Networked Security Observer-Based Reference Tracking Control of Stochastic Quadrotor UAV System Under Cyber-Attack:T-S Fuzzy Approach

In this study, a robust $H_{\infty }$ networked security observer-based reference tracking control scheme is proposed for the stochastic quadrotor unmanned aerial vehicle (UAV) system under malicious attacks on the actuator and sensor of network control system (NCS). To reduce the computational burden on UAV system, the UAV system is connected with a remote computing unit and the complicated tracking control command can be calculated by remote computing unit. By using the novel discrete smoothed model, the model of attack signals on actuator and sensor can be embedded in system state of UAV and thus the attack signals as well as the quadrotor system state can be simultaneously estimated through a conventional Luenberger observer. Further, the corruption of attack signals on state estimation of UAV is also avoided. To eliminate the effect of unavailable external disturbance and intrinsic fluctuation during the reference tracking control process, a robust $H_{\infty }$ networked security observer-based reference tracking control scheme is introduced to attenuate their effects on the NCS of quadrotor UAV. By using the characteristic of convex Lyapunov function, the design condition of robust $H_{\infty }$ networked security observer-based tracking control is derived in terms of the nonlinear functional inequalities. Since the nonlinear functional inequalities are not easy to be solved analytically or numerically, the Takagi-Sugeno (T-S) fuzzy interpolation technique is employed to interpolate the nonlinear stochastic quadrotor NCS by a set of linear local systems via fuzzy bases. In this case, the nonlinear functional inequalities can be converted to a set of linear matrix inequalities (LMIs) which can be easily solved by the MATLAB LMI TOOLBOX. Simulation results are provided to validate the effectiveness of the proposed method in comparison with conventional robust observer-based T-S fuzzy tracking control scheme.


I. INTRODUCTION
In recent years, the unmanned aerial vehicle (UAV) has attracted more attention from researchers due to its wide utilization and convenience of use. Through these advantages, UAV has extensive applications such as humanitarian relief [1], topographic survey [2], and military reconnaissance [3]. To successfully complete the tasks mentioned above, an UAV is required to track a desired trajectory. Therein, the trajectory tracking control is a popular issue for The associate editor coordinating the review of this manuscript and approving it for publication was Shen Yin. the control of UAV's flight [4], [5]. Among several types of UAV, quadrotor UAV has the capability of vertical take-off and landing (VTOL) [6]. Based on this maneuverability, a quadrotor UAV can track more kinds of trajectories than other types of UAV. Though quadrotor UAV has many appealing advantages in practical applications, the power consumption of quadrotor UAV during the flight process is a critical problem needed to be further considered. Clearly, highly power consumption limits the flight distance and using time of quadrotor UAV which may restrict its applications [7], [8].
Recently, along with the advance of communication techniques, e.g., 5G and 6G in the future smart cities [9], [10], the authors address on the estimation of malicious attack signals and the consequent attack-tolerant tracking control for the stochastic network-based quadrotor UAV system under cyber-attack and external disturbances.
In this study, the robust H ∞ networked security observerbased reference tracking control scheme is proposed to guarantee that the networked control quadrotor UAV can gradually approach to the desired attitude and reference path under the influence of malicious attacks, external disturbances and intrinsic random perturbations. By introducing the structure of network system, the stochastic dynamic model of networked control quadrotor UAV is constructed. With the help of discrete smoothed model, malicious attacks on actuator and sensor of quadrotor UAV through network communication channels can be effectively described and embedded in the augmented states of an augmented system consisted of quadrotor model and smoothed models of attack signals. Therefore, the attack signals and states of quadrotor can be estimated by a conventional Luenberger observer. Then, the estimated state variables and attack signals are used for the networked security tracking control of stochastic NCS of quadrotor UAV under cyber-attack. By utilizing the convex Lyapunov function, the reference tracking control design is transformed to equivalent nonlinear functional inequalities problem. Since the nonlinear functional inequalities problem for robust H ∞ networked security reference tracking control design problem is not easily to be solved, the Takagi-Sugeno (T-S) fuzzy model [27] is utilized to approximate the nonlinear stochastic system of quadrotor by interpolating a set of several linearized local systems. After applying the T-S fuzzy model, the H ∞ networked security reference tracking control problem can be transformed to a linear matrix inequalities (LMIs)-constrained problem, which can be efficiently solved by the MATLAB LMI TOOLBOX. A simulation example of a single UAV to track a upwards round-shape trajectory is provided to validate the performance of proposed robust H ∞ networked security observer-based reference tracking scheme. Further, the conventional T-S fuzzy observer-based tracking control scheme is used for performance comparison.
The main contributions of this work are described as follows: 1) A novel discrete smoothed model is utilized for modeling unavailable cyber-attack signals on the network control system. As a result, through a conventional Luenberger observer, the attack signals on NCS of quadrotor UAV can be estimated with the system states simultaneously and thus the networked security observer-based reference tracking control scheme of quadrotor UAV NCS can be guaranteed.
2) A H ∞ networked security observer-based tracking control design is proposed for quadrotor NCS through the estimated attack signals. In order to avoid solving highly complicate nonlinear observer-based H ∞ network reference tracking control design problem of stochastic quadrotor NCS, T-S fuzzy model is proposed to interpolate the nonlinear stochastic UAV networked system via several local linearized systems to simplify the design of controller and observer in VOLUME 10, 2022 the H ∞ networked observer-based security tracking control of quadrotor UAV system under cyber-attack and external disturbance. We only need to solve a set of LMIs via MATLAB LMI TOOLBOX to obtain fuzzy control gains and observer gains of H ∞ fuzzy networked security observer-based reference tracking control strategy of quadrotor UAV NCS.
