Adaptive resilient control for cyber-physical systems against unknown injection attacks in sensor networks

An adaptive resilient control is concerned for a class of cyber-physical systems (CPSs) in the presence of stealthy false data injection attacks in sensor networks and strict-feedback nonlinear dynamics. As the sensors are attacked by ill-disposed hackers, the exactly measured state information is unavailable for state feedback control. After theory ratiocinations, the initial issue of false data injection attacks is transformed into nonlinear uncertainty dynamics and unknown control directions at the last step. At each step in the recursive backstepping control, extended state observers (ESOs) in active disturbance rejection control (ADRC) are investigated to approximate the lumped system uncertainties. Specially, the Nussbaum functions are introduced at the last step in the adaptive control. All the closed-loop signals are proved to be semi-globally uniformly ultimately bounded by Lyapunov theory. Finally, numerical simulations verify that the proposed control can afford favorable stabilization performance and counter false data injection attack.

According to different attack targets, the cyber-attacks can be classified into denial-of-service (DoS) attacks and deception attacks, which may deteriorate the system performance, and even cause instability. The DoS attacks attempt to congest data transmission and intercept accurate information availability [16][17][18][19]. The deception attacks signify to undermine the integrity of transmitted information, such as replay attacks [20][21][22], false data injection attack [23,24]. Both the measurement faults and adversary attacks by hacker may contribute to the unexpected time-varying injection data in sensor network. Therefore, the resilient control should be studied to defend against the unknown false data injection attack and achieve secure operation of CPSs.
To mitigate the effects generated by the false data injection attacks, different schemes are addressed for multiple theoretical CPSs, such as strict-feedback systems, multi-agent systems, and networked systems. In the strict-feedback CPSs, a new type of coordinate transformation is proposed to convert the sensor attacks to the multiple time-varying state feedback gains. A novel Nussbaum function is developed in the adaptive backstepping control [25]. For nonlinear strict-feedback systems under false injection data and unknown nonlinearities, injection data compensators via neural networks are designed and an adaptive approximation recursive event-triggered control is investigated [26]. In timevarying multi-agent systems, the random stealthy false data injection attacks on sensor and actuator channels are modeled as the Bernoulli processes. By stochastic analysis and recursive linear matrix inequalities, the prescribed H ? criteria are guaranteed and the eventtriggered security consensus is achieved [27]. Considering multi-agent systems with false data injection attacks and noises, the graphical theory and Kalman filter are adopted to construct distributed state estimators. Then, the mean square consensus is studied to accomplish the security mechanism [28]. In networked systems, for linear stochastic discrete form, a Kalman filter-based detection is designed to compensate for the feedback/forward channel delays and a predictive control scheme is formulated [29]. For networked switched T-S fuzzy systems, delay system transformation and average dwell time technique are both utilized to develop eventtriggering mechanism and adaptive controller [30].
In practice CPSs, automatic voltage control or load frequency control of a power system is investigated in the presence of false data injection attacks. To accomplish the automatic voltage control, the optimal stealthy attack strategy is modeled as a partial observable Markov decision process. Then, a Q-learning algorithm in nearest sequence memory is developed to learn and analogy the attack online. Finally, the kernel density estimation is conducted to detect the bad data and offset the disruptive attack impacts [31]. In smart grids, petri nets are introduced to model the cyber-attacks, and the event trigger control is proposed to suppress the voltage fluctuation in power system [32]. In automatic generation control to keep the grid frequency fixed at a nominal value, an optimal attack is modeled and analyzed to guide the protection of sensor data links, then efficient algorithms are provided to estimate and mitigate attack impact [33]. A novel resilient load frequency control is proposed, where artificial neural network and Luenberger observer are combined to detect anomalies online and eliminate the adverse attack effects [34].
The adversarial attackers supervise the measured state information furtively and send hostile deception attacks/false injection attacks depended on the state variables, which make the exactly measured state variables unavailable. As only the corrupted system states can be provided to accomplish state feedback control design, the corrupted system dynamic is deduced tentatively. After theoretical analytic deduction, the injected false attack information in the original CPSs generates nonlinear system uncertainties and unknown control direction in the corrupted system dynamic, which is the motivation for this manuscript.
