plant and the plant with flexible mode are
2.6 × 10 7 × 2.3
s 2 + 131.2s +1.328 × 10 5 ,
G p (s)=
s 2 + 2312s +1.305 × 10 9 ]
respectively. The designed controller with one lag section and one lead section
s 2 + 131.2s +1.328 × 10 5 +
G p (s)=2.6 × 10 7 × [
G c (s)= 1.8258 × 10 −8 (s + 1047)(s + 748)
(s + 9425)(s +62.83)
with crossover frequency chosen to be f v = 500 Hz or 3142 rad/s.
The bode plots of the following transfer functions are shown here and in
the subsequent sections:
• Open loop transfer function: L(s)=G c (s)G p (s),
• Controller transfer function: G c (s),
• Sensitivity transfer function: S(s)=
1+G c (s)G p (s) ,
• Complementary sensitivity transfer function: T (s)= G c (s)G p (s)
1+G c (s)G p (s) ,and
G p (s)
1+G c (s)G p (s) .
• Shock transfer function: S h (s)=
The step response of the closed-loop system is shown in Figure 3.6.
It is clearly evident from these figures that a simple lag-lead compensator
is able to achieve desired shape of the open loop transfer function. The high
frequency actuator resonance (compared with the open-loop servo bandwidth)
does not adversely affect the system's behavior.
Remark 3.1: When ω 1 is suﬃciently small, the lag-lead compensator (3.9)
can be written as the PI-Lead compensator form:
G c (s)=k c T c1 s +1
T c2 s +1
T c3 s +1 .
Reference  suggests T c1 = T c2 =1/(2π250), and T c3 =1/(2π10k)inone
It has been shown above that by using a simple lag-lead compensator, ap-
proximately 40 ◦ phase lead is achieved at the the crossover frequency f v in the
case of a double integrator plant. One may be tempted to increase the open
loop crossover frequency f v in order to achieve higher servo bandwidth. How-
ever, in reality, the effects due to the actuator resonances, sampling frequency