Assessment and Future Directions of Nonlinear Model Predictive ...

Robustness and Robust Design of MPC 249One way to avoid this impasse is given in [3] where the MPC approach isapplied to an already robust stable system, so that Assumption 7.2 is satisfiedwith κ f (·) ≡ 0. In this case a feasible control sequence isũ t+1,t+N := [ u o t+1,t+N−1, 0 ]In order to obtain a system with a-priori robustness properties with respectto the considered class of disturbances, in [3] it has been suggested to precompensatethe system under control by means of an inner feedback loop designedfor example with the H ∞ approach.7.2 Closed-Loop Min-Max MPCThe limitations of the open-loop min-max approach can be overcome by explicitlyaccounting for the intrinsic feedback nature of any RH implementation ofMPC, see e.g. [41] for the linear case and [29] for nonlinear systems. In thisapproach, at any time instant the controller chooses the input u as a functionof the current state x, so as to guarantee that the effect of the disturbance w iscompensated for any choice made by the “nature”. Hence, instead of optimizingwith respect to a control sequence, at any time t the controller has to choose asequence of control laws κ t,t+N−1 =[κ 0 (x(t)) κ 1 (x(t+1) ...κ N−1 (x(t+N − 1)].Then, the following optimal min-max problem can be stated.Definition 10 (FHCDG). Consider a stabilizing auxiliary control law κ f (·)and an associated output admissible set X f . Then, given the positive integer N,the stage cost l(·, ·) − l w (·) and the terminal penalty V f (·), the Finite HorizonClosed-loop Differential Game (FHCDG) problem consists in minimizing, withrespect to κ t,t+N−1 and maximizing with respect to w t,t+N−1 the cost functionJ(¯x, κ t,t+N−1 ,w t,t+N−1 ,N)=subject to:t+N−1∑k=t(i) the state dynamics (19) with x(t) =¯x;(ii) the constraints (4), k ∈ [t, t + N − 1];(iii) the terminal state constraint x(t + N) ∈ X f .{l(x(k),u(k)) − l w (w(k))} + V f (x(t + N))Finally, letting κ o t,t+N−1 ,wo t,t+N−1 the solution of the FHCDG the feedbackcontrol law u = κ MPC (x) is obtained by settingκ MPC (x) =κ o 0 (x) (20)where κ o 0(x) is the first element of κ o t,t+N−1 .In order to derive the main stability and performance properties associatedto the solution of FHCDG, the following assumption is introduced.Assumption 7.3. l w (·) is such that α w (|w|) ≤ l w (w) ≤ β w (|w|) where α w andβ w are K functions.