Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems

ISA Transactions xxx (xxxx) xxx

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Research article

Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems ∗

Daxing Xu a,c , Bo Chen a,b , , Li Yu a,b , Wen-An Zhang a a

Department of Automation, Zhejiang University of Technology, Hangzhou 310023, China Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China c College of Electrical and Information Engineering, Quzhou University, Quzhou 324000, China b

article

info

Article history: Received 30 October 2018 Received in revised form 4 November 2019 Accepted 5 November 2019 Available online xxxx Keywords: Fusion estimation Dimensionality reduction Artificial noise Eavesdropping Cyber-physical systems

a b s t r a c t This paper studies the distributed dimensionality reduction fusion estimation problem for cyberphysical systems with limited bandwidth in presence of eavesdroppers. Since wireless communication is implemented by broadcasting, the eavesdroppers can collude to collect the data through anther communication networks. To protect data privacy, based on the physical processes and local estimation error covariance (EEC) matrix, an insertion method of artificial noise (AN) is developed such that only eavesdroppers’ fusion EEC becomes worse. Meanwhile, the fusion center needs to decode the received signal due to the noise interference, while the successful decoding probability varies with signal to noise ratio. Subsequently, some criteria for the selection probabilities and the successful decoding probabilities are given to guarantee the effectiveness of the AN insertion strategy. Moreover, a sufficient condition of the designed AN power is derived to guarantee the confidentiality. Simulation examples are given to show the effectiveness of the proposed methods. © 2019 Published by Elsevier Ltd on behalf of ISA.

1. Introduction Cyber-physical systems (CPSs) are systems with close integration and cooperation by computing, cyber and physical elements. A variety of applications of CPSs are found in electricity, environment monitoring and remote medical treatment [1–3]. Real-time state estimation based on sensor observations is an important issue in CPSs, and it has attracted considerable attention in recent years. Particularly, multi-sensor distributed information fusion estimation (DIFE) utilizes multiple sets of data to estimate a quantity/a parameter in a process. It can improve the accuracy of the estimation while increasing reliability and robustness. Since CPSs are integrated in a variety of important infrastructures and communicate over wired or wireless networks, thus one crucial issue of DIFE in CPSs have to face is limited communication bandwidth. Therefore, studying the DIFE problem under communication constraints for the CPSs has great important theoretical and practical significance. The problem of limited communication capabilities for distributed fusion estimation has been well-studied. Dimensionality reduction method and quantization method can be found in most of existing works. Bandwidth constraint means that the ∗ Corresponding author at: Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China. E-mail addresses: [email protected] (D. Xu), [email protected] (B. Chen), [email protected] (L. Yu), [email protected] (W.-A. Zhang).

sensor can only transmit some binary bits to destination at each data transmission moment from the perspective of communication in [4–8]. Considering the bandwidth constraints by limited number of real-valued messages, distributed fusion estimation algorithms have been developed by using different dimensionality reduction strategies in [9–12]. Additionally, to satisfy certain communication rate from sensor to fusion center (FC), distributed fusion estimation problems have been investigated according to different judgment criteria in [13,14]. However, CPSs are vulnerable to malicious agents due to the high degree of openness [15– 17]. In particular, confidentiality is a fundamental vulnerability. Since the wireless communication is implemented by broadcasting, the transmitted data is easily intercepted by placing the receiving antenna in the area through which the radio broadcast signal passes. Therefore, the confidentiality becomes a basic requirement and fundamental issue for CPSs. Conventional cryptography based on information security has been studied in [18,19]. Since the computing resources at sensor side are limited, strong encryption for confidential communication is hard to use. In addition, weak encryption scheme may lead to information leakage if eavesdroppers have sufficiently large computation power. Alternatively, secure estimation problem has been investigated in [20–25] from the perspective of signal processing recently, where physical layer information and AN are exploited to implement confidentiality. A secret transmit beamforming approach based on quality of service has been proposed in [20], which prevents eavesdropping by a certain

https://doi.org/10.1016/j.isatra.2019.11.009 0019-0578/© 2019 Published by Elsevier Ltd on behalf of ISA.

