Assessment and Future Directions of Nonlinear Model Predictive ...

320 D. Limon et al.Definition 1 (Sequence of reachable sets). Consider a system given by (1),consider also that the set of states at sample time k is X k and that a sequenceof control inputs {u(k + i|k)} is given, then the reachable sets {X(k|k), X(k +1|k),...,X(k + N|k)} are obtained from the recursion: X(k + j|k) =f(X(k + j −1|k),u(k + j − 1|k),W) where X(k|k) =X k .The set of states X k might be either a singleton x k , in the case that the state iscertainly known, or a set, in the case that this is unknown but bounded (this iswhat happen when the state is not fully measurable).It is clear that the computation of this sequence of sets is not possiblein general; fortunately, a guaranteed estimation of this sequence can be obtainedif for a given control input u, a guaranteed estimator of f(X, u, W),ψ(X, u, W) is used for the computation. The guaranteed estimation must satisfythat ψ(X, u, W) ⊇ f(X, u, W), for all X, u, W . Thus the sequence of guaranteedestimation of the reachable set is given by the following recursion:ˆX(k + j|k) =ψ(ˆX(k + j − 1|k),u(k + j − 1|k),W)with ˆX(k|k) =X k . In order to emphasize the parameters of ˆX(k + j|k), this willbe denoted as X(k + j|k) =Ψ(j; X k , u,W).In the previous section, two methods to obtain a guaranteed estimator of afunction were introduced: the interval extension of the function f(X, u, W) andthe one based on Kühn’s method. Both procedures can be used to approximatethe sequence of reachable sets.The interval extension method provides a guaranteed box considering thatX and W are boxes. Its main advantages are that this procedure is easy toimplement and its computational burden is similar to the one corresponding toevaluate the model. Moreover, the bounding operator based on interval extensionis monotonic, that is, if A ⊆ B, thenψ(A, u, W ) ⊆ ψ(B,u,W). Its maindrawback is that it may be very conservative.The procedure based on Kühn’s method calculates zonotopic approximations,assuming that X and W are zonotopes. This method provides more accurateapproximation than the one obtained by the interval extension at expense ofa more involved procedure with a bigger computational burden. Moreover thisprocedure may not be a monotonic procedure.The monotonicity property of the estimation procedure ψ(·, ·, ·) providesaninteresting property to the estimated reachable set: consider the sequence ofestimated reachable sets ˆX(k + j|k) =Ψ(j; X k , u,W). Consider a set X k+1 ⊆ˆX(k +1|k), then for all j ≥ 1, Ψ(j − 1; X k+1 , u,W) ⊆ Ψ(j; X k , u,W). Thisproperty will be exploited in the following sections.Based on this estimation of the sequence of reachable sets, a robust MPCcontroller is presented in the following section.4 Robust MPC Based on Guaranteed Reachable SetsWhile in a nominal framework of MPC, the future trajectories are obtainedby means of the nominal model of the system, when uncertainties are present,