Smart city power supply system safety control method for aperiodic DoS attack

Abstract

The invention discloses a smart city power supply system safety control method for aperiodic DoS attacks. The method comprises the following steps: firstly, establishing a state space model of a smart city power supply system based on positive Markov jump system modeling; modeling is carried out on the non-periodic DoS attack, a dynamic event triggering mechanism of the system is designed, and output is quantified; secondly, designing a positive Markov jump system to meet a positive condition under the DoS attack; and finally, designing a positive Markov jump system to meet a random stability condition under DoS attack, verifying the positive Markov jump system, and solving controller gain. According to the invention, the output feedback controller meeting the performance is designed, the safe and stable operation under the DoS attack of the subsystem is ensured, the consumption of communication resources is reduced, and the control benefit of the smart city power supply system is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of automation and modern control, and relates to the security control of the power supply system in a smart city. Specifically, it relates to a security control method for the power supply system in a smart city against non-periodic DoS attacks. Background Art

[0002] In a networked smart city power supply system, the communication network provides important support for realizing remote and flexible control. However, with the continuous increase of sampled data, network resources become increasingly limited, resulting in a decline in system performance. The traditional periodic sampling mechanism needs to collect and transmit data at fixed time intervals, regardless of whether there are significant changes in the system state. This leads to the transmission of a large amount of redundant data, occupying valuable communication bandwidth and increasing network load. For feedback control, the state feedback control method depends on all state information of the system. In practical applications, some state variables may be difficult to directly measure or obtain, resulting in limitations in the engineering implementation of state feedback control.

[0003] With the expansion of the system scale and the improvement of the degree of networking and intelligence, the complexity and vulnerability of the power supply system also increase. Especially in a network communication environment, it is vulnerable to the threat of network attacks, resulting in communication interruptions, data transmission delays, and even control failures, affecting the stability and security of the system. Non-periodic DoS attacks are particularly prominent. Different from periodic DoS attacks, the duration and interval of non-periodic DoS attacks are random, which may lead to serious transmission delays and packet losses, further exacerbating the waste of network resources.

[0004] In addition, the operating environment of the power supply system is complex and changeable, and may randomly switch between different operating modes. This randomness of the mode increases the difficulty of system control. At the same time, the physical quantities in the power supply system have non-negative characteristics, which add positive constraints in actual control, making the design of control strategies more complex.

[0005] In summary, in the case of non-periodic DoS attacks and limited communication resources, studying the security control method for the power supply system in a smart city is still a challenging problem. Summary of the Invention

[0006] Aiming at the deficiencies of the existing technology, the present invention proposes a security control method for the power supply system in a smart city against non-periodic DoS attacks. Using modern control theory technology, a state-space model of a positive Markov jump system under non-periodic DoS attacks is established. A dynamic event-triggering mechanism and a logarithmic quantizer are designed to reduce the consumption of communication resources. And an output feedback controller that satisfies positivity and stochastic stability is designed for the power supply system in a smart city, providing a new solution for the security control of the power supply system in a smart city.

[0007] A security control method for a smart city power supply system to cope with aperiodic DoS attacks, and the specific steps are as follows:

[0008] Step 1: Establish a positive Markov jump system state space model of the smart city power supply system

[0009] In the smart city power supply system, multi-source energy fluctuations, equipment failures, and load changes cause the system to frequently switch operating modes. Traditional deterministic models are difficult to characterize the characteristics of such discontinuous state spaces and random switching rules. The positive Markov jump system predicts the mode switching law through a Markov chain and combines positive constraints such as non-negativity of voltage and power to ensure that the system always remains in the feasible region. Establish the following state space model of a continuous-time positive Markov jump system to describe the smart city power supply system:

[0010]

[0011] where, x(t) ∈ R w represents the state variable of the system, where t is time and w is the dimension of the state variable; u(t) ∈ R m represents the control instruction generated by the controller, which is used to represent the control input of the power supply system, and m is the dimension of the control input; y(t) ∈ R r represents the system output measured by the sensor, and r is the dimension of the output variable. x0 represents the initial state of the power supply system at the initial time t0. {ρ(t), t ≥ t0} represents the discrete Markov process of the multi-modal operation of the power supply system, taking values in the finite set S = {1, 2,... N}, and the state transition probability matrix Π Ψ = {Ψ ab} satisfies the following transition probabilities:

[0012]

[0013] where, ρ(t) = a means that the a-th system working state is activated, a, b ∈ S; Δt > 0 is the mode sojourn time and satisfies Ψ ab > 0 represents the transition probability from mode a to mode b, and N is the number of modes. A ρ(t) , B ρ(t) , C ρ(t) represent known constant matrices with appropriate dimensions in the power supply system.

