Small gain theorem
Undergraduate (linear feedback systems)
Consider two stable systems $H_1$ and $H_2$ connected in a feedback loop, so that the output of each system is the input of the other. The small gain theorem says that if the loop gain is less than $1$, that is,
\[\gamma_1 \gamma_2 < 1,\]where $\gamma_1$ and $\gamma_2$ are the gains of $H_1$ and $H_2$, then the feedback loop is stable.
Graduate (nonlinear systems)
Consider two input-to-state stable systems connected in feedback, whose states $x_1$ and $x_2$ satisfy ISS bounds with gain functions $\gamma_1$ and $\gamma_2$ of class $\mathcal{K}$, where $x_2$ is the input of the first system and $x_1$ is the input of the second system. The nonlinear small gain theorem says that the interconnection is globally asymptotically stable if
\[\gamma_1\bigl(\gamma_2(r)\bigr) < r \quad \text{for every } r > 0.\]Properties
- The small gain theorem is conservative: the theorem uses only the size of each system’s gain, not the phase, so many stable feedback loops violate its condition.
- In a cascade, the output of the first system feeds the second system but nothing feeds back, so no small gain condition is needed: a cascade of ISS systems is ISS.