Design of a control system

SE4

Prof. Davide Manca – Politecnico di Milano

Dynamics and Control of Chemical Processes

Solution to Lab #4

Design of a control system

System representation

Data:

3

2 1

2 2

2

9.4 m s 30 m

50 m 1.2 s m F i

A A r









A 2

h 2

A 1

h 1

F i

r 1

LC

F o

I.C.:

2 6.6 m h 

New set-point

2 8.6 m

h 

Design of the proportional-integral controller

Performance criteria:

     

2 2

, ,

0 0

C I C I

K K

ISE t dt Min ISE Min t dt

 

 

   

    

 

 

     

, ,

0 0

C I C I

K K

IAE t dt Min IAE Min t dt

 

 

   

    

 

 

     

, ,

0 0

C I C I

K K

ITAE t t dt Min ITAE Min t t dt

 

 

   

    

 

 

Minimum search: brute-force method

 _ max

 _ min max

K C _

min

K C _

Points where to evaluate the objective function

K C

 _

MATLAB implementation

for iIter = 1:nPoints

Kc = KcMin + (iIter - 1) * dKc;

for jIter = 1:nPoints

tauI = tauIMin + (jIter - 1) * dtauI;

... Obtaining dynamics with PI controller

... Minimum ISE (or a different performance index) computing

... Comparison between ISE and its minimum value end

end

Optimizer use

  ²  

, ,

C I C I

0

K K

Min ISE Min t dt

   ^  ^  ^ 

  

   

n , n

C I

K 

Evaluation of the dynamics

Objective function evaluation

Evaluation of the new controller parameters



 



n

,

K



^{ }

 ^

^    ^{ }

  ^

^  

time

Initial steady state

New set-point

MATLAB optimizer

A constrained optimization is performed in order to avoid that the optimizer chooses unfeasible values of the variables respect to which the optimization is being done:

x0 = [Kc tauI];

xMin = [KcMin tauIMin]; % Lower limits xMax = [KcMax tauIMax]; % Upper limits

[x] = fmincon(@(x)funct(x,params),x0,[],[],[],[],xMin,xMax);

KcOptimum = x(1);

tauIOptimum = x(2);