[0014] Step 2: Establish an aperiodic DoS attack model, a dynamic event-triggering mechanism for the system, and a logarithmic quantizer

[0015] s2.1: Establish an aperiodic DoS attack model:

[0016] In the smart city power supply system, the aperiodic DoS attack signal can affect the communication between the monitoring system, the control center, and distributed devices through devices such as network intrusion detection systems, firewalls, and communication modules. To more intuitively depict the aperiodic DoS attack signal, the aperiodic DoS attack signal suffered by the system is divided into a dormant sub-interval and an active sub-interval in the time series:

[0017]

[0018] Among them, the time intervals and represent the nth DoS attack dormant sub-interval and the DoS attack active sub-interval respectively, where n ∈ Z + ; indicates that the DoS attack is in a dormant state. At this time, the monitoring data of the power supply system can be normally transmitted to the control center, and the control instructions issued by the control center can be normally sent to the execution device. indicates that the DoS attack is in an active state. At this time, the monitoring data of the power supply system cannot be transmitted to the control center, and the operating state of the system cannot be adjusted in time, which may lead to problems such as voltage fluctuations and frequency deviations.

[0019] Suppose that within the duration of the nth DoS attack dormancy, the shortest time u min required for the power supply system to restore communication and adjust its operating state, and within the duration of the nth DoS attack activity, the maximum time v max for the power supply system to stably operate under communication interruption, then:

[0020]

[0021] Among them, sup{} represents the supremum in the function values, and inf{} represents the infimum in the function values.

[0022] To ensure that within any time interval (t1, t2), the active and dormant switching times F(t1, t2) of the DoS attack are not too frequent, considering the basic frequency κ > 0 of the DoS attack switching and the time scale τ f > 0, there is the following relationship:

[0023]

[0024] s2.2. Establish a dynamic event trigger mechanism:

[0025] In the smart city power supply system, real-time monitoring and control are crucial for ensuring the stable operation of the power system. However, the operating environment of the power network is complex and variable, and the load demand, renewable energy generation, and network topology may change at any time. For example, load fluctuations, intermittent power generation from distributed energy sources, and sudden faults will all cause the state of the power supply system to change continuously. In addition, sensors, communication modules, and control devices in the power supply system need to continuously monitor and adjust the system state, thus consuming a large amount of communication resources and energy. When the power supply system changes rapidly, traditional periodic time-triggered strategies cannot flexibly adjust the data transmission frequency according to preset trigger conditions. Therefore, a dynamic event-triggered mechanism based on periodic sampling is designed to avoid waste of communication resources:

[0026]

[0027] where h represents the sampling period of the sensor. Considering that when the system is under a DoS attack, data transmission is blocked and event triggering will not occur, during the nth DoS attack dormancy period and at the end of the attack, the set of times of the dth event trigger and represent the output signal received by the controller and the output signal sent by the trigger module respectively, ζ(t) is an internal dynamic variable used to dynamically adjust the signal sampling interval to ensure timely capture of key information when the system state changes significantly. δ > 0 is used to adjust the weight of the internal dynamic variable ζ(t). σ ρ(t) ∈[0,1) represents the event trigger threshold; Ω ρ(t) is the dynamic event trigger vector to be solved. The superscript T represents transpose, and min{} represents taking the minimum value.

[0028] In the smart city power supply system, since all operating state variables of the power network are positive, considering the positivity of the system, a constant is introduced to define the following error equation:

[0029]

[0030] Due to the existence of sampling intervals and network delays and satisfying where h0 represents the upper bound of the change rate of the network delay. Denote the interval between two data releases of the event trigger generator as Adopt the time-delay analysis method to represent the output signal Convert the dynamic event-triggered mechanism into:

[0031]

[0032] The internal dynamic variable ζ(t) satisfies:

[0033]

[0034] Among them, ζ > 0 is the decay coefficient of the internal dynamic variable ζ(t).