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

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amount of AN to the transmitted signal. Motivated by the idea of physical layer security, an optimal transmission scheduling with feedback has been derived by minimizing the EEC, while keeping the eavesdropper’s EEC above some level in [21]. Similar results were derived by the data transmission rate without feedback in [22]. But the performance for the legitimate user was degraded because of the random transmission. Meanwhile, the encryption method by adding AN to information bearing signal was proposed in [23], where the intended receiver’s communication channel was not degraded. Moreover, similar results were given for multiple transmit antennas in [24]. It proved that one can keep the expected eavesdropper EEC unbounded, while the legitimate user’s expected EEC remains bounded by injecting noise into sensor transmissions. Recently, for distributed secure fusion estimation problem, perfect expected secrecy was achieved in [25] by using AN based on the DIFE-EEC matrix. The above-mentioned works assumed that the bandwidth resources were sufficient, however, the problem of distributed secure fusion estimation against eavesdroppers under the finite bandwidth was not fully investigated. It has been pointed out in [26–29] that the dimensionality reduction method has more advantages than vector quantization method [30] in solving the bandwidth-constrained problem. Under the dimensionality reduction fusion framework, a sensor only transmits partial components of local sensor estimate to the FC. Therefore, the confidentiality problem of dimensionality reduction fusion estimation for CPSs with bandwidth constraints is challenging. In this paper, based on physical process and system information, we design the AN injection method, which makes the expected EEC unbounded to the eavesdropper but bounded to the legitimate user under some conditions. To ensure the effective injection of the AN, some sufficient conditions and the power of AN are derived. Finally, Simulations of different AN levels are employed to show that the proposed encryption strategy can achieve perfect encryption. Notations. The trace operator is represented by ‘‘tr’’. The identity matrix with dimension n is denoted by ‘‘In ’’ . Mathematical expectation is represented by E { · }. The superscript ‘‘T ’’ stands for the transpose, while ‘‘lim’’ denotes the limit. diag { · } denotes a block diagonal matrix. ‘‘sup’’ represents an upper bound, while λmax {A} represents the maximum eigenvalue of the matrix A. 2. Problem formulations 2.1. System model Fig. 1 describes the physical process we considered, which can be represented by the following mathematical model: x(t + 1) = Ax(t) + w (t),

(1)

yi (t) = Ci x(t) + vi (t)

(2)

(i = 1, 2, . . . , L),

Fig. 1. Block diagram of the distributed dimensionality reduction fusion estimation.

and the local optimal EEC can be obtained as follows: Pi (t) = GKi (t)Pi− (t),

(6)

Pi− (t) = APi (t − 1)AT + Q .

(7)

Moreover, according to (3)–(7), the error cross-covariance Pij (t) is calculated by [34]: Pij (t) = [In − Ki (t)Ci ][APijs (t − 1)AT + Q ][In − Kj (t)Cj ]T .

(8)

As mentioned in Introduction, the communication bandwidth and sensor energy are limited in CPSs. Dimensionality reduction method based on local estimation components transmission has been proposed to satisfy the limited bandwidth [3,12]. Specifically, after the sensor i computes the local estimate xˆ i (t), only ri (1 ≤ ri < n) components of xˆ i (t) are sent to the FC at time t denoting by xˆ si (t). This transmission mechanism is called ‘‘random transmission" as shown in Fig. 1. The reorganized state estimate (RSE) xˆ ri (t) at the FC has ∆i possible cases in the following set: ∆

i Si (t) = {H1i xˆ i (t), . . . , Hhi i xˆ i (t), . . . , H∆ xˆ i (t)}, i (t)

(9)

where x(t) ∈ Rn is the system state, yi (t) ∈ Rqi is the measurement from the sensor i. w (t) and vi (t) are mutually independent Gauss white noise with zero-mean, and their variances are Q and Ri , respectively. L is the number of observation equations. A and Ci are system matrices and observation matrices, and it is considered that (A, Q 1/2 ) is controllable. In our scenario, the smart sensor i collects the measurement yi (t). The local estimate at ith sensor can be obtained by the standard Kalman filter [31–33]:

where ∆i (t) = (n(n − 1) · · · (n − ri + 1))/(ri (ri − 1) · · · 1), hi = 1, 2, . . . , ∆i (t), and each Hhi represents a diagonal matrix, which i contains ri elements ‘1’ and n − ri elements ‘0’ on the diagonal. For the convenience of description, the indicator functions are introduced as follows:

xˆ i (t) = GKi (t)Axˆ i (t − 1) + Ki (t)yi (t),

(3)

Ki (t) = Pi− (t)Ci T (Ci Pi− (t)CiT + Ri )−1 ,

(4)

Due to the limited energy, the sensor i may not transmit i message to the FC, i.e., α1i (t) = α2i (t) = · · · α∆ (t) = 0. Thus i (t) i each variable αh (t) satisfies

where GKi (t) =In − Ki (t)Ci ,

αhi i (t) =

1 if xˆ ri (t) = Hhi xˆ i (t) i 0 if xˆ ri (t) ̸ = Hhi xˆ i (t)

hi ∈ {1, 2, . . . , ∆i (t)}.

