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

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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]
DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biology1 ______________2 Final results [single-optimization; Q=0.0]:3 ............. Problem name: Lee-RamirezBioreactor4 ...... NLP or MINLP solver: FMINCON5 . Number of time intervals: 256 ... IVP relative tolerance: 1.000000e-0077 ... IVP absolute tolerance: 1.000000e-0078 . Sens. absolute tolerance: 1.000000e-0079 ............ NLP tolerance: 1.000000e-00510 ....... Final state values: 4.182486e+000 6.413670e+000 3.893872e+001 1.470736e+000 1.367472e+000 7.134128e-001 2.865872e-001 1.429858e+00011 ...... 1th optimal control: ...12 ...... 2th optimal control: ...13 ______________14 ............ Final CPUtime: ... seconds15 . Cost function [max(J_0)]: 6.151233551617 ______________18 Final results [single-optimization; Q=2.5]:19 ............. Problem name: Lee-RamirezBioreactor20 ...... NLP or MINLP solver: FMINCON21 . Number of time intervals: 2522 ... IVP relative tolerance: 1.000000e-00723 ... IVP absolute tolerance: 1.000000e-00724 . Sens. absolute tolerance: 1.000000e-00725 ............ NLP tolerance: 1.000000e-00526 ....... Final state values: 1.910086e+000 1.479996e+001 3.497523e+001 3.131412e+000 1.878273e-001 7.134005e-001 2.865995e-001 8.969161e-00227 ...... 1th optimal control: ...28 ...... 2th optimal control: ...29 ______________30 ............ Final CPUtime: ... seconds31 . Cost function [max(J_0)]: 5.756940211u 11u 10.90.90.80.80.70.7Control variables0.60.50.4u 20u 21 2 3 4 5 6 7 8 9 10Control variables0.60.50.40.30.30.20.20.10.1000 1 2 3 4 5 6 7 8 9 10TimeTimeFigure 9.1 – Optimal control trajectories for the Lee-Ramirez bioreactor, left for Q = 0 and right for Q = 2.5case.9.2 OPTIMAL PRODUCTION OF PROTEIN IN THE FED-BATCH REACTORDOTcvp: cdop_OptimalProductionOfSecretedProtein.mConsider a fed-batch reactor where the goal of the optimization is to achieve the maximum amount of thesecreted protein at the end of the batch time. This optimal control problem has been studied by many authors [1;23; 6]. The cost function is defined as followsmaxu iJ 0 = x 1 (t F )x 5 (t F ) (9.12)where x 1 is the concentration of the protein (L −1 ) and x 5 is the culture volume (L) at the final time t F = 15 h.The optimal control problem is solved subject tox˙1 = g 1 (x 2 −x 1 )− u x 1 (9.13)x 5x˙2 = g 2 x 3 − u x 2 (9.14)x 5x˙3 = g 3 x 3 − u x 3 (9.15)x 5x˙4 = −7.3g 3 x 3 + u (20−x 4 ) (9.16)x 5x˙5 = u (9.17)Page – 42
Page 1 and 2: TECHNICAL REPORTDOTcvpSB: a Matlab
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