Operational amplifier circuit safety control method based on WTOD protocol

Abstract

The invention discloses an operational amplifier circuit safety control method based on a WTOD protocol, and belongs to the technical field of network safety control. The method comprises the following steps: establishing a networked Markov jump system based on a state equation of an operational amplifier circuit; a logarithm quantizer and a WTOD protocol are introduced; a redundant channel is designed for the random DoS attack; establishing an output feedback controller depending on a mapping mode and a scheduling signal; establishing a closed-loop system based on a Markov jump system, a logarithm quantizer, a WTOD protocol, a redundant channel and an output feedback controller; based on a Lyapunov function method, generating a stability criterion of the closed-loop system; and based on the stability criterion, solving the feedback gain of the output feedback controller. The stability of the operational amplifier circuit under random DoS attacks can be improved, unnecessary data transmission can be reduced under the condition that communication resources are limited, and the transmission burden of a communication channel is effectively reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of network security control, and particularly relates to a method for securely controlling an operational amplifier circuit based on the WTOD protocol. Background Art

[0002] With the booming development of modern control technologies, the modeling and control of complex systems have become the core issues in the scientific research field. Among many system models, the Markov jump system, due to its characteristic of depicting the random jump of the system state over time, is widely used in control theory and engineering practice. As a hybrid system, the Markov jump system consists of multiple subsystems, where the operating state of each subsystem is randomly switched through a Markov chain, with a certain degree of randomness and probability. Such a system can better simulate the phenomena of structural changes and parameter fluctuations in reality, and is particularly suitable for describing dynamic systems affected by external disturbances or internal randomness. In practical applications, operational amplifier circuits are often affected by external environments and internal factors, resulting in sudden changes in circuit parameters or adjustments to circuit structures. Therefore, in order to more accurately describe the behavioral characteristics of operational amplifier circuits under different operating conditions, the operational amplifier circuit is modeled as a networked Markov jump system by means of the Markov jump theory, thereby providing a more reliable theoretical basis for system design and optimization.

[0003] In a networked Markov jump system, ensuring efficient and accurate information interaction is crucial for system stability. However, due to the limited number and bandwidth of communication network transmission channels, problems such as transmission delays, packet losses, and channel congestion frequently occur, thereby affecting system performance. To solve these problems, quantization techniques and communication scheduling protocols have been introduced in existing research. The quantization technique reduces the packet size by converting continuous signals into discrete values, thereby alleviating the bandwidth pressure. The communication scheduling protocol, on the other hand, allocates the right to use channels through a reasonable scheduling method to relieve channel congestion, optimize the timing and order of data transmission, and improve resource utilization efficiency. Therefore, when designing a networked Markov jump system, it is necessary to comprehensively consider quantization techniques and communication scheduling protocols. On the one hand, the packet size is reduced through quantization to relieve the bandwidth burden; on the other hand, channel congestion is avoided through a reasonable communication scheduling protocol to ensure accurate information transmission. In this way, the system efficiency can be maximized under resource constraints and complex network environments, and the stable and efficient operation of the system can be guaranteed.

[0004] However, although these technical measures have played an important role in improving the performance of networked control systems, when realizing the interconnection between nodes in a communication network, conducting data exchange and cooperative control, they also bring many security risks. Among them, network attacks are one of the important security risks. Common attack types include spoofing attacks, replay attacks, and DoS attacks (Denial of Service attacks). It is worth noting that DoS attacks prevent signal transmission by exhausting network resources or overloading computing resources, resulting in a sharp decline in system performance, and then causing resource exhaustion, network congestion, and even system paralysis, thus seriously affecting the stability of the control system. Especially when the communication between the controller and the controlled object is limited or the network condition is unstable, the harm of DoS attacks is particularly prominent. In order to effectively resist DoS attacks, the design of networked Markov jump systems must take communication security into consideration and adopt defense strategies such as event-triggered mechanisms and redundant channels to build an impregnable security line. Therefore, the integration of communication security and protection measures plays an indispensable role in system optimization and design.

