Security Control Method for Industrial Cyber-Physical Systems under DoS Attacks Based on Markov Model

Abstract

The present invention is a security control method for industrial cyber-physical systems under DoS attacks based on the Markov model, including the following steps: Based on the industrial cyber-physical system, establish a linear model between the state vector and the control vector, and determine the parameter matrix in the linearized model; Considering the situation of parameter matrix changes caused by known corresponding attack and defense strategies under DoS attacks, establish an industrial cyber-physical system model based on the Markov model with parameter matrix jumps occurring according to attack and defense game changes under DoS attacks; Considering the engineering practice, given the hard constraints of the system state and control input; Considering the interference of noise signals, construct an elastic robust model predictive controller, and substitute the expression of the elastic robust model predictive controller into the model; Given a cost function, obtain the elastic robust model predictive controller gain by solving the optimization problem. Enable the industrial cyber-physical system to maintain stable performance under DoS, polyhedral uncertainties, random noise, and hard constraints on inputs and states.

Description

Technical Field

[0001] The present invention belongs to the field of network control, and specifically relates to a security control method for industrial cyber-physical systems under DoS attacks based on the Markov model. Background Art

[0002] Industrial cyber-physical systems are a new generation of intelligent systems that integrate computing, communication, and control by connecting intelligent manufacturing devices through the Internet of Things and information technology for data interaction. Compared with traditional control systems, they have significant advantages in terms of cost, scalability, and collaboration, and are thus widely used in fields such as Industry 4.0, intelligent healthcare, and network security. However, with the introduction of networking and the increase in connectivity, industrial cyber-physical systems face both severe interference in the industrial environment and network security issues in data transmission, severely affecting system performance and stability. Among them, DoS (Denial of Service) attacks, as the most frequent and harmful network attacks, have received extensive attention in recent years. DoS attacks disrupt communication security by blocking data transmission channels and occupying network bandwidth. In existing literature, there have been related studies on security state analysis and intelligent defense decision-making through attack and defense games for DoS attacks. However, there has been no related research on model predictive control analysis of the system considering the influence of system input and state constraints under DoS attacks. At the same time, no related literature has considered the influence of random noise on the design of elastic robust model predictive controllers. And due to the complexity of industrial systems, traditional model predictive control is difficult to apply. Introducing polyhedral uncertainty can more accurately realize system modeling and further construct an elastic robust model predictive controller. In terms of attack modeling, this paper considers data packet loss caused by attack and defense games and establishes a Markov model, thereby performing modal transformation on the parameter matrix of the system. This modeling scheme is introduced into the control design of industrial cyber-physical systems for the first time. Therefore, how to ensure the security and reliability of industrial cyber-physical systems under DoS attacks, polyhedral uncertainty, random noise, and input and state hard constraints is an important research topic. Summary of the Invention

[0003] The purpose of the present invention is to provide a security control method for industrial cyber-physical systems under DoS attacks based on the Markov model, so that the industrial cyber-physical system can maintain stable performance under DoS, polyhedral uncertainty, random noise, and input and state hard constraints.

[0004] To achieve the above purpose, the technical solution of the present invention is as follows:

[0005] A security control method for industrial cyber-physical systems under DoS attacks based on Markov models. The method is applied to the security control of industrial cyber-physical systems and is characterized in that the method comprises the following steps:

[0006] Step 1: Based on the industrial cyber-physical system, establish a linear model between the state vector and the control vector, and determine the parameter matrix in the linearized model;

[0007] Step 2: Considering the impact of data packet loss caused by the attack and defense game between the attacker and the defender and the influence of data packet loss on system modeling under DoS attacks, based on the known parameter matrix changes caused by corresponding attack and defense strategies, establish an industrial cyber-physical system model based on Markov models with parameter matrix jumps occurring according to the changes of the attack and defense game;

[0008] Step 3: Considering the actual engineering situation, give the hard constraints of the system state and control input;

[0009] Step 4: Considering the interference of noise signals, construct an elastic robust model predictive controller, and substitute the expression of the elastic robust model predictive controller into the industrial cyber-physical system model based on Markov models with parameter matrix jumps occurring according to the changes of the attack and defense game under DoS attacks;

[0010] Step 5: Given the cost function, obtain the elastic robust model predictive controller gain by solving the optimization problem.

[0011] Considering the case of data packet loss caused by known attack and defense strategies, model the changes in the parameter matrix affected by data packet loss as a parameter matrix that jumps according to the random process r k Perform a jump, and characterize the transition probability of the attack and defense strategy changes as the modal jump probability of r k Thus, establish the connection between the parameter matrix of the system and the changes in the attack and defense game strategies under DoS attacks, construct a modal-related elastic robust model predictive controller, and achieve accurate modeling and control of the system.

