Assessment and Future Directions of Nonlinear Model Predictive ...

Discrete-Time Non-smooth Nonlinear MPC: Stability and Robustness 97Theorem 1. (Stability of Non-smooth Nonlinear MPC) Fix N ≥ 1 andsuppose that either Assumption 1 holds or Assumption 2 holds. Then:(i) If Problem 1 is feasible at time k ∈ Z + for state x k ∈ X, Problem 1 isfeasible at time k +1 for state x k+1 = g(x k ,u MPC (x k )). Moreover, X T ⊆ X f (N);(ii) The origin of the MPC closed-loop system (2a)-(4) is asymptotically stablein the Lyapunov sense for initial conditions in X f (N);(iii) If Assumption 1 or Assumption 2 holds with α 1 (s) as λ , α 2 (s) bs λfor some constants a, b, λ > 0, the origin of the MPC closed-loop system (2a)-(4)is exponentially stable in X f (N).The interested reader can find the proof of Theorem 1 in [20]. Next, we statesufficient conditions for ISS (in the sense of Definition 2) of discrete-time nonsmoothnonlinear MPC.Theorem 2. (ISS of Non-smooth Nonlinear MPC) Let W be a compactsubset of IR l that contains the origin and let X be a robustly positively invariant(RPI) set [20] for the MPC closed-loop system (2b)-(4) and disturbances in W,with 0 ∈ int(X). Letα 1 (s) as λ , α 2 (s) bs λ , α 3 (s) cs λ for some positiveconstants a, b, c, λ and let σ ∈K.SupposeL(x, u) ≥ α 1 (‖x‖) for all x ∈ X andall u ∈ U, V MPC (x) ≤ α 2 (‖x‖) for all x ∈ X and that:V MPC (˜g(x, u MPC (x),w)) − V MPC (x) ≤−α 3 (‖x‖)+σ(‖w‖), ∀x ∈ X, ∀w ∈ W.(6)Then, the perturbed system (2b) in closed-loop with the MPC control (4) obtainedby solving Problem 1 at each sampling-instant is ISS for initial conditions in Xand disturbance inputs in W. Moreover, the ISS property of Definition 2 holdsfor β(s, k) α −11 (2ρk α 2 (s)) and γ(s) α −11(2σ(s)1−ρ),whereρ 1 − c b∈ [0, 1).For a proof of Theorem 2 we refer the reader to [20]. Note that the hypotheses ofTheorem 1 and Theorem 2 allow g(·, ·), ˜g(·, ·, ·) andV MPC (·) to be discontinuouswhen x ≠0.Theyonly imply continuity at the point x =0,andnot necessarilyon a neighborhood of x =0.5 A Robust MPC Scheme for Discontinuous PWASystemsIn this section we consider the class of discrete-time piecewise affine systems, i.e.x k+1 = g(x k ,u k ) A j x k + B j u k + f j if x k ∈ Ω j , (7a)˜x k+1 =˜g(˜x k ,u k ,w k ) A j ˜x k + B j u k + f j + w k if ˜x k ∈ Ω j , (7b)where w k ∈ W ⊂ IR n , k ∈ Z + , A j ∈ IR n×n , B j ∈ IR n×m , f j ∈ IR n , j ∈Swith S {1, 2,...,s} a finite set of indices. The collection {Ω j | j ∈S}definesa partition of X, meaning that ∪ j∈S Ω j = X and int(Ω i ) ∩ int(Ω j ) = ∅ fori ≠ j. EachΩ j is assumed to be a polyhedron (not necessarily closed). Let