Towards Distributed Accommodation of Covert Attacks in Interconnected Systems
aa r X i v : . [ ee ss . S Y ] J u l Towards Distributed Accommodation of CovertAttacks in Interconnected Systems
Angelo Barboni
Department of Electrical and Electronic Engineering,Imperial College London [email protected]
Thomas Parisini
Department of Electrical and Electronic Engineering,Imperial College London, UK, andKIOS Research and Innovation Centre of Excellence,University of Cyprus, andDepartment of Engineering and Architecture, University of Trieste, Italy [email protected]
Abstract
The problem of mitigating maliciously injected signals in interconnected systemsis dealt with in this paper. We consider the class of covert attacks, as they arestealthy and cannot be detected by conventional means in centralized settings.Distributed architectures can be leveraged for revealing such stealthy attacks byexploiting communication and local model knowledge. We show how such detec-tion schemes can be improved to estimate the action of an attacker and we proposean accommodation scheme in order to mitigate or neutralize abnormal behaviorof a system under attack.
Many systems of critical importance consist nowadays of tightly integrated physical and compu-tational components, which may perform control and safety-critical tasks with high reliability re-quirements. Additionally, such systems are often composed of several physically interconnectedsubsystems that exchange information over a network for a number of reasons, ranging from dataanalysis to design convenience, or simply because the physical system is itself geographically spreadover a large area. As a downside, however, these systems are potentially vulnerable to cyber-attackswhich may entail tangible consequences on the physical layer, if not disruption of the system it-self. As observed recently Lee et al. (2016); Sobczak (2019), attacks constitute a realistic threat,and being able to detect and counteract them to preserve some level of functionality is then of greatimportance. In fact, this problem has attracted the interest of the control community over the lastdecade; see for instance Cheng et al. (2017) and Dibaji et al. (2019) for recent surveys. However,in the majority of cases, the centralized scenario is considered, with only a few works tackling theissue from a distributed perspective Anguluri et al. (2019); Gallo et al. (2020).Compared to other types of attacks, covert attacks are a class of particularly dangerous attacks,which are undetectable by design in the centralized case Smith (2015). In Barboni et al. (2019),it was shown that a specific residual generation scheme allows to detect such attacks, while theyremain stealthy within the attacked subsystem. The distributed detection strategy is inspired bymodel-based fault detection (see Shames et al. (2011) and Boem et al. (2017) for instance), with
Preprint. Under review. igure 1: Diagram of a single subsystem’s architecture. Details about the accommodation architec-ture, with attack detection and accommodation measures.a novel design that accounts for the stealthiness of the attacks (which is not an issue for faults ingeneral).In this paper, we extend the detection architecture in Barboni et al. (2019) with the objective ofneutralizing (or at least mitigating) the attacker’s malicious effect on the system, that is to say we aimto design a control law that steers the system as close as possible to the desired equilibrium regardlessof how the attacker manipulates the control actions (input injection). To the best of the authors’knowledge, this is the first time that an active countermeasure methodology is proposed in the areaof control security, and even more in relation with distributed systems. However, accommodation offaults has received considerable attention from the control community (see Zhang and Jiang (2008)for a comprehensive review on the topic). One way to accomplish this relies on fault estimation inorder to cancel their unwanted effect on the system’s dynamics via a suitable change of the controlaction Polycarpou and Vemuri (1995); Blanke et al. (2006). This is effective in case of actuator ormatching faults, and since input-injection attacks satisfy these same conditions, we gather fromthis idea in order to compensate the attacks in an additive way. In dealing with this task from adistributed perspective, a number of issues may arise in case of sparse interconnections betweensubsystems. This fundamentally ties with the problem of (partial) input reconstruction Bejarano(2011), as discussed later in the paper. Additionally, results are hereby presented in discrete time, asopposed to the continuous-time case of Barboni et al. (2019).The paper is structured as follows: in Section 2, the problem is formulated and the attack model ispresented; Section 3 provides a short recap of useful results and equations for the detection strategy;in Section 4, the accommodation strategy is presented. Finally, in Section 5 an academic example isgiven to show the effectiveness of the proposed distributed accommodation strategy.
