PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 155 1968

CALCULATION OF THE RELIABILITY OF LIMIT CONTROL

FOR CERTAIN DISTRIBUTION LAWS OF THE TESTED

PARAMETER AND THE ERROR OF CONTROL EQUIPMENT

N. P. FEFËLOV

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In [1] general formulas were derived for calculating the probabilities of the results of limit control in the case when the tested parameter is bounded on only one side. In the present work they are considered for the most frequently encountered distribution laws of the controlled parameter and of the error of the checking device.

The case of uniform distribution laws

Let us consider the probabilities of the control results for uniform distribution laws of \(X\) and \(Y\). Although this type of law is encountered practically very rarely, it is of interest because an error distributed according to a uniform law has the most “destructive” properties. In this case the reliability of control is the smallest in comparison with other distribution laws.

Let the tested parameter be uniformly distributed on the interval \(x_{\min} \div x_{\max}\) of width \(L\) (Fig. 1).

\[ f(x)= \begin{cases} 0, & X \leq x_{\min},\\[4pt] \dfrac{1}{L}, & x_{\min}<X\leq x_{\max},\\[4pt] 0, & X>x_{\max}. \end{cases} \]

The device being tested is considered serviceable if \(X\) lies in the interval \(X_0 \div X_{\max}\). The probability of this event before control is

\[ P_{\mathrm{n}}=\frac{X_{\max}-X_0}{L}. \]

The checking device issues a random control signal \(Y\), which is uniformly distributed on the interval \(y_{\min} \div y_{\max}\) of width \(2\gamma\) (Fig. 1), i.e.

\[ f(y)= \begin{cases} 0, & Y\leq y_{\min},\\[4pt] \dfrac{1}{2\gamma}, & y_{\min}<Y\leq y_{\max},\\[4pt] 0, & Y>y_{\max}. \end{cases} \tag{2} \]

The conditional distribution densities of the parameter \(X\) after inspection, for uniform laws, have the form:

\[ f(x \mid Z>0)= \begin{cases} 0, & X \leqslant x_0-\gamma,\\[4pt] \dfrac{1}{2\gamma}\cdot\dfrac{X-x_0+\gamma}{x_{\max}-x_0}, & x_0-\gamma<X\leqslant x_0+\gamma,\\[8pt] \dfrac{1}{x_{\max}-x_0}, & x_0+\gamma<X\leqslant x_{\max},\\[8pt] 0, & X>x_{\max}. \end{cases} \tag{3} \]

Fig. 1

\[ f(x \mid Z<0)= \begin{cases} 0, & X\leqslant x_{\min},\\[4pt] \dfrac{1}{x_0-x_{\min}}, & x_{\min}<X\leqslant x_0-\gamma,\\[8pt] \dfrac{1}{2\gamma}\cdot\dfrac{x_0+\gamma-X}{x_0-x_{\min}}, & x_0-\gamma<X\leqslant x_0+\gamma,\\[8pt] 0, & X>x_0+\gamma, \end{cases} \tag{4} \]

Integrating (3) and (4), we obtain the probabilities of the inspection results:

\[ \begin{aligned} \text{a) }\quad P_{\mathrm{oi}}&=P(X>x_0\mid Z>0) =1-\frac{\gamma}{4(x_{\max}-x_0)} &&\text{— detected malfunction;}\\[6pt] \text{b) }\quad P_{\mathrm{nd}}&=P(X<x_0\mid Z>0) =\frac{\gamma}{4(x_{\max}-x_0)} &&\text{— undetected defect;}\\[6pt] \text{c) }\quad P_{\mathrm{lo}}&=P(X>x_0\mid Z<0) =\frac{\gamma}{4(x_0-x_{\min})} &&\text{— false rejection;}\\[6pt] \text{d) }\quad P_{\mathrm{od}}&=P(X<x_0\mid Z<0) =1-\frac{\gamma}{4(x_0-x_{\min})} &&\text{— detected defect.} \end{aligned} \tag{5} \]

Figure 2 shows curves constructed from formulas 5(a) and 5(b). Along the abscissa axis the values of \(P_{\mathrm{n}}\) are plotted. The parameter of the curves is the relative width of the distribution of the control signal

\[ \delta = \frac{\gamma}{L}. \]

Fig. 2

The curves for the probabilities of a detected defect and a false rejection have an analogous form; they can be obtained by replacing the designations \(P_{\mathrm{n}}\) by \(1 - P_{\mathrm{n}}\), \(P_{\mathrm{oi}}\) by \(P_{\mathrm{od}}\), and \(P_{\mathrm{nd}}\) by \(P_{\mathrm{lo}}\).

