Optimal energy allocation for remote state estimate under Denial-of-Service attack

This paper considers an energy-constrained sensor power allocation problem for remote estimation under Denial-of-Service attack. Due to power constraints, the sensor needs to make a decision on how much power it should employ to optimise the state estimation performance under Denial-of-Servic attacks without using real-time acknowledgment information. Different from the existing works concerning non-zero power value with discrete values or one power value, the authors consider the multilevel continuous-valued energy allocation policy containing zero power value in this work. By analysing the effects on performance index under different energy allocation policies, an optimal off-line power scheduling policy with non-decreasing structure is derived. To obtain the optimal power scheduling sequences, the authors formulate the problem as non-linear programming after analysing the probability distribution of the estimation error covariance matrix. Moreover, the two power values energy allocation policy is investigated as a special case. At last, simulation examples are presented to verify the effectiveness of their theoretical results.

finite time horizon. Briefly, the contributions of this paper consist of the following threefold: (i) We derive the optimal structure of the energy allocation strategy that minimised the terminal expected estimation error. Different from the existing works concerning nonzero power value in [15] and one power value in [17], the multiple power levels scheme containing zero power value is adopted to optimise estimation quality against the DoS attacker in this work, and the transmission structure is nondecreasing with respect to time. To the best of our knowledge, this strategy is novel in the sense of off-line strategy. (ii) Solutions to the power allocation problem under two power levels case and multiple power levels case are presented, respectively. The specific transmission power value can be obtained directly by solving non-linear programming for the two cases. Different from the discrete solution space, the candidate solution in the proposed transmission scheme are both continuous for the two cases. (iii) We also investigate the monotonicity between estimation error and scheduling power based on the SINR model, and the index of terminal estimation error is analysed based on this monotonicity. The estimation performance of two power levels scheme and one power scheme are also compared.
The remainder of this paper is organised as follows. The system model is described and the optimal transmission problem is formulated in Section 2. Some preliminaries are given in Section 3. In Section 4, the optimal energy allocation schemes for two different situations are derived. Numerical examples are provided to illustrate the theoretical results in Section 5,while Section 6 concludes this paper.
Notation: ℤ + , ℝ + , ℝ m , ℝ m×m + and ℝ m×m ++ represent for the set of positive integers, positive real numbers, vectors of mdimension with the real number, m × m positive semi-definite matrices, m × m positive definite matrices, respectively. Denote by X 1 ≤ X 2 if (X 2 − X 1 ) ∈ R m×m + , and Y > 0 if Y ∈ R m×m ++ . The mean of random variable Y is E [Y ]. Denote by Pr(⋅) the probability. Tr(⋅) is the trace of a matrix. The transposition is denoted by a superscript '. Define the function h and g ℝ n + → ℝ n + as: h(X ) ≜ AXA ′ + Q, g(X ) ≜ X − XC ′ [CXC ′ + R] −1 CX . The symbol g h stands for g(h(⋅)), which is the composition of function g and h with an appropriate domain. Denote g k h (X ) ≜ g h (g k−1 h (X )) with initial value g 0 h (X ) ≜ X . Denote by h k (X ) ≜ h(h k−1 (X )) with h 0 (X ) ≜ X .

