Output Tracking Control Performance Analysis Method for Cyber-Physical Systems under Hybrid Attacks

Abstract

The present application provides a method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks, which relates to the field of cyber-physical system performance analysis. The method includes: using Bernoulli random distribution to model DoS attacks and spoofing attacks, and constructing a cyber-physical system under hybrid attacks; based on the single-degree-of-freedom controller of the cyber-physical system, using co-prime factorization, inner-outer factorization, and Youla parameterization method in the frequency domain to define the output tracking performance index of the cyber-physical system; calculating an explicit expression of the optimal output tracking performance of the cyber-physical system affected by hybrid attacks through all-pass decomposition, space decomposition technology, and the Youla parameterization method of the controller; realizing the analysis of the output tracking performance of the cyber-physical system through the explicit expression of the optimal output tracking performance, and obtaining the optimal tracking performance of the cyber-physical system.

Description

Technical Field

[0001] This application relates to the field of cyber-physical system performance analysis, and particularly to a method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks. Background Art

[0002] Two existing networked cascaded control systems (NCCSs) with different event-triggering mechanisms (ETMs) consider a method combining random denial-of-service (DoS) attacks and spoofing attacks in communication networks. Control design conditions are derived and established using linear matrix inequalities (LMIs).

[0003] Although this system considers the hybrid attack constraints combining DoS attacks and spoofing attacks, this method cannot obtain specific expression results. To study the impact of the above network attacks on the performance limit of the information system, it is necessary to analyze the output tracking performance limit of the cyber-physical system with the above constraints through frequency-domain analysis. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks to fill the technical gap in the research on the performance limit of networked cascaded control systems.

[0005] The above object of this application is achieved through the following technical solutions:

[0006] S1: Use Bernoulli random distribution to model DoS attacks and spoofing attacks, and construct a cyber-physical system under hybrid attacks;

[0007] S2: Based on the single-degree-of-freedom controller of the cyber-physical system, use coprime factorization, inner-outer factor factorization, and Youla parameterization method in the frequency domain to define the output tracking performance index of the cyber-physical system;

[0008] S3: Through all-pass decomposition, space decomposition technology and the Youla parameterization method of the controller, calculate the explicit expression of the optimal output tracking performance of the cyber-physical system affected by hybrid attacks;

[0009] S4: Through the explicit expression of the optimal output tracking performance, realize the analysis of the output tracking performance of the cyber-physical system, and obtain the optimal tracking performance of the cyber-physical system.

[0010] Optionally, step S1 includes:

[0011] S11: The cyber-physical system includes: reference input , output of the cyber-physical system , input of the single-degree-of-freedom controller , the controlled object , single-degree-of-freedom controller , the output of the noise channel ;

[0012] represents the reference input, , represents the direction vector of the reference input; Let be a unit vector, that is , of is transformed into , with variance ;

[0013] S12: Let represent the set of stable, proper, real-rational transfer functions or matrices; The Youla parameterization form of the single-degree-of-freedom controller is , represents the single-degree-of-freedom controller; represents the matrix that satisfies the double Bezout identity; represents the parameter for the free design of the controller; represents the factor regarding the zeros; represents the matrix that satisfies the double Bezout identity; represents the factor regarding the poles;

[0014] S13: Use the Bernoulli random distribution to describe the DoS attack and spoofing attack respectively;

[0015] The DoS attack is represented as the parameter , and its probability distribution function is , represents the probability of data loss;

[0016] The spoofing attack is represented as , and its probability distribution function is ; represents the occurrence probability of the spoofing attack;

[0017] Let the error signal of the cyber-physical system be ;

[0018] Let the tracking performance index of the cyber-physical system , where represent the variance of the tracking error and the variance of the system output respectively;

[0019] Since the control energy input to the communication channel is limited, there exists such that , is the expectation operator, is the trade-off parameter between the tracking error and the control energy input, where .

