STUDY OF A FIRST-ORDER ASTATIC SYSTEM WITH INTEGRAL CORRECTION BASED ON THE DEVIATION OF THE ROOT-MEAN-SQUARE VALUE OF THE CONTROLLED VARIABLE

I. N. Trofimuk

(Recommended by the scientific seminar of the Department of EPP)

In a closed automatic-control system, under the continuous action of random disturbances, an error appears in the root-mean-square value of the controlled variable. In some cases—for example, in a system for controlling the operating regime of an electric arc steel-melting furnace—such an error is inadmissible. The absence of complete information about the character of the disturbances makes it necessary to use, for controlling the operating regime of the arc furnace, a closed self-adjusting system with integral correction. The study of such a system and the derivation of criteria for assessing the quality of control are of practical interest; nevertheless, similar systems have not been considered in the literature.

Since the integral correcting element changes the parameters of the control system very slowly, the inertia of the main controller may be neglected, and the main control loop may be regarded as containing a single integrating element—the actuating motor.

The structural diagram of the system under consideration is shown in Fig. 1. In this system, in addition to the controlled variable \(X\), one must simultaneously consider the variable gain coefficient \(Y\) of the feedback circuit of the main loop.

Under the condition that all elements are linear, the behavior of the system is described, in relative units, by differential equations of the form:

\[ \begin{aligned} \frac{dx}{d\tau} &= a(1 - xy),\\ \frac{dy}{d\tau} &= x^2 - 1 \end{aligned} \tag{1} \]

where \(\tau\) is the reduced time, and \(a\) is the reduced gain coefficient of the open-loop system.

The method of reducing to relative units is widely used for a clear identification of the mathematical structure of a control system. Without considering the details of the reduction method, we indicate that the value of the reduced gain coefficient \(a\) is determined by the quanti-

of the quotient obtained by dividing the gain coefficient of the open loop of the main circuit by the gain coefficient of the integral correcting device, taking into account the base values of the controlled quantities.

Unlike control systems that are nonlinear because of the characteristics of the transfer functions of individual elements, the system of the form (1) has a variable structure. The characteristic equation for

Fig. 1.

the controlled quantity in this system is not reducible to a linear polynomial in powers. In this connection, the usual methods of piecewise-linear approximation prove inapplicable in the analysis, and the phase plane \((x,\dot{x})\) cannot be used. The analysis can be carried out in a special phase plane, where the values \(x\) of the controlled quantity and \(y\) of the variable gain coefficient are plotted along the coordinate axes. To undisturbed motion in the plane \((x,y)\) there corresponds the singular point \((1,1)\). From the character of the perturbed motion of the representative point one can judge the behavior of the system; a closed limit cycle in the plane \((x,y)\) will correspond to the presence of self-oscillations in the control system. A drawback of analysis in the plane \((x,y)\) is the need, from the whole set of initial conditions, to choose the condition with zero initial values of the rates of change of the quantities \(x\) and \(y\).

Mathematical analysis of system (1) in the first quadrant of the phase plane, in the small and in the large (by the Bendixson criterion), showed that the given system is stable and contains no closed solutions. Further investigations were carried out by a graphical-analytical method and on the MN-7 analog computer. The character of the motion of the representative point in the presence of backlash, limitation of the speed of the actuator motor, and the presence of upper and lower limits for the gain coefficient \(y\), is shown in Fig. 2.

In the part of the phase plane located between the hyperbola \(xy = 1\) (2) and the backlash line \(y = f(x,a,\Delta)\), the process of taking up the backlash occurs; the representative point moves vertically. In the remaining part of the phase plane, located between the hyperbolas,

\[ \begin{aligned} xy_r &= 1 + \frac{|\dot{x}_M|}{a},\\ xy_r &= 1 - \frac{|\dot{x}_M|}{a}, \end{aligned} \tag{3} \]

where \(\dot{x}_M\) is the maximum motor speed, the path of the integral curves is determined by the equation

\[ \frac{dy}{dx}=\frac{x^2-1}{a(1-xy)}. \tag{4} \]

The backlash line \(y=f(x,a,\Delta)\) between the hyperbolas (3) is determined by the equation

\[ y=\frac{1}{x}\left(1+\operatorname{sign}\alpha\cdot \sqrt{\frac{2\Delta|\alpha|}{a}}\right), \tag{5} \]

where \(\alpha=x(x^2-1)\), \(\Delta\) is the reduced backlash. Outside the hyperbolas (3), the equation of the backlash line takes the form

Fig. 2.

\[ y=y_{\Gamma}+\frac{\operatorname{sign}\alpha}{x|x_M|} \left(\Delta|\alpha|-\frac{x_M^2}{2a}\right), \tag{6} \]

where \(y_{\Gamma}\) is determined from (3).

