CN108268025B - Elasticity evaluation method for networked control system under random disturbance - Google Patents

Elasticity evaluation method for networked control system under random disturbance Download PDF

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CN108268025B
CN108268025B CN201810003685.3A CN201810003685A CN108268025B CN 108268025 B CN108268025 B CN 108268025B CN 201810003685 A CN201810003685 A CN 201810003685A CN 108268025 B CN108268025 B CN 108268025B
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李瑞莹
马文停
康锐
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Beijing University of Aeronautics and Astronautics
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Abstract

The invention discloses an elasticity evaluation method for a networked control system under random disturbance, and belongs to the technical field of reliability. The method comprises four steps of determining disturbance sample amount, disturbance identification and sample distribution, disturbance sample injection and key performance parameter testing, and comprehensive evaluation of system elasticity. The invention provides a whole set of elasticity evaluation method aiming at random disturbance, which can be used for determining the elasticity level of a system, finding weak links in the system and a recovery strategy and improving the elasticity of the system; the invention considers the sample size determination method and the comprehensive evaluation method in the given confidence degree and error requirement range aiming at the probability type elasticity measurement of the elasticity mean value, and the evaluation result has statistical significance.

Description

Elasticity evaluation method for networked control system under random disturbance
Technical Field
The invention belongs to the technical field of reliability, and particularly relates to an elasticity evaluation method of a networked control system under random disturbance.
Background
The system inevitably suffers from a variety of disturbances, including external disturbances originating from natural disasters, man-made attacks, and systematic disturbances originating from internal faults. The functional interruption or performance degradation that occurs after the system is disturbed may result in considerable loss if not timely and effective recovery is achieved. The prediction, resistance, absorption, adaptation and recovery capability of the system to disturbance determine the performance degradation and recovery process, and the 'elasticity' summarizes the capability of the system to bear the disturbance and the recovery capability after the disturbance.
Assessment is an important means of determining the level of system elasticity. Because the disturbance type, disturbance intensity, system response after disturbance (system performance reduction and recovery process) and the like of the system belong to random events, a system elasticity evaluation method under random disturbance is provided by taking a networked control system as a research object. Previous elasticity analysis and assessment studies tend to be spread around a given perturbation (reference [1 ]]：Wenting Ma,Ruiying Li,Chong Jin,Rui Kang.“Resilience test and evaluation of networked control systems for given disturbances,”The2ndInternational Conference on Reliability Systems Engineering,2017), there are few methods for elasticity assessment under the effect of random perturbations. Although the U.S. department of energy has proposed an elasticity analysis framework for energy network systems that includes seven steps of "define elasticity targets-define systems and elasticity indicators-characterize threats-define disturbance levels-define and apply system models-compute outcomes-evaluate elasticity improvements" (ref.)[2]: j. p.watson, r.guttromson, c.silva-Monroy, r.jeffers, k.jones, j.ellison, c.rath, j.gearhart, d.jones, t.corbet, c.hanley and l.t.walker, "joint frame for devivelingresience measurements for the elasticity, oil, and gas segments in the United States," United States decision of elasticity.2015) by a system elasticity behavior test under various disturbance events, however, it is not clear how the disturbance behavior in the elasticity test under random disturbance should be identified and analyzed, how the sample amount and sample response should be determined, the elasticity test results should be comprehensively analyzed, and so on.
Disclosure of Invention
The invention aims to solve the disturbance identification problem, the sample amount determination problem and the sample distribution problem in the system elasticity evaluation under random disturbance, and a set of elasticity evaluation method provided by a networked control system for elasticity comprehensive analysis and evaluation to solve the evaluation problem of probability elasticity measurement such as an elasticity mean value.
The invention provides a method for evaluating the elasticity of a networked control system under random disturbance, which comprises the following steps:
the method comprises the following steps: determining the amount of a disturbance sample;
assumptions require elastic mean estimators
Figure BDA0001537883610000011
And elastic mean value
Figure BDA0001537883610000012
With an absolute error limit of δ and a confidence of 1- α, i.e. the requirement
Figure BDA0001537883610000021
When the sample capacity satisfies the large sample condition, there areApproximately obey a standard normal distribution N (0,1), where d2Is composed of
Figure BDA0001537883610000023
N is the sample size, which is determined as follows:
in the formula, zpIs the lower side p quantile of the standard normal distribution;
step two: disturbance identification and sample allocation.