The study is organized as follows. Preliminaries and the problem formulation of the network observer-based tracking control system of UAV under cyber-attack are presented in Section II. In Section III, the stochastic H ∞ networked observer-based security tracking analysis is provided. T-S fuzzy model is introduced to cope with the nonlinear stochastic H ∞ networked observer-based security tracking control problem in Section IV. The stochastic robust H ∞ fuzzy networked observer-based security tracking control design scheme is also proposed in this section. A simulation example of quadrotor UAV flighting along a reference path under attack signals and external disturbances is given to validate the effectiveness of the proposed robust H ∞ fuzzy networked security observer-based tracking control scheme in Section V. The conclusion is made in Section VI.
Notation: A T : the transpose of matrix A; A ≥ 0(A > 0): symmetric positive semi-definite (symmetric positive definite) matrix A, respectively; I n : the n-dimensional identity matrix; x 2 : the Euclidean norm for the given vec-
In the real world, the quadrotor UAV will be disturbed by not only external disturbances but also internal fluctuations. To make the quadrotor UAV system more practical, the Wiener process is used to formulate the internal random parametric fluctuations in the quadrotor UAV. Thus, the nonlinear stochastic quadrotor UAV can be written as follows: where W (t) denotes the 1-D Wiener process and σ (X (t))dW (t) denotes the nonlinear state-dependent parametric fluctuations. The Wiener process W (t) is defined on the complete filtered probability space ( , F, {F t } t∈R ≥0 , P) where denotes the sample space, σ -field F t is generated by Wiener process W (s) for s < t, F = ∪ t≥0 F t and P is the probability measure [37]. It is assumed that the nonlinear functions f (X (t)), g(X (t)) and σ (X (t)) are local-Lipschiz [37].

B. NETWORK CONTROL SYSTEM
Under the framework of network control system (NCS), the quadrotor is controlled through a remote computing unit (controller). The schematic of NCS of quadrotor UAV is shown in Fig. 2. Through the sensor, output data of quadrotor UAV is sampled and transmitted to the observer on the remote side. Then, the control commands are computed at each computing unit and transmitted via wireless network channel to zero-order holder (ZOH) at every sampling period. ZOH will transmit the control commands to actuator and this makes the control commands become a continuous control signal within a sampling period. It is worth to point out that the control signals stored in ZOH will be updated until new control commands are received. Based on the NCS framework in Fig. 2, the remote computing unit only receives the plant information at each sampling period. In this case, instead of using a continuous-time observer, a discrete-time observer is more appealing to be utilized to save the computation resource in the remote side. However, due to the fact that the local UAV plant in (2) is a continuous-time system and the observer is a discretetime system, it is almost impossible to simultaneously analyze the estimation performance of observer and tracking performance of local UAV system in (2). In this situation, the digital redesign [31] should be utilized to discretilize the continuous plant for the simplicity of design. Nevertheless, due to the stochastic Wiener process and nonlinearity system characteristics in UAV system, it is impossible to obtain the exact form of system matrices of the UAV system by using conventional discretization method. As a result, the Euler-Maruyama method in [30] is applied to discretilize the UAV system and the nonlinear stochastic UAV dynamic system can be approximately represented by the following difference equation where  [32]. By applying these advanced mechanisms, it can further improve the control performance and save the limited communication resource. At the same time, it also increases the analysis difficulty during the controller/observer design.
For NCS, time-varying delay is an important issue that needs to be considered. According to Fig. 2, there are uplink and downlink networked communication channels in the network control system. As signals transmit through these two channels, signals may suffer influence from time-varying delays. These delays can be represented as follows: τ 1 (k) is the downlink delay from controller to ZOH and τ 2 (k) is the uplink delay from sensor to observer. By taking these two time-varying delays into consideration, the NCS-based control system of quadrotor UAV can be formulated as [33]: 10,2022 and U p (kh) = (1 − δ(kh))U (kh) + δ(kh)U (kh − τ 1 (k)) (5) From above equations, Y p (kh) denotes the signal of system output which is received by the observer. Similarly, U p (kh) is the signal of control command which is received by ZOH. Both of them are transmitted through the wireless network channel and may be influenced by time-varying delays. To formulate the influence of delay effect, the occurrence of time-varying delays is assumed to satisfy with Bernoulli sequence. Thus, δ(kh) and β(kh) are Bernoulli processes as follows: whereδ > 0 andβ > 0 are the known probability of time-varying delay occurring in the signal transmission from controller to ZOH and sensor to observer through wireless network channels, respectively.
Remark 3: As time-varying delay τ 1 (k) occurs, i.e., δ(kh) = 1, the control signal U (kh) can not be transmitted on time. Since the quadrotor UAV can not be out of control, the previously received control signal is used to compensate the delayed control signal. If time-varying delay τ 1 (k) does not occur, δ(kh) = 0, U (kh) can be transmitted through the wireless network successfully and the system plant can receive control command on time. According to the analysis above, U p (kh) can be constructed in the form of (5). Similarly, Y p (kh) has the same form in (4).
