Hardware Reference
In-Depth Information
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)=
(3.12)
and
−1
s
2
+ 2312s +1.305 × 10
9
]
(3.13)
respectively. The designed controller with one lag section and one lead section
is
2.3
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)
(3.14)
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),
1
• 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
s
T
c2
s +1
T
c3
s +1
.
(3.15)
Reference [113] suggests T
c1
= T
c2
=1/(2π250), and T
c3
=1/(2π10k)inone
example.
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