[0020] Optionally, step S2 includes:

[0021] The output of the noise channel , where represents the parameter by which the spoofing attack is modeled by a Bernoulli random distribution; represents the data injected into the cyber-physical system during the attack, and ; and the transfer function between;

[0022] According to , , , let the coprime factorization , and be the factors of after right coprime factorization with respect to zeros and poles respectively, and be the factors of after left coprime factorization with respect to zeros and poles respectively;

[0023] Let the coprime factorization , and be the factors of after right coprime factorization with respect to zeros and poles respectively, and be the factors of after left coprime factorization with respect to zeros and poles respectively;

[0024] Combined with the double Bezout identity and the Youla parameterization of the single-degree-of-freedom controller, where represents the matrix that satisfies the Bezout identity; represents the matrix that satisfies the Bezout identity; represents the matrix that satisfies the Bezout identity; represents the matrix that satisfies the Bezout identity; represents the factor with respect to zeros; represents the factor with respect to zeros; represents the factor with respect to poles; represents the factor with respect to poles; is the identity matrix, and the output tracking performance index of the cyber-physical system is defined as follows:

[0025]

[0026] wherein , represents a matrix satisfying the Bezout equation; represents a matrix satisfying the Bezout equation; is a parameter preset by the controller; represents the variance of

[0027] Optionally, step S3 includes:

[0028] Decompose the output tracking performance index into the following four parts:

[0029]

[0030]

[0031] .

[0032] Optionally, step S3 further includes:

[0033] Calculate , as follows:

[0034] For the controlled object has an all-pass decomposition , ; is the minimum-phase factor; and are all-pass factors, including the non-minimum-phase zeros and unstable poles ; and represent the total number of non-minimum-phase zeros and unstable poles; represents the conjugate of and the conjugate of

[0035] Define , , then

[0036]

[0037] wherein represents the all-pass factor; ; represents a subspace of Hilbert; ; represents another subspace of Hilbert, and Mutually orthogonal;

[0038] Decompose into

[0039]

[0040]

[0041]

[0042] There is Lemma 1: For any matrix , there exists a matrix such that the following equation holds

[0043]

[0044]

[0045] where , , , represents the product of all-pass factors , represents the product of all-pass factors , represents the total number of unstable poles, represents the product of all-pass factors , represents the product of all-pass factors , represents the unstable poles, represents the reciprocal of the all-pass factor; the all-pass factor , the sign in represents or ;

[0046] There is

[0047]

[0048] where , ; is the partial product of , is the partial product of ;

[0049] Obtain

[0050]

[0051] wherein, ; An expression containing z An expression containing z;

[0052] Determine in the Youla expression such that and ;

[0053] Rewrite as follows:

[0054]

[0055] According to and get as follows:

[0056] .

[0057] Optionally, step S3 includes:

[0058] By defining , , , where , combined with Lemma 1, get:

[0059]

[0060]

[0061]

[0062] where represents partial product of; represents partial product of; represents partial product of; represents partial product of; represents partial product of; represents partial product of, represents conjugate transpose of; represents conjugate transpose of; is the case when the unstable poles start from , is the case when the non-minimum phase zeros start from ;

[0063] The explicit expression of the optimal output tracking performance is .

[0064] An electronic device includes a processor, a memory, a user interface, and a network interface. The memory is used to store instructions. The user interface and the network interface are used to communicate with other devices. The processor is used to execute the instructions stored in the memory so that the electronic device executes an output tracking control performance analysis method for a cyber-physical system under hybrid attacks.

[0065] A computer-readable storage medium stores instructions that, when executed, perform an output tracking control performance analysis method for a cyber-physical system under hybrid attacks.

[0066] The beneficial effects brought by the technical solution provided in this application are as follows:

[0067] The Bernoulli random distribution is used to model the DoS attack and the spoofing attack, and a cyber-physical system under hybrid attacks is constructed. The specific form of the spoofing attack with a spoofing signal is described by using a transfer function. Based on a single-degree-of-freedom controller, an optimal controller is designed by using coprime factorization, inner-outer factorization, and the Youla parameterization method of the controller in the frequency domain. While ensuring the stability of the system, the output tracking performance of a single-input single-output discrete cyber-physical system under hybrid attacks is improved.

