Project notes

Dynamical Equations (19/11/25)

The event-triggered dynamics discussed on 19/11/25 were written in agent form as:

$$ \begin{cases} \dot{x} = -\nabla f - Lx - Lz - L(\hat{x}-x) - L(\hat{z}-z),\\ \dot{z} = Lz + L(\hat{z}-z) \end{cases} $$

$$ \Rightarrow\quad \begin{cases} \dot{x} = -\nabla f - Lx - Lz - L e_x - L e_z,\\ \dot{z} = Lz + L e_z \end{cases} $$

These are the equations exactly as written on the board on 19/11/25.

Correction in Equations (10/12/25)

It was later identified that the expression for $\dot{z}$ was incorrect for the intended purpose.

The corrected form is (error as a tigger is studied here) :

$$\quad \begin{cases} \dot{x} = -\nabla f - Lx - L e_x - Lz - L e_z,\\ \dot{z} = Lx + L e_x \end{cases} $$

Here:

• $z$ now behaves as an integrator of disagreement in $x$,
• the term $-z$ in $\dot{x}$ provides the required integral correction,
• the system now structurally matches the role of $v$ in the original algorithm.

This correction restores the intended convergence behaviour.

Results for the above dynamics

1. 
2. 

• the only caveat is that the there is some oscillation when agents converge
• number of broadcasts and time of convergence for this system trinagular ring):
  • Triggers for z broadcasts per agent:
    • Agent 1: 81
    • Agent 2: 74
    • Agent 3: 46
    • Time of convergence: 14.220 s

Simulation of the 2014 and 2015 systems to see thier time of convergences:

1. for 2014 system (cortes) :

• number of broadcasts and time of convergence for this system trinagular ring):
  • Triggers for z broadcasts per agent:
    • Agent 1: 20
    • Agent 2: 15
    • Agent 3: 17
    • Time of convergence: 6.070 s

2. for the 2015 system (S.S Kia and corets):

• number of broadcasts and time of convergence for this system trinagular ring):
  • Triggers for z broadcasts per agent:
    • Agent 1: 13
    • Agent 2: 10
    • Agent 3: 12
    • Time of convergence: 8.480 s

code for both the simulations in this repositroy

Progress in the project till now

1. studied the form given by Cortes and S.S Kia
2. tried to prove S.S Kia's form in a way similar to Cortes, but cannot yet make a definitive comment about the rates as that requires a nice lypunov candidate, and im haveing trouble comming up wiht one, currently the work i did is under correction by the supervisor and phds.
3. the hypothesis is that the form by kia is garuneented to converge faster than cortes even when event triggered communication is not used, need to prove/disprove it.
4. need to do a same comparitive study for our form, also need to define $e_z$

Sem 6 proj progress:

1. found the conditions on the constant for S.S Kia using W.Ren 2006 as the matricies turned out the be similar, which concludes the comeparitive study as we simulated the event triggered communication for Cortes (explicit proof not needed)
2. Changing the direction of the project to work on adaptive gains for the S.S Kia type algotrihtm, trying to establish its convergence and seeing whether or not its feasible under event-triggered communication. A similar study is already done in MuSIC lab recently (here)
3. the algorithm, combining S.S Kia and Zhenhong Li, $$ \begin{aligned} \dot{v}^i &= \gamma_1 (\alpha_i + \beta_i) \sum_{j=1}^{N} a_{ij}(x^i - x^j), \ \dot{x}^i &= -\gamma_2 \nabla f_i(x^i) - \gamma_1 (\alpha_i + \beta_i) \sum_{j=1}^{N} a_{ij}(x^i - x^j) - v^i, \ \dot{\alpha}_i &= \beta_i. \end{aligned} $$ where

$$ \beta_i = e_i^T e_i, \qquad e_i = \sum_{j=1}^{N} a_{ij}(x^i - x^j). $$

got proved, it works, and the code for both the simlations can be found here (and so can be the report, although i do not like this version, re-editing it!!)

4. Next step is to find the conditions on the constant encoutnered during the lyapunov analysis in the proof of convergence, and prove it for the event triggered communication (which by simulation we know mostly works).

recent advancments in this field :

1. using PI controller instead of event-triggered contorl (the integral control essentially).
2. subgradient methods
3. Distributed Predefined-time Zero-gradient-sum Optimization for Multi-agent Systems: From Continuous-time to Event-triggered Communication (recent most 2025)
4. recent most study related to this from MuSIC lab
5. Distributed Adaptive Convex Optimization on Directed Graphs via Continuous-Time Algorithms (2018)