Safe Hierarchical Model Predictive Control and Planning for Autonomous Systems

The paper proves recursive feasibility in two steps: (i) feasibility propagation within the lower layer between planning instants via a shifted candidate using the tube error recursion E_i(j+1) = (A+BK)E_i(j) ⊕ W_i and the tightened constraints (17a–d), and (ii) feasibility of the planning problem at the next planning instant by shifting the previous planning solution and closing with the terminal control law under Assumption 4; see Proposition 1 and Proposition 3 (parts 1–3) . The candidate solution mirrors this structure: it constructs the same lower-layer “shift-and-recenter” candidate using Δ_{j+1}=(A+BK)Δ_j and verifies the tightened constraints with standard Pontryagin-difference identities, matching the intent of part 2 in the paper’s proof . At planning instants, it shifts the planning optimizer and applies the terminal controller, invoking Assumption 4 exactly as in Proposition 1 . One small difference is that the model spells out the set-calculus details (PD1/PD2) that the paper leaves implicit. Overall, both arguments align technically and logically with the same construction and assumptions (Assumptions 1–4, constraints (7), (12), (17), and contract (6)) .