Detecting Integrity Attacks on Control Systems using a Moving Target Approach
DDetecting Integrity Attacks on Control Systems using a Moving TargetApproach
Sean Weerakkody Bruno Sinopoli
Abstract — Maintaining the security of control systems inthe presence of integrity attacks is a significant challenge. Inliterature, several possible attacks against control systems havebeen formulated including replay, false data injection, and zerodynamics attacks. The detection and prevention of these attacksmay require the defender to possess a particular subset oftrusted communication channels. Alternatively, these attackscan be prevented by keeping the system model secret fromthe adversary. In this paper, we consider an adversary whohas the ability to modify and read all sensor and actuatorchannels. To thwart this adversary, we introduce external statesdependent on the state of the control system, with linear time-varying dynamics unknown to the adversary. We also includesensors to measure these states. The presence of unknown time-varying dynamics is leveraged to detect an adversary whosimultaneously aims to identify the system and inject stealthyoutputs. Potential attack strategies and bounds on the attacker’sperformance are provided.
I. I
NTRODUCTION
Cyber-Physical systems (CPSs), referring to the tightinterconnection of sensing, communication, and control inphysical spaces, are becoming widespread in today’s soci-ety. Indeed, these systems will serve a significant role inseveral applications including transportation, water distri-bution, medical technologies, manufacturing, and of coursethe smart grid. Due to the proliferation of CPSs in criticalinfrastructures, their safety and security are of paramountimportance. There have already been several powerful attacksagainst CPSs. One major example is Stuxnet, which targetedSupervisory Control and Data Acquisition (SCADA) sys-tems at uranium enrichment facilities in Iran [1], [2]. Here,the adversary was able to appropriate controllers runningcentrifuges at the plant, and avoid detection by replayingprevious measurements to the system operator. An additionalexample is the Maroochy Shire incident where a disgruntledemployee performed an attack on a SCADA based sewagecontrol system [3].Previous work [4] has suggested that existing tools incyber security are insufficient to address attacks on CPSsdue to the underlying physical system. Two main classes ofattacks defined by [4] are denial of service attacks where an
S. Weerakkody and B. Sinopoli are with the Department of Elec-trical and Computer Engineering, Carnegie Mellon University, Pitts-burgh, PA, USA 15213. Email: [email protected],[email protected]
S. Weerakkody is supported in part by the Department of Defense (DoD)through the National Defense Science & Engineering Graduate Fellow-ship (NDSEG) Program. The work by S. Weerakkody, and B. Sinopoliis supported by NSF grant CNS-1329936 CPS: Synergy: CollaborativeResearch: Event-Based Information Acquisition, Learning, and Control inHigh-Dimensional Cyber-Physical Systems attacker restricts the flow of information between the plantand control center, and integrity attacks where an adversarycan alter control inputs and sensor outputs. An intelligentadversary can potentially cause physical damage to a systemusing access to control inputs while manipulating sensormeasurements to avoid detection. As such, integrity attacksare the main focus of this paper.Several integrity attacks have been investigated in theliterature. For instance, [5], [6] analyze zero dynamics attackswhere an adversary injects inputs into both the actuatorsand sensors so as to bias the state without inserting a netbias on the sensor measurements. False data injection attackson measurements, where an adversary alters a subset ofsensor measurements to induce destabilizing control inputsfrom the defender have also been studied. Liu et. al. [7]first studied false data injection attacks in the context ofelectricity grids. Furthermore, in [8], the authors considerfalse data injection in control systems, providing sufficientand necessary conditions for an attacker to destabilize asystem while introducing a bounded bias on measurementresidues. Finally, replay attacks where an adversary repeatsa sequence of past measurements are analyzed in [9], [10].The detection and prevention of integrity attacks on controlsystems against adversaries who are aware of the systemmodel rely on the presence of one or more secure commu-nication channels between the operator and the plant. Forinstance, [6] provides sufficient and necessary conditions forzero dynamics attacks based on the actuators and sensorsin possession of the adversary. If the adversary has accessto all sensors and actuators, a trivial zero dynamics attackis to subtract ones influence from the true measurements.To prevent false data injection attacks in control systems, aparticular subset of measurements must be secure from theadversary [8]. Moreover, [11] proposes assigning securityindices to each sensor to quantify the effort required for anadversary to introduce a successful false data injection attack.Physical watermarking, used to detect replay attacks in [9],[10] and robust attacks defined in [12], relies on the ability toinject secret noisy inputs into the control system. Also, [13]which considers the problem of robust estimation and controlin the presence of integrity attacks, relies on the assumptionthat the attacker is only able to manipulate less