Comparison between SOS and (S)DSOS Lyapunov functions for nonlinear systems

Apolo Silva Marton¹; Alexandre Monteiro Ribeiro¹; André Ricardo Fioravanti¹

¹ School of Mechanical Engineering, UNICAMP

doi:10.20906/CPS/COB-2015-1678

Resumo

The continuous search for new Lyapunov functions for nonlinear system has brought several approaches and techniques. Generally, dealing with a nonlinear system is a difficult problem. However, during the last decade, the sum of squares (SOS) approach has presented relevant results, transforming the polynomial positivity problem into the search for polynomials that can be represented as a sum of squares of polynomials, which can be more easily treated by means of semidefinite optimization. In this paper, we will present a numerical comparison between different approaches of search for Lyapunov functions while maximizing the region of attraction (ROA) using SOS. For high-dimensional systems, SOS constraints may demand an excessive computational requirement, precluding the use the method. Alternatively, it is possible to relax the semidefinite cone by means of stronger conditions but computationally more tractable optimization problems. With the properties of diagonally dominant matrix theory, we may rewrite SOS constraints as diagonally-dominant-sum-of-squares (DSOS) constraints. Accordingly, for Lyapunov functions of the same order, the ROA obtained will be smaller. In a resembling way, the use of the scaled diagonal dominance (SDSOS) approach presents less conservatism when compared with DSOS conditions, and therefore a better approximation for the ROA. These new approaches compromise conservativeness of the results with less computational effort. Nevertheless, high dimensional systems or high order Lyapunov functions that reached the limits of available solvers are now solvable with these alternatives approaches. For illustration, we use the reverse-time Van der Pol Oscillator as example, presented both in continuous and discrete-time.

Palavras-chave: Lyapunov function; sum of squares; diagonally dominant; region of attraction