PROBLEM FORMULATION
A linear time-invariant system (LTI) process is run by a plant and the sensor measures the state in the plant, which can be formulated as follows, shown in Figure 1: x k+1 = Ax k + w k , System architecture: The sensor locally estimates the system state x k and sends packets x s k to the remote estimator over a wireless channel. Packets may drop out due to the malicious attacks k where k ∈ ℤ + , x k , w k ∈ ℝ n and y k , v k ∈ ℝ m are the system state, the process noise that is zero-mean Gaussian noise with covariance Q ≥ 0, the measurement taken by the sensor, as well as the measurement noise which is zero-mean Gaussian noise with covariance R > 0, respectively, at time k. Besides, it is assumed that w k and v k are uncorrelated, the pair (A, C ) is detectable and (A, √ Q) is controllable. As shown in Figure 1, at each time k, the sensor that has sufficient computation ability uses the measurement data y k to compute the local estimate. The minimum mean square error (MMSE) state estimate at the sensor side can be computed by Kalman filter:x s k = E[x k |y 1 , … , y k ], and the corresponding error covariance matrix is P s After that, the local estimatex s k is transmitted to a remote terminal over an unreliable wireless channel that may suffer from malicious attacks k . The transmitted packets may drop out due to the malicious attack. Denote k as the packet arrival indicator, k = 1 if the data packet arrives at the remote terminal at time k and k = 0, otherwise. Denote s ≜ { 1 , 2 , … , T } as the power scheduling sequence at the sensor side, where k is the transmission power value at time k. Denote by k the transmission indicator, that is, k = 1 represents that the sensor uses energy k to transmit data packet at time k, and k = 0 if the sensor does not transmit and uses zero power at time k. The remote estimator updates its state estimationx k based on whether it successfully receives the data from the sensor at time k [15]: where k ( s ) means k is a function of s , that is, the value of k is affected by the sensor power k . We sometimes write k ( s ) as k for simplicity if there is no ambiguity. The sensor has a limited power budget, which can be formulated as ∑ T k=1 k ≤ △. When s is given, k are assumed to be i.i.d Bernoulli random variables, and the corresponding probability distribution follows where k is the attack indicator such that k = 1 if the malicious attack is launched at time k and k = 0 otherwise. k and k are the dropout probabilities in the presence/absence of malicious attack at time k, respectively. With abusing notation, we denotep k ( k ) = 1 − k as the packet received probability in the presence of malicious attacks when transmission energy k is employed. We sometimes writep k ( k ) asp k if there is no ambiguity. In this work, it is assumed that the communication channel between the sensor and the remote estimator is an Additive White Gaussian Noise (AWGN) network using Quadrature Amplitude Modulation (QAM) (similar to [17]). We consider that the malicious attacker is of "jamming" type [21,22]; that is, the attacker can launch DoS attacks to jam the communication channel between the sensor and the remote estimator to degrade the system performance. According to digital communication theory [23], the relationship between the symbol error rate (SER) and signal to noise ratio (SNR) is: where When a DoS interference attacker is considered, the SNR for the communication model can be rewritten as SINR: where k , G s , G a , k , a , 2 are the SINR at time k, wireless channel gains for the sensor and the attacker, the power used by the sensor and attacker as well as the noise power, respectively. We assume that a ≡ a 0 within finite horizon T . Assume that the data packet is composed of l bits, and the bit error rate (BER) for each bit is identical. We suppose that the communication network is a memoryless lossy channel, that is, the transmission error is bit-to-bit independent. The data packet will be dropped if there is one bit received by mistake. Then from [20],p where 1 −p k = k . According to Kalman filter theory, the estimation error covariance matrix P s k will converge exponentially to the steadystate value denoted byP [24], and the evolution of error covariance matrix P k follows: It is assumed through this paper that the data packet at initial time k = 0 is received successfully by the remote estimator, that is, P 0 =P. In this work, we aim to design an optimal energy allocation policy so that the estimation error can be minimised at the end of finite time T . Hence, the terminal estimation error covariance J T = E[P T ( s )] is considered to measure the performance of remote estimation. A similar index can be found in [25] and [26]. In our scenario, the sensor with limited energy budget desires to transmit the data packets using an optimal strategy, under which the performance index Tr(J T ) will be minimised. The corresponding problem can be formulated as follows.

Tr
[ where ,̄, and Θ are the given lower bound and upper bound of transmission power, all admissible transmission policies, respectively. The sensor should choose appropriate power from the continuous-valued power levels to minimize the expected estimation error at time T .
Remark 1. In this work, we aim to derive the optimal transmission power allocations in an off-line mode by assuming the attacker using constant power. In an off-line transmission scheme, the power schedule sequence should be designed before the system runs without real-time acknowledgment information.

PRELIMINARIES
This section presents some preliminary knowledge as far as terminal error is concerned. An analysis of the monotonicity between the terminal error and transmission energy is also given.

