Robust PID design for non-minimum systems

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 10 | Oct -2017 www.irjet.net p-ISSN: 2395-0072
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Robust PID Controller Design for Non-Minimum Phase Systems Using
Magnitude Optimum and Multiple Integration and Numerical
Optimization Methods
M. Anil Kumar1, B. Amarendra Reddy2
1PG Student, EEE-Dept. Andhra University, Visakhapatnam, India.
2Assistant Professor, EEE-Dept. Andhra University, Visakhapatnam, India.
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Abstract - In this paper the controller design for non
minimum phase systems are obtained by using magnitude
optimum and multiple integration method and numerical
optimization approach. There are many ways to design a
proper controller for a specific system. This paper mainly
focused on the optimization approach methods. In this paper
non minimum phase systems are used for design controller.
Key Words: Magnitude optimumandmultipleintegration
method,Numericaloptimization approach,Non minimum
phase systems, Controller parameters.
1. INTRODUCTION
There have been great amount of research work on the
tuning of PID controllers. In this paper, two approaches are
given for the controller design of non minimum phase
systems. PID controllers have been used for a long time.
Taylor developed the first PID controller. But the problem is
occurred in how to tune a PID controller. at that time Ziegler
and Nichols discover the famous Ziegler and Nichols tuning
rules. These rules are still widely used. This is why so many
different tuning rules have been developed which are based
on the same tuning procedure. by this way In this paper
discussed about the controller design of the non minimum
phase systems by using magnitude optimum and multiple
integration and numerical optimization approach.Generally
a non minimum phase system is meant byamongall systems
having the same magnitude plot, those with the least phase
shift range are called minimum phase. Remainingareall non
minimum phase. The transfer functionofa MPsystemcan be
determined from the magnitude alone. systems having rhp
zeros are non minimum phase. but they’re not the onlyones.
every rational transfer function has a high frequency
magnitude asymptote with slope -20(n-m)dB/dec, where n
is the number of poles and m is the no of zeros. every
rational, minimum phase transfer function has high
frequency phase asymptote at-90(n-m).use these facts to
detect non minimum phase systems from theirbodeplot. ifa
transfer function has poles and /or zeros in the right half
plane then the system shows non minimum phasebehavior.
In this paper find the controller parameters of same non
minimum phase system by using magnitude optimum and
multiple integration andnumerical optimizationapproaches
by using their controller design process.
2. MAGNITUDE OPTIMUM AND MULTIPLE
INTEGRATION METHOD
ROBUST CONTROLLER DESIGN:
The problems with original MO tuning method just
mentioned can be avoided by using the concept of
‘moments’. This can be done by repetitive (multiple)
integrating the input (u) this method is called magnitude
optimum multiple integration (MOMI) tuning method
Derivation of PID controller parameters:
Assume that the rational transfer function of the actual
process be:
1
2
1 2
2
2
1 ....
( ) ;
1 ....
del
m
sTm
P PR n
n
b s b s b s
G s K e
a s a s a s
   

   
(1)
Here Kpr denotes the process steady-state gain, and 1a to
na and 1b to mb are the parameters (m  n) of the process
transfer function, and here delT represents the process
pure time delay.
The following transfer function is describes the filtered PID
controller:
( )
( )
( )
C
U s
G s
E s
  ik
s
 k 
1
d
f
sK
sT
(2)
Where U and E denotes the Laplace transform of controller
output and the controller output, and the controller error
(e=w-y), respectively. Here iK is represented as integral
gain, K is represented as proportional gain, d
K is
represented as derivative gain and fT is called as filter time
constant respectively. The filtered PID controller in a closed
loop configuration with the process is shown in following
figure. In this d denotes the load disturbance. From given
Magnitude optimum method the PID controller parameters
are derived by taking the open-loop transfer function has to
given in a polynomial form. Theexpression(1)containspure

time delay, by using Taylor series the time delay will
transformed into polynomial form.
iK
s
1
d
f
sK
sT
+ +
+
+
+
+
K
-
-
w e
process
d
u
a
Fig. 1. The closed loop system with PID controller
The open- loop system transfer function can be given as
follows:
1
2 3
0 1 2 3
2 3
0 2 3
....
( ) ( ) ;
....
c P
d d s d s d s
G s G s
c c s c s c s
   

