Algorithm 1. A KF detector that uses an authentication signal [95]. 1) The state-space dynamics of a linear time-invariant system are provided as follows: x (k + 1) = Ax (k ) + Bu (k ) + w (k ) y (k ) = Cx (k ) + v (k ), (6) where x (k ) ! R n denotes the system state, y (k ) ! R m indicates a vector of sensor measurements, u (k ) ! R p implies the control input, and w (k ) ! R n and v (k ) ! R m represent process noise and measurement noise, respectively. It is assumed that x(0), w(k), and v(k) are independent Gaussian random variables. Moreover, A, B, and C show system, input, and output matrices, respectively. 2) The state is estimated by the KF: xt (k + 1|k) = Axt (k |k) + Bu (k) P (k + 1|k) = AP (k |k) A T + Q K (k) = P (k |k - 1) C T (CP (k |k - 1) C T + R ) -1 tx (k |k) = xt (k |k - 1) + K (k) (y (k) - Cxt (k |k - 1)) (7) P (k |k) = P (k |k - 1) - K (k) CP (k |k - 1), where xt (k |k ) and P (k |k ) indicate the estimated state and the covariance matrix of the state, respectively. In addition, xt (k + 1|k ) and P (k + 1|k ) denote the predicted state and the expected covariance of the state, respectively; K(k) implies the Kalman gain. It is assumed that the values of Q and R are constant and known. We also put xt (0|- 1) = xr (0), and P (0|- 1) = R , as initial conditions. For a detectable system, the Kalman gain converges in a few steps, and we define P / lim k " 3 P (k |k - 1), K / PC T (CPC T + R )-1. Therefore, the state is recursively computed by xt (k |k ) = xt (k |k - 1) + K (y (k ) - Cxt (k |k - 1)). 3) The optimal control is augmented using the cost function J = Min lim T " 3 E T -1 1| (x (k )T Wx (k ) + u (k )T Uu (k )), (8) T k =0 where matrices W and U are semidefinite. The optimal control is driven as u * = Lxt (k |k ). and L = -(B T SB + U )-1 B T SA. The variable S is determined by the Riccati equation S = A T SA + W - A T SB (B T SB + U )-1 B T SA. 4) An | 2 detector is formulated as follows: g (k ) = k | i +k -g +1 0.4 0.35 Detector With an Authentication Signal 0.3 0.25 0.2 0.15 0.1 0.05 Detector (y (k ) - Cxt (i |i - 1)T g -1(y (i ) - Cxt (i |i - 1)), (9) where g is a sliding window in which g(k)is calculated. If o #|g (k )|, an attack is detected. Parameter o is a predefined threshold. The final control signal is calculated: u (k ) = u * (k ) + Tu (k ), where Tu (k ) is an authentication signal with zero mean. 40 intended damage. Simulations on an unmanned ground vehicle (UGV) illustrate the attack's effectiveness. A model-based attack detector is developed in [95] to isolate a replay attack. A KF-based estimator is presented to approximate the system state, and an LQG controller is designed to obtain an optimal control law. An |2 detector based on residual estimation is employed to detect system abnormality. The proposed |2 detector is independent identically distributed (IID) Gaussian with low probability. Therefore, if the system under the attack remains stable, the detector's Gaussian distribution converges to one similar to that of normal system operation. The detection rate is equal to the false alarm rate, and the detector cannot distinguish the replay attack. Later, an authentication control signal is added to the optimal control signal to address this issue, and the new detector succeeds. The procedure for designing the control law is explained in Algorithm 1. Figure 3 illustrates an |2 detector and a detector with an authentication signal added to the optimal controller. Note that the capability of the detector sharply improves when an authentication signal is added to the controller. However, the controller is not optimal, and as a result, an extra cost is forced upon the system. To address optimality and detectability problems in systems under attack, authentication signal optimization is performed in [96]. The detectability of the |2 detector is maximized, and concurrently, the LQG performance loss is constrained to be less than a certain value. This problem can be solved by the Lagrangian method. Figure 4 presents an |2 detector with nonoptimal and optimal authentication signals. The importance of optimizing the authentication signal is obvious from Figure 4. The power of the optimal detector is almost 15 times greater than the nonoptimal detector. In [9], a model-based approach using an extended KF (EKF)-based estimator is developed to isolate cyberattacks for a class of stochastic nonlinear systems. DoS and false data injections are considered. The EKF is applied to Detection Rate applied to detect attacks on sensors. A linear attack strategy is adopted, and the operation proves to be stealthy, providing an optimal closed-loop form for a successful attack. Moreover, simulations show that the attack cannot be detected. In [94], an optimal deception strategy is introduced against a CPS. First, an |2 detector that uses a Kalman filtering method is discussed. Then, an optimal attack is formulated using the Kullback-Leibler divergence, and the singular value decomposition method is employed to divide the residual into two parts to investigate the attack's stealthiness. An attack policy is proposed from a hacker's perspective to obtain an upper bound for the IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2021 0 10 11 12 13 14 15 16 17 18 19 20 Time (s) Figure 3. The | 2 detector and a detector with an authentication signal added to the optimal controller [95].

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