[0005] Currently, there are patents publicly available on security control methods for networked systems against DoS attacks. For example, the Chinese invention patent with an application publication date of September 10, 2024, and an application publication number of CN118631504A discloses an event-triggered quantization consensus control method for unmanned boats under DoS attacks. The method includes: establishing a state-space model of an unmanned boat considering external environmental disturbances and input delays caused by communication network transmission delays, and transforming it. Using the transformed model without input delays, an output feedback state observer of the unmanned boat is established. For the case where the state cannot be directly measured, the state of the unmanned boat is reconstructed using the state observer. At the same time, a uniform quantizer is established to effectively reduce the communication frequency and bandwidth occupancy. Based on the communication network topology diagram of the unmanned boat state-space model, the attack error of the unmanned boat during the duration of the DoS network attack is obtained, an event-triggered condition is constructed, and a consensus protocol based on the sufficient condition of event-triggered quantization consensus of the unmanned boat is established, which can achieve the multi-unmanned boat to maintain consensus under various complex communication situations. This application case greatly improves the control performance of unmanned boats and can resist the situation of multiple channels being independently attacked by DoS, providing important technical support for the wide application of unmanned boats.

[0006] For another example, a Chinese invention patent with the application publication number CN113014605A discloses a self-triggered quantization control method for coping with denial-of-service attacks. The method includes: setting the dynamic equation of the system to be stabilized, discretizing the dynamic equation to obtain a discretized system equation; defining self-triggering moments that satisfy preset conditions. If the current moment is a self-triggering moment, the output value of the system to be stabilized is sent to the encoder in the quantizer at the current moment, and the encoder encodes the output value to obtain an output code. If the output channel is not under a DoS attack at the current moment, the output code is successfully sent to the decoder on the controller side and the self-triggering module, and the decoder decodes the output code to obtain a quantization value. If the output channel is under a DoS attack at the current moment, a default value is sent to the decoder on the controller side and the self-triggering module instead of the output code, the next triggering moment is calculated, and the calculation mechanism is updated, and the calculation is repeated until the quantization value is obtained.

[0007] However, although the influence of limited network transmission channel bandwidth and DoS attacks has been considered in the prior art, the influence of limited number of transmission channels on system performance is generally ignored, and the design complexity of the controller is relatively high when there are multiple random variables in the system, and the utilization rate of communication resources also needs to be further improved. Summary of the Invention

[0008] The present invention provides a safety control method for an operational amplifier circuit based on the WTOD protocol, and combines the dependence mapping mode with the scheduling signal to jointly construct an output feedback controller, so that not only the stability of the operational amplifier circuit under random DoS attacks can be improved, but also unnecessary data transmission can be reduced in the case of limited communication resources, effectively reducing the transmission burden of the communication channel and improving the utilization rate of communication resources.

[0009] To achieve the above object, the technical solution provided by the present invention is as follows:

[0010] The present invention provides a safety control method for an operational amplifier circuit based on the WTOD protocol, including:

[0011] S1. Based on the state equation of the operational amplifier circuit, establish a networked Markov jump system;

[0012] S2. Introduce a logarithmic quantizer and the WTOD protocol;

[0013] S3. Design redundant channels for random DoS attacks;

[0014] S4. Establish an output feedback controller that depends on the mapping mode and the scheduling signal;

[0015] S5. Based on the networked Markov jump system under random DoS attacks, the logarithmic quantizer, the WTOD protocol, the redundant channels, and the output feedback controller that depends on the mapping mode and the scheduling signal, establish a closed-loop system;

[0016] S6. Based on the Lyapunov function method, generate the stability criterion for the closed-loop system;

[0017] S7. Based on the stability criterion, solve the feedback gain of the output feedback controller.

[0018] Furthermore, in S1, assume that the operational amplifier circuit is affected by the disturbance ω(h) and C2 is a jump parameter controlled by the Markov chain r(h), that is, C2 = ε r(h) , then the obtained networked Markov jump system is expressed as follows:

[0019] x(h + 1) = A r(h) x(h) + B r(h) u(h) + D r(h) ω(h),

[0020]

[0021] z(h) = E r(h) x(h);

[0022] where x(h), y α (h), z(h), u(h) and ω(h) represent the system state, the measured output, the control output, the control input, and the external disturbance respectively; D r(h) and E r(h) are known system matrices; α ∈ {1, 2}, 1 and 2 represent the main channel and the redundant channel respectively; r(h) depends on the Markov chain taking values in a finite set S = {1, 2, θN}, N is a positive integer; R1, R2, R3, R4 are resistors, C1, C2 are capacitors; d is the time step.

