xi_p)$ and from (6)-(7), it follows

$$ \dot{\xi}_c - \dot{\xi}_p = A_d(\xi_c - \xi_p) - L_n S_p C x_a \quad (29) $$

Define $X = [\tilde{x}_p^T (\xi_c - \xi_p)^T x_a^T]^T$, and augment (24), (28) with (29) to get

$$ \dot{X} = A_x X + B_x u_a + E_x d, \quad \tilde{y}_c = C_x X \quad (30) $$

where $C_x = [C \quad 0 \quad -C]$, $E_x = [E^T \quad 0 \quad 0]^T$,

$$ A_x = \begin{bmatrix} A_o & BK_n & L_n C \\ \mathbf{0} & A_d & -L_n S_p C \\ \mathbf{0} & \mathbf{0} & A \end{bmatrix}, \quad B_x = \begin{bmatrix} B \\ \mathbf{0} \\ B \alpha^{-1} \end{bmatrix}. $$

The corresponding transfer function is $\tilde{y}_c = W'_a(\lambda)u_a + C(\lambda I_n - A_o)^{-1}Ed$, and from Remark 3 $\|C(\lambda I_n - A_o)^{-1}Ed\| \le \gamma_s\|d\|$, $W'_a(\lambda) = (s/\alpha)C(\lambda I_n - A)^{-1}B - sC(\lambda I_n - A_o)^{-1}BK_n(\lambda I_n - A_d)^{-1}L_n C(\lambda I_n - A)^{-1}B$. When $s \ne 0$ and $s \ne -1$ (when $s = -1$, $\alpha = 0$), $W'_a(\lambda) \ne 0$, and in this case, there exists a time $T_d \ge T_a$ such that for $t \ge T_d$, $\|\tilde{y}_c(t)\| > \gamma_s\|d\|$ and then an alarm unveiling an attack will be triggered and the attack detection time is $T_d$.

To better show the effectiveness on attack detection of the proposed signal generators, an analysis on the performance without signal generators is conducted here. Without signal generators $\mathcal{E}_c$ and $\mathcal{E}_p$, in Phase 1, the signals propagated through the networks namely $y_t$ and $u_t$ are $y_t = y_p$ and $u_t = u_c$. Since the attack signal $u_a$ has not been injected into the network, the control signal received by the plant is $u_p = u_t = u_c$. Hence an approximated model of the plant can be established by the covert agent as follows:

$$ \dot{x}_a = Ax_a + Bu_a, \quad y_a = Cx_a \quad (31) $$

In Phase 2, without the signal generator, $\eta_c = \eta_p = 0$, (27) becomes $u_p = u_c + u_a$ and $y_c = Cx_p - Cx_a$, and then (28) becomes

$$ \dot{x}_p = A_o \tilde{x}_p + B u_a + Ed + L_n C x_a, \quad \tilde{y}_c = C \tilde{x}_p - C x_a \quad (32) $$
Combine (31) with (32) to get

$$ 
\begin{bmatrix} \dot{\tilde{x}}_p \\ \dot{x}_a \end{bmatrix} = \begin{bmatrix} A_o & L_n C \\ \mathbf{0} & A \end{bmatrix} \begin{bmatrix} \tilde{x}_p \\ x_a \end{bmatrix} + \begin{bmatrix} E \\ \mathbf{0} \end{bmatrix} d + \begin{bmatrix} B \\ B \end{bmatrix} u_a, \\
\tilde{y}_c = \begin{bmatrix} C & -C \end{bmatrix} \begin{bmatrix} \tilde{x}_p \\ x_a \end{bmatrix}
$$

and the transfer function is $\tilde{y}_c = C(\lambda \mathbf{I}_n - A_o)^{-1}Ed + W_a(\lambda)u_a$ where

$$ 
\begin{aligned}
W_a(\lambda) &= \begin{bmatrix} C & -C \end{bmatrix} \left( \lambda \mathbf{I}_{2n} - \begin{bmatrix} A_o & L_n C \\ \mathbf{0} & A \end{bmatrix} \right)^{-1} \begin{bmatrix} B \\ B \end{bmatrix} \\
&= C(\lambda \mathbf{I}_n - A_o)^{-1} \left[ \begin{aligned} & (\lambda \mathbf{I}_n - A)(\lambda \mathbf{I}_n - A)^{-1} \\ & + L_n C(\lambda \mathbf{I}_n - A)^{-1} \end{aligned} \right] B - C(\lambda \mathbf{I}_n - A)^{-1} B \\
&= \mathbf{0}
\end{aligned}
$$

