Assessment and Future Directions of Nonlinear Model Predictive ...

Hard Constraints for Prioritized Objective Nonlinear MPC 225Table 1. Model Parameters for the Simulated Network of Pressure TanksV 1 =8V 3 =8c CV 1 =0.25 c 12 =0.02 c 22 =0.06 c 44 =0.06 γ 1 =0.5V 2 =5V 4 =5c CV 2 =0.25 c 14 =0.05 c 34 =0.02 c 32 =0.05 γ 2 =0.36 Closed-Loop ResultsThe performance of the proposed control algorithm is tested on the simulatedpressure tank network. Its ability to appropriately handle control objective prioritizationthrough a number of reference transitions and in the presence ofdisturbance loads is demonstrated. A number of control objectives are definedand assigned a relative priority. These are summarized in Table 2. Assume thatfor safety concerns, it is important that the pressure in the upstream tanks arekept below a pressure of 60 psig. These constraints are given highest priority.A secondary goal is the regulation of the pressure in second tank (P 2 ). It isdesirable for this tank pressure to closely track setpoint, and thus a setpointconstraint as well as tight upper and lower bounds (±2 psigfromsetpoint)areimposed. Note that the lower and upper bounds are assigned to be priority 3and 4 respectively, while the setpoint constraint is not considered for objectiveprioritization and use of hard constraints. Subsequent control objectives includea lower bound on the pressure in tank 1 and bounds on the pressure in tank 4.The pressure in tank 3 is left unconstrained. All constraints include a 15 minutedelay for enforcement.Table 2. Summary of Prioritized Control Objectives (* Note that a hard constraintcorresponding to the setpoint control objective is not used.)Relative Variable Constraint Constraint Relative Variable Constraint ConstraintPriority Constrained Type Value Priority Constrained Type Value1 P 1 UB 60 6 P 4 UB 302 P 3 UB 60 7 P 4 LB 203 P 2 LB 25/20/25 8 P 4 LB 234 P 2 UB 29/24/29 9 ∗ P 2 SP 27/22/275 P 1 LB 55The controller is tuned to with m =2andp =20. Each control objective is assigneda weight of Γ e = 100 and the input movements are not penalized, Γ u =0.These weights are used to determine the tradeoffs between soft constraint violationsof the various control objectives for which the hard constrained problemcannot be solved. For this example, at each time step, the appropriate NLP issolved using a stochastic approach followed by a local gradient based search.Specifically, 1000 points in the solution space are considered, the best of whichis used as the starting point for the gradient-based solution. This stochastic andgradient-based solution process is repeated 3 times and the best solution is takenas the optimal control sequence to be implemented.At t = 150 minutes, a setpoint change for the pressure in tank 2 steps fromits initial value of 27 psig to 22 psig. The controller recognizes the change and