Simultaneous Input and State Set-Valued Observers with Applications to Attack-Resilient Estimation

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1 Simultaneous Input and State Set-Valued Observers with Applications to Attac-Resilient Estimation Sze Zheng Yong Abstract In this paper, we present a fixed-order set-valued observer for linear discrete-time bounded-error systems that simultaneously finds bounded sets of compatible states and unnown inputs that are optimal in the minimum H -norm sense, i.e., with minimum average power amplification. We also analyze the necessary and sufficient conditions for the stability of the observer and its connection to a system property nown as strong detectability. Next, we show that the proposed setvalued observer can be used for attac-resilient estimation of state and attac signals when cyber-physical systems are subject to false data injection attacs on both actuator and sensor signals. Moreover, we discuss the implication of strong detectability on resilient state estimation and attac identification. Finally, the effectiveness of our set-valued observer is demonstrated in simulation, including on an IEEE 14-bus electric power system. I. INTRODUCTION Cyber-physical systems (CPS) are systems in which computational and communication elements collaborate to control physical entities. Such systems include the power grid, autonomous vehicles, medical devices, etc. Most of these systems are safety-critical and if compromised or malfunctioning, can cause serious harm to the controlled physical entities and the people operating or utilizing them. Recent incidents of attacs on CPS, e.g., the Urainian power grid, the Maroochy water service and an Iranian nuclear plant 1 3 highlight a need for CPS security and for new designs of resilient estimation and control. In particular, false data injection attac is one of the most serious forms of attacs on CPS, where malicious and strategic attacers intrude and inject fae data signals into the sensor measurements and actuator signals with the goal of causing harm, energy theft etc. Given the strategic nature of these false data injection signals, they are not wellmodeled by a zero-mean, Gaussian white noise nor by signals with nown bounds. Hence, traditional Kalman filtering and unnown input observers do not apply. Nevertheless, reliable estimates of states and unnown inputs are valuable and needed for purposes of resilient control, attac identification, etc. Similar state and input estimation problems can be found across a wide range of disciplines, from the estimation of mean areal precipitation 4 to fault detection and diagnosis 5 to input estimation in physiological systems 6. Literature review. Much of the research focus on simultaneous input and state estimation has been on obtaining point estimates for deterministic systems with unnown inputs via asymptotic and sliding mode observers (e.g., 7 9) S.Z. Yong is with the School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA ( szyong@asu.edu). or for stochastic systems with unnown inputs via unbiased minimum-variance estimation (e.g., 10 14). These methods do not directly apply to bounded-error models, i.e., uncertain dynamic systems with set-valued uncertainties, where instead, the sets of states and unnown inputs that are compatible/consistent with sensor observations are desired. Similarly, while H, H, L 1 filters (e.g., 15 17) can deal with bounded modeling errors, only point estimates of states are obtained in addition to the fact that it is unsuited to handle large unnown inputs. In contrast, set-membership or set-valued state observers are capable of estimating the set of compatible states and are preferable to stochastic estimation when hard accuracy bounds are important 18, e.g., to guarantee safety. Since its conception, it was apparent that characterizing the set of states that are compatible with measurements is in general computationally intensive. The complexity of optimal observers 19 grows with time, and also for methods based on an l 1 model matching problem 0 and polyhedral set computation using Fourier-Motzin elimination 1. Thus, fixedorder recursive filters were designed with equalized performance (i.e., with invariant estimation errors) for superstable systems in 18,. However, all these set-membership approaches can only compute the set of compatible states and do not apply when the unnown input signals have unnown bounds, as is often required in attac-resilient estimation where the attac signals are malicious and strategic. In the context of attac-resilient estimation against false data injection attacs, numerous approaches were proposed for deterministic systems (e.g., 3 6), stochastic systems (e.g., 7 9) and bounded-error systems 30 3, but they share the common theme of only obtaining point estimates. In particular, error bounds were computed in 30 for only the initial state and in 31 with the assumption of zero initial state and without optimality considerations. More importantly, only sensor attacs are considered and setvalued estimates of state and attac signals are not computed. Contributions. The goal of this paper is to bridge the gap between set-valued state estimation without unnown inputs and point-valued state and unnown input estimation. We propose a fixed-order set-valued observer for linear discrete-time bounded-error systems that simultaneously finds bounded sets of states and unnown inputs that contain the true state and unnown input, are compatible/consistent with measurement outputs and are optimal in the minimum H -norm sense, i.e., with minimum average power amplification. In addition, we provide the necessary and sufficient

