Broadcast Event-Triggered Control Scheme for Multi-Agent Rendezvous Problem in a Mixed Communication Environment

Featured Application: In this work, a broadcast event-triggered control approach is used to effec-tively solve the rendezvous problem and to improve the performance of a multi-agent system in a mixed communication environment. Abstract: This paper addresses the communication issue encountered by a hybrid controller when ﬁnding consensus in terms of the rendezvous target point in a broadcast and communication environment. This issue may result in a high level of computation and the utilization of agent resources when a continuous communication is required by agents to meet convergence requirements. Thus, an event-triggered system was integrated into the design of a broadcast and distributed consensus linear controller using the simultaneous perturbation stochastic algorithm (SPSA). The agent’s movement towards the rendezvous point is based on the broadcast value, whereas the next agent’s state position depends on the distributed local controller output. The communication error obtained during communication between the agent and neighbors is only added to the gradient approximation error of the SPSA if the event-triggered function is violated. As a result, in our model, the number of channel utilizations was lower and the agents’ performances were preserved. The efﬁciencies and effectiveness of the proposed controller have been compared with the traditional sampling broadcast time-triggered (BTT) approach. The time and iterations required by the broadcast event-triggered (BET) system were less than 40.42% and 21% on average as compared to BTT. The trajectory was not the same—the BET showed scattered movements at the initial stage, whereas BTT showed a linear movement. In terms of the number of channels, 28.91% of channels were preserved during the few hundred iterations. Consequently, a variety of hybrid controllers with event-triggered mechanisms can be proposed for other multi-agent motion coordination tasks.


Introduction
Multi-agent robot research has been expanding due to the effectiveness, robustness, flexibility and operational efficiency involved in accomplishing tasks with many agents, compared to the use of a single agent [1]. The advantages of load-sharing, enable complex tasks to be simplified for agents, which has allowed cooperative multi-agent robot research to be more actively explored [2][3][4]. The multi-agent system is specifically designed to work in hazardous environments and in very limited spaces, which are impossible for humans to reach. Multiagent systems have applications in medicine and nanotechnology [5]-magnetic [6] robots have been used to send medicine directly to human organs, planetary rovers have been used to accomplish missions on other planets [7,8], inspection robots have been used to inspect and clean the pipelines in oil and gas industries [9,10] and researchers have explored the use of unmanned aerial vehicles and swarm robot applications [11][12][13]. Due to the importance of these and dynamic role assignment [53]. Not only could this save communication resources, it could also save energy. As the multi-agent robot is normally embedded with the digital microcontroller, which has limited resources, the research on reducing energy usage resulting from communication [54,55], actuators [56] and trajectory [57] have become recent hot topics in multi-agent research, and this area is also known as "energy aware" [58] or "energy efficient" [12,59] research.
Motivated by the aforementioned works, this study was carried out to solve the homogenous agent communication issues when finding a consensus for rendezvous applications in the broadcast and communication environment. Although a few stochastic controllers have been applied to finding a consensus for this application, the issue of communication has not been taken into consideration. Therefore, in this study, we propose an integration of an event-triggered system into the simultaneous perturbation stochastic algorithm (SPSA) and a distributed controller to obtain the minimum utilization of the channel, as well as preserving agent performances. This study can be considered an advancement in the existing method, aiming to highlight the importance of reducing energy resources from communication, which will guarantee the practicality of controllers. The effectiveness of the proposed controller was evaluated and observed in terms of trajectory, time, iteration, and number of channels taken to reach consensus for rendezvous purposes. In addition, a conventional sampling system, known as the time-triggered system, was applied in this case study as a benchmark for comparison to the event-triggered system. The obtained results are compared with traditional sampling systems in terms of channel utilization and agent performances to show the effectiveness and robustness of the proposed consensus controller. The rest of this article is structured as follows. In Section 2, the formulation of the problem is explained. Section 3 describes the system, including the global and local consensus controller designs and the proposed theorems. In Section 4, results from the simulation setup are presented and discussed, whereas Section 5 concludes this paper.