hence $\tilde{y}_c$ can be rewritten as

$$ \tilde{y}_c = C(\lambda \mathbf{I}_n - A_o)^{-1}Ed \quad (33) $$

From Remark 3, it follows that $\|\tilde{y}_c\| \le \gamma_s \|d\|$. Hence such attacks easily evade classical anomaly detectors and security defense. As already mentioned, this is because covert attacker can calculate the effect (namely $y_a$) of attack signal $u_a$ on the measured plant output $y_p$ and then mitigate it by injecting $-y_a$ into $y_p$. As a consequence, the signal received by the controller will appear unchanged from the nominal case.

**Remark 4:** To unmask the attacker, we propose to insert external signals $\eta_c$ and $\eta_p$ from (6) and (7) into the network at the beginning of system operation as shown in Fig. 1. This operation confounds the adversary, leading to establish an erroneous system model. It's worth noting that this approach differs from the moving target-based method [8], which relies on rapidly altering system dynamics to prevent the adversary from accurately recognizing the system model.

## B. Secure Control

In this section a secure controller based on an adaptive observer is proposed to replace the nominal controller as soon as a covert attack is detected. This controller will ensure the output tracking performance (11).

Before designing the secure controller, the following proposition is firstly given.

**Proposition 1:** The system (30) satisfies

* $(A_x, B_x, C_x)$ is minimum phase;

* $\text{rank}(C_x B_x) = \text{rank}(B_x)$.

*Proof:* The proof is reported in Appendix A.

The secure controller is designed as

$$ u_{c,s} = K_n \hat{x}_p + r + \bar{u}_c, \bar{u}_c = -\hat{u}_a - K_x \hat{X} \quad (34) $$

where $r$ is from the reference model (5), $\hat{x}_p$ is from (3), $K_x = [-K_n K_n 0]$, $K_n$ satisfies (17), $\hat{X}$ and $\hat{u}_a$ are from the following observer

$$ \dot{\hat{X}} = A_x \hat{X} + B_x \hat{u}_a + L_x e_{y,c} \quad (35a) $$

$$ e_{y,c} = \tilde{y}_c - C_x \hat{X} \quad (35b) $$

$$ \hat{u}_a(t) = K_1 \hat{u}_a(t - \tau) + K_2 e_{y,c} \quad (35c) $$

where $L_x$, $K_1$ and $K_2$ are design matrices, $\tau > 0$ is called the 'learning interval'.

Let $\tilde{X} = X - \hat{X}$ and $\tilde{u}_a = u_a - \hat{u}_a$, then the error dynamics is given by subtracting (35) from (30)

$$ \dot{\tilde{X}} = (A_x - L_x C_x) \tilde{X} + E_x d + B_x \tilde{u}_a, \quad e_{y,c} = C_x \tilde{X} \quad (36) $$

After the attack is detected (for $t \ge T_d$), from (2), $u_c = u_{c,s}$. Combine (27) with (34), then

$$ u_p = K_n \hat{x}_p + r - K_x \hat{X} + \tilde{u}_a + \eta_c - \eta_p \quad (37) $$

Substitute (37) into (1) and the closed-loop system is given by

$$ \dot{x}_p = A_c x_p + B K_x \tilde{x} + B \tilde{u}_a + Ed + Br \quad (38) $$

Subtract (5) from (38) to get

$$ \dot{e}_p = A_c e_p + B K_x \tilde{X} + B \tilde{u}_a + Ed \quad (39) $$

Define $\xi = \begin{bmatrix} \tilde{X} \\ e_p \end{bmatrix}$ and combine (36) with (39) to get

$$ \dot{\xi} = (A_{\xi} - L_{\xi} C_{\xi}) \xi + E_{\xi} d + B_{\xi} \tilde{u}_{a}, \quad e_{y,c} = C_{\xi} \xi \quad (40) $$

where

$$ 
\begin{aligned}
A_{\xi} &= \begin{bmatrix} A_x & \mathbf{0} \\ B K_x & A_c \end{bmatrix}, \quad L_{\xi} = \begin{bmatrix} L_x \\ \mathbf{0} \end{bmatrix}, \quad E_{\xi} = \begin{bmatrix} E_x \\ E \end{bmatrix}, \\
B_{\xi} &= \begin{bmatrix} B_x \\ B \end{bmatrix}, \quad C_{\xi} = \begin{bmatrix} C_x & \mathbf{0} \end{bmatrix}.
\end{aligned}
\quad (41) $$