2 conditions for observer stability and boundedness of the setvalued estimates, which we show is closely related to a system property nown as strong detectability. We further show that the proposed set-valued observer is applicable for achieving attac-resiliency in cyber-physical systems against false data injection attacs on both actuator and sensor signals. Specifically, the set-valued observer can compute the sets of states and attac signals that are compatible with measurements, where the latter enables not only attac detection but also identification. Moreover, we discuss the implication of strong detectability on resilient state estimation and attac identification. Finally, the effectiveness of our set-valued observer is demonstrated using a benchmar system and an IEEE 14-bus electric power system. Notation. R n denotes the n-dimensional Euclidean space, C the field of complex numbers and N nonnegative integers. For a vector v R n and a matrix M R p q, v v v and M denote their (induced) -norm. Moreover, the transpose, inverse, Moore-Penrose pseudoinverse and ran of M are given by M, M 1, M and r(m). For a symmetric matrix S, S 0 (S 0) is positive (semi-) definite. II. PROBLEM STATEMENT System Assumptions. Consider the linear time-invariant discrete-time bounded-error system x +1 = Ax + Bu + Gd + W w, y = Cx + Du + Hd + v, where x R n is the state vector at time N, u R m is a nown input vector, d R p is an unnown input vector, and y R l is the measurement vector. The process noise w R n and the measurement noise v R l are assumed to be bounded, with w η w and v η v (thus, they are l sequences). We also assume an estimate ˆx 0 of the initial state x 0 is available, where ˆx 0 x 0 δ x 0. The matrices A, B, C, D, G, H and W are nown and of appropriate dimensions, where G and H are matrices that encode the locations through which the unnown input or attac signal can affect the system dynamics and measurements. Note that no assumption is made on H to be either the zero matrix (no direct feedthrough), or to have full column ran when there is direct feedthrough. Without loss of generality, we assume that rg H = p, n l 1, l p 0 and m 0. Unnown Input (or Attac) Signal Assumptions. The unnown inputs d are not constrained to be a signal of any type (random or strategic) nor to follow any model, thus no prior useful nowledge of the dynamics of d is available (independent of {d l } l, {w l } and {v l } l). We also do not assume that d is bounded or has nown bounds and thus, d is suitable for representing adversarial attac signals. The simultaneous input and state set-valued observer design problem is twofold and can be stated as follows: 1) Given a linear discrete-time bounded-error system with unnown inputs (1), design an optimal and stable filter that simultaneously finds bounded sets of compatible states and unnown inputs in the minimum H -norm sense, i.e., with minimum average power amplification. (1) ) Develop an attac-resilient set-valued observer for system (1) that computes a bounded set of state estimates that contains the true state and identifies the set of compatible attac signals irrespective of the magnitude of false data injection attacs on its actuators and sensors. In addition, recommend preventative attac mitigation strategies based on detectability conditions. A. System Transformation III. PRELIMINARY MATERIAL We first carry out a transformation of the system to decouple the output equation into two components, one with a full ran direct feedthrough matrix and the other without direct feedthrough. Note, however, that this similarity transformation is different from the one in 14, which is no longer applicable as it was based on the noise error covariance. Let p H r(h). Using singular value decomposition, we rewrite the direct feedthrough matrix H as H = Σ 0 V U1 U V, where Σ R p H p H is a diagonal matrix of full ran, U 1 R l p H, U R l (l ph), V 1 R p p H and V R p (p ph), while U U 1 U and V V 1 V are unitary matrices. When there is no direct feedthrough, Σ, U 1 and V 1 are empty matrices a, and U and V are arbitrary unitary matrices. Then, we define two orthogonal components of the unnown input given by d 1, = V 1 d, d, = V d. () Since V is unitary, d = V 1 d 1, + V d, and the system (1) can be rewritten as x +1 = Ax + Bu + GV 1 d 1, + GV d, + W w = Ax + Bu + G 1 d 1, + G d, + W w, (3) y = Cx + Du + HV 1 d 1, + HV d, + v = Cx + Du + H 1 d 1, + v, (4) where G 1 GV 1, G GV and H 1 HV 1 = U 1 Σ. Next, we decouple the output y using a nonsingular transformation T = T1 T U = U 1 U to obtain z 1, R p H and z, R l p H given by z 1, T 1 y = U 1 y = C 1 x + D 1 u + Σd 1, + v 1, z, T y = U y = C x + D u + v, (5) where C 1 U 1 C, C U C, D 1 U 1 D, D U D, v 1, U 1 v and v, U v. This transform is also chosen such that v 1, v, = U v = v. B. Input and State Detectability (a..a. Strong Detectability) Similar to the stability of the deterministic and stochastic input and state observers/filters, we will show in Section IV-B that the stability of the set-valued observer is directly related to the notion of strong detectability. Without loss of generality, we assume that B = 0 and D = 0 in this section, since u is nown. a We adopt the convention that the inverse of an empty matrix is also an empty matrix and assume that operations with empty matrices are possible.