   
(3)
Parameters ic and id (i=1,2,…….) can be expressed as
functions of the parameters in the transfer function(1),PID
controller(2) parameters, and parameters of Taylor’s
expansion:
1 ,
1 2[ ( ) ( ) ],
i i i f
i pr i i i f i f d i
c a a T
d K K K T K KT K  

 
 
    
(4)
Where
0
0 0
( 1) ,
( )!
1,
0, 0.
i ki
i k del
i
k k
i i
T
b
i k
a b
a b i




 

 
  

(5)
Let us first assume that filter time constant ( fT ) is given. in
order to determine three control parameters ( , ,i dK K K ),as
required by the given magnitude optimumcriterion,thefirst
three equations( 0 2n   ) from the following set of
equations(Hanus,1975)must hold:
2 1 2
2 1 2
0 0
1
( 1) ( 1) .
2
n n
i i
i n i i n i
i i
d c c c

  
 
    (6)
To calculate the parameters , ,iK K and dK of the controller,
the expression (4) into (6) then applying n=0,1,and2. Then
the result occurred is
1 11, 2 5 , 2 5 ,( , ,...., , ,...., , ),deli pr fK f K a a a b b b T T (7)
12 1, 2 5 , 2 5 ,( , ,...., , ,...., , ),delpr fK f K a a a b b b T T (8)
13 1, 2 5 , 2 5 ,( , ,...., , ,...., , ).deld pr fK f K a a a b b b T T (9)
Hanes (1975) was defined the necessary stability condition
as
0
0
0.
d
c

The above inequality verifies thatthenyquistcurve starts (at
 =0) bellow the real axis ( mI < 0).
The control parameters are depends on 12 process
parameters from (7) to (9). The accurate estimation of the
12 parameters from real measurements could be very
problematic. To avoid this problem use the concept of
repetitive integrationtechnique. Thisconceptisbasedon the
measurement of areas which arecalculatedfromtheprocess
open-loop step response. The areas iA (i=0,1,….) can be
expressed by integrating the process input (u(t) and the
process output(y(t)) after applyingthestepchange∆Uatthe
process input:
0 0 ( ) ,
( )
pr
k k
A y K
A y
  
 
1
1
(( 1) ( ) ( 1) )
!
ik
k k i del k i
pr k k
i
T b
K a b
i
  

    +
1
1
1
( 1) ,
k
k i
i k i
i
Aa

 



(10)
Integrals are defined as follows:
0
( ) (0)
( ) ,
y t y
y t
U



1 1
0
( ) [ ( )] .
t
k k ky t A y d    (11)
By using the expression (10) the all areas are coming
Using the expression (10) it is possibletoeliminateall the12
process parameters from (7)-(9).he controller parameters
iK , ,K and dK are:
2 2 3
2 3 1 4 2 0 4 1 2 0 2( )
2
f f f
i
A A A A T A A A T A A T A A
K
    


(12)
2 2 3
3 1 5 2 3 0 5 1 3 0 3( )
2
f f fA A A T A A A A T A A T A A
K
    


(13)
3 4 2 5
,
2
d
A A A A
K



(14)
Where

2 2
1 2 3 0 1 5 1 4 0 3
2 2
1 2 0 5 0 1 4 0 2 3
2 2 3 2
1 2 0 1 3 0 1 2 0 3
( )
( ) ( ).
f
f f
A A A A A A A A A A
T A A A A A A A A A A
T A A A A A T A A A A A
    
   
   
(15)
The above expression can be write as:
i
d
K
K
K
 
 
 
  
1
1 0
3 2 1 0
2 3
5 4 3 1 0
0 0.5
0
0
f
f f
A A
A A A T A
A A A T A T A

    
           
           
(16)
By using this expression controllerparameters , ,i dK K andK
are calculated.
3. NUMERICAL OPTIMIZATION APPROACH
Numerical optimization presentsa comprehensiveandupto
date description of the most effective methods in
continuous optimization it responds to the going inters in
the optimization engineering. In this paper design the
controller for a no minimum phase system using numerical
optimization method.
Due to its simplicity, robustness and wide ranges of
applicability in the regulatory control layer, the (PID)
controller is use widely. However, a very broad class is
characterized by a periodic response. The commonly used
category of industrial systems can be represented by a first-
order plus dead time model given as,
0
( )
1
t s
ke
G s
s



