Design Case Studies

MIMO LQG Design

Start with the complete two-axis state-space model Pc derived above. The model inputs and outputs are

• Pc.inputname
  
  ans = 
      'u-x'
      'u-y'
      'w-ex'
      'w-ix'
      'w_ey'
      'w_iy'
  
  P.outputname
  
  ans = 
      'x-gap'
      'y-gap'
      'x-force'
      'y-force'
  

As earlier, add low-pass filters in series with the 'x-gap' and 'y-gap' outputs to penalize only low-frequency thickness variations.

• Pdes = append(lpf,lpf,eye(2)) * Pc
  Pdes.outputn = Pc.outputn
  

Next, design the LQ gain and state estimator as before (there are now two commands and two measurements).

• k = lqry(Pdes(1:2,1:2),eye(2),1e-4*eye(2))     % LQ gain
  est = kalman(Pdes(3:4,:),eye(4),1e3*eye(2))    % Kalman estimator
  
  RegMIMO = lqgreg(est,k)    % form MIMO LQG regulator
  

The resulting LQG regulator RegMIMO has two inputs and two outputs.

• RegMIMO.inputname
  
  ans = 
      'x-force'
      'y-force'
  
  RegMIMO.outputname
  
  ans = 
      'u-x'
      'u-y'
  

Plot its singular value response (principal gains).

• sigma(RegMIMO)
  
  
  
  

Next, plot the open- and closed-loop time responses to the white noise inputs (using the MIMO LQG regulator for feedback).

• % Form the closed-loop model
  cl = feedback(Pc,RegMIMO,1:2,3:4,+1);
  
  % Simulate with lsim using same noise inputs
  lsim(Pc(1:2,3:6),':',cl(1:2,3:6),'-',wxy,t)
  
  
  
  

The MIMO design is a clear improvement over the separate SISO designs for each axis. In particular, the level of / thickness variation is now comparable to that obtained in the decoupled case. This example illustrates the benefits of direct MIMO design for multivariable systems.

Cross-Coupling Between Axes

Kalman Filtering