[0012] Furthermore, in Step 1, the linear model between the state vector and the control vector is:

[0013] x(k + 1) = A (σ) x(k) + B (σ) u(k), (1)

[0014] where k is the step size of the discrete system, and are the system state and control input respectively, and represent n x and n uEuclidean space, σ represents uncertainty, and the parameter matrix A of the system (σ) , B (σ) are uncertain matrices obtained from the linearized model of an industrial continuous stirred tank reactor. The parameter matrices A (σ) , B (σ) are contained in the polyhedron set Σ composed of L vertices:

[0015]

[0016] where denotes equivalence, and the parameter matrix of each system is weighted and represented by the vertices in the polyhedron set as:

[0017]

[0018] where α l is an uncertain parameter, α l ≥0, and A (l) , B (l) represent a vertex in the polyhedron set. k is the control step of the discrete system, generally a positive integer, and L represents the number of vertices in the polyhedron set.

[0019] Furthermore, in step 2, the modeling method of the DoS attack is as follows:

[0020] Considering that there is an attacker between the sensor and the controller in the industrial cyber-physical system, the attack strategy is denoted as A = {a1, a2,..., a M}, and the defender adopts the corresponding defense strategy denoted as D = {d1, d2,..., d M}, where M represents the number of attack and defense strategies under the energy constraint of the DoS attack. r k represents the situation of consecutive data packet loss when the attacker adopts the attack strategy a i and the defender adopts the defense strategy d i at the k-th step. r k+1 represents the situation of consecutive data packet loss when the attacker adopts the attack strategy a j and the defender adopts the defense strategy d j at the (k + 1)-th step. i, j ∈ S = {1, 2,... M}, where S is a finite mode set. Considering mapping different attack and defense strategies to different modes, assume that the attack and defense strategies satisfy the following transition probability:

[0021] π ij = Pr(r k+1 = j|r k = i),

[0022] where, Pr(rk+1 = j|r k = i) represents the known r k = i condition, r k+1 = j probability; π ij represents that the attacker adopts the attack strategy a i , the defender adopts the defense strategy d i transfers to the attacker adopting the attack strategy a j , the defender adopts the defense strategy d j probability, 0 ≤ π ij ≤ 1,

[0023] Thus, the stochastic process {r k , k ≥ 0} is regarded as a time-discrete homogeneous Markov stochastic process taking values on the finite mode set S = {1, 2,..., M};

[0024] Define the transition probability π of the attack and defense strategy change ij as the mode jump probability of the Markov model. Considering the impact of data packet loss caused by the change of attack and defense strategies on the parameter matrix of the system, the parameter matrix under the influence of the change of attack and defense strategies is denoted as Then establish an industrial cyber-physical system model with parameter matrix jump occurring according to the change of attack and defense game based on the Markov model under DoS attack as:

[0025]

[0026] where, the stochastic process {r k , k ≥ 0} is a time-discrete homogeneous Markov stochastic process taking values on the finite mode set S = {1, 2,..., M}, and the mode jump probability is expressed as:

[0027] Pr(r k+1 = j|r k = i) = π ij ,

[0028] where, π ij at this time represents the mode jump probability of the system jumping from mode i to mode j, 0 ≤ π ij ≤ 1,

[0029] In formula (1), the influence of DoS attack is not considered, and the parameter matrices A and B therein do not change with the attack and have no subscript r k , and are only affected by polyhedral uncertainty. The utilization of the Markov model is reflected in the mode change of the parameter matrix with the attack, that is, when the attack and defense strategies are different, the stochastic process r kCorresponding changes are made, and the corresponding parameter matrix changes synchronously.

[0030] Furthermore, in step 3, the hard constraints on the system state and control input are as follows:

[0031]

[0032]

[0033] Among them, the one with ~ represents the upper bound, that is, the known upper bounds of the control input and system state (constrained by actual conditions), which need to be given in advance; the one without ~ represents the actual values of the control input and system state; n u represents the dimension of the system control input, and n ψ represents the dimension of the matrix ψ. represents all n ψ ×n x real matrix sets, [·] p ([·] q ) represents the p(q)-th row of the matrix or the p(q)-th element of the vector, |[·] p |(|[·] q |) represents the norm of the p(q)-th row of the matrix or the absolute value of the p(q)-th element of the vector. and are known vectors, representing the upper bounds of the control input and system state respectively. respectively represent the total number of rows of the matrix or the total number of elements of the vector of the control input and system state.

[0034] Furthermore, in step 4, the elastic robust model predictive controller is as follows:

[0035]

[0036] Among them represents the elastic robust model predictive controller gain matrix, v ke , v kf (e,f∈{1,2,...n x}, e≠f) are independent Gaussian white noises and satisfy E[v ke = 0, and E[v ke v kf = 0, E[X] represents the mathematical expectation of the random variable X, s 2 represents the variance of the Gaussian white noise, and n x represents the dimension of the system state vector.

