Control Systems (K-Wiki_Control Systems Design).pdf

Control Systems (Control Systems Design)
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Control Systems Design
Objectives
Upon completion of this chapter you will be able to:
 Design different compensators to meet the frequency domain specifications of any
Control System.
 Design different controllers and understand their effects on time domain
specifications of any Control System.
Introduction
For any system, the specifications of output can be described in terms of time domain
parameters like t ,t ,t ,t ,M & e
d r p s p ss
and in frequency domain using parameters like
M ,
r r
 , Gain margin, phase margin, Bandwidth. To meet these specifications, process
parameters are not altered but rather we add additional components that can modify these
characteristics to meet the specifications.
Controllers or compensators are additional components that will be added to existing design
to improve the time domain & frequency domain specifications respectively.
Compensators
Compensator is an electrical component which will be introduced in series with the forward
path transfer function in order to introduce certain phase angle into steady state sinusoidal
response of the system.
This property helps us change the phase of the Transfer Function and hence alter the Phase
margin of the system to impact Stability of the system.

Lead Compensators
Lead Compensators introduce phase lead or positive phase shift into sinusoidal response of
the system.
  1
Pole P
c



,   1
Zero Z
c a



Zero is closer to origin than pole, so a 1

The pole-zero plot of the Lead Compensator is shown
in the adjoining figure
Transfer Function  
 
 
c
1
s
s Z a
c
G s
1
s P s
c
 

 
 
 
 
 
 
 

 
P
c
a 1, 0
Z
c
   
  1 a s 1
G s
c a s 1
  
  
  
  
 

 
  ja 1
1
G j
c a j 1
 
 
 
  
  

 
 
   
1 1
tan a tan
 
    
Since, a 1
 , so Ф is always positive and thus it provides lead angle.
Maximum phase angle
m
 
Frequency for maximum phase angle =
m

For maximum phase shift
d
0
d




a
0
2 2 2 2 2
1 a 1
 
 
     
1 1
m 2 a
a
  


a 1 a 1
1 1
sin tan
m a 1 2 a
 
 
 
 
    
 

   
Bode Plot for this system is shown below:
The gain at high frequencies is higher as compared to low frequencies as observed from
Bode Plot so it acts as a High Pass Filter.
Electrical Network
 
 
 
R Cs 1
E s R R
1
0 2 2
R R R
E s R R C
1 1 2
i 1 2
s 1
sC R R
R 1 2
2 1
R
1 sC

 
 
 
 
  
 
   
  
 
 

  
 

R R
1 2
a 1 ; R C
1
R
2

   

Effect of lead compensator
1. Improves the transient response, makes the system response faster.
2. Increases the margin of stability, that means improve the stability.
3. Increases the bandwidth.
4. The noise level in output increases, due to which SNR decreases    
o i
S / N S / N

5. It helps to increase the error constant up to some extent, but doesn’t affect steady state
error much.
Lag Compensator
Lag compensator introduces phase lag or negative phase shift into sinusoidal response of
the system.
  1
Pole P
c



,   1
Zero Z
c a



Pole is closer to origin than zero & hencea 1
 .
The pole-zero plot of lag Compensator is shown
in the adjoining figure.
Transfer Function  
 
 
c
1
s
s Z a
c
G s
1
s P s
c
 

 
 
 
 
 
 
 

 
P
c
a 1
Z
c
 
  1 a s 1
G s
c a s 1
  
  
  
  
 

 
  ja 1
1
G j
c a j 1
 
 
 
 
  
  

 
 

   
1 1
tan a tan
 
    
Since, a 1,
  is always negative and thus it provides a lag angle.
Frequency for maximum phase shift is given by:
1
m a
 

= GM of both corner frequencies.
The maximum phase shift is given by the expression:
a 1 a 1
1 1
sin tan
m a 1 2 a
 
 
 
 
    
 

   
The Bode Plot for this system is shown below:
The gain at lower frequencies is higher as compared to high frequencies as observed from
Bode Plot so it acts as a Low Pass Filter.
Electric Network
 
   
1
R
E s R Cs 1
2
0 sC 2
1
E s R R Cs 1
R R
i 1 2
1 2 sC
 
 
 
 
 
R
2
a 1
R R
1 2
 


Effect of lag compensator
1. Improves the steady state response, it increase error constant and decreases steady state
error.
2. Decrease bandwidth.
3. Reduces effect of noise which means it increases SNR.
4. Reduce stability margin so makes the system less stable.
5. Does not affect transient response.
Lag – Lead Compensator
It is a compensation of lag & lead compensator which provides lag compensation at lower
frequencies & lead compensation at higher frequencies.
The pole-zero plot for Lag-Lead Compensator is shown below:
The Transfer Function for this system can be written as:
 
c
2 1
2 1
1 1
s s
b a
G s
1 1
s s
   
   
   
   
   
   
   
   
 
 
 
 
 
a>1 and b<1

The Bode Plot for this system is shown below:
The gain at edges is more than the gain at middle of band so it acts as Band Stop Filter.
Electric Network
Remember, Lag-Lead compensator cannot be realized using lag and lead compensators
connected in cascade.
If we interchange the location of poles and zeroes we can realize a Lead-Lag Compensator
which provides Phase Lead at lower frequencies and Phase Lag at higher frequencies.