(10)

i

i

{ (5)

{

α ∑

α α

i i hm (t) hn (t) ∆i (t) i hi =1 hi (t)

= 0 (hm ̸= hn ) ∈ {0, 1} (i = 1, 2, . . . , L),

(11)

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

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∑∆i (t)

i where hi =1 αhi (t) = 1 indicates that some local estimation components will be transmitted to the FC from the sensor , while ∑∆i (t) i hi =1 αhi (t) = 0 means that the sensor does not send any message to FC. Thus, it is concluded from (9)–(11) that the RSE xˆ ri (t) can be described as

xˆ ri (t) = Hi (t)xˆ i (t), ∆

(12)

∑∆i (t)

i i where Hi (t) = hi =1 αhi (t)Hhi is a diagonal matrix, and its elements on the diagonal are 0 or 1 i.e.,

Hi (t) = diag {β1i (t), . . . , βni (t)},

i

Prob{αhi i (t) = 1} = πhi i , Prob{αhi i (t) = 0} = 1 − πhi i ,

(14)

where {αhi (t)} (hi = 1, 2, . . . , ∆i (t)) obeys independent and i identical distribution, which is of independent w (t), and vi (t). Moreover, πhi in (14) is a positive scalar satisfying i

∆i (t)

0≤

∑

πhi i ≤ 1,

(15)

hi =1

In addition, according to (13), βji (t) obeys Bernoulli distribution. We assume that Prob{βji (t) = 1} = βji and Prob{βji (t) = 0} = 1 − βji , then it is concluded from (14) that ∆i (t) n ∑ ∑ E{ βji (t)} = ri πhi i ,

(16)

where βji (t) = 0 indicates that the j-th component of xˆ i (t) is not transmitted from the sensor side to the FC. Combining the RSE (12), the compensating state estimate (CSE) of x(t) is computed by: xˆ ci (t) = Hi (t)xˆ i (t) + [In − Hi (t)]Axˆ (t − 1),

(17)

where [In −Hi (t)]Axˆ (t −1) is the compensation for the components of xˆ i (t) that are not transmitted. 2.2. Distributed dimensionality reduction fusion estimation Define ei (t) = x(t) − xˆ ci (t). Then, based on the xˆ ci (t) in (17), the optimal fusion estimate xˆ (k) is calculated by xˆ (t) =

L ∑

∆

diag and ⎡ {z1 , . . . , zn }, ⎤ u11 · · · u1n

.. .

a

stochastic

matrix

∆

U

=

. ⎥ . .. ⎦ . Let product ⊙ for G and Z is defined by un1 · · · unn ⎡ ⎤ g1 z1 · · · g1 zn ⎢ . .. ⎥ . .. G ⊙ Z = ⎣ .. . . ⎦

⎢ ⎣

..

gn z1

···

gn zn

Then the following formula is established. E {GUZ } = (G ⊙ Z ) ⊗ E {U } , where ⊗ represents Hadamard product. Then, let us define

⎧ ˆ = E {diag {H1 (t), . . . , HL (t)}} H ⎪ ⎪ ⎪ ⎪ ⎪ ¯ = E {[(In − H1 (t))T , . . . , (In − HL (t))T ]T } ⎪ H ⎪ ⎪ ⎪ ⎪ Λij = E {Hi (t) ⊙ Hj (t)} ⎪ ⎪ ⎪ ⎨ Ξij = E {(In − Hi (t)) ⊙ (In − Hj (t))} . ⎪ Vij = E {Hi (t) ⊙ (In − Hj (t))} ⎪ ⎪ ∑ ⎪ − 1 ˆ (t − 1) = P(t − 1)IaT ⎪ (t − 1) ⎪ Σ ⎪ ⎪ ⎪ ⎪ Aˆ i (t − 1) = GKi (t)A ⎪ ⎪ ⎪ ⎩ T Pˆ i (t − 1) = [P T (t − 1) · · · P T (t − 1)] Li

1i

Denote the covariance matrix Σij (t) = E {ei (t)eTj (t)}, which is calculated by [11]

∑

ij

= Λij ⊗ Pij (t) + Vij ⊗ [ΦiT AT + GKi (t)Q ] + VjiT ⊗ [AΦj (t) + QGTKj (t)] + Ξij ⊗ [AP(t − 1)AT + Q ],

