Passive Attack Detection for a Class of Stealthy Intermittent Integrity Attacks

I. Introduction

CYBER-PHYSICAL systems (CPS) integrate control, computation and communication techniques with physical engineered control systems [1]. Due to the emergence of such a complex integration, more security vulnerabilities in CPS arise and more malicious cyber threats greatly endanger various key aspects of CPS operation. A series of cyber attack events, such as the Stuxnet worm attack on the Iranian nuclear facilities, the attack on the Ukrainian power distribution network, and the recent colonial oil pipeline attack in USA, have taken place in recent years (more details and examples can be found in [2]–[5]). Therefore, state-of-the-art cyber attack diagnostic technologies are required to safeguard the operation of CPS against possible malicious attacks.

A State of the Art

Integrity in computer science refers to the trustworthiness of data, whereas in the context of CPS, integrity attacks compromise the integrity of the transmitted data of the CPS [1]. Replay attacks [6], covert attacks [7], [8], zero-dynamics attacks [9] and false-data injection attacks [10] are the most commonly studied stealthy integrity attacks. Several survey papers provide overviews of integrity attacks from a system control perspective (see, e.g., [2], [9], [11] and [12]). Recently, some new types of integrity attacks have also been investigated. For instance, in order to achieve perfect stealthiness, a self-generated approach is developed in [13] for generating particular false data, and [14] considers a class of closed-loop nonlinear systems and develops a stealthy integrity attack formulation approach for such systems. In the aforementioned integrity attack studies, false data are injected into the CPS continuously, whereas the problem of intermittent data-injection is overseen. It is important to note that the intermittent mode of attack injection may greatly affect both the stealthiness of an attack event and the power energy consumed by an attacker.

Intermittent attacks are characterized by piece-wise attack signals. Denial-of-service attacks (DOS) [15]–[18] are typical examples of intermittent attacks. The available power energy to the attacker is optimally managed by scheduling the DoS attack application time instants in [16], [17], whereas the stealthiness of the attack is not considered. In this paper, we consider integrity attacks performed in an intermittent mode; these are referred to as intermittent integrity attacks. Unlike the strategy for typical continuous integrity attacks, the strategy for generating intermittent integrity attacks includes two steps: 1) constructing a stealthy attack model, and 2) scheduling attack-activating and attack-pausing time instants. In terms of attack models, typical stealthy integrity attack models such as zero-dynamics attack models [14], [19], replay attack models [6] and covert attack models [20], [21], can also act as models for intermittent integrity attacks since they are undetectable by typical anomaly detectors. The scheduling of intermittent attacks (activation and pausing time instants) can improve their stealthiness through activating the attacks for a limited amount of time such that anomaly detectors do not have sufficient time to detect them. Using attacks with specially scheduled activating and pausing time instants can also save the adversary’s instantaneous power energy by avoiding attack signal divergence requirements. A counterexample to this is the zero-dynamics attack case, in which the divergence attack signals are used (e.g., [9]), and may consume large instantaneous power energy. Moreover, intermittent integrity attacks can overcome the attack defense strategy such as topology switching in the case of multi-agent systems [22]. Particularly, the adversary may pause the attack prior to the switch of the topology, and then resume and update the attack to maintain the stealthiness in the new topology.

In the past decade, several methods for detecting stealthy integrity attacks have been proposed by the research community, which fell into two categories: active and passive detection methods. In active detection methods, such as the watermarking and moving-target approaches, either authentication signals are injected into the information flows of CPS or secret modules are embedded to CPS loops and series-connected to physical plants (see, e.g., [23]–[27]). On the other hand, in passive attack detection methods, only analytical redundancy approaches are used to detect integrity attacks, without using any authentication signals or secret modules [28]. However, typical passive anomaly detectors are not able to provide desirable attack detection performance. Note that attack detectors are a special type of anomaly detectors specifically designed for detecting attacks. For example, fault detection schemes in [29]–[33] may not be able to detect stealthy integrity attacks (such as [6]–[9] and [13]). Generally, the reason for this is that, in the presence of a stealthy integrity attack, the information resources (sensory measurements and control inputs) of analytical redundancy approaches remain either unchanged or slightly altered and therefore, analytical redundancy approaches that are inherently not sufficiently sensitive to such slight changes, are unable to detect stealthy integrity attacks. In [34], multiple filters are combined to formulate a type of analytical redundancy-based passive attack detector, which can detect various types of stealthy integrity attacks. However, such a detector requires additional physical communication channels, which may not be feasible or realistic. Traditional analytical redundancy-based passive anomaly detectors are enhanced in [35] and [36] for detecting stealthy integrity attacks by using a backward-in-time signal processor. In these studies, even a small change due to a stealthy integrity attack is amplified by a backward-in-time signal processor such that the amplified change becomes sufficiently “large” to be detected. However, stealthy intermittent integrity attacks are not considered in [34]–[36].

The detection of stealthy intermittent integrity attacks remains an open problem and few research works have been published. For example, [37] combines an analytical redundancy-based passive detector and a set-theoretic detector for detecting intermittent integrity attacks. In this method, the transient overshoots of the analytical redundancy and the strict detection guarantees of the set-theoretic detector are integrated to detect promptly the intermittent attacks. However, the stealthiness of the attacks is not considered in [37], which prevents the detection method from applied effectively in the case of stealthy intermittent integrity attacks.

B Main Contributions

This paper utilizes the backward-in-time approach in the context of intermittent integrity attacks, and proposes an analytical redundancy-based passive detection methodology for detecting a class of stealthy intermittent integrity attacks. Specifically, the contributions of this paper are summarized as follows:

1) A stealthy intermittent integrity attack generation strategy is formulated, which does not require that the adversary has precise knowledge of the system states. A backward-in-time detection residual is formulated, which increases in magnitude each time a new attack occurs;

2) An optimal fixed-point smoother with covariance matrix resetting is proposed to implement the aforementioned backward-in-time residual. Such a smoother guarantees robustness to both disturbances and noise, and can also reset the covariance matrix to maintain sensitivity to intermittent integrity attacks;

3) The corresponding adaptive threshold is designed, and an attack detectability analysis is carried out to characterize quantitatively the class of detectable stealthy intermittent integrity attacks.

In terms of the stealthy intermittent integrity attacks, compared to [22], the attack generation proposed in this paper addresses the practical issue that the adversary does not have precise knowledge of the system states. In addition, in contrast with [38] in which the pausing and resuming time instants of intermittent integrity attacks are scheduled for saving power energy, this paper focuses on designing the attack generation strategy such that the generated attacks are stealthy, regardless of the pausing and resuming time instants.

Compared to the authors’ previous work [36], the intermittency feature of stealthy integrity attacks is considered in this paper. Moreover, the designed smoother in this paper introduces the covariance matrix resetting technique, which is shown to guarantee robustness to both disturbances and noise, and simultaneously guarantee sensitivity to stealthy intermittent integrity attacks.

C Notations

Consider a vector signal $x(t):{\mathbb{R}}_{+}\rightarrow {\mathbb{R}}^{n}$. Then, $x(t)=0$ for $t\in[t_1,t_2]\subset{\mathbb{R}}_{+}$ means that $x(t) = 0$ identically for all $t\in[t_1,t_2]$; $x(t)\not=0$ for $t\in[t_1,t_2]\subset{\mathbb{R}}_{+}$ means that $x(t)\neq0$ for at least one time instant $t\in[t_1,t_2]$. The notation $|\cdot|$ is used in this paper to represent the absolute value for scalars, and the 2-norm for vectors and matrices. For a set S, $|S|$ represents the number of the elements in S. A vector $x(t)\in {\cal{L}}_2[t_1,t_2]$ if $\int_{t_1}^{t_2} {x^T(\tau){x}(\tau)d{\tau}}$ is finite. For a signal $x(t)$ in the finite time interval $[t_1,t_2]$ and a given matrix $R\geq0$ with proper dimensions, we define $\left\|x(t) \right\|^2_{R} = \int_{t_1}^{t_2} {x^T(\tau)R{x}(\tau)d{\tau}}$. For a constant vector x and a given matrix $R\geq0$ with proper dimensions, we define $\left\|x \right\|^2_{R} = x^TRx$. For a matrix A, $\bar{\sigma}(A)$ and $\underline{\sigma}(A)$ represent the maximum and minimum singular values of A respectively. Italics in the paper are used to highlight important associated sentences and terminologies.

The rest of this paper is organized as follows. In Section II, the problem is formulated. Section III analyzes the stealthiness of the intermittent integrity attacks formulated in this work. In Section IV, the backward-in-time detection residual is introduced, and its theoretical feasibility to indicate the stealthy intermittent integrity attacks is presented. The design details of the implementable backward-in-time detection methodology are presented in Section V and the attack detectability analysis is shown in Section VI. Section VII presents a simulation example and finally, the conclusions are drawn in Section VIII.

II. Problem Formulation

A general structure of a CPS subject to integrity cyber attacks is depicted in Fig. 1. It consists of a physical plant ${\cal{P}}$, a feedback controller ${\cal{C}}$, an anomaly detector ${\cal{D}}$, an actuator communication network ${\cal{N}}_a$ and a sensor communication network ${\cal{N}}_s$.

Figure  1.  Schematic diagram of CPS in the presence of integrity cyber attacks.