Step 2.1, disturbance identification and analysis, namely, a disturbance mode, a disturbance occurrence rate and disturbance intensity possibly suffered by the system are analyzed through system-level disturbance identification and analysis;
step 2.2 sample distribution: distributing the sample size in the step one to a specific disturbance mode, and determining the intensity of disturbance injection;
(2.2.1) distributing the samples to single-point disturbance and common cause disturbance according to the disturbance occurrence frequency, wherein the details are as follows:
Figure BDA0001537883610000025
and nc＝n-ns
Where n is the sample size, ns is the sample size assigned to a single point disturbance, ncTo allocate the amount of samples to the common cause disturbance,<·>denotes rounding off, γs,iAnd ts,iRespectively representing the times of single-point disturbance of the unit i in unit time and the working time coefficient of the unit i; gamma rayc,jAnd tc,jRespectively determining the frequency of single-point disturbance of the common cause disturbance event j in unit time and the working time coefficient of an object subjected to the common cause disturbance;
(2.2.2) for single-point disturbance, continuously distributing the samples to system composition units according to the relative occurrence frequency of the disturbance, wherein the relative occurrence frequency of the disturbance of the unit i is calculated as follows:
in the formula, QiIs the number of cells i, the disturbance sample size of the cells i is distributed as follows:
ns,i＝<fs,ins>
after the distribution is finished, if
Figure BDA0001537883610000027
Increasing or decreasing the sample size of the corresponding unit according to the size of the out-of-tolerance by rounding the difference;
(2.2.3) assigning single point perturbations and co-cause perturbations to specific perturbation patterns. Distributing the sample size distribution result of the single-point disturbance in each unit and the distribution result of the system common cause disturbance through random sampling according to the relative occurrence frequency of the disturbance mode;
(2.2.4) determining the intensity of the perturbed sample distribution result: obtaining a disturbance intensity distribution result by simple random sampling by adopting an inverse function method according to the probability distribution of the disturbance intensity;
step three: disturbance sample injection and key performance parameter testing: according to the perturbation samples distributed in the steps, the corresponding definite type elasticity measurement is obtained by one-by-one evaluation
Step four: and (4) comprehensively evaluating the elasticity of the system. Elastic mean point estimates and interval estimates at confidence 1-alpha are calculated.
The invention has the advantages and positive effects that:
(1) the invention provides a whole set of elasticity evaluation method aiming at random disturbance, which can be used for determining the elasticity level of a system, finding weak links in the system and a recovery strategy and improving the elasticity of the system;
(2) according to the invention, the elasticity evaluation is taken as a target, a disturbance identification and analysis method is designed, and the method is beneficial to sample selection in system elasticity evaluation under random disturbance;
(3) the invention considers the sample size determination method and the comprehensive evaluation method in the given confidence degree and error requirement range aiming at the probability type elasticity measurement of the elasticity mean value, and the evaluation result has statistical significance.
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FIG. 1 is a schematic illustration of system elastic behavior;
FIG. 2 is a schematic overall flow chart of the networked control system elasticity evaluation method under random disturbance according to the present invention;
fig. 3 is a functional block diagram of a wireless controlled dc servo motor system.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The invention provides an elasticity evaluation method of a networked control system under random disturbance, which firstly explains the probability type elasticity measurement-elasticity mean value related to the invention
Figure BDA0001537883610000032
It is the average performance of system elasticity under random disturbance, and reflects the determined elasticity measurement of the system
Figure BDA0001537883610000033
Average level of (d). To facilitate the elasticity measurement, firstly, according to the characteristics of the performance parameters of the expectation-maximization, the expectation-minimization and the expectation-maximization, a normalization method for defining the performance P (t) of the system at the time t is as follows:
inspection of the small size:
Figure BDA0001537883610000034
inspection of the large scale:
eye type:
Figure BDA0001537883610000036
where P (t) is the system performance at time t, PminAnd PmaxAre the minimum and maximum values specified by the key performance parameters; pthIs a threshold value of a key performance parameter, Pmin<Pth<PmaxFor the eye type key performance parameter, Pth,LAnd Pth,URespectively, the lower threshold value of the eye type parameterUpper threshold and has Pmin<Pth,L<Pth,U<Pmax。
And then, defining the deterministic elastic measurement of the system aiming at the single disturbance behavior and the system response according to the performance normalization result. As shown in FIG. 1, the performance of the system without disturbance is assumed to be Q0(t) after a perturbation event r occurs, the performance of the system is represented as Qr(t), then the system determines a type elasticity metricThe following can be calculated:
in the formula, t0T is the required recovery time of the system for the start of the disturbance.
