Assessment and Future Directions of Nonlinear Model Predictive ...

324 D. Limon et al.5.2 Robust Output Feedback MPCIn the previous section, some robust MPC controllers based on a guaranteed estimationof the reachable sets were presented. This estimation is achieved by usinga guaranteed estimator ψ(·, ·, ·) of the model function f(·, ·, ·). These controllersare able to robustly steer the system to a target set under assumption that thefull state variables are measurable. In the case that only the output signals aremeasurable, then a set of estimated states can be calculated by means of theproposed guaranteed state estimator estimator. Then the state feedback MPCcontroller can be modified to deal with the output feedback case by consideringthat the predicted sequence of reachable sets starts from a given set instead ofa single state initial state.Consider that the control law u = h(x) is an admissible robustly stabilizingcontrol law for system (1) in a neighborhood of the origin. Assume that system(1) is locally detectable for the corresponding dynamic output feedback controllerˆx k+1 = κ(ˆx k ,u k ,y k ) (8)u k = h(ˆx k )in such a way that the closed loop systemx k+1 = f(x k ,h(ˆx k ),w k ) (9)ˆx k+1 = κ(ˆx k ,h(ˆx k ),g(x k ,h(ˆx k ),v k )) (10)is robustly stable in (x, ˆx) ∈ Γ ,whereΓ is a polyhedral robust invariant setfor system (9), i.e. ∀(x k , ˆx k ) ∈ Γ , x k ∈ X, h(ˆx k ) ∈ U, and(x k+1 , ˆx k+1 ) ∈ Γ ,∀w ∈ W, v ∈ V . It is clear that Γ contains the origin in its interior.Assume that there is available a procedure to implement the proposed estimationalgorithm that provides ˆX k at each sampling time from the measurementy k . Then the robust output feedback MPC controller first estimates the set ofstates ˆX k and then calculates the control input based on this set by minimizingan optimization problem PN o (ˆX k ).The cost to minimize J N (ˆX k , u) must be calculated from the set of initial statesˆX k . This can be done for instance based on the nominal prediction consideringas the initial state the center of the zonotope ˆX k or calculating the sequenceof reachable sets and computing the worst case cost, in a similar way to themin-max paradigm. Thus, the optimization problem PN o (ˆX k )isgivenbyminu,ˆxJ N (ˆX k , u)s.t. ˆX(k + j|k) =Ψ(j; ˆX k , u,W) j =1, ··· ,Nu(k + j|k) ∈ U, j =0, ··· ,N − 1ˆX(k + j|k) ⊆ X j =0, ··· ,N(x, ˆx) ∈ Γ,∀x ∈ ˆX(k + N|k)The extra decision variable ˆx has been added to force that for any statecontained in ˆX(k + N|k), the dynamic controller (8) stabilizes the system