Observability of linear systems under adversarial attacks*

Transcription

1 Observabiity of inear systems under adversaria attacks* Michee S. Chong 1, Masashi Wakaiki 1 and João P. Hespanha 1 Abstract We address the probem of state estimation for muti-output continuous-time inear systems, for which an attacker may have contro over some of the sensors and inject (potentiay unbounded) additive noise into some of the measured outputs. To characterize the resiience of a system against such sensor attacks, we introduce a new notion of observabiity termed observabiity under attacks that addresses the question of whether or not it is possibe to uniquey reconstruct the state of the system by observing its inputs and outputs over a period of time, with the understanding that some of the avaiabe system s outputs may have been corrupted by the opponent. We provide computationay efficient tests for observabiity under attacks that amount to testing the (standard) observabiity for an appropriate finite set of systems. In addition, we propose two state estimation agorithms that permit the state reconstruction in spite of the attacks. One of these agorithms uses observabiity Gramians and a finite window of measurements to reconstruct the initia state. The second agorithm, takes the form of a switched observer that asymptoticay converges to the correct state estimate in the absence of additive noise and disturbances, or to a neighborhood of the correct state estimate in the presence of bounded noise and disturbances. I. INTRODUCTION This paper is motivated by the observation that computer contro systems can be especiay vunerabe to cyber attacks; most particuary remote sensors that can be infitrated and reprogrammed to report erroneous measurements. The issue of security is not new to the contro fied, in particuar in the areas of faut detection and identification (FDI) [7] and game theory [5]. Some of the recent work on the cyber security of contro systems has been focused on the effect of specific types of attacks on stabiity and/or estimation [12], such as fase data injection attacks [6], [3], denia-of-service attacks [1], [11] and integrity attacks [8]. Coser to the work presented here, there has aso been an effort to derive resuts that are independent of the attack type in works such as [9] and [2]. In [9], the authors mode the attacked system as a continuous-time descriptor system and view the attack signa as an unknown input. The authors then propose an agorithm that detects the presence of an attack. Another reated work focused on robust state estimation appeared in [2]. The authors of [2] characterized This materia is based upon work supported by the Nationa Science Foundation under Grant No. CNS and by the U.S. Army Research Laboratory and the U.S. Army Research Office under MURI grant No. W911NF M. Chong acknowedges the American Austraian Association for their support of this work. M. Wakaiki is supported by The Kyoto University Foundation. 1 The authors are with the Center for Contro, Dynamicasystems and Computation (CCDC), University of Caifornia, Santa Barbara, CA USA. {mchong, mwakaiki, hespanha}@ece.ucsb.edu the resiience of a discrete-time LTI system against attacks by the number of attacked sensors aowed for accurate state reconstruction. They aso proposed an error correction agorithm which exacty reconstructs the state and is made computationay efficient by transforming the optimization probem into a convex one, which is possibe ony under certain conditions. The scenario considered in this paper considers a continuous-time LTI system with N outputs, each measured by a potentiay vunerabe sensor. One then asks whether or not it is possibe to reconstruct the initia state of the system from an input/output time series if M ď N of these sensors have been taken over by an adversary that has fu contro over the measurement reported by the M infitrated sensors, with the understanding that we do not know which of the M sensors have been infitrated