[0068] Through the frequency-domain optimal control method, the infimum of the tracking performance of a single-input single-output discrete cyber-physical system is derived (i.e., the influence of the hybrid attack on the cyber-physical system is described by an explicit expression), revealing the internal relationship between the output tracking performance of the cyber-physical system, the characteristics of the system itself, and the hybrid attack, and better designing the optimal controller for the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] The present application will be further described below in conjunction with the drawings and embodiments. In the drawings:

[0070] Figure 1 is a discrete-time cyber-physical system diagram in an embodiment of the present application;

[0071] Figure 2 is a schematic diagram of the output tracking control performance limit of a cyber-physical system under hybrid attacks in an embodiment of the present application;

[0072] Figure 3 is a schematic diagram for comparing the output tracking control performance limits of a cyber-physical system under different hybrid attacks in an embodiment of the present application;

[0073] Figure 4 is a schematic diagram of the structure of an electronic device in an embodiment of the present application. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0074] To have a clearer understanding of the technical features, objectives, and effects of this application, the specific implementation manners of this application will now be described in detail with reference to the accompanying drawings.

[0075] An embodiment of this application provides a method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks.

[0076] Please refer to Figure 1 , Figure 1 which is a discrete-time cyber-physical system diagram of a method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks in an embodiment of this application, including:

[0077] S1: Use the Bernoulli random distribution to model the DoS attack and the spoofing attack, and construct a cyber-physical system under hybrid attacks;

[0078] Specifically, the cyber-physical system includes: a communication cyber-physical system, an intelligent transportation system, and an intelligent medical system.

[0079] Step S1 includes:

[0080] S11: The cyber-physical system includes: a reference input , the output of the cyber-physical system, the input of the single-degree-of-freedom controller, the controlled object , the single-degree-of-freedom controller , and the output of the noise channel;

[0081] represents the reference input, , , and is a unit vector, that is, , of is transformed into , and the variance is ;

[0082] S12: Let represent the set of stable, proper, real rational transfer functions or matrices; the Youla parameterization form of the single-degree-of-freedom controller is , represents the single-degree-of-freedom controller; represents the matrix that satisfies the double Bezout identity; represents the parameter for free design of the controller; represents the factor regarding the zero point; represents the matrix that satisfies the double Bezout identity; Denote the factor with respect to the pole;

[0083] S13: Describe the DoS attack and spoofing attack respectively using the Bernoulli random distribution;

[0084] The DoS attack is represented as the parameter , and its probability distribution function is , denotes the probability of data loss occurring;

[0085] The spoofing attack is represented as , and its probability distribution function is ; denotes the occurrence probability of the spoofing attack;

[0086] Let the error signal of the cyber - physical system be ;

[0087] Let the tracking performance index of the cyber - physical system be , where denote the variance of the tracking error and the variance of the system output respectively;

[0088] Due to the limited control energy input to the communication channel, there exists such that , is the expectation operator, is the trade - off parameter between the tracking error and the control energy input, where .

[0089] Specifically, the present invention discloses an analysis method for the output tracking performance limit of a single - input single - output discrete - time cyber - physical system based on a single - degree - of - freedom controller under hybrid attacks. The structure of the cyber - physical system is as Figure 1 shown.

[0090] S2: Based on the single - degree - of - freedom controller of the cyber - physical system, use coprime factorization, inner - outer factorization, and Youla parameterization method in the frequency domain to define the output tracking performance index of the cyber - physical system;

[0091] Step S2 includes:

[0092] The output of the noise channel , where denotes the parameter by which the spoofing attack is modeled using the Bernoulli random distribution; denotes the data injected into the cyber - physical system during the attack, and ; and the transfer function between;

[0093] According to , , , perform co - prime factorization , and be factors of the right co - prime factorization with respect to zeros and poles respectively, and be factors of the left co - prime factorization with respect to zeros and poles respectively;