Assumption 1: In this study, the time-varying delays satisfy the following boundary conditions: As signals transmitting in the wireless network channels, there may exist some malicious attacks from attackers and it tries to degrade the performance of NCS and even let NCS become unstable. Thus, these attacks need to be considered in NCS. In Fig. 2, two malicious attacks f a (kh) and f s (kh) will influence on the reference tracking performance and the state estimation of NCS of quadrotor UAV via wireless network, respectively. In this situation, sensor and actuator will receive wrong information from network channel and may cause whole system not being controlled correctly. That is, these malicious attacks can be equivalently regarded as sensor fault and actuator fault in quadrotor UAV NCS. Then, the malicious attacks should be considered in the security control design of NCS and the NCS of quadrotor UAV should be modified as: where f a (kh) is the actuator attack signal, f s (kh) is the sensor attack signal and D(X (kh)) is the effect matrix of sensor attack signal f s (kh). Remark 4: Since actuator attack signal f a (kh) disturbs the transmission of control signal U p (kh) through the wireless network channel, actuator will receive f a (kh) and U (kh) simultaneously, i.e., the effect matrix of actuator attack signal f a (kh) is the same as the input matrix g(X (kh)).
Since the actuator attack f a (kh) and sensor attack f s (kh) are unavailable signals, they can not be estimated by the conventional estimator from the discrete stochastic NCS in (6) directly. To simplify the attack signal estimation, the discrete-time smoothed models of attack signals f a (kh) and f s (kh) are constructed. By the discrete-time smoothed models of attack signals f a (kh) and f s (kh), the conventional Luenberger observer can be designed to estimate system states and attack signals simultaneously. Similar to [34], the discretetime smoothed model of actuator attack signal f a (kh) is proposed as follows: where are the extrapolation coefficients and a 0 I n a a 1 I n a a 2 I n a · · · 0 I n a 0 · · · 0 0 I n a . . .
Similar to (7), the discrete-time smoothed model of sensor attack signal f s (kh) can be described as follows: are the extrapolation coefficients and Based on the extrapolation method, one practical selection of extrapolation parameters for two smoothed signal models in (7) and (9) can be given as: For the first rule, it is expected that f a (kh) and f s (kh) are more related to f a ((k − 1)h) and f s ((k − 1)h), respectively, and thus the extrapolation parameters {a i , b i } d i=0 should be chosen as positive decreasing sequences. On the other side, to avoid the over-extrapolation, the summation of these extrapolation coefficients should be one.
Remark 6: The proposed smoothed signal models in (7) and (9) are the modification conventional Kalman fix-laged smoothed model [24], [25], in which the future signals are the combination of current signals and extrapolation errors, i.e., a 0 = b 0 = 1 and (7) and (9), respectively. To further use the lag information, the modified smoothed signal models in (7) and (9) include the extrapolation coefficients which can be selected by designer. In this case, two-side information of actuator/sensor attack signals can be utilized to achieve the better estimation of actuator/sensor attack signals in (12). Further, the more precise estimationF a (kh) andF s (kh) can be employed for the observer-based reference tracking control to efficiently eliminate the effect of attack signals by the proposed H ∞ observer-based reference control strategy of UAV Remark 7: In general, the selection of these extrapolation is not unique and there exist several adaptive parameter estimation methods (e.g., recursive least squares filter [26]) to estimate these extrapolation coefficients. However, there exists some difficulties to be overcome. At first, since the attack signals are unavailable, it is not easy to update these extrapolation coefficients via their adaptive estimation error for the conventional recursive algorithms. Secondly, in the networked-based control structure, the feedback information by the parameter adaptive algorithm for the UAV system of extrapolation coefficients will be delayed and it will make the proposed H ∞ network observer-based reference control strategy become more difficult to design. Moreover, it may destroy the performance of H ∞ observerbased reference control.
By combining (6), (7), and (9), we define the augmented NCS of quadrotor UAV asX (kh) = [X T (kh) F T a (kh) F T s (kh)] T and the corresponding augmented system state is formulated as follows: with the following system matrices: In general, due to the nonlinear system funtions in (11), the observability of augmented system in (11) can not be easily ensured. To faciliate the observer design in sequel, the following assumption is made.
Through the assumption above, the following nonlinear Luenberger observer is employed to estimate the states of the augmented system in (11): X ((k + 1)h) =f (X (kh)) +ḡ(X (kh))U (kh) whereX (kh) is the estimated state of augmented quadrotor NCS (11),Ŷ (kh) is estimated measurement output and L(X (kh)) is the nonlinear observer gain. observer gain. Remark 8: If we estimate quadrotor state X (kh) from quadrotor NCS in (6), the attack signals f a (kh) and f s (kh) will deteriorate the state estimation. While attack signals f a (kh) and f s (kh) are embedded in the state of augmented system in (11), we not only estimate X (kh), f a (kh) and f s (kh) simultaneously by the observer in (12) but also avoid the corruption of attack signals f a (kh) and f s (kh) on state estimation. This is the main merit of the proposed discrete-time smoothed model of attack signals in (7) and (9).
Remark 9: Since the control command can be directly calculated and transmitted to observer system in (12), the control command U (kh) in observer system is delay-free. However, due to the effect of networked induced delay in the network communication link in Fig. 2, the control command U p (kh) in the augmented NCS of quadrotor UAV in (11) is delay-dependent. Similarly, the estimated measurement out-putŶ (kh) is delay-free for the observer system in (12) and the real measurement output Y p (kh) received by the observer is delay-dependent.
