We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancybased interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.
In the future grids, to reduce greenhouse gas emissions Electric Vehicles (EVs) seems to be an important means of transportation. One of the major disadvantages of the future grid is the demand-supply mismatch which can be mitigated by incorporating the EVs into the grid. The paper introduces the concept of the Distributed Resource Allocation (DRA) approach for incorporating a large number of Plug-in EV (PEVs) with the power grid utilizing the concept of achieving output consensus. The charging/discharging time of all the participating PEVs are separated with respect to time slots and are considered as strategies. The major aim of the paper is to obtain a favorable charging strategy for each grid-connected PEVs in such a way that it satisfies both grid objectives in terms of load profile smoothening and minimizing of load shifting as well as economic and social interests of vehicle owners i.e. a fair share of the rate of charging for all connected PEVs. The threefold contribution of the paper in smoothening of load profile, load shifting minimization, and fair charging rate is validated using a representative case study. The results confirm improvement in load profile and also highlight a fair deal in the charging rate for each PEV.
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