A reliable control method for cyber-physical systems under malicious attacks

Abstract

The present invention provides a reliable control method for an information-physical system under malicious attacks, and belongs to the field of information security technology. The method comprises: establishing a control-oriented dynamic model for an information-physical system with external disturbances under malicious attacks; based on the established dynamic model, determining the tracking error according to the actual output and ideal output of the system, and designing the system switching surface required in the variable structure control method; constructing a Gaussian radial basis function neural network, and based on the determined tracking error, performing online real-time prediction and reconstruction of malicious attacks and external disturbances; based on the Lyapunov stability theorem, determining the adaptive law of the Gaussian radial basis function neural network, and designing an adaptive neural network finite time controller according to the determined adaptive law and the switching surface, so as to realize reliable control of the information-physical system under malicious attacks. The present invention can ensure that the system output accurately tracks the ideal output within a limited time.

Description

Technical Field

[0001] The present invention relates to the field of information security technology, and in particular to a reliable control method for an information-physical system under malicious attacks. Background Art

[0002] At present, with the rapid development of perception, computing, communication and control technologies, cyber-physical systems that integrate elements such as people, machines, objects, environment and information in physical space and cyber space have emerged and become the research frontier in the field of automation today. They are also widely used in practical engineering fields, such as smart grids, smart connected vehicles, industrial Internet, smart medical care, etc. The typical feature of cyber-physical systems is that the control layer and the perception and execution layer need to communicate through the data transmission layer. Therefore, in order to attack cyber-physical systems, malicious attackers can launch multiple and complex cyber-physical cross-domain attacks, aiming to detect, invade and hijack cyber systems, thereby causing serious non-contact damage to physical systems.

[0003] In recent years, scholars at home and abroad have proposed a series of ideas for reliable control of cyber-physical systems under malicious attacks from different perspectives. Although progress and results have been achieved, there are still shortcomings: most methods only achieve asymptotic stability of cyber-physical systems under malicious attacks; few studies can achieve online real-time prediction and reconstruction of malicious attacks and external disturbances. At the same time, due to the simultaneous existence of malicious attacks and external disturbances, it is difficult to achieve finite-time stability of cyber-physical systems under malicious attacks with external disturbances. Therefore, it is urgent to design a finite-time control method for cyber-physical systems under malicious attacks with external disturbances. Summary of the invention

[0004] The embodiment of the present invention provides a reliable control method for a cyber-physical system under malicious attacks, which can ensure that the system output accurately tracks the ideal output within a limited time. The technical solution is as follows:

[0005] The embodiment of the present invention provides a reliable control method for a cyber-physical system under malicious attacks.

[0006] Establish a control-oriented dynamic model for cyber-physical systems with external disturbances under malicious attacks;

[0007] Based on the established dynamic model, the tracking error is determined according to the actual output and ideal output of the system, and the system switching surface required in the variable structure control method is designed;

[0008] Construct a Gaussian radial basis function neural network to perform online real-time estimation and reconstruction of malicious attacks and external disturbances based on the determined tracking error;

[0009] Based on Lyapunov stability theorem, the adaptive law of Gaussian radial basis function neural network is determined. According to the determined adaptive law and switching surface, an adaptive neural network finite time controller is designed to achieve reliable control of cyber-physical systems under malicious attacks.

[0010] Furthermore, the control-oriented dynamic model of the cyber-physical system with external disturbances under malicious attacks is established as follows:

[0011]

[0012]

[0013] Where t represents time, ξ(t) represents the state variable of the cyber-physical system, …、ξ (n-1) (t), ξ (n) (t) represents the first-order derivative, second-order derivative, third-order derivative, …, n-1-order, and n-order derivative of ξ(t) with respect to time t, n represents the system is an n-order nonlinear system, p(.) and q(.) are known real continuous nonlinear functions of the system state variables, y(t) represents the actual output of the system, C represents the coefficient matrix, and u a (t) represents a malicious attack, u(t) represents the system input, and the malicious attacker intends to add u(t) to the system input. a (t) to achieve the purpose of malicious attack, d(t) represents the external disturbance existing in the control object, |d(t)|<β, β is a positive constant;

