A key element of cooperative driving systems for the longitudinal control of vehicles fleet is the design of algorithms able to impose a common motion to the ensemble by acting at a single vehicle level on the base of neighbours behaviour. The aim is to regulate speed and relative position of each vehicle in the platoon to that of the respective predecessor and of the leading
3.1 Introduction
vehicle (usually the first vehicle in the string) according to a required spacing policy. As information on the surrounding vehicles can be obtained both via sensors and/or via V2V wireless communication, different time-varying delays are associated to the different types of interconnections to account for the different devices/communication tools used to gather information. To guarantee platooning in the presence of time-varying delays, a novel control algorithm based on distributed consensus has been recently proposed from authors in [31,32]. Note that platoon maintenance and its string stability (i.e robustness in the presence of periodic disturbance acting on the leader motion) may be compromised by the presence of time varying delays affecting the information received via wireless communication [76]. Specifically, following the paradigm of dynamical networks, the connected vehicle system in the presence of delays is represented as a network where each node is a vehicle characterized by its own dynamics; the presence/absence of edges mimics the presence/absence of interconnections among vehicles; the structure of vehicles communication is encoded in the network topology; communication delays may be associated to links (depending on the specific link features). In so doing the designed algorithm are able to cope with different platoon topologies originated from the specific communication infrastructure, that hence are not restricted to common pairwise interactions (e.g. predecessor- following [100]) as well as with the integration of both sensor-based and communication-based vehicle technologies.
In what follow we will present two approach specifically designed to take into account spacing policy constraints, communication logical topology, as well as impairments as time-varying delays. The approaches allows the inte- gration of both sensor-based and communication-based vehicle technologies and, if the delay affecting information gathered via on-board sensors (like, e.g., radars) is smaller than the wireless communication delay, it might be considered negligible. In Sec. 3.2 we present a distributed control algorithm for cooperative driving system that solves a third-order consensus problem, while in Sec. 3.3 we present an algorithm that solves an adaptive synchro- nization problem in a vehicular network in the presence of time-varying heterogeneous delays. In consensus problem, we assume that the goal is to make that every vehicle have to reach the same velocity of the leader vehicle (set-point problem). In synchronization problem, the target is to synchronize the dynamics of all agent of the platoon to the leader dynam-
ics (tracking-problem). The problem essentially consists in leader tracking manoeuvres, whose velocity is not constant as in consensus problem. In both strategies, vehicles within the platoon are supposed to be equipped with on-board sensors (measuring relative and absolute position, speed and acceleration) as well as wireless V2V-DSRC/WAVE [2, 3] access network with beaconing messages to share information with neighbours and to receive reference signals.
3.1.1 Network model for Inter-Vehicular Communication
To model the dynamical network we exploit the complex network theory (see Sec. A.3). The inter-vehicle communication structure can be modelled by a graph where every vehicle is a node. Hence, a platoon of N vehicles is represented as a directed graph G (digraph) G = (V, E, A) of order N characterized by the set of nodes V = {1, . . . , N } and the set of edges E ⊆ V × V. The topology of the graph is associated to an adjacency matrix with non-negative elements A = [αij]N ×N. In what follows we assume
αij = 1 in the presence of a communication link from node j to node i,
otherwise αij = 0. Moreover, αii = 0 (i.e., self-edges (i, i) are not allowed
unless otherwise indicated). The presence of edge (i, j), (i, j) ∈ E, means that vehicle i can obtain information from vehicle j, but not necessarily vice versa, according to the classical definition used for example in [94]). Note that the existence of edge (i, j) means that i uses the information received by j and not only that i is within the communication range of j. Arrows indicating the direction of the information flow among vehicles and arrows indicating the direction of edges in the associated network graphs have opposite directions since we adopted for the network edges the definition used in the network literature (e.g. [94, 138]). Note that, defining the degree matrix as ∆ = diag{∆1, ∆2, ..., ∆N}, with ∆i =
X
j∈V
αij, the Laplacian of the
directed graph G can be defined as L = ∆−A. We say that j is reachable from i if there exists a path from node i to node j. In the rest of the chapter we consider N vehicles together with a leader vehicle taken as an additional agent labelled with the index zero, i.e., node 0. We use an augmented weighted directed graph GN +1 to model the resulting network topology. We assume
node 0 is globally reachable in GN +1 if there is a path in GN +1 from every