Page 412 - Mechanical Engineers' Handbook (Volume 2)

5 Control Laws 403

5.3 Proportional-plus-Integral Control
Integral control raised the order of the system by 1 in the preceding examples but did not
give a characteristic equation with enough flexibility to achieve acceptable transient behavior.
The instantaneous response of proportional control action might introduce enough variability
into the coefficients of the characteristic equation to allow both steady-state and transient
specifications to be satisfied. This is the basis for using proportional-plus-integral control
(PI control). The algorithm for this two-mode control is

K I
F(s) KE(s) E(s) (19)
P
s
The integral action provides an automatic, not manual, reset of the controller in the presence
of a disturbance. For this reason, it is often called reset action.
The algorithm is sometimes expressed as
F(s) K 1
1
P
Ts E(s) (20)
I
where T is the reset time. The reset time is the time required for the integral action signal
I
to equal that of the proportional term if a constant error exists (a hypothetical situation). The
reciprocal of reset time is expressed as repeats per minute and is the frequency with which
the integral action repeats the proportional correction signal.
The proportional control gain must be reduced when used with integral action. The
integral term does not react instantaneously to a zero-error signal but continues to correct,
which tends to cause oscillations if the designer does not take this effect into account.

PI Control of a First-Order System
PI action applied to the speed controller of Fig. 20 gives the diagram shown in Fig. 21 with
G(s) K K /s. The gains K and K are related to the component gains, as before. The
P I P I
system is stable for positive values of K and K .For (s) 1/s, 1, and the offset
P I r ss
error is zero, as with integral action only. Similarly, the deviation due to a unit step distur-
bance is zero at steady state. The damping ratio is (c K )/2 IK . The presence of
P I
K allows the damping ratio to be selected without fixing the value of the dominant time
P
constant. For example, if the system is underdamped ( 1), the time constant is 2I/
(c K ). The gain K can be picked to obtain the desired time constant, while K is used
P P I
to set the damping ratio. A similar flexibility exists if 1. Complete description of the
transient response requires that the numerator dynamics present in the transfer functions be
accounted for. 1,2
PI Control of a Second-Order System
Integral control for the position servomechanism of Fig. 23 resulted in a third-order system
that is unstable. With proportional action, the diagram becomes that of Fig. 22, with G(s)
K K /s. The steady-state performance is acceptable, as before, if the system is assumed
P
I
to be stable. This is true if the Routh criterion is satisfied, that is, if I, c, K , and K are
P
I
positive and cK IK 0. The difficulty here occurs when the damping is slight. For small
P
I
c, the gain K must be large in order to satisfy the last condition, and this can be difficult
P
to implement physically. Such a condition can also result in an unsatisfactory time constant.
The root-locus method of Section 9 provides the tools for analyzing this design further.