Hybridization of a class of "second order" models of traffic flow

Despite the simultaneous progress of traffic modelling both on the macroscopic and microscopic front, recent works [E. Bourrel, J.B. Lessort, Mixing micro and macro representation of traffic flow: a hybrid model based on the LWR theory, Transport. Res. Rec. 1852 (2003) 193–200; D. Helbing, M. Treiber, Critical discussion of “synchronized flow”, Coop. Transport. Dyn. 1 (2002) 2.1–2.24; A. Hennecke, M. Treiber, D. Helbing, Macroscopic simulations of open systems and micro–macro link, in: D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow ’99, Springer, Berlin, 2000, pp. 383–388] highlighted that one of the most promising way to simulate efficiently traffic flow on large road networks is a clever combination of both traffic representations: the hybrid modelling. Our focus in this paper is to propose two hybrid models for which the macroscopic (resp. mesoscopic) part is based on a class of second order model [A. Aw, M. Rascle, Resurection of second order models of traffic flow?, SIAM J. Appl. Math. 60 (2000) 916–938] whereas the microscopic part is a Follow-the Leader type model [D.C. Gazis, R. Herman, R.W. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res. 9 (1961) 545–567; R. Herman, I. Prigogine, Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971]. For the first hybrid model, we define precisely the translation of boundary conditions at interfaces and for the second one we explain the synchronization processes. Furthermore, through some numerical simulations we show that the waves propagation is not disturbed and the mass is accurately conserved when passing from one traffic representation to another.

Modelling road traffic accidents using macroscopic second-order models of traffic flow

In this paper, we introduce a macroscopic model for road traffic accidents along highway sections. We discuss the motivation and the derivation of such a model, and we present its mathematical properties. The results are presented by means of examples where a section of a crowded one-way highway contains in the middle a cluster of drivers whose dynamics are prone to road traffic accidents. We discuss the coupling conditions and present some existence results of weak solutions to the associated Riemann Problems. Furthermore, we illustrate some features of the proposed model through some numerical simulations. © The authors 2012.

A Lagrangian approach for modeling road collisions using second-order models of traffic flow

In [M. Herty, A. Klein, S. Moutari, V. Schleper, and G. Steinaur, IMA J. Appl. Math., 78(5), 1087–1108, 2013] and [M. Herty and V. Schleper, ZAMM J. Appl. Math. Mech., 91, 763–776, 2011], a macroscopic approach, derived from fluid-dynamics models, has been introduced to infer traffic conditions prone to road traffic collisions along highways’ sections. In these studies, the governing equations are coupled within an Eulerian framework, which assumes fixed interfaces between the models. A coupling in Lagrangian coordinates would enable us to get rid of this (not very realistic) assumption. In this paper, we investigate the well-posedness and the suitability of the coupling of the governing equations within the Lagrangian framework. Further, we illustrate some features of the proposed approach through some numerical simulations.

A new approach to reduced-order modelling

An error polynomial is defined, the coefficients of which indicate the difference at any instant between a system and a model of lower order approximating the system. It is shown how Markov parameters and time series proportionals of the model can be matched with those of the system by setting error polynomial coefficients to zero. Also discussed is the way in which the error between system and model can be considered as being a filtered form of an error input function specified by means of model parameter selection.

Reduced-order output feedback controllers for discrete-time systems

This article considers the stabilization by output feedback controllers for discrete-time systems. The controller can place all of the closed-loop poles within a specified disk D(-α, 1/β), centred at (-α,0) with radius 1/β, where | - α|  + 1/β < 1. The design method involves the decomposition of the system into two portions. The first portion comprises of all of the poles that are lying outside of the specified disk. A reduced-order model is constructed for this portion. The second portion comprises of all of the remaining poles of the system and is characterized by an H_∞-norm bound. The controller design is then accomplished by using H_∞-control theory. It is shown that, subject to the solvability of an algebraic Riccati equation, output feedback controllers can be systematically derived. The order of the controller is low, and can be as low as the number of the open-loop poles that are lying outside of the specified disk. A step-by-step design algorithm is provided. Numerical examples are given to illustrate the attractiveness of the design method.

Design of reduced-order functional observers for linear systems with unknown inputs

This brief paper presents new conditions for the existence and design of reduced-order linear functional state observers for linear systems with unknown inputs. Systematic procedures for the synthesis of reduced-order functional observers are given. Numerical examples are given to illustrate the attractiveness and simplicity of the new design procedures.

Design of reduced-order scalar functional observers

In the existing literature, the existence conditions and design procedures for scalar functional observers are available for the cases where the observers’ order p is either p=1 or p=(v-1), where v is the observability index of the matrix pair (C,A). Therefore, if an observer with an order p=1 does not exist, the other available option is to use a higher order observer with p=(v-1). This paper shows that there exists another option that can be used to design scalar linear functional observers of the order lower than the well-known upper bound (v-1). The paper provides the existence conditions and a design procedure for scalar functional observers of order 0≤ p ≤2, and demonstrates the presented results with a numerical example.

Existence and design of reduced-order scalar functional observers

In the existing literature, the existence conditions and design procedures for scalar functional observers are available for the cases where the observers’ order p is either p=1 or p=(v-1), where v is the observability index of the matrix pair (C,A). Therefore, if an observer with an order p=1 does not exist, the other option is to use a high-order observer with p=(v-1). This paper provides the existence conditions and a design procedure for scalar functional observers of order 0≤p≤2, and demonstrates the presented results with a numerical example. where K, M, E, H and G are constant matrices to be designed. The problem of observing a scalar functional or multi functionals (z(t)∈Rk , k>1) of the state vector has been the subject of numerous papers, and different algorithms have been proposed (see, [1]-[13] and references therein). There are also papers that deal with the order reduction of multi-dimensional functional observers [9,10,12,13]. For scalar functional observers, a well-known Luenberger’s classic result [1] provides an upper bound on the order with p=(v-1). It is interesting to note here that, except for a recent result of Darouach [12,13], little results have been reported on the order reduction for scalar functional observers.