Distributed $H_\infty$ Estimation Resilient to Biasing Attacks

TL;DR

This paper develops a distributed $H_ abla$ estimation method that detects and compensates biasing attacks, enabling each node to produce unbiased, robust estimates while maintaining decentralized gain computation for enhanced security.

Contribution

It introduces a novel distributed observer design with integrated attack detection and compensation, ensuring unbiased estimates and decentralized gain computation for resilience against biasing attacks.

Findings

Distributed observer achieves unbiased, robust estimates.

Gains can be computed in a decentralized manner.

Network vulnerability is reduced through attack compensation.

Abstract

We consider the distributed $H_\infty$ estimation problem with an additional requirement of resilience to biasing attacks. An attack scenario is considered where an adversary misappropriates some of the observer nodes and injects biasing signals into observer dynamics. The paper proposes a procedure for the derivation of a distributed observer which endows each node with an attack detector which also functions as an attack compensating feedback controller for the main observer. Connecting these controlled observers into a network results in a distributed observer whose nodes produce unbiased robust estimates of the plant. We show that the gains for each controlled observer in the network can be computed in a decentralized fashion, thus reducing vulnerability of the network.

Equations

$$\dot{x}=A(t)x+B(t)w,x(0)=x_0,$$

$$y_i=C_i(t)x+D_i(t)v_i,y_i\in {\mathbf{R}}^{p_i}.$$

$$c_{ij}=W_{ij}{\hat{x}}_j+H_{ij}{v}_{ij},$$

$${\dot{\hat{x}}}_i$$

$$f(t)=f_1(t)+f_2(t),$$

$$\hat{f}(s)=\frac{1}{s}(sI+G(s){)}^{-1}G(s)f(s).$$

$${\dot{\hat{x}}}_i$$

$${\dot{\chi }}_i$$

$$u_i$$

$${\zeta }_i$$

$${\eta }_{ij}$$

$${\dot{e}}_i$$

$${\dot{ϵ}}_i={\Omega }_i{ϵ}_i+{\Gamma }_i{\nu }_i,{ϵ}_i(0)=0,$$

$${\hat{f}}_i={{\rm Y}}_i{ϵ}_i,$$

$${\dot{e}}_i$$

$${\dot{ϵ}}_i$$

$${\zeta }_i$$

$${\phi }_i=f_i-u_i={{\rm Y}}_i{ϵ}_i-u_i-{\nu }_i.$$

$${\dot{\hat{e}}}_i$$

$${\dot{\widehat{ϵ}}}_i$$

$${\eta }_{ij}$$

$$u_i$$

$${\dot{z}}_i$$

$${\dot{\delta }}_i$$

$${\hat{L}}_i^r(t)=L_i^r(t)+{\bar{L}}_i^r(t),{\hat{K}}_{ij}^r(t)=K_{ij}^r(t)+{\bar{K}}_{ij}^r(t),$$

$${\hat{F}}_i=F_i+{\bar{F}}_i,{\check{\Gamma }}_i={\Gamma }_i+{\check{F}}_i,{\hat{H}}_{ij}=[H_{ij}{H}_{c,ij}].$$

$$\displaystyle\sum_{i=1}^{N}\int_{0}^{\infty}\|\Upsilon_{i}\delta_{i}\|^{2}dt\leq\gamma^{2}\sum_{i=1}^{N}\bigg{(}\|x_{0}-\xi_{i}\|_{X_{i}^{-1}}^{2}$$

$${\dot{\tilde{e}}}_i$$

$${\tilde{\Delta }}_i$$

$${\tilde{\Phi }}_{ij}$$

$${\dot{\tilde{Y}}}_i$$

$$\tilde{R}>\tilde{P}-{\tilde{\gamma }}^2(\tilde{\Phi }+{\tilde{\Phi }}^{\prime }-\tilde{\Delta }),$$

$${\tilde{L}}_i(t)$$

Full text

I Introduction

The topic of understanding possible attacker’s strategies and developing suitable countermeasures receives considerable attention in the literature dedicated to designing control and filtering systems resilient to malicious interference. The approaches of the game and optimization theories are conventional in this area [1, 8, 27, 30, 5, 32], although they often tend to overestimate the attacker, assuming that the adversary has sufficient resources to collect detailed measurements and implement a sophisticated attack policy. For instance, the recent Byzantine models of the adversarial behavior allow the adversarial nodes to possess complete knowledge about the graph topology and the plant dynamics, i.e., an adversarial node knows the measurements received by the healthy nodes [11]. This has undesirable consequences of having to endow the network with a substantial redundancy, in order to guarantee that it is able to sustain multiple simultaneous attacks [3, 7, 11].

There have been suggestions that robust control techniques could provide an alternative framework for optimizing the system performance in the presence of an attacker [4]. Indeed, robust control models are quite flexible and can handle large numbers of uncertainty inputs. In this regard they may offer a certain flexibility over the Byzantine model in that they are generally less restrictive about the number of nodes affected by the adversary. However, robustness specifications differ significantly from requirements for resilience to strategic attacks [32], which could make the robust control approach too conservative as the means for accommodating strategic intentions of a malicious attacker [7].

In this paper we show that this drawback of the robustness methodology can be overcome if the features of the attack which make it distinct from benign disturbances are properly accounted. To this end, we consider the so-called biasing attacks [21] which admit a simple mathematical description, and yet capture a nontrivial type of misappropriation attacks on distributed networks. To be concrete, the paper focuses on the problem of resilient distributed estimation using sensor networks in the presence of a strategic adversary who misappropriates a number of sensing nodes and injects a biasing signal into the observers [6, 21]. To interfere with operation of the network the intruder first infiltrates the network, by breaching the cyber layer of security. Once the adversary has penetrated the cyber security barrier and has gained access to the estimation algorithm at the misappropriated node, it modifies it by injecting biasing signals. In this regard, our attack scenario resembles that of the Stuxnet computer worm [18]. The injected biasing signals cause the misappropriated observers to produce erroneous estimates of the plant. Also, the bias spreads through the network interconnections causing a ‘domino’ effect of cascading errors, forcing the entire observer network to fail. To achieve this, the attacker does not need any knowledge of the system or the plant.

We show that information collected by sensors within a typical network of state observers coupled in a diffusive manner [13] is sufficient for discovering and disarming such biasing misappropriation attacks. To demonstrate this, we propose a distributed observer architecture which augments the observer network with an additional layer consisting of a network of interconnected attack detecting controllers. These controllers utilize the same noisy innovation information which is used for state estimation. based on this information, they estimate an extended vector of observer errors and use it for feedback. We show that this feedback control policy has an asymptotically negligible effect on the dynamics of healthy observers. However at the compromised nodes, the controller outputs track the biasing attack inputs; this enables to negate the attack. The controllers are constructed within the $H_{\infty}$ control framework. The choice of the framework is consistent with the original state estimation objective and facilitates the performance analysis of the resilient estimation algorithm.

Controlling the plant observers via feedback to make them resilient to the biasing behavior of misappropriated nodes is the main distinction of this paper from our previous work [6]. As another distinct feature, the paper considers the most general biasing attack scenario in which the attacker seeks to bias both the state observer and the attack detector. Indeed, the attacker who has gained access to the data processing algorithm will likely to temper with both the state observer and security devices. Hence, the attack detection algorithm must also be made resilient to attacks, in the same manner as the plant observer. Although the paper considers this problem in the context specific to misappropriation attacks on distributed observers, we believe the problem is important in general since security devices cannot not be assumed to be immune to malicious interference.

Starting from the seminal work [15], distributed computations are widely used in the literature on distributed estimation. In addition to solving the main task of obtaining state estimates of the plant, in many algorithms, sensor nodes must communicate to compute parameters of their observers, such as error covariance matrices, innovation gains, etc [15]. Such distributed computations present obvious security risk, and hardening of distributed algorithms is a subject of current research [20]. In this paper, we approach this issue differently, aiming to achieve a computational autonomy of the observers. Namely, in the proposed method, each observer node computes its observer in a decentralized fashion, without communicating with other nodes. The method is based on the distributed observer design approach proposed in [29]. It involves an initial setup step during which certain auxiliary parameters are computed which are then assigned to the nodes. This initial step is carried out centrally when the network is offline, hence it does not jeopardize the system security. Also, the knowledge of the plant observed is not required at this stage, since these parameters are computed only using characteristics of the communication network and the desired performance characteristics. Each sensor node then utilizes these auxiliary parameters to compute its node observer autonomously. The plant model must be used at this stage, of course. This methodology contrasts the methodology used in our previous work [6] which relied on solving certain linear matrix inequalities coupled among the nodes to compute attack detectors.

