A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

Abstract

<p style='text-indent:20px;'>The paper proposes a Lyapunov theory-based method to compute inner estimates of the <i>region of attraction</i> (ROA) of stable limit cycles. The approach is based on a transformation of the system to transverse coordinates, defined on a <i>moving orthonormal coordinate system</i> (MOC) for which a novel construction is presented. The proposed <i>center point MOC</i> (cp-MOC) is associated with a user-defined center point and provides flexibility to the construction of the transverse coordinates. In particular, compared to the standard approach based on hyperplanes orthogonal to the flow, the new construction allows the analyst to obtain larger regions of the state space where the well-definedness property of the transformation is satisfied. This has important benefits when using transverse coordinates to compute inner estimates of the ROA. To demonstrate these improvements, a sum-of-squares optimization-based formulation is proposed for computing inner estimates of the ROA of limit cycles for polynomial dynamics described in transverse coordinates. Different algorithmic options are explored, taking into account computational and accuracy aspects. Results are shown for three different systems exhibiting increasing complexity. The presented algorithms are extensively compared, and the newly cp-MOC is shown to markedly outperform existing approaches.</p>

Kernel-Based Identification of Local Limit Cycle Dynamics with Linear Periodically Parameter-Varying Models

“…This condition restricts the region where the transformation to transverse coordinates is well-defined. An alternative set of surfaces is considered, originally proposed in [11]. These surfaces, referred to as center surfaces, connect x ⋆ (τ ) with a fixed center (e.g., the geometric center of the limit cycle).…”