Preliminaries
In this section, a generic system description, covering multi-agent environment, agent control and agent connection, is presented with notations and assumptions as outlined below: Notations. Assume that R is a real number, R + is a positive real number set, and N is a non-negative integer set. The n × m zero matrix is denoted as 0 n×m . The Euclidean vector x is represented by x i (t) = [x 1 , x 2 , x 3 . . . . . . .x n ] T ∈ R n with non-zero elements in Euclidean coordinates representing the positions of the agent. The agent is represented with i, neighbours with j, discrete time with t, local controller L i , global controller G and broadcast signal B ∈ R. The function f (e, x) contains the variables of e and x. Assumption 1. The feedback control system presented in Figure 1 has been applied in a broadcast communication environment. The global controller G will keep broadcasting the scalar value β(t) to agents until consensus is achieved when it reaches its desired target x d . The local controller L i will determine the updated agent's state position based on the input received by the controller, which consists of random errors and deterministic errors with ET at every discrete time t. The camera will capture and update the state position of agent at G if there are changes to agent movements. This closed-loop process will repeat until the solution converges where the agent performance achievement of P(x(t)) is equal to 0. Assumption 2. The physical dynamics of homogenous agent i's linear discrete time model is represented as where the initial state location is x i (t) ∈ R n and u i (t) is the input of the controller at time t. The output of the controller, representing the next location of the state agent x i (t + 1), depends on the changes in the input of the controller, also known as the local controller error.
Assumption 3. It is assumed that agent i has its own neighbor set j ∈ N i which is strongly linked to the radius as an undirected graph; r. r represents the radius where a relation can be formed by the agent. In undirected graph theory, G T = (V, E) where V is the vertex that represents the agent and E is the edge of each vertex representing the connection between agent i and its neighbours j or (i, j) ∈ E, as depicted in Figure 2. N i = {(j|(i, j))} ∈ E denotes a set of all the neighbors of the agent. The representation of the connection/topology of the graph and the number of edges E for each vertex V is expressed in matrix form. An adjacency matrix A = a ij ∈ R NXN is used to describe the graph topology, where a ij = 1 if (j, i) ∈ E and a ij = 0, or otherwise. The degree matrix of the system is defined as

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 N i 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.
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.

Definition 2.
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.

Proposed Broadcast Event-Triggered Consensus Control Scheme
The feedback system shown in Figure 3 has been designed using an optimization algorithm known as the simultaneous perturbation stochastic algorithm (SPSA). The local controller was designed by adopting a stochastic and deterministic element of SPSA, which will determine the linear dynamic agent's next state position x i (t + 1) at time t. At every movement of the agent, a global controller will update the information of its state position to determine the scalar value based on the objective function calculation, which represents agent achievement. As long as the local controller receives the scalar value, it means that the agent has yet to reach the desired target. Therefore, the process will be continuous until consensus is achieved, when the termination criteria is satisfied, that is, when the agent reaches the rendezvous at the desired target point. The uniqueness of this hybrid consensus controller is that an event-triggered function (ETF) is embedded in the distributed-agent consensus controller. The ETF is specifically designed with a certain threshold value, which will limit the number of data transmissions and control updates. With event-triggered control (ETC), the value of gradient approximation of the SPSA is affected by the communication error, which has an effect on the agent's state position in the next iteration. Details of the design are explained in Section 3.1 for the global controller, in Section 3.2 for the local controller, in Section 3.3 for the event-triggered controller and in Section 3.4 for related theorems.

Global Controller, G
A global controller is used to represent the agent performances in reaching consensus as to the rendezvous target given in Equation (7), where B(t) ∈ R is the output of G, x(t) ∈ R nN is the state position of all agents and P(x) is an objective function of rendezvous,

Local Controller, L
Local controller L i , which is the distributed control integrated with the event-triggered process, will determine the agents' movement A i at every even or odd time t is based on input from the broadcast signal, B(t) ∈ R. The local controller is denoted by Equation (11), α is the state function, β is the random and deterministic function of the SPSA and γ is the standard consensus protocol with ET. The column of vector of Equation (9) represents the variables in the control system, where δ i1 (t) is the state position, δ i2 (t) is the broadcast signal, δ i3 (t) the even time and δ i4 (t) is the odd time.
Based on Figure 4, the scalar value obtained from the global controller will be fed into the local controller until consensus as to the rendezvous point is achieved. The output from the local controller consists of u R , determined by β of the SPSA Bernoulli distribution error during even times and the sum of u D1 and u D2 , obtained from β of the SPSA gradient approximation error and γ of the communication error, during odd times. The communication between agent i and neighbor j occurs only when ETF is violated. Details of the ET design are discussed in the next section. The output from u i (t) will exert an effect on the next state's position, as expressed in Equation (1).