In summary, the main contribution and features of the paper are as follows:

1. (a)

We introduce a new class of networked attack detectors that use feedback to suppress biasing attacks on distributed observer networks. 2. (b)

Despite the proposed methodology of resilient estimation is based on the approach of $H_{\infty}$ control, it distinguishes between benign disturbances and attack signals. 3. (c)

Formally, our approach does not limit the number of nodes subjected to simultaneous biasing attacks. Essentially, every node is treated conservatively as a potential target. However, trusted nodes can be easily accommodated in our formulation. This will only simplify the system model used in the derivation of the attack detector and controller. 4. (d)

The proposed methodology identifies the nodes subjected to biasing attacks, regardless whether the adversary targets the state observer or its attack detecting feedback controller, or both. 5. (e)

The node observers compute their parameters in decentralized manner in real time. Only the information about their states and states of their controllers needs to be shared through communication channels. 6. (f)

In the special case of an LTI plant and observers, we are able to provide a deeper insight into feasibility and performance optimization of the proposed resilient observers, including selecting the best network topology from a set of candidate topologies.

The preliminary version of the paper has been presented at the 2018 American Control Conference [26]. Compared with the conference version, the current version contains a substantial amount of new material concerned with security of the proposed attack detecting controllers and performance optimization over network graphs. Also, the paper includes an illustrating example which was left out from the conference version due to space constraints.

The paper begins with presenting a background on distributed filtering in Section II. Also, in that section we describe the class of biasing attacks which will be used in the derivation of our main results. The formal problem statement is given in Section II-B. The procedure to construct a resilient distributed observer is described in Section III, and its feasibility and optimization aspects are discussed in Section III-D. It is illustrated in Section IV with an example. The conclusions are given in Section V.

Notation: $\mathbf{R}^{n}$ denotes the real Euclidean $n$-dimensional vector space, with the norm $\|x\|=(x^{\prime}x)^{1/2}$; here the symbol ′ denotes the transpose of a matrix or a vector. The symbol $I$ denotes the identity matrix. For real symmetric $n\times n$ matrices $X$ and $Y$, $Y>X$ (respectively, $Y\geq X$) means the matrix $Y-X$ is positive definite (respectively, positive semidefinite). $\otimes$ is the Kronecker product of matrices. $\|x\|_{X}$ denotes the weighted norm of $x$: $\|x\|_{X}=(x^{\prime}Xx)^{1/2}$. $\mathrm{diag}\left[C_{1},\ldots,C_{N}\right]$ denotes the block-diagonal matrix with the matrices $C_{1}$, …, $C_{N}$ as its diagonal blocks. The notation $L_{2}[0,\infty)$ refers to the Lebesgue space of $\mathbf{R}^{n}$-valued vector-functions $z(.)$, defined on the time interval $[0,\infty)$, with the norm $\|z\|_{2}\triangleq\left(\int_{0}^{\infty}\|z(t)\|^{2}dt\right)^{1/2}$ and the inner product $\int_{0}^{\infty}z_{1}^{\prime}(t)z_{2}(t)dt$. For a causal signal $f(t)$, $f(s)$ denotes the Laplace transform of $f(t)$.

II Biasing misappropriation attacks on distributed observers and

resilient distributed estimation

II-A Biasing misappropriation attacks on observer networks

Consider a linear time-varying plant

[TABLE]

subject to an unknown disturbance $w\in L_{2}[0,\infty)$. The plant is observed by a network of $N$ sensors. The sensor at node $i$ collects measurements of the plant, corrupted by disturbances $v_{i}$:

[TABLE]

It also exchanges information with other nodes. The communication network forms a directed graph $\mathbf{G}$. Without loss of generality, the graph is assumed to be connected but not necessarily strongly connected111 This assumption is justified due to Proposition 1 in [22]. The sufficiency part of that proposition can be easily extended to distributed estimation problems with a general cost considered in this paper. It implies that if the network is disconnected, to obtain a solution to the distributed $H_{\infty}$ filtering problem over such network, it suffices to obtain a solution to the distributed $H_{\infty}$ filtering problem over each connected network component.. For each node $i$, let $\mathbf{V}_{i}$ denotes the set of its neighbors supplying information to that node. The information received by node $i$ from its neighbor $j\in\mathbf{V}_{i}$ is a $p_{ij}$-dimensional signal

[TABLE]

which contains information about the neighbor $j$’s estimate $\hat{x}_{j}$ of the plant state $x$. The $p_{ij}\times n$ matrix $W_{ij}$ determines what information about $\hat{x}_{j}(t)$ node $j$ shares with node $i$. The signal $v_{ij}\in L_{2}[0,\infty)$ in (3) represents a channel disturbance. We think of the matrices $W_{ij}$, $H_{ij}$ as characteristics of the network which is considered to be fixed (the motivation for this will be explained later) and independent of the plant. For that reason, these matrices are assumed to be constant; cf. [12].

We now make standing assumptions about the coefficients of the system (1), (2), (3). Throughout the paper, it will be assumed that the matrix-valued functions $A(t)$, $B(t)$, $C_{i}(t)$, $D_{i}(t)$, $i=1,\ldots,N$, are bounded on the interval $[0,\infty)$. Also, it will be assumed that $(D_{i}(t)D_{i}(t)^{\prime})^{-1}$ exists and is bounded on $[0,\infty)$ for all $i$.

In the distributed estimation scenario, the measurements and the communicated information are processed at the sensor nodes rather than centrally. Following [15], the following observers are often considered for this [10, 22, 19, 9]:

[TABLE]

Since the plant is time-varying, the coefficients $L_{i}$, $K_{ij}$ in (4) are assumed to be time-varying in general. The observers (4) are coupled. The coupling is especially useful when some of the pairs $(A(t),C_{i}(t))$ are not detectable222Here we follow the definition of detectability in [14] and say that the pair $(A(t),C_{i}(t))$ is detectable if there exists a gain $\tilde{L}(t)$ such that the system $\dot{e}=(A(t)-\tilde{L}(t)C_{i}(t))e$ is exponentially stable.. In this case, the corresponding nodes face the situation that their filters may not be able to track the plant if they rely on local measurements alone. The interconnections provide those nodes with an information about $x(t)$ which cannot be obtained from the local measurement $y_{i}$ but can be extracted from the neighbors’ messages $c_{ij}$ [22, 23, 12]. Distributed observers of the form (4) of course require the system to be detectable as whole; i.e., the pair $(A(t),[C_{1}^{\prime},\ldots,C_{N}^{\prime}]^{\prime})$ must be detectable.

A common problem of distributed estimation is to ensure that the node estimates $\hat{x}_{i}(t)$ track $x(t)$ (or a part thereof) with acceptable accuracy and robustness against disturbances in the plant model, measurements and interconnection channels; e.g., see [15, 16, 22]. However, the dependency on information sharing makes a network of observers (4) vulnerable to attacks seeking to disrupt this task. Common scenarios of such attacks involve an adversary injecting false signals into sensor measurements or communicated data [17] or behaving as a Byzantine fault [11]. The latter behavior assumes that the adversary can misappropriate some network nodes and force them to deviate arbitrarily from the prescribed estimation algorithm and transmit different false state estimates to different neighbors. In this paper, we consider a similar adversarial behavior, however we assume that the adversary strategically substitutes the node observers (4) with their biased versions [6],

[TABLE]

Here $f_{i}\in\mathbf{R}^{n_{f_{i}}}$ is an unknown signal representing the attacker’s input.

When node $i$ is forced to use the observer (5) in lieu of (4), it generates biased estimates of $x(t)$ closely resembling those that could be obtained using the true observer (4). These biased estimates $\hat{x}_{i}$ are then broadcast across the network and will bias other nodes. To disrupt the network, the adversary does not need to know the plant dynamics, the measurements or the communication graph of the system; cf. [11]. Different from the Byzantine attack modeling, we will not have to impose a limit on the number of misappropriated nodes, and can consider the worst-case situation where every node of the observer network can be biased.

