CN111830829A - Optimal control method for repair reliability time of excitation type over-damping RLC circuit - Google Patents

Optimal control method for repair reliability time of excitation type over-damping RLC circuit Download PDF

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CN111830829A
CN111830829A CN202010650066.0A CN202010650066A CN111830829A CN 111830829 A CN111830829 A CN 111830829A CN 202010650066 A CN202010650066 A CN 202010650066A CN 111830829 A CN111830829 A CN 111830829A
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金婷
夏红萱
刘海蓉
田宇
保进烽
陈昊
张锐
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Nanjing Forestry University
Maintenance Branch of State Grid Jiangsu Electric Power Co Ltd
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Maintenance Branch of State Grid Jiangsu Electric Power Co Ltd
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Abstract

The invention discloses an excitation type over-damping second orderRLCThe optimal control method for the circuit fault restoration reliable time converts the uncertain time optimal control model into a corresponding deterministic optimal control problem according to the definition of the uncertain time optimal control model. Provides a brand-new fault repair reliable time criterion based on first-arrival time, and applies the fault repair reliable time criterion to excitation type over-damping second-orderRLCIn the uncertain second-order optimization control model of the circuit, an analytic expression about uncertainty distribution of first-arrival time is obtained, meanwhile, sufficient conditions of the optimal solution of the model are obtained, and then the optimal solution of the model under the appointed trust degree and the corresponding fault repairing reliable time are given through a dichotomy. The invention can provide an optimal strategy more conforming to the actual situation, solve the result error caused by neglecting the artificial uncertain factors and improve the excitation type over-damping second orderRLCThe circuit controls the practical application capability of the model.

Description

Optimal control method for repair reliability time of excitation type over-damping RLC circuit
Technical Field
The invention belongs to the field of optimization of circuit system control, and particularly relates to a control method for excitation type over-damping second-order circuit fault repair reliable time under an uncertain state.
Background
As one of the main branches of modern control theory, the optimization control theory has been rapidly developed in recent 60 years, and the status is increasingly prominent. In the 50 s, Pontelagin first applied the maximum principle to the optimal control theory. In the sixties, based on a new concept of controllable observability, Kalman proposed an optimal estimation theory. The series of works lay a foundation as an optimal control theory. At present, the optimal control theory is more and more emphasized in the field of mathematics due to the wide application value and the great development prospect of advanced subjects such as mathematics, engineering, computer science and the like.
Many studies are now being conducted in specific environments. However, with the deepening of understanding of nature and human society, the existing classical model cannot completely solve the optimal control problem in real life. The model is influenced by uncertain factors and has various unpredictabilities. For this reason, in order to measure the feasibility of the occurrence of an event, studies based on the degree of confidence are increasingly spotlighted. In order to quantify the trust of the people, the professor of Liu Baoding introduces an uncertainty theory for the first time in 2007 to reasonably describe the trust of an uncertain event, and corrects the common and chemical definition of the uncertainty measure in 2009. Meanwhile, a corresponding differential equation is provided and is called an uncertainty differential equation. The Sheng and the Ju Yuan professor put forward a new optimal control model in 2013, and a corresponding optimal control equation is established according to an optimal value principle. Because the uncertain second-order differential equation has memorability and heredity, establishing an uncertain dynamic system model based on the uncertain second-order differential equation is a better choice. In 2015, professor Zhuyuanhu initially defined UFDE types Riemann-Liouville and Caputo. In addition, the first arrival time, which is a special numerical characteristic of the uncertain process, was preliminarily defined by professor Liu Baobai in 2013. Subsequently, yaoka studied the uncertain distribution in 2013 and proposed a numerical method of first arrival time. Meanwhile, the first arrival time is widely applied to the hotspot problems in the international automatic control fields of intelligent fault diagnosis, reliability (risk) analysis, operation and maintenance technology, power control and the like.
The second-order circuit system is a commonly used system and is widely applied to control systems in the field of power systems, such as a room temperature adjusting system of a transformer substation protection room, secondary equipment test constant temperature equipment, a cable layer water level monitoring system and the like. The existing second-order circuit control system is generally based on a deterministic system, and has poor capability of realizing an artificial uncertain system with memorability and heredity in reality. The differential equation of motion (zero state response) of the resistor-inductor-capacitor circuit is not suitable to be described by simple determinants or random variables due to the influence of some uncertainties, such as manufacturing consistency differences of the capacitor and the resistor element, circuit connection stability, working time errors of operators, external electromagnetic environment interference and the like. Therefore, the existing method for optimizing the fault repair reliability time of the second-order RLC circuit based on the random differential equation has the defect that the result is often too ideal, so that the situation outside the design is easy to occur in the actual use, and even accidents are caused. In addition, the model adopted by the existing control method is difficult to optimize a circuit system lacking historical data, and for some systems with management lag or lacking data storage conditions, the traditional optimization method is difficult to meet the requirement of shortening the reliable time of system fault repair.