Event-Triggered Controller, γ
The event-triggered controller was designed as part of the agent distributed consensus protocol (Equation (14)) to reduce the number of communication channels while reaching an average consensus (Equation (6)) among connected agents. Since agent i is connected to its neighbour j ∈ N i via a strongly connected undirected graph topology (Figure 2), the agent will then exchange its current state value x i with its neighbour's state value x j if it violates the ETF, which leads to the consensus of the average point (Equation (6)) and the desired target point (Equation (8)). The ETF of this system is represented by Equation (14), where f (.) represents the error function, whereas σ(.) represents the threshold function. The state measurement error (Equation (15)) depends on the difference between the agent state's position, where t ∈ t i k , t i k+1 denotes the event instant of agent t i k (k = 1, 2, . . . . . . . . . .) and t refers to the previous event. The threshold function (Equation (16)) is known as state-dependent since the value depends on the agent's state position.
The sampling process, transmission of information and updating of the controller of agents i and j with ET is shown in Figure 5. The sampler will first sample n the state of x i (t), which becomes x i (nt) at every periodic odd sampling time t. Based on the given samples, the event detector will monitor the condition of the event, where the agent state of x i (t) need to be transmitted to its neighbour or not, based on Equation (14). If the event detector i detects an event, k, in sample n kj , the new updated sample state x i (n k+1 t) of agent i is sent to neighbours j giving sample state x j (n k t). When neighbours j receive the state sent by agent i, the neighbours j update the agent i state information and store the newly received state values of agent i, x i (nt). This state will then be used by the controller and event detector of agent j until the next event is triggered from agent i. When no event occurs, this means that it fulfils the condition of Equation (14), then x i (n k t) is directly fed into the controller, which means there is no transmission and controller update (Equation (1)) that occurs at this time. The zero-order hold will ensure that the control signal is kept constant until the next event occurs.

Proposed Theorems
Theorem 1. Given the linear discrete system dynamics (Equation (1)) with the control input of communication as represented in Equation (16), if the system satisfies ETF (Equation (14)), where the measurement error is represented by Equation (15), the information will not be sent to the agents unless it violates the function. The information will be sent to the neighbors to update the control input. The average consensus (Equation (6)) can asymptotically marginally stabilize, which is achieved at t → ∞, and reach a steady state error when the eigenvalue of Perron matrix is equal to λ, which is shown in the Gershgorin circle (Proof 1) and Lyapunoz stability, indicating V < 0, which is a definite negative (Proof 2).

Remark 1.
The uniqueness of this ET is that the event is triggered at the periodic sampling time, which is during odd time intervals. This is known as the sampled-data ET system, where the sampled data of the agent's state position will determine the condition of the event (Equation (14)). The difference between this study and [51,52] is the integration of ET into the broadcast system, whereas in comparison with [60], the communication issue is taken into consideration for multi-agent consensus in a broadcast communication environment.

Remark 2.
The threshold value σ i of the triggering function (Equation (14)) is state-dependent, where the agent's state position depends on its neighbor's state position.

Assumption 4.
The network of topology is assumed to be strongly connected among inter-agents under the undirected graph to achieve a marginally stable consensus.
Proof of Theorem 1. Refer to Appendix A.

Theorem 2.
For BET communication, suppose that the objective function P is given and assumed that P is changing satisfying ∇P(x d ) = 0 with ETF, f (e, x) (Equation (14)) applied in the agent and neighbors' communication system using a standard distributed protocol (Equation (16)). Let L i and G be given by Equations (7), (8) and (11). If the BET satisfies the condition of Theorem 1 and Lemma 1, then lim t→∞ x(t) = x d w.p.1 is achieved.
Proof of Theorem 2. Refer to Appendix B.

Results and Discussion
In this section, a series of simulations that include ten connected homogeneous agents (Assumption 3) were assumed to be initially located at the planar coordinates (refer to Table 1). With optimal SPSA parameter settings and ET settings, as presented in Table 2, in a feedback control system in a broadcast and communication environment (Assumption 1), the agent was expected to reach consensus and meet at a rendezvous point at the desired target point located at (80, 80), as shown in Figure 6. The SPSA gain of a k and c k depended on Equations (17) and (18). The agent worked based on the "objectives" function determined in Equation (2) and the termination criteria, based on the measurement state error value, which should reach zero. The results obtained specifically in relation to the efficiencies of the time and iterations of this research were based on results obtained using MATLAB 2015B simulation software with the Intel ® Core™ i5-2410M CPU @ 2.3GHz processor, running on a 64-bit operating system.  Table 2. Optimal simultaneous perturbation stochastic algorithm (SPSA) and event-triggered (ET) parameter settings for the broadcast event-triggered (BET) controller. The evaluation of agent performances in terms of time, iteration, trajectory and number of channels taken by the multi-agent to reach the rendezvous point in a few time runs were then recorded. This started with evaluation of the BET consensus controller, and the BTT consensus controller was then observed in order to compare the effectiveness of the proposed controller with the conventional sampling system. The results are divided into two sections, starting with the BET and followed by the performance comparison of the BET and BTT in finding the consensus in relation to the desired rendezvous target.

