CN109358509B - Rapid parameter identification method for chaotic ferromagnetic resonance system of coal mine power grid - Google Patents

Abstract

The invention provides a rapid parameter identification method for a chaotic ferromagnetic resonance system of a coal mine power grid, which is suitable for the field of electrical engineering. According to a fixed time stability theory and in combination with a traditional self-adaptive control method, a nonlinear controller is designed to enable a controlled variable to reach a neighborhood with an arbitrarily small reference value within a limited time, and the method specifically comprises the following steps: the method can identify the uncertain parameters of the chaotic ferromagnetic resonance system of the coal mine power grid in a limited time under the condition of not depending on an initial value; the time range of parameter identification can be ensured to have an upper bound, and the upper bound of the identification time can be calculated by the method provided by the invention; the method can quickly and accurately obtain the ferroresonance parameters, provide parameter support for preventing and controlling abnormal operation phenomena such as overvoltage and overcurrent of the transformer, and provide theoretical basis and technical support for safe operation research of a coal mine power grid.

Description

Fast parameter identification method for chaotic ferromagnetic resonance system in coal mine power grid

technical field

The invention designs a fast parameter identification method, which is especially suitable for the fast parameter identification method of the chaotic ferromagnetic resonance system of the coal mine power grid used in the field of electrical engineering.

Background technique

Ferromagnetic resonance is a kind of oscillation phenomenon of nonlinear inductance and capacitance in power system. It belongs to a complex electrical phenomenon. It may occur in power transformers or electromagnetic voltage transformers (PTs) and reactors with iron cores. In the loop composed of non-linear inductive elements and capacitive elements. So far, scholars at home and abroad have studied ferromagnetic resonance for a hundred years, but ferromagnetic resonance still occurs frequently in power systems, and various harmonic elimination measures cannot completely eliminate ferromagnetic resonance. The main reason is that the generating circuit of ferromagnetic resonance is extremely diverse, and its excitation conditions are also extremely complex. The complexity of ferromagnetic resonance is not only manifested in the complexity of its generating circuit, but also in very complex manifestations. In recent years, some scholars have found that ferromagnetic resonance with chaotic effect may occur in power systems, which is also one of the manifestations of the complexity of ferromagnetic resonance. one. This resonance state is very harmful to the coal mine power grid. It can destabilize the interconnected power system, cause circuit voltage and current waveform distortion, especially the generation of overvoltage, resulting in serious local instability of the system. The power grid brings serious impacts and hazards. And in the normal long-term operation of the transformer, the parameters may change with the change of temperature and humidity in the working environment. Through the parameter identification of the chaotic ferromagnetic resonance model, the ferromagnetic resonance parameters can be obtained in time, which can provide parameter support for the prevention and control of abnormal operation phenomena such as overvoltage and overcurrent of transformers, and provide theoretical basis and technology for the study of safe operation of coal mine power grids. Therefore, it is very necessary to study the fast parameter identification method of the chaotic ferromagnetic resonance system of coal mine power grid.

Most chaotic control methods are based on the premise that the system parameters are known. The structure of coal mine power grid is complex, and some parameters are difficult to be accurately determined by on-line measurement, or due to the influence of the environment, the system parameters are prone to drift and disturbance during operation. In order to realize the chaotic control and synchronization of the system, the commonly used control methods all have certain limitations. Therefore, the problem of parameter identification is an important topic in the field of chaotic system control and synchronization. In recent years, the problem of parameter identification of chaotic systems has attracted great attention of many scholars at home and abroad. At present, the commonly used parameter identification methods mainly include adaptive synchronization parameter identification method, parameter identification method based on synchronization parameter observer, and model identification method based on fuzzy theory. Among the above parameter identification methods, the identification method based on the synchronous observer does not need to copy the original chaotic system, the controller is simple and easy to implement, and has certain advantages. However, in the work of multi-parameter synchronous identification of high-power terms, sine terms, cosine terms in structures and complex systems, this method has problems such as inaccurate identification and identification of oscillations, and there are certain limitations in application. Therefore, improving the existing chaotic system parameter identification methods and finding a better parameter identification scheme has become the main subject of chaotic parameter identification. Therefore, scholars began to apply finite-time stability theory and fixed-time stability theory to chaos identification and combine them with traditional identification methods.

