Hausdorff Dimension Estimates for Interconnected Systems With Variable Metrics

In this letter, we develop a framework for estimating the Hausdorff dimension of a compact invariant set for both autonomous and interconnected systems. We first generalize Smith's method for Hausdorff dimension estimates by using variable metrics in linear matrix inequalities. Then, we study open systems with a characterization similar to the differential dissipativity theory. For linear time-invariant systems, we show that our characterization can be considered as a pure input/output property. This fact would be important because it is independent of internal model representations. Finally, we provide an estimate of the attractor dimension for feedback and interconnected systems. Our estimation is scalable in the sense that the components in an interconnected system can be analyzed independently.