Assessment and Future Directions of Nonlinear Model Predictive ...

322 D. Limon et al.problem may be lost. In this case, admissibility can be ensured applyingthe tail of the last computed optimal sequence. [5].Convergence: this is achieved by means of a dual mode strategy ensuring thatthe MPC steers the system to the terminal set Ω and then switching to thelocal robust control law u k = K·x k . The convergence to the terminal set canbe guaranteed by using two different techniques: the first one, used in [6],consists in reducing the prediction horizon at each sampling time. Hence,the system reaches Ω in N steps or less.A drawback of this approach is that the reduction of the prediction horizonmay provide a worse closed loop behavior than when the prediction horizon isconstant. In order to mitigate this and to provide convergence for a constantprediction horizon, an stabilizing constraint can be added to the problem.This stabilizing constraint is based on functionJ E (k) =N−1∑i=0‖ˆX(k + i|k)‖ βΩwhere β is a parameter contained in (0, 1) and ‖A‖ B denotes a measure ofthe maximum distance between sets A and B. Clearly, if A ⊆ B, then thismeasure is zero. The proposed stabilizing constraint is [5]N−1∑i=0‖ˆX(k + i|k)‖ Ω − J E (k − 1) < − 1 − ββ(7)which does not reduce the feasibility region and ensures that J E (k) tends tozero, or equivalently, x k tends to Ω.This controller requires that the state is fully measurable to be implemented.If this is not possible, then an output feedback approach could be used. In thefollowing section an output feedback MPC is presented.5 Robust Output Feedback MPC Based on GuaranteedEstimationIn this section we present a robust MPC controller based on the measurement ofthe outputs. This controller computes the control input considering an estimationof the set of states calculated at each sampling time by an interval arithmeticbased algorithm. This procedure is presented in what follows.5.1 Guaranteed State EstimationConsider an uncertain non-linear discrete-time system (1) such that the appliedcontrol inputs are known and only the outputs are measurable. The impossibilityto calculate accurately the states from the outputs together with the uncertainties