The finite-time identification method can identify the system parameters in a finite time, but it is limited by the limitation of finite-time stability, which is affected by the initial conditions of the system, which are generally difficult to obtain accurate parameters in coal mine power grids. Fixed-time stability is an extension of finite-time stability. Compared with the above identification methods, the fixed-time identification method can not only ensure the upper bound of the identification time range, but also identify the system parameters without relying on the initial conditions.

SUMMARY OF THE INVENTION

Aiming at the above technical problems, an error response system based on the synchronization idea to construct a chaotic ferromagnetic resonance system is provided. According to the fixed-time stability theory combined with the traditional adaptive control method, an adaptive fixed-time nonlinear controller is designed to make the control The quantity reaches its reference value in the neighborhood of arbitrarily small in a limited time independent of the initial value, realizes the synchronization of the master-slave system, and realizes the fast parameter identification method of the chaotic ferromagnetic resonance system of the coal mine power grid more quickly and accurately.

In order to achieve the above purpose, the method for fast parameter identification of a chaotic ferromagnetic resonance system of a coal mine power grid of the present invention includes the following steps:

a. Collect various data of coal mine power grid, including transformer excitation inductance flux information, excitation inductance voltage information, resistance and capacitance information in chaotic ferromagnetic resonance equivalent circuit, power supply voltage angular frequency and power supply voltage and magnetization of transformer excitation inductance Characteristic parameters, use the above information to establish the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

b. Based on the idea of synchronization, aiming at the error variables and uncertain parameters, the error response system of the chaotic ferromagnetic resonance of the coal mine power grid is established through the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

c. For the error response system, the nonlinear control law and the adaptive law of the uncertain parameters are designed according to the fixed time stability theory, so that the synchronization error between the controlled model and the reference model is arbitrarily small in a fixed time;

d. Design a fixed-time stable control rate and an adaptive law to achieve parameter identification and the stability of the controlled model within a limited time, and the upper bound of the identification time does not depend on the initial value;

e. According to the Lyapunov stability theory, confirm the control parameters of the design control rate.

The second-order differential mathematical model I of the chaotic ferromagnetic resonance system of the coal mine power grid is as follows:

In the formula: Ф is the magnetic flux of the transformer excitation inductance, V is the excitation inductance voltage; R is the resistance in the chaotic ferromagnetic resonance equivalent circuit, C is the capacitance in the chaotic ferromagnetic resonance equivalent circuit; ω and E are the power supply voltages respectively Angular frequency and power supply voltage; a ₀ and b ₀ are the magnetization characteristic parameters of the transformer excitation inductance, t is the time, and the model I is dimensionless, so that the coal mine power grid chaotic ferromagnetic resonance system model is equivalent to the following system, as Main system model II:

In the formula, a, b, c, and d are the uncertain parameters of coal mine power grid operation, but under the condition that the magnetization characteristic parameters a ₀ and b ₀ of the transformer excitation inductance are known, through the identification of the parameter c in the system model II, the Determine the specific values of the resistance R and the capacitance C of the chaotic dynamic behavior of the transformer ferromagnetic resonance model, then the parameters a and b can be obtained, so that the main purpose of the parameter identification work is the identification of the parameter c.

The method to realize the synchronization of uncertain parameters is: According to formula II, the following slave system model III is obtained:

分别为系统状态变量的估计值，

分别为不确定参数的估计值，u为煤矿电网混沌铁磁谐振从系统的控制输入；in,

are the estimated values of the system state variables, respectively,

are the estimated values of the uncertain parameters, respectively, and u is the control input of the chaotic ferromagnetic resonance slave system of the coal mine power grid;

作为误差变量，

作为不确定参数的估计误差，则求式Ⅲ与式Ⅱ之差得到下面的误差响应系统Ⅳ：definition

As the error variable,

As the estimation error of the uncertain parameter, the difference between Equation III and Equation II can be obtained to obtain the following error response system IV:

为关于

和Ф的多项式。In the formula,

for about

and a polynomial of Ф.