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A Closed-Loop CPS

In order to simplify the notation, the closed-loop CPS including ${\cal{C}}$, ${\cal{P}}$, ${\cal{N}}_a$ and ${\cal{N}}_s$ are jointly denoted by ${\cal{W}}$ throughout the paper. The closed-loop system ${\cal{W}}$ is described by

where $x\in {\mathbb{R}}^{n_x}$ is the state vector, $u\in{\mathbb{R}}^{n_u}$ is the vector of control data generated by the controller ${\cal{C}}$, $y\in{\mathbb{R}}^{n_y}$ is the vector of sensor measurements received by the controller and the anomaly detector. The integers $n_x$ and $n_y$ satisfy $n_y<n_x$. Moreover, $y_{ {{\rm{ref}}}}(t)\in{\mathbb{R}}^{n_y}$ denotes the output reference signals, and $\omega(t)\in{\mathbb{R}}^{n_\omega}$ and $v(t)\in{\mathbb{R}}^{n_y}$ represent the vectors of process disturbances and measurement noise, respectively. Let $K_u\subseteq\{1,\ldots,n_u\}$ and $K_y\subseteq\{1,\ldots,n_y\}$ represent the disruption resources available to the attacker, i.e., the sets of actuator and sensor communication channels respectively that can be affected by the attacker. The distribution matrices $\Gamma_u\in{\mathbb{B}}^{n_u\times |K_u|}$ and $\Gamma_y\in{\mathbb{B}}^{n_y\times |K_y|}$ (${\mathbb{B}}=\{0,1\}$) are the binary incidence matrices mapping the attack signal to the respective channels. The attack signals are $a_u(t) = [a_{u,1}(t),\ldots,a_{u,|K_u|}(t)]^T\in{\mathbb{R}}^{|K_u|}$ and $a_y(t) = [a_{y,1}(t),\ldots,a_{y,|K_y|}(t)]^T\in{\mathbb{R}}^{|K_y|}$. For each $i\in\{1,\ldots,{|K_u|}\}$, $a_{u,i}(t)=0$ for $t\in{\mathbb{R}}_{+}$ if no attack occurs on the i-th transmission channel of ${\cal{N}}_a$, and similarly, for each $j\in\{1,\ldots,{|K_y|}\}$, $a_{y,j}(t) =0$ for $t\in{\mathbb{R}}_{+}$ if the j-th transmission channel of ${\cal{N}}_s$ is not under attack. Throughout this paper, we denote jointly the attack signals $a_u$ and $a_y$ as $a(t)=[a_u^T(t),a_y^T(t)]^T$. We consider that the attack starts at an unknown time $t = T_0$ and hence $a(t)=0$ for $0\leq t< T_0$. By letting $B_a=[B\Gamma_u,0_{n_x\times |K_y|}]$ and $D_a= [0_{n_y\times |K_u| },\Gamma_y]$, then $B\Gamma_ua_u(t) = B_aa(t)$ and $\Gamma_ya_y(t) = D_aa(t)$.

In addition, the matrices $A\in {\mathbb{R}}^{n_x\times n_x}$, $B\in {\mathbb{R}}^{n_x\times n_u}$, $C\in {\mathbb{R}}^{n_y\times n_x}$ and $D\in {\mathbb{R}}^{n_x\times n_\omega}$ are system matrices known by the defender. The pair $(A,D)$ is assumed to be stabilizable and $(C,A)$ is assumed to be observable. The control gain $K\in{\mathbb{R}}^{n_u\times n_x}$ is designed to stabilize the system. In the nominal case ($a(t)=0$ for $t\in{\mathbb{R}}_{+}$), the closed-loop system ${\cal{W}}^n$ is given by

where $x^n$, $u^n$ and $y^n$ represent the state, the control input and the output, respectively, in the nominal case.

B Anomaly Detector

We consider that the CPS is equipped with a typical anomaly detector ${\cal{D}}$ (see Fig. 1) for detecting some normal anomalies (faults or attacks). Specifically, the anomaly detector ${\cal{D}}$ contains a detection residual $r(t)$ and a constant threshold $J_{th}$. Without loss of generality, based on [39], we consider that the residual has the following form:

where ${\cal{D}}: {\mathbb{R}}^{n_u}\times{\mathbb{R}}^{n_y}\times{\mathbb{R}}^{n_y}\rightarrow {\mathbb{R}}^{n_y}$ is an observer (robust Luenberger observer [39], Kalman filter [29], adaptive observer [40] or sliding mode observer [41]) designed based on the analytical redundancy concept to generate an estimate of $y(t)$ in the nominal case. By using the evaluation function $J(t) = |r(t)|$, the occurrence of an anomaly (fault or attack) is ascertained if at some time $t>T_0$, $J(t)$ exceeds the threshold $J_{th}$, i.e.,

It should be noted that the residual $r(t)$ contributing to indicate the occurrence of the anomaly is based on the estimates of the system outputs at the time posterior to the anomaly occurrence time. Hence, ${\cal{D}}$ is referred to as a forward-in-time detector in this paper. Most fault detectors in the literature such as [29], [30], [39], belong to the class of forward-in-time detectors.

C Stealthy Intermittent Integrity Attacks

Throughout this work, to simplify the presentation, stealthiness is used to refer to stealthiness with respect to the typical standard anomaly detector ${\cal{D}}$ characterized by the residual r in (3) and the threshold $J_{th}$ in (4). The attacker has the following available resources: partial model knowledge ($A$, $B\Gamma_u$, $C$) and disruption resources ($K_y$, $K_u$), where $B\Gamma_u$ represents the partial columns of $B$ corresponding to $K_u$. The strategy for generating stealthy intermittent integrity attacks includes two steps: 1) constructing attack models, and 2) scheduling attack-activating and attack-pausing time instants. For Step 1), the model used for generating stealthy integrity attack is given in Section III. For Step 2), in this paper, we consider that the activating and pausing time instants have been scheduled by the attacker, and the attack model is activated at the following time instants:

In addition, we consider that the k-th attack is active for a time length $\tau_k$ (dwell time) where $0<\tau_k\leq t_{k+1}-t_k$ and hence, it is inactive (attack silence) for $t\geq t_k+\tau_k$. Then, the activating time interval $\Omega^{\mathrm{ac}}_k$ (attack active) and the silence time interval $\Omega^{\mathrm{si}}_k$ (attack silence) for the k-th attack can be given respectively as follows:

Also, we define an action time interval $\Omega^0_k$ that the k-th attack affects the system as follows:

In addition, an auxiliary time interval “attack slot $\Omega_k$” is defined as follows:

A schematic of $\Omega_k$, $\Omega^{\mathrm{ac}}_k$, $\Omega^{\mathrm{si}}_k$ and $\Omega^0_k$ is given in Fig. 2.

Figure  2.  Schematic diagram of $\Omega_k$, $\Omega^{\mathrm{ac}}_k$, $\Omega^{\mathrm{si}}_k$ and $\Omega^0_k$.

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D Objective

The first aim is to propose an intermittent attack model and then analyze rigorously the stealthiness of the generated intermittent integrity attacks by the model with respect to the typical anomaly detector ${\cal{D}}$. The second objective of this paper is to design an attack detection methodology, based on a passive analytical redundancy approach (i.e., only using the system output y and the control input u in the system ${\cal{W}}$). Such a methodology is able to overcome the stealthiness of the intermittent integrity attacks and thus, can detect their occurrences.

III. Stealthiness of the Generated Intermittent Integrity Attacks

This section proposes an intermittent integrity attack model and analyzes the stealthiness (with respect to the typical anomaly detector ${\cal{D}}$) of the generated attacks. In this paper, the zero-dynamics attack model in [19] is modified to perform intermittent integrity attacks. Given the attack activating time interval $\Omega^{\mathrm{ac}}_{k}$ and the silence time interval $\Omega^{\mathrm{si}}_{k}$, the attack model for the k-th attack slot, $k\in\{1,\ldots,N_a\}$, is proposed as follows:

where the design parameters $F_k\in{\mathbb{R}}^{(n_u+n_y)\times n_x}$ and $\Delta {\textit{z}}_k\in{\mathbb{R}}^{ n_x}$ are discussed in detail in the sequel. In addition, $a_{y,k}(t) = [0,$ $I_{|K_y|}]a_k(t)$ and $a_{y,k}((t_k+\tau_k)^-) = \lim_{t\uparrow (t_k+\tau_k)}a_{y,k}(t_k+\tau_k)$. The purpose of the value $a_k(t)$ for $t\in\Omega^{\mathrm{si}}_k$ is to guarantee the continuity of the output $y(t)$ at the pausing time instant $t_k+\tau_k$. This design of $a_k(t)$ guarantees that the generated attacks can pass through some statistical anomaly detectors (such as some detectors in [42] targeting abrupt value jumps) without being detected. The continuity issue and the corresponding design of $a_k(t)$ in (8b) arise due to the intermittent attack implementation mode, whereas typical continuous zero-dynamics attacks such as the ones in [9] and [19] do not have this issue.

The divergence of $a(t)$ generated by attack model (8) can be avoided through the intermittency performing manner described in (5). Hence, compared with the typical zero-dynamics attack in [19], the intermittent integrity attack $a(t)$ generated by attack model (8) can save the adversary’s instantaneous power energy. It must be noted that the initial condition $\zeta_k(t_k) = -\Delta {\textit{z}}_{k}$ in (8a), where $\Delta {\textit{z}}_{k}$ is not required to be zero, is more practical than the initial conditions used in [9] and [22]. Specifically, the initial condition $\Delta {\textit{z}}_{k}$ in (8a) can represent the difference between the true value of the system state x and the value known by the attacker. In other words, the attack model (8) does not require the knowledge of the true value of the system states, something that is implied by the nonzero $\Delta {\textit{z}}_{k}$ in (8a), whereas the attack models in [9] and [22] require that. In addition, as shown in the following section, similarly with [9] and [22], the intermittent integrity attacks generated by (8) are stealthy with respect to the standard anomaly detector ${\cal{D}}$ if $\Delta {\textit{z}}_{k}$ is sufficiently small. We assume that $\Delta {\textit{z}}_{k}$ satisfies the following assumption.

Assumption 1: There exist two scalars $\underline{\delta}>0$ and $\bar{\delta}>0$ such that the initial condition $\Delta {\textit{z}}_{k}$ in (8a) is bounded as follows:

where $\underline{\delta}$ and $\bar{\delta}$ are sufficiently small positive scalars that are not required to be known by the defender.

Remark 1: The lower bound $\underline{\delta}$ implies that the attacker does not need to know the true value of the system state, which is a practical assumption since the true value of the system state is hard to be precisely measured in practice due to the presence of measurement noise. The bound $\underline{\delta}$ is also not required to be known by the defender in this paper for developing the attack detection methodology. Furthermore, the (sufficiently small) upper bound $\bar{\delta}$ guarantees that the generated intermittent attacks can pass through the anomaly detector ${\cal{D}}$ without any alarms, which is analyzed in detail in the next section. In addition, the next section provides guidance on selecting a suitable $\bar{\delta}$.