and, in fact, if they have been infitrated at a. When it is possibe to do state reconstruction under this scenario, we say that the LTI system is observabe under M attacks. The first key resut of this paper is a necessary and sufficient condition presented in Section II for observabiity under M attacks. This condition requires the number of sensors N to be arger than 2M and aso that a famiy of LTI systems (derived from our origina system) is observabe, under the usua notion of observabiity. It was expected for N ą M to be necessary for an N-output system to be observabe under M attacks since with N ď M there woud be no attack-free measurements eft to use for estimation. However, it is somewhat unexpected to see that N ą 2M is actuay necessary for an N-output system to be observabe under M attacks. The second key resut of the paper is an agorithm presented in Section III-A that ooks at the vaues of the system s input and a N measured outputs over a finite interva r, T s, T ă 8 and provides a correct estimate of the system s initia state, in spite of the fact that M of the N measured outputs may have been compromised by an attacker. As expected, this agorithm is ony appicabe to systems that are observabe under M attacks. In essence, the agorithm proposed constructs mutipe state estimates using observabiity Gramians and utiizes a consistency condition to seect the correct estimate. The third key resut is an observer-ike agorithm presented in Section III-B that (causay) creates an asymptoticay correct estimate of the system s current state based on the vaues of its past input and N (potentiay compromised) outputs. This agorithm is appicabe when the system is observabe under M attacks, but it actuay requires ess than that. In practice, it ony requires a notion of detectabiity

2 under M attacks. This is not surprising in view of the fact that this observer-ike agorithm does not promise state reconstruction in finite time (ony asymptoticay). For this agorithm, we actuay prove more than just asymptoticay correct state reconstruction, as we aso show that an additive bounded disturbance and additive bounded measurement noise to a N outputs wi resut in a bounded estimation error, by providing an input-to-state stabiity-ike bound on the estimation error, in terms of bounds on the disturbance and measurement noise [1]. Notation We denote the cardinaity of a set S as cardpsq. x denotes the Eucidean norm of a vector x P R n. }z} T denotes the supremum norm of a signa z on an interva T Ă r, 8q. The binomia coefficient is denoted `a b, where a, b are nonnegative integers. II. OBSERVABILITY UNDER ATTACKS Consider the foowing continuous-time LTI system with N outputs: 9x Ax ` Bu y i C i x ` D i u ` η i, i P t1,..., Nu, where the state vector is x P R nx, the input vector is u P R nu, the measured outputs are y i P R ni, and the η i P R ni denote additive, possiby unbounded attack signas that cannot be measured. We denote the soution to (1) for the input u and initia condition xpq x as xptq xpt; x, uq and the corresponding measured outputs as y i ptq y i pt; x, u, η i P t1,..., Nu. We seek to derive conditions under which the initia condition xpq of (1) can be reconstructed from the measured outputs y However, we are interested in the the possibiity that a subset y i, i P I Ă t1,..., Nu of the sensor outputs have been attacked, but we do not know which. Specificay, we assume that there is an unknown subset I Ă t1,..., Nu with at most M eements for which the corresponding η i, i P I are nonzero and coud be unbounded. This motivates the foowing definition of observabiity under attacks. Definition 1: The system (1) is observabe under M- attacks on the interva r, T s, T ă 8 if for every initia conditions xpq, xpq P R nx, input uptq, t ě, sets I, Ī Ă t1,..., Nu with at most M eements, and attack vectors η pη 1,..., η N q P N I, η p η 1,..., η N q P N Ī we have y i pt; xpq, u, η i q y i pt; xpq, u, η i q, P r, T s, i P t1,..., Nu ùñ xpq xpq. (2) The notation N I denotes