3 Definition 1 (Strong detectability). The linear system (1) is strongly detectable if y = 0 0 implies x 0 as for all initial states and input sequences {d i } i N. Definition (Invariant Zeros). The invariant zeros z of zi A G the Rosenbroc system matrix R S (z) := of C H system (1) are the finite values of z for which R S (z) drops ran, i.e., r(r S (z)) < nran(r S ), where nran(r S ) is the normal ran (maximum ran over z C) of R S (z). Theorem 1 (Strong detectability). A linear time-invariant discrete-time system is strongly detectable if and only if either of the following holds for all z C, z 1: zi A G (i) r R S (z) r = n + p, C H (ii) r ˆR zi  G S (z) r = n + p p C 0 H, zi A G (iii) r R S (z) r = n + p p C 0 H, (iv) r R zi A G S(z) r = n + p p C A C G H, where  A G 1Σ 1 C 1 and A (I G M C ) for any M R (p p H) (p p H ). The above conditions are equivalent to the system being minimum-phase (i.e., the invariant zeros of R S (z) in Condition (i) are stable). Moreover, strong detectability implies that the pairs (A, C), (Â, C ), (A, C ) and (A, C A) are detectable; and if l = p, then strong detectability implies that the pairs (A, G), (Â, G ) and (A, G ) are stabilizable. Proof. The equivalence of Conditions (i) and (ii) with strong detectability in Definition 1 can be found in 14. Thus, it is sufficient to show the equivalence of Conditions (ii), (iii) and (iv) using the following identity for all z C, z 0: zi  G zi  G I 0 r = r C 0 C 0 zi A G I 0 = r = r C 0 C zi zi A G = r. C A C G M C  I zi A G C 0 Finally, comparing Conditions (i) (iv) to the PBH ran test for detectability (and stabilizability), we see that strong detectability implies that (A, C), (Â, C ), (A, C ) and (A, C A) are detectable (and that (A, G), (Â, G ), (A, G ) are stabilizable if l = p). IV. FIXED-ORDER SIMULTANEOUS INPUT AND STATE SET-VALUED OBSERVERS A. Set-Valued Observer Design We consider a recursive three-step set-valued observer design (similar to 1, 14), composed of an unnown input estimation step that uses the current measurement and the set of compatible states to estimate the set of compatible unnown inputs, a time update step that propagates the compatible set of states based on the system dynamics, and a measurement update step that updates the set of compatible states using the current measurement. In brief, our goal is to design a recursive three-step set-valued observer of the form: Unnown Input Estimation: ˆD 1 = F d ( ˆX 1, u ), Time Update: ˆX = F x( ˆX 1, ˆD 1, u ), Measurement Update: ˆX = F x ( ˆX, u, y ), where F d, F x and F x are the to-be-designed set mappings while ˆD 1, ˆX and ˆX are the sets of compatible unnown inputs at time 1, propagated and updated states at time, respectively. Note that we have a (one-step) delayed estimate of ˆD 1 because it is the only estimate we can obtain in light of (5) since d, 1 does not affect z 1, and z,, and hence, cannot be estimated from y. The reader is referred to a previous wor 13 for a detailed discussion on when a delay is absent or when further delays are expected. Since the complexity of optimal observers grows with time 19 1, we will only consider fixed-order recursive filters as in 18,, where set-valued estimates are of the form: ˆD 1 = {d R p : d 1 ˆd 1 δ d 1}, (6) ˆX = {x R n : x ˆx δx, }, (7) ˆX = {x R n : x ˆx δ}, x (8) i.e., we confine the estimation errors to balls of norm δ. In this case, the observer design problem is reduced to finding the centroids ˆd 1, ˆx and ˆx as well as the radii δ d 1, δ x, and δ x of the sets ˆD 1, ˆX and ˆX, respectively. We further limit our attention to observers for the centroids ˆd 1, ˆx and ˆx that belong to the class of three-step recursive filters given in 1, 14, defined as follows for each time (with ˆx 0 0 = ˆx 0 ): Unnown Input Estimation: ˆd 1, = M 1 (z 1, C 1ˆx D 1 u ), (9) ˆd, 1 = M (z, C ˆx 1 D u ), (10) ˆd 1 = V 1 ˆd1, 1 + V ˆd, 1, (11) Time Update: ˆx 1 = Aˆx Bu 1 + G 1 ˆd1, 1, (1) ˆx = ˆx 1 + G ˆd, 1, (13) Measurement Update: ˆx = ˆx + L(y C ˆx Du ) = ˆx + L(z, C ˆx D u ), (14) where L R n l, L LU R n (l ph), M 1 R p H p H and M R (p p H) (l p H ) are observer gain matrices that are chosen in the following lemma and theorem to minimize the volume of the set of compatible states and unnown inputs, quantified by the radii δ 1 d, δx, and δ x. Their proofs will be provided in the appendix. Note also that we applied L = LU U = LU from the following lemma into (14). Lemma 1 (Necessary Conditions for Boundedness of Set Valued Estimates). The input and state estimation errors, d 1 d 1 ˆd 1 and x x ˆx, are bounded for