[0035] S2.3. Establish a logarithmic quantizer to quantize the output signal:

[0036] In the smart city power supply system, the data collected by sensors needs to be transmitted to the control center through a communication network. The data acquisition and processing unit is usually used in conjunction with sensors to monitor the operating state of the power network in real time. However, the bandwidth and energy resources of the communication network are limited. Directly transmitting high-precision continuous signals may lead to excessive communication burden and high energy consumption. To solve this problem, a logarithmic quantizer is introduced to convert the output signal released by event-triggered sampling into a discrete quantized output signal before being transmitted to the actuator thus reducing the data transmission volume:

[0037]

[0038] The quantized output is obtained through the sector bound method

[0039]

[0040] where Δ q represents the quantization error, and max{} represents taking the maximum value. I represents the identity matrix with appropriate dimensions.

[0041] S2.4. Based on S2.1 - S2.3, design the output feedback controller u(t) of the positive Markov system under non-periodic DoS attacks:

[0042]

[0043] where, K ρ(t) is the gain of the output feedback controller of the power supply system within the DoS attack dormant interval when the trigger condition is met. When the DoS attack is in the active interval, the attack signal completely blocks the signal transmission, and the controller does not work at this time. Based on this, a new closed-loop system can be obtained:

[0044]

[0045] where, represents the first derivative of x(t). K a ∈R m×r and satisfies and respectively represent the sets of non - negative real matrices and non - positive real matrices of size \(m\times r\).

[0046] Step 3: Construct the condition for the power supply system to be positive:

[0047] When a DoS attack occurs, Design \(A\) a is a Metzler matrix, and \(C\) a \(\geq0\). At this time, the open - loop system is a positive system. When a DoS attack does not occur, Design \(A\) a is a Metzler matrix, \(C\) a \(\geq0\). At this time, the closed - loop system is a positive system.

[0048] Therefore, design the matrix \(A\) a to be a Metzler matrix and ensure that the matrix \(C\) a \(\geq0\), which is the condition for the power supply system to be positive.

[0049] Step 4: Solve the output feedback controller gain:

[0050] When \(h > 0\), \(0\leq h_0<1\), \(\xi>0\), \(\delta>0\), \(0\leq\sigma\) a <1\), \(u\) min \(\geq h\), \(v\) max \(\geq h\), \(k\geq0\), \(\tau\) f \(>0\), \(v\) s \(>1\), \(\mu\) s \(\geq0\), then, if there exist constants \(\chi>0\), \(\lambda>0\) and a vector satisfying:

[0051] \(A\) a +\(\chi I\geq0\), \(C\) a \(\geq0\)

[0052]

[0053]

[0054] then the closed - loop system in s2.4 satisfies positivity and stochastic stability under the aperiodic DoS attack. Therefore, the output feedback controller gain satisfies:

[0055]

[0056] where, is the controller gain vector; \(1\)m Denote an \(m\)-dimensional column vector with all elements being 1; Denote an \(m\)-dimensional column vector with the \(g\)-th row element being 1 and the remaining elements being 0.

[0057] The present invention has the following beneficial effects:

[0058] (1) Aiming at the multi-modal operation characteristics of the power supply system, the present invention introduces a positive Markov jump system for modeling, uses a Markov chain to describe the random switching of system modes, and ensures the non-negativity of state variables through positive system theory. On this basis, a new discrete co-positive Lyapunov function is designed. This mode-dependent linear piecewise Lyapunov function, combined with a dynamic event-triggering mechanism and quantization error constraints, can ensure the robustness of the positive Markov jump system during DoS attacks.

[0059] (2) Different from traditional periodic sampling, the dynamic event-triggering mechanism based on periodic sampling proposed by the present invention can not only effectively avoid Zeno behavior, but also introduce an internal dynamic variable on the basis of the static event-triggering mechanism, and further impose a positivity constraint on the output error, which can meet the positivity requirements of the positive Markov jump system while reducing the sampling frequency.

[0060] (3) Aiming at the randomness of aperiodic DoS attacks, the present invention further reduces the consumption of communication resources through the combination of a dynamic event-triggering mechanism and quantization technology. At the same time, a trigger condition is constructed based on the output quantity and the output is quantized and analyzed. This output feedback design method avoids the problems of unmeasurable states or missing partial state information.

