Lyapunov Function Synthesis using Handelman Representations.

Abstract

We investigate linear programming relaxations to synthesize Lyapunov functions that establish the stability of a given system over a bounded region. In particular, we attempt to discover functions that are more readily useful inside symbolic verification tools for proving the correctness of control systems. Our approach searches for a Lyapunov function, given a parametric form with unknown coefficients, by constructing a system of linear inequality constraints over the unknown parameters. We examine two complementary ideas for the linear programming relaxation, including interval evaluation of the polynomial form and "Handelman representations" for positive polynomials over polyhedral sets. Our approach is implemented as part of a branch-and-relax scheme for discovering Lyapunov functions. We evaluate our approach using a prototype implementation, comparing it with techniques based on Sum-of-Squares (SOS) programming. A comparison with SOSTOOLS is carried out over a set of benchmarks gathered from the related work. The evaluation suggests that our approach using Simplex is generally fast, and discovers Lyapunov functions that are simpler and easy to check. They are suitable for use inside symbolic formal verification tools for reasoning about continuous systems.

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

“…Ratschan and She use interval arithmetic relaxations with branch-and-bound to discover Lyapunov like functions to prove a notion of region stability of polynomial systems [49]. This is extended in our previous work to find LP relaxations using the notion of Handelman representations [50]. In practice, the interval arithmetic approach is known to be quite coarse for proving polynomial positivity, especially for intervals that contain 0.…”

“…Furthermore, the coarseness of interval relaxation is remedied by resorting to branch-and-bound over the domain. A detailed comparison between interval and Handelman approach is provided in our previous work [50], wherein we conclude that both approaches have complementary strengths. A combined approach is thus formulated.…”

“…Handelman Representations: We now briefly summarize our previous work that uses Handelman representations for Lyapunov function synthesis [50].…”

“…The Handelman approach reduces the synthesis to solving a set of linear programs, and was presented in our previous work [50]. The Putinar representation approach uses SOS programming, and was presented by Papachristadoulou et al [42].…”

“…As a first contribution of this paper, we extend the so-called Handelman representations, considered in our previous work [50], using the idea of Bernstein polynomials from approximation theory [6,16,40]. Bernstein polynomials are a special basis of polynomials that have many rich properties, especially over the unit interval [0,1].…”