Circuit Model Reduction with Scaled Relative Graphs

Abstract

Continued fractions are classical representations of complex objects (for example, real numbers) as sums and inverses of simpler objects (for example, integers). The analogy in linear circuit theory is a chain of series/parallel one-ports: the port behavior is a continued fraction containing the port behaviors of its elements. Truncating a continued fraction is a classical method of approximation, which corresponds to deleting the circuit elements furthest from the port. We apply this idea to chains of series/parallel one-ports composed of arbitrary nonlinear relations. This gives a model reduction method which automatically preserves properties such as incremental positivity. The Scaled Relative Graph (SRG) gives a graphical representation of the original and truncated port behaviors. The difference of these SRGs gives a bound on the approximation error, which is shown to be competitive with existing methods.

Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs

“…Follow-up work has extended the theory and applied it to analyze nonlinear operators: Huang, Ryu, and Yin characterized the SRG of normal matrices [18], Pates further characterized the SRG of linear operators using the Toeplitz-Hausdorff theorem [23], and Huang, Ryu, and Yin [19] established the tightness of the averagedness coefficient of the composition of averaged operators [21] and the DYS operator [13]. SRG has also found applications in control theory: Chaffey, Forni, and Rodolphe utilized the SRG to analyze input-output properties of feedback systems [7,8], and Chaffey and Sepulchre furthermore used it as an experimental tool to determine properties of a given model [6,9,10].…”