**Remark 5:** From the structure of (41), it is clear that $\text{rank}(C_{\xi} B_{\xi}) = \text{rank}(C_x B_x)$ and $\text{rank}(B_{\xi}) = \text{rank}(B_x)$, which when combined with Proposition 1, it follows that $\text{rank}(C_{\xi} B_{\xi}) = \text{rank}(B_{\xi})$. Again from the structure of (41), the Rosenbrock matrix of $(A_{\xi}, B_{\xi}, C_{\xi})$ is

$$ \mathcal{R}_{\xi} = \begin{bmatrix} \lambda \mathbf{I}_{3n} - A_x & \mathbf{0} & B_x \\ -B K_x & \lambda \mathbf{I}_n - A_c & B \\ C_x & \mathbf{0} & \mathbf{0} \end{bmatrix} \quad (42) $$

Define

$$ S_R = \begin{bmatrix} \mathbf{I}_{3n} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{I}_n \\ \mathbf{0} & \mathbf{I}_m & \mathbf{0} \end{bmatrix}, \quad S_L = \begin{bmatrix} \mathbf{I}_{3n} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{I}_p \\ \mathbf{0} & \mathbf{I}_n & \mathbf{0} \end{bmatrix} \quad (43) $$

and then

$$ 
\begin{aligned}
\text{rank}(\mathcal{R}_{\xi}) &= \text{rank}(S_L \mathcal{R}_{\xi} S_R) \\
&= \begin{bmatrix} \lambda \mathbf{I}_{3n} - A_x & B_x \\ C_x & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix} \\
&\quad - B K_x \begin{bmatrix} \mathbf{0} \\ B \end{bmatrix} \begin{bmatrix} \lambda \mathbf{I}_n - A_c \end{bmatrix}
\end{aligned}
\quad (44) $$

From Proposition 1, it is known that $(A_{\xi}, B_{\xi}, C_{\xi})$ is minimum phase.

**Proposition 2:** From Remark 5, there exist a s.p.d. matrix $P_{\xi}$, matrices $L_{\xi}$ and $K_{\xi}$ such that

$$ (A_{\xi} - L_{\xi} C_{\xi})^{T} P_{\xi} + P_{\xi} (A_{\xi} - L_{\xi} C_{\xi}) < 0, \quad (45a) $$

$$ B_{\xi}^{T} P_{\xi} = K_{\xi} C_{\xi} \quad (45b) $$

The proof of Proposition 2 is directly from Lemma 1.

Then we have the following main result.

**Theorem 1:** The proposed secure controller (34)-(35) can ensure the acceptable performance (11) if
* there exist matrices $L_\xi$, $K_\xi$ and a s.p.d. matrix $P_\xi$ such that

$$ \Omega_{\xi} = \begin{bmatrix} \Psi_{\xi} & P_{\xi}E_{\xi} \\ E_{\xi}^{T}P_{\xi} & -(\gamma_{s}^{2} - \gamma_{f}^{2})\mathbf{I} \end{bmatrix} < 0 \quad (46) $$

and (45b) are satisfied simultaneously, where $\Psi_{\xi} = (A_{\xi} - L_{\xi}C_{\xi})^{T}P_{\xi} + P_{\xi}(A_{\xi} - L_{\xi}C_{\xi}) + [\mathbf{0} \ \mathbf{I}_{n}]^{T}C^{T}C[\mathbf{0} \ \mathbf{I}_{n}], 0 < \gamma_{f} < \gamma_{s}$,

$$ K_2 = K_\xi \quad (47) $$

* $K_1$ is chosen as

$$ K_1 = k_1 \mathbf{I} \quad \text{and} \quad 0 < k_1 < \frac{1}{\sqrt{\gamma + 1}} < 1 \quad (48) $$

where $0 < \gamma < 1$ is an arbitrary constant,

* $\tau$ is selected appropriately such that

$$ \|\Lambda_{\tau}\| \le k_{f} \quad (49) $$

where $\Lambda_{\tau} = u_{a}(t) - K_{1}u_{a}(t-\tau)$, $k_{f} > 0$ is an arbitrarily small constant.

