A Study Of Optimal False Information Injection Attack On Dynamic State Estimation in Multi-Sensor Systems

1 A Study Of Optimal F alse Inf ormation Injection Attack On Dynamic State Estimation in Multi-Sensor Systems Jingyang Lu and Ruixin Niu Abstract In this pap er , the impact of false info rmation injection is investigated for linear dynam ic systems with multiple sensors. It is assumed that the system is unsuspecting the existence of false information and the adversary is trying to maximize the negati ve effect of the false information on Kalman fil ter ’ s estimation perfor mance. The false in formation attack und er different cond itions is mathem atically characteriz e d . For th e adversary , many closed-for m results f or th e optimal attack strategies that maximize Kalman filter’ s estimation error are theo retically derived. It is shown that by ch oosing the optimal corr elation coefficients am ong the bias no ises and allocating power optimally amo ng sensor s, the adversary cou ld significantly incre ase Kalman filter’ s estimation errors. T o be co n crete, a target tracking sy stem is used as an examp le in the paper . From the adversary’ s point o f view , the best attack stra tegies are obtained und er different scenarios, inclu ding a single-sen sor system with bo th position and velocity measuremen ts, and a multi-sensor system with position an d velo c ity measurem ents. Under a constrain t on the total power of the injected bias noises, the optimal solu tions ar e solved fr o m two perspectives: trace and d eterminant. Nu merical resu lts are also provided in o rder to illustrate the effecti veness of the propo sed a ttac k strategies. I . I N T R O D U C T I O N System state estimation in the presence of an adver sary that injects false information into sensor re adin gs h as attracted much attention in wide app l ication areas, such as target tracking with compromised sensors, secure monitoring o f dynamic electric power systems and radar tracking and detection in t he presence of jammers [1]. Thi s topic h as been studi ed in [2]–[9]. In [2], the problem of t aking adv antage of the power system configuration t o introduce ar bi trary bias to the system without being detected was in vestigated and insp i red many researchers further study false information i n jection along this di rectio n . [3] shows the impact of malicious attacks September 24, 2018 DRAFT 2 on real-time e l ectricity mark et concerning the locational marginal price and how the attacker s can m ake profit by mani pulating certain values of the measurements. Some certain strategies are also provied to find the optim al single attack vector . The relationship between the attackers and the control center was discus sed in [4], where both the adversary’ s attacking st rategies and the control center’ s detection algorithms have been propos ed. Refer to [5] and [6] for m ore about false information att acks on the electricity m arket. Ins p ired by [2], the data frame attack in which deleting the comprised sensors the defender system detects will m ake the system unobs ervable was formulated as a quadraticall y cons trained quadratic program (QCQP) in [7]. In [8], [9], the relation between a tar get and a MIMO radar was characterized as a t wo-person zero-sum game. Howe ver , in the aforementioned publi cations, only the problem of static system state estimation has been considered. In this paper , for a lin ear dynamic system, we analyze the impact of the injected false information on Kalman filter’ s state esti mation performance over time, which has no t got much attention in the literature. Some related publications e xi st o n sensor management [10]–[12], where the problem o f arra ng i ng the sensors to min i mize the co variance o f the state es t imation error so t hat a more accurate stat e estimate can b e obtain ed is in vestigated. Th is problem is clearly opposite to the problem we study in the p aper , where the goal for the adversary i s to maximize the mean s quare st ate estim at i on error matrix, and to confuse Kalman filt er . In [13], the prob lem of sensor bias estimati on and comp ensation for target tracking has been add ressed. Interested readers are referred to [13] and the references therein for details. In [14], we ha ve studi ed the imp act of the injected bi ases on a Kalman filter’ s estimation performance, showing t hat if the f als e in form ation is injected at a single time, its impact con ver ges to zero as tim e goes on; if the false information is injected into t h e system continuously , the estimation error tends to reach a st eady state. In [15], we have found the best strategies for the adversary to attack Kalman filter system from the perspecti ve of the t race of the mean squared error (M SE) matrix, and obtained so me close-form resul ts. Also in [16], based on th e pre vi ous work, the problem is furt her refined regarding the determinant of MSE matrix, which the correlation among the elements is taken into consideration . In [17], [18], the Kalman filter system has been in vestigated regarding t h e system robustness in the case where sens or reading is continuously j am med b y the false informati on using Greedy search and dynam ic programm ing. Howe ver , the it is of great challenge to find the closed form solut ion in term of determinant of the MSE matrix and opt imal solution to th e case in which the Kalman filter sy s tem i s com promised September 24, 2018 DRAFT 3 by the false information cont inuously . Considering the problems mentio n ed above, in this p aper , our go al is to find the closed form optim al at t ack strategy for the adversary , which maximizes the impact of th e false inform ation inj ecti on on Kalman filter’ s state esti mation from the determinant perspectiv e. By adopting the objective function as the determinant of the M SE matrix, we change the problem significantly . As shown later in the paper , the optimal attack strategy that maxi m izes the determinant of the MSE matrix i s a functi on of Kalman filter’ s state estimatio n covariance and hence ”adapti ve” t o Kalman filter; wh ereas that maximizing t he trace of the MSE matrix is not a functi o n of Kalman filter’ s st ate estimation cov ariance. Previous works concentrated more on the si t uation where the adversary attacks the system by a single shot . In this paper , the problem of continuous attack i s also