An a-L∞ stability Small-Gain Theorem for Nonlinear Interconnection Systems

This paper presents a small-gain result for interconnected large-scale system based on a- ${\mathcal{L}}_{\infty}$ stability. It is assumed that the output of the subsystem satisfies the a- ${\mathcal{L}}_{\infty}$ stability bound and the small gain condition holds, then the composite output of the interconnected system can be proved to satisfy an a- ${\mathcal{L}}_{\infty}$ stability bound. Comparing with the input-output stability, the $\mathrm{a}-\mathrm{L}_{\infty}$ stability only requires that the system state is bounded and the output converges. This theorem expounds the stability problem of a more general nonlinear system with less restrictive conditions.