Compared with the circuit control system under the random system, almost all sample paths of the Liu process are continuous functions of Lipschitz, so that the Liu process is considered to be a better choice for describing the motion process.
Disclosure of Invention
The invention aims to solve the technical problem of how to optimize the fault repair reliability time of a second-order circuit system while ensuring the reliability of the system in the control process of the fault repair reliability time of an excitation type over-damping second-order circuit system.
The principle of the invention is as follows: the invention provides a brand-new fault repair reliable time criterion based on first-arrival time, and the fault repair reliable time criterion is applied to an uncertain second-order optimization control model of an excitation type over-damping second-order RLC circuit, so that an analytical expression about the uncertain distribution of the first-arrival time is obtained, meanwhile, sufficient conditions of an optimal solution of the model are obtained, further, the optimal solution of the model under the appointed trust degree is given through a dichotomy, and the corresponding fault repair reliable time is obtained.
In order to solve the technical problem, the invention provides an optimal control method for the fault repair reliability time of an excitation type over-damping second-order RLC circuit, which comprises the following steps:
(1) giving out a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, and converting the uncertain second-order differential equation into a corresponding confirmed second-order differential equation by an alpha path (alpha-path) method; the alpha-path (alpha-path) method is a method proposed by Yaoqian in A numerical method for dissolving uncurtain differential equations;
(2) giving out a fault repairing reliable time constraint condition based on an uncertain first arrival time theorem, and introducing the fault repairing reliable time criterion into an uncertain excitation type over-damping second-order RLC circuit state equation to obtain an analytical expression of a first arrival time distribution function;
(3) and establishing an uncertain second-order optimization control model to obtain sufficient conditions of an optimal solution of the uncertain second-order optimization control model, and finally giving the optimal solution through a numerical algorithm of a dichotomy.
In the step (1), based on an uncertain theory and an alpha-path method, an uncertain second-order differential equation based on an excitation type over-damping second-order RLC circuit zero-state response equation is provided, and specifically comprises the following steps: based on the self-inductance L, the resistance coefficient R, the capacitance coefficient C, the variance sigma and the external excitation w of the actual circuit, the corresponding time is obtainedt zero state response equation of change and going to capacitor voltage XtLeading in uncertainty to obtain corresponding definite solution of uncertain differential equation, namely alpha path
Figure BDA0002574607140000041
Thus, the state equation of the uncertain excitation type over-damping second-order RLC circuit is given as follows:
Figure BDA0002574607140000042
in which the inverse distribution function is not determined
Figure BDA0002574607140000043
In the step (2), a fault repair reliable time criterion based on an uncertain first-arrival time theorem is given, the fault repair reliable time criterion is introduced into an uncertain excitation type over-damping second-order RLC circuit state equation, and an analytical expression of a first-arrival time distribution function is obtained, and the method specifically comprises the following steps:
(2.1) utility function J (X)t) Is a function of the increment of X when J (X)t) The time τ to reach a given level z for the first time is of the form:
τ＝min{t≥0|J(Xt)≥z}
let T be the expiration time, then the fault repair reliability time [ tau ]z]max(beta) is T epsilon [0, T ∈]The minimum value of the system running time s that can be reached while ensuring the system reliability is obtained, so that the following fault repair reliability time criterion is obtained:
Figure BDA0002574607140000044
the beta value indicates the confidence of the user on the system reaching the specified state, namely an optimistic value, and represents the confidence degree of the user on the system recovery in the process of slowly maintaining a second-order circuit system;
(2.2) preferred utility function J (X)t)＝Xt-w3To find the first arrivalA distribution function of time tau is selected to satisfy
Figure BDA0002574607140000051
α of (1), I1＝{α∈(0,1)|-w-σΦ-1(α)≥0}，I2＝{α∈(0,1)|-w-σΦ-1(α) < 0}, obviously, (0,1) ∈ I2Therefore I is2Is a non-empty set; when alpha is epsilon to I1，
Figure BDA0002574607140000052