than half thesensors.In this paper, we consider the scenario where an adversaryhas access to all communication channels. Thus, to preventan attack, an adversary must not be aware of the fullsystem model. [14] considers the problem of altering systemmatrices to avoid zero dynamics attacks. However, in practice a r X i v : . [ c s . S Y ] J un n adversary can use his access to both inputs and outputsto identify the system. Moreover, a malicious insider such asthe attacker in the Maroochy Shire incident might be awareof the system model. Consequently, we propose introducingextraneous states correlated to the ordinary states of thesystem so that modification of the original states will impactthe extraneous states. The extraneous states will have lineartime-varying dynamics, known to the system operator andhidden from the adversary. The dynamics act as a movingtarget, changing fast enough so the adversary does not haveadequate opportunity to identify the extraneous system. Inthis scenario, we propose attacks for the adversary and obtaindetection bounds.The rest of the paper is organized as follows. In SectionII, we introduce our system model and control strategy.In Section III, we propose the moving target approach todetect integrity attacks on control systems. In Section IV, wesummarize the attacker’s capabilities and propose two attackmodels. In Section V, we analyze bounds on the attacker’sperformance. Section VI concludes the paper.II. S YSTEM M ODEL
In this section, we introduce the model for our system.In particular, we assume our cyber-physical system can bemodeled as a discrete time control system where x k +1 = Ax k + Bu k + w k , (1) y k = Cx k + v k . (2)Here x k ∈ R n is the state vector at time k and u k ∈ R p is a collection of control inputs. A suite of sensors are usedto monitor the state. Here y k ∈ R m is a vector of sensormeasurements taken at time k . w k is the independent andidentically distributed (IID) process noise with probabilitydistribution given by N (0 , Q ) where Q (cid:31) . Meanwhile, v k is the IID measurement noise with distribution given by v k ∼ N (0 , R ) where R (cid:31) . We assume that ( A, C ) isdetectable. Additionally, ( A, B ) and ( A, Q ) are assumedto be stabilizable.The set of measurements y k are sent to the SCADAcenter in order to compute the optimal control input. For ourpurposes, we assume that the operator wishes to minimize aquadratic function of the states and inputs as follows J = lim T →∞ T + 1 E (cid:34) T (cid:88) k =0 x Tk W x k + u Tk U u k (cid:35) , (3)where W ∈ R n × n , U ∈ R p × p are positive definite matricesdefining the relative cost of each state and input. The optimalcontrol input for the given cost function is a combination ofa Kalman filter and a linear state feedback controller [15].The Kalman filter computes the minimum mean squarederror state estimate ˆ x rk | k given the previous set of measure-ments up to y k denoted by y k . We assume that the system The superscript r is used to distinguish the ordinary state estimate fromthe state estimate obtained through the moving target model. has been running for a long time so that the Kalman filterhas converged to a fixed gain linear estimator. ˆ x rk +1 | k = A ˆ x rk | k + Bu k , (4) ˆ x rk | k = ( I − KC )ˆ x rk | k − + Ky k , (5) K = P C T ( CP C T + R ) − , (6) P = AP A T + Q − AP C T ( CP C T + R ) − CP A T . (7)The optimal control input with respect to (3) is given by u ∗ k = L ˆ x rk | k , L = − ( B T SB + U ) − B T SA, (8)and S satisfies the following Riccati equation S = A T SA + W − A T SB ( B T SB + U ) − B T SA. (9)A bad data detector can be utilized to determine whether amalicious attack is occurring. Typically, the bad data detectorcan be written as a threshold-based detector where g k ( I k ) H ≷ H η k . (10)Here, I k is the information available to the defender. Thenull hypothesis H is that the system is operating normallywhile the alternate hypothesis H is that the system is underattack. A more specific detector will be discussed later in thearticle. We furthermore define the probability of detection β k and false alarm α as β k = Pr ( g k ( I k ) > η k |H ) , α = Pr ( g k ( I k ) > η k |H ) . (11)Observe that α is independent of k since the system isstationary under H . Regardless of the information availableto a system operator, an attacker with knowledge of the inputto output model as well as the ability to manipulate sensormeasurements and control inputs, can generate undetectableattacks [16].For instance, an adversary can simply subtract the influ-ence he inserts through the control inputs from the systemoutputs as follows x k +1 = Ax k + B ( u ∗ k + u ak ) + w k , (12) y k = Cx k + v k + s ak , (13)where s ak is given by x ak +1 = Ax ak + Bu ak , (14) s ak = − Cx ak . (15)In this case, the attack has zero net effect on the outputs andas a result β k = α .III. T HE M OVING T ARGET
As discussed in the previous section, an adversary whois both aware of the system model and has access to allchannels can generate undetectable attacks. In this work, wepropose introducing linear time-varying dynamics, unknownto the adversary, but known to the defender, into the system.The defender can leverage his knowledge of the systemto detect integrity attacks by the adversary. Moreover, byintroducing time-varying dynamics, the defender limits thedversary’s ability to identify the system using his access tomeasurements and inputs. The time-varying dynamics act asa moving target.