For an ordered index set I , and a family of matrices { M i ∈ R n × m , i ∈ I} , col i ∈I ( M i ) denotesthe vertical concatenation of said matrices. For brevity, if x ( k ) is the value of a vector signal x attime k , x + . = x ( k + 1) denotes the value at the next time step. Similarly, x − . = x ( k − denotesthe value at the previous time step. For a vector v ∈ R n , v [ m ] denotes its m -th component. For amatrix M ∈ R n × m , M † is its pseudo-inverse. Let R ⊂ X be vector spaces, then X / R denotesthe quotient space of X by R . We consider a linear time-invariant (LTI) system that can be partitioned into N subsystems, eachdenoted as SS i , i ∈ { , . . . , N } . For each subsystem, let N i denote the index set of neighbors of SS i . We model each subsystem as a discrete-time LTI system: S i : x + i = A i x i + B i ˜ u i + X j ∈N i A ij x j y i = C i x i , (1)2here x i ∈ R n i , ˜ u i ∈ R m i , and y i ∈ R p i are the local states, inputs, and outputs, respectively. x j , j ∈ N i are the neighbors’ states which enter the dynamics of SS i through the interconnectionmatrix A ij ∈ R n i × n j .Each subsystem is managed by a local unit LU i which contains a diagnoser implementing the pro-posed attack-detection strategy and a given controller C i . Additionally, these units are interconnectedalong the same topology of the physical interconnections. As shown on the left-hand side of Fig. 1,the attacker is represented by an interconnection block A i , which injects signals η i and γ i in thecontrol and measurement channels, respectively, according to: ˜ u i = u i + η i , ˜ y i = y i − γ i . Hence, LU i receives possibly attacked measurements ˜ y i , and yields a control action u i computedaccordingly; on the other hand, the plant SS i receives the counterfeit control action ˜ u i to whichcorresponds an actual output response y i .The attacker’s objective is to remain undetected while steering the system’s state to a trajectorydifferent from the nominal one, to its own advantage. A particular instance of attacks that are stealthyby design are covert attacks, firstly introduced in Smith (2015) and investigated in the distributedcase in the time domain in Barboni et al. (2019). Definition 1.
The attacker A i is covert if the outputs ˜ y i are indistinguishable from the nominalresponse y i . ⊳ To perform a covert attack, the attacker implements a model ˜ SS i given by ˜ S i : ( ˜ x + i = ˜ A i ˜ x i + ˜ B i η i γ i = ˜ C i ˜ x i , (2)which is used to compute a “canceling” signal γ i , for a prescribed input injection η i . Let k a be theattack onset instant and let the following assumption hold. Assumption 1.
The attacker has perfect knowledge of the subsystem model, i.e. ( ˜ A i , ˜ B i , ˜ C i ) =( A i , B i , C i ) . ⊳ It is shown in (Barboni et al., 2019, Proposition 1) that under Assumption 1, if ˜ x i ( k a ) = 0 , theattacker A i is covert at all times. Therefore, any residual generator exploiting only u i and ˜ y i cannotbe used for detection.We point out that Assumption 1, while being difficult to achieve, represents the worst-case scenarioof an omniscient attacker who is perfectly stealthy. Such a condition frames the detection problemas the most difficult, hence the derived results will also hold for easier cases.For the scope of the present work, the problem is not limited to attack detection – which has beenalready covered in the referenced works – but rather it focuses on the design of a control input u i which attenuates the attack’s effects. To the best of the authors’ knowledge, this is the first time thata solution to this problem in this distributed flavor is proposed. Due to the early stage nature of thisbranch of research, we consider the ideal case in order to obtain basic conditions under which theproposed strategy is effective, and ignore hereby other issues such as robustness to noise. For sake of completeness, we recap the detection architecture presented in Barboni et al. (2019), aswell as some important results that are needed for presentation. Extended results including disturbances and detection bounds have been presented in a journal articlecurrently under review.