[0014] The kinetic model is transformed into the following form:

[0015]

[0016] in, represents the n-dimensional state vector of the cyber-physical system after transformation. ξ1(t) and ξ(t) have the same meaning, both representing the first dimension of the n-dimensional state variable of the cyber-physical system. ξ2(t), ξ3(t), ξ4(t), …, ξ n (t) respectively represent the first-order derivative, second-order derivative, third-order derivative, ..., n-1-order derivative of ξ(t) with respect to time t, A(t)=Ψ(Ξ,t)=q(Ξ)u a (t) Characterize the energy of malicious attacks suffered by cyber-physical systems;

[0017] According to the bounded attack energy theory, we get:

[0018]

[0019] Among them, φ i>0,i=1,2,…,n represents the upper bound of the energy of malicious attacks suffered by cyber-physical systems, …、Ψ (n) (Ξ, t) represents the first-order derivative, second-order derivative, third-order derivative, ..., n-order derivative of Ψ(Ξ, t) with respect to time t.

[0020] Furthermore, the method of determining the tracking error based on the established dynamic model according to the actual output and the ideal output of the system and designing the system switching surface required in the variable structure control method includes:

[0021] Based on the established dynamic model, the tracking error of the system is determined according to the actual output and ideal output of the system:

[0022]

[0023] in, represents the tracking error of the system, y d (t) represents the ideal output of the system, and y(t) represents the actual output of the system;

[0024] The tracking error dynamics of the system is described as:

[0025]

[0026] in, represents the ideal output y d (t) is the nth-order derivative of time t, Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative, n derivative at time t, Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative with respect to time t;

[0027] The control objective of the system is described as:

[0028]

[0029] Among them, Π(t) is a short form, ||.|| represents the Euclidean norm of the vector, T represents the adjustment time of the system, and ε represents the preset tracking accuracy;

[0030] Define the switching surface as:

[0031]

[0032] Where W(t) represents the switching surface, and υ represents the conversion rate of the switching surface;

[0033] According to Newton's expansion formula, the switching surface is specifically written as follows:

[0034]

[0035] in, Respectively The first derivative, second derivative, ..., n-3, n-2, and n-1 derivatives with respect to time t;

[0036] According to the tracking error dynamics of the system, the switching surface is transformed into:

[0037]

[0038] in, Represents the tracking error of the system;

[0039] After taking the first-order derivative of the switching surface with respect to time t, we obtain:

[0040]

[0041] Furthermore, the construction of a Gaussian radial basis function neural network to perform online real-time prediction and reconstruction of malicious attacks and external disturbances based on the determined tracking error includes:

[0042] Construct a Gaussian radial basis function neural network;

[0043] Tracking Error As input, the weight Q between the hidden layer and the output layer in the constructed Gaussian radial basis function neural network is trained so that the trained Gaussian radial basis function neural network can estimate the sum of malicious attacks and external disturbances.

[0044]

[0045] Among them, S(.) represents the Gaussian function, They represent the estimated weights between the hidden layer and the output layer, the center of the hidden layer neurons, and the standard deviation of the hidden layer neurons; X(t) represents the input vector of the Gaussian radial basis function neural network,

[0046] Define the estimated error of the weight Q between the hidden layer and the output layer of the Gaussian radial basis function neural network for:

[0047]

[0048] in, is the weight estimated by the Gaussian radial basis function neural network constructed in this embodiment, Q *is the optimal weight between the hidden layer and the output layer.

[0049] Furthermore, the adaptive law of the Gaussian radial basis function neural network is expressed as:

[0050] Based on Lyapunov's stability theorem, the adaptive law of Gaussian radial basis function neural network is determined:

[0051]

[0052] in, express The first derivative with respect to time t, μ = μ * , τ=τ * , μ,τ represent the center μ of the optimal hidden layer neurons respectively * and the standard deviation τ of the optimal hidden layer neurons * .

