adequate phase margin. The crossover frequency is the frequency at which the open-loop gain
of the system is unity (0 db). Third, a small gain at higher frequencies will help to attenuate
the effects of noise or mechanical vibrations. All of these characteristics proved to be
applicable to the controller design for the orange-picking robot. Thus, controllers were desired
which provided flexibility to shape the phase and gain curves to achieve the necessary phase
and gain margins.
Lag and lead compensators furnish this flexibility by providing the ability to adjust
the phase and gain in a wide variety of frequencies. When tuned properly, a lag-lead
compensator improves the steady-state performance while also improving the transient
response (Palm, 1983; Ogata, 1970). The lag-lead compensator is represented by the transfer
function
HP s) Kc rdS+ 1
Hp (S)=
Tis+l (3-7)
where s = Laplace operator (sec-1)
Kc = controller gain (appropriate units),
dra lead time constant (sec), and
zi lag time constant (sec).
The design of the controllers involves the placement of the controller gain and the pole and zero
of the controller so that an acceptable response is achieved while meeting the steady-state
requirements. The open-loop system gain can be increased by increasing the controller gain, Kc.
The use of the lag-lead compensator increases the order of the system by one in between the lag
and lead factors. In other words, the addition of the lag or lead compensator causes an
attenuation of the system response between the frequencies of the lag and lead factors (1/zi and
1/Td). Thus, the value of Ti determines the frequency at which this attenuation takes place
while the value of rd determines the frequency at which the attenuation is canceled. The slope
of the gain curve is decreased by 1 between the lag and lead factors.