In [19], it is shown that the stealthiness of zero-dynamics attacks can be violated if $\Delta {\textit{z}}_{k}$ is nonzero. However, [19] does not take into consideration the effects of the controller on the convergence of the system outputs in the attack scenario. In this section, both of the nonzero $\Delta {\textit{z}}_k(t)$ and the controller $u(t)$ are considered in analyzing the stealthiness of the generated attacks by (8). To this end, a system splitting is presented. During the attack slot $\Omega_k$, ${\cal{W}}^n$ in (2) is split into

where the initial conditions are $x^n_{1}(t_k) = 0$ and $x_2^n(t_k) = x^n(t_k)$. Such a system splitting guarantees that $x^n(t) = x^n_{1}(t)+x_2^n(t)$ and $y^n(t) = y^n_1(t)+y^n_2(t)$. Using a similar splitting approach, ${\cal{W}}$ in (1) is split into ${\cal{W}}_1$ and ${\cal{W}}_2$ where

where the initial conditions are $x_{1}(t_k) = -\Delta {\textit{z}}_{k}$ and $x_2(t_k) = x(t_k)+ \Delta {\textit{z}}_{k}$. Such a splitting also guarantees that $x(t) = x_{1}(t)+x_2(t)$ and $y(t) = y_1(t)+y_2(t)$.

Throughout this paper, the notation $\Delta$ is used to represent the change of any variable due to an attack. For example, $\Delta x$ is the change of $x^n$ due to an attack, i.e., $\Delta x = x-x^n$. Then, the stealthiness of the intermittent integrity attacks generated by (8) is presented.

Theorem 1 (Stealthiness): Consider the weakly unobservable subspace of ${\cal{W}}_1$ in (11a) (denoted by ${\cal{V}}({\cal{W}}_1)$), the unobservable subspace of the pair $(CA,A)$ (denoted by ${\cal{H}}$), and the largest controlled invariant subspace of ${\cal{W}}_1$ contained in ${\cal{H}}$ (denoted by ${\cal{V}}({\cal{H}})$). By letting ${\cal{V}}_0 = {\cal{V}}({\cal{W}}_1)\cap {\cal{V}}({\cal{H}})$, if the following conditions are satisfied:

then the change of output y of ${\cal{W}}$ in (1) due to the intermittent integrity attack $a(t)$ generated by (8) is written as

where $\Delta y_{2,i}(t)$ is generated by the following system:

Proof: The proof is presented in Appendix A.■

Remark 2: Comparing the space ${\cal{V}}_0$ with the controlled invariant space for generating the continuous zero-dynamics attack in [19], an additional restriction (the subspace ${\cal{V}}({\cal{H}})$) is introduced to guarantee that $\Delta y$ is continuous at the time instant $t_k+\tau_k$. Given the observable pair $(C,A)$, ${\cal{V}}({\cal{H}})$ is nonempty if and only if the matrix A is singular.

Remark 3: According to the incremental system (14) in Theorem 1, the output change $\Delta y_{2,k}$ due to the k-th attack converges to zero exponentially with the initial nonzero condition $\Delta {\textit{z}}_{k}$, since $A+BKC$ is a Hurwitz matrix. Hence, we can conclude that the stealthiness of the generated attacks by model (8) under Assumption 1 with respect to the typical anomaly detector ${\cal{D}}$, can be guaranteed by choosing a sufficiently small initial condition $\Delta {\textit{z}}_{k}$. In addition, a system with all states measurable (i.e., C is column full rank) does not possess a weakly unobservable subspace ${\cal{V}}({\cal{W}}_1)$. Hence, no subspace ${\cal{V}}_0$ satisfying (12) exists and no stealthy intermittent integrity attack characterized by (8) exists for such a system.

Theorem 1 shows that $\Delta {\textit{z}}_{k}$ satisfying Assumption 1 causes a nonzero change $\Delta y_{2,k}$ at the initial time $t_k$, which provides an evidence that the attacks generated by (8) can be detected by some well designed passive analytical redundancy-based detection methodology. Hence, the objective of the rest of this paper is to design a passive attack detection methodology, namely by using only y and u of the system ${\cal{W}}$ in (1), for detecting the intermittent integrity attacks generated by (8) under Assumption 1.

IV. Backward-in-Time Residual Design and Analysis

In this section, an equivalent quantity of the system state change at a fixed time prior to the attack occurrence time is introduced, which is referred to as backward-in-time equivalent quantity in this paper. Also, its properties in the context of intermittent integrity attacks, are also rigorously investigated. By using the proposed backward-in-time equivalent quantity, a backward-in-time residual is designed, and the theoretical feasibility of this residual to capture the considered stealthy intermittent integrity attacks is analyzed.

At first, we suppose that the attacker selects a sufficiently large dwell time $\tau_k$ such that the k-th attack has sufficient time to cause significant damages to the system. Also, the attacker is supposed to set the time length between $t_k$ and $t_{k+1}$ to be sufficiently long such that sufficient energy is recovered for activating the $(k+1)$-th attack. Then, since the change $\Delta y_{2,k}$ due to the k-th attack converges to zero with an exponential speed rate (see $\Delta {\cal{W}}_{2,i}$ in Theorem 1), it is reasonable to consider that $\Delta y_{2,k}(t)\approx0$ for $t\geq t_{k+1}$ in the design of the attack detection methodology. Therefore, based on (13) and (14), we have $\Delta y(t) = \Delta y_2(t)$, and is described by

where $k\in\left\{1,\ldots,N_a\right\}$, and $\Delta {\textit{z}}(t) = \Delta x_{2,k}(t)$ for $t\in\Omega_k$. In the rest of this paper, the detection methodology is developed based on system (15).

Intuitively, the backward-in-time equivalent quantity (mathematically defined later) is a virtual quantity of the system state change, due to the attack, at a time prior to the attack occurrence time, which is recovered from the change of the system state posterior to the attack occurrence time. Based on the backward-in-time equivalent quantity for nonlinear systems given in [36], we define a backward-in-time equivalent quantity of $\Delta {\textit{z}}(t)$ in (15) at a time $t_b$, recovered based on $\Delta {\textit{z}}(t)$ at the time $t$ ($t\geq t_b$).

Definition 1: The backward-in-time equivalent quantity of the state ${\Delta {\textit{z}}}(t)$ of the incremental system (15) at a time $t_b$ ($t_b\leq t$), denoted by ${\Delta {\textit{z}}}(t_b|t)$, is defined as

where Φ is the transition matrix associated with $A+BKC$, i.e.,

The properties of ${\Delta {\textit{z}}}(t_b|t)$ are summarized in the following lemma.

Lemma 1: Consider the attack generated by (8) satisfying Assumption 1, and the backward-in-time equivalent quantity ${\Delta {\textit{z}}}(t_b|t)$ defined in (16). Then, ${\Delta {\textit{z}}}(t_b|t)$ has the following properties:

1) The vector ${\Delta {\textit{z}}}(t_b|t)$ is constant for $t\in\Omega_k$, i.e.,

2) The vector ${\Delta {\textit{z}}}(t_b|t)$ is nonzero for $t\in\Omega_k$, i.e.,

3) (Accumulation property) For the k-th and $(k+1)$-th attack slots, their recovered backward-in-time equivalent quantities satisfy $|{\Delta {\textit{z}}}(t_b|t)|_{t\in\Omega_{k+1}}\geq|{\Delta {\textit{z}}}(t_b|t)|_{t\in\Omega_{k}}$ if $\Delta {\textit{z}}_{k+1}$ and $\Delta {\textit{z}}_k$ satisfy the following inequality:

where the transition matrix Φ is given in (17).

Proof: The proof is provided in Appendix B.■

It is worth pointing out that under Assumption 1, inequality (20) is easy to satisfy. Due to the exponential convergence of the transition matrix $\Phi(t_{k+1},t_k)$ and the sufficiently large time length $t_{k+1}-t_k$ of the k-th attack slot, $\Phi(t_{k+1},t_k)$ is almost zero, so the right hand side of (20) is almost zero as well. Therefore, in the rest of this paper, inequality (20) is considered to hold.

Lemma 1 implies that the stealthy intermittent integrity attack can be indicated by using a residual based on $\Delta {\textit{z}}(t_b|t)$. This is because $\Delta {\textit{z}}(t_b|t)$ is monotonically increasing with respect to the attack slot $k\in\{1,\ldots,N_a\}$ given that (20) holds (see result 3)). Next, the backward-in-time residual is proposed by using ${\Delta {\textit{z}}}(t_b|t)$ and is rigorously investigated. Based on $\Delta {\textit{z}}({t_b}|t)$, the backward-in-time equivalent quantity of $\Delta y(t)$ in (15) at the time $t_b$ is constructed as

Then, the backward-in-time residual, denoted by $r\left(t_b|t\right)$, is defined as follows:

where the value $r(t_b)$ is given in (3). The threshold with respect to $r\left(t_b|t\right)$ is chosen as $J_{th}$ in (4). Then, by using the residual evaluation $J(t_b|t) = |r\left(t_b|t\right)|$, the occurrence of an anomaly (fault or attack) is ascertained if at some $t>T_0$, $J(t_b|t)$ exceeds the threshold $J_{th}$, namely

Note that the backward-in-time residual $r\left(t_b|t\right)$ can indicate both faults and attacks, whereas in this paper, we mainly focus on the characteristics of $r\left(t_b|t\right)$ in terms of detecting the considered stealthy intermittent integrity attacks.

The detection methodology that integrates the residual $r(t)$ in (3) and the newly proposed backward-in-time residual $r\left(t_b|t\right)$ in (22), is referred to as ideal backward-in-time detection methodology in this paper. The terminology ideal is used since $r(t_b|t)$ is unknown due to the unknown $\Delta y(t_b|t)$. For theoretical analysis purposes at this stage, we consider $r(t_b|t)$ to be known. The ideal backward-in-time detection methodology can be found in the left hand side of Fig. 3, which includes the residual generations of $r(t)$ and $r\left(t_b|t\right)$, their evaluations, and the anomaly detection decision logic. The theoretical feasibility of $r\left(t_b|t\right)$ to detect stealthy intermittent integrity attacks is presented in the following theorem.

Figure  3.  Schematic diagram of the ideal and implementable backward-in-time detection methodologies.