(21)

where

hi =1

j=1

∆

Lemma 1 ([3]). Define two matrices G = diag {g1 , . . . , gn }, Z =

(13)

and βji (t) ∈ {0, 1} (j = 1, 2, . . . , n). Then, it is deduced from (10) ∑n i and (11) that j=1 βj (t) ∈ {0, ri } (i = 1, . . . , L). For the aboveproposed dimensionality reduction strategy, the sensor randomly transmits each component of xˆ i (t) to the FC. The diagonal matrix Hi (t) can be constructed at the FC according to the identification process of RSE [11]. Then, the practical data transmission situation can be known by αhi (t) according to (9)–(12). The value of i αhi (t) is 1 or 0, and the occurrence probabilities are as follows:

3

Wi (t)xˆ ci (t),

(18)

¯ Φi (t − 1)Aˆ T (t − 1) ˆ (t − 1)HA Φi (t) = Σ i ˆ (t − 1)Hˆ Pˆ i (t − 1)Aˆ Ti (t − 1) +Σ ¯ T (t)Aˆ T (t − 1). ˆ (t − 1)HQG +Σ K i

(22)

i

Therefore, the optimal weighting matrices Wi (t) and EEC matrix Σij (t) are obtained by substituting (21) into (19) and (20). Moreover, the optimal fusion estimate xˆ (k) is calculated by (18). Remark 1. It follows from (18)–(22) that the calculation for P(t) is dependent on all parameters information, including subsystem parameters, the noise characteristics and the probabilities of the binary variables αhi (t). Meanwhile, the calculation of the matrix i P(t) does not rely on real-time measurements of physical system. In this case, the error covariance matrix can be computed off-line due to that the parameters are known in priori.

i=1

where W1 (t), W2 (t),∑ . . . ., WL (t) are the weighting matrices to be L designed satisfying i=1 Wi (t) = In . Thus, according to [34], the optimal weighting matrices can be calculated by the following equation:

[W1 (t), . . . , WL (t)] = [IaT (Σ (t))−1 Ia ]−1 IaT (Σ (t))−1 ,

(19)

where Ia = [In , In , . . . , In ] ∈ R , Σ (t) = E {[eT1 (t) T T T T · · · eL (t)] [e1 (t) · · · eL (t)]}. Then, the EEC of optimal fusion estimate can be obtained by T

P(t) = [IaT (Σ (t))−1 Ia ]−1 .

nL×n

(20)

According to the above analysis, it can be concluded that the key of fusion estimation is the computation of Σ (t).

2.3. Problem of interest Definition 1 ([21]). A secrecy mechanism can be called perfect expected secrecy if and only if the following conditions hold simultaneously: lim sup trE {P(t)} < ∞,

(23)

lim trE {P e (t)} = ∞,

(24)

t →∞

t →∞

where P e (t) represents the EEC computed by the eavesdropper. Consequently, this paper will consider two problems of interest, which are described as follows:

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

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• For the DIFE based on dimensionality reduction, the first aim is to design efficient fusion criteria to achieve the perfect expected secrecy. • Under the constraints of bandwidth and energy, the second aim is to derive noise-energy-dependent and probabilitydependent conditions to guarantee the effectiveness of the secure estimation strategy proposed. 3. Main results In order to ensure confidentiality, an AN injecting approach will be proposed in this section, and then the AN energy condition will be derived to achieve the perfect expected secrecy. 3.1. Artificial noise injecting approach As mentioned before, the inserting AN technique is introduced for secure data transmission in this paper. In detail, at time t, we assume that the sensor i send the selected components xˆ si (t) to the FC. Then, a vector ai (t) will be added to xˆ si (t) before being transmitted; Otherwise, it is not necessary to add the vector ai (t). Under this condition, we can obtain the transmitted signal through ith communication channel, which is written as zi (t) = xˆ si (t) + ai (t),

(25)

∆

Pis (t) = Hi (t)Pi (t) to form a row vector Θi (t) ∈ R1×ri . Let N(Θi (t)) represent the null space of the matrix Θi (t). We choose an AN ai (t) such that ai (t) lies in N(Θi (t)) and Θi (t)ai (t) = 0. Then, r −1 denote Φi (t) = [θi1 (t), θi2 (t), . . . , θi i (t)] as an orthonormal basis matrix for N(Θi (t)), where Φi (t) ∈ Rri ×(ri −1) satisfies (26)

Under this case, the AN ai (t) can be design by: ai (t) = Φi (t)ςi (t),

(27)

where the components of ςi (t) ∈ Rri −1 are zero-mean Gauss white noise with variance σi2 (t). Notice that the quantities σi2 (t) can be designed at the FC to ensure the effectiveness of the AN approach. Then, the message arrived to the FC is described as follows: zi (t) = xˆ si (t) + Φi (t)ςi (t).

