Design of driver-assist systems under probabilistic safety constraints near stop signs: Proofs of technical results

This document contains the statements and proofs of the main theoretical results presented in the paper by the same authors with the title “Design of driver-assist system under probabilistic safety specifications near stop signs”, which will appear in the special issue on Human Centered Automation of the IEEE Transactions on Automation Science and Engineering, [1]. The notations, definitions and the numbering of theorems, propositions, assumptions and corollaries follows [1]. One key definition is provided first, followed by the assumptions. Then we provide the technical results with the corresponding proofs. Definition 5. Let Σ1 = (X1, U, ∅, O1, f1, h1) and Σ2 = (X2, ∅,∆, O2, f2, h2) be continuous systems. The parallel composition Σ∗ := Σ1 ‖ Σ2 is an order preserving system with stochastic disturbance (OPSD) if the following conditions are satisfied: i) Σ1 is input/output order preserving with respect to the control; ii) Σ2 is input/output order preserving with respect to the disturbance; iii) There exists um ∈ U such that um ≤ u for all u ∈ U ; iv) The disturbance input d of the system is a ∆-valued random variable with unimodal, invertible, cumulative distribution function Φ: R→ [0, 1]. Let us also define um : [0, τend]→ U, um(t) = um ∀t ∈ [0, τend]. (1) ∗Department of Mechanical Engineering, MIT, 77 Massachusetts Avenue, Cambridge, MA †Toyota Technical Center, 1555 Woodridge Avenue, Ann Arbor, MI ‡Department of Mechanical Engineering, MIT, 77 Massachusetts Avenue, Cambridge, MA §Department of Mechanical Engineering, MIT, 77 Massachusetts Avenue, Cambridge, MA