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Theorem 2: Consider system (1), the intermittent integrity attack generated by (8) with $\Delta {\textit{z}}_k$ satisfying Assumption 1, and a fixed time instant $t_b$. Then, residual $r(t_b|t)$ in (22), its evaluation $J(t_b|t) = |r\left(t_b|t\right)|$, and threshold $J_{th}$ in (4) satisfy:

1) In the absence of the intermittent integrity attack

2) In the presence of the intermittent stealthy integrity attack, $J(t_b|t)> J_{th}$ for all $t\in\bigcup_{i = k}^{N_a}\Omega_i$ if condition (20) holds and the k-th attack slot satisfies

where $\lambda_{0}>0$ and $k_{0}>0$ are scalars satisfying $|\Phi(t_{1},t_2)|\leq$ $k_{0} e^{-\lambda_{0}(t_1-t_2)}$ for any $t_1\geq t_2$.

Proof: The proof can be found in Appendix C.■

Remark 4: Theorem 1 and Lemma 1 provide theoretical results for the ideal case that $r(t_b|t)$ is known. These theoretical findings prove rigorously that the backward-in-time residual $r({t_b}|t)$ is able to trigger alarms in the presence of a stealthy intermittent integrity attack generated by (8) under Assumption 1. Result 2) in Theorem 1 is derived from result 3) in Lemma 1. Intuitively, these results imply that during an intermittent integrity attack event with various attack slots, $|\Delta {\textit{z}}(t_b|t)|$ increases each time a new attack occurs, and maintains its new value until the next attack occurs (corresponding to (20)). As the number of intermittent attacks accumulates, at some attack occurrence time instant, $|r(t_b|t)|$ (see (21) and (22)) exceeds the detection threshold $J_{th}$ and thus, the intermittent integrity attack is successfully detected (corresponding to (25)).

In practice, the backward-in-time residual $r(t_b|t)$ is unknown due to the unknown $\Delta y(t_b|t)$ (see (21) and (22)). Therefore, one task of the next section is to implement a procedure for estimating optimally the unknown equivalent quantity $\Delta y(t_b|t)$ so that the residual $r(t_b|t)$ given by (22) can be implemented in practice and accordingly, a new adaptive threshold is developed based on the implemented estimation procedure.

V. Implementable Backward-in-Time Detection Methodology Design

In this section, the implementable backward-in-time detection methodology is designed. The right hand side of Fig. 3 shows the structure of the detection methodology. The optimal fixed-point smoother in Fig. 3 is first developed to estimate the unknown $\Delta y(t_b|t)$. The new backward-in-time residual $\hat{r}(t_b|t)$ and the corresponding adaptive threshold $\hat{J}_{th}(t)$ are then formulated based on the estimation results provided by the smoother. The details are given in the sequel.

A Optimal Fixed-Point Smoother

A fixed-point smoother provides a backward-in-time estimation procedure, which produces an estimate for a signal using the past time measurements at the first stage, and then updates it using the new measurements as time progresses. Next, a fixed-point smoother in a finite time horizon $[t_b,T]$ is designed for estimating ${\Delta y}(t_b|t)$. Note that the fixed point $t_b$ can be arbitrarily selected by the defender, and so it is available in the detector design.

Recalling the system splitting given in (11), in the nominal phase (i.e., $t<T_0$), we have $x(t) = {\textit{z}}(t)$ and $y(t) = y_2(t)$ where ${\textit{z}}(t)$ and $y_2(t)$ are the state and output of ${\cal{W}}_2$ given in (11) respectively. In addition, in the attack phase, $\Delta y(t)$ equals the output change of ${\cal{W}}_2$ (i.e., $\Delta y(t) = \Delta y_2(t)$, see Theorem 1 and $\Delta{\cal{W}}_2$ in (15)), and therefore, the smoother is designed based on the system ${\cal{W}}_2$ in (11). Let ${\textit{z}}(t_b|t)$ denote the backward-in-time equivalent quantity of ${\textit{z}}(t)$. Then, we have

Thus, instead of estimating $\Delta {\textit{z}}(t_b|t)$ directly, we first design a fixed-point smoother to estimate ${\textit{z}}(t_b|t)$ and then, we use relation (26) to reconstruct $\Delta {\textit{z}}(t_b|t)$ and relation (21) to reconstruct $\Delta y(t_b|t)$.

Now, we start by constructing the fixed-point smoother using the state augmentation approach given in [43]. To perform this task, a new state variable $\phi(t)= {\textit{z}}(t_b|t)$ for $t\geq t_b$ is introduced, where, from result 1) in Lemma 1 and (26), $\phi(t)$ satisfies

By letting $\hat{\phi}(t)$ be the estimate of $\phi(t)$, it follows from (40) and (26) that:

Thus, based on (22), the new backward-in-time residual, denoted by $\hat{r}\left(t_b|t\right)$, is proposed as

Motivated by the optimal residual design methodology in [29], $\hat{r}\left(t_b|t\right)$ is to be optimized for achieving robustness with respect to the disturbance ω and the measurement noise v, and sensitivity with respect to the system changes due to attacks. In the sequel, the specific objectives associated with the robustness and sensitivity are presented.

1) Robustness: The robustness considered in this paper is achieved by minimizing a piece-wise linear quadratic (LQ) cost function. Suppose that a set of smoother switching time instants have been determined, which are given as follows:

where $t_{s,0} = t_b$ and $t_{s,N_s+1} = T$. Correspondingly, a bank of covariance matrices^① are introduced and given as follows:

where $\bar{\Theta}_k\in{\mathbb{R}}^{2n\times 2n}$ and $\bar{\Theta}_k\geq0$ for all $k\in\{0,\ldots,N_s\}$. The objective function associated with robustness is presented in the context of linear fractional transformation (LFT) (see LFT in [44]) in the sequel.

Consider the system ${\cal{W}}_2$ in (11b), and let $\hat{{\textit{z}}}$ be the estimate of z, and $\hat{y}({t_b}|t) = C\hat{\phi}(t)$ be the estimate of $y(t_b|t)$ in (21). Then, the smoother design is formulated to find a system K such that the LFT ${\cal{F}}_{l}({\boldsymbol{P}},{\boldsymbol{K}})$ can satisfy the robustness requirement where ${\cal{F}}_{l}({\boldsymbol{P}},{\boldsymbol{K}})$ is given by

In the above system, $q(t) = [{\textit{z}}^T(t),\phi^T(t)]^T$, P represents the system from $[\omega^T, v^T]^T$ and ${\hat y\left( {{t_b}|t} \right)}$ to ${ y\left( {{t_b}|t} \right) - \hat{y}\left( {{t_b}|t} \right)}$ and y, which can be written in the following matrix form:

with $\bar A = {{\rm{diag}}}(A,0),\; \bar{C} = [C,\; 0],\; \bar{L} = [0,\; C]$ and $\bar{D} = [D^T,0]^T$. Note that the 2-norm of a system and the 2-norm of its adjoint are equal and also, note that the adjoint of ${\cal{F}}_{l}({\boldsymbol{P}},{\boldsymbol{K}})$ is ${\cal{F}}_{l}({\boldsymbol{P}}^{\sim },{\boldsymbol{K}}^{\sim })$, in which the notation $(\cdot)^{\sim }$ represents the adjoint^①. The adjoint ${\cal{F}}_{l}({\boldsymbol{P}}^{\sim },{\boldsymbol{K}}^{\sim })$ is given as follows:

where $\tau$ is a time-to-go variable $\tau = T+t_b-t$ with $p(t_b) = 0$, and ${\boldsymbol{P}}^{\sim }$ is given as follows:

Hence, the piece-wise LQ cost function is readily proposed in the context of the adjoint system (33) as follows:

where $\tau_{s,k} = T+t_b-t_{s,k}$ for $k\in\{0,\ldots,N_s+1\}$, $R>0$, and the weighting matrix $\bar{\Theta}_k$ is given in (31) and satisfies some structural requirements. Let $P_k$ and $\Omega_k$ denote the covariance matrices of ${\textit{z}}(t_{s,k})-\hat{{\textit{z}}}(t_{s,k})$ and $\phi(t_{s,k})-\hat{\phi}(t_{s,k})$ respectively, and $\Sigma_k$ denote the mean value of the cross term $({\textit{z}}(t_{s,k})-\hat{{\textit{z}}}(t_{s,k}))(\phi(t_{s,k})- \hat{\phi}(t_{s,k}))^T$. Then, $P_k>0$ and $\Omega_k>0$, and the covariance matrix $\bar{\Theta}_k$ has the following structure:

Note that by choosing $\hat{\textit{z}}(t_b) = \hat{\phi}(t_b) = 0$, ${\textit{z}}(t_b)-\hat{\textit{z}}(t_b) = \phi(t_b)- \hat{\phi}(t_b)$, $P_0$, $\Sigma_0$ and $\Omega_0$ satisfy

Furthermore, note also that the estimation accuracy for ϕ is higher than the one for z due to the smoothing process (see [45], [46]). Therefore, the covariance matrix of $\phi(t_{s,k})-\hat{\phi}(t_{s,k})$ is smaller than the one of ${\textit{z}}(t_{s,k})-\hat{\textit{z}}(t_{s,k})$, namely, $P_k$ and $\Omega_{k}$ must satisfy

2) Sensitivity: The $H_-$ performance is introduced to quantitatively scale the sensitivity to attacks. Based on the definition of $H_-$ in [47], the $H_-$ performance from ${\textit{z}}-\hat{\textit{z}}$ to $\hat{y}(t_b|t)$ is defined as follows:

Thus, to guarantee the sensitivity requirement, the inequality

must be satisfied, where $\alpha>0$ is a user-defined performance goal.

Remark 5: It is worth pointing out that typical optimal LQ fixed-point smoothers in [45] and [46] are obtained by minimizing the following LQ cost function given in the context of the adjoint system (33):

Such typical LQ smoothers possess inherently the covariance matrix wind-up problem for estimating $y(t_b|t)$. As a result, the sensitivity from ${\textit{z}}-\hat{\textit{z}}$ to $\hat{y}(t_b|t)$ vanishes as time proceeds, thereby resulting in a conflict to the $H_-$ performance requirement in (39). More technical details in terms of this issue are given later. In this work, the fixed-point smoother, designed by minimizing the piece-wise LQ cost function (34), can reset the covariance matrix $\bar{\Theta}_k$, and is able to guarantee the sensitivity requirement (39) by designing suitable covariance matrices $\bar{\Theta}_k$ for all $k\in\{0,1,\ldots,N_s\}$, i.e., $P_k$, $\Sigma_k$ and $\Omega_{k}$ in (35).

