Network Small Gain Theorem

The network stabilization problem in a heterogeneous multi-agent system with diffusive connections is investigated in this paper. It is demonstrated that, under the assumption that the agents are finite-gain
$\mathcal{L}_{2}$
-stable and zero state observable, the interconnection is zero input asymptotically stable, and all agents' dynamics converge to the origin. The stability analysis is based on small-gain theory. We provide the maximum bound on the
$\mathcal{L}_{2}$
-gain of the controller using various optimization methods when the
$\mathcal{L}_{2}$
-gain of each agent is known. A study on the variation of
$\mathcal{L}_{2}$
-gain bound of the controller with the Laplacian eigenvalues of the underlying graph is provided. Numerical examples, which support and illustrate the analytical results, are also included.