ON THE PROCEDURE FOR POLLING PARAMETERS IN CENTRALIZED MONITORING SYSTEMS

Yu. M. Ageev, V. I. Konovalov

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

One of the important problems requiring solution in the development of centralized monitoring systems is increasing the accuracy of the results obtained. If the monitoring system includes a computer, then the functional diagram of the measuring channel may be represented as follows (Fig. 1):

Fig. 1

\(\sigma_{\text{d}}\) — the error of the sensor, taking into account the possibility of introducing corrections due to changing operating conditions;

\(\sigma_{\text{vp}}\) — the error of the secondary instrument (in the case where information is input into the computer from secondary measuring instruments);

\(\sigma_{\text{kv}}\) — the quantization error of the ideal commutator, taking into account the possibility of one or another method of extrapolation or interpolation;

\(\sigma_{\text{kom}}\) — the switching error of the real commutator;

\(\sigma_{\text{ADC}}\) — the error of the ADC;

\(\sigma_{\text{VU}}\) — the error of the methods of primary information processing carried out in the computer.

Let us consider the error \(\sigma_{\text{kv}}\), whose magnitude is affected by the statistical characteristics of the monitored parameters, the extrapolation and interpolation method used, the response speed of the commutator, and the order in which the parameters are polled, or the commutator operating program.

In modern monitoring systems the number of monitored parameters can be very large. In view of the fact that the response speed of commutators and of other functional units of the monitoring system is always limited, it is not possible to ensure uniform-in-time quantization of all parameters. Owing to the nonuniform quantization step, an additional error \(\sigma_{\text{kv}}\) arises, which depends substantially on the order in which the parameters are polled.

In the present paper a simple method is proposed for determining the polling order of the monitored parameters, making it possible to reduce \(\sigma_{\mathrm{qv}}\), as well as certain relations needed for estimating the speed of operation of the computer and of the commutator. Similar problems have been considered in [1] and [2]. However, the relations obtained in [1] make it possible to determine only the capacity of the commutator. In [2] an algorithm is considered for the optimal order of servicing several independent subsystems by a digital computer with minimum allowable speed. The results obtained, however, presuppose the presence of free positions in the commutator, which naturally reduces the value of the method.

Let us first introduce some notation:

\(n\) — the number of monitored parameters;

\(f_i\) — the polling frequency of the \(i\)-th parameter, \(f_{\max}\), \(f_{\min}\) — the maximum and minimum values of \(f_i\);

\(T_{\mathrm{c}}\) — the duration of one polling cycle;

\(N_0\) — the number of commutator positions;

\(t_k\) — the time during which the commutator remains in the \(k\)-th position;

\(\nu_{\mathrm{v}}\) — the conditional speed of the computer;

\(\nu_{\mathrm{k}}\) — the speed of the commutator;

\(h_{ij}\) — the commutator position occupied by the \(i\)-th parameter when polled for the \(j\)-th time;

\(\Delta h_{i0}\) — the interval under uniform quantization, expressed as a number of commutator positions;

\(\Delta h_{ij}\) — the quantization interval of the \(i\)-th parameter for \(j=\overline{1,\gamma_i}\);

\(\Delta \tau_{ij}\) — the instability of the quantization interval of the \(i\)-th parameter within one polling cycle \([j=\overline{1,\gamma_i}]\);

\(\Theta_i\) — the servicing time by the computer for the \(i\)-th parameter.

The number of commutator positions is determined by the formula

\[ N_0=\sum_{i=1}^{n}\gamma_i, \]

where

\[ \gamma_i=\frac{f_i}{f_{\min}}, \]

\(\gamma_i\) are integers;

\[ 1\leq \gamma_i<\gamma_{\max}. \]

If the \(\gamma_i\) are fractions, one may pass to integers by finding the least common denominator and multiplying the numerators of the fractions by the corresponding factors. In this case \(\gamma_{\min}>1\), and the number of commutator positions increases. Proceeding from the condition of uniformity of all quantization intervals for all parameters, the latter must be connected to the commutator positions \(K_{ij}\), which are calculated by the formulas collected in Table 1. The parameter numbers \(i\) are entered in the table in increasing order of the frequency \(f_i\).

If the values \(K_{ij}\) calculated in this way are arranged in increasing order, and then the indices \(i\) corresponding to the respective \(K_{ij}\) are written in the same order, then the required polling order is obtained, characterized by the following.

