Assessment and Future Directions of Nonlinear Model Predictive ...

A Minimum-Time Optimal Recharging Controller 451where x is the tank temperature or pressure, the partials are the sensitivities,σx,P 2 Cis the estimate using the pressure measurement variance σP 2 C, σx,u 2 is theestimate using the control variance σu 2, F α is the F statistic at a confidencelevel α, λ is a tuning parameter to account for measurement noise and normalvariation in the inlet line pressure, and F > F α implies σx,P 2 C>σx,u 2 [9]. Ifthe variance estimated from the inlet line pressure measurement exceeds thatestimated from the control, the constraint control is terminated to a fail-safeoperation. This operation can either shut off the gas flow completely, where thetank pressure could then be determined by the inlet line pressure sensor, or canreduce the gas flow to a predetermined minimum safe value.5 ExampleThe control strategy is illustrated using a nitrogen–helium gas mixture. Wechoose this system because the sign of the Joule-Thompson coefficient is differentfor each component; negative for helium and positive for nitrogen. Thereare also significant differences in the intermolecular potentials leading to largedeviations from ideal behavior. The coefficients in Eqs. 2–3 are affine functionsof temperature taken from [10]. We consider a 1/4 He/N 2 gas blend in a 100lit storage tank where the component source pressures are both 175 bar, theambient temperature is 300 K, and the initial tank pressure is 10 bar. Figure 3presents the predicted tank temperature profiles for a series of flow rates whichclearly demonstrate the nonideal behavior and path independence of the tankstate. The predicted pressure profiles behave in a similar manner.3635345 mol/min10 mol/min20 mol/min50 mol/minTank Temperature (C)333231302928270 10 20 30 40 50 60 70 80 90 100 110Time (min)Fig. 3. Predicted tank temperature profiles