[0023] Furthermore, in S4, the mapping technology is used to map {r(h), θ1(h), θ2(h)} to a new random variable ξ(h), and the specific mapping process includes:

[0024] Step 1. The mapping of the DoS attack random variable, that is, {θ1(h), θ2(h)} is mapped to the variable through Θ(·) 0 means selecting the main channel for transmission, 1 means selecting the redundant channel for transmission, and 2 means unable to transmit:

[0025]

[0026] Among them, and ⊕ respectively represent the NOT operation and the XOR operation; θ α (h) ∈ {0, 1}. When θ α (h) = 0, it indicates that a DoS attack has occurred. When θ α (h) = 1, it indicates that no DoS attack has occurred;

[0027] Step 2, Map it to the variable ξ(h) through Ξ(·):

[0028]

[0029] Among them, ξ(h) takes values from the set S = {1, 2, …, 3N};

[0030] Step 3, Establish the relationship between {r(h), θ1(h), θ2(h)} and the variable ξ(h) according to the mapping processes of the first two steps. When ξ(h) is given, we get:

[0031]

[0032] Among them, mod(·) represents the modulo operation, represents rounding down, and ⊕ respectively represent the NOT operation and the XOR operation.

[0033] Furthermore, the output feedback controller that depends on the mapping mode and the scheduling signal is:

[0034]

[0035] Among them, K νj is the feedback gain; x(h) is the system state; Δ = diag{Δ1, …, Δ ny}, Δ i ∈ [-μ, μ], representing the quantization error range; represents the signal received by the controller at time h;

[0036]

[0037] Ψ σα(h) = diag{δ(σ α (h) - 1), δ(σ α (h) - 2), …, δ(σ α (h) - n y )}, δ(·) ∈ {0, 1} is the Kronecker function.

[0038] Furthermore, before the signal is transmitted to the communication network, use the logarithmic quantizer q α (·) for the measurement output y α(h) is quantized, and the quantized output is expressed as:

[0039] q α (y α (h)) = (I + Δ)y α (h), α ∈ {1, 2},

[0040] where Δ = diag{Δ1, …, Δ ny}, Δ i ∈ [-μ, μ], representing the quantization error range; represents the sector boundary, represents the quantization density, and

[0041] Furthermore, according to the WTOD protocol, the actual signal transmitted in the channel is:

[0042]

[0043] where I represents the identity matrix.

[0044] Furthermore, after introducing the redundant channel, the calculation of the obtained output signal is as follows:

[0045]

[0046] Furthermore, in S5, let Then the resulting closed-loop system is represented as:

[0047]

[0048] where,

[0049] Furthermore, the process of solving the feedback gain of the output feedback controller in S7 is as follows:

[0050] When the given quantization parameter δ ∈ (0, 1), if there exist matrices P ν > 0, scalar ∈ > 0, γ > 0, θ 11 > 0, θ 12 > 0, θ 21 > 0, θ 22 > 0, such that the linear matrix inequality

[0051]

[0052] holds, then the feedback gain K νj can be obtained. Where:

[0053]

[0054] β vιis the transition probability of the variable ξ(h).

[0055] Adopting the technical solution provided by the present invention, compared with the prior art, the following beneficial effects can be achieved:

[0056] (1) Through the design of the mapping mode, the present invention maps multiple different random variables into a new random variable, thereby effectively reducing the dimension of the problem, reducing the computational complexity, and helping to construct a more analytical and operable control strategy, thus facilitating the design of the controller and subsequent performance analysis; and by combining the mapping mode with the scheduling signal to jointly construct an output feedback controller, not only effectively reduces the design complexity under the condition of multiple random variables, but also can optimize the channel resource allocation and improve the utilization rate of communication resources.

[0057] (2) During the signal transmission process, in view of the problem of limited communication resources, the present invention introduces a logarithmic quantizer and the WTOD protocol, thereby reducing unnecessary resource waste; at the same time, considering the insecure factors existing in the communication network transmission process, in view of the stability problem of the operational amplifier circuit under random DoS attacks, further by adopting the redundant channel method, the reliability of network communication can be improved.