[0037] Furthermore, in step 5, the cost function is as follows:

[0038]

[0039] Among them x(k + n|k) and u(k + n|k) respectively refer to the predicted state and control input at the future time k + n obtained based on the available information at the current time k. n is the prediction step length, and there are denotes equivalent to Q and R are given weight matrices; J is derived from the Jacobian matrix (the matrix composed of derivatives in all directions), used to represent the cost function, and E{X} represents the mathematical expectation of the random variable X.

[0040] Furthermore, in step 5, the optimization problem is as follows:

[0041]

[0042] Among them represents the mode-dependent time-varying terminal constraint set of the system.

[0043] Furthermore, in step 5, considering that the optimization problem OP1 cannot be directly solved, the auxiliary optimization problems OP2 and OP3 are introduced.

[0044]

[0045] s.t. (7), (8), (9), (19).

[0046]

[0047] s.t. (7)-(9), (18), (22)-(25).

[0048] Furthermore, the solution process of the elastic robust model predictive controller gain matrix is as follows: For the given weight matrices Q and R, the vector and the matrix ψ, if there exist positive definite matrices L i X i Z i W i the matrix Y i and the positive scalar γ, the following linear matrix inequalities hold:

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] where [·] pp ([·] qq ) represents the p(q)-th diagonal element of the matrix “·”, W i T , Y i T , ψ T represent the transposes of the matrices W i , Y i , ψ respectively, and * is the symmetric part in the symmetric matrix; and there is,

[0058]

[0059]

[0060] Then, the elastic robust model predictive controller gain matrix is calculated according to the above linear matrix inequality where, the matrix Y i , W i can be obtained by solving the above linear matrix inequality using the existing matrix inequality toolbox, is the inverse matrix of W i .

[0061] The term (1 + s 2 ) in the above formula is a noise processing term, which is considered in the design of the elastic robust model predictive controller for the industrial cyber-physical system for the first time; for the constraints of the control input and the system state, equations (19)-(25) are given for processing, and such methods are conventional methods of model predictive control.

[0062] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0063] The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model of the present invention, based on the data packet loss caused by the attack-defense game under DoS attacks, considers the data packet loss situation of the system when the attacker uses different attack strategies and the defender uses corresponding defense strategies. Based on the number of lost packets, the parameter matrix of the system is modeled as a packet loss number-related matrix that jumps with the Markov random process. This modeling process is introduced into the security control of industrial cyber-physical systems for the first time, and then the industrial cyber-physical system under DoS attacks is modeled as an industrial cyber-physical system model based on the Markov model with parameter matrix jumps according to the attack-defense game under DoS attacks.

[0064] The method of the present invention considers polyhedral uncertainty, random noise interference, and hard constraints on control inputs and system states. Further considering the influence of random noise on the resilient robust model predictive controller, the corresponding linear matrix inequality conditions are obtained, and the conditions for the mean-square stability of the system are derived; then, with the help of stochastic analysis techniques (stochastic analysis techniques based on Lyapunov functionals, which need to consider the polyhedral uncertainty, Markov system parameter matrix jumps, interference of random noise on the resilient robust model predictive controller, and the influence of control inputs and system states in this paper, and the following conditions and upper bounds are obtained by synthesizing the above problems), the sufficient conditions for meeting the requirements of the terminal invariant set are derived, and the upper bound of the infinite-time horizon cost function in the worst case is obtained; finally, by solving the auxiliary optimization problem, the gain of the resilient robust model predictive controller is obtained. The present invention considers the attack-defense game between the attacker and the defender under DoS attacks, probabilistically describes the system mode, and then defends against DoS attacks, and can ensure the mean-square stability of the system under DoS attacks, polyhedral uncertainty, random noise, and hard constraints on inputs and states.

[0065] In addition to the continuous stirred tank reactor used in the simulation of the present invention, the method of the present invention can also be applied to urban water cycle real-time management systems, intelligent ship operation and maintenance systems, smart grid systems, intelligent factory construction systems, intelligent cargo management systems, intelligent transportation management systems, etc. Brief Description of the Drawings

[0066] In order to make the content of the present invention easier to be clearly understood, the present invention will be further described in detail below according to specific embodiments and in conjunction with the drawings.

[0067] Figure 1 It is a schematic diagram of a continuous stirred tank reactor;

[0068] Figure 2 It is the experimental result of the system state response;

[0069] Figure 3 It is the experimental result of the control input response;

[0070] Figure 4 These are the results of the system modal jump experiment. Specific implementation manners

[0071] In order to enable researchers in the technical field to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. The preferred embodiments of the present invention are given in the drawings. The present invention can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the understanding of the disclosure content of the present invention more thorough and comprehensive. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0072] The present invention relates to a security control method for an industrial cyber-physical system under DoS attacks based on a Markov model, so that the industrial cyber-physical system can maintain stable performance under DoS, polyhedral uncertainty, random noise, and hard constraints on inputs and states. The specific method includes the following steps:

[0073] S1: Based on the industrial cyber-physical system, establish a linear model between the state vector and the control vector, and determine the parameter matrix in the linearized model;

[0074] The industrial cyber-physical system is a continuous stirred tank reactor. Establish a mathematical model of the continuous stirred tank reactor. Considering an irreversible exothermic reaction occurring in the reactor, and the control objective is to make the output concentration as close as possible to its steady-state point. Then the energy and material balance equations are

[0075]

[0076]

[0077] where C A represents the outlet concentration, T represents the outlet temperature, T c represents the coolant temperature, C Af represents the inlet concentration, represents the derivative of C A , represents the derivative of T. The control objective is to adjust C Af and T c to regulate C A and T.