[0061] (4) The present invention proposes a new output feedback controller for the proposed positive Markov closed-loop system and factorizes the gain matrix, representing the stability condition in LP form, which reduces the computational cost and complexity compared with the widely used LMI format. Description of the Drawings

[0062] Figure 1 It is the structure diagram of the smart city power supply system.

[0063] Figure 2 It is the model diagram of the smart city power supply system.

[0064] Figure 3 It is the curve diagram of system mode jumps

[0065] Figure 4 It is the system state under aperiodic DoS attacks

[0066] Figure 5 It is the quantization curve of the system output \(y_1(t)\)

[0067] Figure 6 For the quantization curve of the system output y2(t)

[0068] Figure 7 For the release time and trigger interval triggered by dynamic events Specific implementation manner

[0069] The present invention will be further explained below with reference to the accompanying drawings;

[0070] For the security control method of the smart city power supply system against non-periodic DoS attacks, a positive Markov jump system with dynamic event triggering and quantization is introduced into the smart city power supply system to perform security control on the smart city power supply system. The specific steps are as follows:

[0071] Step 1: Establish the state space model of the positive Markov jump system of the smart city power supply system

[0072] Establish the smart city power supply system as shown in Figure 1 The power supply system samples the output state y(t) of the sensor sampling system and sends the relevant data of the obtained output state into the sampler to generate a discrete signal y(ih); the discrete signal is screened by the dynamic event trigger generator to generate an output sampling signal and is input into the logarithmic quantizer for quantization processing; the logarithmic quantizer generates an output quantization signal Considering that the process of taking this signal as the input of the controller to the controller may be affected by non-periodic DoS attacks, resulting in some signals not being able to be sent to the controller; then the output feedback controller inputs the obtained control quantity into the actuator, and finally feeds the data back to the original system and repeats the above process. For this smart city power supply system, establish the state space model of the continuous-time positive Markov jump system:

[0073]

[0074] where, x(t) ∈ R w represents the state variable of the system, t is time, w is the dimension of the state variable, including bus voltage, line current and load power; u(t) ∈ R m represents the control command generated by the controller, used to represent the control input of the power supply system, and m is the dimension of the control input; y(t) ∈ R r represents the system output measured by the sensor, r is the dimension of the output variable; x0 represents the initial state of the power supply system at the initial time t0; {ρ(t), t ≥ t0} represents the Markov process of discrete modes, taking values in the finite set S = {1, 2,..., N}, N is the number of modes, and has the following transition probabilities:

[0075]

[0076] where \(a,b\in S\); the modal sojourn time \(\Delta t>0\), \(\Psi\) ab >0 represents the transition probability from mode \(a\) to mode \(b\), and \(A\) ρ(t) , \(B\) ρ(t) , \(C\) ρ(t) , are constant matrices with appropriate dimensions. When \(\rho(t)=a\), it means the \(a\)-th mode is activated.

[0077] Step 2: Establish an aperiodic DoS attack model, a dynamic event-triggering mechanism for the system, and a logarithmic quantizer

[0078] As Figure 2 shown, the smart city power supply system is mainly centrally powered by power plants. As the key nodes of the power supply network, sensors and actuators are required to send status information or control commands to each regional substation, and then the substation conveys the acquired valid status information to each user-side device, thus realizing functions such as demand response. Due to the complex and changeable operating environment of the power supply system, the power plant, as the signal transmitter, may switch between different operating modes, and its corresponding subsystems correspond to the links from the power plant to each user-side device, indicating that the positive Markov jump system established in Step 1 is very suitable for describing the smart city power supply system. To avoid wasting resources during information transmission, a dynamic event-triggering mechanism and quantization means need to be adopted at a certain link to limit the signal transmission speed of this channel; and in the networked environment of information transmission, it may also be subject to aperiodic DoS attack signals, and thus a suitable controller needs to be designed for regulation to achieve secure control.