4 all (i.e., the set-valued estimates are bounded with radii δ 1 d, δx,, δx < ), only if M 1Σ = I, M C G = I and LU 1 = 0. Consequently, r(c G ) = p p H, M 1 = Σ 1, M = (C G ) and L = LU U = LU. Theorem. Suppose Lemma 1 holds, and let T x,w,v denote the transfer function matrix that maps the noise signals w w v to the updated state estimation error x x ˆx. Moreover, assume that the following hold: (A.1) (A, C ) is detectable, (A.) D D 0 and A e (A.3) r jω I B = n + l p H, ω 0, π, C D with  A G 1M 1 C 1, Φ I G M C, A ΦÂ, A A, B ΦW ΦG 1 M 1 G M 0, C C A and D C ΦW C ΦG 1 M 1 C G M I. Then, there exists an H -observer that satisfy T x,w,v γ, i.e., the maximum average signal power amplification is upper-bounded by γ : lim 1 +1 x i i x i i lim 1 +1 w i w i = T x,w,v γ, (15) if and only if there exists P = P 0 that satisfies the following discrete-time algebraic Riccati equation (DARE) b : P = (A P + Cl + B )(C l P Cl + R l ) 1 (A P + Cl + B ) + B B + A P A, (16) C D D D with C l γ 1, D I l and R 0 l 0 0 I such that U I γ P 0 and Ă A (A P C + B Dl )(C lp Cl + R l ) 1 C l is asymptotically stable, i.e., λ i (Ă) < 1 for all eigenvalues λi(ă) of Ă. When such a P matrix exists, the filter gain L is given by L = (B D +A V C )(C V C +D D ) 1 (17) with V = P + γ P U 1 P. Moreover, A e (I LC )A and A e A(I LC ) are asymptotically stable. Thus, we can search over γ (e.g., via bisection) to find the smallest γ and the corresponding optimal observer gain L in the minimum H -norm sense. Further, by upperbounding the estimation errors, we find the radii of the sets of compatible inputs and states to be (cf. proof in the appendix): δ 1 d = δx 0 V e Ae 1 + η w ( V M C 3 + V e A i eb e,w ) + η v ( V e A i e(b e,v1 + A e B e,v ) δ x, + V M T + V e B e,v + (V 1 V M C G 1 )M 1 T 1 + V e Ae B e,v1 ), (18) = δ0 x AAe 1 + η w ( AA i eb e,w + Be,w ) 3 +η v ( AA i e(b e,v1 + A e B e,v ) + Be,v + AA e B e,v1 + Be,v1 + AB e,v ), (19) b The DARE equation in (16) can be solved with control system software. For example, in MATLAB s Control System Toolbox, we can use the command DARE(A, C l, Φ(I + G 1M 1 M 1 G 1 )Φ, (C C 1 γ I) 1 ). δ x = δx 0 A e + η v ( B e,v + A 1 e B e,v1 1 + A i e(b e,v1 + A e B e,v ) ) + η w A i eb e,w, (0) where  A G 1M 1 C 1, Φ I G M C, A ΦÂ, V e V 1 M 1 C 1 + V M C Â, A e (I LC )A, Be,w ΦW, Be,v1 ΦG 1 M 1 T 1, Be,v G M T, B e,w (I LC )Be,w, B e,v1 (I LC )Be,v1 and B e,v (I LC )Be,v LT. Moreover, since A e is stable as a consequence of Theorem (hence, lim A e = 0), it is straightforward to see that the radii converge to finite steady-state values given by lim δ 1 d = δd, lim δ x, = δ x, and lim δ x = δx, where δ d η w ( V M C + lim V ea i eb e,w ) 3 +η v ( lim V ea i e(b e,v1 + A e B e,v ) + V M T + V e B e,v + (V 1 V M C G 1 )M 1 T 1 ) <, δ x, η w ( lim AAi eb e,w + Be,w ) 3 +η v ( lim AAi e(b e,v1 + A e B e,v ) + Be,v + AA e B e,v1 + Be,v1 + AB e,v ) <, δ x η w lim 1 + lim Ai eb e,w + η v ( B e,v Ai e(b e,v1 + A e B e,v ) ) <. Algorithm 1 summarizes the three steps of the fixed-order input and state set-valued observer, in which d, 1 is estimated before the time update, followed by the measurement update and finally, the estimation of d 1,. Note that we did not include δ x, and ˆX in the algorithm for conciseness. B. Observer Stability and Strong Detectability Next, we provide the relationship between observer stability and strong detectability (cf. Definition 1), whose proof will be provided in the appendix. Note that r(c G ) = p p H is a necessary condition for the boundedness of the set-valued estimates by Lemma 1 and this is assumed in the following results. It is also noteworthy that C G is the first invertibility matrix in 13 (similar to a Marov parameter). Lemma. All non-zero invariant zeros of the system (1) are eigenvalues/poles of the state matrices A e A(I LC ) and A e (I LC )A of the propagated and updated state estimation error dynamics x and x, respectively, for any observer gain L, where A Φ and Φ I G M C. Theorem 3 (Strong Detectability Observer Stability). Suppose Lemma 1 holds. Then, strong detectability is necessary and sufficient for asymptotic stability of the observer dynamics with A e and A e, and for the boundedness of the set-valued input and state estimates for any non-zero δ x 0. Theorem 4 (Strong Detectability and Existence of an H -Observer). Suppose Lemma 1 holds. Then, strong detectability guarantees that the Assumptions (A.1) and (A.) in Theorem hold. If p = l, then strong detectability also satisfies Assumption (A.3). Otherwise, that (A, G ) or (A, B ) is stabilizable on the unit circle also satisfies

