Domain-of-attraction estimation for uncertain non-polynomial systems

Abstract

In this paper, we consider the problem of computing estimates of the domain-ofattraction for non-polynomial systems. A polynomial approximation technique, based on multivariate polynomial interpolation and error analysis for remaining functions, is applied to compute an uncertain polynomial system, whose set of trajectories contains that of the original non-polynomial system. Experiments on the benchmark nonpolynomial systems show that our approach gives better estimates of the domain-ofattraction.

Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation

“…In particular, the LEDA found in [1] is compatible with the enclosure found here. However, this enclosure is not compatible with the LEDA found in [6]. The following feasible point has been found by Ibex using a relative precision of 10 −4 :…”

“…x f = (0.460239373, 0.330548554). This point has a positive Lie derivative and belongs to the LEDA computed in [6], which is therefore slightly larger than the LEDA.…”

“…In [1,6] only fixed points that do not depend on the uncertainty are considered. Notice that in the case where the fixed point actually depends on the uncertainty, the RDA may be empty although each uncertain system has a nonempty DA.…”

“…The origin is an attractive fixed point, which is used as the center of the computed EDA. The following quadratic Lyapunov function is used in [5,6]…”

“…Finding such SOS polynomials amounts to solve linear matrix inequalities (LMIs), LMIs being convex and solved by efficient solvers today. This way, SOS polynomials have been applied to prove that functions are Lyapunov functions for polynomial systems, see, e.g., [2,3,1], and further extended to non-polynomial systems [4,5,6] and to hybrid systems [7]. In spite of its powerfulness, this approach presents several drawbacks: Its computational complexity turns out to be sensitive to both the dimension of the system and the degree of the polynomials involved in the description of the system (see Subsection 5.7).…”