Design Case Studies

Process and Disturbance Models

The rolling mill is used to shape rectangular beams of hot metal. The desired outgoing shape is sketched below.

This shape is impressed by two pairs of rolling cylinders (one per axis) positioned by hydraulic actuators. The gap between the two cylinders is called the roll gap.

The objective is to maintain the beam thickness along the x- and y-axes within the quality assurance tolerances. Variations in output thickness can arise from the following:

• Variations in the thickness/hardness of the incoming beam
• Eccentricity in the rolling cylinders

Feedback control is necessary to reduce the effect of these disturbances. Because the roll gap cannot be measured close to the mill stand, the rolling force is used instead for feedback.

The input thickness disturbance is modeled as a low pass filter driven by white noise. The eccentricity disturbance is approximately periodic and its frequency is a function of the rolling speed. A reasonable model for this disturbance is a second-order bandpass filter driven by white noise.

This leads to the following generic model for each axis of the rolling process.

The measured rolling force variation is a combination of the incremental force delivered by the hydraulic actuator and of the disturbance forces due to eccentricity and input thickness variation. Note that:

• The outputs of , and are the incremental forces delivered by each component.
• An increase in hydraulic or eccentricity force reduces the output thickness gap .
• An increase in input thickness increases this gap.

The model data for each axis is summarized below.

Model Data for the x-Axis

Model Data for the y-Axis

LQG Regulation: Rolling Mill Example

LQG Design for the x-Axis