3t a glance, the detection architecture comprises two observers – O di and O ci – and an alarm mech-anism that compares a specially constructed residual to a threshold. Observer O di is decentral-ized and computes an estimate ˆ x di of x i insensitive to the neighboring states x j , j ∈ N i ; O ci in-stead is distributed and accounts for the neighboring coupling by including communicated estimates ˆ x dj , j ∈ N i .Let us define the estimation errors, where we distinguish between the actual errors and the received (or attacked ) ones as follows: ǫ di . = x i − ˆ x di , ˜ ǫ di . = x i − ˜ x i − ˆ x di ,ǫ ci . = x i − ˆ x ci , ˜ ǫ ci . = x i − ˜ x i − ˆ x ci . (3)Note that the attacked errors are in fact those that a diagnoser can compute, as ˜ y i are the availablemeasurements. In fact, the actual errors are not available in any way, but they are still useful foranalysis and to quantify the attacker’s impact.By construction, the signal ˜ r ci = C i ˜ ǫ ci is sensitive to attacks in neighboring systems. Suppose thatSubsystem i is under attack, the detection logic is as follows. • Each LU j , j ∈ N i computes ˜ r ci and compares it to a threshold ¯ r i . An alarm δ j = 0 israised if k ˜ r ci k > ¯ r i . • δ i is broadcast to each neighbor; conversely, SS i receives δ j from all its neighbors. • If δ j = 0 , ∀ j ∈ N i , then LU i decides that SS i is under attack. Remark 1.
In Barboni et al. (2019), the alarm variable δ i was binary. In the present work, instead, δ i ∈ R n i , as it will take part into the attack accommodation algorithm, as shown in Section 4. ⊳ O d The decentralized observer consists of a discrete-time Unknown Input Observer (UIO): O d : ( z + i = F i z i + T i B i u i + ( K (1) i + K (2) i )˜ y i ˆ x di = z i + H i ˜ y i . (4)If the existence conditions in Chen et al. (1996) hold, (4) is an UIO for subsystem (1). As such, theerror dynamics is described by: ǫ d + i = F i ǫ di . (5) Result 1 (Barboni et al. (2019)) . Let S i be under attack, if the UIO conditions and Assumption 1are satisfied, the following equations hold. The error dynamic of the observer (4) is ǫ d + i = F i ǫ di + ( A i − F i )˜ x i + B i η i , (6) while the attacked estimation error defined in (3) is given by ˜ ǫ d + i = F i ˜ ǫ di . (7) (cid:3) A consequence of (7) is that the estimate does not converge to the actual state of the system, butrather to the difference x i − ˜ x i , as can be seen from (3). O c The distributed observer O ci relies on decentralized estimates received over a communication net-work, and its dynamics is defined as: O c : ˆ x c + i = A i ˆ x ci + B i u i + L i (˜ y i − C i ˆ x ci )+ X j ∈N i A ij ˆ x dj . (8)Let F ci = A i − L i C i , if both F i and F ci are stable, then the estimate ˆ x ci converges to the subsystem’sstate in attack-free conditions. In particular, the following result holds.4 esult 2 (Barboni et al. (2019)) . Let S i be under attack, if Assumption 1 is satisfied. The errordynamics of the observer (8) is ǫ c + i = ( A i − L i C i ) ǫ ci + B i η i + L i γ i + X j ∈N i A ij ǫ dj , where ǫ dj is given by (5) , (6) . Conversely, the attacked estimation error is given by ˜ ǫ c + i = ( A i − L i C i )˜ ǫ ci + X j ∈N i A ij ǫ dj . (9) (cid:3) Eq. (9) holds also when the system is not under attack. It can be seen that the received error (andhence the residual) depends on the actual neighbors’ decentralized errors ǫ dj . Since under attack ǫ dj evolves according to (6), it follows that under reachability conditions of the pairs ( F ci , A ij ) , for all i ∈ N j , the error ˜ ǫ ci does not converge to . This section is devoted to the design of a control action u i , which compensates for the effect of theattacker in an attacked subsystem SS i . This strategy is triggered after the attack has been success-fully detected and isolated. The accommodation strategy is based on the following observations: • Since the estimation errors converge in nominal conditions, the error ˜ ǫ cj , j ∈ N i can beused to define d j . = X l ∈N j A jl ǫ dl = ˜ ǫ c + j − F cj ˜ ǫ cj , (10)which we have written in forward form for the sake of convenience. The variable d j repre-sents the aggregate actual error of neighboring systems. • From (3) we have that: ǫ di = ˜ ǫ di + ˜ x i . Given that ˜ ǫ di converges to , the actual error ǫ di depends directly on the state of the at-tacker’s state ˜ x i , which can be seen as superimposed to the nominal system state x ni . In-deed, with initial condition ˜ x i ( k a ) = 0 , the dynamics of SS i can be decomposed as ( x ni + ˜ x i ) + = A i ( x ni + ˜ x i ) + B i ˜ u i + X j ∈N i A ij x j ,y i = C i x ni ,γ i = C i ˜ x i . In view of these observations, the developed strategy aims at constructing an estimate of the at-tacker’s state using neighboring errors. For this purpose, we redefine the alarm signal introduced inSection 3 using (10) as: δ j = ( d − j if k ˜ ǫ cj k > θ j , otherwise , for some suitable threshold θ j . Suppose that SS i is under attack; if d j = 0 , ∀ j ∈ N i , then the attackis detected in the same way as previously presented.A one-step lag estimate of ˆ˜ x − i of ˜ x i can be obtained by solving the following Least Squares (LS)problem: ˆ˜ x − i = arg min ξ N i X j ∈N i k δ j − A ji ξ k . (11)The solution to (11) is given by: ˆ˜ x − i = X j ∈N i A ⊤ ji A ji − X j ∈N i A ⊤ ji δ j . emark 2. With respect to just performing attack detection, the designed accommodation strategyrequires that the local diagnoser also knows the outbound interconnection matrices A ji . ⊳ Remark 3.
Let δ i . = col j ∈N i ( δ j ) and A i . = col j ∈N i ( A ji ) . Then problem (11) can be equivalentlyreformulated in matrix form as: ˆ˜ x − i = arg min ξ k δ i − A i ξ k , for which standard solution techniques can be applied. ⊳ Uniqueness of the estimate depends on well-known rank conditions on A i , which we refer to as the aggregate interconnection matrix . We defer discussion to the respective subsections.To further develop our analysis, consider (1) and a control action of the form u i = K i (ˆ x di + ˆ˜ x − i ) + X j ∈N i K ij ˆ x dj − ˆ η i . (12)Matrices K ij can be optimally chosen ˇSiljak (1978) to minimize the effects of neighbors on the localdynamics. Without loss of generality, since our analysis focuses on attack compensation on SS i , wecan assume exact “cancellation” of neighboring states. Let ǫ ai . = ˜ x i − ˆ˜ x − i and ǫ ηi . = η i − ˆ η i , we havethat: x + i = A i x i + B i K i (ˆ x di + ˆ˜ x − i ) + B i ( η i − ˆ η i )+ X j ∈N i ( A ij + B i K ij ) x j − X j ∈N i B i K ij ǫ dj = A i x i + B i K i ( x i − ˜ ǫ di − ˜ x i + ˆ˜ x − i ) + B i ǫ ηi + X j ∈N i ( A ij + B i K ij ) x j − X j ∈N i B i K ij ǫ dj = ( A i + B i K i ) x i + X j ∈N i ( A ij + B i K ij ) x j + B i ǫ ηi − B i K i ǫ ai − B i K i ˜ ǫ di − X j ∈N i B i K ij ǫ dj . (13)In (13), K i can be designed to achieve asymptotic closed-loop stability, and, with proper designof the UIOs, the respective error terms are asymptotically vanishing. In this case, the attackedsubsystem is still driven not only by the attacker’s internal state, but also by the injected input η i . This entails the necessity of reconstructing such an input from the estimate ˆ˜ x − i that has beencomputed. Remark 4.
Although no assumptions are made on the controller, (12) is presented to simplify theanalysis in the case of linear control. In fact, the proposed accommodation strategy consists of anadditive input ˆ η i and an additive estimate compensation ˆ˜ x i , which can in principle be used in severalcontrol designs. ⊳ If rank A i = n i , then the solution to (11) is unique, and we have that ˆ˜ x − i = ˜ x − i . As a result, the error term ǫ ai obeys the dynamics ǫ a + i = ˜ A i ǫ ai + ˜ B i ∆ η i , where ∆ η i = η i − η − i . It is possible to obtain an estimate ˆ η i which solves the input reconstructionproblem relative to the dynamics (2) for a known ˜ x − i .6 efinition 2 (Relative degree) . Consider a linear system of the form (1) . Let c l be the l -th row ofmatrix C i ; if there exist integers r l such that C i A ki B i = 0 , C i A r l − i B i = 0 , ∀ k < r l − and rank c A r − i B i ... c p A r pi − i B i = m i then r = [ r , . . . , r p i ] is called the relative degree of system (1) . ⊳ Let us define the stacked vector of the attacker’s state and inject input estimates ˆ˜ x i . = col t ∈{ ,...,r } (ˆ˜ x i ( k − t )) , ˆ η i . = col t ∈{ ,...,r } (ˆ η i ( k − t − , respectively, and the input-to-state dynamic matrix Ψ i Edelmayer et al. (2004): Ψ i . = B i . . . A i B i B i . . . ... ... . . . ... A r − i B i A r − i B i . . . B i , where r ≥ n i . Then, we can state the following. Proposition 1.
The injected signal η i ( k ) can be estimated in finite time if and only if system (2) isleft-invertible. Furthermore, ˆ η i = Ψ † i ˆ˜ x i (14) and ˆ η i ( k ) = η i ( k − r −
1) = ˆ η i [ r ] (15) (cid:3) Proof.
If system (2) is left-invertible, then ˆ˜ x i = Ψ i ˆ η i admits a unique solution, given by (14). Thedelay in the input estimate (15) follows by Definition 2, with c i taken as the canonical euclideanbasis vectors. In this, r is the largest component of the vector relative degree, i.e. r = max l ∈{ ,...,r pi } ( r l ) . The additional lag step is given by the intrinsic delay in the estimate ˆ˜ x i . Remark 5.
The left-invertibility condition necessary to obtain ˆ η i is implied by existence conditionsfor the UIO O di Hou and Patton (1998), hence no further assumptions are made on the problemsetting. ⊳ In this case, from an analytical point of view, the compensation mismatch depends on differences ofthe attacker’s input signal because of the intrinsic delay of the estimation procedure. For constantinjected inputs, e.g. steady offsets, it follows that attack compensation is exact.
Remark 6.
Notice that rank A ji = n i , ∀ j ∈ N i ⇒ rank A i = n i , but the converse is not true ingeneral. ⊳ In this subsection, we consider the case rank A i < n i . This is attained when a subset of the com-ponents of x i does not influence the neighbors. More formally, the aggregate interconnection islow-rank if ∃ g i ∈ N such that dim \ j ∈N i ker A ji = g i > .
7e can introduce a decomposition of the state space R n i into a non-interacting subspace X ⊥ i . =ker A i and an interacting one X || i . = R n i /X ⊥ i . Clearly, we have that dim X || = n i − g i . Conse-quently, we can define respective canonical projections of R n i onto these subspaces, and in particularwe consider P i : R n i → X || i .With this projection, the solution of the LS problem will be exact only on the interacting subspace.In particular, we have that ˆ˜ x ||− i . = P i ˆ˜ x − i = P i ˜ x − i , ˆ˜ x ⊥− i . = ( I − P i )ˆ˜ x − i = 0 . By means of this projection, it is possible to reframe the problem as the input reconstruction for thesystem ˜ SS || i : ( ˜ x + i = ˜ A i ˜ x i + ˜ B i η i ˜ x || i = P i ˜ x i , (16)where the second equation is considered as the system output. The input-output matrix can berewritten as Ψ || i . = P i A r − i B i . . . P i A r i B i P i A r − i B i . . . ... ... . . . ... P i A r − i B i P i A r − i B i . . . P i A r − i B i , where r ≥ n i and r = max l ∈{ ,...,r pi } ( r l ) is the maximum relative degree of (16). Definition 3 (Trentelman et al. (2012)) . The constant λ ∈ C is an ( A, C ) -unobservable eigenvalueif rank (cid:20) λI − AC (cid:21) < n , with n = dim A . ⊳ Proposition 2.
The input η i ( k ) in (16) can be estimated in finite time if and only if the set of invariantzeros of ˜ SS || i is identical to the set of ( ˜ A i , P i ) -unobservable eigenvalues. Furthermore, ˆ η i = (cid:16) Ψ || i (cid:17) † ˆ˜ x || i (17) and ˆ η i ( k ) = η i ( k − r −
1) = ˆ η i [ r ] (18) (cid:3) Proof.
