Modeling Integrals with Variable Limits on Analog Computing Machines (ACMs)

V. V. Tsygankov

(Presented by the scientific seminar of the computing laboratory of TPI)

For the study of control systems, ACMs have found wide application. In this case the problem consists in modeling the controlled object and the regulator. Most often such problems are solved for the case when the controlled object is described by differential equations. However, this by no means exhausts the full variety of controlled objects, and, in particular, the latter may be described by an integral with variable limits.

For example, one encounters such a problem when considering an accelerator of charged particles—a synchrotron—as a controlled object. The output parameter of a synchrotron—the number of accelerated particles—is a function of such parameters as the moment of particle injection, the moment of switching on the high-frequency accelerating field, etc., and is proportional to a product of integrals whose limits depend on the above-mentioned parameters [1].

Suppose it is required to implement on an ACM the integral:

\[ I(t)=\int_{x_1(t)}^{x_2(t)} f(x)\,dx, \tag{1} \]

where

\[ f(x) \]

is a function specified in the form of a graph, the integral of which cannot be computed analytically;

\[ x_1(t),\ x_2(t) \]

are variable limits having a finite derivative with respect to the parameter \(t\).

We shall use the expression for the derivative of an integral [2]:

\[ \frac{d}{dt}\int_{x_1(t)}^{x_2(t)} f(x,t)\,dx = \int_{x_1(t)}^{x_2(t)} \frac{\partial f(x,t)}{\partial t}\,dx + f(x_2,t)\frac{dx_2}{dt} - f(x_1,t)\frac{dx_1}{dt}. \tag{2} \]

For the case (1), formula (2) has the form

\[ \frac{d}{dt}\int_{x_1(t)}^{x_2(t)} f(x)\,dx = f(x_2)\cdot\frac{dx_2}{dt} - f(x_1)\frac{dx_1}{dt}. \tag{3} \]

From expression (3) follows the scheme for modeling the integral (1), Fig. 1. The value \(I(0)\) is specified by the initial conditions of the integrator, if \(I(0)\) is known. If \(I(0)\) is unknown, it is easy to obtain it on the out-

the integrator input, by virtue of the latter’s property of remembering the voltage at the output. For this it is sufficient to change \(x_1\) and \(x_2\) from zero to \(x_1(0)\) and \(x_2(0)\), respectively. Moreover, the order and the law of variation of \(x_1\) and \(x_2\) are not of essential importance. It is important that the operation of differentiating \(x_1\) and \(x_2\) be performed without large error.

Fig. 1. Block diagram of continuous simulation of an integral with variable limits

The upper frequency range of variation of the limits \(\omega_{x_1}\) and \(\omega_{x_2}\) when simulating (1) according to the scheme of Fig. 1 is limited by the dynamic error of the multiplication block, which amounts to \(1\%\) in the range up to \(100\ \text{Hz}\). The lower frequency range of variation of \(\omega_{x_1}\) and \(\omega_{x_2}\) is limited by the error of the differentiation operation. The scheme of Fig. 1 corresponds to a continuous model of the integral (1), in contrast to the pulse model proposed below.

Indeed, integrating expression (3) for \(x_1=0\), we obtain the well-known scheme of integration on an analog computer with respect to a variable that is not time:

\[ \int f(x)\,dx=\int\left[f(x)\frac{dx}{dt}\right]dt. \tag{4} \]

If, according to (4), \(x\) is varied from \(x_1\) to \(x_2\), then we obtain the scheme of pulse simulation of the integral (1), Fig. 2, since the voltage at the output of the integrator is equal to the integral (1) at the instants \(t_1,t_2,\ldots\). The relations between the frequencies \(\omega_{x_1}\), \(\omega_{x_2}\) and the frequency of the oscillating function \(\omega_x\) must satisfy the Kotelnikov theorem; therefore the scheme of Fig. 2 may be recommended for simulating the integral (1) with low-frequency limits. Both schemes of Figs. 1 and 2 cover the entire frequency range of analog computers.

If in (2) \(\omega_f \ll \omega_{x_1}, \omega_{x_2}\), where \(\omega_f\) is the frequency of variation of the integrand function, then it is possible to compute also the integral

\[ \int_{x_1(t)}^{x_2(t)} f(x,t)\,dt. \tag{5} \]

If the moduli of the derivatives

\[ \left|\frac{\partial f}{\partial x}\right|,\quad \left|\frac{\partial x_1}{\partial t}\right|,\quad \left|\frac{\partial x_2}{\partial t}\right| \]

are of the same order, and \(\omega_f \ll \omega_{x_1}, \omega_{x_2}\), then the error of simulating expression (5) according to the scheme of Fig. 1 exists at the first instant of time after

change of \(f(x,t)\), i.e., the modeling of integral (5) is carried out with some delay. When, however, integral (5) is modeled according to the scheme of Fig. 2, the error is absent altogether if the change of the integrand with respect to the argument \(t\) is approximated by a step line, i.e., \(f(x,t_i)=\mathrm{const}\) at the moment of integration with respect to \(x\). In this case the frequencies \(\omega_f\), \(\omega_{x_1}\), and \(\omega_{x_2}\) may be of the same order.

Fig. 2. Block diagram and timing diagram of pulse modeling of an integral with variable limits

The question of the applicability of the scheme of Fig. 1 or the scheme of Fig. 2 must be decided under specific conditions, when \(\omega_f\), \(\omega_{x_1}\), \(\omega_{x_2}\), the control loop, etc. are specified.

With a somewhat different approach to deriving the relations for particle capture into acceleration than in [1], the number of particles accelerated to a finite energy is proportional to the integral

\[ \int_a^b \int_c^{d-y} f(x)\,dx\,dy, \tag{6} \]

where \(a,b,c,d\) are functions of time.

Consider the inner integral

\[ I(y,t)=\int_c^{d-y} f(x)\,dx. \]

Writing expression (3) for it, we obtain a scheme analogous to Fig. 1, which may be regarded as a functional block. Integral (6) will be equal to

\[ \int_{a(t)}^{b(t)} I(y,t)\,dy. \tag{7} \]

The last expression is analogous to (5), and, taking into account the limitations relating to the modeling of (5) according to the scheme of Fig. 1, we obtain a scheme for modeling integral (6).

Conclusion

The article gives methods and schemes for modeling integrals with variable limits, for the case when the integrand does not depend on time, and also when this dependence is present. In particular, these methods may be applied in modeling a synchrotron as a control object, if the beam of injected particles has no angular spread and the distribution of particles over the amplitudes of betatron oscillations is nonuniform.

REFERENCES

1. I. S. Danilkin, M. S. Rabinovich. ZhTF, vol. 28, no. 2, 1958.
2. V. I. Smirnov. A Course of Higher Mathematics, vol. 2, GITTL, Moscow, 1957.