ABSTRACT
Perhaps the main reason for defining speed of response as we have done is that the inputs to our systems are often in the nature of commands which the output quantity is to reproduce or follow. In a differential equation, the command input appears on the right-hand side and is the term which produces the particular solution portion of the output. This portion of the response is sometimes called the forced part, while the complementary solution (which is present whether there is a driving input or not) is called the natural part of the response. If the forced portion of the response corresponds to the command input, and the natural part is of a form that eventually disappears [such as e-t/r in Eq. (7-21)], then the faster the natural part disappears the faster the system response conforms to the command input. In a firstorder system this leads to the conclusion that a small time constant denotes a fast system.
