DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems ...

CHAPTER 9Biochemical engineering problems9.1 LEE-RAMIREZ BIOREACTORDOTcvp: cdop_LeeRamirezBioreactor.mConsidering a bioreactor, which was first solved in [22] and later slightly modified in [32]. The objectiveis to maximize the profitability of the process using the nutrient u 1 and the inducer feeding rates u 2 . Differentscenarios with the various value ofQare considered.The mathematical formulation of the problem is as follow: find the control trajectories that maximize thecost function at the final timesubject tomaxJ 0 = x 1 (t F )x 4 (t F )−Qu i∫ tFt o(u 2 )dt (9.1)x˙1 = u 1 +u 2 (9.2)x˙2 = g 1 x 2 − u 1 +u 2x 2 (9.3)x 1x˙3 = 100u 1− u 1 +u 2x 3 − g 1x 1 x 1 0.51 x 2 (9.4)x˙4 = R fp x 2 − u 1 +u 2x 4 (9.5)x 1x˙5 = 4u 2− u 1 +u 2x 5x 1 x 1(9.6)x˙6 = −k 1 x 6 (9.7)x˙7 = k 2 (1−x 7 ) (9.8)where()(x 3g 1 =x 6 + 0.22x )714.35+x 3 (1+x 3 /111.5) 0.22+x 5()( )0.233x 3 0.0005+x5R fp =14.35+x 3 (1+x 3 /111.5) 0.022+x 5(9.9)(9.10)k 1 = k 2 = 0.09x 50.034+x 5(9.11)The scenario withQ = 0 and2.5 is considered. The final time is specified as10 h and the vector process anddecision initial conditions as x(0) = [1;0.1;40;0;0;1;0;0] and u 1,2 (0) = [0.5;0.5], respectively. The additionalconstrains on the decision variables are the following: u 1,2 ∈ [0;1]