[0078] Under standard operating conditions, when the parameters satisfy the following conditions, the system is in equilibrium:

[0079] T eq = 350K,

[0080] where the subscript eq represents the standard operation.

[0081] Table 1 Symbols and Parameters of Continuous Stirred Tank Reactor

[0082]

[0083]

[0084] Define the state vector as x = [C A T] T , and the control vector as u = [T c C Af T , where [C A T] T and [T c C Af T represent the transposes of the matrices [C A T] and [T c C Af respectively. The state equation of the continuous stirred tank reactor can be obtained, that is, the linearized model of the industrial continuous stirred tank reactor is:

[0085] x(k + 1) = A (σ) x(k) + B (σ) u(k), (1)

[0086] where k is the step size of the discrete system, and are the system state and control input respectively, and represent the n x - and n u - dimensional Euclidean spaces respectively, σ represents the uncertainty, and the parameter matrices A (σ) , B (σ) are uncertain matrices. The parameter matrices A (σ) , B (σ) are included in the polyhedron set Σ composed of L vertices:

[0087]

[0088] where means equivalent to, and the parameter matrix of each system can be represented by the weighted vertices in the polyhedron set:

[0089]

[0090] where α l is the uncertain parameter, α l ≥ 0, and A (l) and (l) represent a vertex in the set of polyhedra.

[0091] S2: Considering the impact of the attack - defense game between the attacker and the defender on data packet loss and system modeling under DoS attacks, and based on the changes in the parameter matrix caused by known corresponding attack - defense strategies, establish an industrial cyber - physical system model with parameter matrix jumps based on the Markov model under DoS attacks that change according to the attack - defense game.

[0092] Under DoS attacks, the signal transmission of the sensor is blocked by the attacker, and the controller cannot receive complete data packet information, which may lead to deviations in the modeling of the system model. Considering that there is an attacker between the sensor and the controller, who adopts the attack strategy A = {a1, a2,..., a M}, and the defender adopts the corresponding defense strategy D = {d1, d2,..., d M}, where M represents the number of attack and defense strategies under the energy constraint of DoS attacks. Let r k represent the situation where, at the k - th step, the attacker adopts the attack strategy a i and the defender adopts the defense strategy d i resulting in consecutive data packet losses. Let r k+1 represent the situation where, at the (k + 1)-th step, the attacker adopts the attack strategy a j and the defender adopts the defense strategy d j resulting in consecutive data packet losses. i, j ∈ S = {1, 2,... M}, where S is a finite modal set. Map different attack - defense strategies to different modes. Assume that the attack - defense strategies satisfy the following transition probabilities:

[0093] π ij = Pr(r k+1 = j|r k = i),

[0094] where Pr(r k+1 = j|r k = i) represents the probability of j given that r k = i). π ij represents the probability that the attacker adopts the attack strategy a i and the defender adopts the defense strategy d i transfers to the situation where the attacker adopts the attack strategy a j and the defender adopts the defense strategy d j . 0 ≤ π ij ≤ 1, Thus, the stochastic process {r k, k≥0} can be regarded as a time-discrete homogeneous Markov random process taking values on the finite mode set S = {1, 2,..., M}, and the mode transition probability matrix under the attack-defense game can be expressed as:

[0095]

[0096] Among them, M represents the number of modes, that is, the number of attack-defense strategies.

[0097] Define the transition probability π of the change of attack-defense strategies ij as the mode transition probability of the Markov model. Considering the impact of data packet loss caused by the change of attack-defense strategies on the parameter matrix of the system, the parameter matrix under the influence of the change of attack-defense strategies is denoted as Thus, the industrial cyber-physical system under DoS attack can be modeled as the following Markov system:

[0098]

[0099] Among them, the parameter matrix is an uncertain matrix under the influence of attack-defense strategy transfer. The random process {r k , k≥0} is a time-discrete homogeneous Markov random process taking values on the finite mode set S = {1, 2,..., M}, and its mode transition probability can be expressed as:

[0100] Pr(r k+1 = j|r k = i) = π ij ,

[0101] Among them, π ij at this time represents the mode transition probability of the system jumping from mode i to mode j, 0 ≤ π ij ≤ 1,

[0102] Then formula (2) is the industrial cyber-physical system model based on the Markov model with parameter matrix jump occurring according to the attack-defense game under DoS attack.

