Design and comprehensive analysis of improved Proportional-Integral-Retarded protocol for second-order multi-agent systems

Background: The attractive characteristics of an innovative delay-based controller called Proportional-Integral-Retarded (PIR) controller are reported in many fields. However, a detailed investigation of the merits of the PIR controller for multi-agent systems (MASs) is still missing. Methods: To bridge this gap, we propose an improved PIR protocol for a general class of second-order MASs. This protocol, using multiple intentional delays and heterogeneous gains as tunable parameters, realizes the optimal spectrum assignment of each subsystem in a completely independent way without affecting the stability of other subsystems. We then present a systematic and flexible parameter tuning strategy for the PIR protocol such that the desired exponential decay rate of the consensus system can be achieved. Specifically, this strategy achieves an arbitrary dominant root assignment in the domain of non-negative real numbers and enables enhanced control flexibility by adjusting an additionally introduced free variable in the analytical parameter formulae. To unveil the superiority of our proposed PIR protocol, we develop a comprehensive comparison of the PIR protocol with the Proportional-Integral-Derivative (PID) scheme and the Proportional-Retarded (PR) protocol. Significant findings: The numerical simulations show that our PIR protocol outperforms the PID scheme regarding the noise attenuation property. Besides, this protocol can attain faster consensus, stronger robustness against parameter uncertainties, and superior disturbance resistance than the PR protocol. Our paper provides a valuable reference for the design of an effective and implementable multi-agent consensus protocol.