A Convex Optimization Approach to Compute Trapping Regions for Lossless Quadratic Systems
CoRR(2024)
摘要
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.
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