Assessment and Future Directions of Nonlinear Model Predictive ...

Interval Arithmetic in Robust Nonlinear MPC 323and noises present in the system makes that only a region where the actual stateis confined is estimated. Thus, based on the measured outputs, an estimation ofthe set of possible states at each sample time is obtained assuming that the initialstate is confined in a known compact sets x o ∈ X 0 .For a measured output y k and input u k , the consistent state set at time k isdefined as X yk = { x ∈ R n : y k ∈ g(x, u k ,V) }. Thus, the set of possible statesat sample time k, X k , is given by the recursion X k = f(X k−1 ,u k−1 ,W) ⋂ X ykfor a given bounding set of initial states X 0 and for a given sequence of inputs uand outputs y. It is clear that the exact computation of these sets is a difficulttask for a general nonlinear system. Fortunately, these sets can be estimated byusing a guaranteed estimator ψ(·, ·, ·) of the model f(·, ·, ·) [3]. The estimationis not obtained by merely using ψ(·, ·, ·) instead of f(·, ·, ·) in the recursion, andsome problems must be solved.The first issue stems from the fact that consistent state set X yk is typicallydifficult to compute. Therefore, this can be replaced by a tractable outer approximation¯X yk . In [3] an ad-hoc procedure is proposed to obtain an outerapproximation set ¯X yk as the intersection of the p strip-type sets. This procedurerequires the measured output y k and a region ¯X k where X yk is contained;thus this will be denoted as ¯X yk = Υ (y k , ¯X k ).The second problem to solve is derived from the fact that the procedureψ(A, u, W ) requires that A has an appropriate shape. For instance, if the naturalinterval extension of f(·, ·, ·) is being used, then A must be a box, while if it isbasedonKühn’s method, then A must be a zonotope. Assume that X k has theappropriate shape, a zonotope for instance, then the set ψ(X k ,u k ,W)isalsoa zonotope, but the intersection ψ(X k ,u k ,W) ∩ ¯X yk+1 may be not a zonotope.Then, it is compulsory to obtain a procedure to calculate a zonotope whichcontains this intersection. This procedure is such that for a given zonotope (box)A, andagivensetB, defined as the intersection of strips, calculates a zonotope(box) C = Θ(A, B) such that C ⊇ A ∩ B. An optimized procedure for zonotopesis proposed in [3]. In case of boxes, a simple algorithm can also be obtainedbasedonthis.Considering these points, the following algorithm to obtain a sequence ofguaranteed state estimation sets X k is proposed:1. For k = 0 and for a given zonotope X 0 ,takeˆX 0 = X 0 .2. For k ≥ 1, makea) ¯X k = ψ(ˆX k−1 ,u k−1 ,W).b) ¯X yk = Υ (y k , ¯X k ).c) X k = ¯X k ∩ ¯X ykd) ˆX k = Θ(¯X k , ¯X yk ).From this algorithm it is clear that both X k and ˆX k are guaranteed stateestimation sets. Based on this algorithm, a robust output feedback MPC isproposed in the following section.