Assessment and Future Directions of Nonlinear Model Predictive ...

Interval Arithmetic in Robust Nonlinear MPC 325considering ˆx as initial dynamic state. This constraint can be easily implementedthanks to the polyhedral nature of Γ . Assume that ˆX(k + N|k) =p ⊕ HB Nand Γ = {(x, ˆx) :T 1 x + T 2ˆx ≤ t} then the terminal constraint can be posed asT 2ˆx ≤ t−T 1 p−‖T 1 H‖ 1 ,where‖T 1 H‖ 1 denotes the vector which i−th componentis the 1-norm of the i−th row of the matrix T 1 H.The admissibility of the controller can be ensured by means of the methodsused for the case of full-state measurement presented in the previous section.However, the addition of the state estimator may introduce feasibility loss ofthe optimization problem due to the fact that the set ˆX k+1 may be not containedin the set ψ( ˆX k ,u k ,W) because of the outer approximation used in thecomputation of ˆX k+1 . The probability that this happens is low but if so, the admissibilityof the problem can be ensured by solving the problem P N (ˆX(k +1|k))instead or by merely applying u ∗ (k +1|k). Convergence of the real state to theset Ω = Proj x (Γ ) can be ensured by the previously proposed method: shrinkingthe prediction horizon or considering an stabilizing constraint. Once thatˆX k ⊂ Ω, then the controller switches to the local dynamic controller consideringas the initial dynamic state ˆx ∗ .6 ConclusionsThis paper summarizes some results on interval arithmetic applied to the designof robust MPC controllers. First it is shown how the keystone is the procedureto bound the range of a function and how this can be used to approximate thesequence of reachable sets. This sequence can be used to design the robust MPCcontroller in a natural way replacing the predicted trajectory by the sequence ofreachable sets. Admissibility and convergence of this controller can be guaranteedby several methods.The bounding procedure allows us to present an algorithm to estimate theset of states based on the measure of the outputs. Based on this, a robust MPCcontroller based on output feedback is presented. This controller ensures theadmissible evolution of the system to a neighborhood of the origin under theexistence of a local detector.References[1] Moore, R.E. “Interval Analysis”, Prentice-Hall, Englewood Cliffs, NJ., (1966).[2] Kühn, W. “Rigorous computed orbits of dynamical systems without the wrappingeffect”, Computing, 61, 47-67, (1998).[3] Alamo, T. and Bravo, J.M. and Camacho, E.F. , “Guaranteed state estimation byzonotopes”, Automatica, 41, 1035-1043, (2005).[4] Bravo, J.M. and Limon, D. and Alamo, T. and Camacho, E.F. , “Robust MPC ofconstrained descrete-time nonlinear systems based on zonotopes”, European ControlConference, (2003).