0195136047.pdf

790 BASIC CONTROL SYSTEMS

• Settling time tsis the time required for the step response to decrease and stay within a
  specified percentage (usually 5%) of its final value.
  While these quantities are relatively easy to measure once a step response is plotted, they
  cannot easily be determined analytically, except for simple cases.

Steady-State Error of Linear Systems

If the steady-state response of the output does not agree exactly with the steady state of the input,

the system is said to have asteady-state error. Steady-state errors in practical control systems are

almost unavoidable because of friction, other imperfections, and the nature of the system. The

objective is then to keep the error to a minimum, or below a certain value. The steady-state error

is a measure of system accuracy when a specific type of input is applied to a control system.

Referring to Figure 16.2.3, assuming the input and output signals are of the same dimension

and are at the same level before subtraction, with a nonunity elementH(s) incorporated in the

feedback path, the error of the feedback control system is defined as

e(t)=r(t)−b(t) or E(s)=R(s)−B(s)=R(s)−H(s)C(s)

or using Equation (16.2.2),

E(s)=

R(s)

1 +G(s)H (s)

(16.2.18)

Applying the final-value theorem, the steady-state error of the system is

eSS=lim

t→∞

e(t)=lim

s→ 0

sE(s) (16.2.19)

in whichsE(s) is to have no poles that lie on the imaginary axis and in the right half of thes-plane.

Substituting Equation (16.2.18) into Equation (16.2.19), we get

eSS=lim

s→ 0

sR(s)

1 +G(s)H (s)

(16.2.20)

which apparently depends on the reference inputR(s) and the loop gain (loop transfer function)

G(s)H(s).

Thetypeof feedback control system is decided by the order of the pole ofG(s)H(s)ats=0.

Thus, if the loop gain is expressed as

G(s)H (s)=

KN (s)

sqD(s)

=

K( 1 +T 1 s)( 1 +T 2 s)···( 1 +Tms)

sq( 1 +Tas)( 1 +Tbs)···( 1 +Tns)

(16.2.21)

whereKand all of theTare constants, the exponent ofs, i.e.,q, in the denominator represents

the number of integrations in the open loop, and the exponentqdefines the system type. Withq

=0, 1, or 2, the system is classified asposition,velocity,oraccelerationsystem, respectively.

Table 16.2.2 summarizes the error response for different unit inputs and three system types. The

development of these results is left as Problem 16.2.32 at the end of this chapter.

Classification of Feedback Control Systems by Control Action

A more common means of describing industrial and process controllers is by the way in which

the error signalE(s) is used in the forward loop of Figure 16.2.3. The basic control elements are

• Aproportional device,such as an amplifier


• Adifferentiating device,such as an inductor