[0094] Perform co - prime factorization , and be factors of the right co - prime factorization with respect to zeros and poles respectively, and be factors of the left co - prime factorization with respect to zeros and poles respectively;

[0095] Combine the double Bezout identity and the Youla parameterization of the single - degree - of - freedom controller, where represents the matrix satisfying the Bezout identity; represents the matrix satisfying the Bezout identity; represents the matrix satisfying the Bezout identity; represents the matrix satisfying the Bezout identity; represents the factor with respect to zeros; represents the factor with respect to zeros; represents the factor with respect to poles; represents the factor with respect to poles; is the identity matrix, define the output tracking performance index of the cyber - physical system as follows:

[0096]

[0097] where , represents the matrix satisfying the Bezout identity; represents the matrix satisfying the Bezout identity; is the parameter preset by the controller; represents variance of.

[0098] S3: Through all - pass decomposition, space decomposition technology and the Youla parameterization method of the controller, calculate the explicit expression of the optimal output tracking performance of the cyber - physical system under the influence of hybrid attacks;

[0099] Step S3 includes:

[0100] The output tracking performance index is decomposed into the following four parts:

[0101]

[0102]

[0103] .

[0104] Step S3 further includes:

[0105] Calculate , as follows:

[0106] For the controlled object There is an all-pass decomposition , ; Is the minimum-phase factor; And Are all-pass factors, including the non-minimum-phase zeros of the controlled object And unstable poles ; And Represent the total number of non-minimum-phase zeros and unstable poles; Represents The conjugate of and The conjugate of;

[0107] Define , , then

[0108]

[0109] Where Represents the all-pass factor; ; Represents Represents a subspace of Hilbert; ; Represents another subspace of Hilbert, And Are orthogonal to each other;

[0110] Decompose Into

[0111]

[0112]

[0113]

[0114] Existence Lemma 1: For any matrix , there exists a matrix such that the following equation holds

[0115]

[0116]

[0117] where , , , represents the product of all-pass factors , represents the product of all-pass factors , represents the total number of unstable poles, represents the product of all-pass factors , represents the product of all-pass factors , represents the unstable poles, represents the reciprocal of the all-pass factor; the all-pass factor , in the symbol represents or ;

[0118] has

[0119]

[0120] where , ; is 's partial product, is 's partial product;

[0121] obtain

[0122]

[0123] where, ; represents an expression containing z, represents an expression containing z;

[0124] Determine in the Youla expression such that and ;

[0125] Rewrite as follows:

[0126]

[0127] According to and , obtain as follows:

[0128] .

[0129] Step S3 includes:

[0130] By defining , , , where , combined with Lemma 1, obtain:

[0131]

[0132]

[0133]

[0134] where represents partial product of; represents partial product of; represents partial product of; represents partial product of; represents partial product of; represents partial product of, represents conjugate transpose of; represents conjugate transpose of; is the case when unstable poles start from , is the case when non-minimum phase zeros start from ;

[0135] The explicit expression of the optimal output tracking performance is .

[0136] S4: Through the explicit expression of the optimal output tracking performance, perform the analysis of the output tracking performance of the cyber-physical system, and obtain the optimal tracking performance of the cyber-physical system.

[0137] Experimental data and conclusions:

[0138] Consider a discrete single-input single-output controlled plant, whose transfer function model is

[0139]

[0140] It can be seen from the transfer function matrix that the model contains a non-minimum phase zero , and contains an unstable pole . The trade-off parameter is 0.5, select , the channel input power .

[0141] For the hybrid attack, as shown by Figure 2 , as the probability of the hybrid attack increases, the output tracking control performance of the cyber-physical system becomes worse. When the probability of the hybrid attack approaches zero, the performance of the output tracking control will be improved. In addition Figure 3 compares two types of attacks. When the non-minimum phase zero and the unstable pole are close to each other, the output tracking control limit of the system will tend to infinity, resulting in a sharp deterioration of the output tracking control performance. With the probability of the control DoS attack remaining unchanged, the output tracking control performance of the cyber-physical system deteriorates as the probability of the spoofing attack increases. On the contrary, when the hybrid attack does not occur, the performance of the system will be better

