Assessment and Future Directions of Nonlinear Model Predictive ...

244 L. Magni and R. ScattoliniDefinition 6 (ISS-Lyapunov function). AfunctionV (·) is called an ISS-Lyapunov function for system (16) if there exist a set Ξ, K functions α 1 ,α 2 ,α 3 , and σ such thatV (x) ≥ α 1 (|x|), ∀x ∈ ΞV (x) ≤ α 2 (|x|), ∀x ∈ Ξ (17)∆V (x, w) =V (f(x, w)) − V (x) < −α 3 (|x|)+σ(|w|), ∀x ∈ Ξ,∀w ∈M WNote that if the condition on ∆V is fulfilled with σ(·) =0, then the origin isasymptotically stable for any considered disturbance w.Lemma 2. [21] Let Ξ be a positive invariant set for system (16) that containsthe origin and let V (·) be a ISS-Lyapunov function for system (16), then thesystem (16) is ISS in Ξ.5 Inherent Robustness of Nominal MPCIn this section, the robustness properties of nominal MPC algorithms are reviewedunder the fundamental assumption that the presence of uncertaintiesand disturbances do not cause any loss of feasibility. This holds true when theproblem formulation does not include state and control constraints and whenany terminal constraint used to guarantee nominal stability can be satisfied alsoin perturbed conditions.5.1 Inverse OptimalityIt is well known that the control law solving an unconstrained optimal InfiniteHorizon (IH) problem guarantees robustness properties both in the continuousand in the discrete time cases, see [12], [43], [10], [1]. Hence, the same robustnesscharacteristics can be proven for MPC regulators provided that they can beviewed as the solution of a suitable IH problem. For continuous time systems,this has been proven in [34], while in the discrete time case, from the optimalityprinciple we havewithV (x, N) =¯l(x(k),κ MPC (x(k))) + V (f(x, κ MPC (x)),N)¯l(x(k),κ MPC (x(k))) : = l(x(k),κ MPC (x(k))) − V (f(x, κ MPC (x)),N)+V (f(x, κ MPC (x)),N − 1)Then κ MPC (x(k)) is the solution of the Hamilton-Jacobi-Bellman equation forthe IH optimal control problem with stage cost ¯l(x, u). In view of Assumption3.2 and (10) it follows that¯l(x(k),κ MPC (x(k))) >l(x(k),κ MPC (x(k)))