Assessment and Future Directions of Nonlinear Model Predictive ...

122 F.A.C.C. Fontes, L. Magni, and É. GyurkovicsV (t, x) :=V ⌊t⌋π (t, x)where V ti (t, x t ) is the value function for the optimal control problem P(t, x t ,T c −(t − t i ),T c − (t − t i )) (the optimal control problem defined where the horizon isshrank in its initial part by t − t i ).From (7) we can then write that for any t ≥ t 0∫ t0 ≤ V (t, x ∗ (t)) ≤ V (t 0 ,x ∗ (t 0 )) −t 0M(x ∗ (s))ds.Since V (t 0 ,x ∗ (t 0 )) is finite, we conclude that the function t ↦→ V (t, x ∗ (t)) isbounded and then that t ↦→ ∫ tt 0M(x ∗ (s))ds is also bounded. Therefore t ↦→ x ∗ (t)is bounded and, since f is continuous and takes values on bounded sets of (x, u),t ↦→ ẋ ∗ (t) is also bounded. All the conditions to apply Barbalat’s lemma 2 aremet, yielding that the trajectory asymptotically converges to the origin. Notethat this notion of stability does not necessarily include the Lyapunov stabilityproperty as is usual in other notions of stability; see [8] for a discussion.6 Robust StabilityIn the last years the synthesis of robust MPC laws is considered in differentworks [14].The framework described below is based on the one in [9], extended to timevaryingsystems.Our objective is to drive to a given target set Θ (⊂ IR n ) the state of thenonlinear system subject to bounded disturbancesẋ(t) =f(t, x(t),u(t),d(t)) a.e. t≥ t 0 , (8a)x(t 0 )=x 0 ∈ X 0 ,(8b)x(t) ∈ X for all t ≥ t 0 , (8c)u(t) ∈ U a.e. t≥ t 0 , (8d)d(t) ∈ D a.e. t≥ t 0 , (8e)where X 0 ⊂ IR n is the set of possible initial states, X ⊂ IR n is the set of possiblestates of the trajectory, U ⊂ IR m is a bounded set of possible control values,D ⊂ IR p is a bounded set of possible disturbance values, and f :IR× IR n ×IR m × IR p → IR n is a given function. The state at time t from the trajectory x,starting from x 0 at t 0 , and solving (8a) is denoted x(t; t 0 ,x 0 ,u,d)whenwewantto make explicit the dependence on the initial state, control and disturbance. Itis also convenient to define, for t 1 , t 2 ≥ t 0 , the function spacesU([t 1 ,t 2 ]) := {u :[t 1 ,t 2 ] → IR m : u(s) ∈ U, s ∈ [t 1 ,t 2 ]},D([t 1 ,t 2 ]) := {d :[t 1 ,t 2 ] → IR p : d(s) ∈ D, s ∈ [t 1 ,t 2 ]}.