Assumption 1.
We assume that the Kalman filter at the sensor side has entered into steady-state at initial time instant and the data packet at initial time k = 0 is received successfully by the remote estimator, that is, P s k =P, k ≥ 0, and P 0 =P.
Based on Assumption 1 and the evolution procedure of the error covariance matrix P k in (6), the state space of P k is where function h k (⋅) is defined in Notations of Section 1. In order to derive the optimal structure of Problem 1, first an analysis of the expression of J T ( s ) is given. Note that J T is a welldetermined value when the data packets arrival sequence is given.

Lemma 1. For function h defined in Section 1, the following inequality holdsP
According to [17], when the sensor has only one available energy level, the optimal power schedule is ).
And the corresponding trace of the expected terminal estimation error covariance matrix can be computed as where Equation (8) follows from Equation (16) of the work [17] after some simple algebra transformations for easier to derive the results in Proposition 1 below. When the transmission power 0 is given, the transmission times can be computed by n = ⌊Δ∕ 0 ⌋. Equation (8) is a performance index measuring the quality of estimation when the transmission power is given, and the packet dropout probability is related to some environmental conditions, for example, change of transmission power, channel state, and other malicious attack. If only the transmission power is considered, Tr(J T ) has monotonicity with it. This can be shown as the following proposition.
Proof. We denote and the shorthand of k and k here, and Δ 1 ≤ Δ 2 means Δ 1 ∕n ≤ Δ 2 ∕n when k ≠ 0, then we verify the monotonicity of Tr(J T ). From (8), there holds Take the partial derivative of J T in (9), the following equation is true, (11) one can see that M < 0. Substituting (11) into (10), there holds From Lemma 1, the following inequality is true: Remark 2. Proposition 1 shows that the estimation process will be more effective when more total energy is available. Since more transmission power leads to lower packet dropout probability, estimation error covariance will be smaller.
It is observed from (8) that Tr(J min ) is the sum of polynomial, furthermore, h i (P ) is a constant value for the given i and system parameters. Most of the existing work concerning the off-line energy allocation focus on using one power value at the last several time instants. While h i (P ) can be regarded as a welldetermined value and ( i − i+1 ) as the weighted coefficient determined by the transmission power. When multiple power values are employed, the weighted coefficients will be adjusted. Accordingly, the off-line transmission scheduling with limited energy budget can be further optimised.

OPTIMAL OFF-LINE ENERGY ALLOCATION STRATEGY
In this section, the optimal energy allocation structure is proposed. Then the optimal off-line energy allocation strategy for different cases are presented, that is, the power schedule sequence will be determined before the system runs. Comparison between one power level scheme and two power levels is given. We also analyse the implementation cost and computation complexity.