[0058] That is, the present invention can not only improve the stability of the operational amplifier circuit under random DoS attacks, but also reduce unnecessary data transmission in the case of limited communication resources, effectively reducing the transmission burden of the communication channel. Description of the Drawings

[0059] Figure 1 is a flowchart of the safety control method for the operational amplifier circuit based on the WTOD protocol in the embodiment of the present invention;

[0060] Figure 2 is a circuit diagram of the operational amplifier in the embodiment of the present invention;

[0061] Figure 3 is a structural diagram of the networked Markov jump system in the embodiment of the present invention;

[0062] Figure 4 is a modal switching diagram of the networked Markov jump system in the embodiment of the present invention;

[0063] Figure 5 is a diagram of the situation where the main and redundant channels are under DoS attacks and the channel selection situation in the embodiment of the present invention.

[0064] Figure 6 is H in the embodiment of the present invention ∞ is a trajectory diagram of the performance index γ(h);

[0065] Figure 7It is the state trajectory diagram of the networked Markov jump system in the embodiment of the present invention;

[0066] Figure 8 It is the trajectory diagram of the control input in the embodiment of the present invention. Specific implementation manner

[0067] As Figure 1 shown, the operational amplifier circuit security control method based on the WTOD protocol of the present invention specifically includes:

[0068] S1: Based on the state equation of the operational amplifier circuit, establish a networked Markov jump system;

[0069] S2: In view of the problem of limited communication resources, introduce a logarithmic quantizer and the WTOD protocol;

[0070] S3: In view of the stability problem of the operational amplifier circuit under random DoS attacks, adopt a redundant channel method to improve the reliability of network communication;

[0071] S4: In view of the switching behavior of the controlled object and DoS attacks, through a new mapping technology, map multiple random variables into a single modal variable, simplify the controller design, and on this basis, design an output feedback controller that depends on the mapping mode and the scheduling signal;

[0072] S5: Based on the networked Markov jump system, logarithmic quantizer, WTOD protocol, redundant channels, and output feedback controller that depends on the mapping mode and the scheduling signal under random DoS attacks, establish a closed-loop system;

[0073] S6: Based on the Lyapunov function method, generate the stability criterion of the closed-loop system;

[0074] S7: Based on the stability criterion, solve the feedback gain of the output feedback controller.

[0075] The present invention can not only improve the stability of the operational amplifier circuit under random DoS attacks, but also reduce unnecessary data transmission in the case of limited communication resources, effectively reducing the transmission burden of the communication channel, and providing an innovative solution for the security and efficiency of networked control systems.

[0076] For the control method according to any technical solution of the present invention, specifically, the specific process of establishing the networked Markov jump system in S1 is as follows:

[0077] Combined with Figure 2 the circuit diagram of the operational amplifier in, establish the state equation of the operational amplifier circuit:

[0078]

[0079] Among them, represents the rate of change of voltage v1 with respect to time, represents the rate of change of voltage v2 with respect to time, R1, R2, R3, and R4 are resistors, C1 and C2 are capacitors, and u represents the input voltage externally applied to the circuit.

[0080] Select x1(t) = v1 and x2(t) = v2. According to the above continuous-time state equation, it can be written in the standard state-space form:

[0081]

[0082] Then, based on the first-order Euler approximation formula and the step size d = 0.1, and H ∈ {1, 2,..., n}, where H is a positive integer, discretize the state-space equation of the operational amplifier circuit to obtain:

[0083]

[0084] Assume that the operational amplifier circuit is affected by the disturbance ω(h), and C2 is a jumping parameter controlled by the Markov chain r(h), that is, C2 = ε r(h) , and adding redundant channels for the impact of random DoS attacks in network communication, then the above equation can be re-expressed as a discretized networked Markov jump system:

[0085] x(h + 1) = A r(h) x(h) + B r(h) u(h) + D r(h) ω(h),

[0086]

[0087] z(h) = E r(h) x(h);

