0195136047.pdf

16.2 FEEDBACK CONTROL SYSTEMS 789

With the development and widespread use of digital (discrete) control systems and the advent
of relatively inexpensive digital computers, time-response methods have become more necessary
and available. These may be divided into two broad methodologies:

1. The actual simulation or modelingof the system differential equations by either analog
   or digital computers.


2. The state-variable formulationof the system state equations and their solution by a digital
   computer. State-variable methods offer probably the most general approach to system
   analysis and are useful in the solution of both linear and nonlinear system equations.

The transient portion of the time response of a stable control system is that part which goes to
zero as time increases and becomes sufficiently large. The transient behavior of a control system
is usually characterized by the use of a unit-step input. Typical performance criteria that are used
to characterize the transient response to a unit-step input includeovershoot, delay time, rise time,
andsettling time. Figure 16.2.6 illustrates a typical unit-step response of a linear control system.
The following four quantities give a direct measure of the transient characteristics of the step
response:

• Maximum(orpeak)overshoot Ais the largest deviation of the output over the step input
  during the transient state. It is used as a measure of the relative stability of the system. The
  percentage maximum overshoot is given by the ratio of maximum overshoot to final value,
  expressed in percent.


• Delay time tdis defined as the time required for the step response to reach 90% of its final
  value.


• Rise time tris defined as the time needed for the step response to rise from 10% to 90% of
  its final value.

c(t)

Response

Rise time tr

Delay

time

td

Unit-step input

Maximum

overshoot A

Decrement = A/B

Steady-state

error

(t → ∞)

B

0 t

0.10

0.50

1.00

0.95

0.90

1.05

Settling time ts

tmax

Figure 16.2.6Typical unit-step response of a control system.