Learning-Based Switched Reliable Control of Cyber-Physical Systems With Intermittent Communication Faults

Xin Huang and Jiuxiang Dong,

Abstract—This study deals with reliable control problems in data-driven cyber-physical systems (CPSs) with intermittent communication faults, where the faults may be caused by bad or broken communication devices and/or cyber attackers. To solve them, a watermark-based anomaly detector is proposed, where the faults are divided to be either detectable or undetectable.Secondly, the fault’s intermittent characteristic is described by the average dwell-time (ADT)-like concept, and then the reliable control issues, under the undetectable faults to the detector, are converted into stabilization issues of switched systems. Furthermore,based on the identifier-critic-structure learning algorithm, a datadriven switched controller with a prescribed-performance-based switching law is proposed, and by the ADT approach, a tolerated fault set is given. Additionally, it is shown that the presented switching laws can improve the system performance degradation in asynchronous intervals, where the degradation is caused by the fault-maker-triggered switching rule, which is unknown for CPS operators. Finally, an illustrative example validates the proposed method.

I. Introduction

DRIVEN by networking, computing and control techniques, cyber-physical system (CPS) can greatly improve the work efficiency of existing industrial systems[1]–[5]. On account of the dependence of the CPS’s operation on data transmitted via the communication devices and networks, the system/control performance of the CPS relies heavily on the quality of the transmitted data [6]. In some practical control applications, bad or broken communication, sudden environmental disturbances, or malfunction of either software or hardware often corrupt the transmitted data. Thus, the characteristics of the communication devices may change over time, and there may be partial or complete system failure [7],which can deteriorate performance and even diverge systems in certain cases.

In addition, since communication networks are vulnerable to attacks, and such attacks compromise transmitted data, cyber attacks are also one of the main reasons behind communication faults [8]–[11]. Recently, typical cyber attacks in CPSs, such as denial-of-service (DoS) attacks [12], [13],false data injection attacks [14], [15], replay attacks [16], [17],and other attacks [18]–[20], have been reported. To defend them, some works about secure estimation and detection have been reported in [21]–[28]. In secure control aspects, [29]introduced in great detail and provided available techniques for secure control designs against cyber-physical attacks,while [30] gives a defense analysis against malicious threats on cloud control systems via a Stackelberg game. To mitigate sensor and actuator attacks, [31] presents a moving target defense control framework. Against the adversarial attacks,[32] proposes an event-triggered secure observer-based control policy. In [33], an adaptive reliable control policy was presented to stabilise the CPSs under the frequencyconstrained sensor and actuator attacks. In [34] and [35],resilient control methods against frequency- and durationconstrained DoS attacks were proposed. In [36], adaptive control architectures were presented for sensor attacks in CPSs to recover system performance. In [18] and [37], which consider simultaneous sensor and actuator attacks, adaptive control policies were proposed to ensure stability of the systems. In addition, for actuator attacks, [38] presented an adaptive integral sliding-mode control strategy such that datadriven CPSs are stable with an optimal performance.

On the other hand, it is costly and difficult to attain a system’s accurate model due to the complexity of industrial systems. Furthermore, for many industrial systems, data is pervasive in every aspect of industrial production, and should be fully utilized. Due to this, data-driven controls have gained much attention [39]. In the research of data-driven controls,the adaptive dynamic programming (ADP) technique (or policy iteration (PI) algorithm), as an effective approach of solving the algebraic Riccati equation (ARE), is widely applied in order to obtain a data-driven (or model-free)controller [40]–[44]. Both [45] and [46] studied the datadriven control problems via the zero-sum game. In order to achieve a model-free optimal controller, [47] developed a data-based policy gradient ADP algorithm. Reference [48]presented an identifier-critic-based ADP structure to online solveH∞control problem of nonlinear continuous-time systems. Recently, [49] developed a new model-free eventtriggered optimal control algorithm for continuous-time linear systems. Finally, [50] presented an event-triggered robust control policy for unknown nonlinear systems via the neural network (NN) and ADP techniques.