Remark 1

Biasing attacks of this type can occur as a result of a strategic network intrusion, when the adversary breaches the cyber security layer and gains access to the observer algorithm at the misappropriated nodes. In this sense, our attack model is conceptually different from models of benign disturbances. Also unlike benign disturbances, in our model the biasing inputs disturb the observer dynamics, rather than sensor measurements and/or plant dynamics. $\Box$

Since the aim is to obtain an algorithm for fending such biasing attacks, we assume that the matrices $F_{i}$ are known to the defender. That is, the observer (5) is regarded as a model of the misappropriated attack perceived by the defender. The defender has several choices for the matrices $F_{i}$. For instance, one can assume $F_{i}=[1,\ldots,1]^{\prime}$; this choice captures the attack model where the attacker injects a scalar biasing input into the observer dynamics. Alternatively, we can consider $F_{i}=I$; this means that the adversary may bias each component of the observer separately.

We assume that the attacker does not seek to change the network topology. Although it is conceivable that the adversary may attempt to do so, this type of attack will imminently expose the attacker. It may not be suitable for the adversary who wishes to remain covert.

Definition 1** ([6])**

A class of admissible biasing inputs, denoted $\mathscr{F}_{a}$, consists of causal signals $f(t)\in\mathbf{R}^{n_{f}}$,

[TABLE]

where the Laplace transform of $f_{1}(t)$, $f_{1}(s)$, has the property $f_{1}^{\infty}=\sup_{\omega}|\omega f_{1}(j\omega)|^{2}<\infty$, and $f_{2}\in L_{2}[0,\infty)$. In the decomposition (6), $f_{1}(t)$ represents the biasing component, and $f_{2}(t)$ is a masking signal; cf. [21].

Note that the attack set $\mathscr{F}_{a}$ includes as a special case biasing attack inputs which consist of an unknown steady-state component and an exponentially vanishing masking signal generated by a low pass filter [21].

The following lemma characterizes the properties of admissible biasing inputs. Its proof is given in [6].

Lemma 1

Let $N(s)$, $D(s)$ be arbitrary real polynomials with the following properties:

1. (a)

The degree of $N(s)$ is not greater than the degree of $D(s)$, and so the transfer function $G(s)=\frac{N(s)}{D(s)}I$ is proper; 2. (b)

The transfer function $(sI+G(s))^{-1}G(s)$ is stable, and hence $g_{1}\triangleq\frac{1}{2\pi}\int_{-\infty}^{\infty}\left|\frac{D(j\omega)}{j\omega D(j\omega)+N(j\omega)}\right|^{2}d\omega<\infty$ and $g_{2}\triangleq\sup_{\omega}\left|\frac{j\omega D(j\omega)}{j\omega D(j\omega)+N(j\omega)}\right|^{2}<\infty$.

Then for any signal $f\in\mathscr{F}_{a}$, it holds that $\nu\triangleq\hat{f}-f\in L_{2}[0,\infty)$, where $\hat{f}$ is the signal whose Laplace transform is

[TABLE]

The idea behind the decomposition (6) is to separate ‘slow’ parts of $f(t)$ responsible for biasing the observer (denoted $f_{1}(t)$) from disturbance-like components (denoted $f_{2}(t)$) whose impact can be attenuated provided the observer is made sufficiently robust to disturbances. According to Lemma 1, sufficiently slow biasing inputs can be approximated with a dynamic model, up to a bounded energy error. This dynamic model will be used in the derivation of a distributed observer. The idea is to endow the observer with a capacity to filter out slow biases and be robust against bounded-energy perturbations and attack approximation errors alike.

Remark 2

In this paper, the attacker is assumed to have no knowledge of the system, the observers or the network. This forces the adversary to structure the biasing signal to include both slow-varying biasing components and disturbance-like components into the signal $f(t)$. If the slow biasing component is not included in $f(t)$, and $f(t)$ acts as a disturbance, the observer will likely attenuate its effect along with the effect of other noises and disturbances present in the system, since it is designed to be robust against disturbances. This will likely reduce effectiveness of the attack. This motivates our assumption that the admissible biasing inputs must have the form (6).

In contrast, the Byzantine model does not prescribe attack inputs to have a particular structure. However, within the Byzantine model, the attacker is assumed to have knowledge of the system which is not required in our model. For instance in [11], the adversarial nodes are assumed “to possess complete knowledge about the graph topology and the plant dynamics, i.e., an adversarial node knows the measurements received by the normal nodes at every time step.” The attack model (6) does not use this assumption. Also, the Byzantine model limits the number of adversarial nodes that can be tolerated. If this number exceeds a certain threshold, this model cannot guarantee a successful attack detection. In contrast, our model does not impose such a threshold, it allows for the situation when all nodes are corrupted with biasing inputs. This indicates that both models have place in the theory of resilient estimation, with their advantages and limitations. $\Box$

II-B Resilient distributed estimation problem

Our proposal is to modify the observers (4) to include additional control inputs to suppress biasing misappropriation attacks. The corresponding model of a misappropriated observer will then be as follows

[TABLE]

The gains $L_{i}^{r}(t)$, $K_{ij}^{r}(t)$ of the observer will be computed to ensure that each $\hat{x}_{i}$ converges to $x(t)$ even in the presence of admissible attacks. To achieve this resilience property, the control inputs $u_{i}$ will need to be generated so that when node $i$ is under attack, $u_{i}$ counters the biasing input $f_{i}$. Also, when the node $i$ is not attacked directly, the control $u_{i}$ must not impede its observer from producing unbiased estimates of $x(t)$. Since the signals $\hat{f}_{i}$ approximate $f_{i}$ up to an $L_{2}$-integrable error $\nu_{i}$, this will be achieved by forcing $u_{i}$ to track $\hat{f}_{i}$ instead of $f_{i}$ while attenuating the errors $\nu_{i}$ along with the system disturbances $w$, $v_{i}$, $v_{ij}$.

To generate suitable control inputs, each node observer will be augmented with an output feedback controller

[TABLE]

The second last term in (9) captures the situation where the attacker interferes with the operation of both the state observer and the defence layer at the misappropriated node, and the last term is included to compensate this interference, in the same manner as this is done for the main observer. The matrices $F_{c,i}$ are selected by the defender. Similar to the matrices $F_{i}$, they describe the attack pattern anticipated by the defender.

The inputs to the controller (9) are the innovation signals

[TABLE]

which capture the new information contained in the local measurements and obtained through communications, respectively. These signals are readily available at node $i$ and will be used for both estimating the plant and detecting and compensating biasing attacks. Also, controllers (9) will be allowed to communicate, and the signals

[TABLE]

describe the information received from the neighbors’ controllers through imperfect channels containing disturbances $v_{c,ij}\in L_{2}[0,\infty)$. For simplicity, we assume that these communications replicate the topology of the original network $\mathbf{G}$, because in practice the same channels will likely be used to transmit both $c_{ij}$ and $\eta_{ij}$.

The problem of resilient estimation is now formally stated. Let $e_{i}(t)=x(t)-\hat{x}_{i}(t)$ be the state estimation error of the observer (8) at node $i$, and define the vector of observer errors $\mathbf{e}=[e_{1}^{\prime}~{}\ldots~{}e_{N}^{\prime}]^{\prime}$.

Definition 2

The problem of resilient estimation in this paper is concerned with constructing a network of controlled observers (8), (9), to achieve the following:

1. (i)

In the absence of disturbances and when the system is not under attack, $e_{i}$ and $u_{i}$ must converge to 0 exponentially as $t\to\infty$ at every node $i$. 2. (ii)

In the presence of uncertainties and/or when the system is subjected to an attack of class $\mathscr{F}_{a}$, the network of controlled observers (8), (9) must ensure that $\int_{0}^{\infty}\|f_{i}-u_{i}\|^{2}dt<\infty$ $\forall i=1,\ldots,N$, and also $\int_{0}^{\infty}\mathbf{e}^{\prime}P\mathbf{e}dt<\infty$, Here $P=P^{\prime}\geq 0$ is a given $nN\times nN$ matrix.

We will subsequently show that when $f_{i}=0$ and the node $i$ is not under attack, then the signals $u_{i}(t)$ vanish asymptotically even in the presence of disturbances. Thus, the asymptotic behavior of the control signals $u_{i}$ at the compromised nodes differs from the behavior of similar signals at healthy nodes, which makes the signals $u_{i}$ suitable to be used as the attack indicators. Also according to (ii), in the event of attack, the estimates produced by the network of controlled observers (8), (9) will remain unbiased, up to an $L_{2}$ integrable weighted error $P^{1/2}\mathbf{e}$. We will subsequently guarantee a certain level of disturbance attenuation with respect to this weighted error.

