Combined Coulomb and Viscous Damping Estimation Using Topological Signal Processing

In this work we develop a novel time-domain approach for the simultaneous estimation of the damping parameters for a single degree of freedom oscillator with both viscous and coulomb damping. Our approach leverages zero-dimensional sublevel set persistence — a tool from Topological Signal Processing (TSP) — to analyze the ring down vibration of the signal. Sublevel set persistence is used as it alleviates the need for peak selection when analyzing the time-domain of the signal and provides an alternative noise-robust method for visualizing the damping envelope. We are able to successfully estimate the damping parameters using both a direct approach and a function fitting method. We show that the direct approach is only appropriate for low levels of additive noise, but allows for a less computationally demanding estimation of the parameters. Alternatively, the function fitting method provides accurate estimates for significantly higher levels of additive noise. The results are provided through a numerically simulated example with mixed coulomb and viscous damping. We demonstrate the robustness of our method for accurately estimating both damping parameters for various levels of additive noise, a wide range of sampling frequencies, and both high and low levels of damping. This analysis includes providing suggested limitations of the method when applied to real-world signals.

Nonnegative forms with sublevel sets of minimal volume

We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even degree) complement its numerous intrinsic geometric properties. We also provide a probabilistic interpretation of the results.

Estimation of the regions of attraction for autonomous nonlinear systems

A methodology for estimating the region of attraction for autonomous nonlinear systems is developed. The methodology is based on a proof that the region of attraction can be estimated accurately by the zero sublevel set of an implicit function which is the viscosity solution of a time-dependent Hamilton–Jacobi equation. The methodology starts with a given initial domain and yields a sequence of region of attraction estimates by tracking the evolution of the implicit function. The resulting sequence is contained in and converges to the exact region of attraction. While alternative iterative methods for estimating the region of attraction have been proposed, the methodology proposed in this paper can compute the region of attraction to achieve any desired accuracy in a dimensionally independent and efficient way. An implementation of the proposed methodology has been developed in the Matlab environment. The correctness and efficiency of the methodology are verified through a few examples.

Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

We investigate various properties of the sublevel set G = {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions (and in particular homogeneous polynomials). For instance, the latter integral reduces to integrating h exp (-g) on the whole space ℝn (a nonGaussian integral) and when g is a polynomial, then the volume of G is a convex function of the coefficients of g. We also provide a numerical approximation scheme to compute the volume of G or integrate h on G (or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.