Stabilization of Nonlinear Multiple Time Scale Systems by Nonlinear $\mathcal{H}_{\infty}$ Control Based on Cyclic-Small-Gain Theorem

This paper (in the absence of exogenous disturbances) proposes an approach that stabilizes nonlinear multiple time scale systems. First, a general class of nonlinear systems containing a finite number of subsystems is considered, and a cyclic-small-gain theorem is proposed. Next, a controller design approach is introduced and an existence theorem is expressed. After that, by considering a general class of nonlinear multiple time scale systems, it is shown that the application of the proposed method will be independent of the positive values of the perturbation parameters. This result means that the perturbation parameters can be removed in the design procedure. But, in the closed-loop system, the equations without inherent perturbation parameters can be multiplied by additional perturbation parameters, and all perturbation parameters can take any positive values. Finally, the proposed approach is applied to the angular velocities system of the coupled satellites in a loop. It stabilizes the system globally exponentially for any positive values of the perturbation parameters.