Example 7.10
Given the system of Figure, calculate the sensitivity of the closed-loop transfer function to changes in the parameter a.
How would you reduce the sensitivity?
Figure: Feedback control system
SOLUTION:
G(s) =
K
s(s+a)
The clocked-loop transfer function is
T(s) =
K
𝑠 2 +as+K
The sensitivity of transfer function
ST:a
=
=
=
a 𝛿𝑇
T 𝛿𝑎
a
K
𝑠2 +as+K
×
−𝐾𝑠
(𝑠 2 +as+K)2
-as
(𝑠 2 +as+K)2
Example 7.11
For the system of Figure, find the sensitivity of the steady-state error to changes in parameter K and parameter a with
ramp inputs.
Figure: Feedback control system
SOLUTION:
G(s) =
K
s(s+a)
Velocity constant, Kv
= lims→0 sG(s)
sK
= lims→0 s(s+a)
=
K
a
The steady-state error for this system is
e(∞) =
=
1
Kv
1
a
=
K
a
K
The sensitivity of e(∞) to changes in parameter a is
Se:a
=
=
a 𝛿𝑒
e 𝛿𝑎
a
a
K
×
1
𝐾
=1
The sensitivity of e(∞) to changes in parameter K is
K 𝛿𝑒
Se:K = e
𝛿𝐾
=
K
a
K
×
−𝑎
𝐾2
= -1
Example 7.12
Find the sensitivity of the steady-state error to changes in parameter K and parameter a for the system shown in figure
with a step input.
Figure: Feedback control system
SOLUTION:
G(s) =
K
(s+a)(s+b)
Position constant Kp = lims→0 G(s)
K
= lims→0 (s+a)(s+b)
=
K
ab
The steady-state error for this Type 0 system is
e(∞) =
1
1+Kp
=
1
K
1+
ab
=
ab
ab+K
The sensitivity of e(∞) to changes in parameter a is
Se:a
=
=
=
a 𝛿𝑒
e 𝛿𝑎
a
ab
ab+K
×
(ab+K)-ab2
(ab+K)2
K
ab+K
The sensitivity of e(∞) to changes in parameter K is
Se:K =
=
=
K 𝛿𝑒
e 𝛿𝐾
K
ab
ab+K
×
−𝑎𝑏
(ab+K)2
-K
ab+K
Skill-Assessment Exercise 7.6
Find the sensitivity of the steady-state error to changes in K for the system of Figure.
Figure: Feedback control system
SOLUTION:
G(s) =
K(s+7)
𝑠 2 +2𝑠+10
Position constant Kp = lims→0 G(s)
= lims→0
=
7K
10
K(s+7)
𝑠2 +2𝑠+10
= 0.7K
The steady-state error for this Type 0 system is
e(∞) =
1
1+Kp
=
1
1+0.7K
The sensitivity of e(∞) to changes in parameter K is
Se:K =
=
K 𝛿𝑒
e 𝛿𝐾
K
1
1+0.7K
×
−0.7
(1+0.7K)
2
=
-0.7K
1+0.7K
=
-7K
10+7K