Based on the above discussions, it is noted that some important problems need to be further investigated for CPS security. For instance, 1) most secure control results including[18], [31]–[34], [36], and [37], are model-based; however, the exact system knowledge for the industrial systems is usually difficult to obtain in practice. 2) Optimal/robust control problems of model-free systems without adversarial environments have been well solved via ADP-based methods[40]–[50], yet the secure control issue for data-driven CPSs remains open. 3) The communication faults in [18], [36], and[37] are supposed to be on a single transmission channels;nonetheless, the multi-transmission-channel situation has been not fully studied.

Motivated by these observation, this paper, which is written from the CPS operator’s viewpoint, studies reliable control problems for data-driven CPSs under communication faults,and focuses on the intermittent faults of the multitransmission-channel situation. The main contributions of this paper are summarized as below:

1) According to the theory of the describing function, a watermark-based anomaly detector is presented, so that the faults are classified as either detectable or undetectable to the detector. It can contribute to the effective execution of the proposed learning-based switched control policy.

2) Based on the identifier-critic-structure learning algorithm, a data-driven switch controller with a prescribedperformance-based switching law is proposed, and with the aid of the average dwell-time (ADT) approach, a fault set,which the closed-loop systems can tolerate, is given.

3) The advantages of the presented method are:

i) Different from the model-based secure control results[18], [31]–[34], [36], [37], ours is data-driven;

ii) Compared with most ADP-based model-free methods[40]–[50], ours guarantees the reliability for the case of a class of intermittent communication faults;

iii) Contrary to the switched controls under the intermittent faults [51], [52], the system knowledge and switching rule are unknown for the CPS operators in this paper;

iv) Distinct from [18], [36] and [37], the presented approach focuses on the faults in multi-transmission-channel case.

The remainder of the paper is organized as follows. Section II states problem formulations. Section III presents a new data-driven control policy to tolerate a class of intermittent communication faults. Simulation results are shown in Section IV. Section V concludes this paper.Rn∥·∥Notations:is then-dimensional vector space.indicates the 2-norm of vectors or induced 2-norm of matrices.For a matrixX,XTrepresents its transpose.X≥0(X> 0)means thatXis a symmetric positive semi-definite (positive definite) matrix with an appropriate dimension.diag{x1,x2,...,xm}denotes a diagonal matrix withx1,x2,...,xmin its main diagonal.Iindicates an identity matrix with a suitable dimension. ? means the Kronecker product.wherexiis theith column ofX∈Rn×m. For a square matrixX,He(X)=XT+X.w∈L2(0,∞) means

II. Problem Formulation

This paper considers the architecture of data-driven CPSs,as shown in Fig. 1, and the models of the physical system and communication faults are depicted as follows.

A. Physical System

The physical system can be described by the following linear time-invariant system

wherex(t)∈Rn,u(t)∈Rmandd(t)∈Rpare the measurable system state, control input and external disturbance, respectively.A,BuandBdare all unknown matrices with appropriate dimensions. It is assumed that the pair (A,Bu) is controllable,d∈L2(0,∞)and there is no packet dropout and delay in the transmission channels (or network layer). For the sake of simplicity, the dependence of the functions (e.g.,x˙,x,u(t),d(t),etc) ontis omitted in some cases.

Remark 1:This paper mainly focuses on reliable controls under the case of no packet dropout and delay in the transmission channel. Such an assumption was found in [18],[31]–[33], [36] and [37], and contributes to simplifying the system models such that we are only concerned with the reliable controller design under the faults. Actually, in practice, the packet dropout and delay are common in the networked control systems. They may influence the control performance. Nevertheless, [29] and [30] have developed some effective methods to solve them, thus, in future work,we will further investigate CPS reliable controls for packetdropout and delay cases with the aid of the nice results in [29]and [30].