[0103] S3: Considering the engineering practice, given the hard constraints on the system state and control input:

[0104] Considering the engineering practice, the present invention introduces the following hard constraints on the input and state:

[0105]

[0106]

[0107] Among them, nu Denotes the dimension of the system control input, n ψ Denotes the dimension of the matrix ψ Denotes all n ψ ×n x Set of real matrices, [·] p ([·] q ) denotes the p(q)-th row of the matrix or the p(q)-th element of the vector, |[·] p |(|[·] q |) denotes the norm of the p(q)-th row of the matrix or the absolute value of the p(q)-th element of the vector And Are known vectors, representing the upper bounds of the control input and the system state respectively Respectively represent the total number of rows of the matrix or the total number of elements of the vector of the control input and the system state

[0108] S4: Considering the interference of the noise signal, construct an elastic robust model predictive controller, and substitute the expression of the elastic robust model predictive controller into the industrial cyber-physical system model with parameter matrix jumps based on the Markov model under DoS attacks

[0109] Considering that in the actual signal transmission process, it is inevitable to be interfered by noise signals, construct the following elastic robust model predictive controller with random noise interference as:

[0110]

[0111] Where Denotes the elastic robust model predictive controller gain matrix, v ke ,v kf (e,f∈{1,2,...n x},e≠f) are independent Gaussian white noises and satisfy E[v ke =0 And E[v ke v kf =0, E[X] represents the mathematical expectation of the random variable X, s 2 Denotes the variance of the Gaussian white noise, n x Denotes the dimension of the system state vector

[0112] Substitute the elastic robust model predictive controller shown in formula (3) into the model of formula (2), and the closed-loop system can be written as:

[0113]

[0114] S5: Given the cost function, obtain the elastic robust model predictive controller gain by solving the optimization problem

[0115] Step 5.1. First, the cost function of this paper is given as follows:

[0116]

[0117] where x(k + n|k) and u(k + n|k) respectively refer to the predicted state and control input at the future time k + n obtained based on the available information at the current time k. n is the prediction step length, and there is denotes equivalence to, Q and R are given weight matrices; J is derived from the Jacobian matrix (the matrix composed of derivatives in all directions) and is used to represent the cost function. E{X} represents the mathematical expectation of the random variable X.

[0118] Thus, by solving the following online optimization problem, the feedback gain under the model predictive control scheme can be obtained:

[0119]

[0120] where represents the mode - dependent time - varying terminal constraint set of the system.

[0121] Step 5.2. For the convenience of subsequent derivation, the following definitions and lemmas are introduced:

[0122] Definition 1: If for any and r0 ∈ S, the following conditions hold, then the stochastic Markov system with u(k) ≡ 0 is mean - square stable.

[0123]

[0124] Definition 2: For the system (2), if the system state at time k belongs to the positive set and its future states also belong to such a set ( is the set of positive integers), then the set is called a positive invariant set.

[0125] Lemma 1: If there exists a matrix H i > 0 satisfying or then if and only if there exist matrices H i > 0, G i > 0, the following conditions hold.

[0126]

[0127]

[0128] Next, to further process the feedback gain, this paper will give a set a sufficient condition for the terminal constraint set and obtain an upper bound of the worst-case infinite-horizon cost function from the auxiliary optimization problem of OP1. Let i = r k+nk , if the following two conditions are satisfied simultaneously, then the set Φ i is a terminal constraint set in the mean-square sense.

[0129] Condition 1: The mode-dependent stochastic Lyapunov function

[0130] V(x(k + n|k), r k+n|k = i) = x T (k + n|k)P i x(k + n|k) (5)

[0131] satisfies the following inequality.

[0132]

[0133] Condition 2: The set Φ defined by the following formula i is a positive invariant set.

[0134]

[0135] Step 5.3: First, give the sufficient condition for the mean-square stability of the system without constraints:

[0136] Theorem 1: For MJSs (4), given the weight matrices Q and R, if there exist positive definite matrices L i , X i , Z i , W i , matrix Y i and positive scalar γ such that the following conditions hold, then for any i ∈ S, the system satisfies equation (6), and the state feedback gain in the control law is given by K i = Y i W i -1 .

[0137]

[0138]

[0139]

[0140] Where,

[0141]

[0142]

[0143] The proof process is as follows:

[0144] Let E[X] denote the expectation of the random variable X. Taking the difference of Equation (5) and taking the corresponding expectation, we have

[0145]

[0146] where

[0147] The uncertainty of the state is expressed as where F = F T , then from E[v ke = 0, D[v ke = s 2 , D[v ke represents the variance of the random variable v ke , the uncertainty of the state can be expressed as E(R T R) = diag[1 + s 2 , 1 + s 2 ,..., 1 + s 2 . Based on the above analysis, for each positive definite symmetric matrix P j > 0, the expectation of the corresponding mode can be expressed by the following formula:

[0148]

[0149] Moving the terms on the right side of Equation (6) to the left side, then the system satisfies Equation (6) if and only if the following equation holds.

