New spectral bounds on H2-norm of linear dynamical networks

In this paper, we obtain new lower and upper bounds for the H2-norm of a class of linear time-invariant systems subject to exogenous noise inputs. We show that the H2-norm, as a performance measure, can be tightly bounded from below and above by some spectral functions of state and output matrices of the system. In order to show the usefulness of our results, we calculate bounds for the H2-norm of some network models with specific coupling or graph structures, e.g., systems with normal state matrices, linear consensus networks with directed graphs, and cyclic linear networks. As a specific example, the H2-norm of a linear consensus network over a directed cycle graph is computed and shown how its performance scales with the network size. Our proposed spectral bounds reveal the important role and contribution of fast and slow dynamic modes of a system in the best and worst achievable performance bounds under white noise excitation. Finally, we use several numerical simulations to show the superiority of our bounds over the existing bounds in the literature.