[0079] s2.1: Establish an aperiodic DoS attack model:

[0080] In the power supply system, DoS attacks will affect the communication between the monitoring system, the control center, and distributed devices. The aperiodic DoS attack signals suffered by the system are divided into a dormant sub-interval and an active sub-interval in the time series:

[0081]

[0082] where the time intervals and represent the \(n\)-th DoS attack dormant sub-interval and the DoS attack active sub-interval respectively, \(n\in Z\) + ; indicates that the DoS attack is in a dormant state. At this time, the monitoring data of the power supply system can be normally transmitted to the control center, so that the control instructions can be normally issued to the execution devices; Indicates that the DoS attack is active. At this time, the monitoring data of the power supply system cannot be transmitted to the control center, and the operating state of the system cannot be adjusted in a timely manner, resulting in problems such as voltage fluctuations and frequency deviations.

[0083] Suppose the duration of the nth DoS attack dormancy Within, the shortest time required for the power supply system to resume communication and adjust its operating state is u min , during the duration of the nth DoS attack being active Within, the maximum time for the stable operation of the power supply system under communication interruption is v max , then:

[0084]

[0085] Among them, sup{} represents the supremum in the function values, and inf{} represents the infimum in the function values.

[0086] To ensure that the active and dormant switching times F(t1,t2) of the DoS attack within any time period (t1,t2) are not too frequent, considering the basic frequency κ>0 of the DoS attack switching and the time scale τ f >0, there is the following relationship:

[0087]

[0088] S2.2. Establish a dynamic event-triggering mechanism:

[0089] Establish the following dynamic event-triggering mechanism based on periodic sampling:

[0090]

[0091] Among them, h represents the sampling period of the sensor. Considering that when the system is under a DoS attack, data transmission is blocked and no event triggering will occur. During the nth DoS attack dormancy period and at the end of the attack, the set of times of the dth event trigger is expressed as And Represent the output signals received by the controller and the output signals sent by the trigger module respectively, The internal dynamic variable ζ(t) is used to dynamically adjust the signal sampling interval to ensure that key information is captured in a timely manner when the system state changes significantly. δ>0 is used to adjust the weight of ζ(t); σ ρ(t) ∈[0,1) represents the event-triggering threshold; Ω ρ(t) Is the dynamic event-triggering vector to be solved. The superscript T represents the transpose, and min{} represents finding the minimum value.

[0092] Considering the network delay existing between the triggering module and the controller for the sampling signal, denote the moment when it reaches the controller as Then for adjacent sampling signals and the time interval between their arrivals at the controller is Therefore, in each event-triggering interval, there exists:

[0093]

[0094] wherein, Therefore, we can divide the event-triggering interval into

[0095]

[0096] The time-delay function is:

[0097]

[0098] Satisfying wherein represents the first derivative of, and h0 represents the upper bound of the change rate of the network delay.

[0099] Since the operating state variables of the power network are all positive, considering the positivity of the system, introduce the constant Define the error equation as follows:

[0100]

[0101] Therefore, the event-triggered output sampling signal is:

[0102]

[0103] Meanwhile, the dynamic event-triggering mechanism can be transformed into:

[0104]

[0105] The internal dynamic variable ζ(t) satisfies:

[0106]

[0107] wherein, ξ>0 is the decay coefficient of the dynamic variable ζ(t).

[0108] s2.3. Establish a logarithmic quantizer to quantize the output signal:

[0109] The output signal released by the event-triggered sampling is first converted into a quantized output signal by the quantizer before being transmitted to the actuator

[0110]

[0111] Define the quantization rule of the \(i\)-th sub-quantizer as follows:

[0112]

[0113] where \(i = 1, 2, \cdots, r\). According to the quantization rule, map the input signal to the nearest quantization level The set of quantization levels is:

[0114]

[0115] where and represent the initial quantization parameter and quantization density of the \(i\)-th sub-quantizer respectively. For any quantization error \(\Delta\) qi \(= q\) i (y i (t d h)) - y i (t d h) and \(|\Delta\) qi | \(\leq \epsilon\) i , define \(\epsilon=\max\{\epsilon_1,\epsilon_2,\cdots,\epsilon\) r \}, so there is \((1 - \epsilon)I \leq I+\Delta\) q \(\leq (1 + \epsilon)I\), where \(I\) represents the identity matrix with appropriate dimensions.

[0116] Obtain the quantization output through the sector bound method

[0117]

[0118] where \(\Delta\) q \(= \text{diag}\{\Delta\) q1 ,\Delta\) q2 ,\cdots,\Delta\) qr \}, and \(\max\{\}\) represents taking the maximum value.