[0150]

[0151] where

[0152] From the Schur complement lemma and Lemma 1, Equation (10) can be satisfied by the following conditions:

[0153]

[0154]

[0155]

[0156] Multiplying both sides of Equation (11) by γ, we have

[0157]

[0158] Let X i = γP i -1 , According to the Schur complement lemma, Equation (14) can be guaranteed by Equation (7).

[0159] Multiply both the left and right sides of Equation (12) by We get

[0160]

[0161] From Let Y i = K i W i Then Equation (15) can be guaranteed by Equation (8).

[0162] Multiply both the left and right sides of Equation (13) by We get

[0163]

[0164] From According to the Schur complement lemma and and Y i = K i W i Equation (16) can be guaranteed by Equation (9), and the feedback gain is given by K i = Y i W i -1 Q.E.D.

[0165] Step 5.4. Next, verify that the set Φ i is a positively invariant set, that is, the set Φ i satisfies the following two conditions:

[0166] (1) For any x(k + n|k) ∈ Φ i , the following equation holds;

[0167] x T (k + n|k)P i x(k + n|k) ≤ γ. (17)

[0168] (2) For being the set of integers, the state at a future time x(k + n + f|k) ∈ Φ i .

[0169] The proof process is as follows:

[0170] According to the Schur complement lemma, Equation (17) holds if and only if the following equation holds:

[0171]

[0172] Multiply both the left and right sides of Equation (18) by Let X i = γP i -1 , there is

[0173]

[0174] The following proves that when x(k + n|k) ∈ Φ i , there is x(k + n + 1|k) ∈ Φ i .

[0175] From equation (6), there is From P i ≥ 0 and π ij ≥ 0, there is E{x T (k + n + 1|k)P j x(k + n + 1k)} ≤ x T (k + n|k)P i x(k + n|k). Then according to mathematical induction, if and only if x(k + n|k) ∈ Φ i , for there is x(k + n + f|k) ∈ Φ i . Based on the above analysis, it can be seen that Φ i is a positive invariant set of system (4).

[0176] Step 5.5. Introduce an auxiliary optimization problem to solve the original optimization problem:

[0177] Due to the existence of polyhedral uncertainty, the optimization problem OP1 cannot be directly solved. Therefore, consider introducing an upper bound γ to approximate the desired cost function. Take the expectation of both sides of equation (6) conditional on the available information at time k, and sum both sides of the above inequality from n = 0 to ∞. Since V(x(∞|k)) = 0, there is

[0178]

[0179] Thus, the upper bound of the optimization problem OP1 can be obtained from the following auxiliary optimization problem OP2, and then the optimization problem can be solved:

[0180]

[0181] s.t. (7), (8), (9), (19).

[0182] Step 5.6. Consider the recursive feasibility and robust stability of the optimization problem. Under the condition that the optimization problem at time k = 0 is solvable, it is shown that the algorithm in this paper is solvable at time k > 0 and the closed-loop system is mean-square stable:

[0183] The proof process is divided into two parts: recursive feasibility and robust stability.

[0184] (1) Recursive feasibility: In OP2, only the inequality in Equation (18) explicitly depends on the measurement state x(k). Therefore, it is only necessary to prove that this inequality is feasible for all future measured states. This can be easily proven from the following equation and Condition 2.

[0185]

[0186] (2) Robust stability: Select the quadratic function where is obtained by solving OP2. Define and as the optimal solutions at time k and k + 1, respectively. From the properties of the optimal solution and the feasibility at time k + 1, we have:

[0187]

[0188] From Condition 2, we have:

[0189]

[0190] Also, from By combining the two equations, we have:

[0191]

[0192] Thus, V k (x(k)) is strictly decreasing, and when k → ∞, x(k) → 0. Q.E.D.

[0193] Step 5.7. Give the sufficient condition for the mean-square stability of the system under constraints:

[0194] Theorem 2: For MJSs with hard constraints, given the vector and the matrix ψ, if there exist positive definite matrices X i , W i , the matrix K i , Y i and the positive scalar γ such that the following conditions hold, where [·] pp ([·] qq ) represents the p(q)-th diagonal element of the matrix “·”, then the hard constraint conditions can be satisfied.

[0195]

[0196]

[0197]

[0198]

[0199] The proof process is as follows:

[0200] From the constraint conditions, we have:

[0201]

[0202] According to the Schur complement lemma, Equation (26) can be guaranteed by the following equation and Equation (23).

[0203]

[0204] Multiply both sides of Equation (27) by diag{I, W i T}, and from and X i = γP i -1 , it can be known that Equation (26) can be guaranteed by Equation (22) and Equation (23).