A. Extended Model
We extend the state x k to include extraneous states ˜ x k ∈ R ˜ n as follows (cid:20) ˜ x k +1 x k +1 (cid:21) = A k (cid:20) ˜ x k x k (cid:21) + B k u k + (cid:20) ˜ w k w k (cid:21) , (16)where A k (cid:44) (cid:20) A ,k A ,k A (cid:21) , B k (cid:44) (cid:20) B k B (cid:21) . (17)Moreover, we introduce additional sensors ˜ y k ∈ R ˜ m tomeasure the extraneous states. (cid:20) ˜ y k y k (cid:21) = C k (cid:20) ˜ x k x k (cid:21) + (cid:20) ˜ v k v k (cid:21) , C k (cid:44) (cid:20) C k C (cid:21) . (18)The matrices are assumed to be IID random variables whichare independent of the sensor and process noise with distri-bution A ,k , A ,k , B k , C k +1 ∼ f A ,k ,A ,k ,B k ,C k +1 ( A , A , B, C ) . (19)Furthermore, we also assume that (cid:20) ˜ w k w k (cid:21) ∼ N (0 , Q ) , (cid:20) ˜ v k v k (cid:21) ∼ N (0 , R ) , (20)where Q = (cid:20) ˜ Q ˜ Q ˜ Q T Q (cid:21) (cid:31) , R = (cid:20) ˜ R ˜ R ˜ R T R (cid:21) (cid:31) . (21) Remark 1:
While we assume the structure of the systemintroduced above with IID matrices A ,k , A ,k , B k , C k +1 ,the moving target design can still be effective in otherscenarios. For instance, the dynamics need not be linearas long as the defender can accurately model the system.Moreover, the system parameters do not have to evolve ateach time step, though the longer the target remains in place,the easier it is for the adversary to identify the system. Inaddition, the matrices A ,k , A ,k , or B k may be sparse, aslong as there exists adequate coupling between x k and ˜ x k . Remark 2:
The defender must be able to introduce ex-traneous states with time-varying dynamics correlated tothe original state of the system. The extraneous states areapplication dependent and are to be decided by the systemoperator. Nonetheless, the system operator can leverageexisting waste products of the system, for instance the heatdissipated by a reaction or process. The dynamics can bemade time-varying by changing conditions at the plant.Alternatively, the defender can introduce dynamics into thesystem. For instance, the defender can introduce RLC circuitswhich measure the states. Time varying dynamics can beincorporated by including variable resistors or capacitors. Byvarying the components of the circuit according to some IIDdistribution at each time step, the defender can generate IIDsystem matrices.
Remark 3:
In the above formulation we assume that thedefender is aware of the real time system matrices althoughthey are random. In general, this information should notbe sent over the network since doing so amounts to theexistence of a secure communication channel. The securecommunication channel could be leveraged to detect anattack without considering a moving target approach, forinstance through physical watermarking [12]. Alternatively,we can generate pseudo random system matrices using apseudo random number generator (PRNG). In this case, theseed of the PRNG will be known to the defender and kepthidden from the attacker.
