Control Theory
Session 5 – Transfer Functions
Transfer function of
A.
B.
C.
D.
dz (t )
6
 2 z (t )  5u (t )
dt
A
B
C
None of the above
A
3
6s  2
B
5
6s  1
C
2.5
3s  1
Step response of
4
6s  2
z(t)
A
t
B
Definition of step response: Δz(t) if Δu(t) is a step of size 1
A, B on previous graph?
A.
B.
C.
D.
A=2, B=3
A=2, B=6
A=4, B=6
None of the above
[Default]
[MC Any]
[MC All]
Standard form of first order TF
dz (t )

 z (t )  K p ,u u (t )  c te
dt

Z  ( s ) K p ,u
T p ,u ( s ) 

U  (s)  s  1
t


Step response: z (t )  K p ,u 1  e 





Second order processes
Typical example: mass-spring-damper
z(t)
u(t)
d 2 z (t )
dz (t )
m
c
 kz(t )  u (t )
2
dt
dt
(set-up in a horizontal plane, spring in rest position when x=0)
The step response of the m-c-k
A. Will oscillate
B. Will not oscillate
C. Might oscillate, depending on the values of
m,c and k
[Default]
[MC Any]
[MC All]
The step response will oscillate if
A. c  2 km
B. c  2 km
C. c  2 km
D. That doesn’t depend on
[Default]
[MC Any]
[MC All]
km
Standard form of second order TF
d 2 z (t )
dz (t )
2
te

2



z
(
t
)

K
u
(
t
)

c
n
n
p ,u
dt 2
dt

K p ,u 
Z  (s)
T p ,u ( s ) 
 2
U  ( s ) s  2 n s  n2
2
n
Step respones of 2nd order processes
>1: Overdamped

t
t
K
1
2
1
T p ,u ( s ) 
 z step (t )  K 1   2 1 e   2 1 e  2
( 1s  1)( 2 s  1)
=1: Critically damped = fastest without oscillations

zstep (t )  K 1  ent  ntent
<1: Underdamped: Oscillations!
z step (t ) 
+


The step response of an underdamped
2nd order system
A. Shows no overshoot
B. Shows overshoot of which the size depends
on n but not on 
C. Shows overshoot of which the size depends
on  but not on n
[Default]
D.
Shows overshoot of which the size depends
[MC Any]
[MC All]
on  and n
Overshoot in 2nd order systems
Overshoot in 2nd order systems
Tosc 
2
n 1  
2

 P.O.  100e
1 2
Tpeak 

n 1  
2
Group Task
m=1 [kg]
k=1 [N/m]
Find the TF and
plot the step response for
1) c= 4 [Ns/m]
2) c=2 [Ns/m]
3) c=1 [Ns/m]
Group Task 2
m=1 [kg]
k=1 [N/m]
Can we now add a P controller and calculate the
transfer function of the closed loop?
(by the way, what’s the transfer function of a P controller?)