Formula Sheets: Basics of Control System & Transfer Function | Control Systems - Electrical Engineering (EE) PDF Download

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Control S ystems F ormula Sheet: Basics &
Tr ansfer Function
1. Basics of Control S ystems
• Open-Loop S ystem : Output does not affect input. Tr ansfer function: G(s) =
Y(s)
U(s)
.
• Closed-Loop S ystem : Output affects input via feedback. Tr ansfer function
with negative feedback:
T(s)=
G(s)
1+G(s)H(s)
whereG(s) is the forward path gain,H(s) is the feedback path gain.
• S ystem Order : Highest power ofs in the denominator of the tr ansfer function.
• Type of S ystem : Number of poles at the origin (s =0 ) in the open-loop tr ansfer
functionG(s)H(s) .
2. Tr ansfer Function
• Definition : Ratio of Laplace tr ansform of output to input, assuming zero initial
conditions:
T(s)=
Y(s)
R(s)
• Gener al F orm :
T(s)=
b
m
s
m
+b
m-1
s
m-1
+···+b
0
a
n
s
n
+a
n-1
s
n-1
+···+a
0
wheren=m (system is proper).
• Poles and Zeros :
– Poles: Roots of denominator polynomial (seta
n
s
n
+···+a
0
=0 ).
– Zeros: Roots of numer ator polynomial (setb
m
s
m
+···+b
0
=0 ).
• Char acteristic Equation : Denominator of tr ansfer function set to zero:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
=0
1
Page 2


Control S ystems F ormula Sheet: Basics &
Tr ansfer Function
1. Basics of Control S ystems
• Open-Loop S ystem : Output does not affect input. Tr ansfer function: G(s) =
Y(s)
U(s)
.
• Closed-Loop S ystem : Output affects input via feedback. Tr ansfer function
with negative feedback:
T(s)=
G(s)
1+G(s)H(s)
whereG(s) is the forward path gain,H(s) is the feedback path gain.
• S ystem Order : Highest power ofs in the denominator of the tr ansfer function.
• Type of S ystem : Number of poles at the origin (s =0 ) in the open-loop tr ansfer
functionG(s)H(s) .
2. Tr ansfer Function
• Definition : Ratio of Laplace tr ansform of output to input, assuming zero initial
conditions:
T(s)=
Y(s)
R(s)
• Gener al F orm :
T(s)=
b
m
s
m
+b
m-1
s
m-1
+···+b
0
a
n
s
n
+a
n-1
s
n-1
+···+a
0
wheren=m (system is proper).
• Poles and Zeros :
– Poles: Roots of denominator polynomial (seta
n
s
n
+···+a
0
=0 ).
– Zeros: Roots of numer ator polynomial (setb
m
s
m
+···+b
0
=0 ).
• Char acteristic Equation : Denominator of tr ansfer function set to zero:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
=0
1
3. 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 .
4. Time Domain Specifications
• Standard Second-Order S ystem :
T(s)=
?
2
n
s
2
+2??
n
s+?
2
n
where?
n
= natur al frequency ,? = damping r atio.
• Damping Ratio : ? =
Damping coefficient
2
v
Spring constant· Mass
.
• Natur al Frequency : ?
n
=
v
Spring constant
Mass
.
• Damped Frequency : ?
d
=?
n
v
1-?
2
.
• Rise Time (t
r
):
t
r
˜
1.8
?
n
(for underdamped systems)
• Peak Time (t
p
):
t
p
=
p
?
d
• Peak Overshoot (M
p
):
M
p
=e
-
?p
v
1-?
2
• Settling Time (t
s
):
t
s
˜
4
??
n
(for 2% criterion)
5. Steady-State Error
• Error Constants :
– Position Error Constant: K
p
= lim
s?0
G(s)H(s) .
– V elocity Error Constant: K
v
= lim
s?0
sG(s)H(s) .
– A cceler ation Error Constant: K
a
= lim
s?0
s
2
G(s)H(s) .
• Steady-State Error (e
ss
):
2
Page 3


Control S ystems F ormula Sheet: Basics &
Tr ansfer Function
1. Basics of Control S ystems
• Open-Loop S ystem : Output does not affect input. Tr ansfer function: G(s) =
Y(s)
U(s)
.
• Closed-Loop S ystem : Output affects input via feedback. Tr ansfer function
with negative feedback:
T(s)=
G(s)
1+G(s)H(s)
whereG(s) is the forward path gain,H(s) is the feedback path gain.
• S ystem Order : Highest power ofs in the denominator of the tr ansfer function.
• Type of S ystem : Number of poles at the origin (s =0 ) in the open-loop tr ansfer
functionG(s)H(s) .
2. Tr ansfer Function
• Definition : Ratio of Laplace tr ansform of output to input, assuming zero initial
conditions:
T(s)=
Y(s)
R(s)
• Gener al F orm :
T(s)=
b
m
s
m
+b
m-1
s
m-1
+···+b
0
a
n
s
n
+a
n-1
s
n-1
+···+a
0
wheren=m (system is proper).
• Poles and Zeros :
– Poles: Roots of denominator polynomial (seta
n
s
n
+···+a
0
=0 ).
– Zeros: Roots of numer ator polynomial (setb
m
s
m
+···+b
0
=0 ).
• Char acteristic Equation : Denominator of tr ansfer function set to zero:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
=0
1
3. 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 .
4. Time Domain Specifications
• Standard Second-Order S ystem :
T(s)=
?
2
n
s
2
+2??
n
s+?
2
n
where?
n
= natur al frequency ,? = damping r atio.
• Damping Ratio : ? =
Damping coefficient
2
v
Spring constant· Mass
.
• Natur al Frequency : ?
n
=
v
Spring constant
Mass
.
• Damped Frequency : ?
d
=?
n
v
1-?
2
.
• Rise Time (t
r
):
t
r
˜
1.8
?
n
(for underdamped systems)
• Peak Time (t
p
):
t
p
=
p
?
d
• Peak Overshoot (M
p
):
M
p
=e
-
?p
v
1-?
2
• Settling Time (t
s
):
t
s
˜
4
??
n
(for 2% criterion)
5. Steady-State Error
• Error Constants :
– Position Error Constant: K
p
= lim
s?0
G(s)H(s) .
– V elocity Error Constant: K
v
= lim
s?0
sG(s)H(s) .
– A cceler ation Error Constant: K
a
= lim
s?0
s
2
G(s)H(s) .
• Steady-State Error (e
ss
):
2
– Step Input: e
ss
=
1
1+Kp
.
– Ramp Input: e
ss
=
1
Kv
.
– Par abolic Input: e
ss
=
1
Ka
.
6. Block Diagr am Reduction
• Series Connection : G
1
(s)G
2
(s) .
• Par allel Connection : G
1
(s)+G
2
(s) .
• F eedback Loop (negative feedback):
T(s)=
G(s)
1+G(s)H(s)
• Moving a Summing Point (before a blockG(s) ):
Input to summing point becomes
R(s)
G(s)
.
• Moving a T ak e-off Point (after a blockG(s) ):
Output becomesY(s)G(s).
7. Signal Flow Gr aph
• Mason’ s Gain F ormula :
T =
?
P
k
?
k
?
where:
– P
k
= gain of thek -th forward path.
– ? = 1-
?
L
i
+
?
L
i
L
j
-
?
L
i
L
j
L
k
+··· (determinant).
– ?
k
= determinant of gr aph excluding loops tou chingk -th forward path.
– L
i
= gain of thei -th loop.
3