III Design of a resilient distributed observer

III-A Observer error dynamics

Consider the dynamics of the estimation errors of the controlled observers (8),

[TABLE]

Also, for every node $i$, consider a class of admissible biasing inputs $\mathscr{F}_{a}$, of dimension $n_{f_{i}}$. According to Lemma 1, a proper $n_{f_{i}}\times n_{f_{i}}$ transfer function $G_{i}(s)$ can be associated with each class of admissible biasing inputs. Let $\hat{f}_{i}(t)$, $\nu_{i}(t)$ denote the corresponding approximation signal defined by (7) and the corresponding approximation error $\nu_{i}(t)=\hat{f}_{i}(t)-f_{i}(t)$. From (7), $\nu_{i}$ and $\hat{f}_{i}$ are related as $\hat{f}_{i}=-\frac{1}{s}G_{i}(s)\nu_{i}$. Consider the minimal realization of the transfer function $-\frac{1}{s}G_{i}(s)$:

[TABLE]

Next, let us substitute $f_{i}=\Upsilon_{i}\epsilon_{i}-\nu_{i}$ into equation (12) and combine the dynamics of the systems (12) and (13) into a system with $(e_{i}^{\prime},\epsilon_{i}^{\prime})^{\prime}$ as a state vector,

[TABLE]

The system (14) is driven by the bounded energy disturbances $w$, $v_{i}$, $v_{ij}$ and the error signals $\nu_{i}$ which also have bounded energy, according to Lemma 1. Each such system is controlled via the control input $u_{i}$. The innovation signals (10) can be written in terms of the variables of the systems (14) as

[TABLE]

Therefore, they can be regarded as outputs of the interconnected large-scale uncertain system comprised of systems (14). The attack tracking error $f_{i}-u_{i}$ can also be expressed in terms of this system variables,

[TABLE]

It will be regarded as the system performance output. This allows to carry out the derivation of the observers (8) and controllers of the form (9) within the $H_{\infty}$ disturbance attenuation framework. This approach will be pursued in the remainder of the paper.

III-B Feedback attack detector and compensator

Since the biasing inputs $f_{i}$ are not available directly, we first introduce a distributed filter to estimate the state of the extended system (14) from the information available at the nodes, namely from the outputs (10) and interconnections $\eta_{ij}$:

[TABLE]

Each filter (III-B) has the desired form of the system (9); i.e., $\chi_{i}=[\hat{e}_{i}^{\prime},\hat{\epsilon}_{i}^{\prime}]^{\prime}$. To complete the similarity with (9), define the interconnections (11) between the filters (III-B) and their outputs $u_{i}$ as follows:

[TABLE]

i.e., $W_{c,ij}=[W_{ij}~{}0]$, $C_{c,i}=[0~{}\Upsilon_{i}]$.

The distributed filter comprised of the systems (III-B)–(19) will be used to generate control inputs for the underlying observer (8). As explained previously, it accounts for the possibility that the adversary may launch a biasing attack on the controller, along with attacking the original observer. Also, the feedback is included to compensate for these attacks.

Remark 3

The use of feedback for compensating biasing attack inputs is the main distinction between the systems (III-B)–(19) and similar filters proposed in [6, 25] for detecting biasing attacks. Thus, the role of the systems (III-B)–(19) in this paper shifts from signaling biasing attacks to countering them.

III-C The system design procedure

Consider the errors of the filters (III-B), $z_{i}=e_{i}-\hat{e}_{i}$, $\delta_{i}=\epsilon_{i}-\hat{\epsilon}_{i}$. It follows from (14) and (16)–(19) that

[TABLE]

Here we have combined all channel disturbances into the vectors $\hat{v}_{ij}=[v_{ij}^{\prime}~{}v_{c,ij}^{\prime}]^{\prime}$ and have also used the following notation:

[TABLE]

The signal $\varphi_{i}$ defined in (16) will serve as a performance output for the system (20), since $\varphi_{i}=\Upsilon_{i}\delta_{i}-\nu_{i}$.

This notation reveals that the disturbance attenuation properties of the system comprised of systems (20) can be analyzed separately from the underlying resilient estimation problem. This leads us to propose the procedure for obtaining the coefficients for the observers (8) and the controllers (III-B)–(19):

First, the coefficients $\hat{L}_{i}^{r}(t)$, $\hat{K}_{ij}^{r}(t)$, $\check{L}_{i}^{r}(t)$, $\check{K}_{ij}^{r}(t)$ for each system (20) will be derived. The aim of this derivation is to guarantee that the filters (III-B)–(19) are able to track the signals $\hat{f}_{i}$. The following specific properties are sought:

1. (i)

When $w=0$, $v_{i}=0$, and $\hat{v}_{ij}=0$, $j\in\mathbf{V}_{i}$, $i=1,\ldots,N$, and the system is not under attack, i.e., when $f_{i}=0$ for all $i=1,\ldots,N$, all signals $z_{i}$, $\delta_{i}$ must decay to 0 exponentially. 2. (ii)

When at least one of the signals $w$, $v_{i}$, $\hat{v}_{ij}$ or $f_{i}$ is nonzero, all signals $z_{i}$, $\delta_{i}$ must vanish asymptotically, and the following disturbance attenuation property must hold

[TABLE]

Here $\gamma^{2}$ denotes the level of disturbance attenuation which reflects a desired tracking accuracy. 2. 2.

Next, the coefficients $L_{i}^{r}(t)$, $K_{ij}^{r}(t)$ for the controlled distributed observer (8) will be computed. With the parameters $\hat{L}_{i}^{r}(t)$, $\hat{K}_{ij}^{r}(t)$, $\check{L}_{i}^{r}(t)$, $\check{K}_{ij}^{r}(t)$ obtained in the previous step, it follows from (22) that the tracking error $\varphi_{i}=\Upsilon_{i}\delta_{i}-\nu_{i}$ is $L_{2}$ integrable for every attack input $f_{i}\in\mathscr{F}_{a}$, therefore the coefficients $L_{i}^{r}(t)$, $K_{ij}^{r}(t)$ will be obtained so that the observer (8) attenuates this additional disturbance, along with $w$, $v_{i}$, $\hat{v}_{ij}$. Essentially, we redesign the original unbiased observers (4) to endow them with robustness against attack tracking errors. This additional robustness property is needed to ensure the observers can withstand the transients generated by the attack canceling controllers. 3. 3.

Finally, the remaining coefficients $\bar{L}_{i}^{r}(t)$, $\bar{K}_{ij}^{r}(t)$ of the attack detector (III-B) will be obtained from (21), using the values of $\hat{L}_{i}^{r}(t)$, $\hat{K}_{ij}^{r}(t)$ and $L_{i}^{r}(t)$, $K_{ij}^{r}(t)$ obtained at the previous steps.

To implement this procedure, we employ a decentralized observer synthesis technique from [29]. It has an advantage in that it the parameters of the nodes observers are computed in a decentralized fashion. This feature is particularly attractive in the presence of adversaries, since each node observer can be computed on-line independently from other nodes, after a certain initial centralized setup.

The following technical result is adapted from [29]. Consider an interconnected system comprised of the systems

[TABLE]

subject to $L_{2}$-integrable disturbances $\tilde{w}_{i}$, $v_{i}$, $\tilde{v}_{ij}$. The matrix-valued functions $\tilde{A}_{i}$, $\tilde{C}_{i}$, and $\tilde{B}_{i}$ are bounded on $[0,\infty)$.

Let $\tilde{X}_{i}$, $\tilde{R}_{i}$, be positive definite symmetric $n\times n$ matrices, and $\tilde{Z}_{ij}$, $j\in\mathbf{V}_{i}$, $i=1,\ldots,N$, be positive definite symmetric $p_{ij}\times p_{ij}$ matrices. Also, define the following matrices: $\tilde{U}_{ij}=\tilde{H}_{ij}\tilde{H}_{ij}^{\prime}+\tilde{Z}_{ij}>0$,

[TABLE]

Let $\tilde{R}\triangleq\mbox{diag}[\tilde{R}_{1},\ldots,\tilde{R}_{N}]$, $\tilde{\Delta}\triangleq\mbox{diag}[\tilde{\Delta}_{1},\ldots,\tilde{\Delta}_{N}]$ and $\tilde{\Phi}\triangleq[\tilde{\Phi}_{ij}]_{i,j=1,\ldots,N}$. Next, consider $N$ differential Riccati equations:

[TABLE]

Each equation (25) only depends on the parameters associated with node $i$, and can be solved by node $i$ without interacting with its neighbors.

