Power system random sampling control method based on attack compensation

Abstract

The invention discloses a power system random sampling control method based on attack compensation. The method comprises the following steps: constructing an adaptive random switching model, establishing a lightweight DoS attack detection and event trigger compensation mechanism, and optimizing a PD controller fusing compensation data, the method combines the random switching model, a network security defense mechanism and an optimization control strategy, and aims at a dynamic multi-mode power system containing an electric vehicle EV. The technical bottleneck of traditional load frequency control in the problems of random disturbance, network attack and mode switching dependency is solved, the method is suitable for improving the frequency stability and the anti-attack ability of the power system, and the anti-interference ability and the operation efficiency of the power system can be improved.

Description

Technical Field

[0001] The present invention relates to the automatic control technology of power systems, and particularly to a random sampling switching power system anti-Denial of Service (DoS) attack compensation load frequency control technology based on a proportional derivative (PD) controller, specifically a random sampling control method for power systems based on attack compensation. Background Art

[0002] Power system load frequency control (LFC) is a core technology for maintaining the stability of the power grid frequency. Traditional methods usually based on fixed sampling periods and Markov jump system models have the following problems:

[0003] 1) Insufficient anti-disturbance ability: Signal transmission with random sampling intervals is vulnerable to noise and external interference, resulting in a decrease in frequency regulation accuracy;

[0004] 2) Vulnerability to network attacks: DoS attacks cause data loss by blocking communication channels. Traditional compensation mechanisms are inefficient, and error accumulation leads to system instability;

[0005] 3) Strong model dependence: Multi-mode switching requires accurate modeling of Markov transition probabilities, but it is difficult to accurately obtain transition probabilities in complex dynamic environments, limiting the practicality of the model;

[0006] 4) Slow response speed: Conventional state feedback controllers have a long adjustment time and are difficult to quickly suppress frequency fluctuations caused by attacks.

[0007] Although existing technology improvement schemes introduce event-triggered mechanisms or reinforcement learning strategies, there are still problems such as high model complexity, poor compensation real-time performance, and unoptimized control gains. For example, Markov models require prior transition probabilities and are difficult to adapt to random attack scenarios; traditional integral controllers have insufficient ability to fuse compensation data, resulting in an extended adjustment time. Summary of the Invention

[0008] The object of the present invention is to provide a random sampling control method for power systems based on attack compensation in view of the deficiencies of the prior art. This method combines a random switching model, a network security defense mechanism, and an optimized control strategy, and aims to solve the technical bottlenecks of traditional load frequency control in random disturbances, network attacks, and mode switching dependence problems for a multi-mode power system containing electric vehicle (EV) dynamics. It is applicable to improving the frequency stability and anti-attack ability of power systems and can enhance the anti-interference ability and operating efficiency of power systems.

[0009] The technical solution for achieving the object of the present invention is as follows:

[0010] A random sampling control method for power systems based on attack compensation, comprising the following steps:

[0011] 1) Adaptive random switching model construction: Construct a multi-mode switching power system model that includes the dynamics of electric vehicle EVs, characterize the mode jumps caused by load fluctuations and attacks. The load frequency regulation system of the multi-mode power system involves loads, generators, prime movers, speed governors, and the LFC loop part. The multi-mode switching power system is modeled as shown in Equation (1):

[0012]

[0013] where f(t) is the frequency deviation, M(φ(t)) and D(φ(t)) are the inertia constant and the load damping coefficient respectively; P e (t), P g (t), P d (t) are the increments of the output power of electric vehicles, steam turbines, and the load disturbance respectively; X g (t), T g (φ(t)), α g (φ(t)), R g (φ(t)), u(t) are the governor valve position, governor speed, participation factor of the steam turbine, governor droop characteristic, and control input respectively; P g (t), T t (φ(t)) are the output power of the steam turbine and the steam turbine time constant respectively; P e (t), T e (φ(t)), ρ e (φ(t)), α e (φ(t)), are the increment, time constant, electric vehicle droop characteristic, electric vehicle participation factor, and electric vehicle gain in the electric vehicle respectively;

[0014] Let x T (t)=[f(t) X g (t) P g (t) P e (t) △(t)], △(t)=∫ACEdt, where ACE is the area control error and ACE = bf(t), y T (t)=[f(t) △(t)] is the output vector, w(t)=P d (t) is the system disturbance and satisfies w T (t) w(t) < ρ, ρ is a finite positive real number, then the following state space representation form of the power system can be obtained:

[0015]

[0016] Matrices A, B, C, and B w are system matrices, and these matrices satisfy the following values:

[0017]

[0018] In traditional research, power systems are usually modeled as Markov jump systems, and their stability depends on accurate transition probability parameters. However, the complex and dynamic characteristics of the power system environment mean that mode switching is often affected by random interference and external attacks. Therefore, it is difficult to accurately obtain the transition probability through prior knowledge. This limitation restricts the practicality of traditional models. Therefore, a switching rule based on dwell time (DT) and dwell probability (DP) is proposed;

[0019] 2) Establish a multimodal switching electric vehicle power system based on modal dwell time and dwell probability: The process includes:

[0020] Assume that there are m types of switching mode signals φ(t), and the parameter φ(t) is associated with the random variable where t l represents the l-th transition time from mode to mode , and t 0 =0. Here, is the mode reached after the l-th transition, and is the dwell time in mode , satisfying The function represents the probability distribution function. For t≥0, where φ(t) takes values in the set is governed by a Markov process. For any and the distribution function depends on the dwell time relative to the specified mode and is derived as follows:

[0021]

[0022] where θ j represents the average dwell probability of the j-th submode, and the probability distribution function is:

[0023]

[0024] 3) Random sampling: To reduce the pressure on network signal transmission and improve the stability of the system, random sampling of the power system state x(t) is considered before transmitting the data. Random sampling means that the sampling period h is no longer a fixed positive number but belongs to a sampling set. Assume the sampling instances of the system are:

[0025] t 0 =0 < t 1 < t 2 <… < t k <…,

[0026] This means that the transmitted data control signal will be updated at instant t k and then the zero-level hold control signal will be maintained for a period of time until the control signal data is updated. The sampling interval is defined as: Then s k ∈P = {p 1 , p 2 , …, p M}, where p 1 < p 2 < … < p M . They are M different positive numbers representing possible sampling time intervals, from which the following probability distribution can be obtained:

[0027] Prob{s k = p i} = α i , i = 1, 2, …, M, (5),

[0028] where α i ∈[0, 1], and satisfies

[0029] 4) DoS attack monitoring algorithm and data compensation mechanism: Network attacks may occur in the power grid, leading to system instability. DoS attack is one of the most common and destructive attacks. Therefore, an energy-constrained DoS attack is considered here, that is, the frequency and duration of the attack are both limited. To reduce the harm caused by DoS attacks, a DoS attack monitoring algorithm and data compensation mechanism are considered to maintain the stability of the system and reduce information loss. Specifically:

[0030]

[0031]

[0032] Once the initial compensation time for each period is established, the following event-triggered compensation mechanism can be adopted to determine the order of compensation time:

[0033]

[0034] where, e(l k ) = x(t m,n ) - x(l k );

[0035] 5) Design of a PD-like controller: A general proportional-integral controller is expressed as shown in formula (7):

[0036]

[0037] where \(K = [K 1 ,K 2 \). Considering the multi-modal characteristics of the system, the design of the PD controller based on the data compensation mechanism is shown in Equation (8):

[0038]

[0039] where is the interval where no attack is detected, while is the interval where an attack is detected. Therefore, the power system can be reconstructed as shown in Equation (9):

[0040]

[0041] 6) Solve the proportional-integral controller gains: Based on Lyapunov theory, sufficient conditions for the asymptotic stability of the system are given and the \(H ∞ \) performance is satisfied. Use the LMI toolbox of MTALAB to solve the proportional-integral controller gains. The process is as follows:

[0042] Based on Lyapunov theory, the sufficient conditions to ensure the asymptotic stability of the system and satisfy the \(H ∞ \) performance are:

[0043] For given compensation parameter \(\rho j > 0\) and scalar \(b 1 ,b 2 ,\) in the case where the system shown in Equation (9) has a symmetric matrix \(Z 1 ,Z 2 ,\) a positive definite matrix \(\Theta j ,\) and matrices \(K j1 ,K j2 ,K jd1 ,K jd2 ,\) asymptotic stability can be achieved and the \(H ∞ \) performance index \(\gamma\) is obtained if the inequalities and

[0044] where and is the probability density function. Use linear matrix inequalities to solve the PD integral observer gains and controller gains:

[0045] For given compensation parameter \(\rho j > 0\) and scalar \(b 1 ,b 2 ,\) in the case where the system shown in Equation (9) has a symmetric matrix \(Z 1 ,Z 2 ,\) a positive definite matrix Θ j , and matrix Y j1 , Y j2 , Y jd1 and Y jd2 In the case of, asymptotic stability can be achieved and it has H ∞ performance index γ. If the matrix inequality and where

[0046]

[0047] control gain matrix K j1 can be designed as Other gain matrices can be obtained in the same way.