(28)

At the legitimate user’s FC, in order to eliminate the AN interference, the received message can be multiplied by known Θi (t) when decoding, i.e., ziu (t) = Θi (t)zi (t) = Θi (t)xˆ si (t).

3.2. Perfect expected secrecy under constrained bandwidth and energy We will design suitable AN power so that the perfect expected secrecy conditions (23) and (24) are satisfied. First, one important property in wireless channel is that the signal is subject to a lot of interference, such as natural noise, fading, and so on [35, 36]. When the FC receives the transmitted message, it decodes the data, and the decoding success probability depends on SNR from [37]. Let δi (t) be the energy emitted by the smart sensor i, then the SNR of the FC can be calculated by

δ (t)G (t) εi (t) = i∑ i ,

(30)

ai (t)

where ai (t) denotes the inserting AN. Apparently, the transmitted signal xˆ si (t) can be disturbed by ai (t). For the distributed sensor systems, since the eavesdropper is hard to get all system information, one may first think of a solution that uses all dynamic system information to design AN. However, it is difficult for us to use so much dynamic information to design suitable AN. Alternatively, an attractive choice is the fusion EEC Pi (t) which contains all information on systems from (6). In addition, we notice that Hi (t) is a random diagonal matrix that is determined by random real-time transmitted local estimate components. Thus, the embedment of information Hi (t) to the designed AN can enhance confidentiality. Specifically, it is proposed to take the diagonal elements of

ΦiT (t)Φi (t) = Iri −1 .

Meanwhile, the random matrix can be constructed by the identification process of RSE in the FC, while Φi (t) can be obtained according to local parameters information at each sensor. Moreover, it is known from (29) that the transmitted message do not affected by the injected AN. But for the powerful eavesdropper, it is difficult to get the accurate matrix Pi (t). Although all those prior parameter information are usurped by the eavesdroppers, the accurate transmitted signal still cannot be obtained as the existence of the random matrix Hi (t) generated by real-time physical processes. Consequently, we can find appropriate AN variances σi2 (t) to satisfy the transmission confidentiality.

(29)

Therefore, it is concluded from (29) that the transmitted components xˆ si (t) is not affected by the AN ai (t). Remark 2. Pi (t) can be computed by the sensor and the legitimate user’s FC according to known system parameter information.

∑

where Gi (t) represents the channel gain, while ai (t) denotes the channel noise power. We use binary variable γi (t) represent whether the FC decode the received message successfully, i.e. γi (t) = 1 represents that the received message is decoded successfully, γi (t) = 0 denotes that the decoding failed and the FC do not get any useful information. Additive White Gaussian Noise (AWGN) network is the most basic noise and interference channel model. According to references [38,39], the successful decoding probability function of a message based on SNR can be written as

√

p(γi (t) = 1|εi (t)) = f (εi (t)) = [1 − ξ ( 2εi (t))]m ,

∫∞

(31) t2

where the Gauss ξ - function ξ (x) = x √1 exp(− 2 )dt, and 2π m is the packet length. Therefore, it follows from (29) and (30) that SNR of the FC for the legitimate user is ∞. However, for eavesdropper, SNR of the FC is computed as follows in terms of (30):

εie (t) =

δi (t)Gi (t) , σi2 (t)

i = 1, 2, . . . , L.

(32)

Remark 3. It is concluded from (31) that the data successful decoding probability at the FC is 1 as εie (t) goes to ∞, and thus the estimation performance of the legitimate user is always optimal. According to (31) and (32), due to the effects of AN, the SNR of the received message by the eavesdropper is reduced, which will lead to lower probability of its successful decoding. Next, we will show the eavesdropping EEC matrix satisfying the perfect expected secrecy condition (24). For a stable system, without even eavesdropping, the eavesdropper can always get a bounded error covariance by prediction. Therefore, the data privacy protection problem of the stable system is more challenging and it is our future work. An unstable system is considered in this paper, whose spectral radius satisfies ρ (A) > 1. From (32), the SNR for the eavesdropper is related to δi (t), Gi (t) and σi2 (t), we can achieve the perfect encryption by designing these three parameters and fed back to each sensors. ∆

i T Define ρi = [π1i , π2i , . . . , π∆ ] (i = 1, 2, . . . , L), and γie (t) i represents whether the received message can be successfully

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

D. Xu, B. Chen, L. Yu et al. / ISA Transactions xxx (xxxx) xxx

decoded for the eavesdropper with Prob{γie (t) = 1} = γie . Then according to (13) there exist a positive constant χji ∈ R1×∆i such that