B. Intermittent Communication Fault

In this paper, the communication channels used to transmit sensor output signals are assumed to be vulnerable to the faults. In reality, communication faults are common in CPSs,and have been investigated in [18], [36] and [37]. Reference[36] has also reported that faults exist in some practical systems such as control systems of unmanned air vehicles.The model is usually represented by δs(x). In [18], [36] and[37], it is parameterized as δs(x)=ωs(t)x, with the gain ωs(t)∈R subject toThe system state compromised by the faults (i.e., that the controller receives) is given by

Fig. 1. Subsystem architecture of cyber-physical systems under communication faults.

where the compromised system stateand the parameter Λ =diagλ1,λ2,...,λn.

Additionally, as reported in [51] and [52], the fault usually has intermittent and random characteristics. Then, to describe the intermittent characteristic of fault (2), we introduce the concept of average dwell-time (ADT), which has been used to describe intermittent DoS attacks [29], [34], [35]. Inspired by definitions in [29], [34] and [35], let {ti}i∈Nwitht0≥0 denote the sequence of the fault on/off and off/on transitions. Without loss of generality, letTk:=[t2k+1,t2k+2) (fork∈N) mean thekth time interval where the fault is active. Then, define

where Υ (τ,t) andrepresent the subsets of [ τ,t], in which there is the fault and no fault, respectively. Meanwhile, letandindicate the total lengths of the occurring fault and no fault within the interval [ τ,t]. Thus, the gain in (2)can be denoted as

Letn(τ,t) represent the number of the faults occurring in the interval [τ,t). Thus, the following assumptions are given to describe the intermittent characteristic of the fault in (2).

Assumption 1 (Fault frequency [34], [35]):There exist constants ηf≥0 and τD>0 such that

Assumption 2 (Fault duration [34], [35]):There exist constants κ ≥0,T1≥1 and 1 ≥1/T1≥T2≥0 such that

Remark 2:In our work, Assumptions 1 and 2 are similar to those in [34] and [35], and are used to constrain the fault signal in terms of its average frequency and duration.Following [34] and [35], τDindicates the average dwell-time between consecutive fault off/on transitions; ηfmeans the chattering bound; 1/T1andT2respectively provide an upper bound and a lower bound on the average duration of the fault per unit time; analogous to ηf, κ plays the role of a regularization term. Note that the considered fault may be always active, thenT1in Assumption 2 satisfiesT1≥1 notT1>1.

The objective in this paper is to give a reliable control scheme such that the systems under the communication faults are stable.

III. Reliable Control Scheme of Data-Driven CPSs With Communication Faults

From the Laplace transform of (2), it follows thatwith frequency ω. This means that the fault only damages the value of the sensor data, and does not change its frequency. According to the characteristic, a periodic oscillation with a known fixed frequency and amplitude is inserted into sensor transmission channels, and then, based on the amplitude change of the oscillation, the system checks whether it is under faults or not; meanwhile,according to the characteristic of the fixed frequency, a filter,equipped at the position of the controller side, is used to eliminate the effect of the introduced oscillation. In reality, the frequency of the introduced oscillation can be designed such that the oscillation, and, the input and output of the system (1)are in different frequency domains, thus, in such a way, a filter is used to effectively eliminate the oscillation. However, in theory, it is simple to produce a periodic oscillation. By contrast, in practice, the oscillation is very sensitive to parameter changes. Hence, according to the describing function theory [53], we can design an oscillation which is robust to disturbance, noise and uncertainty. Then, the expected oscillation is generated and injected into communication channels by the way shown in Fig. 2. In Fig. 2,we consider a smart sensori, which performs a process measurement (i.e., sensorito measure the system state), and a“closed-loop” to generate a desired oscillation to check whether the sensor communication channel is faulty, whereG(s)is the transfer function with the sufficient low-pass behavior, andN(A) is the describing function of a nonlinear function. In “closed-loop”, ?y2andvare the input and output ofN(A), respectively.y1is the output ofG(s), andris zeroreference input of “closed-loop”. In addition, sincey1is needed to be fed-back for “closed-loop”, a acknowledgementbased protocol is needed. It is noted that, for no fault case,y2=y1; otherwise,y2=λiy1. To facilitate the analysis,suppose that the system statexand the desired oscillationy1are low- and high-frequency signals, respectively, and,ReceiverIiin “closed-loop” has a high-pass behavior. Thus,in this setting, after the signaly1+xienters ReceiverIi, the output of the receiver approximates to the signaly1. For the Fourier series of the periodic signalv(t) with periodT, the input and output of transfer functionG(s) can be written as