Using (Bejarano et al., 2009, Theorem 4.10), we ensure that (16) is left invertible. In thatcase, (17) admits a unique solution. The remaining considerations follow those in Proposition 1.Finally, reusing ˆ η i in (16) allows for a forward estimation of ˜ x ⊥ i . In fact, by definition, such partof the state does not directly affect the neighboring dynamics and hence the communicated errors.As a result, the only way of reconstructing the entire ˆ˜ x i is via the state update equation of the localattacker’s model (2). Remark 7.
The decomposition method presented in this subsection can be related to Bejarano(2011), where however the problem statement is different. Despite the differences in the setting, inthe cited work, the input matrix (in the present case, the outbound interconnection) is decomposedonto orthogonal subspaces, and partial left-invertibility conditions are developed for the decom-posed system. ⊳ We show the effectiveness of the proposed method on a simple numerical example on regulation.This example is meant to illustrate the practicality of the proposed procedure.8igure 2: State trajectories of SS (full rank interconnection).The overall system comprises N = 5 subsystems and the topology is described the following neigh-bors sets: N = { , } , N = { , , } , N = { , } , N = { , , } , N = { } . We consider thefull and low rank interconnection cases, and for both we use the following subsystem’s dynamics. ∀ i ∈ { , . . . , N } : A i = (cid:20) . .
20 0 . (cid:21) B i = (cid:20) (cid:21) C i = I D i = 0 . Choice of A ij will be presented in two separate examples in the following subsections. Individualpairs of observers are designed for the two systems, as presented in Section 3. Each system im-plements a control law of the form (12), which optimally decouples the neighboring dynamics andachieves a prescribed rate of convergence.For simplicity and without loss of generality, all subsystems implement the same dynamics and thesame interconnection. At time k a = 20 , SS is covertly attacked according to the model presentedin Section 2. The attacker’s objective is to introduce a steady-state error into the regulator. For this case, the interconnection matrix is chosen as A ij = (cid:20) . − . (cid:21) . Results for this example can be seen in Fig. 2 and 3. In the former, components of the true stateare shown, and it is possible to see that the effect of the attack is compensated according to thecontroller’s dynamics with a certain delay. This delay is particularly evident in Fig. 3: in the left-hand side figure, the one-step delay intrinsic in the computation of ˆ˜ x is shown. This translates intoan r + 1 delay for the computation of ˆ η , as depicted on the right-hand side of Fig. 2. In this case, the interconnection matrix is chosen as A ij = (cid:20) . − . (cid:21) . As in the previous subsection, the results can be seen in Fig. 4 and 5. It can be noticed how theperformance of the accommodated system is not particularly different from that obtained in the full9igure 3: (left) Attacker’s state trajectories and LS estimates; (right) Actual injected signal η andreconstructed estimate ˆ η (full rank interconnection).Figure 4: State trajectories of SS (low rank interconnection).rank case. On the other hand, the estimation of the attacker’s state ˜ x is affected by a longer delay,as shown in the left-hand side of Fig. 5. These estimates are obtained using the reconstructed inputas computed by (18), and which are shown in the right-hand side of the picture. In this paper, a novel distributed methodology for accommodation of stealthy local attacks in in-terconnected systems is presented. To the best of the authors’ knowledge, this is the first time astep is done in this direction, and the approach can be in principle applied to other cases where lo-cally unobservable states have no effects on residuals. Given the early stage nature of the presentedmethodology, additional work is being done on characterizing robustness and the impact of noiseand disturbances on the estimates. 10igure 5: (left) Attacker’s state trajectories and LS estimates; (right) Actual injected signal η andreconstructed estimate ˆ η . Notice the longer delay needed due to the projection procedure (low rankinterconnection). Acknowledgments
This work has been partially supported by the EPSRC Centre for Doctoral Training in High Per-formance Embedded and Distributed Systems (HiPEDS, Grant Reference EP/L016796/1), the Eu-ropean Union’s Horizon 2020 Research and Innovation Programme under grant agreement No.739551 (KIOS CoE), and by the Italian Ministry for Research in the framework of the 2017 Programfor Research Projects of National Interest (PRIN), Grant no. 2017YKXYXJ.
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