In the determination of elastic measurement
Figure BDA0001537883610000043
Based on the system probability type measurement-elastic mean value
Figure BDA0001537883610000044
The following can be calculated:
wherein E (-) is the mean value.
The overall steps of the method for evaluating the elasticity of the networked control system under random disturbance provided by the invention are shown in fig. 2, and the implementation of each step is described below.
The method comprises the following steps: and determining the amount of the disturbance sample.
And the system elasticity evaluation under random disturbance adopts a statistical inference method. Assumptions require elastic mean estimators
Figure BDA0001537883610000046
And elastic mean value
Figure BDA0001537883610000047
With an absolute error limit of δ and a confidence of 1- α, i.e. the requirement
Figure BDA0001537883610000048
When the sample volume satisfies the large sample condition (sufficiently large), there are
Figure BDA0001537883610000049
Approximately obey a standard normal distribution N (0,1), where d2Is composed of
Figure BDA00015378836100000410
Variance of (2)n is the sample size, which is determined as follows:
Figure BDA00015378836100000412
in the formula, zpThe lower p quantile of the standard normal distribution. In practical application, if the variance d is2If the value is unknown, an initial sample can be selected by a test investigation method, and the sample standard deviation of the sample is used as the estimated value of the d value.
Step two: disturbance identification and sample allocation.
And 2.1, disturbance identification and analysis. The perturbation behavior suffered by a system can be divided into two categories: external disturbances (including natural disasters, man-made attacks, and human factors) and systematic disturbances (internal faults). The natural disasters and the artificial attacks can be used as data sources through a database/statistical dictionary and a statistical model induced by literature, and the artificial factors and the internal faults can be used for providing the data sources through artificial factor analysis and fault mode influence analysis. Disturbance sample identification is carried out according to an appointed hierarchy, and a single-point disturbance and common cause disturbance specific analysis framework is shown in tables 1 and 2. And (5) iterating layer by layer to realize iterative analysis from a low convention level to a high convention level.
TABLE 1 Single-Point disturbance identification and analysis framework
Figure BDA00015378836100000413
TABLE 2 Co-cause disturbance identification and analysis framework
Figure BDA0001537883610000051
Step 2.2 sample distribution. And (4) according to the system-level disturbance identification and analysis result, distributing the sample size in the step one to a specific disturbance action object and a disturbance mode, and determining the intensity of disturbance injection.
(2.2.1) distributing the samples to single-point disturbance and common cause disturbance according to the disturbance occurrence frequency, wherein the details are as follows:
Figure BDA0001537883610000052
and nc＝n-ns(7) Wherein n is the sample size, nsTo assign the sample size to a single point disturbance, ncIs the amount of samples assigned to the common cause disturbance.<·>Denotes rounding off, γs,iAnd ts,iRespectively the frequency of single-point disturbance of the unit i in unit time and the working time coefficient (i.e. the ratio of the working time of the unit to the working time of the whole process); gamma rayc,jAnd tc,jThe number of times of single-point disturbance of the common cause disturbance event j in a unit time and the working time coefficient of the common cause disturbance object (the ratio of the longest working time of the common cause disturbance object to the whole working time) are respectively.
And (2.2.2) for single-point disturbance, continuously distributing the samples to the system composition units according to the relative occurrence frequency of the disturbance. The relative frequency of occurrence of the perturbations for cell i is calculated as follows:
Figure BDA0001537883610000053
in the formula, QiIs the number of cells i. The perturbed sample size for cell i is assigned as follows:
ns,i＝<fs,ins> (9)
after the distribution is finished, if
Figure BDA0001537883610000054
The sample size of the corresponding cell is increased or decreased in a round-robin order according to the magnitude of the out-of-tolerance.
(2.2.3) assigning single point perturbations and co-cause perturbations to specific perturbation patterns. And distributing the sample size distribution result of the single-point disturbance in each unit and the distribution result of the system common cause disturbance through random sampling according to the relative occurrence frequency of the disturbance mode. Table 3 shows a sample allocation selection manner of single-point disturbance or system common cause disturbance of the unit i, thereby determining a disturbance sample allocation result.