Then there is a minimum α, that satisfies the utility function by z0
Figure BDA0002574607140000053
(2.3) giving out a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, introducing a fault repairing reliable time constraint condition based on a first-arrival time theorem into the uncertain second-order differential equation, and obtaining a distribution function U (s, w) of the first-arrival time tau as follows:
Figure BDA0002574607140000054
wherein the intermediate variable
Figure BDA0002574607140000061
In the step (3), an uncertain second-order optimization control model is established, sufficient conditions of an optimal solution of the uncertain second-order optimization control model are obtained, and finally the optimal solution of the model under the appointed trust degree and the corresponding fault repair reliable time are given through a numerical algorithm of a dichotomy, and the method specifically comprises the following steps:
(3.1) substituting the distribution function U (s, w) of the first arrival time tau into the formula (2) to obtain the fault repair reliable time
Figure BDA0002574607140000062
Wherein
Figure BDA0002574607140000063
Is a measure, τzFor the first arrival time of the utility function to z, | represents as a condition symbol that the variable on its left side should satisfy its condition;
establishing a fault repair reliable time model of the uncertain excitation type over-damping second-order RLC circuit system based on the formula (1) as follows:
Figure BDA0002574607140000064
(3.2) to make the optimal solution exist, and the optimal solution can be solved by dichotomy in the range, it is required to satisfy
[τz]max(β)＝U-1(1-β)
U(s,w)＝1-β
Wherein U is-1The inverse of the first arrival time distribution function. The above conditions are equivalent to:
Figure BDA0002574607140000071
by the formula as above, the compound has the advantages of high purity,
Figure BDA0002574607140000072
for convenience of solution, let equal sign right be Q (t):
Figure BDA0002574607140000073
easily obtain the first derivative thereof
Figure BDA0002574607140000074
And is
0＜Q(t)＜1
Therefore, it is necessary to satisfy
Figure BDA0002574607140000075
β＞1/2；
Order to
Figure BDA0002574607140000076
Is easily obtained by the above conditions
0 < S (w) < 1 formula (6)
The guiding rule is
Figure BDA0002574607140000081
Inscription of the molecule is
Figure BDA0002574607140000082
Is easy to obtain
Figure BDA0002574607140000083
And is
Figure BDA0002574607140000084
Therefore, it is sufficient that V (w) has at least one zero point
Figure BDA0002574607140000085
β＞1/2，w∈(0,1)
In addition, the first derivative of V (w)
Figure BDA0002574607140000086
Due to the monotonic decrease of V (w), V (w) has only one zero point w in the interval (0,1)*I.e. the optimal solution of the model;
(3.3) it was then confirmed that the failure recovery reliability time corresponding to the optimal solution in this range can be found by the dichotomy, and the value is obtained,
from equation (6), for a given zero point w*Must have
0＜S(w*)＜1
Is prepared from (4) formula (I), is easy to obtain
Figure BDA0002574607140000091
Figure BDA0002574607140000092
By combining the above with the formula (5), a compound of formula (5) can be obtained
Figure BDA0002574607140000093
Figure BDA0002574607140000094
In summary, Q (t) -S (w)*) Is a monotonically decreasing function with and without a zero
Figure BDA0002574607140000095
Therefore, the optimal solution for solving the control can be obtained by a dichotomy;
finally solving Q (t) -S (w) by dichotomy*) And obtaining the fault repairing reliable time corresponding to the model optimal solution under the appointed trust degree at the zero point of 0.
The invention has the beneficial effects that:
according to the control method, the uncertain optimal control model can be converted into a corresponding uncertain model through a distribution function of the first arrival time tau; the uncertain fault repair reliable time model ensures that the system reaches a better state z under the excitation w after the specified expiration time T, so that the fault repair reliable time of the system is better than the fault repair time corresponding to other non-optimal excitations under the specified trust degree. The invention is widely applied to repair systems and has better application value to systems including a second-order RLC circuit.
Drawings
FIG. 1 is a flow chart of a control method of the present invention;
FIG. 2 is a schematic diagram of a deterministic time response;
FIG. 3 is a schematic view of an indeterminate time response;
FIG. 4 is a diagram of sensitivity analysis of optimistic β and variance σ;
FIG. 5 is a graph showing sensitivity analysis of an optimistic value β with a given level z;
FIG. 6 is a diagram of a sensitivity analysis of variance σ with a given level z.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of the invention.
