Trade-off performance evaluation method for discrete networked control systems under cyber attacks

Abstract

The application discloses a kind of trade-off performance evaluation methods of discrete networked control system under network attack, which comprises: considering forward channel network attack, additive white Gaussian noise, data packet loss multiple communication constraints, the discrete networked control system model under multiple communication constraints is established, based on two degrees of freedom controller, using the tools such as coprime factorization, internal and external factor decomposition and Youla parameterization method to design optimal controller, on the basis of ensuring system stability, greatly improve the optimal trade-off performance expression of multiple-input multiple-output discrete networked control system.The application profoundly studies and reveals the relationship between the trade-off performance of networked control system and the characteristics of the system itself and each communication constraint, and better optimally designs the controller of this kind of system.

Description

Technical Field

[0001] This invention relates to the field of network system control, and more specifically to a method for evaluating the trade-off performance of discrete networked control systems under network attacks. Background Technology

[0002] The paper "Performance Analysis of MIMO Information Time-Delay System Under Bandwidth Cyber-Attack and Gaussian White Noise" introduces a system model for a multi-input multi-output (MIMO) information time-delay system. It investigates the trade-off performance of a networked control system constrained by network attacks, bandwidth limitations, and additive white Gaussian noise (AWGN). This performance is based on a trade-off between tracking error and channel energy limitations. The network parameters mainly include network attacks, bandwidth constraints, and AWGN in the backhaul channel. Based on a two-degree-of-freedom controller, using spectral decomposition and Youla parameterization, the optimal two-parameter structure is selected, yielding an explicit expression for the system's performance limits. Although this system simultaneously considers network attacks, AWGN, and bandwidth constraints, real-world network communication channels also present constraints such as interference and packet loss in the feedforward channel. Therefore, the model's considerations are not comprehensive enough, necessitating further research into other performance limits of the networked control system in this model. Summary of the Invention

[0003] The main technical problem to be solved by this invention is to analyze the performance limits of other trade-offs in a networked control system that takes into account various interferences, packet loss and other constraints, so as to perform optimal control of the system based on the optimal trade-off performance.

[0004] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0005] A method for evaluating the performance trade-offs of discrete networked control systems under network attacks includes the following steps:

[0006] A multi-input multi-output discrete networked control system model based on a two-degree-of-freedom controller is established. The control energy input of the networked control system model is expressed by the first expression:

[0007]

[0008] The system output of the networked control system model is the second expression:

[0009]

[0010] Where n represents additive white Gaussian noise, Denotes its Z-transform, d r Indicates packet loss. Its probability distribution function is ζ represents the probability of packet loss, 0 ≤ ζ ≤ 1, and r represents the reference input. Represents the Z-transform of r. in v represents the direction vector of the reference input, and y and u represent the system output and control energy input, respectively. Let be the Z-transforms corresponding to y and u, respectively, and G be the controlled object. Represents a stable, regular, real rational transfer function or matrix set; [K1K2] is a two-degree-of-freedom controller, where: Represents a set of controllers. and Let N(z) and M(z) be the factors of the rational transfer function matrix (1-ζ)G with respect to the zeros and poles, respectively, obtained by coprime decomposition. X(z), R and Q are matrices that satisfy the two Bezout equations respectively, and are the parameters of the two-degree-of-freedom controller designed freely.

[0011] The tracking error is expressed by the third expression:

[0012]

[0013] The tracking performance index J of the networked control system model is defined as the fourth expression:

[0014]

[0015] Where E is the expectation operator and ε is the tracking error. With control of energy input The trade-off between them satisfies 0 ≤ ε ≤ 1;

[0016] Based on the fourth expression, the optimal performance expression for the networked control system model is obtained as the fifth expression:

[0017]

[0018] Where, V = diag(a1) 2 ,...,a m 2 ),W=diag(β1 2 ,...,β m 2), where a and β are the power spectral densities of the reference input r and Gaussian white noise n, respectively, and P is the performance consumed by the intrusion detection system IDS;

[0019] Coprime decomposition based on rational transfer function matrix Double Bezout Equation The optimal performance expression is solved by the Youla parameterized form of the two-degree-of-freedom controller, and the optimal trade-off performance of the networked control system is finally obtained to achieve optimal control.

[0020] Furthermore, in the coprime decomposition based on the rational transfer function matrix... Double Bezout Equation Before the step of solving for the optimal performance expression using the Youla parameterized form of the two-degree-of-freedom controller, the following steps are also included:

[0021] The expression for calculating the performance P consumed by the intrusion detection system (IDS) is as follows:

[0022] For a network attack, if an attacker has n strategies to launch an attack, then an IDS also has n strategies to detect and respond to the attack. Let the attack strategies be S = {s1, s2, ... s}. n}, then its corresponding probability expression is: Let the defense strategy be D, and the same That is, when the attacker uses attack strategy s j At that time, IDS uses response strategy d i The utility matrices of the attacker and the IDS are obtained based on the Bayesian Nash equilibrium, i.e. and and Let A and B represent the utility of the attacker and the IDS, respectively. The performance consumed by the Intrusion Detection System (IDS) satisfies: where l∈R + R + Let be the set of positive real numbers, according to By selecting a linear function in the S strategy Make Derivation

[0023] Furthermore, the coprime decomposition based on the rational transfer function matrix... Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include:

[0024] definition The sixth, seventh, and eighth expressions are respectively:

[0025]

[0026]

[0027]

[0028] in, This is the first part of the optimal performance expression. This is the second part of the optimal performance expression. This is the third part of the optimal performance expression.

[0029] Furthermore, the coprime decomposition based on the rational transfer function matrix... Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include: calculating...

[0030] Controlled object The all-pass decomposition is N(z) = L(z)N m (z) and L(z) and B(z) are all-pass factors, which respectively contain the non-minimum phase zeros s of the controlled object. i (i = 1, ..., N) s and unstable poles p i (i = 1, ..., N) p ), N m and M m As the minimum phase factor, L(z) can be decomposed into: Therefore, the first part of the optimal performance expression is simplified to the ninth expression:

[0031]

[0032] Furthermore, according to Define a subspace of the Hilbert space. and have in For the tenth expression:

[0033]

[0034] For the eleventh expression:

[0035]

[0036] Furthermore, calculate according to the eleventh expression.

[0037] Defining internal and external in set up Among them ΨΨ H =I, can be calculated Substitute have to:

[0038]

[0039] in:

[0040]

[0041]

[0042] Calculations show that: make And because Therefore, we can calculate them separately:

[0043]

[0044] Where, δ(jw)=B(e jw )-2(1-ε) 2 Re{f(e jw )}+(1-ε) 2 f(1),

[0045] Calculate W2, because So:

[0046]

[0047] in Since A(z) + B(z) = 1 - ε, further calculation yields:

[0048]

[0049] Then the calculation was... For the twelfth expression:

[0050]

[0051] Furthermore, the coprime decomposition based on the rational transfer function matrix... Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include: calculating...

[0052] Define an internal-external decomposition Assumption but get

[0053]

[0054] definition By the lemma: For any exist Makes the following equation true

[0055]

[0056] Therefore, it exists. make From the double Bezout equation, we can obtain And N(s) i ) = 0, therefore Therefore c i Can be written as c i =Λ o (s i M -1 (s i W can be further calculated to obtain

[0057]

[0058] because By choosing a suitable R and performing simple calculations, we can obtain the thirteenth expression:

[0059]

[0060] Furthermore, the coprime decomposition based on the rational transfer function matrix... Double Bezout Equation The optimal performance expression is solved using the Youla parameterized form of the two-degree-of-freedom controller, including: calculation

[0061] calculate Methods and calculations The methods are similar:

[0062] definition and because After with A similar calculation process yields the fourteenth expression:

[0063]

[0064] Furthermore, based on expressions 12, 13, and 14, the optimal trade-off performance expression for this networked control system model is obtained as expression 15:

[0065]

[0066] in

[0067]

[0068]

[0069]

[0070]

[0071]

[0072] The beneficial effects of the technical solution provided by this invention are:

[0073] (1) Taking into account various communication constraints such as forward channel network attacks, additive white Gaussian noise, and data packet loss, a networked control system model under multiple communication constraints was established.

[0074] (2) Based on a two-degree-of-freedom controller, the optimal controller is designed using tools such as coprime decomposition, internal and external factor decomposition and Youla parameterization method. While ensuring system stability, the optimal trade-off performance of the multi-input multi-output discrete networked control system is greatly improved.

[0075] (3) By using the frequency domain optimal control method, the lower bound of the performance trade-off of the multi-input multi-output discrete networked control system is obtained. This study deeply reveals the relationship between the performance trade-off of the networked control system and the characteristics of the system itself, as well as the relationship between various communication constraints, so as to better design the controller for this type of system. Attached Figure Description

[0076] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0077] Figure 1 This is a structural diagram of a multi-input multi-output discrete networked control system model based on a two-degree-of-freedom controller in an embodiment of the present invention.

