Individual Prediction of Gradual Failures in Automation Devices

R. N. Lyublinsky, F. F. Idrisov

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In connection with the increasing complexity of automation devices and the responsibility of the tasks they perform, ensuring the reliable operation of these devices over long periods of time is a very difficult problem. The effectiveness of solving this problem is determined to a considerable degree by the reliability of predicting the failure-free operation of elements and of the device as a whole.

By prediction of failure-free operation is meant the determination of probabilistic characteristics of the operable state, i.e., the probability of failure-free operation of a device over a certain time interval, the distribution law of the time of failure-free operation, its individual moments—the mean time of failure-free operation, the variance of the mean time, etc.

Methods for predicting failure-free operation are classified primarily according to the a priori information used for prediction. The following are used as a priori information:

a) statistics of element failures;

b) logical and functional relationships between the states of individual elements and the output parameters of the device;

c) statistics of the processes of approach to failure of elements;

d) data from monitoring the processes of approach to failure of elements in the individual device for which the prediction is being made;

e) the relationship between disturbing factors and the output parameters of the device.

A priori information determines the accuracy of prediction, the mathematical apparatus used, and the form of the final results of the prediction.

Methods for predicting failure-free operation that are based on the statistics of element failures and on logical relationships in the device, also called methods for calculating sudden failures, require the least a priori information and are the least labor-intensive. Therefore, they have received the greatest development. But the reliability of prediction by these methods is low, since information is used only about the moments at which failures occur. Meanwhile, any failure of a device is preceded by accumulating changes within the elements, leading to failures—the so-called processes of approach to failure.

The processes by which failures approach are characterized by a number of parameters whose variation constitutes continuous random functions of time. Methods for predicting operability that are based on the statistics of the processes of parameter variation and on logical or functional relationships, also called methods for calculating gradual failures, make it possible to forecast a set of identical devices and are widely used at the design stage.

At the storage and operation stages, an essential means of ensuring operability is the individual servicing of devices. For effective servicing, the state of each device is diagnosed. Parameter monitoring carried out during diagnosis makes it possible to accumulate information on the features of the processes by which failures arise in each device. This information, together with information on the statistical properties of the processes by which failures arise, makes it possible to perform forecasting for each individual device—so-called individual prediction.

Individual prediction of operability (IPO) has the greatest reliability of forecast, since it is carried out with allowance for the features of the processes leading to failures in each individual device.

IPO is of especially great importance in automation devices, since they consist of a much smaller number, as compared with radio-electronic systems, of large elements; they are serviceable devices; in automation elements the processes by which failures arise are, as a rule, controllable; the processes of accumulating changes in the elements lead to deregulation of the automatically operating device. For IPO of automatic devices it is necessary to solve the following problems:

1. Identify the processes leading to device failures.
2. Select the monitored parameters describing these processes.
3. Identify the parameters whose departure beyond the permissible limits constitutes a device failure, and establish the functional relationships of these parameters with the monitored parameters.
4. Calculate the region of permissible parameter values.
5. Select a method for individual prediction of gradual failures (IPGF).

The urgency of the IPO problem, as well as related forecasting problems in other applications—for example, in forecasting the parameters of a technological process, in geology, medicine, meteorology, etc.—has given rise to a large number of prediction methods.

In the present work an attempt is made to carry out a comparative analysis of IPGF methods; the work does not claim to be a complete survey of all methods.

Mathematically, IPGF is the problem of determining the probabilistic characteristics of the first passage, by a one-dimensional or vector random process whose segment of realization is known, of a specified level or surface bounding the region of permissible values of this process.

The principal IPGF methods are:

1. Methods of mathematical modeling.
2. Methods of pattern recognition.
3. Methods of the general theory of random processes.
4. Methods of the theory of Markov chains.
5. Methods of special tests.

IPFO by means of mathematical modeling consists in determining the probabilistic properties of a posteriori random processes of parameter variation, i.e., processes that are continuations of segments of realizations obtained by individual monitoring, modeling these processes, and recording the instants at which the process models go beyond the permissible values.

