Biomedical Engineering Reference
In-Depth Information
Using the following relationship,
ω
d
and
ω
n
can be found:
2
π
t
,t
ω
d
=
=
peak-to-peak time
(11.14)
2
π
t
=
2
π
0.3
=
6.67
π
t
=
0.3 seconds
⇒
ω
d
=
(11.15)
ω
d
1
−
6.67
π
1
−
0.53
2
ω
n
=
=
=
24.7
(11.16)
ζ
2
24.7
2
610
G(s)
=
=
(11.17)
+
24.7
2
s
2
+
26.2
s
+
610
s
2
+
2
(
0.53
)(
24.7
) s
Now the transfer function is calculated:
K
p
a(s)
b (s)
G(s)
=
(11.18)
+
K
p
a(s)
K
p
a (s)
=
361
,K
p
=
7
⇒
a (s)
=
87
(11.19)
a(s)
b(s)
=
87
H (s)
=
(11.20)
s
2
+
26.2
s
Relationships 11.11 and 11.20 were derived from two different experiments and are
very close to each other. The error that exists between the two relations is small and is
considered to be normal due to the nonlinearities in the linear motor.
11.6.3 Design of the PID Controller
A PID controller can be designed using the following methods: the classical synthetic
open-loop design, analytic state-space methods, and parameter optimizing [2].
11.6.3.1 Classical Synthetic Open-Loop Design
In this method, the designer starts with a simple controller and continues to add more
terms to the controller action in the expectation that, after further additions, the controller
will meet the design goals such that the controller is constructed in a compartmentalized
manner in which a variety of tools are used at different stages [3 - 6].
11.6.3.2 Analytic State-Space Methods
This method, on the other hand, is based on an analytic solution of some optimal controller
design problem, such as the linear quadratic Gaussian (LQG) problem [7 - 9].