FIG. 1 is a flow chart of a control method of the present invention;
the invention discloses an optimal control method of an excitation type over-damping second-order RLC circuit, which is characterized by comprising the following steps:
(1) giving out a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, and converting the uncertain second-order differential equation into a corresponding confirmed second-order differential equation by an alpha path method;
(2) giving out a fault repairing reliable time constraint condition based on an uncertain first arrival time theorem, and introducing the fault repairing reliable time criterion into an uncertain excitation type over-damping second-order RLC circuit state equation to obtain an analytical expression of a first arrival time distribution function;
(3) and finally, giving the optimal solution of the model under the appointed trust degree and the corresponding fault repair reliable time through a numerical algorithm of a dichotomy.
Considering a criterion based on the reliability time of fault recoveryThe excitation type over-damping second-order RLC circuit model, in which the resistance R is 4, the self-inductance L is 1, the capacitance C is 2, the optimistic value β is 0.9, the variance σ is 0.5, and the given state level z is 0.2. Due to the fact that
Figure BDA0002574607140000111
The model therefore operates specifically as follows:
in the step (1), based on an uncertain theory and an alpha-path method, an uncertain second-order differential equation based on an excitation type over-damping second-order RLC circuit zero-state response equation is provided, and specifically comprises the following steps: based on the self-inductance L, the capacitance C, the variance sigma and the external excitation w of the actual circuit, a corresponding zero-state response equation is obtained, and the capacitance voltage X is converted into the capacitance voltagetLeading in uncertainty to obtain corresponding alpha path
Figure BDA0002574607140000112
Thus, the state equation of the uncertain excitation type over-damping second-order RLC circuit is given as follows:
Figure BDA0002574607140000113
in which the inverse distribution is not determined
Figure BDA0002574607140000114
In the step (2), a fault repair reliable time criterion based on the uncertain first arrival time theorem is provided, and an uncertain excitation type over-damping second-order RLC circuit state equation is introduced to obtain an analytical expression of a first arrival time distribution function, and the method specifically comprises the following steps:
(2.1)J(Xt) Is a function of the increment of X when J (X)t) The time τ to reach a given level z for the first time is of the form:
τ＝min{t≥0|J(Xt)≥z}
then, the fault repair reliability time [ tau ]z]max(beta) is T epsilon [0, T ∈]The minimum value of the system running time s that can be achieved while ensuring the system reliability is obtainedThe following fail-over reliability times are reached:
Figure BDA0002574607140000115
(2.2) preferred utility function J (X)t)＝Xt-w3. To find the distribution function of the first arrival time tau, a selection is made which satisfies
Figure BDA0002574607140000121
α of (a). Let I1＝{α∈(0,1)|-w-σΦ-1(α)≥0}，I2＝{α∈(0,1)|-w-σΦ-1(α) < 0 }. Obviously, (0,1) ∈ I2Therefore I is2≠φ。
When alpha is epsilon to I1，
Figure BDA0002574607140000122
Then there is
Figure BDA0002574607140000123
(2.3) giving a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, and introducing a fault repair reliable time criterion based on a first-arrival time theorem into the uncertain second-order differential equation, thereby obtaining a distribution function of the first-arrival time tau as follows:
Figure BDA0002574607140000124
wherein
Figure BDA0002574607140000125
In the step (3), an uncertain second-order optimization control model is established to obtain sufficient conditions of the optimal solution of the model, and finally the optimal solution is given through a numerical algorithm of a dichotomy, and the method specifically comprises the following steps:
(3.1) substituting the distribution function U (s, w) of the first arrival time tau into the formula (2) to obtain
Figure BDA0002574607140000131
Establishing a fault repair reliable time model of the uncertain excitation type over-damping second-order RLC circuit system based on the formula (1) as follows:
Figure BDA0002574607140000132
(3.2) to make the optimal solution exist, and the optimal solution can be solved by dichotomy in the range, it is required to satisfy
[τz]max(β)＝U-1(1-β)
U(s,w)＝1-β
Wherein U is-1The inverse of the first arrival time distribution function. That is to
Figure BDA0002574607140000133
By the formula as above, the compound has the advantages of high purity,
Figure BDA0002574607140000134
order to
Figure BDA0002574607140000141
Satisfy the requirement of
0＜S(w)＜1#(6)
The guiding rule is
Figure BDA0002574607140000142
Inscription of the molecule is
Figure BDA0002574607140000143
Is easy to obtain
Figure BDA0002574607140000144
And is
Figure BDA0002574607140000145
Therefore, it is sufficient that V (w) has at least one zero point
Figure BDA0002574607140000146
β＜1/2，w∈(0,1)
In addition, the first and second substrates are,
Figure BDA0002574607140000147
due to the monotonic decrease of V (w), V (w) has only one zero point w in the interval (0,1)*I.e. the optimal solution of the model.
