Assessment and Future Directions of Nonlinear Model Predictive ...

394 K. Naidoo et al.1. Maintain process safety ...2. ... while satisfying the product quality objectives ...3. ... at the required production rate ...4. ... and minimizing energy consumption.The controller uses two different schemes for this multi-objective optimization:lexicographic ordering of constraints, and weighting of constraints. Lexicographicordering implies that one constraint can be made infinitely more important thana lower ”ranked” constraint. Several constraints can also be assigned to the samerank, and weighting is then used for constraints at the same priority (rank).Lexicographic ordering is achieved by solving a series of nonlinear optimization(or feasibility) problems. This can be written in general form as:min rk T Wr ks.t. g(x) − r k ≤ dh(x) ≤ ewhere W is a weighting matrix, g(x) are the constraints at rank k (soft constraints)k is the current rank, and h(x) are the constraints at ranks higher thank (hard constraints). Let r k (opt) be the solution to rank k. Then the next rank(k+1) can be written as:min rk+1 T Wr k+1s.t. g k (x) ≤ d + r k (opt)g k+1 (x) − r k+1 ≤ d kh(x) ≤ eNote that the soft constraints from rank k have been promoted to hard constraints.This resulting scheme has been used for a number of years in the industrialcommunity and has recently been attracting academic interest [1].A full optimization problem must be solved at each rank. A sequentialquadratic programming approach is used. Trust regions are placed on the inputs.A primal-dual interior point algorithm is used to solve the QP sub-problems [8].Artificial variables are added to the QP sub-problems to ensure feasibility.Usually 3 to 5 QP sub-problem iterations are sufficient. Although there is noproof of convergence to a local optimum, the cyclical nature of the controllerensures that a local optimum will eventually be reached. Options are availablewithin the controller to activate other algorithms with formal convergence proofs,but these algorithms do not appear to be as consistent.The limits, rankings, and economic values are the main tuning values for thesteady state optimization.6.3 Ranking for High Pressure Tube exampleIn the high-pressure tube example the highest priority objective is to maintainthe melt index and gloss within their limits. However a secondary objective is