in vestigated. The rest of paper is organized as follows. Section II generally describes the discrete-time linear dyn amic s y s tem. Section III m athematically characterizes the impact of determined or random false informati on on Kalman filter’ s system. Section IV and V analyze ho w to get the best strategy to attack Kalman filter’ s system from trace and determinant cases standing from the perspective of the adversary . Und er the constraint on the adversary’ s total sens or b ias noise po wer , dif ferent strategies are proposed to maximize Kalm an filt er’ s mean squared state estimation error for dif ferent s cenarios. Section VI pro vides t h e simulati on results and Section VII concludes the paper . I I . K A L M A N FI L T E R S Y S T E M I I I . S Y S T E M M O D E L The discrete-time linear dynamic system can be described as below , x k +1 = F k x k + G k u k + v k (1) where F k is the system state transition matrix, x k is the system state vector at time k , u k is a known input vec t o r , G k is the input gain matrix, and v k is a zero-mean white Gaussi an process noise with covariance matrix E [ v k v T k ] = Q k . Let us assu m e that M sensors are used b y the linear syst em. Th e measurement at time k collected by sensor i is z k ,i = H k ,i x k ,i + w k ,i (2) with H k ,i being the measurement matrix, and w k ,i a zero-mean white Gaussian measurement noise with covariance matrix E [ w k ,i w T k ,i ] = R k ,i , for i = 1 , · · · , M . W e further assume t hat September 24, 2018 DRAFT 4 the measurement noises are independent across sensors. The matrices F k , G k , H k ,i , Q k , and R k ,i are ass umed to be k n own with proper dimensions . In this paper , we as s ume that a bias b k ,i is injected by the adversary into th e measurement of the i th s ensor at time k intentionally . Therefore, the m easurement equation (2) becomes z ′ k ,i = H k ,i x k + w k ,i + b k ,i = z k ,i + b k ,i (3) where z ′ k ,i is the corrupted measurement, b k ,i is either an unknown cons tant or a random variable independent of { v k ,i } and { w k ,i } . For com pactness, let us denote the system sensor observ ation as z k = [ z T k 1 , · · · , z T k M ] T , which contains the o bserva t ions from all the M sensors. Similarly , let us denote t he system b ias ve ctor as b k = [ b T k 1 , · · · , b T k M ] T which includes the biases at all t he M sens ors. Correspondingly , the measurement matrix becom es H k = [ H T k 1 , · · · , H T k M ] T (4) W ith these notations, it is easy to con vert (2) and (3) into the foll owing equations respectively . z k = H k x k + w k (5) and z ′ k = z k + b k (6) Further , we have the measurement error cov ariance matrix correspond i ng to w k is R k =      R k , 1 · · · 0 . . . . . . . . . 0 · · · R k ,M      (7) which i s obt ained by using the assumptio n th at measurement noi ses are independent across sensors. I V . I M P A C T O F F A L S E I N F O R M A T I O N I N J E C T I O N In th i s paper , let us assume that the adversary attacks t he system by i n jecting false information into the sens o rs while unawar e of such attacks. W e start with the case where b iases ( b k ) are continuously inj ected i nto the system starti n g from a certain time K . Note that single injection is just a special case of continuous injectio n when b k are s et to be nonzero at time K and zero otherwise. September 24, 2018 DRAFT 5 In the cont i nuous injection case, Kalman filter’ e x t ra state estim ation error , which is caused by the cont inuous bias i njection alone, is deri ved in [19] and provided as foll ows. Pr o position 1. Kalman filter’ s s t ate estimatio n err or a t time K + N is ˆ x ′ K + N | K + N − x K + N = ˆ x K + N | K + N − x K + N + N X m =0 m − 1 Y i =0 B K + N − i ! W K + N − m b K + N − m (8) wher e ˆ x ′ K + N | K + N is K al man filter’ s st ate est imate in the pr esence of the bias s equ ence { b k } , ˆ x K + N | K + N is Kalman filter’ s state esti mate in the absence of t h e bias, B K , ( I − W K H K ) F K − 1 , (9) I is the identity matrix, an d W K is Kalman filter gain [20] at time K . As a re s u lt, the extra state estimatio n err or at time K + N due to t he continuou s bias b k injected at and af ter time K is N X m =0 m − 1 Y i =0 B K + N − i ! W K + N − m b K + N − m , (10) If { b k } is a zero-mean, random, and independent sequence, the extra m ean squared error (EMSE) at a particular time instant K + N due to t he bi as alone is provided in the following proposition . Usin g the results from Proposition 1 , the proof of Proposition 2 is provided as well. Pr o position 2. When the bias sequence { b k } is zer o mean, r an d om, and independent over ti me, the E M S E at time K + N due to t he biases injected at and after time K , denoted as A K + N , is A K + N = N X m =0 D m Σ K + N − m D T m (11) wher e D m = m − 1 Y i =0 B K + N − i ! W K + N − m (12) Q − 1 i =0 B K + N − i = I is an identity matrix, a n d Σ K + N − m is the covariance matr ix of b K + N − m . Pr oof Sketche s : Let us d enote ˜ x K + N | K + N = ˆ x K + N | K + N − x K + N as Kalman filter’ s state e s t i- mation error i n the absence of any false information, and a m = m − 1 Y i =0 B K + N − i ! W K + N − m b k + N − m (13) September 24, 2018 DRAFT 6 From (8), we can get A K + N = E   ˜ x K + N | K + N + N X m =0 a m ! ˜ x K + N | K + N + N X n =0 a n ! T   − E  ˜ x K + N | K + N ˜ x T K + N | K + N  = E ˜ x K + N | K + N N X n =0 a T n ! + E N X m =0 a m ˜ x T K + N | K + N ! + E N X m =0 N X n =0 a m a T n ! = E N X m =0 N X n =0 a m a T n ! where the last line is due t o t h e fa ct that a m and a n hav e zero mean, are independent from each other when m 6 = n , and are independent from ˜ x K + N | K + N . Using this fac t again, we further ha ve E N X m =0 N X n =0 a m a T n ! = E N X m =0 a m a T m ! (14) = N X m =0 D m Σ K + N − m D T m where D m has been defined in Proposit i on 2 . V . T H E O P T I M A L A T T A C K S T R A T E G Y 1) Pr oblem F ormulation for a General Linear System : In this paper , we i n vestigate the optimal attack strategy that an adversary can adopt to maximize th e system estimator’ s estim ati on error . This problem can be formulated as a constrained opti m ization problem. W ithout loss of generality , let us consider that the attacker is in terested in maximi zing the system st ate estimatio n error at t ime K ri g ht after a single false bi as is