In real application, the quadrotor UAV is controlled to track the desired path to complete some tasks. To generate the desired tracking control trajectory, the reference tracking model is used to generate desired reference state [35]: where X r (kh) is the reference state to be tracked, A r denotes a specific stable matrix and r(kh) is the reference input. Remark 10: At the steady state, X r ((k +1)h) = X r (kh) and the reference model will become If the eigenvalues of matrix A r are all inside unit circle in the z-complex domain, i.e., |z| < 1, the desired trajectory X r (kh) is equal to the reference input r(kh) at the steady state, i.e., X r (kh) will approach to r(kh) at the steady state. Besides, A r specifies the transient characteristics of X r (kh) to approach the desired reference r(kh).

C. PROBLEM FORMULATION
In this study, the nonlinear Luenberger observer in (12) is employed to estimate quadrotor states X (kh) as well as malicious attacks f a (kh) and f s (kh) in (11). If quadrotor states and malicious attack signals on actuator and sensor can be effectively estimated, their effects on the system can be reduced or even eliminated. On the other hand, in order to make the quadrotor UAV track the desired trajectory and finish its task efficiently, an observer-based tracking controller U (kh) = K (X (kh), X r (kh)) based onX (kh) and X r (kh) is needed. Since there are some disturbances and measurement noises during the flight of UAV which are unavoidable, the following robust H ∞ networked security observer-based reference tracking control design strategy of stochastic quadrotor NCS is proposed to effectively attenuate these undesired effects on the tracking control performance below a prescribed disturbance attenuation level ρ: where V 1 (X (0),X (0), X r (0)) denotes the effect of initial condition of augmented system in (11) to be deduced on the state estimation of the observed augmented system and the tracking of reference system, Q 1 ≥ 0 and Q 2 ≥ 0 denote the weighting matrix of the estimation error and the tracking error, respectively, k f ∈ N denotes the terminal time, and υ 1 (kh) = [ῡ T (kh) r T (kh) ] T . If one could specify a tracking controller U (kh) and a nonlinear observer gain L(X (kh)) such that (14) holds, then the worst-case effect of external disturbanceῡ 1 (kh) on the estimation errorX (kh) −X (kh) and tracking error X (kh)−X r (kh) can be attenuated to a prescribed disturbance attenuation level ρ from the viewpoint of energy.

III. H ∞ NETWORKED SECURITY OBSERVER-BASED REFERENCE TRACKING CONTROL DESIGN OF QUADROTOR UAV NCS
In this section, the robust networked security observer-based tracking control design for nonlinear quadrotor UAV NCS is investigated. To begin with, we define the augmented state estimation error vector as e(kh) =X (kh) −X (kh) and the corresponding dynamic of e(kh) can be derived as: Next, since the augmented stateX (kh) includes X (kh), F a (kh) and F s (kh), to be consistent in dimension, the reference tracking model in (13) should be extended as follows: Then, to simplify the design procedure of robust H ∞ networked security observer-based tracking control design in (14), the dynamic system in (11), estimation error dynamic in (15) and the reference dynamic system in (16) are augmented as Based on the augmented system (17), the robust H ∞ networked security observer-based reference tracking control strategy can be rewritten as Based on the above analysis, the robust H ∞ networked security observer-based reference tracking control design problem in (14) for the quadrotor NCS under attack signals and external disturbances becomes how to specify U (kh) and L(X (kh)) to achieve the robust H ∞ stabilization problem in (18) for the augmented system in (17) to simplify the design procedure. On the other hand, ifῡ 1 (kh) is vanished in (17), the following definition of mean square stability in probability is given to address the stability criterion in (17) Definition 1: The nonlinear stochastic augmented UAV system in (17) satisfies the mean-square stability in probability if the following condition holds Due to the nonlinear system functions and the delayed external disturbanceῡ 1 (kh − τ 2 (k)) with corresponding statedependent matrixõ 2 (X (kh)), the robust tracking control design is more difficult than the conventional case (e.g., the linear system matrix with non-delay external disturbance). Thus, the following assumption and definition are made to facilitate the robust tracking control design in this study.
Assumption 4: The state trajectory of augmented system X (kh) in (17) lies in a compact domain , i.e.,X (kh) ∈ .
With the help of the convex Lyapunov function V (·) in above definition and Assumption 4, the design method of robust H ∞ networked security observer-based reference tracking controller is given as follows: Theorem 1: For the observer-based security tracking control design of quadrotor UAV in (11)- (13) or in the augmented system in (17), if there exist observer gain L(X (kh)), tracking controller U (kh), a prescribed disturbance attenuation level ρ and convex Lyapunov function V (·) satisfying the following nonlinear functional inequalities , kh))}, then the robust H ∞ networked security observer-based reference tracking control strategy in (18) of quadrotor NCS is guaranteed for a prescribed disturbance attenuation level ρ. Besides, if the external disturbanceῡ 1 (kh) is vanished, the mean square stability ofX (kh) is achieved, i.e., the mean square state estimation error and mean square tracking error will converge to 0 in probability.
Proof: Please refer to Appendix A. Due to the characteristics of discrete-time nonlinear system and nonlinear Lyapunov function, the system characteristic which includes deterministic parts and stochastic parts are embedded in the Lyapunov function in (20). In this situation, it is not easy to decouple the deterministic parts and the stochastic parts. Thus, it can be seen that the design condition of Theorem 1 in (20) involves the increment of Wiener process W (kh). In general, it is not easy to directly find a convex Lyapunov function V (·) with the corresponding observer gain L(X (kh)) and tracking controllerŪ (kh) to meet the design conditions in Theorem 1.
