Chapter 1: Understanding the Rendezvous Problem

In the following article, we delve into the complexities of the Rendezvous problem within networked control systems.

Recently, I presented an analytical approach to solving the simplified case of two agents (N=2) and speculated on how this could extend to a larger number of agents, using a centroid-seeking control law. This article will offer code for simulating rendezvous for any number of agents across multiple dimensions, though I will primarily demonstrate rendezvous in one-dimensional (1D) and two-dimensional (2D) spaces.

As illustrated, we observe that the agents successfully converge at a designated point, confirming the predictions of our mathematical model. In particular, both agents meet at the midpoint, demonstrating their convergence towards the origin.

In the 2D case, the convergence is similarly evident. The agents rapidly approach the origin, with their positions clustering together as they draw nearer. By adjusting the parameter gamma to a higher value, the speed of convergence can be significantly enhanced.

For a simulation involving three agents (N=3), we can also observe the specific 2D point where these agents converge. Returning to the phase portrait will reveal how the points cluster and validate the agents' successful rendezvous.

Chapter 2: Rendezvous Simulation Code

Below is the code you can use for performing rendezvous simulations:

def rendezvous(num_agents=10, num_dims=2, num_steps=100):

"""

Simulates the trajectory of agents as they converge

in a defined state space over a specified number of steps.

"""

# Initialize random positions for all agents:

X_inits = np.random.rand(num_agents, num_dims)

# Prepare to store state trajectories:

Xs = np.zeros((num_agents, num_dims, num_steps))

# Save the initial positions

Xs[:,:,0] = X_inits

# Set gamma as the inverse of the number of neighbors:

gamma = 1/(num_agents-1)

for k in range(1, num_steps):

# Employ a double loop for the Rendezvous process:

for i in range(num_agents):

for j in range(num_agents):

if (j != i):

Xs[i,:,k] += gamma * Xs[j,:,k-1]

return Xs

The video titled "Moving Self Hosted Connectwise Control / Screenconnect From Linux to Windows & HA Proxy" provides further insights into implementing rendezvous strategies in multi-agent systems. It details practical applications and configurations that enhance understanding of the concepts discussed here.

Until next time,

Caleb. Level Up Coding

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