Disorder in large-scale networks with uni-directional feedback

This work investigates local and global measures of disorder in large-scale directed networks of double-integrator systems connected over a multi-dimensional torus. We quantify these performance measures in systems subjected to distributed disturbances using an H2 norm with outputs corresponding to local state errors or deviations from the global average. We consider two directed uni-directional state feedback interconnections that correspond to relative position and relative velocity feedback in vehicle network applications. Our main result reveals that absolute state feedback plays a critical role in system robustness when local state measurements are unidirectional. Specifically, if absolute measurements of either state variable are available, then systems with uni-directional relative feedback perform as well as their symmetric bi-directional counterparts but have the advantage of reduced communication requirements. However in the absence of absolute feedback their performance is worse; in fact, it is impossible to maintain stability (i.e. a finite H2 norm) with uni-directional state measurements for arbitrarily large networks. Numerical examples illustrate the theory.