The main difficulty is determining the probabilistic properties of a posteriori processes.

For a broad class of random processes, to determine the probabilistic properties of a posteriori processes we shall apply the method of canonical representation of random functions [16, 17, 20].

The canonical expansion of the a priori random process \(X(t_k)\) is sought in the following form:

\[ X(t_k)=m[X(t_k)]+\sum_{i=0}^{k} V_i X_i(t_k), \tag{1} \]

where: \(m[X(t_k)]\) is the mathematical expectation of the process \(X(t_k)\);

\(V_i\) are random coefficients possessing the properties

\[ m[V_i]=0;\qquad m[V_iV_j]=0 \quad \text{for } i\ne j; \]

\(X_i(t_k)\) are nonrandom functions satisfying the requirements

\[ X_i(t_i)=1 \quad \text{and} \quad X_i(t_j)=0 \quad \text{for } i>j. \]

The canonical expansion of the a posteriori random process is sought by successively replacing the coefficients \(V_i\) of the a priori expansion (1). As a result, a recurrent expression is obtained for the a posteriori random process:

\[ X_n^{ps}(t_k)=m[X(t_k)]+\sum_{i=0}^{k} W_i X_i^n(t_k), \]

where:

\(n\) is the number of known values of the process;

\[ W_i= \begin{cases} \xi(t_i), & \text{for } i\in R;\\ V_i, & \text{for } i\notin R, \end{cases} \]

\(R\) is the set of time instants for which the values of the realization of the random process are known;

\[ X_i^{j}(t_k)= \begin{cases} X_i^{(j-1)}(t_k)-X_i^{(j-1)}(t_{k_j})\cdot X_{k_j}^{(j-1)}(t_k), & \text{for } i<k_j;\\ X_i^{(j-1)}(t_k), & \text{for } i\ge k_j, \end{cases} \]

\(j=0, 1, 2,\ldots\) is the number of known values used.

The method of canonical representation of random functions gives a good approximation of processes leading to failures; however, it requires considerable work to obtain the coefficients and functions of the expansion. Therefore, in some cases the a posteriori random processes are obtained by the method of statistical linearization [10, 11, 12].

The processes of parameter variation are divided into two components:

\[ X(t)=\xi(t)+\gamma(t), \]

where: \(\gamma(t)\) denotes short-term reversible changes, approximated by a stationary random process with \(m[\gamma(t)]=0\) and a rapidly decaying correlation function;

\(\xi(t)\)—irreversible changes approximated by a piecewise linear model:

\[ \xi(t)=a+bt, \]

where \(a\) and \(b\) are random coefficients.

The posterior random process is determined by calculating the coefficients \(a\) and \(b\) from the known values of a realization of the random process. The posterior process is determined analogously for the model

\[ \xi(t)=a+bt^q, \]

where \(q\) is a nonrandom coefficient.

Statistical linearization, as a rule, is a rough approximation, but it substantially simplifies both obtaining the posterior process and modeling. The modeling of random processes is carried out analogously to the modeling of a priori processes and, consequently, the method subsequently does not differ in any way from the method of modeling gradual failures on the average for a set of devices. Modeling may be performed on analog computers and digital computers.

Modeling on analog computers is preferable if the equations relating the parameters of the elements and the output parameters of the device are unknown, and simple models of the linear type are used to approximate the processes occurring in the elements. The operation of the device is modeled on a real physical model, i.e., instead of individual elements, their analog models are included in the overall structure of the real device [5].

Tracking systems [9] may be used to coordinate the models of the elements and the real device. The modeling process is readily automated when additional equipment of the type of distributing devices and electric-pulse counters is used.

Modeling on a digital computer is preferable if the entire device as a whole is being modeled. In this case the complexity of the models of the individual elements is not decisive. For modeling, the method of statistical trials is used [16, 21].

Modeling on a digital computer offers greater possibilities in comparison with analog modeling. At present, considerable experience in such modeling has been accumulated. The difficulty lies in finding equations relating the parameters of the elements and the output parameters of the device.

