Small-gain theory is a strong result to prove stability of uncertain systems. It is also a powerful tool to prove stability of nonlinear systems, see this wonderful book "nonlinear systems" by Khalil. But this result is handy in proving stability of interconnected systems. Let's dig in!
Commonly used Small-gain theorem:
Consider a stable open-loop transfer function L(s). The closed-loop system is stable if || L || < 1. where || . || denotes any matrix norm.
This result is commonly used along with the L_2- and L_\{infty}-norm to prove stability. Cool! right? Now, let's see a cooler result from "Multi-variable Feedback Control: Analysis and Design" by Skogestad and Postlethwaite
Spectral radius Small-gain Theorem:
Consider a stable open-loop transfer function L(s). The closed-loop system is stable if \rho( L ) < 1. where \rho(.)=max | \lambda(L) | is the maximum eigenvalue magnitude of L in all frequencies. This is more general than the common small gain theorem since \rho(L) \leq || L ||.
How to use this result in linear multi-agent systems?
It is simple: Find the closed-loop transfer function of the closed-loop system and write the characteristic equation of the closed-loop system. It will probably have a structure similar to
\Delta = Det ( Diag{s I -A_{cl}} F )where
F= I - Diag {f_i} Zwhere f_i is the single-agent dynamics and Z is the normalized adjacency matrix. For stability it is enough to have
|| f_{i}||_{\infty} < 1\rho(Z).To see more details, please see the following papers. Make sure to cite them if you use this result!