Optimal off-line transmission structure analysis
Reviewing the origin problem in Section 2, one can see that the total energy in the whole time horizon is limited and there is upper and lower bound over it. The energy constraint in Problem 1 can be transformed into the constraint of transmission times. That is, A similar power scheduling problem was studied in [15]. The authors considered that the sensor has L power levels with 1 , … , L which satisfy 0 < 1 ≤ ⋯ ≤ L , and the optimal allocation is s * = { 1 , 2 , … , L }, which means the transmission power should have the non-decreasing form satisfying Note that the authors in [15] considered that transmission power will be used at every time step within time horizon T , that is, k ≠ 0, ∀k ≥ 1. However, they did not show how to obtain the specific power value and how to utilize the zero power to optimise the performance. One can see that there will be N transmission using energy and T − N transmission using zero power in Problem 2.
When zero power value is needed in a scheduling scheme, we show that the solution is of the non-decreasing form. This can be seen in the following theorem. Theorem 1. The optimal structure of the solution to Problem 2 is given by Proof. For brevity, denote T ⋯1 as the packet arrival sequence ( T , … , 1 ) and T …1⧵i as the packet arrival sequence except i , . Let function f k (X ) be the estimation error covariance matrix associated with the packet reception result at time k, then we have Furthermore, denote f T …0 = f T ⋯ f 0 as the function composition like Notations in Section 1, and let Pr( T …1 | s ) be the conditional probability of the occurrence of the packet arrival sequence ( T , … , 1 ) under a given energy allocation policy s . Denote byp i = 1 − p i the probability of packet loss when power value is employed at time i, where i and i + 1 are two adjacent time instants for some 1 To illustrate the reasonability of the optimal structure briefly, first consider that there are two similar scheduling strategies s 0 and s 1 . The two schemes are only different in adjacent time instants such that i ( s 0 ) = 0 and i ( s 1 ) = 1, as well as i+1 ( s 0 ) = 1 and i+1 ( s 1 ) = 0, where i ( s 1 ) = 0 means i = 0 given energy allocation policy s 1 . Similar to [15], the expected terminal error covariance matrix can be expressed as Pr( T ⋯1⧵i+1⧵i | s 0 ) means sum of all possible probabilities of packet dropout sequences except for i and i+1 under power allocation policy s0 . Similarly, we have Since the two scheduling strategies s 0 and s 1 are only different at time i and i + 1, ∑ T ⋯1⧵i+1⧵i In a power scheduling sequence, two similar scheduling schemes discussed aforementioned can be found. Then exchange the two power level values in order from smallest to largest, which results that the scheme arranging zero power in front is superior to that non-zero power in front as far as the terminal error is concerned. After several such operations, the optimal scheduling scheme is derived as (12). The above process is similar to one step in the Bubble sort algorithm [27]. The reasonability of such exchange operation between zero power value and lower power level is given here. The case lower power level and higher level are exchanged is similar to this, which can be seen in [15]. This completes the proof. □ Remark 3. Theorem 1 shows that the optimal power scheduling sequence will be a non-decreasing structure and zero power value should be considered. Different from [17], the continuous candidate transmission power including zero power is considered, which results that it is difficult to find a closed-form of the optimal transmission power. Fortunately, the non-linear programming can be used to obtain the optimal energy allocation (see Theorem 2).
Remark 4. Since the estimation error covariance matrix P k is computed iteratively, P 0 =P is chosen as the initial estimation error covariance matrix for ease of calculation, which is also considered in the existing works, for example, [17, 20 28]. We note that any given initial value P 0 will not affect our results.
As can be seen in the proof in Theorem 1, the non-decreasing energy allocation structure exists due to the property h(X ) ≥ g(X ), ∀X ≥ 0, rather than the initial value P 0 . More details can be found in Section 5.1.
Remark 5. In this work, our main result in Theorem 1 is based on the assumption that k = 1 and a k ≡ a 0 , 1 ≤ k ≤ T . We note that if there exists an arbitrary attack sequence k (i.e. i ≠ j with i ≠ j ), the energy allocation policy still can be obtained but the non-decreasing structure in Theorem 1 may not hold. This is because, in the case of arbitrary attacks, larger transmission k may lead to smaller packet reception rate due to the jamming attack which decreases the SINR. In this case, the formulated non-linear programming in Theorem 2 can be applied to get the optimal energy allocation policy. Numerical experiments are provided in Section 5.1 to further illustrate this issue.

Multiple-level power scheduling study
The energy allocation policy is considered in the case of multilevel in this subsection. The sensor chooses power value at time k from continues-valued power levels within the given upper bound and lower bound. In this case, we will formulate Problem 1 as non-linear programming and thus the optimal scheduling sequence can be obtained. From (4) and (5) one can see that the data packet arrival probability is a function of transmission power. Considering the estimation error covariance matrix h i (P ), which will appear when the datax s k is received successfully and datã [28]. According to [7], when only one power level is accessible, the distribution of the estimation error covariance matrix is Note that different transmission power leads to different dropout rates and estimation error. When multiple power is accessible at the sensor side, the distribution of estimation error covariance matrix will be determined by the scheduled power. Denote by k the probability distribution of estimation error covariance matrix at time T , that is, k = Pr(P T = h k (P )). It can be seen that k follows where k satisfies ∑ T k=0 k = 1. It should be noted that distribution (14) will reduce to (13) when only one power level is available. When multiple power levels are available at the sensor side, we can transform Problem 1 into a non-linear programming problem.