βji = χji ρi ,

j = 1, 2, . . . , n,

(33)

where an arbitrary element of χji is 1 or 0. Theorem 1. Consider an unstable system for system (1). The encryption mechanism (28) is used in wireless transmission. The perfect expected secrecy for the designed fusion estimator can be achieved, if the following conditions are simultaneously satisfied: (C1) For the L measurement equations (2), at least one local subsystem i is observable. (C2) There exists at least a set of selected transmission probabilities πhi , at every moment t, hi = 1, 2, . . . , ∆i (t) satisfying i

{ } λmax AT Mi A < 1,

(34)

or

{

⎤⎫ β1i (t)β1i (t) · · · β1i (t)βni (t) ⎪ ⎬ ⎢ ⎥ .. .. .. Λeii = E γie (t) ⎣ ⎦ . . . ⎪ ⎪ ⎩ ⎭ βni (t)β1j (t) · · · βni (t)βni (t) ⎤ ⎡ e i γi β1 · · · γie β1i βni ⎢ .. ⎥ .. .. =⎣ . ⎦. . . ⎧ ⎪ ⎨

{ Nie ⊗ Pii (t) + Mie ⊗ AP e (t − 1)AT + Q −

]

e ∑

} (t − 1)

>o

ii

(36) e i e i e e i where Mie = diag ∑{(1 − γi β1 ), . . . , (1 − γi βn )}, Ni = diag {γi β1 , . . . , γie βni }, and eii (t − 1) is the covariance matrix for the eavesdropper.

Proof. As pointed out in Remarks 2 and 3, the AN has no effect on legitimate user. Then, we can obtain the result (23) based on (C1) and (C2) by the similar derivations in [11], and thus the corresponding proof is omitted here.

γie βni β1i

Next, we show another perfect expected secrecy result (24) under the above conditions. For brevity, all variables with superscripts are used to represent the corresponding variables for the eavesdropper in the following derivation processes. According to (21), when i = j, Σiie (t) for the eavesdropper can be obtained as: ii (t)

= Λeii ⊗ Pii (t) + Viie ⊗ [ΦiT (t)AT + (In − Ki (t)Ci )Q ] + (Viie )T ⊗ [AΦi (t) + Q (In − Ki (t)Ci )T ] + Ξiie ⊗ [AP e (t − 1)AT + Q ].

m1 (t) = Tr(Nie ⊗ Pii (t)).

(41)

Similar to (40), combining the definition of Vii yields that

⎤⎫ β1i (t)(1 − β1i (t)) · · · β1i (t)(1 − βni (t)) ⎪ ⎬ ⎢ ⎥ .. .. .. Viie = E γie (t) ⎣ ⎦ . . . ⎪ ⎪ ⎩ ⎭ βni (t)(1 − β1i (t)) · · · βni (t)(1 − βni (t)) ⎤ ⎡ 0 · · · γie β1i (1 − βni ) ⎥ ⎢ . . .. .. .. =⎣ ⎦. . ⎧ ⎪ ⎨

⎡

0

···

(42) Using the properties of matrix trace operations, one has m2 (t) = 0.

(43)

Ξij , Ξiie

Meanwhile, combining the definition of

is calculated by:

Ξiie ⎧ ⎤⎫ ⎡ (1 − γie (t)β1i (t))(1 − γie (t)β1i (t)) · · · (1 − γie (t)β1i (t))(1 − γie (t)βni (t)) ⎪ ⎪ ⎪ ⎪ ⎨⎢ ⎥⎬ . . . ⎥ . . . =E ⎢ ⎦⎪ ⎣ . . . ⎪ ⎪ ⎪ ⎭ ⎩ (1 − γie (t)βni (t))(1 − γie (t)β1i (t)) · · · (1 − γie (t)βni (t))(1 − γie (t)βni (t)) ] [ · · · (1 − γie β1i )(1 − γie βni ) (1 − γie β1i ) . = (1 − γie βni ) (1 − γie βni )(1 − γie β1i ) · · ·

(44) The following equation is obtained according to the trace operator property: m3 (t) = Tr(Mi ⊗ [AP(t − 1)AT + Q ]).

(45)

Σiie (t)

Then, from (39), (41), (43) and (45), Tr {

Σiie (t)

Nie

} = Tr {

⊗ Pii (t) +

Mie

} is rewritten as

⊗ [AP(t − 1)AT + Q ]}.