[0088] Among them, x(h), y α (h), z(h), u(h), and ω(h) represent the system state, measurement output, control output, control input, and external disturbance, respectively. D r(h) and E r(h) are known system matrices; α ∈ {1, 2}, where 1 and 2 represent the main channel and the redundant channel, respectively. r(h) depends on a Markov chain taking values in a finite set S = {1, 2,..., N} (N is a positive integer), and its transition probability matrix is defined as Λ = [λ mn N×N (m, n ∈ S), where λ mn = Pr{r(h + 1) = n, |r(h) = m}, λ​mn ∈[0, 1],

[0089] In some embodiments, the specific process of introducing the logarithmic quantizer and the WTOD protocol in S2 is as follows:

[0090] Assume that there are n y sensor nodes during the signal transmission process, representing the i-th node of the measurement output. Before the signal is transmitted to the communication network, the logarithmic quantizer q α (·) is used to quantize the measurement output y α (h):

[0091]

[0092] For each define the quantization level as:

[0093]

[0094] where Q represents the set of quantization levels, represents the initial value of, represents the quantization density. The higher the quantization density, the more accurate the quantization and the better the performance; represents the quantization level. Then, the logarithmic quantizer can be expressed as:

[0095]

[0096] where represents the sector boundary; using the sector boundary method, we can obtain:

[0097] q α (y α (h)) = (I + Δ)y α (h), α ∈ {1, 2},

[0098] where Δ i ∈ [-μ, μ], representing the quantization error range.

[0099] For the i-th sensor node, define as:

[0100]

[0101] where, is the signal of the i-th node received by the controller at time h - 1, θ i > 0 represents the given weight, Represents the error between the weighted control signal at the previous moment and the output signal at the current moment. When h = 0, the given

[0102] According to the WTOD protocol, the scheduling signal is defined as:

[0103]

[0104] According to the WTOD protocol rules, if The above equation is satisfied, that is, when i = σ α (h), the actually transmitted signal is If the above equation is not satisfied, that is, when i ≠ σ α (h), the signal received by the controller at the previous moment is retained Then the actual signal transmitted in the channel is:

[0105]

[0106] Among them, I represents the identity matrix; Ψ σα(h) = diag{δ(σ α (h)-1), δ(σ α (h)-2), …, δ(σ α (h)-n y )}, δ(·) ∈ {0, 1} is the Kronecker function.

[0107] According to the control method described in any embodiment of the present invention, the specific process of designing redundant channels for random DoS attacks in S3 is as follows:

[0108] During the network transmission process, there are many insecure factors, and attacks are one of them, and DoS attacks are the most destructive. Therefore, the present invention considers that random DoS attacks will be encountered during the network transmission process and uses a Markov chain to characterize the attack process, that is, whether an attack occurs at the next moment only depends on the attack situation at the current moment. The transition probability of the attack can be expressed as:

[0109]

[0110] Among them, θ α (h) ∈ {0, 1}, when θ α (h) = 0, it means that a DoS attack occurs, and when θ α (h) = 1, it means that no DoS attack occurs. Therefore, it can be obtained that:

[0111] To address the stability issue of the operational amplifier circuit under random DoS attacks, the present invention adopts a redundant channel method to improve network communication. When the primary channel is not under attack, the information in the primary channel is transmitted; if the primary channel is attacked, the signal in the redundant channel is transmitted; if both channels are attacked simultaneously, the signal at the current moment cannot be transmitted. Therefore, it can be obtained that:

[0112]

[0113] As a further improvement to the technical solution of any embodiment of the present invention, the specific process of designing the output feedback controller that depends on the mapping mode and scheduling signal is as follows:

[0114] Since there are multiple random variables in the control method of the present invention, namely r(h), θ1(h), and θ2(h). Therefore, it is very complex to consider all the above random variables separately for the controller design. To facilitate the controller design and subsequent performance analysis, the present invention adopts a new mapping technique to map {r(h), θ1(h), θ2(h)} to a new random variable ξ(h). This mapping method can effectively reduce the dimension of the problem, reduce the computational complexity, and help construct a more analytical and operable control strategy. The main steps of this technique are as follows:

[0115] The first step is the mapping of the DoS attack random variables, that is, {θ1(h), θ2(h)} can be mapped to the variable 0 indicates selecting the primary channel for transmission, 1 indicates selecting the redundant channel for transmission, and 2 indicates unable to transmit:

[0116]

[0117] where and ⊕ represent the NOT operation and the exclusive OR operation respectively.