Time and Iteration
The average time the agent took to reach the rendezvous point for the desired target point in 10 time runs was an average of 84.3413 s and 676.6 iterations as shown in Table 3 (Proof of Theorem 2 and Proof 4). The shortest time and iterations taken by an agent to converge was at least 35.996 s with 312 iterations when the state measurement error, which indicates the performance index, showed a reading of 0 after a certain number of iterations as shown in Figure 7.   Figure 8 shows the agent's movement obtained from BET, demonstrating that the rendezvous was reached at 225.424 s with 1104 iterations. The first 300 iterations showed that the agents moved in a scattered manner and were not too close with one another. However, when it reached more than 500 iterations, the communication error caused the agents to meet at the average consensus point (proving Theorem 1) while heading to the goal point. Figure 8 shows that the agent's movement was synchronized, with a reduction of error.

Utilization of Channels
The channel usage was recorded as shown in Table 4 for the BET controller. The number of channels (NOC) was not constant during ET, and it depended on the value of ETF whether to stop or to allow agent communication among agents. Based on Table 4, the agent successfully reduced its channel utilization by 21.79%. Details of the calculations of the agent are presented below. Each iteration is equivalent to 10 channels, which carried a total of 552 × 10 = 5520 channels.

Time and Iteration Efficiencies
The average readings of time and iteration taken by the multi-agent system to reach consensus in relation to the rendezvous in ten time runs are shown in Table 5. BET was proven to lead BTT by 57.153 s, with 177 iterations. The efficiency of BET and BTT in reaching consensus depended on the effectiveness of the broadcast consensus controller in working with the distributed controller with either event sampling or conventional time sampling.

Agent Trajectory
The trajectory or agent movement with BET and BTT were observed at every 50 iterations to estimate the trajectory patterns obtained by both controllers. With BET, the 10 agents' movement was scattered during the first 200 iterations, before the agents showed movement heading to the rendezvous point when they reached 200 to 900 iterations. The movement can be seen clearly in Figure 9, in which the agents' state position accumulates within the range of x-y coordinates when it reaches 80 < x < 100 and 70 < y < 90. This was unlike the BTT system, in which the agents' movement looked consistent, as illustrated in Figure 10. The agents were pulled among one another (Assumption 3) to reach an average consensus while heading to the rendezvous point.   Tables 6 and 7 show the NOC used by BET and BTT in a selected time run. The BET met the rendezvous target at 225.2 s with 1104 iterations, whereas BTT converged at 100.8 s with 840 iterations. The NOC was recorded per 100 iterations to observe how the NOC was involved in communication between the agent and neighbors along the way until the agent reached the rendezvous point. BET showed a total of 71.09% NOC usage as compared to BTT, which utilized 100% of the channels. This was due to the effect of implementing event-triggering in the BET system, which resulted in the usage of channels which were not full for the first 500 iterations, as shown in Figure 11. As a result, there were at least 28.9% usage of channels reserved for the agent with the use of the BET controller as compared to none of the channels being reserved in the BTT system. The NOC of the first 10 iterations were recorded to observe the patterns of agent communication at each iteration. Referring to Table 8 and Figure 11, there were only a few agents that would send the information of the agent's state position at each iteration, which depended on the ETF condition concerning whether there were any violations. As a result of this, the channel utilization can be reduced in the BET system as compared to the BTT system, which fully utilized the channel, as shown in Table 9 and Figure 12.

