A Convex Optimization Approach to Compute Trapping Regions for Lossless Quadratic Systems

[Paper on arXiv] / [GitHub]

Abstract

Quadratic systems with lossless quadratic terms arise in many applications, including models of atmosphere and incompressible fluid flows. Such systems have a trapping region if all trajectories eventually converge to and stay within a bounded set. Conditions for the existence and characterization of trapping regions have been established in prior works for boundedness analysis. However, prior solutions have used non-convex optimization methods, resulting in conservative estimates. In this paper, we build on this prior work and provide a convex semidefinite programming condition for the existence of a trapping region. The condition allows precise verification or falsification of the existence of a trapping region. If a trapping region exists, then we provide a second semidefinite program to compute the least conservative trapping region in the form of a ball. Two low-dimensional systems are provided as examples to illustrate the results. A third high-dimensional example is also included to demonstrate that the computation required for the analysis can be scaled to systems of up to $∼O(100)$ states. The proposed method provides a precise and computationally efficient numerical approach for computing trapping regions. We anticipate this work will benefit future studies on modeling and control of lossless quadratic dynamical systems.

Recommended Citation

@misc{liao2024convex,
      title={A Convex Optimization Approach to Compute Trapping Regions for Lossless Quadratic Systems}, 
      author={Shih-Chi Liao and A. Leonid Heide and Maziar S. Hemati and Peter J. Seiler},
      year={2024},
      eprint={2401.04787},
      archivePrefix={arXiv},
      primaryClass={math.OC}
}