Investigating the effect of edge modifications on networked control systems

This paper investigates the impact of addition/removal of edges in a complex networked control system, for the purposes of improving its controllability, system performances or robustness to external disturbances. The transfer function formulation we obtain allows to quantify the impact of an edge modification with the $\hinf$ and $\htwo$ norms. For stable networks with positive edge weights, we show that the $\hinf$ norm can be computed exactly for each possible single edge modification, as well as the associated stability margin. For the $\htwo$ norm we instead obtain a lower bound. Since this bound is linked to the trace of the controllability Gramian, it can be used for instance to reduce the energy needed for control. When instead the dynamics is of Laplacian type, then the norms become unbounded. However, the associated displacement systems are stable and for them the effect of edge modifications can be quantified. In particular, in this case we provide an upper bound on the $\hinf$ norm and compute the exact value of the $\htwo$ norm for arbitrary edge additions.