Assessment and Future Directions of Nonlinear Model Predictive ...

450 K.R. Muske, A.E. Witmer, and R.D. Weinsteinconstant at their current values until the tank is refilled. The target final molesof gas in the tank is determined at each sample period k by[]n F D(k) =min n ⋆ D,n| TD =T(k), n|Dmax TD =T(k), n|Dmin PD =P(k) (26)Dmaxwhere n F D (k) is the current target final moles of gas at sample period k, n⋆ D is thedesired final moles of gas, n| TD =Tis the current predicted moles of gas suchDmaxthat the tank temperature reaches its maximum constraint limit, n| TD =TisDminthe current predicted moles of gas such that the tank temperature reaches itsminimum constraint limit, n| PD =Pis the current predicted moles of gas suchDmaxthat the tank pressure reaches its maximum constraint limit, and the min operatorselects the most limiting model-predicted constraint. The current predictedmoles of gas required to reach a tank constraint is determined directly from thepredicted future tank state profile. The length of the prediction horizon is alwaysthe time required to obtain n ⋆ D moles of gas in the tank. If a constraint violationis not predicted within this horizon, it is not considered by the min operator inEq. 26. A first-order approximation to the control move required to achieve themost limiting constraint in minimum time is then determined from the currentpredicted tank state and target by[u(k) =min u max , nF D (k) − n ]D(k)(27)∆twhere u(k) is the current input, u max is the maximum flow rate constraint, andn D (k) is the current prediction of the moles of gas in the storage tank.This dynamic constraint prediction is computed at every sample period afterthe initial start-up phase. The start up is carried out at a minimum safe gas flowrate to ensure that the system is operating properly. The constraint predictionis updated by the incorporation of the most recent inlet pressure measurementat the current sample time. We note that on-line optimization is not required todetermine the control input because of the assumption that the optimal operationis at an active constraint (motivated by the minimum-time optimal controltrajectory). Because the DAE system and the state sensitivities, required forthe uncertainty estimate described in the next section, can be computed veryquickly, the sample period ∆t is not limited by computational issues as is oftenthe case for nonlinear predictive control implementations.4.2 Fail-Safe OperationBecause there is no direct measurement of the actual tank state, some mechanismto monitor the uncertainty in this state estimate is required for the safe implementationof the proposed controller. Linear approximations to the variance ofthe tank state can be obtained from the first-order sensitivities [8] between thetank state and the control and inlet line pressure as followsσ 2 x,P C=( ∂x∂P C) 2σ 2 P C, σ 2 x,u =( ∂x∂u) 2σu, 2 F = σ2 x,P Cλ + σx,u2 >F α (28)