Assessment and Future Directions of Nonlinear Model Predictive ...

Interval Arithmetic in Robust Nonlinear MPC 321the predictions depends on the future realization of the uncertainties. This effectof the uncertainties can be considered in the controlled design by replacingthe sequence of predicted states by the sequence of reachable sets. As it wascommented before, the exact computation of this sequence is very difficult andtractable methods to compute a guaranteed estimation of this sequence can beused instead.In order to enhance the accuracy of the estimation of the reachable sets, apre-compensation of the system can be used. This consists in parametrizing theinput as u k = K·x k + v k where v k is the new (artificial) control input. By doingthis, system (1) can be rewritten as f(x, K·x + v, w) =f K (x, v, w). This is apractical way to provide some amount of feedback to the predictions as well asa technique to enhance the structure of the system to obtain better interval orzonotopic approximations.The proposed MPC is based on the solution of an optimization problem suchthat a performance cost of the predicted nominal trajectory is minimized (althoughother cost functions depending on the uncertainties might be considered,such as the cost of the worst case in the min-max framework). This cost functionfor a given sequence of N control inputs v(k) ={v(k|k), ··· ,v(k + N − 1|k)}, isgiven byJ N (x k , v) =N−1∑j=0L(ˆx(k + j|k),v(k + j|k)) + V (ˆx(k + N|k)) (6)where ˆx(k + j +1|k) =f K (ˆx(k + j|k),v(k + j|k), 0), with ˆx(k|k) =x k . The stagecost function L(·, ·) is a positive definite function of the state and input and theterminal cost function F (·) is typically chosen as a Lyapunov function of systemx + = f K (x, 0, 0). Thus, the optimization problem P N (x k )tosolveisminvJ N (x k , v)s.t. ˆX(k + j|k) =Ψ K (j; x k , v,W) j =1, ··· ,Nv(k + j|k) ⊕ K ˆX(k + j|k) ⊆ U, j =0, ··· ,N − 1ˆX(k + j|k) ⊆ X j =0, ··· ,NˆX(k + N|k) ⊆ Ωwhere Ω is an admissible robust invariant set for the system x + = f K (x, 0,w),and ψ K (·, ·, ·) denotes a guaranteed estimator of f K (·, ·, ·).The stabilizing design of the controller arises two questions: the admissibilityof the obtained trajectories and the convergence to a neighborhood of the origin.Admissibility: this is ensured if the initial state is feasible. In effect, if the estimationoperator ψ K (·, ·, ·) is monotonic (as for instance, the one based oninterval extension of the model), then the optimization problem is feasibleall the time [6] and hence the closed loop system is admissible.If the monotonicity is not ensured (as may occur, for instance, whenKühn’s method approach is used), then feasibility of the optimization