[0205] Similarly, from the constraint conditions, we have:

[0206]

[0207] According to the Schur complement lemma, Equation (28) can be guaranteed by the following equation and Equation (25).

[0208]

[0209] Multiply both sides of Equation (29) by diag{I, W i T}, and from and X i = γP i -1 , it can be known that Equation (28) can be guaranteed by Equation (24) and Equation (25). Q.E.D.

[0210] From the optimization problem OP2 and Theorem 2, the optimization problem of the elastic robust model predictive controller under state and input constraints can be expressed as:

[0211]

[0212] s.t. (7)-(9), (18), (22)-(25).

[0213] By analogy with the proof method in Step 5.6, the recursive feasibility and robust stability of OP3 can be easily guaranteed.

[0214] Step 5.8: Considering the coupling term in the optimization problem OP3, the offline part and online part algorithms of the elastic robust model predictive controller under the initial mode r0 are given:

[0215]

[0216] S5: Simulation Example Analysis

[0217] By writing a Matlab program to solve the elastic robust model predictive controller gain and plot the simulation curve, a simulation example is used to prove the effectiveness of the present invention:

[0218] Consider as Figure 1 shown continuous stirred tank reactor:

[0219] Use the Euler method for simulation, set the sampling time as T s = 0.1s, assume there exists an attack strategy A = {a1, a2}, a defense strategy D = {d1, d2}, with two modes, considering the influence of polytopic uncertainty and DOS attack, the parameter matrices of the following system are obtained from the linearized model of the industrial continuous stirred tank reactor:

[0220]

[0221]

[0222]

[0223]

[0224] The corresponding mode transition probability matrix is:

[0225]

[0226] The upper limits of the system state and control input are set as ψ = I2, Set the initial state as x(0) = [0.31 0.25] T , the initial mode is r0 = 1. Select the weight matrices as Q = 0.7I2, R = 1, and the elastic robust model predictive controller gain obtained by Algorithm 1 is (since it is difficult to present all the results of time-varying variables, only the elastic robust model predictive controller gain when k = 1 is listed here):

[0227]

[0228] The simulation results are as Figures 2 - 4 shown. Figure 2 The state trajectory of the system is given. Obviously, the elastic robust model predictive controller proposed in this paper can make the system achieve the expected stability under the existence of polytopic uncertainty and DoS attack in the parameter matrix. Figure 3 The motion trajectory of the system control input is given, Figure 4It is the change of the system mode over time. The above simulation results show that a security control method for industrial cyber-physical systems under DoS attacks based on the Markov model proposed by the present invention can meet the hard constraints of the system state and control input while ensuring the stability of the system and effectively coping with network security threats.

[0229] The method of this application introduces the input and state constraint conditions existing in engineering practice, considers the influence of random noise on the elastic robust model predictive controller and the influence of polyhedral uncertainty on system modeling, and models the DoS attack by the Markov model, thereby constructing an elastic robust model predictive controller to realize the security control of industrial cyber-physical systems under DoS attacks. Among them, random noise is considered in the design of the elastic robust model predictive controller of industrial cyber-physical systems for the first time, using the Markov model to model the DoS attack is first introduced into the security design of industrial cyber-physical systems, and comprehensively considering DoS attacks, polyhedral uncertainty, random noise and input and state hard constraints, the collaborative design of the elastic robust model predictive controller of industrial cyber-physical systems is proposed for the first time.

[0230] Matters not described in the present invention apply to the prior art.

