A Feedback Optimal Control Algorithm with Optimal Measurement Time Points
Abstract
:1. Introduction
2. On the Estimation, Control, and Design Problems
2.1. Nonlinear Dynamic Systems
2.2. State and Parameter Estimation Problems
2.3. Optimal Control Problems
2.4. Optimal Experimental Design Problems
Algorithm 1 OED 
Input: Fixed p and u, initial values $x({t}_{0}),G({t}_{0})$, possible measurement times $\{{t}_{1},\dots ,{t}_{N}\}\subset [{t}_{0},{t}_{\mathrm{f}}]$

3. A Feedback Optimal Control Algorithm With Optimal Measurement Times
Algorithm 2 FOCoed 
Input: Initial guess $\widehat{p}$, initial values $x({t}_{0}),G({t}_{0})$, possible measurement times $\{{t}_{1},\dots ,{t}_{N}\}\subset [{t}_{0},{t}_{\mathrm{f}}]$ Initialize sampling counter $i=0$, measurement grid counter $k=0$ and “current time” ${\tau}_{0}={t}_{0}$ while stopping criterion not fulfilled do

3.1. Finite Support Designs
3.2. Robustification
4. Numerical Examples
4.1. LotkaVolterra Fishing Benchmark Problem
4.2. Software and Experimental Settings
4.3. Three Versions of Algorithm FOCoed applied to the LotkaVolterra fishing problem
 with_OED. This is Algorithm 2, i.e., using measurement time points from nonrobust OED.
 without_OED. The OED problem in Step 2 of Algorithm 2 is omitted, and an equidistant time grid is used for measurements.
 with_r_OED. The OC problem in Step 1 and the OED problem in Step 2 of Algorithm 2 are replaced with their robust counterparts as described in Section 3.2.
4.4. Analyzing Finite Support Designs of Optimal Experimental Design Problems
4.5. Discussion
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
FIM  Fisher Information Matrix 
MPC  Model Predictive Control 
NLP  Nonlinear Program 
ODE  Ordinary Differential Equation 
OC  Optimal Control 
OED  Optimal Experimental Design 
SPE  State and Parameter Estimation 
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At t = 15  
OC  with_r_OED (A)  with_OED (B)  without_OED (C)  
value  value  ${\sigma}^{2}$  value  ${\sigma}^{2}$  value  ${\sigma}^{2}$  ${I}_{AC}$  ${I}_{BC}$  ${I}_{AB}$  
${p}_{1}$  1.000  1.0074  0.0003377  0.9925  0.0003300  1.0293  0.0005090  33.65  35.17  2.33 
${p}_{2}$  1.000  1.0085  0.0005540  0.9954  0.0005404  1.0267  0.0005313  4.27  1.71  2.52 
${p}_{3}$  1.000  0.9935  0.0005861  1.0073  0.0006063  0.9758  0.0006139  4.53  1.24  3.33 
${p}_{4}$  1.000  0.9959  0.0006466  1.0053  0.0006635  0.9762  0.0008780  26.36  24.43  2.55 
At t = 30  
with_r_OED (A)  with_OED (B)  without_OED (C)  
value  value  ${\sigma}^{2}$  value  ${\sigma}^{2}$  value  ${\sigma}^{2}$  ${I}_{AC}$  ${I}_{BC}$  ${I}_{AB}$  
${p}_{1}$  1.000  1.0066  0.0002414  0.9974  0.0002418  1.0082  0.0004214  42.71  42.62  0.17 
${p}_{2}$  1.000  1.0065  0.0003639  1.0004  0.0003706  1.0069  0.0004624  21.30  19.85  1.81 
${p}_{3}$  1.000  0.9936  0.0003472  1.0029  0.0003582  0.9924  0.0005068  31.49  29.32  3.07 
${p}_{4}$  1.000  0.9958  0.0003575  1.0014  0.0003764  0.9937  0.0006837  47.71  44.95  5.02 
${M}_{LV}$  0.714  0.724  0.727  0.790  9.62  7.97  0.41 
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Jost, F.; Sager, S.; Le, T.T.T. A Feedback Optimal Control Algorithm with Optimal Measurement Time Points. Processes 2017, 5, 10. https://doi.org/10.3390/pr5010010
Jost F, Sager S, Le TTT. A Feedback Optimal Control Algorithm with Optimal Measurement Time Points. Processes. 2017; 5(1):10. https://doi.org/10.3390/pr5010010
Chicago/Turabian StyleJost, Felix, Sebastian Sager, and Thuy ThiThien Le. 2017. "A Feedback Optimal Control Algorithm with Optimal Measurement Time Points" Processes 5, no. 1: 10. https://doi.org/10.3390/pr5010010