Lemma 2** (cf. [29])**

For a given a positive semidefinite $nN\times nN$ matrix $\tilde{P}=\tilde{P}^{\prime}$ and a constant $\tilde{\gamma}>0$, suppose the collection of matrices $\tilde{X}_{i}$, $\tilde{R}_{i}$ and $\tilde{Z}_{ij}$, $j\in\mathbf{V}_{i}$, $i=1,\ldots,N$, is found which satisfies the conditions:

1. (a)

It holds that

[TABLE] 2. (b)

The equations (25) have positive definite bounded solutions on $[0,\infty)$; i.e., $\alpha_{1i}I\leq\tilde{Y}_{i}(t)\leq\alpha_{2i}I$ ($\exists\alpha_{1i},\alpha_{2i}>0$).

Then, the interconnected system comprised of systems (III-C) equipped with the coefficients

[TABLE]

has the following properties:

1. (i)

In the absence of disturbances, $\tilde{e}_{i}$ vanish exponentially. 2. (ii)

In the presence of disturbances, each $\tilde{e}_{i}(t)$ vanishes asymptotically, and it holds that

[TABLE] 3. (iii)

In addition, each system (III-C) with the coefficients defined in (27) attenuates the local disturbances $\tilde{w}_{i}(t)$, $v_{i}$, $\tilde{v}_{ij}$ and its neighbors inputs $\tilde{e}_{j}$, $j\in\mathbf{V}_{i}$:

[TABLE]

The proof of the lemma is similar to the proof of Theorem 1 in [29]; it is omitted for brevity.

Remark 4

In Lemma 2, each node exercises its own $H_{\infty}$ filter, and uses its own differential Riccati equation (25) to compute the filter. The existence of this filter is equivalent to the Riccati equation (25) having a bounded solution. A standard proof of this fact [14] uses a certain detectability assumption. In our case, it requires the pair $(\tilde{A}(t),[\tilde{C}_{i}(t)^{\prime},\tilde{W}_{i1}^{\prime},\ldots,\tilde{W}_{ij_{d_{i}}}^{\prime}]^{\prime})$ to be detectable; $d_{i}$ is the cardinality of $\mathbf{V}_{i}$. However, this detectability property is only needed to prove the necessity of the Riccati equation condition. Since the conditions of Lemma 2 are sufficient, the above detectability assumption does not need to be stated explicitly; it is subsumed in the requirement of the lemma that the Riccati equation (25) must have a bounded solution. When this true, each local node error dynamics system is dissipative with the storage function $\tilde{V}_{i}=\tilde{e}_{i}^{\prime}\tilde{Y}_{i}^{-1}\tilde{e}_{i}$ and the supply rate $\tilde{\gamma}^{2}\left(\|\tilde{w}_{i}\|^{2}+\|v_{i}\|^{2}+\sum_{j\in\mathbf{V}_{i}}(\|\tilde{v}_{ij}\|^{2}+\|\tilde{e}_{j}\|^{2}_{Z_{ij}^{-1}})\right)-\tilde{e}_{i}^{\prime}\tilde{R}_{i}\tilde{e}_{i}.$ The inequality (26) balances these dissipativity characteristics of individual nodes in such a way that the error dynamics of the whole network are dissipative, with the storage function $\tilde{V}=\sum_{i=1}^{N}\tilde{e}_{i}^{\prime}\tilde{Y}_{i}^{-1}\tilde{e}_{i}$ and the supply rate $\tilde{\gamma}^{2}\sum_{i=1}^{N}(\tilde{w}_{i}\|^{2}+\|v_{i}\|^{2}+\sum_{j\in\mathbf{V}_{i}}\|\tilde{v}_{ij}\|^{2})-\tilde{e}^{\prime}\tilde{P}\tilde{e};$ also, see [29].

Lemma 2 will play an instrumental role in the procedure for the design of resilient filters proposed in this paper. We are now in a position to describe this procedure in detail.

Step 1. Stabilization of the detector error dynamics

(20) via output injection

Introduce the following notation:

[TABLE]

Also, consider positive definite $(n+n_{f_{i}})\times(n+n_{f_{i}})$ block-diagonal matrices $\mathbf{X}_{i}$, $i=1\ldots,N$, partitioned as \mathbf{X}_{i}=\left[\begin{array}[]{cc}X_{i}&0\\ 0&X_{i,0}\end{array}\right], where $X_{i}=X_{i}^{\prime}>0$, $X_{i,0}^{\prime}=X_{i,0}>0$ are respectively $n\times n$ and $n_{f_{i}}\times n_{f_{i}}$ matrices. In addition, consider the matrices $\Phi$ and $\Delta$ of the form (24), (23):

[TABLE]

where $Z_{ij}^{\prime}=Z_{ij}$ and $U_{ij}=\hat{H}_{ij}\hat{H}_{ij}^{\prime}+Z_{ij}$.

Theorem 1

Let a constant $\gamma>0$ and symmetric matrices $R_{i}>0$, $\check{R}_{i}>0$, $Z_{ij}>0$ of dimensions $n\times n$, $n_{f_{i}}\times n_{f_{i}}$, $p_{ij}\times p_{ij}$, respectively, $j\in\mathbf{V}_{i}$, $i=1,\ldots N$, be selected so that the following conditions hold:

1. (a)

It holds that

[TABLE]

where $R=\mathrm{diag}[R_{1},\ldots,R_{N}]$. 2. (b)

Each differential Riccati equation

[TABLE]

with \mathbf{R}_{i}\triangleq\left[\begin{array}[]{cc}R_{i}&0\\ 0&\check{R}_{i}\end{array}\right], has a positive definite symmetric bounded solution $\mathbf{Y}_{i}(t)$ on the interval $[0,\infty)$.

Then the system comprised of systems (20) with the coefficients $\hat{L}_{i}^{r}$, $\hat{K}_{ij}^{r}$, $\check{L}_{i}^{r}$, $\check{K}_{ij}^{r}$, obtained by partitioning the matrices

[TABLE]

according to (42), has the following properties:

1. (i)

In the absence of disturbances and attacks on the controlled observers (8), (III-B), $z_{i}(t)$, $\delta_{i}(t)$ vanish exponentially as $t\to\infty$, for all $i=1,\ldots,N$. 2. (ii)

When the network of controlled observers (8), (III-B) is subjected to disturbances and/or admissible biasing attacks of class $\mathscr{F}_{a}$, all $z_{i}(t)$, $\delta_{i}(t)$ decay asymptotically to 0 as $t\to\infty$, and (22) holds.

*Proof: * Using the notation (42) and letting $\mu_{i}=[z_{i}^{\prime}~{}\delta_{i}^{\prime}]^{\prime}$, $\mathbf{w}_{i}=[w^{\prime}~{}\nu_{i}^{\prime}]^{\prime}$, the system (20) can be written as

[TABLE]

This system is precisely of the form of the system (III-C) considered in Lemma 2. Note that in the absence of disturbances and attacks, $\nu_{i}=0$ and $\mathbf{w}_{i}=0$; see (7). Therefore, claims (i) and (ii) of the theorem can be inferred from the corresponding claims of Lemma 2. For this, we need to validate the conditions of that lemma.