Fig. 2. A mechanism of producing a period oscillation with a fixed frequency and amplitude in ith sensor transmission channel.

According to the theory of the describing function [53], the describing function of the nonlinear function is denoted as

whereey(·)=?y2(·), and A denotes the oscillation amplitude received by the detector. From the Nyquist criterion, it is known that if the open loop transfer functionG0(A,jω)=N(A)G(jω) crosses the critical point (?1+j0), a periodic oscillation with the fixed frequency ω and amplitude Awill be produced, where ω and A satisfy the following equalities

whereNI(A) represents the inverse describing function,which depends on the signal amplitude A. So, by designingG(s) and N(A), a period oscillation with the expected ω and A can be obtained.

Based on (4), the detection criterion is given in the following form

Remark 3:The detection mechanism (5) can detect whether theith transmission channel is under faults or not, and the faults are divided into the detectable faults satisfyingorand relatively undetectable faults satisfyingBy applying this detection method to all (or unreliable) channels, it is checked whether the overall sensor transmission channels are under faults, and it is restricted that parameter Λ of the communication faults is invertible.

According to the detection mechanism (5), it is known that for no-alarm cases, the communication channels might also be under faults successfully bypassing the proposed detector,thus, the reliability of the system may not be guaranteed. In the following sections, for this situation, a reliable control policy is given to stabilise the data-driven CPSs under the communication faults or no faults.

A. Switching-Based Reliable Control Strategy of CPSs Under Communication Faults

When system (1) suffers the communication fault which successfully bypasses the proposed watermark-based anomaly detector, the dynamics of the compromised system states can be depicted in the following form

Lemma 1:Considering systems (1) and (6), and the invertible matrix Λ, the pairis controllable if and only if the pair (A,Bu) is controllable.

Proof:According to Theorem 9.5 in [54], it is easy to prove Lemma 1, thus, it is omitted.

From Lemma 1 and the controllable pair (A,Bu), system (6)is controllable. Thus, to ensure the stability of system (6), a control policyuand a disturbance inputdare given in the following forms

The gainsK1andL1of control and disturbance inputs are determined in the linear quadratic zero-sum game framework,where control and disturbance inputs are viewed as two players. It is known from the optimal theory [55] thatuanddsatisfy the following two-player zero-sum differential game

where

is theH∞performance index,Q=QT>0 andR=RT>0.The pairis observable and γ>γ?is theH-infinity gain. For the controllable pairand accurate parametersandthere exists a uniqueby solving the following game algebraic Riccati equation (GARE)

The opti mal feedback gain matrixin (10) andin (11)can be determined by

andu?andd?satisfy the following inequalities

On the other hand, it is noted from (6) that if Λ=I, (6)reduces to system (1); thus, in that case, the control policyuand the worst disturbance inputdare designed by the abovementioned method, and are given by

and

Based on the aforementioned description and the intermittency of the communication fault (2), the reliable control problem of system (1) under fault (2) can be converted into a problem to stabilise the following virtual switched system, which is switching between subsystems (1) and (6)

where σ(t)∈{0,1} is the switched signal of the system (9),which is determined by the fault maker. σ′(t)∈{0,1} is the switched law ofuσ′(t)anddσ′(t), to be determined, where

where

σ(t)=0indicates that subsystem (1) is active, i.e., the system is under normal operation, otherwise, σ(t)=1 means that the subsystem (6) is active. The corresponding system parameters are given in the following:

The fault maker may be a smarter attacker and their aims may be to degrade the system/control performance as much as possible. On account of their intelligence, the fault-makert riggered switching law σ(t) is generally unknown for the CPS operators. So, it is a challenging problem to give a suitable switching law σ′(t) of the controller such that the system (9) is stable. Fig. 3 describes the relationship of σ(t) and σ′(t).Without loss of generality, it is assumed thatfor anyk. By Fig. 3, there always exists an intervalorcaused by the wrong switching, in which the system may be unstable. To address this problem, a new switching law based on a prescribed performance function is proposed in this paper, and it is given in the following form:

Fig. 3. Relationship between σ (t) and σ ′(t). on/off transitions are represented as ↓, while off/on transitions are represented as ↑. For k =0,1,···,(a) off/on transitions occurring at t 2k+1 indicate that the faults occur and on/off transitions occurring at t 2k+2 mean that the faults stop; (b) off/on transitions occurring at indicate that the mode 0 of the controller switches the mode 1 (i.e., the CPS operator thinks that the faults occur), and on/off transitions occurring at mean that the mode 1 of the controller switches the mode 0(i.e., the CPS operator thinks that the faults stop).

where η0, η∞are positive constants and satisfy η0≥1, η∞≥0.satisfieswhereis a known constant, andfori∈{0,1}.

Lemma 2:Denote the time sequence consisted of the no fault time intervaland fault-activated time intervalasand the switching time sequence generated by the switched law (12)under the sequence ι asthus, the following inequalities are satisfied

and

where

and

which implies that

On the other hand, from the switched law (12), forwe have

and, it is manipulated into

Furthermore, the above equation can be written as

In addition, fort=t2k+1, one has

the above inequality can be manipulated into

thus, by some mathematical operations, we have

Based on (14) and (15), we have

Remark 4:Note that, in the time intervalsandthe time derivative of the Lyapunov function(13) is greater than zero; thus, decreasing the intervals can improve the system performance. From Lemma 2, it follows that under the switched law proposed in this paper,andcan be guaranteed, andTτ,2k+1andTτ,2k+2are reduced by decreasing η0and η∞.

In the sequel, a main result can be depicted as follows.

Theorem 1:Consider system (9) with communication faults(2) satisfying Assumptions 1 and 2 with arbitrary ηfand κ.Denoteas the solution of GARE with the mode σ′(t)∈{0,1}, then,

1) for no controller-switched case,H∞control pair (10) and(11) with the switching law (12) guaranteeswhere

2) for the controller-switched case, if the following conditions are satisfied

Proof:Consider the following Lyapunov function candidate

The proof starts from two cases: one is where there are no switching controller, and the other is the case of switching controller fort∈[0,∞].

Case 1 (No controller-switched case):In this situation, it is assumed that the mode of the controller isi, i.e., σ′(t)=i.From the switched law (9), it is known that

Case 2 (Controller-switched case):We considerin which the mode of the controller is assumed to bei, i.e., σ′(t)=i, and, in this case, there exists the intervalin which the time derivative of the Lyapunov function is greater than zero, so, it is assumed thatForthe time derivative of (17) is

further, one has

From Lemma 2, it is known thatandfork=0,1,..., where

andthus, let(in fact,forbeing any positive constant, just replace 0 with the correspondingand the following proof still holds), we have

thus, one gets

which implies that system (9) is global asymptotically stable.

Remark 5:Note thatis equivalent toand it is known from Theorem 1 that the proposed control policy can ensure that the virtual switched system (9) is stable, which implies that the system states in CPSs with the controller (10) under the intermittent communication faults (2) are stable. On the other hand,parameters τD,T1andT2help to depict the fault frequency and duration; hence, condition (16) also means a fault set that can be tolerated by controller (10).