B. Estimation and Detection
The presence of additional sensors allows us to improveour estimate of the state. In particular, we can incorporatean additional Kalman filter to estimate the state as follows. (cid:20) ˆ˜ x k +1 | k ˆ x k +1 | k (cid:21) = A k (cid:18) ( I − K k C k ) (cid:20) ˆ˜ x k | k − ˆ x k | k − (cid:21) + K k (cid:20) ˜ y k y k (cid:21)(cid:19) + B k L ˆ x rk | k , (22) K k = P k C Tk (cid:0) C k P k C Tk + R (cid:1) − , (23) P k +1 = A k ( P k − K k C k P k ) A Tk + Q . (24)Observe that we use the state estimate ˆ x rk | k to compute theinput u ∗ k as opposed to an estimate derived from (22). Weassume the defender does not care about controlling ˜ x k . Inthis case, adding the moving target does not change J . Sucha strategy also prevents the attacker from using informationfrom the input to learn about the system model. In fact, wehave the following result. Theorem 1:
The input u ∗ k = L ˆ x rk | k is independent fromthe system matrices A ,k − , A ,k − , B k − , C k for all k . Proof:
The input u ∗ k is given by l (ˆ x r | , x , A, B, C, K, L, w . . . w k − , v . . . v k ) , (25)where l is some deterministic function of variables which byassumption are independent from A ,k − , A ,k − , B k − , C k for all k . The result immediately follows.A similar result can be obtained under attack where u ∗ k is conditionally independent of the system matrices A ,k − , A ,k − , B k − , C k for all k , given the adversary’sattack inputs.We assume that a residue based detector is incorporatedwhere the residue z k is given by z k (cid:44) (cid:20) ˜ y k y k (cid:21) − C k (cid:20) ˆ˜ x k | k − ˆ x k | k − (cid:21) ∼ N (cid:0) , C k P k C Tk + R (cid:1) . (26)We can leverage knowledge of the distribution of z k undernormal operation to design a detector. In particular weconsider a χ detector where g k in (10) is given by g k ( z k ) = z Tk ( ¯ P k ) − z k , (27)where ¯ P k = C k P k C k + R . Under normal operation g k hasa χ distribution. In general, the window for the detectorcan be extended to consider past measurements. In Figure 1,we include a diagram of the moving target system operatingnormally. ig. 1. Diagram of system under normal operation IV. A
TTACK M ODEL
In this section we describe a near omnipotent attackerin terms of his capabilities, access to information, andpotential strategies. On one hand, the adversary may acquirehis knowledge and resources through a highly sophisticatedattack strategy as done in Stuxnet. On the other hand, an ad-versary can obtain his resources through insider informationand access as done in the Maroochy Shire incident.
A. Attack Capabilities
1) The attacker can insert arbitrary inputs into the systemand can arbitrarily alter the sensor measurements. As a result,when under attack, the system has dynamics given by (cid:20) ˜ x k +1 x k +1 (cid:21) = A k (cid:20) ˜ x k x k (cid:21) + B k ( u k + u ak ) + (cid:20) ˜ w k w k (cid:21) , (28) (cid:20) ˜ y ak y ak (cid:21) = (cid:20) ˜ y k y k (cid:21) + (cid:20) ˜ s ak s ak (cid:21) . (29)where u ak is the attacker’s control input and ˜ s ak and s ak arethe biases injected on the extraneous sensors and ordinarysensors respectively.2) The attacker can read the true outputs of the system ˜ y k , y k and the inputs being sent by the defender to the plant u k for all time k . Remark 4:
The attacker essentially performs a man in themiddle attack between the plant and system operator so thathe can manipulate and read all communication channelsarbitrarily. A malicious insider can do this by breakingencryption schemes. Furthermore, physical attacks can beused to change sensor measurements and control inputs. Forinstance, locally heating or cooling a temperature sensorwould change the sensor measurements without violating theintegrity or authenticity of data from a cyber perspective.3) The attacker has full knowledge of the system model S (cid:44) { A, B, C, K, L, Q , R} . Moreover, the adversary knowsthe probability density function (pdf) of random matrices A ,k , A ,k , B k , C k +1 . Remark 5:
While conservative, the adversary can obtainhis knowledge of the system model by observing the com-munication channels for an extended period of time and performing system identification. Moreover, observe thatsince the attacker is aware of the original system model andall outputs, he can asymptotically predict the state estimate ˆ x rk | k if the matrix ( A + BL )( I − KC ) is stable [9]. Remark 6:
The attacker can leverage his probabilisticknowledge of the system model as well as the true outputs ofthe system to generate stealthy attack inputs s ak , ˜ s ak . In par-ticular, the adversary can attempt to simultaneously identifythe moving target and generate convincing counterfeit sensoroutputs.Based on the above definitions we can define the privateinformation available to the attacker and defender at time k I Ak , I Dk and the public information I Pk available to both as I Ak (cid:44) { ˜ y j , y j , ˜ s aj − , s aj − , u aj − } ∀ j ≤ k, (30) I Dk (cid:44) { A ,j − , A ,j − , B j − , C j } ∀ j, (31) I Pk (cid:44) {S , f ( A , A , B, C ) , u j − , ˜ y aj − , y aj − } ∀ j ≤ k. (32)In Figure 2, we include a diagram of the system under attack. Fig. 2. Diagram of system under attack
B. Attack Strategy
In this subsection we propose two main attack strategies.Without loss of generality we assume any attack begins at k = 0 .