the set tpη 1,..., η N q : η i P r, R Iu. In essence, this definition means that, when a system is observabe under M-attacks, there is at most one initia condition that is compatibe with the input signa u and the measured outputs y i P t1,..., Nu on the interva r, T s, regardess of which M of the N sensors have been attacked and the corresponding attack signas η i seected by the opponent. The foowing resut provides a necessary and sufficient condition for system (1) to be observabe under M-attacks, which permits checking whether a system is observabe under attacks using standard observabiity tests [4, Section 15.9]. The proof is provided in Section II-A. Theorem 1: For every integer M ě, the foowing statements are equivaent: (i) System (1) is observabe under M-attacks on the time interva r, T s, T ă 8. (ii) N ą 2M and, for every set J Ă t1,..., Nu with cardpj q ě N 2M, the pair pa, C J q is observabe, where C J is a matrix obtained by stacking a the output matrices C i, i P J from system (1). Theorem 1 impicity restricts the number of attacked outputs M to be ess than haf of the number of outputs N, which is consistent with the resut in [2] for the state estimation of discrete-time LTI systems under attacks. Remark 1: Since condition (ii) in Theorem 1 does not depend on T, we concude that if a system is observabe under attack on the interva r, T 1 s, T 1 ă 8, it is aso observabe under attack on r, T 2 s, for every T 2 ă 8. This means that xpq can be determined from future inputs uptq and outputs y i ptq, i P t1,..., Nu over an arbitrariy sma time interva r, T s. Remark 2: By defining M-attack observabiity for a cass of noninear systems of this form: 9x Ax ` φpuq, y i C i x ` ψ i puq ` η i for i P t1,..., Nu, in the same manner as Definition 1, the resuts of Theorem 1 can be extended to this cass of noninear systems under the assumption that the system is forward compete, i.e. the soution xptq exists for a t ě, for any initia condition xpq, input u and attack signa η i. In this case, this system is M-attack observabe if and ony if N ą 2M and, for every set J Ă t1,..., Nu with cardpj q ě N 2M, the pair pa, C J q is observabe in the usua sense. The foowing simpe exampes iustrate the use of Theorem 1 in checking the observabiity of the system (1) when M of the N outputs are under attack. Exampe 1: Consider the system 9x 1 x 2 ` u 9x 2 a 2 x 1 2ax 2, a ą, with N 3 outputs y i x T 1, x T T 2 ` ηi, for i P t1, 2, 3u. (4) The system (3) with outputs (4) is observabe in the usua sense. Since there are N 3 outputs, the maximum aowabe number of attacked outputs is M 1. We wi see that it is 1-attack observabe by writing system (3)-(4) in the form of (1) and appying Theorem 1. There are ony 3 sets J with N 2M 1 eement: t1u, t2u, t3u and the pairs pa, C 1 q, pa, C 2 q and pa, C 3 q are a observabe. Hence, the system is 1-attack observabe. However, we wi see in the next exampe that observabiity through every output does not necessariy coincide with M-attack observabiity. (3)

3 We now consider the same system (3), but with N 6 outputs defined as foows y i x 1 ` η i, for i P t1, 2, 3u, y i x 2 ` η i, for i P t4, 5, 6u. First, observe that this system is observabe in the usua sense. With N 6, the maximum aowabe number of attacked outputs is M 2. However, we wi see that this system is not 2-attack observabe, it is ony 1-attack observabe. By writing system (3) and (5) in the form of (1) and checking condition (ii) of Theorem 1, when M 1, we obtain ` N N 2M 15 combinations of J t3, 4, 5, 6u, t2, 4, 5, 6u, t1, 4, 5, 6u etc. where cardpj q 4 (ě N 2M) and we need to check the observabiity of the pairs `A, rc3 T, C4 T, C5 T, C6 T s, `A, T rc T 2, C4 T, C5 T, C6 T s, `A, T rc T 1, C4 T, C5 T, C6 T s, T etc. Since a such pairs are observabe, the system (3) with outputs defined in (5) is 1- attack observabe. However, when there are M 2 attacked outputs, we obtain 15 combinations of J t1, 2u, t1, 3u, t1, 4u, t1, 5u etc. where cardpj q 2 (ě N 2M) and we see that not a