(17)
For the purpose of simplified analysis, this process model
can only be used.. But the actual mayhavemultiplelags,non-
minimum phase zero, etc. Similarly, another industrial
process is characterized as non-a periodic response. This is
represented by a second-order plus dead-time model given
as,
0
2
1 0
( )
t s
ke
G s
s a s a


 
(18)
Robust PI/PID Controller design
Assume classical and very well known feedback control
system show in figure, here G(s) represents the transfer
function model and K(s) is the transfer function of standard
PI/PID controller
Σ Σ




R(S)
U(S)E(S) K(S)
D(S)
G(S)
Y(S)
Fig. 2. PID feedback control system
PID: ( ) i
p d
k
K s k k s
s
   (19)
The transfer function of the closed loop system is
respectively defined as Sensitivity Function S(s)
1 1
( )
1 ( ) ( ) 1 ( )
S s
K s G s L s
 
 
(20)
Where,      L S K s G S is the open-loop transfer function,
and Complementary sensitivity function C(S)
( )
( ) 1 ( )
1 ( )
L s
C s S s
L s
  

(21)
SECOND ORDER PID CONTROLLER DERIVATION
The open loop transfer function of standardPID controller is
02
0
2
1 0
(1 )(1 )
( )
( )
t s
p i d i
i
k k T s k T s P s e
L s
T s s a s a

  

 
, with 1i iT k . And using
the approximation 0
0
1
(1 )
t s
e
t s



.
Now the open loop transfer function is given by
02
0
2
1 0
(1 )(1 )
( )
( )
t s
p i d i
i
k k T s k T s P s e
L s
T s s a s a

  

 
Now the closed loop transfer function is given by
0
0
2
0
2
1 0
2
0
2
1 0
(1 )(1 )
( ) ( )
(1 )(1 )1 ( )
1
( )
t s
p i d i
i
t s
p i d i
i
k k T s k T s P s e
L s T s s a s a
k k T s k T s P s eL s
T s s a s a


  
 

  

 
(22)
Therefore, the polynomial characteristic equation f the
closed loop system is given by
1 0 0 04 3 21 0 0
0 0
0 0
0 0
1
( ) ( )
( ) ( )
d pd
p i
i i
a a t kk kk Pa t kk P
s s s
t t
kk T kP a k
s s
Tt Tt

   
  
 
 
Which is in the form of
2 2
0 0( ) ( )( 2 )s s a s s      i.e.

4 3 2 2 2
0 0 0
2 2 2 2
0 0 0
( ) (2 2 ) ( 4 )
(2 2 )
s s s a s a a
s a a a
   
  
     
  
(23)
By comparing (21) and (22) we get
0
1 0
0 0
1 1
2 2 2
dkk P
a a
t t
   
2 2
0 0 0 0 0 0 0(2( ) )
p
a a t a Pt a
k
k
     

2 2
0 0
i
a t
k
k


1 0 0 0 0
0
1 2 2
d
a t t at
k
kP
  

The closed-loop stability impose a > 0 which is verified if
0
1
0 0 0
1 1 1
( ) 1
2 2 2
dkk P
a
t t
  
The above inequality is satisfied for
0
1
0 0 0
1 1 1
( )
2 2 2
dkk P
a b
t t
  
With b > 1. Taking into account the first constraint one can
choose m  which gives
0
0 1
0 0
1 1 1
( )
2 2 2
d
m
kk P
a
b t t


  
0
1 0
0 0
1 1
2 2 2
d
m
kk P
a a
t t
    
Therefore, the optimization problem is then written as
0
1
2
0
2
1 0
0
0 1
0 0
0
1 0
0 0
2 2
0 0 0 0 0 0 0
2 2
0 0
1 0 0 0 0
0
max {min 1 ( , )}
(1 )(1 )
( )
( )
1 1 1
( )
2 2 2
1 1
2 2 2
(2( ) )
1 2 2
b
t s
p i d i
i
d
m
d
m
m
p
i
m
d
L j b
k k T s k T s P s e
L s
T s s a s a
kk P
a
b t t
kk P
a a
t t
a a t a P t a
k
k
a t
k
k
a t t at
k
kP
 


