On Predicting the Operability of Automatic Control Systems at the Design Stage by the Generalized Parameter Method
Abstract Generated abstract
The paper addresses prediction of the operability and reliability of automatic control systems at the design stage under aging and environmental influences. It argues that the degree of stability, defined through the characteristic equation root nearest the imaginary axis, can serve as a generalized parameter, and presents an approximate method for estimating it without solving for all roots by using ratios of characteristic equation coefficients. This formulation identifies which element parameters most affect stability and, with specified tolerances, provides a simple way to estimate the probability that the system will retain a prescribed degree of stability during long-term operation.
Full Text
Vol. 230
1972
ON THE QUESTION OF PREDICTING THE OPERABILITY OF ACS AT THE DESIGN STAGE BY THE METHOD OF A GENERALIZED PARAMETER
F. F. IDRISOV
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
When designing a complex and responsible automatic control system (ACS) intended to operate for a long time under conditions in which the effects of aging and environmental influences make themselves felt, it is necessary to have such a generalized parameter (or indicator) that would characterize the reliability of the system as fully as possible. Such an approach to determining the reliability of designed systems of any complexity makes it possible to simplify the problem substantially, since it is based on predicting the probability that the ACS will conform not to all its characteristics, but only to the generalized parameter. In this case it is possible to avoid the use of multidimensional distributions and correlation relations [1].
In [2], the system gain coefficient is used as the generalized parameter. However, a more general parameter is the degree of stability \(H\). The larger \(H\) is, the more reliable the system. Therefore it is necessary to have methods convenient for the engineer that would quickly determine the value \(H\) of the system and those parameters entering into the complex mathematical model that have the strongest influence on the stability of the ACS.
According to [3], the degree of stability \(H\) is the absolute value of the real part of the root of the characteristic equation closest to the imaginary axis
\[ A_n X^n + A_{n-1} X^{n-1} + \ldots + A_1 X + A_0 = 0. \tag{1} \]
In the case of high degrees of equation (1), it is necessary to be able to determine \(H\) without computing the roots. The procedure set forth in [3] leads to cumbersome calculations. A very convenient method of determination is proposed in [4].
Following the procedure set forth in [4], for the characteristic equation the following ratios are determined:
\[ H_1 = \frac{A_0}{1A_1}, \quad H_2 = \frac{A_1}{2A_2}, \quad H_3 = \frac{A_2}{3A_3}, \ldots, \quad H_n = \frac{A_{n-1}}{nA_n}. \tag{2} \]
From this the value of the degree of stability is found:
\[ H \simeq \min \{H_1,\ H_2,\ \ldots,\ H_n\}. \tag{3} \]
Thus, we have obtained
\[ H=H_k=\frac{A_{k-1}}{kA_k},\quad k\in \overline{1,n}. \tag{4} \]
Since
\[ A_{k-1}=f(M_{k-1}),\quad A_k=\varphi(M_k), \]
where \(M_{k-1}\), \(M_k\) are intersecting sets of parameters of the system elements, the degree of stability is the ratio of certain system parameters:
\[ H=\frac{f(M_{k-1})}{k\varphi(M_k)}. \tag{5} \]
It follows from formula (5) that one can determine which parameters of the ACS elements should be changed in order to increase the degree of stability.
By specifying tolerances for the influence of the external environment and natural aging, we determine the resulting tolerance of the degree of stability. The probability that, in the course of prolonged operation, the system will possess the prescribed degree of stability is determined from the relation
\[ P_c(H)=1-\frac{\Delta H}{H+\Delta H}. \tag{6} \]
Example.
If the characteristic equation of the system is
\[ T_1T_2X^3+(T_1+T_2)X^2+(1+k_1k_2)X+k_1k_2k_3=0 \tag{7} \]
and
\[ T_1=0.06;\quad T_2=0.2;\quad k_1=5.2;\quad k_2=0.5;\quad k_3=15, \tag{8} \]
then equation (7) takes the form:
\[ 0.012X^3+0.26X^2+3.6X+3.9=0, \]
and the quantity
\[ H \simeq \min\left\{\frac{39}{1\cdot 3.6},\ \frac{3.6}{2\cdot 0.26},\ \frac{0.26}{3\cdot 0.012}\right\}, \]
that is,
\[ H=\frac{1+k_1k_2}{2(T_1+T_2)}=\frac{3.6}{2\cdot 0.26}=6.9. \]
Hence it is seen that the quantity \(H\) is most strongly affected by the product \(k_1\) and \(k_2\). Consequently, when designing an ACS it is necessary to ensure the maximum invariance of the parameters \(k_1\) and \(k_2\) to disturbances of the external environment and to natural aging.
Conclusions. This approach makes it possible, at the design stage, to determine simply changes in which parameters of the ACS elements primarily affect stability, i.e., operability. On the other hand, by specifying tolerances for the influence of the external environment and natural aging, one can also simply determine the probability that the ACS will possess the prescribed degree of stability.
References
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A. A. Bessonov. On the accuracy of the noncorrelation method for calculating the probability of failure-free operation of automatic control systems. LMI, Collected Works No. 29, “Automatic Control,” issue 3, 1963.
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A. A. Bessonov. Predicting the reliability of automatic control systems by the generalized-parameter method. Izvestiya LETI, issue 56, part 3, 1966.
-
A. A. Voronov. Fundamentals of the Theory of Automatic Control. Energiya Publishing House, Moscow–Leningrad, 1965.
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A. N. Lebedev. A rapid approximate calculation of the degree of stability of linear automatic control systems. Izvestiya LETI, issue 67, 1968.