Remark 11: In general, due to the state-dependent nonlinear matricesõ 1 (X (kh), kh) andõ 2 (X (kh), kh) w.r.t. the noise terms in (17), it is not easy to decouple the noise terms and its' system matrices during the derivation. However, with the utilization of Assumption 4, these time-varying nonlinear matrices can be bounded by its' operator norm. In this case, the design conditions of robust H ∞ networked security observer-based reference tracking control strategy in (18) can be derived in terms of nonlinear difference inequality problem with operator constraints in (20)- (22).

IV. T-S FUZZY H ∞ OBSERVER-BASED SECURITY REFERENCE TRACKING CONTROL FOR NONLINEAR QUADROTOR UAV NCS UNDER CYBER-ATTACK
In general, the observer-based controller design of robust H ∞ networked security observer-based tracking control for nonlinear stochastic quadrotor UAV NCS needs to solve nonlinear functional inequalities in (20), which are not easy to be solved. Therefore, the T-S fuzzy interpolation method is introduced to represent the nonlinear stochastic NCS of quadrotor UAV by the convex combination of a set of specific local linearized systems [38]. By designing the observerbased controller of each local system, the H ∞ observerbased security tracking controller of whole nonlinear NCS of quadrotor UAV under cyber-attack can be constructed by the combination of these T-S fuzzy local observer-based controllers. To begin with, the ith T-S fuzzy rule of stochastic quadrotor UAV NCS in (6) is described as follows [38]- [40]: where are the premise variables, G iq denotes the ith fuzzy set of the qth premise variable, for i = 1, . . . , M and q = 1, . . . , g, M is the number of fuzzy rules and g is the number of premise variables. The local linearized matri- are with appropriate dimensions. By the defuzzification process, the overall T-S fuzzy NCS of quadrotor UAV can be inferred as follows [38]: where z(kh) , which satisfies In order to estimate X (kh), f a (kh) and f s (kh) simultaneously, the dynamic models of cyber-attack signals in (7) and (9) need to be augmented with T-S fuzzy NCS of quadrotor UAV in (24) as the following augmented NCS of quadrotor UAV: Before the further discussion, the following theorem is proposed to address the observability of local linear system in the augmented fuzzy system in (25): Theorem 2: In the augmented NCS of quadrotor UAV in (25) where A F i I 12 + hA i and the following conditions are satisfied where Z 1 is the set that collects the complex number with norm less than 1, then the ith local linearized system Proof: Please refer to Appendix B. Then, in order to estimate the augmented NCS of quadrotor UAV in (25), the ith fuzzy Luenberger observer rule is defined as: where L i is the fuzzy observer gain for i = 1, · · · , M . Then, the overall fuzzy observer can be represented as: According to above analysis, the observer-based fuzzy controller of the nonlinear NCS of quadrotor UAV in (25) can be inferred as follows: The overall fuzzy controller can be formulated as: Then, the dynamic stateX (kh) of augmented NCS of quadrotor UAV in (25) can be obtained as follows: Also, the nonlinear estimation error system in (15) can be rewritten as follows: Finally, the augmented fuzzy observer-based reference tracking control system can be formulated as follows: ×X (kh − τ 2 (k)) + (D ijlv,D +D ijlv,R (kh))ῡ 1 (kh) The system matrices in (35) are given as follows: Before the further discussion, the following assumption is made to deal with the fuzzy approximation error: Assumption 5: There exists some scalars { i ≥ 0} 6 i=1 such that the fuzzy approximation errors are bounded as follows Besides, the following Lemmas are introduced to facilitate the design procedure: Lemma 1 [42]: For any matrix X and Y with appropriate dimensions, the following inequality holds: where P is any positive definite symmetric matrix.
Lemma 2 [29]: Given a set of matrices {A i } n i=1 with appropriate dimension, a positive matrix P and a set of series By using the above two Lemmas, the main theorem is proposed as follows: Theorem 3: Given a set of scalars with the following matrices for i, j, l, v = 1, · · · , M , then the H ∞ stochastic fuzzy security observer-based reference tracking control strategy in (14) is achieved with the disturbance attenuation level ρ. Besides, if the external disturbanceῡ 1 (kh) is vanished, the mean square stability ofX (kh) in (35) is achieved, i.e., the mean square state estimation error and mean square tracking error will converge to 0 in probability. Proof: Please refer to Appendix C. According to the analysis above, the design procedure of H ∞ fuzzy networked security observer-based reference tracking control of quadrotor NCS is given as follows: Step 1: Select fuzzy plant rules and membership function for nonlinear system in (24).
Step 4: Select the weighting matrices Q 1 , Q 2 and disturbance attenuation level ρ for robust H ∞ networked security observer-based reference tracking strategy in (14).
Step 5: Select the boundary parameters { i > 0} 8 i=1 and P , and calculate the fuzzy approximation errors Step 6: Solve the LMIs in (36)- (39) to obtain W > 0, Then, construct the fuzzy observer in (32) and construct the fuzzy controller in (34).

V. SIMULATION RESULTS
In this section, to validate the effectiveness of proposed robust H ∞ security observer-based reference tracking controller of stochastic quadrotor NCS under cyber-attack and external disturbances, a simulation example of a quadrotor NCS under cyber-attack and disturbances is given for the illustration of design procedure and validation of the desired tracking performance. On the other hand, the conventional discrete-time robust H ∞ observer-based tracking control scheme in [43] is also provided for the performance comparison.
