# Distributed Average Tracking of Heterogeneous Physical Second-order Agents With No Input Signals Constraint

## TL;DR

This paper develops a novel distributed control algorithm for heterogeneous second-order agents that can track average inputs without input signal constraints, using adaptive filters and state-dependent gains.

## Contribution

It introduces a new control and filtering scheme for heterogeneous nonlinear agents with unbounded inputs, extending average tracking capabilities.

## Key findings

- Successfully tracks average input signals and velocities in simulations.
- Handles unbounded nonlinearities with state-dependent time-varying gains.
- Achieves improved results for double-integrator agents without input constraints.

## Abstract

This paper addresses distributed average tracking of physical second-order agents with heterogeneous nonlinear dynamics, where there is no constraint on input signals. The nonlinear terms in agents' dynamics are heterogeneous, satisfying a Lipschitz-like condition that will be defined later and is more general than the Lipschitz condition. In the proposed algorithm, a control input and a filter are designed for each agent. Each agent's filter has two outputs and the idea is that the first output estimates the average of the input signals and the second output estimates the average of the input velocities asymptotically. In parallel, each agent's position and velocity are driven to track, respectively, the first and the second outputs. Having heterogeneous nonlinear terms in agents' dynamics necessitates designing the filters for agents. Since the nonlinear terms in agents' dynamics can be unbounded and the input signals are arbitrary, novel state-dependent time-varying gains are employed in agents' filters and control inputs to overcome these unboundedness effects. Finally the results are improved to achieve the distributed average tracking for a group of double-integrator agents, where there is no constraint on input signals and the filter is not required anymore. Numerical simulations are also presented to illustrate the theoretical results.