Contraction Theory and Timescale Separation for Interconnected Nonlinear Systems

Timescale separation is a powerful tool for analyzing interconnected dynamical systems. Meanwhile, operator theory provides a general framework for studying the convergence of iterative methods, including algorithms in optimization, learning, and control. In this paper, we bridge these two areas by establishing timescale separation results for a class of interconnected fixed-point iterations. As is customary in timescale separation, our results involve auxiliary systems that separately capture the dynamics induced by the slow and fast operators, arising from the original interconnection in the limit as the timescale parameter tends to zero. In particular, by assuming contractivity and paracontractivity of the respective auxiliary systems, we derive explicit bounds on the timescale parameter that guarantee linear convergence of the original, interconnected system. To illustrate the applicability of our result, we employ it to prove the convergence of a feedback optimization scheme.