Assessment and Future Directions of Nonlinear Model Predictive ...

224 C.E. Long and E.P. GatzkeVd3CV2V34V44TANK 3TANK 4High PressureAir FeedP3V32P4P1V14P2V22TANK 1TANK 2CV1V12Vd1Fig. 1. Schematic of the Network of Pressure TanksSupply air is fed to the system at 60 psig through two control valves whichact as the manipulated variables (u 1 and u 2 ) in the control problem. Pressuregradients drive the flow of air through the system. The air flows through thefour tanks that are interconnected, while the numerous valves throughout thesystem dictate the direction of the flow. After traveling through the system,the air ultimately exits the downstream tanks to the atmosphere. It is assumedthat the pressure in each of the four tanks can be measured. These will act asthe process outputs to be controlled. The non-square nature of this configuration(2×4) lends itself well to demonstrating the ability of this specific controller as allfour measurements cannot be maintained at setpoint using only two manipulatedvariables, thus forcing the controller to decide the appropriate trade-off basedon the prioritized objectives.For this work, it is assumed that the flow of air across a given valve (v i )canbe defined as:√f i (k) =c i ∆Pi (k) (4)where f i is a molar flow, k is the sample time, c i is a proportionality constantrelated to the valve coefficient, and ∆P i is the pressure drop across the valve.For this study, it is assumed that there is no reverse flow across a valve, forcing∆P i non-negative. Under ideal conditions, the discrete time governing equationsdefining the pressures in each tank are taken as:P 1 (k +1)= hP1(k)V 1(u 1 f CV 1 − γ 1 f 12 − (1 − γ 1 )f 14 )+P 1 (k)P 2 (k +1)=hP2(k)V 2(γ 1 f 12 +(1− γ 2 )f 32 − f 22 )+P 2 (k)P 3 (k +1)= hP3(k)V 3(u 2 f CV 2 − γ 2 f 34 − (1 − γ 2 )f 32 )+P 3 (k)P 4 (k +1)= (γ 2 f 34 +(1− γ 1 )f 14 − f 44 )+P 4 (k)hP4(k)V 4where γ 1 and γ 2 define the fractional split of air leaving the upstream tanks.Here, this set of equations represents both the nonlinear process and the modelused for control purposes. The sampling period (h) used was 3 minutes. This isimportant as it defines the time limit in which each optimization problem mustbe solved for real-time operation. The parameter values used in this study aresummarized in Table 1.(5)