By specifying the value of \(P_{\mathrm{oi}}\) (or any other) and knowing \(P_{\mathrm{n}}\), one can determine \(\delta\) from the graph and, consequently, the permissible error of the test equipment.

Case in which the tested parameter has a uniform distribution law, and the error of the control device is normally distributed

Let \(X\) have a distribution density of the form (1), and let the control signal \(Y\) have a normal distribution law

\[ f(y)=\frac{1}{\sqrt{2\pi}\,\sigma_y}\exp\left[-\frac{(y-x_0)^2}{2\sigma_y^2}\right]. \tag{6} \]

For this case, expressions (5) [1] take the form:

\[ P_{\mathrm{oi}} = \frac{ P_{\mathrm{n}}+P_{\mathrm{n}}\Phi\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) - }{ 1+P_{\mathrm{n}}\Phi\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) -(1-P_{\mathrm{n}})\Phi\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) + } \to \]

\[ \to \frac{ -\sqrt{\frac{2}{\pi}}\delta \left\{ 1-\exp\left[-\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] \right\} }{ +\sqrt{\frac{2}{\pi}}\delta \left\{ \exp\left[-\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] - \exp\left[-\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] \right\} }, \tag{7} \]

\[ P_{\mathrm{nd}}=1-P_{\mathrm{oi}}, \]

\[ P_{\mathrm{od}} = \frac{ (1-P_{\mathrm{n}})+(1-P_{\mathrm{n}})\Phi\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) - }{ 1-P_{\mathrm{n}}\Phi\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) +(1-P_{\mathrm{n}})\Phi\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right) + } \to \]

\[ \to \frac{ -\sqrt{\frac{2}{\pi}}\delta \left\{ 1-\exp\left[-\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] \right\} }{ +\sqrt{\frac{2}{\pi}}\delta \left\{ \exp\left[-\left(\frac{1-P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] - \exp\left[-\left(\frac{P_{\mathrm{n}}}{\sqrt{2}\sigma_y}\right)^2\right] \right\} }, \]

\[ P_{\mathrm{lo}}=1-P_{\mathrm{od}}, \]

where

\[ P_{\mathrm{n}}=\frac{x_{\max}-x_0}{L} \]

is the initial (before control) probability of serviceable operation with respect to the parameter \(X\);

\[ \delta=\frac{\sigma_y}{L} \]

is the relative root-mean-square deviation of the control signal \(Y\);

\[ \Phi(u)=\frac{2}{\sqrt{\pi}}\int_0^u \exp[-t^2]\,dt \]

is the Laplace function.

The Laplace function and \(\exp[-t^2]\) asymptotically rapidly approach constant values; therefore formulas (7) can be simplified:

\[ P_{\mathrm{oi}}=1-\frac{\delta}{\sqrt{2\pi}P_{\mathrm{n}}}, \]

\[ P_{\mathrm{nd}}=\frac{\delta}{\sqrt{2\pi}P_{\mathrm{n}}}, \]

\[ P_{\mathrm{od}}=1-\frac{\delta}{\sqrt{2\pi}(1-P_{\mathrm{n}})}, \tag{8} \]

\[ P_{\mathrm{lo}}=\frac{\delta}{\sqrt{2\pi}(1-P_{\mathrm{n}})} . \]

The simplified formulas give four significant figures after the decimal point when \(\dfrac{P_{\mathrm{n}}}{\delta}>4.25\), and five significant figures when \(\dfrac{P_{\mathrm{n}}}{\delta}>4.8\).

Figure 3 shows graphs for \(P_{\mathrm{on}}\) and \(P_{\mathrm{nd}}\). In order that the graphs in Figs. 2 and 3 may be compared with one another, the value of the parameter

Fig. 3

\(\delta\) was chosen from the relation \(3\sigma_y=\gamma\), so that the width of the distribution of the control signal would be equal to its width for

uniform law. This corresponds to the same magnitude of the error of the measuring equipment under uniform and normal distribution laws.