TABLE 3 selection of perturbed sample distribution
Figure BDA0001537883610000055
Figure BDA0001537883610000061
(2.2.4) determining the intensity of the perturbed sample distribution result. And obtaining the distribution result of the disturbance intensity by simple random sampling by adopting an inverse function method according to the probability distribution of the disturbance intensity given in the tables 1 and 2.
Step three: disturbance sample injection and key performance parameter testing. According to the perturbation samples distributed in the steps, the corresponding definite type elasticity measurement is obtained by one-by-one evaluation
Figure BDA0001537883610000062
Step 3.1, determining u typical test scenes according to the use requirements, and recording the u typical test scenes as Ben1,Ben2,…,Benu。
Step 3.2 determining key performance parameters of the system, respectively P1,P2,…,PmAnd m is a positive integer.
Step 3.3, the test is carried out under the normal condition, and the test scenes Ben are sequentially tested1～BenuRun time T*Measuring the m key performance parameters at intervals of delta t, and obtaining the key performance normalization result Q under each test scene through normalizationi,j,k,0. Wherein i represents a test scenario sequence number, j represents a key performance parameter sequence number, and k represents the number of measurements, i.e., Qi,j,k,0Represented in a test scenario BeniLower key performance parameter PjThe results are normalized for performance at time ktat.
Step 3.4 injecting disturbance sample r and carrying out test, and sequentially carrying out test in test scene Ben1～BenuRun time T*Measuring m key performance parameters at intervals of delta t time, and obtaining each key performance normalization result under each test scene under disturbance r through normalization
Figure BDA0001537883610000063
And 3.5, calculating the system elasticity measurement under the disturbance sample r.
(3.5.1) calculating by using the formula (1) and a trapezoidal formula to obtain the system elasticity of the system based on each key performance parameter, wherein the disturbance r acts on each test scene
(3.5.2) the integration of the elasticity measurement results under different test scenes is calculated as follows:
Figure BDA0001537883610000065
in the formula, deltaiIs BeniThe scene occupies a time length proportion in the whole operation process. On the basis, if various key performance parameters are integrated, the weight occupied by the key performance parameters in the system performance evaluation is considered to be carried out:
Figure BDA0001537883610000066
in the formula, ωjThe key performance parameter j is the weight of the system performance evaluation. Or the elasticity of each key performance parameter can be analyzed independently without the synthesis of the key performance parameters.
Step four: and (4) comprehensively evaluating the elasticity of the system. And calculating the elasticity mean value point estimation and the interval estimation at the confidence coefficient 1-alpha according to the system elasticity evaluation result under each sample disturbance.
(4.1) estimating the system elasticity point:
Figure BDA0001537883610000067
(4.2) interval estimation of confidence 1-alpha, unilateral lower limit:
Figure BDA0001537883610000071
(4.3) interval estimation of confidence coefficient 1-alpha, upper and lower limits of two sides:
Figure BDA0001537883610000072
and
Figure BDA0001537883610000073
when d is2When unknown, available
Figure BDA0001537883610000074
Instead.
Example (b):
the embodiment of the invention is realized by a wireless-controlled direct-current servo motor, and a functional schematic diagram of a system is shown in fig. 3, aiming at enabling the direct-current servo motor to work according to the rotating speed requirement predefined by a user. In the working process of the system, the rotating speed (physical signal) of the direct current motor is periodically acquired by the sensor, the sensing signal acquired by the sensor is transmitted to the controller by the wireless network unit, then the controller generates a control signal according to the sensing signal and a reference signal (namely the user requirement), and the generated control signal is transmitted to the execution unit by the wireless network unitThe actuator is converted into a driving signal to act on the direct current motor so as to control the rotating speed of the motor. In this embodiment, Simulink/TrueTime is used to perform simulation evaluation of the elastic mean value, where the confidence 1- α is 95%, the allowable error δ is 0.1, and the operating time of each component unit of the system is the same. User-defined maximum allowable recovery time T for a system*10 seconds.
The method comprises the following steps: and determining the amount of the disturbance sample.
By preliminary analysis, the estimated sample standard deviation d is 0.3, which can be calculated from equation (6):
Figure BDA0001537883610000075
from this, the sample volume is disturbed 35.