To show that condition (a) of Lemma 2 is satisfied, consider the block-diagonal matrix $\tilde{P}$ composed of $N$ diagonal $(n+n_{f_{i}})\times(n+n_{f_{i}})$ blocks \tilde{P}_{i}=\left[\begin{array}[]{cc}0&0\\ 0&\Upsilon_{i}^{\prime}\Upsilon_{i}\end{array}\right]; i.e., $\tilde{P}=\mathrm{diag}[\tilde{P}_{1},\ldots,\tilde{P}_{N}]$. Let $\tilde{W}_{ij}=\mathbf{W}_{ij}$. Then from the definition of $\mathbf{W}_{ij}$, the matrices $\tilde{\Delta}_{i}$, $\tilde{\Phi}_{ij}$ in (24), (23) are \tilde{\Delta}_{i}=\left[\begin{array}[]{cc}\Delta_{i}&0\\ 0&0\end{array}\right], \tilde{\Phi}_{ij}=\left[\begin{array}[]{cc}\Phi_{ij}&0\\ 0&0\end{array}\right]. The satisfaction of condition (26) readily follows from (45) if we let $\tilde{R}_{i}=\mathbf{R}_{i}$. Condition (b) of Lemma 2 trivially follows from condition (b) of the theorem if we let $\tilde{A}_{i}=\mathbf{A}_{i}$, $\tilde{C}_{i}=\mathbf{C}_{i}$, $\tilde{X}_{i}=\mathbf{X}_{i}$. $\Box$

According to Theorem 1, each node can compute the matrices $\mathbf{L}_{i}^{r}$, $\mathbf{K}_{ij}^{r}$ and their components $\hat{L}_{i}^{r}$, $\hat{K}_{ij}^{r}$, $\check{L}_{i}^{r}$, $\check{K}_{ij}^{r}$ on-line, by solving the respective Riccati equations (b). This allows to compute the coefficients of the observer (8) and the controller (III-B) in real time. The nodes do not need to communicate to solve these Riccati equations.

Even though statement (iii) of Lemma 2 is not used explicitly in the proof of Theorem 1, together with the second inequality (45) $\check{R}_{i}>\Upsilon_{i}^{\prime}\Upsilon_{i}$ it yields

[TABLE]

This inequality characterizes the capacity of the attack detector at node $i$ to attenuate disturbances as well as the impact of the neighbours’ errors on the detector $i$’s accuracy. In addition, the first inequality (45), $R+\gamma^{2}(\Phi+\Phi^{\prime}-\Delta)>0$, ensures that the interconnections between the detector nodes are balanced so that the overall system is able to absorb and dissipate the impact of these errors.

Also note that the first inequality (45) only involves constant parameters of the interconnections between the systems (III-B). It does not involve parameters of the plant and the sensors, while the second inequality in (45) only involves the output matrices of the minimal realization of $-\frac{1}{s}G_{i}(s)$ in (13). This allows to solve the inequality (45) off-line in advance, and then use its solutions $R_{i}$, $\check{R}_{i}$, $Z_{ij}$ in (b). Even if some of the sensors at node $i$ have failed, and the dimension of the local sensor matrices $C_{i}$, $D_{i}$ has changed as a result of this failure, one can continue using the same matrices $R_{i}$, $\check{R}_{i}$, $Z_{ij}$. The Riccati equation (b) at that node will have to be updated to include the modified $C_{i}$, $D_{i}$. However, the computation of the filter gains at other nodes will not be affected. This is a significant advantage, in comparison with some existing distributed estimation techniques which require that the entire network of observers must be recomputed should one of the sensors have failed. To use this advantage, the matrix inequality (45) must be solved centrally to assign each node with suitable matrices $R_{i}$, $\check{R}_{i}$333An obvious solution to (45) is $R_{i}=(\gamma^{2}\lambda_{\max}(\Phi+\Phi^{\prime}-\Delta)+\alpha)I$ and $\check{R}_{i}=(\lambda_{\max}(\Upsilon_{i}^{\prime}\Upsilon_{i})+\alpha)I$, where $\alpha>0$ is a constant and $\lambda_{\max}(\cdot)$ is the largest eigenvalue of a matrix. However not every solution of (45) is suitable. As noted previously, the matrices $R_{i}$ and $\check{R}_{i}$ must also be selected so that the differential Riccati equations (b) have bounded solutions.. This needs to be done only once at time $t=0$; this step can be regarded as an initialization of the algorithm. Once the matrices $Z_{ij}$ are selected, the inequality (45) becomes a linear matrix inequality with respect to $R_{i}$, $i=1,\ldots,N$ and $\gamma^{2}$. It can be solved numerically using the existing software. This feature facilitates tuning the local and global performance of the proposed resilient filter.

Step 2. Design of the resilient distributed observer

(8)

From (8), it is clear that the signals $\varphi_{i}$ in (16) affect the accuracy of the proposed resilient observer (8). However, under conditions of Theorem 1, $\tilde{\nu}_{i}=\Upsilon_{i}\delta_{i}$ is $L_{2}$-integrable. Also, $\nu_{i}$ is $L_{2}$-integrable, according to Definition 1. Therefore, the effect of $\varphi_{i}$ can be attenuated, along with the effects of $L_{2}$-integrable disturbances present in the system. The coefficients $L_{i}^{r}$, $K_{ij}^{r}$ of the observer (8) which accomplish this task can also be computed using Lemma 2. To derive these coefficients, let us re-write the error dynamics (12) of the observer (8) as

[TABLE]

here we used the notation B_{1,i}(t)=\left[\begin{array}[]{ccc}B(t)&-F_{i}\end{array}\right], w_{i}=\left[\begin{array}[]{ccc}w^{\prime}&\varphi_{i}^{\prime}\end{array}\right]^{\prime}. Then Lemma 2 can be applied to the interconnection of systems (50). As before, consider symmetric matrices $\bar{Z}_{ij}>0$, $\bar{U}_{ij}=H_{ij}H_{ij}^{\prime}+\bar{Z}_{ij}>0$, $j\in\mathbf{V}_{i}$, $i=1,\ldots N$, and define the matrices $\bar{\Phi}=\left[\bar{\Phi}_{ij}\right]_{i,j=1}^{N}$, $\bar{\Delta}=\mathrm{diag}[\bar{\Delta}_{1},\ldots,\bar{\Delta}_{N}]$ of the form (24), (23):

[TABLE]

Theorem 2

Suppose condition of Theorem 1 are satisfied. Let a symmetric positive semidefinite $Nn\times Nn$ matrix $P=P^{\prime}$ and a constant $\bar{\gamma}>0$ be such that there exit symmetric matrices $\bar{R}_{i}>0$, $\bar{X}_{i}>0$, $\bar{Z}_{ij}>0$, $j\in\mathbf{V}_{i}$, $i=1,\ldots N$, such that

1. (a)

The following matrix inequality is satisfied

[TABLE]

where $\bar{R}=\mathrm{diag}[\bar{R}_{1},\ldots,\bar{R}_{N}]$; 2. (b)

Each differential Riccati equation

[TABLE]

has a positive definite symmetric bounded solution $\bar{Y}_{i}(t)$ on the interval $[0,\infty)$.

Then the network of systems (12), with the coefficients $L_{i}^{r}$, $K_{ij}^{r}$, obtained as

[TABLE]

has the following properties:

1. (i)

When $w=0$, $v_{i}=0$, and $v_{ij}=0$ for all $j\in\mathbf{V}_{i}$, $i\in\mathbf{V}$, and the system is not under attack, i.e., $f_{i}=0$ for all $i\in\mathbf{V}$, every error $e_{i}(t)$ of the distributed observer (12) vanishes exponentially. 2. (ii)

In the presence of disturbances and/or when the system is subjected to a biasing attack of class $\mathscr{F}_{a}$, $e_{i}(t)$ converge to 0 asymptotically. Furthermore, it holds that

[TABLE]

*Proof: * First consider the case where $w_{i}=0$, $v_{i}=0$, $v_{ij}=0$, and $f_{i}=0$. Note that the latter assumption implies $\nu_{i}=0$ due to (7). Therefore, in this case $w_{i}$ reduces to $w_{i}=[0^{\prime}~{}\tilde{\nu}_{i}^{\prime}]^{\prime}$. Let $\Psi(t,\tau)$ be the state transition matrix of the large scale system comprised of subsystems (50). Using Lemma 2, we obtain that $\|\Psi(t,t_{0})\|\leq\beta_{0}e^{-\lambda_{0}(t-t_{0})}$. Also, in Theorem 1 we have established that the signal $\tilde{\nu}(t)$ decays exponentially to 0 when $w_{i}=0$, $v_{i}=0$, $v_{ij}=0$, and $f_{i}=0$ for all $i$, $j\in\mathbf{V}_{i}$. Together these observations imply that the response of the system to $\tilde{\nu}=[\tilde{\nu}_{1}^{\prime}\ldots\tilde{\nu}_{N}^{\prime}]^{\prime}$ vanishes exponentially, i.e., claim (i) of the theorem holds. Statement (ii) follows directly from Lemma 2. $\Box$

Step 3. The complete controlled observer

We now complete the last step of the design procedure and obtain the remaining coefficients $\bar{L}_{i}^{r}$, $\bar{K}_{ij}^{r}$ for the controller (III-B) from (21).