Conclusions
A hybrid controller in a broadcast and communication environment with the BET system produced very promising results in terms of finding a consensus as to the rendezvous point. The agent was able to reach consensus regarding the rendezvous for the desired target in a minimum time and number of iterations, while reducing the utilization of channels. A lower number of channels will reduce communication resources, which will simultaneously reduce the number of control updates and save energy. In fact, BET was much better than BTT in terms of time, iterations, number of channels and trajectory movement. Furthermore, the trajectory of agents showed less communication effects towards the gradient value of SPSA, which would have apparent effects on the agent's next state position. Thus, BET has been proven to be efficient and practical in order to find a consensus as to a rendezvous point in a broadcast and communication environment.
In future, this research can be validated using a real multi-agent robot which uses broadcast and distributed control supported by a network wireless system. The effectiveness and robustness of the proposed controller to achieve convergence with probability one, w.p.1 is expected from the results obtained in our modeling. Additional physical tests can be conducted to measure agent efficiencies by recording the time, channel utilization and energy usage. In addition, instead of reducing the communication channels to limit the usage of communication resources, an extension of this research towards energy-aware and energy efficient systems can be proposed. To this end, the total energy obtained from communication and motion can be calculated, which will solve the issue of limitations in agent resources in agent microcontrollers. Finally, the idea of embedding event-triggered systems in other complex systems and in relation to various dynamics of agents is recommended in order to guarantee the practicality and viability of multi-agent controllers.
If the conditions of Equation (14) are satisfied, there is no event occurring at this time with the error of Equation (15), and e i (t) is equal to 0, which means that there is no transmission and control update happening with the agent and its neighbours. This means that the sampled data are not updated, and the agent's state value will remain the same. The opposite occurs for the situation where the conditions of Equation (14) are violated. In this scenario, an event occurs, in which the error is equal to Equation (15) and the new state position of Equation (A3) is sent to the connected neighbors for a control update. Even though the state measurement error gives a significant effect to the next agent's state, as stated in Equation (A2), the results of the closed loop system are guaranteed asymptotically to reach average consensus, Equation (6), among agents in the group when the eigenvalues of square matrix P are strictly contained in the Gershgorin circle criterion. The eigenvalues of the Perron matrix for a topology of 10 agents are strongly connected with an undirected graph-these are λ 1 = 0 , λ 2 = 0.5 ± 0.5i and λ 10 = 1. The eigenvalues inside the Gershgorin circle unit are expected to be λ 10 = 1.
With the Laplacian matrix, Equation (A5) will be For a > 0, · V can bound using inequality |xy| ≤ a 2 x 2 + 1 2a y 2 , thus Since the communication graph is undirected, which shows that the system is stable when the derivative of the Lypunov function is negative definite for 0 < σ i < 1.

Lemma 1.
Each state position of the agent, x i is sampled at every specific odd time, t bounded by t + 1. The channel utilization in each sampling time is limited to a maximum number of agents N i per sampling, which shows that Zeno behavior does not exist and can be excluded in this case, as proven in Proof 3.

Proof 3 of Theorem 1.
The agent deterministic movement (Equation (11)) is determined by the sum of the ET consensus control input (Equation (13)) with the SPSA deterministic input of the local distributed agent controller, which occurs during odd time intervals (Equation (12)). The agent movement is alternated with the stochastic movement at even times. x(t + 2) = x(t) − a(t) Substitution of Equations (A9) and (A10) into Equation (A11) will obtain Equation (A12), which can be simplified to Equation (A13).
E(d(x(t), ∆ i (t), c(t)|x(t)) = ∇ P(x(t) + O(c(t)(c(t) → 0) Thus, the agent state of x(t + 2) is represented by Equation (A15), in which the gradient descent method has been integrated with the sum of the agent communi-cation error between the connected agents. With these, x(t) will converge to a local minimum point of P. E((x(t + 2)|x(t)) ∼ = x(t) − a(t) P(x(t)) − k a ij ∑ The collective dynamics movements of agent BET at |x(1) − x(0)|, |x(3) − x(2)|, |x(5) − x(4)|, . . . . . . . and |x(n + 1) − x(n)| show that the agents will converge to x d or 0 w.p.1. The dynamics are determined with the aim of asymptotically achieving consensus locally and globally, as highlighted in Theorems 1 and 2. (5) If (ET1), (ET2), (A3) and (A4) hold, the dynamics in Equation (1) will converge to a minimum point of function P, represented by the quadratic function (Equation (8)) that measures the degree of achievement when the agents move from the initial state x 0 to the target point x d w.p.1. The settings of broadcast (A3) and (A4) and eventtriggered (ET1) and (ET2) are imposed for tuning L i and G as defined in Equations (7) and (11).
, then e i (t) = 0, else e i (t) =x i(t) − x i(t) (6) If the conditions of SPSA convergence satisfy (A1)-(A7) and the conditions of ET satisfy (ET1) and (ET2), the system will converge when all agents reach the desired target x d .