*Proof:* It follows from (35c) that

$$ \begin{aligned} \tilde{u}_a(t) &= u_a(t) - \hat{u}_a(t) \\ &= u_a(t) - K_1 \hat{u}_a(t - \tau) - K_2 e_{y,c} \\ &= u_a(t) - K_1 u_a(t - \tau) + K_1 \tilde{u}_a(t - \tau) - K_2 e_{y,c} \\ &= \Lambda_{\tau} + K_1 \tilde{u}_a(t - \tau) - K_2 e_{y,c} \end{aligned} \quad (50) $$

Construct a positive function as

$$ V = \xi^T P_\xi \xi + \int_{t-\tau}^{t} \tilde{u}_a^T(\omega) \tilde{u}_a(\omega) d\omega \quad (51) $$

where $P_\xi$ is from (46), and differentiate $V$ with the use of (40), then

$$ \begin{aligned} \dot{V} ={}& \xi^T [(A_\xi - L_\xi C_\xi)^T P_\xi + P_\xi (A_\xi - L_\xi C_\xi)] \xi + \\ & 2\xi^T P_\xi E_\xi d + 2\xi^T P_\xi B_\xi \tilde{u}_a + \tilde{u}_a^T(t) \tilde{u}_a(t) \\ & - \tilde{u}_a^T(t - \tau) \tilde{u}_a(t - \tau) \end{aligned} \quad (52) $$

Let Term1 = $2\xi^T P_\xi B_\xi \tilde{u}_a + \tilde{u}_a^T(t) \tilde{u}_a(t)$, and use (45b), (47) and (50) to get

$$ \begin{aligned} \text{Term1} &= 2\xi^T C_\xi^T K_\xi^T (\Lambda_\tau + K_1 \tilde{u}_a(t-\tau) - K_2 e_{y,c}) + (\Lambda_\tau \\ &\quad + K_1 \tilde{u}_a(t-\tau) - K_2 e_{y,c})^T (\Lambda_\tau + K_1 \tilde{u}_a(t-\tau) - K_2 e_{y,c}) \\ &= 2e_{y,c}^T K_\xi^T (\Lambda_\tau + K_1 \tilde{u}_a(t-\tau) - K_2 e_{y,c}) + (\Lambda_\tau + K_1 \\ &\quad \tilde{u}_a(t-\tau) - K_2 e_{y,c})^T (\Lambda_\tau + K_1 \tilde{u}_a(t-\tau) - K_2 e_{y,c}) \\ &= -e_{y,c}^T K_2^T K_2 e_{y,c} + \underbrace{\Lambda_\tau^T K_1 \tilde{u}_a(t-\tau) + \tilde{u}_a^T(t-\tau) K_1^T \Lambda_\tau}_{\text{Term2}} \\ &\quad + \Lambda_\tau^T \Lambda_\tau + \tilde{u}_a^T(t-\tau) K_1^T K_1 \tilde{u}_a(t-\tau) \end{aligned} \quad (53) $$

From the Young's inequality [16], it follows that

$$ \text{Term2} \le \frac{1}{\gamma} \Lambda_{\tau}^{T} \Lambda_{\tau} + \gamma \tilde{u}_{a}^{T}(t-\tau) K_{1}^{T} K_{1} \tilde{u}_{a}(t-\tau), \quad (54) $$

where $0 < \gamma < 1$. By substituting (54) and (53) into (52), and using (48), $\dot{V}$ can be further written as

$$ \begin{aligned} \dot{V} \le {}& \xi^T [(A_\xi - L_\xi C_\xi)^T P_\xi + P_\xi (A_\xi - L_\xi C_\xi)] \xi + \\ & 2\xi^T P_\xi E_\xi d - e_{y,c}^T K_2^T K_2 e_{y,c} + \Lambda_\tau^T (\mathbf{I} + 1/\gamma \mathbf{I}) \Lambda_\tau \\ & + \tilde{u}_a^T(t-\tau)(\gamma K_1^T K_1 + K_1^T K_1 - \mathbf{I}) \tilde{u}_a(t-\tau) \\ \le {}& \xi^T [(A_\xi - L_\xi C_\xi)^T P_\xi + P_\xi (A_\xi - L_\xi C_\xi)] \xi \\ & + 2\xi^T P_\xi E_\xi d + (1+1/\gamma)k_f^2 \end{aligned} $$

