2 research outputs found
Sum of squares certificates for stability of planar, homogeneous, and switched systems
We show that existence of a global polynomial Lyapunov function for a
homogeneous polynomial vector field or a planar polynomial vector field (under
a mild condition) implies existence of a polynomial Lyapunov function that is a
sum of squares (sos) and that the negative of its derivative is also a sum of
squares. This result is extended to show that such sos-based certificates of
stability are guaranteed to exist for all stable switched linear systems. For
this class of systems, we further show that if the derivative inequality of the
Lyapunov function has an sos certificate, then the Lyapunov function itself is
automatically a sum of squares. These converse results establish cases where
semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov
inequalities. Finally, we demonstrate some merits of replacing the sos
requirement on a polynomial Lyapunov function with an sos requirement on its
top homogeneous component. In particular, we show that this is a weaker
algebraic requirement in addition to being cheaper to impose computationally.Comment: 12 pages. The arxiv version includes some more details than the
published versio
SOS-Convex Lyapunov Functions and Stability of Difference Inclusions
We introduce the concept of sos-convex Lyapunov functions for stability
analysis of both linear and nonlinear difference inclusions (also known as
discrete-time switched systems). These are polynomial Lyapunov functions that
have an algebraic certificate of convexity and that can be efficiently found
via semidefinite programming. We prove that sos-convex Lyapunov functions are
universal (i.e., necessary and sufficient) for stability analysis of switched
linear systems. We show via an explicit example however that the minimum degree
of a convex polynomial Lyapunov function can be arbitrarily higher than a
non-convex polynomial Lyapunov function. In the case of switched nonlinear
systems, we prove that existence of a common non-convex Lyapunov function does
not imply stability, but existence of a common convex Lyapunov function does.
We then provide a semidefinite programming-based procedure for computing a
full-dimensional subset of the region of attraction of equilibrium points of
switched polynomial systems, under the condition that their linearization be
stable. We conclude by showing that our semidefinite program can be extended to
search for Lyapunov functions that are pointwise maxima of sos-convex
polynomials