(46)

Therefore, it is concluded from (C3) that there exists a positive number ki (t − 1) greater than 1 for t ≥ N Tr

∑e

∑ ≥ ki (t − 1) · Tr( eii (t − 1)) ≥ ··· t∏ −N ∑e ≥ ki (t − τ ) · Tr( ii (N)).

ii (t)

(47)

Then, according to (20) and (47), the eavesdropping EEC P e (t) is derived by

∑e

Tr {P e (t)} = Tr(IaT (

= Tr(

(38)

(39)

−1

(t))

(

Ia )

−1

−1

L ∑ ∑ e

−1

ii (t))

)

(48)

i=1

≥ 1L Tr(

Then, it follows from (37) and (38) that the trace of Σiie (t) can be calculated by Tr {Σiie (t)} = m1 (t) + 2m2 (t) + m3 (t).

γie βni

···

τ =1

(37)

For simplicity, let us define

⎧ ⎨ m1 (t) = tr(Λeii ⊗ Pii (t)) m2 (t) = tr(V e ⊗ [Φ T (t)AT + (In − Ki (t)Ci )Q ]) ⎩ m (t) = tr(Ξiie ⊗ [APi e (t − 1)AT + Q ]). 3 ii

(40)

Moreover, it is deduced from the trace operator property that

Tr {

∑e

⎡

(35)

where Mi = diag {1 − χ1i ρi , . . . , 1 − χni ρi }. (C3) There exists a positive real number o > 0 such that each set of the selected transmission probabilities and the successful decoding probability at each time satisfy

[

Combining the definition of Λij , one has

γie βni (1 − β1i )

{ } λmax {AT Mi A} = 1 { } , λmax AT Mi A ̸= λmin AT Mi A

Tr

5

∑e

ii

(t)),

for some i.

Moreover, applying the limit on both sides of Eq. (48), we can get lim Tr {P e (t)} ≥ lim

t →∞

t →∞

t −N 1∏

L

ki (t − τ ) · Tr {Σiie (N)} → ∞.

(49)

τ =1

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

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D. Xu, B. Chen, L. Yu et al. / ISA Transactions xxx (xxxx) xxx

Thus, we obtain perfect expected secrecy condition (24), i.e., limtt →∞ Tr {E {P e (t)}} = ∞. Remark 4. The observability condition C1 is easily satisfied, because it is only determined by system parameters. As pointed out in [11], it is easy to give a set of random transmission probabilities to ensure convergence of mean squared errors for the user. On the other hand, according to the proof of above theorem, the mean squared errors of the estimator for the eavesdropper will diverge to infinity if the condition C1 and inequality (34)–(36) are satisfied. Notice that the conditions are dependent ∑(C2)–(C3) e on the variables Pii (t), P e (t − 1) and ii (t − 1), so an effective probability selecting range is not easy to find such that the mean squared errors is divergent. From (34)–(36), the conditions are independent of the observations, and thus the selected transmission probabilities πhi and the successful decoding probability i γie can be obtained off-line. Selecting πhi i and γie to satisfy both conditions (C2) and (C3) may be time consuming. Alternatively, probabilities πhi can be selected first to satisfy (C2), and then i adjust the decoding success probability γie to satisfy condition (C3). It is concluded from (31), (32) and (36) that the high energy noise results in a low probability of successful decoding, thus condition (C3) is more easily satisfied. In what follows, how to choose AN power σi2 (t) will be given by Theorem 2. Theorem 2. Consider an unstable system, if the condition (36) is established, then energy of AN should satisfy the following formula:

σi2 (t) >

1

ε

δ

(t)Gi (t) e∗ i i

,

(50)

where ε is the SNR for the eavesdropper corresponding to the probability of successful decoding γie . e∗ i

Proof. It is concluded from (31) and (32) that the probability function monotonically increase with the SNR, the SNR for the eavesdropper should satisfy

εie (t) =

δi (t)Gi (t) < εie∗ . σi2 (t)

(51)

Then, the result (50) can be directly obtained from the above inequality. Remark 5. From above theorem, we can conclude that the larger the quantity σi2 (t) is, the easier the inequality (36) is satisfied. This means that the sensor needs to consume more energy. Notice that σi2 (t) needs to be sent to the sensor through a feedback channel. However, if the sensors have sufficient supply of energy, sufficiently large constant energies of the AN σi2 (t) can be chosen in order to satisfy power condition (50). Under this case, the feedback of the time varying power σi2 (t) is not required. In fact, the constant AN energy often can achieve perfect encryption without much energy. The simulation below will demonstrate this result. 4. Examples Consider the following parameters for the systems (1)–(2):

[ A=

1.1 0

0.5 1

]

, C1 = C2 =

[

1 0

0 1

]

,Q =

[

0.5 0.25 0.25 1

]

R1 = diag {0.5, 0.1}, R2 = diag {0.1, 0.3}. Notice that rank([Ci , Ci A]) = 2, we can easily find that the condition (C1) is satisfied by verification. However, due to the limited bandwidth and energy, at each time, only one component

Fig. 2. The estimation performance with σi2 (t) = 0.9077.

of xˆ i (t) is sent through the wireless channel, i.e., r1 = r2 = 1. Then according to (9), one can obtain that: H11 (t) = H12 (t) =

[

1 0

0 0

]

, H21 (t) = H22 (t) =

[

0 0

0 1

]

.