1) Attack 1: Subtract Influence:
In the first attack strategythe attacker aims to estimate his influence on the controlsystem and subtract it. Define ¯ s ak (cid:44) [˜ s a Tk s a Tk ] T . Observethat if ¯ x ak +1 = A k ¯ x ak + B k u ak , ∆¯ y ak = C k ¯ x ak , (33)with initial state ¯ x a = 0 and ¯ s ak = − ∆¯ y ak , an attack iscompletely stealthy. As the adversary does not know thetime varying matrices, we assume he computes an estimateof ∆¯ y ak and uses that to subtract his influence on the sensormeasurements. Thus, we would have ¯ s ak = − E [∆¯ y ak |I Ak ∪ I Pk ] . (34) emark 7: Observe that the adversary can exactly subtracthis influence from measurements y k due to his knowledge ofthe system model. However, the adversary should be unableto completely subtract his bias from the extraneous sensors ˜ y k . Optimal Theoretical Estimation
Define ¯ y ak (cid:44) [˜ y aTk y aTk ] T , ¯ x k (cid:44) [˜ x Tk x Tk ] T , ¯ w k (cid:44) [ ˜ w Tk w Tk ] T , ¯ v k (cid:44) [˜ v Tk v Tk ] T , and ¯ y k (cid:44) [˜ y Tk y Tk ] T . The adversary’sobservations can be formulated through the following lineartime-varying system, (cid:20) ¯ x k +1 ¯ x ak +1 (cid:21) = (cid:20) A k A k (cid:21) (cid:20) ¯ x k ¯ x ak (cid:21) + (cid:20) B k B k B k (cid:21) (cid:20) u k u ak (cid:21) + (cid:20) ¯ w k (cid:21) , (35) ¯ y k = (cid:2) C k (cid:3) (cid:20) ¯ x k ¯ x ak (cid:21) + ¯ v k . (36)To estimate ∆¯ y ak at time k , assume the adversary has ac-cess to the following distribution f (¯ x k , ¯ x ak , C k |I A ∪ Pk ) where I A ∪ Pk = I Ak ∪ I Pk Then we have ¯ s ak = − (cid:90) ¯ x k (cid:90) ¯ x ak (cid:90) C k C k ¯ x ak f (¯ x k , ¯ x ak , C k |I A ∪ Pk ) d ¯ x k d ¯ x ak d C k . (37)We show that the pdf can be recursively computed at eachstep. Letting ζ k +1 = { ¯ x k +1 , ¯ x ak +1 , C k +1 } we have f ( ζ k +1 |I A ∪ Pk +1 ) = f ( ζ k +1 |I A ∪ Pk , ¯ y ak , ¯ y k +1 , ¯ s ak , u ak , u k ) , = f ( ζ k +1 |I A ∪ Pk , ¯ y k +1 , u ak , u k ) , = f (¯ y k +1 |I A ∪ Pk , ζ k +1 ) f ( ζ k +1 |I A ∪ Pk , u k , u ak ) f (¯ y k +1 |I A ∪ Pk , u k , u ak ) . (38)The second equality follows from the conditional indepen-dence of ζ k +1 and ¯ y ak , ¯ s ak given ¯ y k and u k . The last equalityfollows from Bayes rule and the conditional independence of ¯ y k +1 and u k , u ak given ζ k +1 . We note that this distributioncan be theoretically computed given the attacker’s informa-tion. That is, we know that f (¯ y k +1 |I A ∪ Pk , ζ k +1 ) ∼ N ( C k +1 ¯ x k +1 , R ) . (39)Moreover, ζ k +1 and ¯ y k +1 are deterministic functions of ζ k , u k , u ak and random variables A ,k , A ,k , B k , C k +1 , ¯ w k , ¯ v k +1 which are independent of ζ k given I A ∪ Pk . Thus,theoretically, f ( ζ k +1 |I A ∪ Pk +1 ) can be recursively computedfrom f ( ζ k |I A ∪ Pk ) . Remark 8:
If the attacker subtracts his influence, he mightbe susceptible to a growing cancellation error if he attemptsto excite the system’s unstable dynamics. Instead of subtract-ing his influence the attacker can instead directly estimatewhat the defender expects to see as summarized in the nextsection.