Since $k_f$ can be arbitrarily small, there must exist $0 < \gamma_f < \gamma_s$ such that $(1+1/\gamma)k_f^2 = \gamma_f^2 d^T d$.

$$ \text{Define } W = \dot{V} + \xi^T \begin{bmatrix} \mathbf{0} \\ \mathbf{I}_n \end{bmatrix} C^T \underbrace{C[\mathbf{0} \ \mathbf{I}_n]}_{y_p - y_{ref}} \xi - \gamma_s^2 d^T d $$

and it follows that $W = [\xi^T d^T] \Omega_\xi [\xi \ d]$, where $\Omega_\xi$ is from (46). Since $\Omega_\xi < 0$, $W < 0$. There are two cases: if $\dot{V} \ge 0$, then $\|y_p - y_{ref}\| \le \gamma_s \|d\|$; if $\dot{V} < 0$, then $V$ will decrease until $\dot{V} \ge 0$ holds. Hence $\|y_p - y_{ref}\| \le \gamma_s \|d\|$ can be ensured. Here completes the proof of Theorem 1. ■

**Remark 6:** With the proposed secure controller (34)-(35), the output tracking performance $\|y_p - y_{ref}\| \le \gamma_s \|d\|$ can be achieved. It is noted that the adaptive observer is used to suppress the effect of attacks, not to provide the estimates of states and attack signals. Hence the estimation precision has no influences on the control effect, different, for example, from the observer-based scheme reported in [19]. Moreover, there is no assumption on the amplitude of the attack signal $u_a$, as for example in [20] where the attack signal was assumed to be bounded.

**Remark 7:** Define a new variable $\bar{L}_\xi = P_\xi L_\xi$, then (46) can be transformed into an LMI with respect to the variables $P_\xi$ and $\bar{L}_\xi$ as follows:

$$ \Omega_{\xi} = \begin{bmatrix} \text{He}(P_{\xi}A_{\xi} - \bar{L}_{\xi}C_{\xi}) + C* & P_{\xi}E_{\xi} \\ E_{\xi}^{T}P_{\xi} & -(\gamma_{s}^{2} - \gamma_{f}^{2})\mathbf{I} \end{bmatrix} < 0 \quad (55) $$

where $C* = [\mathbf{0} \ \mathbf{I}_n]^T C^T C [\mathbf{0} \ \mathbf{I}_n]$. However, it is not a trivial work to find a systematic solution satisfying (55) and (45b) simultaneously. To solve this difficult problem, similar to [17], we rewrite linear equation (45b) as

$$ \text{Trace}[(B_{\xi}^{T} P_{\xi} - K_{\xi} C_{\xi})^{T} (B_{\xi}^{T} P_{\xi} - K_{\xi} C_{\xi})] = 0 \quad (56) $$

Further, we introduce the following condition

$$ (B_{\xi}^{T} P_{\xi} - K_{\xi} C_{\xi})^{T} (B_{\xi}^{T} P_{\xi} - K_{\xi} C_{\xi}) < \varepsilon I \quad (57) $$

where $\varepsilon$ is a small enough positive scalar. By Schur complement [15], (57) is equivalent to

$$ \begin{bmatrix} -\varepsilon \mathbf{I}_{3n} & B_{\xi}^{T} P_{\xi} - K_{\xi} C_{\xi} \\ \star & -\mathbf{I}_{3n} \end{bmatrix} < 0 \quad (58) $$

which is an LMI with respect to variable $P_\xi$ and $K_\xi$. Now, standard LMI toolbox in Matlab can be used to solve (55) and (58). The feasibility of (55) and (58) is ensured in Proposition 2.

## IV. NUMERICAL EXAMPLE

In this section, simulations will be conducted to verify the effectiveness of the proposed scheme.

