Links for the YouTube Video: https://www.youtube.com/watch?v=TaMO9UIF-Oc&feature=youtu.be Links for the journal papers: http://bit.ly/constr_opt_PI_reg_uns_ISA http://bit.ly/constr_opt_PI_servo_uns_KJCHE http://bit.ly/constr_opt_PI_reg_stable_Tw ------------------------------------------------------- In this video, I am going to show you explicit solutions of the Optimal Control of PI controller, subject to constraints on the process variable y, the manipulated (control) variable u, and the rate of change of the manipulated (control) variable, u'. This video shows the results of the following paper, that proposes a novel optimization-based approach for the design of an industrial proportional-integral(PI)controller for the optimal regulatory control of unstable processes subjected to three common operational constraints related to the process variable, manipulated variable and its rate of change. The proposed analytical design method explicitly takes into account the operational constraints in the controller design stage and also provides useful insights into the optimal controller design. Practical procedures for designing optimal PI parameters and a feasible constraint set exclusive of complex optimization steps are also proposed. The proposed controller was compared with several other PI controllers to illustrate its performance. The robustness of the proposed controller against plant-model mismatch has also been investigated.

2. Analytical constrained optimal Control
for Regulatory and Servo systems
Rodrigue Tchamna, Hien Cao Thi Thu, Moonyong Lee*

3. Optimal control is good, but constrained optimal
control is superior
The most important things are not necessarily the
most complicated
Shck Ca᷅mnà'

4. Problem
Formulation

5.  
2 2
0
min ( ) y ( ) ( '( ))y sp uy t t u t dt 


      
max( )y t y
max'( ) 'u t u
max( )u t u
PI System with proportional quick cancelled : Type-C Controller (I-P controller)

6. Note on the
structure of PID used

7. Fig.1. Block diagram of the PID feedback control of
unstable first order process.
     
 c
c c d
I
de tK
u t K e t e t dt K
dt


  
   
 
 c
c c d
I
E sK
U s K E s K sE s
s


  
Textbook/Ideal PID
     spe t y t y t 
     spE s Y s Y s 
Fig.2. Block diagram of the type C PID (or I-PD) feedback
control of unstable first order process.
Type C PID
Remove the setpoint from Proportional and derivative
     
 c
c c d
I
d y tK
u t K y t e t dt K
dt


        
   
 
 c
c c d
I
E sK
U s K Y s K s Y s
s


          
     spe t y t y t 
 spy t
 spy t

8. -
PI System with proportional quick cancelled : Type-C Controller (I-P controller)

9. Optimal Unconstraint case: Regulatory Unstable
 
2
2
;
2
2 u
y
K
D D



  
  
 
  
Variables Constraint level
Moderate
Moderate
Moderate
maxy
maxu
maxu
1
1 12
2 4
† †
2 †
1 1 1
;
2 42
c
 
 
  
    
      
     
2
u
y K
 

  
  
 
 
opt
†
†
2† †
I
1
1
4 1
c
c
opt c
c
K
K



  

 
 
 
 
  
 


10. Optimal constraint case: Regulatory Unstable
 
2
2
;
2
2 u
y
K
D D



  
  
 
  
Variables Constraint level
Moderate
Moderate
Tight
maxy
maxu
maxu
max
h
u
D




2
2
for 0 1
1
for 1
x 

 
  

 
 

 
 
c
2
c3 2
c 2 2
c
1
,
4
 
   
  
 
    
 
 
   
 
 
 
2
2 2 1
c 012
01
41 1
, 4 exp tan
2 4 3 2
0
c
c c h
c c c
c
h
c
h x for
x
u
for
D
   
        
      
 
 


   
            
 
  

 h
1
, 0
0
c
c c
h
h h
  
   

  

     
   
    
 01
4
h
c


 



11. Optimal constraint case: Regulatory Unstable
Variables Constraint level
Tight
Moderate
Moderate
maxy
maxu
maxu
 
2
2
;
2
2 u
y
K
D D



  
  
 
  
2
2
for 0 1
1
for 1
x 

 
  

 
 

max
g
y
K D




 
 
c
2
c3 2
c 2 2
c
1
,
4
 
   
  
 
    
 
 
 
1
2
1
2
2 tan
exp for 0 1
1
2exp( 1) for 1
2 tanh
exp for 1
1
x
g
xx
x
xx
 




 
    
  
  
 
   
  
g ( ) 0
0
c
c
c
g
g
g
  

  
 

  
   

12. Optimal constraint case: Regulatory Unstable
max max
g ; h
y u
K D D

 


 

Variables Constraint level
Tight
Moderate
Tight
maxy
maxu
maxu
 
1
2
1
2
2 tan
exp for 0 1
1
2exp( 1) for 1
2 tanh
exp for 1
1
x
g
xx
x
xx
 




 
    
