Assessment and Future Directions of Nonlinear Model Predictive ...

72 J.A. Rossiter, B. Pluymers, and B. De Moor4.4 SummaryWe summarize the changes required to extend nominal interpolation algorithmsto the LPV case.1. The simplest ONEDOF interpolations can make use of a robust MAS, inminimal form, and apart from this no changes from the nominal algorithmare needed. The simplest GIMPC algorithm is similar except that the costneeds to be represented as a minimum upper bound.2. More involved ONEDOF interpolations require non-minimal representationsof the robust MAS to ensure consistency between respective S i , and hencerequire many more inequalities. The need to compute these simultaneouslyalso adds significantly to the offline computational load.3. The GIMPC2 algorithm requires both mutual consistency of the MAS andthe cost to be replaced by a minimum upper bound.4. Interpolation MPQP requires the robust MCAS which can be determinedusing an autonomous model representation, although this gives a large increasein the dimension of the invariant set algorithm. It also needs an upperbound on the predicted cost.It should be noted that recent results [14] indicate that in the LPV casethe number of additional constraints can often be reduced significantly with amodest decrease in feasibility.5 Numerical ExampleThis section uses a double integrator example with non-linear dynamics, todemonstrate the various interpolation algorithms, for the LPV case only. Thealgorithm of [19] (denoted OMPC) but modified to make use of robust MCAS[13] is used as a benchmark.5.1 Model and ConstraintsWe consider the nonlinear model and constraints:x 1,k+1 = x 1,k +0.1(1 + (0.1x 2,k ) 2 )x 2,k ,x 2,k+1 = x 2,k +(1+0.005x 2 2,k )u k,(28a)−0.5 ≤ u k ≤ 1, [−10 − 10] T ≤ x k ≤ [8 8] T , ∀k. (28b)An LPV system bounding the non-linear behaviour is given as:[ ] [ ][ ] [ ]10.1 0 10.20A 1 = ,B 1 = , A 2 = ,B 2 = . (29)0 11 0 11.5