Summary of Compensators
Compensator Pole-Zero location Type of filter
Lead High Pass
Lag Low Pass
Lead-Lag Band Pass
Lag-Lead Band Reject
Solved Examples
Problem: The TF of a compensator is
s 4
s 16


.The maximum phase lead frequency and the
corresponding phase is?
Solution: TF=
1 0.25s
4
1 .0.0625s
 

 

 
.
Comparing with
1 a s
1 s
 
 
We get 0.0625 and a =0.25 =>a=4
  
m
m
1 1
8
a 0.0625 4
8
   

 
1 1
m m
s 4
tan tan
s 16 4 16
 
 

  
 

 
 
;
0
m 36.87
 

Problem: Consider the open loop transfer function of unity feedback system as
21s 97
G(s)
(s 33)



. The maximum phase angle provided by this system is?
Solution:
21s
1
97 97
G(s)
s
33
1
33
 

 
   
 

 
 
Comparing with standard transfer function
1 a s
k
1 s
 
 
 
 
 
We get,
1 21 21 33
= and a =>a= 7.1443
33 97 97

   
a 1 i.e. Lead Compensator

Maximum phase lead angle
1 1 0
a 1 7.1443 1
sin sin 48.97
a 1 7.1443 1
 
   
 
   
   
 
   
Controllers
Controllers are the components that are connected in the forward path to change the steady
state response and transient response of the system.
There are different kind of controllers which have different impact on characteristics of the
system as described below:

Proportional (P)
In proportional controller, the actuating controller signal is proportional to applied input
error signal.
   
y t e t

   
y t K e t
p

 
 
Y s
K
p
E s

Without P controller With P controller
 
 
1
G s
s s 10


Characteristic equation:
2
s 10s 10 0
  
10 rad s
n
  and 10
2
 
 
 
k
p
G s
s s 10


Characteristic equation:
2
s 10s K 0
p
  
K rad s
n p
  and
10
K
p
 
Using proportional controller, the transient response can be improved but e
ss cannot be
altered.
Derivative (D)
In D controller the actuating signal is proportional to derivative of input error signal.
   
d
y t e t
dt

   
d
y t k e t
d dt
 
 
 
 

 
  d
Y s
k s
E s


Without D Controller With D Controller
 
 
1
G s
2
s s 10


e
ss = constant for parabolic i/p.
G(s) = unstable
 
   
d d
K s k
G s
2 s s 10
s s 10
 


Characteristic equation: d
2
s 10s k 0
  
For stability k 0
d
 always
G (s) = stable and e
ss
  for parabolic
input
Using derivative controller stability can be improved but steady state response gets poorer.
Integral (I)
An integral controller gives an output proportional to the integral of input error signal.
   
y t e t dt
 
   
y t k e t dt
i
 
 
 
K
Y s i
E s s

Without I Controller With I Controller
 
 
1
G s
s s 10


System is stable.
e
ss = constant for ramp i/p
 
 
K
i
G s
2
s s 10


Characteristic equation: = 3 2
s 10s K 0
i
  
It is unstable
e 0
ss
 for ramp input.

Using integral control, steady state response can be improved but stability is reduced.
PI Controller
In PI controller, the actuating output is proportional to input error signal and integral of
input error signal.
     
 
y t e t e t dt
  
     
y t K e t K e t dt
p i
  
 
 
K
Y s i
K
p
E s s
 
Without PI Controller With PI Controller
 
 
1
G s
s s 10


System is stable
e
ss = constant for ramp input
 
 
K
i
K
p s
G s
s s 10



:
p
Characteristic Equation:
3 2
s 10s k s K 0
i
   
System is stable if 10k K 0
p i
 
e 0
ss
 for ramp input
PI controller improves steady state response without affecting stability of the system.
PD Controller
In PD controller, the control action is proportional to error signal as well as its derivatives.
     
d
y t e t e t
dt
 
 
 
 

     
d
y t K e t K e t
p d dt
 
 
 
Y s
K K s
p d
E s
 
 
 
 
Without PD Controller With PD controller
 
 
1
G s
2
s s 10


It is unstable
e
ss = constant for parabolic input
 
 
K K s
p d
G s
2
s s 10



Charact. Equation
3 2
s 10s K s K 0
d p
   
Stable if 10k k 0
d p
 
e
ss = constant for parabolic input
PD controller improves stability without affecting steady state response of system.
PID Controller
In PID controller, all 3 control actions are incorporated so it can improve transient & steady
state response as well as stability of the system.

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