In the celebrated active disturbance rejection control (ADRC), extended state observers (ESOs) can estimate and compensate total disturbances such as system nonlinearities, uncertainties, and external disturbances, to improve the performance and robustness. Inspired by the above analysis, ESOs are introduced to implement the recursive backstepping control, for strict-feedback CPSs in the presence of false injection attacks in sensor networks. Specifically, a ESO is formulated to estimate the nonlinear system uncertainty term induced by false injection attacks at each step. At the last step, an extra ESO is explored owing to the unknown control direction, while Nussbaum function is exploited to achieve resilient control.
The innovation and novelty of the paper can be summarized as (i) the false injection attacks in the original strict-feedback CPSs are ingeniously modeled as the nonlinear system uncertainties and unknown control direction in the corrupted system dynamics for the first time. (ii) In the resilient backstepping control, ESOs are formulated to compensate the nonlinear uncertainties induced by false injection attacks at each step, while an extra ESO and Nussbaum function are combined to cope with the issue of unknown control direction at the last step.
The paper is arranged as following. Section 2 elaborates problem formulation and preliminaries. An adaptive recursive control is constructed in Sect. 3. Section 4 provides the stability analysis of closed-loop system. Simulation examples are included to illustrate the effectiveness of the proposed control schemes in Sect. 5. Finally, Sect. 6 states concluding remarks.

Problem formulation and preliminaries
Consider the strict-feedback CPSs under false data injection attacks in sensor network, described by where i ¼ 2; 3; :::n À 1; k ¼ 1; 2; :::n; x i ðtÞ ¼ ½x 1 ðtÞ; x 2 ðtÞ; :::x i ðtÞ 2 R i is system state vector, uðtÞ 2 R is control law to be specified later, x c k ðtÞ 2 R is the available corrupted system state, g k ðx k ðtÞ; tÞ : ðR k Â RÞ7 !R is the unknown false data injection attack term in the kth sensor. The C 1 nonlinear function f k ðx k ðtÞÞ : R k 7 !R is assumed to be unknown. g k is known positive system parameter. Assumption 1 [25] The false data injection attack g k ðx k ðtÞ; tÞ can be parameterized as g k ðx k ðtÞ; tÞ ¼ kðtÞx k ðtÞ, where kðtÞ is unknown time-varying gain. Assumption 2 [35,36] The unknown gain kðtÞ, its first derivate _ kðtÞ and second derivate € kðtÞ are bounded, i.e., M are unknown positive constants. Assumption 3 [25] The unknown gain kðtÞ satisfies that 1 þ kðtÞ 6 ¼ 0, subsequently, a general assumption is made that 1 þ kðtÞ [ 0.
The real system state that is unavailable owing to the false data injection attack can be denoted as where jðtÞ ¼ 1=ð1 þ kðtÞÞ. From Assumptions 2 and 3, the term jðtÞ, its first derivate _ jðtÞ and second derivate € jðtÞ are bounded, that is, M are unknown positive constants. Assumption 4 [37] The nonlinear real-valued continuous function f k ðÁÞ and its first derivate _ f k ðÁÞ are available and bounded. Definition 1 [38] If the continuous function NðsÞ satisfies that then NðsÞ is a Nussbaum function. Lemma 1 [39] V N ðÁÞ and vðÁÞ are two smooth functions on ½0; t f Þ;V N ðtÞ ! 0; 8t 2 ½0; t f Þ; NðsÞ is a Nussbaum-type function; h 0 is a nonzero constant. If the following formula (4) holds, with l 0 is a proper constant and l 1 [ 0, then V N ðtÞ,vðtÞ and R t 0 ðh 0 NðvðsÞÞ À 1Þ _ vðsÞds are bounded on ½0; t f Þ.