[0078] Figure 2 This represents the optimal performance limit of the networked control system in this embodiment of the invention, regardless of whether there is a network attack.

[0079] Figure 3 This represents the optimal performance limit of the networked control system under different network attacks in the embodiments of the present invention;

[0080] Figure 4 This represents the optimal performance limit of the networked control system under different packet loss conditions in the embodiments of the present invention. Detailed Implementation

[0081] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0082] This invention provides a method for evaluating the trade-off performance of discrete networked control systems under network attacks. First, a multi-input multi-output discrete networked control system model based on a two-degree-of-freedom controller is established, with the network structure as follows: Figure 1 As shown, its control energy input is expressed by the first expression:

[0083]

[0084] Its system output is the second expression:

[0085]

[0086] Where n represents additive white Gaussian noise, Denotes its Z-transform, d r Indicates packet loss. Its probability distribution function is ζ represents the probability of packet loss, 0 ≤ ζ ≤ 1, and r represents the reference input. Represents the Z-transform of r. in v represents the direction vector of the reference input, and y and u represent the system output and control energy input, respectively. Let be the Z-transforms corresponding to y and u, respectively, and G be the controlled object. Represents a stable, regular, real rational transfer function or matrix set; [K1K2] is a two-degree-of-freedom controller, where: Represents a set of controllers. and Let N(z) and M(z) be the factors of the rational transfer function matrix (1-ζ)G with respect to the zeros and poles, respectively, obtained by coprime decomposition. X(z), R and Q are matrices that satisfy the two Bezout equations respectively, and are the parameters of the two-degree-of-freedom controller designed freely.

[0087] Its tracking error is expressed by the third expression:

[0088]

[0089] The tracking performance index J of the networked control system model is defined as the fourth expression:

[0090]

[0091] Where E is the expectation operator and ε is the tracking error. With control of energy input The trade-off between them satisfies 0 ≤ ε ≤ 1;

[0092] Therefore, the optimal performance expression of this model is the fifth expression:

[0093]

[0094] Where, V = diag(a1) 2 ,...,a m 2 ),W=diag(β1 2 ,...,β m 2 ), where a and β are the power spectral densities of the reference input r and Gaussian white noise n, respectively, and P is the performance consumed by the intrusion detection system IDS;

[0095] For a network attack, if an attacker has n strategies to launch an attack, then an IDS also has n strategies to detect and respond to the attack. Let the attack strategies be S = {s1, s2, ... s}. n}, then its corresponding probability expression is: Let the defense strategy be D, and the same That is, when the attacker uses attack strategy s j At that time, IDS uses response strategy d i The utility matrices of the attacker and the IDS are obtained based on the Bayesian Nash equilibrium, i.e. and and Let A and B represent the utility of the attacker and the IDS, respectively. The performance consumed by the Intrusion Detection System (IDS) satisfies: where l∈R + R + Let be the set of positive real numbers, according to By selecting a linear function in the S strategy Make Derivation

[0096] Coprime decomposition based on rational transfer function matrix Double Bezout Equation The optimal performance expression is solved by the Youla parameterization form of the two-degree-of-freedom controller, and finally the optimal trade-off performance of the networked control system is obtained.

[0097] To calculate the optimal performance trade-off mentioned above, the optimal performance expression is divided into three parts, namely... And respectively defined as For the sixth, seventh, and eighth expressions:

[0098]

[0099]

[0100]

[0101] in, This is the first part of the optimal performance expression. This is the second part of the optimal performance expression. This is the third part of the optimal performance expression.

[0102] First calculate

[0103] Due to the controlled object The all-pass decomposition is N(z) = L(z)N m (z) and L(z) and B(z) are all-pass factors, which respectively contain the non-minimum phase zeros s of the controlled object. i (i = 1, ..., N) s and unstable poles p i (i = 1, ..., N) p ), N m and M m As the minimum phase factor, L(z) can be decomposed into: Therefore, the first part of the optimal performance expression is simplified to the ninth expression:

[0104]

[0105] because For a subspace of the Hilbert space, in order to better compute... definition and have in For the tenth expression:

[0106]

[0107] For the eleventh expression:

[0108]

[0109] Next, further calculations are performed based on the eleventh expression.