[0119] s2.4. On the basis of s2.1, s2.2 and s2.3, design the output feedback controller \(u(t)\) of the positive Markov system under the non-periodic DoS attack as follows:

[0120]

[0121] where \(K\) ρ(t)The gain of the output feedback controller of the power supply system within the DoS attack dormancy interval to meet the triggering conditions. When the DoS attack is in the active interval, the attack signal completely blocks the signal transmission, and the controller does not work at this time. Based on this, a new closed-loop system can be obtained:

[0122]

[0123] where, K a ∈R m×r , and satisfies

[0124] Step 3: Construct the condition for the power supply system to be positive:

[0125] s3.1. Design the positivity condition of the power supply system when a DoS attack occurs:

[0126] When That is At this time, the DoS attack is in the active sub-interval, and the closed-loop system is transformed into an open-loop system:

[0127]

[0128] When A a is a Metzler matrix, and C a ≥0, the open-loop system is a positive system.

[0129] s3.2. Design the positivity condition of the power supply system when a DoS attack does not occur:

[0130] When That is At this time, the DoS attack is in the dormancy sub-interval, and the closed-loop system is:

[0131]

[0132] Since (I + Δ q ) ≥ (1 - ∈)I ≥ 0, so when A a is a Metzler matrix, C a ≥0, the closed-loop system is a positive system. In addition, for the convenience of subsequent writing, and are rewritten as τ d (t) and e d (t).

[0133] s3.3. The conditions for the system to satisfy positivity:

[0134] Design the matrix Aa is a Metzler matrix and ensure that the matrix C a ≥0,

[0135] Step 4. Solve the output feedback controller gain:

[0136] s4.1. Design the stochastic stability condition of the closed-loop system under the aperiodic DoS attack in s2.4: When h > 0, 0 ≤ h0 < 1, ξ > 0, δ > 0, 0 ≤ σ a < 1, u min ≥ h, v max ≥ h, κ ≥ 0, τ f > 0, v s > 1, μ s ≥ 0, if there exist constants χ > 0, λ > 0 and vectors

[0137] satisfy:

[0138] A a + χI ≥ 0, C a ≥ 0

[0139]

[0140] the output feedback controller gain satisfies:

[0141]

[0142] s4.2. The verification of the closed-loop positive Markov jump system satisfying the stochastic stability condition under the aperiodic DoS attack is as follows:

[0143] First, verify the positivity of the system based on the existing positivity conditions. It should be noted that can ensure From the conditions in s4.1, we can obtain:

[0144]

[0145] Substitute it into the output feedback controller gain, and we can obtain From A a + χI ≥ 0, it can be concluded that A a is a Metzler matrix. Also, C a ≥ 0, then according to s3.3, the closed-loop system satisfies positivity.

[0146] Since the power supply system of the smart city under discussion has suffered an aperiodic DoS attack, according to whether the DoS attack occurs or not, two forms of mode-dependent linear co-positive Lyapunov functions will be designed below to discuss the stochastic stability of the system.

[0147] First, design a Lyapunov function in the following form:

[0148]

[0149] Where:

[0150]

[0151] When the DoS attack is in the dormant interval, Taking the derivative of the Lyapunov function with respect to time t, the following relationship exists:

[0152]

[0153] And according to the output feedback controller gain matrix and the preset conditions in S4.1, the following inequality holds:

[0154]

[0155] Through the definition of the logarithmic quantizer in S2.3, the following inequality can be obtained:

[0156]

[0157] Furthermore, substituting the output feedback controller gain into the above inequality about the Lyapunov function can be written in the following form:

[0158]

[0159] Combined with the conditions in S4.1, it can be obtained that:

[0160]

[0161] Therefore, it can be obtained that:

[0162]

[0163] Similarly, when the DoS attack is in the active interval, Taking the derivative of the Lyapunov function with respect to time t, the following relationship exists:

[0164]

[0165] Combined with the conditions in S4.1, we can obtain:

[0166]

[0167] Therefore, we can get:

[0168]

[0169] In summary, we can get:

[0170]

[0171] Combined with the Lyapunov function and 's definition and the conditions in S4.1, we can obtain the relationship between and at time and :