[0053] Furthermore, the designed adaptive neural network finite time controller is expressed as:

[0054]

[0055] where sgn(·) represents a sign function, k is a positive constant, k ≥ α + β, β is the upper bound of the external disturbance d(t) in the system, α is a positive constant, and q(Ξ) ≠ 0.

[0056] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0057] 1) For cyber-physical systems with external disturbances under malicious attacks, a reliable control method for cyber-physical systems is proposed to ensure that the system output accurately tracks the ideal output within a limited time;

[0058] 2) The Gaussian radial basis function neural network constructed in this embodiment is an adaptive neural network predictor that can realize online prediction and reconstruction of external disturbances and malicious attacks in the system;

[0059] 3) The reliable control method of the cyber-physical system provided in this embodiment is designed for n-order nonlinear cyber-physical systems, and can therefore be applied to the safety control of various nonlinear cyber-physical systems;

[0060] 4) This embodiment integrates Lyapunov stability theory, which ensures the stability of the information-physical system while effectively improving the control performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0062] Figure 1 A schematic diagram of the structure of the information-physical system provided by an embodiment of the present invention;

[0063] Figure 2 A schematic diagram of a flow chart of a reliable control method for a cyber-physical system under malicious attacks provided by an embodiment of the present invention;

[0064] Figure 3 A schematic diagram of an adaptive neural network finite time control strategy provided by an embodiment of the present invention;

[0065] Figure 4 A schematic diagram of a malicious attack provided by an embodiment of the present invention;

[0066] Figure 5 A schematic diagram of an external disturbance provided by an embodiment of the present invention;

[0067] Figure 6 A schematic diagram of the prediction effect of the adaptive neural network predictor provided by an embodiment of the present invention;

[0068] Figure 7 A schematic diagram of a vehicle speed simulation result provided by an embodiment of the present invention;

[0069] Figure 8 A schematic diagram of a switching surface simulation result provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0070] In order to make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0071] In this embodiment, the structure diagram of the cyber-physical system with external disturbance under malicious attack is as follows: Figure 1 As shown in the figure, the communication mode in the system is wireless communication. The controller sends control instructions to the actuator through the wireless communication network. The actuator acts on the controlled object. There is always external disturbance in the controlled object. The sensor measures the signal in the controlled object and then sends it to the controller through the wireless communication network. The malicious attack intends to add the malicious attack signal u to the control instruction u(t) (i.e., system input) sent by the controller to the actuator through the wireless communication network. a(t), thereby achieving the purpose of malicious attack. The embodiment of the present invention provides a reliable control method for the information-physical system under malicious attack, such as Figure 2 As shown, the following steps may be specifically included:

[0072] S101, establish a control-oriented dynamic model of a cyber-physical system with external disturbances under malicious attacks, so as to mathematically characterize the malicious attacks and external disturbances suffered by the cyber-physical system;

[0073] In this embodiment, the control-oriented dynamic model of the cyber-physical system with external disturbance under malicious attack is established as follows:

[0074]

[0075]

[0076] Where t represents time, ξ(t) represents the state variables of the cyber-physical system, such as vehicle speed, distance between vehicles, etc. …、ξ (n-1) (t), ξ (n) (t) represents the first-order derivative, second-order derivative, third-order derivative, …, n-1-order, and n-order derivative of ξ(t) with respect to time t, respectively. n represents that the system is an n-order nonlinear system. p(.) and q(.) are known real continuous nonlinear functions of the system state variables. y(t) represents the actual output of the system, such as vehicle speed, vehicle acceleration, etc. C represents the coefficient matrix, C = [1, 0, 0, …, 0] T ,u a (t) represents a malicious attack, u(t) represents the system input, and the malicious attacker intends to add u(t) to the system input. a (t) to achieve the purpose of malicious attack, d(t) represents the external disturbance existing in the control object, |d(t)|<β, β is a positive constant;

[0077] The kinetic model is transformed into the following form:

[0078]