[0110] Defining internal and external in set up Among them ΨΨ H =I, can be calculated Substitute have to:

[0111]

[0112] in:

[0113]

[0114]

[0115] Calculations show that: We ordered And because Therefore, we can calculate them separately:

[0116]

[0117] Where, δ(jw)=B(e jw )-2(1-ε) 2 Re{f(e jw )}+(1-ε) 2 f(1),

[0118] Calculate W2, because So:

[0119]

[0120] in Since A(z) + B(z) = 1 - ε, further calculation yields:

[0121]

[0122] Then we have calculated... For the twelfth expression:

[0123]

[0124] calculate

[0125] Define another internal-external decomposition Assumption but get

[0126]

[0127] definition By the lemma: For any exist This makes the following equation true:

[0128]

[0129] Therefore, it exists. make From the double Bezout equation, we can obtain And N(s) i ) = 0, therefore Therefore c i Can be written as c i =Λ o (s i M -1 (s i W can be further calculated to obtain:

[0130]

[0131] because By choosing a suitable R and performing simple calculations, we can obtain the thirteenth expression:

[0132]

[0133] calculate Methods and calculations The methods are similar:

[0134] definition and because After with A similar calculation process yields the fourteenth expression:

[0135]

[0136] Therefore, based on expressions 12, 13, and 14, the optimal trade-off performance expression for this networked control system model is expressed as expression 15:

[0137]

[0138] in

[0139]

[0140]

[0141]

[0142]

[0143]

[0144] The following experimental data demonstrates the outstanding optimization effect that this embodiment can produce:

[0145] Experimental data and conclusions:

[0146] Considering a discrete-time multi-input multi-output controlled object, its transfer function matrix model is as follows:

[0147]

[0148] The transfer function G(z) shows that it contains a non-minimum phase zero z = k, and the direction of the output zero is η = (1,0). T It contains an unstable pole p = 3, whose direction is ω = (0, 1). T Define V = diag(1,1) T W = diag(1,1) T Select the all-pass vector but Through coprime decomposition, we can obtain...

[0149] The comparison chart between having a network attack and not having a network attack, with a packet loss probability ζ = 0.5, is shown below. Figure 2 As shown, it can be observed that the optimal performance of the system when it is not under network attack is significantly better than when it is under network attack; when selecting At that time, the corresponding F1 = 10, F2 = 5, F3 = 2. The impact of different F values ​​on the optimal performance of the network system is as follows: Figure 3 As shown, the larger F is, the less performance the IDS consumes, and the optimal performance trade-off of the system; the impact of different packet loss scenarios on network system performance is as follows: Figure 4 As shown, the higher the packet loss probability ζ, the greater the optimal performance limit J of the system. * The larger the value, the worse the optimal trade-off performance of the system.

[0150] This invention provides and implements a method for evaluating the trade-off performance of discrete networked control systems under network attacks. It proposes a novel optimal trade-off performance by adding network attack interference to the forward channel of the networked control system, using a binary stochastic process to simulate packet loss, and assuming that the channel noise is additive white Gaussian noise, thereby establishing a discrete networked control system model under multiple communication constraints. Based on a two-degree-of-freedom controller, the model is derived using all-pass decomposition, spatial decomposition techniques, and the Youla parameterization method of the controller. This significantly improves the expression for the optimal trade-off performance of multi-input multi-output discrete networked control systems while ensuring system stability. This invention deeply studies and reveals the relationship between the trade-off performance of networked control systems and the characteristics of the system itself, as well as the relationships between various communication constraints, enabling better optimal design of controllers for such systems.

[0151] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0152] The sequence numbers of the above embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments. In the unit claims listing several devices, several of these devices may be embodied by the same hardware item. The use of the terms first, second, and third, etc., does not indicate any order and can be interpreted as identifiers.

[0153] The above are merely preferred embodiments of the present invention and do not limit the scope of the patent. Any equivalent structural or procedural transformations made based on the description and drawings of the present invention, or direct or indirect applications in other related technical fields, are similarly included within the scope of patent protection of the present invention.