Claims

1. A security control method for industrial cyber-physical systems under DoS attacks based on the Markov model, the method is applied to the security control of industrial cyber-physical systems, and is characterized in that, The method includes the following steps: Step 1: Based on the industrial cyber-physical system, establish a linear model between the state vector and the control vector, and determine the parameter matrix in the linearized model; Step 2: Considering the attack and defense game between the attacker and the defender under DoS attack, which leads to data packet loss and the impact of data packet loss on system modeling, based on the change of the parameter matrix caused by the known corresponding attack and defense strategies, establish an industrial cyber-physical system model with parameter matrix jumps based on the Markov model under DoS attack according to the change of the attack and defense game; The specific process of Step 2 is: Considering that there are attackers between sensors and controllers in industrial cyber-physical systems, the attack strategies are denoted as Α = {a1, a2,..., a M}, and the corresponding defense strategies adopted by the defender are denoted as D = {d1, d2,..., d M}, where M represents the number of attack and defense strategies under the energy constraint of DoS attacks, and r k represents that at the k-th step, the attacker adopts the attack strategy a i and the defender adopts the defense strategy d i , resulting in the situation of continuous packet loss. r k+1 represents that at the (k + 1)-th step, the attacker adopts the attack strategy a j and the defender adopts the defense strategy d j , resulting in the situation of continuous packet loss. i, j ∈ S = {1, 2,... M}, where S is a finite mode set. Considering mapping different attack and defense strategies to different modes, it is assumed that the attack and defense strategies satisfy the following transition probabilities: π ij = Pr(r k+1 = j | r k = i), where Pr(r k+1 = j|r k = i) represents the probability that r k = i given that r k+1 = j; π ij represents the probability that the attacker adopts attack strategy a i and the defender adopts defense strategy d i transfers to the situation where the attacker adopts attack strategy a j and the defender adopts defense strategy d j , 0 ≤ π ij ≤ 1, Thus, the stochastic process {r k , k≥0} is regarded as a time-discrete homogeneous Markov stochastic process taking values in the finite state set S = {1, 2, ..., M}; Define the transition probability π of the attack and defense strategy change ij as the mode jump probability of the Markov model. Considering the impact of data packet loss caused by the change of attack and defense strategy on the parameter matrix of the system, the parameter matrix under the influence of the change of attack and defense strategy is denoted as Then, the industrial cyber-physical system model based on the Markov model with parameter matrix jump occurring according to the change of attack and defense game under DoS attack is established as: where the stochastic process {r k , k≥0} is a time-discrete homogeneous Markov stochastic process taking values in a finite mode set S = {1, 2, ..., M}, and the mode transition probability is expressed as: Pr(r k+1 = j|r k = i) = π ij , where, π ij represents the modal transition probability from mode i to mode j of the system at this time, 0 ≤ π ij ≤ 1, Step 3: Considering the actual engineering, give the hard constraints of the system state and control input; Step 4: Considering the interference of the noise signal, construct an elastic robust model predictive controller, and substitute the expression of the elastic robust model predictive controller into the industrial cyber-physical system model with parameter matrix jumps based on the Markov model under DoS attack according to the change of the attack and defense game; Step 5: Given the cost function, obtain the elastic robust model predictive controller gain by solving the optimization problem.

2. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, wherein, Considering the case where known attack and defense strategies cause packet loss, the change of the parameter matrix affected by data packet loss is modeled as a stochastic process r k For the parameter matrix that undergoes jumps, the transition probability of the change in the attack and defense strategy is characterized as r k The modal jump probability of r, thereby establishing the connection between the parameter matrix of the system and the change in the attack and defense game strategy under DoS attacks, constructing a modal-related elastic robust model predictive controller, and realizing the accurate modeling and control of the system.

3. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, characterized in that In Step 1, the linear model between the state vector and the control vector is: x(k + 1)=A (σ) x(k)+B (σ) u(k), (1) where k is the step size, and are the system state and control input respectively, and represent the Euclidean spaces of dimension n x and n u respectively, σ represents the uncertainty, and A (σ) , B (σ) are the parameter matrices of the system. The parameter matrices A (σ) , B (σ) are included in the polyhedron set Σ composed of L vertices: wherein denotes equivalence, and the parameter matrix of each system is weighted by vertices in the polyhedron set and represented as: where α l is an uncertain parameter, α l ≥ 0, and A (l) and B (l) represent a vertex in the polyhedron set; L represents the number of vertices in the polyhedron set.

4. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, wherein, In Step 3, the hard constraints of the system state and control input are: where, n u represents the dimension of the system control input, n ψ represents the dimension of the matrix ψ, n x represents the dimension of the system state vector; [·] p ([·] q ) represents the p(q)-th row of the matrix or the p(q)-th element of the vector, |[·] p |(|[·] q |) represents the norm of the p(q)-th row of the matrix or the absolute value of the p(q)-th element of the vector, and are known vectors, representing the upper bounds of the control input and the system state respectively, represent the total number of rows of the matrix or the total number of elements of the vector of the control input and the system state respectively.

5. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, wherein In Step 4, the elastic robust model predictive controller is: where denotes the elastic robust model predictive controller gain matrix, v ke , v kf (e, f ∈ {1, 2,... n x}, e ≠ f) are independent Gaussian white noises and satisfy E[v ke = 0, and E[v ke v kf = 0, E[X] represents the mathematical expectation of the random variable X, s 2 denotes the variance of the Gaussian white noise, n x denotes the dimension of the system state vector.

6. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, wherein In Step 5, the cost function is: Among them x(k + n|k) and u(k + n|k) respectively refer to the predicted state and control input at the future time k + n obtained based on the available information at the current time k. n is the prediction step length, and there is denotes equivalent to Q and R are given weight matrices; J is derived from the Jacobian matrix and is used to represent the cost function. E{X} represents the mathematical expectation of the random variable X.

7. The security control method for industrial cyber-physical systems under DoS attacks based on the Markov model according to claim 1, characterized in that, The industrial cyber-physical system is a continuous stirred tank reactor, a real-time urban water cycle management system, an intelligent ship operation and maintenance system, a smart grid system, a smart factory construction system, a smart cargo management system or a smart traffic management system.

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