[0079] in, represents the n-dimensional state vector of the cyber-physical system after transformation, ξ1(t) and ξ(t) have the same meaning, both representing the first dimension of the n-dimensional state variable of the cyber-physical system, ξ2(t), ξ3(t), ξ4(t), ..., ξ n (t) respectively represent the first-order derivative, second-order derivative, third-order derivative, ..., n-1-order derivative of ξ(t) with respect to time t, A(t)=Ψ(Ξ,t)=q(Ξ)u a(t) Characterize the energy of malicious attacks suffered by cyber-physical systems;

[0080] According to the bounded attack energy theory, we get:

[0081]

[0082] Among them, φ i >0,i=1,2,…,n represents the upper bound of the energy of malicious attacks suffered by cyber-physical systems, …、Ψ (n) (Ξ, t) represents the first-order derivative, second-order derivative, third-order derivative, ..., n-order derivative of Ψ(Ξ, t) with respect to time t.

[0083] S102, based on the established dynamic model, determining the tracking error according to the actual output and the ideal output of the system, and designing the system switching surface required in the variable structure control method;

[0084] In this embodiment, the tracking error of the system is:

[0085]

[0086] in, represents the tracking error of the system, y d (t) represents the ideal output of the system, and y(t) represents the actual output of the system;

[0087] The tracking error dynamics of the system is described as:

[0088]

[0089] in, represents the ideal output y d (t) is the nth-order derivative of time t, …、 Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative, n derivative at time t, Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative with respect to time t;

[0090] The control objective of this embodiment is to ensure that for any system initial state, the system output can accurately track the ideal output within a limited time. The control objective of the system can be written as:

[0091]

[0092] Among them, Π(t) is a short form, ||.|| represents the Euclidean norm of the vector, T>0 represents the adjustment time of the system, and ε>0 represents the preset tracking accuracy;

[0093] Define the switching surface as:

[0094]

[0095] Where W(t) represents the switching surface, υ>0 represents the conversion rate of the switching surface;

[0096] According to Newton's expansion formula, the switching surface is specifically written as follows:

[0097]

[0098] in, Respectively The first derivative, second derivative, ..., n-3, n-2, and n-1 derivatives with respect to time t;

[0099] According to the tracking error dynamics of the system, the switching surface is transformed into:

[0100]

[0101] in, Represents the tracking error of the system;

[0102] After taking the first-order derivative of the switching surface with respect to time t, we obtain:

[0103]

[0104] S103, constructing a Gaussian radial basis function neural network, and based on the determined tracking error, performing online real-time prediction and reconstruction of malicious attacks and external disturbances, thereby achieving the purpose of online prediction and reconstruction of external disturbances and malicious attacks on the system;

[0105] In this embodiment, according to the global approximation theorem, for the sum of external disturbance and malicious attack: A d (t) = A(t) + d(t), there is always a Gaussian radial basis function neural network such that for any approximation accuracy, the following equation is always satisfied:

[0106]

[0107] Among them, S(.) represents the Gaussian function, represents the result estimated by the Gaussian radial basis function neural network constructed in this embodiment, specifically: the sum of external disturbance and malicious attack, χ>0 represents the approximation accuracy, Q * , μ * , τ *They represent the optimal weights between the hidden layer and the output layer, the center of the hidden layer neurons, and the standard deviation of the hidden layer neurons, respectively. X(t) represents the input vector of the Gaussian radial basis function neural network.

[0108] The above formula shows that there is always an optimal Q * ,μ * ,τ * So that the above formula satisfies, for the Gaussian radial basis function neural network constructed in this embodiment, A d The estimated results of (t) Can be expressed as follows:

[0109]

[0110] in, They represent the estimated weights between the hidden layer and the output layer, the center of the hidden layer neurons, and the standard deviation of the hidden layer neurons respectively;

[0111] It is worth noting that the Gaussian radial basis function neural network constructed in this embodiment is only trained for the weight Q between the hidden layer and the output layer, and the designed Gaussian radial basis function neural network is used as an adaptive neural network predictor, the purpose of which is to achieve online prediction and reconstruction of external disturbances and malicious attacks in the system.

