TEMA 4: EDUCANDOS Y DOCENTES
4.5 Perfeccionamiento profesional permanente
In this chapter, we present novel edge-event-triggered consensus algorithms (based on edge-measurement-based scheme reviewed inSection 2.2) to achieve multi-agent consensus with Zeno-free triggers under both synchronized and unsynchronized clocks. The agent’s dynamics are modelled by single integrators and the graph topology is assumed to be fixed, undirected and connected. The contributions of this chapter is two-fold.
Firstly, as compared to [Xiao et al., 2012, 2015], the synchronized clock case stud- ied in Section 4.2 provides another point of view with much simpler trigger condi- tions. In our framework, agents only use relative information measured in its own local coordinate frame to achieveaverage consensus. This is in contrast to prior work [Xiao et al., 2012; Seyboth et al., 2013; Nowzari and Cortés, 2016] (a global coordinate frame is required for all agents) and [Fan et al., 2013, 2015] (average consensus cannot be achieved). We also apply the time regulation idea from [Fan et al., 2015] to guar- antee Zeno-free triggers, which differs from the time-dependent trigger condition used in [Wei et al., 2017].
Secondly, the unsynchronized clock case studied in Section 4.3 provides a gen- eralised framework for edge-measurement-based trigger scheme. We note that the edge-measurement-based trigger scheme reviewed inSection2.2 requires synchronous controller updates for two linked agents. To achieve this synchronous requirement, all agents in the network have to share a global clock and are activated simultane- ously, i.e. ti0 = 0,i = 1,· · · ,n. In the generalised framework, each agent measures the relative information and updates the control input under its own isolated clock. An edge event is defined over an individual agent rather than two linked agents, i.e. two agents linked by one edge do not update their control inputs synchronously.
4.1.1 Problem formulation
The MAS we study in this chapter consists of n single integrators that are labelled from 1 to n. The n agents are connected by medges (sensing links). Let xi(t) ∈ R denote the state of agenti, i=1, 2, . . .n. The dynamics of agent iare described by
˙
xi(t) =ui(t), i=1,· · · ,n (4.1) where ui(t)is the control input. We assume that each agent is only equipped with relative position sensors, e.g. sonar or ToF (time-of-flight) camera, to measure the relative states between its neighbours and itself, in its own local coordinate frame. The sensing topology is captured by a fixed, undirected and connected graphG with corresponding incidence matrixH, Laplacian matrixLand adjacency matrixA. For each edge er connecting agent iand agent j, both agent i and agent j measure the relative statezrcontinuously.
In the synchronized clock case, we further assume that all agents share a global clock t, i.e. each agent in the MAS are activated simultaneously. The sequence of event-triggered executions for edge er is t0r = 0,t1r, . . . ,tkr, . . .. At tkr, agent i and agent j(correspond to vertexes vi andvj in graphG) linked by edge er update their control input simultaneously. For agenti, which is one agent of the agent pair (i,j) linked by edgeer, the control input is designed as follows:
ui(t) =
∑
j∈Ni xj(tkr)−xi(tkr) (4.2) fort∈ [tkr,tkr+1).In the unsynchronized clock case, we let t, t(0) =0 denote a global clock. How- ever, each agent i has its own isolated, local clock ti, i = 1, 2, . . . ,n. Let ti(0) ≥ 0 denote the initial value for each ti and ti(0),∀i is not necessarily identical. That is to say, agentsiandjlinked by edgeer start to measure the relative information and update their control inputs under their own clocks with non-identical initial time. Because of this, agent i and j linked by er do not update their control inputs syn- chronously.
Since the trigger times of agents iandjlinked byerare non-identical, we define two time sequences of event-triggered executions for agents i and j, respectively, which areti0i r,t i 1i r, . . . ,t i ki
r, . . . for agentiundert iandtj 0jr ,tj 1rj , . . . ,tj kjr
, . . . for agentjunder
tj. tiki
r denotes the time ofk-th edge event of agentitriggered over edgeerunder agent
i’s clock. Both agents update their control inputs at their own edge event times. For agent i, which is one agent of the agent pair (i,j) linked by er, the control input is designed as follows: ui(ti) =
∑
j∈Ni xj(tiki r)−xi(t i ki r) , i=1, 2, . . . ,n (4.3) forti ∈[tiki r,t i ki r+1).Problem 3. Consider the multi-agent system consisting of n>1agents whose dynamics are described by(4.1). We assume each agent is driven by(4.2)in the synchronized clock case. In the unsynchronized clock case, the control input is designed as(4.3). For both cases, the aims are to find triggering conditions for each agent, such that complete consensus can be achieved and each agent does not exhibit Zeno behaviour.