injected at ti me K . In this case, we are interested in designing the injected random bias’ cov ariance matrix such that max Σ K T r  P K | K + A K ( Σ K )  s.t. T r( Σ K ) = a 2 (15) where a is a constant , T r( · ) is the m at ri x trace operator , and P K | K is Kalman filt er’ s stat e estimation error cova riance matrix at time K in the absence of any false information. Note th at September 24, 2018 DRAFT 7 it is m eaningful to ha ve a con straint on t he trace of Σ K , since it can be deemed as the po wer of injected sensor bi as b K , and a s m aller power for b K reduces the probabilit y that the adver s ary is detected by the system estimator usi ng an innova t i on based detector . Note that t he opt i mization problem is equ iv alent to one that m axi mizes T r ( A K ( Σ K )) , since P K | K is not a function of Σ K , and trace is a linear operator . If one is more interested in the determinant of the estimation MSE matrix, a s i milar optim ization probl em can be easil y formulated as follows. max Σ K   P K | K + A K ( Σ K )   s.t. T r( Σ K ) = a 2 (16) 2) Equivalent Meas ur ement in Multi-Senso r Syst ems: T o s i mplify the mathematical analysis, it is helpful to deriv e the equiv alent sensor measurement, which is a lin ear combi nation o f t h e observations from all t he sensors, and is a su f ficient s tatistic contain ing all the i nformation about the s ystems s t ate. The equi valent sensor measurement vector and its corresponding cov ariance matrix should ha ve much sm aller dimens ion than the original measurement vector and its cov ariance, making the math em atical manipulation and deriv ation later in the paper much simpl er . In a i nformation filter recursion [20], which i s equiv alent to K al m an filter recursion, we hav e ˆ y k | k = ˆ y k | k − 1 + H T k R − 1 k z k (17) where ˆ y k | k = P − 1 k | k x k | k and ˆ y k | k − 1 = P − 1 k | k − 1 x k | k − 1 . It is clear that ˆ y k | k − 1 represents t h e prior knowledge about th e system state based on past sens o r data, and the second t erm in (17) represents the new i nformation from the new sensor data z k , which can be e xpanded by usin g (4) and (7) as fol l ows. H T k R − 1 k z k = [ H T k 1 , · · · , H T k M ]      R − 1 k 1 · · · 0 . . . . . . . . . 0 · · · R − 1 k M           z k 1 . . . z k M      = M X i =1 H T k i R − 1 k i z k i (18) In the following deriv ation s, we skip the time index k for simplicit y . Our purpose is to find an equiv alent measurement z e such that z e = H e x + w e (19) September 24, 2018 DRAFT 8 where w e ∼ N ( 0 , R e ) , and H T e R − 1 e z e = M X i =1 H T i R − 1 i z i (20) Let us consider t wo cases. First, suppos e all the H i s are the sam e ( H i = H ) , then i t i s natu ral to set H e = H . Note that a su fficient condition for (20) to be true is z e = R e M X i =1 R − 1 i z i (21) T aking t h e cov ariance on the both si d es of (21), we get R e = R e co v M X i =1 R − 1 i z i ! R T e = R e " M X i =1 R − 1 i R i ( R − 1 i ) T # R T e (22) This impl ies t h at R e = M X i =1 R − 1 i ! − 1 (23) In the second case, let us assum e that the system state x is ob s erv able based on the observ ations from all the sensors, meaning t hat the Fisher information m atri x P M i =1 H T i R − 1 i H i is inv ertible. In thi s case, by setting H e = I , using (20), and following a similar procedure as in the first case, we have z e = R e M X i =1 H T i R − 1 i z i (24) and R e = M X i =1 H T i R − 1 i H i ! − 1 (25) V I . A T A R G E T T RA C K I N G E X A M P L E In this paper , we give a concrete tar get tracking e xam p le. W e assume that the tar get moves in a 1-dimensional space according to a discrete w h i te no ise acceler ati on model [20], which can still be d escrib ed by the pl ant and measurement equ at i ons given in (1) and (2). In such a system, the state is defined as x k = [ ξ k ˙ ξ k ] T , where ξ k and ˙ ξ k denote the target’ s posi t ion and velocity at tim e k respectively . The in p ut u k is a zero sequence. The s tate transit i on m atrix is F =   1 T 0 1   (26) September 24, 2018 DRAFT 9 where T is the time between measurements. The process noise is v k = Γ v k , where v k is a zero mean white accelera t i on noise, with va riance σ 2 v , and the vector gain multi p lying the scalar process noise is given by Γ T = [ T 2 / 2 T ] . The cov ariance matrix of the process noise is therefore Q = σ 2 v ΓΓ T . In thi s paper , we in vestigate the attack st rategies for t wo scenarios. In t he first scenario, only position measurements are ava i l able to the senso rs, whereas in t he second s cenario, t he sensors measure both position and velocity of the target. A. Att a ck Strag eti es Anal ysis F r om T race perspective 1) Attack Strate gy F or Mul t iple P osition Sensors: In this case, it i s assu med t hat at each sensor , only the position measurement is a vailable, so that H i = [1 0] . At each sensor , the measurement noise process is zero-mean, whi te, and with variance, σ 2 w i . In order to si m plify the problem, we think of z e k as th e equiv alent measurement, which i s a lin ear combination o f the m easurements from all the sensors. Using the result s we deri ved in Section V -2 for the first case, namely (21) and (23), the measurement equati o n (3) becomes z ′ k = z ek + b ek (27) where z ek = M X m =0 c i z k i (28) b ek = M X m =0 c i b k i (29) and c i = 1 /σ 2 w i P M j =1  1 /σ 2 w j  (30) which is the corresponding coef ficient/weight for the i th sensor . In th i s tar get tracking problem, let us first consider the strate gy that maximi zes the trace of the Kalamn filter est i mation error , which is the solution of (15) in Section V -1. In t h is case, Σ K =        σ 2 b 1 ρ 12 σ b 1 σ b 2 · · · ρ 1 M σ b 1 σ b M ρ 12 σ b 1 σ b 2 σ 2 b 2 · · · ρ 2 M σ b 2 σ b M . . . . . . . . . . . . ρ 1 M σ b 1 σ b M ρ 2 M σ b 2 σ b M · · · σ 2 b M        (31) September 24, 2018 DRAFT 10 where σ 2 b i is the v ariance of the random bias injected at the i th s ensor ( b i ), and ρ ij is t he correlation coeffi cient between b i and b j . Therefore, (15) is equ ivalent to max T r [ A K ] s.t. M X i =1 σ 2 b i = a 2 − 1 ≤ ρ ij ≤ 1 , for 1 ≤ i, j ≤ M (32) T o simplify this problem , we first use t h e equiva l ent measurement to con vert th e multi -sens or problem to a single sensor problem. Namely , in Propositi o n 2 by replacing H k =      1 0 . . . . . . 