[0118] The second step, is mapped to the variable ξ(h) through Ξ(·):

[0119] ξ(h) = Ξ(r(h),

[0120] where ξ(h) takes values from the set S = {1, 2,..., 3N}.

[0121] The third step is to establish the relationship between {r(h), θ1(h), θ2(h)} and the variable ξ(h) according to the mapping processes of the first two steps;

[0122] When ξ(h) is given, it can be obtained that:

[0123]

[0124] where mod(·) represents the modulo operation, represents rounding down, and ⊕ represent the NOT operation and the exclusive OR operation respectively. Therefore, the transition probability of the variable ξ(h) can be calculated as:

[0125]

[0126] According to the above mapping mode ξ(h), we can get:

[0127]

[0128] where:

[0129]

[0130]

[0131] ξ(h) = ν, σ1(h), σ2(h) = j.

[0132] Next, based on the above mapping technology and the WTOD protocol, the output feedback controller depending on the mapping mode and the scheduling signal is designed as:

[0133]

[0134] where K νj is the feedback gain.

[0135] The design strategy of this controller effectively reduces the design complexity under the condition of multiple random variables through the mapping technology. At the same time, the scheduling mechanism adopted by it can optimize the channel resource allocation and improve the utilization rate of communication resources. By organically combining the mapping technology and the scheduling mechanism, not only the controller design process is simplified, but also the efficient utilization of communication resources is achieved.

[0136] In some embodiments, the specific process of establishing a closed-loop system in S5 is as follows:

[0137] Substitute the output feedback controller u(h) obtained in S4 into the networked Markov jump system obtained in S1 to get the closed-loop system:

[0138]

[0139] To simplify the expression, let Then the closed-loop system can be re-expressed as:

[0140]

[0141] where

[0142] Furthermore, the specific process of generating the stability criterion of the closed-loop system in S6 is as follows:

[0143] When ω(h) = 0,

[0144] If: holds, then the closed-loop system is said to be asymptotically stable;

[0145] Under zero initial conditions, if there exists a scalar γ > 0 such that: holds, then the closed-loop system is said to have H ∞ performance, where E{·} represents the expected value.

[0146] Based on the Lyapunov function: V(h, η(h), ξ(h)) = η T (h)P ξ(h) η(h), we get:

[0147]

[0148] where

[0149] According to the WTOD protocol principle, we can obtain:

[0150]

[0151] Then, under zero initial conditions, we can obtain:

[0152]

[0153] where:

[0154]

[0155] If there exist matrices P ν > 0, scalar γ > 0, θ 11 > 0, θ 12 > 0, θ 21 > 0, θ 22 > 0 such that:

[0156]

[0157] holds, then the closed-loop system is said to have H ∞ performance.

[0158] In addition, when ω(h) = 0, if there exist matrices P ν > 0, scalar θ 11 > 0, θ 12 > 0, θ 21 > 0, θ 22 > 0 such that:

[0159]

[0160] If it holds, the closed-loop system is said to be asymptotically stable.

[0161] Furthermore, in the embodiments of the present invention, the specific process of solving the feedback gain of the output feedback controller is as follows:

[0162] The first step is to perform the Schur complement on the matrix Σ νj to obtain:

[0163]

[0164] where:

[0165]

[0166] The second step, since the matrix P ν > 0, we can get That is

[0167] Introduce the matrix X ν > 0, and let Then:

[0168]

[0169] where

[0170] The third step, since the matrices and in it have the uncertain term Δ introduced by the logarithmic quantizer. To solve the uncertain term Δ, introduce a scalar ∈ > 0, and we can deduce:

[0171]

[0172] Since Δ 2 ≤ δ 2 , thus we can get:

[0173]

[0174] In summary, perform the Schur complement on and finally we can get:

[0175]

[0176] where

[0177]

[0178] Then, when the quantization parameter δ ∈ (0, 1) is given, if there exists a matrix Pν > 0, scalar ∈ > 0, γ > 0, θ 11 > 0, θ 12 > 0, θ 21 > 0, θ 22 > 0 such that the linear matrix inequality holds, and the feedback gain K νj value can be obtained.

[0179] To further understand the technical solution of the present invention, the present invention will now be described in detail with reference to specific embodiments.