Theorem 1 indicates that theH∞control pair (10) and (11)can ensure the stability of (9), where the accurate parametersandneed to be used in (10) and (11).However, for the data-driven CPSs, the system parameters are usually unknown. In the next section, an identifier-critic learning algorithm is used to obtain the available parameters.Then, the above-mentioned result is extended to the datadriven case.

B. Learning-Based Switched Reliable Control Policy of Data-Driven CPSs

In this section, we first assume that the unknown parameters(in (10) and (11)) for the systems under the considered faults and no faults can be estimated. Then, (6) can be written as

whereand. In this paper, the filtered variablesxfand Ψ1fin [56] are introduced to identify the system parameters, and the corresponding filters are denoted asand,wherexf(0)=0 and ψ1f(0)=0. ?>0 is the “bandwidth” of the filter (·)f=(·)/(?s+1), and should be set small to retain robustness [48].xfand ψ1fare obtained by applying a stable filter operation (·)f=(·)/(?s+1) onand ψ1. By this filter operation, (18) can be written as

Furthermore, the following auxiliary matricesP1∈andare defined as

where P1(0)=0 and Q1(0)=0.is a forgetting factor,and should be chosen to trade off the convergence speed and the robustness [48]. In light of (19)–(21), one gets

Letandbe the estimate and estimation error ofW1,i.e.,then, the weight updating law is given by

where Γ1>0 is a learning gain matrix.

Lemma 3:Consider system (6). Under the adaptive law(22), the estimation errorconverges to zero exponentially if ψ1is persistently exciting (PE).

Proof:Similar to the proof of Theorem 2 in [56], thus, the corresponding proof is omitted.

Based on Lemma 3, system (6) can be written in the following form

whereandare the estimations ofandrespectively, and the corresponding estimation errors areandis the system identification error.According to the optimal theory [55] and performance index(7), it is known that the value function is

The Hamiltonian for the system (23) can be written as

where

By using the critic network, the value function can be reconstructed as

and, its derivative is

whereW2is the weight of the critic NN. ψ2is the activation function of the criticNN,e2is the NN reconstruction error,and ?ψ2and ?e2are the derivatives of ψ2ande2with respect torespectively. It is assumed thatandwhereandare positive constants [57]. In the sequel, (25) is approximated by the following critic NN

Based on (27) and (28), Hamilton-Jacobilsaacs (HJI) can be rewritten as

where P2(0)=0 and Q2(0)=0.l2is a positive constant. Thus,we have

Lemma 4:Consider critic NN (26) with adaptive law (29). If Ξis PE, then the critic NN weight errorconverges to a compact set around zero forMoreover, foreHJI=0,W?2converges to zero exponentially.

Proof:Similar to the proofs of Theorem 2 in [56] and Theorem 4.1 in [48], thus, the corresponding proof is omitted.

Next, the stability of the closed-loop system (6) with (27)and (28) is analysed. By substituting (27) and (28) into (6),one gets

Lemma 5:Consider the system (6) equipped with (27) and(28) with adaptive laws (22) and (29). If ψ1and Ξ are PE,then, the system stateidentifer errorand critic NN weight errorare uniformly ultimately bounded (UUB).Furthermore,uin (27) anddin (28) converge to a bounded set around the idealH∞control solutionsu??andd??.

Proof.See Appendix A.

Remark 6:From the optimal theory [55], it is known that,for the linear system, the value function and its derivative with respect toare denoted asandrespectively. In addition, the value function also can be rewritten asIf the activation function of the critic NN is selected asthus, the corresponding weight iswhich implies thatand there is no NN reconstruction error, i.e.,e2=0. So, in this case, according to the proof of Lemma 5, it is shown that, ife2=0, the system states can converge to zero and the proposedH∞control pair can converge to their ideal solution in ideal case.

Remark 7:Note that if Λ=I, the system (6) can be reduced to (1), thus, by implementing the above identifier-critic based method, the accurate estimations of the parametersA,Bu,Bd,andL0can be obtained.

