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Control S ystems F ormula Sheet: Time
Response Analysis
1. Time Response Components
• Time Response : c(t) = c
t
(t) +c
ss
(t) , where c
t
(t) is tr ansient response, c
ss
(t) is
steady-state response.
• Tr ansient Response : T empor ary response that deca ys to zero ast?8 .
• Steady-State Response : Response ast?8 .
2. Standard T est Inputs
• Step Input : u(t) =A fort= 0 , Laplace:
A
s
.
• Ramp Input : u(t) =At fort= 0 , Laplace:
A
s
2
.
• Par abolic Input : u(t) =At
2
fort= 0 , Laplace:
2A
s
3
.
• Impulse Input : u(t) =d(t) , Laplace: 1 .
3. First-Order S ystem
• Tr ansfer Function :
T(s) =
K
ts+1
whereK is the steady-state gain,t is the time constant.
• Step Response (to unit step input):
c(t) =K
(
1-e
-t/t
)
, t= 0
• Time Constant (t ): Time for response to reach 63.2 % of final value ( 1-e
-1
˜
0.632 ).
• Settling Time (t
s
, for 2% criterion):
t
s
˜ 4t
1
Page 2


Control S ystems F ormula Sheet: Time
Response Analysis
1. Time Response Components
• Time Response : c(t) = c
t
(t) +c
ss
(t) , where c
t
(t) is tr ansient response, c
ss
(t) is
steady-state response.
• Tr ansient Response : T empor ary response that deca ys to zero ast?8 .
• Steady-State Response : Response ast?8 .
2. Standard T est Inputs
• Step Input : u(t) =A fort= 0 , Laplace:
A
s
.
• Ramp Input : u(t) =At fort= 0 , Laplace:
A
s
2
.
• Par abolic Input : u(t) =At
2
fort= 0 , Laplace:
2A
s
3
.
• Impulse Input : u(t) =d(t) , Laplace: 1 .
3. First-Order S ystem
• Tr ansfer Function :
T(s) =
K
ts+1
whereK is the steady-state gain,t is the time constant.
• Step Response (to unit step input):
c(t) =K
(
1-e
-t/t
)
, t= 0
• Time Constant (t ): Time for response to reach 63.2 % of final value ( 1-e
-1
˜
0.632 ).
• Settling Time (t
s
, for 2% criterion):
t
s
˜ 4t
1
4. Second-Order S ystem
• Standard Tr ansfer Function :
T(s) =
?
2
n
s
2
+2??
n
s+?
2
n
where?
n
is natur al frequency ,? is damping r a tio.
• Char acteristic Equation :
s
2
+2??
n
s+?
2
n
= 0
• Poles :
s =-??
n
±?
n
v
?
2
-1
• Damped Natur al Frequency :
?
d
=?
n
v
1-?
2
• Step Response (underdamped, 0<? < 1 ):
c(t) = 1-
e
-??nt
v
1-?
2
sin
(
?
d
t+ tan
-1
v
1-?
2
?
)
5. Time Domain Specifications
• Rise Time (t
r
, time from 10% to 90% of final value):
t
r
˜
1.8
?
n
(approximate for underdamped systems)
• Peak Time (t
p
, time to first peak):
t
p
=
p
?
d
=
p
?
n
v
1-?
2
• Maximum Overshoot (M
p
):
M
p
=e
-
?p
v
1-?
2
• Percentage Overshoot (%OS):
%OS =M
p
×100 =e
-
?p
v
1-?
2
×100
• Settling Time (t
s
, for 2% criterion):
t
s
˜
4
??
n
• Settling Time (for 5% criterion):
t
s
˜
3
??
n
2
Page 3


Control S ystems F ormula Sheet: Time
Response Analysis
1. Time Response Components
• Time Response : c(t) = c
t
(t) +c
ss
(t) , where c
t
(t) is tr ansient response, c
ss
(t) is
steady-state response.
• Tr ansient Response : T empor ary response that deca ys to zero ast?8 .
• Steady-State Response : Response ast?8 .
2. Standard T est Inputs
• Step Input : u(t) =A fort= 0 , Laplace:
A
s
.
• Ramp Input : u(t) =At fort= 0 , Laplace:
A
s
2
.
• Par abolic Input : u(t) =At
2
fort= 0 , Laplace:
2A
s
3
.
• Impulse Input : u(t) =d(t) , Laplace: 1 .
3. First-Order S ystem
• Tr ansfer Function :
T(s) =
K
ts+1
whereK is the steady-state gain,t is the time constant.
• Step Response (to unit step input):
c(t) =K
(
1-e
-t/t
)
, t= 0
• Time Constant (t ): Time for response to reach 63.2 % of final value ( 1-e
-1
˜
0.632 ).
• Settling Time (t
s
, for 2% criterion):
t
s
˜ 4t
1
4. Second-Order S ystem
• Standard Tr ansfer Function :
T(s) =
?
2
n
s
2
+2??
n
s+?
2
n
where?
n
is natur al frequency ,? is damping r a tio.
• Char acteristic Equation :
s
2
+2??
n
s+?
2
n
= 0
• Poles :
s =-??
n
±?
n
v
?
2
-1
• Damped Natur al Frequency :
?
d
=?
n
v
1-?
2
• Step Response (underdamped, 0<? < 1 ):
c(t) = 1-
e
-??nt
v
1-?
2
sin
(
?
d
t+ tan
-1
v
1-?
2
?
)
5. Time Domain Specifications
• Rise Time (t
r
, time from 10% to 90% of final value):
t
r
˜
1.8
?
n
(approximate for underdamped systems)
• Peak Time (t
p
, time to first peak):
t
p
=
p
?
d
=
p
?
n
v
1-?
2
• Maximum Overshoot (M
p
):
M
p
=e
-
?p
v
1-?
2
• Percentage Overshoot (%OS):
%OS =M
p
×100 =e
-
?p
v
1-?
2
×100
• Settling Time (t
s
, for 2% criterion):
t
s
˜
4
??
n
• Settling Time (for 5% criterion):
t
s
˜
3
??
n
2
6. Steady-State Error
• Error Constants (for unity feedback,H(s) = 1 ):
– Position Error Constant: K
p
= lim
s?0
G(s) .
– V elocity Error Constant: K
v
= lim
s?0
sG(s) .
– A cceler ation Error Constant: K
a
= lim
s?0
s
2
G(s) .
• Steady-State Error (e
ss
):
– Step Input: e
ss
=
1
1+Kp
.
– Ramp Input: e
ss
=
1
Kv
.
– Par abolic Input: e
ss
=
1
Ka
.
3
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