2) Attack 2: Estimate Expected Measurement:
In the nextstrategy, the adversary aims to track the system operator’sstate estimate. Using the system operator’s state estimate, theadversary attempts to generate stealthy outputs. Let ˆ¯ x k = [ˆ˜ x Tk | k − ˆ x Tk | k − ] T . The attacker’s observations and strategycan be formulated as follows (cid:20) ¯ x k +1 ˆ¯ x k +1 (cid:21) = (cid:20) A k A k ( I − K k C k ) (cid:21) (cid:20) ¯ x k ˆ¯ x k (cid:21) + (cid:20) ¯ w k (cid:21) , + (cid:20) B k B k B k A k K k (cid:21) u k u ak ¯ y ak , (40) ¯ y k = (cid:2) C k (cid:3) (cid:20) ¯ x k ˆ¯ x k (cid:21) + ¯ v k , ¯ s ak = E [ C k ˆ¯ x k |I A ∪ Pk ] − ¯ y k . (41)The attacker wishes to track ζ k = { ¯ x k , ˆ¯ x k , C k , P k } . The useof the preceding attack design is motivated by the ensuingresult which states that the chosen attack vector minimizesa fixed quadratic function of the measurement residues. Theorem 2:
Let Σ (cid:23) be a positive semidefinite matrix. E [ C k ˆ¯ x k |I A ∪ Pk ] − ¯ y k = arg min ¯ s ak E [ z Tk Σ z k |I A ∪ Pk ] . (42) Proof:
Observe that E [ z Tk Σ z k |I A ∪ Pk ] = (cid:90) ζ k z Tk Σ z k f ( ζ k |I A ∪ Pk ) d ζ k . (43)Taking the gradient with respect to ¯ s ak and setting theresulting expression equal to 0, we obtain (cid:90) ζ k Σ(¯ y k + ¯ s ak − C k ˆ¯ x k ) f ( ζ k |I A ∪ Pk ) d ζ k = 0 . (44)Solving gives ¯ s ak = − ¯ y k + (cid:90) ζ k C k ˆ¯ x k f ( ζ k |I A ∪ Pk ) d ζ k , (45)and the result holds.To determine ¯ s ak at time k assume the adversary hasaccess to the following distribution f ( ζ k |I A ∪ Pk ) . As donebefore, the attacker can theoretically compute ¯ s ak by takinga conditional expectation. Additionally, similar to (38) wehave f ( ζ k +1 |I A ∪ Pk +1 )= f (¯ y k +1 |I A ∪ Pk , ζ k +1 ) f ( ζ k +1 |I A ∪ Pk , u k , u ak , ¯ y ak ) f (¯ y k +1 |I A ∪ Pk , u k , u ak , ¯ y ak ) . (46)Moreover, by similar analysis as in attack 1, we candemonstrate that f ( ζ k +1 |I A ∪ Pk +1 ) can be recursively computedfrom f ( ζ k |I A ∪ Pk ) . The main difference here is that theadversary must also estimate P k . Note that in practice theproposed attacks are difficult to execute for an adversarysince it is likely a challenge to compute the necessary distri-bution functions and expected values. As a result, in the nextsection we aim to provide bounds on the attacker’s estimationperformance in terms of mean square error matrices.. B OUNDS ON A TTACKER ’ S P ERFORMANCE
A. Bounds on Attacker’s State Estimation
In this section we attempt to characterize lower boundson the error matrices associated with the states ζ k definedin attack strategy 1 and 2. From there, we can attempt tocharacterize how well the adversary can design ¯ s ak to foolthe bad data detector.We leverage conditional posterior Cramer-Rao lowerbounds for Bayesian sequences derived by [17]. The authorshere make use of the Bayesian Cramer-Rao lower boundor Van Trees bound derived in [18] which states that forobservations y and states ζ the mean squared error matrix isbounded by the Fisher information as follows E f ( ζ,y ) (cid:104) [ˆ ζ ( y ) − ζ ][ˆ ζ ( y ) − ζ ] T (cid:105) ≥ I − , (47)where the Fisher information matrix I is given by I = E f ( ζ,y ) (cid:104) −(cid:52) ζζ log f ( ζ, y ) (cid:105) . (48)Note that (cid:52) yx g ( x, y ) (cid:44) (cid:79) x (cid:79) Ty g ( x, y ) . In [17], this result is extended to nonlinear Bayesian se-quences with dynamics given by ζ k +1 = F k ( ζ k , ω k ) , ¯ y k = G k ( ζ k , ¯ v k ) , (49)where ω k and ¯ v k are independent process and sensor noiserespectively. In our case, we slightly adapt these results toaccount