Complete Parameter-Space Stability Characterization of Neutral-Type Consensus Dynamics in First-Order Multi-Agent Systems with Delayed PD Protocols

This paper studies consensus of continuous-time first-order linear multi-agent systems (MASs) with identical agent dynamics characterized by a scalar self-dynamics coefficient, under proportional derivative (PD) protocols with a common constant time delay over connected undirected graphs. Unlike existing parameter-space characterization studies, delayed derivative feedback gives rise to neutral-type dynamics, introducing an additional stability requirement associated with the neutral operator. To account for both this neutral operator constraint and the delay-induced stability variations, a unified framework combining frequency-sweeping techniques and parameter-space decomposition is developed, with the proportional gain, derivative gain, and time delay treated as free parameters. For fixed controller gains, delay-dependent stability is first characterized and the corresponding consensus delay sets are explicitly determined. The analysis is then extended to the controller gain plane, which is analytically partitioned according to the neutral operator constraint and the stability properties of the associated quasipolynomials. Algebraic conditions are established to classify the resulting regions. Based on this partition, delay-independent consensus, delay-dependent consensus, and non-consensus regions can be identified before quantitative critical-delay computation. For delay-dependent regions, the corresponding consensus sets are subsequently constructed over the joint controller-delay parameter space. Illustrative examples demonstrate the proposed characterization.