1. With a variable polling step of the commutator \((t_k \ne \mathrm{const})\), the component of the error \(\sigma_{\mathrm{qv}}\) decreases to zero if \(t_k \to 0\) at a bounded average operating speed of the commutator, i.e.,

\[ T_{\mathrm{c}}=\sum_{1}^{N_0} t_k \ne 0. \]

1. For \(t_k=\mathrm{const}\) within one polling cycle, the order under consideration is the best. Over several adjacent cycles, however, it is not such, since in this case the parameter with \(f_{\max}\) will always be polled first and last in each cycle and, consequently, the minimum quantization interval of this parameter will always be equal to \(t_k\).

Let us consider several possible variants of the joint operation of the CU and the commutator. Suppose that, in the time interval \(t_k\), the CU performs the mathematical processing of the measurement results for each parameter, spending time \(\Theta_j\) on this.

Denote the utilization factor of the CU by \(\xi\):

\[ \xi = \frac{ \displaystyle \sum_{i=1}^{n}\sum_{j=1}^{\gamma_i}\Theta_{ij} }{ T_{\mathrm{c}} }. \]

Let us consider three possible cases.

1. The basic requirement is \(t_k=t_0=\mathrm{const}\). Then the necessary speed of the commutator and the CU is determined by the formulas:

\[ \nu_k \geq N_0 f_{\min}, \]

\[ \Theta_l < t_0;\qquad \nu_{\mathrm{v}} \geq l_{i\max}\nu_k, \tag{1} \]

\[ \xi \leq \frac{f_{\min}}{\nu_{\mathrm{v}}} \sum_{i=1}^{n}\sum_{j=1}^{\gamma_i} l_{ij}, \]

where \(l_i\) is the number of elementary operations performed by the CU in processing the \(i\)-th parameter, \(l_{i\max}=\max(l_i)\). The estimate of the variance of the quantization step is

\[ \bar{\sigma}_i= \frac{1}{\nu_k} \sqrt{ \frac{ \displaystyle \sum_{j=1}^{\gamma_i}(\Delta h_{ij}-\Delta h_{i0})^2 }{ \gamma_i-1 } }. \]

1. The basic requirement is \(\xi \leq \xi_0\); \(\nu_{k\,\mathrm{av}}=\nu_{k0}=N_0 f_{\min}\). In this case

\[ \nu_{\mathrm{v}} \geq \frac{ \displaystyle \sum_{i=1}^{n}\sum_{j=1}^{\gamma_i} l_{ij} }{ \xi_0 } \, f_{\min} \]

and, if

\[ \frac{l_{i\max}}{\nu_{\mathrm{v}}}\leq \frac{1}{\nu_{k0}}, \]

then the commutator operates with a constant step \((t_k=t_0)\). Otherwise one more component of the quantization error arises because of the nonconstancy of \(t_k\), and the estimate of the variance of the quantization step is

\[ \bar{\sigma}_i= \frac{1}{\nu_{k0}} \sqrt{ \frac{ \displaystyle \sum_{j=1}^{\gamma_i} (\Delta h_{ij}-\Delta h_{i0}+\Delta \tau_{ij}-\Delta \tau_{ij-1})^2 }{ \gamma_i-1 } }. \]

1. The basic requirement is minimum nonuniformity of quantization. This is possible only if the commutator operates with a variable step and the instants at which the commutator switches on the specified positions strictly correspond to the polling order described above. Strict fulfillment of this requirement is possible in ob-

Table 1

case only when $\nu_{\mathrm{в}}\to\infty$, $t_k\to0$. In reality, however, it is always the case that $\nu_{\mathrm{в}}\ll\nu_{\mathrm{в}0}$, $t_k\gg t_{k0}$. Therefore it is possible that some $\Theta_i$ will be larger than their corresponding $t_k$, and the quantization becomes nonuniform.

For $\xi$ we have (1), and

\[ \bar{\sigma}_i=\frac{1}{\nu_{k.\mathrm{cp}}} \sqrt{ \frac{ \sum_{j=1}^{r_i}\left(\Delta\tau_{ij}-\Delta\tau_{ij-1}\right)^2 }{ r_i-1 } }; \qquad \nu_{k.\mathrm{cp}}=\nu_{k0}=f_{\min}\cdot N_0 . \]

The maximum required speed of the commutator is

\[ \nu_{k\max}=\frac{1}{t_{k0}} . \]

Let us consider an example.