[0048] This technical solution aims to solve the frequency instability problem of traditional power systems under scenarios of random disturbances, cyberattacks, and multi-mode switching. This technical solution is based on an adaptive random switching model of dwell time (DT) and dwell probability (DP). By dynamically generating switching parameters, it reduces the modeling complexity; designs a lightweight DoS attack detection algorithm and an event-triggered compensation mechanism to reconstruct lost signals in real time to reduce errors; combines linear matrix inequality (LMI) to optimize the PD controller gain, significantly improving the response speed and anti-disturbance performance. Through the synergistic effects of dynamic modeling, attack detection and compensation, and control optimization, this technical solution can significantly enhance the anti-interference ability and operating efficiency of the power system. Compared with the existing technologies, the advantages of this technical solution are as follows:

[0049] 1) The adaptive random switching model significantly reduces complexity: By constructing a dynamic switching rule based on dwell time (DT) and dwell probability (DP), and using real-time operation data statistics to generate switching parameters, without relying on the exact transition probabilities of traditional Markov models, it effectively reduces the model construction complexity and improves the system's adaptability to random disturbances and cyberattacks at the same time;

[0050] 2) Lightweight DoS attack detection and efficient compensation mechanism: The lightweight DoS attack detection algorithm (Algorithm 1) and the event-triggered compensation mechanism (Algorithm 2) can monitor attacks in real time and reconstruct lost signals;

[0051] 3) The PD controller integrates LMI optimization to achieve fast response: The PD controller that integrates compensation data and state differential information, namely formula (8), combines linear matrix inequality (LMI) theory to optimize the control gain.

[0052] Compared with the existing technologies, this technical solution has significant advantages in reducing complexity, enhancing network security, and shortening the adjustment time.

[0053] This method combines a random switching model, a network security defense mechanism, and an optimization control strategy. For a multi-mode power system with the dynamics of electric vehicles (EVs), it solves the technical bottlenecks of traditional load frequency control in random disturbances, cyberattacks, and mode switching dependence problems, and is applicable to improving the frequency stability and anti-attack ability of the power system, and can enhance the anti-interference ability and operating efficiency of the power system. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 is a schematic flowchart of the method of the embodiment;

[0055] Figure 2 is a schematic diagram of an electric vehicle power system with a DoS in the embodiment;

[0056] Figure 3 is a state response state trace diagram of a dual-mode switching power system with a controller in the embodiment;

[0057] Figure 4 are the signals of the DoS attack and the detected DoS attack signals in the embodiment;

[0058] Figure 5 is the signal of random sampling of a dual-mode switching power system in the embodiment;

[0059] Figure 6 is the compensation time of a dual-mode switching power system in the embodiment;

[0060] Figure 7 is a modal change diagram of a dual-mode switching power system in the embodiment;

[0061] Figure 8 is the state response of a dual-mode switching power system of Scheme 2 in the embodiment;

[0062] Figure 9 is a comparison of the state responses of a dual-mode switching power system between Scheme 1 and Scheme 2 in the embodiment;

[0063] Figure 10 is the state response of a dual-mode switching power system of Scheme 3 in the embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0064] The following further elaborates on the content of the present invention in conjunction with the drawings and embodiments, but does not limit the present invention.

[0065] Embodiment:

[0066] Referring to Figure 1 , a random sampling control method for a power system based on attack compensation includes the following steps:

[0067] 1) Construction of an adaptive random switching model: AsFigure 2 As shown, a multi - mode switching power system model including the dynamics of an electric vehicle EV is constructed to characterize the mode jumps caused by load fluctuations and attacks. The load frequency regulation system of the multi - mode power system involves loads, generators, prime movers, governors, and the LFC loop part. The multi - mode switching power system is modeled as shown in Equation (1):

[0068]