[0142] This application also discloses an electronic device. Referring to Figure 4 , Figure 4 is a schematic structural diagram of an electronic device disclosed in an embodiment of this application. The electronic device 500 may include: at least one processor 501, at least one network interface 504, a user interface 503, a memory 505, and at least one communication bus 502

[0143] Among them, the communication bus 502 is used to realize the connection and communication between these components

[0144] Among them, the user interface 503 may include a display screen. Optionally, the user interface 503 may further include a standard wired interface and a wireless interface

[0145] Among them, the network interface 504 may optionally include a standard wired interface and a wireless interface (such as a WI-FI interface)

[0146] This application also discloses a computer-readable storage medium. The computer-readable storage medium stores multiple instructions, and the instructions are suitable for being loaded by a processor to execute the above-mentioned method for analyzing the output tracking control performance of a cyber-physical system under a hybrid attack

[0147] The above are only exemplary embodiments of the present disclosure, and the scope of the present disclosure cannot be limited thereby. That is, any equivalent changes and modifications made in accordance with the teachings of the present disclosure still fall within the scope covered by the present disclosure. Those skilled in the art will readily think of other implementation schemes of the present disclosure after considering the specification and the practice of the present disclosure

[0148] This application is intended to cover any variations, uses, or adaptations of the present disclosure, which follow the general principles of the present disclosure and include well-known knowledge or conventional technical means in the technical field not recorded in the present disclosure. The description and examples are only regarded as exemplary, and the scope and spirit of the present disclosure are defined by the claims.

Claims

1. A method for analyzing the output tracking control performance of a cyber-physical system under hybrid attacks, characterized in that, The method includes the following steps: S1: Use the Bernoulli random distribution to model the DoS attack and spoofing attack, and construct a cyber-physical system under the hybrid attack; S2: Based on the single-degree-of-freedom controller of the cyber-physical system, define the output tracking performance index of the cyber-physical system by using coprime factorization, inner-outer factorization, and Youla parameterization method in the frequency domain; S3: Calculate the explicit expression of the optimal output tracking performance of the cyber-physical system under the influence of the hybrid attack by using all-pass factorization, H2 space factorization technology, and the Youla parameterization method of the controller; S4: Through the explicit expression of the optimal output tracking performance, realize the analysis of the output tracking performance of the cyber-physical system, and obtain the optimal tracking performance of the cyber-physical system.

2. The output tracking control performance analysis method for a cyber-physical system under hybrid attack according to claim 1, characterized in that Step S1 includes: S11: The cyber-physical system includes: a reference input r, an output y of the cyber-physical system, an input u of a single-degree-of-freedom controller, a plant P, a single-degree-of-freedom controller K, and an output y of the noise channel a ; r represents the reference input, v represents the direction vector of the reference input; let v be a unit vector, i.e., ‖v‖ = 1, and the transformation of variance is S12: Set denote the set of stable, proper, real rational transfer functions or matrices; the Youla parameterization form of the single-degree-of-freedom controller K is as follows: where \(K(z)\) represents a single-degree-of-freedom controller; represents a matrix that satisfies the double Bezout identity; \(Q(z)\) represents the parameters of the controller's free design; represents a factor regarding zeros; represents a matrix that satisfies the double Bezout identity; represents a factor regarding poles; S13: Use the Bernoulli random distribution to describe the DoS attack and spoofing attack respectively; The DoS attack is represented as a parameter whose probability distribution function is indicating the probability of data loss; The spoofing attack is represented as whose probability distribution function is indicating the occurrence probability of the spoofing attack; Let the error signal of the cyber-physical system be e = r - y; Set the tracking performance index of the cyber-physical system where respectively represent the variance of the tracking error and the variance of the system output; Since the control energy input to the communication channel is limited, there exists Γ > 0 such that E is the expectation operator, and ε is the trade-off parameter between the tracking error and the control energy input, where 0 ≤ ε ≤ 1.

