Design Case Studies

Cross-Coupling Between Axes

The / thickness regulation, is a MIMO problem. So far you have treated each axis separately and closed one SISO loop at a time. This design is valid as long as the two axes are fairly decoupled. Unfortunately, this rolling mill process exhibits some degree of cross-coupling between axes. Physically, an increase in hydraulic force along the -axis compresses the material, which in turn boosts the repelling force on the -axis cylinders. The result is an increase in -thickness and an equivalent (relative) decrease in hydraulic force along the -axis.

The figure below shows the coupling.

Accordingly, the thickness gaps and rolling forces are related to the outputs of the - and -axis models by

Let's see how the previous "decoupled" LQG design fares when cross-coupling is taken into account. To build the two-axes model, shown in "Coupling Between the x- and y-axes" above, append the models Px and Py for the - and -axes.

• P = append(Px,Py)
  

For convenience, reorder the inputs and outputs so that the commands and thickness gaps appear first.

• P = P([1 3 2 4],[1 4 2 3 5 6])
  P.outputname
  
  ans = 
      'x-gap'
      'y-gap'
      'x-force'
      'y-force'
  

Finally, place the cross-coupling matrix in series with the outputs.

• gxy = 0.1; gyx = 0.4;
  CCmat = [eye(2) [0 gyx*gx;gxy*gy 0] ; zeros(2) [1 -gyx;-gxy 1]]
  Pc = CCmat * P
  Pc.outputname = P.outputname
  

To simulate the closed-loop response, also form the closed-loop model by

• feedin = 1:2    % first two inputs of Pc are the commands
  feedout = 3:4   % last two outputs of Pc are the measurements
  cl = feedback(Pc,append(Regx,Regy),feedin,feedout,+1)
  

You are now ready to simulate the open- and closed-loop responses to the driving white noises wx (for the -axis) and wy (for the -axis).

• wxy = [wx ; wy]
  lsim(Pc(1:2,3:6),':',cl(1:2,3:6),'-',wxy,t)
  
  
  
  

The response reveals a severe deterioration in regulation performance along the -axis (the peak thickness variation is about four times larger than in the simulation without cross-coupling). Hence, designing for one loop at a time is inadequate for this level of cross-coupling, and you must perform a joint-axis MIMO design to correctly handle coupling effects.

LQG Design for the y-Axis

MIMO LQG Design