APPLICATION OF THE METHOD OF MATHEMATICAL STATISTICS TO THE STUDY OF COMMUTATION

E. K. STRELBITSKY, V. S. STUKACH, A. Ya. TSIRULNIK

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

The presence of technological deviations leads to unequal commutation quality in machines of the same standard size. To determine the influence of technology on the mean level and variance of sparking, to assign a well-founded system of tolerances, and to predict the stability of commutation over time, it is necessary to know the dependence of sparking on technological and mechanical factors. We shall call such a dependence the mathematical model of sparking.

The existing models of commutation at present do not include mechanical and technological factors and therefore do not explain the variance of sparking in machines of the same series. Therefore, in order to solve the above-mentioned problems, it is first of all necessary to create a sufficiently adequate model of commutation.

Here two possible paths for seeking the model present themselves. The classical path assumes an investigation of the mechanism of the process, a description of the elementary processes in the system by integro-differential equations, and, on this basis, the construction of a theoretical model of commutation that takes into account technological and mechanical factors.

This path does not lead to a solution of the problem within a reasonable time.

It is expedient to follow the path of an empirical search for a statistical model of commutation. The scheme of such a search is shown in Fig. 1, and the essence of the method is as follows.

Fig. 1.

At the input of the system there act \(K\) factors, varied at different levels. The system is an operator that transforms the incoming information according to an algorithm unknown to us and issues at the output the quantity

\[ y=\varphi(x_1, x_2, \ldots, x_K). \]

This function is called the response function, and the corresponding geometric image in the space of the variables \(x_i\) is called the response surface.

The determination of the response function is carried out by methods of regression analysis. The form of the response function is usually unknown; therefore its

are represented in the form of a polynomial, which is thus an expansion of the actual response function:

\[ \hat{y}=b_0+\sum_1^k b_i x_i+\sum_{i<j}^k b_{ij}x_i x_j+\sum_1^k b_{ii}x_i^2+\ldots \tag{1} \]

Here \(\hat{y}\) is an estimate of \(y\), and \(x_i\) are the factors taken into account. Although the polynomial form does not make it possible to obtain complete heuristic information about the process immediately, this method does provide a technical solution within a reasonable time when the mechanism of the process is not fully known.

For most problems it is sufficient to restrict oneself to the linear or quadratic part of polynomial (1). The quadratic form is used to increase the adequacy of the representation of the response surface when it deviates substantially from linearity. In particular cases, the adequacy of the representation can be increased by replacing the variables \(x_i\) by their functions on the basis of a priori information about the process. In studying the commutation of machines of the P series, a preliminary analysis of two-dimensional dependences \(y=\varphi(x)\) showed the expediency of representing the response function in a multiplicative-functional form, which is easily transformed into a polynomial form after taking logarithms:

\[ \hat{y}=b_0\cdot \prod_{i=1}^m x_i^{b_i}\cdot \exp\left[\sum_{i=m+1}^k b_i\cdot x_i\right], \tag{2} \]

\[ \ln \hat{y}=b_0^1+\sum_{i=1}^m b_i\cdot \ln x_i+\sum_{i=m+1}^k b_i\cdot x_i. \tag{3} \]

Here \(x_i(i\le m)\) are factors that reduce \(y\) to zero when \(x_i=0\);
\(x_i(i>m)\) are factors that minimize (maximize) \(y\) when \(x_i=0\).

At the first stage of the investigation, in order to develop the method and to determine the substantially influencing factors (a screening experiment), we confined ourselves to the linear model (3) and included the following factors:

\(x_1\) — armature current, \(I\), \(a\);
\(x_2\) — voltage on the armature, \(U\), \(b\);
\(x_3\) — rotational speed, \(n\), \(1000\) rpm;
\(x_4\) — pressure on the brush, \(p\), \(100\) g;
\(x_5\) — eccentricity of the commutator, \(e\), microns;
\(x_6\) — ellipticity of the commutator, \(E\), microns;
\(x_7\) — root-mean-square deviation of the lamella step differences, \(\sigma\), microns.

As the output quantity \(y\), characterizing the degree of sparking, we took the voltage under the trailing edge of the brush, measured by the auxiliary-brush method.

Mathematically, the problem consists in determining the regression coefficients \(b_i\) by solving the system of equations

\[ X^*X\cdot B=X^*\cdot Y, \tag{4} \]

obtained by differentiating the quadratic form

\[ \sum_{u=1}^N (y_u-\hat{y}_u)^2 \tag{5} \]

with respect to all regression coefficients and equating them to zero in order to minimize form (5) (the method of least squares). Here \(N\) is the number of experiments (variants) in which the variables are varied. In the matrix \(X^*X\) of the coefficients of the system of equations, the element of the \(i\)-th row and \(j\)-th column is equal to

\[ \sum_{u=1}^N x_{iu}\cdot x_{ju} \]

—the scalar product of the column vectors \(X_i\) and \(X_j\). In the matrix of free terms \(X^*Y\), the \(i\)-th element is equal

\[ \sum_{u=1}^{N} x_{iu}\cdot y_u . \]

The solution of system (4) in matrix form is written as expression (6):

\[ B=(X^*X)^{-1}(X^*y) \tag{6} \]

or, in algebraic form:

\[ b_i=\sum_{j=0}^{K} c_{ij}\sum_{u=1}^{N}x_{ju}\cdot y_u, \tag{7} \]

where \(c_{ij}\) is the element of the \(i\)-th row and \(j\)-th column of the matrix \((X^*X)^{-1}\).

