Operational Methods in Servomechanism Design

Abstract

An important class of automatic control devices whose use is not restricted to any one field is that known as servomechanisms. Defined technically, a servomechanism is an automatic control system having a controlling element actuated by some function of the difference between the actual and desired response of the controlled element. By the use of operational methods in handling physical transients, the servo systems are readily analyzed and quantitative design characteristics obtained. I t is the purpose of this paper to show the use of the Laplace transformation in the analysis and design of servomechanisms. The treatment is divided into three main par ts : (a) the fundamental theory of servomechanisms; (b) the Laplace transformation; and (c) application to a simple servomechanism. Because of military secrecy, the most interesting applications are restricted and cannot be treated publicly. For this reason, the subject matter of this paper is based entirely on accessible literature, and the example chosen is typical of applications in various fields including the military. THEORY OF SERVOMECHANISMS IN A BROAD SENSE, automatic control devices are used to replace human labor by supplying power of proper form and accurate amount to perform a given task with little or no direction. In fact, they are usually more reliable and accurate and economically cheaper than their human counterpart. Automatic control mechanisms may be classified on the basis of the type of control requirements. In the open-cycle type the controlling quantity is independent of the controlled quantity—for example, a time-actuated traffic signal. When the controlling quantity is dependent upon the controlled quantity, the control is of the closed-cycle type. A simple example would be a thermostat. Combinations of both types are also used, such as in a clock-set thermostat. The power requirements of the two types are quite different. Open-cycle devices are usually quite simple and rugged, since the controlling mechanism has sufficient power to operate the device directly. Closedcycle devices are usually more complex, since the controlling mechanism is some form of measuring instrument and has insufficient power to operate the device. Hence, some sort of power or torque amplifier is required to give instrument indications at higher power levels. Presented at the National Fall Meeting, I.A.S., Dayton, Ohio, November 9-10, 1944. * Assistant Professor of Mathematics and Mechanics, College of Engineering and Commerce. Now Ensign, Aircraft Electrical Division, Naval Research Laboratory. For example, in the automatic piloting of ships or aircraft, a measuring instrument sensitive to compass settings must eventually control the ship itself. For purposes of technical definition, a servomechanism is an automatic control system having a controlling element actuated by some function of the difference between the actual and desired response of the controlled element. The power-amplifying device that tends to. make this difference, or error, zero is called a servomotor or servo. The component parts can be visualized better with the help of the block diagram in Fig. 1. If, in particular, it is desired to keep the corrected input constant, the servomechanism is called a regulator. The physical behavior of the servo and controlled system is always such as to affect the quantity under consideration and its successive rates of change with respect to time. The interrelation between the different quantities and the effects of the system components on them is expressed conveniently in the following manner. If F(p) is the transfer polynomial operator for the servo and if G(p) is the transfer polynomial operator for the system, where p = d/dt, then e(t) = i(t) o(t) c(t) = F{p)e{t) o{t) = G{p)[c{t) + d(t)] Solving for the error as a function of the operators on the input e{t) = [*(*) -G(p)d(t)]/[l + F(p)G(p)] and o{t) = [F(p)G(p)m + G(p)d(t)]/[1 + F(p)G(p)] Since G(p) is a function of the system and changes only with it, the basic problem of design in automatic control is to determine a servo having the required transfer operator such that the system is stable and the error lies within specified limits.