4.6. MODELING THE TRANSFER FUNCTIONS OF CONTROL SYSTEMS 
In the previous material presented, we designate transfer functions to various elements of the control system. For example in Figure 2.9, some of the elements shown in this block diagram are designated G1(s), G2(s), G3(s), G4(s), etc. How do we determine their actual transfer functions? If these elements are simple networks as described in Table 2.4, we can then use their corresponding theoretical transfer functions as shown in Table 2.4. If they are electromechanical motors, we can use the transfer functions for these devices that are derived in Section 3.4. However, in the case of motors and other devices, how do we know whether the actual transfer function agrees with the theoretical transfer function? What if we have an unknown device, or “black box,” and we do not know what its transfer function is at all?
To address this problem, we can apply a test signal to the device and measure its output. From knowledge of the input test signal and the measured output, we can then determine the device’s actual transfer function. The most commonly used test signals are a unit step and a sinusoidal signal. We use the unit step to obtain a transient response of the device. This approach is presented in this section. The frequency-response approach for testing the frequency reponse of a device to a sinusoidal input signal is presented in Chapter 6.
Suppose we have a linear control system consisting of only one element, G(s), as ...