Given:

\[ l_1=100,\qquad l_2=200,\qquad l_3=500,\qquad l_4=100, \]

\[ f_1=1\ \frac{1}{\mathrm{sec}},\qquad f_2=2\ \frac{1}{\mathrm{sec}},\qquad f_3=3\ \frac{1}{\mathrm{sec}},\qquad f_4=4\ \frac{1}{\mathrm{sec}} . \]

It is required to determine the polling order and the nonuniformity of the quantization step for some parameter.

\[ N_0=r_1+r_2+r_3+r_4=10. \]

The polling order of the parameters adopted according to the method described above is given in Table 2.

Table 2

\[ \nu_k\geqslant 10\cdot1=10\left[\frac{\text{points}}{\mathrm{sec}}\right]; \]

\[ \nu_{\mathrm{в}}\geqslant 500\cdot10=5000\left[\frac{\text{el. op}}{\mathrm{sec}}\right]; \]

\[ \xi\leqslant \frac{1}{5000}\left|100+2\cdot200+3\cdot500+4\cdot100\right|=0,48; \]

\[ \bar{\sigma}_4=\frac{1}{10} \sqrt{ \frac{ (3-2,5)^2+(2-2,5)^2+(3-2,5)^2+(2-2,5)^2 }{ 4-1 } } = \]

\[ =0,058\ [\mathrm{sec}]=5,8\ [\%]. \]

\[ \nu_{k0}=10\cdot1=10\left[\frac{\text{points}}{\mathrm{sec}}\right]; \qquad \xi_0=0,8. \]

Then

\[ \nu_{\mathrm{в}}\geqslant \frac{100+200\cdot2+500\cdot3+100\cdot4}{0,8}\cdot1 =3000\left[\frac{\text{op}}{\mathrm{sec}}\right]. \]

\[ \frac{500}{3000}>\frac{1}{10},\quad \text{therefore } t_k\ne\mathrm{const}. \]

Estimate the variance of the quantization step for the 4th parameter

\[ \bar{\sigma}_{4}=\frac{1}{10} \sqrt{ \frac{ \left(3-2\frac{1}{2}+\frac{1}{3}-0\right)^2+ \left(2-2\frac{1}{2}+\frac{2}{3}-\frac{1}{3}\right)^2 }{4-1} + \frac{ \left(3-2\frac{1}{2}+\frac{2}{3}-\frac{2}{3}\right)^2+ \left(2-2\frac{1}{2}+0-\frac{2}{3}\right)^2 }{4-1} } =0.067\ [\mathrm{sec}]=6.7\ [\%]. \]

1. \(\nu_{\mathrm{v}}=5000\) op/sec.

Determine the values \(\Delta h_{ij}\) and \(t_k\), and also estimate whether the values \(\Theta_i\) correspond to the \(t_k\)’s.

\[ \Delta h_{ij\min}=\Delta h_{41}=1.66-1.25=0.41;\qquad t_{41}=\frac{1}{10}\cdot 0.41=0.041\ [\mathrm{sec}]. \]

\[ \Theta_{4}=\frac{100\ \mathrm{op}}{5000\ \mathrm{op/sec}}=0.02\ \mathrm{sec}.\qquad \Theta_{4}<t_{41}. \]

The smallest of the remaining intervals is the interval \(t_{33}\).

\[ \Delta h_{33}=0.42;\qquad t_{33}=0.042\ \mathrm{sec};\qquad \Theta_{3}=\frac{500\ \mathrm{op}}{5000\ \mathrm{op/sec}}=0.1\ \mathrm{sec}; \]

\[ \Theta_{3}>t_{33}. \]

Consequently, during the time interval \(t_{33}\) the computing unit will not have time to process the monitored parameter. During the time \(t_{33}\),

\[ 5000\ \mathrm{op/sec}\cdot 0.042\ \mathrm{sec}=210 \]

operations will be performed instead of the required 500; therefore the start of processing of the next parameter will shift by

\[ \frac{500-210}{500}=0.58 \]

positions of the commutator.

In the remaining cases \(t>\Theta_i\).

\[ \bar{\sigma}_{4}=\frac{1}{10} \sqrt{ \frac{(0-0.58)^2+(0.58-0)^2}{4-1} } =0.041\ [\mathrm{sec}]=4.1\ [\%]. \]

LITERATURE

1. Yu. V. Fedyukin. Central regulator of production processes. Transactions of MEI, issue 52, Moscow, 1963.

2. G. F. Lugovoi. Selection of variants of the sequence for servicing several independent automatic-control systems by a control computer. Izvestiya AN SSSR, Technical Cybernetics, No. 4, 1968.