Outside the part of the phase plane lying between the hyperbolas (3), the representative point moves along the integral curves

\[ y=\frac{x^2}{3\dot{x}_M}-\frac{x}{\dot{x}_M}+C, \tag{7} \]

where the constant \(C\) is determined from the initial conditions.

To take into account the boundedness of the limits of variation of the variable coefficient \(y\), an auxiliary variable quantity \(y^*\) was introduced according to the following relation:

\[ \left. \begin{aligned} y^*&=y \quad \text{for } y_{\min}<y<y_{\max},\\ \frac{dy^*}{d\tau}&=x^2-1 \quad \text{for } y=y_{\text{ext}}, \end{aligned} \right\} \tag{8} \]

where \(y_{\text{ext}}\) is one of the extreme values of the gain coefficient.

In the phase plane \((x,y^*)\), when \(y=y_{\text{ext}}\), the representative point in the region between the hyperbolas (3) moves along integral curves of the form

\[ y=\frac{1}{a}\left[ -\frac{x^2}{2y_{\text{ext}}} -\frac{x}{y_{\text{ext}}^2} +\frac{y_{\text{ext}}^2-1}{y_{\text{ext}}^3} \cdot \ln|y_{\text{ext}}\cdot x-1| \right]+C_1. \tag{9} \]

Outside the region bounded by the hyperbolas (3), the representative point moves according to (7) until it meets the vertical line \(x=1\); further motion takes place according to (9).

The investigation has shown that the presence of backlash causes the appearance of a stable limit cycle. The parameters of the limit cycles were found by the method of point transformations. The amplitude of self-oscillations, without taking into account the velocity limitation, can be calculated from the approximate relation:

\[ (x_{n,\max}-1)\simeq \frac{0.7\cdot \Delta}{\sqrt{a}\cdot \sqrt[3]{e^a}} . \tag{10} \]

with an error of no more than \(5\%\) of the relative deviation.

Expression (10) is also valid when the velocity is limited to \(\dot{x}=0.375\) and the backlash is up to \(\Delta=0.25\). A greater velocity limitation may cause self-oscillations with variation of the controlled quantity from

\[ x_{\min}=\frac{1}{y_{\max}} \quad \text{to} \quad x_{\max}=\frac{1}{y_{\min}} . \]

It has been found that increasing the reduced parameter \(a\) promotes stabilization of the correction system. Physically, this is explained by the fact that an increase in \(a\) corresponds to a decrease in the rate of action of the integral correcting device. It should be noted that increasing \(a\) by increasing the gain coefficient of the main loop is undesirable, since it will reduce the stability of the main system.

For values \(a>8\), the process of integral correction has no more than one overshoot.

Conclusions

1. Elimination of the systematic error of the root-mean-square value of the controlled quantity requires the presence of an integral correction system.

2. The integral correction system can be investigated in the phase plane \((x,y)\), where \(x\) is the controlled quantity and \(y\) is the variable gain coefficient.

3. The correction system is structurally stable, but the presence of backlash and limitations causes self-oscillations. The amplitude of the self-oscillations can be determined analytically in general form from (10).

4. Increasing the parameter \(a\) of the system increases the stability of the correction process.

5. The analysis carried out makes it possible to select the system parameters in engineering design calculations.

References

1. Fundamentals of Automatic Control. Edited by V. V. Solodovnikov. Mashgiz, 1954.

2. A. A. Feldbaum. Computing Devices in Automatic Control Systems. Fizmatgiz, 1959.

3. V. V. Nemitskii and V. V. Stepanov. Qualitative Theory of Differential Equations. GITTL, 1949.

4. N. M. Matveev. Methods for Integrating Ordinary Differential Equations. Leningrad University Press, 1955.