### A. Example and simulation conditions

We consider the continuous-time linearized model of the F-404 engine [21], described as $\dot{x} = Ax + Ed(t)$ where $d(t) = 0.001 \sin(t) + 0.002 \sin(2t)$,

$$ A = \begin{bmatrix} -1.46 & 0 & 0.248 \\ 0.1643 & -0.4 & -0.3788 \\ 0.3107 & 0 & -2.23 \end{bmatrix}, E = \begin{bmatrix} 0.2 \\ 0.8 \\ 0 \end{bmatrix}. $$
Assume that the input and output matrices are $B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}$

$C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$; the measurement noise $\nu(t)$ is the white noise vector, the attack signal is formulated as $u_a = \begin{bmatrix} 0.7316e^{0.0127t} \\ 0.01 \sin(t) \end{bmatrix}$ which starts at $T_a = 15s$. The initial states of the plant and the attacker are $x_p(0) = [-1 \ 1 \ 1]^T$ and $x_a(0) = [0 \ 0 \ 0]^T$, respectively. The reference command is $r = [1 \ 1]^T$.

The simulations are done for the following four cases:

Case 1 Covert attacks have not compromised the plant;

Case 2 Covert attacks compromise the plant without the signal generator;

Case 3 Covert attacks compromise the plant with the signal generator;

Case 4 Covert attacks have been detected, and the secure controller is operating.

## B. Simulation parameters

The design parameters are chosen as $\tau = 0.001$ (which is the simulation step), $s = 2$, $\gamma_s = 1$,

$$L_n = \begin{bmatrix} 14.07 & 163.48 \\ 60.28 & 653.48 \\ 0.85 & -0.14 \end{bmatrix}, K_2 = \begin{bmatrix} 738.44 & -180.32 \\ -180.32 & 53.02 \end{bmatrix},$$

$$K_1 = 0.9988 \mathbf{I}_2, K_n = \begin{bmatrix} -72.92 & -0.38 & -0.34 \\ -0.37 & -74.66 & 0.37 \end{bmatrix},$$

$$L_x = \begin{bmatrix} 860.27 & -147.13 \\ -486.29 & 373.18 \\ -5.86 & 1.33 \\ -1874.23 & 632.24 \\ 482.72 & 564.16 \\ -24.15 & 6.16 \\ 914.44 & -132.97 \\ -253.55 & 429.37 \\ 12.31 & -3.00 \end{bmatrix}.$$

(59)

## C. Simulation results

Simulation results for the four cases are presented in Fig. 2-Fig. 5. The simulation plots including the system outputs $y_{p,i}$, $i = 1, 2$, the controller inputs $y_{c,i}$, $i = 1, 2$, the controller outputs $u_{c,i}$, $i = 1, 2$, the system inputs $u_{p,i}$, $i = 1, 2$, the evaluation function $\|\tilde{y}_c\|$, the output tracking error $\|y_p - y_{ref}\|$ and threshold $\gamma_s \|d\|$.

The simulation results for Case 1 are presented in Fig. 2, and it is clear that $y_p = y_c$, $u_p = u_c$, $|\tilde{y}_c| \le \gamma_s \|d\|$ and $\|y_p - y_{ref}\| \le \gamma_s \|d\|$. Hence the nominal controller (8) can guarantee the acceptable performance (11) and there are no false alarms.

The simulation results for Case 2 are presented in Fig. 3. Comparing with Fig. 2, it can be seen that the controller inputs $y_{c,i}$, $i = 1, 2$ and the controller outputs $u_{c,i}$, $i = 1, 2$ have no changes, hence $|\tilde{y}_c| \le \gamma_s \|d\|$ namely a missing alarm occurs (covert attacks are masked); but the system outputs $y_{p,i}$, $i = 1, 2$ and the system inputs $u_{p,i}$, $i = 1, 2$ undergo substantial

changes for $t \ge T_a$ hence $\|y_p - y_{ref}\| > \gamma_s \|d\|$ for $t > T_a$ and the acceptable performance (11) cannot be ensured.

To reveal covert attacks, the signal generators are constructed in Case 3 whose simulation curves correspond to Fig. 4. Comparing Fig. 4 to Fig. 3, it is clear that both the controller inputs $y_{c,i}$, $i = 1, 2$ and the controller outputs $u_{c,i}$, $i = 1, 2$ change substantially for $t \ge T_a$; $|\tilde{y}_c| > \gamma_s \|d\|$ for $t > T_d = 15.01s$ (namely covert attacks are unmasked); however $\|y_p - y_{ref}\| > \gamma_s \|d\|$ since the nominal controller is still used.

In Case 4, the proposed secure controller is adopted and simulation plots are presented in Fig. 5, where $\|y_p - y_{ref}\| \le \gamma_s \|d\|$, which shows the effectiveness of the proposed secure controller.

