Optimization based nonlinear feedback control for pedestrian evacuation from a network of corridors

An organized evacuation of pedestrians entrapped in a building in emergencies such as those caused by fire, bomb blast, or intentional or unintentional release of toxins in the environment is of utmost importance. The focus of this study is on the development of a formal control methodology for an orderly and jam-free evacuation of pedestrians entrapped in a network of corridors in buildings in such emergency situations. To develop an effective solution, the pedestrian evacuation from a network of corridors is divided into two basic problems of flow routing and flow control. In this study the solutions of these two basic problems are developed. The proposed solution for the flow routing problem is based on the concept of shortest paths on a graph wherein the shortest paths are determined by the dynamic programming approach. The proposed approach can be used to determine the static shortest paths that are commonly displayed as evacuation routes in the buildings. The approach can also be used to determine the real-time evacuation routes if the real-time changes in the passage conditions are monitored. Such monitoring will usually consist of quantitative assessments of the congestion enroute and of the corridor availabilities along the evacuation paths. Based on these assessments one could compute either the shortest-distance-to-exit paths using the actual corridor lengths or shortest-time-to-exit paths using the congestion based corridor traverse time estimates. The application of the proposed shortest path routing scheme is demonstrated on a realistic problem of evacuation of pedestrians from a large network of corridors in a building consisting of multiple floors and multiple exits. Next the problems of pedestrian flow in a long corridor as well as in a network of corridors are considered. Analytical formulations are developed to define the flow conditions in the two cases. The time evolution of pedestrian flow in a corridor is defined by ordinary differential equation with the average pedestrian density in the corridor as the state variable. To accommodate the possible spatial variations in the pedestrian densities, a long corridor can be divided into several subsections with the flow in each subsection defined by an ordinary differential equation. With this, the flow evolution in a long corridor or a network of corridors is defined by a system of differential equations. The pedestrian in-flow consisting of rear input discharge as well room input discharges are considered in the formulation. For a long corridor flow control, analytical models for three flow conditions representing the end, middle and beginning phases of an evacuation in an exit corridor are developed. This is followed by the development of an analytical framework for the flow control in a network of corridors. The state variables representing the flow conditions and the control variables to regulate the pedestrian flow are introduced. For both the corridor and the network models, first the control variable values are defined to simulate the uncontrolled situation. These values are typically those that can be expected in a panic situation. Simulation of such uncontrolled scenarios for different models clearly indicates serious interruptions in evacuation caused by the jamming of corridors at different locations. To avoid the jam problem associated with the uncontrolled flow, the feedback control schemes are developed. An optimization-based approach is proposed for designing the feedback control. The control scheme is designed to ensure that the flow states continuously track a certain optimal state and that a certain objective function of the control variables is optimized. From the implementation point of view, an important aspect of the proposed control methodology is that the control actions always remain within the desired bounds. Numerical simulations for the uncontrolled and controlled flow scenarios are performed for an evacuation from a complex network of corridors representing a realistic evacuation problem. A comparison of the numerical results of the controlled and the uncontrolled flow scenarios clearly demonstrates the superiority of the controlled case in terms of smooth evolution of the flow parameters with continuous outflow of pedestrians from the network at some optimum level.