Claims

1. A method for evaluating the trade-off performance of a discrete networked control system under network attacks, the method considering multiple communication constraints such as forward-channel network attacks, additive white Gaussian noise, and data packet loss, characterized in that... Includes the following steps: A multi-input multi-output discrete networked control system model based on a two-degree-of-freedom controller is established. This networked control system model comprehensively considers various communication constraints, including forward-channel network attacks, additive white Gaussian noise, and data packet loss. The control energy input of the networked control system model is expressed by the first expression: The system output of the networked control system model is the second expression: Where n represents additive white Gaussian noise, Denotes its Z-transform, d r Indicates packet loss. Its probability distribution function is ζ represents the probability of packet loss, 0 ≤ ζ ≤ 1, and r represents the reference input. Represents the Z-transform of r. in v represents the direction vector of the reference input, and y and u represent the system output and control energy input, respectively. Let be the Z-transforms corresponding to y and u, respectively, and G be the controlled object. Represents a stable, regular, real rational transfer function or matrix set; [K1 K2] is a two-degree-of-freedom controller, where: Represents a set of controllers. and Let N(z) and M(z) be the factors of the rational transfer function matrix (1-ζ)G with respect to the zeros and poles, respectively, obtained by coprime decomposition. R and Q are matrices that satisfy the two Bezout equations respectively, and are the parameters of the two-degree-of-freedom controller designed freely. The tracking error is expressed by the third expression: The tracking performance index J of the networked control system model is defined as the fourth expression: Where E is the expectation operator and ε is the tracking error. With control of energy input The trade-off between them satisfies 0 ≤ ε ≤ 1; Based on the fourth expression, the optimal performance expression for the networked control system model is obtained as the fifth expression: in, a and β are the power spectral densities of the reference input r and Gaussian white noise n, respectively, and P is the performance consumed by the intrusion detection system (IDS). The expression for calculating the performance P consumed by the intrusion detection system (IDS) is as follows: For a network attack, if an attacker has n strategies to launch an attack, then an IDS also has n strategies to detect and respond to the attack. Let the attack strategies be S = {s1, s2, ... s}. n }, then its corresponding probability expression is: Let the defense strategy be D, and the same That is, when the attacker uses attack strategy s j At that time, IDS uses response strategy d i The utility matrices of the attacker and the IDS are obtained based on the Bayesian Nash equilibrium, i.e. and Let A and B represent the utility of the attacker and the IDS, respectively. The performance consumed by the Intrusion Detection System (IDS) satisfies: where l∈R + R + Let be the set of positive real numbers, according to By selecting a linear function in the S strategy Make Derivation Coprime decomposition based on rational transfer function matrix Double Bezout Equation The optimal performance expression is solved by the Youla parameterization form of the two-degree-of-freedom controller, and the optimal trade-off performance of the networked control system is finally obtained to achieve optimal control and improve the optimal trade-off performance of the multi-input multi-output discrete networked control system while ensuring system stability.

2. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 1, characterized in that, The coprime decomposition based on the rational transfer function matrix Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include: definition The sixth, seventh, and eighth expressions are respectively: in, This is the first part of the optimal performance expression. This is the second part of the optimal performance expression. This is the third part of the optimal performance expression.

3. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 2, characterized in that, The coprime decomposition based on the rational transfer function matrix Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include: calculating... : Controlled object The all-pass decomposition is N(z) = L(z)N m (z) and L(z) and B(z) are all-pass factors, which respectively contain the non-minimum phase zeros s of the controlled object. i (i = 1, ..., N) s and unstable poles p i (i = 1, ..., N) p ), N m and M m As the minimum phase factor, L(z) can be decomposed into: Therefore, the first part of the optimal performance expression is simplified to the ninth expression:

4. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 3, characterized in that, according to Define a subspace of the Hilbert space. and have in For the tenth expression: For the eleventh expression:

5. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 4, characterized in that, Calculated according to the eleventh expression : Defining internal and external in set up Among them ΨΨ H =I, can be calculated Substitute have to: in: Calculations show that: make And because Therefore, we can calculate them separately: where δ(jw) = B(e jw ) - 2(1 - ε) 2 Re{f(e jw )} + (1 - ε) 2 f(1), Calculate W2, because So: in Since A(z) + B(z) = 1 - ε, further calculation yields: Then the calculation was... For the twelfth expression:

6. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 5, characterized in that, The coprime decomposition based on the rational transfer function matrix Double Bezout Equation The steps for solving the optimal performance expression using the Youla parameterized form of a two-degree-of-freedom controller include: calculating... : Define an internal-external decomposition Assumption but get definition By the lemma: For any exist Makes the following equation true Therefore, it exists. make From the double Bezout equation, we can obtain And N(s) i ) = 0, therefore Therefore c i Can be written as c i =Λ o (s i M -1 (s i W can be further calculated to obtain because By choosing a suitable R and performing simple calculations, we can obtain the thirteenth expression:

7. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 6, characterized in that, The coprime decomposition based on the rational transfer function matrix Double Bezout Equation The optimal performance expression is solved using the Youla parameterized form of the two-degree-of-freedom controller, including: calculation : calculate Methods and calculations The methods are similar: definition and because After with A similar calculation process yields the fourteenth expression:

8. The method for evaluating the trade-off performance of discrete networked control systems under network attacks as described in claim 7, characterized in that, Based on expressions 12, 13, and 14, the optimal trade-off performance expression for this networked control system model is expressed as expression 15: in

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