for the fact there is feedback in our system so that ζ k +1 = F k ( ζ k , ¯ y k , ω k ) , ¯ y k = G k ( ζ k , ¯ v k ) . (50)The inputs u k , u ak and ¯ s ak are incorporated into the definitionof F k , while uncertainty in the model ( A ,k , A ,k , B k , C k +1 ) can be incorporated in the process noise ω k . It can shownthat the following posterior Cramer-Rao lower bound holds E f ck +1 (cid:2) e k +1 e T k +1 | ¯ y k (cid:3) ≥ I − ( ζ k +1 | ¯ y k ) , (51)where e k +1 (cid:44) ζ k +1 − ˆ ζ k +1 (¯ y k +1 | ¯ y k ) , (52) f ck +1 (cid:44) f ( ζ k +1 , ¯ y k +1 | ¯ y k ) , (53) I ( ζ k +1 | ¯ y k ) (cid:44) E f ck +1 (cid:104) −(cid:52) ζ k +1 ζ k +1 log f ck +1 | ¯ y k (cid:105) . (54) Remark 9:
We remark that since F k is defined by in-puts u k , u ak and ¯ s ak , f ck +1 is implicitly conditioned on u k , ¯ s a k , u a k . Moreover, f ck +1 is defined given the adver-sary’s knowledge of S , f ( A , A , B, C ) .Observe that (51) gives us an expected lower bound for theerror matrix associated with the entire state history ζ k +1 with knowledge of measurements ¯ y k . This expectation istaken over the state history as well the measurement ¯ y k +1 sothat ˆ ζ k +1 is a function of the measurement ¯ y k +1 . Observethat unlike the traditional Cramer-Rao bound which is limitedto unbiased estimators, the Bayesian Cramer-Rao bound hereconsiders both biased and unbiased estimators ˆ ζ . While the lower bound given here applies to the entirestate history ζ k +1 , in practice we care about estimating alower bound on the current state ζ k +1 . Nonetheless, it canbe easily shown that E f ck +1 (cid:2) e k +1 e Tk +1 | ¯ y k (cid:3) ≥ I − ( ζ k +1 | ¯ y k ) , (55)where I − ( ζ k +1 | ¯ y k ) is the dim ( ζ k ) × dim ( ζ k ) lowerright submatrix of I − ( ζ k +1 | ¯ y k ) . In practice, computing I − ( ζ k +1 | ¯ y k ) from I − ( ζ k +1 | ¯ y k ) is impractical sinceit requires computing and taking the inverse of a Fisherinformation matrix which grows in dimension at each timestep. As a result, we would like a recursion to compute I − ( ζ k +1 | ¯ y k ) . From [17] we have the following result, I ( ζ k +1 | ¯ y k ) = D k − D k (cid:2) D k + I A ( ζ k | ¯ y k ) (cid:3) − D k , (56)where D k = E f ck +1 (cid:104) −(cid:52) ζ k ζ k log f ( ζ k +1 | ζ k , ¯ y k ) (cid:105) ,D k = E f ck +1 (cid:104) −(cid:52) ζ k +1 ζ k log f ( ζ k +1 | ζ k , ¯ y k ) (cid:105) = ( D k ) T ,D k = E f ck +1 (cid:104) −(cid:52) ζ k +1 ζ k +1 log f ( ζ k +1 | ζ k , ¯ y k ) f (¯ y k +1 | ζ k +1 ) (cid:105) . In addition, I A ( ζ k | ¯ y k ) = E k − E k (cid:0) E k (cid:1) − E k , (57)where E k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k − ζ k − log f ( ζ k | ¯ y k ) (cid:105) ,E k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k ζ k − log f ( ζ k | ¯ y k ) (cid:105) = ( E k ) T ,E k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k ζ k log f ( ζ k | ¯ y k ) (cid:105) . We observe that it is still difficult to obtain matrices E k , E k , E k , E k so [17] introduces the following approx-imate recursion I A ( ζ k | ¯ y k ) ≈ S k − S Tk (cid:2) S k + I A ( ζ k − | ¯ y k − ) (cid:3) − S k , (58)where S k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k − ζ k − log f ( ζ k | ζ k − , ¯ y k − ) (cid:105) ,S k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k ζ k − log f ( ζ k | ζ k − , ¯ y k − ) (cid:105) ,S k = E f ( ζ k | ¯ y k ) (cid:104) −(cid:52) ζ k ζ k log f ( ζ k | ζ k − , ¯ y k − ) f (¯ y k | ζ k ) (cid:105) . We observe that in practice it may still be difficult tocompute the exact expectations because high dimensionalintegration is generally involved. Nonetheless, particle filtersas described in [19] can be used to approximate theseexpectations. Alternative approximations for the conditionalposterior Cramer-Rao lower bound can be found in [20].Unconditional bounds can be found in [21]. . Bounds on Detection