1 0      with H e = [1 0] , and replacing Σ K with Σ e K = E [ b 2 e K ] (33) = E   M X i =1 c i b i ! 2   = M X i =1 c 2 i σ 2 b i + X i X j 6 = i 2 ρ ij c i c j σ b i σ b j we can easily show that A K = D 0 Σ e K D T 0 . Since Σ e K is a scalar a nd D 0 is not a function of Σ K , maxim izing the trace of A K is equiv alent to maxim izing Σ e K . First, let us consider t he case where the random biases at different s ensors are independent , meaning th at ρ i,j = 0 for 1 ≤ i, j ≤ M . The opt imal strategy for the adv ersary in this case is clearly to put all the bias power to the sensor with the largest coefficient c i : Pr o position 3. F or a system with M sensors, if the adversary in j ects i n dependent random n o ises, the best s trate g y is to allo cat e all the power to the sensor with sma llest noise vari ance. Next, let us consider the more general case where the random biases are dependent. By inspecting (33), it is clear that to maximize Σ e K , we need to set all the ρ ij s to 1. As a resul t, (33) becomes Σ e K = M X i =1 c i σ b i ! 2 (34) September 24, 2018 DRAFT 11 Now , the op t imization problem i n (32) has been con verted to the following problem: max M X i =1 c i σ b i ! 2 s.t. M X i =1 σ 2 b i = a 2 (35) The above problem can be solved by using standard constrained optimization techniques [21] based on gradient and H ess ian, which ar e rather inv olved. Here we solve the problem using a much simpler geometric solutio n, wh i ch has been shown to gi ve the same solution as that by the standard optimi zation techniques. W e start with the simplest case with two sensors, i n which we need t o solve the following opt i mization problem. max c 1 σ b 1 + c 2 σ b 2 (36) s.t. σ 2 b 1 + σ 2 b 2 = a 2 W e can get the optimal solution by analyzin g t he problem geometri call y wi th the norm vector ( c 1 , c 2 ) T of the objective functi o n as sho wn in the Fig. 1. The constraint of the problem is represented by the circle with a radius of a . W e move the l ine l 1 with the slo p e − c 1 c 2 to get the largest i ntercept between l 1 and σ 2 axis under th e constraint t hat t h ere is an intersection between the circle and the line l 1 . The corresponding optimal solu t ion is foun d when l 1 becomes a tangent li ne to th e circle, which i s σ 1 = c 1 a p c 2 1 + c 2 2 σ 2 = c 2 a p c 2 1 + c 2 2 (37) For a system with arbitrary n u m ber of sens o rs, we can repeat the same procedure to find the optimal solution by usi ng h y perplanes and hyperspheres. In general, the optimal attack strategy can be foun d and sum marized as follows. Theor em 1. F or a system w it h M sensors, the optimal s t rate gy for the adversary is to inject dependent random n oises with a pairwise correlation coefficient of 1 . The r and om bias power is allocated s u ch tha t σ b i = c i a q P M j =1 c 2 j , fo r i = 1 , · · · , M . (38) September 24, 2018 DRAFT 12 Fig. 1. Geometric solution for systems with two sensors. 2) Attack Strate gy F or A Single P osition And V elocity Sensor: In this case, let us assume that the sensors collect both po sition and velocity measurements of the target. Therefore, th e measurement matri x for the i th sensor is H i = I 2 , where I 2 is a 2 × 2 identity matrix. At th e i th sensor , the adversary injects the bias n oise vector b k i to t he sensor measurement z k i , where b k i = [ b p i b v i ] T consists biases in pos ition and velocity m easurements. Let us assume that the system bias vector b k = [ b T k 1 , · · · , b T k M ] T is zero-mean and has a 2 M × 2 M cov ariance mat ri x Σ K . Further , the ( i, j ) th 2 × 2 subm atrix for Σ K is defined as Σ K ( i, j ) =   ρ b p i ,b p j σ b p i σ b p j ρ b p i ,b v j σ b p i σ b v j ρ b v i ,b p j σ b v i σ b p j ρ b v i ,b v j σ b v i σ b v j   (39) for 1 ≤ i, j ≤ M . σ b p i and σ b v i are the p osition and velocity bias noise standard deviations at the i th sensor respectiv ely . The ρ s are defined as the proper correlation coef ficients between components of the bias vector , and ρ b p i ,b p i = ρ b v i ,b v i = 1 , for 1 ≤ i ≤ M . Since the positio n bias b p and v elo ci t y bias b v hav e dif ferent un i ts, we need an appropriate constraint for bias noi s e power . Here we assum e t hat the to t al noise p ower is defined as M X i =1 σ 2 b p i + T 2 σ 2 b v i (40) Note that thi s is a meaningful power definition, s ince the two terms in the abo ve equati o n has the same unit. Recall that according to the target trac ki ng system plant equation and ignoring the system process noise, we have ξ k +1 = ξ k + T ˙ ξ k . Therefore, the po wer defined i n (40) can be interpreted as the summatio n of the extra mean squared errors for the position estim ate caused September 24, 2018 DRAFT 13 by independent bias injections. W e can see that the best attack strategy deriv ed under a constraint on power defined in (40) can be easil y adjusted and extended for ot her power definitions, as long as i n the new definition, the second term i s proportional to T 2 σ 2 b v i . As we can u se the equiv alent sensor to represent the m ultiple s ens o rs, we focus on the single- sensor case first. If we are i nterested in the case of N = 0 , maximizing the trace of A K is equiv alent to maximi ze the W K Σ K W T K . W e assume that t he adversary knows the system models and the prior inform ation P 0 | 0 at ti m e zero, so that he/she can calculate the o f fline Kalman filter gain matrix W k recursiv ely . Th erefore, the best strategy t he adversary can adopt to attack th e system is the soluti on t o the foll owing optimi zati on prob lem: max Σ K T r  W K Σ K W T K  s.t. σ 2 b p + T 2 σ 2 b v = a 2 − 1 ≤ ρ b p ,b v ≤ 1 σ b p , σ b v > 0 (41) where Σ K =   σ 2 b p ρ b p ,b v σ b p σ b v ρ b p ,b v σ b p σ b v σ 2 b v   (42) and W K =   w 11 w 12 w 21 w 22   (43) It is easy to show that T r  W K Σ K W T K  = T r  W T K W K Σ K  = ( w 2 11 + w 2 21 ) σ 2 b p + ( w 2 12 + w 2 22 ) σ 2 b v + 2( w 11 w 12 + w 21 w 22 ) ρ b p ,b v σ b p σ b v (44) According to the s ign of ( w 11 w 12 + w 21 w 22 ) , we can set the value of t h e ρ b p ,b v to maximize the objective function. For example, if ( w 11 w 12 + w 21 w 22 ) is posit ive, we set ρ b p ,b v = 1 and the optimizatio n problem becomes max( w 11 σ b p + w 12 σ b v ) 2 + ( w 21 σ b p + w 22 σ b v ) 2 s.t. σ 2 b p + T 2 σ 2 b v = a 2 (45) σ b p , σ b v ≥ 0 September 24, 2018 DRAFT 14 T o solve this constrained optim i zation problem , let us first