pairs pa, C J q are observabe, e.g. the pairs pa, rc1 T, C2 T s T q, pa, rc1 T, C3 T s T q, pa, rc2 T, C3 T s T q etc. are not observabe. Therefore, this system is not 2-attack observabe. A. Proof of Theorem 1 We first note that, in view of [4, Definition 15.2], condition (ii) in Theorem 1 can be equivaenty re-stated as (5) (ii) N ą 2M and for every set J Ă t1,..., Nu with cardpj q ě N 2M, and for every initia condition xpq P R nx, we have C i e At P J, t P r, T s ùñ xpq. (6) We wi thus prove Theorem 1 by showing that condition (i) is equivaent to (ii) above. (i) ùñ (ii) : Suppose by contradiction that (i) hods, but (ii) is fase, i.e., N ď 2M or there exists a set J Ă t1,..., Nu with cardpj q ě N 2M and an initia condition xpq P R nx, such that C i e At P J, t P r, T s and xpq. (7) First note that if N ď 2M, the empty set J H and an arbitrary non-zero initia condition satisfy (7). Henceforth, when (ii) is fase it is aways true that there exists a set J Ă t1,..., Nu with cardpj q ě N 2M and an initia condition xpq P R nx, such that (7) hods. We sha prove that this contradicts (i). To this effect, seect two disjoint sets I, Ī Ă t1,..., Nu, each with at most M eements, so that J t1, 2,..., NuzpI Y Īq. Next define attack vectors η pη 1,..., η N q P N I, η p η 1,..., η N q P N Ī so that η i ptq C i e At P I, η i ptq C i e At P where xpq is the non-zero initia condition from (7). Since η i ě, i R I and η i ě, i R Ī, this choice for the attack vectors eads to C i e At xpq ` η i ptq, η i ptq, C i e At xpq η i ptq, η i ptq, C i e At xpq, η i ptq η i ptq, and P J t1, 2,..., NuzpI Y P P Ī C i e At xpq ` η i ptq η i P t1,..., Nu, t P r, T s, (8) for some xpq. However, we can view the eft-hand side expression C i e At xpq ` η i ptq as the output y i ptq associated with the initia condition xpq, the zero input, and the attack η P N I ; whereas the right-hand side expression η i ptq can be considered as the output y i ptq associated with the zero initia condition, zero input, and attack η P N Ī. We have thus found two distinct initia conditions compatibe with the same outputs, which contradicts observabiity under M attacks and thus (i). (ii) ùñ (i): Suppose by contradiction that (ii) hods, but that (i) does not, and therefore that there exist initia conditions xpq, xpq P R nx, an input uptq, t ě, sets I, Ī Ă t1,..., Nu with at most M eements, and attack vectors η pη 1,..., η N q P N I, η p η 1,..., η N q P N Ī such that y i pt; xpq, u, η i q y i pt; xpq, u, η i P t1,..., Nu, t P r, T s and xpq xpq, which, using the variation of constants formua, means that C i e At xpq ` µ i ptq ` η i ptq C i e At xpq ` µ i ptq ` η i P t1,..., Nu, t P r, T s and xpq xpq, ş t where µ i ptq C eapt sq i Bupsqds. Since η i ě, i R I and η i ě, i R Ī, we concude that C i e At`xpq P J, t P r, T s and xpq xpq, where J t1, 2,..., NuzpI Y Īq is a set with no ess than N 2M eements, which is in contradiction with (ii). Therefore, we have shown that (ii) impies (i) and we have competed the proof. III. ESTIMATION ALGORITHMS From Theorem 1, we know that an N-output system (1) is M-attack observabe on r, T s if and ony if for every subset J of t1, 2,..., Nu with at east N 2M eements, the pair pa, C J q is observabe. In this case, we can construct state estimators based on measurements from N 2M or more outputs on the interva r, T s, which woud provide accurate state estimates in the absence of the attack signas η i. An essentia observation behind the design of the state estimators proposed here is that, for each combination of the