 
   

 



  

 
  
   
  


  

(24)
4. EXAMPLES.
To show the effectiveness of these PID controller design
methods for non minimum phasesystemsareconsideredfor
simulation in MATLAB.
Example 1:
Case (a) : magnitude optimum and multiple integration
method:
Consider the second-order system described using a
transfer function 1( )G s . The PID controller parameters are
obtained using magnitude optimum multiple integration
method the detailed step-by step computation procedure is
given as below.
1 2
(1 3 )
( )
2 1
s
s e
G S
s s



 
Here in this problem areas which are obtained from the
expression (10) is:
0 1 2 3
4 5
1, 1,
1, 6, 14.50000, 21.6667
29.2083, 36.8833
pr delK T
A A A A
A A
 
   
 
0.01fT 
By substituting these values of areas in expression (16) the
parameters values are obtained as
0.1186, 0.2114, 0.0826i p dK K K  
Case(b):
Numerical optimization method:
1 2
(1 3 )
( )
2 1
s
s e
G S
s s



 
From the expressions (24) in The controller parameters
are:
0 1 0 0
0
1; 2; 3; 1; 1;
0.5; 3; 0.75; 0.7500
0.2188; 0.1406; 0.2500p i d
a a p t k
b a
K K K
 
     
   
  

0 10 20 30 40 50 60 70 80 90 100
-1
-0.5
0
0.5
1
1.5
Step Response Comparison of (1-3s)exp(-s)/(s2
+2s+1)
Time response
y
MOMI
NA
Fig.3. Time response of with MOMI and NA controllers
Example 2:
Case (a): magnitude optimum and multiple integration
method:
Consider the second-order system described using a
transfer function 2 ( )G s . The PID controller parameters are
obtained using magnitude optimum multiple integration
method the detailed step-by step computation procedure is
given as below.
0.2
2 2
1.5( 0.2 1)
( )
2 1
s
s e
G s
s s

 

 
expression (10) is:
0 1 2 3
4 5
1.5, 0.2,
1.5, 3.6, 5.75, 8.0
10.19, 12.4
pr delK T
A A A A
A A
 
   
 
0.01fT 
0.8081, 1.6060, 0.7894i p dK K K  
Case (b):
0.2
2 2
1.5( 0.2 1)
( )
2 1
s
s e
G s
s s

 

 
From the expressions (24) The controller parameters are:
0 1 0 0
0
1; 2; 0.2; 0.2; 1.5;
0.9; 4.6; 0.75; 2.2325
0.8203; 0.5383; 0.7900p i d
a a p t k
b a
K K K
 
    
   
  
0 10 20 30 40 50 60 70 80 90 100
-0.5
0
0.5
1
1.5
Step Response Comparison1.5(1-0.2s)exp(-0.2s)/(s2
+2s+1)
Time response
y
MOMI
NAOPT
Fig.4. Time response with MOMI and NA
controllers
Example 3:
Case (a) : magnitude optimum and multiple integration
method:
0.2
3 2
(1 4 )
( )
3 2
s
s e
G S
s s



 
expression (10) is:
0 1 2 3
4 5
1, 0.2,
1, 7.200, 21.4200, 56.983
149.5291, 391.6061
pr delK T
A A A A
A A
 
   
 
0.01fT 
0.1180, 0.3496, 0.1061i p dK K K  
Case(b):
0.2
3 2
(1 4 )
( )
3 2
s
s e
G S
s s



 
From the expressions (24) in The controller parameters
are:
0 1 0 0
0
2; 3; 4; 0.2; 1;
0.5; 7.5; 0.75; 2.4375
0.232; 0.2971; 0.1188p i d
a a p t k
b a
K K K
 
    
   
  

0 20 40 60 80 100 120 140 160
-1
-0.5
0
0.5
1
1.5
Step Response Comparison Of (1-4s)exp(-0.2s)/s2
+3s+2
Time response
y
MOMI
NA
Fig.5. Time response with MOMI and NA controllers
CONCLUSION.
This paper deals with the two approaches for controller
design for PID controllers. Which are obtained by MOMIand
Numerical optimization approaches. It is observed that
MOMI method is giving accurate results when compared
with Numerical optimization method. second ordersystems
are considered for simulation in mat lab.
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