denote w 2 11 + w 2 21 = β 1 w 2 12 + w 2 22 = β 2 w 11 w 12 = α 1 w 21 w 22 = α 2 (46) The constraint in (41) can be written as σ 2 b p T 2 + σ 2 b v = a 2 T 2 = a 2 1 (47) Now w e set σ b p = a 1 T sin( θ ) and σ b v = a 1 cos( θ ) . Plugging σ b p and σ b v into the objectiv e function, we h a ve the following equiv alent optim ization probl em max θ a 2 1  β 1 T 2 1 + β 2 2 + A sin (2 θ + φ )  s.t. 0 ≤ θ ≤ π 2 (48) where A = r 1 4 ( β 2 − β 1 T 2 ) 2 + T 2 ( α 1 + α 2 ) 2 (49) tan( φ ) = β 2 − β 1 T 2 2 T ( α 1 + α 2 ) (50) Clearly , the opt imal solut ion is θ ∗ = π 4 − φ 2 (51) W e su m marize this result in the foll owing theorem. Theor em 2. F or a system with one sensor observing po sition and velocity o f the tar get, the optimal strate gy f or the adversary is to inject random noise th at has dependent positio n an d September 24, 2018 DRAFT 15 velocity component s . If w 11 w 12 + w 21 w 22 > 0 , t he corr elation coefficient ρ b p ,b v should be set as 1 , and the random bias power is allocated s u ch tha t σ b p = a sin( θ ∗ ) (52) σ b v = a T cos( θ ∗ ) θ ∗ = π 4 − φ 2 φ = arctan  β 2 − β 1 T 2 2 T ( α 1 + α 2 )  w 2 11 + w 2 21 = β 1 w 2 12 + w 2 22 = β 2 w 11 w 12 = α 1 w 21 w 22 = α 2 When w 11 w 12 + w 21 w 22 < 0 , we s h ould set ρ b p ,b v = − 1 and s et α 1 = − w 11 w 12 and α 2 = − w 21 w 22 . The r est of the equa tions in f o rmula (52) r emains the same. 3) Attack St rate gy F or Multipl e P osition And V elocity Senors: In this case, M = 2 , and t h e measurement matrix is H = [ I 2 I 2 ] T . The m easurement covariance matrix for th e i th sensor is assumed to be R i =   σ 2 p i 0 0 σ 2 v i   (53) Now , according to (25), we h a ve R e = [ R − 1 1 + R − 1 2 ] − 1 =    σ − 2 p 1 + σ − 2 p 2  − 1 0 0  σ − 2 v 1 + σ − 2 v 2  − 1   (54) According to (24), we d efine C i = R e H T i R − 1 i =   σ − 2 p i σ − 2 p 1 + σ − 2 p 2 0 0 σ − 2 v i σ − 2 v 1 + σ − 2 v 2   (55) as the weight ing matrix for the i th s ensor’ s obs erv ati o n z i . Further , we define c p i = C i (1 , 1) c v i = C i (2 , 2) (56) September 24, 2018 DRAFT 16 both of w h ich are po s itive numbers. Th e equiv alent noise in j ection is therefore b eK = 2 X i =1 C i b K i (57) So the cov ariance matrix of the equivalent bias vector is Σ eK = 2 X i =1 2 X i = j C i Σ K ( i, j ) C T j (58) where Σ K ( i, j ) has been defined in (39). It can be shown that Σ eK =   s 1 s 2 s 2 s 3   (59) Where s 1 = c 2 p 1 σ 2 b p 1 + c 2 p 2 σ 2 b p 2 + 2 ρ b p 1 ,b p 2 c p 1 c p 2 σ b p 1 σ b p 2 s 3 = c 2 v 1 σ 2 b v 1 + c 2 v 2 σ 2 b v 2 + 2 ρ b v 1 ,b v 2 c v 1 c v 2 σ b v 1 σ b v 2 (60) s 2 = c p 1 c v 1 ρ b p 1 ,b v 1 σ b p 1 σ b v 1 + c p 1 c v 2 ρ b p 1 ,b v 2 σ b p 1 σ b v 2 + c p 2 c v 1 ρ b p 2 ,b v 1 σ b p 2 σ b v 1 + c p 2 c v 2 ρ b p 2 ,b v 2 σ b p 2 σ b v 2 (61) The opti m ization probl em can be written as fol l ows. max Σ eK T r  W eK Σ eK W T eK  (62) s.t. σ 2 b p 1 + σ 2 b p 2 + T 2 σ 2 b v 1 + T 2 σ 2 b v 2 = a 2 , − 1 ≤ ρ p i ,v j ≤ 1 , − 1 ≤ ρ v i ,v j ≤ 1 , − 1 ≤ ρ p i ,p j ≤ 1 , σ p i , σ v i ≥ 0 , ∀ i, j ∈ { 1 , 2 } where W eK =   w 11 w 12 w 21 w 22   (63) is Kalman filter gain calculated using the equivalent measurement covariance matrix R e and equiv alent measurement matrix H e . It is easy to show that T r  W K Σ K W T K  = T r  W T K W K Σ K  (64) = ( w 2 11 + w 2 21 ) 2 s 1 + ( w 2 12 + w 2 22 ) 2 s 3 +2( w 11 w 12 + w 21 w 22 ) s 2 September 24, 2018 DRAFT 17 Clearly , all the ρ s that appear i n s 1 and s 3 should be set as 1 to maximize the obj ectiv e functio n. The optim al v alues for ρ s in s 2 depend on Kalman filter gai n W eK . More specifically , when w 11 w 12 + w 21 w 22 > 0 , all the ρ s that appear in s 2 should be set to 1 ; otherwise, they s hould be set as − 1 . Let us first suppose t h at w 11 w 12 + w 21 w 22 > 0 is true, then we hav e T r  W K Σ K W T K  = ( w 2 11 + w 2 21 ) 2 ( c p 1 σ p 1 + c p 2 σ p 2 ) 2 + ( w 2 12 + w 2 22 ) 2 ( c v 1 σ v 1 + c v 2 σ v 2 ) 2 + 2( w 11 w 12 + w 21 w 22 )( c p 1 c v 1 σ p 1 σ v 1 + c p 1 c v 2 σ p 1 σ v 2 + c p 2 c v 1 σ p 2 σ v 1 + c p 2 c v 2 σ p 2 σ v 2 ) (65) So far , we hav e con verted the obj ectiv e function in (62), which in volves 1 0 variables to one that i n volve s only 4 va riabl es. Consid erin g that the power constraint reduces one de g ree of freedom, we only need to solve an optim ization problem in a 3-dimensional space. 4) Strate gy F or A Single S ens or W i t h Mul tiple T i me Attack: Based on Propositi o n 2, we get the extra mean squ are matrix, A K + N = N X m =0 D m Σ K + N − m D T m Suppose at the time K , the adversary wants to attack the system continuously from tim e K to K + N , the weigh t for different time is α i , i ∈ N , as sh own below , A ′ K +0 = α 0 ( D 0 Σ K D T 0 ) A ′ K +1 = α 1 ( D 0 Σ K +1 D T 0 + D 1 Σ K D T 1 ) (66) ... A ′ K + N = α N ( D 0 Σ K + N D T 0 + ... + D N Σ K D T N ) where P N m =0 α m = 1 . So the objective function in the multi-s h ot attack case is t h e trace of the weighted sum of t he EMSE matrices at dif ferent tim e points that is P N m =0 α m A K + m = P N m =0 A ′ K + m . It is equ iv alent to maximize the t race of the weigh t ed sum of the MSE matrices of th e state estim ates, because once t h e sys t em reaches its steady state, P K + m | K + m becomes constant, and the weighted sum of P K + m | K + m will remain the same. First we study the case where the system has posi t ion sensors which are being attacked, s o all the i tems above are September 24, 2018 DRAFT 18 scalars. Using lower case d, σ 2 p to denote D , Σ , we can formulat e the o p timization problem below , max σ p K , ··· ,σ p K + N N X m =0 α m A K + m = N X m =0 A ′ K + m (67) = σ 2 p K ( α 0 d 2 0 + α 1 d 2 1 + ... + α N d 2 N ) + σ 2 p K +1 ( α 1 d 2 0 + α 2 d 2 1 + ... + α N d 2 N − 1 ) + σ 2 p K +2 ( α 2 d 2 0 + α 3 d 2 1 + ... + α N d 2 N − 2 ) + ... + σ 2 p K + N ( α N d 2 0 ) s.t. K + N X m = K σ 2 p m ≤ a 2 N X m =0 α m = 1 The adversary can allocate the power based on the coef ficients of the var i ance v ariables at diffe rent tim e. For example, if t h e weight s α ′ m s are all the same, the best strategy is to allocate all the power to the sensors at the first beginning (at ti m e K) because the coef ficient for σ 2 p K is the lar gest . Second, if the sensors measure both po s ition and velocity , and the att acker aims to attack th e system with pos ition and velocity false informati on, the op timization probl