5 Algorithm 1 Fixed-Order Input & State Set-Valued Observer 1: Initialize: M 1 = Σ 1 ; M = (C G ) ; Â = A G 1M 1C 1; Φ = I G M C ; A = ΦÂ; V e = V 1M 1C 1 + V M C Â; Compute L via Theorem and perform bisection to minimize γ. A e = (I LC )A; B e,w = (I LC )ΦW ; B e,v1 = (I LC )ΦG 1M 1T 1; B e,v = ((I LC )G M + L)T ; ˆx 0 0 = ˆx 0 = centroid( ˆX 0); δ0 x = min{ x ˆx 0 0 δ, x ˆX 0}; δ ˆd 1,0 = M 1(z 1,0 C 1ˆx 0 0 D 1u 0); : for = 1 to N do Estimation of d, 1 and d 1 3: ˆx 1 = Aˆx Bu 1 + G 1 ˆd1, 1 ; 4: ˆd, 1 = M (z, C ˆx 1 D u ); 5: ˆd 1 = V 1 ˆd1, 1 + V ˆd, 1 ; + η w( + V M C ) + η v( V M T + V eae + V eb e,v + (V 1 V M C G 1)M 1T 1 6: δ d 1 = δ x 0 V ea 1 e VeA i e B e,w B e,v1 + i=1 VeA i e (B e,v1 + A eb e,v) ); 7: ˆD 1 = {d R l : d ˆd 1 δ 1}; d Time update 8: ˆx = ˆx 1 + G ˆd, 1 ; Measurement update 9: ˆx = ˆx + L(z, C ˆx D u ); 10: δ x = δ0 x A e + η 1 w B e,v1 + + A 1 e 11: ˆX = {x R n : x ˆx δ x }; Estimation of d 1, 1: ˆd1, = M 1(z 1, C 1ˆx D 1u ); 13: end for Ai eb e,w + η v( B e,v Ai e(b e,v1 + A eb e,v) ); Assumption (A.3) in Theorem. Then, an H -observer with cost γ exist if U 0 and Ă is asymptotically stable (with U and Ă as defined in Theorem ). V. APPLICATION TO ATTACK-RESILIENT ESTIMATION In this section, we show that the proposed set-valued observer can be applied for attac-resilient estimation of state and attac signals when a system is subject to false data injection attacs on both the actuator and sensor signals. Note that we are not considering sparse false data injection attacs, which is a subject of ongoing research. But rather, we assume that the attac locations are nown (i.e., the G and H matrices that encode the attac locations are given) while the attac magnitudes d at any time are unnown. As previously discussed, the false data injection attac magnitudes d on the actuator and sensor signals are adversarial and strategic. Hence, we ought not mae any assumptions on the attac model (random or deterministic) because a strategic adversary could otherwise simply select a different attac model than is assumed. This non-assumption aligns perfectly with the unnown input signal model in Section II. Therefore, for any linear time-invariant bounded-error models of a cyber-physical system, the system description in (1) can capture false data injection attacs on the actuator and sensor signals on the system without any limitations. Moreover, the fixed-order simultaneous input and state setvalued observer proposed in Section IV is capable not only of reliably estimating the set of all compatible states, ˆX, but also of identifying the attac signals via the estimation of the set of all compatible unnown inputs, ˆD 1. A. Implications on Attac-Resilient Estimation Having established that the proposed set-valued observer is applicable to attac-resilient estimation, we now discuss the implication of the relationship between observer stability and strong detectability on resilient state estimation and attac identification. First, we introduce the following definitions: Definition 3 (Resilient Set of Compatible States). We say that the estimated set of compatible states is resilient, if for any initial state x 0 R n and signal attac sequence {d j } j N in R p, the true state x is contained in the set estimates ˆX and ˆX, and these sets remain bounded for all. Definition 4 (Data Injection Attac Identification). A false data injection attac is identified, if for any initial state x 0 R n and signal attac sequence {d j } j N in R p, the true attac signal d 1 is contained in the set estimate ˆD 1 and this set remains bounded for all. Definition 5 (Correctable false data injection attacs c ). We say that false data injection attacs on p actuators and sensors are correctable, if for any initial state x 0 R n and signal attac sequence {d j } j N in R p, we have a set-valued observer to compute the resilient set of compatible states and to identify the false data injection attac signals. Based on these definitions as well as the observer design in Section IV, we have the following conclusions: Proposition 1 (Resilient Set-Valued State Estimation). A resilient set of compatible sets can be obtained if and only if the system (1) is strongly detectable and Lemma 1 holds. Proposition (Attac Identification). A false data injection attac is identified if and only if the system (1) is strongly detectable and Lemma 1 holds. Note that strong detectability is a system property that is independent of the observer design. Hence, the necessity of strong detectability can serve as a guide to determine and recommend which actuators or sensors need to be safeguarded to guarantee resilient estimation as a preventative attac mitigation method (cf. Section VI-B for an example). Moreover, we can derive an upper bound on the maximum number of false data injection attacs that can be asymptotically corrected based on strong detectability. Theorem 5. The maximum number of correctable actuators and sensors signal attacs, p, for system (1) is equal to the number of sensors, l, i.e., p l (upper bound is achievable) d. Proof. The theorem follows immediately as an implication of Theorem 3 and 7, Theorem 1, 8, Theorem 4.3. c This definition is distinct from 5, Definition 1 that is defined for exact finite-time point estimation (after n steps) and requires strong observability 5. Instead, it is related to boundedness in infinite time, similar to infinitetime point estimation that only requires strong detectability 7, 8. d By contrast, the stronger requirement of strong observability in 5 (implies strong detectability 14) leads to a maximum of p l 1.