Theorem 3

Suppose the conditions of Theorems 1 and 2 are satisfied. Let the coefficients $\hat{L}_{i}^{r}$, $\hat{K}_{ij}^{r}$, $\check{L}_{i}^{r}$, $\check{K}_{ij}^{r}$ of the attack detecting controllers (III-B) be obtained from (1), and let $L_{i}^{r}$, $K_{ij}^{r}$ be the matrices defined in (2). Define $\bar{L}_{i}^{r}$, $\bar{K}_{ij}^{r}$ as

[TABLE]

Then, the network of state observers (8), augmented with the network of attack detectors (III-B)–(19) produces state estimates $\hat{x}_{i}$ which have the following convergence properties:

1. (i)

In the absence of disturbances and biasing attacks, $\|x-\hat{x}_{i}\|\to 0$ exponentially as $t\to\infty$. 2. (ii)

When the plant and/or the network is subject to $L_{2}$-integrable disturbances and/or admissible biasing attacks, the estimates $\hat{x}_{i}$ converge to $x$ asymptotically as $t\to\infty$, and the resilient performance of this observer network is characterized by the condition

[TABLE]

here $P$ is the matrix from condition (53) of Theorem 2. 3. (iii)

Also, the outputs $u_{i}$ of the distributed controllers (III-B) have the following properties:

1. (a)

If the node $i$ is not under attack, the signal $u_{i}$ generated by the controller (III-B) at this node vanishes asymptotically, even in the presence of disturbances. 2. (b)

If node $i$ is subjected to a biasing attack $f_{i}$ of the class $\mathscr{F}_{a}$, then $\int_{0}^{\infty}\|f_{i}-u_{i}\|^{2}dt<\infty$.

*Proof: * Statements (i) and (ii) of the theorem follow from Theorems 1 and 2. The inequality (58) is proved by combining (22) and (56) using the inequality $\|\varphi_{i}\|^{2}\leq 2(\|\nu_{i}\|^{2}+\|\Upsilon_{i}\delta_{i}\|^{2})$.

To prove statement (iii)a, we note that if node $i$ is not under attack, we have $\hat{f}_{i}(t)=0$ and $\nu_{i}(t)=0$, therefore $\epsilon_{i}(t)=0$. Further, we have established in Theorem 1 that in the presence of disturbances, $\delta_{i}(t)\to 0$ as $t\to\infty$ asymptotically. Therefore in this case, $u_{i}=\Upsilon_{i}\hat{\epsilon}_{i}=-\Upsilon_{i}\delta_{i}$ vanishes to $f_{i}=0$ asymptotically as $t\to\infty$.

When the observer at node $i$ is subjected to a biasing attack and $f_{i}\neq 0$, then $\nu_{i}\neq 0$, however $\nu_{i}\in L_{2}[0,\infty)$ according to Definition 1. Also, it has been established in Theorem 1 that $\Upsilon_{i}\delta_{i}\in L_{2}[0,\infty)$. Thus, $\int_{0}^{\infty}\|f_{i}-u_{i}\|^{2}dt<\infty$, i.e., (iii)b holds. $\Box$

III-D Performance optimization over communication graphs: an LTI case

As mentioned, the proposed procedure allows each node to compute the coefficients $L_{i}^{r}$, $K_{ij}^{r}$ of its observers and the parameters $\bar{L}_{i}^{r}$, $\bar{K}_{ij}^{r}$, $\check{L}_{i}^{r}$, $\check{K}_{ij}^{r}$ of its controllers without communicating with other nodes. This is because each node solves its Riccati equations (b) and (b) locally. For this, each node must be assigned with constants $\gamma^{2}$, $\bar{\gamma}^{2}$ and matrices $\mathbf{R}_{i}$, $\bar{R}_{i}$, $Z_{ij}$, $\bar{Z}_{ij}$. A deeper look into the matrix inequalities (45), (53) reveals that the selection of these matrices and constants is constrained by the network topology. The simplest way to see this is to restrict attention to case where the matrices $A$, $B$, $C_{i}$ and $D_{i}$ are constant, the communications between the network nodes are noise-free, and the node observers transmit their complete estimates $\hat{x}_{i}$. In this case, the communication model (3), (18) simplifies to $c_{ij}=\hat{x}_{j}$, $\eta_{ij}=\hat{e}_{j}$ for all $j\in\mathbf{V}_{i}$, $i=1,\ldots,N$; i.e., $W_{ij}=I$, $\hat{H}_{ij}=0$. Also, let $Z_{ij}=\bar{Z}_{ij}=I$. With these parameters, we have

[TABLE]

where $d_{i}$, $\mathscr{L}$ denote the in-degree of node $i$ and the Laplace matrix of the network graph, respectively. Let $\mathscr{D}=\mathrm{diag}[d_{1},\ldots,d_{N}]$ be the in-degree matrix of the network graph. Conditions (45), (53) can then be explicitly expressed in terms of the Laplace and in-degree matrices of the graph $\mathbf{G}$:

[TABLE]

These constraints can be used to select the communication topology which endows the controlled distributed observer (8), (III-B)–(19) with an optimized estimation accuracy or an optimized biasing attack detection performance.

Optimization of the attack detection performance

Let $\{\mathbf{G}_{m},m=1,\ldots,M\}$ be a given collection of admissible communication graphs. From now, we will use the subscript m to denote quantities corresponding to the graph $\mathbf{G}_{m}$ from this set; i.e., $\mathscr{L}_{m}$ will denote the Laplace matrix of $\mathbf{G}_{m}$, etc. The optimized disturbance attenuation among attack detectors (III-B)–(19) interconnected over the admissible graphs $\mathbf{G}_{m}$ is expressed as

[TABLE]

where the infimum is taken over the set of matrices $\bar{R}_{i}$, $R_{i}$, $\check{R}_{i}$, $i=1,\ldots,N$, and constants $\gamma^{2}$, $\bar{\gamma}^{2}$ which satisfy the conditions of Theorem 3 stated for the graph $\mathbf{G}_{m}$. In the $H_{\infty}$ theory, a filter delivering optimal disturbance attenuation is usually difficult to compute, and a standard practice is to use suboptimal $H_{\infty}$ filters [2, 31]. Therefore, we seek to find a resilient distributed observer with an optimized suboptimal attack detection performance.

Initially, let us restrict attention to the simplified time-invariant case. In addition to the assumptions made above, we assume that $(A,B)$ is stabilizable. With this additional assumption,  [29, Theorem 2] states that the network of observers connected over an admissible graph $\mathbf{G}_{m}$ satisfies the conditions of Theorem 3 provided the following Linear Matrix Inequalities (LMIs) in the variables $(\{\bar{R}_{i},R_{i},\check{R}_{i},\mathbf{Q}_{i},\bar{Q}_{i}\}_{i=1}^{M},\gamma^{2},\bar{\gamma}^{2})$ are feasible:

[TABLE]

where $d_{i,m}$ is the cardinality of $i$’s neighborhood $\mathbf{V}_{i}^{m}$ in the graph $\mathbf{G}_{m}$. Indeed, the first two lines are the inequalities (III-D) particularized for the graph $\mathbf{G}_{m}$, and the remaining conditions guarantee that the corresponding differential Riccati equations (b), (b) do not have a conjugate point [2].

The following sequential optimization procedure provides a tractable upper bound on (61) which yields a desired suboptimal attack detector. Let $\gamma_{0}^{\circ}=+\infty$, and for every $m=1,\ldots,M$, define $\gamma_{m}^{2}=\inf\gamma^{2}$ where the infimum is taken over the feasible set of the LMIs (74). If these LMIs are not feasible for a particular $m$, we set $\gamma_{m}^{2}=+\infty$. Then let

[TABLE]

Since the sequence $\{\gamma^{\circ}_{m}\}$ is monotone decreasing, the recursion (75) terminates at the graph $\mathbf{G}_{m}$ which attains $\min_{\mathbf{G}_{m}}(\gamma_{m}^{\circ})^{2}=\min_{\mathbf{G}_{m}}\gamma_{m}^{2}$. Also, since feasibility of the LMIs (74) is only a sufficient condition for the conditions of Theorem 3 to be satisfied, then for every $m$, $\gamma_{m}^{2}$ is greater than or equal to the inner infimum value in (61). Therefore, the graph $\mathbf{G}_{m}$ which attains $\min_{\mathbf{G}_{m}}(\gamma_{m}^{\circ})^{2}$ is the most favorable graph among the candidate graphs $\{\mathbf{G}_{m}\}$, from the view point of suboptimal attack detection performance.

