Modern Control Engineering

Section 7–13 / Lag–Lead Compensation 519

ments in both the transient response and steady-state response can be expected. The

response curves shown depict the nature of improvements that may be expected from

using different types of compensators.

Feedback Compensation. A tachometer is one of the rate feedback devices.

Another common rate feedback device is the rate gyro. Rate gyros are commonly used

in aircraft autopilot systems.

Velocity feedback using a tachometer is very commonly used in positional servo

systems. It is noted that, if the system is subjected to noise signals, velocity feedback

may generate some difficulty if a particular velocity feedback scheme performs

differentiation of the output signal. (The result is the accentuation of the noise

effects.)

Cancellation of Undesirable Poles. Since the transfer function of elements in

cascade is the product of their individual transfer functions, it is possible to cancel some

undesirable poles or zeros by placing a compensating element in cascade, with its poles

and zeros being adjusted to cancel the undesirable poles or zeros of the original system.

For example, a large time constant may be canceled by use of the lead network

as follows:

If is much smaller than we can effectively eliminate the large time constant

Figure 7–116 shows the effect of canceling a large time constant in step transient

response.

If an undesirable pole in the original system lies in the right-half splane, this com-

pensation scheme should not be used since, although mathematically it is possible to

cancel the undesirable pole with an added zero, exact cancellation is physically impos-

sible because of inaccuracies involved in the location of the poles and zeros. A pole in

the right-half splane not exactly canceled by the compensator zero will eventually lead

to unstable operation, because the response will involve an exponential term that in-

creases with time.

It is noted that if a left-half plane pole is almost canceled but not exactly can-

celed, as is almost always the case, the uncanceled pole-zero combination will cause

the response to have a small amplitude but long-lasting transient-response compo-

nent. If the cancellation is not exact but is reasonably good, then this component will

be small.

It should be noted that the ideal control system is not the one that has a transfer

function of unity. Physically, such a control system cannot be built since it cannot

T 2 T 1 , T 1.

a

1

T 1 s+ 1

ba

T 1 s+ 1

T 2 s+ 1

b =

1

T 2 s+ 1

AT 1 s+ 1 BAT 2 s+ 1 B

T 1

x

xyz

yz

ttt 1

T 1 s+ 1

T 1 s+ 1

T 2 s+ 1

Figure 7–116
Step-response curves
showing the effect of
canceling a large
time constant.