Control Strategy
Single input and single output (SISO) with unit feedback
This is a particular type of control extensively studied and used. The
purpose is to design the controller in the following control process. A
controller works on the error (the difference between the desired output
and the current output), which varies with time as control is actuated.
Controller
Plant
X(s)
R(s)
+
-
G(s)
C(s)
E(s)
F(s)
H(s) =1
Strategy One: proportional (to the present error)  C ( s)  k p
It is shown already this strategy produces response (output) that oscillates
many times around the desired input. As the output gets closer to the
desired input, the control ‘force’ becomes smaller and hence ineffective.
Strategy Two: proportional and derivative (the future)  C ( s)  k p  kd s
This should reduce overshoot and speed up the convergence of the output
to the desired input. However, if the desired input drifts, this strategy will
not work.
Strategy Three: proportional, integral (the past) and derivative
C (s)  k p 
ki
 kd s
s
Such a controller is called PID controller and is widely used.
An example of using the above strategy
Suppose a radar is trying to track a satellite in the sky. Once the satellite
moves to a new place, a sensor measures the distance between the radar’s
current orientation and send the signal to a computer, which commands the
controller to orient the radar towards the new position of the satellite. This
force from the controller is proportional to the distance between the
current orientation of the radar and the new position of the satellite in a
dynamic manner. As the radar turns towards the satellite, the control force
become smaller and smaller and due to friction the force will never be able
to align the radar with the satellite. If derivative control is also used, the
force is proportional to the rate of the error and hence will still be effective
when the error itself is small. However, if the satellite keeps moving about,
then the (proportional and derivative) controller will keep trying to follow
a moving target and will never quite catch up. When integral control is
added, the controller will make use of the time history of the target to track
it and should do a better job.
bm ( s  z1 )( s  z2 )       ( s  zm )
Open-loop transfer function G ( s )  a ( s  p )( s  p )       ( s  p )
n
1
2
n
The characteristic equation of the closed-loop transfer function with
negative unity feedback is T ( s) 
CG

1  CG
bm ( s  z1 )( s  z2 )    ( s  zm )(kp s  ki  kd s 2 )
an ( s  p1 )( s  p2 )    (s  pn )s  bm (s  z1 )( s  z2 )    ( s  zm )( kp s  ki  kd s 2 )
The error is E ( s)  R( s)  X ( s)  [1  T ( s)]R( s) 
R( s )

1  CG
an ( s  p1 )( s  p2 )    ( s  pn ) sR(s)
an ( s  p1 )( s  p2 )    (s  pn )s  bm (s  z1 )( s  z2 )    (s  zm )(kp s  ki  kd s 2 )
[e(t )]  lim [ sE ( s)] 
The steady-state error due to a unit-step input is lim
t 
s 0







n
(1) an  pi
n
i 1
n
(1) an  pi  k p bm (1)
n
i 1
0
P or PD
m
n
z
i
i 1
I (PI or PID)
Example: A close-loop system has C ( s) 
0.5
s
and G ( s) 
1
. Determine
s 1
the response under a unit-step input.
Solution: T ( s ) 
0.5
1
. R ( s )  . The output is
s ( s  1)  0.5
s
0.5
1 1
s 1
  2
s( s  1)  0.5 s s s  s  0.5
1
s 1
1
s  0.5
0.5
 



s ( s  0.5) 2  0.25 s ( s  0.5) 2  0.25 ( s  0.5) 2  0.25
X ( s )  T ( s ) R( s ) 
The final output value
x()  lim [ sT ( s) R( S )]  lim [
s 0
s 0
0.5
] 1
s ( s  1)  0.5
The response (output) is
x(t )  1  exp( 0.5t )(cos 0.5t  0.5 sin 0.5)
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Effects of Increasing Parameters
Parameter
Rise Time
Overshoot
Settling Time
S.S. Error
kp
Decrease
Increase
Small Change
Decrease
ki
Decrease
Increase
Increase
Eliminate
kd
Small Change
Decrease
Decrease
None
The table summarises some of the features of some widely-used control
strategies. Among them, PID control is arguably the most versatile one.
Strategy
C(s)
f (t)
P
kp
k p e(t )
simple,
cheap
e (t )  0
PD
k p  kd s
k p e  kd e
improve
damping
may generate
noise
PI
kp 
k p e  ki  edt
e (t )  0
may cause
instability
k p e  ki  edt  k d e
requires no
plant model
for design,
accurate,
more stable
may yield
poor response
most
advantages
of PI or PD
more
complicated
PID
Lag/Lead
kp 
ki
s
ki
 kd s
s
k l (1  Ts)
1  Ts
kle
1

(k l e  f )dt
 T 
Advantages Disadvantages