  
  
 
   
  
   
 
 
 
2
2 2 1
c 012
01
41 1
, 4 exp tan
2 4 3 2
0
c
c c h
c c c
c
h
c
h x for
x
u
for
D
   
        
      
 
 


   
            
 
  

2
2
for 0 1
1
for 1
x 

 
  

 
 

 
 h
0
, 0
g c
c
g
h
  
  
  

 
 01
4
h
c


 



13. Optimal constraint case: Regulatory Unstable
 
2
2
;
2
2 u
y
K
D D



  
  
 
  
Variables Constraint level
Moderate
Tight
Moderate
maxy
maxu
maxu
 
   
 
   
 
 
 
2 2 2 2
1
c 2 2
2 2 2 2
1
2 2
2 2
4 21
, 1 exp tan 0
2
4 21
1 exp tan 1
2
22
1 exp 1
2
4
1
c c c
f
c
c c c
f
c
cc
c
c c
f x for
x x
x for
x
for
       
   
    
      
 
    
  

  
   


     
       
    
     
      
    
 
    
 
 
 
 
 
2 2
1
2 2
21
exp tanh 1
2
c
c
x for
x
  

    

   
   
    
2
2
for 0 1
1
for 1
x 

 
  

 
 

max
f
u
D
 

 
 
c
2
c3 2
c 2 2
c
1
,
4
 
   
  
 
    
 
 
f ( , ) 0
0
c
c c
f
f f

   
  

   
    
 2
f
c


 



14.  f
g
, 0
( ) 0
c
c
f
g
  
  
  

 
Optimal constraint case: Regulatory Unstable
max max
fg ;
y u
K D D

  
 

Variables Constraint level
Tight
Tight
Moderate
maxy
maxu
maxu
 
1
2
1
2
2 tan
exp for 0 1
1
2exp( 1) for 1
2 tanh
exp for 1
1
x
g
xx
x
xx
 




 
    
  
  
 
   
  
2
2
for 0 1
1
for 1
x 

 
  

 
 

 
   
 
   
 
 
 
2 2 2 2
1
c 2 2
2 2 2 2
1
2 2
2 2
4 21
, 1 exp tan 0
2
4 21
1 exp tan 1
2
22
1 exp 1
2
4
1
c c c
f
c
c c c
f
c
cc
c
c c
f x for
x x
x for
x
for
       
   
    
      
 
    
  

  
   


     
       
    
     
      
    
 
    
 
 
 
 
 
2 2
1
2 2
21
exp tanh 1
2
c
c
x for
x
  

    

   
   
    
   01;
2 4
f h
c c
 
 
   
 
 

15. Optimal constraint case: Regulatory Unstable
Variables Constraint level
Moderate
Tight
Tight
maxy
maxu
maxu
   01;
2 4
f h
c c
 
 
   
 
 
max max
f ; h
u u
D D
 

 
 
2
2
for 0 1
1
for 1
x 

 
  

 
 

 
   
 
   
 
 
 
2 2 2 2
1
c 2 2
2 2 2 2
1
2 2
2 2
4 21
, 1 exp tan 0
2
4 21
1 exp tan 1
2
22
1 exp 1
2
4
1
c c c
f
c
c c c
f
c
cc
c
c c
f x for
x x
x for
x
for
       
   
    
      
 
    
  

  
   


     
       
    
     
      
    
 
    
 
 
 
 
 
2 2
1
2 2
21
exp tanh 1
2
c
c
x for
x
  

    

   
   
    
   
 
 
 
2
2 2 1
c 012
01
41 1
, 4 exp tan
2 4 3 2
0
c
c c h
c c c
c
h
c
h x for
x
u
for
D
   
        
      
 
 


   
            
 
  

 
 
f
h
, 0
, 0
c
c
f
h






  

 

16. opt
opt
c
opt
opt 2 optc
I c
1
1
4 1 ( )
c
opt
K
K



  

 
 
 
 
  
 

Servo plus proportional quick cancelled : Type-C Controller
-

17. Typical Example

18. Example case
constraint specification PI parameter
1 A 0.70 2.70 2.70 1.52 1.52
2 B 0.70 2.70 1.11 1.11 1.11
3 C 0.36 2.70 2.70 2.03 1.59
4 D 0.285 2.70 2.10 2.10 0.62
5 E 0.70 1.105 2.70 2.40 4.14
6 F 0.30 1.20 2.70 2.36 1.19
maxy maxu maxu CK I
10K  
p
10
( )
10 1
G s
s



21. Analytical constrained optimal Control
for Regulatory and Servo systems
Rodrigue Tchamna, Hien Cao Thi Thu, Moonyong Lee*