[Paper Review] Reachability of Delayed Hybrid Systems Using Level-set Methods

Giovanni Granato|arXiv (Cornell University)|Sep 26, 2012

Electric and Hybrid Vehicle Technologies6 references3 citations

TL;DR

This paper proposes a level-set method based on optimal control and dynamic programming to compute the maximum driving range of range extender electric vehicles (REEVs) under unknown or long driving routes. By modeling the REEV as a discrete-time hybrid system with switch-time constraints, the approach computes the reachable set via a value function, enabling controllers that extend vehicle range by up to 104.7% compared to purely electric operation.

ABSTRACT

This study proposes an algorithm to synthesize controllers for the power management on board hybrid vehicles that allows the vehicle to reach its maximum range along a given route. The algorithm stems from a level-set approach that computes the reachable set of the system, i.e., the collection of states reachable from a certain initial condition via the computation of the value function of an optimal control problem. The discrete-time vehicle model is one of a particular class of hybrid vehicles, namely, range extender electric vehicles (REEV). This kind of hybridization departures from a full electric vehicle that has an additional module -- the range extender (RE) -- as an extra energy source in addition to its main energy source -- a high voltage battery. As an important feature, our model allows for the switching on and off of the range extender and includes a decision lag constraint, i.e., imposes two consecutive switches to be separated by a positive time interval. The approach consists in the introduction of an adequate optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a dynamic programming algorithm. This algorithm is implemented and some numerical examples are presented.

Motivation & Objective

• To address the lack of optimal control strategies that maximize driving range in hybrid electric vehicles when no prior route information is available.
• To develop a controller synthesis method for range extender electric vehicles (REEVs) that operates under decision lag constraints on the range extender switch.
• To characterize the reachable set of states for a delayed hybrid system without relying on backward recursion from a known endpoint.
• To enable autonomous range maximization in REEVs using real-time driving profile data from navigation systems.
• To provide a generalizable framework applicable to series and parallel hybrid architectures beyond REEVs.

Proposed method

• Formulates the REEV power management problem as a hybrid optimal control problem with lag constraints on the range extender switch.
• Introduces an obstacle-constrained optimal control problem whose value function characterizes the reachable set as the negative sublevel set.
• Applies the dynamic programming principle to the value function, enabling backward computation of the minimum time function.
• Implements a deterministic dynamic programming algorithm with fixed time and state discretization (Δt = 0.4, Δp = 0.5) to compute the value function numerically.
• Uses a level-set approach to extract the reachable set from the value function, with contour levels indicating reachable states.
• Applies the algorithm to both a toy model and a realistic REEV model, using driving profiles with variable link speeds and distances to compute autonomy.

Experimental results

Research questions

• RQ1 How can the reachable set of a delayed hybrid system with switch-time constraints be computed without prior knowledge of the final route point?
• RQ2 What optimal control strategy maximizes the driving range of a range extender electric vehicle when the vehicle cannot complete a long route on battery alone?
• RQ3 How does the inclusion of a decision lag constraint affect the reachable set and the resulting control policy in REEV power management?
• RQ4 To what extent can the range extender extend the vehicle’s range compared to purely electric operation, under realistic driving profiles?
• RQ5 Can the proposed level-set method with dynamic programming be effectively applied to real-world REEV models with non-autonomous dynamics?

Key findings

• The algorithm successfully computes the reachable set of a delayed hybrid system using a level-set approach based on the value function of an optimal control problem.
• For an initial state with 30% battery and 30% fuel, the maximum range reached was 45.126 km, representing a 104.7% increase over the purely electric range of 22.045 km.
• With an initial state of 60% battery and 40% fuel, the maximum range was 78.405 km, a 62.64% improvement over the 48.209 km achievable in purely electric mode.
• The range extender operating cost was estimated at 11.70 €/100 km for the low-initial-state case and 11.92 €/100 km for the high-initial-state case, normalized to range extender utilization.
• Numerical simulations confirmed the method's effectiveness on both toy and realistic models, with convergence observed under fixed discretization (Δt = 0.4, Δp = 0.5).
• The controller synthesis achieved full autonomy extension by optimally scheduling the range extender's on/off states, with the optimal control sequence visualized in Figure 2.