The nonlinear control law designed according to the fixed-time stability theory is:

In the formula, α, β are the parameters to be designed for the system, satisfying 0<α<1, β>1, k is the terminal attractor feedback gain coefficient of the tuning parameter, satisfying,

The adaptive law for designing uncertain parameters according to the adaptive control principle is:

where g is any positive real number.

The calculation method of the upper bound of the identification time range is: using the fixed time stability theory to design the nonlinear system as follows:

且T_max与状态变量x无关,使得

且t＞T_max时，x(t)≡0。则此时的非线性系统Ⅶ为全局固定时间稳定；In the formula, ^x∈Rn is the state variable of the nonlinear system VII, and f is an unknown smooth nonlinear function. If the initial value of the nonlinear system is arbitrary, the abstract nonlinear system cannot be specifically described T(x ) where and the specific expression exists a local bounded stable time function T(x), and the upper bound value T _max of the bounded stable time function T(x) has nothing to do with the state variable x, that is, when the initial conditions of the system take any value ,

And _Tmax is independent of the state variable x, such that

And when t>T _max , x(t)≡0. Then the nonlinear system VII at this time is globally fixed-time stable;

Lemma 1: For any non-negative real numbers ξ ₁ , ξ ₂ ,...,ξ _n and 0<p≤1, the following inequality holds:

非线性系统Ⅶ的任意解x(t)满足不等式：

For nonlinear system VII, suppose there is a continuous radially unbounded function V: R ⁿ → R ₊ ∪{0} satisfying:

Any solution x(t) of nonlinear system VII satisfies the inequality:

Among them, α, β, p, q, k are real numbers greater than 0, pk<1, qk>1 are simulation parameters, at this time, for the nonlinear system VII origin global fixed time stability, within fixed time T must make V(x )≡0, and its settling time is:

The above description is the cornerstone of designing an adaptive fixed-time algorithm to identify the uncertain parameters of the system in a finite time. The Lyapunov function is constructed according to the Lyapunov function stability theory:

Using the designed controller u and the corresponding tuning parameters, the derivative of the Lyapunov candidate function of the error response system is obtained

in,

From this, the upper bound of the system stability time can be obtained as:

主系统Ⅱ和从系统Ⅲ实现同步，得到不确定参数辨识结果。This means that when t≥t ₁ ,

The master system II and the slave system III are synchronized, and the identification results of uncertain parameters are obtained.

Beneficial effects:

The fast parameter identification method of the coal mine power grid chaotic ferromagnetic resonance system disclosed by the invention firstly establishes the master-slave system model and the error response system model of the coal mine power grid chaotic ferromagnetic resonance system based on the synchronization idea, and uses the adaptive control and fixed time Combined with the stability theory, the uncertain parameter identification can be realized in a limited time without relying on the initial value of the system parameters, and the upper bound of the identification time can be obtained quickly, and the overvoltage and overcurrent phenomena can be prevented or controlled by identifying the parameters. The purpose of suppressing ferromagnetic resonance.

Description of drawings

Fig. 1 is the coal mine power grid chaotic ferromagnetic resonance system structure schematic diagram that the present invention adopts;

Fig. 2 is the equivalent circuit of the coal mine power grid chaotic ferromagnetic resonance system of the present invention;

Fig. 3 is the flow chart of the fast parameter identification method of the coal mine power grid chaotic ferromagnetic resonance system of the present invention;

Fig. 4 (a) is the error system control result of the chaotic ferromagnetic resonance system of the present invention;

Fig. 4 (b) is the error system control result of the chaotic ferromagnetic resonance system of the present invention;

Fig. 5 is the parameter c identification result of the chaotic ferromagnetic resonance model of coal mine power grid of the present invention;

Detailed ways

Embodiments of the present invention will be further described below in conjunction with the accompanying drawings:

As shown in Figure 3, the method for fast parameter identification of the coal mine power grid chaotic ferromagnetic resonance system of the present invention, the steps are:

a. Collect various data of coal mine power grid, including transformer excitation inductance flux information, excitation inductance voltage information, resistance and capacitance information in chaotic ferromagnetic resonance equivalent circuit, power supply voltage angular frequency and power supply voltage and magnetization of transformer excitation inductance Characteristic parameters, use the above information to establish the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

b. Based on the idea of synchronization, aiming at the error variables and uncertain parameters, the error response system of the chaotic ferromagnetic resonance of the coal mine power grid is established through the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

c. For the error response system, the nonlinear control law and the adaptive law of the uncertain parameters are designed according to the fixed time stability theory, so that the synchronization error between the controlled model and the reference model is arbitrarily small in a fixed time;

d. Design a fixed-time stable control rate and an adaptive law to achieve parameter identification and the stability of the controlled model within a limited time, and the upper bound of the identification time does not depend on the initial value;

e. According to the Lyapunov stability theory, confirm the control parameters of the design control rate.

Specifically include:

a. Collect various data of coal mine power grid, including transformer excitation inductance flux information, excitation inductance voltage information, resistance and capacitance information in chaotic ferromagnetic resonance equivalent circuit, power supply voltage angular frequency and power supply voltage and magnetization of transformer excitation inductance Characteristic parameters, use the above information to establish the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

The chaotic ferromagnetic resonance system of the coal mine power grid shown and the equivalent circuit of the chaotic ferromagnetic resonance system of the coal mine power grid shown in Fig. 2 are mathematically modeled. The second-order differential mathematical model I of the chaotic ferromagnetic resonance system of the coal mine power grid is as follows:

In the formula: Ф is the magnetic flux of the transformer excitation inductance, V is the excitation inductance voltage; R is the resistance in the chaotic ferromagnetic resonance equivalent circuit, C is the capacitance in the chaotic ferromagnetic resonance equivalent circuit; ω and E are the power supply voltages respectively Angular frequency and power supply voltage; a ₀ and b ₀ are the magnetization characteristic parameters of the transformer excitation inductance, t is the time, and the model I is dimensionless, so that the coal mine power grid chaotic ferromagnetic resonance system model is equivalent to the following system, as Main system model II:

In the formula, a, b, c, and d are the uncertain parameters of coal mine power grid operation, but under the condition that the magnetization characteristic parameters a ₀ and b ₀ of the transformer excitation inductance are known, through the identification of the parameter c in the system model II, the Determine the specific values of the resistance R and the capacitance C of the chaotic dynamic behavior of the transformer ferromagnetic resonance model, then the parameters a and b can be obtained, so that the main purpose of the parameter identification work is the identification of the parameter c;

b. Based on the idea of synchronization, aiming at the error variables and uncertain parameters, the error response system of the chaotic ferromagnetic resonance of the coal mine power grid is established through the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid;

In order to realize the synchronization of uncertain parameters in this embodiment, the following slave system model III is obtained according to formula II:

分别为系统状态变量的估计值，

分别为不确定参数的估计值，u为煤矿电网混沌铁磁谐振从系统的控制输入；in,

are the estimated values of the system state variables, respectively,

are the estimated values of the uncertain parameters, respectively, and u is the control input of the chaotic ferromagnetic resonance slave system of the coal mine power grid;

作为误差变量，

作为不确定参数的估计误差，则求式Ⅲ与式Ⅱ之差得到下面的误差响应系统Ⅳ：definition

As the error variable,

As the estimation error of the uncertain parameter, the difference between Equation III and Equation II can be obtained to obtain the following error response system IV:

为关于

和Ф的多项式；In the formula,

for about

and the polynomial of Ф;

c. For the error response system, the nonlinear control law and the adaptive law of the uncertain parameters are designed according to the fixed time stability theory, so that the synchronization error between the controlled model and the reference model is arbitrarily small in a fixed time;

According to the fixed-time stability theory, the nonlinear control law is designed as:

