Simplified Design Approach for Fast Consensus in a Double-Integrator Multi-Agent System using a Proportional-Retarded Controller

This paper presents a new design approach that utilizes delay-based controllers to enhance the speed of response in a double-integrator multi-agent system (MAS). It proposes a modified version of the recently introduced Proportional-Retarded (PR) consensus protocol that simplifies the control architecture by using only a single delay and two control gains, eliminating the need for multiple delays and heterogeneous gains. We demonstrate that the stability of the overall MAS can be addressed in a decomposed manner, breaking down the characteristic equation into N − 1 simplified factors. However, designing for N − 1 factors with only three parameters can be challenging, especially for large values of N. Considering undirected graphs, here we show that achieving fast consensus is equivalent to minimizing the spectral abscissa associated with the product of only two of these factors, simplifying the design. Two numerical examples are provided to illustrate the results, and an optimization algorithm is proposed to solve the problem.