The physical model of quadrotor UAV is illustrated in (1) and the specific simulation parameters of quadrotor UAV are given as [36]: m = 2 (kg), g = 9.8 (m/s 2 ), J x = J y = 1.25 (Ns 2 /rad) For the desired reference tracking trajectory in this simulation, the quadrotor is asked to track a spiral trajectory with constant velocity on the z-axis. As a result, the position of reference input r(kh) in (13) is specified as follows: where h = 0.01 denotes the sampling period. In general, for a real quadrotor system, the attitude of quadrotor is closely related to the corresponding position trajectory. Hence, according to the desired position trajectory in (40), the attitude of reference signal r(kh) in (13) is given as: tively, which mean the reference of acceleration. The matrix A r in the reference tracking model in (13) is specified as: From the network point of view, the observation on position will suffer the influence from sensor attack f s (kh) through the wireless communication channels. On the other hand, actuator attack signal f a (kh) will be transmitted into the actuator with control commands to corrupt the system plant of quadrotor. Thus, the effect matrix of sensor attack D(X (kh)) and measurement output matrix C(X (kh)) in (6) are given as: For the first actuator attack signal, it will provide additional forces on the change of x 2 (kh), y 2 (kh) and z 2 (kh) within the specified time interval during the flight process. Also, three actuator attack signals {f a,2 (kh), f a,3 (kh), f a,4 (kh)} are used to describe the random noise on three torques. Due to the limitation of torque forces {τ 1 (kh), τ 2 (kh), τ 2 (kh)}, the amplitudes of these attack signals are chosen as reasonable values. On the other hand, for the sensor attack signal, it causes oscillation effect on sensor and makes the estimation of UAV become much harder. Besides, the stochastic term σ (X (kh)) = 0.05[0 x 2 (kh) 0 x 4 (kh) 0 x 6 (kh) 0 0 0 0 0 0] T is formulated to describe the effect of time-varying air viscosity on three velocities.
To construct the fuzzy system in (24), the observer state variables z 2 , φ 1 , φ 2 , θ 1 , θ 2 are selected as premise variables and the operation points of premise variables are given as follows:

The fuzzy approximation errors and boundary parameters
, P are selected as follows: The quadrotor UAV NCS may suffer interferences such as external disturbance υ(kh) on system plant and measurement noise n(kh) on measurement output. Hence, in this study, the external disturbance is set as υ(kh) = 0.01[0 sin(kh) 0 sin(kh) 0 sin(kh) 0 0.01 cos(kh) 0 0.01 cos(kh) 0 0.01 cos(kh)] T and measurement noise n(kh) is assumed to be zero mean white noise with unit variance. Since the values of angular velocity of quadrotor UAV are very small, the external disturbances of angular velocity are also given with some very small values. By solving the LMIs in Theorem 3 with the prescribed disturbance attenuation level ρ = 10, we can obtain the corresponding fuzzy tracking controller gains {K 1j , K 2j } 32 j=1 and observer gain {L i } 32 i=1 . The simulation results are shown in Figs. 3-10. Fig. 3 shows the malicious attack signals f s (kh) and f a (kh) = [ f a,1 (kh) f a,2 (kh) f a, 3 (kh) f a,4 (kh)] T on the quadrotor NCS and their estimation. From Fig. 3, by the effect of the quadrotor's transient state response due to the initial condition, the estimation of first actuator attack signal f a,1 (kh) also has a large transient response. After that, the malicious attack signals f s (kh) and f a,1 (kh) can be estimated quite well by the proposed H ∞ fuzzy security Luenberger observer. Also, while these two attack signals have suddenly jumps, they cause some fluctuation on the estimation. After that, the changed attack signals can be estimated well, e.g., the sensor attack signal jumps at 20s and can be estimated precisely by the proposed observer. Since the malicious attack signals f s (kh)    and f a,1 (kh) are estimated together in the augmented system in (11), their effects may impact with each other. For example, the estimated signal f a,1 (kh) oscillates due to the high frequency signal f s (kh). By the fact that f s (kh) corrupts Y p (kh) in (12) directly, Y p (kh) could influence on the estimated signal f a,1 (kh) indirectly, which can be seen from the structure of Luenberger observer in (12) or (32). Besides, since three actuator attack signals {f a,2 (kh), f a,3 (kh), f a,4 (kh)} are selected as Gaussian white noises, the corresponding estimations can not be done due to its' strong randomness. However, by the proposed H ∞ fuzzy networked security reference tracking control strategy in (14), these effects of δ a (kh) and δ s (kh) in v 1 (kh) on the tracking can be passively attenuated during the reference tracking control process.   Since the attack signal f s (kh) is estimated precisely, its estimation can be used to compensate the effect of real sensor attack signal on the sensor. Thus, the system state estimation can be more precisely achieved by  the proposed H ∞ fuzzy networked security reference tracking control scheme than the conventional H ∞ fuzzy observerbased tracking control method in [43].