Probabilities of Control Results under Normal Laws

Let us now consider the case when the parameter being tested and the control signal have normal distribution laws.

Fig. 4

\[ f(x)=\frac{1}{\sqrt{2\pi}\sigma_x}\exp\left[-\left(\frac{x-m_x}{\sqrt{2}\sigma_x}\right)^2\right], \tag{9} \]

\[ f(y)=\frac{1}{\sqrt{2\pi}\sigma_y}\exp\left[-\left(\frac{y-x_n}{\sqrt{2}\sigma_y}\right)^2\right]. \tag{10} \]

In the case of normal laws, the probabilities of the control results are best calculated as conditional probabilities using the general formulas [1]. Thus, for example, the probability of a good-

of operation \((X > x_0)\), under the condition of obtaining a positive control result \((Z > 0)\), will be equal to:

\[ P(X>x_0\mid Z>0)= \frac{ \displaystyle \int_{x_0}^{\infty}\int_{-\infty}^{x} \frac{1}{2\pi\sigma_x\sigma_y} \exp\left[ -\left(\frac{x-m_x}{\sqrt{2}\sigma_x}\right)^2 -\left(\frac{y-x_0}{\sqrt{2}\sigma_y}\right)^2 \right]\,dy\,dx }{ \displaystyle \int_{-\infty}^{\infty}\int_{-\infty}^{x} \frac{1}{2\pi\sigma_x\sigma_y} \exp\left[ -\left(\frac{x-m_x}{\sqrt{2}\sigma_x}\right)^2 -\left(\frac{y-x_0}{\sqrt{2}\sigma_y}\right)^2 \right]\,dy\,dx }. \]

The integrals in this expression cannot be evaluated by ordinary methods. However, there are tables [2] in which integrals of the two-dimensional normal density are tabulated. The tables make it possible to calculate numerically the conditional probabilities of the control results. We note that the denominator is the probability

\[ P(Z>0)=\int_{0}^{\infty} f(z)\,dz. \]

The distribution law of \(Z\) is a composition of the normal distribution laws of the random variables \(X\) and \(Y\). In this case the distribution law of \(Z\) is also normal [3]:

\[ f(z)=\frac{1}{\sqrt{2\pi}\sigma_z} \exp\left[ -\left(\frac{z-Lm_z}{\sqrt{2}\sigma_z}\right)^2 \right], \]

where \(\sigma_z=\sqrt{\sigma_x^2+\sigma_y^2}\), \(m_z=m_x-m_y=m_x-x_0\).

This makes it possible to simplify the expressions for the control results. The formulas for the probabilities of the other control results can be brought to an analogous form. The probabilities were calculated from the tables [2], and graphs were constructed from the results (Fig. 4). The parameter of the curves is the ratio of the root-mean-square deviations of the quantities \(Y\) and \(X\), i.e.,

\[ \delta=\frac{\sigma_y}{\sigma_x}. \]

All the remarks made concerning the graphs in Figs. 2 and 3 are also valid for Fig. 4 on p. 34.

Use of the Results Obtained in Practice

The problem of the reliability of control, depending on the specific conditions, can be considered from two sides:

1) the distribution density of the parameter \(f(x)\) and the probability of proper operation of the device before control \(P_{\mathrm{n}}\) are specified, as well as the distribution law of the error of the control equipment; it is necessary to determine the probabilities of the control results, in particular the probability of proper operation after control;

2) the probability of proper operation of the device after control \(P_{\mathrm{ok}}\) with respect to the parameter \(X\) is specified, and its distribution law is known; it is necessary to determine the maximum permissible error of the control equipment that ensures the required value of \(P_{\mathrm{ok}}\).

The graphs presented make it possible to solve the stated problems easily.

REFERENCES

1. N. P. Fefelov. Reliability of limit control with one-sided limitation of the parameter being tested (in the present collection).

2. N. V. Smirnov, A. N. Bolshev. Tables for Calculating the Bivariate Normal Distribution Function. Academy of Sciences of the USSR, 1962.

3. E. S. Venttsel. Probability Theory. Fizmatgiz, 1962.

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