Step two: disturbance identification and sample allocation.
And 2.1, disturbance identification and analysis. The disturbance analysis distinguishes single-point disturbance and common cause disturbance, the lowest convention level is set as a unit level, and the disturbance identification and analysis results are shown in tables 4 and 5.
Table 4 wireless control direct current servo motor single-point disturbance identification and analysis result
Figure BDA0001537883610000076
Figure BDA0001537883610000081
TABLE 5 Wireless control DC servo motor common cause disturbance identification and analysis framework
Figure BDA0001537883610000082
Step 2.2 sample distribution.
Firstly, allocating a sample to single-point disturbance and common cause disturbance according to the disturbance occurrence frequency, and obtaining the disturbance according to the formula (2): n iss32 and nc＝3。
Then, the single-point disturbed sample is distributed to the system component unit, and the distribution result is shown in table 6:
TABLE 6 distribution of DC Servo Motor disturbance samples to cells
Figure BDA0001537883610000083
Then, single-point disturbance and common cause disturbance are allocated to a specific disturbance mode, the allocation result is shown in a 5 th column of table 7, and the intensity of the allocation result of the disturbance sample is determined according to the probability distribution of the disturbance intensity, and the result is shown in a 6 th column of table 7.
TABLE 7 distribution of DC servo motor disturbance to disturbance mode and intensity
Step three: disturbance sample injection and key performance parameter testing.
The present embodiment controls the performance parameter P only considering a typical test scenario and a key performance parameter1. Thus, for each perturbed sample, T is run under normal and perturbed conditions, respectively*Recording the performance parameter value every 10s and every 10ms, and normalizing the performance normalization result corresponding to each perturbation sampleThen, the determined elasticity under each disturbance action is obtained through testing
Figure BDA0001537883610000093
The results are shown in Table 8.
TABLE 8 elasticity evaluation results for each disturbance sample of DC servo motor
Figure BDA0001537883610000094
Step four: and (4) comprehensively evaluating the elasticity of the system. Wherein, the system elasticity point estimation:
Figure BDA0001537883610000102
estimating the interval when the confidence coefficient is 0.95, and performing unilateral lower limit;
Figure BDA0001537883610000103
interval estimation when the confidence coefficient is 0.95, upper and lower limits on two sides;
Figure BDA0001537883610000104
Figure BDA0001537883610000105
。

Claims (4)

1. the method for evaluating the elasticity of the networked control system under random disturbance is characterized by comprising the following steps:
the method comprises the following steps: determining the amount of a disturbance sample;
assumptions require elastic mean estimators
Figure FDA0002273421590000011
And elastic mean value
Figure FDA0002273421590000012
With an absolute error limit of δ and a confidence of 1- α, i.e. the requirement
Figure FDA0002273421590000013
When the sample capacity satisfies the large sample condition, there are
Figure FDA0002273421590000014
Approximately obey a standard normal distribution N (0,1), in whichd2Is composed of
Figure FDA0002273421590000015
N is the sample size, which is determined as follows:
Figure FDA0002273421590000016
in the formula, zpIs the lower side p quantile of the standard normal distribution;
step two: disturbance identification and sample allocation;
step 2.1, disturbance identification and analysis, namely, a disturbance mode, a disturbance occurrence rate and disturbance intensity possibly suffered by the system are analyzed through system-level disturbance identification and analysis;
step 2.2 sample distribution: distributing the sample size in the step one to a specific disturbance mode, and determining the intensity of disturbance injection;
(2.2.1) distributing the samples to single-point disturbance and common cause disturbance according to the disturbance occurrence frequency, wherein the details are as follows:
Figure FDA0002273421590000017
and nc＝n-ns
Wherein n is the sample size, nsTo assign the sample size to a single point disturbance, ncTo allocate the amount of samples to the common cause disturbance,<·>denotes rounding off, γs,iAnd ts,iRespectively representing the times of single-point disturbance of the unit i in unit time and the working time coefficient of the unit i; gamma rayc,jAnd tc,jRespectively determining the frequency of single-point disturbance of the common cause disturbance event j in unit time and the working time coefficient of an object subjected to the common cause disturbance;
(2.2.2) for single-point disturbance, continuously distributing the samples to system composition units according to the relative occurrence frequency of the disturbance, wherein the relative occurrence frequency of the disturbance of the unit i is calculated as follows:
Figure FDA0002273421590000018
in the formula, QiIs the number of cells i, the disturbance sample size of the cells i is distributed as follows:
ns,i＝〈fs,ins〉
after the distribution is finished, if
Figure FDA0002273421590000019
Increasing or decreasing the sample size of the corresponding unit according to the size of the out-of-tolerance by rounding the difference;
(2.2.3) assigning single point perturbations and common cause perturbations to specific perturbation patterns; distributing the sample size distribution result of the single-point disturbance in each unit and the distribution result of the system common cause disturbance through random sampling according to the relative occurrence frequency of the disturbance mode;
(2.2.4) determining the intensity of the perturbed sample distribution result: obtaining a disturbance intensity distribution result by simple random sampling by adopting an inverse function method according to the probability distribution of the disturbance intensity;
step three: disturbance sample injection and key performance parameter testing: according to the perturbation samples distributed in the steps, the corresponding definite type elasticity measurement is obtained by one-by-one evaluation
Figure FDA0002273421590000021
Step four: comprehensively evaluating the elasticity of the system; elastic mean point estimates and interval estimates at confidence 1-alpha are calculated.