The algorithm described allows us to compute an approx-imate lower bound on the mean square error matrix of theattacker’s state ζ k for a given set of inputs u a k , ¯ s a k andobservation history ¯ y k . This allows us to obtain a lowerbound on the value of g k ( z k ) as follows. Theorem 3:
Consider the special case that { C j } is knownto the adversary for all j ∈ Z . Suppose an attacker attemptsto estimate ζ k = { ¯ x k , ˆ¯ x k , P k } as in attack strategy 2. Let ˆ¯ x ek (¯ y k ) be an estimate of ˆ¯ x k as a function of ¯ y k given ¯ y k − and ˆ e k = ˆ¯ x k − ˆ¯ x ek (¯ y k ) . Suppose a lower bound Z on the errormatrix of ˆ¯ x k is obtained so that E f ck (cid:2) ˆ e k ˆ e Tk (cid:3) ≥ Z k . (59)Then we havemin ¯ y ak E f ∗ [ g k ( z k )] ≥ tr ( C Tk ¯ P − k C k Z k ) , (60)where f ∗ = f (ˆ¯ x k , ¯ y k |I A ∪ Pk − , u ak − , ¯ s ak − , u k − ) . Proof:
First, observe from remark 9 f ( ζ k , ¯ y k |I A ∪ Pk − , u ak − , ¯ s ak − , u k − ) = f ck . (61)We now have the following.min ¯ y ak E f ∗ [ g k ( z k )] (62) = min ¯ y ak E f ∗ (cid:2) tr (cid:0) (¯ y ak − C k ˆ¯ x k )(¯ y ak − C k ˆ¯ x k ) T ¯ P − k (cid:1)(cid:3) , = min ¯ y ak tr (cid:0) E f ∗ (cid:2) (¯ y ak − C k ˆ¯ x k )(¯ y ak − C k ˆ¯ x k ) T ¯ P − k (cid:3)(cid:1) , = tr (cid:18) min ¯ y ak (cid:0) E f ∗ (cid:2) (¯ y ak − C k ˆ¯ x k )(¯ y ak − C k ˆ¯ x k ) T (cid:3)(cid:1) ¯ P − k (cid:19) , = tr (cid:32) min ˆ¯ x ek (cid:0) E f ∗ (cid:2) (ˆ¯ x ek − ˆ¯ x k )(ˆ¯ x ek − ˆ¯ x k ) T (cid:3)(cid:1) C Tk ¯ P − k C k (cid:33) , = tr (cid:32) min ˆ¯ x ek (cid:0) E f ck (cid:2) (ˆ¯ x ek − ˆ¯ x k )(ˆ¯ x ek − ˆ¯ x k ) T (cid:3)(cid:1) C Tk ¯ P − k C k (cid:33) , ≥ tr ( C Tk ¯ P − k C k Z k ) . The first two equalities follow from properties of the traceand expectation. The third equality follows from monotonic-ity properties of the trace function and the fact that ¯ P − k isconstant with respect to f ∗ . The fourth equality is based onthe fact that given C k , a minimizer lies in the range spaceof C k . The fifth equality is due to (61). The final inequalityfollows from (59). Remark 10:
In general, the adversary’s ability to estimate { ζ k } is dependent on the inputs { u ak } , { ¯ s ak } . For instance, themore the adversary biases the state away from its expectedregion of operation, the more challenging it is to performestimation. Thus, if the system operator wishes to analyzehow well an adversary can generate stealthy outputs, he mustconsider a particular sequence of attack inputs u ak , ¯ s ak . Remark 11:
In practice, it may be difficult to performperformance analysis when assuming P k is an unknownstate. However, one can still approximate a lower bound onthe error matrix by assuming that the adversary has an oraclewhich allows him to know P k , K k , I − K k C k . VI. C ONCLUSION
In this paper, we have considered attacks on controlsystems where an adversary has access to all channels ina communication network. In order to counter such anadversary, we propose introducing time-varying dynamicsinto the system which are unknown to the adversary andcan in turn be leveraged to detect attacks. Future work willconsider sufficient conditions for the design of these matricesto prevent zero-dynamic attacks and the analysis of optimalidentification techniques for the adversary.R
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