Theorem 2.
When the sensor can access multiple transmission power values, the optimal solution to Problem 1 can be obtained by solving the following problem.
where k are given in (14).
Problem 3 is a typical non-linear programming with constraint, the solution of which can be obtained by MATLAB programming functions.
Remark 6. We shall explain how we select the power levels within the lower bound and upper bound̄. The optimal policy (also the power levels) is solved automatically by the optimiser when we set the times of using zero power level. For the zero power level, the exhaustive method can be applied to obtain the global optimal energy allocation policy, where the times of using zero power level take values from {0, 1, 2, … , T − 1}.
After obtaining the solution to Problem 1, we shall evaluate the performance of remote estimation under the proposed energy allocation policy. According to Theorem 2, we have the following corollary about the bounds of the terminal error Tr(J T ) = TrE[P T ( s )].

Corollary 1. The terminal error is bounded by Tr(P ) ≤ Tr(J T ) ≤ Tr(h T (P )).
Proof. By Equation (14) and Theorem 2 one can see that the value of J T depends on the probability distribution of estimation error covariance matrix k . How we allocate the total transmission energy Δ will affect the distribution. Two extreme circumstances are considered: 1) If we have sufficient total energy Δ such that k ≈ 0 and allocate total power Δ at the terminal time slot T , there holds 0 → 1 and k → 0, 1 ≤ k ≤ T . From Lemma 1, Equation (14) and Theorem 2, this will lead to J T →P; 2) On the contrary, if we have little total energy Δ and allocate them uniformly, there holds k → 0, 0 ≤ k ≤ T − 1 and T → 1, this leads to J T → h T (P ). Moreover, by Lemma 1, Equation (14), and Theorem 2, we have Tr(h T (P )) k .

Since
∑ T k=0 k = 1, we also obtain This completes the proof. □

Special case on two power levels scheduling
Due to the coupling of the transmission parameters Δ, T , and system parameters A, Q, it is difficult to give a closed-form solution to Problem 1 with multiple power levels, and we cannot directly compare the multilevel scheme with one power level policy. As a special case of the multilevel power allocation policy, the optimal structure of two power levels transmission scheme can be directly obtained from the result in Theorem 2. Moreover, the simpler structure makes it easier to derive the closedform expression of the terminal error and design the algorithm to obtain the optimal policy, which allows us to compare the two power levels power allocation policy with the single power level one.
According to Theorem 1, the structure of the optimal offline energy allocation should be the non-decreasing form. When only two power levels are available at the sensor side, we have the following proposition which presents the optimal structure for two power levels.
where n 1 and n 2 are two given real power value with satisfying n 1 < n 2 , corresponding to the packet loss rate 1 , 2 , respectively, and the trace of terminal error covariance matrix under policy (15) is where the value of transmission times n 0 , n 1 and n 2 can be obtained by the following non-linear programming. where n 1 , n 2 are the given real number with n 1 < n 2 , respectively, and the trace of estimation error Tr(J ) min can be calculated by (16).

Proof. A direct result from Theorem 1. □
Now we are ready to compare one power scheme and two different power levels scheme, after which a sufficient condition is obtained. When the condition is satisfied, the performance of two power levels scheme is better than that of one power level.
Define H i ≜ h i (P ) − h i−1 (P ), suppose there exists a positive integer n such that packet-drop probabilities satisfȳ0 = 0 ,̄1 <̄0,̄1 >̄0, wherē0 and̄0 are the packet-drop probabilities under jamming attacks when only one power level is available for time instants 1 to n and n + 1 to T , respectively.̄1 and̄1 are the packet-drop probabilities under jamming attacks while using low power value for 1 to n and high power value for n + 1 to T , respectively. We denote J T 1 as the value of terminal error performance index under two power levels scheme, J T 0 as the value when only one power is employed in a strategy, and the corresponding two schemes are denoted as s 1 and s 0 , respectively. By comparing s 0 and s 1 , the following result can be obtained. i) ∀t 1 , t 2 ∈ ℤ + , and t 1 ≤ t 2 , there holds Tr(H t 2 ) ≥ Tr(H t 1 ) ≥ 0; ii) the probability satisfȳn Proof. First subtracting J T 0 from J T 1 , there holds From the condition i) in Theorem 3, the following inequality is true: As the consequence of condition (i) in Theorem 3, Tr(H t 1 ) ≥ 0, the following inequality is obtained: Sincē1 <̄0, it suffices to prove that The proof is completed. □ In [20], the authors gave a sufficient condition to make Tr(H t 2 ≥ H t 1 ) set up, which is not sufficient to prove H t 1 ≥ 0. The following corollary is the supplement for that.

