RELIABILITY OF BOUNDARY CONTROL UNDER A ONE-SIDED CONSTRAINT ON THE PARAMETER BEING CHECKED

N. P. Fefelov

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

An indicator of the effectiveness of complex systems may be a quantity determined by the probability that the parameters of the system lie in the region of admissible values. Checking the parameters of complex equipment before use makes it possible to increase effectiveness substantially. Often, in checking, it is sufficient to ascertain only that the parameters lie within the admissible region; therefore, control of parameters by tolerances is used. For some parameters the region of admissible values may be bounded only on one side. In this case, in order to determine the region in which the parameter lies, it is necessary to compare its value with the single boundary value. Such a check is called boundary control.

The presence of error in the measuring equipment may lead to errors in boundary control. In the present work an estimate is made of the reliability of boundary control performed by instruments with known error. All the arguments are valid for the control of a one-dimensional parameter.

Formulation of the problem

Let the parameter \(X\) being checked have distribution density \(f(x)\). The boundary value \(x_0\) divides the domain of existence of \(f(x)\) into two parts. The probabilities that the parameter lies in one of the two regions are equal (Fig. 1):

\[ \begin{aligned} \text{Region I}\quad & P_1=\int_{-\infty}^{x_0} f(x)\,dx,\quad \text{the parameter is less than } x_0,\\[4pt] \text{Region II}\quad & P_2=\int_{x_0}^{\infty} f(x)\,dx,\quad \text{the parameter is greater than } x_0. \end{aligned} \tag{1} \]

We shall stipulate that the second region corresponds to the region of normal operation of the device. Control is carried out in the following way. The testing device produces a control signal \(Y\); its magnitude, owing to the error of the controlling device, is random with distribution density \(f(y)\), and almost always \(M(Y)=x_0\). The recording device is constructed so that it gives the operator only the sign of the quantity \(Z=X-Y\). We shall call the readings of the recording device the outcomes of control, and the act—

states of the parameter being checked—as the results of control.

In the case considered, two outcomes of control are possible: \(Z>0\) and \(Z\leqslant 0\). The outcome \(Z>0\) logically corresponds to the state \(X>x_0\), and \(Z\leqslant 0\) to the state \(X\leqslant x_0\). However, since \(Y\) is a random variable,

Fig. 1

then two results may correspond to each outcome of control, i.e.,

\[ X>x_0 \text{ with } Z>0 \text{ — it has been found that } X>x_0, \]

\[ X\leqslant x_0 \text{ with } Z>0 \text{ — it has not been found that } X\leqslant x_0 \]
\[ \text{(undetected defect),} \]

\[ X\leqslant x_0 \text{ with } Z\leqslant 0 \text{ — it has been found that } X\leqslant x_0 \]
\[ \text{(detected defect),} \]

\[ X>x_0 \text{ with } Z\leqslant 0 \text{ — it has not been found that } X>x_0 \]
\[ \text{(false rejection).} \]

Errors of the type of an undetected defect and a false rejection reduce the quality of control. We shall take as an indicator of the quality of control the probability that the true result corresponds to the obtained outcome of control; therefore, to calculate reliability it is necessary to determine the probabilities of the results of control.

Derivation of expressions for the probabilities of the results of control

The probabilities of the results of boundary control can be represented as conditional probabilities [1]:

\[ P(X>x_0\mid Z>0)=\frac{P(X>x_0,\ Z>0)}{P(Z>0)}, \]

\[ P(X\leqslant x_0\mid Z>0)=\frac{P(X\leqslant x_0,\ Z>0)}{P(Z>0)}, \]

\[ P(X>x_0\mid Z\leqslant 0)=\frac{P(X>x_0,\ Z\leqslant 0)}{P(Z\leqslant 0)}, \tag{2} \]

\[ P(X\leqslant x_0\mid Z\leqslant 0)=\frac{P(X\leqslant x_0,\ Z\leqslant 0)}{P(Z\leqslant 0)}. \]

The probabilities of the results of control are determined by the independent random variables \(X\) and \(Y\), which in the plane \(xOy\) are distributed with density \(f(x,y)=f(x)f(y)\).

The boundaries separating the results of the check are the nonrandom quantities \(Z=0\) and \(X=x_0\), which in the coordinate system \(xOy\) are represented by the straight lines \(X-Y=0\) and \(X=x_0\) (Fig. 2). The straight lines divide the entire domain of existence of \(f(x,y)\) into four parts, corresponding to definite control results. Integrating \(f(x,y)\)

Fig. 2

over the regions given in expressions (2), one can obtain the probabilities of the control results.

It is convenient to introduce the conditional distribution densities of the parameter \(X\) for different outcomes of the control. On the basis of the general expression for the conditional density we have:

\[ f(x\mid Z>0)= \frac{f(x,Z>0)} {\displaystyle \int_{-\infty}^{\infty} f(x,Z>0)\,dx} = \frac{\displaystyle f(x)\int_{-\infty}^{x} f(y)\,dy} {\displaystyle \int_{-\infty}^{\infty} f(x)\int_{-\infty}^{x} f(y)\,dy\,dx} \tag{3} \]

and

\[ f(x\mid Z\leq 0)= \frac{\displaystyle f(x)\int_{x}^{\infty} f(y)\,dy} {\displaystyle \int_{-\infty}^{\infty} f(x)\int_{x}^{\infty} f(y)\,dy\,dx}. \tag{4} \]

Using (3) and (4), one can obtain the probabilities of the control outcomes in the following form:

\[ P(X>x_0\mid Z>0)= \int_{x_0}^{\infty} f(x\mid Z>0)\,dx \quad \text{— the parameter is in region II,} \]

\[ P(X\leq x_0,\ Z>0)= \int_{-\infty}^{x_0} f(x\mid Z>0)\,dx \quad \text{— an undetected fault,} \tag{5} \]

\[ P(X>x_0\mid Z\leq 0)= \int_{x_0}^{\infty} f(x\mid Z\leq 0)\,dx \quad \text{— a false defect,} \]

\[ P(X \leqslant x_0 \mid Z \leqslant 0) = \int_{-\infty}^{x_0} f(x \mid Z \leqslant 0)\,dx \]
—the detected fault.

Expressions (5) make it possible to compute the probabilities of the outcomes of boundary control. They are valid for arbitrary distribution laws of \(X\) and \(Y\). Knowledge of the probability of proper operation makes it possible to assess the effectiveness of the technical device after testing.

REFERENCES

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