Approximating the Frequency Response of Contractive Systems
For researchers studying periodic behaviors in contractive systems, this work offers a practical approximation framework with theoretical guarantees, though it is an incremental extension of known stability properties.
The paper develops an approximation method for the unique periodic trajectory of contractive systems with periodic vector fields, providing explicit error bounds based on input-to-state stability. It further proves that such systems act as low-pass filters and demonstrates the approach on systems biology examples.
We consider contractive systems whose trajectories evolve on a compact and convex state-space. It is well-known that if the time-varying vector field of the system is periodic then the system admits a unique globally asymptotically stable periodic solution. Obtaining explicit information on this periodic solution and its dependence on various parameters is important both theoretically and in numerous applications. We develop an approach for approximating such a periodic trajectory using the periodic trajectory of a simpler system (e.g. an LTI system). Our approximation includes an error bound that is based on the input-to-state stability property of contractive systems. We show that in some cases this error bound can be computed explicitly. We also use the bound to derive a new theoretical result, namely, that a contractive system with an additive periodic input behaves like a low pass filter. We demonstrate our results using several examples from systems biology.