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Clarifying ‘minimizing’ and ‘optimizing’ for wet end and dry end controls

I enjoyed reading Coordinating wet end and dry end controls on a fine paper machine in the August 200l issue of Pulp and Paper Canada.

However, a few errors have crept into Fig. 3 in the article, and should be pointed out for the sake of accuracy. These are as follows:

l. The past portion of the trend line is shown, and is labelled ‘minimize’. Trend line is the actual post value of the process variable and minimizing it does not have meaning. It should be a solid trend line.

2. Future value of control action is shaded and is labelled ‘minimize’ also. It is the cost function which has to be minimized. Control action should be optimized. Minimizing the control action will require changing the horizon, making it short will make more aggressive control action where as lengthening it will make control action more sluggish.

3. For a stationary process the set point trajectory into the future is normally a straight line. In the graph, it is shown discontinuous, as a ramp and a straight line. Using set point trajectory instead of exponentially (first order) smoothed reference line may result in overshoot. Usually the ramp is never used in any calculations.

Sincerely,

Y.P. B. Sharma

Fort Frances, Ontario

Dear Mr. Sharma:

The figure Mr. Sharma refers to is a standard picture used in textbooks as a conceptual explanation of predictive control. As such, it should not be taken totally literally. However, I believe it is an accurate conceptual depiction of how a single variable MPC controller operates. Taking Mr. Sharma’s points in order.

1. Although hard to see in the small magazine figure, the “past portion of the trend line” is, in fact, two lines with a shaded area between them. One is a trend of actual values. The other is a trend of past predictions. The annotation says “Adjust model to minimize this area”. i.e. the area between the two curves. Since both trends are available, there is an opportunity to compare model predictions with reality and improve model accuracy. Most implementations (including ours) do only rudimentary model adjustments to the model. In our case, adjust a bias between the model and the measurement and the use of some special filtering in the measurement. Since model errors are common, even this simple correction towards reality has a beneficial effect.

2. The cost function minimized consists of a quadratic term in process error and one in actuator movements. Figure 3 implies that the cost function is computed as an area. For a conceptual diagram, we felt this was a forgivable simplification to increase graphical clarity. The time horizons over which this minimization of the objective function occurs and the weighting factors for each term are chosen as tuning parameters. Normally, one would choose the control action horizon to begin at the present time rather than when the controlled variable horizon starts (at some future time).

3. While most of the time, the control will operate with constant setpoints, one of the most exciting potentials of MPC controllers is that they can control the transient as well as the steady state. On paper machines, speed changes and grade changes are normally made using a ramped setpoint as shown in Figure 3. More complex setpoint trajectories such as the “S-curve” used in some winder controls could also be handled. So we felt we should use the setpoint trajectory in our implementation. As to whether or not there will be overshoot, this is a matter of choosing sensible setpoint trajectories and adequate tuning of the controller.

From the authors,

David Lang, Risto Kuusisto, Mika Kosonen