With arbitrary variation of the parameters \(x_i\) (the so-called “passive experiment”), the following inconveniences arise:

1. Since it is usually customary to vary the variables one at a time, only a small part of the experiments carries information about each coefficient \(b_i\). Therefore the regression coefficients are estimated with a large variance.

2. Since, under arbitrary planning of the experiment, the parameters \(x_i\) turn out to be pairwise correlated

\[ \left(\sum_{u}^N x_{iu}\cdot x_{ju}\ne 0\right), \]

the regression coefficients are determined dependently on one another. It is impossible to separate individual effects and interaction effects. It is impossible to obtain a mathematical model of the process.

This statement does not apply to the case when the true response function is linear.

1. The calculations are laborious.

Guided by modern methods of mathematical experimental design, we planned the experiment “orthogonally,” i.e., varied the variables according to such a plan that the scalar products of differently named column vectors in the design matrix \(X\) were equal to zero

\[ \left(\sum_{\substack{u=1\\ i\ne j}} x_{iu}\cdot x_{ju}=0\right). \]

Then, obviously, the coefficient matrix \(X^*X\) of the system of equations becomes diagonal

\[ \left(\sum_u^N x_{iu}^{2}\ne 0\right) \]

and the system breaks up into \(K+1\) independent equalities.

The advantages of orthogonal experimental design are:

1. Information about each regression coefficient is carried by all \(N\) experiments; therefore the coefficients are estimated with minimal and equal variance.

2. Each regression coefficient \(b_i\) is determined independently of the others by the formula

\[ b_i=\frac{\sum_{u=1}^{N}x_{iu}\cdot y_u}{\sum_{u=1}^{N}x_{iu}^{2}}. \tag{8} \]

This means that linear effects are determined independently of paired interactions; the degree of influence of each factor and of each interaction of two or several factors is determined, i.e., we obtain a mathematical model of the process.

1. The calculations are simplified, since there is no need to solve a system of equations.

2. The design is rotatable, i.e., information is supplied uniformly from all regions of the response surface.

We obtained the following dependence for machines of the 2nd size of series P:

\[ \hat{y}=7.4\cdot 10^{-2}\cdot 10^{0.80}\cdot \exp[-0.0955\cdot p+0.0087\cdot e+0.72E+1.64\sigma],\ \text{V}. \tag{9} \]

Here the voltage and speed have been incorporated into the constant term by their nominal values with the corresponding regression coefficients, owing to the insignificance of the latter.

To estimate the residual variance we determine the residual sum of squares from the formula

\[ S_R=\sum_{u=1}^{N}(y_u-\hat{y}_u)^2 =\sum_{u=1}^{N}y_u^2-\sum_{i=0}^{k}b_i\sum_{u=1}^{N}x_{iu}\cdot y_u=33.7. \tag{10} \]

The multiple correlation coefficient is

\[ R=\sqrt{1-\frac{S_R}{\displaystyle\sum_{u=1}^{N}(y_u-\bar{y})^2}}=0.802 \tag{11} \]

with a lower estimate \(R_{\mathrm{l}}=0.67\) and an upper estimate \(R_{\mathrm{u}}=0.88\) at the 95-percent significance level.

The relative residual variance is

\[ \delta_R=\frac{D(y)-\displaystyle\sum_{i}^{k}b_i^2\cdot D(x_i)}{D(y)}=0.358. \tag{12} \]

The high value of \(R\) and the comparatively small value of \(\delta_R\) indicate that a significant portion of the essential factors has been taken into account, explaining a substantial part of the variance of sparking.

In further work we intend to include in the model parameters characterizing deviations in the geometry of the magnetic system and the position of the brushes relative to the geometric neutral, the grade of the brushes, and other factors.

When regression equation (2) has been brought to the required degree of adequacy with the response function, it can be used to solve the following problems:

1. Estimating the mean value and scatter of sparking, if the actual values \(x_i\) and variances \(D(x_i)\) are known for a given level of technology.

2. Assigning tolerances for the parameters \(x_i\), if a tolerance for sparking is specified. This problem is not unique; only a multidimensional region of permissible values \(D(x_i)\) can be determined. If optimization of the tolerance system by cost is necessary, the use of mathematical programming is inevitable.

3. Forecasting the share of machines with impermissible sparking both at the testing station of the manufacturing plant and during operation. In the latter case it is necessary to know the mean rates of change of \(x_i\) over time.

REFERENCES

1. I. P. Klepikov, S. N. Sokolov. Analysis and Planning of Experiments by the Maximum Likelihood Method, Nauka Publishing House, Moscow, 1964.

2. V. V. Nalimov, N. A. Chernova. Statistical Planning of Extremal Experiments, Nauka Publishing House, Moscow, 1965.

3. L. N. Bol’shov, N. E. Smirnov. Tables of Mathematical Statistics, Nauka Publishing House, Moscow, 1965.