A procedure for the computation of accurate PWL approximations of non‐linear dynamical systems

Abstract

SUMMARYIn this paper we propose a variational method to ÿnd out piecewise-linear (PWL) approximations of non-linear dynamical systems in view of their circuit implementations. The method is based on some signiÿcant trajectories of the dynamical system and provides reasonably accurate PWL approximations with a relatively low number of parameters. The e ectiveness of the method is validated by applying it to the approximation of limit cycles (both stable and unstable) in the Bautin system.

Towards Accurate PWL Approximations of Parameter-Dependent Nonlinear Dynamical Systems With Equilibria and Limit Cycles

“…Some examples of basis functions and inner products can be found in [5], [23]. Here we adopt the -basis proposed in [5], which is orthonormal with respect to the inner product directly related to the functional with , i.e., to , which is the only invariant term of the proposed method.…”

“…In order to estimate the optimal value also for , we define the following quality factor: (23) where is the number of sample values of are the Lyapunov exponents of the original system's cycle for , whereas are the corresponding Lyapunov exponents of the PWL system's cycle. In our simulations, we fixed 50 linearly spaced values .…”

“…As a general statement, to obtain accurate dynamical behaviors, the involved nonlinear functions have to be approximated by linear affine functions over very small subregions of the considered domain, thus requiring very large numbers of subregions and then of approximation parameters. In [23], PWL approximations of dynamical systems with fixed parameters have been derived by applying variational criteria and model reduction strategies based on the method proposed in [5], [24] for the PWL "static" approximation of functions.…”