[0112] In this embodiment, the estimated error of the weight Q between the hidden layer and the output layer of the Gaussian radial basis function neural network is defined as follows: for:

[0113]

[0114] in, is the weight estimated by the Gaussian radial basis function neural network constructed in this embodiment, Q * is the optimal weight between the hidden layer and the output layer.

[0115] S104, based on the Lyapunov stability theorem, determines the adaptive law of the Gaussian radial basis function neural network, and designs an adaptive neural network finite time controller based on the determined adaptive law and switching surface to achieve reliable control of the cyber-physical system under malicious attacks. In this way, the Lyapunov stability theory is integrated to ensure the stability of the cyber-physical system while effectively improving the control performance.

[0116] In this embodiment, based on Lyapunov's stability theorem, the adaptive rule of the Gaussian radial basis function neural network is determined:

[0117]

[0118] in, express For the first-order derivative at time t, since this embodiment only trains the weights from the hidden layer to the output layer, μ and τ represent the center of the optimal hidden layer neuron and the standard deviation of the optimal hidden layer neuron, respectively, that is, μ = μ * , τ=τ * ;

[0119] According to the above global approximation theorem, there is always an optimal Gaussian radial basis function neural network that can accurately approximate A d (t), that is, the following formula holds:

[0120] A d (t) = Q *T S(X(t),μ,τ)

[0121] Therefore, the following adaptive neural network finite time controller is designed:

[0122]

[0123] where sgn(·) represents a sign function, k is a positive constant, k ≥ α + β, β is the upper bound of the external disturbance d(t) in the system, α is a positive constant, and q(Ξ) ≠ 0.

[0124] The system stability proof process is as follows:

[0125] Choose a Lyapunov function of the form:

[0126]

[0127] Taking the derivative of the Lyapunov function, we get:

[0128]

[0129] Will Substituting into the above formula:

[0130]

[0131] Substituting the adaptive neural network finite time controller constructed in this embodiment into the above formula, we can obtain:

[0132]

[0133] Further simplifying the above formula yields:

[0134]

[0135] According to the global approximation theorem, we can get:

[0136]

[0137] Then Simplified to:

[0138]

[0139] Substituting the adaptive rule determined in this embodiment into the above formula, we can obtain:

[0140]

[0141] Further simplification yields:

[0142]

[0143] According to Lyapunov's theorem, the system output can accurately track the ideal output within a finite time.

[0144] The reliable control method of a cyber-physical system under malicious attacks described in the embodiment of the present invention has at least the following advantages:

[0145] 1) For cyber-physical systems with external disturbances under malicious attacks, a reliable control method for cyber-physical systems is proposed to ensure that the system output accurately tracks the ideal output within a limited time;

[0146] 2) The Gaussian radial basis function neural network constructed in this embodiment is an adaptive neural network predictor that can realize online prediction and reconstruction of external disturbances and malicious attacks in the system;

[0147] 3) The reliable control method of the cyber-physical system provided in this embodiment is designed for n-order nonlinear cyber-physical systems, and can therefore be applied to the safety control of various nonlinear cyber-physical systems;

[0148] 4) This embodiment integrates Lyapunov stability theory, which ensures the stability of the information-physical system while effectively improving the control performance.

[0149] In order to better understand the present invention, the reliable control method of the cyber-physical system under malicious attacks provided by the embodiment of the present invention is described by taking ξ(t)=v(t), where v(t) represents the vehicle speed of the heavy vehicle, such as Figure 3 As shown, in this embodiment,

[0150]

[0151]

[0152] y(t)=v(t),u(t)=T e (t)