6 A. Benchmar System VI. SIMULATION EXAMPLES In this example, we consider a system that has been used as a benchmar for many state and input filters (e.g., 14): A = ; G = ; H = ; B = 0 5 1; C = I 5; D = The unnown inputs used in this example are as given in Fig. 1, while the initial state estimate and noise signals (drawn uniformly) have bounds δ x = 0.5, η w = 0.0 and η v = The invariant zeros of the system matrix R S (z) are {0.3, 0.8}. Thus, this system is strongly detectable. We observe from Fig. 1 that the proposed algorithm is able to find set-valued estimates of the states and unnown inputs. The actual estimation errors are also within the predicted upper bounds (cf. Fig. ), which converges to a steady-state as established in Section IV-A. Moreover, with the gain L of the H observer, the eigenvalues of A e = A(I LC ) and A e = (I LC )A are {0, 0.107, 0.3, 0.407, 0.8}. Hence, as is predicted in Lemma, all invariant zeros of the system are eigenvalues of the set-valued observer. Fig. 1: Actual states x 1, x, x 3, x 4, x 5 and their estimates, as well as unnown inputs d 1, d and d 3 and their estimates. Fig. : Actual estimation errors and radii of set-valued estimates of states, x, δ x, and unnown inputs, d, δ d. B. Attac-Resilient Estimation We now demonstrate the effectiveness of our attacresilient H observer using an IEEE 14-bus system 33 that is subject to data injection attacs. The system consists of 5 synchronous generators and 14 buses, with secure phasor measurement units (PMUs) being installed on the buses depicted in Fig. 3. It is represented by n = 10 states comprising the rotor angles and frequencies of each Fig. 3: IEEE 14-bus electric power system 33. Fig. 4: Resilient estimation errors and radii of set-valued estimates of states, x, δ x, and attac signals, d, δ d. generator. The dynamics of the system can be represented by an uncertain LTI model 3, 5 that is discretized with a sampling interval of dt = 0.05s to obtain the model in (1), where p = 35 sensors are deployed to measure the real power injections at every bus, the real power flows along every branch and the rotor angle of generator 1. The initial state estimate and noise signals (drawn uniformly) have bounds δ x = 1, η w = 0.03 and η v = In this example, we assume that all unsecured PMU measurements are attaced (sensor attacs with nown locations). Nevertheless, we observe from Fig. 4 that the proposed algorithm is able to find set-valued state estimates that contain the true state, as well as to identify a bounded set that contains the actual attac signals. Individual state and attac signals as in Fig. 1 can also be estimated but are omitted for brevity. Moreover, performing strong detectability tests (necessary condition by Propositions 1 and ) for various attac locations, i.e., G and H, we found that false data injection attacs on the sensors are correctable if at least one sensor from among sensors 10, 14 and 15 is protected. VII. CONCLUSION We presented an optimal fixed-order set-valued observer that simultaneously computes the set of compatible states and unnown inputs for linear discrete-time bounded-error systems in the minimum H -norm sense. Necessary and sufficient conditions for the stability of the observer and its connection to strong detectability were also derived. Then, we showed that the proposed set-valued observer is applicable for attac-resilient estimation of state and attac signals when cyber-physical systems are subject to malicious false data injection attacs on both actuator and sensor signals, as well as discussed the implication of strong detectability on resilient state estimation and attac identification. REFERENCES 1 A.A. Cárdenas, S. Amin, and S. Sastry. Research challenges for the security of control systems. In Conference on Hot Topics in Security, pages 6:1 6:6, 008. J.P. Farwell and R. Rohozinsi. Stuxnet and the future of cyber war. Survival, 53(1):3 40, 011.

7 3 K. Zetter. Inside the cunning, unprecedented hac of Uraine s power grid. Wired Magazine, P.K. Kitanidis. Unbiased minimum-variance linear state estimation. Automatica, 3(6): , November R. Patton, R. Clar, and P.M. Fran. Fault diagnosis in dynamic systems: theory and applications. Prentice Hall, G. De Nicolao, G. Sparacino, and C. Cobelli. Nonparametric input estimation in physiological systems: Problems, methods, and case studies. Automatica, 33(5): , M. Corless and J. Tu. State and input estimation for a class of uncertain systems. Automatica, 34(6): , C.P. Tan and C. Edwards. Sliding mode observers for detection and reconstruction of sensor faults. Automatica, 38(10): , K. Kalsi, J. Lian, S. Hui, and S.H. Za. Sliding-mode observers for systems with unnown inputs: A high-gain approach. Automatica, 46(): , H. Fang, Y. Shi, and J. Yi. On stable simultaneous input and state estimation for discrete-time linear systems. International Journal of Adaptive Control and Signal Processing, 5(8): , S. Gillijns and B. De Moor. Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, 43(1): , January S. Gillijns and B. De Moor. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43(5): , S.Z. Yong, M. Zhu, and E. Frazzoli. On strong detectability and simultaneous input and state estimation with a delay. In IEEE Conference on Decision and Control (CDC), pages , S.Z. Yong, M. Zhu, and E. Frazzoli. A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. Automatica, 63:31 39, I. Yaesh and U. Shaed. A transfer function approach to the problems of discrete-time systems: H -optimal linear control and filtering. IEEE Transactions on Automatic Control, 36(11): , P.P. Khargonear and M.A. Rotea. Mixed H //H filtering. In IEEE Conference on Decision and Control, pages , K. Zhou, J. C. Doyle, and K. Glover. Robust and optimal control, volume 40. Prentice Hall New Jersey, F. Blanchini and M. Sznaier. A convex optimization approach to synthesizing bounded complexity l filters. IEEE Transactions on Automatic Control, 57(1):16 1, M. Milanese and A. Vicino. Optimal estimation theory for dynamic systems with set membership uncertainty: An overview. Automatica, 7(6): , M.A. Dahleh and I.J. Diaz-Bobillo. Control of uncertain systems: a linear programming approach. Prentice-Hall, Inc., J.S. Shamma and K. Tu. Set-valued observers and optimal disturbance rejection. IEEE Trans. on Automatic Control, 44():53 64, J. Chen and C.M. Lagoa. Observer design for a class of switched systems. In IEEE Conference on Decision and Control European Control Conference, pages , F. Pasqualetti, F. Dörfler, and F. Bullo. Attac detection and identification in cyber-physical systems. IEEE Transactions on Automatic Control, 58(11):715 79, November M.S. Chong, M. Waaii, and J.P. Hespanha. Observability of linear systems under adversarial attacs. In IEEE American Control Conference (ACC), pages , H. Fawzi, P. Tabuada, and S. Diggavi. Secure estimation and control for cyber-physical systems under adversarial attacs. IEEE Transactions on Automatic Control, 59(6): , June Y. Shoury, P. Nuzzo, A. Puggelli, A.L. Sangiovanni-Vincentelli, S.A. Seshia, M. Srivastava, and P. Tabuada. Imhotep-SMT: A satisfiability modulo theory solver for secure state estimation. In 13th International Worshop on Satisfiability Modulo Theories (SMT), S.Z. Yong, M. Zhu, and E. Frazzoli. Resilient state estimation against switching attacs on stochastic cyber-physical systems. In IEEE Conference on Decision and Control, pages , S.Z. Yong, M. Zhu, and E. Frazzoli. Switching and data injection attacs on stochastic cyber-physical systems: Modeling, resilient estimation and attac mitigation. arxiv: , H. Kim, P. Guo, M. Zhu, and P. Liu. Attac-resilient estimation of switched nonlinear cyber-physical systems. arxiv: , M. Pajic, P. Tabuada, I. Lee, and G.J. Pappas. Attac-resilient state estimation in the presence of noise. In IEEE Conference on Decision and Control (CDC), pages , Y. Naahira and Y. Mo. Dynamic state estimation in the presence of compromised sensory data. In IEEE Conference on Decision and Control (CDC), pages , S.Z. Yong, M.Q. Foo, and E. Frazzoli. Robust and resilient estimation for cyber-physical systems under adversarial attacs. In IEEE American Control Conference (ACC), pages , J. Kim and L. Tong. On phasor measurement unit placement against state and topology attacs. In IEEE International Conference on Smart Grid Communications, pages , L. Xie, C. E. de Souza, and M. Fu. H estimation for discretetime linear uncertain systems. International Journal of Robust and Nonlinear Control, 1():111 13, C. De Souza, M. Gevers, and G. Goodwin. Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices. IEEE Trans. on Automatic Control, 31(9): , R.A. Horn and C.R. Johnson. Matrix analysis. Cambridge University Press, 01. A. Proof of Lemma 1 APPENDIX: PROOFS We observe from (5), (9) and (10) that ˆd 1, = M 1 (C 1 x + Σd 1, + v 1, ), (1) ˆd, 1 = M (C (A 1 x G 1 d1, 1 + w 1 ) + v, + C G d, 1 ). () Then, from (1) and (13), as well as (4) and (14), the propagated and updated state estimate errors are found as x = A x G 1 d1, 1 + G d, 1 + w 1, (3) x = (I LC) x LU 1Σd 1, Lv. (4) We show by induction that the estimates ˆd, ˆx and ˆx are bounded, assuming at the moment that their dynamics are stable (which we will show to hold in Theorem 3). For the base case, since ˆx 0 0, ˆx 0 0 and the noise signals are bounded by assumption, from (1) and (), we find that ˆd1,0 and ˆd,0 are bounded, only if M 1 Σ = I and M C G = I, since we assumed that d 1,0 and d,0 can be unbounded. Hence, ˆd0 is bounded. In the inductive step, we assume that x 1 1 and x 1 1 are bounded. Then, the input estimates d 1, 1 and d, 1 are bounded, only if M 1 Σ = I and M C G = I, since the unnown inputs d 1, and d, 1 can be unbounded although the noise signals are bounded. Similarly, from (4) with a bounded measurement noise, we need the constraint LU 1 = 0 such that x remains bounded. Thus, by induction, x and x are bounded for all. Since we require M C G = I for the existence of bounded input estimates, it follows that r(c G ) = p p H is a necessary condition. Furthermore, L = LUU = LU U = LU since LU 1 = 0. B. Proof of Theorem First, we reduce the system with unnown inputs to one without unnown inputs. From (14) and (5), we have x = x L(C x + v,). Then, substituting (1) with M 1 = Σ 1 into (3) and the above, and rearranging, we obtain x = A x w 1 L (C A x C w 1 + v, ), (5) with the state matrix A (I G M C )Â and noise signal w 1 (I G M C )(G 1 M 1 v 1, 1 w 1 )