The adaptive resilient control problem tolerating sensor attack is to accomplish an adaptive control strategy, such that the strict-feedback CPSs (1) can be stabilized, and all the signals in the resulting closedloop system remain bounded.
Notations. 'ðÁÞ denote the set of continuous differentiable functions, Á k k denotes the Euclid norm of R n .

Adaptive recursive control
Considering the false data injection attack model (2), the strict-feedback CPSs (1) can be rewritten as The resilient control problem of CPSs in the presence of false data injection attacks (1) can be converted into the stabilization problem of CPSs (1) subjected to lumped system uncertainties 1 . . .; n À 1 and unknown control direction 1 j g n , as the false data injection attacks term jðtÞ and nonlinear function f i ðx i Þ are unknown previously.
The backstepping and ADRC are adopted to implement the recursive control with the corrupted state information in the section. At each step in the recursive design process, the celebrated ADRC scheme is introduced and the extended state observer (ESO) is designed to approximate the lumped system uncertainties. At the last step, the Nussbaum function is introduced to tackle unknown control direction.
The following coordinate transformation is defined as where z i ðtÞ is the tracking error, u f i ðtÞ is the filtered signal of smooth virtual control u i ðtÞ passing by a firstorder filter: where s is the filter gain 0\s\ minf 2 1þgi ; 1g. The filter error is defined as To facilitate the expression, the suffixes ðtÞ are omitted in the later section.
Step 1: The dynamic of z 1 is A ESO is constructed to estimate the system uncertainty term wherex c 1 andf 1 are estimations of corrupted system state x c 1 and lumped uncertainty f 1 , pertinent chosen functions w 11 ðÁÞ; w 12 ðÁÞ 2 'ðR; RÞ,e is a small positive constant.
The notationsx c 1 ¼ x c 1 Àx c 1 andf 1 ¼ f 1 Àf 1 are the estimation errors of system state and lumped uncertainty, which satisfy that From Assumptions 2-4, there is a positive constant f d 1M , such that the first derivate of lumped uncertainty The similar conditions hold for the remaining n À 1 subsystem.
To stabilize the z 1 -dynamic, the virtual control u 1 is defined as Eq. (9) yields that Step 2 i n À 1: The dynamic of z i is where f i is the lumped system uncertainty where the functions w i1 ðÁÞ; w i2 ðÁÞ 2 'ðR; RÞ,x c i andf i are estimations of x c i and f i , To stabilize the z i -dynamic, the ith virtual control u i is defined as with k i is the positive controller gain. Eq. (14) can be deduced as Define the scale estimation error vector of system state and lumped uncertainty as g i ¼ g i1 ; g i2 ½ T 2 R 2 ; i ¼ 1; 2:::n À 1 and g i1 ¼ 1 The g i -dynamic satisfies the following Assumption 5.
Assumption 5. There exist the positive define, continuous differentiable functions where r i1 ; r i2 ; r i3 ; r i4 ; # i are positive constants.
Step n: The dynamic of x c n is with f n is the lumped system uncertainty f n , 1 j f n ðjx c n Þ À _ j j x c n . Two ESOs are proposed to approximate x c n and f n as where i i;j ; i; j ¼ 1; 2 is the state of ESOs. Define the estimation error vector as p ¼ ½p 1 ; p 2 T and p 1 ¼ x c n À ði 1;1 þ 1 j i 2;1 Þ; p 2 ¼ f n À ði 1;2 þ 1 j i 2;2 Þ yielding that The p-dynamic can be expressed in vector form: with A 0 , Àc 1 1 As long as f n ¼ _ f n f d nM and matrix A 0 is Hurwitz, p can convergent to the neighborhood near zero ð0; f d nM k min ðA 0 ÞÞ asymptotically. The estimation equation of p-dynamic can be defined as Then, the nth corrupted system state and lumped system uncertainty are The dynamic of z n is where U is the reconstructed control input U,g n u þ i 2;2 .
Considering the issue of false data injection attack, the control direction 1 j is unknown. Nussbaum function is utilized to construct the control input U: where k n is the positive controller gain, NðÁÞ is Nussbaum-type function chosen as NðsÞ ¼ s 2 cosðsÞ herein.

