Passivity-based PI control of first-order systems with I/O communication delays: A complete sigma-stability analysis

The PI control of first-order linear passive systems through a delayed communication channel is revisited in light of the relative stability concept called σ-stability. Treating the delayed communication channel as a transport PDE, the passivity of the overall control-loop is guaranteed, resulting in a closed-loop system of neutral nature. Spectral methods are then applied to the system to obtain a complete stability map. In particular, we perform the D-subdivision method to declare the exact σ-stability regions in the space of PI parameters. This framework is then utilized to analytically determine the maximum achievable exponential decay rate σ ∗ d of the system while achieving the PI tuning as explicit function of σ ∗ d and system parameters.