Assessment and Future Directions of Nonlinear Model Predictive ...

Robust NMPC for a Benchmark Fed-Batch Reactor 4614 A Min-Max NMPC FormulationThis paper has presented an open-loop optimization problem for NMPC withsafety constraints for a given process model with exact parameters. In reality,at least some of the parameters will not be known exactly and methods foroptimization under uncertainty become important (see [16] for an overview).It is clear that a changed parameter can deteriorate the performance of asupposedly optimal (open-loop) solution. It seems natural to see parameters asadverse players that try to disturb any control efforts as strongly as possible.One remedy is to minimize the success of parameters to maximally worse theoptimization results. In the context of robust batch optimization, such a minmaxformulation has been used by Nagy and Braatz [15] who also point outthe possibility to extend these ideas to NMPC. Such min-max formulations leadto a semi-infinite programming problem, which is numerically intractable in thereal-time context. To overcome this obstacle, Körkel et al. [10] propose an approximatedmin-max formulation which is also applied in this paper.The min-max formulation considered here reads as the semi-infinite programmingproblemminu∈Us.t.max‖p−¯p‖ 2,Σ −1 ≤γmax‖p−¯p‖ 2,Σ −1 ≤γJ(x(u, p)) (4)r i (x(u, p)) ≤ 0, i =1,...,n r .The parameters p are assumed to lie within a given confidence region withthe symmetric covariance matrix Σ and an arbitrary confidence level γ. Thestate x implicitly depends on the discretized control u and parameters p as asolution to the system equations (1,2). The cost function J is the same as in thenominal problem (3), while r i (x(u, p)) summarizes those constraints in (3) thatare considered critical with respect to uncertainty and shall be robustified. Allother constraints are treated as in the nominal problem.First order Taylor expansion of the inner maximization part yields a convexoptimization problem that has an analytical solution (cf. [10] for this problemand [8] for a more general problem class). Using this closed form, we finallyobtain a minimization problem that can efficiently be solved numerically:minu∈Us.t.∥ ∥ ∥∥∥ J(x(u, ¯p)) + γ d ∥∥∥2,Σdp J(x(u, ¯p)) (5)r i (x(u, ¯p)) + γd∥dp r i(x(u, ¯p))∥ ≤ 0, i =1,...,n r ,2,Σwhere the bar denotes the nominal value of the parameters as assumed for thenominal optimization problem.In an NMPC scheme, the robust version (5) can replace the nominal controlproblem (3). In the next section, numerical results for such a robust NMPC arepresented and compared to the nominal NMPC.