MPEC Problem Formulations in Chemical Engineering Applications

the following inner minimization problem and associated equation [21]:minys.t.N ∑i=1(x − a i )(x − a i−1 )y i (38a)N∑y i = 1i=1y i ≥ 0(38b)(38c)z =N∑f i (x)y i (39)i=1where N is the number of piecewise segments, f i (x) is the function over theinterval x ∈ [a i−1 , a i ], and z represents the value of the piecewise function. ThisLP will set y i = 1 and y j≠i = 0 when x ∈ [a i−1 , a i ]. The associated equationwill then set z = f i (x), which is the function value on the interval. The NLP(38) can be rewritten as the following complementarity system:N∑y i = 1 (40a)i=1(x − a i )(x − a i−1 ) − γ − s i = 0 (40b)0 ≤ y i ⊥ s i ≥ 0 (40c)In some cases, the function of interest may only be piecewise continuousand y i can “cheat” by taking fractional values. For instance, cost per unit mayincrease stepwise over different ranges, and jumps in the costs may occur at a i .A way around this problem would be to define f i (x) as the total cost (i.e., unitcost times quantity), which is continuous everywhere, but not differentiable ata i .5.5 PI Controller SaturationPI controller saturation has been studied using complementarity formulations[30]. The PI control law takes the following form:(u(t) = K c e(t) + 1 ∫ t)e(t ′ )dt ′ (41)τ Iwhere u(t) is the control law output, e(t) is the error in the measured variable,K c is the controller gain and τ I is the integral time constant. The controlleroutput v(t) is typically subject to upper and lower bounds, v up and v lo , i.e.,v = max(v lo , min(v up , u(t))). The following inner minimization relaxes thecontroller output to take into account the saturation effects:0miny up,y lo(v up − u(t))y up + (u(t) − v lo )y lo (42)s.t. 0 ≤ y lo , y up ≤ 118