Remark 6: The adjoint system (33) facilitates the presentation of the piece-wise LQ cost function given by (34). Note that $\bar{\Theta}_k$, with the structure in (35), is a nonnegative definite matrix and nonsingular, which cannot be used for weighting the initial condition (see [48]). However, in the context of the adjoint system (33), the penalty for the initial condition becomes a penalty for the terminal condition (see the term $\frac{1}{2}\sum_{k = 0}^{N_s}\|p(\tau_{s,k})\|^2_{\bar{\Theta}_k}$ in (34)), and in addition, the weighting matrix is not required to be invertible. Hence, the nonsingular matrix $\bar{\Theta}_k$ in (35) can be used as the weighting matrix in the context of the adjoint system (33).

By synthesizing the objectives (34) and (39), and the requirements for the covariance matrices given in (35)−(37), a feasible way to solve the optimization problem is given in the following steps:

1) Minimize ${\cal{J}}$ for any $\bar{\Theta}_k\geq0$;

2) Restrict $P_k$, $\Sigma_k$ and $\Omega_{k}$ for all $k\in\{0,\ldots,N_s\}$ such that (36), (37) and (39) are satisfied.

In the sequel, two lemmas are rigorously derived to realize the aforementioned Steps 1) and 2) respectively. The optimal solution to (34), i.e., Step 1), is first presented.

Lemma 2: Consider linear system ${\cal{W}}_2$ in (11b) and state ϕ in (27). An optimal fixed-point smoother that minimizes (34) in the finite horizon $[t_b,T]$ is described by the following dynamics:

where $\hat{\textit{z}}(t_b) = 0$ and $\hat{\phi}(t_b) = 0$. The matrices $P(t) = P^T(t)>0$, $\Sigma(t)$ and an auxiliary matrix $\Omega(t) = \Omega^T(t)>0$ are obtained from the solution of the following differential equations:

In the above differential equations, at each time $t_{s,k}$ with $k\in\{0,\ldots,N_s\}$, the following switch occurs:

Moreover, if (37) holds, then the matrices P and Ω satisfy

Proof: Consider the adjoint system (33) and the cost function ${\cal{J}}$ in (34). According to LQ control theory (see Section 5.3 in [44]), the cost function ${\cal{J}}$ is minimized by the following optimal control law $\tilde{u}(\tau)$:

where the covariance matrix $\bar{\Theta}(\tau)$ is generated by

with the following switches:

Then, the optimal smoother is obtained as the adjoint system of $\tilde{u}(\tau)$ and is given as follows:

where $\hat{q} = [\hat{\textit{z}}^T,\hat{\phi}^T]^T$ and $\hat{q}(t_b) = 0$, and the covariance matrix $\bar{\Theta}(\tau)$ is generated by

with the following switches:

Thus, from (44), the smoother (40) can be obtained.

We now proceed to derive the differential equations given in (41) and switches in (42). By letting

It follows from (45) that:

By simplifying the above differential Riccati equation, the differential equations (41) can be obtained. In addition, from (35) and (46), the switches in (42) can also be obtained.

Regarding the result (43), by letting $Y(t) = P(t)-\Omega(t)$ and $A_Y = A-\Omega C^TR^{-1}C-\frac{1}{2}Y C^TR^{-1}C$, it follows from (41a) and (41c) that:

The above equation indicates that

Under the condition (37), we can obtain that $Y(t_{s,k})\geq0$. Therefore, based on Theorem 4.1.2 in [49], $Y(t)\geq0$ for $t\in[t_{s,k},t_{s,k+1})$ and $k\in\{0,\ldots,N_s\}$ can be obtained and the result (43) follows.■

It can be observed from (40b) and (41b) that in the case of a typical LQ smoother (obtained by minimizing the cost function ${\cal{J}}_1$ and without matrix resetting (42)), the sensitivity from $\hat{y}(t_b|t)$ to ${\textit{z}}-\hat{\textit{z}}$, characterized by the matrix $\Sigma(t)$, decreases as time proceeds. The reason for this is that the solution $\Sigma(t)$ of the differential equation (41b) converges to zero as the time progresses, which is the aforementioned covariance matrix wind-up problem. The matrix resetting at the time instant $t_k$ characterized in (42) provides an alternative way to maintain the sensitivity. In the following lemma, feasible resetting matrices $P_k$, $\Sigma_k$ and $\Omega_{k}$ that satisfy (36), (37) and (39) are presented.

Lemma 3: Consider the piece-wise fixed-point smoother given in Lemma 2 and the switching time instants given in (30). Consider also the differential equations (41) without the switches (42). The requirements (36) and (37) are guaranteed if

where $P(t_{s,k})$, $\Sigma(t_{s,k})$ and $\Omega(t_{s,k})$ are the values at $t_{s,k}$ of the solutions to the differential equations (41) without the switches (42) and under the following initial conditions:

Moreover, the $H_-$ performance index requirement in (39) is guaranteed if $\Theta_{k}$ in (48) satisfies

Proof: According to (47), $P_k$ and $\Omega_k$ are the matrices associated with a typical LQ fixed-point smoother without the matrix resetting. Hence, (37) can be guaranteed directly.

According to (40b), we can obtain that

which indicates that the $H_-$ performance requirement in (39) can be guaranteed if

Based on the inequality $\underline{\sigma}(C\Sigma^T(t)C^TR^{-1}C)\geq \underline{\sigma}(C^TR^{-1}C)\times$ $\underline{\sigma}(\Sigma(t))\underline{\sigma}(C)$, a sufficient condition to guarantee the above inequality is obtained as

Let $X(t) = \Sigma(t)\Sigma^T(t)$ where $X(t_b) = \Theta_k\Theta_k^T$. Then, it follows from (41b) without the switches (42) that:

where the system $\dot{x} = (A-P(t)C^TR^{-1}C)x$ is exponentially stable. Moreover, $X(t)$ can be written as

which indicates that $X(t)$ is monotonically decreasing with respect to time and $X(t)\leq X(t_b) = \Theta_k\Theta_k^T$ for $t\geq t_b$. Thus, the inequality (50) can be guaranteed, if at the end of k-th time interval (i.e., $t_{s,k+1}$), the following condition is satisfied:

It then follows from (51) and $X(t_b) = \Theta_k\Theta_k^T$ that the above inequality can be guaranteed by (49).■

Subsequently, by synthesizing the results in Lemmas 2 and 3, a feasible solution to minimize ${\cal{J}}$ in (34) and to satisfy the restrictions (36), (37) and (39) is presented by the following theorem.

Theorem 3: Consider linear system ${\cal{W}}_2$ in (11b) and state ϕ in (27). The fixed-point smoother (40) with $P(t)$, $\Sigma(t)$ and $\Omega(t)$ determined by (41) and the matrix resetting given by (42), minimizes the cost function ${\cal{J}}$ in (34) in the finite time horizon $[t_b,T]$. Moreover, by constructing $P_k$, $\Sigma_k$ and $\Omega_{k}$ as in (47) and (48), and by choosing $\Theta_{k}$ to satisfy (49) for $k\in\{0,$ $1,\ldots,N_s\}$, the $H_-$ performance requirement in (39) and the requirements (36) and (37) are guaranteed simultaneously.

B Residual Evaluation and Threshold Generation

In this section, the detection residual $\hat{r}(t_b|t)$ in (29) is evaluated and an adaptive threshold is generated. We start by evaluating $\hat{r}\left(t_b|t\right)$. Let $e_{\textit{z}}(t)= {\textit{z}}(t)-\hat{\textit{z}}(t)$, $e_{\phi}(t)=\phi(t)-\hat{\phi}(t)$ and $e_{y}(t)= y ({t_b}|t)-\hat{y}({t_b}|t)$ be the estimation errors of x, ϕ and $y ({t_b}|t)$, respectively. Then, from (11b), (27) and (40), the error system is obtained as follows:

where the initial conditions are $e_{\textit{z}}(t_b) = {\textit{z}}(t_b)$ and $e_{\phi}(t_b) = \phi(t_b)$. In addition, from $\hat{y}({t_b}|t) = y({t_b}|t)-e_{y}(t)$, $\Delta \hat{y}(t_b|t) = \hat{y}(t_b|t)-y(t_b)$ and $\Delta y(t_b|t) = y(t_b|t)-y(t_b)$, $\hat{r}\left(t_b|t\right)$ in (29) can be split into

As in the case of fault diagnosis literature, the residual evaluation should ensure that in the non-attack case, the value of the evaluation function is close or equal to zero under ideal conditions (e.g., no disturbance, no noise and no modeling uncertainty). Thus, from (53), the evaluation function is proposed as follows:

where the correction term $-|r\left(t_b\right)|$ guarantees that $\hat{J}\left(t_b|t\right)$ is close to zero in the absence of attacks.

Next, a bound $\hat{J}_{th}(t)$ (adaptive threshold) of the residual evaluation function $\hat{J}\left(t_b|t\right)$ is derived. To this end, a lemma required to generate the threshold is given.

Lemma 4: Let $\Phi_k(t,\tau)$ be the transition matrix associated with the matrix $A-P(t)C^TR^{-1}C$ for $t,\tau\in[t_{s,k},t_{s,k+1})$. Then, there exist known scalars $\beta_k>0$ and $\lambda_k>0$ such that

Proof: Note that since the pair $(A,D)$ is stabilizable and $(C,A)$ is observable, the system $\dot{x} = (A-P(t)C^TR^{-1}C)x$ is exponentially stable. Thus, $x(t) = \Phi_k(t,\tau)x(\tau)$ converges exponentially to zero during the time interval $t\in[t_{s,k},t_{s,k+1})$. Hence, such scalars $\beta_k$ and $\lambda_k$ that satisfy (55) exist.■

In addition, the disturbance $\omega(t)$ and the measurement noise $v(t)$ are supposed to satisfy the following assumption.

Assumption 2: The disturbance $\omega(t)$ and the measurement noise $v(t)$ are bounded by $\bar{\omega}>0$ and $\bar{v}>0$, respectively, i.e.,

where $\bar{\omega}$ and $\bar{v}$ are known scalars by the defender.

Remark 7: Such an assumption is commonly used in anomaly diagnosis literature (see, e.g., [29], [32], [40]) for guaranteeing robustness and avoiding false alarms. The bound $\bar{\omega}$ can be obtained a priori by the defender by some experimental tests to the considered CPS, while $\bar{v}$ can be obtained by exploiting a priori knowledge of the sensor bias deviation based on the technical characteristics of the sensors.

In the following theorem, the adaptive threshold is presented.

Theorem 4 (Robustness): Consider closed-loop CPS ${\cal{W}}$ in (1) and anomaly detector described in (3) and (4). Also, consider the piece-wise fixed-point smoother (40) described in Theorem 3, the residual (29) and the residual evaluation function (54). Under Assumptions 1 and 2, and in the non-attack case (no intermittent integrity attacks), the residual evaluation function $\hat{J}\left(t_b|t\right)$ in (54) is bounded by the adaptive threshold $\hat{J}_{th}(t)$ as follows:

where $\hat{J}_{th}(t)=|C|\varepsilon_{\phi,k}\left(t\right)$ with

In the above equations, $\beta_k$ and $\lambda_k$ are specified in Lemma 4.

Proof: Based on (53) and by using the triangle inequality, the evaluation $\hat{J}\left(t_b|t\right)$ in (54) satisfies

Note that in the non-attack case, $\Delta {\textit{z}}(t_b|t) = 0$ and $\Delta y(t_b|t) = 0$. It then follows from (60) that the threshold $\hat{J}_{th}(t)$ is chosen as:

By solving the differential equation (52a), $e_{\textit{z}}(t)$ can be explicitly expressed as

Note that based on Theorem 3.1.1 in [44], $P(t)$ in (41a) is monotonically decreasing with respect to t and thus, $P(t)\leq P(t_{s,k}) = P_k$ for $t\in[t_{s,k},t_{s,k+1})$. Thus, based on Assumption 2 and Lemma 4, we can obtain

Thus, $\varepsilon_{{\textit{z}},k}(\cdot)$ in (59) is obtained. In addition, by solving the differential equation (52b), we can obtain

Note that it follows from (51) that $\Sigma(t)$ satisfies:

Thus, from $|e_{\textit{z}}(t)|\leq\varepsilon_{{\textit{z}},k}(\cdot)$ and $|v(t)|\leq\bar{v}$ in Assumption 2, and by using the triangle inequality, we can obtain

Since $|e_{\phi}(t^-_{s,k})|\leq\varepsilon_{\phi,k-1}(t^-_{s,k})$, then $\varepsilon_{\phi,k}(\cdot)$ in (58) is obtained. Hence, from $|e_{y}|\leq|C|\cdot|e_{\phi}(t)|$, (57) is obtained.■

Remark 8: The threshold $\hat{J}_{th}(t)$ in (57) is calculated iteratively since $\varepsilon_{{\textit{z}},k}$ and $\varepsilon_{\phi,k}$ for the time interval $[t_{s,k},t_{s,k+1})$ rely on $e_{\textit{z}}(t^-_{s,k})$ and $\varepsilon_{\phi,k-1}(t^-_{s,k})$ respectively from the previous time interval $[t_{s,k-1},t_{s,k})$. In addition, $\varepsilon_{{\textit{z}},k}$ in (59) cannot be used directly since $|e_{\textit{z}}(t^-_{s,k})|$ is not available to the defender. To overcome this, it is reasonable to suppose that there exists a scalar $\delta_0>0$ such that $|e_{\textit{z}}(t_{s,k})|\leq \delta_0$. Such $\delta_0$ does not affect significantly the final detection result since the term $\beta_ke^{-\lambda_k(t-t_{s,k})}|e_{\textit{z}}(t^-_{s,k})|\leq\beta_ke^{-\lambda_k(t-t_{s,k})}\delta_0$ converges to zero exponentially. Therefore, the designer can select a sufficiently large $\delta_0$ for implementing the threshold $\hat{J}_{th}(t)$.

Based on the residual $\hat{r}(t_b|t)$ in (29), the evaluation function $\hat{J}(t_b|t)$ in (54), and the threshold $\hat{J}_{th}(t)$ in (57), the attack occurrence decision principle is given as follows: if there exists a time $t>t_b$ such that $\hat{J}\left(t_b|t\right)$ in (54) exceeds the threshold $\hat{J}_{th}(t)$, i.e., $\hat{J}\left(t_b|t\right)> \hat{J}_{th}(t)$, then an alarm is triggered to indicate the presence of an attack. The detection time $T_{d}$ of the attack is defined as the first time instant when the inequality $\hat{J}({t_b|t})>\hat{J}_{th}(t)$ holds for a given $t_b$, i.e.,

In addition, as Fig. 3 shows, by combining the residuals $r(t)$ and $\hat{r}(t_b|t)$, the evaluation functions $J(t)$ and $\hat{J}(t_b|t)$, and the thresholds $J_{th}$ and $\hat{J}_{th}(t)$, the occurrence of an anomaly (fault or attack) is decided if there exists a time $t>t_b$ such that $J(t)> J_{th}$ or $\hat{J}\left(t_b|t\right)> \hat{J}_{th}(t)$.

Algorithm 1 provides in concise form the steps required for implementing the smoother, generating the backward-in-time residual $\hat{r}(t_b|t)$ and threshold $\hat{J}_{th}(t)$, and the decision principle for detecting the considered stealthy intermittent integrity attacks.

Algorithm 1 Backward-in-Time Attack Detection Algorithm

1: procedure SMOOTHER($t_b$, T, $t_{s,k}$, $\alpha$, R) 　　// Theorem 3;

2: 　　$k\gets0$;

3: 　　repeat　　　　　　　　// Lemma 3;

4: 　　　Choose $\Theta_{k}$ based on (49);

5: 　　　$P(t_b)\gets\Theta_{k}$, $\Sigma(t_b)\gets\Theta_{k}$, $\Omega(t_b)\gets\Theta_{k}$; 　　// (48);

6: 　　　$P_k\gets P(t_{s,k})$, $\Sigma_k\gets\Sigma(t_{s,k})$, $\Omega_{k}\gets\Omega(t_{s,k})$; //solve the differential equations in (41); return $P_k$, $\Sigma_k$, $\Omega_{k}$;

7: 　　　$k\gets k+1$;

8: 　　until $k=N_s$

9: 　　$P(t_{s,0})\gets\Theta_{0}$, $\Sigma(t_{s,0})\gets\Theta_{0}$, $\Omega(t_{s,0})\gets\Theta_{0}$; // Lemma 2;

10: 　　Solve differential equations in (41) for $t\in[t_{s,0},t_{s,1})$;

11: 　　return $P(t)$, $\Sigma(t)$, $\Omega(t)$ for $t\in[t_{s,0},t_{s,1})$;

12: 　　$k\gets1$;

13: 　　repeat

14: 　　　$P(t_{s,k})\gets P_{k}$, $\Sigma(t_{s,k})\gets\Sigma_{k}$, $\Omega(t_{s,k})\gets\Omega_{k}$; // (42);

15: 　　　Solve differential equations in (41) for $t\in[t_{s,k},t_{s,k+1})$;

16: 　　　return $P(t)$, $\Sigma(t)$, $\Omega(t)$ for $t\in[t_{s,k},t_{s,k+1})$;

17: 　　　$k\gets k+1$;

18: 　　until $k=N_s$

19: 　　Construct the smoother as follows: 　　　　// (40);

　　　　$\begin{align*} \dot {\hat {\textit{z}}}(t) & = A\hat {\textit{z}}(t) + Bu(t) + P(t)C^TR^{-1}\left( {y(t) - C\hat {\textit{z}}(t)} \right) \\ \dot {\hat \phi}(t) & = \Sigma^T(t) C^TR^{-1}\left( {y(t)- C\hat {\textit{z}}(t)} \right) \end{align*}$

20: end procedure

21: //

22: procedure RESIDUAL($r(t_b),y(t_b),\hat{\phi}(t)$)

23: 　　Residual $\hat{r}\left(t_b|t\right)= r(t_b)+C\hat {\phi} (t)-y(t_b)$; 　　// (29) and $\Delta\hat{y}({t_b}|t)= C\hat{\phi}(t)-y(t_b)$;

24: 　　Evaluation $\hat{J}\left(t_b|t\right)= |\hat{r}\left(t_b|t\right)|-|r\left(t_b\right)|$; 　　// (54);

25: end procedure

26: //

27: procedure THRESHOLD($P_k,\Sigma_k,\bar{\omega},\bar{v}$) 　　　// Theorem 4;

28: 　　$k\gets0$

29: 　　repeat

30: 　　　Determine $\beta_k$, $\lambda_k$ satisfying (55); // Lemma 4;

31: 　　　Determine $\delta_0$ satisfying $|e_{\textit{z}}(t_{s,k})|\leq \delta_0$;

32: 　　　Calculate $\varepsilon_{z,k}(\cdot)$, $\varepsilon_{\phi,k}(\cdot)$ for $t\in[t_{s,k},t_{s,k+1})$; // (59) and (58);

33: 　　until $k=N_s$

34: 　　Threshold $\hat{J}_{th}(t)=|C|\varepsilon_{\phi,k}(t)$ for $t\in[t_{s,k},t_{s,k+1})$; // (57);

35: end procedure

36: //

37: procedure DECISION PRINCIPLE($\hat{J}(t_b|t),\hat{J}_{th}(t)$)

38: 　　if $\hat{J}(t_b|t)>\hat{J}_{th}(t)$ then an alarm is triggered;

39: 　　else no attack is detected;

40: 　　end if

41: end procedure

VI. Attack Detectability Analysis

In this section, the attack detectability of the developed backward-in-time detection methodology characterized by the residual $\hat{r}(t_b|t)$ in (29), the evaluation function $\hat{J}(t_b|t)$ in (54), and the threshold $\hat{J}_{th}(t)$ in (57), is investigated rigorously, characterizing quantitatively the class of detectable intermittent integrity attacks.

Theorem 5 (Detectability): Consider closed-loop CPS ${\cal{W}}$ in (1) and anomaly detector described in (3) and (4). The attack detection decision scheme, characterized by the piece-wise fixed-point smoother (40) described in Theorem 3, the residual (29), residual evaluation function (54) and detection threshold (57), guarantees that an intermittent integrity attack generated by the model (8) can be detected at a time $T_d\geq T_0>t_b$, i.e., $\hat{J}({t_b|T_d})>\hat{J}_{th}(T_d)$, if for the given fixed time instant $t_b$, there exists an attack slot k such that

where $\underline{\delta}$ is given in Assumption 1, $k_0$ and $\lambda_0$ are specified in Theorem 2, $J_{th}$ is given in (4) and $\hat{J}_{th}$ is given in (57).