[0069] where f(t) is the frequency deviation, M(φ(t)) and D(φ(t)) are the inertia constant and the load damping coefficient respectively; P e (t), P g (t), P d (t) are the increments of the output power of the electric vehicle, the steam turbine, and the load disturbance respectively; X g (t), T g (φ(t)), α g (φ(t)), R g (φ(t)), u(t) are the governor valve position, the governor speed, the participation factor of the steam turbine, the governor droop characteristic, and the control input respectively; P g (t), T t (φ(t)) are the output power of the steam turbine and the steam turbine time constant respectively; P e (t), T e (φ(t)), ρ e (φ(t)), α e (φ(t)), are the increment, time constant, electric - vehicle droop characteristic, electric - vehicle participation factor, and electric - vehicle gain in the electric vehicle respectively;

[0070] Let x T (t)=[f(t) X g (t) P g (t) P e (t) △(t)], where △(t)=∫ACEdt, ACE is the area control error, and ACE = bf(t), y T (t)=[f(t) △(t)] is the output vector, w(t)=P d (t) is the system disturbance, and it satisfies w T (t) w(t)<ρ, ρ is a finite positive real number. Then the following state - space representation form of the power system can be obtained:

[0071]

[0072] Matrices A, B, C, and B w are system matrices, and these matrices satisfy the following values:

[0073]

[0074] In traditional research, power systems are usually modeled as Markov jump systems, and their stability depends on accurate transition probability parameters. However, the complex and dynamic characteristics of the power system environment mean that mode switching is often affected by random disturbances and external attacks. Therefore, it is difficult to accurately obtain the transition probability through prior knowledge. This limitation restricts the practicality of traditional models. This example proposes a switching rule based on dwell time (DT) and dwell probability (DP);

[0075] 2) Establish a multimodal switching electric vehicle power system based on modal dwell time and dwell probability: The process includes:

[0076] Assume that there are m types of switching mode signals φ(t), and the parameter φ(t) is associated with the random variable where t l represents the l-th transition time from mode to mode , and t 0 = 0. Here, is the mode reached after the l-th transition, and is the dwell time in mode , satisfying The function represents the probability distribution function. For t ≥ 0, where φ(t) takes values in the set is governed by a Markov process. For any and the distribution function depends on the dwell time relative to the specified mode and is derived as follows:

[0077]

[0078] where θ j represents the average dwell probability of the j-th submode, and the probability distribution function is:

[0079]

[0080] 3) Random sampling: To reduce the pressure of network signal transmission and improve the stability of the system, random sampling of the power system state x(t) is considered before transmitting data. Random sampling means that the sampling period h is no longer a fixed positive number but belongs to a sampling set. Assume the sampling instances of the system are:

[0081] t 0 = 0 < t 1 < t 2 <… < t k <…,

[0082] This means that the transmitted data control signal will be updated at instant t k and then the zero-order hold control signal will be maintained for a period of time until the control signal data is updated. The sampling interval is defined as: Then s k ∈P = {p 1 , p 2 , …, p M}, where p 1 < p 2 < … < p M . They are M different positive numbers representing possible sampling time intervals. From this, the following probability distribution can be obtained:

[0083] Prob{s k = p i} = α i , i = 1, 2, …, M, (5),

[0084] where α i ∈[0, 1] and satisfies

[0085] 4) DoS attack monitoring algorithm and data compensation mechanism: Network attacks may occur in the power grid, leading to system instability. DoS attack is one of the most common and destructive attacks. Therefore, in this example, an energy-limited DoS attack is considered, that is, the frequency and duration of the attack are both limited. To reduce the harm caused by DoS attacks, a DoS attack monitoring algorithm and data compensation mechanism are considered to maintain the stability of the system and reduce information loss. Specifically:

[0086]

[0087]

[0088] , once the initial compensation time for each period is established, the following event-triggered compensation mechanism is adopted in this example to determine the order of compensation time:

[0089]

[0090] where e(l k ) = x(t m,n ) - x(l k );

[0091] 5) Design of a PD-like controller: A general proportional-integral controller is expressed as shown in formula (7):

[0092]

[0093] where K = [K 1,K 2 , considering the multi-modal characteristics of the system, the design of the PD controller based on the data compensation mechanism is shown in Equation (8):

[0094]

[0095] where is the interval where no attack is detected, while is the interval where an attack is detected. Therefore, the power system reconstruction in this example is shown in Equation (9):

[0096]

