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

CHAPTER 5Sucessive Re-optimizationFor reaching a high level of the control discretization in a very effective way, sucessive re-optimization module wasimplemented into the toolbox. The basis of this module is secured by a modified mesh refinement algorithm, whichwas first proposed and described in detail in [1]. This algorithm allows to use a high level of control discretizationwith relative low computational cost.5.1 APPLICATION OF THE MESH REFINEMENT ALGORITHMDOTcvp: cdop_VanDerPolOscillator_reoptimization.mThe problem of the van der Pol oscillator, investigated in the previous section 4.1, is in this case solvedwithout constraints. The re-optimization was run with the default settings presented in the section 11.2.The final control trajectories for 10, 30, 90, and 270 piecewise time constants are shown in Figure 5.1. Itshould be noted that the increased number of time intervals smooths the control profile and this has an influence onthe cost function value. From the above mentioned default file all options are taken, where ’data.option.Mesh_NRO’,’data.option.Mesh_Increasing’ represent the number of mesh refinings and how quickly will the number of timeintervals be increased – the actual number of time intervals is multiplied by this number. The default file alsocontains all tolerances needed for the re-optimization. At the beginning of the re-optimization these tolerances aretaken into account and at the end, when the last re-optimization is running, the tolerances are taken from the userinput file. Between the first and the last re-optimization, the tolerances are interpolated as it is possible to see inthe box below.1 ______________2 Final results [1/4 re-optimization]:3 ............. Problem name: VanDerPolOscillator4 ...... NLP or MINLP solver: IPOPT5 . Number of time intervals: 106 ... IVP relative tolerance: 1.000000e-0057 ... IVP absolute tolerance: 1.000000e-0058 . Sens. absolute tolerance: 1.000000e-0059 ............ NLP tolerance: 1.000000e-00310 ....... Final state values: 7.120680e-002 -5.283990e-002 2.926158e+00011 ...... 1th optimal control: ...12 ______________13 ............ Final CPUtime: ... seconds14 . Cost function [min(J_0)]: 2.926232921516 ______________17 Final results [2/4 re-optimization]:18 ............. Problem name: VanDerPolOscillator19 ...... NLP or MINLP solver: IPOPT20 . Number of time intervals: 3021 ... IVP relative tolerance: 2.154435e-00622 ... IVP absolute tolerance: 2.154435e-00623 . Sens. absolute tolerance: 2.154435e-00624 ............ NLP tolerance: 2.154435e-00425 ....... Final state values: 6.712704e-002 -5.209653e-002 2.873413e+00026 ...... 1th optimal control: ...27 ______________28 ............ Final CPUtime: ... seconds29 . Cost function [min(J_0)]: 2.8735082430