From Remarks 6 and 7, it is known that, if the activation function of the critic NN is selected asthe unknown parameters in (10) and (11) can converge to the ideal ones. So, one has that the followingH∞control pairs,which can ensure that the time derivatives ofandare less than 0, respectively,

Based on the above-obtained available parameters, the method proposed in Section III-A is extended to the datadriven case. Thus, for the system (9) with the unknown parameters, a switched controller is given by

where σ′(t)∈{0,1}. The available parameters used in the controller are

A learning-based switching law is given in the following form

Theorem 2:Consider system (9), whose system parameters are unknown. Suppose that the system parameter estimations andPσ′, σ′∈{0,1} can be obtained from Lemmas 4 and 5:

1) for no controller-switched case,H∞control pair (31) and(32) with the switching law (33) guarantees,where

2) for the controller-switched case, if the following conditions are satisfied

wherethus,H∞control pair (31) and (32) with the switching law (33) guarantees that the system (9) is global asymptotically stable.

Proof:The proof is analogous to that of Theorem 1 and omitted.

Remark 8:Notice that the existing results on the switched controls against the intermittent attacks mainly include 1) [29]and [34] focus on the intermittent DoS (I-DoS) attack case,and 2) [51] and [52] are concerned with the intermittent actuator failure (I-AF) case. Then, the differences between the existing approached and ours can be described by Table I.Clearly, due to unknown system knowledge and the switching law σ(t), under the considered faults, the switched controls in[29], [34], [51] and [52] lose efficacy.

From the above analysis, the proposed reliable control policy can be described in Fig. 4. In Fig. 4, ADP-based learning module indicates the identification-critic based controllers (27), and switched controller represents controller(31) with the switching law (33). The corresponding implementing process can be depicted as follows:

Step 1:Activate the ADP-based learning module. Let σ =0.During the learning process, decision module exportsuσof the ADP-based learning module. It stops until the available parametersare gained, and the obtained available parameters are sent to switched controller.

Step 2:After switched controller receives the available parameters, the decision module exports the control signaluσof switched controller. Until the system is abnormal, ADPbased learning module is reactivated. Let σ=1, and repeat Step 1.

TABLE I Comparisons of the Existing Switched Controls Against the Intermittent Attacks

Fig. 4. Diagrammatic sketch of the learning-based switched reliable control policy.

IV. Illustrative Example

In this section, the DC motor speed control system is given to validate the presented method, where the control system is shown in Fig. 5. It can be seen that the rotating speed and armature current signals of the motor are transmitted via the network channels 1 and 2, respectively, and the transmission channels are assumed to be vulnerable to cyber attacks.Furthermore, the system parameters are borrowed from [58],where they are obtained from actual measurements and hardware equipments. These parameters are summarized in Table II. Additionally, through the working principle of the DC motor, the dynamics of the motor are represented by

whereIis the armature current; ? is the rotating speed of the motor; andUrepresents the terminal voltage.

The desired rotating speed and the steady-state current of

Fig. 5. Block diagram of a DC motor speed control system.

TABLE II DC Motor Speed Control System Parameters

the DC motor are defined as ?dandrespectively. Thus, the tracking errors are ??=???dandThe dynamics of the errors are given by

In order to verify the proposed algorithm, it is assumed that the system parameters are unknown, the control system is equipped with the detector (5) withandand the communication channels are subjected to a fault/attack in (2)with Λ=diag{?1.5,?0.5} , ηf=0, κ=0, τD=75,T1=1 andT2=0.6.

Next, the ADP-based learning algorithm in this paper is first implemented to obtain the system parameters for (1) and (6) and the corresponding critic weights. Then we consider theH∞performance index (7) with theH∞gain γ=2,R=1 andQ=diag{1,1}, and the activation functionof the critic NN. In the simulation experiment, to accurately estimate the unknown parameters, a probing noise is introduced before 2 s. The initial critic weights are randomly chosen and all the initial values for the parameters in the adaptive laws are set to zero. The ADP-based learning process is shown in Fig.6. Due to similarity to (1), the learning process for the system(6) is omitted. The obtained optimal estimations of the system parameters and the critic weights are given by

Fig. 6. The ADP-based learning process for the system (1): (a) System parameters; (b) Critic NN weights; (c) System states; (d) Control input signal.

and

Fig. 7. Simulation results using the proposed method. (a) The system and controller switching signals σ and σ′; (b) Prescribed performance and system performance The system performance (d) The norm ∥x∥of the system states.