## V. CONCLUSION

In this paper, an active secure control scheme including attack detection and secure control has been proposed. It has been shown that covert attacks cannot be detected with the classical methods since the signals received by the controller undergo no changes before and after covert attacks. In order to unmask such attacks, signal generators are constructed in both layers, and judicious extra signals have been inserted into the network as soon as the Cyber-Physical System (CPS) begins operation. A secure controller, based on an adaptive observer, has been designed to maintain a certain level of tracking performance even in the presence of attacks. The learning observer's estimation precision does not impact control effectiveness; its role is to mitigate the impact of covert attacks. The paper also includes numerical simulations that demonstrate the effectiveness of the proposed scheme. Future research will explore managing the transition between nominal and secure controllers, consider more complex control performance indicators, and system modeling. These topics are currently under investigation.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant 61973197 and the Shaanxi Provincial Natural Science Basic Research Program Project under Grant 2019JLZ-06.

## APPENDIX

### THE PROOF OF PROPOSITION 1

The invariant zeros of $(A_x, B_x, C_x)$ are given by the values of $\lambda$ that make its Rosenbrock matrix (denoted by $\mathcal{R}_x := \begin{bmatrix} \lambda \mathbf{I}_{3n} - A_x & B_x \\ C_x & \mathbf{0} \end{bmatrix}$) lose rank [13]. Using the partitions in (30), it follows that

$$\mathcal{R}_x = \begin{bmatrix} \lambda \mathbf{I}_n - A_o & -B K_n & -L_n C & B \\ \mathbf{0} & \lambda \mathbf{I}_n - A_d & L_n S_p C & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \lambda \mathbf{I}_n - A & B \alpha^{-1} \\ C & \mathbf{0} & -C & \mathbf{0} \end{bmatrix} \quad (60)$$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Simulation curves for Case 1.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Simulation curves for Case 2.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Simulation curves for Case 3.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. Simulation curves for Case 4.</figcaption></figure>

$$ \text{Define } T_R^1 = \begin{bmatrix} \mathbf{I}_n & 0 & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ \mathbf{I}_n & 0 & \mathbf{I}_n & 0 \\ 0 & 0 & 0 & \mathbf{I}_m \end{bmatrix}, $$

$$ T_R^2 = \begin{bmatrix} \mathbf{I}_n & 0 & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ 0 & 0 & \mathbf{I}_n & 0 \\ 0 & K_n \alpha / s & 0 & \mathbf{I}_m \end{bmatrix}, \quad T_R^3 = $$

$$ \begin{bmatrix} \mathbf{I}_n & -1/s\mathbf{I}_n & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ 0 & 0 & \mathbf{I}_n & 0 \\ 0 & 0 & 0 & \mathbf{I}_m \end{bmatrix}, T_L^1 = \begin{bmatrix} \mathbf{I}_n & 0 & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ -\mathbf{I}_n & 0 & \mathbf{I}_n & 0 \\ 0 & 0 & 0 & \mathbf{I}_p \end{bmatrix}, $$

$$ T_L^2 = \begin{bmatrix} \mathbf{I}_n & 1/s\mathbf{I}_n & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ 0 & 0 & \mathbf{I}_n & 0 \\ 0 & 0 & 0 & \mathbf{I}_p \end{bmatrix}, \quad T_L^3 = $$

$$ \begin{bmatrix} \mathbf{I}_n & 0 & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & 0 \\ 0 & 0 & \mathbf{I}_n & L_n \\ 0 & 0 & 0 & \mathbf{I}_p \end{bmatrix}, \quad T_L^4 = \begin{bmatrix} \mathbf{I}_n & 0 & 0 & 0 \\ 0 & \mathbf{I}_n & 0 & sL_n \\ 0 & 0 & \mathbf{I}_n & 0 \\ 0 & 0 & 0 & \mathbf{I}_p \end{bmatrix} $$

then

$$ = \operatorname{rank}(\mathcal{R}_x) = \operatorname{rank}(T_L^4 T_L^3 T_L^2 T_L^1 \mathcal{R}_x T_R^1 T_R^2 T_R^3) $$

$$ = \operatorname{rank} \left( \begin{bmatrix} \lambda \mathbf{I}_n - A_o & \mathbf{0} & 0 & B \\ sL_n C & \lambda \mathbf{I}_n - A_c & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \lambda \mathbf{I}_n - A & -Bs/\alpha \\ \mathbf{0} & \mathbf{0} & -C & 0 \end{bmatrix} \right) $$

