Polynomial Lyapunov Functions and Invariant Sets from a New Hierarchy of Quadratic Lyapunov Functions for LTV Systems
CoRR(2024)
Abstract
We introduce a new class of quadratic functions based on a hierarchy of
linear time-varying (LTV) dynamical systems. These quadratic functions in the
higher order space can be also seen as a non-homogeneous polynomial Lyapunov
functions for the original system, i.e the first system in the hierarchy. These
non-homogeneous polynomials are used to obtain accurate outer approximation for
the reachable set given the initial condition and less conservative bounds for
the impulse response peak of linear, possibly time-varying systems. In
addition, we pose an extension to the presented approach to construct invariant
sets that are not necessarily Lyapunov functions. The introduced methods are
based on elementary linear systems theory and offer very much flexibility in
defining arbitrary polynomial Lyapunov functions and invariant sets for LTV
systems.
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