Stability analysis
The main results of the paper are summarized in the following theorem.
Proof Firstly, the convergence of extended state observers (10) and (15) in the first n-1th subsystems is discussed. By Assumption 5, the first derivate dynamic of Lyapunov function V i ðg i Þ along the trajectory (19) is Considering Assumption 5, it follows that Recall the definition of g i1 ; g i2 yielding that uniformly in t 2 ½t 0 ; 1Þ as e ! 0. Follow from Eq. (33), the Theorem 1(ii) holds. Then, the boundedness of the filter error (8) is discussed. The Lyapunov function candidate is chosen as V e ¼ 1 2 P nÀ1 i¼1 e 2 i , then the first derivate is with a general assumption provided that _ u i j j u d iM . Finally, the boundedness of extended state observers (22) in the nth subsystem and the tracking error (6) is discussed.
The proof is completed. h

Simulation examples
Two simulation examples are studied to illustrate the effectiveness of the proposed control scheme to resist the false data injection in full state measurements.
Example 1: A nonlinear second-order system is considered: The initial conditions are set to x 1 ð0Þ ¼ 1:5; x 2 ð0Þ ¼ À1. The linear ESOs are utilized in Eqs. (10) and (15) as w 11 ðsÞ ¼ w 12 ðsÞ ¼ s. The controller, ESOs and filter gains are chosen as The comparative simulations are carried out between the proposed resilient control and approximation adaptive control in Ref. [26]. Figures 1 and 2 depict the stabilization results and state trajectories of x 1 ; x 2 , which can converge to the origin after a transient response.
As plotted in Fig. 3, the corrupted state x c 1 can estimate its estimation statex c 1 about 2 s. The ESOs states i i;j ; i; j ¼ 1; 2 in Fig. 4 are smooth and converge to the corresponding steady-state values about 2 s.
Compared to the approximation adaptive control in Ref. [26], the proposed resilient control can provide a better stabilization performance, while refrain from designing NN-based function approximations and injection data compensator.
Example 2 An IEEE 6 bus power system with three generators in Kron-reduced form is considered: where x 1 ; x 2 2 R 3 is the rotor angle vector and rotor frequency vector of generator, u denotes the equivalent power input, the generator inertial matrix and damping matrix are M g ¼ diagf0:125; 0:034; 0:016g 2 R 3Â3 ; D g ¼diagf0:125; 0:068; 0:48g 2 R 3Â3 , other physical parameters are The corrupted system state dynamics are segmented as The initial conditions are xð0Þ ¼ 0:5 0:5 0:5 0:5 ½ 0:50:5 T . In Example 2, a similar linear ESOs are adopted as in Example 1. The controller, ESOs and filter gains are selected as  x c 1 ;x c 2 are plotted in Fig. 7. The ESOs state vectors i i;j ; i; j ¼ 1; 2 are plotted in Figs. 8 and 9.
The simulation results can demonstrate that the proposed ESOs can estimate the lumped system uncertainty terms induced by false injection attacks, and the resilient control can achieve a satisfactory stabilization performance, although only the corrupted system states can be measured. The results in both numerical example 1 and practical simulation 2 can validate the superiority of the proposed adaptive recursive control scheme, which can stabilize the strict-feedback CPSs and accommodate unknown injection attack in sensor networks.     For a class of nonlinear strict-feedback cyber-physical system, the backstepping and ADRC are combined to develop a nonlinear resilient control framework to circumvent the malicious attacks through sensor networks. ESOs are constructed to compensate for the false injection data effects and unknown nonlinear dynamics via the corrupted states. The semi-globally uniformly ultimately boundedness of all the closed-loop signals can be achieved by Lyapunov analysis. Finally, the numerical examples are provided to illustrate the effectiveness of the proposed control. Further research can be extended to the resilient control problem of CPSs resisting DOS attacks and replay attacks. Data Availability Statement The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.