Assessment and Future Directions of Nonlinear Model Predictive ...

A New Real-Time Method for Nonlinear Model Predictive Control 549A Proof of Lemma 2It can be shown from (11a) that ∇ t J = −L a (x p ,u φ ) −〈∇ x J, f(x p ,u φ )〉. From(12a),dJdt = ∇ tJ + ∇ z J ż=−L a (x p ,u φ )− 〈 ∇ t θJ, Proj{k t αΓ t ∇ t θJ T ,Ξ} 〉 − 〈 ∇ θ J, Proj{k θ Γ θ ∇ θ J T , Θ N } 〉≤−γ L (‖x p ,u φ ‖ Σ)The conditions of the lemma guarantee that J(t 0 ,z 0 ) is bounded (although notuniformly), and the above ensures that J(t, z) ≤ J(t 0 ,z 0 ), for all t ∈ [t 0 ,t 1 ].Since all dynamics in (12a) are locally Lipschitz on the set Z = { z : (t θ ,θ) ∈Φ(t, x(t), P) }, continuity of the solution implies that the states can only exit Zby either 1) reaching the boundary A = cl{Z} \ Z (i.e. the set where x ∈ ∂X,u φ ∈ ∂U, orx p (t θ N ) ∈ ∂X f ), or 2) passing through the boundary B = Z\˚Z. Thefirst case is impossible given the decreasing nature of J and lim z→A J(t, z) =∞,while the second case is prevented by the parameter projection in (12a).B Proof of Lemma 3The first claim follows fromJ(t, z + )−J(t, z) ==δ(x p f )∫0t θ+∫Nt θ NL a (x p (τ,t,z + ,φ), u φ (τ,z + ,φ)) dτ + W a (x p+f) − W a (x p f )L(x κ (τ), φ(τ,κ(x p f )) + µ (B x (x κ (τ)) + B u (φ(τ,κ(x p f ))) )dτ()+ W (x p+f) − W (x p f )+µ B xf (x p+f) − B xf (x p f )≤ 0 (by (4) and (9))where x p f xp (t θ N ,t,z,φ), xp+f x p (t θ+N ,t,z+ ,φ), and x κ (·) is the solution toẋ κ = f(x κ ,φ(t, κ(x p f ))), x κ(0) = x p f. The second claim follows by the propertiesof κ(x) guaranteed by Assumption 2, since the portion of the x p (τ) andu φ (τ)trajectories defined on τ ∈ (t, t θ N ] are unaffected by (14).C Proof of Theorem 1Using the cost J(z π ) as an energy function (where J(z π ) ≡ J(s, x, t θ − s, θ) from(11a), with s arbitrary), the result follows from the Invariance principle in [15,Thm IV.1]. The conditions of [15, Thm IV.1] are guaranteed by Lemmas 2, 3,