Assessment and Future Directions of Nonlinear Model Predictive ...

488 R. Lepore et al.3 Control Strategy3.1 Control Objectives and NMPC SchemeThe feed flow rate and the classifier selectivity can be used as manipulated variables.Two mass fractions are used as controlled variables. The first one, denotedwP 3 , corresponds to the fine particles in the product flow. Experimental studies[10] demonstrate that this variable is highly correlated with the compressivestrength of the cement, if the upper size of interval 3 is chosen around 30 µm.The second one, denoted wM 2 , corresponds to the mid-size particles in the milloutflow, which can be directly related to the grinding efficiency of the mill (toofine particles correspond to overgrinding whereas too coarse particles correspondto undergrinding).The use of these several variables is illustrated by the steady-state diagramwM 2 = f(q P ,wP 3 ) of Figure 2, where the curve ABC represents all the operatingpoints with wP 3 =0.86. Clearly, point B corresponds to a maximum product flowrate and, as demonstrated in [3], the arcs AB and BC correspond to stable andunstable process behaviours, respectively. By setting, for example, wM 2 =0.35on arc AB, a single operating point (point 1) is defined. This corresponds toproducing cement of a given fineness (wP 3 =0.86) at near maximum product flowrate in the stable region. A significant advantage of these controlled variables isthat the measurement of mass fractions is simple and inexpensive. A classicalsieving technique is used instead of sophisticated (and costly) laser technology.The design of the NMPC scheme [1, 8] is based on a nonlinear optimizationproblem, which has to be solved at each sampling time t k = kT s (where T s isthe sampling period). More specifically, a cost function measuring the deviationof the controlled variables from the setpoint over the prediction horizon has tobe minimized. Denoting y =[wP 3 w2 M ]T the controlled variable, the optimizationproblem is stated as follows:∑Npmin {y s − ŷ(t k+i )} T Q i {y s − ŷ(t k+i )} (2){u i} Nu−10 i=1where Nu and Np are the control and prediction horizon lengths, respectively(number of sampling periods with Nu < Np). {u i } Nu−10is the sequence ofcontrol moves with u i = u Nu−1 for i ≥ Nu (u i is the input applied to theprocess model from t k+i to t k+i+1 ). ŷ(t k+i ) is the output value at time t k+i ,as predicted by the model. {Q i } Np1are matrices of dimension 2, weighting thecoincidence points. y s is the reference trajectory (a piecewise constant setpointin our study).In addition, the optimization problem is subject to the following constraints:u min ≤ u i ≤ u max (3a)−∆u max ≤ ∆u i ≤ +∆u max(3b)qM min ≤ q M ≤ qMmax(3c)