IPGF by pattern-recognition methods consists of two stages: training (classification) and recognition.

At the training stage, realizations of the vector random process of parameter variation are divided into classes according to the time at which the process is located in the region of admissible values.

On the basis of a sufficient number of realizations of the process, the probabilities of each class are determined and, from the initial segments of the realization, the conditional probability of their falling into one class or another.

Determination of the probability of each class is, in essence, a prediction of operability for a set of similar devices.

At the recognition stage, the initial segment of a realization of the vector random process of variation of device parameters is analyzed, and the conditional probability that this realization falls into each of ...

classes. These conditional probabilities also constitute the distribution of the time of failure-free operation of an individual device.

Individual prediction of gradual failures (IPGF) by the pattern-recognition method has been developed for devices described by one or two parameters [3, 7]. The accuracy of the method is determined by the number of realizations used for training. For one and the same accuracy, the required number of realizations increases approximately in proportion to the square of the number of parameters. Therefore, IPGF by this method is possible after a completed operating cycle of a large number of devices.

The recognition process is carried out on a digital computer, and IPGF of devices described by several parameters requires a large amount of machine memory.

IPGF by methods of the theory of random processes and the theory of Markov chains consists in solving a vector stochastic equation:

\[ \overline{X}(t)-\overline{S}_{\partial}=0, \tag{2} \]

i.e., the problem of the first passage by realizations of the vector random process of changes in the device parameters \(\overline{X}(t)\) of the surface \(S_{\partial}\), which bounds the region of admissible values.

In the general case, equation (2) is unsolvable. For monotone random processes of parameter variation, Pugachev’s method [1] is used. The condition for applicability of the method is the requirement that the number of realizations crossing the boundary of the region of admissible parameter values twice tend to zero.

The time of failure-free operation of a device described by one parameter is determined by the following equation:

\[ T[X_{\min}, X_{\max}] = \int_{0}^{\infty} \{1[X(t)-X_{\min}]-1[X(t)-X_{\max}]\}\,dt, \tag{3} \]

where

\[ 1[X(t)-X_{\max}] = \begin{cases} 0 & \text{when } X(t)<X_{\max};\\ 1 & \text{when } X(t)>X_{\max}; \end{cases} \]

\[ 1[X(t)-X_{\min}] = \begin{cases} 0 & \text{when } X(t)<X_{\min};\\ 1 & \text{when } X(t)>X_{\min}. \end{cases} \]

Expression (3) represents a nonlinear integral transformation of the random process \(X(t)\), reducible to a linear transformation [20]. The probability density of the distribution of the time of failure-free operation is determined through the conditional density of the distribution of the parameter at a future point in time. The conditional density of the distribution of the parameter, i.e., taking into account data from its monitoring, is determined by Bayes’ formula. V. S. Pugachev’s method requires knowledge of three-dimensional probability distribution densities of the parameter values. When it is used, repeated numerical integration must be carried out. Therefore its practical use is possible when computations are performed on a digital computer. The computations lend themselves well to automation [20].

V. S. Pugachev’s method can also be used for IPGF of a device in which the processes of occurrence of failures are described by a multivariate random process. The computation procedure is the same as for a one-dimensional random process, but the amount of computation increases sharply.

The probability of failure-free operation of a device during a certain time interval can be calculated directly from the conditional

... to probability distribution densities of the value of the parameter \(X(t)\). In [19] a similar method is considered when data are available from monitoring the process \(X(t)\) at two instants of time \(t_{k-1}\) and \(t_k\).

The probability of failure-free operation over the time interval \([t_k,\, t_{k+1}]\) is determined by the formula:

\[ P(T>t_{k+1})= \int_{X_{\min}}^{X_{\max}} f\left[X(t_{k+1})/X(t_k),\, X(t_{k-1})\right]\,dX(t_{k+1}) \]

The conditional probability distribution density \(f[X(t_{k+1})/X(t_k), X(t_{k-1})]\) is determined through the two- and three-dimensional densities of the joint probability distribution of the values of the parameter \(X(t)\) at the instants of time \(t_{k-1}\), \(t_k\), and \(t_{k+1}\). The expressions for the distribution densities are not integrated analytically; therefore such a method requires a large amount of numerical integration work. The principal difficulty of the method is obtaining the three-dimensional probability distribution density \(X(t_{k-1}), X(t_k), X(t_{k+1})\). If at the instant \(t_{k+1}\) statistical data on the parameter are unknown, then this distribution density can be specified only on the basis of indirect investigations.

Most parameters describing failure-occurrence processes can be represented in the form of Markov processes. The application of the theory of simple Markov chains to problems of IPGF is considered in [13, 18, etc.].

The distribution density of the parameter \(X(t_k)\) at the instant \(t>t\) is determined only by the value \(X_k(t_k)\) of the parameter at the instant of time \(t_k\). The Markov process is specified by the intensity coefficients

\[ K_s[X(t)] = \lim_{\Delta t\to 0} \frac{M\{[X(t+\Delta t)-X^t]^s\}}{\Delta t}; \qquad s=1,\,2. \]

To solve the prediction problem it is necessary to determine the two-dimensional probability distribution density \(f[X(t_k),\, X(t)]\), or the conditional probability distribution density \(f[X(t)/X_k(t)]\). If measurement errors are neglected, then the conditional probability distribution density of the transition \(f[X(t)/X_k(t_k)]\) of a continuous Markov process satisfies the forward and backward diffusion equations, also called the equations of A. N. Kolmogorov; i.e., the diffusion equations describe trajectories of the process that have not once touched the boundaries \(X_m\) and \(X_{\max}\) during the time \([t_k, t]\). Therefore the following are adopted as boundary conditions:

\[ P[X_{\min}(t)/X_k(t_k)] = P[X_{\max}(t)/X_k(t_k)] = 0. \]

The conditional distribution of failure-free operating time for a Markov process of parameter variation is determined by the formula:

\[ P[T>t/X_k(t_k)] = \int_{X_{\min}}^{X_{\max}} f[X(t)/X_k(t_k)]\,dX. \]

The principal difficulty of the prediction problem lies in the need to solve nonstationary differential diffusion equations.

For stationary Markov processes of parameter variation, the intensity coefficients do not depend on the reference instant, and the diffusion equations are stationary. In this case the solution of the equations is sought by expansion in eigenfunctions, each of which depends on only one argument:

\[ f\left[X,\,T/X_k\right]=\theta(T)\cdot X(t). \]

The solution takes the form

\[ f\left[X,\,T/X_k\right]=\theta_k X_k(X)+\sum_{m=1}^{\infty}\theta_m e^{-\lambda_m T}\cdot X_m(X). \]

Here \(\theta_m\) is chosen in accordance with the initial conditions, and \(X_m\) \((m=0,1,2,\ldots)\) are obtained for definite eigenvalues \(\lambda=\Theta'(T)/\Theta(T)\) and boundary conditions.

B. V. Vasil’ev proposed a simplified prediction method that makes it possible not to solve nonstationary diffusion equations [4]. The diffusion equation is transformed into a linear differential equation of the second order, relating two successive conditional initial moments of lifetime. Its solution \(\alpha_m[X_k(t_k)]\) is obtained through the intensity coefficients of the initial Markov process and the initial moment \(\alpha_{m+1}[X_k(t_k)]\); thus, successively, all conditional initial moments of the lifetime distribution are found, which completely determine the random variable \(T\).

The probability density of the quantity \(T\), on the basis of the initial moments, is sought in the form of a series:

\[ f[T]=f_0(T)\left[c_0P_0(T)+c_1P_1(T)+\ldots\right], \]

where

\(f_0(T)\) is a certain reference density;

\(P_0(T),\ P_1(T)\) are a system of polynomials orthonormal with respect to the reference distribution;

\(c_0,\ c_1\) are coefficients determined through the initial moments of the random variable.

The problem reduces to choosing the reference density, on which the number of terms in the expansion \(f[T]\) depends. For \(f_0[T]\) corresponding to the normal law, the polynomials \(P_n[T]\) are expressed through Hermite polynomials, and for \(f_0[T]=e^{-T}\cdot T^\alpha\Gamma(\alpha+1)\) through Laguerre polynomials.

If the initial statistical data show nonstationarity of the process being predicted, then it is necessary either to reduce the process to a stationary one, or to consider it over limited time intervals where the intensity coefficients are approximately constant.

Diffusion equations are the most accurate and most universal mathematical apparatus describing the problem of a random process reaching a fixed boundary of the tolerance field. The main drawbacks of the method are the error of approximating the real process by a Markov process, the laboriousness of computing the process intensities and the coefficients of the diffusion equation, and also the unwieldiness of the final solution of these equations. The use of diffusion equations for individual prediction of a device whose failure processes are described by multidimensional random processes is hindered, above all, by the difficulty of determining the intensity coefficients of a multidimensional Markov process from the initial statistics of the parameter-change processes.

Prediction of the serviceability of a device by means of special tests consists in physically modeling the influence of the principal disturbing factors, constructing a surface bounding the region of permissible deviations of the disturbing factors, and estimating the reserve of serviceability. For this it is necessary:

1) to identify the principal disturbing factors;

2) to create artificial conditions in which these factors can be regulated;

3) conduct tests and calculate the margin of operability.

The disturbing factors are classified according to the sensitivity of the device to them and the variance of the deviations of these factors from their nominal values. The choice of the number of disturbing factors by which the test is carried out determines the accuracy of prediction and depends on the equipment capabilities and the testing time [14]. If changes in the output parameters under the action of various disturbing factors are independent of one another, then the tests are carried out as follows. Each of the disturbing factors, with the remaining factors at their nominal values, is varied until the output parameter of the device goes beyond the tolerance limits. The value of the disturbing factor at which this departure occurs is recorded. By such tests over all disturbing factors, the region of admissible values in the space of disturbing factors is formed. Tests in this case are called boundary tests [15, 24].

Matrix tests consist in the sequential variation of each of the disturbing factors with a specified step according to a special test program. In this case, a large number of combinations of disturbing factors is simulated, which makes it possible to construct the surface limiting the region of admissible deviations of the disturbing factors more accurately [5]. However, carrying out matrix tests is considerably more labor-intensive than carrying out boundary tests.

With aging and wear of elements, their properties deteriorate. This is expressed in the corresponding narrowing of the region of admissible deviations of disturbing factors from their nominal values. Individual prediction of reliability is carried out by estimating the position of the point of the nominal values of the disturbing factors, i.e., the origin of coordinates, relative to the surface limiting the region of admissible deviations of these factors. In this way the margin of operability is determined.

Most often the margin of operability is estimated by the length of the shortest perpendicular from the origin of coordinates to the limiting surface [8, 14, 24].

When the form of the limiting surface is complex and the variances of the disturbing factors differ significantly, it is expedient, for estimating the margin of operability, to introduce weighting coefficients for each disturbing factor.

Such estimates of the margin of operability agree well with probabilistic characteristics of operability [22].

Individual prediction of reliability by means of special tests differs from other methods of individual prediction of reliability in its a priori information, the information obtained in the course of operation, and the form of the forecast results. The main difficulty in carrying out special tests is the physical modeling of adjustable disturbing factors, which requires the use of complex test installations. In addition, when special tests are carried out, there is an increased consumption of service life. Therefore, individual prediction of reliability by means of special tests has found application only for certain units of electronic circuits.

Analogous estimates of the margin of operability can be made in the space of parameters describing the processes of failure occurrence [6]. The tests reduce to the usual variation of parameters. With such prediction it is possible to get by with a small number of check points of the vector process of change in system parameters and with limited information about the probable properties of this process, but the reliability of the forecast is low.

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