[0180] The circuit diagram of the operational amplifier according to the embodiment of the present invention is as Figure 2 shown. First, consider the state - space equation of the operational amplifier circuit:

[0181]

[0182] where, represents the rate of change of voltage v1 with respect to time, represents the rate of change of voltage v2 with respect to time, R1, R2, R3, R4 are resistors, C1, C2 are capacitors, and u represents the input voltage externally applied to the circuit.

[0183] Secondly, assume that the state is affected by the disturbance ω(h) and C2 is a jump parameter controlled by the Markov chain r(h), i.e., C2 = ε r(h) (r(h) = {1, 2}). Through discretization, the following networked Markov jump - system is obtained:

[0184] x(h + 1)=A r(h) x(h)+B r(h) u(h)+D r(h) ω(h),

[0185]

[0186] z(h)=E r(h) x(h);

[0187] where the system parameters in this embodiment are specifically:

[0188]

[0189] E2 = [0.1 0.1], R1 = R4 = 2Ω, R2 = 3Ω, R3 = 1Ω, C1 = 0.3F, d = 0.1, ε1 = 1, ε2 = 0.5.

[0190] In addition, the transition probability matrices Λ, Π1 and Π2 are respectively:

[0191]

[0192] Set the quantization parameters according to the introduced logarithmic quantizer And set the corresponding weight matrix Φ = diag{0.1, 0.1} according to the WTOD protocol.

[0193] Use MATLAB and the YALMIP and MOSEK toolboxes to solve the linear matrix inequality The optimal H can be obtained ∞ Performance index γ min = 0.4854 and the feedback gain of the corresponding output feedback controller is:

[0194] K 11 = [393.5945 -1.0376], K 12 = [-1.5158 394.0727],

[0195] K 21 = [202.0499 -4.4157], K 22 = [-4.1598 201.8008],

[0196] K 31 = [209.9080 -0.1814], K 32 = [0.0463 229.8899],

[0197] K 41 = [229.1275 -0.0437], K 42 = [-2.6948 230.1611],

[0198] K 51 = [-0.2268 0.2813], K 52 = [-0.2268 0.2813],

[0199] K 61 = [-0.1040 0.3051], K 62 = [-0.1040 0.3051].

[0200] Based on the above parameters, in the simulation, set the initial value of the system state as x(0) = [0 0] T . In addition, the external disturbance is set as ω(h) = e -0.5h cos(0.5h).

[0201] Figure 3 Figure shows the structure diagram of the networked Markov jump system Figure 4 Figure depicts the mode switching diagram of the system. Figure 5It depicts the situations of single-channel and dual-channel encountering DoS attacks. It can be seen from the figure that the moment of simultaneous attack in the dual-channel scenario is significantly lower than that in the single-channel scenario. Figure 6 Shows H ∞ The trajectory of the performance index γ(h). Figure 7 And Figure 8 Show the trajectories of the state and control input of the closed-loop system respectively. Obviously, as time h goes from 0→∞, its trajectory tends to zero, which means that the designed output feedback controller depending on the mapping mode and scheduling signal can ensure the stability of the system and have H ∞ Performance.

[0202] To sum up, the present invention adopts the WTOD protocol and reasonably allocates limited transmission channel resources through a dynamic scheduling mechanism, significantly alleviating the limitation of insufficient channel quantity on system performance. Regarding the dynamic characteristics of DoS attacks, a method for modeling attack behaviors based on the Markov model is proposed, and combined with a new mapping technology, the switching behaviors of the Markov jump system and DoS attacks are combined, thus more accurately describing the impact of attack behaviors on the system. On this basis, the present invention also proposes an output feedback controller depending on the mapping mode and scheduling signal. By combining the mapping mode with the scheduling signal of the WTOD protocol, precise control of the system state and efficient utilization of resources are achieved. Compared with the prior art, the present invention effectively alleviates the problem of limited network resources through the application of the WTOD protocol. At the same time, through the design of redundant channels and the synergistic effect of the output feedback controller depending on the mapping mode and scheduling signal, the stability and reliability of the system are improved under the influence of random DoS attacks, providing a new solution for the security and performance optimization of networked control systems.