In the formula, α, β are the parameters to be designed for the system, satisfying 0<α<1, β>1, k is the terminal attractor feedback gain coefficient of the tuning parameter, satisfying,

The adaptive law for designing uncertain parameters according to the adaptive control principle is:

In the formula, g is any positive real number;

d. Design a fixed-time stable control rate and an adaptive law to achieve parameter identification and the stability of the controlled model within a limited time, and the upper bound of the identification time does not depend on the initial value;

The calculation method of the upper bound of the identification time range is: using the fixed time stability theory to design the nonlinear system as:

且T_max与状态变量x无关,使得

且t＞T_max时，x(t)≡0。则此时的非线性系统Ⅶ为全局固定时间稳定；In the formula, ^x∈Rn is the state variable of the nonlinear system VII, and f is an unknown smooth nonlinear function. If the initial value of the nonlinear system is arbitrary, the abstract nonlinear system cannot be specifically described T(x ) where and the specific expression exists a local bounded stable time function T(x), and the upper bound value T _max of the bounded stable time function T(x) has nothing to do with the state variable x, that is, when the initial conditions of the system take any value ,

And _Tmax is independent of the state variable x, such that

And when t>T _max , x(t)≡0. Then the nonlinear system VII at this time is globally fixed-time stable;

Lemma 1: For any non-negative real numbers ξ ₁ , ξ ₂ ,...,ξ _n and 0<p≤1, the following inequality holds:

非线性系统Ⅶ的任意解x(t)满足不等式：

For nonlinear system VII, suppose there is a continuous radially unbounded function V: R ⁿ → R ₊ ∪{0} satisfying:

Any solution x(t) of nonlinear system VII satisfies the inequality:

Among them, α, β, p, q, k are real numbers greater than 0, pk<1, qk>1 are simulation parameters, at this time, for the nonlinear system VII origin global fixed time stability, within fixed time T must make V(x )≡0, and its settling time is:

The above description is the cornerstone of designing an adaptive fixed-time algorithm to identify the uncertain parameters of the system in a finite time. The Lyapunov function is constructed according to the Lyapunov function stability theory:

Using the designed controller u and the corresponding tuning parameters, the derivative of the Lyapunov candidate function of the error response system is obtained

in,

From this, the upper bound of the system stability time can be obtained as:

主系统Ⅱ和从系统Ⅲ实现同步，得到不确定参数辨识结果；This means that when t≥t ₁ , e ₁ ≡ 0,

The master system II and the slave system III are synchronized, and the identification results of uncertain parameters are obtained;

e. According to the Lyapunov stability theory, confirm the control parameters of the design control rate.

Example: Fast parameter identification method of chaotic ferromagnetic resonance system in coal mine power grid

Mathematical modeling is carried out according to the chaotic ferromagnetic resonance system of coal mine power grid shown in Figure 1 and the equivalent circuit of the chaotic ferromagnetic resonance system of coal mine power grid shown in Figure 2, as follows:

Among them, Ф is the magnetic flux of the transformer excitation inductance, V is the voltage of the excitation inductance; R and C are the resistance and capacitance in the chaotic ferromagnetic resonance equivalent circuit, respectively; ω and E are the power supply voltage angular frequency and power supply voltage, respectively; a ₀ and b ₀ is the magnetization characteristic parameter of the transformer magnetizing inductance, and t is the time. In order to facilitate the derivation, we perform dimensionless processing on the above model, so that the chaotic ferromagnetic resonance system model of the coal mine power grid is equivalent to the following system as the main system:

In the formula, a, b, c, d are uncertain parameters of the system operation. However, under the condition that the magnetization characteristic parameters a ₀ and b ₀ of the transformer magnetizing inductance are known, through the identification of the parameter c in the system (2), the resistance R and the capacitance C of the chaotic dynamic behavior of the transformer ferromagnetic resonance model can be determined. The specific value of , the parameters a and b can be obtained. Therefore, in the parameter identification work of the present invention, the identification of the parameter c is mainly realized.