em can be characterized as below , September 24, 2018 DRAFT 19 max Σ K , ··· , Σ K + N T r " N X m =0 α m A K + m # = T r " N X m =0 A ′ K + m # (68) = T r  Σ K ( α 0 D T 0 D 0 + ... + α N D T N D N )  +T r  Σ K +1 ( α 1 D T 0 D 0 + ... + α N D T N − 1 D N − 1 )  +T r  Σ K +2 ( α 2 D T 0 D 0 + ... + α N D T N − 2 D N − 2 )  + ... +T r  Σ K + N ( α N D T 0 D 0 )  s.t. K + N X m = K σ 2 p m + T 2 σ 2 v m ≤ a 2 N X m =0 α m = 1 where Σ m and D T j D j are positive semidefinite matrices, so T r  Σ m ( D T j D j )  ≥ 0 all t he t ime. The trace function T r( · ) is a monotonically increasing function of the positive sem idefinite matrix. So the best strategy for the adversary to attack the sy stem is t o put all t he power at the time with t he largest p o sitive semidefinite matrix. B. Att a ck Strate gies f rom Determinant P erspective 1) Attack Strate gy F o r Multipl e P osition Senso rs: W e are also interested i n the ef fect of bias information on Kalman filter’ s estimation MSE from th e determinant perspective. By usi ng the equiv alent measurement approach as in Section VI-A1, we ha ve | P K | K + A K | = | P K | K + Σ eK D 0 D T 0 | = | P K | K || I + Σ eK D 0 P − 1 K | K D T 0 | (69) where D 0 is defined in Proposition ?? . Σ eK is defined in (33). As P K | K is constant and positive definite, D 0 P − 1 K | K D T 0 is positive semidefinite meaning that all t h e eigen values of the D 0 P − 1 K | K D T 0 are non-negativ e. First , let us denote C as a square matrix whos e col u mns are the eigen vectors of D 0 P − 1 K | K D T 0 . Then throu gh eigendecomposit ion, (69) can be wri t ten concisely as, | P K | K || CIC − 1 + Σ eK CΛC − 1 | = | P K | K || I + Σ eK Λ | (70) September 24, 2018 DRAFT 20 where Λ is a d i agonal m atrix whose diagonal elements are the eigen values of the D 0 P − 1 K | K D T 0 . So we j ust need to maximize Σ eK in order to maximize t he determinant of P K | K + A K . This is equiv alent to maximizing the trace of P K | K + A K as discuss ed in Section VI-A1. 2) Attack Strate gy F or A S i ngle P ositio n And V elocity Sensor: W e assume t hat th e adversary knows the s ystem mod el and the prio r information P 0 | 0 at tim e zero, so that he/she can calculate the offline Kalman filter gain m atrix W k recursiv ely . The best attack s t rategy is the so l ution to the following opt i mization problem. max Σ K   P K | K + W K Σ K W T K   s.t. σ 2 b p + T 2 σ 2 b v = a 2 (71) − 1 ≤ ρ b p ,b v ≤ 1 σ b p , σ b v > 0 where W K Σ K W T K = A K , and Σ K =   σ 2 b p ρ b p ,b v σ b p σ b v ρ b p ,b v σ b p σ b v σ 2 b v   (72) Using the properties of t h e determinant, we g et the formul a as follows. | P K | K + W K Σ K W T K | = | P K | K || I n + Σ K W T K P − 1 K | K W K | (73) Since P K | K is independent of Σ K , the optimizatio n problem can be further written as: max Σ K    I n + Σ K W T K P − 1 K | K W K    s.t. σ 2 b p + T 2 σ 2 b v = a 2 (74) − 1 ≤ ρ b p ,b v ≤ 1 σ b p , σ b v > 0 By defining W T K P − 1 K | K W K =   m 1 m 2 m 2 m 3   (75) September 24, 2018 DRAFT 21 and after si mplifying (74), the objective function becomes    I n + Σ K W T K P − 1 K | K W K    = 1 + (1 − ρ 2 b p ,b v ) σ 2 b p σ 2 b v ( m 1 m 3 − m 2 2 ) (76) + σ 2 b p m 1 + σ 2 b v m 3 + 2 ρ b p ,b v σ b p σ b v m 2 The opti m al s o lution to the problem will be the best strategy to attack t h e system. W e denot e Σ K = R T R and since Σ K is inv ertible, we h ave    I n + Σ K W T K P − 1 K | K W K    =    I n + R T R W T K P − 1 K | K W K    (77) =    I n + R W T K P − 1 K | K W K R T    In order t o obtain the opt imal solut ion, two u s eful lemmas [22] are int roduced as follows, Lemma 1. Suppose A and B are n × n positi ve semidefini t e matrices with eigendecomposit i on A = Ψ A Σ A Ψ T A and B = Ψ B Σ B Ψ T B , the eigen values of A and B satisfy th a t α 1 ≥ α 2 ≥ · · · ≥ α n and β 1 ≥ β 2 ≥ · · · ≥ β n , then Π n i =1 ( α i + β i ) ≤ det( A + B ) ≤ Π n i =1 ( α i + β n +1 − i ) (78) wher e the upper bound is achieved if and only if Ψ A = Ψ B Θ , the lo wer bound is ac hieved if and only if Ψ A = Ψ B , and Θ is the matrix defined below ,        0 0 · · · 1 0 · · · 1 0 . . . . . . . . . . . . 1 0 · · · 0        (79) Readers are refe rred to [22] for the proof of Lemma 1. The optimal sol u tion to find the upper bound is the best strategy to attack the s ystem with the most ef fect on Kalman filter system and the lower b ound is the least attack ef fect t he adversary can get. Lemma 2. Given a n × n matrix V 1 and a n × n positive semidefinite matrix Ξ 1 with V 1 Ξ 1 V T 1 being a diagonal matrix with diagonal elements in incr easi ng or der , it is al ways poss ible to fin d another n × n ma trix ¯ V 1 such that ¯ V 1 Ξ 1 ¯ V T 1 = β V 1 Ξ 1 V T 1 with T r ( V 1 V T 1 ) = T r ( ¯ V 1 ¯ V T 1 ) wher e β ≥ 1 . ¯ V 1 can be written as Σ Ξ Ψ T 1 , wher e Ψ 1 is the unitary matri x whose columns ar e the September 24, 2018 DRAFT 22 eigen vectors corr espond i ng t o the eigen values of Ξ 1 in incre as ing or der , and Σ Ξ is a d i agonal matrix. By com bining t h e two lemm as togeth er , we can get the final optimal sol ution to the o ptimiza- tion probl em above. It is obvious that I n and R W T K P − 1 K | K W K R T are b o th posit ive sem i definite matrices, and t heir eigendecompos ition can be writt en as follows, I n = Ψ 1 Σ 1 Ψ T 1 R W T K P − 1 K | K W K R T = Ψ 2 Σ 2 Ψ T 2 (80) with identity matrix Σ 1 = d iag ([ σ 1 , 1 , · · · , σ 1 ,n ]) and Σ 2 = d iag ([ σ 2 , 1 , · · · , σ 2 ,n ]) , where σ 2 ,i , i ∈ { 1 , · · · , n } is the di agonal element of the matrix Σ 2 . Based on Lemma 1, we can get,    I n + R W T K P − 1 K | K W K R T    ≤ Π n i =1 ( σ 2 ,i + 1) (81) where Ψ 1 = Ψ 2 Θ . | I n + R W T K P − 1 K | K W K R T | = | Ψ T 1 || I n + R W T K P − 1 K | K W K R T || Ψ 1 | (82) = | I n + Ψ T 1 R W T K P − 1 K | K W K R T Ψ 1 | Set R 1 = Ψ T 1 R and Σ 3 = ΘΣ 2 Θ T with the eigen values of increasing order and T r ( R R T ) = T r ( R 1 R T 1 ) . So the optimizati o n probl em can be written as bel ow , max | I n + R 1 W T K P − 1 K | K W K R T 1 | s.t. T r ( R 1 R T 1 ) ≤ a 2 (83) R 1 W T K P − 1 K | K W K R T 1 = Σ 3 Setting W T K P − 1 K | K W K = ˜ Ξ , we hav e R 1 ˜ ΞR T 1 = Σ 3 . Based on Lemma 2, we can surely find a matrix ¯ R su ch that ¯ R 1 ˜ Ξ ¯ R T 1 = β R 1 ˜ ΞR T 1 , with β ≥ 1 . Note th at det( · ) is a monotoni c increasing function of the positive semidefinite matrix. So | I n + R 1 ˜ ΞR T 1 | ≤ | I n + ¯ R 1 ˜ Ξ ¯ R T 1 | (84) So the optim al so lution ¯ R should be in t he form of ¯ V . T h e eigendecompostion of ˜ Ξ is as follows, ˜ Ξ = V Ξ Σ Ξ V T Ξ (85) September 24, 2018 DRAFT 23 where Σ Ξ = diag ([ σ ξ , 1 , σ ξ , 2 , · · · , σ ξ ,n ]) in increasing o rder . V Ξ is a uni tary m atrix whose column vectors correspond s to t h e eigen values of ˜ Ξ . The probl em can be written as max σ 2 b,i n X i =1 lo g ( σ 2 b,i σ ξ ,i + 1) (86) s.t. n X i =1 ( σ 2 b,i ) ≤ a 2 The objectiv e functio n abov e is a concave and increasing functi on. The optimal soluti on i s achie ved through Lagrangian multipl iers yi elding the water- filli ng strategy , σ 2 b,i =  1 λ − 1 σ ξ ,i  + (87) where the value of λ can be obtained by solv i ng n X i =1  1 λ − 1 σ ξ ,i  + = a 2 (88) The solu t ion is R opt = Ψ 1 [ Σ 1 / 2 b ] T V T Ξ (89) Finally , the opt imal solut i on o f (74 ) is, Σ K = V Ξ Σ b V T Ξ (90) 3) Attack Strate gy F or Mult iple P osi t ion and V elocity Sensors: For a sys tem with m ultiple sensors, the best strategy to allocate the bias noise power and set the correlation coeffi cients among the bias noises at different sensors is als o inv estig at ed. Let us denote the number of sensors as M , and the measurem ent matrix as H = [ I 2 , · · · , I 2 ] T . The measurement covariance matrix for t he i th sensor is assumed t o be R i =   σ 2 p i 0 0 σ 2 v i   (91) Now , according to (23), we h a ve R e = M X i =1 R − 1 i ! − 1 =    P M i =1 σ − 2 p i  − 1 0 0  P M i =1 σ − 2 v i  − 1   (92) September 24, 2018 DRAFT 24 According to (21), we d efine C i = R e R − 1 i =    σ − 2 p i P M j =1 σ − 2 p j 0 0 σ − 2 v i P M j =1 σ − 2 v j    (93) as the weight ing matrix for the i th s ensor’ s obs erv ati o n z i . The equiv alent injected b i as noise is therefore b eK = M X i =1 C i b K i (94) and the covariance m at ri x of the equiva l ent bias vector is Σ eK = M X i =1 M X j =1 C i E ( b i b T j ) C T j (95) Now the optimization problem can be formulated as follows. max Σ eK   P K | K + W eK Σ eK W T eK   (96) s.t. N X i =1 σ 2 b p i + T 2 N X j =1 σ 2 b v i = a 2 , − 1 ≤ ρ b p i ,b v j ≤ 1 , − 1 ≤ ρ b v i ,b v j ≤ 1 , − 1 ≤ ρ b p i ,b p j ≤ 1 , σ b p i , σ b v i ≥ 0 , ∀ i, j ∈ { 1 , M } where W eK is Kalman filter gain calculated u sing H e and R e . The optimal solution of (96 ) can be obtained n umerically as s hown later in t h e paper . V I I . N U M E R I C A L R E S U L T S Some numerical resul t s are presented in this sectio n to il l ustrate t he theoretical result s. A. System with P o s ition Sensors The parameters used in the tar get tracking example are provided bel ow . The system sampling interval is T = 1 . The adversary injects bias information to two sensors with σ 2 w 1 = 3 and σ 2 w 2 = 4 , respectiv ely . The v ariance of the system process no i se is σ 2 v = 0 . 2 5 . The biases b i s are September 24, 2018 DRAFT 25 zero-mean Gaussian random variables with variances σ 2 b i s. For the power const raint we discussed earlier , we set the sum of σ 2 b i to be 3000. The effe ct of t he bias injection on Kalman filter is measured by a Chi-s q uared test. More specifically , we us e the sum o f the no rmalized MSE over N m Monte-Carlo runs q k = N m X j =1 h ˆ x ′ j k | k − x j k i T P − 1 k | k h ˆ x ′ j k | k − x j k i (97) where at time k , P k | k is the nomi n al state cov ariance matrix calculated by Kalm an filter , ˆ x ′ j k | k is the st ate esti m ate, and x j k is the true state, during the j th Monte-Carlo run. First, if t h e random biases injected to different sensors are independent, we should allocate all th e bias power to the sensor with the smallest measurement no i se variance. This is clearly true as demon s trated in Fig. 2, where allocating all the po wer to sensor 1 causes the maximum estim at i on MSE. In Fig. 95 100 105 110 0 2 4 6 8 10 12 x 10 4 Iteration Number k q k σ b1 =0.8a; σ b2 =0.6a; ρ =0 σ b1 =0.7a; σ b2 =0.7a; ρ =0 σ b1 =a; σ b2 =0; ρ =0 Fig. 2. The normalized MSE when independent biases are used. σ 2 b 1 + σ 2 b 2 = a 2 for each case. Fig. 3. The normalized MSE for dependen t biase s. σ 2 b 1 + σ 2 b 2 = a 2 for each case. 3, t h ree d epend ent -noise attack strategies are compared, in cluding the opti mal one according to (37), allocating the power equall y amo n g the sensors, and allocating all the power to the sensor with smallest measurement error variance. It is clear that the opti mal solution h as the lar g est im pact on the estimation performance, and it outperforms the best independent-noi s e attack strategy signi ficantly . September 24, 2018 DRAFT 26 B. Systems wit h P osit ion and V elocit y Sensors W e now consider the case where t he adversary att acks Kalman filtering system with a vector sensor observation containi ng both position and velocity measurements . W e first consider a single-sensor system, and the sensor h as a posit ion measurement variance of 3 and a velocity measurement variance of 4. W e set the sum of σ 2 b p 1 and T 2 σ 2 b v 1 to be 3000. In this particular case, w 11 w 12 + w 21 w 22 > 0 , so the opti mal choi ce is ρ b p ,b v = 1 . Based on Theorem 2, th e best strategy is to set σ b p = 5 2 . 3 and σ b v = 16 . 2 . It is clear from Fig. 4 that t he strategy provided in Theorem 2 m aximizes the MSE of Kalman filt er system by injecting vector bias i n formation. 95 100 105 110 0 1 2 3 4 5 x 10 5 Iteration Number k q k σ b p =52.3; σ b v =16.2; ρ b p , b v =1 σ b p =38.7; σ b v =38.7; ρ b p , b v =1 σ b p =54.8; σ b v =0; ρ b p , b v =0 Fig. 4. The normalized MSE for a system with a single sensors. σ 2 p 1 + T 2 σ 2 v 1 = a 2 for each case. Next we consider a system wi th t wo sensors. The first sensor is the same as the one described above, and the second one i s with positi o n measurement variance 4 and velocity measurement var i ance 5. In thi s particular case, again we hav e w 11 w 12 + w 21 w 22 > 0 , s o all the ρ s in s 1 , s 2 , and s 3 should be set as 1 . W e first use a systematic grid search to find an approxi m ate globally optimal solution and then we us e t h e FMINCON function in Matlab, a local search algorithm , to refine t his approximate globally op t imal soluti o n. The o ptimal solut ion we have obtained is σ 2 b p 1 = 1826 , σ 2 b p 2 = 1023 , σ 2 b v 1 = 81 , σ 2 b v 2 = 68 . For comp arison purpose, we also implement an attack strategy th at allocate power equally among the ob serv ati on component s and