自动化专业英语Chapter41b

Chapter 4 Response — Classical Method

4-1 INRODUCTION

Having represented control systems using block diagrams as well as state variables, we turn our attention to system response, i.e. how does a system respond as a function of time when subjected to various types of stimuli? Here we are interested in the system output without regard to the behavior of variables inside the control system. When this is the case, we can work with the system transfer function. If we desire C (s ) we can work with C (s )/R (s ) and specify R (s ) and obtain the output. On the other hand, if we need E (s ) we should work with E (s )/R (s ) and specify R (s ). In any event it is important to recognize that when the response to a single input is required without regard to the behavior of variables inside a control system, we speak of applying the classical approach. This can be most readily achieved by employing techniques. This technique involves representing the output (or desired variable) as the ratio of two polynomials and then expanding the expression in partial fractions. The constants of the partial fraction are calculated by the residue theorem. The output in the time domain is the obtained by taking the inverse Laplace transform. A detailed discussion of Laplace transforms is given in Appendix A and should be reviewed by those that do not have a good working knowledge in the use of Laplace transforms.

In general, the input excitation to a control system is not known ahead of time. However, for purposes of analysis it is necessary that we assume some simple types of excitation and obtain system response to at least these types of signals. In general, there are three types * of excitations used in obtaining the response of linear feedback control systems. They are the step input, ramp input, and the parabolic input. These are typical test or reference inputs. In practice, the input is generally never exactly specifiable.

Step input

A step input consists of a sudden change of reference input at t = 0. Mathematically it is

A t r =)( )0(>t

0)(=t r )0(

The function shown in Fig. 4-1a is not defined for t = 0. The Laplace transform of the step input is A/s .

Ramp Input (Step V elocity)

A ramp input is a constant velocity and is represented as

At t r =)( )0(≥t

0)(=t r )0(

The function is shown in Fig. 4-1b and has a Laplace transform of A/s 2.

Parabolic Input (Step Acceleration)

In this case the input is a constant acceleration,

* In many control systems the input may be a sinusoidally varying signal. When this is so, and we know the system is linear, then the output also consists of a sinusoidally varying signal but having a different magnitude and a phase shift which may be functions of the input frequency . W e shall consider this in more detail in a later chapter .