[0097] 6) Solve the proportional-integral controller gain: Based on Lyapunov theory, sufficient conditions for the asymptotic stability of the system are given and the H ∞ performance is satisfied. Use the LMI toolbox of MTALAB to solve the proportional-integral controller gain. The process is as follows:

[0098] Based on Lyapunov theory, the sufficient conditions to ensure the asymptotic stability of the system and satisfy the H ∞ performance are:

[0099] For a given compensation parameter ρ j > 0 and a scalar b 1 , b 2 , in the system shown in Equation (9), there exists a symmetric matrix Z 1 , Z 2 , a positive definite matrix Θ j , and a matrix K j1 , K j2 , K jd1 , K jd2 such that asymptotic stability can be achieved and the H ∞ performance index γ is satisfied if the inequalities and

[0100] where and is the probability density function. Use linear matrix inequalities to solve the PD integral observer gain and the controller gain:

[0101] For a given compensation parameter ρ j > 0 and a scalar b 1 , b 2 , in the system shown in Equation (9), there exists a symmetric matrix Z 1 , Z 2 , a positive definite matrix Θ j, and matrix Y j1 , Y j2 , Y jd1 and Y jd2 , asymptotic stability can be achieved and it has H ∞ performance index γ, if the matrix inequalities and where

[0102]

[0103] control gain matrix K j1 can be designed as Other gain matrices can be obtained in the same way.

[0104] To verify the effectiveness of the method in this example, a simulation example of a dynamic dual-mode switching power system model for an electric vehicle is given, and its system parameters are shown in Table 1 below:

[0105] Table 1. Parameters of the electric vehicle power system model

[0106]

[0107] , and other parameters are as follows:

[0108] x(0) = col{5, 1, 2, 3, -0.58}, b 1 = 10, b 2 = 15, ρ 1 = ρ 2 = 0.6, τ f = 1, γ = 100, for M = 3, the given sampling intervals are p 1 = 0.1, p 2 = 0.2, p 3 = 0.25, and the probabilities of the corresponding sampling intervals are α 1 = 0.3, α 2 = 0.3, α 3 = 0.4. In this example, the dwell time function follows the Weibull distribution, and its probability density function is: Parameter is defined as: Specifically, when l = 1, the scale parameter a is selected as 1 and the shape parameter b is selected as 2; for l = 2, the scale parameter a is 2 and the shape parameter b is also 2;

[0109] Using the LMI toolbox of MATLAB, the proportional-derivative controller gain and the event compensation mechanism trigger matrix are obtained as:

[0110] K 11 = K 12= [0.0089, 0.0023], K 21 = K 22 = [0.0179, 0.0033],

[0111] K 1d1 = K 1d2 = [0.0018, 0.0001], K 2d1 = K 2d2 = [0.0029, 0.0001].

[0112]

[0113] To highlight the superiority of the method in this example, two other schemes are selected to control the power system. The method in this example is denoted as Scheme 1;

[0114] Scheme 2: Design a proportional - derivative controller for the power system without data compensation. The obtained controller gains are:

[0115] K 11 = [-0.2262, -0.0927], K 12 = [0.9090, -0.0057],

[0116] K 21 = [0.0044, 0.0007], K 22 = [-0.0040, 0.0003];

[0117] Scheme 3: Research on a power system with a proportional controller. The obtained controller gains and the gain matrix of the data compensation mechanism are:

[0118] K 1 = [-0.0009, -0.0002], K 2 = [-3.2603, 0.1438],

[0119] K 1d = [-0.0009, -0.0002], K 2d = [0.0011, -0.0002].

[0120]

[0121] The simulation results are as Figure 3 、 Figure 4 、 Figure 5 、 Figure 6 、 Figure 7 、 Figure 8 、 Figure 9 、 Figure 10 shown. The simulation time is 25 s. From Figure 9It can be seen that although Solution 2 can also stabilize the electric vehicle power system, the amplitude fluctuation of the state response in Solution 1 is smaller, and the power system can be stabilized faster. And Figure 10 It reflects the important role of the differential term in the controller. This shows that under the same parameter conditions, compared with the traditional method, the method in this example is more general and has the advantage of a short stabilization time. Therefore, the method in this example can obtain better control effects.

