Lyapunov Differential Equation Hierarchy and Polynomial Lyapunov Functions for Switched Linear Systems

Performance Analysis and Non-Quadratic Lyapunov Functions for Linear Time-Varying Systems

“…Phase portrait of the impulse response for the system (9). The impulse response of ( 9) is shown in blue, and x(0) = b is shown as a blue dot.…”

“…An essential observation made in [9] is that a quadratic Lyapunov function for the i th level system H i identifies a homogeneous polynomial Lyapunov function for ẋ = Ax; that is, if P i ∈ S n i ++ satisfies…”

“…As illustrated in Example 3, the accuracy of the bound (13) will generally increase as i increases. This is due to the fact that the homogeneous polynomial Lyapunov functions generalize quadratic Lyapunov functions [9].…”

“…Assume there exists a P i ∈ S n i ++ that satisfies (11) for both A+ ∆ and A− ∆ at the i th level. Then, the system ẋ = A(t)x is stable with a homogeneous polynomial Lyapunov function V (x) = (⊗ i x) T P i (⊗ i x) [9]. Therefore |h(t)| ≤ h, where h(t) is the impulse response of ( 21) and h the a point on the level set {x ∈ R n | V (x) = b T i P i b i } in the direction c T .…”

“…The topic addressed in this paper relies on the recent observation in [9] that the search for homogeneous polynomial Lyapunov functions for LTV systems can be recast as the search for quadratic Lyapunov functions for a related hierarchy of Lyapunov differential equations. Indeed, every stable LTV system induces a homogeneous polynomial Lyapunov function [10], [11], and the search for such a Lyapunov function is easily expressed as sum-of-squares and found by solving a convex, semi-definite feasibility program.…”