Optimization of resilient estimation performance

The foregoing procedure is readily modified to obtain a network topology yielding a suboptimal level of disturbance attenuation $\bar{\gamma}^{2}$ in (56). Define $\bar{\gamma}_{m}^{2}=\inf\bar{\gamma}^{2}$ where the infimum is taken over the feasible set of the LMIs (74). Then the recursion

[TABLE]

terminates at the graph which attains the desired suboptimal network configuration.

General time-invariant case

The optimization procedures proposed above are applicable in the general case where the communications between the nodes are described by the general model (3), (18). We still assume that the matrices $Z_{ij}$, $\bar{Z}_{ij}$ are given and that the pair $(A,B)$ is stabilizable. Then $\gamma^{2}$ and $\bar{\gamma}^{2}$ can be optimized subject to a series of linear matrix inequalities, similar to (74). In lieu of the first three inequalities in (74), this series of LMIs includes the original inequalities (45) and (53) involving the appropriately defined matrices $\Phi_{m}+\Phi_{m}-\Delta_{m}$ and $\bar{\Phi}_{m}+\bar{\Phi}_{m}-\bar{\Delta}_{m}$ which are associated with the admissible candidate network topology $\mathbf{G}_{m}$. Also, the remaining matrix inequalities in (74) are replaced with the corresponding more general LMIs reflecting the general structure of communications

[TABLE]

The optimization over the set of graphs can then be performed recursively, in the same way as in the previous cases.

Suitable candidate graphs

Our final remarks are concerned with selecting suitable candidate graphs $\mathbf{G}_{m}$ for optimization.

The feasibility of the proposed optimization procedure relies on $H_{\infty}$ stabilizability of the large-scale interconnected systems (20) and (50) via output injection. In turn, this stabilizability property requires the corresponding noise-free large-scale systems to have a basic distributed detectability property; see [23] for a more detailed discussion. For observer networks with identical matrices $W_{ij}=W$, this property requires that the plant must be detectable from the combined outputs associated with each maximal subgraphs spanned by a tree [23] or with each strongly connected subgraph that does not have incoming edges [16, 12, 28]. In addition, within each such subgraph, every state of the plant must be either detectable from the measurements or observable through interconnections (or both)444Observability through interconnections holds trivially in [16, 28, 9] since these references assume $W=I$. In [12], the matrices $W_{ij}$ were selected to ensure that each node receives from its neighbors the part of the state vector undetectable from its local measurements.. Although we did not state these properties explicitly in this paper, they are necessary for stabilizability of the large-scale interconnected systems (20) and (50) via output injection. In particular, a necessary condition for detectability of biasing attacks by detectors of the form (III-B) obtained in [6] is based on these properties. Therefore, each candidate graph $\mathbf{G}_{m}$ must satisfy these necessary conditions for stablizability via output injection, at least when the system is time-invariant, and the matrices $W_{ij}$ are identical555The extension of the results of [23] for the case where $W_{ij}=W_{i}$ can be found in [24].. We are not aware of similar necessary conditions for distributed detectability of time-varying systems, and in this case the problem of characterizing distributed detectability as well as detectability of biasing attacks appears to remain open.

IV Illustrative example

The example is based on the example from [22], where a distributed observer was constructed for the system of the form (1), with $B=0.1I$ and

[TABLE]

The plant is observed by 6 sensors. Sensor $i$ measures the $i$-th and $(i+1)$-th coordinates of the state vector, with sensor 6 measuring the 6th and 1st coordinates; see [22] for the definitions of the matrices $C_{i}$. All 6 pairs $(A,C_{i})$ are not detectable in this example. Also, $D_{i}=0.01I$ $\forall i$. As in [22], suppose that the nodes broadcast the full vector $\hat{x}_{i}$, i.e., $W=I$, however the communications between the nodes are subject to disturbances, so we let $H_{ij}=H_{c,ij}=(0.1/\sqrt{2})\times[1~{}1~{}1~{}1~{}1~{}1]^{\prime}$.

The problem in [22] was to achieve a robust $H_{\infty}$ consensus performance of the observers, which corresponds to $P=(\mathscr{L}+\mathscr{L}_{\mathrm{T}})\otimes I$; here $\mathscr{L}$, $\mathscr{L}_{\mathrm{T}}$ are the Laplace matrices of $\mathbf{G}$ and its transpose graph, respectively. We now consider a resilient version of that problem. As in [6], suppose that $F_{i}=[1~{}1~{}1~{}1~{}1~{}1]^{\prime}$, $\hat{F}_{i}=0$ $\forall i$. Next we selected $G_{i}(s)=\frac{410}{s+40}$. Formally, the algorithm imposes mild requirements on these transfer functions — any proper rational transfer function of the form $\frac{N(s)}{D(s)}I$ which renders the transfer function $(sI+G_{i}(s))^{-1}G(s)$ stable can be selected as $G_{i}(s)$. Some additional considerations in regard to selecting these transfer functions are as follows. Firstly, it is reasonable to keep the energy in the approximation error $\nu_{i}$ to a minimum if possible, since it appears on the right hand side of the performance inequality (56). For this, the system (7) should be sufficiently fast and have no overshoots. Secondly, the differential Riccati equation (b) must admit a bounded solution with an acceptably small $\gamma$, to guarantee an acceptable robustness of attack detection; see (22). This can be accomplished by either simulating those equations offline or, in the time-invariant case, by testing that the third LMI condition (74) is feasible. In accordance with these recommendations, in this example we let $Z_{ij}=\bar{Z}_{ij}=0.01\times I$ and solved the LMI conditions (74) to obtain the smallest $\gamma^{2}$, $\bar{\gamma}^{2}$ for which the LMIs (74) were feasible, $\gamma^{2}=6.9113\times 10^{-3}$, $\bar{\gamma}^{2}=3.4511\times 10^{-2}$, and obtained the corresponding matrices $R_{i}$, $\check{R}_{i}$ and $\bar{R}_{i}$ from (74) to be used in the Riccati equations (b) and (b). To be consistent with the original example from [22], we consider only one network topology, hence we did not need to perform optimization over a collection of graphs.

Next, the plant and the resilient observers (8) endowed with the controllers (III-B)–(19) were jointly simulated. An attack input of amplitude 5 and duration 3 seconds was applied to the plant observer at node 2 at time $t=4$. No other disturbances were applied; this made the comparison between the errors of the original observer (4) from [22] and the errors of the observers (8) operating under the attack most vivid; see plots in Fig. 1. As can be seen from these plots, the observers from [22] designed without consideration for resilience were adversely affected by the attack. In contrast, the controlled observers (8) were able to successfully negate the biasing effect of the attack. Their errors have only short transients at the beginning and the end of the attack interval, which decay quite quickly; the decay rate was adjusted by choosing the transfer functions $G_{i}(s)$. Figure 2 shows the corresponding outputs $u_{i}$ of the controllers. One can see that the attack at node 2 has been detected successfully.

V Conclusion

The paper has proposed a novel class of controlled distributed observers for robust estimation of a linear plant, which are resilient to biasing misappropriation attacks. To counter the attacks, we introduce an auxiliary network of distributed output feedback controllers which provide an attack correcting action to the main observer. The filter components of these controllers can also serve as attack detectors. Our problem formulation is quite general in that it captures biasing attack scenarios which target both the observer and the attack detector at the compromised nodes.

The proposed design method is based on the methodology of distributed $H_{\infty}$ filtering; it allows to obtain observers which attenuate benign disturbances, while discovering and suppressing biasing inputs. This approach allows to consider the worst-case situation where every node of the observer network can be subjected to an attack. When the defender knows a priori that certain nodes are secure (e.g., as a result of a security audit), this information can be easily incorporated into the design procedure by assigning zero values to the corresponding attack input matrices $F_{i}$, $\check{F}_{i}$.

Another feature of our approach is that the nodes compute their observers and attack detectors independently from each other. This decentralization of computation enhances security of the network since the computation does not rely on potentially vulnerable communications.

VI Acknowledgment

The author is grateful to the anonymous referees whose comments helped to improve the presentation of the paper.