For the main performance index, we define the range on which it should be. That would be our target parameter Y_{p}, and P(Y_{p} = [y_{pm}i_{n}y_{pmax}]). So, we need to find the slices in the data for the control and observable variables that would maximize P(Y_{p} = [y_{pmin}y_{pmax}]slices). These slices will compose a certain rule of the type “if

^{Y [y}min ^{y}man^{] and U [u}min ^{u}max^{] then Y}p ^{[y}pmin ^{y}pmax^{] with probability p} .

Since we are interested only on the desired performance index Y_{p}, we can leave only the desirable values, reducing the rule to the form “if Y = [y_{min}y_{max}] then U = [u_{min}u_{max}] with a probability P". The rules can be determined using information criteria applied via association rules and decision trees techniques (Figs. 4 and 5).

Fig. 6 Response time of Flame to Calcine Temperature according to Saeman [16]

Dynamic Part of the System

To handle the delays of the system, we applied two approaches. Firstly, we used the Saeman [16] residence time equation to determine proportionally the response time to control according to kiln’s rotation.

That resulted in the following relation with the response time (Fig. 6).

Another additional strategy to determine time differences is by tracking the influences via the theoretical causal relations. It is known by the work of Naud & Emond [7], that the fuel combustion causes the flame to heat up the lining, which is a refractory material, i.e. it retains the heat. The material then receives the heat from the lining, besides the flame itself. So the more accurate way to investigate the temporal response is by determining the influence of the fuel valve on a temperature point nearest to the flame’s highest temperature, and then the influence of this point to the calcine’s temperature. In our case, this point is 75 m from the cold end (Figs. 7 and 8).

By performing correlation analyses and filtering out noisy data, we found the following relation.

Fig. 7 Dependency relation between Fuel, Material and Calcine Temperature

Fig. 8 a Response time of calcine temp. to temp. 75 m b- Response time of flame to temp. 75 m

Finally, to find the rules, we collected 4 months (April thru August 2016) of data from our process plant and sliced up this dataset into 10 zones corresponding to 10 rotation speeds (from 50 T/h to 140 T/h). The total size of the database was about 152000 records sampled by minute. The variables included in the control were: