# Exercise: Exploring the dynamics of rolling mill

Lars Cremean, September 2002

Download the file “hw1rollmill.mdl”, which contains a SIMULINK model of a rolling mill, which is a machine used to create material (paper, steel, etc) of a given thickness (see documentation from web page for more details).

1. Run the simulation with default parameters and create a plot of the output thickness versus time.

2. While keeping the proportional gain  fixed at its default value, vary the integral gain  from zero to twice the default simulation value. Give the value for which the system begins to go unstable. Plot the settling time as a function of Ki for the stable cases.

3. While keeping  fixed at its default value, vary  from zero to twice the default simulation value. Give the value for which the system begins to go unstable. Plot the settling time as a function of  for the stable cases.

All plots should included a title, labeled axes (with units), and reasonable axis limits.

Note: It is not necessary for you to understand the dynamics of the process and control law at this point. This model makes use of some more advanced features in MATLAB and SIMULINK, which you may need to spend some time learning.

### Rolling mill dynamics

The rolling mill example is a simplified model of a steel rolling mill. The control problem is to control the exit thickness of a sheet of steel that is fed between two rollers, as shown below The exit thickness h of the steel is related to the gap between the rollers, which depends on the unloaded position of backup rollers (also called the screw position),  and the input thickness . The force on the rollers is reasonably modeled by



as well as



where  and  are constants.

Combining these two gives



Measurement and actuator dynamics are modeled with time delays of 1 second and 0.01 seconds, respectively. A proportional-integral control law is incorporated into the model whose job is to try and regulate the output thickness to the desired thickness.

### Notes

1. The output thickness for each run is saved to the MATLAB workspace as a structure called "rollmill". You can access the elements of a structure using a period (.), e.g. rollmill.signals...
2. The settling time is the time it takes for the system error to settle to within the desired thickness plus or minus 5%. Any method of finding these values to produce your plots is acceptable, but you may like to investigate the MATLAB "find" command.
3. Include at least ten values in each of your settling time versus  plots.
4. Use the MATLAB "plot" command to produce your plots. Type "help plot" in the MATLAB command window for help on using this function.
5. Include a title, axis labels (with units), and reasonable axis limits on all plots. See MATLAB commands "title", "axis", "ylabel" and "xlabel".