Design Case Studies

LQG Design for the y-Axis

The LQG design for the -axis (regulation of the thickness) follows the exact same steps as for the -axis.

• % Specify model components
  Hy = tf(7.8e8,[1 71 88^2],'inputn','u-y') 
  Fiy = tf(2e4,[1 0.05],'inputn','w-iy')    
  Fey = tf([1e5 0],[1 0.19 9.4^2],'inputn','w-ey')
  gy = 0.5e-6     % force-to-gap gain
  
  % Build open-loop model
  Py = append([ss(Hy) Fey],Fiy)
  Py = [-gy gy;1 1] * Py
  set(Py,'outputn',{'y-gap' 'y-force'})
  
  % State-feedback gain design
  Pydes = append(lpf,1) * Py      % Add low-freq. weigthing
  set(Pydes,'outputn',{'y-gap*' 'y-force'})
  ky = lqry(Pydes(1,1),1,1e-4)
  
  % Kalman estimator design
  esty = kalman(Pydes(2,:),eye(2),1e3)
  
  % Form SISO LQG regulator for y-axis and close the loop
  Regy = lqgreg(esty,ky)
  cly = feedback(Py,Regy,1,2,+1)
  

Compare the open- and closed-loop response to the white noise input disturbances.

• dt = 0.01
  t = 0:dt:50
  wy = sqrt(1/dt) * randn(2,length(t))
  
  lsim(Py(1,2:3),':',cly(1,2:3),'-',wy,t)
  
  
  
  

The dotted lines correspond to the open-loop response. The simulation results are comparable to those for the -axis.

LQG Design for the x-Axis

Cross-Coupling Between Axes