## 2 Problem Formulation
### Problem Formulation  
Having an environment as shown in Figure 1 (Assumption 1) and a linear dynamic of agent \(i\) (Assumption 2) strongly connected via an undirected graph (Assumption 3), the broadcast event-triggered (BET) consensus controller is required to satisfy the motion coordination task with the rendezvous objectives function (Equation (2)) when all agents  
Figure 1: Multi-agent broadcast communication environment.  
Figure 2: Agent \(i\) and neighbours \(j\) connected in a graph topology network.  
meet at a rendezvous point at the desired target \(x_{d}\) (Equation (3)) with minimum utilization of resources while preserving multi-agent performances at \(t\) time.  
\[P(x)\ =\ \sum_{i=1}^{N}||x_{i}-x_{d}|| \tag{2}\]  
\[lim_{t\rightarrow\infty}\ P(x(t))=0 \tag{3}\]  
Several remarks on the problem are given in the definitions below:  
The limitation of the agent to reach consensus for the rendezvous in the scope of the broadcast and communication environment. Each agent has a limited knowledge of its state position--it has the information concerning the relative position between agent \(i\) and \(j\) but does not know its position in the world's coordinates. The agent will rely on the feedback broadcasted by the \(G\) controller to determine its current position in order to the desired target point and the \(L_{i}\) controller to determine the next updated agent's state position. The communication between agent \(i\) and neighbours \(j\) in reaching an average consensus (Equation (6)) while heading to the desired target point may cause a high level of computation and utilization of communication resources. With the simultaneous perturbation stochastic algorithm (SPSA) consensus controller design, the agents' movement is determined by a local distributed controller (Equation (4)). The output of the local controller, consisting of stochastic and gradient error values, will be alternately added during even and odd times to obtain the updated agent's state position. Since communication error is taken into account during the SPSA gradient calculation, continuous data communication is thus required for the agent to meet the rendezvous point at time \(t\).  
In order to obtain the communication error at every odd time, the agent has to exchange its state position with its connected neighbor (Equation (5)) continuously. This is to ensure that the agent moves consistently with its neighbors while avoiding the stochastic effects of SPSA during the movement. However, continuous communication by the agent and neighbors at every time-triggered interval results in a high level of computation and utilization of communication channels, especially when the target point is far from the agent and when it involves multiple agents within the environment. Without proper communication control, communication resources, i.e., the number of channels (NOC) and bandwidth of the agent will increase, which will simultaneously affect the resources used by the agent. Thus, the time, iteration and trajectory will also be affected, since there is a correlation between communication, local controller output, global controller output and the next agent's state position, as represented in Equations (1), (2), (4) and (5). Therefore, the communication factor must be taken into consideration as it will guarantee the feasibility and practicality of the controller, especially when the agent has limited power from the digital embedded microcontroller.  
\[L_{i};\ \left\{\begin{array}{l}\delta_{i}(t+1)=\alpha(\delta_{i}(t),B(t))\\ u_{i}(t)=\beta(\delta_{i}(t),B(t))+\gamma(\delta_{i}(t),B(t))\end{array}\right. \tag{4}\]  
\[\gamma(\delta_{i}(t),B(t))=k\sum_{j\in N_{i}}a_{ij}\big{|}x_{i}\delta_{i4}(t)-x _{j}\ \delta_{i4}(t)\big{|}\ \text{when}\ \delta_{i4}(t)\in\{1,3,5\} \tag{5}\]  
\[\alpha_{c}=\frac{1}{n}\sum_{i}x_{i}(0) \tag{6}\]