3. The output tracking control performance analysis method for a cyber-physical system under hybrid attacks according to claim 2, characterized in that, Step S2 includes: Output y of the noise channel a = β k y+(1 - β k )h k , where β k represents the parameter by which the spoofing attack is modeled by a Bernoulli random distribution; h k represents the data injected into the cyber - physical system during the attack, and represents the transfer function between y and h k . According to y = Pu, u = K(r - y a ), e = r - y, assume the co-prime factorization N δ and M δ are the factors of the right co-prime factorization with respect to zeros and poles respectively, and are the factors of the left co-prime factorization with respect to zeros and poles respectively; Set the coprime factorization and be the factors of (1 - α)(1 - β)P with respect to zeros and poles respectively after right coprime factorization, and be the factors of (1 - α)(1 - β)P with respect to zeros and poles respectively after left coprime factorization; Combined with the double Bezout identity and the Youla parameterization of a single-degree-of-freedom controller, where denotes a matrix that satisfies the Bezout identity; X denotes a matrix that satisfies the Bezout identity; denotes a matrix that satisfies the Bezout identity; Y denotes a matrix that satisfies the Bezout identity; denotes a factor with respect to zeros; N denotes a factor with respect to zeros; denotes a factor with respect to poles; M denotes a factor with respect to poles; I is the identity matrix. Define the output tracking performance index of the cyber-physical system as follows: where X(z) represents a matrix satisfying the Bezout identity; Y(z) represents a matrix satisfying the Bezout identity; Q is a parameter preset by the controller; represents the variance of r(k).

4. The output tracking control performance analysis method for a cyber-physical system under hybrid attacks according to claim 3, wherein Step S3 includes: Decompose the output tracking performance index into the following four parts:

5. The output tracking control performance analysis method for a cyber-physical system under hybrid attacks according to claim 4, characterized in that Step S3 also includes: Calculation is as follows: For the controlled object There is an all-pass decomposition \(N\) δ = \(L\) δ \(N\) δm , \(N\) δm , is the minimum-phase factor; and are all-pass factors, containing the non-minimum-phase zeros and the unstable poles \(\rho\) j , \(j = 1,\cdots,N\) p ; \(N\) s and \(N\) p represent the total number of non-minimum-phase zeros and unstable poles; represents the conjugate of \(\zeta\) i and the conjugate of \(\rho\) j ; Definition Then wherein represents an all-pass factor; represents a subspace of Hilbert; H2 represents another subspace of Hilbert, and H2 and are orthogonal to each other; Decompose into Existence Lemma 1: For any matrix there exists a matrix such that the following equation holds Among them represents the product of all-pass factors of represents the product of all-pass factors of, N p represents the total number of unstable poles, represents the product of all-pass factors of represents the product of all-pass factors of, ρ i represents an unstable pole, represents the reciprocal of the all-pass factor; the all-pass factor in which the symbol * represents or δ; There is Among them P mi is the partial product of mi and H is the partial product of Obtain Among them, Τ1(z) represents an expression containing z, and Τ1(∞) represents an expression containing z; Determine in the Youla expression such that and Rewrite as follows: According to and obtain as follows:

6. The output tracking control performance analysis method for a cyber-physical system under hybrid attacks as described in claim 5, characterized in that, Step S3 includes: By defining wherein Combined with Lemma 1, we get: Among them represents partial product; represents partial product; represents partial product; represents partial product; represents partial product; represents partial product, represents conjugate transpose, represents the conjugate transpose of P δj ; is the case when the unstable poles are counted from i, and P δi is the case when the non-minimum phase zeros are counted from i; The explicit expression for the optimal output tracking performance is 7. An electronic device, characterized in that, It includes a processor (501), a memory (505), a user interface (503), and a network interface (504). The memory (505) is used to store instructions. The user interface (503) and the network interface (504) are used to communicate with other devices. The processor (501) is used to execute the instructions stored in the memory (505) so that the electronic device executes the method according to any one of claims 1-6.

8. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores instructions, and when the instructions are executed by a computer, the method according to any one of claims 1-6 is executed.

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