[Paper Review] Distributed Average Tracking for Second-order Agents with Nonlinear Dynamics

Sheida Ghapani, Salar Rahili+1 more|arXiv (Cornell University)|Mar 23, 2016

Distributed Control Multi-Agent Systems15 references3 citations

TL;DR

This paper proposes a distributed average tracking control law for second-order physical agents with nonlinear dynamics, using a non-smooth filter and state-dependent time-varying gains to track the average of time-varying reference inputs and velocities. The method ensures asymptotic convergence despite unbounded nonlinearities by employing novel adaptive gains, and is simplified to constant gains under bounded nonlinear terms, achieving exact average tracking with minimal information exchange.

ABSTRACT

This paper addresses distributed average tracking of physical second-order agents with nonlinear dynamics, where the interaction among the agents is described by an undirected graph. In both agents' and reference inputs' dynamics, there is a nonlinear term that satisfying the Lipschitz-type condition. To achieve the distributed average tracking problem in the presence of nonlinear term, a non-smooth filter and a control input are designed for each agent. The idea is that each filter outputs converge to the average of the reference inputs and the reference velocities asymptotically and in parallel each agent's position and velocity are driven to track its filter outputs. To overcome the nonlinear term unboundedness effect, novel state-dependent time varying gains are employed in each agent's filter and control input. In the proposed algorithm, each agent needs its neighbors' filters outputs besides its own filter outputs, absolute position and absolute velocity and its neighbors' reference inputs and reference velocities. Finally, the algorithm is simplified to achieve the distributed average tracking of physical second-order agents in the presence of an unknown bounded term in both agents' and reference inputs' dynamics.

Motivation & Objective

• To solve the distributed average tracking problem for physical second-order agents with nonlinear dynamics, where agents must track the average of time-varying reference inputs and velocities.
• To address the challenge of unbounded nonlinear terms in both agent and reference dynamics that disrupt standard linear or double-integrator control laws.
• To design a control and filtering scheme that enables agents to track the average without requiring full knowledge of reference inputs or velocity measurements.
• To simplify the algorithm to constant gains under bounded nonlinearities while preserving asymptotic average tracking performance.

Proposed method

• A local non-smooth filter is designed for each agent to estimate the average of reference inputs and velocities, using neighbor information and state-dependent time-varying gains.
• A non-smooth control input is introduced to drive each agent’s position and velocity to track its filter output, with gains adapted to counteract unbounded nonlinear effects.
• The filter dynamics are governed by a signed sum of relative differences between agent and neighbor filter outputs, ensuring convergence to the average under undirected communication graphs.
• The control law combines a discontinuous term based on position and velocity tracking errors with the filtered acceleration command to ensure robustness.
• For bounded nonlinear terms, the algorithm is simplified to constant gains, reducing implementation complexity while maintaining asymptotic tracking.
• The design requires each agent to access its own and neighbors’ filter outputs, absolute position and velocity, and neighbors’ reference inputs and velocities.

Experimental results

Research questions

• RQ1 How can distributed average tracking be achieved for second-order physical agents when both agent and reference dynamics include unbounded nonlinear terms?
• RQ2 What control and filtering architecture enables asymptotic convergence to the average of time-varying reference inputs and velocities under Lipschitz-type nonlinearities?
• RQ3 How can state-dependent time-varying gains be designed to counteract the destabilizing effect of unbounded nonlinearities in agent dynamics?
• RQ4 Under what conditions can the proposed algorithm be simplified to use constant gains while preserving tracking performance?
• RQ5 What information exchange requirements are necessary for the algorithm to achieve distributed average tracking without requiring global knowledge?

Key findings

• The proposed filter dynamics ensure that each agent’s filter output asymptotically converges to the average of all reference inputs and reference velocities.
• The control input drives each agent’s position and velocity to asymptotically track its filter output, achieving distributed average tracking despite nonlinear dynamics.
• With unbounded nonlinear terms, the use of state-dependent time-varying gains in both filter and control input successfully stabilizes the system and ensures convergence.
• When the nonlinear term is bounded, the algorithm can be simplified to constant gains, maintaining asymptotic average tracking with reduced complexity.
• The control law does not require correct initialization of physical states, only the initialization of filter auxiliary variables, which is practically feasible.
• The method achieves exact asymptotic tracking under the conditions that the nonlinear terms satisfy a Lipschitz-type condition and the communication graph is undirected and connected.