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 H-2 norm with outputs corresponding to local state errors or deviations from the global average. We consider two directed uni-directional state feedback inter-connections 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 uni-directional. 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 H-2 norm) with uni-directional state measurements for arbitrarily large networks. Numerical examples illustrate the theory.