[0172]

[0173] For and two cases, analyze the relationship between the expected value of the Lyapunov function at the initial time and the expected value of the Lyapunov function Ev ρ(t) (t) at time t:

[0174] When , combined with the two inequality relationships obtained above and the constraint conditions of the DoS attack for scaling, we can get:

[0175]

[0176] Then, combined with the constraint conditions of the DoS attack in S2.1 and the conditions in S4.1, we can further obtain:

[0177]

[0178] Similarly, when :

[0179]

[0180] Combined with the constraint conditions of the DoS attack in S2.1 and the conditions in S4.1, we can get:

[0181]

[0182] Define ω = κ(ln(ν1ν2)+(μ1 + μ2)h - u min μ1 + v max μ2), can be constructed uniformly and EV ρ(t) (t)'s relationship:

[0183]

[0184] wherein it is defined λ(x) and respectively represent the maximum element and the minimum element in the vector x. Then, through the constructed Lyapunov function V ρ(t) (t)'s definition, the following inequality can be obtained:

[0185] EV ρ(t) (t) ≥ θ1E||x(t)||1

[0186]

[0187] Substitute it into and EV ρ(t) (t)'s relationship inequality to obtain:

[0188]

[0189] Through the dynamic event-triggering mechanism, it can be obtained Also, because π > 0, λ > 0, and θ1 > 0, so when t → +∞, there is:

[0190]

[0191] Integrate the above formula from t = 0 to +∞, and then there is:

[0192]

[0193] To sum up, the positive Markov jump system satisfies stochastic stability under the non-periodic DoS attack.

[0194] s4.4. To verify the effectiveness of this method, the output feedback controller designed by the present invention will be simulated and verified using MATLAB software below. Set the system matrices:

[0195]

[0196] where the system matrix A a reflects the dynamic characteristics of the power supply system in the current working mode, the input matrix B a represents the influence of control instructions such as generator output adjustment and reactive power compensation instructions on the state of the power supply system, and the output matrix C a represents the influence of state variables on the bus voltage and line current. At the same time, set the state transition probability matrix of the positive Markov jump system as:

[0197]

[0198] Set the parameters of the dynamic event triggering mechanism as σ1 = σ2 = 0.3, ξ = 1.5, δ = 0.4, The quantization densities ε1 = ε2 = 0.3, and other parameters μ1 = 1.1, μ2 = 1.2, ν1 = 2, ν2 = 3, u min = 1.6, v max = 1, the sampling period h = 0.05, h0 = 0.1. Solving the conditions in s4.1 through the feasp solver gives the controller vector:

[0199]

[0200] The dynamic event triggering matrix is:

[0201] From this, the controller gain matrix can be obtained:

[0202]

[0203] Set the initial state variable of the positive Markov jump system as x(t) = [5, 3] T , and through MATLAB simulation, the mode jump curve of the system can be obtained as Figure 3 shown; the system state image under the aperiodic DoS attack is as Figure 4 shown. It can be seen from Figure 4 that the system can remain stable and satisfy positivity under the designed output feedback controller; the quantization curves of the system outputs y1(t) and y2(t) are respectively as Figure 5 and Figure 6 shown; the release time and trigger interval curves of the dynamic event triggering under the output feedback controller are as Figure 7 shown. It can be seen that the system is triggered only 22 times within a finite time, which significantly reduces the sampling frequency and effectively saves network bandwidth. Therefore, the output feedback controller designed for the positive Markov jump system in the present invention is effective.

Claims

1. A smart city power supply system security control method for non-periodic DoS attacks, characterized by: The specific steps include: Step 1: Establish a positive Markov jump system state space model of the smart city power supply system; Step 2: Send the non-periodic DoS attack signal suffered by the system The time series is divided into dormant subintervals and active subintervals, and a non-periodic DoS attack model is established; internal dynamic variables are introduced, a dynamic event trigger mechanism based on periodic sampling is designed, and a logarithmic quantizer is established to quantize the output signal; the output feedback controller u(t) of the positive Markov system under non-periodic DoS attack is designed; Step 3: Construct the conditions for the positive Markov jump system state space model of the power supply system to be positive under DoS attack; Step 4: Express the stability condition in LP form, design the modal-dependent linear piecewise Lyapunov function, and solve the dynamic event trigger vector and output feedback controller gain.

2. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 1, characterized in that: The state space model of the positive Markov jump system of the smart city power supply system is: Where t represents time, x(t) represents the state variable of the system, u(t) represents the control input of the power supply system, and y(t) represents the system output measured by the sensor; x0 represents the initial state of the power supply system at the initial time t0; {ρ(t), t≥t0} represents the discrete Markov process of the multimodal operation of the power supply system, which takes values ​​in the finite set S = {1, 2, ... N}, and the state transition probability matrix Π Ψ ={Ψ ab }Satisfies the following transition probability: Where N is the number of modes; ρ(t) = a means that the a-th system working state is activated, a,b∈S; Δt>0 is the mode residence time and satisfies Δt>0, Ψ ab >0 indicates the transition probability from mode a to mode b, and A ρ(t) , B ρ(t) , C ρ(t) Represents the known constant matrix in the power supply system.

3. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 1, characterized in that: The state variables of the system include bus voltage, line current and load power.

4. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 1, characterized in that: The non-periodic DoS attack model is: Among them, the time interval and They represent the nth DoS attack dormant subinterval and DoS attack active subinterval, n∈Z + ; Indicates that the DoS attack is dormant. Indicates that the DoS attack is active; Assume that the duration of sleep during the nth DoS attack is The shortest time required for the power supply system to restore communication and adjust the operating status is u min , the duration of the active DoS attack in the nth time The maximum stable operation time of the power supply system under communication interruption is v max ,but: Among them, sup{} represents the supremum of the function value, and inf{} represents the infimum of the function value; Set the basic frequency of DoS attack switching κ>0 and the time scale τ that reflects the regularity of attack time f >0The following relationship exists: F(t1, t2) represents the number of active and dormant switches of the DoS attack at any time (t1, t2).

5. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 2, characterized in that: Introducing the internal dynamic variable ζ(t), the following dynamic event triggering mechanism based on periodic sampling is designed: Where h represents the sampling period of the sensor; considering that when the system is attacked by DoS, data transmission is blocked and no event triggering occurs, the time set of the dth event triggering during the nth DoS attack dormancy period and the attack end time is and They represent the output signal received by the controller and the output signal sent by the trigger module. δ>0, used to adjust the weight of the internal dynamic variable ζ(t); σ ρ(t) ∈[0,1), indicating the event triggering threshold; Ω ρ(t) is the dynamic event trigger vector to be found; the superscript T indicates transposition, and min{} indicates finding the minimum value; Introducing constants Define the following error equation: Consider sampling interval and network delay Where h0 represents the upper bound of the rate of change of network delay, satisfy The interval between two releases of data by the event trigger generator is recorded as Use time delay analysis method to represent the output signal Convert the dynamic event trigger mechanism to: The internal dynamic variable ζ(t) satisfies: Among them, ξ>0 is the attenuation coefficient of the internal dynamic variable ζ(t); The logarithmic quantizer will trigger the sampling of the output signal released by the event Quantized to Where r represents the number of sub-logarithm quantizers; the quantized output is obtained by the sector bound method where Δ q represents the quantization error, max{} represents the maximum value, and I represents the unit matrix.

6. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 5, characterized in that: The output feedback controller u(t) of the positive Markov system under the non-periodic DoS attack is: Among them, K ρ(t) The output feedback controller gain of the power supply system in the dormant period of the DoS attack when the trigger condition is met; They represent the nth DoS attack dormant sub-interval and the DoS attack active sub-interval respectively.

7. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 6, characterized in that: Design Matrix A a is a Metzler matrix, and ensure that the matrix That is the condition for the power supply system to be positive.

8. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 6, characterized in that: The Lyapunov function of the modal-dependent linear segmentation is: in:

9. The smart city power supply system security control method for non-periodic DoS attacks as claimed in claim 8, characterized in that: Design the random stability conditions of the closed-loop system under non-periodic DoS attacks and solve the output feedback controller gain K a : in, is the controller gain vector; 1 m represents an m-dimensional column vector whose elements are all 1; represents an m-dimensional column vector with the g-th row element being 1 and the rest being 0; m is the dimension of the control input u(t).

10. A computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to execute the method according to any one of claims 1 to 9.

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