8 G M v,. As it turns out, the updated state estimate error dynamics above is the same for an a posteriori H filter 15, Eq. (5.) for a linear system without unnown inputs x e, +1 = Axe, + w, y e, = C x e, + v,. Since the objective for both systems is the same, i.e., to obtain the observer gain L with an a posteriori H filter, they are equivalent systems from the perspective of estimation. Furthermore, from the analysis in 15, Eq. (5.3), it can be seen that an equivalent observer gain L can be obtained with the a priori H filter for the following system x e +1 = A x e + B w e, y e = C x e + D w e, w e 1 v,+1 1 v, with A A, B ΦW ΦG 1 M 1 G M 0, C C A, D I + C B and, w v 1, where the fac- 1 tors are included such that lim we i = lim wi vi. The design of the observer gain L is then a direct application of the a priori H filter (e.g., 34, Theorem.). Next, we consider the following identity ω 0, π (equivalently, z C, z = 1) A e r jω I B A zi B = r C D C D A zi ΦW ΦG1 M = r 1 G M 0 C A I A zi ΦW ΦG1 M = r 1 (6) G M I = r A zi ΦW ΦG 1 M 1 G M + l ph, where the third equality is obtained by subtracting C times the first row from the second row, and then subtracting z times the first column to the final column. Hence, Assumption (A.3) implies that (A, B ) is stabilizable on the unit circle, which along with the Assumption (A.1) are are necessary and sufficient conditions for the stability of A e 35 and by extension, A e, since A e and A e have the same eigenvalues 36, Theorem C. Proof of Error Bounds/Set Radii (18), (19) and (0) From (1) (4), we find the state estimation errors as x = A x 1 1 +B e,ww 1 +B e,v1v 1 +B e,vv, x = (I LC ) x Lv,, (7) where  A G 1M 1 C 1, Φ I G M C, A ΦÂ, B e,w ΦW, Be,v1 ΦG 1 M 1 T 1 and Be,v G M T. Then, via substitution, we obtain the estimation errors in terms of the initial state error x 0 0 and noise signals: x = A e x Ae 1 Be,w B e,v1 w0 + B e,v v + 1 i=1 A 1 i e Be,w B e,v1 + A e B e,v wi, x = AA 1 e x AAe Be,w B e,v1 w0 +Be,ww 1 + (Be,v1 + AB e,v )v 1 + Be,vv + i=1 AA 1 i e Be,w B e,v1 + A e B e,v wi, with w i =, wi vi Ae (I LC )A, B e,w (I LC )Be,w, B e,v1 (I LC )Be,v1 and B e,v (I LC )Be,v LT. Moreover, we find the unnown input estimation error as d 1 = V 1 d1, 1 + V d, 1 = V e Ae 1 x 0 V e Ae Be,w B e,v1 w0 V M C w 1 V M T v +(V e B e,v1 + (V 1 V M C G 1 )M 1 T 1 )v 1 + i=1 V ea 1 i e Be,w B e,v1 + A e B e,v wi, with V e V 1 M 1 C 1 + V M C Â. Finally, the error bounds (18), (19), (0) can be found using triangle inequalities. D. Proof of Lemma Let z be any invariant zero of system (1), i.e., when R S (z) drops ran (cf. Theorem 1). Then, there exists ν µ 0 such that (zi A)ν G µ = 0, C ν = 0. Premultiplying the former with (I G M C ) and applying the latter as well as the fact that M C G = I, we have (I G M C )(zi Â)ν +(I G M C )G µ = 0 = (zi A)ν = (zi A)ν +A LC ν = (zi A(I LC ))ν. If ν = 0, then G µ = 0 and, in turn, µ = 0, which is a contradiction. Hence, ν 0 and the determinant of zi (A A LC ) is zero, i.e., any invariant zero of R S (z) is also an eigenvalue of the propagated state estimation error dynamics of x, which can be found from (7) to be x = A(I LC ) x B e,ww 1 +(B e,v1 A LT )v 1 + B e,vv, (8) and by extension, the state matrix A e in (5), since A e and A e have the same eigenvalues 36, Theorem E. Proof of Theorem 3 Let Lemma 1 hold. We will prove that strong detectability implies that state estimation errors with A e = A(I LC ) and A e = (I LC )A (they have same eigenvalues 36) are asymptotically stable and bounded, and vice versa. ( ): By Theorem 1, strong detectability implies that (A, C A) is detectable. Hence, there exists L such that A e, and by extension A e, are asymptotically stable, which in turn implies bounded-input, bounded-state stability of (5), (8). ( ): We will show this by contraposition. By Lemma, we now that (1) being not strongly detectable implies that A e, and by extension A e, are unstable. This implies that lim A e = lim (A e) =. Thus, from (19) and (0), the estimation errors are unbounded for any δ x 0 0. F. Proof of Theorem 4 By Theorem 1, strong detectability implies that Assumption (A.1) holds. Moreover, D D = C (G M M G + Φ(W W +G 1 M 1 M1 G 1 )Φ )C +I 0 readily satisfies Assumption (A.). Finally, for Assumption (A.3) to hold, we require that r A zi ΦW ΦG 1 M 1 G M = n (cf. (6)), which is satisfied if (A, G ) or (A, B ) is stabilizable on the unit circle. Moreover, if p = l, from Theorem 1, strong detectability satisfies the ran condition above and thus Assumption (A.3).