2. The networked control system elasticity evaluation method under random disturbance according to claim 1, characterized in that: the perturbation behavior suffered by the system is divided into two categories: external disturbances and systematic disturbances; disturbance identification is carried out according to an appointed hierarchy, and a single-point disturbance and common cause disturbance specific analysis framework is shown in tables 1 and 2; iteration is carried out layer by layer, and iterative analysis from a low convention level to a high convention level is realized;
TABLE 1 Single-Point disturbance identification and analysis framework
Figure FDA0002273421590000022
TABLE 2 Co-cause disturbance identification and analysis framework
Figure FDA0002273421590000023
3. The networked control system elasticity evaluation method under random disturbance according to claim 1, characterized in that: the third step is specifically as follows:
step 3.1, determining u typical test scenes according to the use requirements, and recording the u typical test scenes as Ben1,Ben2,…,Benu；
Step 3.2 determining key performance parameters of the system, respectively P1,P2,…,PmM is a positive integer;
step 3.3, the test is carried out under the normal condition, and the test scenes Ben are sequentially tested1～BenuRun time T*Measuring the m key performance parameters at intervals of delta t, and obtaining the key performance normalization result Q under each test scene through normalizationi,j,k,0(ii) a Wherein i represents a test scenario sequence number, j represents a key performance parameter sequence number, and k represents the number of measurements, i.e., Qi,j,k,0Represented in a test scenario BeniLower key performance parameter PjNormalizing the performance result at the kth delta t moment;
step 3.4 injecting disturbance sample r and carrying out test, and sequentially carrying out test in test scene Ben1～BenuRun time T*Measuring m key performance parameters at intervals of delta t time, and obtaining each key performance normalization result under each test scene under disturbance r through normalization
Figure FDA0002273421590000024
Step 3.5, calculating the elasticity measurement of the disturbance sample r;
(3.5.1) calculating the action of the disturbance r in each test sceneSystem elasticity based on key performance parameters
Figure FDA0002273421590000025
(3.5.2) the integration of the elasticity measurement results under different test scenes is calculated as follows:
Figure FDA0002273421590000031
in the formula, deltaiIs BeniThe scene occupies a time length proportion in the whole operation process; if various key performance parameters are to be integrated, the integrated result of the elastic measurement under each different test scene is considered
Figure FDA0002273421590000032
And performing comprehensive calculation after the weights respectively occupied in the system performance evaluation:
Figure FDA0002273421590000033
in the formula, ωjThe key performance parameter j is the weight of the system performance evaluation.
4. The networked control system elasticity evaluation method under random disturbance according to claim 1, characterized in that: the fourth step is specifically as follows:
(4.1) estimating the system elasticity point:
Figure FDA0002273421590000034
(4.2) interval estimation of confidence 1-alpha, unilateral lower limit:
Figure FDA0002273421590000035
(4.3) interval estimation of confidence coefficient 1-alpha, upper and lower limits of two sides:
Figure FDA0002273421590000036
and
Figure FDA0002273421590000037
when d is2When unknown, use
Figure FDA0002273421590000038
Instead.
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