The control signals U (kh) = [F(kh) τ φ (kh) τ θ (kh) τ ψ (kh)] T of the proposed method for the NCS of quadrotor UAV are displayed in Fig. 9. To cancel out the effect of actuator attack signal f a (kh), F(kh) changes its control effort when the cyber-attack signal f a (kh) attacks the actuator of NCS of quadrotor UAV. For example, as attack signal f a (kh) occurs at 13s in Fig. 3, F(kh) gives a corresponding signal to cancel the attack signal f a (kh) on the NCS of quadrotor UAV by using the estimated fault signalf a (kh) in Fig. 9. On the other hand, the control signals of the NCS in (34) of quadrotor UAV will also suffer from the sensor attack signal f s (kh) through Y p (kh) in (32). Since Y p (kh) is directly transmitted to the observer through wireless network channel, it suffers from the effect of sensor attack signal f s (kh). Then, the sensor attack signal f s (kh) will affect observer stateX (kh) and then affectŪ (kh). For example, τ ψ (kh) shows its high frequency oscillation caused by the high frequency sensor attack signal f s (kh) in Fig. 9. However, due to the good sensor attack signal estimation for compensation, these effects on control signals are attenuated efficiently by the proposed H ∞ security observer-based reference tracking control strategy.
The time sequences, due to the occurrence of the timevarying delays, τ 1 (k), τ 2 (k) are displayed in Fig. 10,. When the time-varying delay τ 1 (k) occurs, the value of Bernoulli process δ(kh) is equal to 1. Similarly, when the time-varying delay τ 2 (k) occurs, β(kh) is equal to 1. Even the control inputs and measurement outputs suffer from these delay effects, the proposed H ∞ fuzzy networked security reference tracking controller can achieve the desired tracking control performance with the robustness for the attenuation of these network-induced effects.
For the performance comparison, the conventional robust H ∞ fuzzy discrete-time observer-based tracking control scheme in [43] is carried out and the results are also shown in Figs. 4-8. Without the attack signal estimation, these attack signals are regarded as a kind of disturbance and their effects on the state estimation and reference tracking performance are passively attenuated by the conventional robust H ∞ fuzzy discrete-time observer-based tracking control strategy in [43]. Due to the large influence of attack signals, the estimation error of position/velocity variables in Fig. 7 will not converge and it also deteriorate the performance of observer-based controller. However, due to the selection of D(X (kh)), the effect of sensor attack signal on three angular positions is relatively minor and thus the estimation error of angular velocity/angular position is relatively small. Even the three attitudes are well controlled and estimated in Figs. 6 and 8, the sinusoidal sensor attack signal and actuator attack signal severely influence the control strategy on three position velocities. Moreover, since the networked-induced effects (e.g., delay effect) are not well considered, the conventional control strategy is more deteriorated and thus it cause some fluctuations on the velocity/position variables. Thus, for the 3-D reference tracking trajectory of UAV controlled by the conventional robust H ∞ fuzzy observer-based tracking controller in [43], the reference tracking trajectory is up-anddown around the desired trajectory as shown in Fig. 4. In this situation, once the UAV reference tracking trajectory is deviated from the desired trajectory, it may fail to complete the pre-designed task or even cause collision during the flight process.
Also, the following performance indices about the modified integral square error (MISE), and the modified integral time square error (MITSE) (i.e., discrete version of ISE and ITSE) are respectively given in the following for the tracking/estimation performance evaluations about our method and the method in [43] where X (kh) is the UAV state in (3), X r (kh) is the reference trajectory to be tracked in (13),X (kh) is the estimation of UAV state in the fuzzy Luenberger-observer in (32), F att (k) = [F T a (kh) F T s (kh)] T is the augmented attack signal with augmented actuator attack signal F a (kh) in (7), and augmented sensor attack signal F s (kh) in (9),F att (k) is the estimation of augmented attack signal F att (k) in fuzzy Luenberger-observer in (32), T = 60 is the terminal time and h = 0.01s is the sampling time. It is worth to point out that the attack signal estimation is not considered in the observer design in [43]. For equality, we only compare the performances of state estimation between our method and the estimation method in [43]. On the other hand, the performance of attack signal estimation is independently calculated.
The evaluation results are shown in Figs. 11-13. For the tracking performance evaluation, due to the effect of actuator/sensor attack signals, the MISE of the conventional robust H ∞ fuzzy discrete-time observer-based tracking control scheme in [43] is increasing as time increase as shown in Fig. 11-(a). This fact implies the effect of actuator/sensor attack signals on tracking performance can not be passively attenuated by the conventional robust H ∞ fuzzy discretetime observer-based tracking control scheme. On the contrary, since the actuator/sensor attack signals are embedded   in the augmented state to avoid their corruption on state estimation, the MISE of the proposed H ∞ fuzzy networked security reference tracking control scheme almost converges to a constant at 10s in Fig. 11-(a), i.e., the energy of tracking error almost approaches to zero at 10s. However, due to the external disturbance, the measurement noise and the Wiener process in UAV system in (3), there exist small fluctuations in the MISE of the proposed H ∞ fuzzy networked security reference tracking control design. According to MISE results of two methods, for the MITSE tracking performance evaluation in Fig. 11-(b), the MITSE of the conventional robust H ∞ fuzzy discrete-time observer-based tracking control scheme exponentially increases and the MITSE of the proposed H ∞ fuzzy networked security reference tracking control design increases slightly. From Fig. 11-(a), the final value of the MISE of the conventional robust H ∞ fuzzy discrete-time observer-based tracking control scheme is about the ten times of the final value of the MISE of proposed method. Similar to the discussion of the tracking performance evaluation, from Fig. 12-(a),(b), it reveals the UAV state estimation performance by the proposed method is better than the state estimation performance by the conventional robust H ∞ fuzzy discrete-time observer-based tracking control scheme.
