Distributed Feedback Control 2-D

A mesoscopic model for the effect of density on pedestrian group dynamics

“…2: (Colour on-line) Black circles and dotted line: empirical average value of v (1) . Red squares (confidence interval in continuous lines): empirical average value of v (2) . Dashed blue line and triangles: average value of v (2) obtained by a numerical integration of the 2-person group system using for each ρ the value of η that provides the best agreement with the empirical velocity in the ρ → 0 limit.…”

“…The parameter η weakly affects the pedestrian position probability distribution, but it has an important effect on the difference between the velocity of individuals and 2-person groups, v (1) − v (2) [14]. By numerically solving 9 the 2-person Langevin equation ( 3) with the addition of the ρ-dependent term, eq.…”

“…( 20). Dotted black line and circles show the empirical v (2) (ρ). interaction" approach described in [14], and compare the empirical and predicted ρ-dependence for these observables in figs.…”

Analysis of Heterogeneous Vehicular Traffic: Using Proportional Densities

“…The Roe decomposition scheme is used because it is capable of capturing abrupt changes or discontinuities in the data variables and providing accurate numerical solutions for traffic flow models [2]. Roe's scheme is employed purposely to transform the proposed model ( 11) into quasi-linear form and linearize locally by approximating the Jacobian matrix, B(U ) with Roe averages and in every time step, the procedure is repeated [18]. The resulting system can then approximate speed found from the eigenvalues of the averaged Jacobian matrix, as described below [18].…”

“…Roe's scheme is employed purposely to transform the proposed model ( 11) into quasi-linear form and linearize locally by approximating the Jacobian matrix, B(U ) with Roe averages and in every time step, the procedure is repeated [18]. The resulting system can then approximate speed found from the eigenvalues of the averaged Jacobian matrix, as described below [18].…”