Modern Control System Theory and Design, 2nd Edition

4.2.  CHARACTERISTIC RESPONSES OF SECOND-ORDER CONTROL SYSTEMS

The purpose of this section is to describe the transient response of a typical feedback control system. We consider a very common configuratuon in which a two-phase ac servomotor, whose transfer function is given by Eq. (3.101) is enclosed by a simple unity feedback loop. Figure 4.1 illustrates the block diagram of this second-order system. For purposes of simplicity, the gain of the amplifier driving the motor is assumed to be unity.

The closed-loop transfer function of this system is given by

By defining the undamped natural frequency ω_n and the dimensionless damping ratio ζ as

Eq. (4.1) can be rewritten as

The parameters ω_n and ζ are very important for characterizing a system’s response. Note from Eq. (4.3) that ω_n turns out to be the radian frequency of oscillation when ζ = 0. As ζ increases from 0, the oscillation decays exponentially and becomes more damped. When ζ 1, an oscillation does not occur.

We assume that the initial conditions are zero and the input is a unit step. Therefore, R(s) = 1/s, and the Laplace transform ...