[0153] in, m a =40000kg,m a represents the mass of the car, m t Represents the total mass of the heavy vehicle system, J w =32.9kgm 2 , J w represents the rotational inertia of the car tire, r w =0.5m, r w represents the wheel radius of the vehicle, Related to the wheel inertia, λ G =1,λ G represents the gearbox, λ F =2.71,λ F Indicates the transmission ratio of the main drive gear, η G =0.97,η G represents the gearbox, η F =0.97,η F represents the main transmission efficiency coefficient, Related to the flywheel inertia, represents the transfer coefficient of the force generated by the engine, z a Characterizes the air resistance coefficient, c d represents the drag coefficient, A a =10m 2 , A a represents the front area of ​​the vehicle, ρ a =1.29kg / m 3 , ρ a represents the air density, z r =c r m a g, c r =0.007, c r represents the sliding friction coefficient, g = 9.82, g represents the gravity constant, z r represents the drag coefficient due to vehicle gravity, z g =m a g,z g represents the vehicle gravity, J e =3.5kgm 2 , α(s) represents the slope road angle (a function of the vehicle's position s on the road), T e (t) represents the engine torque, which is the power input of the vehicle, v(t) represents the speed of the vehicle, y(t) = v(t) represents the speed of the vehicle as the system output, A(t) and d(t) represent the malicious attack and external disturbance respectively;

[0154] The first step is to establish a control-oriented dynamic model of the cyber-physical system with external disturbances under malicious attacks:

[0155]

[0156] y(t)=v(t)

[0157] Among them, A(t) and d(t) are respectively Figure 4 and Figure 5 shown.

[0158] The second step is to determine the tracking error:

[0159]

[0160] Among them, y d (t) represents the ideal speed, y d (t) = 22.2 m / s;

[0161] Then, determine the switching surface of the system as:

[0162]

[0163] The third step is to construct a Gaussian radial basis function neural network to estimate the sum of malicious attacks and external disturbances, where the input of the neural network is The output is The number of neurons in the hidden layer of the designed neural network is 5. The Gaussian radial basis function neural network constructed in this embodiment is only trained for the weight Q between the hidden layer and the output layer, and the designed Gaussian radial basis neural network is used as an adaptive neural network estimator to realize online prediction and reconstruction of external disturbances and malicious attacks in the system.

[0164] The fourth step is to determine the following adaptive rules:

[0165]

[0166] Then, the following adaptive neural network finite-time controller is designed:

[0167]

[0168] Where k ≥ α + β, β is the upper bound of the external disturbance d(t) in the system, α = 0.5, and q(Ξ) ≠ 0.

[0169] In this embodiment, the constructed adaptive neural network predictor performs online prediction of external disturbances and malicious attacks in the system. The prediction results are as follows: Figure 6As shown, it is obvious that the adaptive neural network predictor constructed in this embodiment can accurately estimate and reconstruct malicious attacks and external disturbances. In addition, the vehicle speed v(t) and the switching surface W(t) operation curve results obtained by simulating the heavy-duty vehicle system implementation case of the present invention are as follows Figure 7 , Figure 8 As shown by Figure 7 , Figure 8 It can be seen that the car speed v(t) tracks the ideal speed v in a limited time. cc =22.2 m / s, and the switching surface W(t) is converted to 0 in a finite time.

