Polynomial Lyapunov Functions and Invariant Sets from a New Hierarchy of Quadratic Lyapunov Functions for LTV Systems

TL;DR

This work introduces a hierarchical lifting framework for linear time-varying (LTV) systems in which a quadratic Lyapunov function at any level serves as a non-homogeneous polynomial Lyapunov function for the base system. The authors develop a diagonal-assembly (tilde) hierarchy and show that stable lifted systems yield non-homogeneous polynomial certificates that improve reachability-sets and impulse-response bounds, while extending to polynomial invariant sets via S-procedure. They also connect these quadratic certificates to sum-of-squares relaxations and discuss dimension-reduction strategies to curb computational complexity. The approach provides a flexible, accessible tool for stability analysis and performance guarantees for LTV systems using standard linear-system and convex optimization techniques. Overall, the framework broadens the set of tractable Lyapunov-like functions and invariants beyond homogeneous quadratic forms, with practical impact on safety envelopes and robust performance in time-varying contexts.

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.

Figures (4)

• Figure 1: Simulated response of system (\ref{[' LTV system']}) with parameters (\ref{['LTV system parameters']}). The light yellow region represents the set of reachable states from $x_0$, the cyan point. The red, green and blue regions represent the approximation of the reachable set using $4^{\text{th}}$, $8^{\text{th}}$ and $12^{\text{th}}$ order non-homogeneous polynomials computed by solving (\ref{['reachable set optimization']}) respectively.
• Figure 2: The black line represents the simulated response of system (\ref{['LTI system']}) . The red, green and blue regions represent the approximation of the reachable set using $4^{\text{th}}$, $6^{\text{th}}$ and $8^{\text{th}}$ non-homgenuous polynomials computed by solving (\ref{['reachable set optimization']}) respectively.
• Figure 3: In the left figure , the black line represents the phase portrait of the impulse response of system (\ref{['stiff system']}). The red, green and blue level sets represent the invariant sets produced by solving (\ref{['impulse response semidefinite program']}) for $i=1,3 \text{ and } 5$ respectively. In the right figure, the black line is the impulse response of the system (\ref{['stiff system']}). The red, green and blue lines present the bounds obtained by theorem 2 at $i=1,3,\text{ and } 5$ respectively.
• Figure 4: Simulated response for system presented in example 1. The light yellow region represents the set of reachable states from the initial condition $x_0$. The red and blue regions represents the non-homogeneous polynomial invariant sets resulted from solving (\ref{['invariant set semi-definite program']}) for $i=5 \text{ and } 7$ respectively and for $\lambda =-0.05$.

Theorems & Definitions (11)

• Remark 1
• Theorem 1
• proof
• Example 1
• Remark 2
• Example 2
• Theorem 2
• proof
• Example 3
• Definition 1