Claims

1. A safety control method for an operational amplifier circuit based on the WTOD protocol, characterized in that: include: S1. Establish a networked Markov jump system based on the state equation of the operational amplifier circuit; S2, introduce the logarithmic quantizer and WTOD protocol; S3. Design redundant channels to prevent random DoS attacks; S4, establishing an output feedback controller that depends on the mapping mode and the scheduling signal; S5. Establish a closed-loop system based on a networked Markov jump system under random DoS attack, a logarithmic quantizer, a WTOD protocol, redundant channels, and an output feedback controller that depends on the mapping mode and the scheduling signal; S6. Generate the stability criterion of the closed-loop system based on the Lyapunov function method; S7. Based on the stability criterion, solve the feedback gain of the output feedback controller.

2. The operational amplifier circuit safety control method according to claim 1, characterized in that: In S1, it is assumed that the operational amplifier circuit is affected by the disturbance ω(h) and C2 is a jump parameter controlled by the Markov chain r(h), that is, C2 = ε r(h) , then the established networked Markov jump system is expressed as follows: x(h+1)=A r(h) x(h)+B r(h) u(h)+D r(h) ω(h), z(h)=E r(h) x(h); Among them, x(h), y α (h), z(h), u(h) and ω(h) represent the system state, measured output, control output, control input and external disturbance respectively; D r(h) and E r(h) is a known system matrix; α∈{1,2}, 1, 2 represent the main channel and redundant channel respectively; r(h) depends on a Markov chain taking values ​​in a finite set S={1,2,…N}, N is a positive integer; R1, R2, R3, R4 are resistors, C1, C2 are capacitors; d is the time step.

3. The operational amplifier circuit safety control method according to claim 2, characterized in that: In S4, mapping technology is used to Mapped to a new random variable ξ(h), the specific mapping process includes: Step 1: Mapping of DoS attack random variables, namely Mapped to variables via Θ(·) 0 means selecting the main channel for transmission, 1 means selecting the redundant channel for transmission, and 2 means that transmission cannot be performed: in, and ⊕ represent the negation and XOR operations respectively; when It indicates a DoS attack. When it is, it means that no DoS attack has occurred; Step 2 Mapping to variable ξ(h) through Ξ(·): Where ξ(h) takes values ​​from the set S = {1,2,…,3N}; Step 3: Create a The relationship between and variable ξ(h), when ξ(h) is given, we get: Where mod(·) represents the modular operation, Indicates rounding down. and ⊕ represent the negation and exclusive-or operations respectively.

4. The operational amplifier circuit safety control method according to claim 3, characterized in that: The output feedback controller that depends on the mapping mode and the scheduling signal is: Among them, K νj is the feedback gain; x(h) is the system state; Δ i ∈[-μ,μ], represents the error range of quantization; Represents the signal received by the controller at time h; ξ(h)=ν,σ1(h),σ2(h)=j, δ(·)∈{0,1} is the Kronecker function.

5. The operational amplifier circuit safety control method according to claim 4, characterized in that: Before the signal is transmitted to the communication network, a logarithmic quantizer q is used. α (·) for the measured output y α (h) quantization is performed, and the quantization output is expressed as: q α (y α (h))=(I+Δ)y α (h),α∈{1,2}, in Δ i ∈[-μ,μ], represents the error range of quantization; represents the sector boundary, represents the quantized density, and 6. The operational amplifier circuit safety control method according to claim 5, characterized in that: According to the WTOD protocol, the actual signal transmitted in the channel is: Where I represents the identity matrix.

7. The operational amplifier circuit safety control method according to claim 6, characterized in that: After introducing redundant channels, the output signal is calculated as follows:

8. The operational amplifier circuit safety control method according to claim 7, characterized in that: In S5, let The resulting closed-loop system is expressed as: in, 9. The operational amplifier circuit safety control method according to claim 8, characterized in that: The process of solving the feedback gain of the output feedback controller in S7 is as follows: When a quantization parameter δ∈(0,1) is given, if there exists a matrix P ν >0, scalar∈>0,γ>0,θ 11 >0,θ 12 >0,θ 21 >0,θ 22 >0, so that the linear matrix inequality Established, that is, the feedback gain K can be obtained νj The value of ; where: β vι is the transition probability of variable ξ(h).

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