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

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DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biology63 data -> data.nlp -> data.nlp.ineq.status64 data -> data.nlp -> data.nlp.ineq.NEC65 data -> data.nlp -> data.nlp.ineq.InNUM66 data -> data.nlp -> data.nlp.ineq.eq67 data -> data.nlp -> data.nlp.ineq.Tol68 data -> data.nlp -> data.nlp.ineq.PenaltyFun69 data -> data.nlp -> data.nlp.ineq.PenaltyCoe70 data -> data.nlp -> data.nlp.yPointConstraint71 data -> data.nlp -> data.nlp.uPointConstraint7273 data -> data.options74 data -> data.options -> data.options.intermediate75 data -> data.options -> data.options.display76 data -> data.options -> data.options.title77 data -> data.options -> data.options.state78 data -> data.options -> data.options.control79 data -> data.options -> data.options.ConvergCurve80 data -> data.options -> data.options.Pict_Format81 data -> data.options -> data.options.report82 data -> data.options -> data.options.commands83 data -> data.options -> data.options.trajectories84 data -> data.options -> data.options.profiler85 data -> data.options -> data.options.multistart86 data -> data.options -> data.options.action87 data -> data.options -> data.options.IVPsimulation88 data -> data.options -> data.options.cd89 data -> data.options -> data.options.path90 data -> data.options -> data.options.ic91 data -> data.options -> data.options.ic.switchdata92 data -> data.options -> data.options.ic.rehash93 data -> data.options -> data.options.ic.ic94 data -> data.options -> data.options.set95 data -> data.options -> data.options.set.Constraint96 data -> data.options -> data.options.fmincon97 data -> data.options -> data.options.fmincon.MaxTime98 data -> data.options -> data.options.fmincon.GradObj99 data -> data.options -> data.options.fmincon.GradConstr100 data -> data.options -> data.options.fmincon.TolFun101 data -> data.options -> data.options.fmincon.TolCon102 data -> data.options -> data.options.fmincon.TolX103 data -> data.options -> data.options.fmincon.Hessian104 data -> data.options -> data.options.fmincon.DerivativeCheck105 data -> data.options -> data.options.fmincon.MaxFunEvals106107 data -> data.output108 data -> data.output -> data.output.iter109 data -> data.output -> data.output.J0110 data -> data.output -> data.output.time111 data -> data.output -> data.output.state112 data -> data.output -> data.output.control113 data -> data.output -> data.output.StateVariables114 data -> data.output -> data.output.CPUtimeStart115 data -> data.output -> data.output.CPUtime116 data -> data.output -> data.output.fmincon117 data -> data.output -> data.output.CPUtimeFINAL118 data -> data.output -> data.output.RHO119 data -> data.output -> data.output.ObjFinal120 data -> data.output -> data.output.DecisionVariables12.2 SENSITIVITIES AND GRADIENTS DERIVATIONThe simple batch reactor given in subsection 10.1 is considered. Firstly, the scenario with fixed time intervalsis solved. The problem of the optimization is to find the optimal control trajectory which minimizessubject tominu iJ 0 = −x 2 (t F ) (12.1)ẋ 1 = −u i x 1 (12.2)ẋ 2 = u i x 1 −cu α i x 2 (12.3)For simplicity and demonstrative purposes only 3 piecewise constant control variables with fixed time werechosen. The gradients with respect to the control are defined as follows∂J 0∂u i= −s 2i (t F ), i = 1,3 (12.4)To obtain the necessary gradients (12.4) it is needed to integrate sensitivity equations over the whole time(t F ). The sensitivity coefficients fori = 1,3 are defined ass 1i = ∂x 1∂u i(12.5)s 2i = ∂x 2∂u i(12.6)Page – 57
DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biologytogether with the sensitivity equations⎧ṡ 11 = ∂(−u1x1) ∂x 1∂x 1 ∂u 1+ ∂(−u1x1) ∂x 2∂x 2 ∂u 1+ ∂(−u1x1) ∂u 1∂u 1 ∂u 1= −u 1 s 11 −x 1 , t ∈ [t 0 ,t 1 ]ṡ 11 = ∂(−u2x1) ∂x 1∂x 1 ∂u 1+ ∂(−u2x1) ∂x 2∂x 2 ∂u 1+ ∂(−u2x1) ∂u 2∂u 2 ∂u 1= −u 2 s 11 , t ∈ [t 1 ,t 2 ]ṡ 11 = ∂(−u3x1) ∂x 1∂x 1 ∂u 1+ ∂(−u3x1) ∂x 2∂x 2 ∂u 1+ ∂(−u3x1) ∂u 3∂u 3 ∂u 1= −u 3 s 11 , t ∈ [t 2 ,t F ]⎪⎨ ṡ 12 = −u 1 s 12 , t ∈ [t 0 ,t 1 ]ṡ 12 = −u 2 s 12 −x 1 , t ∈ [t 1 ,t 2 ]ṡ 12 = −u 3 s 12 , t ∈ [t 2 ,t F ]ṡ 13 = −u 1 s 13 , t ∈ [t 0 ,t 1 ]ṡ ⎪⎩ 13 = −u 2 s 13 , t ∈ [t 1 ,t 2 ]ṡ 13 = −u 3 s 13 −x 1 , t ∈ [t 2 ,t F ]⎧⎪⎨⎪⎩ṡ 21 =ṡ 21 =( ) ( )∂(u1x 1−cu α 1 x2) ∂x 1 ∂(u1x∂x 1 ∂u 1− 1−cu α 1 x2)∂x 2= u 1 s 11 −cu α 1s 21 +x 1 −αcu α−1( )∂(u2x 1−cu α 2 x2)∂x 1= u 2 s 11 −cu α 2s 21∂x 1∂u 1−( 1 x 2 )∂(u2x 1−cu α 2 x2)∂x 2( )∂x 2 ∂(u1x∂u 1+ 1−cu α 1 x2) ∂u 1∂u 1 ∂u 1, t ∈ [t 0 ,t 1 ]( )∂x 2 ∂(u2x∂u 1+ 1−cu α 2 x2) ∂u 2∂u 2 ∂u 1, t ∈ [t 1 ,t 2 ]( ) ( ) ( )∂(u3xṡ 21 = 1−cu α 3 x2) ∂x 1 ∂(u3x∂x 1 ∂u 1− 1−cu α 3 x2) ∂x 2 ∂(u3x∂x 2 ∂u 1+ 1−cu α 3 x2) ∂u 3∂u 3 ∂u 1, t ∈ [t 2 ,t F ]= u 3 s 11 −cu α 3s 21ṡ 22 = u 1 s 12 −cu α 1s 22 , t ∈ [t 0 ,t 1 ]ṡ 22 = u 2 s 12 −cu α 2s 22 +x 1 −αcu α−12 x 2 , t ∈ [t 1 ,t 2 ]ṡ 22 = u 3 s 12 −cu α 3s 22 , t ∈ [t 2 ,t F ]ṡ 23 = u 1 s 13 −cu α 1s 23 , t ∈ [t 0 ,t 1 ]ṡ 23 = u 2 s 13 −cu α 2s 23 , t ∈ [t 1 ,t 2 ]ṡ 23 = u 3 s 13 −cu α 3s 23 +x 1 −αcu α−13 x 2 , t ∈ [t 2 ,t F ]or in a more compact form⎧⎨⎩⎧⎨⎩ṡ 11 = −us 11 −x 1 R 1 , t ∈ [t 0 ,t F ]ṡ 12 = −us 12 −x 1 R 2 , t ∈ [t 0 ,t F ]ṡ 13 = −us 13 −x 1 R 3 , t ∈ [t 0 ,t F ]ṡ 21 = us 11 −cu α s 21 +(x 1 −αcu α−1 x 2 )R 1 , t ∈ [t 0 ,t F ]ṡ 22 = us 12 −cu α s 22 +(x 1 −αcu α−1 x 2 )R 2 , t ∈ [t 0 ,t F ]ṡ 23 = us 13 −cu α s 23 +(x 1 −αcu α−1 x 2 )R 3 , t ∈ [t 0 ,t F ](12.7)(12.8)(12.9)(12.10)whereR i is equal to1or0depending on the time intervalt i and relevant control value. Note that not all equationsneed to be integrated in each time interval, because the optimized variable in the interval t i+1 does not affect onthe interval t i .Secondly, is the scenario with the free time intervals considered. In this case it is required to integrate newsensitivities with respect to time{ṡ14 = −us 14 , t ∈ [t 0 ,t F ](12.11)ṡ 15 = −us 15 , t ∈ [t 0 ,t F ]{ṡ24 = us 14 −cu α s 24 , t ∈ [t 0 ,t F ]ṡ 25 = us 15 −cu α (12.12)s 25 , t ∈ [t 0 ,t F ]At the end of the time interval it is needed to compute discontinuity of sensitivity equations (12.11), (12.12)following (3.15)s 14 (t + 1 ) = s 14(t − 1 )+[f 1(t,x,u 1 ,p)−f 1 (t,x,u 2 ,p)] t1(12.13)s 15 (t + 2 ) = s 15(t − 2 )+[f 1(t,x,u 2 ,p)−f 1 (t,x,u 3 ,p)] t2(12.14)s 24 (t + 1 ) = s 24(t − 1 )+[f 2(t,x,u 1 ,p)−f 2 (t,x,u 2 ,p)] t1(12.15)s 25 (t + 2 ) = s 25(t − 2 )+[f 2(t,x,u 2 ,p)−f 2 (t,x,u 3 ,p)] t2(12.16)Page – 58
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TECHNICAL REPORTDOTcvpSB: a Matlab
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Main Web PageIt is possible to get
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DOTcvpSB: a Matlab Toolbox for Dyna
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Page 13 and 14: CHAPTER 2Optimal control problem2.1
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