4 N M (greater than N 2M) outputs, we can construct one state estimator that woud produce a correct state estimate based on measurements from those outputs in the interva r, T s, in the absence of attacks η i on the chosen outputs. Moreover, assuming that at most M sensors have been attacked, for each of these sets of N M outputs, there is at east one subset of N 2M outputs that consists of attack-free outputs. Hence, a state estimator based on this subset of N 2M outputs wi resut in an accurate estimate. We expoit this fact by proposing agorithms that choose wisey among severa potentia estimates to obtain good state estimates for the system (1). We propose an estimator that uses observabiity Gramians for state reconstruction in finitetime in Section III-A and an observer-based estimator in Section III-B, which we prove to be robust with respect to noise and disturbances. A. A Gramian-based estimator Assume that the system (1) is M-attack observabe on r, T s. Given a set J Ă t1, 2,..., Nu with N 2M or more eements, we denote by ˆx J pq the initia state estimate produced by the observabiity Gramians using the input uptq and the outputs y i ptq, for a i P J coected in the interva r, T s, that woud be accurate if η i ptq, for every i P J and t P r, T s. One can show that such estimate is given by ˆx J pq W J p, T ş q 1 T s C T eat J ỹj psqds, (9) where ỹ J psq y J psq şs C J e Ar Buprqdr D J upsq (where y J and D J denotes the stacking of a y i and D i for i P J, respectivey) and W J p, T q ş T s C T eat J C J e As ds is the observabiity Gramian (see [4, Section 15.5]), which is invertibe because the pair pa, C J q is observabe (by Theorem 1). For each subset J Ă t1, 2,..., Nu with N M (ě N 2M) eements, define π J to be the argest deviation between the estimate ˆx J pq and any estimate that uses an N 2M subset P Ă J of the outputs used to construct ˆx J pq: π J max ˆx J pq ˆx P pq. (1) PĂJ :cardppq N 2M When a the η i, i P J are equa to zero, a the estimates that appear in the definition of π J wi be consistent and we have π J. This motivates the foowing state estimate: ˆxpq ˆx σ pq, σ arg min π J. (11) J Ăt1,2,...,Nu:cardpJ q N M When more than one π J achieve the minimum simutaneousy, we can choose σ to be any of them. We ca this scheme a finite-time Gramian-based estimator. The foowing can be proved about this state estimator. Theorem 2: Assume that the N-output system (1) is M- attack observabe and that the attack vector η beongs to N I for some set I Ă t1,..., Nu with cardpiq ď M. For every initia conditions xpq P R nx and input u, the foowing hods ˆxpq xpq, (12) where ˆxpq is the estimate produced by the Gramian-based estimator (9)-(11). Proof of Theorem 2. Since system (1) is M-attack observabe, we have from Theorem 1 that for any set J Ă t1, 2,..., Nu with cardpj q ě N 2M, the pair pa, C J q is observabe. Foowing standard deveopments for Gramianbased reconstruction (see Section 15.6 of [4]), we rewrite the estimate ˆx J pq of the initia condition (9) in terms of the true initia condition xpq as foows ˆx J pq W J p, T q 1 e AT s CJ T ỹ J psqds W J p, T q 1 e AT s CJ T C J xpsqds W J p, T q 1 e AT s CJ T ż s W J p, T q 1 e AT s CJ T D J upsqds ` W J p, T q 1 e AT s CJ T η J psqds, C J e Ar Buprqdrds xpq ` W J p, T q 1 e AT s CJ T η J psqds, (13) where η J denotes the stacking of a η i, for i P J. We obtain the ast equaity since the first three (attack-free) terms reconstruct the true initia condition xpq exacty according to [4, Section 15.6]. Since η pη 1, η 2,..., η N q P N I, we concude from (13) with J Ī Ă t1,..., NuzI with cardpīq N M and aso with J P Ă Ī, cardppq N 2M that ˆx Ī pq ˆx P pq xpq (14) which means that π Ī. Since π Ī and σ arg min π J, we have that π σ and therefore, J ˆx σ pq ˆx P Ă σ : cardppq N 2M. (15) Most importanty, since we are removing an additiona M eements from σ to obtain the sets P, regardess of what σ turns out to be, there is aways one set P Ă σ, with cardppq N 2M for which η i ptq, for a i P P, t ě. For this set ˆx P pq xpq and therefore we must necessariy have ˆx σ pq xpq, because of (15). Once we obtain an estimate of the initia condition ˆxpq, we can then generate the state estimate for system (1) at any time t ě using ˆxptq e Atˆxpq ` ż t e Apt sq Bupsqds. (16) Since we obtain ˆxpq xpq using the data uptq and yptq on the interva r, T s, we achieve a correct estimate in finite-time, which is an advantage over the observerbased estimator introduced in the next section. However, the impementation of the Gramian-based estimator requires the