From Assumption 3, $[\lambda \mathbf{I}_n - A - Bs/\alpha \quad -C \quad 0]$ is full of column rank for any $\lambda$ that lies in $C_+$ namely the right complex plane. From Remark 3, it is known that the values of $\lambda$ that make $\lambda \mathbf{I}_n - A_o$ and $\lambda \mathbf{I}_n - A_c$ lose rank are both lie in $C_-$ namely the left complex plane. Hence $(A_x, B_x, C_x)$ is minimum phase. From $C_x B_x = (s/\alpha) C B$ and Assumption 2, it is known that when $s \neq 0$, $\operatorname{rank}(C_x B_x) = \operatorname{rank}(B_x)$. $\square$

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<figure><img src="image_8.png" /></figure>

**Xianghua Wang** received the B.E. degree in Automation from Northwestern Polytechnical University, Xi'an, China, in 2010 and the Ph.D. degree in Mechanical System and Control from Peking University, Beijing, China, in 2015. She is currently an associate professor at Beijing University of Posts and Telecommunications. Her current research interests include fault diagnosis, fault tolerant control, guidance and control. She authored or co-authored 1 book and more than 60 papers.

<figure><img src="image_9.png" /></figure>

**Ali Zolghadri** received his PhD from the University of Bordeaux, France and has been a Full Professor of Control Systems Engineering there since 2003. His current research interests are centered around autonomy, resilience and safety/(cyber)security of cyber-physical systems. He is member of International Technical Committees "SafeProcess" and "Aerospace" of IFAC C IEEE senior member, and TC member of EuroGNC (Council of European Aerospace societies). He has served as IPC member for various international conferences, and has delivered a number of plenary lectures and other invited talks at venues worldwide. He is an Associate Editor of the "Journal of the Franklin Institute" (Elsevier, USA) and "Complex Engineering Systems" journal C and Editorial Board member of "Aerospace Science and Engineering", MDPI (Switzerland). He is author/co-author of more than 260 publications in archive journals, refereed conference proceedings and technical book chapters, and co-holder of 15 patents in aerospace. He is the recipient of CNRS Medal of Innovation 2016 which rewards C considering all fields and subfields of research C "outstanding scientific research with innovative applications in the technological and societal fields".

<figure><img src="image_10.png" /></figure>

**Changqing Wang** received the B.E. degree in Mechanical Design and Automation from Northwestern Polytechnical University, Xi'an, China, in 1996 and the Ph.D. degree in System Analysis and Control from Moscow Power Engineering Institute, Moscow, Russia, in 2006. He is currently a professor at Northwestern Polytechnical University. His current research directions include fault-tolerant control, navigation, and control. He has authored or co-authored 1 book and more than 40 papers.

<footnote>[16] T. Yucelen, A. J. Calise. "Derivative-free model reference adaptive control," *J. Guidance, Control, and Dynamics*, vol. 34, no. 4, pp. 933-950, 2011.</footnote>

<footnote>[17] Q. Jia, W. Chen, Y. Zhang, H. Li. "Integrated design of fault reconstruction and fault-tolerant control against actuator faults using learning observers," *Int. J. Systems Science*, vol. 47, no. 16, pp. 3749-3761, 2016.</footnote>

<footnote>[18] F. Hou, J. Sun. "Covert attacks against output tracking control of cyber-physical systems," *Proc. the 43rd Annual Conference of the IEEE Industrial Electronics Society (IECON 2017)*, Beijing, China, 2017, pp. 5743-5748.</footnote>

<footnote>[19] S. B. Rebai, H. Voos, M. Darouach. "Attack-tolerant control and observer-based trajectory tracking for Cyber-Physical Systems," *European J. Control*, vol. 47, pp. 30-36, 2019.</footnote>

<footnote>[20] A. Barboni, H. Rezaee, F. Boem, et al. "Detection of covert cyber-attacks in interconnected systems: A distributed model-based approach," *IEEE Trans. Automatic Control*, vol. 65, no. 9, pp. 3728-3741, 2020.</footnote>

<footnote>[21] X. Ge, Q. L. Han, M. Zhong, et al. "Distributed Krein space-based attack detection over sensor networks under deception attacks," *Automatica*, vol. 109, no. 108557, 2019.</footnote>