amo ng the t wo sensors, which i s σ 2 b p 1 = σ 2 b p 2 = σ 2 b v 1 = σ 2 b v 2 = 750 . Th e sim ulation result is shown in Fig. 5. As we can see, the opt imal attack strategy has a much greater impact th an the o ne t hat allocates power equally . Based on the optimal solutio n , we can find that allocating more power to the m easurement with lower va riance will have a greater effe ct on K alm an filter syst em. September 24, 2018 DRAFT 27 95 100 105 110 2000 3000 4000 5000 6000 7000 8000 Iteration Number k q k Best Strategy Power Equally Allocated Fig. 5. The normalized MSE for a system with two sensors. σ 2 p 1 + σ 2 p 2 + T 2 σ 2 v 1 + T 2 σ 2 v 2 = a 2 for each case. C. Determinant case Numerical results are present ed in this s ection to ill ustrate the ef fectiveness of t he proposed attack st rategies. As suming that the i njected bias noise b k is zero-mean and Gaussian di stributed, we can show that the posterior probabi l ity d ensity function (PDF) of the target st ate conditioned on the past observations and the current corrupted observation i s p ( x K | z 1: K − 1 , z ′ K ) = N ( ˆ x K | K , P K | K + A K ) (98) where ˆ x K | K is the updated state esti mate calculated by Kalman filter , whi ch is u n aw are of the presence of the injected false i nformation. Then th e tar get state x K will be in the fol l owing confidence region (or error ellipse)  x : ( x − ˆ x K | K ) T ( P K | K + A K ) − 1 ( x − ˆ x K | K ) ≤ γ  (99) with probability determined by the threshold γ [23]. The volume of the confidence region defined by (99) correspondi ng to t he threshold γ is V ( K ) = c n x | γ ( P K | K + A K ) | 1 / 2 (100) where n x is th e d i mension of the target state x , c n = π n/ 2 Γ( n/ 2 + 1) (101) September 24, 2018 DRAFT 28 and Γ( · ) is the gamm a function. First, let us consi der a single-sensor case, where the sensor has a position measurement with noise var i ance of 3 , which is independent of the velocity m easurement with noise var i ance o f 4 . W e set the bias nois e power constraint as σ 2 b p + T 2 σ 2 b v = 3 0 00 . W e solve the optimization problem form ulated in Section ?? numerically , and the optimal s o lution to (71) is σ 2 b p = 1500 , σ 2 b v = 1 5 00 , ρ b p,v = 0 . 0 63 . In Fig. 6, error ellipsi s for different attack s trategies are plotted. F or all the dif ferent attack st rategies, we set ρ b p,v = 0 . 0 63 . As we can see, under normal −40 −30 −20 −10 0 10 20 30 40 −15 −10 −5 0 5 10 15 Position a xis V elocity axis Normal condition σ 2 b p =1500, σ 2 b v =1500 σ 2 b p =0, σ 2 b v =3000 σ 2 b p =1000, σ 2 b v =2000 σ 2 b p =3000, σ 2 b v =0 Fig. 6. Error ellipsis for different power allocation strategies condition without false information injection , the error ellipse has the sm al l est area, while the optimal attack st rategy leads to an error ellipse with the lar gest area . In Figs. 7 and 8, th e volume (area) of the error ellips e is provided as a function of ρ b p,v and the ratio κ = σ b p σ b v T . W e can see that when t he κ = σ b p σ b v T = 1 , the area of t he elli pse is maximi zed. Also from Figs. 7 and 8, it is clear that the area of ellipse increases as the absolute value of ρ d ecreases. In Fig. 9, th e t rend of the error ellipsis as the ρ changes from − 1 to +1 is illust rated. In this particul ar case, since σ 2 b p + T 2 σ 2 b v = 3000 , Σ K is lar ge and in (71) th e second term ( W K Σ K W T K ) dominates. Therefore, in (76) the identit y matrix in the objective function is relativ ely small com paring to the second item, and approximately we ha ve    I n + Σ K W T K P − 1 K | K W K    ≈ | Σ K |    W T K P − 1 K | K W K    (102) The second term in the second line of the above equation is a constant. Hence, i n order to get the maxim u m determinant, we should set σ 2 b p = σ 2 b v T 2 and ρ b p ,b v = 0 . This is almost the same s olution as we ha ve obtained numerically . Next we consider a system with tw o sensors. September 24, 2018 DRAFT 29 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 κ V olu me ρ =0 ρ =0.2 ρ =0.4 ρ =0.6 ρ =0.8 ρ =1 Fig. 7. Error ellipse volume 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 κ V olu me ρ =−1 ρ =−0.8 ρ =−0.6 ρ =−0.4 ρ =−0.2 ρ =0 Fig. 8. Error ellipse volume The first senso r is th e same as t he one described above, and the second one is w i th position measurement variance 4 and velocity m easurement var i ance 5 . T o sol ve the opt imization problem formulated in (96), we first use a systematic grid search to find an approximate globally optimal solution and then we use the FMINCON functi o n in Matlab, a local search algorithm, t o refine this approximate globally optimal solu t ion. The opti m al solut ion we hav e obtained is σ 2 b p 1 = 1100 , σ 2 b p 2 = 600 , σ 2 b v 1 = 750 , σ 2 b v 2 = 550 , ρ b p 1 ,p 2 = 0 . 99 , ρ b p 1 ,v 1 = − 0 . 8 3 , ρ b p 1 ,v 2 = 0 . 75 , ρ b v 1 ,p 2 = 0 . 8 9 , ρ b p 2 ,v 2 = − 0 . 23 , ρ b v 1 ,v 2 = 0 . 9 5 . For comparis o n purpose, we int roduce three sub-opti mal attack s t rategies: Strategy I with all the ρ s bein g 0 s, and σ 2 b p 1 = 1100 , σ 2 b p 2 = 600 , σ 2 b v 1 = 750 , σ 2 b v 2 = 550 ; Strategy II with all th e ρ s being 1 s, and σ 2 b p 1 = 1100 , σ 2 b p 2 = September 24, 2018 DRAFT 30 −30 −20 −10 0 10 20 30 −15 −10 −5 0 5 10 15 Position a xis V elocity axis Normal condition ρ =0.0663 ρ =−1 ρ =−0.5 ρ =0.5 ρ =1 Fig. 9. Error ellipsis for different ρ s 600 , σ 2 b v 1 = 750 , σ 2 b v 2 = 550 ; and Strategy II with the ρ s being the same as those for the opt i mal strategy , and σ 2 b p 1 = σ 2 b p 2 = σ 2 b v 1 = σ 2 b v 2 = 750 . The numerical results are shown in Fig. 1 0. As we can see, the opt imal attack strategy has a greater imp act than those sub-opt imal attack strategies, resulting in the lar gest error elli pse. −8 −6 −4 −2 0 2 4 6 8 −4 −3 −2 −1 0 1 2 3 4 Position a xis V elocity axis Normal condition Optimal attack Attack strategy I Attack strategy II Attack strategy III Fig. 10. Error ell i psis for different po wer allocation strategies V I I I . C O N C L U S I O N S In this paper , we deri ved the EMSE due to th e i n jected random biases for a Kalman filter in a linear dynamic sys tem. This allows us to find how to allocate the bias power among multipl e sensors in order to maximize the effect of the false inform ation on Kalman filter from two perspectiv es: t race and determinant. 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