(52)

Moreover, considering the failure of decoding, the CSEs can be computed from (13), (17) and (51): xci (t)

ˆ

] γie (t)β1i (t) 0 = xˆ i (t) 0 γie (t)β2i (t) ] [ 1 − γie (t)β1i (t) 0 Axˆ (t − 1). + 0 1 − γie (t)β2i (t) [

(53)

where Prob{β1i (t) = 1} = β1i , Prob{β2i (t) = 1} = β2i , and Prob{γie (t) = 1} = γie . In the simulation, β11 , β12 , β21 , β22 are all taken as 0.5, thus it is concluded from (33) and (34) that

{ } λmax AT Mi A = 0.8902 < 1,

(54)

which means that the condition (C2) holds. Assume that the transmission energy and channel gain are fixed with δi (t), Gi (t). The initial state is taken as x(0) = [0.15, 0.25]T , the initial local estimate and fusion estimate all chosen as [0.1, 0.2]T , and are taken as P(0) = [ other parameters ] [ ] 0.02 0.03 0.04 0.02 diag {0.2, 0.3}, Φ1 (0) = , Φ2 (0) = . 0.01 0.02 0.05 0.02 The packet length is m = 5. First, we consider a constant AN power σi2 (t) = 0.9077. Then, according to (31) and (32), the probability of successful packet decoding can be computed as Prob{γie (t) = 1} = 0.7. Furthermore, it follows from (36) that Mie = diag {0.65, 0.65}, Nie = diag {0.35, 0.35}, the condition (C3) is not satisfied. The estimation performance of local CSEs xˆ ci (t)(i = 1, 2) and fusion estimate xˆ e (t) for eavesdropper, and optimal estimate xˆ (t) for user, are shown in Fig. 2. One can see that each local CSE performance is worse than that of the fusion estimation. This implies that the eavesdropper can achieve a better performance by eavesdropping more information from local sensors. However, the fusion estimation performance for eavesdropper is worse than that of the optimal estimate for user because there is no noise interference. On the other hand, due to the high successful decoding rate, a bounded trace of EEC can be obtained by combining eavesdropped message from two wireless networks for the eavesdropper. Next, the AN power is chosen as σi2 (t) = 5.6097, and then the probability of successful packet decoding can be calculated as

Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.

D. Xu, B. Chen, L. Yu et al. / ISA Transactions xxx (xxxx) xxx

7

physical process and all system parameter information, was proposed to maintain perfect expected secrecy using fusion EEC. Several sufficient conditions, which were dependent on the transmitting probabilities and the successful decoding probabilities, have been derived such that the trace of the eavesdropper’s expected EEC was unbounded. Moreover, the designed AN power was derived to guarantee the confidentiality. The simulation results illustrated the effectiveness of the proposed methods. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Fig. 3. The estimation performance for eavesdropper with σi2 (t) = 5.6097.

This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61973277, 61673351. References

Fig. 4. The trajectories of Υ (γie ).

Prob{γie (t) = 1} = 0.2. the condition (C3) is satisfied by verification. The estimation performance for eavesdropper are shown in Fig. 3. If the proposed encryption mechanism is embedded in the communication data, it can be concluded that the trace of both local CSEs and fusion EEC for eavesdropper grow unbounded with exponential rates. Moreover, define

Υ (γie ) = Tr {Nie ⊗ Pii (t) + Mie ⊗[AP e (t − 1)AT + Q ]−

e ∑

(t − 1)}. (55)

ii

Then, Υ (γie ) associated with different reception probabilities are shown in Fig. 4. In this case, the condition (C3) is satisfied with low reception probabilities by verification, which implies that the trace of the fusion EEC is unbounded. However, Υ (γie )(i = 1, 2) go to 0 with exponential rates when the eavesdropper has high successful decoding possibility, and the inequality (36) does not hold. Thus, the judgment criterion (C3) is effective.

5. Conclusion This paper has studied the confidentiality problem of secure dimensionality reduction fusion estimation. An AN injecting approach, which contained random information generated by the

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Please cite this article as: D. Xu, B. Chen, L. Yu et al., Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.009.