For the attack signal estimation in Fig. 13-(a), the quickly increasing phenomenon of MISE for attack signal estimation at the initial is caused by the first actuator signal estimation f a,1 (k) in Fig. 3. After that, due to the amplitude change of actuator/sensor attack signals, the MISE for attack signal estimation is smoothly increased. At 50s, the MISE almost converges to a constant and it implies the first actuator signal f a,1 (k) and sensor attack signal f s (k) are efficiently estimated. However, since three actuator attack signals {f a,2 (k), f a,3 (k), f a,4 (k)} are Gaussian white noises and they can not be precisely estimated, it casues a slight increment in MITSE of attack signal estimation for t > 50 in Fig. 13-(b).

VI. CONCLUSION
In this study, a robust H ∞ networked security observer-based reference tracking control scheme is proposed for quadrotor UAV NCS under cyber-attack. To estimate the cyber-attack on the actuator and sensor of quadrotor UAV NCS, a discrete smoothed dynamic model is introduced to describe these unavailable attack signals to simplify the state and attack signal estimation for the observer-based tracking control design of quadrotor UAV NCS. Then, by embedding the smoothed model of cyber-attack signals in quadrotor system, a conventional Lunberger observer can be applied to estimate attack signals as well as quadrotor UAV state simultaneously. By the robust H ∞ networked observer-based security reference tracking control scheme, the proposed security controller and observer could also minimize the effect of external disturbances and measurement noise to achieve the robust tracking performance and observation performance of quadrotor UAV NCS. By using the convex Lyapunov functional, the observer-based security tracking control design problem in the quadrotor UAV NCS is transformed to an equivalent nonlinear functional inequality. Further, by using T-S fuzzy interpolation method, the nonlinear quadrotor UAV NCS can be interpolated by a set of local linearized systems via fuzzy bases. Then, the robust H ∞ fuzzy observerbased security reference tracking control design of quadrotor UAV NCS can be transformed to solving a set of LMIs. A simulation example of model reference tracking control of quadrotor UAV NCS under actuator and sensor attack is given to validate the effectiveness of the proposed method in comparison with the conventional H ∞ fuzzy discrete-time robust observer-based tracking control method in [43]. Since the attack signals could be estimated effectively by the proposed discrete smoothed model, their effects on the system and sensor can be cancelled out by their estimation to improve the performance of observer-based security reference tracking controller of quadrotor UAV NCS. As a result, the security for the networked observer-based reference tracking control of the network-based quadrotor UAV NCS under actuator and sensor attack can be guaranteed. In future researches, if we want to adapt the parameters of two smoothed models by some adaptive algorithm in every time step to improve the two smoothed models, then the assumption of persistence of excitation on the input signals in (7) and (9) (i.e., δ a (kh),δ s (kh)) should be addressed to support the proposed approach.
Based on the inequalities in (43), (44), (45), the following inequalities hold By the inequalities in (46), (42) can be relaxed as follows , kh) +g(X (kh))Ū (kh) +g 1 (X (kh), kh)U (kh − τ 1 (k)) +σ (X (kh)) W (kh)))} − E{V (X (kh)) Further, if the following condition holds the inequality in (47) can be written as follows By performing summation on both sides of the above inequality from k = 0 to k = k f , we have By Assumption 2 thatῡ 1 (kh) = 0, for k < 0, the following inequality holds As a result, by (50) with the fact that V (X (k f h)) ≥ 0, we immediately obtain the following inequality As a result, if the inequalities in (20)- (22) hold, then the robust H ∞ networked security observer-based tracking performance in (18) of stochastic quadrotor NCS can be guaranteed for a prescribed level ρ.
On the other hand, ifῡ 1 (kh) = 0, the above inequality implies E{X T (kh)X (kh)} → 0 as k → ∞ due to the fact that the initial value E{V (X (0))} is bounded, i.e., the mean square state/estimation error and mean square tracking error will converge to 0 in probability. The proof is completed.

APPENDIX B PROOF OF THEOREM 2
By the rank test, the ith augmented system in (25) where Z 1 collects the elements in the unit circle of complex z domain and A F i = I 12 + A i . To prove the rank conditions above, the proof is separated into following two cases: In the first case (i), the following rank conditions can be obtained immediately: From the result above, (52) is satisfied for z ∈ Z 1 \ (eig{A F i } ∪ eig{A fa } ∪ eig{A fs }), i.e., In the second case (ii), by the assumptions in (27) and (28), the rank condition in (52) can be decoupled as By applying the rank conditions in (26)- (30), the rank condition in (55) can be rewritten as: Therefore, the observability for the ith local augmented quadrotor NCS in (25) is guaranteed.

APPENDIX C PROOF OF THEOREM 3
The proof of Theorem 3 is separated as two parts. In the first part, we derive the sufficient condition for the robust H ∞ networked security observer-based tracking control design of stochastic quadrotor NCS. Then, in the second part, the sufficient conditions for the design are transformed to solvable linear matrix inequalities (LMIs).

B. TRANSFORMATION OF BILINEAR MATRIX INEQUALITIES
From the above discussion, the H ∞ observer-based security tracking control design is transformed to equivalent matrix inequalities in (62), (65), (71). However, due to the coupling of design variables, these matrix inequalities are hard to be solved via current optimization methods. To deal with this problem, some matrix inequality formulas are used to transform these matrix inequalities into linear matrix inequalities (LMIs) and it can be easily solved via MATLAB LMI TOOL-BOX. Furthermore, some slack variables are introduced to reduce the conservative in the derived LMIs.