Some related files:

mlokal\matlab\bin\nonlinpi.m : Nonlinear example (also copy here!)
mlokal\latex\work\talk_feedbacktrick.tex
mlokal\matlab\fb\feedback.m : Distillation example with FF and FB (NOT used)

%  BIG FEEDBACK/FEEDFORWARD EXAMPLE START HERE (file tunepidd)

Kc=0;taui=999,taud=0;td=0;tstop=50;tys=999; tol=0.1; d=1;
graphics;

k=10;kd=10; tau=10; theta=0.001;

Kc=0; uff=0; sim('tunepidd'); t0=Tid; y0=y; 
plot(t0,y0,'--'); 
xlabel('Time');ylabel('Output y');text(20,8,'No control')
hold; 

% FB CONTROL:
Kc=1; taui=4; uff=0;
%sim('tunepidd'); plot(Tid,y,'red',Tid,ys,'--'); text(15,0.3,'Output y (PI control)')
%sim('tunepidd'); plot(Tid,u); text(15,-1.5,'Input u')
%Kc=1; taui=999;
%sim('tunepidd'); plot(Tid,y,'red'); text(15,1.2,'Output y (P control only)')
%sim('tunepidd'); plot(Tid,u); 

k=5; theta=0; sim('tunepidd'); plot(Tid,y,'red'); %text(10,0.5,'k=20,10,5')
k=10; theta=0; sim('tunepidd'); plot(Tid,y,'red'); 
k=20; theta=0; sim('tunepidd'); plot(Tid,y,'red'); 

k=10; theta=0; tau=10;sim('tunepidd'); plot(Tid,y,'red'); %text(10,0.5,'\tau=5,10,20')
k=10; theta=0; tau=5; sim('tunepidd'); plot(Tid,y,'red'); 
k=10; theta=0; tau=20;sim('tunepidd'); plot(Tid,y,'red'); 

k=3; tau=k; sim('tunepidd'); plot(Tid,y,'red'); %text(20,0.5,'k=20,10,3')
k=10; tau=k; sim('tunepidd'); plot(Tid,y,'red'); 
k=20; tau=k; sim('tunepidd'); plot(Tid,y,'red'); 

k=10; theta=0; sim('tunepidd'); plot(Tid,y,'red'); %text(0.7,-6,'\theta=0,1,2,3')
k=10; theta=1; sim('tunepidd'); plot(Tid,y,'r:'); 
k=10; theta=2; sim('tunepidd'); plot(Tid,y,'r:'); 
k=10; theta=3; sim('tunepidd'); plot(Tid,y,'r:'); 
axis([0 50 -10 10])

% FF CONTROL:
uff=1; Kc=0;
%sim('tunepidd'); plot(Tid,y,'red'); text(20,0.5,'Perfect FF control')
  
k=5; theta=0; sim('tunepidd'); plot(Tid,y,'red'); %text(20,5.5,'k=5')
k=10; theta=0; sim('tunepidd'); plot(Tid,y,'red'); %text(20,0.7,'k=10')
k=20; theta=0; sim('tunepidd'); plot(Tid,y,'red'); %text(20,-8,'k=20')

k=10; theta=0; tau=10;sim('tunepidd'); plot(Tid,y,'red'); %text(20,0.5,'\tau=10')
k=10; theta=0; tau=5; sim('tunepidd'); plot(Tid,y,'red'); %text(20,3,'\tau=5')
k=10; theta=0; tau=20;sim('tunepidd'); plot(Tid,y,'red'); %text(20,-2,'\tau=20')

k=3; tau=k; sim('tunepidd'); plot(Tid,y,'red'); %text(20,5.5,'k=3')
k=10; tau=k; sim('tunepidd'); plot(Tid,y,'red'); %text(20,0.7,'k=10')
k=20; tau=k; sim('tunepidd'); plot(Tid,y,'red'); %text(20,-6,'k=20')

k=10; theta=0; sim('tunepidd'); plot(Tid,y,'red'); 
k=10; theta=1; sim('tunepidd'); plot(Tid,y,'red'); %text(14,1,'\theta=0,1,2,3')
k=10; theta=2; sim('tunepidd'); plot(Tid,y,'red'); 
k=10; theta=3; sim('tunepidd'); plot(Tid,y,'red'); 

% ENDS HERE

% CASCADE EXAMPLE
tys=1;td=50;tstop=100; tol=0.1; d=6;
theta=1; numg=[-0.6 1]; deng = [6 1]; num2=1; den2 = conv([6 1],[0.4 1]); 
graphics;

% Without cascade. thetaeff = 1 + 0.6 + 6/2 + 0.4 = 5, tau1 = 9
%    Kc = 0.5 * tau1 / thetaeff = 0.90; taui = 9
Kc=0.90; taui=9; taud=0;
Kc2=1; taui2=9999; k2 = 0;
sim('tunepidc'); t1=Tid; y1=y; 
plot(t1,y1,'red',[0 100],[1 1],'--'),axis([0 100 -0.2 5])

% With cascade.   thetaeff2 = 0.2; tau12 = 6; Kc = 0.5*6/0.2 = 12.5 -> 12; taui2 = 6
% gives L = resulting closed-loop transfer function is  1 / (0.2 s^2 + 0.5 s + 1)
%  which I approximate as effective delay sqrt(0.2) = 0.44 -> 0.4
% Overall outer transfer function then has tau1=6 and thetaeff = 1 + 0.6 + 0.4 =2
%which gives Kc=1.5 and taui=6
Kc=1.50; taui=6; taud=0;
Kc2=12; taui2=6; k2 = 1;
sim('tunepidc');  
plot(t1,y1,':',[0 100],[1 1],'--',Tid,y,'red')
ylabel('Output y');axis([0 100 -0.2 5])

----------

Noen stikkord:


Self-optimizing control

Running 10 km : constant speed? 
           easy to measure with clock.
     constant heart beat? 
    constant level of sugar?
    lactic acid (melkesyre) 

skiing: disturbance = hill. Constant speed  no longer optimal.
   Could have a mix depending on disturbance (constant feed when flat,
    lactic acid in hill?), max speed downhill turn
¨
Cell?

Optimize cell growth? 
   constant oxygen?