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

DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biologysubject to( )x1ẋ 1 = g 1 x 1 −ux 4( ) 150−x2ẋ 2 = −10g 1 x 1 +ux 4( )x3ẋ 3 = g 2 x 1 −ux 4(9.32)(9.33)(9.34)ẋ 4 = u (9.35)where( )( )0.408 x 2g 1 =1+x 3 /16 0.22+x 2( )( )1 x 2g 2 =1+x 3 /71.5 0.44+x 2(9.36)(9.37)The state values x 1 ,x 2 ,x 3 , andx 4 are the cell mass, the substrate, the ethanol concentration (g/L), and thevolume (L), respectively. The initial conditions for the process and decision variables were initially set at the valueof: x(0) = [1;150;0;10] and u(0) = [6]. The lower and upper bound of the decision variables (feed rate) wasdefined as follows: u ∈ [0;12]. The volume of the container (x 4 ) is bound with the equation as follows0 ≤ x 4 (t) ≤ 200 (9.38)1 ______________2 Final results [single-optimization]:3 ............. Problem name: FedBatchReactorForEthanolProduction4 ...... 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: 1.504808e+001 1.916873e-002 1.021811e+002 2.000084e+00211 ...... 1th optimal control: ...12 ______________13 ............ Final CPUtime: ... seconds14 . Cost function [max(J_0)]: 20437.07128407220200180x 1x 2x 312101601408State variables120100x 40 5 10 15 20 25 30 35 40 45 50Control variable6804604022000 5 10 15 20 25 30 35 40 45 50Time0TimeFigure 9.4 – Optimal state trajectories (left) and the control profile (right) for the fed-batch reactor for ethanolproduction.Page – 45