Proof: For $\hat{r}\left(t_b|t\right) = r\left(t_b|t\right)-e_{y}(t)$ given in (53), by using the reverse triangle inequality, we can obtain

From (54) and $|r(t_b)|< J_{th}$, to detect an attack at the time instant $T_d$, i.e., $\hat{J}({t_b|T_d})>\hat{J}_{th}(T_d)$, the following inequality must hold:

Then, from (63) and the fact that $|e_{y}(t)|\leq \hat{J}_{th}(t)$ for $t\leq T_d$, a sufficient condition to guarantee (64) can be obtained as follows:

By using the same reasoning logic with the proof of Theorem 2, we can obtain that for the fixed time instant $t_b$, if there exists an attack slot k satisfying (62), then the inequality (65) can be guaranteed for any $T_d\in\bigcup_{i = k}^{N_a}\Omega_i$. Hence, the result is proved.■

Theorem 5 is a theoretical result that cannot be checked a priori. It is important to note that according to Theorem 5, the attack detection by the developed implementable backward-in-time detection methodology, characterized by the residual $\hat{r}(t_b|t)$ in (29), the evaluation function $\hat{J}(t_b|t)$ in (54), and the threshold $\hat{J}_{th}(t)$ in (57), is guaranteed by selecting a small $t_b$ (i.e., close to zero) so that condition (62) can hold. In other words, as time progresses and at each attack activation time $t_k$, the left side of (62) increases and at some time, will exceed the right hand side of (62), leading to the detection of the intermittent integrity attack.

VII. Simulation

In this section, a numerical simulation example based on a linear time-invariant system in the form of system (1) is presented. The system matrices are given as follows:

where the pair ($C,A$) is observable and $(A,D)$ is stabilizable. The output control gain is given by $K = -4.7333$ and the reference signal $y_{ {{\rm{ref}}}} = 0$. In addition, for the simulation purpose, the disturbance $\omega(t)$ is given by

and the measurement noise $v(t)$ is chosen at each time instant as a uniformly distributed random number ranging from $-0.3$ to $0.3$. The anomaly detector ${\cal{D}}$ in (3) is designed based on [29]. The residual is designed to satisfy the optimal ${\cal{H}}_{\infty}$ performance, i.e., $\int_{0}^{15}r^T(t)r(t)dt\;\leq \;2\int_{0}^{15}[\omega^T(t), v^T(t)]^T[\omega^T(t), v^T(t)]dt$ and the threshold is chosen as $J_{th} = 5$ based on the fault and attack free operation.

A Intermittent Integrity Attack

In this part, the intermittent integrity attack used in this simulation is given. The attacker is supposed to know the system matrices A, B and C, and is able to compromise all the sensors and actuators, i.e., $\Gamma_u = 1$ and $\Gamma_y = 1$. The attack activating time instants $t_k$ are given first as follows:

and the same dwell time is used for all attacks, i.e., $\tau_k = 1\;{{\rm{s}}}$ for all $k\in\{1,\ldots,6\}$. In the sequel, the design parameters of the attack model (8) are calculated. Based on Theorem 1, and by using the geometric approach toolbox in [50], we can obtain that the subspace ${\cal{V}}_0$ satisfying Theorem 1 is ${\cal{V}}_0 = [0,0,-1]^T$ and further, a feasible $F_k$ satisfying (12a) is calculated as

According to the obtained ${\cal{V}}_0$, $\Delta {\textit{z}}_k$ satisfying (12b) is chosen as

Thus, the design parameters of the attack model (8) have been selected and the attack signals for the attack activating time interval $\Omega^{\mathrm{ac}}_k$ can be generated. Next, the attack signals during the attack silence time interval $\Omega^{\mathrm{si}}_k$, i.e., $a_{y,k}((t_k+\tau_k)^{-})$, are given based on (8b) as follows:

Thus, the intermittent integrity attack signal for the attack activating time interval $\Omega^{\mathrm{ac}}_k$ and the attack silence time interval $\Omega^{\mathrm{si}}_k$ is generated.

The attack signal and its effects on the system are shown in Figs. 4-6, respectively. As it is shown in Fig. 4, the attack signal $a_y(t)$ is continuous at the attack pausing time instants $\{2 \,{{\rm{s}}},\;4\, {{\rm{s}}},\;6\, {{\rm{s}}},\;8 \,{{\rm{s}}},\;10\, {{\rm{s}}},\;12\, {{\rm{s}}}\}$. Moreover, the resulting system output y and its change in Fig. 6 have no jump (abrupt change) at each of these attack pausing time instants. By comparing the system states x in the attack case with the state $x^n$ in the nominal case in Fig. 5, we can observe that the system state $x_3$ is significantly affected by the injected intermittent integrity attacks. However, in contrast to $y^n$ in the nominal case, the system output y in Fig. 6 has a very small increment $\Delta y$ at each of the attack resuming time instants $\{1\, {{\rm{s}}},\;3\, {{\rm{s}}},\;5 \,{{\rm{s}}},\;7\, {{\rm{s}}},\;9 \,{{\rm{s}}}, 11\, {{\rm{s}}}\}$, and such an increment converges to zero exponentially. Fig. 6 also shows that the increments caused by the attack are relatively small and hidden by the disturbances and noise, which is a result of the selected relatively “large” process disturbances and measurement noise used in the simulation. This particularly created simulation scenario is used for verifying that the designed backward-in-time detector is robust to the process disturbances and measurement noise, and also sensitive to the stealthy intermittent integrity attacks.

Figure  4.  Time responses of the attack signals $a_u(t)$ and $a_y(t)$.

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Figure  6.  Time responses of the sensor measurements $y(t)$ and $y^n(t)$ in the attack case and nominal case respectively, and the output change $\Delta{y}(t)$.

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Figure  5.  Time responses of the state vector $x(t)$ in the attack case and $x^n(t)$ in the nominal case.

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Fig. 7 illustrates the anomaly detection results using the equipped anomaly detector ${\cal{D}}$, in which the residual $r(t)$ remains far below the threshold $J_{th}$ during the attack event. Hence, the attack is not detected by ${\cal{D}}$. This indicates that the change $\Delta y$ due to the intermittent attack is sufficiently small to maintain the stealthiness of the attack with respect to the detector ${\cal{D}}$.

Figure  7.  Time responses of the forward-in-time residual $r(t)$, the evaluation function $J(t) = |r(t)|$ and the threshold $J_{th}$ of the anomaly detector ${\cal{D}}$.

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B Attack Detection

Following Algorithm 1, the parameters for the SMOOTHER procedure are given as follows: $t_b = 2,\; T = 15,\; t_{s,0} = t_b = 2,\; t_{s,1} = 9,\; t_{s,2} = 16$, $\alpha = 0.009$ and $R = 0.1$. Moreover, $\Theta_k$ satisfying (49) is given by $\Theta_{0} = 8.5I_3,\Theta_{1} = 43.5I_3$. By solving the differential equations in (41), $P_1$ and $\Sigma_1$ are obtained as follows:

Furthermore, $P(t)$ and $\Sigma(t)$ for $[2 \;{{\rm{s}}},\;9 \;{{\rm{s}}})$ and $[9 \;{{\rm{s}}},\;16\; {{\rm{s}}})$ can be obtained respectively by solving the differential equations in (41), and thus, the fixed-point smoother can be implemented. Then, given $y(t_b) = 0.1985$, $r(t_b) = -0.1332$ and the estimate $\hat{\phi}(t)$, the RESIDUAL procedure can be completed by following Algorithm 1, and the residual $\hat{r}(t_b|t)$ from (29) and its evaluation function $\hat{J}(t_b|t)$ from (54) can be obtained. We proceed with the THRESHOLD procedure based on Algorithm 1. The scalars $\bar{\omega}$ and $\bar{v}$ in Assumption 2 are given by $\bar{\omega} = 0.3$ and $\bar{v} = 0.2$. Moreover, $\beta_0 = 0.25$, $\lambda_{0} = 0.64$, $\beta_1 = 0.1$ and $\lambda_1 = 0.63$, and $\delta_0 = 100$. The initial value of $\varepsilon_{\phi,k}$ is chosen as $\varepsilon_{\phi,0} = 5$. By following the THRESHOLD procedure in Algorithm 1, the threshold $\hat{J}_{th}(t)$ in (57) can be calculated.

The residual $\hat{r}(2|t)$, the evaluation function $\hat{J}(2|t)$ and the threshold $\hat{J}_{th}(t)$ are shown in Fig. 8. It is shown that at each of the attack resuming time instants $\{3\, {{\rm{s}}},\;5\, {{\rm{s}}},\;7 \,{{\rm{s}}},\;9\, {{\rm{s}}},\;11\, {{\rm{s}}}\}$, the residual $\hat{r}(2|t)$ and its corresponding evaluation $\hat{J}(2|t)$ have a jump in magnitude. Thus, the accumulation property described in Lemma 1 is satisfied. Furthermore, note also that in Fig. 8, the jump that occurs at $t = 9 \;{{\rm{s}}}$ is much larger than the one at $t = 7\; {{\rm{s}}}$ due to covariance matrix resetting, even though the increment $\Delta y$ at $t = 9 \;{{\rm{s}}}$ is similar with the one at $t = 7 \;{{\rm{s}}}$ (see $\Delta {\textit{z}}_4 = \Delta {\textit{z}}_5$). Moreover, based on the DECISION PRINCIPLE specified in Algorithm 1, we can conclude from Fig. 8 that the injected intermittent stealthy integrity attack is successfully detected at the time $T_d\approx9.5 \;{{\rm{s}}}$ when the evaluation function $\hat{J}(2|t)$ exceeds the threshold $\hat{J}_{th}(t)$.

Figure  8.  Time responses of the backward-in-time residual $\hat{r}(2|t)$, the evaluation function $\hat{J}(2|t)$ and the threshold $\hat{J}_{th}(t)$.

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VIII. Conclusion

In this paper, stealthy intermittent integrity attacks being stealthy with respect to typical anomaly detectors have been formulated. A backward-in-time detection residual that can accumulate at each attack activation time and is able to indicate the stealthy intermittent integrity attacks has been introduced. A fixed-point smoother has been designed as the backward-in-time estimator for estimating the unknown backward-in-time detection residual. A covariance matrix resetting technique has been applied in the design of the smoother to guarantee the required sensitivity to the attacks. The corresponding adaptive threshold generation scheme for detecting the stealthy intermittent integrity attacks has been designed, and the attack detectability has also been investigated rigorously. Some future research works are given as follows:

1) One of our studies work focuses on scheduling the attack pausing and resuming time instants such that the attack can cause significant damage to the system, and at the same time, achieve the power energy saving aim. Game theory may provide a way to solve the trade-off between damaging effects and energy saving [38].

2) Another future research direction involves the modification of typical fixed-point smoothers to improve the sensitivity to stealthy integrity attacks. The forgetting factor for the covariance matrix used in [51] is a potential way to achieve this improvement.

3) Resilience control against intermittent integrity attacks is also one potential research direction. Some control issues, such as quantized sensor measurements well handled by the model reference control methodologies in [52], [53] and high order nonlinearities in [54], [55] will be investigated.

4) Stealthy intermittent integrity attack generation and detection issues for nonlinear systems and large-scale systems such as in [56] will be considered in our future research. Note that a way for generating stealthy intermittent attacks for nonlinear systems and large-scale systems is to use the geometric approach proposed in [14].

Appendix A. Proof of Theorem 1

Proof : Let $\Delta x_1$, $\Delta x_2$, $\Delta y_1$, $\Delta y_2$ and $\Delta u$ represent the changes of $x_1$, $x_2$, $y_1$, $y_2$ and u respectively due to the attack $a(t)$, i.e., $\Delta x= x-x^n$, $\Delta x_2= x_2-x_2^n$, $\Delta y_1= y_1-y_1^n$, $\Delta y_2= y_2-y_2^n$ and $\Delta u= u-u^n$. Then, from (10) and (11), the incremental systems between ${\cal{W}}_{j}$ and ${\cal{W}}^n_{j}$ for $j = 1,2$, denoted as $\Delta{\cal{W}}_{j}$, can be obtained as follows:

where $\Delta x_{1}(t_k) = -\Delta {\textit{z}}_k$ and $\Delta x_2(t_k) = \Delta {\textit{z}}_k$. Thus, the state change $\Delta x$ and the output change $\Delta y$ can be written as

Consider ${\cal{W}}_j$ for $j = 1,2$ given in (11). Denote ${\cal{W}}^n_{j,k}$ as the “nominal” system of ${\cal{W}}_j$ in the absence of the attack signal $a_k(t)$ but in the presence of $a_1(t),\ldots,a_{k-1}(t)$, where $k\geq1$. By letting $x^n_{1,k}$ and $y^n_{1,k}$ represent the state and the output of ${\cal{W}}^n_{1,k}$, and letting $x^n_{2,k}$, $y^n_{2,k}$ and $u^n_{k}$ represent the state, the output and the control of ${\cal{W}}^n_{2,k}$, respectively, it then follows from (11) that ${\cal{W}}^n_{j,k}$ can be written as:

where the initial conditions can be chosen as $x_{1}(t_k)-x^n_{1,k}(t_k) = -\Delta {\textit{z}}_k$ and $x_2(t_k)-x^n_{2,k}(t_k) = \Delta {\textit{z}}_k$ such that $x_{1}(t_k)+x_2(t_k) = x^n_{1,k}(t_k)+ x^n_{2,k}(t_k)$.

Let $\Delta x_{1,k}$ and $\Delta y_{1,k}$ represent the changes of $x_{1,k}^n$, and $y_{1,k}^n$, and $\Delta x_{2,k}$, $\Delta y_{2,k}$ and $\Delta u_{k}$ represent the changes of $x_{2,k}^n$, $y_{2,k}^n$ and $u_{k}^n$ respectively due to the attack $a_k(t)$, i.e., $\Delta x_{1,k}= x_{1}-x_{1,k}^n$, $\Delta y_{1,k}= y_{1}-y_{1,k}^n$, $\Delta x_{2,k}= x_2-x_{2,k}^n$, $\Delta y_{2,k}= y_{2}-y_{2,k}^n$ and $\Delta u_{k}= u- u_{k}^n$. Then, based on (70), (71) and (11), the incremental systems, denoted as $\Delta{\cal{W}}_{j,k}$, are obtained as

where $\Delta x_{1,k}(t_k) = -\Delta {\textit{z}}_k$ and $\Delta x_{2,k}(t_k) = \Delta {\textit{z}}_k$.

Therefore, based on the superposition principle for linear systems ($a(t)$ is the sum of $a_k(t)$ in (8c)), $\Delta x_1$, $\Delta {\textit{z}}$ and $\Delta y$ can be written as

In the sequel, the responses of $\Delta{\cal{W}}_{j,k}$ in the attack activating time interval and the attack silence time interval are analyzed.

1) Activating Time Interval $\Omega^{\mathrm{ac}}_k$: In this time interval, $\Delta {\cal{W}}_{1,k}$ and $a_k(t)$ described by (8a) and (8b) respectively can be equivalently written in the coordinates $(\Delta\bar{x}_{1,k},\zeta_k)$ with $\Delta\bar{x}_{1,k}=$ $\Delta x_{1,k}-\zeta_k$ as follows:

where $\Delta\bar{x}_{1,k}(t_k) = \Delta x_{1,k}(t_k)-\zeta_k(t_k) = 0$. Since $\Delta\bar{x}_{1,k}(t_k) = 0$, $\Delta\bar{x}_{1,k}(t) = 0$ for $t\in\Omega^{\mathrm{ac}}_k$, and hence, $\Delta x_{1,k}(t) = \zeta_k(t)$ for $t\in\Omega^{\mathrm{ac}}_k$. For $\Delta {\textit{z}}_{k}\in{\cal{V}}_0$ and $F_k$ satisfying (12a), we have

Thus, we obtain

2) Silence Time Interval $\Omega^{\mathrm{si}}_k$: In this time interval, the initial condition of $\Delta {\cal{W}}_{1,k}$ is $\Delta x_{1,k}(t_k+\tau_k) = \Delta x_{1,k}((t_k+\tau_k)^{-})\in{\cal{V}}_0$. Based on $a_{k}(t)$ in (8b) during $t\in\Omega^{\mathrm{si}}_k$, $\Delta x_{1,k}$ satisfies

Also, $\Delta y_{1,k}$ at the time instant $t_k+\tau_k$ satisfies

Thus, it follows from $\Delta x_{1,k}(t_k+\tau_k)\in{\cal{V}}_0$ and ${\cal{V}}_0\subset{\cal{H}}$ with ${\cal{H}}$ being the unobservable subspace of the pair $(CA,A)$ that $\Delta x_{1,k}(t_k+\tau_k)\in{\cal{H}}$, which indicates:

Thus, by combining (77) and (78), we have

Hence, from the result (76) in the activating time interval and the result (79) in the silence time interval, we can conclude

Therefore, it follows from (75) that the change $\Delta y$ can be written as in (13) with $\Delta y_{2,k}(t)$ being generated by $\Delta{\cal{W}}_{2,k}$ in (79). In addition, since $\Delta u_k = K\Delta y_k = K\Delta y_{2,k} = KC\Delta {\textit{z}}_k$, (79) can be written as (14).■

Appendix B. Proof of Lemma 1

Proof:

1) By using the transition matrix Φ in (17), the solution of the system $\Delta {\cal{W}}$ in (15) with the initial condition $\Delta {\textit{z}}(t_k) = \Delta {\textit{z}}_{k}$ can be written as

Thus, from Definition 1, ${\Delta {\textit{z}}}(t_b|t)$ can be written as

Since both $\Phi(t_b,t_k)$ and $\Delta {\textit{z}}_{k}$ are independent of time t, ${\Delta {\textit{z}}}(t_b|t)$ is also independent of time t, then ${\Delta {\textit{z}}}(t_b|t)$ is a constant vector with respect to time. Hence, (18) follows.

2) Since $\Delta {\textit{z}}_{k}\neq0$ and $\Phi(t_b,t_k)\neq0$, then the result (19) follows directly from (80).

3) From (80), for the consecutive attack slots $\Omega_k$ and $\Omega_{k+1}$, we have

By using $\Phi(t_b,t_{k}) = \Phi(t_b,t_{k+1})\Phi(t_{k+1},t_{k})$, $\Delta {\textit{z}}(t_b|t_{k})$ can be equivalently written as

Then, we can derive

Hence, $|{\Delta {\textit{z}}}(t_b|t)|_{t\in\Omega_{k+1}}\geq|{\Delta {\textit{z}}}(t_b|t)|_{t\in\Omega_{k}}$ if (20) is satisfied.■

Appendix C. Proof of Theorem 3

Proof:

1) Based on Lemma 1, in the absence of the attack, ${\Delta {\textit{z}}}(t_b|t) = 0$ and from (21), $\Delta y({t_b}|t) = 0$, and thus, $r\left(t_b|t\right) = r(t_b)$. Since $|r(t_b)|\leq J_{th}$, result 1) follows.

2) By using the reverse triangle inequality and based on (21) and (22), a sufficient condition to guarantee $|r(t_b|t)|> J_{th}$ for all $t\in\bigcup_{i = k}^{N_a}\Omega_i$ can be obtained as

Note that based on result 1) in Lemma 1, $\Delta {\textit{z}}({t_b}|t)$ is a constant vector during an attack slot $\Omega_k$. Thus, $|\Delta {\textit{z}}({t_b}|t)| =$ $|\Delta {\textit{z}}({t_b}|t_k)|$ for $t\in\Omega_k$. Note also that under Assumption 1 and based on result 3) in Lemma 1, $|\Delta {\textit{z}}({t_b}|t_i)|\geq|\Delta {\textit{z}}({t_b}|t_k)|$ for any $i\geq k$ since (20) is considered to hold. Thus, a sufficient condition to guarantee (81) is obtained as follows:

Based on Definition 1, we can write $\Delta {\textit{z}}({t_b}|t_k) = \Phi^{-1}(t_k,t_b)\Delta {\textit{z}}_k$. Thus, a sufficient condition to guarantee (82) can be obtained as

Note that according to [40], for the Hurwitz matrix $A+BKC$, there exist $k_0>0$ and $\lambda_0>0$ such that

Thus, the time $t_k$ satisfying (25) can guarantee the sufficient condition (83). Hence, result 2) follows.■