In order to realize the synchronization of uncertain parameters in this embodiment, the following slave system is considered according to formula (2):

为系统状态变量的估计值，

为不确定参数的估计值，u是煤矿电网混沌铁磁谐振从系统的控制输入。in,

is the estimated value of the system state variable,

is the estimated value of the uncertain parameter, u is the control input of the coal mine power grid chaotic ferromagnetic resonance slave system.

作为误差变量，

作为不确定参数的估计误差，那么由式(3)与式(2)做差可以得到下面的误差响应系统definition

As the error variable,

As the estimation error of the uncertain parameters, then the following error response system can be obtained by making the difference between equation (3) and equation (2)

为关于

和Ф的多项式。in the formula

for about

and a polynomial of Ф.

In order to achieve the identification goal, the nonlinear control law of the error response system and the adaptive law of the uncertain parameters are designed as:

According to the fixed-time stability theory, the nonlinear control law is designed as:

Among them, α, β are the parameters to be designed for the system, satisfying 0<α<1, β>1, and k is the terminal attractor feedback gain coefficient of the tuning parameter, satisfying k>0. In this embodiment, parameters α=0.5 and β=1.5 are taken.

The adaptive law for designing uncertain parameters according to the adaptive control principle is:

where g is any positive constant. This embodiment takes the parameter g=0.3.

The upper bound of the stability time range is determined according to the stability analysis of the Lyapunov function:

Construct the Lyapunov function:

Using the designed controller u and the corresponding tuning parameters, the derivative of the Lyapunov function of the error-response system is obtained:

in,

From this, the upper bound of the system stability time can be obtained as:

主从系统实现同步，得到不确定参数辨识结果。This means that when t≥t ₁ , e ₁ ≡ 0,

The master-slave system is synchronized, and the uncertain parameter identification result is obtained.

Taking the parameters taken in this embodiment into it, it is obtained that t ₁ ≤30.35, that is to say, the upper bound of the identification time of the uncertain parameters of the system is within 30.35 s after the controller is applied.

The flowchart of the provided fast parameter identification method for the chaotic ferromagnetic resonance system of the coal mine power grid is shown in FIG. 3 . 5. Perform data simulation on the MATLAB simulation platform of this embodiment to verify the identification effect. The initial value in this embodiment is taken as (Φ, V)=(0, 1.4142). The identification results of the errors e ₁ , e ₂ and the uncertain parameter c in the chaotic ferromagnetic resonance system of the coal mine power grid are shown in Figure 4(a), Figure 4(b) and Figure 5. As shown in Figures 4(a) and 4(a)(b), the errors e ₁ and e ₂ are stable to 0, realizing the synchronization of the master-slave system. As shown in Fig. 5, the fixed time method based on the synchronization idea can identify the parameter c to the parameter target value c=-0.00122. In the experiment with a duration of 50s, the parameter identification curve is forced to be stable within ±3% of the parameter target value. The adjustment time required within the region is t _c =8.7s, the overshoot of the identification curve is small, and the identification curve is stable without chattering.

Claims (1)

1. a fast parameter identification method of coal mine power grid chaos ferromagnetic resonance system, is characterized in that step is as follows: a. Collect various data of coal mine power grid, including transformer excitation inductance flux information, excitation inductance voltage information, resistance and capacitance information in chaotic ferromagnetic resonance equivalent circuit, power supply voltage angular frequency and power supply voltage and magnetization of transformer excitation inductance Characteristic parameters, use the above information to establish the second-order differential mathematical model I of the chaotic ferromagnetic resonance system of the coal mine power grid; The second-order differential mathematical model I of the chaotic ferromagnetic resonance system of the coal mine power grid is as follows:

In the formula: Ф is the magnetic flux of the transformer excitation inductance, V is the excitation inductance voltage; R is the resistance in the chaotic ferromagnetic resonance equivalent circuit, C is the capacitance in the chaotic ferromagnetic resonance equivalent circuit; ω and E are the power supply voltages respectively Angular frequency and power supply voltage; a ₀ and b ₀ are the magnetization characteristic parameters of the transformer excitation inductance, t is the time, and the model I is dimensionless, so that the coal mine power grid chaotic ferromagnetic resonance system model is equivalent to the following system, as Main system model II:

In the formula, a, b, c, and d are the uncertain parameters of coal mine power grid operation, but under the condition that the magnetization characteristic parameters a ₀ and b ₀ of the transformer excitation inductance are known, through the identification of the parameter c in the system model II, the Determine the specific values of the resistance R and the capacitance C for the chaotic dynamic behavior of the transformer ferromagnetic resonance model, then the parameters a and b can be obtained; b. Based on the idea of synchronization, aiming at the error variables and uncertain parameters, the error response system IV of the chaotic ferromagnetic resonance of the coal mine power grid is established through the second-order differential mathematical model of the chaotic ferromagnetic resonance system of the coal mine power grid; specific: The method to realize the synchronization of uncertain parameters is: obtain the following slave system model III according to the master system model II:

in,

are the estimated values of the system state variables, respectively,

are the estimated values of the uncertain parameters, respectively, and u is the control input of the chaotic ferromagnetic resonance slave system of the coal mine power grid; definition

As the error variable,

As the estimation error of the uncertain parameter, the following error response system IV is obtained from the difference between the system model III and the system model II:

In the formula,

for about

and a polynomial of Φ; c. For the error response system, the nonlinear control law and the adaptive law of the uncertain parameters are designed according to the fixed time stability theory, so that the synchronization error between the controlled model and the reference model is arbitrarily small in a fixed time; The nonlinear control law designed according to the fixed-time stability theory is:

In the formula, α, β are the parameters to be designed for the system, satisfying 0<α<1, β>1, k is the terminal attractor feedback gain coefficient of the tuning parameter, The adaptive law for designing uncertain parameters according to the adaptive control principle is:

In the formula, g is any positive real number; d. Design a fixed-time stable control rate and an adaptive law to achieve parameter identification and the stability of the controlled model within a limited time, and the upper bound of the identification time does not depend on the initial value; The calculation method of the upper bound of the identification time is: using the fixed-time stability theory to design the nonlinear system as follows:

In the formula, ^x∈Rn is the state variable of the nonlinear system VII, and f is an unknown smooth nonlinear function. If the initial value of the nonlinear system is arbitrary, the abstract nonlinear system cannot be specifically described T(x ) where and the specific expression exists a local bounded stable time function T(x), and the upper bound value T _max of the bounded stable time function T(x) has nothing to do with the state variable x, that is, when the initial conditions of the system take any value ,

And _Tmax is independent of the state variable x, such that

And when t>T _max , x(t)≡0, then the nonlinear system VII at this time is stable in global fixed time; Lemma 1: For any non-negative real numbers ξ ₁ , ξ ₂ , ..., ξ _n and 0<p≤1, the following inequality holds:

For nonlinear system VII, suppose there is a continuous radially unbounded function V: R ⁿ → R ₊ ∪{0} satisfying:

Any solution x(t) of nonlinear system VII satisfies the inequality:

V:R ⁿ →R ₊ ∪{0}, indicating that the constructed function V maps from an n-dimensional vector to a non-negative real number; Among them, α, β, p, q, k are real numbers greater than 0, pk<1, qk>1 are simulation parameters, at this time, for the nonlinear system VII origin global fixed time stability, within fixed time T must make V(x )≡0, and its settling time is:

The above description is the cornerstone of designing an adaptive fixed-time algorithm to identify the uncertain parameters of the system in a finite time. The Lyapunov function is constructed according to the Lyapunov function stability theory:

Using the designed controller u and the corresponding tuning parameters, the derivative of the Lyapunov candidate function of the error response system is obtained

in,

is the estimated value of the parameter; From this, the upper bound of the system stability time can be obtained as:

This means that when t≥t ₁ , e ₁ ≡ 0,

The master system II and the slave system III are synchronized, and the identification results of uncertain parameters are obtained; e. According to the Lyapunov stability theory, confirm the control parameters of the design control rate.

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