Order to
V(w)＝0
Get it solved
w*＝0.3630
(3.3) it is next proved that the failure recovery reliability time corresponding to the optimal solution in this range can be found by the dichotomy, and the value is obtained.
From the formula (6) for a given w*Must have
0＜S(w*)＜1
Is prepared from (4) formula (I), is easy to obtain
Figure BDA0002574607140000151
Figure BDA0002574607140000152
By combining the above with the formula (5), a compound of formula (5) can be obtained
Figure BDA0002574607140000153
Figure BDA0002574607140000154
In summary, Q (t) -S (w)*) Is a monotonically decreasing function with and without a zero
Figure BDA0002574607140000155
After the syndrome is confirmed. Finally solving Q (t) -S (w) by dichotomy*) And (3) obtaining the fault repairing reliable time corresponding to the model optimal solution under the appointed trust degree:
Figure BDA0002574607140000156
note that this result is consistent with the response first arrival z at α ═ 0.9 in fig. 3, which is less than the first arrival time in the ideal case of fig. 2, indicating that model repair and maintenance have occurred under the influence of human uncertainty, resulting in a shorter fail-safe time.
FIGS. 4 to 6 show the sensitivity analysis results of model fault repair reliability time with respect to different parameters under a specified confidence level, and it can be seen that the fault repair reliability time
Figure BDA0002574607140000161
Inversely proportional to the optimistic value β and the variance σ, and proportional to the given level z, indicate that the method is consistent with the actual situation.
In the first analysis, the optimal control modeling is carried out by introducing a fault repair reliable time criterion, and finally the optimal solution of the model under the appointed confidence level and the corresponding fault repair reliable time are obtained by solving. In the present embodiment, compared with the existing ideal method, the optimal control method for the excitation type over-damping second-order RLC circuit provided by the present invention is more in line with the actual situation, and can meet the user requirements of different belief degrees.

Claims (6)

1. An optimal control method for excitation type over-damping second-order RLC circuit fault repair reliable time is characterized by comprising the following steps:
(1) giving out a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, and converting the uncertain second-order differential equation into a corresponding confirmed second-order differential equation by an alpha path (alpha-path) method;
(2) giving out a fault repairing reliable time constraint condition based on an uncertain first arrival time theorem, and introducing the fault repairing reliable time criterion into an uncertain excitation type over-damping second-order RLC circuit state equation to obtain an analytical expression of a first arrival time distribution function;
(3) and establishing an uncertain second-order optimization control model to obtain sufficient conditions of an optimal solution of the uncertain second-order optimization control model, and finally giving the optimal solution through a numerical algorithm of a dichotomy.
2. The optimal control method for the fault repair reliability time of the excitation type over-damping second-order RLC circuit according to claim 1, wherein in the step (1), the specific process is as follows: based on the self-inductance L, the resistance coefficient R, the capacitance coefficient C, the variance sigma and the external excitation w of the actual circuit, a corresponding zero state response equation changing along with the time t is obtained, and the zero state response equation is converted into the capacitance voltage XtLeading in uncertainty to obtain corresponding definite solution of uncertain differential equation, namely alpha path
Figure FDA0002574607130000011
Thus, the state equation of the uncertain excitation type over-damping second-order RLC circuit is given as follows:
Figure FDA0002574607130000012
in which the inverse distribution function is not determined
Figure FDA0002574607130000013
3. The optimal control method for the fault recovery reliability time of the excitation type over-damping second-order RLC circuit as claimed in claim 1, wherein the step (2) specifically comprises the following steps:
(2.1) utility function J (X)t) Is a function of the increment of X when J (X)t) The time τ to first reach the repaired state z has the following form:
τ＝min{t≥0|J(Xt)≥z}
let T be the expiration time, then the fault repair reliability time [ tau ]z]max(beta) is T epsilon [0, T ∈]Then, the minimum value of the system running time s that can be reached under the condition of ensuring the system reliability is obtained, so that the following fault repairing reliable time is obtained:
Figure FDA0002574607130000021
the beta value indicates the confidence of the user on the system reaching a specified state, namely an optimistic value, and represents the confidence degree of the user on the system repair in the process of slow maintenance of a second-order circuit system;
(2.2) preferred utility function J (X)t)＝Xt-w3To find the distribution function of the first arrival time tau, a value satisfying
Figure FDA0002574607130000022
α of (1), I1＝{α∈(0，1)|-w-σΦ-1(α)≥0}，I2＝{α∈(0，1)|-w-σΦ-1(α) < 0}, obviously, (0,1) ∈ I2Therefore I is2Is a non-empty set; when alpha is epsilon to I1，
Figure FDA0002574607130000023
Then there is a minimum α, that satisfies the utility function by z0
Figure FDA0002574607130000024
Figure FDA0002574607130000031
(2.3) giving out a corresponding uncertain second-order differential equation based on a zero state response equation of a second-order RLC circuit system, introducing a fault repairing reliable time constraint condition based on a first-arrival time theorem into the uncertain second-order differential equation, and obtaining a distribution function U (s, w) of the first-arrival time tau as follows:
Figure FDA0002574607130000032
wherein the intermediate variable
Figure FDA0002574607130000033
4. The optimal control method for the fault repair reliability time of the excitation type over-damping second-order RLC circuit as claimed in claim 1, wherein in the step (3), the specific process of establishing the uncertain second-order optimal control model comprises the following steps:
substituting the distribution function U (s, w) of the first arrival time tau into the formula (2) to obtain the fault repair reliable time
Figure FDA0002574607130000034
Wherein
Figure FDA0002574607130000035
Is a measure, τzFor the first arrival time of the utility function to reach the better state z, | represents that the variable on the left side thereof should satisfy the condition thereof as a condition symbol;
establishing a fault repair reliable time model of the uncertain excitation type over-damping second-order RLC circuit system based on the formula (1) as follows:
Figure FDA0002574607130000041
5. the optimal control method for the fault repair reliability time of the excitation type over-damping second-order RLC circuit as claimed in claim 1, wherein in step (3), the process of obtaining the sufficient condition of the optimal solution of the uncertain second-order optimal control model is as follows:
to make the optimal solution exist and the optimal solution can be solved by dichotomy in the range, it needs to satisfy
[τz]max(β)＝U-1(1-β)
U(s，w)＝1-β
Wherein U is-1The above condition is equivalent to:
Figure FDA0002574607130000042
by the formula as above, the compound has the advantages of high purity,
Figure FDA0002574607130000043
for convenience of solution, let equal sign right be Q (t):
Figure FDA0002574607130000044
easily obtain the first derivative thereof
Figure FDA0002574607130000051
And is
0＜Q(t)＜1
Therefore, it is necessary to satisfy
Figure FDA0002574607130000052
β＞1/2；
Order to
Figure FDA0002574607130000053
Is easily obtained by the above conditions
0 < S (w) < 1 formula (6)
The guiding rule is
Figure FDA0002574607130000054
Inscription of the molecule is
Figure FDA0002574607130000055
Is easy to obtain
Figure FDA0002574607130000056
And is
Figure FDA0002574607130000061
Therefore, it is sufficient that V (w) has at least one zero point
Figure FDA0002574607130000062
β＞1/2，w∈(0，1)
In addition, the first derivative of V (w)
Figure FDA0002574607130000063
Due to the monotonic decrease of V (w), V (w) has only one zero point w in the interval (0,1)*I.e. the optimal solution of the model.
6. The optimal control method for the fault repair reliability time of the excitation type over-damping second-order RLC circuit according to claim 1, wherein in the step (3), the optimal solution of the model under the specified confidence level is given by a numerical algorithm of dichotomy, and the specific process is as follows:
from equation (6), for a given zero point w*Must have
0＜S(w*)＜1
Is prepared from (4) formula (I), is easy to obtain
Figure FDA0002574607130000064
Figure FDA0002574607130000065
By combining the above with the formula (5), a compound of formula (5) can be obtained
Figure FDA0002574607130000066
Figure FDA0002574607130000067
In summary, Q (t) -S (w)*) Is a monotonically decreasing function with and without a zero
Figure FDA0002574607130000068
Therefore, the optimal solution for solving the control can be obtained by a dichotomy;
finally solving Q (t) -S (w) by dichotomy*) And obtaining the fault repairing reliable time corresponding to the model optimal solution under the appointed trust degree at the zero point of 0.
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