In the sequel, we test the effectiveness of the reliable switched controller (31) under the intermittent communication faults in the simulation experiment. In the simulation, the disturbance occurs from the initial time to 500 s, and a potential switching signal generated from the fault marker satisfying Assumptions 1 and 2 is considered and plotted by the blue solid line in Fig. 7(a). And then, the parameters of the prescribed performance function used in switching law (33)are chosen as η0=1.1, η∞=0.05,a0=2.6 anda1=1.7. Then,under the communication faults, the switching signal of the controller generated by switching law (33) is denoted by the red dash line in Fig. 7(a). It can be observed thatTτ,M0≤0.353 andTτ,M1≤0.353. Furthermore, byW2,0andW2,1, we have α0=2.6451, α1=1.7171, β0=94.4412 and β1=109.6481. We chose μ=5, thus, by computing, it is obtained that λ?>Tμ/τD=1.0240 , andλ?<α0(1?1/T1)+α1T2=1.0302, which implies that condition (34) is satisfied.Under the proposed switching law (33), the system performances are exhibited by Fig. 7(b)?(d). It can been seen that whenVσ′ violates the prescribed performance, the switching of the controllers is triggered, and prior to switching controllers, the performanceVσ′ is inside the prescribed performance. Also, observe that there are some chatters in the trajectory of the norm of the compromised system states. They are caused by passively switching the controllers. These chatters can be improved by reducing the parameters of the prescribed performance. In order to clearly illustrate this, we consider the trajectories of the norm of the compromised system states under four cases: 1) η∞=50 ; 2) η∞=5; 3)η∞=0.5 and 4) η∞=0.05. The simulation results are plotted in Fig. 8. It verifies the chatters can be reduced through decreasing η∞.

Fig. 8. The norm of the system states under (a) η ∞=50; (b) η ∞=5; (c)η∞=0.5; (a) η ∞=0.05.

In the following simulation experiment, we consider the same intermittent communication faults as those mentioned above. Then, under S-ADPC and ET-ADPC, the compromised system performances are shown in Fig. 9. It can be seen that the norms ∥x∥ of the compromised system states under the two

Fig. 9. The system performances: (a) the norm of the system states under S-ADPC; (b) under S-ADPC; (c) the norm of the system states under EV-ADPC; and (d) under EV-ADPC.

V. Conclusion

This paper from the CPS operator’s viewpoint studied reliable control problems of data-driven cyber-physical systems with communication faults on multiple channels.Based on our previous result [11], a data-driven switched controller with a prescribed-performance-based switching law was proposed, and through the ADT approach, a fault set, which the closed-loop systems can tolerate, was given. Then, driven by the cooperation of the proposed watermark-based anomaly detector and learning-based switched control policy, the reliability of the systems under the faults was guaranteed. An illustrative example verified the effectiveness of the presented method.

Appendix A The Proof of Lemma 5

Proof:Consider the following Lyapunov function candidate

where

From Lemmas 3 and 4, it follows that

wherec1,c2>0 are constants. The time derivative of L3is

From the value function (24), it is clear thatand then, substituting(30) into (38) yields

The time derivative ofL4is

Substituting (36), (37), (39) and (40) into (35) yields

where

Hence, if the designed parameters Γ,Kand η satisfies the following conditions

thus, f rom (41), it is deduced thatandv2are UUB.

Next, the errors between (27), (28) and the ideal ones are analysed. The corresponding errors are

According to the above analysis, one has

whereeuandedare positive constants, which are determined by the NN approximation errors.