5 inversion of the observabiity Gramians for each interva of time considered, which woud be computationay very intensive if we wanted to construct a time series of state estimates. We wi see that the observer-based estimator in the foowing section, ony invoves the soution of an ordinary differentia equations (ODEs), for which numericay efficient sovers are widey avaiabe. B. An observer-based estimator We now consider an augmented version of system (1) with a process disturbance d : r, T s Ñ R nx and measurement noise m i : r, T s Ñ R ny that enter the system in the foowing manner: 9x Ax ` Bu ` Bd y i C i x ` D i u ` η i ` m i, i P t1,..., Nu, (17) Opposite to the attack signas η i, a the measurement noise signas m i may be nonzero, but are typicay bounded. Our goa is to show that the observer-based estimated proposed beow is robust with respect to the process disturbance d and the measurement noise m i. Foowing the same framework as the Gramian-based estimator in Section III-A, we assume that the N-output system (17) is observabe through any N 2M outputs and construct an observer for every set J Ă t1,..., Nu with N M (ě N 2M) eements as foows: 9ˆx J Aˆx J ` Bu ` L J pŷ J y J q (18) ŷ J C J ˆx J ` D J u, where the matrix L J is chosen such that A ` L J C J is Hurwitz, which is aways possibe since every pair pa, C J q is observabe (and therefore detectabe) in view of Theorem 1. From the bank of ` N N M estimates ˆxJ, we choose the state estimate aong the ines foowed by the Gramian-based estimator in Section III-A: ˆxptq ˆx σptq ptq, (19) σptq π J ptq arg min π J ptq, (2) J Ăt1,2,...,Nu:cardpJ q N M max ˆx J ptq ˆx P ptq, (21) PĂJ :cardppq N 2M where the state estimate ˆx P for P Ă J with N 2M eements is generated in the same manner as (18). The foowing resut states that the proposed estimator is robust with respect to the disturbance d and measurement noise m i, where we denote the stacking of a m i, i P J as m J. Theorem 3: Assume that the augmented N-output system (17) is M-attack observabe and η i in (17) beongs to N I for some set I Ă t1,..., Nu with cardpiq ď M. There exist constants k, ᾱ, γ x and γ y ą such that for every initia condition xpq P R nx and input uptq, t ě, the foowing inequaity hods: xptq ˆxptq ď k expp ᾱtq max J ` γ x }d} r,ts ` γ y ˆmax }m J } r,ts J p xpq ˆx J pq q, t ě, (22) for any initia conditions xpq, ˆx J pq, ˆx P pq P R nx, for every P Ă J Ă t1,..., Nu where cardppq N 2M and cardpj q N M, as we as bounded signas d and m i, i P t1,..., Nu. Proof of Theorem 3. For an arbitrary set J Ă t1,..., Nu with cardpj q ě N 2M, the state estimation error x J : x ˆx J has the foowing error dynamics aong soutions to the process (17) and the observer (18): 9 x J pa ` L J C J q x J L J η J L J m J ` Bd. (23) Since A ` L J C J is Hurwitz, the soution to (23) satisfies x J ptq ď k J expp α J tq x J pq ` γ η }η J } r,ts `γ y }m J } r,ts ` γ x }d} ě, (24) where k J, α J, γ η, γ y and γ x ą. Since η i ptq, for a i P t1,..., NuzI and t ě, we concude from (24) with J Ī Ď t1,..., NuzI with cardpīq N M that x Ī ptq ď k Ī expp α Ī tq x Ī pq ` γ y }m Ī } r,ts `γ x }d} r,ts, t ě. (25) and aso for any set P Ă Ī with cardppq N 2M, we have from (24) with J P that x P ptq ď k P expp α P tq x P pq ` γ y }m P } r,ts `γ x }d} r,ts, t ě. (26) Recaing the definition of π Ī from (1), we have that π Ī ptq max ˆx Ī ptq ˆx P ptq max ď ˆx Ī ptq xptq ` xptq ˆx P ptq x Ī ptq ` max From (25) and (26), we obtain x P ptq. (27) π Ī ptq ď 2k expp αtq} x Ī pq} ` 2γ y }m Ī } r,ts ` 2γ x }d} r,ts, t ě, (28) where k : max that since P Ă tk Ī, k P u and α : min tα Ī, α P u. Observe Ī with cardppq N 2M, we have from Theorem 1 that there is at east 1 set of P, denoted by P, which satisfies x Pptq ď k expp αtq x Ppq `γ y }m P} r,ts ` γ x }d} r,ts, t ě. (29) Reca from (21) that ˆxptq ˆx σptq ptq where σptq arg min π J ptq, hence π σptq ptq ď π Ī ptq. Using the J :cardpj q N M ˇ fact that π σptq ptq : max ˇˆxσptq ptq ˆx P ptqˇˇ ě PĂσ:cardpPq N 2M ˆx σptq ptq ˆx P, we have from triange inequaity that xptq ˆx σptq ptq x σptq ptq xptq ˆx Pptq ` ˆx Pptq ˆx σptq ptq ď x Pptq ` ˆx Pptq ˆx σptq ptq ď x Pptq ` π σptq ptq ď x Pptq ` π Ī ptq, t ě. (3)