Corollary 2. For function h and H defined previously, the condition (i) of Theorem 3 holds if min (A ′ A) ≥ 1 and the following condition is satisfied:
where min is the minimum eigenvalue of matrix A ′ A.
Proof. Note that the function H is the difference form of the discrete value function h. Tr(H t 2 ) ≥ Tr(H t 1 ) ≥ 0 means that the function h is a convex function in its domain. It suffices to prove that H 2 − H 1 ≥ 0, the rest of the work has been completed in [20] and the proof therein. Since based on the fact that Tr(BCD) = Tr(DBC ) and Tr(C + B) = Tr(C ) + Tr(B), the following inequality is true, This completes the proof. □ Remark 7. In fact, Corollary 2 provides a supplementary condition under which the condition (i) of Theorem 3 holds. In [20], the condition min (A ′ A) ≥ 1 guarantees Tr(H t 2 ≥ H t 1 ) set up. Our condition makes H t 1 ≥ 0 set up. This is illustrated by the same example in [20].

Implementation cost and computation complexity analysis
In this subsection, the implementation cost and computation complexity analysis of the system are given. We first focus on the implementation cost. In a realistic environment, the sensor may actively switch to an off-line transmission scheme to reduce the performance loss when the fake-acknowledgment attack [29] (which means the feedback information from the remote estimator is unavailable) is launched by the malicious attacker. Moreover, as can be seen from Theorem 2 and Proposition 2, the parameters needed in the two non-linear programming problems are all non-real-time, and different from the dynamic power allocation as investigated in [30], the real-time estimation error covariance matrix is not necessary for our scenario. Thus, we can obtain the off-line energy allocation policy within time horizon T before the system runs. Therefore, the computational complexity of the overall system lies in determining the power employed by the sensor.
When the sensor has two different power levels, according to Proposition 2 and the optimal structure shown in (15), by solving a non-linear programming as Problem 4, the optimal transmission power value can be obtained, and the transmission times are all non-negative real numbers with 0 ≤ n 0 ≤ T − 2, 0 ≤ n 1 , n 2 ≤ T − 1; hence, the exhaustion method can be applied to solve Problem 4 with the computational complexity not worse than (T 2 ). When the sensor can access multiple power value, the programming function can be used to solve Problem 3.

SIMULATION EXAMPLES
In this section, the optimal transmission policy for multiple power levels case is provided. Then we will focus on the transmission scheme to two power levels case. The following numerical example and parameters are borrowed from [17]