[0170] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A reliable control method for a cyber-physical system under malicious attacks, characterized in that: include: Establish a control-oriented dynamic model for cyber-physical systems with external disturbances under malicious attacks; Based on the established dynamic model, the tracking error is determined according to the actual output and ideal output of the system, and the system switching surface required in the variable structure control method is designed; Construct a Gaussian radial basis function neural network to perform online real-time estimation and reconstruction of malicious attacks and external disturbances based on the determined tracking error; Based on Lyapunov's stability theorem, the adaptive law of Gaussian radial basis function neural network is determined. According to the determined adaptive law and switching surface, an adaptive neural network finite time controller is designed to achieve reliable control of cyber-physical systems under malicious attacks. Among them, the control-oriented dynamic model of the cyber-physical system with external disturbance under malicious attack is established as follows: Where t represents time, ξ(t) represents the state variable of the cyber-physical system, represents the first-order derivative, second-order derivative, third-order derivative, …, n-1-order, and n-order derivative of ξ(t) with respect to time t, n represents the system as an n-order nonlinear system, p(.) and q(.) are known real continuous nonlinear functions of the system state variables, y(t) represents the actual output of the system, C represents the coefficient matrix, and u a (t) represents a malicious attack, u(t) represents the system input, and the malicious attacker intends to add u(t) to the system input. a (t) to achieve the purpose of malicious attack, d(t) represents the external disturbance existing in the control object, |d(t)|<β, β is a positive constant; The kinetic model is transformed into the following form: in, represents the n-dimensional state vector of the cyber-physical system after transformation, ξ1(t) and ξ(t) have the same meaning, both representing the first dimension of the n-dimensional state variable of the cyber-physical system, ξ2(t), ξ3(t), ξ4(t), ..., ξ n (t) respectively represent the first-order derivative, second-order derivative, third-order derivative, ..., n-1-order derivative of ξ(t) with respect to time t, A(t)=Ψ(Ξ,t)=q(Ξ)u a (t) Characterize the energy of malicious attacks suffered by cyber-physical systems; According to the bounded attack energy theory, we get: Among them, φ i >0,i=1,2,…,n represents the upper bound of the energy of malicious attacks suffered by cyber-physical systems, represents the first-order derivative, second-order derivative, third-order derivative, ..., n-order derivative of Ψ(Ξ,t) with respect to time t in turn; Wherein, based on the established dynamic model, determining the tracking error according to the actual output and the ideal output of the system, and designing the system switching surface required in the variable structure control method include: Based on the established dynamic model, the tracking error of the system is determined according to the actual output and ideal output of the system: in, represents the tracking error of the system, y d (t) represents the ideal output of the system, and y(t) represents the actual output of the system; The tracking error dynamics of the system is described as: in, represents the ideal output y d (t) is the nth-order derivative of time t, Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative, n derivative at time t, Respectively The first derivative, second derivative, third derivative, ..., n-1 derivative with respect to time t; The control objective of the system is described as: Among them, Π(t) is a short form, .|| represents the Euclidean norm of the vector, T represents the adjustment time of the system, and ε represents the preset tracking accuracy; Define the switching surface as: Where W(t) represents the switching surface, and υ represents the conversion rate of the switching surface; According to Newton's expansion formula, the switching surface is specifically written as follows: in, Respectively The first derivative, second derivative, ..., n-3, n-2, and n-1 derivatives with respect to time t; According to the tracking error dynamics of the system, the switching surface is transformed into: in, Represents the tracking error of the system; After taking the first-order derivative of the switching surface with respect to time t, we obtain:

2. The reliable control method of a cyber-physical system under malicious attacks according to claim 1 is characterized in that: The Gaussian radial basis function neural network is constructed to perform online real-time estimation and reconstruction of malicious attacks and external disturbances based on the determined tracking error, including: Construct a Gaussian radial basis function neural network; Tracking Error As input, the weight Q between the hidden layer and the output layer in the constructed Gaussian radial basis function neural network is trained so that the trained Gaussian radial basis function neural network can estimate the sum of malicious attacks and external disturbances. Among them, S(.) represents the Gaussian function, They represent the estimated weights between the hidden layer and the output layer, the center of the hidden layer neurons, and the standard deviation of the hidden layer neurons; X(t) represents the input vector of the Gaussian radial basis function neural network, Define the estimated error of the weight Q between the hidden layer and the output layer of the Gaussian radial basis function neural network for: in, is the weight estimated by the constructed Gaussian radial basis function neural network, Q * is the optimal weight between the hidden layer and the output layer.

3. The reliable control method of a cyber-physical system under malicious attacks according to claim 2 is characterized in that: The adaptive rule of the Gaussian radial basis function neural network is expressed as: Based on Lyapunov's stability theorem, the adaptive law of Gaussian radial basis function neural network is determined: in, express The first derivative with respect to time t, μ = μ * , τ=τ * , μ,τ represent the center μ of the optimal hidden layer neurons respectively * and the standard deviation τ of the optimal hidden layer neurons * .

4. The reliable control method of a cyber-physical system under malicious attacks according to claim 2 is characterized in that: The designed adaptive neural network finite time controller is expressed as: where sgn(·) represents a sign function, k is a positive constant, k ≥ α + β, β is the upper bound of the external disturbance d(t) in the system, α is a positive constant, and q(Ξ) ≠ 0.

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