Claims

1. A random sampling control method for a power system based on attack compensation, characterized in that: The steps include: 1) Adaptive random switching model construction: Construct a multi-mode switching power system model that includes the dynamics of electric vehicles (EVs). The load frequency regulation system of the multi-mode power system involves load fluctuations and attack-induced mode jumps. The multi-mode switching power system is modeled as shown in formula (1): Where, f(t) is the frequency deviation, M(φ(t)) and D(φ(t)) are the inertia constant and load damping coefficient respectively; P e (t),P g (t),P d (t) are the increment of output power of electric vehicle and turbine and load disturbance respectively; X g (t), T g (φ(t)),α g (φ(t)), R g (φ(t)), u(t) are respectively the speed control valve position, speed controller speed, turbine participation factor, speed controller droop characteristic and control input; P g (t), T t (φ(t)) are the turbine output power and turbine time constant respectively; P e (t), T e (φ(t)),ρ e (φ(t)),α e (φ(t)), They are incremental change in EV, time constant, EV droop characteristic, EV participation factor, EV gain; Let x T (t) = [f(t)X g (t)P g (t)P e (t)△(t)],△(t)=∫ACEdt, where ACE is the area control error, and ACE=bf(t),y T (t) = [f(t)△(t)] is the output vector, w(t) = P d (t) is the system disturbance and satisfies w T (t)w(t)<ρ, ρ is a finite positive real number, then the following power system state space representation is obtained: Matrices A, B, C, and B w are system matrices, which satisfy the following values: 2) Establish a multi-mode switching electric vehicle power system based on mode dwell time and dwell probability: The process includes: Assume that there are m types of switching pattern signals φ(t), parameter φ(t) and random variable is associated with l Indicates slave mode To Mode The lth conversion time, and t0 = 0, here, is the mode reached after the lth conversion, and is in mode The residence time in function represents the probability distribution function, for t≥0, where φ(t) takes values ​​in the set Governed by a Markov process, for any and The distribution function depends on the dwell time relative to the specified mode and is derived as follows: Among them, θ j represents the average stay probability of the j-th submodule, and the probability distribution function is: 3) Random sampling: Before transmitting data, the state x(t) of the power system is randomly sampled. Assume that the sampling instance of the system is: t0=0 <t1<t2<…<t k <…, This means that the transmitted data control signal will be at instant t k Update, then the zero level holds the control signal for a period of time until the control signal data is updated, and the sampling interval is defined as: Then k ∈P={p1,p2,…,p M }, where p1 <p2<…<p M .They are M different positive numbers, representing possible sampling time intervals, which give the following probability distribution: Prob{s k =p i }=α i ,i=1,2,…,M, (5), where α i ∈[0,1], and satisfies 4) DoS attack monitoring algorithm and data compensation mechanism: Design DoS attack monitoring algorithm and data compensation mechanism to maintain system stability and reduce information loss, specifically: ， Establish the initial compensation time for each cycle and use the compensation mechanism triggered by the following events to determine the order of compensation time: Among them, e(l k )=x(t m,n )-x(l k ); 5) Design of PD-like controller: The proportional-integral controller is expressed as shown in formula (7): Where K = [K1, K2]. Considering the multimodal characteristics of the system, the design of the PD controller based on the data compensation mechanism is shown in formula (8): in is the interval detected without attack, and is the detected attack existence interval, so the power system is reconstructed as shown in formula (9): 6) Solve the proportional-integral controller gain: Based on Lyapunov theory, give the sufficient conditions for the asymptotic stability of the system and satisfy H ∞ Performance, using MTALAB's LMI toolbox to solve the proportional integral controller gain, the process is as follows: Based on Lyapunov theory, the system is guaranteed to be asymptotically stable and satisfy H ∞ The sufficient conditions for performance are: For a given compensation parameter ρ j >0 and scalar b1, b2, in the system shown in formula (9) there are symmetric matrices Z1, Z2, positive definite matrices Θ j , and the matrix K j1 ,K j2 ,K jd1 ,K jd2 In the case of H ∞ The performance index γ, if it satisfies the inequality and in, and is the probability density function, and the linear matrix inequality is used to solve the PD integral observer gain and controller gain: For a given compensation parameter ρ j >0 and scalar b1, b2, the system shown in formula (9) has symmetric matrices Z1, Z2, positive definite matrices Θ j , and the matrix Y j1 ,Y j2 ,Y jd1 and Y jd2 In the case of H ∞ The performance index γ, if it satisfies the matrix inequality and in, Control gain matrix K j1 Designed for The other gain matrices are obtained in the same way.

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