Multiple power levels case
This subsection focuses on the multiple power levels case. Denote n 0 as the times of using zero power value. The nonlinear programming function fmincon in MATLAB is used to obtain the optimal solution. The optimal structure here can be written as s * = (0, 0, … , 0, 1 , 2 , 3 , … , n ). As depicted in Figure 2, it can be seen that the smaller minimum available power value makes Tr(J T ) smaller on each curve, which is similar to the two power levels case. Besides, the is about 50 s. The reason why it takes a lot of time to find the optimal solution is because the candidate transmission power is continuous). The reference terminal error value from [17] is the black solid line on the top.
It can also be seen that the terminal error of the multilevel scheme is sometimes higher than the reference method. This is because the terminal error is affected by the bounds of the available power in our work: if smaller lower bound can be achieved, the optimal policy tends to put the most power on the terminal time slot T such that 0 → 1, which leads to smaller terminal error; The same result occurs when there is a larger allowable upper bound̄. In the following, we will give some numerical results with different upper bounds of the transmission power to illustrate their effects on energy allocation policies. Here, the lower bound ranges from 1 mW to 2mW, and the upper bound takes values from {10, 25, 40 mW}, and zero power is not used, that is, n 0 = 0. From Figure 3, we can see that smaller lower bound leads to better estimation performance in terms of Tr(J T ). It can also be found that larger upper bound̄leads to better estimation performance.
In Figure 4, 50 random schemes for multiple power values are chosen to compare with the optimal transmission strategy for n 0 = 3 case. When only one power level is accessible, the result from [17] showed that Tr(J * min ) = 1224.8192, which is given as a reference value. It can be seen that the estimation error of randomly chosen transmission schemes under multiple power values are greater or less than that of the optimal scheme under one power level. The solution to Problem 3 is the optimal power allocation under multiple power levels, Tr(J * min ) = 1076.7768, which shows the optimality of the proposed scheme.  (2) if the jamming attack is discontinuous within time T (i.e. arbitrary attacks i ≠ j with i ≠ j ), the nondecreasing structure of the energy allocation policy in Theorem 1 may not hold; (3) when the attack is launched discontinuously, the optimal policy tends to use more energy at those no attack time instants comparing with continuous attack circumstance (i.e. k = 1, 1 ≤ k ≤ T ); 4) the initial estimation error covariance matrix P 0 will not affect the non-decreasing structure of the energy allocation policy but leads to larger terminal error Tr(J T ); 5) as we expected, larger channel gains for the attacker will lead to smaller SINR and thus worsen estimation performance.
Al last, a practical system involving the Pendubot is considered, which is a two-link planar robot (see [26] for details). The discrete time model is given as: We do not change the communication parameters comparing with the numerical example below Section 5 except by changing total transmission power Δ = 100 mW. If the sensor can access one power level, according to [17], the optimal power schedule sequence is s = (0, … , 0, 100) (mW) and the corresponding estimation error is 39.4502. While the sensor can access multiple power levels, the optimal transmission power sequence is s * = (2, … , 2, 12, 25, 25) (mW), and the estimation error is 32.0612. We can see that our multiple power levels allocation scheme is superior to the one power level scheme in [17] under different total power (50 and 100 mW).

Two power levels case
In this section, two available power levels are considered at the sensor side. Denote 1 , n 1 as the low power value, times of using low power value in a scheduling policy, respectively, and 2 , n 2 has a similar meaning. Let total energy Δ = 50 mW, which satisfies 1 × n 1 + 2 × n 2 = Δ. In this subsection, first we show the variation of Tr(J T ) with n 1 and 1 to illustrate the sufficient condition in Theorem 3. Only two different power levels are considered and zero power value is ignored in the comparison. The transmission structure can be described as s = ( 1 , 1 , … , 1 , 2 , … , 2 ). Then the optimal solution is presented. Figure 5 shows the variation of Tr(J T ) under different 1 , where 1 is a fixed value on each curve and n 1 ranges from 1 to 24. It should be noted that the condition (ii) in Theorem 3 is a sufficient condition, that is, J T 0 ( s 0 ) ≥ J T 1 ( s 1 ) may hold if In fact, only the points within the interval n 1 ∈ [10, 24] satisfy the condition (ii) on each of the curve in Figure 5.
The result from Figure 6 shows the variation of 1 ranging from 0.5 to 2 mW when n 1 is fixed on each curve. In Figure 6, one can see that larger low power 1 leads to larger Tr(J T ). When the sensor has only one power, the optimal solution can be described as s * = (0, 0, 0, 50∕22, … , 50∕22) (mW) in [17], and the corresponding terminal error is Tr(J T ) = 1224.8192 which is shown as a reference in Figures 5 and 6. It can be seen that when the sufficient condition is satisfied, the performance of the two power levels scheme is better than that of one power level scheme.

CONCLUSIONS
In this paper, we considered the off-line energy allocation strategy when data packets are transmitted at the sensor side via an unreliable wireless network without using real-time acknowledgment information. By taking the terminal error as the per-formance index in the state estimation procedure, the designed optimal transmission strategy should be the nondecreasing structure so that the estimation error can be minimised under DoS attacks. The optimal transmission scheme to multiple power levels case and two power levels case are also investigated, respectively. It is shown that the multiple power levels transmission scheme is better than that of one power level under some conditions.