6 From (28) and (29), we have x σptq ptq ď 3k expp αtq maxt x Ppq, x Ī pq u `3γ y `maxt}m P} r,ts, }m Ī } r,ts u `3γ x }d} r,ts, t ě. (31) We see that (31) satisfies (22) by setting k : 3k, ᾱ : α, γ y : 3γ y and γ x : 3γ x, which concudes the proof. The proposed observer-based estimator provides exponentia convergence of the estimates to a neighborhood of the true states x under the assumption that the perturbed system (17) is M-attack observabe. In other words, the robust observer (18)-(21) generates an error system that is input-to-state stabe (ISS) according to the definition of [1] with respect to the process disturbance d and output measurement noises m i. When there are no disturbances, we obtain exponentia convergence of the estimates to the true states for a initia conditions. Remark 3: The observer-based estimator ony requires detectabiity as opposed to observabiity in the Gramian-based estimator, which is counterpointed by asymptotic, instead of finite-time convergence of the states. [9] F. Pasquaetti, F. Dorfer, and F. Buo. Attack detection and identification in cyber-physica systems. IEEE Transactions on Automatic Contro, 58(11): , November 213. [1] E.D. Sontag. Input to state stabiity: Basic concepts and resuts. Noninear and Optima Contro Theory, 1932:163 22, 28. [11] A. Teixeira, S. Amin, H. Sandberg, K.H. Johansson, and S.S. Sastry. Cyber security anaysis of state estimators in eectric power systems. In Proccedings of the 49th IEEE Conference on Decision and Contro (CDC), pages , 21. [12] A. Teixeira, I. Shames, H. Sandberg, and K.H. Johansson. A secure contro framework for resource-imited adversaries. Automatica, 214. Submitted. IV. CONCLUSIONS We introduced a new notion of observabiity for mutioutput continuous-time LTI systems, for which a subset of the outputs can be attacked by an adversary. A necessary and sufficient condition is derived which aows standard observabiity tests to be empoyed in checking whether a system is observabe under attacks. We propose two stateestimation agorithms: a finite-time Gramian-based estimator and an asymptotic observer-based estimator. For the atter, we show that it provides bounded estimation errors in an ISS-ike manner in the presence of bounded disturbances and measurement noise. Future works incude the consideration of the stabiization probem and reducing the computationa compexity of the proposed estimation agorithms. REFERENCES [1] S. Amin, A.A. Cárdenas, and S.S. Sastry. Safe and secure networked contro systems under denia-of-service attacks. In Hybrid Systems: Computation and Contro, pages Springer, 29. [2] H. Fawzi, P. Tabuada, and S. Diggavi. Secure estimation and contro for cyber-physica systems under adversaria attacks. IEEE Transactions on Automatic Contro, 59(6): , June 214. [3] J.M. Hendrickx, K.H. Johansson, R.M. Jungers, H. Sandberg, and K.C. Sou. Efficient computations of a security index for fase data attacks in power networks. IEEE Transactions on Automatic Contro, 214. Accepted. [4] J.P. Hespanha. Linear systems theory. Princeton University Press, 29. [5] M. Jones, G. Kotsais, and J.S. Shamma. Cyber-attack forecast modeing and compexity reduction using a game-theoretic framework. In Contro of Cyber-Physica Systems, pages Springer, 213. [6] Y. Liu, P. Ning, and M.K. Reiter. Fase data injection attacks against state estimation in eectric power grids. ACM Transactions on Information and System Security (TISSEC), 14(1):13, 211. [7] M.A. Massoumnia, G.C. Verghese, and A.S. Wisky. Faiure detection and identification. IEEE Transactions on Automatic Contro, 34(3): , March [8] Y. Mo, J.P. Hespanha, and B. Sinopoi. Resiient detection in the presence of integrity attacks. IEEE Transactions on Signa Processing, 62(1):31 43, January 214.